Grain boundary segregation and solute drag in multicomponent alloys

Domain: biology
Published: 2026-04-04
Editorial outcome: accept

Abstract
Grain boundary (GB) solute segregation has been proposed as a strategy to tailor the processing pathways and properties of a wide range of polycrystalline materials. Of particular interest is GB solute drag, which results when segregated solutes exert a resistive force hindering GB migration. However, we do not have a mechanistic understanding of the migration kinetics of doped GBs in multicomponent alloys, as most treatments only consider binary systems. In this work, we model GB solute drag in multicomponent alloys through theoretical and computational studies. We show that solute-solute interactions of various alloying element types result in varied GB segregation behaviors, such as synergistic co-segregation and induced desegregation, which greatly influence the characteristics of GB solute drag. Simulation results reveal that in multicomponent alloys GB desegregation can be as effective in inducing boundary solute drag. More broadly, predictions from our modeling framework motivate future research into the migration kinetics of doped interfaces, with the potential to guide the design of advanced alloys with tailored chemistries and properties.

Introduction
Most structural and functional materials are polycrystals, they are composed of crystalline grains that are joined at internal interfaces, termed grain boundaries (GBs). Many of the engineering properties of polycrystalline metals, such as mechanical strength1, corrosion2 and wear3 resistance, and embrittlement4 can be controlled by tailoring the grain size and GB types present in these materials. In addition to microstructure control, alloying offers another strategy by which the properties of polycrystalline metals can be manipulated. Indeed, recent interest in metal physics has been geared towards multicomponent alloys with far-from-dilute concentrations, so-called high entropy alloys5.

In metallic alloys, atoms within the GB region are at higher energy states than those in the perfect crystalline lattice, and, as a result, GBs provide preferential segregation sites for alloying elements. In this context, GB segregation is not viewed as an undesired effect, but rather a strategy that allows for the direct manipulation of materials properties and functionalities6. GB solute segregation has been the subject of active research with studies focused on the dependence of GB energy on boundary concentration7, segregation anisotropy due to the boundary’s geometry8–11, and segregation-induced GB transformations12–16. Recent analytical thermodynamic treatments investigated the impact of multi-element segregation on GB energy17. Experimental studies employing atom probe tomography18–20 and microscopy techniques11,21–24 have been used to reveal and quantify GB chemistry in a wide range of alloys.

The chemistry of GBs dictates boundary motion and, in turn, the evolution of microstructure. For example, solute segregation to GBs has been shown to suppress grain coarsening and enhance the stability of polycrystalline alloys11,24–26; however, in other systems it has been reported to promote abnormal grain growth27,28. In solid-state activated sintering of particle compacts, GB segregation has been used to modulate densification and coarsening rates29,30. The motion of doped GBs is governed by the kinetic phenomenon of solute drag, whereby the GB has to drag the segregated alloying elements during its motion, generating a drag force P that depends on the GB velocity V31,32. Figure 1(a) shows a characteristic solute drag–velocity curve, where the shaded region denotes a regime in which increasing boundary velocity requires overcoming this solute drag. The point (V*, P*) marks the maximum drag pressure P* and its corresponding velocity V*. Beyond this peak, the boundary enters a breakaway regime, escaping its local solute atmosphere and migrating at its intrinsic velocity.

Recent experimental studies on multicomponent alloys revealed sluggish GB migration and grain coarsening rates, which were attributed to GB solute drag33–36. Such observations, however, present several conundrums. First, these studies rely on classical solute drag models to explain the observed sluggish GB migration rates34. Most solute drag treatments consider binary alloys31,32,37. For example, Cahn’s treatment31 assumes ideal and dilute alloys in which GB-solute interactions are characterized by a constant segregation energy. A key result of Cahn’s model is that solute drag scales linearly with the nominal bulk concentration31. Existing studies on GB solute drag in multi-component alloys38–40 extended Cahn’s treatment to ideal multicomponent alloys, and, as a result, these investigations do not account for solute-solute interactions of various alloying element types. For example, Mendelev and Srolovitz38 and Bréchet and Purdy39 examined solute drag in ideal ternary alloys where they assumed constant segregation energy for each solute type. In such models38–40, not accounting for solute-solute interactions in the bulk and GB results in decoupled governing equations, in which the total GB solute drag is the sum of solute drag of the constituent binary alloys. Additionally, these studies38–40, similar to Cahn’s model, predict a solute drag that increases linearly with bulk concentration. There are no GB solute drag models that consider multicomponent alloying effects with solute-solute interactions. Second, it is not clear whether it is the segregation of one type of solute to GBs or the segregation of multiple elemental species that is responsible for the observed sluggish GB migration rates. Third, these experimental studies do not consider the possibility of more complex GB-solute interactions that could give rise to solute drag. Indeed, the interaction of multiple alloying element types with GBs introduces varied segregation behaviors, including competitive segregation, synergistic co-segregation, and induced desegregation, whose impact on GB migration remains unexplored. For example, a study by Li et al.18 on the multicomponent FeMnNiCoCr alloy revealed co-segregation of Ni and Mn and desegregation of Fe and Co. A critical need, therefore, exists for a fundamental and quantitative understanding of GB segregation and boundary solute drag in multicomponent alloys and this is precisely the goal of this study.

In this work, we present a theoretical and computational mesoscale model to elucidate GB solute drag in multicomponent alloys. While atomistic simulations have provided critical insights into GB phenomena, such as boundary energy41 and mobility42, atomic structure and site-specific segregation behavior43,44, and solute transport coefficients45,46, the direct atomistic simulations of concurrent GB segregation and dynamic solute drag remain computationally prohibitive. The pronounced disparity between the time and length scales of atomistic simulations and those governing GB dynamics and chemical re-distribution motivates and highlights the importance of the mesoscale modeling framework developed in this work. Our proposed framework accounts for solute-solute interactions of various alloying element types in the bulk and within the GB, captures mass transport processes of various alloying elements, and provides the means to quantify the impact of complex GB concentration profiles on solute drag. Our simulation results reveal a multitude of solute drag behaviors that are sensitively dependent on the interactions of various solute types and operative mass transport processes. Furthermore, we show that cases with GB solute desegregation can be as effective in inducing GB solute drag effects.

Results
Theory

In this work, we consider ternary alloys because they represent a model system with multicomponent effects while offering computational tractability. We start by defining a ternary alloy bicrystal with a planar GB that consists of a host element, labeled A, and two solutes, labeled B and C. The elemental species are described by the concentration fields ci(x,t), i=A,B,C, where x is the direction measured from the center of the planar GB and t is time. We employ regular solution thermodynamics for the description of the free energy of the bulk, fb, and that of the GB, fgb, which is given by47

f = \sum G\alpha c + k T \sum c ln(c) + \sum \Omega\alpha cc , \alpha = gb, b, (1)
\alpha i i B i i ij i j
i=A,B,C i=A,B,C i,j=A,B,C
j>i

where G\alpha describes the free energy of pure i in bulk, \alpha = b, or GB, \alpha = gb, kB is Boltzmann constant, and T is temperature. The first term on the right-hand side of Eq. 1 corresponds to the free energy of the alloy mixture, while the second term describes the configurational entropy. The last term on the right-hand side of Eq. 1 accounts for the heat of mixing of the multicomponent alloy. Here, \Omega\alpha defines the regular solution interaction parameter between the i and j elemental species in the bulk, \alpha = b, or GB, \alpha = gb. In regular solution thermodynamics, \Omega\alpha \propto [\epsilon - 0.5(\epsilon + \epsilon )] where \epsilon is the i-j bond energy47. Positive \Omega\alpha values describe repulsive i-j interactions and negative values control the degree of mixing of the alloy.

In binary alloys with a miscibility gap, i.e., large values for the regular solution parameter, a recent study revealed that GB phase separation is accompanied by drastic changes in boundary mobility48. Figure 1(b) shows a schematic representation of the bulk fb and GB fgb free energies in the composition space where cA = 1 - cB - cC is invoked. Next, we introduce an indicator function \Upsilon(x) to interpolate the energy between the bulk and GB regions, such that the free energy at any point is given by ftot = (1 - \Upsilon)fb + \Upsilon fgb. In the bulk grain \Upsilon = 0 and in the GB region \Upsilon peaks to a value of one. In Cahn’s original treatment of solute drag in dilute binary alloys31, a triangular function was used to locate the GB region. In this work, we let \Upsilon be a Gaussian function given by

\Upsilon(x) = exp \left( - \frac{4x^2}{\delta^2} \right), (2)

where the GB width \approx 2\delta, i.e., \Upsilon(\delta) is \approx two orders of magnitude smaller than that at the center of the GB. Figure 1(c) shows a plot of \Upsilon(x), where \Upsilon peaks at the center of the GB and decays sharply, yet smoothly to zero in the bulk.

Gradients in the chemical potentials give rise to mass fluxes that can be described using Onsager’s formalism49. Using the Gibbs-Duhem equation [see Methods Section], only the flux equations for solutes B and C are needed to describe the dynamics of ternary alloys, which are written as Ji = -L˜iB \nabla\muB - L˜iC \nabla\muC, i=B,C. Here, L˜ij = Lij - cj \sum k=A,B,C Lik, where Lij are the kinetic transport coefficients with Lij = Lji according to Onsager’s reciprocity theorem49. Further, \mu i is the chemical potential of the ith solute type measured with respect to that of the host A, i.e., diffusion potential, and is given by

\mu i = Gb i - Gb A + kB T ln \left( \frac{c i}{1 - c B - c C} \right) + \Omega b Ai (1 - 2c i) + (\Omega b BC - \Omega b AB - \Omega b AC) c j + Ei(x), i,j=B,C, j \neq i. (3)

Here, E(x) is the position-dependent interaction energy of the ith solute type with the GB and is given by

Ei(x) = [ -\Delta G Ai + (\Omega gb Ai - \Omega b Ai)(1 - 2c) + (\Omega gb BC - \Omega b BC) c j ] \Upsilon(x), i,j=B,C, j \neq i, (4)

where \Omega\alpha BC = \Omega\alpha BC - \Omega\alpha AB - \Omega\alpha AC, \alpha = b, gb, and \Delta G Ai = (Ggb i - Gb i) - (Ggb A - Gb A), i=B,C. Graphically, Ggb i - Gb i defines the difference between the GB free energy and that of the bulk for pure i, which is shown schematically in Fig. 1(b) for pure elements A, B, and C. A close examination of Eq. 4 reveals that \Delta G Ai describes the GB segregation energy of solute i in ideal alloys, i.e., ones with \Omega\alpha ij = 0, \alpha = b, gb, and i,j=A,B,C. However, in regular solutions the segregation energy becomes concentration-dependent, and that the segregation energy of one solute type i is coupled to the concentration of the other solute type j through the heat of mixing model parameters. Xing et al.50 investigated GB segregation in ternary alloys based on the properties of the constituent binary systems and applied their treatment to the Pt-Au-Pd alloy. While we do not consider GB-vacancy interactions during boundary migration51,52, such interactions can be incorporated in our modeling framework by treating GBs as sources/sinks of vacancies with a prescribed interaction energy, similar to Eq. 4, described using a vacancy concentration field. Next, equations for mass fluxes can be expressed in terms of concentration gradients as [see Methods Section]

Ji = - \sum j=B,C Dij \nabla cj and Dij = kB T \left( \frac{L ij}{c i} - \frac{L iA}{c A} \right), i,j=B,C, (5)

where Dij are the relevant diffusion coefficients. Equation 5 results from expressing the diffusion coefficients as Dij = \sum k Lik \partial\mu k / \partial cj, where Lik are the Onsager transport coefficients and \partial\mu k / \partial cj are termed the thermodynamic factors coupling chemical potentials to concentrations49. It is important to note that the Onsager transport coefficients Lik are in general concentration-dependent46,53. Obtaining the thermodynamic factors requires a detailed knowledge of the energetics of the alloy of interest, while the Onsager coefficients can be obtained using techniques such as atomistic simulations in conjunction with the generalized Einstein relation53. As for the off-diagonal diffusion coefficients, their values and algebraic sign are influenced by those of the Onsager coefficients and thermodynamic factors. Herein, we let the diffusion coefficients be constants as this work does not target a specific alloy, but rather examines solute drag trends in alloy design space. Such an assumption has been invoked in many studies involving transport in multicomponent systems in order to deconvolute effects due to thermodynamic driving forces from ones related to transport coefficients54–57. For example, a study on radiation-induced segregation employing a reaction-diffusion modeling framework employed constant transport coefficients that are independent of the thermodynamic factors56. Another study examined dendritic solidification in multicomponent alloys where the diffusion coefficients of various species were assumed to be constant57. Furthermore, we assume that diffusion in the bulk is frozen compared to that in GBs, which act as fast diffusion pathways58, and, as a result, we let Dij take the form

Dij = Dgb ij \Upsilon(x), (6)

where Dgb ij define the GB diffusion coefficients. A future extension of our model to two (2D) or three (3D) dimensional geometries would allow for the incorporation of both diffusion across and along the GB region.

The steady state nonlinear solute transport equations expressed in a frame moving with the migrating GB at a velocity V are given by [see Methods Section]

V \frac{\partial ci}{\partial x} = \nabla \cdot J, i=B,C. (7)

The above transport equations are made non-dimensional using the GB half width \delta and thermal energy kB T as reference length and energy scales, respectively, and we employ a reference GB velocity Vref. This leads to the following non-dimensional quantities: x¯ = x/\delta, E¯ i = Ei/kB T, \Omega¯ = \Omega/kB T, \Delta G¯ Ai = \Delta G Ai/kB T, \mu¯ = \mu/kB T, D¯ ij = Dij/(Vref \delta), and non-dimensional GB velocity V¯ = V/Vref. Numerical simulation of these governing equations provides the means to quantitatively describe the concentration profiles across the GB region, which are then used to obtain GB solute drag P expressed in non-dimensional P¯ form as31

P¯ = \frac{Pv a}{kB T} = - \sum i=B,C \int_{-\infty}^{\infty} (ci - c\infty i) \frac{dE¯ i}{dx¯} dx¯, (8)

where va is the atomic volume. In this work, all computational studies are reported using the non-dimensional parameters discussed above. These model parameters can be mapped onto real metallic alloy values as follows: A representative GB width in a typical metallic alloy is 2\delta = 1 nm and thermal energy kB T \approx 0.08 eV assuming an annealing temperature in the range 800-1000 K, typical in metallic alloys59. At such temperatures, experimentally measured GB velocities are in the range Vref \approx 0.1-10 \mu m s-1 60 and representative values for GB solute diffusion \approx 10-15 m2 s-1 45,61.

The foregoing exposition underscores several key advantages of the proposed model. First, the model requires as input physical parameters that can be directly obtained from atomistic simulations. For example, atomistic methods have been employed to calculate GB diffusivity45, bulk Onsager transport coefficients46, GB chemical free energies via thermodynamic integration62, and bulk heat of mixing derived from alloy phase diagrams63–65. Second, our model enables the direct simulation of coupled diffusive processes governing GB solute drag, thereby establishing a quantitative link between atomic-scale mechanisms and experimentally observed GB phenomena. Here, we note that the GB geometry is described by five macroscopic degrees of freedom66, three for misorientation and two for plane normal, and that each GB geometry has its unique thermodynamic and kinetic properties. We also note that establishing GB structure-property linkages relating the GB’s thermodynamic and transport properties to its geometric degrees of freedom is one of the most active areas of research in GB physics. While this work does not consider anisotropic boundary segregation, it is straightforward to include anisotropy effects in our modeling treatment by allowing the model parameters describing the GB segregation energy and diffusion coefficients to be functions of the boundary geometry. In addition to the dependence of GB segregation on boundary geometry9,10,67, several studies revealed site-specific GB segregation, in which the segregation energy depends on the local atomic environment within the GB68–70. This results in distributions for the GB segregation energy in terms of the atomic environment, even for the same GB geometry43,71,72. Accounting for site-specific segregation for a given GB can be achieved in our treatment by allowing our model parameters to be distributions rather than single-valued quantities.

Model Assumptions

Our proposed modeling treatment considers multi-element segregation and migration of planar GBs using a one-dimensional bicrystal geometry. As a result, the relevant transport coefficients are solute diffusion across the GB. A future extension of our model to 2D/3D geometries would enable the incorporation of a tensorial description of GB diffusion in which both diffusion along and across the GB are incorporated. Indeed, atomistic simulations of Cu-Ag binary alloys, for example, showed that accelerated lateral GB diffusion leads to non uniform redistribution of solute atoms, which in turn influences GB solute drag73. Such a 2D/3D extension of our model will capture the dynamics of GB networks, thereby allowing us to investigate the interplay between GB segregation and phenomena such as abnormal grain growth. Our model treats GBs as regions with thermodynamic properties that are distinct from the bulk, resulting in a segregation energy that is single-valued. It is interesting to extend our treatment to account for site-specific GB segregation, in which the segregation energy is given by a distribution due to the local atomic environment of the GB. This can be incorporated into our treatment by allowing the model parameters that describe the GB segregation energy to be distributions rather than single-valued quantities. Our work considers solid solution alloys in order to deconvolute solute drag effects due to multi-element GB segregation from GB pinning due to second-phase precipitates. It would be interesting to investigate in future work the interplay between GB segregation and phase separation and their net impact on GB migration kinetics. Finally, another extension of our modeling framework is to explicitly consider vacancies and their interactions with GBs, as several studies revealed vacancy generation and emission at migrating boundaries51,52. This can be achieved by treating GBs as sources/sinks of such points defects with prescribed interaction energies.

Multicomponent Segregation Isotherm

We begin our exploration of the model by investigating the equilibrium GB concentrations of stationary boundaries, i.e., V=0. In this case, the GB solute concentration is obtained using the parallel tangent construction74, in which parallel tangents to the bulk and GB free energy surfaces are drawn at the equilibrium bulk c\infty i and GB cgb i concentrations of the ith solute, see Fig. 1(b). This construction results in an analytical solution for the concentration ci(x) of solute i at a distance x from the center of the GB, which is given by

ci(x) = c\infty i + (cgb i - c\infty i) exp [ -E¯ i(x) + 2\Omega¯b Ai (ci(x) - c\infty i) - \Omega¯b(cj(x) - c\infty j) ], i,j=B,C, j \neq i, (9)

Equation 9 defines a multicomponent segregation isotherm describing the GB concentrations as a function of far-field bulk concentrations c\infty i = c(x \to \infty), i = B,C and the alloy model parameters. The segregation isotherm given by Eq. 9 can be mapped onto the treatment by Guttmann75 who investigated equilibrium GB segregation in ternary alloys. However, we note here that Guttmann’s model75 only considered solute-solute interactions in the GB layer as it assumed the bulk phase to be an ideal, dilute alloy. Therefore, our segregation isotherm given by Eq. 9 extends Guttmann’s treatment by introducing both bulk and GB solute-solute interactions. It is interesting to note from Eqs. 4 and 9 that the expressions for the GB solute concentrations are coupled through the solute interaction parameters, where the GB concentration of one solute type depends on the boundary concentration of the other solute types. To demonstrate this multicomponent coupling effect, we consider two ternary alloy classes. The first is an alloy, I where both solutes B and C segregate to the GB. For the second alloy, solute B is chosen to segregate to the GB, while solute C desegregates. Capturing such contrasting segregation behaviors in our model can be achieved in many ways. One possible scenario is through the bulk solute interaction parameter \Omega¯b AC and the results are shown in Fig. 2. For the alloy where both solutes segregate to the GB, we set \Omega¯b AC = 1.5, while for the second alloy, where solute B segregates and solute C desegregates, we let \Omega¯b AC = -1.5. All other models parameters for both alloys are set the same and are shown in Fig. 2(a) and (e), where we assume ideal GBs, i.e., boundary interaction parameters \Omega¯gb AC = \Omega¯gb BC = \Omega¯gb AB = 0.

Panels (a) through (d) of Fig. 2 depict results for the alloy, where both solutes segregate to the GB, while panels (e) through (h) of Fig. 2 show results for the alloy with B(C) segregation(desegregation).

Surface plots of the boundary concentrations at the center of the GB where \Upsilon = 1, cgb B and cgb C, as a function of the bulk alloy concentrations c\infty B and c\infty C are respectively shown in Fig. 2(b) and (f) for the alloy where both solutes segregate to the GB and the second alloy with B(C) segregation(desegregation). These surface plots show that GB concentrations are sensitively dependent on bulk alloy concentrations c\infty B and c\infty C. To highlight such dependencies, we use these surface plots to construct GB concentration line scans as a function of bulk concentrations, and the results are shown in Fig. 2(c), (d), (g), and (h), where the dashed line in black is the unity line. For the alloy where both solutes segregate to the GB, Fig. 2(c) shows a plot of the GB concentration cgb B as a function of c\infty B for various values of c\infty C and Fig. 2(d) shows one of the GB concentration cgb C as a function of c\infty C for various values of c\infty B. For this alloy, it can be seen that the GB solute concentrations cgb B and cgb C increase with the bulk concentrations.

Furthermore, it can be clearly seen that cgb B depends sensitively on the concentration of the second solute c\infty C and a similar trend holds for cgb C, see Fig. 2(d). For the second alloy case, where solute B segregates to the GB and solute C desegregates, Fig. 2(g) shows a plot of the GB concentration cgb B as a function of c\infty B for various values of c\infty C and Fig. 2(h) shows one of the GB concentration cgb C as a function of c\infty C for various values of c\infty B. For this alloy, it can be seen that increasing the bulk concentration c\infty B leads to an increase in GB solute concentration cgb B, and that such an increase is less sensitive to the bulk concentration of C, c\infty C. Also shown in Fig. 2(h) is a plot of GB concentration cgb C as a function of bulk concentration c\infty C, where the GB concentration falls below the bulk values (i.e., below the unity line) indicating desegregation of solute C at the GB. For this alloy with B(C) segregation(desegregation), \Delta G¯ AB = \Delta G¯ AC = 0.5 and the bulk solute interaction parameter between A and C is \Omega¯b AC = -1.5. This indicates a stronger attractive A-C interactions in the bulk compared to the driving force for segregation of C described by \Delta G¯ AC, noting that we set the GB interaction parameters \Omega¯gb AC = \Omega¯gb BC = \Omega¯gb AB = 0. The interested reader is referred to Supplementary_Code.zip for a Python script to plot the multicomponent GB segregation isotherms as a function of the model parameters.

Dynamics of Doped Grain Boundaries

We now direct our attention to GB dynamics and the impact of GB diffusion coefficients on solute drag-velocity profiles. To this end, we employ the two alloy cases discussed in Fig. 2, namely a ternary alloy with \Omega¯b AC = 1.5 resulting in both solutes B and C segregating to the GB, and a second alloy with \Omega¯b AC = -1.5 leading to the segregation of solute B and desegregation of solute C. All other model parameters for both alloys are the same and shown in Fig. 2(a) and (e). In this analysis, we set the cross-diffusion coefficients D¯gb BC = D¯gb CB = 0 and perform simulations varying D¯gb BB and D¯gb CC, which describe the diffusion of solutes B and C, respectively. Figure 3 shows results demonstrating the impact of D¯gb BB/D¯gb CC on GB concentrations and solute drag-velocity profiles. Figure 3(a)–(c) depicts results for the ternary alloy case, where both solutes B and C segregate to the GB, and Fig. 3(d)–(f) shows results for the alloy with B(C) segregation(desegregation).

For the ternary alloy, where both solutes B and C segregate to the GB, Fig. 3(a) shows concentration profiles across the GB region at a boundary velocity V¯ = 1 for various values of D¯gb BB/D¯gb CC. It can be seen that the GB concentration of solute B increases with D¯gb BB/D¯gb CC and that when D¯gb BB/D¯gb CC = 1 the concentrations of both solutes have the same profile. Here, we note that in this alloy case both solutes have the same segregation energy, i.e., \Delta G¯ AB = \Delta G¯ AC = 0.5 and \Omega¯b AB = \Omega¯b AC = 1.5. The corresponding solute drag-velocity profiles for the cases with D¯gb BB/D¯gb CC = 0.1, 1, and 10 are shown in Fig. 3(b). It can be seen that having a fast diffusing segregating solute results in drag-velocity profiles that maintain large values of solute drag beyond the peak point, i.e., a more gradual drop in solute drag past the max solute drag point. For this alloy, the thermodynamic parameters describing GB segregation are symmetric with respect to the exchange of B and C (i.e., \Delta G¯ AB = \Delta G¯ AC and \Omega¯b AB = \Omega¯b AC). However, having different values for D¯gb BB and D¯gb CC breaks this segregation symmetry and leads to concentration profiles of B and C that are different. For this ternary alloy system, where both solutes B and C segregate to the GB, the drag P¯* and GB velocity V¯* values at the peak point of the solute drag-velocity profiles are obtained as a function of D¯gb BB/D¯gb CC and plotted in Fig. 3(c). The peak points in solute drag-velocity profiles shift to larger drag and velocity values with increasing D¯gb BB/D¯gb CC until D¯gb BB/D¯gb CC = 1, which corresponds to alloys where both solutes have the same segregation energies and diffusivities. Then, a gradual decrease in P¯* and V¯* is observed in cases with D¯gb BB/D¯gb CC > 1. While the alloy system depicted in Fig. 3(a)–(c) considers cases where both solutes have the same segregation energy, Fig. 3(c) clearly shows that solute drag effects in multicomponent alloys are sensitively dependant on the operative mass transport processes.

Results for the ternary alloy where solute B segregates to the GB and solute C desegregates are shown in Fig. 3(d)–(f). As discussed above, desegregation of solute C in this ternary alloy case is driven by attractive bulk A-C interactions set by \Omega¯b AC = -1.5. Solute concentration profiles using a GB velocity V¯ = 1 for various values of D¯gb BB/D¯gb CC are shown in Fig. 3(d), where it can be seen that increasing D¯gb BB/D¯gb CC leads to an increase in the GB concentration of solute B and a concomitant decrease in the boundary concentration of solute C. This can be explained as follows: according to Eq. 4, the segregation energy of solute i depends on the heat of mixing parameters of various solute types which are coupled through the concentration fields. For this alloy case, the segregation energy of solute C, Ei (see Eq. 4), depends on the concentration of B through the term \Omega¯b AC cB with \Omega¯b AC = -1.5. As a result, increasing the GB concentration of B reduces the segregation energy of solute C, a trend observed in Fig. 3(d).

Next, Fig. 3(e) shows the corresponding solute drag-velocity profiles for various values of D¯gb BB/D¯gb CC, where it can be seen that the location of the peak point in drag-velocity profiles shifts to larger velocity values with increasing D¯gb BB/D¯gb CC. Using these drag-velocity profiles, the drag P¯* and GB velocity V¯* values at the peak points are plotted in Fig. 3(f) as a function of D¯gb BB/D¯gb CC. A monotonic behavior of P¯* and V¯* is observed, in which P¯* decreases and V¯* shifts to larger values with increasing D¯gb BB/D¯gb CC. This trend for the ternary alloy with B(C) segregation(desegregation) is in contrast to the alloy case where both solutes segregate to the GB, Fig. 3(a), which depicts a non-monotonic behavior for P¯* and V¯* as a function of D¯gb BB/D¯gb CC. The results depicted in Fig. 3 demonstrate the complex GB chemical states and resultant solute drag-velocity profiles that arise due to multicomponent coupling effects. It is interesting to note that while many published studies76,77 suggest GB solute segregation as a strategy to control GB migration, Fig. 3 reveals that in multicomponent systems GB desegregation of a solute type can be as effective in inducing GB solute drag and, therefore, controlling boundary dynamics. For example, comparison of Fig. 3(c) and (f) shows that choosing alloys, where one solute type segregates to the GB and a second that desegregates results in boundary drag values that are comparable in magnitude to alloys where both solutes segregate to the GB. Furthermore, the results depicted in Fig. 3 demonstrate that a quantitative understanding of the impact of GB chemistry on boundary migration in multicomponent alloys requires in addition to the segregation energies of solutes a detailed knowledge of mass transport processes in such alloys.

In Fig. 3, we explored the impact of GB mass transport processes on boundary solute drag using ternary alloys with contrasting segregation behaviors, namely an alloy system where both solutes segregate to the GB and a second alloy with B(C) segregation(desegregation). We now consider the impact of far-field bulk concentration level of both solutes (i.e., c\infty B and c\infty C) on solute drag values. Classical models predict that GB solute drag in binary alloys increases linearly with bulk concentration31. This linear scaling is a direct manifestation of the ideal and dilute solution thermodynamics assumption employed in these modeling treatments. Such assumptions, however, are not valid in multicomponent alloys, especially ones with far-from-dilute concentrations.

To demonstrate the impact of bulk concentration on solute drag, we employ the same ternary alloy classes considered in Figs. 2 and 3, specifically ones with various values for the bulk heat of mixing parameter \Omega¯b AC. We consider cases with \Omega¯b AC = -1.5, 0, and 1.5, while keeping the other alloy model parameters the same, shown in Fig. 4(a), and use D¯gb BB = D¯gb CC = 1.0 and D¯gb BC = D¯gb CB = 0.0. Then, we perform simulations with various bulk concentration values and obtain resultant solute drag-velocity profiles. Figure 4(b) shows contour slices depicting the peak solute drag P¯* as a function of the interaction parameter \Omega¯b AC and bulk concentrations c\infty B and c\infty C. These contour slices reveal several trends. First, the maximum solute drag exhibit a non-monotonic dependence on bulk concentrations c\infty B and c\infty C. For example, the ternary alloy with \Omega¯b AC = 1.5 shows that P¯* increases with c\infty B and c\infty C over some range of concentrations then decreases with increasing bulk concentrations. Second, the ternary alloy with \Omega¯b AC = -1.5, resulting in the segregation of solute B to the GB and desegregation of solute C, yields large solute drag values compared to the ideal alloy case with \Omega¯b AC = 0. Third, for the model parameters used in Fig. 4(a) regimes in bulk concentration space with large solute drag values do not align with lines of equi-concentration of solutes.

In metal physics, multicomponent alloys with equi-atomic concentrations, i.e., so-called high entropy alloys, have been the subject of active research as they exhibit unique engineering and functional properties78,79. Considering ternary alloys with equiatomic solute concentrations, as indicated by the dashed lines in Fig. 4(b), Fig. 4(c) presents the peak solute drag P¯* plotted along the c\infty B = c\infty C concentration line. It can be seen that for the ternary alloys with various values of \Omega¯b AC considered here, the peak solute drag value increases with concentration along this equi-concentration line until it reaches a maximum then it decreases with increasing the bulk concentration. The trends observed in Fig. 4 can be understood by examining Eq. 4, which reveals using the model parameters in Fig. 4(a) that the segregation energies of B and C at the center of the GB are given by EB = -2 + 3cB + (1.5 + \Omega¯b AC)cC and EC = -0.5 + (2cB - 1)\Omega¯b AC + (1.5 + \Omega¯b AC)cB demonstrating that the GB segregation energy of solute i is coupled to the concentration of solute j through the heat of mixing model parameter \Omega¯b AC. This is in contrast to existing studies of solute drag in multicomponent alloys38–40, which assume ideal and dilute alloys, resulting in concentration-independent and decoupled segregation energies for the various species. A key outcome of these assumptions is that GB solute drag scales linearly with bulk composition. The results depicted in Fig. 4 clearly demonstrate that GB solute drag exhibits a non-monotonic dependence on bulk compositions and that such dependence is sensitive to the nature of solute interactions.

To further illustrate the complex GB concentration profiles arising due to multicomponent alloying effects and its impact on boundary solute drag, we now consider the inverse problem. Specifically, we explore the so-called one-to-many relationships in which alloy systems with various thermodynamic properties and GB segregation characteristics result in the same solute drag profiles. As a demonstration, we select three ternary alloys, termed Alloy 1, Alloy 2, and Alloy 3, with contrasting segregation behaviors, yet resulting in near identical GB solute drag-velocity profiles. This is achieved in our model through the three bulk heat of mixing parameters, namely \Omega¯b AB, \Omega¯b AC, and \Omega¯b BC, while fixing all other model parameters, including the dilute-limit segregation energy \Delta G¯ AB = \Delta G¯ AC = 0.1, and the results are shown in Fig. 5. Figure 5(a) shows values of the bulk heat of mixing parameters, \Omega¯b AB, \Omega¯b AC, and \Omega¯b BC, used for these alloys, while Fig. 5(b) shows the rest of the model parameters used in these systems.

We set bulk concentrations c\infty B = c\infty C = 0.1 and let D¯gb BB = D¯gb CC = 1.0 and D¯gb BC = D¯gb CB = 0.0. For Alloy 1, the bulk regular solution parameters for A-B and A-C pairs are \Omega¯b AB = \Omega¯b AC = 1.5, which results in both solutes B and C having the same segregation energy. For Alloy 2, we set \Omega¯b AB = \Omega¯b BC = 0 and \Omega¯b AC = 1.5, which models zero bulk interaction energy for A-B and B-C pairs and repulsive interactions in the bulk for A-C. According to Eq. 4, the additional component to GB segregation energy due to repulsive bulk A-C interactions leads to a competitive segregation effect in which the stronger segregation of solute C result in an induced desegregation of B. For Alloy 3, the bulk regular solution parameters for the A-B, A-C, and B-C are \Omega¯b AB = \Omega¯b AC = \Omega¯b BC = -1.5 favoring bulk alloy mixing which result in desegregation of solutes B and C from the GB.

At a GB velocity V¯ = 1, Fig. 5(c) shows the concentration profiles of solutes B and C across the GB for all three alloys, Alloy 1, Alloy 2, and Alloy 3. As expected, GB segregation of both solutes B and C is observed in Alloy 1, where segregation of solute C and desegregation of solute B is seen in Alloy 2. For Alloy 3, desegregation of both solutes B and C can be observed. Figure 5(d) shows the corresponding GB solute drag-velocity profiles for these three alloy systems, which reveal near identical solute drag profiles, specifically the peak solute drag value and the corresponding velocity at the peak solute drag point. The results depicted in Fig. 5 clearly demonstrate multicomponent effects, in which alloys with contrasting segregation behaviors and GB concentrations can result in identical boundary drag profiles. While studies77,80 proposed GB solute segregation as a strategy to control GB dynamics in binary alloys, Fig. 5(c) and (d) reveals a broader alloy design space in multicomponent systems, where solute desegregation, for example, can be as effective in controlling GB dynamics. Indeed, experimental studies on multicomponent alloys revealed GB segregation of some solute types and desegregation of other alloying element types18,81.

The results depicted in Figs. 2 through 5 explore alloys with various bulk heat of mixing parameters, while assuming ideal GBs with no interactions between elements A, B, and C within the boundary region, i.e., \Omega¯gb AC = \Omega¯gb AB = \Omega¯gb BC = 0. We now direct our attention to the impact of GB heat of mixing on solute drag. To this end, we explore alloys with various values for the GB heat of mixing \Omega¯gb BC while fixing all other model parameters, shown in Fig. 6(a). Again, in regular solution thermodynamics \Omega¯gb BC describes the interaction energy of alloying elements B and C within the GB region, where negative (positive) values for \Omega¯gb BC describe attractive (repulsive) B-C interactions. Here, we let \Delta G¯ AB = 0.1 and \Delta G¯ AC = 0.05 indicating slightly stronger GB segregation of solute B than that of C, recall from Eq. 4 that \Delta G¯ AB and \Delta G¯ AC represent segregation energies in the dilute limit. Also, we set \Omega¯gb AC = \Omega¯gb AB = 0 indicating no A-C and A-B interactions within the GB in order to isolate effects due to B-C interactions described by \Omega¯gb BC. We employ alloys with a bulk concentration of c\infty B = c\infty C = 0.1 and set the diffusion coefficients to D¯gb BB = D¯gb CC = 1.0, D¯gb BC = D¯gb CB = 0.0. Using Eq. 9 for the multicomponent segregation isotherms at zero velocity, Fig. 6(b) shows equilibrium concentrations at the center of the boundary region as a function of the GB heat of mixing \Omega¯gb BC, where it can be seen that GB concentrations of solutes B and C decrease with the increase in \Omega¯gb BC. For the alloy model parameters employed in Fig. 6, the GB concentration of solute C decreases below the bulk level for some range of \Omega¯gb BC values indicating GB desegregation of C. In such a case, the higher GB concentration of solute B due to its larger segregation energy than that of C combined with repulsive B-C interactions set using large values of \Omega¯gb BC result in an induced desegregation case where segregation of B drives desegregation of C. At a GB velocity V¯ = 1, Fig. 6(c) shows concentration profiles across the GB region for various values of \Omega¯gb BC. Decreasing \Omega¯gb BC to negative values, indicating attractive B-C interactions within the boundary, leads to synergistic co-segregation of both solutes. Furthermore, Fig. 6(c) reveals an induced desegregation case for large values of \Omega¯gb BC (i.e., repulsive B-C interactions at the GB), in which segregation of solute B drives desegregation of C. The corresponding GB solute drag-velocity profiles for the ternary alloy systems with various values of \Omega¯gb BC are shown in Fig. 6(d). The markers in magenta identify peak GB solute drag values, which are shown in the inset of Fig. 6(d) as a function of \Omega¯gb BC. It can be clearly seen that GB solute drag is sensitively dependent on solute-solute interactions within the GB region, where an increase in \Omega¯gb BC results in a reduction in the solute content at the GB, see Fig. 6(c), and a decrease in solute drag values. On the other hand, negative values of \Omega¯gb BC indicate attractive GB B-C interactions resulting in synergistic co-segregation of both solutes and an increase in GB solute drag.

All results presented so far do not consider the impact of off-diagonal diffusion coefficients D¯gb BC and D¯gb CB on solute drag, which we investigate now.

To this end, we employ the alloy with the model parameters depicted in Fig. 6(a) and use \Omega¯gb BC = 1.5, i.e., repulsive B-C interactions within the GB. We set D¯gb BB = D¯gb CC = 1.0 and perform simulations systematically varying the off-diagonal diffusion coefficients D¯gb BC and D¯gb CB. Recall that the diffusion coefficients can be expressed as Dij = \sum k Lik \partial\mu k / \partial cj, where Lik are the Onsager coefficients and \partial\mu k / \partial cj are the thermodynamic factors, and that the values and algebraic sign of the off-diagonal diffusion coefficients are influenced by both Lik and \partial\mu k / \partial cj. For example, a negative value for D¯gb BC indicates diffusion of solute B up the concentration gradient of solute C. Figure 7 shows results demonstrating the role of off-diagonal diffusion coefficients in GB solute drag, where the contour plots in Fig. 7(a) and (b) show respectively the dependence of the peak solute drag P¯* and velocity V¯* at peak point on D¯gb BC and D¯gb CB. It can be seen that solute drag is sensitively dependent on the magnitude and algebraic sign of the off-diagonal diffusion coefficients. For the model parameters chosen here which favor the segregation of solute B over C, there exists a combination of D¯gb BC and D¯gb CB values that maximizes GB solute drag. In general, D¯gb BC and D¯gb CB do not have the same values, and Fig. 7(a) shows that solute drag P¯* decreases in regimes where D¯gb BC and D¯gb CB are dissimilar. Figure 7(b) reveals a general trend where the velocity V¯* at peak drag decreases with the increase in both D¯gb BC and D¯gb CB. A close examination of panels (a) and (b) of Fig. 7 reveals that V¯* is much more sensitive to variations in D¯gb BC and D¯gb CB than the peak solute drag P¯*. To demonstrate how the concentration profiles are influenced by the off-diagonal diffusion coefficients D¯gb BC and D¯gb CB, Fig. 7(c) shows GB concentration line scans for solutes B and C in two cases, namely one with D¯gb BC = D¯gb CB = 0 and a second with D¯gb BC = -1 and D¯gb CB = 1.

According to Eqs. 5 and 7, the use of non-zero values for D¯gb BC and D¯gb CB results in coupling of the governing equations for solutes B and C, where the evolution of one solute type is also driven by the chemical potential gradients of the other solute types. This is seen in Fig. 7(c) where the GB concentration profiles are sensitive to the particular values of D¯gb BC and D¯gb CB. Using D¯gb BC = -1 for this alloy with \Omega¯gb BC = 1.5 leads to diffusion of B up the concentration gradient of C while also maintaining repulsive B-C interactions within the GB region.

Figures 3 and 7 show that in addition to the segregation energy of each alloying element, a detailed knowledge of the operative transport processes is equally important as mass transport in multicomponent alloys is characterized by several diffusion coefficients with potentially large contrast in their magnitudes46. For example, even in ternary alloys with equal segregation energies for solutes B and C, our simulation results, see Fig. 7, reveal that varying the diffusion coefficients results in drastic changes to GB solute drag-velocity profiles.

Model Parametrization and Validation

The proposed modeling framework, together with the simulations it produces, provides several pathways for cross-validation with established databases, experimental observations, and atomistic simulations. First, our modeling treatment explicitly accounts for both bulk and GB free energies, parametrized using regular solution thermodynamics. The bulk regular solution model parameters can be obtained from existing thermodynamic databases85. For example, for the PtAuPd50, FeNiZr82, and CuAlY84 ternary alloys, in which experimental observations revealed GB segregation, Table 1 lists reported values for the bulk regular solution model parameters of the constituent binary pairs50,86. On the other hand, GB-solute interactions for each solute type can be computed using atomistic simulations68,87. For example, for the Cu-Ag alloy atomistic simulations employing thermodynamic integration in conjunction with the Gibbs adsorption equation were used to calculate the GB free energy as a function of Ag concentration62. These atomistically-obtained GB free energies can then be fit to regular solution free energy expressions to obtain the GB heat of mixing model parameters. Several studies reported values for GB regular solution model parameters in various alloys, such as \Omega¯gb AlMg = -4.6 in Al-Mg at 327°C88,89 and \Omega¯gb NiP = 6.2 in Fe-Ni-P at 475°C90.

A second avenue for validating our modeling results is comparison with experimental GB segregation data. Here, it is important to note that direct, in situ characterization of concurrent GB migration, solute segregation, and solute drag is constrained by limited spatial and temporal resolution and by the difficulty of isolating individual boundaries in polycrystalline metallic alloys. Nevertheless, recent advances in microscopy have enabled detailed quantification of chemical profiles across static GBs, providing a direct means to validate predictions from our model. To demonstrate this cross-validation strategy, we consider the PtAuPd ternary alloy in which recent experiments revealed GB segregation50. We simulate GB segregation in Pt0.76Au0.18Pd0.06 (in at.%) using reported bulk heat of mixing model parameters50 shown in Table 1 while assuming ideal GBs, and the results are shown in Fig. 8. Our modeling results depicting concentration profiles across a stationary GB are shown in Fig. 8(a), where segregation of Au is observed accompanied by depletion of Pt while the Pd concentration profile does not show any significant enrichment at the GB. Xing et al.50 experimentally investigated GB segregation in the PtAuPd alloy with various compositions and temperatures. For a Pt0.76Au0.18Pd0.06 alloy annealed at 800°C for 24 hrs, Fig. 8(b) shows chemical line scans across a GB using energy-dispersive x-ray spectroscopy (EDS) revealing Au segregation to the Pt GB with no tendency for Pd segregation. Figure 8 reveals that our modeling results [Fig. 8(a)] are in qualitative agreement with experimental EDS line scans [Fig. 8(b)] in terms of segregation profiles. This approach of comparing GB concentration line scans from our model with those obtained experimentally provides avenues to validate our simulation results.

In summary, we presented a theoretical and computational model of GB segregation and solute drag in multicomponent alloys that accounts for both GB and bulk solute-solute interactions of various alloying elements. Ternary alloys were employed as they represent a model system with multicomponent effects while offering computational tractability. Analytical analysis recovered a multicomponent segregation isotherm, describing the GB solute content as a function of the thermodynamic properties the alloy. Simulation results revealed that solute interactions of various elemental pairs resulting in segregation behaviors, such as synergistic co-segregation and induced desegregation, greatly influence the characteristics of GB solute drag-velocity profiles. In addition to the segregation energy of solutes, our work highlights the need to account for the transport coefficients of various alloying elements in order to have a quantitative understanding of GB solute drag in multicomponent systems. Computational studies demonstrated that GB solute drag can exhibit a non-monotonic dependence on bulk concentrations, in contrast to existing solute drag models which only consider binary alloys. Furthermore, it was shown that in multicomponent alloys GB desegregation can be as effective in inducing GB solute drag, thereby expanding the alloy design space in applications where controlling GB dynamical processes is a key microstructural goal. While GB segregation has been extensively studied, prior work has largely been confined to the chemistry of static boundaries. Predictions from our modeling framework motivate future research by shifting the focus to GB dynamics, coupled evolution of boundary segregation and solute drag, as a strategy for designing advanced alloys with tailored microstructures.

Methods
To simplify the presentation of the governing equations describing our GB solute drag model, we use vector and matrix notation in the following sections.

Thermodynamics of Ternary Alloys

We start with a substitutional ternary alloy of host element labeled A and solutes labeled B and C with concentration fields cA, cB, and cC, where cA+cB+cC = 1. Mass fluxes arise due to gradients in the chemical potentials and are given by Ji = - \sum j=A,B,C Lij \nabla\mu˜ j, i=A,B,C, where \mu˜ j is the chemical potential of species j and Lij are the Onsager transport coefficients. Using the Gibbs-Duhem equation, the flux equations for solutes B and C can be expressed respectively as JB = -LBB \nabla\muB - LBC \nabla\muC and JC = -LCB \nabla\muB - LCC \nabla\muC where \mu i = \mu˜ i - \mu˜ A is the chemical potential of species i relative to that of the host A, i.e., the diffusion potential. Considering that \mu B = \mu B(cB,cC) and \mu C = \mu C(cB,cC), we use the relation \nabla\mu B = (\partial\mu B / \partial cB) (\partial cB / \partial x) + (\partial\mu B / \partial cC) (\partial cC / \partial x) and a similar expression for \nabla\mu C to arrive at the following for mass fluxes

{ JB } = - [ LBB LBC ] { \partial\muB / \partial cB \partial\muB / \partial cC } { \nabla cB } = - [ DBB DBC ] { \nabla cB }
{ JC } [ LCB LCC ] { \partial\muC / \partial cB \partial\muC / \partial cC } { \nabla cC } [ DCB DCC ] { \nabla cC }

where Dij are the relevant diffusion coefficients. In this work, we assume bulk diffusion is frozen compared to the one in the GB and we write

Dij = Dgb ij \Upsilon(x). (11)

Assuming an ideal ternary alloy, where the chemical potential difference of the ith species is \mu i = kB T ln(ci/(1 - cB - cC)), one can arrive at analytical expressions for Dij given by

Dij = kB T ( Lij / ci - LiA / cA ), i,j=B,C. (12)

Derivation of the governing equations for solute drag

We consider a semi-infinite bicrystal ternary alloy of host element A and solutes B and C described using composition fields cA(x,t), cB(x,t) and cC(x,t), where x is the direction measured from the GB and t is time. An indicator function \Upsilon(x) given by Eq. 2 is used to interpolate bulk fb(c,T) and GB fgb(c,T) free energies. Here, x \in (- \infty, - \delta) and x \in (+ \delta, + \infty) define the spatial extent of grains 1 and 2, respectively, and the GB is defined in the spatial region x \in (- \delta, + \delta). Regular solution thermodynamics47 are used for the functional form of the bulk and GB free energies leading to expressions for the chemical potentials that are given by Eq. 3.

We express solute transport equations in a frame moving with the migrating GB at a velocity V, where the steady-state solute governing equations are described by

V { \partial cB / \partial x } = { \partial JB / \partial x }
{ \partial cC / \partial x } { \partial JC / \partial x }

Then, we integrate Eq. 13 above using the far-field concentration boundary condition for the ith solute ci = c\infty i as x \to \infty leading to

{ JB } = - [ LBB LBC ] { \nabla\mu B } = V { cB - c\infty B }
{ JC } [ LCB LCC ] { \nabla\mu C } { cC - c\infty C }

Next, using Eq. 3 we re-express gradients in the diffusion potential in terms of concentration gradients as

{ \nabla\mu B } = [ WBB WBC ] { \partial cB / \partial x } + \Upsilon' { UB }
{ \nabla\mu C } [ WCB WCC ] { \partial cC / \partial x } { UC }

where \Upsilon' = d\Upsilon/dx and elements of the matrix [Wij] are given by

Wii = kB T / ci + kB T / cA - 2\Omega b Ai - 2\Upsilon(\Omega gb Ai - \Omega b Ai), i=B,C, (16a)

Wij = Wji = kB T / cA + \Omega b ij + \Upsilon(\Omega gb ij - \Omega b ij), i,j=B,C, j \neq i. (16b)

Also, using the expression for the segregation energy Ei (see Eq. 4) elements of the vector {Ui} are given by

Ui = -\Delta G Ai + (\Omega gb Ai - \Omega b Ai)(1 - 2c) + (\Omega gb BC - \Omega b BC) c j, i,j=B,C, j \neq i. (17)

Finally, we plug Eq. 15 into Eq. 14 and re-arrange to yield the following nonlinear coupled system

{ \partial cB / \partial x } = -[Wij]-1 \Upsilon' {Ui} + V[Lij]-1 {ci - c\infty i}, (18)
{ \partial cC / \partial x }

where [···]-1 is the matrix inverse. Here, we note that [Wij] and {Ui} are functions of the concentrations, and, as a result, the right-hand side of Eq. 18 is only a function of the concentration fields of solutes B and C and the indicator function \Upsilon and its derivative. The first term on the right-hand side of Eq. 18 captures details about alloy thermodynamics and GB segregation energy, and the second term is dynamical related to the GB velocity and transport coefficients. It is straightforward to re-express the Onsager coefficients Lij in terms of the diffusion coefficients Dij as shown in Eq. 12. The last term on the right-hand side of Eq. 18 arises due to the imposed boundary conditions on the simulation domain. Equation 18 above is numerically solved in non-dimensional form using the finite difference method with a grid spacing \Delta x = 0.02\delta and a GB velocity V¯ in the range of 0 and 50 in increments of 0.1 for most cases explored in this work. Once the concentration profiles of solutes B and C are obtained, the non-dimensional GB solute drag P¯ is evaluated numerically using Eq. 8.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The paper presents a computational study of solute segregation at grain boundaries and solute drag in disordered ternary\nalloys. The simulations utilize the phase-field approach based on regular solution thermodynamics. This is a classical\nmetallurgical subject that has been studied for many years. However, most of the previous work has been done on binary\nsystems. To my knowledge, this is the first systematic computational study of solute drag in a ternary system. The parametric\nexploration reveals several important trends related to solute co-segregation and solute competition. The results also\ndemonstrate the sensitive dependence of the solute drag force on the solute interaction parameters and the relative diffusion\ncoefficients of the three components. This work provides a glimpse into the complexity of grain boundary dynamics in\nmulticomponent systems, including high-entropy alloys.\nThe paper makes a significant contribution to the field of grain boundary-solute interactions in multicomponent systems and\npresents the results very clearly. As such, I recommend the paper for publication after the authors address the following\nsuggestions.\n1. Many readers will try to translate the computational results to real-world alloys. Although the simulations were carried out\nusing dimensionless variables, it would help if the authors discussed some typical values of the physical parameters\ncorresponding to simulation conditions. For example, the dimensionless grain boundary velocity varies between 0 and 15.\nWhat are the corresponding physical velocities in meters per second? Similar estimates could be made for the solute\ninteraction parameters and the diffusion coefficients.\n2. The regular solution model predicts a miscibility gap and a critical point of mixing. Such gaps are different in grain\nboundaries and inside the grains. The simulations utilized interaction parameters that kept both solutions above the\nrespective critical points. The authors could mention that, under real conditions, the segregation atmosphere could undergo\na phase separation, which would drastically change the solute drag dynamics. The impact of grain boundary phase\nseparation was studied in binary systems (e.g., https://doi.org/10.1016/j.actamat.2019.08.046) but might be studied in ternary\nsystems by changing the interaction parameters within the present model.\n3. The model studied here is essentially one-dimensional. As a result, solute diffusion along the grain boundary is not\nconsidered. Meanwhile, lateral diffusion plays a significant role in the solute drag (https://doi.org/10.3390/nano11092348).\nThe authors may want to mention this fact as one of the limitations of the model, which could be addressed if the authors\ndecide to extend their model to 2D in the future.\nYuri Mishin.", "reviewer_2": "The authors present a model for grain boundary segregation and solute drag in multicomponent alloys, incorporating solute-\nsolute interactions. While the manuscript is well-written and presents some interesting results, I do not believe it passes the\nbar for publication in Nature Communications.\n1) My biggest criticism is the failure to acknowledge prior literature. Equation (9), presented as a novel multicomponent\nsegregation isotherm, is, in fact, the classical Guttmann isotherm [Guttmann, Surface Science 53.1 (1975)], which the\nauthors should be familiar with. This oversight significantly undermines the perceived novelty of the work.\n2) On the same note, the statement that \"There are no GB solute drag models that consider multicomponent alloying effects\"\nis inaccurate. Models addressing this exist, such as those by Bréchet and Purdy [Acta materialia 51.18 (2003)], Mendelev\nand Srolovitz [Interface Science 10 (2002)], and Hersent et al. [DOI: 10.4236/mnsms.2014.41002]. For example, the work by\nBréchet and Purdy extended the classic Cahn’s treatment to multicomponent systems, and explored the impact of solute-\nsolute interactions on drag in ternary systems. While these models may have limitations (e.g., dilute concentration\nassumptions), a rigorous comparison is essential. The manuscript would benefit significantly from a thorough review of these\nmodels, clearly articulating the advantages of the proposed approach, such as its ability to capture non-dilute behavior, and\ncritically evaluating the limitations of prior work, including what physics, if any, were missed.\n3) Although I think the current model represents an improvement over the prior models, it still retains a main limiting\nassumption of the prior models, specifically the assumption of a single average segregation energy (single site type) at the\ngrain boundaries. This simplification fails to account for: (a) grain boundary segregation anisotropy, a crucial factor in solute\ndrag, as demonstrated by prior work from the corresponding author [Barr et al., Nanoscale 13.6 (2021)]. And, (b) the site-\ndependent nature of segregation, which leads to diverse co-segregation behaviors that cannot be captured by the averaged\napproach. see [Scheiber et al., Scripta materialia 150 (2018)] and [Wagih et al., arXiv:2411.05303].\n4) Can the current model capture experimentally observed abnormal grain growth?\n5) The provided code only solves for the equilibrium segregation isotherm (which is not the main contribution of the paper).\nThe authors should also include the code used to solve the dynamic solute drag equations, which would be more beneficial\nto the community.\nGiven that this work extends the authors' previous model [Physical Review Letters 127.17 (2021)] to multicomponent alloys,\nand primarily focuses on a parametric study, it appears more appropriate, after revision, for a specialized journal in\ncomputational materials science.", "reviewer_3": "The authors have employed theoretical models and computational methods to study the grain boundary (GB) segregation\nbehaviors, and the associated GB solute drag in multicomponent alloys. Their model accounts for thermodynamic solute\ninteractions between different alloying elements in both the bulk and GB regions, as well as the multicomponent diffusion of\nthese alloying elements. Furthermore, the GB solute drag is quantified through the solute concentration profiles near the GB\nunder a constant migration velocity. This study systematically examines the effects of GB diffusion coefficients, bulk solute\nconcentration, bulk solute interactions, and GB solute interactions on the GB solute drag-velocity relationship, contributing to\na deeper understanding of GB solute drag in multicomponent alloys.\nThis work addresses an important topic in alloy design by providing valuable insights into GB solute drag in multicomponent\nalloy systems. The study is well-motivated, and the theoretical approach is appropriate for the investigation. However,\nseveral scientific concerns need to be addressed before the manuscript can be considered for publication. My detailed\ncomments are as follows:\nComments:\n1) Regarding solute segregation thermodynamics, the expressions for free energy and chemical potential in the ternary alloy\nsystem, based on the regular solution model, appear to be a straightforward extension of those for a binary alloy system,\nas described in the authors’ earlier works 1,2. The primary novelty of the current study seems to lie in the investigation of GB\nsolute drag in multicomponent alloys. Nevertheless, as noted in Comment 2, the effect of multicomponent diffusion on GB\nsolute drag remains insufficiently understood in the present work.\n2) In the context of multicomponent diffusion, especially in far-from-dilute alloys, the kinetic coupling between different solute\nfluxes is crucial, as it significantly influences the solute concentration distribution near the GB and, consequently, the GB\nsolute drag. However, the manuscript neglects the cross-diffusion process between solute B and C within the GB by\nassuming D_BC^gb=D_CB^gb=0 in all calculations, meaning that the off-diagonal kinetic transport coefficients in the\nOnsager matrix are set to zero. This assumption of ignoring kinetic coupling between solute fluxes must be justified, and its\nimpact on the results should be clarified.\n3) Related to comment 2, the off-diagonal transport coefficients are closely associated with solute-solute interactions and\nsolute concentrations. Specifically, positive or negative solute-solute interactions can normally lead to distinct solute\ndiffusion behaviors, thereby affecting the GB solute drag effect. Therefore, solute cross-diffusion coefficients and solute-\nsolute interactions should not be considered independent.\n4) What is the temperature value used in the calculations? Generally, an increase in temperature enhances solute diffusion\nwhile reducing solute segregation tendencies and solute interactions. The authors should clarify the role of temperature in\ntheir model and discuss its effect on GB solute drag in multicomponent alloys.\n5) It is widely reported that GB migration is associated with vacancy generation and emission, mediated by vacancy diffusion\nnear the GB. How does the vacancy concentration distribution affect solute diffusion in multicomponent alloys and,\nconsequently, the GB solute drag? The authors should clarify whether vacancy effects are considered in their model and\nmention their potential impact on the results.\n6) Regarding the results in Figure 5, the authors state that solute B preferentially desegregates from the GB in Alloy 2.\nNevertheless, the values of Ω_AB^b=0 and ΔG_AB=0.1 shown in Figure 5(a) suggest a tendency for solute B to segregate\nat the GB, which appears inconsistent with the stated conclusion.\n(2021). https://doi.org:10.1103/PhysRevLett.127.175503\ncomputational and machine learning studies. Acta Materialia 275 (2024). https://doi.org:10.1016/j.actamat.2024.120037"}, "author_response": "We would like to thank the reviewers for their comments. The time spent by the reviewers in\nproviding feedback on our manuscript is very much appreciated. We have taken these comments\nvery seriously and addressed them in a revised manuscript. In what follows, we address the\ncomments/suggestions made by all reviewers. For each reviewer, each comment is taken verbatim\nand shown below in blue font and a response directly follows. All changes in the manuscript are\nhighlighted in blue font.\nReviewer 1\nComment #0: “The paper makes a significant contribution to the field of grain boundary-solute\ninteractions in multicomponent systems and presents the results very clearly. As such, I\nrecommend the paper for publication after the authors address the following suggestions.”\nResponse: We thank the reviewer for the kind comments and for recommending our work for\npublication.\nComment #1: “Many readers will try to translate the computational results to real-world alloys.\nAlthough the simulations were carried out using dimensionless variables, it would help if the\nauthors discussed some typical values of the physical parameters corresponding to simulation\nconditions. For example, the dimensionless grain boundary velocity varies between 0 and 15. What\nare the corresponding physical velocities in meters per second? Similar estimates could be made\nfor the solute interaction parameters and the diffusion coefficients.”\nResponse: We thank the reviewer for the great point. Indeed, we expressed our model in non-\ndimensional form to reduce the number of free model parameters (dimensionality reduction) and\nto easily reveal trends in solute drag behavior. The approach of expressing governing equations in\nnon-dimensional form is common in many fields of science and engineering. We agree with the\nreviewer that relating values of non-dimensional quantities in our study to real alloys will provide\na better exposition of our modeling results and will help readers relate the work to real alloys. To\naddress this comment, we made the following changes in the revised manuscript:\n• We re-expressed the non-dimensional parameters in terms of a characteristic, or reference,\nGB velocity 𝒱 . The sentences following Eq. 7 on page 4 now read: “The above\n!\"#\ntransport equations are made non-dimensional using the GB half width 𝛅 and\nthermal energy k T as reference length and energy scales, respectively, and we\nB\nemploy a reference GB velocity 𝓥 . This leads to the following non-dimensional\n𝒓𝒆𝒇\nquantities: &𝒙 = 𝒙/𝜹, +𝑬++ = 𝑬 /𝒌 𝑻, +𝛀++ = 𝛀 /𝐤 𝐓, 𝚫+++𝐆+++ = 𝚫𝐆 /𝐤 𝐓, 𝛍& = 𝛍 /𝐤 𝐓,\n' 𝒊 𝑩 * 𝐢 𝐁 𝐀* 𝐀𝐢 𝐁 * 𝐢 𝐁\n𝐃++++ = 𝐃 /(𝓥 𝛅), and non-dimensional GB velocity 𝓥& = 𝓥/𝓥 ”\n*. 𝐢𝐣 𝐫𝐞𝐟 𝐫𝐞𝐟\n• On Page 4 and following Eq. 8, we added a discussion on how to convert our non-\ndimensional parameters to real metallic alloys values. Theses sentences now read: “[…] In\nthis work, all computational studies are reported using the non-dimensional\nparameters discussed above. These model parameters can be mapped onto real\nmetallic alloy values as follows: A representative GB width in a typical metallic alloy\nis 2𝛅= 1 nm and thermal energy k T ≈ 0.08 eV assuming an annealing temperature\nB\nin the range 800-1000 K, typical in metallic alloys [49]. At such temperatures,\nexperimentally measured GB velocities are in the range 𝓥 0.1-10 𝛍m/s [50] and\n𝐫𝐞𝐟\nrepresentative values for GB solute diffusion ≈10-15 m2/s [51,52].”\nComment #2: “The regular solution model predicts a miscibility gap and a critical point of\nmixing. Such gaps are different in grain boundaries and inside the grains. The simulations utilized\ninteraction parameters that kept both solutions above the respective critical points. The authors\ncould mention that, under real conditions, the segregation atmosphere could undergo a phase\nseparation, which would drastically change the solute drag dynamics. The impact of grain\nboundary phase separation was studied in binary systems\n(e.g., https://doi.org/10.1016/j.actamat.2019.08.046) but might be studied in ternary systems by\nchanging the interaction parameters within the present model.”\nResponse: The reviewer brings an excellent point. Indeed, according to regular solution\nthermodynamics when the regular solution parameter becomes large (i.e., twice as large as the\nthermal energy) a miscibility gap develops, and the alloy has the tendency to phase separate\n(via nucleation/growth or spinodal decomposition). Having a miscibility gap brings several effects that\ninfluence GB dynamics including pinning due to nucleated domains, competition between\nprecipitation and segregation, and concentration-dependent transport coefficients, among many\nothers. These are certainly beyond the scope of this work which mainly deals with solute drag in\nmulticomponent alloys. However, the reviewer is absolutely right in the need to include a\ndiscussion on how the development of a miscibility gap could influence solute drag. To address\nthis comment, in the revised manuscript we made the following changes:\n• In the first paragraph on Page 3, we added a statement describing phase separation in\nregular solution binary alloys and how GB phase separation has been found to affect GB\nmobility. The statement reads: “In binary alloys with a miscibility gap, i.e., large values\nfor the regular solution parameter, a recent study revealed that GB phase separation\nis accompanied by drastic changes in boundary mobility [41].”\n• In Page 11, we added a short discussion in the first paragraph on phase separation in\nmulticomponent alloys and how it will be an interesting effect to explore in future work.\nThe discussion now reads: “[…] The computational studies presented in this work\nrevealed a wide range of GB segregation effects, such as competitive segregation and\ninduced de-segregation, that are dependent on solute-solute interactions of various\nalloying element types. Other interesting effects to investigate in future work include\nthe interplay between GB segregation and phase separation, driven by repulsive\nsolute-solute interactions […]”\nComment #3: “The model studied here is essentially one-dimensional. As a result, solute diffusion\nalong the grain boundary is not considered. Meanwhile, lateral diffusion plays a significant role\nin the solute drag (https://doi.org/10.3390/nano11092348). The authors may want to mention this\nfact as one of the limitations of the model, which could be addressed if the authors decide to extend\ntheir model to 2D in the future.”\nResponse: We thank the reviewer for the great remark. Yes, our modeling treatment is one-\ndimensional to allow us to fundamentally understand the impact of multicomponent alloying\neffects on GB solute drag without the topological complexity associated with microstructures in\n2D and 3D geometries. Within this 1D geometry, the relevant diffusion coefficients are ones across\nthe GB, which we capture. Indeed, when one extends our model to 2D or 3D, one can employ a\ntensorial description of GB diffusion in which diffusion along and across the GB are accounted\nfor. To address this comment, we made the following changes in the revised manuscript\n• On Page 4 above Eq. 7, the discussion in the top paragraph has been modified to include\nsentences about future extension of the model to account for the full GB diffusion effects.\nThe sentences now read: “A future extension of our model to two (2D) or three (3D)\ndimensional geometries would allow for the incorporation of both diffusion across\nand along the GB region.”\n• In the last paragraph of Page 10, we added several sentences and references to discuss\ndiffusion along and across GB and potential impact on solute drag. These sentences now\nread: “[…] Furthermore, since our modeling treatment is one-dimensional the\nrelevant transport coefficients are solute diffusion across the GB. A future extension\nof our modeling treatment to 2D/3D geometries would enable the incorporation of a\ntensorial description of GB diffusion in which both diffusion along and across the GB\nare accounted for. Indeed, atomistic simulations of Cu-Ag binary alloys, for example,\nshowed that accelerated lateral GB diffusion leads to nonuniform redistribution of\nsolute atoms, which in turn influences GB solute drag [69]. […]”\nReviewer 2\nComment #1: “My biggest criticism is the failure to acknowledge prior literature. Equation (9),\npresented as a novel multicomponent segregation isotherm, is, in fact, the classical Guttmann\nisotherm [Guttmann, Surface Science 53.1 (1975)], which the authors should be familiar with.\nThis oversight significantly undermines the perceived novelty of the work.”\nResponse: We thank the reviewer for the excellent comment. The corresponding author of this\nstudy, F. Abdeljawad, is very familiar with the Guttmann study and frequently refers to it when\ndiscussing segregation isotherms in multicomponent systems. We sincerely apologize about not\ndiscussing Guttmann’s treatment when presenting our segregation isotherm. Here, we should state\none major assumption made in Guttman’s model, in which we relax in our work. Guttman assumed\nthat the bulk is an ideal dilute alloy without any solute-solute interactions and only accounted for\nthose interactions in the GB. In Guttmann’s study, he stated at the bottom of page 216 and top of\npage 217 of his article that: “Since the segregation phenomena we are dealing with occur in dilute\nsolutions, the bulk phase […] can be considered ideal.” We totally agree with the reviewer that a\nproper acknowledgment of Guttman’s study is warranted when presenting the segregation\nisotherm results. To address this comment, we made the following changes in the manuscript:\n• To place our segregation analysis in the proper context, in the last paragraph on Page 4\nwe added a discussion on Guttmann segregation isotherm model and cited his work. The\nnew discussion also deals with how our model compares to Guttmann’s treatment. The\ndiscussion now reads: “The segregation isotherm given by Eq. 9 can be mapped onto\nthe treatment by Guttmann [62] who investigated equilibrium GB segregation in\nternary alloys. However, we note here that Guttmann’s model [62] only considered\nsolute-solute interactions in the GB layer as it assumed the bulk phase to be an ideal,\ndilute alloy. Therefore, our segregation isotherm given by Eq. 9 extends Guttmann's\ntreatment by introducing both bulk and GB solute-solute interactions”\nComment #2: “On the same note, the statement that \"There are no GB solute drag models that\nconsider multicomponent alloying effects\" is inaccurate. Models addressing this exist, such as\nthose by Bréchet and Purdy [Acta materialia 51.18 (2003)], Mendelev and Srolovitz [Interface\nScience 10 (2002)], and Hersent et al. [DOI: 10.4236/mnsms.2014.41002]. For example, the work\nby Bréchet and Purdy extended the classic Cahn’s treatment to multicomponent systems, and\nexplored the impact of solute-solute interactions on drag in ternary systems. While these models\nmay have limitations (e.g., dilute concentration assumptions), a rigorous comparison is essential.\nThe manuscript would benefit significantly from a thorough review of these models, clearly\narticulating the advantages of the proposed approach, such as its ability to capture non-dilute\nbehavior, and critically evaluating the limitations of prior work, including what physics, if any,\nwere missed.”\nResponse: We thank the reviewer for bringing to our attention these studies. Before we address\nthis remark, we would like to comment on these three studies that the reviewer referenced. First,\nthe studies by Mendelev and Srolovitz [Interface Science 10 (2002)] and Bréchet and Purdy [Acta\nmaterialia 51.18 (2003)] extended the classic Cahn model to ideal ternary alloys. As a result, the\ngoverning diffusion equations for solutes are decoupled and the resultant drag is always the sum\nof that of the constituent binary alloys. Further, solute-solute interactions of various alloying\nelements in the bulk and GB are not accounted for in these studies. As a result, these studies recover\na result that Cahn revealed for binary alloys which is: solute drag increases linearly with bulk\ncomposition. In our work, we show that this is not the case when solute-solute interactions are\naccounted for. These interactions lead to effects such as induced de-segregation and competitive\nsegregation which yield non-linear solute drag as a function of bulk concentration [see Fig. 4 of\nour work]. Another minor comment regarding Bréchet and Purdy [Acta materialia 51.18 (2003)],\ntheir study presented the model but provided no results; there are no Figures in the paper aside\nfrom one schematic to show the interaction profile they assumed when deriving the model!\nAs for the study by Hersent et al. [DOI: 10.4236/mnsms.2014.41002] their work follows a\ntreatment similar to the one highlighted in Gottstein and Shvindlerman [“Grain Boundary\nMigration in Metals” Second Edition, Chapter 3, pages 135-151] in which the authors consider\nsolute atomic jumps across the boundary. Each atomic jump is characterized by a diffusion energy\nbarrier and a constant binding energy describing solute-GB interactions. Since the assumed\nactivation energies are constants, their model is only valid in ideal, dilute alloys as the model does\nnot account for solute-solute interactions in GB or bulk. Indeed, the result given by Eq. 25 in\nHersent et al. [DOI: 10.4236/mnsms.2014.41002] shows that the resultant solute drag is always\nthe sum of the solute drag of the binaries, similar to the treatment by Bréchet and Purdy. Finally,\nwe also like to state that the study by Hersent et al. [DOI: 10.4236/mnsms.2014.41002] does not\npresent any results, there are no Figures in the article! To address this comment, we made the\nfollowing changes in the manuscript:\n• Since these existing multicomponent drag models rely on Cahn’s model, we now introduce\nseveral sentences to highlight some key assumptions and results from Cahn’s seminal\nwork. At the bottom of Page 1 and top of page 2, we added the following sentences: “[…]\nMost solute drag treatments consider binary alloys [30,31,36]. For example, Cahn's\ntreatment [30] assumes ideal and dilute alloys in which GB-solute interactions are\ncharacterized by a constant segregation energy. A key result of Cahn's model is that\nsolute drag scales linearly with the nominal bulk concentration [30] […]”\n• In the first paragraph on Page 2, we added a detailed discussion on existing solute drag\nmodels in multicomponent alloy and referenced these works. The discussion highlighted\nthe key assumption of these models. The new discussion now reads: “Existing studies on\nGB solute drag in multi-component alloys [37-39] extended Cahn's treatment to ideal\nmulticomponent alloys, and, as a result, these investigations do not account for solute-\nsolute interactions of various alloying element types. For example, Mendelev and\nSrolovitz [37] and Bréchet and Purdy [38] examined solute drag in ideal ternary alloys\nwhere they assumed constant segregation energy for each solute type. In such models\n[37-39], not accounting for solute-solute interactions in the bulk and GB results in\ndecoupled governing equations, in which the total GB solute drag is the sum of solute\ndrag of the constituent binary alloys. Additionally, these studies [37-39], similar to\nCahn's model, predict a solute drag that increases linearly with bulk concentration.\nThere are no GB solute drag models that consider multicomponent alloying effects\nwith solute-solute interactions.”\n• At the end of the first paragraph on Page 8 where results of our Figure 4, which shows non-\nmonotonic dependence of solute drag on solute concentration, we added a discussion to\nhighlight how existing models do not account for solute-solute interactions in\nmulticomponent alloys. The discussion now reads: “This is in contrast to existing studies\nof solute drag in multicomponent alloys [37-39], which assume ideal and dilute alloys,\nresulting in concentration-independent and decoupled segregation energies for the\nvarious species. A key outcome of these assumptions is that GB solute drag scales\nlinearly with bulk composition. The results depicted in Fig. 4 clearly demonstrate that\nGB solute drag exhibits a non-monotonic dependence on bulk compositions and that\nsuch dependence is sensitive to the nature of solute interactions.”\nComment #3: “Although I think the current model represents an improvement over the prior\nmodels, it still retains a main limiting assumption of the prior models, specifically the assumption\nof a single average segregation energy (single site type) at the grain boundaries. This\nsimplification fails to account for: (a) grain boundary segregation anisotropy, a crucial factor in\nsolute drag, as demonstrated by prior work from the corresponding author [Barr et al., Nanoscale\n13.6 (2021)]. And, (b) the site-dependent nature of segregation, which leads to diverse co-\nsegregation behaviors that cannot be captured by the averaged approach. see [Scheiber et al.,\nScripta materialia 150 (2018)] and [Wagih et al., arXiv:2411.05303].”\nResponse: The reviewer brings two excellent points dealing with segregation anisotropy and site-\nspecific segregation. Here, it is important to note our proposed model is mesoscopic in nature;\nit treats the GB as a region in space with thermodynamic and kinetic properties that are distinct from\nthe bulk. Mesoscale models provide the key advantage of simulating microstructure evolution and\ninterfacial dynamics over diffusive length and time scales that cannot be achieved with pure\natomistics simulations. Yes, it is true that our modeling treatment does not capture the atomic\nnature of GB segregation and associated site-dependency. However, our goal here is to answer\nscientific questions different from the ones a researcher can address with atomistic simulations.\nHere, we argue that our approach is complementary to other computational techniques. For\nexample, one can account for site-specific segregation by using atomistic simulations to capture\nsegregation energy distributions in terms of atomic environments. Then, we can extend our\nmodeling framework by treating all our modeling parameters as distributions rather than single-\nvalued quantities. A key outcome from such an approach is that distributions of model parameters\nwill result in distributions of GB solute drag.\nThe second comment is about anisotropic GB segregation, this is certainly an important aspect.\nRecall that in our modeling treatment, a GB is defined as a region with distinct free energy model\nparameters (described by four model parameters) and kinetic coefficients (the full L Onsager\nij\nmatrix). In general, each GB geometry has its unique thermodynamic and kinetic model\nparameters. Now, a direct mapping between the GB geometric degrees of freedom and our\nthermodynamic free energy model parameters and Onsager transport coefficients is a major task\nand is certainly beyond the scope of this study. However, the results presented in our work\nindirectly provide a glimpse into anisotropy effects. For example, Figure 6 depicts how GB solute\ndrag values are sensitive to the boundary solute-solute interaction parameter, which is also\ndependent on GB geometry.\nTo address this comment, we made the following changes in the revised manuscript:\n• We added a detailed third paragraph and several references on Page 4 of the revised\nmanuscript to address this comment. First, we added sentences describing the mesoscale\nnature of our model and highlighting the GB geometric degrees of freedom. These\nsentences now read: “Our mesoscale model treats GBs as regions with distinct\n𝐠𝐛\nchemical free energies, parameterized using the energy of pure states 𝐆 and regular\n𝐢\n𝐠𝐛 𝐠𝐛\nsolution model parameters 𝛀 , and unique diffusion coefficients 𝐃 . Here, we note\n𝐢𝐣 𝐢𝐣\nthat the GB geometry is described by five macroscopic degrees of freedom [53], three\nfor misorientation and two for plane normal, and that each GB geometry has its\nunique thermodynamic and kinetic properties.”\n• In the same third paragraph on Page 4, we added a discussion how our model can be\nextended to account for anisotropic GB segregation and distributions of GB segregation\nenergy. The discussion now reads: “We also note that establishing GB structure-\nproperty linkages relating the GB's thermodynamic and transport properties to its\ngeometric degrees of freedom is one of the most active areas of research in GB physics.\nWhile this work does not consider anisotropic boundary segregation, it is\nstraightforward to include anisotropy effects in our modeling treatment by allowing\nthe model parameters describing the GB segregation energy and diffusion coefficients\nto be functions of the boundary geometry. In addition to the dependence of GB\nsegregation on boundary geometry [9,10,54], several studies revealed site-specific GB\nsegregation, in which the segregation energy depends on the local atomic environment\nwithin the GB [55-57]. This results in distributions for the GB segregation energy in\nterms of the atomic environment, even for the same GB geometry [58-60]. Accounting\nfor site-specific segregation for a given GB can be achieved in our treatment by\nallowing our model parameters to be distributions rather than single-valued\nquantities.”\n• In the last paragraph on Page 10 toward the end of our study, we added a few sentences to\ndiscuss avenues to extend our model in future work to account for GB site-specific\nsegregation. These sentences now read: “[…] It is also interesting to extend our model\nto account for site-specific segregation, in which the segregation energy is given by a\ndistribution due to the local atomic environment of the GB. This can be incorporated\nin our treatment by allowing the model parameters describing the GB segregation\nenergy to be distributions rather than single-valued quantities.”\nComment #4: “Can the current model capture experimentally observed abnormal grain\ngrowth?”\nResponse: We thank the reviewer for the great question. Abnormal grain growth has been the\nsubject of active research over many years. Numerous studies revealed many mechanisms\nresponsible for abnormal grain growth including anisotropy in GB energy and/or GB mobility,\ninitial grain size advantage, pinning by second-phase particles, etc. From a modeling perspective,\ncapturing abnormal grain growth requires simulating 3D systems with large number of grains in\norder to quantitatively predict the onset of abnormal grain growth and the correct statistics (grain\nsize distributions, GB network topology, etc.) associated with this dynamical process. Very\nrecently, a study revealed that GB segregation could lead to abnormal grain growth in dilute binary\nallows [see: https://doi.org/10.1016/j.matchar.2018.09.047]. While the impact of multi-element\nGB segregation on abnormal grain growth is an interesting effect to study, it is certainly beyond\nthe scope of this work, which mainly focuses on quantifying solute drag-velocity profiles in such\nalloys. Future work extending the model to 2D/3D can certainly be used to investigate whether\nmulti-element GB segregation could lead to abnormal grain growth. To address this comment, we\nmade the following changes in the revised manuscript:\n• In the third paragraph on Page 1, we added a few sentences and references to describe GB\nsegregation as a possible mechanism for abnormal grain growth. These sentences now read:\n“[…] Furthermore, in some alloys GB segregation has been found to lead to abnormal\ngrain growth, in which a small subset of grains grows at a faster rate compared to the\nsurrounding grains [28,29]. […]”\n• In the last paragraph on Page 10, we added a detailed discussion describing future\nextensions of the model. Specifically, we added a few sentences on the interplay of GB\nsegregation and abnormal grain growth. These sentences now read: “[…] Such a 2D/3D\nextension of our modeling treatment will allow us to investigate the interplay between\nGB segregation and phenomena such as abnormal grain growth. […]”\nComment #5: “The provided code only solves for the equilibrium segregation isotherm (which is\nnot the main contribution of the paper). The authors should also include the code used to solve the\ndynamic solute drag equations, which would be more beneficial to the community. Given that this\nwork extends the authors' previous model [Physical Review Letters 127.17 (2021)] to\nmulticomponent alloys, and primarily focuses on a parametric study, it appears more appropriate,\nafter revision, for a specialized journal in computational materials science.”\nResponse: We thank the reviewer for the great remark. In the revised submission, we included the\nMATLAB code we used to solve the governing equations for GB segregation. The code includes\ndocumentation added as comments and is ready to run on any recent version of MATLAB. In the\nrevised manuscript, we made the following changes:\n1) A Code Availability Section has been added in Page 16 to indicate that the numerical codes\nare available with this manuscript. The Section now reads: “The Python script used to\nplot grain boundary segregation isotherms and the MATLAB code used to solve the\ngoverning equations for the evolution of concentration fields.”\nReviewer 3\nComment #0: “This work addresses an important topic in alloy design by providing valuable\ninsights into GB solute drag in multicomponent alloy systems. The study is well-motivated, and the\ntheoretical approach is appropriate for the investigation.”\nResponse: We thank the reviewer for the kind remark.\nComment #1: “Regarding solute segregation thermodynamics, the expressions for free energy\nand chemical potential in the ternary alloy system, based on the regular solution model, appear to\nbe a straightforward extension of those for a binary alloy system, as described in the authors’\nearlier works 1,2. The primary novelty of the current study seems to lie in the investigation of GB\nsolute drag in multicomponent alloys. Nevertheless, as noted in Comment 2, the effect of\nmulticomponent diffusion on GB solute drag remains insufficiently understood in the present\nwork.”\nResponse: The reviewer brings up an excellent point dealing with diffusion in multicomponent\nsystems. Most GB segregation studies focus on the energetics of GB segregation and very little\nattention has been on the GB diffusion part. This is especially important in multicomponent\nsystems due to the presence of several relevant transport coefficients including off-diagonal\ncomponents. When one explores multicomponent systems, the parameter space describing the\nenergetics and kinetics of these alloys expands rapidly as the number of components increases.\nThe first step in our work aimed to develop a theoretical model that will allow researchers to\nexplore all the thermodynamic and kinetic aspects of GB segregation in multicomponent systems.\nNext, we revealed a plethora of segregation effects that arise in such systems, including the impact\nof various diffusion coefficients [Figure 3], bulk concentration effects [Figure 4], the role of\ndesegregation in solute drag [Figure 5], and the impact of solute-solute interactions [Fig. 6]. With\nrespect to multicomponent diffusion effects, we want to direct the attention of the reviewer to\nFigure 3 which clearly demonstrates that a quantitative description of GB solute drag needs to not\nonly account for segregation energetics of each species but also the diffusion of solutes which can\nvary by orders of magnitude. Specifically, we investigated solute diffusion effects in two classes\nof ternary alloys; one where both solutes segregate to the GB and second alloy where one solute\ntype segregates and the other de-segregates.\nTo further highlight the impact of multicomponent diffusion and based on the recommendation of\nthe reviewer, we included a new figure [Figure 7] in the revised manuscript to demonstrate the role\nof off-diagonal diffusion coefficients in solute drag; this is an effect that has not been explored\nbefore. In the revised manuscript, we made the following changes:\n• We included a new figure, Figure 7, to demonstrate the impact of off-diagonal diffusion\ncoefficients. Taken together, Figures 3 and 7 now illustrate the impact of all diffusion\ncoefficients on solute drag.\n• We included a caption for the new Figure 7 which reads: “The impact of off-diagonal\n+++++ +++++\n𝐠𝐛 𝐠𝐛\ndiffusion coefficients 𝐃 and 𝐃 on GB solute drag. For the ternary alloy depicted\n𝐁𝐂 𝐂𝐁\n+++++\nin Fig. 6(a) with 𝛀 𝐠𝐛 = 1.5: Contour plots of (a) peak solute drag +𝐏++∗ and (b) velocity\n𝐁𝐂\n+++++ +++++\n𝓥+++∗ at peak point as a function of 𝐃 𝐠𝐛 and 𝐃 𝐠𝐛 . (c) At a velocity 𝓥& = 0.1, GB\n𝐁𝐂 𝐂𝐁\n+++++ +++++ +++++\n𝐠𝐛 𝐠𝐛 𝐠𝐛\nconcentration profiles for two cases, one with 𝐃 = 𝐃 = 0 and a second with 𝐃 =\n𝐁𝐂 𝐂𝐁 𝐁𝐂\n+++++\n𝐠𝐛\n-1 and 𝐃 = 1.”\n𝐂𝐁\n• We added a new second paragraph on Page 10 to describe the results in Figure 7. The\nparagraph now reads: “All results presented so far do not consider the impact of off-\n+++++ +++++\n𝐠𝐛 𝐠𝐛\ndiagonal diffusion coefficients 𝐃 and 𝐃 on solute drag, which we investigate now.\n𝐁𝐂 𝐂𝐁\nTo this end, we employ the alloy with the model parameters depicted in Fig. 6(a) and\n+++++ +++++ +++++\n𝐠𝐛 𝐠𝐛 𝐠𝐛\nuse 𝛀 = 1.5, i.e., repulsive B-C interactions within the GB. We set 𝐃 = 𝐃 = 1.0\n𝐁𝐂 𝐁𝐁 𝐂𝐂\nand perform simulations systematically varying the off-diagonal diffusion coefficients\n+++++ +++++\n𝐠𝐛 𝐠𝐛\n𝐃 and 𝐃 . Recall that the diffusion coefficients can be expressed as 𝐃 =\n𝐁𝐂 𝐂𝐁 𝐢𝐣\n∑ 𝐋 𝛛𝛍 /𝛛𝐜 , where 𝐋 are the Onsager coefficients and 𝛛𝛍 /𝛛𝐜 are the\n𝐤 𝐢𝐤 𝐤 𝐣 𝐢𝐤 𝐤 𝐣\nthermodynamic factors, and that the values and algebraic sign of the off-diagonal\ndiffusion coefficients are influenced by both 𝐋 and 𝛛𝛍 /𝛛𝐜 . For example, a negative\n𝐢𝐤 𝐤 𝐣\n+++++\n𝐠𝐛\nvalue for 𝐃 indicates diffusion of solute B up the concentration gradient of solute\n𝐁𝐂\nC. Figure 7 shows results demonstrating the role of off-diagonal diffusion coefficients\nin GB solute drag, where the contour plots in Fig. 7(a) and (b) show respectively the\n+++++ +++++\ndependence of the peak solute drag 𝐏+++∗ and velocity +𝓥++∗ at peak point on 𝐃 𝐠𝐛 and 𝐃 𝐠𝐛 .\n𝐁𝐂 𝐂𝐁\nIt can be seen that solute drag is sensitively dependent on the magnitude and algebraic\nsign of the off-diagonal diffusion coefficients. For the model parameters chosen here\n+++++\n𝐠𝐛\nwhich favor the segregation of solute B over C, there exists a combination of 𝐃 and\n𝐁𝐂\n+++++ +++++ +++++\n𝐠𝐛 𝐠𝐛 𝐠𝐛\n𝐃 values that maximizes GB solute drag. In general, 𝐃 and 𝐃 do not have the\n𝐂𝐁 𝐁𝐂 𝐂𝐁\n+++++\nsame values, and Fig. 7(a) shows that solute drag 𝐏+++∗ decreases in regimes where 𝐃 𝐠𝐛\n𝐁𝐂\n+++++\nand 𝐃 𝐠𝐛 are dissimilar. Figure 7(b) reveals a general trend where the velocity 𝓥+++∗ at\n𝐂𝐁\n+++++ +++++\n𝐠𝐛 𝐠𝐛\npeak drag decreases with the increase in both 𝐃 and 𝐃 . A close examination of\n𝐁𝐂 𝐂𝐁\n+++++\npanels (a) and (b) of Fig. 7 reveals that 𝓥+++∗ is much more sensitive to variations in 𝐃 𝐠𝐛\n𝐁𝐂\n+++++\nand 𝐃 𝐠𝐛 than the peak solute drag 𝐏+++∗. To demonstrate how the concentration profiles\n𝐂𝐁\n+++++ +++++\n𝐠𝐛 𝐠𝐛\nare influenced by the off-diagonal diffusion coefficients 𝐃 and 𝐃 , Fig. 7(c) shows\n𝐁𝐂 𝐂𝐁\n+++++\n𝐠𝐛\nGB concentration line scans for solutes B and C in two cases, namely one with 𝐃 =\n𝐁𝐂\n+++++ +++++ +++++\n𝐠𝐛 𝐠𝐛 𝐠𝐛\n𝐃 = 0 and a second with 𝐃 = -1 and 𝐃 = 1. According to Eqs. 5 and 7, the use of\n𝐂𝐁 𝐁𝐂 𝐂𝐁\n+++++ +++++\n𝐠𝐛 𝐠𝐛\nnon-zero values for 𝐃 and 𝐃 results in coupling of the governing equations for\n𝐁𝐂 𝐂𝐁\nsolutes B and C, where the evolution of one solute type is also driven by the chemical\npotential gradients of the other solute types. This is seen in Fig. 7(c) where the GB\n+++++ +++++ +++++\n𝐠𝐛 𝐠𝐛 𝐠𝐛\nconcentration profiles are sensitive to the particular values of 𝐃 and 𝐃 . Using 𝐃\n𝐁𝐂 𝐂𝐁 𝐁𝐂\n+++++\n𝐠𝐛\n= -1 for this alloy with 𝛀 = 1.5 leads to diffusion of B up the concentration gradient\n𝐁𝐂\nof C while also maintaining repulsive B-C interactions within the GB region.”\nComment #2: “In the context of multicomponent diffusion, especially in far-from-dilute alloys,\nthe kinetic coupling between different solute fluxes is crucial, as it significantly influences the\nsolute concentration distribution near the GB and, consequently, the GB solute drag. However,\nthe manuscript neglects the cross-diffusion process between solute B and C within the GB by\nassuming D_BC^gb=D_CB^gb=0 in all calculations, meaning that the off-diagonal kinetic\ntransport coefficients in the Onsager matrix are set to zero. This assumption of ignoring kinetic\ncoupling between solute fluxes must be justified, and its impact on the results should be clarified.”\nResponse: We thank the reviewer for the excellent point. Most certainly, when dealing with\nmulticomponent alloys, one deals with a diffusion matrix. Following Onsager’s formalism, which\nis the starting point of our model, one can express the diffusion coefficients in terms of the Onsager\ncoefficients and the so-called thermodynamic factors as\n𝐷 = >𝐿 𝜕𝜇 /𝜕𝑐\n89 8: : 9\n:\nWhere 𝐿 are the Onsager transport coefficients and 𝜕𝜇 /𝜕𝑐 are the thermodynamic factors\n8: : 9\ncoupling chemical potentials to concentrations. As the reviewer indicated, the diffusion\ncoefficients are inherently dependent on alloy thermodynamics through the derivatives of the\nchemical potential with respect to concentrations (or equivalently the second derivatives of the\nfree energy). Here, we note that in general the chemical potential derivatives and Onsager transport\ncoefficients are concentration-dependent, and, as a result, the diffusion coefficients are also\nconcentration-dependent. However, the functional form of both of these are highly dependent on\nthe alloy of interest¾we do not have a universal expression of 𝐿 in terms of concentrations to\n8:\nmodel all possible alloys. 𝐿 are obtained for a specific alloy with a prescribed concentration using\n8:\natomistic simulations in conjunction with the generalized Einstein relation. We also note that for\nthe off-diagonal components, 𝐿 and 𝜕𝜇 /𝜕𝑐 can be negative.\n8: : 9\nThe reviewer is correct in pointing out that the results we presented considered alloys where we\nset the off-diagonal coefficients to zero, however, it is important to note that our modeling\nframework accounts for these off-diagonal coefficients. To address the reviewer’s comment, we\nincluded a new Figure 7, that probes the role of off-diagonal coefficients in solute drag and\nprovided a discussion related to this new figure. In the revised manuscript, we made the following\nchanges:\n• We added a discussion and references on Page 3 below Eq. 5 describing how the diffusion\ncoefficients are expressed in terms of the Onsager coefficients and thermodynamic factors.\nA few sentences are added to focus on the off-diagonal coefficients. The discussion now\nreads: “Equation 5 results from expressing the diffusion coefficients as 𝑫 =\n𝒊𝒋\n∑ 𝐋 𝛛𝛍 /𝛛𝐜 , where 𝐋 are the Onsager transport coefficients and 𝛛𝛍 /𝛛𝐜 are\n𝐤 𝐢𝐤 𝐤 𝐣 𝐢𝐤 𝐤 𝐣\ntermed the thermodynamic factors coupling chemical potentials to concentrations\n[42]. As for the off-diagonal diffusion coefficients, their values and algebraic sign are\ninfluenced by those of the Onsager coefficients and thermodynamic factors, both of\nwhich are in general concentration dependent [46,47]. Obtaining the thermodynamic\nfactors requires detailed knowledge of the energetics of the alloy of interest, while the\nOnsager coefficients can be obtained using techniques such as atomistic simulations\nin conjunction with the generalized Einstein relation [46].”\n• We included a new figure, Figure 7, to demonstrate the impact of off-diagonal diffusion\ncoefficients.\n• We included a caption for Figure 7 which reads: “The impact of off-diagonal diffusion\n+++++ +++++\n𝐠𝐛 𝐠𝐛\ncoefficients 𝐃 and 𝐃 on GB solute drag. For the ternary alloy depicted in Fig. 6(a)\n𝐁𝐂 𝐂𝐁\n+++++\nwith 𝛀 𝐠𝐛 = 1.5: Contour plots of (a) peak solute drag +𝐏++∗ and (b) velocity 𝓥+++∗ at peak\n𝐁𝐂\n+++++ +++++\npoint as a function of 𝐃 𝐠𝐛 and 𝐃 𝐠𝐛 . (c) At a velocity 𝓥& = 0.1, GB concentration profiles\n𝐁𝐂 𝐂𝐁\n+++++ +++++ +++++ +++++\n𝐠𝐛 𝐠𝐛 𝐠𝐛 𝐠𝐛\nfor two cases, one with 𝐃 = 𝐃 = 0 and a second with 𝐃 = -1 and 𝐃 = 1.”\n𝐁𝐂 𝐂𝐁 𝐁𝐂 𝐂𝐁\n• On Page 10, we added a new second paragraph describing the results in Figure 7. The\nparagraph now reads: “All results presented so far do not consider the impact of off-\n+++++ +++++\n𝐠𝐛 𝐠𝐛\ndiagonal diffusion coefficients 𝐃 and 𝐃 on solute drag, which we investigate now.\n𝐁𝐂 𝐂𝐁\nTo this end, we employ the alloy with the model parameters depicted in Fig. 6(a) and\n+++++ +++++ +++++\n𝐠𝐛 𝐠𝐛 𝐠𝐛\nuse 𝛀 = 1.5, i.e., repulsive B-C interactions within the GB. We set 𝐃 = 𝐃 = 1.0\n𝐁𝐂 𝐁𝐁 𝐂𝐂\nand perform simulations systematically varying the off-diagonal diffusion coefficients\n+++++ +++++\n𝐠𝐛 𝐠𝐛\n𝐃 and 𝐃 . Recall that the diffusion coefficients can be expressed as 𝐃 =\n𝐁𝐂 𝐂𝐁 𝐢𝐣\n∑ 𝐋 𝛛𝛍 /𝛛𝐜 , where 𝐋 are the Onsager coefficients and 𝛛𝛍 /𝛛𝐜 are the\n𝐤 𝐢𝐤 𝐤 𝐣 𝐢𝐤 𝐤 𝐣\nthermodynamic factors, and that the values and algebraic sign of the off-diagonal\ndiffusion coefficients are influenced by both 𝐋 and 𝛛𝛍 /𝛛𝐜 . For example, a negative\n𝐢𝐤 𝐤 𝐣\n+++++\n𝐠𝐛\nvalue for 𝐃 indicates diffusion of solute B up the concentration gradient of solute\n𝐁𝐂\nC. Figure 7 shows results demonstrating the role of off-diagonal diffusion coefficients\nin GB solute drag, where the contour plots in Fig. 7(a) and (b) show respectively the\n+++++ +++++\ndependence of the peak solute drag 𝐏+++∗ and velocity +𝓥++∗ at peak point on 𝐃 𝐠𝐛 and 𝐃 𝐠𝐛 .\n𝐁𝐂 𝐂𝐁\nIt can be seen that solute drag is sensitively dependent on the magnitude and algebraic\nsign of the off-diagonal diffusion coefficients. For the model parameters chosen here\n+++++\n𝐠𝐛\nwhich favor the segregation of solute B over C, there exists a combination of 𝐃 and\n𝐁𝐂\n+++++ +++++ +++++\n𝐠𝐛 𝐠𝐛 𝐠𝐛\n𝐃 values that maximizes GB solute drag. In general, 𝐃 and 𝐃 do not have the\n𝐂𝐁 𝐁𝐂 𝐂𝐁\n+++++\nsame values, and Fig. 7(a) shows that solute drag 𝐏+++∗ decreases in regimes where 𝐃 𝐠𝐛\n𝐁𝐂\n+++++\nand 𝐃 𝐠𝐛 are dissimilar. Figure 7(b) reveals a general trend where the velocity 𝓥+++∗ at\n𝐂𝐁\n+++++ +++++\n𝐠𝐛 𝐠𝐛\npeak drag decreases with the increase in both 𝐃 and 𝐃 . A close examination of\n𝐁𝐂 𝐂𝐁\n+++++\npanels (a) and (b) of Fig. 7 reveals that 𝓥+++∗ is much more sensitive to variations in 𝐃 𝐠𝐛\n𝐁𝐂\n+++++\nand 𝐃 𝐠𝐛 than the peak solute drag 𝐏+++∗. To demonstrate how the concentration profiles\n𝐂𝐁\n+++++ +++++\n𝐠𝐛 𝐠𝐛\nare influenced by the off-diagonal diffusion coefficients 𝐃 and 𝐃 , Fig. 7(c) shows\n𝐁𝐂 𝐂𝐁\n+++++\n𝐠𝐛\nGB concentration line scans for solutes B and C in two cases, namely one with 𝐃 =\n𝐁𝐂\n+++++ +++++ +++++\n𝐠𝐛 𝐠𝐛 𝐠𝐛\n𝐃 = 0 and a second with 𝐃 = -1 and 𝐃 = 1. According to Eqs. 5 and 7, the use of\n𝐂𝐁 𝐁𝐂 𝐂𝐁\n+++++ +++++\n𝐠𝐛 𝐠𝐛\nnon-zero values for 𝐃 and 𝐃 results in coupling of the governing equations for\n𝐁𝐂 𝐂𝐁\nsolutes B and C, where the evolution of one solute type is also driven by the chemical\npotential gradients of the other solute types. This is seen in Fig. 7(c) where the GB\n+++++ +++++ +++++\n𝐠𝐛 𝐠𝐛 𝐠𝐛\nconcentration profiles are sensitive to the particular values of 𝐃 and 𝐃 . Using 𝐃\n𝐁𝐂 𝐂𝐁 𝐁𝐂\n+++++\n𝐠𝐛\n= -1 for this alloy with 𝛀 = 1.5 leads to diffusion of B up the concentration gradient\n𝐁𝐂\nof C while also maintaining repulsive B-C interactions within the GB region.”\nComment #3: “Related to comment 2, the off-diagonal transport coefficients are closely\nassociated with solute-solute interactions and solute concentrations. Specifically, positive or\nnegative solute-solute interactions can normally lead to distinct solute diffusion behaviors, thereby\naffecting the GB solute drag effect. Therefore, solute cross-diffusion coefficients and solute-solute\ninteractions should not be considered independent.”\nResponse: We thank the reviewer for the excellent remark. Since the reviewer indicated that this\ncomment is related to comment 2 above, we addressed these two comments #2 and #3 together.\nPlease refer to our response to comment 2 above for details on the off-diagonal diffusion\ncoefficients.\nNow, due to the lack of data for Onsager coefficients 𝐿 and thermodynamic factors 𝜕𝜇 /𝜕𝑐 for\n8: : 9\nall possible ternary metallic alloys and the fact that our study does not target a specific alloy, but\nrather aims to examine solute drag trends in alloy design space, we let all D ’s be constants and\nij\nvaried them systematically. To address this comment, we made the following changes in the\nrevised manuscript:\n• We added a discussion and references on Page 3 below Eq. 5 describing how the diffusion\ncoefficients are expressed in terms of the Onsager coefficients and the thermodynamic\nfactors. A few sentences are added to focus on the off-diagonal coefficients. The discussion\nnow reads: “Equation 5 results from expressing the diffusion coefficients as 𝑫 =\n𝒊𝒋\n∑ 𝐋 𝛛𝛍 /𝛛𝐜 , where 𝐋 are the Onsager transport coefficients and 𝛛𝛍 /𝛛𝐜 are\n𝐤 𝐢𝐤 𝐤 𝐣 𝐢𝐤 𝐤 𝐣\ntermed the thermodynamic factors coupling chemical potentials to concentrations\n[42]. As for the off-diagonal diffusion coefficients, their values and algebraic sign are\ninfluenced by those of the Onsager coefficients and thermodynamic factors, both of\nwhich are in general concentration dependent [46,47]. Obtaining the thermodynamic\nfactors requires detailed knowledge of the energetics of the alloy of interest, while the\nOnsager coefficients can be obtained using techniques such as atomistic simulations\nin conjunction with the generalized Einstein relation [46].”\n• We added a few sentences on Page 3 below Eq. 5 to indicate that the aim of our work is\nnot to target a specific alloy, but rather explore solute drag trends in alloy design space.\nThese sentences now read: “[…] Obtaining the thermodynamic factors requires\ndetailed knowledge of the energetics of the alloy of interest, while the Onsager\ncoefficients can be obtained using techniques such as atomistic simulations in\nconjunction with the generalized Einstein relation [46]. Herein, we let the diffusion\ncoefficients be constants as this work does not target a specific alloy, but rather\nexamines solute drag trends in alloy design space. Furthermore, we assume that\ndiffusion in the bulk is frozen compared to that in GBs, which act as fast diffusion\npathways [48] […]”\n• As discussed in comment 2 above, we included a new Figure 7 to demonstrate the impact\nof off-diagonal diffusion coefficients on GB solute drag, and added on Page 10 a new\nparagraph describing the results in Figure 7. Please see response to comment 2 above for\ndetails.\nComment #4: “What is the temperature value used in the calculations? Generally, an increase in\ntemperature enhances solute diffusion while reducing solute segregation tendencies and solute\ninteractions. The authors should clarify the role of temperature in their model and discuss its effect\non GB solute drag in multicomponent alloys.”\nResponse: The reviewer brings an excellent point regarding the simulation temperature. We\nexpressed our model in non-dimensional form to reduce the number of free model parameters\n(dimensionality reduction) and to easily reveal trends in solute drag behavior. The approach of\nexpressing governing equations in non-dimensional form is common in many fields of science and\nengineering. More specifically, all energy quantities are expressed in terms of the thermal energy\nof the alloy given by 𝑘 𝑇. For example, the non-dimensional heat of mixing model parameter is\n<\ngiven by:\nΩ\n+Ω+++ = 89\n=> 𝑘 𝑇\n<\nAs a result, varying Ω++++ in our model indicates fixing an alloy type and its interactions Ω and\n=> 89\nvarying the temperature (the denominator in the above equation) or equivalently fixing the\ntemperature and changing the alloy type (changing solute interactions given by Ω while fixing\nstudy to real alloys will provide a better exposition of our modeling results and will help readers\nrelate the work to real alloys. To address this comment, we made the following changes in the\nmanuscript:\n• The sentences following Eq. 7 on page 4 have been re-written to clearly describe our non-\ndimensionalization procedure. These sentences now read: “The above transport\nequations are made non-dimensional using the GB half width 𝛅 and thermal energy\nk T as reference length and energy scales, respectively, and we employ a reference\nB\nGB velocity 𝓥 . This leads to the following non-dimensional quantities: &𝒙 = 𝒙/𝜹,\n𝒓𝒆𝒇\n𝑬+++ = 𝑬 /𝒌 𝑻, 𝛀+++ = 𝛀 /𝐤 𝐓, +𝚫++𝐆+++ = 𝚫𝐆 /𝐤 𝐓, 𝛍& = 𝛍 /𝐤 𝐓, +𝐃+++ = 𝐃 /(𝓥 𝛅), and\n' 𝒊 𝑩 * 𝐢 𝐁 𝐀* 𝐀𝐢 𝐁 * 𝐢 𝐁 *. 𝐢𝐣 𝐫𝐞𝐟\nnon-dimensional GB velocity 𝓥& = 𝓥/𝓥 ”\n𝐫𝐞𝐟\n• On Page 4 and following Eq. 8, we added a discussion on how to convert our non-\ndimensional parameters to real metallic alloys values. Theses sentences now read: “[…] In\nthis work, all computational studies are reported using the non-dimensional\nparameters discussed above. These model parameters can be mapped onto real\nmetallic alloy values as follows: A representative GB width in a typical metallic alloy\nis 2𝛅= 1 nm and thermal energy k T ≈ 0.08 eV assuming an annealing temperature\nB\nin the range 800-1000 K, typical in metallic alloys [49]. At such temperatures,\nexperimentally measured GB velocities are in the range 𝓥 0.1-10 𝛍m s-1 [50] and\n𝐫𝐞𝐟\nrepresentative values for GB solute diffusion ≈10-15 m2s-1 [51,52].”\nComment #5: “It is widely reported that GB migration is associated with vacancy generation and\nemission, mediated by vacancy diffusion near the GB. How does the vacancy concentration\ndistribution affect solute diffusion in multicomponent alloys and, consequently, the GB solute\ndrag? The authors should clarify whether vacancy effects are considered in their model and\nmention their potential impact on the results.”\nResponse: We thank the reviewer for the great question. Indeed, several studies revealed a strong\ninterplay between migrating GBs and various types of defects including vacancies. For example,\nstudies revealed emission of vacancies or interaction with existing ones during GB migration, see\nfor example: https://doi.org/10.1023/A:1008653704896, https://doi.org/10.1146/annurev-matsci-\n080921-091511, https://doi.org/10.1016/j.actamat.2023.119261. Such complex interactions are\ncertainly beyond the scope of this work, as it requires detailed knowledge about the energetics of\nGB-vacancy interactions. However, our modeling framework can be extended to account for such\neffects by: (1) treating vacancies as a species and describing them with a concentration field. So\nan alloy with “n” components is treated as one with “n+1” components to account for vacancies.\nThen, GB-vacancy energetics need to be incorporated by treating the GB as a source/sink of\nvacancy with a sink strength describing the time scale associated with their emission and/or\nannihilation. To address this comment, we made the following changes in the revised manuscript:\n• In the paragraph following Eq. 4 on Page 3, we added a discussion on how our model can\nbe extended to account for GB-vacancy interactions. These new sentences now read: “[…]\nWhile we do not consider GB-vacancy interactions during boundary migration\n[44,45], such interactions can be incorporated in our modeling framework by treating\nGBs as sources/sinks of vacancies with a prescribed interaction energy, similar to Eq.\n4, described using a vacancy concentration field. […]”\n• In the first paragraph on Page 11, we added several sentences and relevant references on\navenues to extend our modeling framework to account for GB-vacancy interactions. These\nsentences now read: “Other interesting effects to investigate in future work include the\ninterplay between GB segregation and phase separation, driven by repulsive solute-\nsolute interactions, and the interaction of GBs with vacancies as several studies\nrevealed vacancy generation from and emission at migrating boundaries [44,45]. Our\nmodel can be extended to account for GB-vacancy effects by treating GBs as\nsources/sinks of such points defects with prescribed interaction energies.”\nComment #6: “Regarding the results in Figure 5, the authors state that solute B preferentially\ndesegregates from the GB in Alloy 2. Nevertheless, the values of Ω_AB^b=0 and ΔG_AB=0.1\nshown in Figure 5(a) suggest a tendency for solute B to segregate at the GB, which appears\ninconsistent with the stated conclusion.”\nResponse: We thank the reviewer for the excellent observation. The main goal of Figure 5 in our\nmanuscript is to demonstrate solute drag in alloys with contrasting thermodynamics and\nsegregation characteristics. As for Alloy 2, we set the dilute-limit segregation energy Δ𝐺̅ =\n?<\nΔ𝐺̅ = 0.1 indicating that both solutes B and C have the same segregation energy. Furthermore,\n?<\nfor this alloy we assumed ideal GBs with\nΩ&AB\n=\nΩ&AB\n=\nΩ&AB\n=\n0 (i.e., no A-B, A-C, and B-C\n?@ ?< <@\ninteractions in the GB). We also assumed that Alloy 2 is characterized by Ω&B = Ω&B = 0 and\n<@ ?<\nΩ&B = 1.5, which models zero bulk interaction energy for A-B and B-C pairs and repulsive\n?@\ninteractions in the bulk for the A-C pair. Using these parameters, both B and C solutes have the\ndilute-limit segregation energy given by Δ𝐺̅ = Δ𝐺̅ = 0.1, however, the strong repulsive A-C\n?< ?<\nbulk interaction introduces an additional component to the driving force for the segregation of\nsolute C (i.e., C wants to leave bulk and segregate to the GB which has no energy penalty for A-C\ninteractions). This additional component to the segregation energy due to repulsive A-C\ninteractions in the bulk (Ω&B = 1.5) is much stronger than the small dilute-limit segregation energy\n?@\nof solute B, and, as a result, C dominates the segregation behavior and results in an induced de-\nsegregation of B. The case of Alloy 2 is one specific example demonstrating the competing\nenergetic effects that result when one accounts for all possible solute-solute interactions in a\nmulticomponent alloy. To address this comment, we made the following changes in the revised\nmanuscript:\n• At the top of Page 9, the paragraph describing the alloys used in Figure 5 has been re-\nwritten to give a more intuitive description of Alloy 1, Alloy 2, and Alloy 3 and their\nsegregation behavior. In the revised manuscript, the paragraph reads: “[…] Figure 5(a)\n+++++ +++++ +++++\nshows values of the bulk heat of mixing parameters, 𝛀𝐛 , 𝛀𝐛 , and 𝛀𝐛 , used for these\n𝐀𝐂 𝐀𝐁 𝐁𝐂\nalloys, while Fig. 5(b) shows the rest of the model parameters used in these systems.\n+++++ +++++ +++++\nWe set bulk concentrations 𝐜C = 𝐜C = 𝟎.𝟏 and let 𝐃 𝐠𝐛 = 𝐃 𝐠𝐛 = 𝟏.𝟎 and 𝐃 𝐠𝐛 =\n𝐁 𝐂 𝐁𝐁 𝐂𝐂 𝐁𝐂\n+++++\n𝐠𝐛\n𝐃 = 𝟎.𝟎. For Alloy1, the bulk regular solution parameters for A-B and A-C pairs\n𝐂𝐁\n+++++ +++++\nare 𝛀𝐛 = 𝛀𝐛 = 𝟏.𝟓, which results in both solutes B and C having the same\n𝐀𝐁 𝐀𝐂\n+++++ +++++ +++++\nsegregation energy. For Alloy 2, we set 𝛀𝐛 = 𝛀𝐛 = 𝟎 and 𝛀𝐛 = 𝟏.𝟓, which models\n𝐀𝐁 𝐁𝐂 𝐀𝐂\nzero bulk interaction energy for A-B and B-C pairs and repulsive interactions in the\nbulk for A-C. According to Eq. 4, the additional component to GB segregation energy\ndue to repulsive bulk A-C interactions leads to a competitive segregation effect in\nwhich the stronger segregation of solute C results in an induced desegregation of B.\n+++++\nFor Alloy 3, the bulk regular solution parameters for the A-B, A-C, and B-C are 𝛀𝐛 =\n𝐀𝐁\n+++++ +++++\n𝛀𝐛 = 𝛀𝐛 = -1.5 favoring bulk alloy mixing which result in desegregation of solutes\n𝐀𝐂 𝐁𝐂\nB and C from the GB.”\nMinor Changes\n2) On the third paragraph on Page 4 above Eq. 9, we added the phrase “see Fig. 1(b)” to refer\nthe reader to the graphic representation of the parallel tangent construction in Fig. 1(b).\n3) The sentence reading: “Recent analytical thermodynamic treatments investigated the\nimpact of multi-element segregation on GB energy [16]” has been moved from the first\nparagraph on Page 2 to the second paragraph on Page 1 for a better flow of the literature\nreview.\n4) In the last paragraph in the Introduction section on Page 1, we added “[…] and operative\nmass transport processes.” when discussing the major highlights of our work.\n5) A Code Availability Section has been added to indicate that the numerical codes are\navailable with this manuscript. The Section now reads: “The Python script used to\nplot grain boundary segregation isotherms and the MATLAB code used to solve the\ngoverning equations for the evolution of concentration fields.”\n6) The Acknowledgments section has been modified to include support from DOE, as the\nwork on the off-diagonal diffusion coefficients was supported by a DOE grant. In the\nrevised manuscript, the statement now reads: “F.A. and M.T. acknowledge support from\nthe National Science Foundation (NSF) DMR-2343682 and the U.S. Department of\nEnergy, Office of Basic Energy Sciences, Materials Science and Engineering Division\nunder Award No. DE-SC0022980.”\n7) In the Data Availability section, we added a sentence on the availability of the data used to\ngenerate Figures 2 through 7. The statement now reads: “The authors declare that the\nsource data for Figures 2 through 7 are provided with the paper.”\nFormatting Changes\n1) In accordance with the author guidelines, we added a short, standalone title for each figure.\nThese now read:\n- Figure 1: “Mesoscale modeling of GB segregation and solute drag in\nmulticomponent alloys.”\n- Figure 2: “GB segregation isotherms in ternary alloys.”\n+++++ +++++\n𝐠𝐛 𝐠𝐛\n- Figure 3: “The impact of diffusion coefficients, 𝐃 and 𝐃 , on GB solute\n𝐁𝐁 𝐂𝐂\ndrag.”\n- Figure 4: “The dependence of GB solute drag on bulk concentrations 𝐜C\n𝐁\nand 𝐜C.”\n𝐂\n- Figure 5: “GB solute drag in ternary alloys with contrasting segregation\ncharacteristics.”\n+++++\n𝐠𝐛\n- Figure 6: “The impact of GB solute-solute interaction parameter 𝛀 on\n𝐁𝐂\nboundary solute drag.”\n+++++ +++++\n𝐠𝐛 𝐠𝐛\n- Figure 7: “The impact of off-diagonal diffusion coefficients 𝐃 and 𝐃 on\n𝐁𝐂 𝐂𝐁\nGB solute drag.”\n2) Following the Nature Communications author guidelines, we added the phrase “In this\nwork” on the fourth line of the abstract.\n3) We added an Excel sheet that has all the data used to generate the figures that support our\nfindings. We included the statement in the Data Availability Statement to reflect this. The\nData Availability Statement now reads: “The authors declare that the source data for\nFigures 2 through 7 are provided with the paper.”\nWe would like to thank the reviewers for their comments. The time spent by the reviewers\nin providing feedback on our manuscript is very much appreciated. We have taken these\ncomments very seriously and addressed them in a revised manuscript. In what follows,\nwe address the comments/suggestions made by all reviewers. For each reviewer, each\ncomment is taken verbatim and shown below in blue font and a response directly follows.\nAll changes in the manuscript are highlighted in blue font.\nREVIEWER #1\nComment #1: “The authors have addressed my comments and suggestions. Regarding\nmy previous comment about the grain boundary miscibility gap, which the authors\nneglect, I would like to point to the recent direct observations of phase separation within\ngrain boundary segregation atmospheres by atomistic computer simulations (O'Brien et\nal., DOI 10.1007/s10853-017-1706-1). I think it would be appropriate to cite this paper.\nAfter the authors consider this suggestion, I recommend the paper for publication.”\nResponse: We thank the reviewer for recommending our manuscript for publication in\nNature Communications. We agree with the reviewer that the atomistic simulations work\nby O'Brien et al. (DOI 10.1007/s10853-017-1706-1) is relevant and should be included.\nIn the revised manuscript, we made the following change:\n§ In the second paragraph of the Introduction Section on page 1, we added the\nreference to the sentence discussing segregation-induced GB transformation. In\nthe revised manuscript, it reads: “[…] segregation-induced GB transformations\n[12-16]”. The study by O'Brien et al. is reference 12.\nComment #0: “I appreciate the authors’ thoughtful and detailed responses, and I\nrecognize the effort that went into the manuscript and revisions. And, while I appreciate\nthe direction of the work, I remain unconvinced of the manuscript’s overall impact and\nnovelty, which I outline below in the spirit of constructive feedback”\nResponse: We thank the reviewer for the feedback. Below, we address in detail all\ncomments raised by the reviewer and make the necessary revisions to highlight the\nnovelty of our work.\nComment #1: “While the revised manuscript now acknowledges the classical Guttmann\nmodel, the presentation of the segregation modes still gives the impression of novelty.\nFor example, the abstract states: “We show that solute-solute interactions of various\nalloying element types result in a plethora of GB segregation behaviors including\nsynergistic co-segregation and induced desegregation.” And similarly, the conclusion\nnotes: “Simulation results revealed that solute interactions of various element pairs lead\nto a plethora of GB concentration profiles and segregation behaviors including synergistic\ncosegregation and induced desegregation.” However, both of these modes are well\nestablished in the segregation community and were already described in Guttmann’s\noriginal 1975 paper. Although the authors demonstrate in Fig. 2 that such modes can\narise due to solute interactions in the bulk, this observation is made using highly simplified\nsystems—zero interaction at the grain boundaries and very strong bulk interactions—\nwhich makes it difficult to generalize to real alloys. As such, the claim of uncovering novel\nsegregation behaviors is overstated.”\nResponse: We thank the reviewer for this comment. In our view, this comment is not a\ntechnical one but rather one dealing with our phrasing of a few sentences that are related\nto a small subset of our results. Here, it is important to highlight that our study has a much\nlarger scope that Guttmann’s 1975 study. First, the work by Guttmann only considered\nstationary GBs; it never dealt with the question of migration of doped GBs and solute\ndrag. Second, Guttmann’s work provides the basic skeleton of a segregation model to\nrecover an isotherm, however, the work does not provide any figures and accompanying\ndiscussion of results, there are no figures in Guttmann’s 1975 published article! Our\nresults in Figure 2 are meant to show the complexity of segregation behavior once one\nconsiders solute-solute interactions. Figure 2 is also meant to recover a segregation\nisotherm, which is a common analysis in the field of interaction segregation. Also, recall\nthat the solution interaction parameters are expressed in our work in nondimensional form\nusing the thermal energy k T. Figure 2 uses values for the solute interaction parameter\nB\nthat do not exceed 1.5, which means 50% larger than the thermal energy, and this is in\nfact a realistic number common in many alloys. Please see our response to comment #3\nbelow for actual values of solute interaction parameters.\nGuttmann’s 1975 work dealt only with isotherms. We do, however, acknowledge the\nreviewer’s point that we should clearly state that these complex segregation profiles play\nan important role in solute drag and not only in isotherms. We strongly believe that this is\na novel contribution. For the sentences in the manuscript that the reviewer mentioned in\ntheir comment, we made changes to clearly state that it is solute drag that is highly\ninfluenced by these complex segregation profiles. To address this comment, we made\nthe following changes in the revised manuscript:\n§ The fifth sentence in the abstract which the reviewer brought up in their comment\nhas been re-worded to change the emphasis to GB solute drag. This change is\nmade to clearly state that our works deals with GB solute drag. The sentence now\nreads “[…] We show that solute-solute interactions of various alloying\nelement types resulting in GB segregation behaviors, such as synergistic\ncosegregation and induced desegregation, greatly influence the\ncharacteristics of GB solute drag […]”.\n§ Similarly, the fifth sentence in the Conclusion Section which the reviewer brought\nup in their comment has been re-worded to change the emphasis to GB solute\ndrag and not only GB segregation behavior. The sentence now reads “Simulation\nresults revealed that solute interactions of various elemental pairs resulting\nin segregation behaviors, such as synergistic cosegregation and induced\ndesegregation, greatly influence the characteristics of GB solute drag-\nvelocity profiles”.\nComment #2: “The model represents an improvement compared to previous extensions\nof Cahn’s model to multicomponent systems, and the formalism is well laid out. That said,\nit’s hard to evaluate its practical value without at least one validation case. The manuscript\ndoes not provide any benchmark comparisons—either to atomistic simulations or to\nexperimental measurements—that would demonstrate the model’s accuracy or predictive\npower, even qualitatively. Even a single illustrative validation case would go a long way\nin reinforcing the manuscript’s contributions.”\nResponse: We thank the reviewer for this important comment. In our view, the most\nrigorous approach to parametrize and validate our model to obtain chemical free energy\nexpressions for both the bulk phase and the GB. Now, free energy expressions for bulk\nalloys can be obtained from experimental and/or computed phase diagrams. Indeed,\nmany thermodynamic data bases of alloys provide chemical free energies as a function\nof composition. Obtaining the chemical free energy of GBs, however, is a much more\ninvolved process. Computationally, GB free energy as a function of composition can be\nobtained using thermodynamic integration in conjunction with the Gibbs absorption\nequation. This has been recently obtained for a small subset of alloys. To address the\nreviewer’s comment, we searched the published literature for ternary alloys that were\nexperimentally characterized to show GB segregation and whose thermodynamic\nproperties are established. We considered the PtAuPd alloy as one representative\nexample and included a new figure in the revised manuscript to show a comparison of\nline scans. To address this comment, we made the following changes in the revised\nmanuscript:\n§ To address the reviewer’s comment about validation, we included a new\nsubsection in the manuscript tilted “Model Parametrization and Validation” on\npage 12. In this subsection, we included a new Figure 8, also on page 12, that\nprovides a comparison between our model and a recent experimental study of GB\nsegregation in a PtAuPd alloy. The experimental panel (b) of Figure 8 shows EDS\nGB line scans which was taken from a study by Xing et al.\n[https://doi.org/10.1016/j.actamat.2018.09.005].\n§ The caption of Figure 8 reads: “For the Pt Au Pd (in at.%) alloy: (a) Our\n0.76 0.18 0.06\nmodeling results depicting GB concentration profiles. The simulations\nemployed reported bulk heat of mixing model parameters for this alloy [44],\nwhich are listed in Table 1, and assumed ideal GBs. (b) For an experimental\nsample annealed at 800oC for 24 hrs, EDS line scans revealing Au\nsegregation to the GB with no tendency for Pd segregation. Panel (b) is\nreprinted with permission from Ref. [44].”\n§ A new paragraph is added at the end of page 12 to describe the modeling-\nexperiment comparison shown in Figure 8. The new paragraph reads: “As a\ndemonstration of how our modeling treatment can be parametrized using\nexisting alloy thermodynamic data and cross-validated with experimental\nstudies, we consider the PtAuPd ternary alloy in which recent experiments\nrevealed GB segregation [44]. We simulate GB segregation in\nPt Au Pd (in at.\\%) using reported bulk heat of mixing model\n0.76 0.18 0.06\nparameters [44] shown in Table 1 while assuming ideal GBs, and the results\nare shown in Fig. 8. Our modeling results depicting concentration profiles\nacross a stationary GB are shown in Fig. 8(a), where segregation of Au is\nobserved accompanied by depletion of Pt while the Pd concentration profile\ndoes not show any significant enrichment at the GB. Xing et al. [44]\nexperimentally investigated GB segregation in the PtAuPd alloy with various\ncompositions and temperatures. For a Pt Au Pd alloy annealed at\n0.76 0.18 0.06\n800oC for 24 hrs, Fig. 8(b) shows chemical line scans across a GB using\nenergy-dispersive x-ray spectroscopy (EDS) revealing Au segregation to the\nPt GB with no tendency for Pd segregation. Figure 8 reveals that our\nmodeling results [Fig. 8(a)] are in qualitative agreement with experimental\nEDS line scans [Fig. 8(b)] in terms of segregation profiles. This approach of\ncomparing GB concentration line scans from our model with those obtained\nexperimentally provides avenues to validate the predictions that emerge\nfrom our theoretical and computational model.”\nComment #3: “Alternatively, another way to increase the manuscript’s impact could be\nto provide a library of fitted parameters (e.g., for a range of transition metal alloys). This\nwould allow others to apply the model more easily and could position the work as a useful\nstarting point for further studies on solute drag in complex alloys. As it stands, its utility\nbeyond the specific toy examples shown is unclear.”\nResponse: We thank the reviewer for the excellent suggestion. Indeed, adding\nparameter sets that are obtained experimentally or through modeling to our manuscript\nclearly demonstrate how our proposed model can be parametrized using existing\ndatasets. To address this comment, we searched the published literature for recent\nstudies of GB segregation in ternary alloys along with their thermodynamic datasets. We\nfound several examples that we now include in the revised manuscript as follows:\n§ First, we included a new subsection in the manuscript on page 12 titled “Model\nParametrization and Validation.” The purpose of this subsection is to\ndemonstrate how our model can be parametrized using published datasets in the\nliterature and/or existing techniques.\n§ We included a new Table 1 on page 13 listing several ternary alloys that were\nexperimentally shown to have GB segregation along with their thermodynamic\nproperties, i.e., regular solution model parameters, from published references.\n§ The first paragraph under subsection “Model Parametrization and Validation”\non page 12 is introduced to demonstrate how our model can be parametrized and\nvalidated using existing experimental and modeling techniques. Used as\nrepresentative examples, several published cases of ternary alloys are listed and\ndiscussed in this paragraph. This new paragraph now reads: “The results\ndepicted in Figs. 3 through 6 demonstrate that a quantitative understanding\nof the migration kinetics of doped GBs in multicomponent alloys should take\ninto account details of solute-solute interactions. Our modeling treatment\nexplicitly accounts for both the bulk and GB free energies, parametrized\nusing regular solution thermodynamics. The bulk regular solution model\nparameters can be obtained from existing thermodynamic databases [78].\nFor example, for the PtAuPd [44], FeNiZr [75], and CuAlY [77] ternary alloys,\nin which experimental observations revealed GB segregation, Table 1 lists\n\"\"\"\"\nreported values for the bulk regular solution model parameters Ω# of the\n!\"\nconstituent binary pairs [44, 79]. On the other hand, GB-solute interactions\nfor each solute type can be computed using atomistic simulations [60, 80].\nFor example, for the Cu-Ag alloy atomistic simulations employing\nthermodynamic integration in conjunction with the Gibbs adsorption\nequation were used to calculate the GB free energy as a function of Ag\nconcentration [81]. These atomistically-obtained GB free energies can then\nbe fit to regular solution free energy expressions to obtain the GB heat of\nmixing model parameters. Several studies reported values for GB regular\nsolution model parameters in various alloys, such as\n𝛀$𝒈𝒃\n= -4.6 in Al-Mg\n𝑨𝒍𝑴𝒈\nat 327oC [82,83] and 𝛀$𝒈𝒃 = 6.2 in Fe-Ni-P at 475oC [84].”\nComment #4: “More broadly, the proposed model—though moving in the right direction—\nstill falls short of capturing key features of real grain boundary behavior—such as site-\nspecific segregation, anisotropy in grain boundary mobility, and abnormal grain growth.\nMoreover, the model does not account for phenomena such as secondary phase\nformation, which are often critical in alloy design. Such limitations make it difficult to see\nhow this model provides a significant advancement over existing theoretical frameworks\nin terms of practical impact on alloy design. I recommend clarifying the framing of the\nmanuscript’s contributions and including some demonstration of utility—whether through\nvalidation or by providing practical parameter sets—to enhance its impact and practical\nrelevance to the field.”\nResponse: We thank the reviewer for the great remark comment. To some extent, this\ncomment re-iterates some of the points raised in comment #3 above. Also, in this\ncomment, the reviewer brings up effects related anisotropy in GB mobility, abnormal grain\ngrowth, and secondary phase formation. Yes, the holy grail in the fields of microstructure\nevolution and GB physics is to capture ALL effects relevant to GB dynamics, including\nmulti-component GB segregation, anisotropic effects due to the GB geometry, and\ninteraction of GB and secondary phases and other microstructural features. However,\ncapturing all these effects is a major undertaking and is certainly beyond the scope of this\nwork which mainly focuses on the largely unexplored effects of multi-element GB\nsegregation on GB solute drag. However, we agree with the reviewer that adding a stand-\nalone paragraph to better highlight model assumptions (framing of the work) would benefit\nthe readers of this work and enhance the impact of our study. In the revised manuscript,\nwe made the following changes:\n§ We added a new subsection titled “Model Assumptions” on page 4 and top of\npage 5 that clearly describes the key assumptions used when deriving this model\nand highlights future extensions aimed at addressing the points that the reviewer\nraised in this comment. The new paragraph now reads: “Our proposed modeling\ntreatment considers multi-element segregation and migration of planar GBs\nusing a one-dimensional bicrystal geometry. As a result, the relevant\ntransport coefficients are solute diffusion across the GB. A future extension\nof our model to 2D/3D geometries would enable the incorporation of a\ntensorial description of GB diffusion in which both diffusion along and\nacross the GB are incorporated. Indeed, atomistic simulations of Cu-Ag\nbinary alloys, for example, showed that accelerated lateral GB diffusion\nleads to nonuniform redistribution of solute atoms, which in turn influences\nGB solute drag [66]. Such a 2D/3D extension of our model will capture the\ndynamics of GB networks, thereby allowing us to investigate the interplay\nbetween GB segregation and phenomena such as abnormal grain growth.\nOur model treats GBs as regions with thermodynamic properties that are\ndistinct from the bulk, resulting in a segregation energy that is single-valued.\nIt is interesting to extend our treatment to account for site-specific GB\nsegregation, in which the segregation energy is given by a distribution due\nto the local atomic environment of the GB. This can be incorporated into our\ntreatment by allowing the model parameters that describe the GB\nsegregation energy to be distributions rather than single-valued quantities.\nOur work considers solid solution alloys in order to deconvolute solute drag\neffects due to multi-element GB segregation from GB pinning due to second-\nphase precipitates. It would be interesting to investigate in future work the\ninterplay between GB segregation and phase separation and their net impact\non GB migration kinetics. Finally, another extension of our modeling\nframework is to explicitly consider vacancies and their interactions with\nGBs, as several studies revealed vacancy generation and emission at\nmigrating boundaries [45,46]. This can be achieved by treating GBs as\nsources/sinks of such points defects with prescribed interaction energies.”\n§ Based on the reviewer’s suggestion, we provided practical parameter sets for\nseveral ternary alloys that have been experimentally demonstrated to show GB\nsegregation. These parameter sets serve to show that the models parameters\nused in our study are in the same order-of-magnitude of and consistent with\nrelevant metallic alloys. This parameter set is included in a new Table 1 on page\n13. The caption of Table 1 reads: “For several ternary alloys in which GB\nsegregation has been observed, reported values for the bulk regular solution\nmodel parameters of the constituent binary pairs.”\nComment #0: The authors have provided a detailed and largely satisfactory response to\nmy previous review, and I appreciate the effort made to clarify the assumptions and\nextend the discussion. The addition of Figure 7 is particularly valuable in demonstrating\nthe impact of cross-diffusion coefficients on solute distribution near the GB and,\nconsequently, on GB solute drag.\nResponse: We thank the reviewer for the kind remark.\nComment #1: “However, one key question remains unresolved. According to Eq. (10),\nthe diffusion coefficients D_ij are determined by both Onsager transport coefficients L_ij\nand thermodynamic factor (∂μ_i)/(∂c_j ). The diffusion potentials μ_i, in turn, are\ninfluenced by the interaction parameters Ω_ij in the alloy, as described by Eq. (3).\nTherefore, D_ij is inherently associated with the thermodynamic interactions among\ndifferent elements in the alloy. In this context, I question whether it is physically consistent\nto treat D_ij and Ω_ij as entirely independent input parameters, as appears to be done in\nthe current study. I would appreciate it if the authors could clarify this point and explain\nwhether and how such decoupling is justified within the framework of their model.”\nResponse: We thank the reviewer for this comment. Indeed, in diffusion science one can\nwrite the diffusion coefficients D_ij in terms of the Onsager coefficients L_ij and\nthermodynamic factors (∂μ_i)/(∂c_j). As a result, D_ij in principle would be concentration-\ndependent and also has information about alloy thermodynamics. Below, we bring up\nthree points to address this comment:\n1) While D_ij has the thermodynamic factor imbedded in it, a very common\napproach in many published studies is to de-convolute effects due to the\nthermodynamic driving forces from ones due to kinetics (i.e., kinetically\nhindered systems). This is achieved by letting the transport coefficients be\nindependent of the thermodynamic parameters, allowing one to examine how\nchemical potential gradients influence the physics of the problem. This is\ncertainly of interest because thermodynamic equilibrium is attained when these\ngradients vanish. Indeed, this approach of de-coupling kinetics from driving\nforces is common in many problems involving transport in multi-component\nsystems. Take for example, a study (DOI: 10.1016/j.jnucmat.2022.153614) on\nradiation-induced segregation using a reaction-diffusion modeling framework\nwhich assumed constant transport coefficients that are independent of the\nthermodynamic factors. Another study (DOI: 10.1016/j.actamat.2011.12.045)\non solidification in ternary alloys where diffusion coefficients of various species\nwere assumed to be constant.\n2) Within the framework of regular solution thermodynamics, adding the\nthermodynamic factor in the D_ij expression change the coefficients by only a\nmultiplicative constant. To clearly demonstrate this effect, we show the case in\n8\nbinary alloy for simplicity where c is the concentration and Ω is the solute\ninteraction parameter. The regular solution contribution to the free energy\n𝑔(𝑐) is given by\n𝑔(𝑐) = Ω𝑐(1−𝑐)\nand the resultant expressions for the chemical potential and thermodynamic\nfactor are given by\n𝜇 = Ω(1−2𝑐)\n𝜕𝜇\n= −2Ω\n𝜕𝑐\nSo, the diffusion coefficient 𝐷 is written as:\n𝜕𝜇\n𝐷 = 𝐿 = −2ΩL\n𝜕𝑐\nIt can be seen that adding the thermodynamic factor within the regular solution\nframework results in changing the diffusion coefficient by a constant\nmultiplicative factor, which can then be absorbed in L itself.\n3) The thermodynamic factor (∂μ_i)/(∂c_j) is linearly dependent on the regular\nsolution parameters, see bullet point above. However, it is important to note\nthat variations in the Onsager parameters L themselves are much larger than\nones due to the linearly dependent (∂μ_i)/(∂c_j) functions of solute interaction\nparameters. Many experimental studies of GB segregation are conducted at\nhigh homologous temperatures (to promote diffusion) resulting in larger values\nfor the Onsager parameters, which are thermally activated, compared to\nchanges in D due to the dependence on solute interaction parameter.\nTo address this comment, we made the following changes in the manuscript:\n§ In the section where we present the model formulation on page 3, we\nadded several sentences to describe how the Onsager parameters L_ij\nand thermodynamic factors (∂μ_i)/(∂c_j) are obtained and their\ndependencies. In the revised manuscript, we added those sentences in\nthe paragraph following Eq. 5 on page 3 which now read: “It is\nimportant to note that the Onsager transport coefficients L are in\nik\ngeneral concentration-dependent [47,48]. Obtaining the\nthermodynamic factors requires a detailed knowledge of the\nenergetics of the alloy of interest, while the Onsager coefficients\ncan be obtained using techniques such as atomistic simulations in\nconjunction with the generalized Einstein relation [47].”\n§ We added several sentences to justify why we considered constant D_ij\nwithout the thermodynamic factor. We also added sentences to state\nthat this is common in several studies in multicomponent systems and\n9\nprovided several examples. In the revised manuscript, we added these\nsentences in the paragraph at the bottom of page 3 and top of page 4\nwhich now read: “Such an assumption has been invoked in many\nstudies involving transport in multicomponent systems in order to\ndeconvolute effects due to thermodynamic driving forces from\nones related to transport coefficients [49-52]. For example, a study\non radiation-induced segregation employing a reaction-diffusion\nmodeling framework employed constant transport coefficients that\nare independent of the thermodynamic factors [51]. Another study\nexamined dendritic solidification in multicomponent alloys where\nthe diffusion coefficients of various species were assumed to be\nconstant [52].”\nWe obtained permission from Elsevier to include the experimental image shown in Figure\n8(b) of our manuscript. The permission form is included with the submission.\nmanuscript. We greatly appreciate the time and effort invested in providing these valuable\ncomments. We have carefully considered all suggestions and revised the manuscript\naccordingly. In the sections below, we address each reviewer’s comments in detail.\nReviewer comments are reproduced verbatim (in blue font), followed by our\ncorresponding responses. All revisions made to the manuscript are highlighted in blue\nfont.\nREVIEWER #1\nWe addressed all comments raised by reviewer1 in prior revisions. Reviewer1\nrecommended our work for publication in Nature Communications.\nREVIEWER #3\nWe addressed all comments raised by reviewer3 in prior revisions. Reviewer3\nrecommended our work for publication in Nature Communications.\nREVIEWER #2\nComment #1: “This revision reads more clearly; thank you for the improvements.”\nResponse: We thank the reviewer for the kind remark.\nComment #2: “I agree with the authors’ clarification that the focus of the manuscript is on"}]

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