Dielectrocapillarity for exquisite control of fluids

Domain: physics
Published: 2026-02-12
Editorial outcome: minor_revision

Abstract
Spatially varying electric fields are prevalent throughout nature, such as in nanoporous materials and biological membranes, and technology, e.g, patterned electrodes and van der Waals heterostructures. While uniform fields cause free ions to migrate, for polar fluids they simply reorient the constituent molecules. In contrast, electric field gradients (EFGs) induce a dielectrophoretic force, offering fine control of polar fluids even in the absence of free charges. Despite their vast potential for optimizing fluid behavior, EFGs remain largely unexplored at the microscopic level due to the absence of a rigorous first-principles theory of electrostriction. By integrating state-of-the-art advances in liquid state theory and deep learning, we reveal how EFGs modulate fluid structure and capillarity. We demonstrate that dielectrophoretic coupling enables tunable control over the liquid–gas phase transition, capillary condensation, and fluid uptake into porous media. Our findings establish “dielectrocapillarity”—the use of EFGs to manipulate confined fluids—as a powerful mechanism for controlling volumetric capacity in nanopores, holding immense potential for energy storage, selective gas separation, and tunable hysteresis in neuromorphic nanofluidics. Furthermore, by linking nanoscale dielectrocapillarity to macroscopic dielectrowetting, we establish a foundation for field-controlled wetting and adsorption phenomena of polar fluids across length scales.

Introduction
Nanoporous materials such as metal-organic frameworks, carbon-based supercapacitors, and nanofluidic devices rely on their ability to uptake and store fluids, in either the gaseous or liquid state, which directly impacts the performance of energy storage, chemical separation, and filtration technologies. The physics of capillarity plays a fundamental role in determining fluid uptake in these systems.

It is well established—initially at the macroscopic scale through the works of Young, Laplace, and Kelvin, and later at the microscopic scale—that adsorption depends not only on the system’s thermodynamic state, but also on the confinement length and the substrate–fluid interaction. These factors are typically intrinsic material properties. As a result, extensive research into enhancing adsorption in porous materials has focused on optimizing these factors, e.g., by tuning porosity or chemical functionalization. However, the potential for manipulation by external means—using applied fields to control confined fluids—remains relatively unexplored.

Electric fields offer a compelling mechanism to control the structure, phase behavior, and interfacial properties of fluids. While a uniform field exerts a direct force only on free charges such as ions, non-uniform fields with electric field gradients (EFGs) generate dielectrophoretic forces on neutral polar molecules. Dielectrowetting experiments have demonstrated that EFGs influence macroscopic contact angles, in a manner consistent with a modified Young’s equation. Whether these effects translate to nanoscale capillarity, however, is unclear. Addressing this issue is important, as porous media and membranes are rarely uniform; surface heterogeneities, defects, and curvature are natural sources of large EFGs. Such inhomogeneous fields can also be engineered using, e.g., patterned electrodes, atomic force microscopes, or layered van der Waals heterostructures.

Such EFGs introduce new length scales that can be comparable to the natural correlation lengths of the confined fluid, posing a severe challenge for a comprehensive theoretical description. With their inherent microscopic resolution, molecular dynamics simulations have provided key insights into the structural response of fluids to electric fields and phase behavior under confinement. However, computational limitations mean that simulation studies typically fix the number of molecules in the system, which introduces mechanical strains in an uncontrolled fashion. As a result, existing approaches—whether experimental, computational, or theoretical—face significant limitations in efficiently resolving both the microscopic restructuring and emergent macroscopic reorganization of fluids in non-uniform electric fields.

Here, we investigate how EFGs on the molecular length scale can be harnessed to manipulate mesoscopic fluid properties and phase behavior. Bringing together the latest advances in liquid state theory, computer simulation, and machine learning, we develop a multiscale framework to study electrostriction in polar fluids—that is, their density response to applied electric fields—within the grand canonical ensemble, representative of real conditions in which fluid molecules can enter and leave a pore. We show that EFGs provide tunable control over the liquid–gas phase transition and directly influence adsorption capability by capillary condensation—we dub this new phenomenon “dielectrocapillarity.” Given the critical role of volumetric fluid uptake in nanoporous materials for energy storage, gas separation, and filtration technologies, our findings establish dielectrocapillarity as a promising avenue in fine-tuning and optimization of such processes. Furthermore, the ability to regulate hysteresis introduces a new level of programmability in nanofluidics systems, where EFG-driven control of adsorption and desorption rates could offer external tunability akin to synaptic plasticity in neuromorphic nanofluidic circuits.

Results
Multiscale approach for electrostriction in fluids
Although molecular simulations cannot fully capture the influence of EFGs on dielectric fluids, they offer vivid qualitative insights into the underlying physics and illustrate how EFGs may arise within nanoscale devices, as illustrated in Fig. 1. Here, we present a simulation snapshot of water at 300 K confined between two hydrophobic substrates that have been patterned with alternating stripes of positive and negative charge—a setup that provides a caricature of an interdigitated electrode. As can be clearly seen in Fig. 1, charging this device induces local wetting near the charged stripes, with pronounced density variations along the direction parallel to the surface (x).

These features arise from the complex interplay between the fluid’s charge and number densities, and their collective response to EFGs, arising from the inhomogeneous electrostatic potential \phi(x,z), where z is along the surface normal. It is instructive to consider the form of \phi far from either surface; here, it will resemble the linear superposition of the asymptotic limit of the two substrates considered independently,

\phi(x,z) \approx \phi_0 \sin \left( \frac{2\pi x}{L_x} \right) \exp \left( -\frac{2\pi z}{L_x} \right),

where z_s indicates the plane of the substrate’s outermost atoms. We therefore observe that inhomogeneity of \phi is characterized by a sinusoidal oscillation of period L_x along x, and an exponential decay from the surface along z. This asymptotic analysis captures the essential behavior of \phi(x, z) computed explicitly from the potential energy of a test charge, as seen in Fig. 1.

The pronounced wetting behavior observed with simulation in Fig. 1 strongly suggests that such EFGs will influence capillarity of the polar fluid; that is, the amount of fluid adsorbed at constant chemical potential. Addressing this issue, however, demands an accurate and efficient framework for determining structure, thermodynamics, and phase behavior in an open system—this lies beyond the practical limits of present day molecular simulations. Instead, we turn to classical density functional theory (cDFT), an exact statistical mechanical framework for inhomogeneous fluids.

Within cDFT, the equilibrium structure and thermodynamics of a fluid can be determined from first principles by its excess intrinsic Helmholtz free energy functional F_{ex}[\rho, T], where \rho(r) is the average inhomogeneous density of the fluid and T is the temperature. This central result, established in ref. 30, places cDFT as the liquid-state generalization of its celebrated electronic structure counterpart—as a modern theory for inhomogeneous fluids. cDFT is naturally formulated in the grand canonical ensemble, where the chemical potential acts as the control variable governing particle exchange in confined systems. The chemical potential, \mu, maps directly onto the relative humidity or vapor pressure for gases, and chemical activity for liquids.

In practice, the exact form of F_{ex}[\rho, T] is generally unknown. Instead of relying on traditional approximations, we leverage state-of-the-art data-driven methodologies to supervise-learn functional mappings directly from quasi-exact referenced data from grand canonical Monte Carlo simulations. This machine-learned cDFT framework has already been successfully applied to liquid–gas coexistence, liquid–liquid phase separation, and the electric double layer. Going beyond established deep-learning approaches to cDFT, we capture electrostriction arising from the coupling between mass and charge density of the fluid by explicitly learning the “hyperfunctional” F_{ex}[\rho, \beta\phi, T] where \phi(r) is the inhomogeneous electrostatic potential and \beta=1/(k_B T) with k_B the Boltzmann constant, as recently introduced in ref. 36. A practical limitation of this cDFT approach is that, at present, only inhomogeneities with planar symmetry can be investigated directly. As a result, the neural-network representation of this functional was trained on simulation data generated under random planar electrostatic potentials \phi(z). Nonetheless, we demonstrate below that calculations in which \phi only varies along a single cartesian direction provide general insight into the influence of EFGs on wetting and capillarity.

The resulting cDFT framework is not only efficiently computable on standard hardware but also unparalleled in its ability to simultaneously capture microscopic fluid structure and mesoscopic phase behavior under arbitrary non-uniform electric fields. Unlike atomistic simulations, which are computationally prohibitive for such a broad exploration, our method achieves orders-of-magnitude speedup, completing each calculation in less than a minute without sacrificing quantitative accuracy. Importantly, this computational efficiency is realized after a one-time training stage: once the functional has been learned, it can be used to perform thousands of free-energy calculations at negligible cost. This amortized advantage is what enables us to map complete phase diagrams, adsorption isotherms, and metastable fluid branches—tasks that would require enhanced sampling and thermodynamic integration if carried out by molecular simulation alone. This efficiency enables an unprecedented, highly accurate mapping of a polar fluid’s response to EFGs of varying strengths and wavelengths across different thermodynamic conditions, from supercritical to subcritical regimes, spanning both bulk and confined environments. With this powerful tool in hand, we now uncover emergent electrostrictive phenomena that arise from the complex interplay of thermodynamics, confinement, and response to EFGs.

To systematically investigate how EFGs influence fluids, we primarily consider a minimal molecular model that incorporates soft-core repulsion, van der Waals attraction, and long-range dipolar interactions. The advantage of using such a simple molecular model is that we can exhaustively explore a broad range of thermodynamic conditions, while potentially uncovering common behaviors among polar fluids, from molecular liquids to colloidal systems. In addition, we also investigate a commonly used simple point charge model for water (SPC/E) that explicitly incorporates hydrogen-bonding, under thermodynamic conditions close to its critical point. Where comparison with ionic fluids is made, we will also show results for a prototypical model comprising oppositely charged hard spheres; in this case, we use a straightforward generalization of cDFT to multicomponent systems.

Dielectrophoretic coupling with EFGs under non-uniform electric fields
While EFGs will be most pronounced near the surfaces that generate them, Fig. 1 demonstrates that they may persist relatively far from the interface. In the specific case considered in Fig. 1, midway between the substrates we observe a sinusoidal electrostatic potential along the x direction. This motivates us to understand the direct influence of such sinusoidal potentials on the fluid, without explicitly considering interfaces. In fact, for L_x larger than a few molecular diameters, we can define bulk response if we consider averages over a thin slice of thickness \Delta z \ll L_x. Although such a clean separation of bulk and interfacial response becomes challenging as L_x approaches molecular length scales, the full potential (as opposed to its asymptotic form) comprises modes of decreasing wavelength—it is therefore instructive to understand bulk-like response across a broad range of wavelengths.

While the behavior of fluids under uniform electric fields has been extensively studied, non-uniform electric fields remain comparatively less explored. Consequently, the effects of EFGs on fluids are less well understood, even in bulk. As a starting point, we characterize the bulk response of water, the simple polar fluid, and the electrolyte, all under supercritical conditions, when subjected to a sinusoidal electric field, E^*(z)=E^*_{max} \sin(2\pi z/\lambda), as shown in Fig. 2a. Quantities labeled with an asterisk are expressed in reduced units, defined in the Methods section.

In the case of water, Fig. 2b, we see that the applied electric field induces a significant structural reorganization along the field direction; its averaged density profile \rho^*(z) is locally depleted in regions of weaker field strength, while molecular reorientation leads to an inhomogeneous average charge density distribution n^*(z). As water is overall neutral and therefore experiences no net electrophoretic force, the observed local reorganization arises instead from dielectrophoretic forces, f_{DEP} \propto \nabla E^2, which push the fluid towards regions of higher electric field strength—an effect termed “dielectrophoretic rise.” This dielectrophoretic force is the same that drives dielectrophoresis, which is widely exploited to manipulate biological cells and colloids, but its role in molecular fluids has received little attention. While water provides an important example of a polar fluid, the observed effects are by no means specific to aqueous systems. As seen in Fig. 2c, the overall picture is the same for the simple polar fluid, aside from the fact that it exhibits a symmetric response, whereas for water, depletion is stronger in regions where \nabla E < 0 than where \nabla E > 0 due to the inherent charge asymmetry of the water molecule.

In contrast, applying a sinusoidal field to an electrolyte induces “electrophoretic rise” in which the fluid migrates toward regions of lower electric field strength (Fig. 2d). This is a result of the electrophoretic force f_{EP} = qE, where E is the local field strength and q is the ionic charge, causing the anions and cations to reorganize, with peaks in their density profiles out of phase due to their opposing charges. In this way, polar fluids and ionic fluids display electromechanical responses that are fundamentally distinct from each other.

Crucially, dielectrophoretic coupling depends on the EFGs as well as the absolute field strength, and therefore offers greater control over the fluid’s response. To illustrate this, we present in Fig. 3a how dielectrophoretic rise can be amplified by controlling the applied field. Owing to their qualitatively similar behavior, in the remainder of the article we focus on the dipolar fluid, with results for water given in the SI, Figs. S13 and S14. As the wavelength of the sinusoidal field decreases, local mass accumulation becomes more pronounced, reflecting the system’s response to larger local EFGs, even as the maximum field amplitude remains unchanged. As a function of field strength, dielectrophoretic rise exhibits strong non-linearity, as evident from the change in maximum local density \Delta\rho^*_{max} = \rho^*_{max} - \rho^*_{max,0} from zero field shown in Fig. 3b, which highlights the complexity and collectiveness of electrostrictive response in fluids.

Dielectrophoretic response is not solely determined by the magnitude of the applied EFGs but also by the fluid’s intermolecular interactions. This distinction is of practical importance when considering colloidal systems in which effective interactions can be tuned. For example, in electrolyte solutions, zwitterionic Janus particles have diameters that far exceed the electrostatic screening length, making their dipolar interactions inherently short-ranged. To explore the effect on colloidal fluids, we consider a nearly identical model fluid whose dipolar interactions are screened, decaying on a length scale comparable to the molecular diameter \sigma. When the wavelength, \lambda, of the electric field is comparable to the particles size, \lambda \approx \sigma, both fluids exhibit identical dielectrophoretic response, shown by the dashed blue line in Fig. 3b. This result reflects that, for wavelengths on the molecular scale, response is dominated by local reordering of individual particles. However, for \lambda \gg \sigma, behaviors differ significantly; for systems with long-ranged interactions, molecular dipoles collectively reorient to screen the applied field, weakening its effect over extended distances. In contrast, the short-ranged colloidal system lacks this screening, leading to a much stronger response (dashed purple lines, Fig. 3b and inset). Such an equilibrium effect could be leveraged for programmable directed self-assembly, where EFGs in combination with tunable Janus particle surfaces may provide a powerful tool for tailoring the assembly of extended structures with dielectrophoretic forces, with additional tunability arising from the solvent and ionic strength.

Fine tuning liquid-vapor coexistence with EFGs
The results so far demonstrate that EFGs cause local reorganization of a supercritical dielectric fluid into regions of low and high density. A natural question then arises: how do EFGs influence the phase behavior of such a single-component fluid? Understanding this fundamental issue will be of central importance to the optimal design of devices with switchable functionality, in a similar spirit to the study of electric-field-induced phase transformations in solid-state materials.

In Fig. 3c, we show the density profiles of the fluid along the direction of an external sinusoidal electric field with \lambda/\sigma = 1.7, at T^* < T^*_{c,0}, where T^*_{c,0} is the critical temperature in the absence of an external field. Results are shown for different chemical potentials. At zero field, the stable solutions separate into homogeneous vapor and liquid states at low and high chemical potential, respectively. As the field strength increases, stronger EFGs (in absolute terms) not only give rise to dielectrophoretic rise but also destabilize liquid–vapor phase separation. Consequently, within the coexistence region, the vapor phase becomes denser, or contracts, while the liquid phase expands. For sufficiently large EFGs (E^*_{max} = 8), the fluid undergoes a transition to a single-phase supercritical fluid. By locating vapor and liquid solutions with equal grand potentials, in Fig. 3d we map out the liquid–vapor binodal curve for the dipolar fluid, both without an external field and under this sinusoidal field with E^*_{max} = 4. Strikingly, we find that the critical temperature T_c shifts downward under the non-uniform field. To our knowledge, this marks the first report of a shift in T_c for a single-component fluid induced by an electric field that varies on the microscopic length scale. Notably, unlike for uniform fields, while the binodal line still represents liquid–vapor equilibrium, the density within each phase is no longer spatially uniform due to dielectrophoretic rise. In this case, the bulk density \rho represents the fluid’s density averaged over a volume large compared to \sigma^3. For water, we observe similar behavior, with a downward shift in T_c of \approx 50 K at E_{max} = 0.4 V \mathring{A}^{-1} (Fig. S14). These observations highlight the exquisite level of control that one can exert over dielectric fluids with EFGs.

Here, we have tuned the EFGs by varying E^*_{max} for fixed \lambda. We could also have varied \lambda for fixed E^*_{max}, which provides additional control over the phase behavior (Fig. S11); importantly, these results demonstrate that the effects we have reported remain at larger wavelengths. Since such phase transitions are fully reversible, they will be particularly relevant for functional nanofluidic and nano-electromechanical devices where dynamic and reconfigurable phase control is advantageous. The act of confinement by itself already leads to fluid behavior that can differ substantially from bulk. EFGs represent an additional powerful tool for tailoring the properties of fluids for the purposes of device design.

Controlling adsorption in porous media through dielectrocapillary phenomena
In a slit pore, uptake of a fluid can be monitored by isothermal adsorption. For fluids below their critical point, of which liquid water at room temperature is an important example, adsorption is governed by capillary effects, such that condensation can occur at chemical potentials below saturation. This phenomenon, known as capillary condensation, underlies the filling of nanochannels, in which fluid uptake is controlled by adjusting the relative humidity of the environment. Such an experimental setup describes an equilibrium between the nanochannel and a reservoir (Fig. 4a), and maps directly onto our theoretical framework, which is formulated in the grand canonical ensemble. In the absence of an applied external field, it is well-established that capillarity is controlled by the chemical potential of the reservoir, the length scale of confinement, the substrate–fluid interaction and temperature. With EFGs, we introduce an additional experimental handle by which to control fluid adsorption behavior; we call this new phenomenon “dielectrocapillarity.”

We investigate a liquid at T^* = T^*_{c,0} = 0.68 confined to a solvophobic slit comprising two repulsive walls separated by a distance H = 6.6\sigma. In Fig. 4b, we show the fluid uptake into the slit as a function of the chemical potential, referenced to its coexistence value at zero field, \Delta\mu = \mu - \mu_{co,0}, typical of an adsorption/desorption isotherm measurement. In the absence of an electric field, the transition is discontinuous, exhibiting a hysteresis loop—this is a well-established hallmark of capillary condensation in nanopores and mesopores, observed across experiments, simulations, and theory. Such hysteresis arises from the metastability of the vapor during condensation and liquid during evaporation.

To gain insight into the influence of EFGs, we perform a “computational experiment” in which we apply a sinusoidal electric field across the slit. While such a setup does not correspond to an EFG established by the substrate walls themselves (see, e.g., Fig. 1), it does allow us to assess the effects of a particular mode. The impact is twofold: (1) condensation shifts to more negative \Delta\mu, i.e., the slit can be filled at even lower humidity; and (2) hysteresis is reduced. Keeping \lambda/\sigma = 1.7 fixed, for sufficiently high field strengths, hysteresis disappears entirely. In other words, we have changed the nature of the transition from first-order to continuous. This behavior directly results from the dielectrophoretic coupling that drives the fluid toward its supercritical state (Fig. 3). Important for our fundamental understanding of fluids under confinement, this result demonstrates that EFGs not only shift the bulk critical temperature, but also influence the capillary critical temperature.

Non-uniform electric fields clearly offer an additional lever for controlling capillary filling. As illustrated in Fig. 4c, adsorption/desorption can be actively controlled by switching the field on or off, offering a precise, tunable means of regulating fluid uptake. Such an ability to tailor hysteresis introduces a new level of programmability in nanofluidic systems, where EFGs can potentially serve as an external control parameter for dynamically altering adsorption and desorption rates. Notably, tunable hysteresis in capillary condensation can also serve as a memory mechanism in neuromorphic nanofluidic circuits, where phase transitions encode state-dependent responses akin to synaptic plasticity.

Connecting to dielectrowetting experiments
We have introduced dielectrocapillarity as an additional mechanism of controlling capillary phenomena, complementing electrocapillarity where electrolytes respond to applied potentials. On the macroscopic scale, these effects manifest in electrowetting and simple dielectrowetting, where the contact angle of a droplet can be tuned with applied potentials. Our nanoscale simulations in Fig. 1 show that a similar phenomenology emerges under confinement: wetting is strongly enhanced directly over the electrode patches, while the electrostatic potential generated by the interdigitated electrodes decays into the slit. Following previous experimental work, this motivates us to employ a simple planar electrostatic potential, \phi(z) = \phi_0 \exp(-2\pi z/L_x) to probe wetting at the nanoscale. Similar to our arguments above, results obtained from such a potential can be considered to report on average behavior in a slice of thickness \Delta x \ll L_x (Fig. S16).

Within a macroscopic model, the resulting contact angle can be described by a modified Young’s law,

\cos \theta(\phi_0) = \cos \theta_0 + \frac{\alpha}{\gamma_{lv}} \phi_0^2,

where \theta_0 is the Young’s contact angle at zero field, \gamma_{lv} the liquid–vapor surface tension, and \alpha a material parameter related to the dielectric permittivity of the liquid. This relationship assumes that the electrostatic energy stored in the liquid droplet is well-described by dielectric continuum theory, and that L_x is sufficiently small so that any changes in energy due to the electric field are effectively localized to the solid-liquid interface. Under these assumptions, the electrostatic free energy per unit area obeys the simple quadratic scaling \gamma_{elec} = -\alpha \phi_0^2.

Our multiscale framework provides a microscopic perspective on this phenomenology, and allows us to test whether, at the nanoscale, EFGs indeed enhance the wetting of dielectric liquids and the extent to which the scaling prescribed by Eq. (2) holds. To this end, we applied \phi(z) with L_x \approx 7\sigma to the confined dipolar model, symmetrically from both walls of a solvophobic slit. While not trained on such electrostatic potentials, as can be seen in Fig. 5b, the neural functional extrapolates well, yielding physically plausible results, aside from some oscillations at high wetting that are likely minor artifacts. As \phi_0 increases, so too does the contact density at the wall, verifying that enhanced wetting occurs. We quantify this effect by computing \gamma_{elec} = \frac{1}{2} \int_{-1}^{+1} dz \phi(z) n(z), which, as can be seen in Fig. 5c, decreases in an approximately quadratic fashion as \phi_0 increases. By identifying \alpha = -\partial \gamma_{elec} / \partial(\phi_0^2), we reconstruct the potential-dependent contact angle, as shown in Fig. 5c. While the results of our microscopic theory are broadly in line the macroscopic model (Eq. (2)) some subtle differences are observed, especially at small \phi_0. These appear to be correlated with significant changes in the local compressibility near the interface, \chi_\mu(z) = \partial \rho(z) / \partial \mu, as \phi_0 increases, which indicates a suppression of density fluctuations. Such microscopic details are lacking in the dielectric continuum model that underpins Eq. (2).

Methods
Reduced units
When reported, reduced units are defined by T^* = k_B T/\epsilon, \rho^* = \rho\sigma^3, and E^* = E\sigma^3/2\epsilon^{-1/2}, where \sigma sets the molecular length scale and \epsilon the energy scale. Specifically for each fluid: (i) SPC/E water, \sigma = 3.166 \mathring{A}, \epsilon = 0.65 kJ mol^{-1}; the dipolar fluid, \sigma = 3.024 \mathring{A}, \epsilon = 1.87 kJ mol^{-1}; and (iii) the electrolyte, \sigma = 2.76 \mathring{A}, \epsilon = e^2/\sigma, where e is the elementary charge.

Hyperdensity functional theory
We employed the recently developed first-principles theory for electromechanics based on hyperdensity functional theory, which provides an exact variational framework for the coupled electromechanical response of fluids. For a single-component fluid, at specified \mu, T, planar external non-electrostatic potential V_{ext}(z), and electrostatic potential \phi(z), the grand potential functional

\Omega[\rho, \beta\phi, T] = F_{id}[\rho, T] + F_{ex}[\rho, \beta\phi, T] + \int dz \rho(z) [V_{ext}(z) - \mu],

where the Rideal intrinsic Helmholtz free energy is F_{id}[\rho, T] = k_B T \int dz \rho(z) [\ln \zeta^{-1} \Lambda^3 \rho(z) - 1], with \Lambda denoting the thermal de Broglie wavelength and \zeta denotes a partition function accounting for intramolecular molecular degrees of freedom of each fluid particle.

The equilibrium number and charge density profiles are obtained by solving the Euler–Lagrange (EL) equation

\rho_{eq}(z) = \frac{\zeta}{\Lambda^3} \exp[-\beta V_{ext}(z) + \beta\mu + c^{(1)}(z; [\rho_{eq}, \beta\phi], T)],

and evaluating

n_{eq}(z) = n^{(1)}(z; [\rho_{eq}, \beta\phi], T),

with the one-body direct correlation functional c^{(1)} and charge density functional n^{(1)} defined as first functional derivatives of the excess intrinsic Helmholtz free energy functional

c^{(1)}(z; [\rho, \beta\phi], T) = -\frac{\delta \beta F_{ex}[\rho, \beta\phi, T]}{\delta \rho(z)},

n^{(1)}(z; [\rho, \beta\phi], T) = \frac{\delta F_{ex}[\rho, \beta\phi, T]}{\delta \beta \phi(z)}.

For fluids whose intermolecular interactions are long-ranged, such as ionic and dielectric fluids, the functional dependence of c^{(1)} and n^{(1)} on \rho and \phi is, in general, non-local. Note that, as we only discuss these functionals evaluated at equilibrium in this work, we drop the “eq” subscript for notational convenience.

In order to learn these functional mappings, we applied the techniques introduced in refs. 34,36 where the local functionals of short-ranged mimic fluids, c^{(1)}(z; [\rho, \beta\phi], T) and n^{(1)}(z; [\rho, \beta\phi], T), were learned from molecular simulation data using the neural functional method. A SR mimic fluid is defined as a system whose Coulomb interactions are replaced with \frac{1}{r} \text{erfc}(\kappa r).

For SPC/E water and the dipolar fluid, we used \kappa^{-1} = 4.5 \mathring{A}. For the electrolyte, we used the functionals trained in ref. 34, with \kappa^{-1} = 5.0 \mathring{A}. The effects of long-ranged interactions are accounted for in a well-controlled mean field fashion using local molecular field theory (LMFT). Implementation details and full derivations are provided in ref. 36 and Section S1 of the SI. We also validated the theory against computer simulations in Section S3 of the SI.

Generation of training data
For the electrolyte, we employed the functional reported in ref. 34 based upon the restricted primitive model. For SPC/E water and the dipolar fluid, we generated training data using a combination of grand canonical Monte Carlo (GCMC) and molecular dynamics (MD) simulations to construct local reference functionals c^{(1)}(z; [\rho, \beta\phi], T) and n^{(1)}(z; [\rho, \beta\phi], T). For the confined fluid, we set the slit height H, and the density \rho(z) was sampled in the slit. Hysteresis loops are mapped by seeding different initial guesses when solving the EL equation. Random inhomogeneous electrostatic potentials of the form \phi(z) = \phi_0 \cos(2\pi kz) were applied. In some cases, planar walls of the form of a 9-3 Lennard–Jones potential were also included. For each condition, GCMC (performed with our own code at https://github.com/annatbui/gcmc) determined the mean particle number, which was then used to initialize MD simulations in the NVT ensemble with LAMMPS, from which the number and charge density profiles are sampled. For water, we defined the molecular center for sampling \rho(z) at the oxygen site, and for the dipolar fluid at the midpoint between the two opposite charges. The charge density was computed as n(z) = \langle \sum_i \sum_\alpha q_\alpha \delta(z - z_{\alpha,i}) \rangle, where the outer sum is over all molecules, and the inner sum is over all sites belonging to molecule i, with z_{\alpha,i} and q_\alpha indicating the z coordinate and charge, respectively, of site \alpha. This hybrid scheme reproduces the accuracy of pure GCMC sampling while significantly accelerating convergence of inhomogeneous structures.

The total computation time for the generation of the entire dataset is on the order of ~10^5 CPU hours.

Training neural functionals
For both SPC/E water and the dipolar fluid, we train two neural networks to represent c^{(1)}([\rho, \beta\phi], T) and n^{(1)}([\rho, \beta\phi], T) following the local learning strategy. The machine learning routine was implemented in Keras/Tensorflow. Inputs consisted of local density and electrostatic potential in a sliding spatial window of size 10 \mathring{A} from the center of the position of interest. To effectively learn spatial variations, the model internally computes the gradient of \beta\phi(z) using a central difference scheme. In addition to these spatially varying inputs, a separate input node encodes T as a scalar. The full architecture details are provided in Figs. S5 and S6. The dataset was split roughly in a 3:1:1 ratio for training, validation, and test sets. Models were trained for 200 epochs with a batch size of 256, using an exponentially decaying learning rate starting at 0.001, achieving errors comparable to the estimated simulation noise. The training of the neural networks was done on a GPU (NVIDIA GeForce RTX 3060) in a few hours. We also verified that the trained functional can recover standard simulation results including bulk equation of state (Fig. S7) the binodal at zero applied field (Fig. S8) as well as number density and charge density response (Fig. S9).

Using the neural functionals
Evaluating the trained neural functionals is fast (~milliseconds) and can be performed on a CPU or GPU. Combining with LMFT give us c^{(1)}([\rho, \beta\phi], T) and n^{(1)}([\rho, \beta\phi], T). The EL equation is solved self-consistently with a mixed Picard iteration scheme, which typically converges within minutes. To determine the liquid–vapor coexistence line at zero electric field, we calculate isotherms of the chemical potential as a function of the bulk density \rho_b from \beta\mu = \ln(\Lambda^3 \zeta^{-1} \rho_b) - c^{(1)}([\rho_b, \beta\phi=0], T), and perform a Maxwell construction to find the coexisting liquid and vapor densities at the binodal. For inhomogeneous systems, i.e, as a result of an applied non-uniform electric field, we set the chemical potential and temperature and find the inhomogeneous solutions by solving the EL equation. The isotherms of the chemical potential, now as a function of the mean density, \rho = L^{-1} \int_0^L dz \rho(z) where L is the total length of the domain over which \rho(z) is defined, now have distinct jumps when the system undergoes phase separation, giving liquid \rho_l and vapor \rho_v solutions. To distinguish stable solutions from metastable solutions, we also calculate the grand potential \Omega_\phi = \Omega([\rho, \beta\phi], T) at a fixed \beta\phi where the excess free energy term can be evaluated via functional line integration. Performing this procedure for \beta\phi = 0 gives results consistent with those obtained by Maxwell construction. Note that the pressure of the bulk fluid at zero field is given by -PV = \beta\Omega_0.

In the dielectrowetting calculations, the electrostatic free energy per contact area is given as \gamma_{elec} = \frac{1}{2} \int_{-1}^{+1} dz \phi(z) n(z), where the factor of a half comes from there being two symmetric confining walls. To construct the dependence of \theta on \phi_0, we obtain \alpha = -\partial \gamma_{elec} / \partial(\phi_0^2) and calculate \gamma_{lv} from a direct coexistence molecular dynamics simulation. We estimate \theta_0 = 160° based on the maximum value of the local compressibility at \phi_0 = 0 and comparing to ref. 69; while more detailed calculations can provide a more accurate estimate, this value is consistent with the solvophobic nature of the slit, and is sufficient for the purposes of demonstrating the increased wetting with \phi_0 shown in Fig. 4.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Overall Evaluation\nThis manuscript introduces the novel concept of \"dielectrocapillarity,\" demonstrating how electric field gradients (EFGs) can\nregulate phase behavior, capillary condensation, and adsorption of confined fluids. By integrating classical density\nfunctional theory (cDFT) with deep learning, the authors develop a multiscale framework that offers new strategies for\napplications in nanoporous energy storage, gas separation, and neuromorphic devices. The study addresses a critical gap\nin understanding electrostrictive effects in fluids, with innovative methodology and clear logical progression. While the\ntheoretical framework and simulations provide robust support for key conclusions, the work could be strengthened by deeper\nexperimental validation, comparative analysis with competing techniques, and expanded discussion of its limitations. The\nresearch holds significant promise for both fundamental science and applied technology, though further refinements are\nrecommended to enhance its broader impact.\nDetailed Comments\nLimited Experimental Validation: The study relies predominantly on theoretical simulations and model derivations without\ndirect experimental data (e.g., observations of dielectrocapillary effects in actual nanofluidic devices). The authors should\nsupplement the manuscript with comparisons to existing experimental techniques (e.g., atomic force microscopy,\ndielectrophoresis experiments) or provide proof-of-concept experimental results to reinforce the credibility of the findings.\nIncomplete Comparison with Traditional Electric Field Methods: While the manuscript distinguishes between uniform electric\nfields and EFGs, it lacks a systematic comparison of dielectrocapillarity with conventional techniques like electrowetting or\ndielectrophoresis in terms of control precision, energy consumption, and applicability. A clear articulation of the method’s\nunique advantages and limitations is necessary.\nGeneralizability to Multicomponent Fluids: The current focus on single-component polar fluids limits relevance to real-world\napplications involving complex mixtures (e.g., electrolyte solutions, multicomponent gases). Simulations or discussions on\nextending the model to multicomponent systems would enhance practical utility.\nClarity of Graphical Information: Some curve annotations in Figures 1 and 2 (e.g., differences between SR/LR interactions)\nand definitions of non-dimensional units (e.g., reduced units in SI) are unclear. Additional explanations in figure captions or\nthe main text would improve accessibility.\nUnderdeveloped Discussion of Application Scenarios: While applications in supercapacitors and neuromorphic devices are\nmentioned, linking the findings to specific case studies (e.g., performance improvements in existing nanoporous materials)\nwould more vividly demonstrate translational potential.\nApplicability of Theoretical Assumptions: The model’s reliance on simplified assumptions (hard-sphere repulsion, van der\nWaals attraction) raises questions about its validity for systems with strong hydrogen bonding (e.g., water) or complex\ninterfacial chemistry. A sensitivity analysis or discussion of these limitations is recommended.\nConclusion\nThis work establishes a groundbreaking framework for fluid manipulation, with the concept of dielectrocapillarity offering\nsignificant promise for nanofluidics and energy technologies. Despite gaps in experimental validation and application-level\nanalysis, the originality of the concept and methodological advancements are compelling. Addressing the above comments\n—particularly by incorporating experimental data, enhancing comparative analyses, and refining graphical clarity—would\nstrengthen the manuscript’s scientific rigor and broader appeal. With these improvements, the study would solidly position\nitself as a high-impact contribution to Nature Communications.", "reviewer_2": "This manuscript presents a numerical investigation of the properties of polar fluids under electric gradients. The starting\npoint is to use dielectrophoresis, a phenomenon under which the polar fluids undergoes local forces in an electric gradient.\nThe central claim is that such forces could be used to compact or dilate the fluids. The manuscript explores the possibility of\nusing these compaction/dilatation forces to tune the liquid-vapor phase diagram of the fluid and then to use that to change\nwetting properties inside confined systems. In essence, the idea is to explore how dielectrophoresis might modify capillary\neffects in a similar way as electrophoresis modifies capillary forces.\nThe idea itself is interesting and creative. Dielectrophoresis being quite subtle, I have no doubt that peculiar phenomena\nmight arise in specific systems, so dielectrocapillary effects are indeed a possibility – that has not been studied so far. The\nmanuscript also includes an investigation of screening of the dielectrophoretic interactions which is captivating too.\nAt this stage, however, the model makes 2 major assumptions which are not well justified and, therefore, make it hard to\nvalidate whether or not the effect might be real. First, an equilibrium AI model is used to study nonequilibrium phenomena.\nSecond, capillarity related to properties near confining surfaces however the response of the surfaces to the electrical\ngradients are not accounted for and may strongly tune the response – as is the case for electrocapillarity. Another main point\nis that it is not clear that the orders of magnitude of changes investigated on a minimal polar fluid model might indeed be\nsignificant and hence accessible in experiments. This is not made easier by the current style of the manuscript which\nsometimes makes unjustified claims. At this stage I rather see the model as a remarkable and creative hypothesis, but its\npotential applicability in realistic systems is not discussed at all, making it hard to see if it could be real.\nI have several detailed comments, that I put below.\n(1) What goes inside the model and what data the AI is trained on is quite opaque, at least from the main text. Is it trained on\nthe different fluids mentioned or on other fluids? How is the functional learned exactly, aka what is the training process?\n(related to one of my main question) What makes us sure that a model that was designed and trained at equilibrium and is\npotentially suitable to investigate driving by uniform fields and then suitable to investigate\ndrivings by non uniform fields? Are some standard results in the field recovered? At this stage it is still not clear to me how the AI training enables one to study the system faster/better so it would be nice to discuss potential limitations/advantages a bit more.\n(2) The electrophoretic rise part could be described and explained more carefully. At this stage, I still don’t understand it.\nI believe this is mainly caused by an absence of details. First, in which direction is the field E imposed? Along the z direction?\nTo me it appears that regions of higher field actually have a locally neutral ion concentration and so there should be no force\non the fluid in these regions, yet this is apparently not what is observed. I can not make sense of the schematic drawing\nbecause it is not clear where the periodic unit is taken along the axis and what the line represents. Eventually I find it would\nbe worthwhile to discuss a little bit how, already at this stage, working in the muVT ensemble instead of NVT leads to\ninteresting differences compared to more traditional simulations setups where the total density varies spatially a lot. This is\nstill surprising to me since I expect fluid to be more or less incompressible and so homogeneous over such lengthscales.\nActually, the relationship between the densities \\rho* and n* is not clear from the images or the caption. Maybe one should\nconsider putting in that information maybe twice. Usually n is used for numbers and so would be used instead of rho, so that\nconvention was puzzling.\n(3) It is equally hard to understand the dielectrophoretic rise for similar reasons. What is n* in the case of the polar fluid? The\nfluid should be locally neutral.\n(4) I do not understand how the phase diagram is constructed – this is information that is completely in the SI but some short\nexplanation would be especially useful. Would it be possible to give some actual illustration of what the separated phases\nwould look like? Do they really have the wiggly interfaces that are shown on the sketches? Because then this creates a\nsurface tension in the fluid that is much bigger than that for a flat interface; so there’s a price to pay for this compacted state…\nwhich makes the story a lot more complicated.\n(5) In the confinement case there is a problem of dynamic entrance effects. There may be a question of how fast you are\nindeed able to recruit more particles within the small opening to contract the fluid etc.\n(6) Often electrical gradients are constructed by using specific/man made surface properties, e.g. patterning electrodes (as\ne.g. in https://pubs.acs.org/doi/full/10.1021/acs.jpclett.4c00278) or others. So there is a deep question there of how the\nelectric gradient is made and how that couples to the fluid properties which was somewhat discarded.\nPotential improvement of clarity/wording/typos:\n(12) : of a fluid or of a polar fluid?\n(general) Electrostrictive is not such a common word so I would suggest defining it.\n(introduction) I do not see how phenomena of gas uptake and separation can be so easily put under the same hat as liquid\nuptake since the electrokinetic properties in such dilute phases may be vastly different.\n(68) “grand canonical ensemble, which is most representative of real conditions in which fluid molecules can dynamically\nenter and leave a pore” It is and it is not. As far as I can tell, there is no great way of simplifying the entrance/exits from a\npore, because the grand canonical ensemble will remove correlations between particles exiting and coming back, which are\ncorrelations that exist. Also, all these things are nonequilibrium so really we don’t know. So the grand canonical ensemble is\na way of representing things, that may have some advantages compared to others, but it is likely not the best simplification of\nthe system.\n(94) “such minimal models uncover universal behaviors that emerge across different scales, from molecular liquids to\ncolloidal systems.“ -> this is a bold claim that is not justified.\n(112) twice the word “directly” and “supervise-learn” is an unusual concatenation\n(169) “dielectrophoretic coupling depends on the EFG rather than the absolute field strength”. They depend both on the\npattern on the field strength.\nIn Fig. 1d, the color code is not given.\nLiterature:\nIn terms of fluid uptake and filling of pores, there is a body of literature related to tailoring wettability properties that could be\ndiscussed (see below). Diffusiophoresis has already been used as an externally tunable force to manipulate filling or\nunfilling of nanopores with imposed chemical gradients (so indeed there already exists some external manipulation –\nalthough not as easy as electric fields) to manipulate particles.\nOn the filling of nanopores:\n- Giacomello, A., Chinappi, M., Meloni, S., & Casciola, C. M. (2012). Metastable Wetting on Superhydrophobic Surfaces:\nContinuum and Atomistic Views of the Cassie-Baxter–Wenzel Transition. Physical review letters, 109(22), 226102.\n- Picard, C., Gérard, V., Michel, L., Cattoën, X., & Charlaix, E. (2021). Dynamics of heterogeneous wetting in periodic hybrid\nnanopores. The Journal of Chemical Physics, 154(16).\nManipulating concentration gradients to fill/unfill dead-end pores\n- Migacz, R. E., Castleberry, M., & Ault, J. T. (2024). Enhanced diffusiophoresis in dead-end pores with time-dependent\nboundary solute concentration. Physical Review Fluids, 9(4), 044203.\n- Marbach, S., & Bocquet, L. (2019). Osmosis, from molecular insights to large-scale applications. Chemical Society\nReviews, 48(11), 3102-3144.\n- Shin, S., Um, E., Sabass, B., Ault, J. T., Rahimi, M., Warren, P. B., & Stone, H. A. (2016). Size-dependent control of\ncolloid transport via solute gradients in dead-end channels. Proceedings of the National Academy of Sciences, 113(2), 257-261.\n- Shin, S., Ault, J. T., Warren, P. B., & Stone, H. A. (2017). Accumulation of colloidal particles in flow junctions induced by\nfluid flow and diffusiophoresis. Physical Review X, 7(4), 041038."}, "author_response": "Response to reviewer comments for NCOMMS-25-37255-T “Dielectrocapillarity for\nexquisite control of fluids”\nAnna T. Bui1,2 and Stephen J. Cox2,a)\n1)Yusuf Hamied Department of Chemistry, University of Cambridge, Lensfield Road,\nCambridge, CB2 1EW, United Kingdom\n2)Department of Chemistry, Durham University, South Road, Durham, DH1 3LE,\nUnited Kingdom\n(Dated: September 26, 2025)\nWewouldliketothanktheeditorsandreviewersfortheirtimeinreadingourmanuscript.\nWhile both reviewers are generally positive, finding our work interesting, innovative and\naddresses a critical gap, they also raised important points regarding: (i) assumptions\nunderlying our theoretical models, (ii) the generality of our approach, and (iii) connec-\ntions to experiments. In response, we have carried out additional analyses and revisions\nto fully address these concerns.\nBelow we detail our responses to all comments from the reviewers point-by-point. The\nreviewers’ comments are reproduced in turquoise. Changes are highlighted in magenta.\nThe reference list from the main paper is replicated in full at the end the document;\nthis is simply so citations in text quoted from the manuscript appear correctly.\na)Electronic mail: stephen.j.cox@durham.ac.uk\nOverall Evaluation\nThis manuscript introduces the novel concept of “dielectrocapillarity,” demonstrating how elec-\ntric field gradients (EFGs) can regulate phase behavior, capillary condensation, and adsorption\nof confined fluids. By integrating classical density functional theory (cDFT) with deep learn-\ning, the authors develop a multiscale framework that offers new strategies for applications in\nnanoporous energy storage, gas separation, and neuromorphic devices. The study addresses\na critical gap in understanding electrostrictive effects in fluids, with innovative methodology\nand clear logical progression. While the theoretical framework and simulations provide robust\nsupport for key conclusions, the work could be strengthened by deeper experimental validation,\ncomparative analysis with competing techniques, and expanded discussion of its limitations.\nThe research holds significant promise for both fundamental science and applied technology,\nthough further refinements are recommended to enhance its broader impact.\nWe thank the reviewer for this positive and thoughtful evaluation. We appreciate their recog-\nnition of the novelty and potential impact of our work, and we agree with their constructive\nsuggestions. Below, we provide a detailed response to the specific points raised.\nDetailed Comments\nLimited Experimental Validation: The study relies predominantly on theoretical simulations\nand model derivations without direct experimental data (e.g., observations of dielectrocapillary\neffects in actual nanofluidic devices). The authors should supplement the manuscript with com-\nparisons to existing experimental techniques (e.g., atomic force microscopy, dielectrophoresis\nexperiments) or provide proof-of-concept experimental results to reinforce the credibility of the\nfindings.\nWe are sympathetic to the reviewer’s criticism that our work is primarily computational and\ntheoretical, focusing on nanoscale effects where direct experimental implementation remains\nchallenging. We are actively collaborating with experimental colleagues using surface force\napparatus, however, such nanoscale experiments are extremely challenging and lie beyond the\nscope of the current study. We can, however, link directly to macroscopic experiments which\nnot only give confidence on the faithfulness of our microscopic approach, but also provide\nindirect support for our findings. In particular, dielectrowetting studies have demonstrated that\nthe contact angle of dielectric liquids can be tuned by applying well-controlled inhomogeneous\nelectric fields\n• G. McHale, C. V. Brown, M. I. Newton, G. G. Wells, and N. Sampara, Dielectrowetting\ndriven spreading of droplets, Phys. Rev. Lett. 107, 186101 (2011)\n• G. McHale, C. V. Brown, and N. Sampara, Voltage-induced spreading and superspreading\nof liquids, Nat. Commun 4, 1605 (2013)\nTheseexperimentsdemonstratethatdielectrophoreticforcescandrivedropletspreadingandthe\ncontact angle observed obeys certain phenomenological scaling laws. To further establish the\nconnection between our theoretical findings and experimental reality, we performed additional\nanalyses using exponentially decaying fields analogous to those generated by interdigitated\nelectrodes in the above experiments. This not only allows us to see whether we observe the\nsame behavior between our microscopic approach and the macroscopic experiments, but also\n2\nthe extent to which macroscopic theory can be expected to hold at the nanoscale. Consistent\nwith experiment and macroscopic theory, our results show enhanced wetting as the voltage is\nincreased, with minor differences observed at low voltages – this we can rationalize in terms of\nsuppressed density fluctuations at the interface (an effect omitted in macroscopic approaches).\nImportantly, these obsevations are consistent with a downward shift of T under the application\nc\nof EFGs, which is also the basis of dielectrocapillarity.\nWe have made substantial changes to the manuscript in light of this comment – we have\nadded a new section, Connection to dielectrowetting experiments, and a new Figure 4, which\ndemonstrate how our models capture these effects from first principles and provide a direct\nmicroscopic foundation for the phenomenological laws observed in experiments:\na (z)\ny\nz\nx\nb\nc\ninterdigitated electrodes, shown schematically in (a), decays exponentially. Applying this\npotentialsymmetricallyfrombothconfiningwallsinaslitgeometrywithH ≈ 16σ enhances\nwettingofthesolid-liquidinterface,ascanbeseeninthedensityprofiles(left)andchangesin\nlocal compressibility (right) in (b) (both quantities are normalized by their bulk values). (c)\nElectrostatic free energy per unit area from cDFT (inset) exhibits a quadratic dependence\non ϕ , enabling reconstruction of the contact angle via Eq. 1 (assuming θ = 160◦ and\n0 0\nusing γ = 0.025Jm−2 computed from direct coexistence simulation), in direct analogy\nlv\nwith dielectrowetting experiments14,50.\n(Lines 339–387)\nConnecting to dielectrowetting experiments\nThus far we have introduced dielectrocapillarity as an additional mechanism of control-\nling capillary phenomena, complementing electrocapillarity51 where electrolytes respond\n3\ntoappliedpotentials. Onthemacroscopicscale,theseeffectsmanifestinelectrowetting52\nand dielectrowetting14,50, where the contact angle of a droplet can be tuned with applied\npotentials. In the case of dielectrowetting, a typical experimental setup employs inter-\ndigitated electrodes, as shown schematically in Fig. 4a, which generate an exponentially\ndecaying electrostatic potential ϕ(z) = ϕ exp(−z/d) that drives a droplet of dielectric\n0\nfluid to spread. The resulting contact angle can be described by a modified Young’s\nlaw14,50,\nα\ncosθ(ϕ ) = cosθ + ϕ2, (1)\n0 0 γ 0\nlv\nwhere θ is the Young’s contact angle at zero field, γ the liquid–vapor surface tension,\n0 lv\nand α a material parameter related to the dielectric response of the liquid. This relation-\nship assumes that the electrostatic energy stored in the liquid droplet is well-described\nby dielectric continuum theory, and that d is sufficiently small so that any changes in\nenergyduetotheelectricfieldareeffectivelylocalizedtothesolid-liquidinterface. Under\nthese assumptions, the electrostatic free energy per unit area obeys the simple quadratic\nscaling γ = −αϕ2.\nelec 0\nOur multiscale framework provides a microscopic perspective on this phenomenology,\nand allows us to test whether, at the nanoscale, EFGs indeed enhance the wetting of\ndielectric liquids and the extent to which the scaling prescribed by Eq. 1 holds. To this\nend, we applied an exponentially decaying electrostatic potential with d = 1.1σ to the\nconfined dipolar model, symmetrically from both walls of a solvophobic slit. While not\ntrained on such electrostatic potentials, as can be seen in Fig. 4b, the neural functional\nextrapolates well, yielding physically plausible results, aside from some oscillations at\nhigh wetting that are likely minor artifacts. As ϕ increases, so too does the contact\n0\ndensity at the wall, verifying that enhanced wetting occurs. We quantify this effect by\ncomputing γ = 1 (cid:82)∞ dzϕ(z)n(z), which, as can be seen in Fig. 4c, decreases in\nelec 2 −∞\nan approximately quadratic fashion as ϕ increases. By identifying α = −∂γ /∂(ϕ2),\n0 elec 0\nwe reconstruct the potential-dependent contact angle, as shown in Fig. 4c. While the\nresults of our microscopic theory are broadly in line the macroscopic model (Eq. 1) some\nsubtledifferencesareobserved,especiallyatsmallϕ . Theseappeartobecorrelatedwith\n0\nsignificant changes in the local compressibility near the interface, χ (z) = ∂ρ(z)/∂µ,\nµ\nas ϕ increases, which indicates a suppression of density fluctuations. Such microscopic\n0\ndetails are lacking in the dielectric continuum model that underpins Eq. 1.\nIncomplete Comparison with Traditional Electric Field Methods: While the manuscript distin-\nguishes between uniform electric fields and EFGs, it lacks a systematic comparison of dielec-\ntrocapillarity with conventional techniques like electrowetting or dielectrophoresis in terms of\ncontrol precision, energy consumption, and applicability. A clear articulation of the methods\nunique advantages and limitations is necessary.\nWe thank the reviewer for highlighting the need to compare the phenomenon we propose with\ntraditional electric-field-based fluid manipulation methods. Dielectrocapillarity can be viewed\nas a parallel to electrocapillarity: for dielectric (polar) fluids, it arises from coupling with electric\nfield gradients, while for conducting (ionic) fluids, electrocapillarity arises from coupling with\nthe field itself. Thus, while they are both capillary phenomena, they apply to distinctly different\nsystems (dielectrics and conductors) – beyond this fact, a systematic comparison in terms of\ncontrol precision, energy consumption and applicability challenging.\nSimilar reasoning applies when trying to contrast electrowetting vs dielectrowetting, and fur-\nther, dielectrowetting vs dielectrocapillarity. Indeed, as we make clear in our response to the\n4\nprevious comment, dielectrocapillarity and dielectrowetting should be seen as complementary\nphenomena.\nRegarding dielectrophoresis, this is a nonequilibrium effect that, although governed by the same\ndielectrophoretic forces, we cannot yet tackle with our framework.\nIn light of this comment, we have changed the manuscript to convey that dielectrocapillarity is\na complementary technique to the more common approaches mentioned by the reviewer:\n(Lines 340–344) ...Thus far we have introduced dielectrocapillarity as an additional\nmechanism of controlling capillary phenomena, complementing electrocapillarity51 where\nelectrolytes respond to applied potentials. On the macroscopic scale, these effects man-\nifest in electrowetting52 and dielectrowetting14,50...\nWe also highlight the limitations regarding, and possible extensions to, non-equilibrium phe-\nnomena in the Conclusions:\n(Lines 402–413) The effects uncovered in this work concern the equilibrium behavior\nof dielectric liquids, and omit potentially important nonequilibrium effects such as pore\nentryandexit53–55,electrokineticphenomena56–58,andcontrolledwettingdynamicssuch\nas rate-dependent droplet spreading50. Nonetheless, the implications of our results for\nnonequilibrium behavior are potentially far-reaching. A natural possible progression from\nthis work is to augment our first-principles framework for electromechanics with dynami-\ncalextensionsofcDFT59, openingapromisingroutetowardamicroscopicunderstanding\nof how EFGs impact non-equilibrium processes....\nGeneralizability to Multicomponent Fluids: The current focus on single-component polar fluids\nlimitsrelevancetoreal-worldapplicationsinvolvingcomplexmixtures(e.g., electrolytesolutions,\nmulticomponent gases). Simulations or discussions on extending the model to multicomponent\nsystems would enhance practical utility.\nWe thank the reviewer for bringing up this important point.\nIndeed, experiments have demonstrated liquid–liquid phase separation in binary mixtures under\nstrong electric field gradients:\n• Y. Tsori, F. Tournilhac, and L. Leibler, Demixing in simple fluids induced by electric field\ngradients, Nature 430, 544 (2004)\n• Y. Tsori and L. Leibler, Phase-separation in ion-containing mixtures in electric fields,\nProc. Natl. Acad. Sci. U.S.A 104, 7348 (2007)\nhighlighting the relevance of multicomponent systems in this context. While our study focuses\non the liquid–vapor transition of single-component polar fluids, the theoretical framework is\ngeneral and naturally extends to multicomponent mixtures (indeed we already included results\nfor a simple electrolyte solution with implicit solvation).\nWe now make it clearer when describing the theory that it is readily generalizable to mixtures\n(and that we use it in the present study).\n(Lines 107–111)...Wherecomparisonwithionicfluidsismade, wewillalsoshowresults\nfor a prototypical model comprising oppositely charged hard spheres24; in this case we\nuse a straightforward generalization of cDFT to multicomponent systems.\napproach to mixtures of dielectric fluids, citing the above papers for context.\n(Lines 413–420) ...Moreover, the current framework naturally accommodates mixtures\nof dielectric liquids – in such cases, the excess intrinsic free energy, Fex ([{ρ },βϕ],T)\nintr ν\nacquires a functional dependence on the density fields of all species present, with ν in-\ndexing each component. This generalization opens the door to investigating more intri-\ncate phase behavior and interfacial phenomena—including liquid–liquid phase separation\ndriven by EFGs12,60.\nClarity of Graphical Information: Some curve annotations in Figures 1 and 2 (e.g., differences\nbetween SR/LR interactions) and definitions of non-dimensional units (e.g., reduced units in SI) are unclear. Additional explanations in figure captions or\nthe main text would improve accessibility.\nTo improve accessibility, we have added detailed information in both the figure captions and\nthe main text:\n(Caption to Figure 1) Reorganization of fluids under non-uniform electric fields.\nAn applied electric field, E∗(z) = E∗ sin(2πz/λ)...Reduced units are described in the\nmax\nMethods section.\n(Lines 151–153) ...Quantities labeled with an asterisk are expressed in reduced units,\ndefined in the Methods section.\n(Lines 462–469) Reduced units. When reported, reduced units are defined by T∗ =\nk T/ε, ρ∗ = ρσ3, and E∗ = Eσ3/2ε−1/2, where σ sets the molecular length scale\nB\nand ε the energy scale. Specifically for each fluid: (i) SPC/E water, σ = 3.166˚A,\nε = 0.65kJmol−1; the dipolar fluid, σ = 3.024˚A, ε = 1.87kJmol−1; and (iii) the\nelectrolyte, σ = 2.76˚A, ε = e2/σ, where e is the elementary charge.\n(Caption to Figure 2) Controlling liquid–vapor equilibrium with EFGs....Solid and\ndashed lines show the response of systems with dipolar interactions that are long-ranged\n(LR), e.g., polar molecules, and screened short-ranged (SR), e.g., colloids, respectively.\nThe effect of LR interactions becomes pronounced for λ ≫ σ. The SR fluid is a nearly\nidentical dipolar fluid, but whose Coulomb potential is replaced by erfc(κr)/r where\nκ−1 = 1.5σ...\nUnderdeveloped Discussion of Application Scenarios: While applications in supercapacitors and\nneuromorphic devices are mentioned, linking the findings to specific case studies (e.g., perfor-\nmance improvements in existing nanoporous materials) would more vividly demonstrate trans-\nlational potential.\nWe thank the reviewer for this comment, but wish to reiterate that our focus in this work is on\nestablishing a fundamental framework for understanding how electric field gradients manipulate\nfluids. To the best of our knowledge, dielectrocapillarity is a new, unreported phenomena – it is\ntoo early to give a detailed specific case study on device performance. However, we note that\n6\nwe now directly link our results to dielectrowetting experiments, which helps to link to potential\nadvances in device design based on this physics. We therefore expand our discussion in the\nrevised manuscript to connect with potential applications in this area. In the Conclusions:\n(Lines 388–398) Our findings establish EFGs as a powerful and versatile tool for ma-\nnipulating fluids. We have revealed their ability to structure fluids, modulate phase\ntransitions, and control capillary effects. Crucially, we demonstrate that EFGs not only\ninfluence a fluid’s behavior in bulk, but also give rise to dielectrocapillarity, a new phe-\nnomenon in which capillary condensation and criticality under confinement can be finely\ntuned. By placing this nanoscale physics in direct correspondence with macroscopic di-\nelectrowetting experiments, our work provides a microscopic foundation for the design of\nEFG-controlled wetting and adsorption phenomena.\n(Lines 421–432) The ability to reversibly control phase behavior and adsorption with\nelectric fields unlocks new avenues for manipulating fluids across multiple length scales,\nfrom adaptive nanofluidic devices, to tunable sorption in porous materials, to colloidal\nassembly. Atthenanoscale, nuclearmagneticresonancetechniques3 candirectlyvalidate\nthese effects. With strong EFGs experimentally accessible via atomic force microscope\ntips, optical tweezers, and patterned electrode configurations, our results lay the founda-\ntion for future experimental exploration, paving the way for new strategies in developing\nenergy storage, selective separation, and responsive fluidic technologies.\nApplicability of Theoretical Assumptions: The models reliance on simplified assumptions (hard-\nsphere repulsion, van der Waals attraction) raises questions about its validity for systems with\nstrong hydrogen bonding (e.g., water) or complex interfacial chemistry. A sensitivity analysis\nor discussion of these limitations is recommended.\nWe thank the reviewer for raising this important point that, which is also raised by Reviewer 2.\nTo assess the robustness of our conclusions beyond simplified dipolar models, we have repeated\nour analysis for SPC/E water. The results confirm our main findings: dielectrophoretic rise\nis observed at supercritical temperature, and the critical temperature shifts downward with\nincreasing EFG strength. Importantly, water also reveals additional features due to its intrinsic\ncharge asymmetry, which we now highlight and discuss. These new results are incorporated\nthroughout the revised manuscript and provided in detail in the SI.\n(Lines 84–94) To investigate how EFGs influence polar fluids, we primarily consider a\nminimal molecular model that incorporates soft-core repulsion, van der Waals attraction,\nand long-range dipolar interactions. The advantage of using such a simple molecular\nmodel is that we can exhaustively explore a broad range of thermodynamic conditions,\nwhilepotentiallyuncoveringcommonbehaviorsamongpolarfluids,frommolecularliquids\nto colloidal systems. In addition, we also investigate a commonly used simple point\ncharge model for water (SPC/E21) that explicitly incorporates hydrogen-bonding, under\nthermodynamic conditions close to its critical point.\n(Lines 148–151) ...As a starting point, we characterize the bulk response of water, the\nsimple polar fluid, and the electrolyte, all under supercritical conditions, when subjected\nto a sinusoidal electric field, E∗(z) = E∗ sin(2πz/λ), as shown in Fig. 1a.\nmax\n7\na\nc\ndielectrophoretic rise (dipolar fluid)\n+ f DEP ~ ∇E2\n-\n|fDEP|>0\n|fDEP|=0\nz\nb dielectrophoretic rise (water) d electrophoretic rise\ntotal + f EP = qE\n-\nanion cation\n|fEP|=0\n|fEP|>0\nFigure 1: Reorganization of fluids under non-uniform electric fields. An applied elec-\ntric field, E∗(z) = E∗ sin(2πz/λ), shown in (a), induces pronounced density variations\nmax\nin bulk supercritical water, as can clearly be seen from a snapshot of a molecular dynamics\nsimulation. (b) Results from cDFT for the number (ρ∗(z), top) and charge (n∗(z), bot-\ntom) densities capture this behavior. It can clearly be seen that number density is locally\ndepleted where the |∇E| is large, and locally enhanced where |∇E| is small. The same\nqualitative behavior is seen in (c) for a supercritical dipolar fluid, except that its response\nis symmetric, in contrast to water where local depletion depends upon the sign of ∇E. (d)\nIn contrast to both water and the dipolar fluid, an electrolyte is locally depleted in regions\nof low field strength due to electrophoretic forces (purple and green lines show cation and\nanion density, respectively, while the blue line shows the total density). Reduced units are\ndescribed in the Methods section.\ndipolar fluid water\nλ = 6.6 σ λ = 2.2 σ λ = 6.3 σ λ = 2.1 σ\nλ = 1 .7\nλ\nλ σ\n=\n= 2\n3\n.\n.\n2\n3\nσ\nσ\nλ=\nλ\nλ\n=\n= 2 .\n3\n1 1\n.\n.\nσ\n2\n6 σ\nσ\nλ = 6.6 σ λ = 6.3 σ\nFigure S13: Dielectrophoretic rise in water under supercritical condition. The left\npanel shows the density response of the simple dipolar fluid (500K, same data as in main\nmanuscript) while the the right panel shows that for SPC/E water (700K).\n(Lines 154–174) In the case of water, Fig. 1b, we see that the applied electric field in-\nduces a significant structural reorganization along the field direction; its average density\nprofile ρ∗(z) is locally depleted in regions of weaker field strength, while molecular reori-\n8\ndipolar fluid water\n0.0 VÅ-1 0.0 VÅ-1\n0.4 VÅ-1 0.4 VÅ-1\nT=300K T=500K\n0.0 VÅ-1 0.4 VÅ-1 0.8 VÅ-1 0.0 VÅ-1 0.4 VÅ-1 0.8 VÅ-1\nFigure S14: Water liquid–vapor equilibrium under EFGs. The intensity of the field\nE is labeled. The bottom panel shows the density response as the chemical potential is\nmax\nvaried continuously.\nentation leads to an inhomogeneous average charge density distribution n∗(z). As water\nis overall neutral and therefore experiences no net electrophoretic force, the observed lo-\ncal reorganization arises instead from dielectrophoretic forces, f ∼ ∇E233,34, which\nDEP\npush the fluid towards regions of higher electric field strength—an effect termed “dielec-\ntrophoretic rise.” This dielectrophoretic force is the same that drives dielectrophoresis,\nwhich is widely exploited to manipulate biological cells35 and colloids36, but its role in\nmolecular fluids has received little attention. While water provides an important example\nof a polar fluid, the observed effects are by no means specific to aqueous systems. As\nseen in Fig. 1c, the overall picture is the same for the simple polar fluid, aside from the\nfact that it exhibits a symmetric response, whereas for water, depletion is stronger in\nregions where ∇E < 0 than where ∇E > 0 due to the inherent charge asymmetry of\nthe water molecule.\n(Lines 186–191) To illustrate this, we present in Fig. 2a how dielectrophoretic rise can\nbeamplifiedbycontrollingtheappliedfield. Owingtotheirqualitativelysimilarbehavior,\nin the remainder of the article we focus on the dipolar fluid, with results for water given\nin the SI, Figs. S13 & S14.\n(Lines 263–267) For water, we observe similar behavior, with a downward shift in T\nc\nof approx. 50K at E = 0.4V˚A−1 (Fig. S14). These observations highlight the\nmax\nexquisite level of control that one can exert over dielectric fluids with EFGs.\nConclusion\nThis work establishes a groundbreaking framework for fluid manipulation, with the concept of\ndielectrocapillarity offering significant promise for nanofluidics and energy technologies. Despite\ngaps in experimental validation and application-level analysis, the originality of the concept and\nmethodological advancements are compelling. Addressing the above commentsparticularly by\n9\nincorporating experimental data, enhancing comparative analyses, and refining graphical clarity-\nwouldstrengthenthemanuscriptsscientificrigorandbroaderappeal. Withtheseimprovements,\nthe study would solidly position itself as a high-impact contribution to Nature Communications.\nWe thank the reviewer for their constructive evaluation and thoughtful suggestions, which have\nsignificantly improved the manuscript. In response, we have:\n1. incorporated new analyses connecting our framework directly to dielectrowetting experi-\nments, including a new section and Figure 4;\n2. added comparative discussion with established electric-field-based methods such as elec-\ntrowetting, electrocapillarity, and dielectrophoresis;\n3. clarified the generalizability of our framework to multicomponent fluids and highlighted\nits implications for future studies;\n4. refined figure captions and the definition of reduced units to improve accessibility; and\n5. repeatedouranalysisforSPC/Ewatertotestrobustnessbeyondsimplifieddipolarmodels,\nwith new results presented in the main text and SI (Figs. S13 and S14).\nThismanuscriptpresentsanumericalinvestigationofthepropertiesofpolarfluidsunderelectric\ngradients. The starting point is to use dielectrophoresis, a phenomenon under which the polar\nfluids undergoes local forces in an electric gradient. The central claim is that such forces could\nbe used to compact or dilate the fluids. The manuscript explores the possibility of using these\ncompaction/dilatation forces to tune the liquid-vapor phase diagram of the fluid and then to\nuse that to change wetting properties inside confined systems. In essence, the idea is to explore\nhow dielectrophoresis might modify capillary effects in a similar way as electrophoresis modifies\ncapillary forces.\nThe idea itself is interesting and creative. Dielectrophoresis being quite subtle, I have no doubt\nthat peculiar phenomena might arise in specific systems, so dielectrocapillary effects are indeed\na possibility that has not been studied so far. The manuscript also includes an investigation\nof screening of the dielectrophoretic interactions which is captivating too.\nWe thank the reviewer for their thoughtful evaluation and for highlighting the novelty and\ninterest of our work.\nAt this stage, however, the model makes 2 major assumptions which are not well justified and,\ntherefore, make it hard to validate whether or not the effect might be real. First, an equilibrium\nAI model is used to study nonequilibrium phenomena. Second, capillarity related to properties\nnear confining surfaces however the response of the surfaces to the electrical gradients are not\naccounted for as is the case for electrocapillarity.\nRegarding the first point, we emphasize that throughout the manuscript we investigate the\nequilibrium response of fluids. The theoretical basis is laid out in recently formulated liquid\nstatetheorybasedonhyperdensity functionaltheory(anextension ofclassical density functional\ntheory) that is capable of describing electromechanics in fluids as an emergent phenomenon:\n• A. T. Bui and S. J. Cox, A first-principles approach to electromechanics in liquids, J.\nPhys.: Condens. Matter 37, 285101 (2025).\nWe have now summarize the key results used in our study in the Methods section.\nWe identified several places where the use of the word “dynamic” or “dynamical” may have\ngiven an incorrect impression that we were investigating nonequilibrium effects. We apologize\nif this was misleading, and we have removed references to “dynamic” or “dynamical.” We also\nnow include a clear statement in the conclusions that we only investigate equilibrium effects,\nand acknowledge that we omit potentially intersting nonequilibrum effects. We also give an\noutlook regarding how our work may be viewed as a platform for investigating such phenomena.\nRegarding the second point, we agree that this is an important consideration. In our framework,\nthe confining walls are modeled as fixed external potentials, and thus their back-response to the\napplied electric gradients is not explicitly captured. In macroscale dielectrowetting experiments,\nthis effect is generally expected to be small, and indeed, experimental observations support\nthis assumption. To address this limitation, we have included a discussion of field gradients\ngenerated by patterned electrodes in dielectrowetting phenomena, which provides insight into\nhow surface responses may influence the fluid behavior. This is further detailed in point 6.\n11\n(Lines 470–508) Hyperdensity functional theory. We employed the recently de-\nveloped first-principles theory for electromechanics27 based on hyperdensity functional\ntheory62, which provides an exact variational framework for the coupled electromechan-\nical response of fluids. For a single-component fluid, at specified µ, T, planar external\nnon-electrostatic potential V (z), and electrostatic potential ϕ(z), the grand potential\next\nfunctional is\nΩ([ρ,βϕ],T) = Fid ([ρ],T)+Fex ([ρ,βϕ],T)\nintr intr\n(cid:90)\n+ dzρ(z)[V (z)−µ],\next\nwheretheidealintrinsicHelmholtzfreeenergyisFid ([ρ],T) = k T (cid:82) dzρ(z)[lnζ−1Λ3ρ(z)−\nintr B\n1], withΛdenoting the thermalde Broglie wavelength andζ denotes a partition function\naccounting for intramolecular molecular degrees of freedom of each fluid particle.\nThe equilibrium number and charge density profiles are obtained by solving the Euler–\nLagrange (EL) equation\nζ (cid:104) (cid:105)\nρ (z) = exp −βV (z)+βµ+c(1)(z;[ρ ,βϕ],T) ,\neq Λ3 ext eq\nand evaluating\nn (z) = n(1)(z;[ρ ,βϕ],T),\neq eq\nwith the one-body direct correlation functional c(1) and charge density functional n(1)\ndefined as first functional derivatives of the excess intrinsic Helmholtz free energy func-\ntional\nδβFex ([ρ,βϕ],T)\nc(1)(z;[ρ,βϕ],T) = − intr ,\nδρ(z)\nδFex ([ρ,βϕ],T)\nn(1)(z;[ρ,βϕ],T) = intr .\nδβϕ(z)\nFor fluids whose intermolecular interactions are long-ranged, such as ionic and dielectric\nfluids, the functional dependence of c(1) and n(1) on ρ and ϕ is, in general, non-local.\nNote that, as we only discuss these functionals evaluated at equilibrium in this work, we\ndrop the “eq” subscript for notational convenience.\nIn order to learn these functional mappings, we applied the techniques introduced in\nRefs.24,27 where the local functionals of short-ranged mimic fluids, c (1) (z;[ρ,βϕ],T)\nR\n(1)\nand n (z;[ρ,βϕ],T), were learned from molecular simulation data using the neural\nR\nfunctionalmethod25. ASRmimicfluidisdefinedasasystemwhoseCoulombinteractions\nare replaced with\n1 erfc(κr)\n→ .\nr r\nFor SPC/E water and the dipolar fluid, we used κ−1 = 4.5˚A. For the electrolyte, we\nused the functionals trained in Ref. 24, with κ−1 = 5.0˚A. The effects of long-ranged\ninteractionsareaccountedforinawell-controlledmeanfieldfashionusinglocalmolecular\nfield theory (LMFT). Implementation details and full derivations are provided in Ref.27\nand Section S1 of the SI. We also validated the theory against computer simulations in\nSection S3 of the SI.\n(Lines 67–69) ...within the grand canonical ensemble, which is most representative of\nreal conditions in which fluid molecules can dynamically enter and leave a pore.\n12\n(Lines 77–79) Furthermore, the ability to dynamically regulate hysteresis introduces a\nnew level of programmability in nanofluidic systems...\n(Lines 393–395) ...dielectrocapillarity, a new phenomenon in which capillary condensa-\ntion and criticality under confinement can be dynamically finely tuned.\n(Lines 402–413) The effects uncovered in this work concern the equilibrium behavior\nof dielectric liquids, and omit potentially important nonequilibrium effects such as pore\nentryandexit53–55,electrokineticphenomena56–58,andcontrolledwettingdynamicssuch\nas rate-dependent droplet spreading50. Nonetheless, the implications of our results for\nnonequilibrium behavior are potentially far-reaching. A natural possible progression from\nthis work is to augment our first-principles framework for electromechanics with dynami-\ncalextensionsofcDFT59, openingapromisingroutetowardamicroscopicunderstanding\nof how EFGs impact non-equilibrium processes.\nAnother main point is that it is not clear that the orders of magnitude of changes investigated\non a minimal polar fluid model might indeed be significant and hence accessible in experiments.\nThisisnotmadeeasierbythecurrentstyleofthemanuscriptwhichsometimesmakesunjustified\nclaims. At this stage I rather see the model as a remarkable and creative hypothesis, but its\npotential applicability in realistic systems is not discussed at all, making it hard to see if it could\nbe real.\nWe thank the reviewer for raising this important point that, which is also raised by Reviewer\n1. As we discuss in response to Reviewer 1, we now include results for model of water with\nexplicit hydrogen bonds and electrostatics, thus demonstrating that the results in this work\nare applicable to a realistic fluid. We reproduce our response to reviewer 1 below for ease of\nreference:\nTo assess the robustness of our conclusions beyond simplified dipolar models, we have repeated\nour analysis for SPC/E water. The results confirm our main findings: dielectrophoretic rise\nis observed at supercritical temperature, and the critical temperature shifts downward with\nincreasing EFG strength. Importantly, water also reveals additional features due to its intrinsic\ncharge asymmetry, which we now highlight and discuss. These new results are incorporated\nthroughout the revised manuscript and provided in detail in the SI.\n(Lines 84–94) To investigate how EFGs influence polar fluids, we primarily consider a\nminimal molecular model that incorporates soft-core repulsion, van der Waals attraction,\nand long-range dipolar interactions. The advantage of using such a simple molecular\nmodel is that we can exhaustively explore a broad range of thermodynamic conditions,\nwhilepotentiallyuncoveringcommonbehaviorsamongpolarfluids,frommolecularliquids\nto colloidal systems. In addition, we also investigate a commonly used simple point\ncharge model for water (SPC/E21) that explicitly incorporates hydrogen-bonding, under\nthermodynamic conditions close to its critical point.\n(Lines 148–151) ...As a starting point, we characterize the bulk response of water, the\nsimple polar fluid, and the electrolyte, all under supercritical conditions, when subjected\nto a sinusoidal electric field, E∗(z) = E∗ sin(2πz/λ), as shown in Fig. 1a.\nmax\n13\na\nc\ndielectrophoretic rise (dipolar fluid)\n+ f DEP ~ ∇E2\n-\n|fDEP|>0\n|fDEP|=0\nz\nb dielectrophoretic rise (water) d electrophoretic rise\ntotal + f EP = qE\n-\nanion cation\n|fEP|=0\n|fEP|>0\nFigure 1: Reorganization of fluids under non-uniform electric fields. An applied elec-\ntric field, E∗(z) = E∗ sin(2πz/λ), shown in (a), induces pronounced density variations\nmax\nin bulk supercritical water, as can clearly be seen from a snapshot of a molecular dynamics\nsimulation. (b) Results from cDFT for the number (ρ∗(z), top) and charge (n∗(z), bot-\ntom) densities capture this behavior. It can clearly be seen that number density is locally\ndepleted where the |∇E| is large, and locally enhanced where |∇E| is small. The same\nqualitative behavior is seen in (c) for a supercritical dipolar fluid, except that its response\nis symmetric, in contrast to water where local depletion depends upon the sign of ∇E. (d)\nIn contrast to both water and the dipolar fluid, an electrolyte is locally depleted in regions\nof low field strength due to electrophoretic forces (purple and green lines show cation and\nanion density, respectively, while the blue line shows the total density). Reduced units are\ndescribed in the Methods section.\ndipolar fluid water\nλ = 6.6 σ λ = 2.2 σ λ = 6.3 σ λ = 2.1 σ\nλ = 1 .7\nλ\nλ σ\n=\n= 2\n3\n.\n.\n2\n3\nσ\nσ\nλ=\nλ\nλ\n=\n= 2 .\n3\n1 1\n.\n.\nσ\n2\n6 σ\nσ\nλ = 6.6 σ λ = 6.3 σ\nFigure S13: Dielectrophoretic rise in water under supercritical condition. The left\npanel shows the density response of the simple dipolar fluid (500K, same data as in main\nmanuscript) while the the right panel shows that for SPC/E water (700K).\n(Lines 154–174) In the case of water, Fig. 1b, we see that the applied electric field in-\nduces a significant structural reorganization along the field direction; its average density\nprofile ρ∗(z) is locally depleted in regions of weaker field strength, while molecular reori-\n14\ndipolar fluid water\n0.0 VÅ-1 0.0 VÅ-1\n0.4 VÅ-1 0.4 VÅ-1\nT=300K T=500K\n0.0 VÅ-1 0.4 VÅ-1 0.8 VÅ-1 0.0 VÅ-1 0.4 VÅ-1 0.8 VÅ-1\nFigure S14: Water liquid–vapor equilibrium under EFGs. The intensity of the field\nE is labeled. The bottom panel shows the density response as the chemical potential is\nmax\nvaried continuously.\nentation leads to an inhomogeneous average charge density distribution n∗(z). As water\nis overall neutral and therefore experiences no net electrophoretic force, the observed lo-\ncal reorganization arises instead from dielectrophoretic forces, f ∼ ∇E233,34, which\nDEP\npush the fluid towards regions of higher electric field strength—an effect termed “dielec-\ntrophoretic rise.” This dielectrophoretic force is the same that drives dielectrophoresis,\nwhich is widely exploited to manipulate biological cells35 and colloids36, but its role in\nmolecular fluids has received little attention. While water provides an important example\nof a polar fluid, the observed effects are by no means specific to aqueous systems. As\nseen in Fig. 1c, the overall picture is the same for the simple polar fluid, aside from the\nfact that it exhibits a symmetric response, whereas for water, depletion is stronger in\nregions where ∇E < 0 than where ∇E > 0 due to the inherent charge asymmetry of\nthe water molecule.\n(Lines 186–191) To illustrate this, we present in Fig. 2a how dielectrophoretic rise can\nbeamplifiedbycontrollingtheappliedfield. Owingtotheirqualitativelysimilarbehavior,\nin the remainder of the article we focus on the dipolar fluid, with results for water given\nin the SI, Figs. S13 & S14.\n(Lines 263–267) For water, we observe similar behavior, with a downward shift in T\nc\nof approx. 50K at E = 0.4V˚A−1 (Fig. S14). These observations highlight the\nmax\nexquisite level of control that one can exert over dielectric fluids with EFGs.\nI have several detailed comments, that I put below.\n(1) What goes inside the model and what data the AI is trained on is quite opaque, at least\nfrom the main text. Is it trained on the different fluids mentioned or on other fluids? How is\nthe functional learned exactly, aka what is the training process? (related to one of my main\nquestion) What makes us sure that a model that was designed and trained at equilibrium and\n15\nis potentially suitable to investigate driving by uniform fields and then suitable to investigate\ndrivings by non uniform fields? Are some standard results in the field recovered? At this stage\nit is still not clear to me how the AI training enables one to study the system faster/better so\nit would be nice to discuss potential limitations/advantages a bit more.\nWethank the reviewer forraising theseimportant points regarding theconstruction and training\nof our model. In the revised manuscript, we have clarified that for each fluid considered (SPC/E\nwater, the dipolar fluid, and the restricted primitive model of an electrolyte) we construct a\nneuralfunctionaltrainedonsimulationdataspecifictothatsystem. Thephenomenaexploredin\nthe manuscript are purely equilibrium in nature. We have verified that our approach reproduces\nstandard simulation results, including the bulk equation of state, the zero-field binodal, and the\nnumber and charge density profiles. Furthermore, we now include results demonstrating that\nthe theory can recover results from dielectrowetting experiments, as discussed in point 6.\nTo address this point, we have moved some content from the supporting information to the\nmain manuscript. Specifically, we have added the construction, training, and validation of the\nAI model in the Methods:\n(Lines 509–534) Generation of training data. For the electrolyte, we employed the\nfunctional reported in Ref.24 based upon the restricted primitive model. For SPC/E\nwater and the dipolar fluid, we generated training data using a combination of grand\ncanonical Monte Carlo (GCMC) and molecular dynamics (MD) simulations to construct\n(1) (1)\nlocal reference functionals c (z;[ρ,βϕ],T) and n (z;[ρ,βϕ],T).\nR R\nFor each fluid, we\nsampled ∼2000 randomized external conditions spanning both subcritical and super-\ncritical regimes. Random inhomogeneous electrostatic potentials of the form ϕ(z) =\nϕ cos(2πkz) were applied. In some cases, planar walls of the form of a 9-3 Lennard–\ncode63) determined the mean particle number, which was then used to initialize MD\nsimulations in the NVT ensemble with LAMMPS64, from which the number and charge\ndensityprofilesaresampled. Forwater,wedefinedthemolecularcenterforsamplingρ(z)\nat the oxygen site, and for the dipolar fluid at the midpoint between the two opposite\ncharges. The charge density was computed as n(z) =\n(cid:80) (cid:80)′\nq ⟨δ(z−z )⟩, where the\ni α α α\nouter sum is over all molecules, and the inner primed sum is over all sites belonging to\nmolecule i, with z and q indicating the z coordinate and charge, respectively, of site α.\nα α\nThis hybrid scheme reproduces the accuracy of pure GCMC sampling while significantly\naccelerating convergence of inhomogeneous structures. The total computation time for\nthe generation of the entire dataset is in the order of ∼ 105 CPU hours.\n(Lines 535–555) Training neural functionals. For both SPC/E water and the dipo-\n(1) (1)\nlar fluid, we train two neural networks to represent c ([ρ,βϕ],T) and n ([ρ,βϕ],T)\nR R\nfollowing the local learning strategy25. The machine learning routine was implemented\nin Keras/Tensorflow65. Inputs consisted of local density and electrostatic potential in a\nsliding spatial window of size 10˚A from the center of the position of interest. To effec-\ntively learn spatial variations, the model internally computes the gradient of βϕ(z) using\nacentraldifferencescheme. Inadditiontothesespatiallyvaryinginputs, aseparateinput\nnode encodes T as a scalar. The full architecture details are provided in Figs. S5 and S6.\nThe dataset was split roughly in a 3:1:1 ratio for training, validation, and test sets. Mod-\nels were trained for 200 epochs with a batch size of 256, using an exponentially decaying\nlearning rate starting at 0.001, achieving errors comparable to the estimated simulation\n16\nnoise. The training of the neural networks was done on a GPU (NVIDIA GeForce RTX\n3060) in a few hours. We also verified that the trained functional can recover standard\nsimulation results including bulk equation of state (Fig. S7) the binodal at zero applied\nfield (Fig. S8) as well as number density and charge density response (Fig. S9).\n(2) The electrophoretic rise part could be described and explained more carefully. At this stage,\nI still dont understand it. I believe this is mainly caused by an absence of details. First, in\nwhich direction is the field E imposed? Along the z direction? To me it appears that regions\nof higher field actually have a locally neutral ion concentration and so there should be no force\non the fluid in these regions, yet this is apparently not what is observed. I can not make\nsense of the schematic drawing because it is not clear where the periodic unit is taken along\nthe axis and what the line represents. Eventually I find it would be worthwhile to discuss a\nlittle bit how, already at this stage, working in the muVT ensemble instead of NVT leads to\ninteresting differences compared to more traditional simulations setups where the total density\nvaries spatially a lot. This is still surprising to me since I expect fluid to be more or less\nincompressible and so homogeneous over such lengthscales. Actually, the relationship between\nthe densities ρ* and n* is not clear from the images or the caption. Maybe one should consider\nputting in that information maybe twice. Usually n is used for numbers and so would be used\ninstead of rho, so that convention was puzzling.\n(3) It is equally hard to understand the dielectrophoretic rise for similar reasons. What is n* in\nthe case of the polar fluid? The fluid should be locally neutral.\nWe address points 2 and 3 together regarding the electrophoretic and dielectrophoretic rises.\nThe applied electric field is oriented along the z direction, and the structural reorganization of\nthe fluid occurs along this axis.\nTo clarify the mechanisms, we have added a molecular depiction through a simulation snapshot\nof bulk water undergoing dielectrophoretic rise (Fig. 1a).\nFor efficient sampling of the condensed phase, training data is acquired using a combina-\ntion of grand canonical Monte Carlo (GCMC) and molecular dynamics (MD) simulations,\nwith randomly generated inhomogeneous potential energy landscapes in a planar ge-\nometry. We first use GCMC to determine the average total number of particles N at\na specified {µ;V;T;V (z);ϕ(z)}, and then perform canonical MD simulations with\next\n{N;V;T;f (z);E(z)}. We have validated that this approach gives results indistinguishable\next\nfrom when we only performed GCMC simulations, yet significantly helps to converge density\nprofiles\n(Lines 503–528) Generation of training data. ... For SPC/E water and the dipolar\nfluid, we generated training data using a combination of grand canonical Monte Carlo\n(GCMC) and molecular dynamics (MD) simulations to construct local reference func-\n(1) (1)\ntionals c (z;[ρ,βϕ],T) and n (z;[ρ,βϕ],T).\nR R\nFor each fluid, we\nsampled ∼2000 ran-\ndomizedexternalconditionsspanningbothsubcriticalandsupercriticalregimes. Random\ninhomogeneouselectrostaticpotentialsoftheformϕ(z) = ϕ cos(2πkz)wereapplied. In\n0\nsomecases, planarwallsoftheformofa9-3Lennard–Jonespotentialwerealsoincluded.\nFor each condition, GCMC (performed with our own code63) determined the mean parti-\ncle number, which was then used to initialize MD simulations in the NVT ensemble with\nLAMMPS64,fromwhichthenumberandchargedensityprofilesaresampled. Thishybrid\n17\nscheme reproduces the accuracy of pure GCMC sampling while significantly accelerating\nconvergence of inhomogeneous structures...\nRegarding the relationship between number and charge densities: The number density ρ(z) is\ndefined at a molecular center: for water, the oxygen site; for the dipolar fluid, the midpoint\nbetween the two point charges; for the electrolyte, at the center of each ion. The charge\ndensity is defined as n(z) =\n(cid:80) (cid:80)′\nq ⟨δ(z −z )⟩, where the outer sum is over all molecules,\ni α α α\nand the inner primed sum is over all sites belonging to molecule i, with z and q indicating\nα α\nthe z coordinate and charge, respectively, of site α. For water and the dipolar fluid, there is no\nsimple analytical relation between n(z) and the molecular center number density; instead, n(z)\nis evaluated through the learned functional n(1)(z;[ρ,βϕ],T). For electrolytes, the relationship\nis straightforward: n(z) = q ρ (z)+q ρ (z). We note that the use of ρ for number density\n+ + − −\nand n for charge density follows conventions in classical DFT and microscopic simulations,\nwhich seems to be the opposite of the typical notation in continuum methods.\nThis information is now in the Methods\n(Lines 475–482) The equilibrium number and charge density profiles are obtained by\nsolving the Euler–Lagrange (EL) equation\nζ (cid:104) (cid:105)\nρ (z) = exp −βV (z)+βµ+c(1)(z;[ρ ,βϕ],T) ,\neq Λ3 ext eq\nand evaluating\nn (z) = n(1)(z;[ρ ,βϕ],T),\neq eq\nwith the one-body direct correlation functional c(1) and charge density functional n(1)\ndefined as first functional derivatives of the excess intrinsic Helmholtz free energy func-\ntional\nδβFex ([ρ,βϕ],T)\nc(1)(z;[ρ,βϕ],T) = − intr ,\nδρ(z)\nδFex ([ρ,βϕ],T)\nn(1)(z;[ρ,βϕ],T) = intr .\nδβϕ(z)\nFor fluids whose intermolecular interactions are long-ranged, such as ionic and dielectric\nfluids, the functional dependence of c(1) and n(1) on ρ and ϕ is, in general, non-local.\nNote that, as we only discuss these functionals evaluated at equilibrium in this work, we\ndrop the “eq” subscript for notational convenience.\nWe also rewrote the description of dielectrophoretic and electrophoretic rise in the main text\nand the caption of Figure 1.\n(Caption to Figure 1) Reorganization of fluids under non-uniform electric fields.\nAn applied electric field, E∗(z) = E∗ sin(2πz/λ), shown in (a), induces pronounced\nmax\ndensity variations in bulk supercritical water, as can clearly be seen from a snapshot of\na molecular dynamics simulation. (b) Results from cDFT for the number (ρ∗(z), top) and charge (n∗(z), bottom) densities capture this behavior. It can clearly be seen that\nnumber density is locally depleted where the |∇E| is large, and locally enhanced where\n|∇E| is small. The same qualitative behavior is seen in (c) for a supercritical dipolar\nfluid, except that its response is symmetric, in contrast to water where local depletion\ndepends upon the sign of ∇E. (d) In contrast to both water and the dipolar fluid, an\n18\nelectrolyte is locally depleted in regions of low field strength due to electrophoretic forces\n(purple and green lines show cation and anion density, respectively, while the blue line\nshows the total density). Reduced units are described in the Methods section.\n(Lines 154–174) In the case of water, Fig. 1b, we see that the applied electric field in-\nduces a significant structural reorganization along the field direction; its average density\nprofile ρ∗(z) is locally depleted in regions of weaker field strength, while molecular reori-\nentation leads to an inhomogeneous average charge density distribution n∗(z). As water\nis overall neutral and therefore experiences no net electrophoretic force, the observed lo-\ncal reorganization arises instead from dielectrophoretic forces, f ∼ ∇E233,34, which\nDEP\npush the fluid towards regions of higher electric field strength—an effect termed “dielec-\ntrophoretic rise.” This dielectrophoretic force is the same that drives dielectrophoresis,\nwhich is widely exploited to manipulate biological cells35 and colloids36, but its role in\nmolecular fluids has received little attention. While water provides an important example\nof a polar fluid, the observed effects are by no means specific to aqueous systems. As\nseen in Fig. 1c, the overall picture is the same for the simple polar fluid, aside from the\nfact that it exhibits a symmetric response, whereas for water, depletion is stronger in\nregions where ∇E < 0 than where ∇E > 0 due to the inherent charge asymmetry of\nthe water molecule.\n(Lines 175–183) In contrast, applying a sinusoidal field to an electrolyte induces “elec-\ntrophoreticrise”inwhichthefluidmigratestowardregionsoflowerelectricfieldstrength\n(Fig. 1d). This is a result of the electrophoretic force f = qE, where E is the local\nEP\nfieldstrengthandq istheioniccharge, causingtheanionsandcationstoreorganize, with\npeaks in their density profiles out of phase due to their opposing charges. In this way,\npolar fluids and ionic fluids display electromechanical responses that are fundamentally\ndistinct from each other.\n(4) I do not understand how the phase diagram is constructed this is information that is\ncompletely in the SI but some short explanation would be especially useful. Would it be\npossible to give some actual illustration of what the separated phases would look like? Do they\nreally have the wiggly interfaces that are shown on the sketches? Because then this creates a\nsurface tension in the fluid that is much bigger than that for a flat interface; so theres a price\nto pay for this compacted state which makes the story a lot more complicated.\nTo determine the liquid–vapor coexistence line at zero field, we compute isotherms of the\nchemical potential as a function of bulk density and apply Maxwell construction to identify the\ncoexisting liquid and vapor densities at the binodal. That is, we explicity feed in a homogeneous\nbulkdensityintotheneuralnetwork. Undernon-uniformelectricfields, duetoelectromechanical\ncoupling,wedonotknowapriori theexactformoftheaveragemicroscopicone-bodydensity. In\nthiscase, weinsteadfixµ,T andobtaininhomogeneoussolutionsbysolvingtheEuler–Lagrange\nequation. The resulting chemical potential isotherms, expressed in terms of the mean density,\nexhibit discontinuities that mark phase separation. Stable branches are distinguished from\nmetastable ones by comparing grand potentials. The binodal is thus constructed by locating\nliquid and vapor solutions with equal grand potentials, rather than by solving for solution with\na liquid–vapor interface explicitly.\nin bulk, not literal depictions of curved interfaces – we appreciate that this may have been\nmisleading. In addition to the changes made in Fig. 1, which hopefully clarify that we are\napplying the sinusoidal field to a bulk system, in Fig. 2c we now explicitly plot both liquid and\nvapor solutions across the chemical potential range, which provides a direct illustration of the\nseparated phases.\nT*<T*\nc, 0\na\nc\nλ = 6.6 σ λ = 2.2 σ *\nd\nb\nλ\n= 1\nλ\n.7\n=\nσ\n2\n.2σ\nλ= 3 .3σ\nλ = 6.6 σ\nxamE\ngnisaercni\nμ\ngnisaercni\nT*>T*\nc, 0\nEm * ax = 0 Em * ax = 4 Em * ax = 8\nT* > Tc*,0\nEm * ax = 0\nEm * ax = 4\nT* < Tc*,0\nFigure 2: Controlling liquid–vapor equilibrium with EFGs. (a) At T∗ > T∗ where\nc,0\nT∗ ≈ 1.97, increasing EFGs by independently varying λ and E amplifies dielec-\nc,0 max\ntrophoretic rise. (b) The local electrostrictive response of the dipolar fluid, as measured by\ntheriseinthemaximumdensitypeakrelativetozerofield∆ρ∗ , ishighlynon-linear. Solid\nmax\nanddashedlinesshowtheresponseofsystemswithdipolarinteractionsthatarelong-ranged\n(LR),e.g., polarmolecules, andscreenedshort-ranged(SR),e.g., colloids, respectively. The\neffect of LR interactions becomes pronounced for λ ≫ σ. The SR fluid is a nearly identical\ndipolar fluid, but whose Coulomb potential is replaced by erfc(κr)/r where κ−1 = 1.5σ...\n(c)AtanisothermwhereT∗ < T∗ , stablesolutionsforthedensityρ∗(z)underasinusoidal\nc,0\nelectricfieldwithλ/σ = 1.7, areshownfordifferentvaluesofthechemicalpotential. These\nresults are used to investigate liquid–vapor coexistence. (d) Results in light pink show the\nbinodal of the dipolar fluid in the absence of an electric field. At E∗ = 4, T∗ shifts\nmax c\nto a lower temperature, as seen in the binodal in dark purple. Solid symbols show results\nobtained from the multiscale cDFT approach, while crosses indicate estimates of T∗ using\nc\nthe law of rectilinear diameters and critical exponents28. Solid lines serve as a guide to the\neye.\nIn the main text we added:\n(Lines 237–254) In Fig. 2c, we show the density profiles of the fluid along the direction\nof an external sinusoidal electric field with λ/σ = 1.7, at T∗ < T∗ , where T∗ is the\nc,0 c,0\ncritical temperature in the absence of an external field. Results are shown for different\nchemical potentials. At zero field, the stable solutions separate into homogeneous vapor\nand liquid states at low and high chemical potential, respectively. As the field strength\nincreases, stronger EFGs (in absolute terms) not only give rise to dielectrophoretic rise\nbut also destabilize liquid–vapor phase separation. Consequently, within the coexistence\nregion, the vapor phase becomes denser, or contracts, while the liquid phase expands.\nmax\nsupercritical fluid. By locating vapor and liquid solutions with equal grand potentials, in\nFig. 2d we map out the liquid–vapor binodal curve for the dipolar fluid, both without an\nexternal field and under this sinusoidal field with E∗ = 4.\nmax\nand more extensively in the Methods\n(Lines 556–595) Using the neural functionals. Evaluating the trained neural func-\ntionals is fast (∼ milliseconds) and can be performed on a CPU or GPU. Combining\nwith LMFT give us c(1)([ρ,βϕ],T) and n(1)([ρ,βϕ],T). The EL equation is solved\nself-consistently with a mixed Picard iteration scheme, which typically converges within\nminutes. Todeterminetheliquid–vaporcoexistencelineatzeroelectricfield,wecalculate\nisotherms of the chemical potential as a function of the bulk density ρ from\nb\nβµ = ln(Λ3ρ )−c(1)([ρ ,βϕ = 0],T),\nb b\nand perform a Maxwell construction to find the coexisting liquid and vapor densities\nat the binodal. For inhomogeneous systems, i.e, as a result of an applied non-uniform\nelectric field, we set the chemical potential and temperature and find the inhomogeneous\nsolutions by solving the EL equation. The isotherms of the chemical potential, now as\na function of the mean density, ρ =\nL−1(cid:82)L\ndzρ(z) where L is the total length of the\n0\ndomain over which ρ(z) is defined, now have distinct jumps when the system undergoes\nphase separation, giving liquid ρ and vapor ρ solutions. To distinguish stable solutions\nl v\nfrom metastable solutions, we also calculate the grand potential Ω = Ω([ρ,βϕ],T)\nϕ\nat a fixed βϕ where the excess free energy term can be evaluated via functional line\nintegration25. Performing this procedure for βϕ = 0 gives results consistent with those\nobtained by Maxwell construction. Note that the pressure of the bulk fluid at zero field\nis given by −PV = βΩ . To investigate capillary condensation in slit pores, adsorption\n0\nisotherms are obtained from the mean density ρ =\nH−1(cid:82)H\ndzρ(z) where H is the slit\n0\nheight. Hysteresis loops are mapped by seeding different initial guesses when solving\nthe EL equation. In the dielectrowetting calculations, the electrostatic free energy per\ncontact area is given as\n1 (cid:90) +∞\nγ = dzϕ(z)n(z),\nelec\n2\n−∞\nwhere the factor of a half comes from there being two symmetric confining walls. To\nconstruct the dependence of θ on ϕ , we obtain α = −∂γ /∂(ϕ2) and calculate γ\n0 elec 0 lv\nfrom a direct coexistence molecular dynamics simulation. We estimate θ = 160◦ based\n0\non the maximum value of the local compressibility at ϕ = 0 and comparing to Ref.66;\n0\nwhile more detailed calculations can provide a more accurate estimate, this value is\nconsistent with the solvophobic nature of the slit, and is sufficient for the purposes of\ndemonstrating the increased wetting with ϕ shown in Fig. 4.\n0\n(5) In the confinement case there is a problem of dynamic entrance effects. There may be a\nquestion of how fast you are indeed able to recruit more particles within the small opening to\ncontract the fluid etc.\nAs stated in our earlier responses, our framework is limited to equilibrium properties and does\nnot consider dynamical entrance effects. While such effects are potentially important, they lie\nbeyond the scope of the present study. We now acknowledge this limitation explicitly in the\nconclusions, and provide an outlook regarding nonequilibirium effects:\nimplications are far-reaching. Our first-principles framework for electromechanics can\nbe augmented with dynamical extensions cDFT59, opening a promising route toward a mi-\ncroscopic understanding of non-equilibrium processes under EFGs. Such advances would\nenable the study of dynamical phenomena including pore entry and exit53–55, electroki-\nnetic transport like diffusiophoresis and electroosmosis56–58 and controlled wetting dy-\nnamics such as rate-dependent droplet spreading50....\n(6)Oftenelectricalgradientsareconstructedbyusingspecific/manmadesurfaceproperties,e.g.\npatterning electrodes (as e.g. in https://pubs.acs.org/doi/full/10.1021/acs.jpclett.4c00278) or\nothers. So there is a deep question there of how the electric gradient is made and how that\ncouples to the fluid properties which was somewhat discarded.\nWe thank the reviewer for raising this important point. Indeed, sinusoidal fields are not realistic\nrepresentations of fields generated in experiments. However, as a theoretical tool, they allow us\nto build our understanding by probing specific normal modes of the fluid response. In practice,\nas the reviewer points out, patterned electrodes are often used generate very specific forms of\nfieldgradients. AsalreadydiscussedinresponsetoReviewer1, thedielectrowettingexperiments\n• G. McHale, C. V. Brown, M. I. Newton, G. G. Wells, and N. Sampara, Dielectrowetting\ndriven spreading of droplets, Phys. Rev. Lett. 107, 186101 (2011)\n• G. McHale, C. V. Brown, and N. Sampara, Voltage-induced spreading and superspreading\nof liquids, Nat. Commun 4, 1605 (2013)\nare particularly relevant to our work. Here, interdigitated electrodes are used to show how di-\nelectrophoretic forces change the contact angle of dielectric droplets on surfaces. The resulting\nfield decays exponentially away from the surface, ϕ(z) = ϕ exp(−z/d).\ndecaying fields analogous to those generated by interdigitated electrodes. This enables us to\ndescribedielectrowettingatthenanoscalewithinthesamemodelingapproach. Ourresultsshow\nenhanced wetting and suppressed density fluctuations in the fluid, consistent with a downward\nshift of T under the application of EFGs, which is also the basis of dielectrocapillarity.\nc\nFollowing the reviewer’ comment, we have added a new section, Connection to dielectrowetting\nexperiments, and a new Figure 4:\n(Lines 339–387)\nConnecting to dielectrowetting experiments\nThus far we have introduced dielectrocapillarity as an additional mechanism of control-\nling capillary phenomena, complementing electrocapillarity51 where electrolytes respond\ntoappliedpotentials. Onthemacroscopicscale,theseeffectsmanifestinelectrowetting52\nand dielectrowetting14,50, where the contact angle of a droplet can be tuned with applied\npotentials. In the case of dielectrowetting, a typical experimental setup employs inter-\ndigitated electrodes, as shown schematically in Fig. 4a, which generate an exponentially\ndecaying electrostatic potential ϕ(z) = ϕ exp(−z/d) that drives a droplet of dielectric\n0\nfluid to spread. The resulting contact angle can be described by a modified Young’s\nlaw14,50,\nα\ncosθ(ϕ ) = cosθ + ϕ2, (2)\n0 0 γ 0\nlv\n22\na (z)\ny\nz\nx\nb\nc\n(left) and the resulting exponentially decaying potential profile (right), which is applied\nsymmetrically from both confining walls in our slit geometry. (b) Density profiles (left)\nand local compressibility (right), both normalized by their bulk values, show enhanced\nwall contact density and suppressed density fluctuations as ϕ increases. (c) Electrostatic\n0\nfree energy per unit contact from cDFT (inset) exhibits a quadratic dependence on ϕ ,\n0\nenabling reconstruction of the contact angle via Eq. 1 (assuming θ = 160◦ and using\n0\nγ = 0.025Jm−2 computed from direct coexistence simulation), in direct analogy with\nlv\ndielectrowetting experiments14,50.\nwhere θ is the Young’s contact angle at zero field, γ the liquid–vapor surface tension,\n0 lv\nand α a material parameter related to the dielectric response of the liquid. This relation-\nship assumes that the electrostatic energy stored in the liquid droplet is well-described\nby dielectric continuum theory, and that d is sufficiently small so that any changes in\nenergyduetotheelectricfieldareeffectivelylocalizedtothesolid-liquidinterface. Under\nthese assumptions, the electrostatic free energy per unit area obeys the simple quadratic\nscaling γ = −αϕ2.\nelec 0\nOur multiscale framework provides a microscopic perspective on this phenomenology,\nand allows us to test whether, at the nanoscale, EFGs indeed enhance the wetting of\ndielectric liquids and the extent to which the scaling prescribed by Eq. 1 holds. To this\nend, we applied an exponentially decaying electrostatic potential with d = 1.1σ to the\nconfined dipolar model, symmetrically from both walls of a solvophobic slit. While not\ntrained on such electrostatic potentials, as can be seen in Fig. 4b, the neural functional\nextrapolates well, yielding physically plausible results, aside from some oscillations at\nhigh wetting that are likely minor artifacts. As ϕ increases, so too does the contact\n0\ndensity at the wall, verifying that enhanced wetting occurs. We quantify this effect by\ncomputing γ = 1 (cid:82)∞ dzϕ(z)n(z), which, as can be seen in Fig. 4c, decreases in\nelec 2 −∞\n23\nan approximately quadratic fashion as ϕ increases. By identifying α = −∂γ /∂(ϕ2),\n0 elec 0\nwe reconstruct the potential-dependent contact angle, as shown in Fig. 4c. While the\nresults of our microscopic theory are broadly in line the macroscopic model (Eq. 1) some\nsubtledifferencesareobserved,especiallyatsmallϕ . Theseappeartobecorrelatedwith\n0\nsignificant changes in the local compressibility near the interface, χ (z) = ∂ρ(z)/∂µ,\nµ\nas ϕ increases, which indicates a suppression of density fluctuations. Such microscopic\n0\ndetails are lacking in the dielectric continuum model that underpins Eq. 1.\nPotential improvement of clarity/wording/typos:\n(12) : of a fluid or of a polar fluid?\nWe clarified the wording to specify polar fluid as intended\nIn contrast, electric field gradients (EFGs) induce a dielectrophoretic force, offering\nexquisite electrokinetic control of polar fluids even in the absence of free charges.\n(general) Electrostrictive is not such a common word so I would suggest defining it.\nWe have defined electrostriction at first use in the introduction\n(Lines)Bringingtogetherthelatestadvancesinliquidstatetheory, computersimulation,\nand machine learning, we develop a multiscale framework to study electrostriction—\nthe density response of a dielectric to applied electric fields—in fluids within the grand\ncanonical ensemble,\n(introduction) I do not see how phenomena of gas uptake and separation can be so easily put\nunder the same hat as liquid uptake since the electrokinetic properties in such dilute phases\nmay be vastly different.\nAs we have clarified in our previous responses, our focus is on equilibrium capillary phenomena,\nwhich provide a unifying framework for both gas and liquid uptake. We therefore did not make\nspecific changes to the introduction, but we now explicitly acknowledge electrokinetic effects\nin the Conclusions in our outlook on nonequilibrium phenomena.\n(Lines) While the effects uncovered here are equilibrium phenomena, their non-\nequilibrium implications are far-reaching. Our first-principles framework for electrome-\nchanics can be augmented with dynamical extensions cDFT59, opening a promising\nroute toward a microscopic understanding of non-equilibrium processes under EFGs.\nSuch advances would enable the study of dynamical phenomena including pore entry\nand exit53–55, electrokinetic transport like diffusiophoresis and electroosmosis56–58 and\ncontrolled wetting dynamics such as rate-dependent droplet spreading50....\n(68) “grand canonical ensemble, which is most representative of real conditions in which fluid\nmolecules can dynamically enter and leave a pore” It is and it is not. As far as I can tell, there is\nno great way of simplifying the entrance/exits from a pore, because the grand canonical ensem-\nble will remove correlations between particles exiting and coming back, which are correlations\nthat exist. Also, all these things are non-equilibrium so really we dont know. So the grand\ncanonical ensemble is a way of representing things, that may have some advantages compared\nto others, but it is likely not the best simplification of the system.\nwork. Use of the word “dynamically” is also potentially misleading in this context. What we\nhoped to emphasize is that with the grand canonical framework, we can directly quantify the\namount of fluid in the system for a given chemical potential (i.e., a pore in equilibrium with a\nreservoir). We have therefore removed “dynamically,” in addition to discussing nonequilibrium\neffects in the Conclusions:\n(Lines 402–413) The effects uncovered in this work concern the equilibrium behavior\nof dielectric liquids, and omit potentially important nonequilibrium effects such as pore\nentryandexit53–55,electrokineticphenomena56–58,andcontrolledwettingdynamicssuch\nas rate-dependent droplet spreading50. Nonetheless, the implications of our results for\nnonequilibrium behavior are potentially far-reaching. A natural possible progression from\nthis work is to augment our first-principles framework for electromechanics with dynami-\ncalextensionsofcDFT59, openingapromisingroutetowardamicroscopicunderstanding\nof how EFGs impact non-equilibrium processes.\nThe introduction now reads:\n(Lines 65–69) ...we develop a multiscale framework to study electrostriction in polar\nfluids—that is, their density response to applied electric fields—within the grand canon-\nical ensemble, which is most representative of real conditions in which fluid molecules\ncan enter and leave a pore.\n(94) “such minimal models uncover universal behaviors that emerge across different scales,\nfrom molecular liquids to colloidal systems.” → this is a bold claim that is not justified.\nWe have now support this claim with additional analyses using a water model (SPC/E),\nas discussed in a previous point. Specifically, the revised text now reads:\n(Lines 84–94) To investigate how EFGs influence polar fluids, we primarily consider a\nminimal molecular model that incorporates soft-core repulsion, van der Waals attraction,\nand long-range dipolar interactions. The advantage of using such a simple molecular\nmodel is that we can exhaustively explore a broad range of thermodynamic conditions,\nwhilepotentiallyuncoveringcommonbehaviorsamongpolarfluids,frommolecularliquids\nto colloidal systems. In addition, we also investigate a commonly used simple point\ncharge model for water (SPC/E21) that explicitly incorporates hydrogen-bonding, under\nthermodynamic conditions close to its critical point.\n(112) twice the word directly and supervise-learn is an unusual concatenation\nWe corrected this typo and streamlined the wording. The revised text now reads:\n(Lines 113–117) Instead of relying on traditional approximations, we leverage state-of-\nthe-art data-driven methodologies to supervise-learn functional mappings directly from\nquasi-exact reference data from grand canonical Monte Carlo simulations\n(169) “dielectrophoretic coupling depends on the EFG rather than the absolute field strengt”.\nThey depend both on the pattern on the field strength.\nWe have revised the sentence accordingly:\n25\n(Lines 184–186) Crucially, dielectrophoretic coupling depends on the EFGs as well as\nthe absolute field strength, and therefore offers greater control over the fluid’s response.\nIn Fig. 1d, the color code is not given.\nThis panel is now Figure 2a. We have added a color bar to clarify the color code indicated\nE∗ .\nmax\nLiterature: In terms of fluid uptake and filling of pores, there is a body of literature related to\ntailoring wettability properties that could be discussed (see below). Diffusiophoresis has already\nbeen used as an externally tunable force to manipulate filling or unfilling of nanopores with im-\nposed chemical gradients (so indeed there already exists some external manipulation although\nnot as easy as electric fields) to manipulate particles.\nOn the filling of nanopores:\n- Giacomello, A., Chinappi, M., Meloni, S., & Casciola, C. M. (2012). Metastable Wetting\non Superhydrophobic Surfaces: Continuum and Atomistic Views of the Cassie-BaxterWenzel\nTransition. Physical review letters, 109(22), 226102.\n- Picard, C., Grard, V., Michel, L., Catto¨en, X., & Charlaix, E. (2021). Dynamics of heteroge-\nneous wetting in periodic hybrid nanopores. The Journal of Chemical Physics, 154(16).\nManipulating concentration gradients to fill/unfill dead-end pores:\n- Migacz, R. E., Castleberry, M., & Ault, J. T. (2024). Enhanced diffusiophoresis in dead-\nend pores with time-dependent boundary solute concentration. Physical Review Fluids, 9(4),\n044203.\n- Marbach, S., & Bocquet, L. (2019). Osmosis, from molecular insights to large-scale applica-\ntions. Chemical Society Reviews, 48(11), 3102-3144.\n- Shin, S., Um, E., Sabass, B., Ault, J. T., Rahimi, M., Warren, P. B., & Stone, H. A. (2016).\nSize-dependent control of colloid transport via solute gradients in dead-end channels. Proceed-\nings of the National Academy of Sciences, 113(2), 257-261.\n- Shin, S., Ault, J. T., Warren, P. B., & Stone, H. A. (2017). Accumulation of colloidal particles\nin flow junctions induced by fluid flow and diffusiophoresis. Physical Review X, 7(4), 041038.\nWe thank the reviewer for pointing out this important body of work. We have expanded the\nConclusion section to place our results in the broader context of externally controlled wetting\nand transport phenomena. The above papers have also been cited. The revised text reads:\n(Lines 402–413) The effects uncovered in this work concern the equilibrium behavior\nof dielectric liquids, and omit potentially important nonequilibrium effects such as pore\nentryandexit53–55,electrokineticphenomena56–58,andcontrolledwettingdynamicssuch\nas rate-dependent droplet spreading50. Nonetheless, the implications of our results for\nnonequilibrium behavior are potentially far-reaching. A natural possible progression from\nthis work is to augment our first-principles framework for electromechanics with dynami-\ncalextensionsofcDFT59, openingapromisingroutetowardamicroscopicunderstanding\nof how EFGs impact non-equilibrium processes.\nWe thank the reviewer for their constructive evaluation and thoughtful suggestions, which have\nsignificantly improved the manuscript. In response, we have:\n1. added further details on our methods to the main text, including theory, training proce-\ndures, and clarification that our focus is on equilibrium capillary processes;\n26\n2. repeated our analysis for SPC/E water to test the robustness of our conclusions beyond\nsimplified dipolar models, with new results included in the main text and SI (Figs. S13\nand S14);\n3. introducedmolecularschematicsandclarifieddefinitions(e.g.,numbervs. chargedensity)\nto better illustrate field direction and distinguish dielectrophoretic from electrophoretic\nrise;\n4. incorporatednewresultsondielectrowettingunderexponentiallydecayingfieldsgenerated\nby interdigitated electrodes, directly linking our framework to experimental setups.\n5. discussed possible extensions of our framework to non-equilibrium phenomena\nREFERENCES\n1Z. Chen, P. Li, R. Anderson, X. Wang, X. Zhang, L. Robison, L. R. Redfern, S. Moribe, T. Islamoglu, D. A.\nG´omez-Gualdr´on, T. Yildirim, J. F. 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I think adding in water and contact angles were good ideas that make the results a bit easier to put in experimental perspective. The authors have also made significant efforts in making their manuscript more pedagogical.\nI would still like to raise a minor comment: In my view, \"out-of-equilibrium\" means that a source of energy is put in the system, in this case, it is the spatially varying electric field. So I was quite worried that the AI had been trained on data that was never under applied external fields. Indeed, that was not the case, as the authors explain in the methods. To me that is something important that enhances the validity (a priori) of the trained model and I would therefore encourage the authors to say that explicitely in the main text.\nI would also like to congratulate the authors on so diligently (and also sensibly) answering both reviewers.\nApart from my minor comment above, I am happy to recommend this manuscript for publication in Nature Comm.", "reviewer_3": "The manuscript deals with the topic of how electrical field gradients induce novel phenomena relating to phase behaviour, capillarity and adsorption in liquids. Focus of the work is in confined conditions and could have implications on porous materials and their use for energy storage applications etc.\nThe manuscript is well structured and provides a well thought of theoretical study. The authors have already addressed most open questions, and the mansucript stands on its own.\nMy concerns are in regards to not being able to come up or or understand an experiment to verify this theory (falsifiability). The example given in Fig. 4 is too simple and does not provide any sort of evidence that the theory is valid except showing that it fits a specific boundary condition. Due to the use of cDFT combined with machine learning methods, the apporoach seems like a black box and it is hard to understand its core assumptions (e.g. the choice of potentials at the start) and their validity. I would recommend publishing in a more specialised journal, and focusing future work to include geometries and conditions where it could be experimentally tested. I would recommend considering the impacts of the theory in also other gemetries, e.g. nanopores and their use for sensing and filtration.", "reviewer_4": "In this article, the authors use ML-based numerical methods to study a novel phenomenon, dielectrocapillarity, where a spatially-varying electric field is used to modulate the density of a dipolar fluid. More precisely, they use a neuron network to learn thermodynamic functionals from MD simulations, and then use these functionals to (1) estimate the density profile of a bulk liquid under a electric field gradient, notably with consequences on the liquid-gas phase transition, (2) fluid entry into a small pore and (3) model electrowetting.\n  The effects reported here are definitely novel and promising, both in terms of technique and in relevance with experiments. My main concerns, as detailed below, regard (1) the clarity of certain aspects of the work (notably figures) and (2) the highly idealized settings considered here, which contrast with the bold claims throughout the article.  Overall, though, I support publication of this work in Nature Communications.\nMajor comments:  \n1) I find it hard to understand the general procedure behind this work, in particular with what relates to geometry, which is only detailed in SI file (note a typo p8 of SI: “L_x = L_z = 80 A” should, I imagine, read “L_x = L_y = 80 A”).  \nIf I understand correctly, the authors generate MD trajectories with a slab geometry (L_z << L_x, L_y), from which they compute thermodynamic functionals, which are then learnt by a neuron network. How generalizable are the results to other geometries? The electrowetting proof-of-concept shows that changing the shape of the gradient is OK; but could, for example, be the learnt functionals be used to cases where the electric field gradient depends on more than one variable? Or non-slab geometries?  \n2) Related to the previous point, the authors state that their method greatly reduces computational time, by comparing a timescale to evaluate the NN to an MD simulation time. This is trivial, since what matters here should instead be the total computational time (ie including training and testing), unless the neural functional can be used directly to cases with, for example, completely different geometries. Can the authors comment on that?  \n3) I am relatively unconvinced by the type of electric field gradients that are considered here. While the numerical results seem fine and interesting, I doubt they really correspond to something technically achievable. They use molecular-scale gradients (wavelength comparable to the molecular size sigma) while comparing to micron-scale experiments (local or interdigitated electrodes, for example). That of course does not mean that considering short wavelengths is wrong, but I doubt the effect survives under molecular confinement as a surface provides interactions with liquid other than through a static electric field (adsorption, chemical defects, roughness, dielectric confinement, polarizability, etc.). \nIn addition, under such confinement, I would expect the gradient to be along the surface (because of natural/engineered defects, for example), rather than normal to it; I do not see any way of replicating experimentally the situation of Fig 1 and 2.   \n4) Similarly, for Fig 3-4 I would expect that chemical effects (eg acid-base reactions with the wall) play a much more important role at the considered lengthscales, and that modelling the walls with a static external electric field is plain wrong. Another thing that comes to my mind is that for this kind of questions (wettability and fluid filling of a small pore), the MD simulations used as training data are typically quite bad - it is for example very hard to correctly estimate a contact angle from simulations of water+LJ walls, and in nanometric slit, slightly changing the LJ parameters of the wall can result in either full wetting or dewetting, at odds with experimental results eg on carbon slits. Here, the authors choose one particular value of the walls LJ coefficients, so I would be very cautious about the results, which could be of second-order magnitude compared to changing interaction parameters.\n  5) Some of the results discussed here are in the “large length scale regime”, eg Janus colloids. I am afraid this limit is quite a dangerous one to take: colloids under an applied field are basically never at thermal equilibrium (because of electrophoresis, for example), so the current framework would not apply, even if it was extended to include dynamical effects (one would need to add hydrodynamics, ions, etc.). While the thought experiment presented in lines 208-226 is conceptually interesting, it is important to signal such limitations.\n Minor/presentation remarks\n  6) Line 28: “Fundamentally, fluid uptake in these systems is driven by capillary forces” \nNot really. Separation and filtration processes are typically out of equilibrium, with (eletro/diffusio)osmotic effects often playing a major role.   \n7) Line 40: “While a uniform field exerts a force only on free charges such as ions” \nThis is misleading; fundamentally this force is cancelled and fully transmitted to the solvent because of viscous dissipation. In almost all nanofluidic experiments, applying a uniform electric field results in a net flow of water (because of electro-osmosis and related phenomena). In typical slit-like geometries like the one considered here, the effect is often dominant compared to other transport phenomena. See for example the 2021 review by Kavokine, Netz and Bocquet.\n  8) I find Fig 2 difficult to follow due to the high number of curves, colors, line styles, etc. In particular, I do not fully understand Fig 2d. What is represented as inset? What are the two blue and red rectangles on the right?   \n9) For all figures, some of the ‘pastel’ colors are difficult to see after printing.  \n10) Line 322: “That is, EFGs impact criticality of polar fluids in both the three-dimensional and confined two-dimensional Ising universality classes” I do not understand this sentence, I would suggest explaining more. I guess that the authors are referring to the fact that the liquid-gas phase transition corresponds to the 3D Ising class; but I am less familiar with the capillary transition and I would imagine that its does not belong to a single universality class but rather would depend on eg. the pore’s geometry. In practice, you would likely need to have a wedge/1D/0D geometry to have a localized electric field in the confined system, so I do not think it is realistic to have the exact 2D regime. \nI also do not understand what this discussion is bringing here; if the observed effect modifies the system’s chemical potential, it is obvious that it will affect both phase transitions, which are defined from it…   It is not particularly surprising that modifying the system's parameters shifts the critical points; what would be more relevant would be to discuss orders of magnitude, like the 50K shift in the case of the liquid-gas transition of water. In addition to this, the authors could discuss what orders of magnitude can be reached in terms of electric field (gradient) experimentally. They list a value of E = 0.4 V/A. \nTo get this kind of field from say a point charge of charge Zq - say surface defects - in water over 1nm (which corresponds roughly to the setting considered here), you would need Z = 10, which is a bit wild. \n11) Related to the previous point, it seems to me that the only two relevant regimes for how to create an electric field gradient within a confined pore are (1) electric field gradient occuring on a molecular lengthscale close to the walls or (2) a gradient with a lengthscale equal to the system size. The authors do not really discuss the impact of the system size in the settings corresponding to Fig 3."}, "author_response": "Response to reviewer comments for NCOMMS-25-37255-T “Dielectrocapillarity for\nexquisite control of fluids”\nAnna T. Bui1,2 and Stephen J. Cox2,a)\n1)Yusuf Hamied Department of Chemistry, University of Cambridge, Lensfield Road,\nCambridge, CB2 1EW, United Kingdom\n2)Department of Chemistry, Durham University, South Road, Durham, DH1 3LE,\nUnited Kingdom\n(Dated: December 20, 2025)\nWe would like to thank all four reviewers for their time in reading our manuscript. In\nthe last round of reviews, we addressed comments from reviewers 1 and 2 only. Aside from\na minor suggestion from reviewer 2 (which we address below), we are delighted that\nboth of the original reviewers recommend publication in Nature Communications. We\nare also grateful to both reviewers for acknowledging our efforts to thoroughly address\ntheir comments.\nOur revised manuscript has also been sent to a further two peer reviewers. Overall, these\nreviews are positive, describing our work as definitely novel and interesting, echoing\nsentiments from the previous round where it was also made clear our work addresses\na critical gap. Reviewers 3 and 4 did, however, raise extra concerns regarding the\nunderlying assumptions of our theoretical models and the generality of our approach.\nIn response to these concerns, we have made further significant changes to our\nmanuscript. This includes a new figure that clearly depicts an example of how the\ninhomogeneous fields that we consider can be realized, and how the cDFT provides\ninsight.\nBelow we detail our responses to all comments from the reviewers point-by-point. The\nreviewers’ comments are reproduced in turquoise. Changes are highlighted in magenta.\nThe reference list from the main paper is replicated in full at the end the document;\nthis is simply so citations in text quoted from the manuscript appear correctly.\na)Electronic mail: stephen.j.cox@durham.ac.uk\nTherevisionisthoroughandcomprehensive, somyrecommendationistoacceptthismanuscript\nfor publication in Nature Communications in its present form.\nWe thank the reviewer for this positive feedback. As reviewer 1 recommends publication as is,\nwe have not made any changes in response to their comment.\nI thank the authors for carefully adressing my comments. I think adding in water and contact\nangles were good ideas that make the results a bit easier to put in experimental perspective.\nThe authors have also made significant efforts in making their manuscript more pedagogical.\nI would still like to raise a minor comment: In my view, ”out-of-equilibrium” means that a\nsource of energy is put in the system, in this case, it is the spatially varying electric field. So I\nwas quite worried that the AI had been trained on data that was never under applied external\nfields. Indeed, that was not the case, as the authors explain in the methods. To me that is\nsomething important that enhances the validity (a priori) of the trained model and I would\ntherefore encourage the authors to say that explicitely in the main text.\nWe thank the reviewer for this positive feedback and their helpful suggestion. In response, we\nnow explicitly state in the main text that the ML model was trained on data generated under\napplied external electrostatic potentials:\n(Lines 153–168) Going beyond established deep-learning approaches to cDFT, we cap-\nture electrostriction arising from the coupling between mass and charge density of the\nfluid by explicitly learning the “hyperfunctional” Fex ([ρ,βϕ],T) where ϕ(r) is the in-\nintr\nhomogeneous electrostatic potential and β = 1/(k T) with k the Boltzmann constant,\nB B\nas recently introduced in Ref.36. A practical limitation of this cDFT approach is that, at\npresent, only inhomogeneities with planar symmetry can be investigated directly. As a\nresult,theneural-networkrepresentationofthisfunctionalwastrainedonsimulationdata\ngenerated under random planar electrostatic potentials ϕ(z). Nonetheless, we demon-\nstrate below that calculations in which ϕ only varies along a single cartesian direction\nprovide general insight into the influence of EFGs on wetting and capillarity. Details of\nthe practical implementation of the theory are given in the Methods section.\nI would also like to congratulate the authors on so diligently (and also sensibly) answering both\nreviewers.\nApartfrommyminorcommentabove, Iamhappytorecommendthismanuscriptforpublication\nin Nature Comm.\nWe would like to thank the reviewer for their careful assessment of our work and constructive\ncriticisms. We are happy that they recommend our work for publication in Nature Communi-\ncations.\nThe manuscript deals with the topic of how electrical field gradients induce novel phenomena\nrelating to phase behaviour, capillarity and adsorption in liquids. Focus of the work is in\nconfined conditions and could have implications on porous materials and their use for energy\nstorage applications etc.\nThe manuscript is well structured and provides a well thought of theoretical study. The authors\nhave already addressed most open questions, and the mansucript stands on its own.\nWe thank the reviewer for the positive assessment and for recognizing the structure, clarity,\nand coherence of our study. We are encouraged that they believe we have addressed most open\nquestions.\nMyconcernsareinregardstonotbeingabletocomeupororunderstandanexperimenttoverify\nthis theory (falsifiability). The example given in Fig. 4 is too simple and does not provide any\nsort of evidence that the theory is valid except showing that it fits a specific boundary condition.\nDue to the use of cDFT combined with machine learning methods, the apporoach seems like\na black box and it is hard to understand its core assumptions (e.g. the choice of potentials\nat the start) and their validity. I would recommend publishing in a more specialised journal,\nand focusing future work to include geometries and conditions where it could be experimentally\ntested. I would recommend considering the impacts of the theory in also other gemetries, e.g.\nnanopores and their use for sensing and filtration.\nThe reviewer essentially raises two issues in their comment: (i) experimental falsifiability; and\n(ii) the “black box” nature of using cDFT combined with ML.\nTo support point (i), they state that the example of dielectrowetting provided by Fig. 4 (now\nFig. 5) is too simple, and that it only shows that the theory fits a specific boundary condition.\nWe believe it is relevant to mention that we added this example at the last round of reviews\nin response to comments from reviewers 1 and 2, both of whom felt it does address the gap\nbetween our theory and experiment. (This is stated explicitly reviewer 2’s comments.) To\nrespond to this comment more directly, within the cDFT framework we do not place bound-\nary conditions on the fluid (in contrast to continuum modeling). Rather, the effects of the\nsubstrate—in this case an interdigitated electrode—are accounted for by the external poten-\ntials. The equilibrium microscopic number and charge densities are then obtained by minimizing\nthe grand potential. We then obtain an estimate for the electrostatic free energy per unit area\nby integrating the product of microscopic charge density and the external electrostatic potential\nfrom the interdigitated electrode. We observe that this microscopic picture is consistent with\nthe macroscopic trends observed experimentally (See Ref. 18). This is not a trivial result—it\nsuggests that the cDFT is capturing the essence of the Maxwell stress tensor—the lens through\nwhich the experimental phenomena are analyzed—as an emergent phenomenon; we have not\nbuilt this into the theory.\nNonetheless, we appreciate that the connection to experimental geometries can be made\nstronger, which is also a concern shared by reviewer 4. To this end, we have introduced a\nnew figure (now Fig. 1). This simulation snapshot not only demonstrates how such inho-\nmogeneous fields may arise, but also the influence they may have on the wetting properties\nof dielectric fluids. Specifically, the simulation snapshot shows water confined between inter-\n4\ndigitated electrodes (Fig. 1). This geometry introduces strong, spatially varying electric fields,\nwhichareconceivablyachievablethroughnanoscalefabricationorvanderWaalsheterostructure\nassembly:\n• Han et al., Dielectric capacitors with three-dimensional nanoscale interdigital electrodes\nfor energy storage, Sci. Adv 1, 9, (2015). https://doi.org/10.1126/sciadv.1500605\n• Castellanos-Gomez, A., Duan, X., Fei, Z. et al. Van der Waals heterostructures. Nat Rev\nMethods Primers 2, 58 (2022). https://doi.org/10.1038/s43586-022-00139-1\nTo clarify the origin and structure of electric-field gradients in this geometry, we analyze the\nelectrostatic potential and show that it naturally decomposes into sinusoidal modes along x and\nexponential decay along z. This directly motivates the understanding the capillarity physics in\nsimplified planar field geometries of sinusoidal nature (Figs. 2, 3, 4) and of exponential nature\nin Fig 5. We also discuss in more detail the physical interpretation of probing the response of\nthe fluid to planar inhomogeneous electric fields.\nWith regard to changes in the manuscript in response to point (i) raised by the reviewer,\nwe have introduced a new Figure 1:\n+ - + -\nz\nx\nz\ny\n+ +\n- -\nFigure 1: Inhomogeneous electric fields arising from interdigitated electrodes strongly\ninfluence water’s wetting behavior. SnapshotofanSPC/Ewatersimulationinahydrophobic\nslit with alternating positive and negative electrode patches. Either holding the electrodes at\na 10V potential difference or attributing a fixed charge of ±0.05e/atom causes the fluid to\nexhibit enhanced wetting at the walls, accompanied by strong lateral density oscillations. Cross-\nsections of the electrostatic potential in the constant charge setup are shown parallel to the\nsurface (top right) and normal to the surface (bottom left).\nWe also make several changes throughout the manuscript:\n(Lines 95–131) Although molecular simulations cannot fully capture the influence of\nEFGs on dielectric fluids, they offer vivid qualitative insights into the underlying physics\n5\nandillustratehowEFGsmayarisewithinnanoscaledevices, asillustratedinFig.1. Here,\nwe present a simulation snapshot of water at 300K confined between two hydrophobic\nsubstrates that have been patterned with alternating stripes of positive and negative\ncharge—a set up that provides a caricature of an interdigitated electrode. As can be\nclearly seen in Fig. 1, charging this device induces local wetting near the charged stripes,\nwith pronounced density variations along the direction parallel to the surface (x).\nThese features arise from the complex interplay between the fluid’s charge and number\ndensities, and their collective response to EFGs, arising from the inhomogeneous electro-\nstatic potential ϕ(x,z), where z is along the surface normal. It is instructive to consider\nthe form of ϕ far from either surface; here, it will resemble the linear superposition of\nthe asymptotic limit of the two substrates considered independently,\n(cid:18) (cid:19) (cid:18) (cid:19)\n2πx −2π|z−z |\ns\nϕ (x,z) ∼ ϕ sin exp , (1)\nsingle 0\nL L\nx x\nwhere z indicates the plane of the substrate’s outermost atoms. We therefore observe\ns\nthat inhomogeneity of ϕ is characterized by a sinusoidal oscillation of period L along\nx\nx, and an exponential decay along from the surface along z. This asymptotic analysis\ncaptures the essential behavior of ϕ(x,z) computed explicitly from the potential energy\nof a test charge, as seen in Fig. 1.\nThe pronounced wetting behavior observed with simulation in Fig. 1 strongly suggests\nthat such EFGs will influence capillarity of the polar fluid; that is, the amount of fluid\nadsorbed at constant chemical potential. Addressing this issue, however, demands an\naccurate and efficient framework for determining structure, thermodynamics, and phase\nbehaviorinanopensystem—thisliesbeyondthepracticallimitsofpresentdaymolecular\nsimulations. Instead, we turn to classical density functional theory (cDFT), an exact\nstatistical mechanical framework for inhomogeneous fluids.\n(Lines 211–226) While EFGs will be most pronounced near the surfaces that generate\nthem, Fig. 1 demonstrates that they may persist relatively far from the interface. In the\nspecific case considered in Fig. 1, midway between the substrates we observe a sinusoidal\nelectrostatic potential along the x direction. This motivates us to understand the direct\ninfluence of such sinusoidal potentials on the fluid, without explicitly considering inter-\nfaces. In fact, for L larger than a few molecular diameters, we can define bulk response\nx\nifweconsideraveragesoverathinsliceofthickness∆z ≪ L (Fig.S16). Althoughsuch\nx\nacleanseparationofbulkandinterfacialresponsebecomeschallengingasL approaches\nx\nmolecular length scales, the full potential (as opposed to its asymptotic form) comprises\nmodes of decreasing wavelength—it is therefore instructive to understand bulk-like re-\nsponse across a broad range of wavelengths.\n(Lines 396–402) To gain insight into the influence of EFGs, we perform a “computa-\ntionalexperiment”inwhichweapplyasinusoidalelectricfieldacrosstheslit. Whilesuch\na set up does not correspond to an EFG established by the substrate walls themselves\n(see, e.g., Fig. 1), it does allow us to assess the effects of a particular mode. The impact\nis twofold: (1) condensation shifts to more negative ∆µ...\n(Lines 429–442) On the macroscopic scale, these effects manifest in electrowetting19\nand dielectrowetting18,20, where the contact angle of a droplet can be tuned with applied\n6\npotentials. Our nanoscale simulations in Fig. 1 show that a similar phenomenology\nemerges under confinement: wetting is strongly enhanced directly over the electrode\npatches,whiletheelectrostaticpotentialgeneratedbytheinterdigitatedelectrodesdecays\ninto the slit. Following previous experimental work18,20, this motivates us employ a\nsimple planar electrostatic potential, ϕ(z) = ϕ exp(−2πz/L ) to probe wetting at the\n0 x\nnanoscale. Similar to our arguments above, results obtained from such a potential can\nbe considered to report on average behavior in a slice of thickness ∆x ≪ L (Fig. S16).\nx\nWith regard to point (ii), the reviewer states that the use cDFT combined with machine learning\nmethods, makes the approach seem like a black-box, and that the core assumptions, e.g., of the\nchoice of potentials, and their validity are hard to understand. On the choice of the external\npotentials that we use, this has been addressed above. Where the cDFT+ML approach is\nconcerned, the theoretical foundations and assumptions are detailed in methodological papers\nthat we have cited:\n• Sammu¨ller et al., Neural functional theory for inhomogeneous fluids: Funda-\nmentals and applications, Proc. Natl. Acad. Sci. U.S.A 120, 50, (2023).\nhttps://doi.org/10.1073/pnas.2312484120\n• Bui, Cox, Learning Classical Density Functionals for Ionic Fluids, Phys. Rev. Lett. 134,\n148001 (2025). https://doi.org/10.1103/PhysRevLett.134.148001\n• Sammu¨ller et al., Neural Density Functional Theory of Liquid-Gas Phase Coexistence ,\nPhys. Rev. X 15, 011013 (2025). https://doi.org/10.1103/PhysRevX.15.011013\n• Bui, Cox, A first-principles approach to electromechanics in liquids, J. Phys.: Condens.\nMatter 37, 285101 (2025). https://doi.org/10.1088/1361-648X/ade7e7\nTo emphasize the theoretical rigor of the cDFT approach, and to guide the reader more directly\nto the relevant methodological papers, we now make these previous works more prominent in\nthe manuscript:\n(Lines 132–168) Within cDFT, the equilibrium structure and thermodynamics of a\nfluid can be determined from first principles by its excess intrinsic Helmholtz free energy\nfunctional Fex ([ρ],T), where ρ(r) is the average inhomogeneous density of the fluid\nintr\nand T is the temperature. This central result, established in Ref.30, places cDFT as\nthe liquid-state generalization of its celebrated electronic structure counterpart31—as a\nmodern theory for inhomogeneous fluids. cDFT is naturally formulated in the grand\ncanonical ensemble, where the chemical potential acts as the control variable governing\nparticle exchange in confined systems. The chemical potential, µ, maps directly onto the\nrelative humidity or vapor pressure for gases, and chemical activity for liquids.\nIn practice, the exact form of Fex ([ρ],T) is generally unknown. Instead of relying\nintr\non traditional approximations, we leverage state-of-the-art data-driven methodologies\nto supervise-learn functional mappings directly from quasi-exact reference data from\ngrand canonical Monte Carlo simulations32–34. This machine-learned cDFT framework\nhas already been successfully applied to liquid–gas coexistence33, liquid–liquid phase\nseparation35, and the electric double layer34. Going beyond established deep-learning ap-\nproachestocDFT,wecaptureelectrostrictionarisingfromthecouplingbetweenmassand\ncharge density of the fluid by explicitly learning the “hyperfunctional” Fex ([ρ,βϕ],T)\nintr\nwhere ϕ(r) is the inhomogeneous electrostatic potential and β = 1/(k T) with k\nB B\nthe Boltzmann constant, as recently introduced in Ref.36. A practical limitation of this\ncDFT approach is that, at present, only inhomogeneities with planar symmetry can be\ninvestigated directly. As a result, the neural-network representation of this functional\nwas trained on simulation data generated under random planar electrostatic potentials\nϕ(z). Nonetheless, we demonstrate below that calculations in which ϕ only varies along\na single cartesian direction provide general insight into the influence of EFGs on wetting\nand capillarity. Details of the practical implementation of the theory are given in the\nMethods section.\nWe again thank the reviewer for their comments. We believe that the resulting discussion\nconcerning the inhomogeneous fields that we apply strengthen the paper. We also believe it\nhas been helpful to provide more context to the cDFT+ML in terms of other work.\nIn this article, the authors use ML-based numerical methods to study a novel phenomenon,\ndielectrocapillarity, where a spatially-varying electric field is used to modulate the density of a\ndipolar fluid. More precisely, they use a neuron network to learn thermodynamic functionals\nfrom MD simulations, and then use these functionals to (1) estimate the density profile of a\nbulk liquid under a electric field gradient, notably with consequences on the liquid-gas phase\ntransition, (2) fluid entry into a small pore and (3) model electrowetting.\nThe effects reported here are definitely novel and promising, both in terms of technique and\nin relevance with experiments. My main concerns, as detailed below, regard (1) the clarity of\ncertain aspects of the work (notably figures) and (2) the highly idealized settings considered\nhere, which contrast with the bold claims throughout the article. Overall, though, I support\npublication of this work in Nature Communications.\nWe thank the reviewer for their careful reading of our manuscript and for the constructive\ncomments. Weappreciatethepositiveassessmentofthenovelty, relevance, andmethodological\ncontributions of the work. Below, we outline our responses to each major point and the changes\nimplemented in the revised manuscript.\n1) I find it hard to understand the general procedure behind this work, in particular with what\nrelates to geometry, which is only detailed in SI file (note a typo p8 of SI: L = L = 80 A\nx z\nshould, I imagine, read L = L = 80 A). If I understand correctly, the authors generate MD\nx y\ntrajectories with a slab geometry (L << L ,L ), from which they compute thermodynamic\nz x y\nfunctionals, which are then learnt by a neuron network. How generalizable are the results to\nother geometries? The electrowetting proof-of-concept shows that changing the shape of the\ngradient is OK; but could, for example, be the learnt functionals be used to cases where the\nelectric field gradient depends on more than one variable? Or non-slab geometries?\nThe issue of the geometry is a fair point. As is hopefully clear from our response to reviewer 3,\nwe have made significant efforts to explain how the results from our cDFT study are relevant\nto realizable geometries. In addition to the general point of geometry, the reviewer raises the\nspecific question of whether the learned functionals can be used where the electric field depends\non more than one variable, or non-slab geometries.\nTo directly address the specific points, we first confirm that the reviewer’s understanding is\ncorrect: wedoindeedtrainthefunctionalsonslabgeometriesinwhichtheelectrostaticpotential\nonly varies along z. The reasons for this are twofold: (i) the typical ML architectures used are\nlimited to planar inhomogeneous geometries; and (ii) by averaging over x and y, we obtain\nsufficiently converged density profiles that form our training data.\nWhile recent work for truly 2D systems demonstrates that the ML architecture is surmountable:\n• Glitschetal., Neuraldensityfunctionaltheoryinhigherdimensionswithconvolutionallay-\ners, Phys. Rev. E 111, 055305, (2025). https://doi.org/10.1103/PhysRevE.111.055305\nObtaining converged density profiles in 3D for a sufficiently large training set remains a daunting\ntask, even if consider “tricks” such as sampling the force (https://doi.org/10.1063/5.0029113).\nIt is important to emphasize, however, that even though we only explicitly sample planar in-\nhomogeneities, we can nonetheless obtain broader insight. The fact that that such functionals\n9\ncan be used to obtain bulk phase diagrams provides one clear example. (We note that obtain-\ning bulk phase diagrams has been demonstrated previously by others for simpler systems33,35.)\nMoreover, by analyzing interfacial free energies in planar geometries, information on wetting\nphenomena in 3D systems can be obtained, as we do in Fig. 5. (Again, analysis in this spirit\nhas previously been performed by others for simpler systems. See, e.g. Ref. 35.)\nOur general conclusions about dielectrocapillarity should, therefore, be generalizable to other\ngeometries. This is supported by our new simulation example of water confined between inter-\ndigitated electrodes (Fig. 1). As discussed in response to reviewer 3, the electrostatic potential\nin this case decomposes into sinusoidal modes in x and exponential decay in z. It is clear from\nthe snapshot that we observe enhanced wetting at the wall. It is also clear from Fig. S15 (new),\nthat the density in a thin slice above the electrodes is very similar to that obtained with a pla-\nnar external electrostatic potential. Furthermore, we also show how simulations with the planar\nsinusoidal potentials report on the bulk response in the interdigitated electrode simulation (see\nFigs. S16 and S17).\nIn light of the reviewer’s comment, we have made several changes throughout the manuscript:\n+ - + -\nz\nx\nz\ny\n+ +\n- -\nFigure 1: Inhomogeneous electric fields arising from interdigitated electrodes strongly\ninfluence water’s wetting behavior. SnapshotofanSPC/Ewatersimulationinahydrophobic\nslit with alternating positive and negative electrode patches. Either holding the electrodes at\na 10V potential difference or attributing a fixed charge of ±0.05e/atom causes the fluid to\nexhibit enhanced wetting at the walls, accompanied by strong lateral density oscillations. Cross-\nsections of the electrostatic potential in the constant charge setup are shown parallel to the\nsurface (top right) and normal to the surface (bottom left).\n(Lines 95–131) Although molecular simulations cannot fully capture the influence of\nEFGs on dielectric fluids, they offer vivid qualitative insights into the underlying physics\n10\nandillustratehowEFGsmayarisewithinnanoscaledevices, asillustratedinFig.1. Here,\nwe present a simulation snapshot of water at 300K confined between two hydrophobic\nsubstrates that have been patterned with alternating stripes of positive and negative\ncharge—a set up that provides a caricature of an interdigitated electrode. As can be\nclearly seen in Fig. 1, charging this device induces local wetting near the charged stripes,\nwith pronounced density variations along the direction parallel to the surface (x).\nThese features arise from the complex interplay between the fluid’s charge and number\ndensities, and their collective response to EFGs, arising from the inhomogeneous electro-\nstatic potential ϕ(x,z), where z is along the surface normal. It is instructive to consider\nthe form of ϕ far from either surface; here, it will resemble the linear superposition of\nthe asymptotic limit of the two substrates considered independently,\n(cid:18) (cid:19) (cid:18) (cid:19)\n2πx −2π|z−z |\ns\nϕ (x,z) ∼ ϕ sin exp , (2)\nsingle 0\nL L\nx x\nwhere z indicates the plane of the substrate’s outermost atoms. We therefore observe\ns\nthat inhomogeneity of ϕ is characterized by a sinusoidal oscillation of period L along\nx\nx, and an exponential decay along from the surface along z. This asymptotic analysis\ncaptures the essential behavior of ϕ(x,z) computed explicitly from the potential energy\nof a test charge, as seen in Fig. 1.\nThe pronounced wetting behavior observed with simulation in Fig. 1 strongly suggests\nthat such EFGs will influence capillarity of the polar fluid; that is, the amount of fluid\nadsorbed at constant chemical potential. Addressing this issue, however, demands an\naccurate and efficient framework for determining structure, thermodynamics, and phase\nbehaviorinanopensystem—thisliesbeyondthepracticallimitsofpresentdaymolecular\nsimulations. Instead, we turn to classical density functional theory (cDFT), an exact\nstatistical mechanical framework for inhomogeneous fluids.\n(Lines 211–226) While EFGs will be most pronounced near the surfaces that generate\nthem, Fig. 1 demonstrates that they may persist relatively far from the interface. In the\nspecific case considered in Fig. 1, midway between the substrates we observe a sinusoidal\nelectrostatic potential along the x direction. This motivates us to understand the direct\ninfluence of such sinusoidal potentials on the fluid, without explicitly considering inter-\nfaces. In fact, for L larger than a few molecular diameters, we can define bulk response\nx\nifweconsideraveragesoverathinsliceofthickness∆z ≪ L (Fig.S16). Althoughsuch\nx\nacleanseparationofbulkandinterfacialresponsebecomeschallengingasL approaches\nx\nmolecular length scales, the full potential (as opposed to its asymptotic form) comprises\nmodes of decreasing wavelength—it is therefore instructive to understand bulk-like re-\nsponse across a broad range of wavelengths.\n(Lines 396–402) To gain insight into the influence of EFGs, we perform a “computa-\ntionalexperiment”inwhichweapplyasinusoidalelectricfieldacrosstheslit. Whilesuch\na set up does not correspond to an EFG established by the substrate walls themselves\n(see, e.g., Fig. 1), it does allow us to assess the effects of a particular mode. The impact\nis twofold: (1) condensation shifts to more negative ∆µ...\n(Lines 429–442) On the macroscopic scale, these effects manifest in electrowetting19\nand dielectrowetting18,20, where the contact angle of a droplet can be tuned with applied\npotentials. Our nanoscale simulations in Fig. 1 show that a similar phenomenology\nemerges under confinement: wetting is strongly enhanced directly over the electrode\n11\npatches,whiletheelectrostaticpotentialgeneratedbytheinterdigitatedelectrodesdecays\ninto the slit. Following previous experimental work18,20, this motivates us employ a\nsimple planar electrostatic potential, ϕ(z) = ϕ exp(−2πz/L ) to probe wetting at the\n0 x\nnanoscale. Similar to our arguments above, results obtained from such a potential can\nbe considered to report on average behavior in a slice of thickness ∆x ≪ L (Fig. S16).\nx\nWe also clarify in the main text the planar geometry used for MD training:\n(Lines 153–168) Going beyond established deep-learning approaches to cDFT, we cap-\nture electrostriction arising from the coupling between mass and charge density of the\nfluid by explicitly learning the “hyperfunctional” Fex ([ρ,βϕ],T) where ϕ(r) is the in-\nintr\nhomogeneous electrostatic potential and β = 1/(k T) with k the Boltzmann constant,\nB B\nas recently introduced in Ref.36. A practical limitation of this cDFT approach is that, at\npresent, only inhomogeneities with planar symmetry can be investigated directly. As a\nresult,theneural-networkrepresentationofthisfunctionalwastrainedonsimulationdata\ngenerated under random planar electrostatic potentials ϕ(z). Nonetheless, we demon-\nstrate below that calculations in which ϕ only varies along a single cartesian direction\nprovide general insight into the influence of EFGs on wetting and capillarity. Details of\nthe practical implementation of the theory are given in the Methods section.\nWe corrected the noted typo in the SI. We thank the reviewer for pointing this out.\n(Page S8) To reduce error due to translating between ensembles, simulation boxes with\nlarger lateral dimensions are used with L = 20˚A and L = L = 80˚A.\nz x y\nWe also added additional simulation results in subsection S7.4. Interdigitated electrode simu-\nlations in the SI:\n80\n40\n0\n]Å[x 0.00 +0.05 e/atom e/atom\n0.00 0.05\ne/atom e/atom\n80\n40\n0\n40 0 40\nz[Å]\n]Å[x\n[Å 3]\n0.00 0.02 0.04 0.06\n0V +5V\n0V 5V\n40 0 40\nz[Å]\nFigure S15: Density profiles of water confined between nanoscale interdigitated elec-\ntrodes. For both fixed-charge (top) and constant-potential (bottom) simulations, imposing a\npotential difference of 10V or a surface charge of ±0.05e/atom enhances wetting at the walls\nand induces strong lateral density oscillations.\n12\n(Pages S19–S22) Interdigitated electrode simulations\nHere we describe the simulation details corresponding to the snapshot in Fig. 1 of the\nmain text. The simulated systems consist of 620 water molecules symmetrically confined\nbetween two solid substrates. Each substrate is composed of 2048 atoms arranged on\na simple cubic lattice (32×8×8, lattice parameter 2.5˚A). The orthorhombic simulation\nbox has lateral dimensions 80×20×100˚A 3 .\nAll simulations were performed with the LAMMPS package68. Water–water interactions\nwere described using the SPC/E model37, with molecular geometry constrained via the\nRATTLE algorithm? . Substrate atoms were held rigid at their lattice positions. Water\noxygenatomsinteractwithsubstrateatomsthrougha12–6Lennard–Jonespotentialwith\nϵ = 0.065kJmol−1 and σ = 3.094˚A, with all Lennard–Jones interactions truncated\nwf wf\nand shifted at 10˚A. Electrostatic interactions were evaluated in real space up to 10˚A,\nwith long-ranged interactions treated using PPPM Ewald summation? , such that the\nRMS force error was 105 smaller than the Coulomb force between two unit charges at\n10˚A separation? .\nSimulations were conducted in the canonical (NVT) ensemble at T = 300K, con-\ntrolled using a Nos´e–Hoover chain with 5 thermostats and a damping constant of 0.1ps.\nDynamics were propagated using the velocity Verlet algorithm with time-step of 1fs.\nWe performed both fixed-charge and constant-potential simulations. Constant-potential\nsimulations employed the fluctuating-charge method? ? ? , in which the electrode atom\ncharges are solved self-consistently at each timestep using the ELECTRODE package? ,\nusing Gaussian charge of width 0.554˚A. Figure S15 shows the resulting water density\ndistributionsbeforeandafterchargingtheelectrodes,forbothfixed-chargeandconstant-\npotential conditions. We observed analogous behavior in simulations comprising 310\nmolecules of the simple dipolar fluid at a temperature of 500K.\nWecanunderstandtheformoftheelectrostaticpotentialarisingfromthesesubstratesby\napproximating their inhomogeneous charge distribution as being confined to two planes\nat z = ±z , where z is along the surface normal. Denoting the direction parallel to\ns\nthe surfaces as x, the electrostatic potential from one of these substrates is a linear\ncombination of functions of the form sin(k x)exp(−|k ||z −z |), where k = 2πn/L\nn n s n x\n(n = ±1,±3...), where L is the period along x. Owing to current practical limitations\nx\nof the neural functional techniques that we employ, we have focused on response to\nelectrostatic potentials with planar symmetry, i.e., either sin(k x) or exp(−|k |z).\nn n\nFar\nfrom the surface, the potential will be determined by the asymptotic form,\n(cid:18) (cid:19) (cid:18) (cid:19)\n2πx 2π|z−z |\ns\nϕ (x,z;z ) ∼ ϕ sin exp − (3)\nsingle s 0\nL L\nx x\nFor our simulation set up, the electrostatic potential midway between the substrates will\nbe well-approximated by ϕ(x,z;−z )+ϕ(x,z;z ).\ns s\nAs discussed in the main text, for L larger than a few molecular diameters, response\nx\nto these planar potentials can be considered to report on averages over a thin slice of\neither ∆z ≪ L or ∆x ≪ L . To verify the extent to which this holds, we first consider\nx x\nthe simple polar fluid. In the central panel of Fig. S16, we compare results from: (i)\nthe average density along z for a slice of thickness ∆x = 10˚A centered above the\nnegatively charged patch (depicted by the black rectangle in the leftmost panel); and\n(ii) the average density along z from a simulation with planar inhomogeneous symmetry,\nwhere the electrostatic potential is approximated by ϕ(z) = ϕ [exp(−2π|z+z |/L )+\n0 s x\nexp(−2π|z −z |/L )],\ns x\nand where the overall density is chosen to be the same as that\n13\nof the thin slice in case (i). As can be seen in the central panel, aside from a slight\ndiscrepancy far from the surface, the two sets of simulations agree well with each other.\nThe rightmost panel shows results for a similar procedure, this time taking a thin slice\ncentered at z = 0 (i.e., far from the surfaces) and with ∆z = 10˚A, as depicted by the\npink rectangle in the leftmost panel. In this case, the planar inhomogeneous simulation\nwas performed with ϕ(x) = ϕ sin(2πx/L ). As can be seen in the rightmost panel, the\n0 x\ntwo sets of simulations give near identical results.\nIn Fig. S17 we show an analogous set of results obtained for water. For the simulations\nwhere we compare the exponentially decaying potentials along z (i.e., the central panel\nin Fig. S17), we again see good agreement between the two sets of simulations. For the\nsinusoidal potentials (i.e., the rightmost panel in Fig. S17), the response to the planar\ninhomogeneous potentials have a pronounced asymmetry that is largely suppressed in\nthe simulation with interdigitated electrodes. This difference in geometries indicates a\ncorrelation between water’s response in the x and z directions that we cannot model\ndirectly within the current limitations of the neural functional approach. Nonetheless,\nthe most salient aspects of electromechanical response are captured and, if anything,\nenhanced in the full simulation with interdigitated electrodes.\n80\n60\n40\n20\n0\n40 20 0 20 40\nz[Å]\n]Å[x\n0.09\n0.06\n0.03\n0.00\n40 0 40\nz[Å]\n]3\nÅ[\nplanar single(z) exp( 2 |z zs|/Lx)\ninterdigitated (horizontal slice)\n0.03\n0.02\n0.01\n0.00\n0 40 80\nx[Å]\n]3\nÅ[\n(x,z)[Å 3] planar single(x) sin(2 x/Lx)\n0.00 0.02 0.04 interdigitated (vertical slice)\nFigure S16: Response of a dipolar fluid to electric field gradients. Left: density profile\nof a supercritical dipolar fluid at T = 500K confined between interdigitated electrodes. Mid-\ndle/right: density responses in simulations with (middle) an exponentially decaying potential or\n(right) a sinusoidal potential. The cross-sections highlighted in the left panel map closely onto\nthe corresponding reduced-potential simulations.\n80\n60\n40\n20\n0\n40 20 0 20 40\nz[Å]\n]Å[x\n0.09\n0.06\n0.03\n0.00\n40 0 40\nz[Å]\n]3\nÅ[\nplanar single(z) exp( 2 |z zs|/Lx)\ninterdigitated (horizontal slice)\n0.03\n0.02\n0.01\n0.00\n0 40 80\nx[Å]\n]3\nÅ[\n(x,z)[Å 3] planar single(x) sin(2 x/Lx)\n0.00 0.02 0.04 interdigitated (vertical slice)\nFigure S16: Response of a water to electric field gradients. Left: densityprofileofasuper-\ncritical SPC/E water at T = 700K confined between interdigitated electrodes. Middle/right:\ndensity responses in simulations with (middle) an exponentially decaying potential or (right) a\nsinusoidal potential.\n2) Related to the previous point, the authors state that their method greatly reduces compu-\ntational time, by comparing a timescale to evaluate the NN to an MD simulation time. This\n14\nis trivial, since what matters here should instead be the total computational time (ie includ-\ning training and testing), unless the neural functional can be used directly to cases with,\nfor example, completely different geometries. Can the authors comment on that?\nThe computational advantage of the neural cDFT approach arises from several factors. First,\nthe functional is trained locally, meaning that once learned, it can be applied to systems that\nare of larger length scales than those used in training. Although we do not fully exploit this\nproperty in the present work, it has been demonstrated in Refs.32 and35.\nFor the purposes of our study, however, the central advantage is that the neural cDFT frame-\nwork operates naturally in the grand canonical ensemble, giving us direct access to the chemical\npotential. More importantly, it also enables rapid and accurate evaluation of free energies, in-\ncluding grand potentials across different fluid configurations. This capability is essential for\naccurately mapping thermodynamics of liquid–vapor coexistence, adsorption, obtaining hys-\nteresis loops by determining metastable phases, and wetting.\nTo put this in perspective, obtaining the results shown in Figs. 3, 4, and 5 (phase diagrams, ad-\nsorption isotherms, and contact-angle behavior) via explicit molecular simulation would require\na combination of thermodynamic integration, enhanced sampling, and large-scale simulations.\nIn contrast, the neural cDFT functional is trained using approximately 2000 molecular simula-\ntions of modest size (a few hundred molecules) with randomly generated external potentials—\ncorresponding to a total computational cost of order ∼ 105 CPU hours. Once trained, the\nfunctional can be reused without retraining to evaluate thousands of free energy calculations\nat near negligible cost. \nWe have now added this point when discussing the method’s advantages\n(Lines 169–192) The resulting cDFT framework is not only efficiently computable on\nstandard hardware but also unparalleled in its ability to simultaneously capture micro-\nscopicfluidstructureandmesoscopicphasebehaviorunderarbitrarynon-uniformelectric\nfields. Unlike atomistic simulations, which are computationally prohibitive for such a\nbroad exploration, our method achieves orders-of-magnitude speedup, completing each\ncalculation in less than a minute without sacrificing quantitative accuracy. Importantly,\nthis computational efficiency is realized after a one-time training stage: once the func-\ntional has been learned, it can be used to perform thousands of free-energy calculations\nat negligible cost. This amortized advantage is what enables us to map complete phase\ndiagrams, adsorption isotherms, and metastable fluid branches—tasks that would require\nenhanced sampling and thermodynamic integration if carried out by molecular simula-\ntion alone. This efficiency enables an unprecedented, highly accurate mapping of a polar\nfluid’s response to EFGs of varying strengths and wavelengths across different thermo-\ndynamic conditions, from supercritical to subcritical regimes, spanning both bulk and\nconfined environments. With this powerful tool in hand, we now uncover emergent\nelectrostrictive phenomena that arise from the complex interplay of thermodynamics,\nconfinement, and response to EFGs.\n3) I am relatively unconvinced by the type of electric field gradients that are considered here.\nWhile the numerical results seem fine and interesting, I doubt they really correspond to some-\nthing technically achievable. They use molecular-scale gradients (wavelength comparable to\nthe molecular size sigma) while comparing to micron-scale experiments (local or interdigitated\n15\nelectrodes, for example). That of course does not mean that considering short wavelengths\nis wrong, but I doubt the effect survives under molecular confinement as a surface provides\ninteractions with liquid other than through a static electric field (adsorption, chemical defects,\nroughness, dielectric confinement, polarizability, etc.). In addition, under such confinement,\nI would expect the gradient to be along the surface (because of natural/engineered defects,\nfor example), rather than normal to it; I do not see any way of replicating experimentally the\nsituation of Fig 1 and 2.\nWe believe that we have addressed this point when answering Point 1 raised by the reviewer.\nTo reiterate, the new simulation shown in Fig. 1 and its associated discussion both motivates\nand clarifies the form the sinusoidal potentials that we use. Regarding whether this is technically\nachievable at the nanoscale, we note that nanoscale interdigiated electrodes do exist. In\naddition, progress in van der Waals heterostructure assembly is rapid, and already being used\nto probe the dielectric properties of fluids under nanoscale confinement.\n• Han et al., Dielectric capacitors with three-dimensional nanoscale interdigital electrodes\nfor energy storage, Sci. Adv 1, 9, (2015). https://doi.org/10.1126/sciadv.1500605\n• Castellanos-Gomez, A., Duan, X., Fei, Z. et al. Van der Waals heterostructures. Nat Rev\nMethods Primers 2, 58 (2022). https://doi.org/10.1038/s43586-022-00139-1\nFor completeness, the changes made to the main text are reproduced below:\n(Lines 95–131) Although molecular simulations cannot fully capture the influence of\nEFGs on dielectric fluids, they offer vivid qualitative insights into the underlying physics\nandillustratehowEFGsmayarisewithinnanoscaledevices, asillustratedinFig.1. Here,\nwe present a simulation snapshot of water at 300K confined between two hydrophobic\nsubstrates that have been patterned with alternating stripes of positive and negative\ncharge—a set up that provides a caricature of an interdigitated electrode. As can be\nclearly seen in Fig. 1, charging this device induces local wetting near the charged stripes,\nwith pronounced density variations along the direction parallel to the surface (x).\nThese features arise from the complex interplay between the fluid’s charge and number\ndensities, and their collective response to EFGs, arising from the inhomogeneous electro-\nstatic potential ϕ(x,z), where z is along the surface normal. It is instructive to consider\nthe form of ϕ far from either surface; here, it will resemble the linear superposition of\nthe asymptotic limit of the two substrates considered independently,\n(cid:18) (cid:19) (cid:18) (cid:19)\n2πx −2π|z−z |\ns\nϕ (x,z) ∼ ϕ sin exp , (4)\nsingle 0\nL L\nx x\nwhere z indicates the plane of the substrate’s outermost atoms. We therefore observe\ns\nthat inhomogeneity of ϕ is characterized by a sinusoidal oscillation of period L along\nx\nx, and an exponential decay along from the surface along z. This asymptotic analysis\ncaptures the essential behavior of ϕ(x,z) computed explicitly from the potential energy\nof a test charge, as seen in Fig. 1.\nThe pronounced wetting behavior observed with simulation in Fig. 1 strongly suggests\nthat such EFGs will influence capillarity of the polar fluid; that is, the amount of fluid\n16\nadsorbed at constant chemical potential. Addressing this issue, however, demands an\naccurate and efficient framework for determining structure, thermodynamics, and phase\nbehaviorinanopensystem—thisliesbeyondthepracticallimitsofpresentdaymolecular\nsimulations. Instead, we turn to classical density functional theory (cDFT), an exact\nstatistical mechanical framework for inhomogeneous fluids.\n(Lines 211–226) While EFGs will be most pronounced near the surfaces that generate\nthem, Fig. 1 demonstrates that they may persist relatively far from the interface. In the\nspecific case considered in Fig. 1, midway between the substrates we observe a sinusoidal\nelectrostatic potential along the x direction. This motivates us to understand the direct\ninfluence of such sinusoidal potentials on the fluid, without explicitly considering inter-\nfaces. In fact, for L larger than a few molecular diameters, we can define bulk response\nx\nifweconsideraveragesoverathinsliceofthickness∆z ≪ L (Fig.S16). Althoughsuch\nx\nacleanseparationofbulkandinterfacialresponsebecomeschallengingasL approaches\nx\nmolecular length scales, the full potential (as opposed to its asymptotic form) comprises\nmodes of decreasing wavelength—it is therefore instructive to understand bulk-like re-\nsponse across a broad range of wavelengths.\n(Lines 396–402) To gain insight into the influence of EFGs, we perform a “computa-\ntionalexperiment”inwhichweapplyasinusoidalelectricfieldacrosstheslit. Whilesuch\na set up does not correspond to an EFG established by the substrate walls themselves\n(see, e.g., Fig. 1), it does allow us to assess the effects of a particular mode. The impact\nis twofold: (1) condensation shifts to more negative ∆µ...\n(Lines 429–442) On the macroscopic scale, these effects manifest in electrowetting19\nand dielectrowetting18,20, where the contact angle of a droplet can be tuned with applied\npotentials. Our nanoscale simulations in Fig. 1 show that a similar phenomenology\nemerges under confinement: wetting is strongly enhanced directly over the electrode\npatches,whiletheelectrostaticpotentialgeneratedbytheinterdigitatedelectrodesdecays\ninto the slit. Following previous experimental work18,20, this motivates us employ a\nsimple planar electrostatic potential, ϕ(z) = ϕ exp(−2πz/L ) to probe wetting at the\n0 x\nnanoscale. Similar to our arguments above, results obtained from such a potential can\nbe considered to report on average behavior in a slice of thickness ∆x ≪ L (Fig. S16).\nx\n4) Similarly, for Fig 3-4 I would expect that chemical effects (eg acid-base reactions with the\nwall) play a much more important role at the considered lengthscales, and that modelling the\nwalls with a static external electric field is plain wrong. Another thing that comes to my\nmind is that for this kind of questions (wettability and fluid filling of a small pore), the MD\nsimulations used as training data are typically quite bad - it is for example very hard to correctly\nestimate a contact angle from simulations of water+LJ walls, and in nanometric slit, slightly\nchanging the LJ parameters of the wall can result in either full wetting or dewetting, at odds\nwith experimental results eg on carbon slits. Here, the authors choose one particular value\nof the walls LJ coefficients, so I would be very cautious about the results, which could be of\nsecond-order magnitude compared to changing interaction parameters.\nand the wall play an important role in determining the contact angle and, as the reviewer\ncorrectly points out, small changes in these parameters can have profound impact on the\nobserved wetting/drying behavior. Establishing precise values for, e.g., the strength of the\nwall-fluid interaction would be essential if our aim was to reproduce, say, the contact angle of\na specific system. But this is not our aim. Rather, within the broader context of capillarity\nphysics, it is well established that wetting and capillary evaporation/condensation are primarily\ngoverned by three factors:\n1. the wall–fluid interaction strength;\n2. the thermodynamic state of the fluid relative to liquid–vapor coexistence;\n3. the confinement length scale.\nThese principles trace back to classical results such as the Kelvin equation, extensive seminal\ncDFT studies, and many modern simulation studies; a particularly clear presentation on the\ntopic can be found at\n• Evans and Marconi, Phase equilibria and solvation forces for fluids confined between par-\nallel walls, J. Chem. Phys. 86, 71380–7148 (1987). https://doi.org/10.1063/1.452363\n• Evans, Fluids adsorbed in narrow pores: phase equilibria and structure, J. Phys. Condens.\nMatter 2, 989 (1990). https://doi.org/10.1088/0953-8984/2/46/001\nOur goal is not to model a specific substrate, but to demonstrate that, for polar fluids, there\nexists an additional and previously underexplored mode of control—namely, dielectrophoretic\ncoupling to electric field gradients—which operates alongside the conventional determinants of\ncapillarity. Framed another way: The influence of wall–fluid interactions on wetting has already\nbeen extensively characterized for both simple liquids and water (see Refs.71 and13), and our\naim here is to build on this foundation by isolating the dielectrophoretic contribution.\nTo address this comment, we now provide broader context to the physics of capillarity at the\nfollowing points in the manuscript:\n(Lines 29–40)Itiswellestablished—initiallyatthemacroscopicscalethroughtheworks\nof Young, Laplace, and Kelvin11,12, and later at the microscopic scale13,14—that adsorp-\ntion depends not only on the system’s thermodynamic state, but also on the confine-\nment length and the substrate–fluid interaction. These factors are typically intrinsic\nmaterial properties. As a result, extensive research into enhancing adsorption in porous\nmaterials has focused on optimizing these factors, e.g., by tuning porosity or chemical\nfunctionalization15. However, the potential for manipulation by external means—using\napplied fields to control confined fluids—remains relatively unexplored.\n(Lines 377–383) In the absence of an applied external field, it is well-established that\ncapillarity is controlled by the chemical potential of the reservoir, the length scale of con-\nfinement, the substrate–fluid interaction and temperature13. With EFGs, we introduce\nan additional experimental handle by which to control fluid adsorption behavior; we call\nthis new phenomenon “dielectrocapillarity.”\n18\n5) Some of the results discussed here are in the large length scale regime, eg Janus colloids.\nI am afraid this limit is quite a dangerous one to take: colloids under an applied field are\nbasically never at thermal equilibrium (because of electrophoresis, for example), so the current\nframeworkwouldnotapply, evenifitwasextendedtoincludedynamicaleffects(onewouldneed\nto add hydrodynamics, ions, etc.). While the thought experiment presented in lines 208-226 is\nconceptually interesting, it is important to signal such limitations.\nWe believe that in the previous version of the manuscript likely gave the impression that the\nsinusoidal inhomogenoeus fields were likely to arise under nonequilibrium conditions, such as\nan AC experiment. We hope that the new discussion surrounding Fig. 1 makes clear that\nsuch inhomogeneous fields can in principle be achieved under equilibrium conditions — for the\nfield arising from a substrate with a static charge distribution, or held at constant potential\ndifference, there is no reason to assume a priori that the fluid will not reach an equilibrium\nstate. (Any steady state current in this system would violate the underlying symmetry.)\nIn light of the reviewer’s comments, we have changed the wording in the manuscript to em-\nphasize that our comment pertains to an equilibrium effect, and we have also softened the\nlanguage.\n(Lines 302–310) In contrast, the short-ranged colloidal system lacks this screening,\nleading to a much stronger response (dashed purple lines, Fig. 3b and inset). Such an\nequilibrium effect could be leveraged for programmable directed self-assembly47, where\nEFGs in combination with tunable Janus particle surfaces48 may provide a powerful\ntool for tailoring the assembly of extended structures with dielectrophoretic forces, with\nadditional tunability arising from the solvent and ionic strength.\n6) Line 28: Fundamentally, fluid uptake in these systems is driven by capillary forces Not really.\nSeparation and filtration processes are typically out of equilibrium, with (eletro/diffusio)osmotic\neffects often playing a major role.\nOur intention was not to imply that practical separation or filtration operates purely under equi-\nlibrium conditions, but rather to highlight the underlying thermodynamic mechanism relevant\nfor our study. We have therefore revised the sentence to read:\n(Lines 27–29) The physics of capillarity plays a fundamental role in determining fluid\nuptake in these systems.\n7) Line 40: While a uniform field exerts a force only on free charges such as ions This is\nmisleading; fundamentally this force is cancelled and fully transmitted to the solvent because\nof viscous dissipation. In almost all nanofluidic experiments, applying a uniform electric field\nresults in a net flow of water (because of electro-osmosis and related phenomena). In typical\nslit-like geometries like the one considered here, the effect is often dominant compared to other\ntransport phenomena. See for example the 2021 review by Kavokine, Netz and Bocquet.\nWe have revised this sentence to clarify that we are referring to the direct interaction between\nthe fluid and the field (i.e., F = qE ):\next ext\n(Lines 43–46) While a uniform field exerts a direct force only on free charges such as\nions, non-uniform fields with electric field gradients (EFGs) generate dielectrophoretic\nforces on neutral polar molecules.\n19\n8) I find Fig 2 difficult to follow due to the high number of curves, colors, line styles, etc.\nIn particular, I do not fully understand Fig 2d. What is represented as inset? What are the two\nblue and red rectangles on the right?\nThe inset in Fig. 2d was originally intended to illustrate the density profiles of the corresponding\ngas and liquid phases under no field and under a field gradient. Since this information is already\npresented in Fig. 2c, we have removed the insets from Fig. 2d. The blue and red boxes indicate\nthe supercritical and subcritical temperatures for which the responses in Figs. 2a-c are shown.\nWe have revised the figure to clarify this distinction. This is now Figure 3 in the revised\nmanuscript.\nT*<T*\nc, 0\na\nc\nλ = 6.6 σ λ = 2.2 σ *\nd\nb\nσ 1.7\nλ\n=\nλ = 2\n.2σ\nλ= 3 .3σ\nλ = 6.6 σ\nxamE\ngnisaercni\nμ\ngnisaercni\nT*>T*\nc, 0\n(ii)\n(iv)\n(i) (iii)\nEm * ax = 0 Em * ax = 4 Em * ax = 8\nT* > Tc*,0\n(i) Em * ax = 0 (ii)\n(iii) Em * ax = 4 (iv)\nT* < Tc*,0\nFigure 3: Controlling liquid–vapor equilibrium with EFGs. (a) At T∗ > T∗ where T∗ ≈\nc,0 c,0\n1.97, increasing EFGs by independently varying λ and E amplifies dielectrophoretic rise.\nmax\n(b) The local electrostrictive response of the dipolar fluid, as measured by the rise in the\nmaximum density peak relative to zero field ∆ρ∗ , is highly non-linear. Solid and dashed\nmax\nlines show the response of systems with dipolar interactions that are long-ranged (LR), e.g.,\npolar molecules, and screened short-ranged (SR), e.g., colloids, respectively. The effect of LR\ninteractions becomes pronounced for λ ≫ σ. The SR fluid is a nearly identical dipolar fluid,\nbut whose Coulomb potential is replaced by erfc(κr)/r where κ−1 = 1.5σ. (c) At an isotherm\nwhere T∗ < T∗ , stable solutions for the density ρ∗(z) under a sinusoidal electric field with\nc,0\nλ/σ = 1.7, are shown for different values of the chemical potential. These results are used to\ninvestigate liquid–vapor coexistence. (d) Results in light pink show the binodal of the dipolar\nfluid in the absence of an electric field. At E∗ = 4, T∗ shifts to a lower temperature, as\nmax c\nseen in the binodal in dark purple. Solid symbols show results obtained from the multiscale\ncDFT approach, while crosses indicate estimates of T∗ using the law of rectilinear diameters\nc\nand critical exponents14. Solid lines serve as a guide to the eye.\n20\n9) For all figures, some of the pastel colors are difficult to see after printing.\nWe updated the figures to use darker colors for better visibility in print.\n10) Line 322: That is, EFGs impact criticality of polar fluids in both the three-dimensional\nand confined two-dimensional Ising universality classes I do not understand this sentence, I\nwould suggest explaining more. I guess that the authors are referring to the fact that the\nliquid-gas phase transition corresponds to the 3D Ising class; but I am less familiar with the\ncapillary transition and I would imagine that its does not belong to a single universality class\nbut rather would depend on eg. the pores geometry. In practice, you would likely need to have\na wedge/1D/0D geometry to have a localized electric field in the confined system, so I do not\nthink it is realistic to have the exact 2D regime.\nI also do not understand what this discussion is bringing here; if the observed effect modifies\nthe systems chemical potential, it is obvious that it will affect both phase transitions, which\nare defined from it It is not particularly surprising that modifying the system’s parameters shifts\nthe critical points; what would be more relevant would be to discuss orders of magnitude, like\nthe 50K shift in the case of the liquid-gas transition of water. In addition to this, the authors\ncould discuss what orders of magnitude can be reached in terms of electric field (gradient)\nexperimentally. They list a value of E = 0.4 V/A.\nTo get this kind of field from say a point charge of charge Zq - say surface defects - in water\nover 1nm (which corresponds roughly to the setting considered here), you would need Z = 10,\nwhich is a bit wild.\nIn the interests of a general readership, we have removed the statement about the 2D/3D\nuniversality. Itisimportanttoemphasize, however, thatwedonotmodifythesystem’schemical\npotential — we are working in the grand canonical ensemble in which the chemical potential is\nheld constant. The EFGs introduce an additional electrostatic contribution to the free energy,\nbut they are conjugate to the charge density, unlike the chemical potential, which is conjugate\nto the one-body density.\nRegarding orders of magnitude, the field gradients used in our study correspond to modest\ncharge densities in the underlying MD simulations: on the order of 0.05e/atom, or potential\ndifference of 10V across nanometric distances. These values are within the range generated\nnear charged surfaces and patterned electrodes at the nanoscale. (Refs.17,18,24)\n(Caption to figure 1) Inhomogeneous electric fields arising from interdigitated\nelectrodes strongly influence water’s wetting behavior. Snapshot of an SPC/E\nwater simulation in a hydrophobic slit with alternating positive and negative electrode\npatches. Either holding the electrodes at a 10V potential difference or attributing a\nfixed charge of ±0.05e/atom causes the fluid to exhibit enhanced wetting at the walls,\naccompanied by strong lateral density oscillations. Cross-sections of the electrostatic\npotential in the constant charge setup are shown parallel to the surface (top right) and\nnormal to the surface (bottom left).\n11) Related to the previous point, it seems to me that the only two relevant regimes for how\nto create an electric field gradient within a confined pore are (1) electric field gradient occuring\non a molecular lengthscale close to the walls or (2) a gradient with a lengthscale equal to the\nsystem size. The authors do not really discuss the impact of the system size in the settings\n21\ncorresponding to Fig 3.\nThe reviewer is correct to point out that the confinement length can play an important role.\nAs discussed in our response to Point 4, the confinement length is indeed one of the key\ndeterminants of capillarity, and its influence on wetting and condensation has been extensively\ncharacterized in prior studies of simple and polar fluids. In the present work, our aim is to\nisolate the role of dielectrophoretic forces, and for this reason we focus on a moderately wide\nsolvophobic slit, H = 6.6σ, where the two walls are sufficiently separated that confinement\neffects do not overwhelm the electric field-induced response.\nIt would be interesting to investigate any interplay between the length and the field; it lies\nbeyond the scope of the current paper, however.\nFor completeness, we list the relevant changes to the manuscript. (They are the same as those\nin Point 4.)\n(Lines 29–40)Itiswellestablished—initiallyatthemacroscopicscalethroughtheworks\nof Young, Laplace, and Kelvin11,12, and later at the microscopic scale13,14—that adsorp-\ntion depends not only on the system’s thermodynamic state, but also on the confine-\nment length and the substrate–fluid interaction. These factors are typically intrinsic\nmaterial properties. As a result, extensive research into enhancing adsorption in porous\nmaterials has focused on optimizing these factors, e.g., by tuning porosity or chemical\nfunctionalization15. However, the potential for manipulation by external means—using\napplied fields to control confined fluids—remains relatively unexplored.\n(Lines 377–383) In the absence of an applied external field, it is well-established that\ncapillarity is controlled by the chemical potential of the reservoir, the length scale of con-\nfinement, the substrate–fluid interaction and temperature13. With EFGs, we introduce\nan additional experimental handle by which to control fluid adsorption behavior; we call\nthis new phenomenon “dielectrocapillarity.”\nWe also clarified the electrostatic potential lengthscale in relation to the interdigitated electrode\nset-up.\n(Lines 430–443) On the macroscopic scale, these effects manifest in electrowetting19\nand dielectrowetting18,20, where the contact angle of a droplet can be tuned with applied\npotentials. Our nanoscale simulations in Fig. 1 show that a similar phenomenology\nemerges under confinement: wetting is strongly enhanced directly over the electrode\npatches,whiletheelectrostaticpotentialgeneratedbytheinterdigitatedelectrodesdecays\ninto the slit. Following previous experimental work18,20, this motivates us employ a\nsimple planar electrostatic potential, ϕ(z) = ϕ exp(−2πz/L ) to probe wetting at the\n0 x\nnanoscale. Similar to our arguments above, results obtained from such a potential can\nbe considered to report on average behavior in a slice of thickness ∆x ≪ L (Fig. S16).\nx\n(Lines 456–462) Our multiscale framework provides a microscopic perspective on this\nphenomenology, and allows us to test whether, at the nanoscale, EFGs indeed enhance\nthe wetting of dielectric liquids and the extent to which the scaling prescribed by Eq. ??\nholds. To this end, we applied ϕ(z) with L ≈ 7σ to the confined dipolar model,\nx\nsymmetrically from both walls of a solvophobic slit.\nmolecular simulations that explicitly connect our model electric field gradients to experimentally\nrelevant geometries, notably interdigitated electrodes. We have also clarified the role of geom-\netry and length scales throughout the manuscript, emphasizing both the scope and limitations\nof the present approach. 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