# Mechanisms of coffee-ring suppression in Picoliter silver nanoparticle ink droplets: Implications for printed electronics

Yahya Rharbi, Vincent Faure, Anne Blayo, Albert Magnin, Aurore Denneulin

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Yahya Rharbi, Vincent Faure, Anne Blayo, Albert Magnin, Aurore Denneulin. Mechanisms of coffee-ring suppression in Picoliter silver nanoparticle ink droplets: Implications for printed electronics. Journal of Colloid and Interface Science, 2026, 707, pp.139703. 10.1016/j.jcis.2025.139703. hal-05421340v2

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# Mechanisms of coffee-ring suppression in Picoliter silver nanoparticle ink droplets: Implications for printed electronics

Yahya Rharbi<sup>a,*</sup>, Vincent Faure<sup>a,b</sup>, Anne Blayo<sup>b</sup>, Albert Magnin<sup>a</sup>, Aurore Denneulin<sup>b,*</sup>

<figure><img src="image_3.png" /></figure>

## HIGHLIGHTS

* Temperature tunes coffee-ring formation in picoliter AgNP printed droplets.

* Thermal control strategy enables high-resolution printed electronics.

* Diffusion and osmotic pressure efficiently homogenize solute distribution.

* Marangoni flow and viscous dissipation play only a minor role in ring suppression.

## GRAPHICAL ABSTRACT

Diffusion and Osmotic Pressure Suppress the Coffee-Ring Effect in Printed Electronics

<figure><img src="image_4.png" /></figure>

## ARTICLE INFO

### Keywords:

- Coffee-ring effect

- Silver nanoparticles

- Picoliter droplets

- Capillary flow

- Marangoni effect

- Fickian diffusion

- Printed electronics

## ABSTRACT

The coffee-ring effect (CRE) remains a major obstacle to achieving uniform functional deposits in high-resolution printed electronics, particularly for picoliter-scale droplets where evaporation dynamics are complex and poorly understood. This study demonstrates that precise control of substrate temperature is a highly effective strategy for suppressing the CRE in inkjet-printed picoliter droplets of a commercial silver nanoparticle ink. We identify a distinct morphological transition: uniform deposits form at low temperatures (20–40 °C), pronounced coffee-rings develop at intermediate temperatures (50–70 °C), and central accumulation emerges at high temperatures (>90 °C). Through a combination of experimental analysis, scaling arguments, and numerical simulations, we systematically rule out Marangoni flows and viscous dissipation as the primary suppression mechanisms at low temperatures. Instead, we show that the extended drying time at low substrate temperatures drastically reduces the Péclet number, shifting the dominant transport mechanism from outward capillary advection to inward particle diffusion. This diffusion-driven homogenization ensures uniform particle redistribution prior to immobilization. Our findings provide a robust, practical, and readily applicable thermal strategy for eliminating capillary-driven inhomogeneities, paving the way toward the reliable fabrication of high-resolution printed electronic devices with superior morphological and functional uniformity.

<footnote><sup>a</sup> Univ. Grenoble Alpes, CNRS, LRP, F-38000 Grenoble, France</footnote>

<footnote><sup>b</sup> Univ. Grenoble Alpes, CNRS, Grenoble INP, LGP2, F-38000 Grenoble, France</footnote>

<footnote>* Corresponding authors.</footnote>

<footnote>E-mail addresses: yahya.rharbi@univ-grenoble-alpes.fr (Y. Rharbi), aurore.denneulin@grenoble-inp.fr (A. Denneulin).</footnote>
# 1. Introduction

Controlling the morphology and uniformity of deposits formed by the evaporation of colloidal suspensions is essential for high-precision manufacturing processes [1–4]. A central challenge in such processes stems from the evaporation-driven coffee-ring effect (CRE) [5,6]. This effect occurs when a sessile droplet containing dispersed particles dries on a substrate, generating capillary flows that transport solutes toward the contact line. The resultant ring-like deposition introduces structural heterogeneity, which is problematic in applications requiring uniform material distribution. This is particularly critical to the context of printed electronics and related technologies [4,7,8] such as transistors [9–11], solar panels [12], batteries [13], and memories [14], in which the non-uniformity represents a major technical bottleneck. For example, inkjet-printed conductive tracks and connectors rely on the controlled deposition of silver nanoparticles (AgNPs) to ensure electrical conductivity and mechanical stability in miniaturized devices [4,15–17]. The CRE-driven assembly of AgNPs at the edges can lead to irregular conductive connectors [15,18,19], degrading device performance. Addressing this limitation becomes crucial for advancing the reliability and scalability of these applications.

The radial capillary flow arises from the coupling of contact-line pinning with edge-enhanced evaporation, which alters the local radius of curvature and establishes a Laplace-pressure gradient; this imbalance drives an outward flux that transports fluid and solutes toward the rim [2,20–28]. Building on this framework, four control levers emerge: (i) capillary flow and its viscous dissipation, which sets the advection speed; (ii) solute redistribution via diffusion, with the Péclet number (Pe) quantifying the competition between advection and diffusion; (iii) contact-line dynamics (pinning strength, stick-slip, wetting hysteresis), which control the residence time of solutes at the rim; and (iv) interfacial stress (thermal or solutal Marangoni effects), which can reverse the outward capillary flow [5,20,26,27,29]. In practice, suppressing non-uniform deposits requires shifting this balance by either reducing advective throughput (increasing viscosity or inducing early gelation), enhancing diffusive leveling (lowering Pe or shortening the drying time at the rim), or actively reversing internal circulation (thermal or solutal Marangoni, tailored surfactant kinetics). Substrate engineering (controlled wettability and roughness) further modulates pinning and heat transfer, thereby tuning local evaporation and curvature, and thus the Laplace-pressure landscape that drives the flow. Additional strategies involve the use of surfactants to regulate interfacial tension and internal flows, as well as external fields such as electric or acoustic forcing to actively control droplet dynamics during evaporation. Altogether, these approaches act on the coupled fields of flow, interfacial tension, and evaporation, enabling transitions from ring-dominated deposits to uniform films or centrally peaked morphologies, depending on operating conditions. 6 [19,25,30–40], Yet, despite two decades of intense research and numerous proposed methods for suppressing the coffee-ring effect, achieving comprehensive and robust control over deposit morphology across a wide range of conditions remains a significant challenge [19,32–40].

In contrast to the extensive investigations on millimeter-scale droplets, the drying dynamics of picoliter-scale droplets are comparatively less explored [41–45]. This knowledge gap is critical, as such microdroplets underpin the fabrication of miniaturized electronic components, including printed circuits, transistors, and sensors, using metallic nanoparticle suspensions such as silver and carbon-based inks [4,7]. At these volumes, interfacial stresses and confined microfluidic effects play a dominant role in the evaporation process, with implications that may differ from those observed at larger scales and *cannot always be* reliably extrapolated [46].

Furthermore, the flow dynamics governing the generation of picoliter droplets in inkjet printing raise the question of how this stage influences particle redistribution and, ultimately, the final deposit morphology. Droplets are expelled at high velocity from the nozzle and

undergo rapid impact on the substrate, followed by spreading and retraction [42,47–49]. These transient hydrodynamic events, dominated by inertial spreading, contact-line motion, and viscous dissipation, can restructure internal flows and redistribute particles well before evaporation sets in as the timescale [47]. Whether such early-stage dynamics suppress or amplify accumulation at the contact line remains an open issue, central to controlling deposition in microscale printing [41].

In printed electronics, these challenges are further amplified at the picoliter scale, where conductive silver nanoparticle inks are widely employed [7,8]. Yet, detailed investigations into the underlying mechanisms remain scarce, with most studies restricted to post-drying morphological characterization [15,18,19]. Addressing this gap requires comprehensive strategies that couple suspension rheology, evaporation dynamics, and jetting-induced hydrodynamics with a clear understanding of how to suppress the coffee-ring effect in printed flexible devices.

Although parameters such as surface wettability, substrate temperature, solvent viscosity, and ink composition have been widely examined, most studies remain phenomenological and focus on final deposit morphologies rather than on developing a unified framework for the underlying mechanisms. In practice, research has focused on modulating drying pathways through solvent formulation, volatility tuning, substrate temperature, and ambient humidity [4,15,18,41,50]. While these approaches help clarify the role of evaporation on deposit morphology, a unified strategy for robust and tunable suppression of the coffee-ring effect in picoliter droplets is still lacking. This gap is particularly critical for picoliter-scale printed electronics, where achieving consensus on controlling capillary-driven flows to tailor deposit morphology remains a central challenge for high-resolution device fabrication [4,8].

This study examines the pattern formation and drying dynamics of picoliter-scale silver nanoparticle droplets produced by piezoelectric drop-on-demand printing, with emphasis on strategies to control or suppress the coffee-ring effect. The objective is to determine how printing and thermal conditions govern evaporation-driven transport and, consequently, the final deposit morphology. The approach is structured around a three-part methodology: (i) control droplet generation to obtain repeatable droplet size and satellite-free ejection with well-defined volume and velocity; (ii) a systematic study of drying dynamics across a range of substrate temperatures to identify regimes that suppress coffee-ring formation and favor uniform, compact morphologies; and (iii) a mechanistic analysis combining scaling arguments and numerical simulations to disentangle the interplay of capillary advective, Marangoni-induced backflow, viscous dissipation, and concentration-gradient-driven transport (Fickian diffusion and osmotic pressure). This framework provides a foundational set of design rules for ink rheology and jet dynamics, facilitating the robust fabrication of high-resolution features essential for printed flexible electronics, with superior morphological and functional uniformity.

# 2. Experimental

## 2.1. Material

Commercial silver nanoparticle ink (Sicrys™ I30G-1; PvNanoCell Ltd.) was used as the stock dispersion. The formulation consists of a light-grey suspension of silver nanoparticles (conductive phase) stabilized by poly(vinylpyrrolidone) (PVP) in ethylene glycol (EG). Ethylene glycol (Sigma-Aldrich, ≥99 %, CAS 107–21-1; boiling point 197.3 °C; viscosity ≈16.1 mPa·s at 25 °C) was used to dilute the ink suspension. The diluted mixture was homogenized on a roller mixer for 1 h and filtered through a 0.45 µm PTFE syringe filter prior to use.

## 2.2. Ink characterization

The nanoparticle size distribution was measured by dynamic light
scattering (Zetasizer Nano, Malvern Instruments, UK). The ink was diluted in ethylene glycol to suppress multiple scattering, and a refractive index of 1.4318 was used. The mean diameter was 74 nm, with a range of 55–95 nm.

The ink composition was determined by thermogravimetric analysis (STA6000, PerkinElmer, USA). A 30 mg sample was heated at 10 °C/min, and the mass was continuously recorded. The analysis indicated 30 wt% silver nanoparticles and 70 wt% ethylene glycol. Based on the respective densities of silver (10,500 kg·m⁻³) and ethylene glycol (1112 kg·m⁻³), the silver volume fraction was calculated as 4.31 vol%.

Ink density was determined gravimetrically at 25, 35, and 45 °C using sealed, temperature-equilibrated 10 mL volumetric flasks. Ten replicate measurements at each temperature yielded values of 1514 ± 15, 1508 ± 21, and 1501 ± 18 kg·m⁻³, respectively. FTIR spectra were collected using a Spectrum 65 spectrophotometer (PerkinElmer, USA). A 50 mg sample was air-dried, ground with KBr (1,4 ratio), and pressed into pellets. Absorption bands at 1664, 1282, and 1267 cm⁻¹ confirmed the presence of polyvinylpyrrolidone (PVP), which acts as a steric stabilizer for the silver nanoparticles.

Surface tension (γ) was measured with a Krüss K10ST tensiometer (Du Noüy ring) at 25, 35, and 45 °C. Ten replicates at each temperature (T) gave values of 48.1 ± 0.1, 46.9 ± 0.2, and 46.4 ± 0.3 mN·m⁻¹, respectively. A linear fit of γ vs. T yielded γ = 50.1–0.085 T (with T in °C) leading to temperature-derivative of surface tension of dγ/dT = -0.085 mN·m⁻¹ °C⁻¹. This behavior is consistent with ethylene glycol data measured over a wider range (beyond ≈120 °C), which extrapolate to γ(0 °C) ≈ 49.6 mN·m⁻¹ with dγ/dT ≈ -0.071 mN·m⁻¹·°C⁻¹ [51]. Marangoni fluxes may be estimated using dγ/dT obtained either from the ink measurements or from neat ethylene glycol.

## 2.3. Printing substrate

A 125 µm polyethylene naphthalate (PEN, TEONEX®) film was used as the substrate, selected for its relevance to printed electronics. According to the manufacturer, the film exhibits a surface energy of 40.2 mJ·m⁻² and a thermal dimensional stability of 0.09 % at 150 °C. Experiments were conducted in the range 20–120 °C.

Surface roughness of the PEN film was measured by AFM (Bruker Dimension Icon®, tapping mode, OTESPA® probe, 300 kHz). Ten 10 × 10 µm² scans over several regions gave an average Ra of 4.3 ± 0.8 nm, confirming suitability for uniform ink deposition.

Contact angles of ink or EG on the PEN substrate were measured using a Digidrop goniometer (GBX, France) equipped with automated drop-shape analysis. Sessile droplets were gently deposited on the substrate, and the contact angle θ was extracted by fitting the droplet profile with a Young–Laplace model. Measurements were repeated at several locations to ensure reproducibility.

## 2.4. Inkjet printing setup

Printing was performed with a piezoelectric drop-on-demand printer (DMP-2831, Fujifilm) using disposable DMC-11610 printheads (16 nozzles, 21.5 × 21.5 µm, 254 µm spacing, ~10 µL droplets). Ejection was controlled by Dimatix Drop Manager software with a four-segment waveform (Fig. S1) allowing adjustment of pulse duration and slope. The actuation voltage was varied between 10 and 40 V, with 20 V used unless otherwise specified. A backpressure of -4.5 mm H₂O was applied to prevent dripping, and the nozzle–substrate distance was fixed at 1 mm. The nozzle temperature was controlled at fixed set points between 20 and 45 °C to regulate ink viscosity and other relevant fluid properties. The printing stage was temperature-controlled with a Peltier system (up to 120 °C). Substrates were equilibrated at the target temperature for 20 min before droplet ejection.

Droplet ejection and flight were monitored with the DropWatcher module in the Dimatix Drop Manager software. A stroboscopic light source synchronized with a high-speed camera captured time-resolved

images over a 1.2 mm flight path, from which droplet velocity was determined (Fig. S2). The volume of a single droplet was estimated by ejecting more than 3 × 10⁶ droplets into a pre-weighed dish and measuring the collected mass. Using the ink density and droplet count, The average volume of a single droplet (V<sub>drop</sub>) is calculated as V<sub>drop</sub> = m<sub>ink</sub>/ρ<sub>ink</sub> where m<sub>ink</sub> is collected ink mass, ρ<sub>ink</sub> is the ink density, and N<sub>drops</sub> is the total number of ejected droplets.

## 2.5. Visualization of droplet morphology during drying

A custom optical imaging system was integrated into the Dimatix platform for in situ visualization of droplets on the substrate (Fig. S3). It consisted of a Sony Alpha6000 camera (25 fps, 1440 × 1080 px) equipped with a 2× lens, a Navitar zoom (3–12×), and a 10× long-working-distance objective, mounted at a 40° angle and illuminated by a fiber-coupled light source. Geometric constraints of the printhead were overcome by repositioning the cartridge (~10 cm offset) and mounting the Peltier stage on a motorized translation table, allowing rapid transfer of the substrate into the optical field. Droplet morphology and drying-front dynamics were monitored from seconds to hours, with tracking software extracting the front position from video frames for quantitative analysis.

## 2.6. Morphology characterization of dried droplet

Three-dimensional surface profiles of dried droplets were measured with an Alicona Infinite Focus microscope (Bruker Alicona, focus variation, 100× objective). Acquisition was performed with 600 µs exposure, contrast level 1, 20 nm vertical resolution, and 1 µm lateral resolution. Two-dimensional profiles were extracted along multiple center-crossing directions, defining the height profile of the dried deposit, denoted as h(r, ∞) or simply h. Acustom Python script was used to compute the base radius (R), deposited volume (V<sub>s</sub>), and a parameter quantifying material redistribution within the footprint. This geometry was then used to back-trace an estimate of the time-dependent height profile, denoted h(r, t).

## 2.7. Rheology measurement

The temperature dependence of ink viscosity was measured with an Ares G2 rheometer (TA Instruments) using a coaxial Couette geometry (28 mm rotor, 30 mm stator, 1 mm gap) and Peltier temperature control. Measurements at shear rates (γ̇) of 0.1 and 30 s⁻¹ were performed from 20 to 120 °C in 20 °C steps, after waiting for 20 min for thermal equilibration. Data were fitted with the Arrhenius equation.

The effect of silver nanoparticle concentration on viscosity (η) was measured with a DHR-3 rheometer (TA Instruments, 25 mm, 1° cone-plate). Concentration was increased by controlled solvent evaporation at 35 °C under stirring, followed by gravimetric determination of silver fraction. Viscosity was then measured at shear rates of 1–100 s⁻¹ at 30 °C, with 180 s acquisition per point to reach steady state.

## 2.8. Evaporation kinetic

Evaporation rate measurements were performed mainly on ethylene glycol, since its behavior dominates that of the ink. Two configurations were tested: flat surfaces and microdroplets.

For the flat surface, 10 mL of ethylene glycol was placed in a 50 mm dish on a temperature-controlled balance (30–110 °C). Evaporation rates were determined by linear regression of mass loss curves (R² > 0.997), and fluxes were obtained by normalizing the rates to the surface area.

For microdroplets, approximately 15 drops (5–40 µL) were deposited into pre-weighed dishes at 30 °C, and evaporation rates were extracted similarly (R² > 0.965).
The evaporation kinetics were characterized for both the ink and its primary constituent, ethylene glycol. Since ethylene glycol constitutes the majority of the ink formulation, its behavior governs the overall evaporation process for most of the drying period. Three complementary experimental configurations were employed to capture evaporation behavior across different scales and conditions: macroscopic flat pools and sessile droplets ranging from microliter to picoliter volumes.

In the macroscopic pool configuration, 10 mL of ethylene glycol was placed in a 50 mm dish on a temperature-controlled balance, with measurements taken across a temperature range of 30–110 °C. Evaporation rates were determined from linear regression of mass loss versus time (R² > 0.997), and corresponding evaporation fluxes were calculated by normalizing these rates to the available surface area. Similar tests were conducted with the ink at selected temperatures to verify that its drying kinetics were consistent with those of ethylene glycol.

The microliter sessile-drop experiments were performed using approximately 15 drops of both ink and ethylene glycol, with volumes ranging from 5 to 40 µL, deposited onto pre-weighed PEN film at a constant substrate temperature of 30 °C. Evaporation rates were extracted via linear regression of mass loss data. This approach served to verify the linear dependence of the drying rate on drop radius, enabling reliable extrapolation of evaporation kinetics to the picoliter regime and facilitating direct comparison with established literature.

The evaporation of picoliter-scale droplets was analyzed using ink suspensions deposited on PEN substrates and monitored with the visualization setup described in Section 2.4. The evaporation kinetics at various substrate temperatures were quantified by in situ tracking of

drop morphology between 20 °C and 40 °C. This approach generated datasets of evaporation kinetics that could be directly compared with the concurrent analysis of microscale deposit morphology at 30 °C.

## 2.9. Simulation of the flows during drying

Axisymmetric thermal-conduction simulations were performed in FEniCS (Python) for a spherical-cap droplet of ethylene glycol (base radius 20 µm, contact angle $θ = 53°$) resting on a PEN substrate (thickness 150 µm, diameter 400 µm). An adaptive mesh with refinement at the three-phase contact line and in the underlying substrate was used to resolve steep gradients. Boundary conditions included: latent-heat removal at the droplet-air interface; a fixed temperature at the bottom of the substrate; convective heat exchange with air on all exposed surfaces; and continuity of temperature and normal heat flux at the liquid-solid interface. These conditions allowed us to resolve the temperature fields in the droplet and the substrate and, in particular, the surface temperature variations, which were used to calculate the Marangoni number and estimate the magnitude of the resulting Marangoni-driven fluxes.

# 3. Results and discussion

## 3.1. Silver ink rheology

Given the central role of ink rheology in drop ejection, spreading, and drying, its dependence on temperature and concentration was

<figure><img src="image_5.png" /><figcaption>**Fig. 1.** (a) Arrhenius representation of viscosity for the silver nanoparticle ink (4.3 vol%, ■) and pure ethylene glycol (▲) at a shear rate of 1 s⁻¹. Solid lines correspond to Arrhenius fits, yielding activation energies of 19.2 kJ·mol⁻¹ for the ink and 28 kJ·mol⁻¹ for ethylene glycol. (b) Steady-shear viscosity curves as a function of shear rate for suspensions with silver volume fractions from bottom to top 2.5 %, 12.5 %, 44.1 % and 57.4 % at 30 °C. Nearly Newtonian behavior is observed up to ~12 vol%, followed by progressive shear-thinning at higher concentrations. Power-law fits are overlaid using $η \propto \dot{\gamma}^{-n}$ (c) Viscosity at 1 s⁻¹ plotted versus silver volume fraction, showing divergence near the close-packing limit ($φ ≈ 0.63$). The data are fitted with the Krieger-Dougherty model ($η_s = 22.2$ mPa·s, $φ_m = 0.605$, $[η] = 6.4$). (d) Flow index n obtained from power-law fits as a function of silver concentration, decreasing continuously with increasing $φ$, consistent with the alignment of anisotropic or branched aggregates under shear.</figcaption></figure>
systematically analyzed. The viscosity (η) of the silver nanoparticle ink at volume fraction φ of 4.3 vol% and pure ethylene glycol at 1 s⁻¹ (Fig. 1a) followed an Arrhenius-type dependence. The ink exhibited a flow activation energy E_a of 19.2 kJ·mol⁻¹, lower than that of ethylene glycol (28 kJ·mol⁻¹), indicating reduced thermal sensitivity due to the presence of the nanoparticles.

At 30 °C, steady shear measurements for suspensions with particle fractions φ from 0.02 to 0.6 showed nearly Newtonian behavior up to ~12 vol% (Fig. 1b). Above this point, shear-thinning appeared and intensified with φ, as reflected by the flow index n from power-law fits η ∝ ˙γ⁻ⁿ with ˙γ the shear rate (Fig. 1b and d), which decreased continuously toward the close-packing limit (φ ≈ 0.63). Viscosity at 1 s⁻¹ (Fig. 1c) rose gradually at low φ but diverged sharply near φ ≈ 0.63, consistent with hard-sphere suspensions [52]. The data were well described by the Krieger-Dougherty model [53]:

$$ \eta = \eta_s \left(1 - \frac{\varphi}{\varphi_m}\right)^{-[\eta]\varphi_m} \quad (1) $$

where η_s is the solvent viscosity, φ_m, is the maximum packing fraction and [η] is the intrinsic viscosity. Fitting the dependence of the viscosity η on the particle volume fraction φ (from the data in Fig. 1c) to the Krieger-Dougherty model (Eq. 1) yielded η_s = 22.2 mPa·s, φ_m = 0.605, and [η] = 6.4. The slightly reduced φ_m compared to random close packing (0.63–0.64) and the high [η] relative to the hard-sphere value of 2.5 indicate extended or branched aggregates. This microstructure explains the progressive shear-thinning, as anisotropic entities align under flow. Extrapolation of the model to zero φ yields η_s ≈ 22.2 mPa·s, higher than pure ethylene glycol, likely due to PVP adsorbed on particle surfaces, which increases the viscosity of the continuous phase.

### 3.2. Jetttability of ink droplet

The morphology of deposited sessile droplets is governed by the reproducibility of the piezo ejection process, which determines size uniformity, monodispersity, and impact velocity. These parameters were systematically controlled through by adjusting the nozzle temperature (25–45 °C) and actuation voltage (20–40 V). Droplet volume and velocity increased with applied voltage due to the greater magnitude of fluid displacement. A similar increase in volume and velocity was observed at elevated nozzle temperature, attributed to reduced ink viscosity (Fig. S4).

Ejected droplets initially elongated, forming a trailing filament, then relaxed into nearly spherical shapes during flight over distances ranging from 0.3 mm to 10 mm from the nozzle. Undesirable satellite droplets formation depended on the breakup mechanism of the trailing filament. These morphologies align with predictions from the Derby diagram, which defines conditions for optimal drop formation [54]. All tested conditions fell within the stable jetting regime defined by the Derby diagram (Fig. S5), satisfying the criteria 1 < Z < 10, We > 4, and We¹/²·Re¹/⁴ < 50 to ensure adequate fluidity, prevent satellites [55], and avoid splashing [54]. Here Re = vρd/η is the Reynolds number, We = v²d/γ the Weber number and Z = Oh⁻¹ = √(ρv²d/η) the inverse of the Ohnesorge number, with v denoting the measured droplet velocity and d the nozzle diameter and ρ the density of ink.

This analysis helped refine the optimal printing parameters for the drying study. At 25 °C, droplet ejection was unstable, attributed to the elevated viscosity of the ink and the resulting low Z values falling near the lower jetttability threshold. In contrast, stable and reproducible jetting was consistently achieved between 35 and 40 °C. However, nozzle clogging occurred at temperatures approaching 40 °C, narrowing the operational window to around 35 °C (Fig. S2). Consequently, 35 °C was selected as the optimal nozzle temperature, where adjusting the actuation voltage from 20 to 40 V enabled precise control of droplet volume, ranging from 6.5 ± 0.2 pL to 11.3 ± 0.5 pL, with corresponding

velocities from 8.5 ± 0.9 to 27.0 ± 0.5 m·s⁻¹, establishing this operating window as optimal for producing stable, well-defined droplets.

### 3.3. Evaporation dynamics of picoliter ink drops

Evaporation times (t<sub>ev</sub>) for picoliter sessile droplets (~6.5 pL) were measured by tracking the inward motion of the drying front using optical imaging (Fig. 2a inset). The front, visible as a contrast change in reflected light, defined the boundary between wet and dry regions. t<sub>ev</sub> was defined as the time when the front reached the center of the droplet. Between 20 °C and 40 °C, t<sub>ev</sub> decreased from ~150 s to ~15 s (Fig. 2a). Above 40 °C, the front advanced too rapidly to be resolved with the available optical setup, setting 40 °C as the maximum experimentally resolved temperature. To extend the temperature range, results were compared with data from Lim et al. [42], who measured drying times of picoliter ethylene glycol droplets diameter ~ 40 µm, initial contact angle θ ≈ 55° between 60 °C and 200 °C using high-speed imaging (Fig. 2a). Despite differences in formulation and method, the two datasets were in good agreement, indicating that evaporation is dominated by the properties of ethylene glycol, with minimal influence from the presence of silver nanoparticles.

The evaporation flux of a sessile picoliter droplet (J<sub>ev</sub> (r)) is described by Eq. 2, where J<sub>0</sub>(θ) represents the contact-angle-dependent evaporation flux at the droplet center, r denotes the radial position within the circular droplet on the substrate, R is the droplet base radius and λ is an angle-dependent parameter [5,56].

$$ J_{ev}(r) = J_0(\theta) \left(1 - \left(\frac{r}{R}\right)^2\right)^{-\lambda}, \quad \text{with} \quad \lambda = 0.5 - \frac{\theta}{\pi} \quad (2) $$

Under diffusion-limited conditions, the evaporation rate of a picoliter sessile droplet shaped as a spherical cap is given by E<sub>V</sub> = 2πRD(C<sub>sat</sub> - C<sub>∞</sub>)K(θ), with K(θ) = 0.135θ² + 0.65, where θ the contact angle, and D the vapor diffusion coefficient. C<sub>sat</sub> the saturation vapor concentration at the droplet surface, and C<sub>∞</sub> is the ambient vapor concentration (C<sub>∞</sub> = 0 for EG). The total evaporation time t<sub>ev</sub> is obtained by solving the temporal evolution of the droplet volume, dV<sub>l</sub>(θ)/dt = -E<sub>V</sub> where spherical cap volume V<sub>l</sub>(θ) = πR³/3 f(θ), f(θ) = (1-cos(θ))²(2+cos(θ))/sin³(θ). The estimate of t<sub>ev</sub> depends on the evaporation mode of the sessile droplet, either variable contact angle (VCA, pinned contact line) or constant contact angle (CCA, receding). This distinction is crucial for reconciling the present results, which likely occur in the VCA regime due to particle-induced contact-line pinning, with the ethylene-glycol data of Lim et al. that proceed in a combined VCA/CCA mode [42]. In the VCA regime, drying ends when θ → 0, while in the CCA regime it ends when R → 0. Accordingly, the total evaporation time (t<sub>ev</sub>) is bounded between the predictions of these two models.

$$ t_{ev}^{\text{CCA}} = \frac{\rho R^2}{4D(C_{\text{sat}} - C_{\infty})} \frac{f(\theta)}{K(\theta)} = \frac{\rho R^2}{D(C_{\text{sat}} - C_{\infty})} \times 0.1953 \quad (3) $$

$$ t_{ev}^{\text{VCA}} = \frac{\rho R^2}{6D(C_{\text{sat}} - C_{\infty})} \int_{0}^{\theta_0} \frac{f(\theta)}{K(\theta)} d\theta = \frac{\rho R^2}{D(C_{\text{sat}} - C_{\infty})} \times 0.2645 \quad (4) $$

with K(θ) and f(θ) as defined above. The predicted t<sub>ev</sub> from both CCA (t<sub>ev</sub><sup>CCA</sup>) and VCA (t<sub>ev</sub><sup>VCA</sup>) models is in good agreement with experimental measurements across the 20–40 °C range (150 s to 15 s) for the silver suspension, as well as with the data for pure EG droplets reported by Lim et al. over 60–200 °C (Fig. 2a) [42]. In both datasets, the calculated evaporation times, or equivalently, the corresponding evaporation rates, follow an Arrhenius dependence, with an effective activation energy of 62.4 kJ/mol. Independent measurements on bulk EG in a 50 mm-diameter container across 30–110 °C yielded an activation energy of ~55 kJ/mol, which is within 15 % of the picoliter-scale value, consistent with the experimental uncertainties inherent to small-volume systems.

Extending evaporation measurements from picoliter droplets (~40
<figure><img src="image_6.png" /><figcaption>**Fig. 2.** (a) Drying time $t_{ev}$ of 40 µm silver ink droplets on PEN substrates (▲) as a function of substrate temperature between 20 °C and 40 °C. The evaporation time was determined from optical visualization as the time when the drying front reached the droplet center. Data are compared with literature values for ethylene glycol droplets of similar size generated by piezoelectric printing, where high-speed imaging allowed access to shorter timescales, particularly above 60 °C [42]. Literature data were extrapolated to estimate evaporation times under pinned (VCA) and receding (CCA) contact line conditions (●, ○). Dashed lines denote extrapolated trends between the two data sets, shown as a guide for the eye. (b) Drying rate of silver ink droplets at 35 °C plotted versus droplet radius, obtained from $V_l/t_{ev}$ in this study and compared with gravimetric measurements on microliter drops (5–40 µL) at the same temperature. The dashed line represents a linear extrapolation.</figcaption></figure>

µm base radius) to millimeter-sized drops using gravimetric analysis reveals an almost linear dependence of the evaporation rate on the base radius R over nearly four orders of magnitude (Fig. 2b). This confirms that sessile droplet evaporation is primarily perimeter-controlled, independent of the presence of silver nanoparticles. The slope of this scaling yields $D \cdot C_{sat} = 6.23 \times 10^{-9} \text{g} \cdot \text{mm}^{-1} \cdot \text{s}^{-1}$ at 30 °C, in close agreement with the theoretical estimate based on EG properties (3.18 × $10^{-9} \text{g} \cdot \text{mm}^{-1} \cdot \text{s}^{-1}$).

## 3.4. Morphological evolution of dried deposit with substrate temperature

### 3.4.1. Morphological profiles of dried Picoliter silver ink droplets

Fig. 3 presents 3D profiles of dried 6.5 pL droplets on substrates held at 20–120 °C, alongside corresponding 2D height profiles ($h(r, \infty)$) extracted along the droplet diameter. Under all conditions, a coffee-ring-like morphology is observed, characterized by the accumulation of material near the contact line. However, the extent and sharpness of this peripheral accumulation strongly depend on the drying temperature. At

<figure><img src="image_7.png" /><figcaption>**Fig. 3.** Three-dimensional and corresponding two-dimensional profiles of printed silver ink droplets on PEN substrates maintained at temperatures between 20 and 120 °C during printing and drying. Droplets were ejected using a piezoelectric printer with the nozzle temperature fixed at 35 °C, and subsequently allowed to dry completely under controlled substrate temperatures. Surface topographies were acquired by profilometry (Alicona Infinite Focus, 100× objective). For each substrate temperature, a representative 3D reconstruction is shown together with a 2D cross-section taken along randomly chosen direction, illustrating a typical droplet profile. The final droplet diameter and dried volume are indicated below each profile.</figcaption></figure>
room temperature (20 °C), the typical coffee-ring profile is markedly reduced. The resulting deposit adopts a more sessile geometry, closely resembling a spherical cap, with only a shallow central depression and minimal material accumulation at the periphery. As the substrate temperature increases, the coffee-ring effect progressively reemerges: a pronounced central depletion zone forms, accompanied by enhanced particle deposition at the edges, consistent with the development and intensification of outward capillary flows during drying. This trend continues up to approximately 70 °C. Beyond this point, a reversal is observed, the central region becomes increasingly elevated, while peripheral accumulation diminishes. Above 100 °C, the deposit morphology becomes heterogeneous, exhibiting irregularities both at the center and the edges. By 120 °C, the edge height is further reduced, yet the ring structure does not vanish completely, indicating that the coffee-ring effect, though attenuated, still persists.

### 3.4.2. Geometrical parameters characterizing deposit evolution with temperature

Quantitative analysis of the dried droplet profiles began with the estimation of the solid deposit volume ($V_s$), expressed as $V_s = 2\pi \int h(r, \infty) r dr$. The corresponding initial liquid volume ($V_l$) was then inferred from $V_l = \frac{q_m V_s}{\varphi}$, where the initial particle volume fraction was $\varphi = 4.31\%$ and the random close-packing fraction $\varphi_m = 0.63$. Across all investigated temperatures, the estimated liquid volume was $6.2 \pm 1$ pL, which agrees well with the gravimetrically measured injection volume of 6.5 pL. This consistency confirms the reliability of both the optical profiling method and the volume estimation procedure based on the assumed packing fraction.

Assuming that, after spreading and retraction, the droplet relaxes into a spherical-cap shape, the initial height apex ($h_0$) can be obtained by solving the geometrical relation: $h_0^3 + 3R^2h_0 - \frac{6V_l}{\pi} = 0$, where R is the final base radius. The initial contact angle follows as $\theta = 2\arctan(\frac{h_0}{R})$.

Over the full temperature range, the calculated contact angle remains consistently around $53^\circ \pm 5^\circ$, which matches the macroscopic contact angle measured from microliter-sized droplets of the same ink (Fig. S6). This indicates that droplets reproducibly relax into spherical caps equivalent to their macroscopic counterparts, establishing a well-defined initial geometry for the drying process.

The agreement between the calculated initial contact angle and its macroscopic equilibrium value, together with the consistency between the measured base radius and the spherical cap volume across all tested temperatures, rules out early contact-line pinning that would arrest droplets in an over-spread state. If pinning occurred before complete retraction after spreading, the apparent contact angle would be smaller and the measured base radius would exceed the theoretical values. This is further supported by the fact that the timescales for thermal equilibration and relaxation are orders of magnitude shorter than those of evaporation. An ethylene glycol droplet equilibrates thermally with the substrate within ~0.5 ms, estimated from $\tau_{th} \approx \frac{\pi^2 h_0^2}{\alpha}$ with a thermal diffusivity of $\alpha = 9.5 \times 10^{-8} \text{ m}^2/\text{s}$, and then relaxes to a spherical cap within ~1 ms. [42] In contrast, evaporation times remain ≥100 ms even at 100 °C [42]. Hence, droplet retraction is likely completed long before evaporation takes place.

To evaluate the role of lateral capillary flows, the final dried profiles were compared to the ideal spherical-cap height profile ($h_{cap}(r)$) expected in their absence. The reference cap was defined by the same base radius and solid volume, as $h_{cap}(r) = \frac{q}{\varphi_m} (\sqrt{R_c^2 - r^2} - (R_c - h_0))$ with the radius of curvature $R_c = \frac{R^2+h_0^2}{2h_0}$ and $h_0$ the apex height calculated using the volume and radius.

For each substrate temperature, the experimental height profiles were compared with the theoretical spherical cap (Fig. 4a–d). An alternative representation is given by the normalized difference between

the two profiles, $\frac{\Delta h(r)}{h_0^{\text{max}}} = \frac{h(r, \infty)-h_{\text{cap}}(r)}{h_0^{\text{max}}}$ where $h_0^{\text{max}}$ is the maximum spherical cap height. This normalized form emphasizes the redistribution of material within the footprint (Fig. 4e). At 20 °C, the deposit remains close to the spherical cap, indicating nearly uniform deposition. With increasing temperature, deviations become more pronounced: the central region is depleted while material accumulates near the contact line, reflecting capillary-driven outward transport. At 70 °C, this redistribution is strongest, with significant peripheral enrichment and a marked central depression, illustrating the clear transition from low- to high-temperature drying regimes. The redistribution of silver particles was quantified using two complementary parameters. The first is the radial average of the normalized absolute deviation, defined as $<\frac{\Delta h(r)}{h_0^{\text{max}}}> =$

$$ \frac{\int_0^R \left| \frac{\Delta h(r)}{h_0^{\text{max}}} \right| 2\pi r dr}{\int_0^R 2\pi r dr} $$

This quantity is nearly zero at 20 °C, increases steadily to about 0.6 at 70 °C, and then decreases again to ~0.2 at 120 °C (Fig. 4f).

A similar non-monotonic trend is observed for the parameter $\rho_{\text{rep}} = \frac{(r)-\langle r_0 \rangle}{R-\langle r_0 \rangle}$ where $\langle r \rangle = \frac{2\pi}{V_s} \int h(r, \infty) r dr$ is the mean radial position of deposited material and $\langle r_0 \rangle$ is the reference value for the spherical cap. At 20 °C, $\rho_{\text{rep}} \approx 0.05$, consistent with weak radial flow and minimal coffee-ring effect. This value increases steadily with temperature, peaking at ~0.65 around 70 °C, before decreasing to ~0.3 at 120 °C (Fig. 4f).

Finally, the ratio of the experimental central thickness to that of the spherical cap decreases with temperature, indicating central depletion, but reverses beyond 70 °C. A maximum accumulation is observed near 90 °C, followed by renewed thinning at higher temperatures (Fig. 4h).

### 3.4.3. Quantitative analysis of fluxes and their role in deposit formation

The peripheral accumulation of nanoparticles arises from a capillary-driven radial flow caused by the radially dependent evaporation rate, $J_{ev}$ (Eq. 2). This enhanced evaporation near the contact line leads to a deviation of the droplet shape from the equilibrium spherical cap. This deviation could induce spatial variations in the radius of curvature of the droplet interface ($R_c$), which generates a gradient in capillary pressure, $\nabla(\gamma/R_c)$. This pressure imbalance drives an outward capillary-driven radial flow ($J_m$) toward the edge of the droplet, carrying particles to the periphery. To quantify the strength of this evaporation-induced radial transport, the underlying fluid flux can be inferred from mass conservation applied to the suspended phase expressed as $\partial(h\varphi)/\partial t = -(1/r) \partial[h \ u_r \varphi r]/\partial r$, where $u_r$ the height-averaged radial fluid velocity and h is the drop height profile. The term $\partial(h\varphi)/\partial t$ can be estimated from the difference between the initial and final droplet profiles, assuming a constant evaporation rate throughout drying $\partial(h\varphi)/\partial t \approx (h(r, \infty)\varphi_f - h_{cap}(r)\varphi_0)/t_{ev}$, where $h_{cap}(r)$ and $h(r, \infty)$ are the initial and final height profiles, $\varphi_0$ and $\varphi_f$ the corresponding particle volume fractions, and $t_{ev}$ the total drying time. The radial particle flux, $h \ u_r \varphi$ is calculated as $h u_r \varphi = \frac{1}{t_{ev}} \int_0^r (h(r, \infty)\varphi_f - h_{cap}(r)\varphi_0) r dr$. Using the experimental values of $h(r, \infty)$, $\varphi_f$ and $\varphi_0$, together with the calculated spherical cap height $h_{cap}(r)$, the spatial distribution of the particle flux $h u_r \varphi$ was computed numerically and is shown in Fig. 5 for various substrate temperatures. The results reveal a clear increase of $h u_r \varphi$ with temperature, indicating that faster evaporation strengthens radial flow within the droplet. This trend is further illustrated by plotting the maximum value of $h u_r \varphi$ as a function of temperature. Above 50 °C, the maximum increases monotonically, reflecting the intensification of outward radial flow under accelerated evaporation. In contrast, at lower temperatures the particle flux is strongly reduced, reflecting the attenuation of advective particle transport within the droplet.

The analytical form of the averaged mass flux can be estimated from the radial evaporation profile $J_{ev}$ (Eq. 2) combined with the mass conservation relation ($\frac{\partial h}{\partial t} = \frac{1}{r} \frac{\partial(h u_r r)}{\partial r} - J_{ev}$). In this analysis, the rate of height change $\frac{\partial h}{\partial t}$ is assumed to be time-independent, and residual
<figure><img src="image_8.png" /><figcaption>Fig. 4. Height profiles of dried silver particle deposits formed from sessile drops evaporated at different substrate temperatures, compared with the corresponding spherical caps estimated from the measured drop base radius and solid volume. (a: 20 °C, b: 30 °C, c: 60 °C, d: 70 °C). (e) Deviations between the experimental height profiles and the calculated spherical caps, normalized by the maximum height of the spherical cap (blue: 20 °C, orange: 30 °C, green: 70 °C). (f) Radial deviation parameter $\sum \frac{\Delta h}{h_0}$, defined as the radial integration of the absolute deviation normalized by the radial integration of the spherical cap profile, plotted as a function of temperature. (g) Parameter $\rho_{rep}$, describing the effective radial distribution of silver particles (definition given in the text), plotted as a function of temperature. (h) Central height deviation $\Delta h^{center}$, defined as the difference between the deposit height at the drop center and the corresponding spherical cap height, normalized by the maximum height of the spherical cap, plotted versus temperature. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

capillary stresses at the droplet surface are neglected. Viscous effects that would counterbalance capillary pressure gradients are therefore not considered. Under these assumptions, the transport is primarily driven by evaporation kinetics, which directly sets the magnitude of the outward flux [5,56].

$$h \cdot u_r = \frac{R^2 h_0}{4 t_{ev} r} \left[ (1 - (r/R)^2)^{1-\lambda} - (1 - (r/R)^2)^2 \right] \quad (5)$$

Using the average particle concentration $\varphi$ over the course of drying, the analytical particle flux $h u_r \varphi$, derived from the evaporation model described above, was compared in Fig. 5 with the flux calculated directly from the measured height profiles. Good agreement was obtained in the temperature range of 50–70 °C. At lower temperatures, however, the analytical form significantly overestimates the measured flux. This

discrepancy is highlighted by the ratio between the maximum values of $h u_r \varphi$ predicted analytically and those obtained experimentally. The ratio remains close to unity between 50 and 70 °C but decreases sharply at lower temperatures, approaching zero near 20 °C.

The agreement between the analytical and experimental radial fluxes observed in the 50–70 °C range indicates that particle transport is predominantly advective, coupled to the underlying fluid flow. This finding suggests that the dynamics are governed by the radial profile of the evaporation rate, $J_{ev}$. Within this regime, other mechanisms, including viscous dissipation counterbalancing capillary pressure, particle diffusion (Fickian or osmotically-driven), or backflow from Marangoni stresses, appear to be secondary and likely negligible. By contrast, the deviations observed at lower temperatures can be attributed to additional mechanisms. First, viscous dissipation may balance capillary pressure during drying, effectively suppressing outward radial flow.
<figure><img src="image_9.png" /><figcaption>**Fig. 5.** (a) Calculated height-averaged radial flux velocity $u_r$ as a function of normalized radius $r/R$ for drying temperatures between 20 and 70 °C (bottom to top). The velocity was obtained from the initial and final solid height profiles (see text for details). (b) Average particle flux $h$ $u_r$ $\varphi$ derived from experimental height profiles, plotted versus normalized radius $r/R$. Temperatures increase from 20 to 70 °C (bottom to top). Symbols represent experimental data, while continuous lines correspond to the analytical form of $h$ $u_r$ $\varphi$ [5,56]. (c) Maximum values of $h$ $u_r$ $\varphi$ as a function of temperature, obtained from experimental profiles (solid circles) and from the analytical form (open circles). (d) Ratio between analytical and experimental maxima of $h$ $u_r$ $\varphi$ as a function of temperature.</figcaption></figure>

Second, osmotic pressure gradients arising from local concentration inhomogeneities could oppose particle transport, even if capillary pressure is sufficient to drive fluid flow. Third, thermally induced Marangoni stresses may promote inward motion toward the droplet center, counteracting capillary-driven transport. To evaluate the relevance of viscous dissipation, the flow timescale was estimated by analyzing the capillary-driven velocity generated by the capillary pressure gradient $\nabla P(r)$, with capillary pressure given by $P(r) = \gamma \nabla(1/R_c)$. The local radius of curvature was obtained from the second derivative of

the height profile $h(r, t)$, according to $R_c(r) = \left[ 1 + \left( \frac{dh}{dr} \right)^2 \right]^{3/2} \sqrt{\frac{g^2 h}{d^2 r^2}}$.

Assuming a local height profile during evaporation given by $h(r, t) = h(r, 0) + J_{ev} \delta t$, where $h(r, 0)$ the initial spherical cap and $\delta t$ is a short time increment during which the evaporation-induced distortion has not yet been compensated by flow. Applying the no-slip condition at the substrate gives the radial velocity $u_r(r) = -(h^3/3\eta) dP/dr$. A comparison between this calculated velocity and the analytical expression given in eq. 5 was then used to estimate the response time $\delta t$ required for the induced flow to reach the same magnitude [5,56]. The estimated $\delta t$ ranges from 10 to 1 µs for substrate temperatures between 20 and 70 °C, which is more than six orders of magnitude shorter than the overall drying time (2–150 s). This means that capillary flow may fail to compensate evaporation-induced deformation only during the very initial instants of drying, within timescales shorter than $10^{-6}$ of the total drying time. Beyond this point, capillary flow rapidly restores the droplet shape and is sufficient to counterbalance all deformations at the suspension viscosities corresponding to the investigated temperatures. Even under conditions of elevated viscosity (up to 1000 mPa·s) and particle concentration (up to 40 vol%), $\delta t$ remains in the range of 25–250 µs, still several orders of magnitude below the drying time. Thus, in most cases, the droplet profile $h(r, t) = h(r, 0) + J_{ev} \delta t$ can effectively be treated as a spherical cap. In other words, even for very small residual deformations, relaxation occurs within only $10^{-6}$ of the drying process, yielding a negligible capillary pressure gradient. Taken together, these results confirm that under all conditions examined, the droplet can be

considered to maintain a spherical-cap geometry throughout drying, and the overall flux velocity can be taken as given by eq. 5.

### 3.5. Marangoni effect and thermal gradients in sessile drops

Marangoni flows are expected to influence particle redistribution during drying by counteracting capillary advection, thereby impacting the formation of coffee-ring patterns. To evaluate their contribution under our experimental conditions, steady-state heat conduction simulations were performed by solving the Laplace equation, $\nabla \cdot (k \nabla T) = 0$ following the strategy of Hu et Larson [57]. The simulations considered the thermal conductivities of ethylene glycol $k_{EG} = 0.252 \text{ Wm}^{-1}\text{K}^{-1}$ for the droplet and $k_{PEN} = 0.24 \text{ Wm}^{-1}\text{K}^{-1}$ for the substrate, together with convective heat exchange at all free surfaces, characterized by a heat-transfer coefficient $h_c = 10 \text{ Wm}^{-2}\text{K}^{-1}$. The droplet geometry was defined by a base radius of 20 µm and a contact angle of 53°, resting on a PEN film of 150 µm thickness and 400 µm radius. The convective heat-transfer terms can be neglected in the energy equation because the contribution is small compared with conduction, as indicated by the ratio $\rho c_p u_r R/k_{EG}$ (with $c_p$ is the heat capacity of ink and $\rho$ its density), which remains below $10^{-3}$ even under the most favorable condition. Thermal transients terms are also negligible because droplet-height changes occur on second timescales whereas thermal equilibration occurs on millisecond timescales [57]. The boundary conditions were chosen to reproduce the drying environment: the bottom of the substrate was fixed at temperatures between 20 and 70 °C, while latent heat removal at the droplet surface was coupled with convective exchange according to $-k_{on} \frac{\partial T}{\partial n} = L_{EG} J_{ev}(r) + h_c(T - T_{\infty})$ with the latent heat coefficient taken as that of EG $L_{EG} = 985 \text{ kJkg}^{-1}$. Additional conditions included convection on the exposed substrate, adiabatic constraints on the lateral boundaries, and continuity of temperature and flux at the liquid-solid interface. The simulation employs an adapted finite-element mesh with targeted refinement near the droplet-substrate interface and the three-phase contact line (Fig. S9a). The model is implemented in FENiCS to solve the steady, axisymmetric energy equation with piecewise thermal conductivity, convective heat
exchange, and a Deegan-type evaporative heat-flux boundary at the free surface (Script S10). The resulting fields yield the temperature distribution along the droplet-air interface, providing the basis for estimating Marangoni numbers and the magnitude of the associated surface-driven flows (Figs. 6a–b, S7, and S9).

The surface temperature profile ($T_{surf}$) was fitted using $T_{surf} = \Delta T_0 (a(r_R)^b + c(r_R)^d + e)$, where $\Delta T_0$ is the temperature difference between the edge and the apex, and a,b,c,d are fitting parameters. These values provide the basis for estimating the thermal Marangoni number (Ma) and the corresponding velocity field, following the approach of Hu and Larson [57]. The Marangoni number expressed as $Ma = \frac{d\gamma/dT}{\mu R} \Delta T_0 t_{ev}$ increases from 587 at 20 °C to 1756 at 70 °C (Fig. 6c), values sufficiently large to induce flow reversal. In all cases, Ma remains positive since $\Delta T_0 > 0$ and $d\gamma/dT < 0$, indicating a Marangoni-driven backflow directed from the edge toward the droplet center along the surface. The contribution of Marangoni stresses to the surface flow was assessed by modifying the formulation of Hu et al. to include the fitted surface temperature profile $T_{surf}$. Eq. (6) provides the total radial velocity at the droplet surface as the sum of the capillary-driven and Marangoni-induced components. To quantify the relative importance of these mechanisms, the ratio of the total velocity to the purely capillary contribution was evaluated as a function of radial position for substrate temperatures ranging from 20 °C to 70 °C (Fig. 6d).

$$u_r(z=h) = -\frac{R}{t_{ev}} \left[ \frac{3}{8\hat{r}} \left( (1-\hat{r}^2) - (1-\hat{r}^2)^{-\lambda(\theta)} \right) + \frac{1}{2}\frac{\hat{r}^2}{R^2}\hat{h}(\hat{J}\lambda(\theta)(1-\hat{r}^2)^{-\lambda(\theta)-1} + 1) + \frac{Ma\hat{h}_0}{4R}\hat{h}(abr^{b-1} + d\,c\,\hat{r}^{d-1}) \right] \quad (6)$$

with $\hat{r} = r/R$, $\hat{h} = h/h_0$, $\hat{J} = J_0 t_{ev}/h_0$.

Across all investigated conditions, the calculated Marangoni term produces a reverse flow directed toward the droplet center, as indicated by ratios below unity. The strength of this contribution increases

systematically with temperature: it is weakest at 20 °C and becomes more pronounced at elevated temperatures. Nevertheless, the capillary-driven flux remains dominant, even at 70 °C, where the surface velocity near the contact line stays positive and directed outward, albeit with a reduced magnitude.

This trend is inconsistent with the hypothesis that Marangoni stresses suppress the coffee-ring effect at low substrate temperatures. If such a mechanism were operative, one would expect the reverse flow to intensify as the temperature decreases, thereby counteracting the outward flux most effectively near 20 °C. Instead, the calculations demonstrate the opposite behavior: Marangoni backflow grows stronger with increasing temperature. Taken together, these results confirm that Marangoni flow does not provide the driving mechanism for the suppression of coffee-ring deposits under the low-temperature conditions examined here. Moreover, the decrease in the contact angle during drying reduces the reverse Marangoni flow, consistent with Hu et al. [57], thereby reinforcing the dominance of convective transport. In addition, solutal Marangoni effects could potentially influence the flow dynamics, particularly through PVP-induced concentration gradients that may create surface tension variations. While the current study focuses on thermal contributions, the possible role of concentration-dependent Marangoni flows warrants further investigation, for example through evaluation of the solutal Marangoni number in future work [57,58].

### 3.6. Role of osmotic pressure

The outward capillary-driven flux that transports particles toward the droplet edge can be counterbalanced by a backward particle diffusion. This diffusive contribution may originate from simple Fickian diffusion, described by the Stokes–Einstein relation, or from an additional contribution induced by osmotic pressure. The conservation of particle mass can be expressed as:

$$\frac{\partial(\varphi h)}{\partial t} = -\frac{1}{r} \frac{\partial}{\partial r} (r \varphi u_r h) + \frac{1}{r} \frac{\partial}{\partial r} (r Dh \frac{\partial \varphi}{\partial r}) \quad (7)$$

where $\varphi$ is the particle volume fraction, $h(r,t)$ the local droplet height,

<figure><img src="image_10.png" /><figcaption>Fig. 6. Temperature distribution along the droplet-air interface as a function of radial position for substrate temperatures of (a) 20 °C, and (b) 70 °C. The temperature profiles were obtained from FEniCS simulations using the thermal properties of the fluid and substrate, together with the radial evaporation flux of the droplet. Continuous lines represent fits of the calculated temperature to Eq. 6. (c) Marangoni number (Ma) calculated from the estimated $\Delta T_0$ shown in Fig. S8 (d) Ratio of the total radial flow velocity (capillary + Marangoni contributions) to the purely capillary-driven flow velocity, both evaluated at the droplet surface ($z = h$), as a function of radial position for substrate temperatures of 20 °C, 40 °C and 70 °C.</figcaption></figure>
and $u_r$ the height-averaged radial velocity. The diffusion coefficient $D$ reduces to Stokes–Einstein form in the dilute limit $D_0 = \frac{k_B T}{6\pi\eta_r R_h}$ (with $R_h$ the particle radius). At higher concentrations, however, diffusion is modified by osmotic effects. Using the Carnahan–Starling prediction for the osmotic pressure $\Pi(\varphi) \propto f(\varphi)$, where $\Pi(\varphi)$ is osmotic pressure and $f(\varphi)$ the Carnahan–Starling function, the diffusion coefficient becomes

$$D(\varphi) = \frac{D_0}{\eta_r(\varphi)} \frac{\partial f(\varphi)}{\partial \varphi}, \quad \text{with} \quad f(\varphi) = \frac{\varphi + \varphi^2 + \varphi^3 - \varphi^4}{(1 - \varphi)^3} \qquad (8)$$

where $\eta_r(\varphi)$ is the relative viscosity ($\eta(\varphi)/\eta_s$) and $\eta(\varphi)$ given by eq. 1.

The competition between particle advection and diffusion is quantified by the Péclet number, $Pe = \frac{\overline{u_r} \varphi L}{D}$ where L is a characteristic length scale. Here, $L \approx 10$ µm, corresponding to the width of the particle accumulation peak near the contact line (or roughly twice this value), provides a relevant estimate. Using a simplified expression of radial velocity $u_r$ of eq. 5, $u_r = \frac{\zeta}{t_{ev}h}$, with geometric and hydrodynamic prefactors grouped into $\zeta$, the Péclet number becomes $Pe = \frac{\zeta \varphi L}{t_{ev}D h}$ where both $t_{ev}$ and $D$ depend on temperature through their connection to viscosity while $\zeta$ and $h$ retain their dependence on r. Fig. 7 shows the variation of $Pe$ with substrate temperature, evaluated at 1 µm and 10 µm from the droplet edge, where particle accumulation is most pronounced. Two approaches were considered: (i) using the initial particle volume fraction at the droplet center, and (ii) correcting the diffusion coefficient for an effective concentration of $\varphi = 0.3$. In the latter case, only the diffusivity is adjusted for concentration effects, while the particle fraction in the advecting flux remains that of the dilute initial state, since the particles transported by the capillary flow originate from the droplet interior.

In both cases, $Pe$ decreases systematically as the substrate temperature is lowered, dropping from values above unity at 70 °C to below 0.05 at 20 °C. These results indicate that at higher temperatures ($T > 40$ °C), capillary advection dominates, producing strong accumulation at the drying front. At lower temperatures, however, diffusion prevails, leading to more uniform particle redistribution across the droplet and thereby suppressing the coffee-ring effect.

These results are consistent with the observed suppression of the coffee-ring effect as the substrate temperature approaches room temperature, and with the enhancement of peripheral deposition as the temperature increases toward 70 °C. At even higher temperatures ($T > 90$ °C), peripheral deposition persists but is accompanied by a growing accumulation in the central region. This behavior is likely associated with the strengthening of Marangoni-induced backflow at higher temperatures. As shown in Fig. 6d, the magnitude of this backflow increases with temperature, and the decreasing ratio of capillary to Marangoni flux suggests that this trend would continue at even higher

<figure><img src="image_11.png" /><figcaption>Fig. 7. Péclet number as a function of substrate temperature, calculated using the Stokes–Einstein diffusion coefficient at 10 µm from the edge (blue triangles) and at 1 µm from the edge (black squares), and compared to values obtained with an effective concentration of $\varphi = 0.3$ (open circles). A dotted line is shown as a visual guide for the limit at $Pe = 1$. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

temperatures. This enhanced backflow promotes particle redistribution toward the droplet center, consistent with the marked increase in central thickness observed for temperatures above 100 °C in Fig. 4h.

Taken together, these results demonstrate that capillary advection is present under all drying conditions of silver nanoparticle droplets, but its relative influence is strongly modulated by temperature (Fig. 8). At low temperatures (20–40 °C), slow evaporation allows diffusive and osmotic redistribution to suppress coffee-ring formation. In the intermediate range (40–70 °C), capillary advection dominates, producing pronounced ring-shaped deposits. At elevated temperatures (70–120 °C), Marangoni-driven backflow intensifies and counterbalances capillary flow, driving central accumulation and generating heterogeneous deposition patterns.

Furthermore, while other mechanisms are known to suppress the coffee-ring effect [3], such as those involving anisotropic particles [33] or large graphene oxide flakes [50], these are considered secondary in the present work. Due to the nanometric size and isotropic, Brownian nature of our particles, such processes are not expected to be the primary cause of the suppression observed in this study. This work establishes a fundamental framework for controlling deposition dynamics in printed electronics, demonstrated using ethylene-glycol-based silver nano-inks on polymer substrates. Although the present study focuses on the EG system printed on PEN substrates, extending the conclusions to other formulations and substrate chemistries, such as aqueous inks on more hydrophilic surfaces, will require dedicated validation in future work.

## 4. Conclusions

This study demonstrates that substrate temperature is a critical and highly effective parameter for controlling the deposit morphology of picoliter silver nanoparticle ink droplets, directly addressing a fundamental challenge in printed electronics. Through a systematic experimental and analytical approach, we identify a clear transition in deposition regimes: from nearly uniform coverage at low temperatures (20–40 °C) to a pronounced coffee-ring formation at intermediate temperatures (50–70 °C), and finally to a more complex morphology with central accumulation at high temperatures (>90 °C).

Though flow processes are deduced from final dried profiles rather than directly observed, this approach successfully provides valuable insights into deposition mechanisms. Our analysis reveals that suppression of the coffee-ring effect at low temperatures is not primarily governed by Marangoni-induced backflow, as is often suggested in the literature. Computational simulations of thermal gradients and surface tension effects confirm that while Marangoni stresses are present and increase with temperature, they are insufficient to reverse the outward capillary flow under these conditions. Furthermore, we rule out viscous dissipation as the main suppression mechanism; although ink viscosity increases at lower temperatures, our calculations show that the timescale for capillary flow to compensate for evaporation-induced deformation (µs) remains orders of magnitude faster than the drying timescale (s), even at the highest viscosities encountered.

Instead, the key mechanism identified is a shift in the competitive balance between capillary advection and particle diffusion. Quantitative analysis of the Péclet number shows that at elevated temperatures, rapid evaporation drives strong outward capillary flow, leading to particle accumulation at the contact line ($Pe > 1$). Conversely, at low substrate temperatures, the significantly longer drying timescale drastically reduces the $Pe$ ($Pe < 1$), allowing backward transport mechanisms, namely Fickian and, more importantly, osmotically driven diffusion, to efficiently homogenize the particle distribution within the droplet before immobilization.

This work provides a robust mechanistic framework for suppressing detrimental capillary-driven assembly without relying on complex ink formulations or external fields. By simply operating within a low, commercially viable substrate temperature window, we achieve uniform, compact deposits essential for the morphological and functional
<figure><img src="image_12.png" /><figcaption>Fig. 8. Temperature-dependent flow regimes and particle accumulation patterns in drying sessile picoliter silver ink droplets.</figcaption></figure>

uniformity of printed conductive features. This finding offers a practical and directly applicable strategy for enhancing the resolution and reliability of inkjet-printed flexible electronics.

### CRediT authorship contribution statement

**Yahya Rharbi:** Writing – review & editing, Writing – original draft, Validation, Supervision, Software, Project administration, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Vincent Faure:** Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation. **Anne Blayo:** Supervision, Project administration, Funding acquisition, Conceptualization. **Albert Magnin:** Supervision, Project administration, Methodology, Conceptualization. **Aurore Denneulin:** Writing – original draft, Validation, Supervision, Resources, Project administration, Methodology, Funding acquisition, Formal analysis, Data curation, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper

### Acknowledgments

LGP2 is part of the LabEx Tec 21 (Investissements d'Avenir – grant agreement n°ANR-11-LABX-0030) and of PolyNat Carnot Institutes Investissements d'Avenir – grant agreement n°ANR-16-CARN-0025-01). This research was made possible thanks to the facilities of the TecLiCell platform funded by the Région Rhône-Alpes (ERDF: European regional development fund). The Laboratoire Rhéologie et Procédés is part of the Labex Tec 21 (Investissement d'Avenir – grant agreement n° ANR-11-LABX-0033) and of the Polynat Carnot Institut (Investissement d'Avenir – grant agreement n°ANR-11-CARN-030-01). The authors gratefully acknowledge Mohamed Karrouch and Didier Bléses for their invaluable assistance in conducting the experiments, and Frédéric Hugenell for his insightful support in building the experimental setup.

## Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.jcis.2025.139703.

### Data availability

Data will be made available on request.

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