The effect of selective surface interaction on polymer phase separation with explicit polydispersity during polymerization

Abstract

In polymerization-induced phase separation, the impact of polymer–substrate interaction on the dynamics of phase separation for polymer blends is important in determining the final morphology and properties of polymer materials as the surface can act as another driving force for phase separation other than polymerization. We modify the previously-developed polymerizing Cahn–Hilliard (pCH) method by adding a surface potential to model the phase separation behavior of a mixture of two species independently undergoing linear step-growth polymerization in the presence of a surface. In our approach, we explicitly model polydispersity by separately considering different molecular-weight components with their own respective diffusion constants, and with the surface potential preferentially acting on only one species. We first show that the surface potential induces faster phase separation of smaller molecules at early stages before the degree of polymerization becomes large enough to drive bulk phase separation. This model is then used to investigate the degree of anisotropic ordering in a direction perpendicular to the surface over various polymerization rates k̃ and strengths of the potential Ṽ. We find that at low k̃, smaller molecules have sufficient time to diffuse and accumulate at the potential surface, resulting in richer production of heavier polymers at the surface without the need for larger polymers to diffuse on their own toward the surface. Conversely, at high k̃, larger polymers first evenly accumulate throughout the system before undergoing phase separation; the concentration wave initiated from the potential surface then propagates into the bulk, resulting in anisotropic phase separation.

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1. Introduction

Surface-directed phase separation (SDPS) describes a phenomenon where a surface exhibits preferential attraction to one component over others in a mixture, thereby inducing phase separation. Consequently, SDPS results in the formation of a wetting layer of the favored component proximal to the surface, accompanied by the propagation of anisotropic concentration waves perpendicular to the surface, extending into the bulk.^1–7 SDPS and the effect of surfaces have been extensively studied in experiments^1,2,8–10 as well as in theoretical and computational research,^{4–7,10–20} due not only to the important role of surfaces in inducing unique phase separation behaviors and morphologies but also to their diverse range of applications spanning from food,²¹ surface-responsive materials,²² to photovoltaic devices.²³ The presence of a surface can significantly influence the phase separation behavior in polymeric systems, playing a crucial role in determining the morphology and properties of the resulting material. First, surfaces introduce different energetic interactions compared to the bulk polymer. The nature of these interactions, whether attractive or repulsive, can affect how polymers adsorb, arrange, and move on the surface. The compatibility between the polymer and the surface material can lead to preferential wetting of one component over another, influencing the phase morphology.^1–7,24 Second, the surface can confine the polymer blend to a limited dimension. This confinement can alter the kinetics and thermodynamics of phase separation, often leading to faster phase separation compared to bulk behavior.^25–27 In addition, while polymers in the bulk phase separate isotropically, the symmetry near a surface is broken, leading to anisotropic domain shapes and orientations.^5,6,28 Hence, it is important to understand molecular-level details of the SDPS process as it provides a means to precisely control phase separation pathways, final properties and structures of materials for targeted specific functions.

In comparison to SDPS, polymerization-induced phase separation (PIPS) describes the phenomenon where initially homogeneous components of a polymer blend or solution spontaneously segregate into separate regions or phases due to chemical disparities as molecular weight increases.^29,30 PIPS presents numerous advantages over conventional phase separation methodologies, notably in its capacity to meticulously regulate the morphology of resultant materials. This includes the attainment of cylindrical,^31,32 lamellar,^33–35 and globular structures,^36–38 each achievable through the evolving morphological processes. Such outcomes are dictated by the intricate interplay between polymerization kinetics and phase separation dynamics, thereby determining the final morphology of the system. PIPS manifests across diverse systems, encompassing polymeric blends, solutions, or colloidal suspensions, typically initiated through thermal^39–42 or photochemical means.^43–46 It finds widespread application in the fabrication of polymer-based materials, notably membranes,^47–49 functional coatings,^50,51 and composites (or adhesives).^52–55 In this sense, comprehending the underlying physics of PIPS and exercising control over this phenomenon are pivotal, given their significant implications across a broad spectrum of industrial and technological applications. Furthermore, combining PIPS and SDPS presents two different time scales of phase separation driven by two competing factors, polymerization and surface potential, for more sophisticated control of phase separating systems.

The classical model of phase separation dynamics in polymer blends integrates two key components: the thermodynamic driving force described by the Flory–Huggins⁵⁶ theory of polymer mixing, and the macromolecular transport described by the Cahn–Hilliard⁵⁷ formalism. The Flory–Huggins theory considers mean-field pairwise interactions between species, characterized by the Flory–Huggins χ parameter, which increases with the degree of immiscibility among different components.⁵⁶ In symmetric, binary polymer blends with a degree of polymerization N, the Flory–Huggins theory predicts phase separation above a critical point at χN = 2, resulting in the formation of two phases; it delineates the thermodynamic state and driving force underlying this phase separation phenomenon.⁵⁶ In contrast, the Cahn–Hilliard formalism captures the dynamics of phase separation in polymer systems at continuum length scales, offering insight into the non-equilibrium process leading to the phase-separated state. It is frequently applied to investigate phenomena like spinodal decomposition, material diffusion, and microstructure formation in a range of materials; alloys,^58–60 polymers,^61–66 and others.^67,68

The Cahn–Hilliard formalism is a form of the diffusion equation integrated with a free energy functional, such as the Flory–Huggins mixing free energy, to simulate the temporal evolution of component densities. In polymer systems, the combination of Flory–Huggins and Cahn–Hilliard models has been extensively utilized for modeling both binary^62–64 and ternary^65,66,69 melt blends, in addition to polymer solutions.^70–76 The Cahn–Hilliard approach offers a notable advantage in its ability to explore larger time and length scales compared to particle-based simulations. Moreover, it has been adapted and widely used to accommodate more complex systems, incorporating features such as reactions^{64,74,75,77–82} or coupling with other equations such as the Stokes equation.^62,83,84 The Cahn–Hilliard formalism has provided key insights into the phase separation dynamics of polymer blends such as quantifying the evolution of concentration fluctuations into larger inhomogeneities during the early stages of phase separation in initially homogeneous and isotropic systems.^85–88 It can also capture the late stage of phase separation driven by interfacial interactions including surface tension,^85,89 where its characteristic scaling of size growth of phase separated domains corresponds to t^1/3 (t is time).^{85,86,89–91}

The Cahn–Hilliard equation is a robust tool for modeling and comprehending phase separation phenomena, and remains an active research field given its broad applicability and adaptability to accommodate complex physical scenarios.^92,93 This versatility renders it particularly promising for modeling surface-directed phase separation (SDPS) and polymerization-induced phase separation (PIPS). In this study, we elucidate our approach of coupling a multicomponent Cahn–Hilliard model with surface potential which decays away from the surface during the process of linear step-growth polymerization involving two partially immiscible polymer species. By modeling the complete molecular weight distribution of both species, we explore the interplay among polymerization kinetics, surface potential and phase separation dynamics across all polymer lengths. This comprehensive approach explicitly accounts for the varying molecular mobilities and contributions to the mixing free energy inherent in different polymer lengths.

2. Methods

In this study, we modify our previously-developed polymerizing Cahn–Hilliard (pCH) model⁹⁴ so that it incorporates the effect of surface and the preferential adsorption of one of the polymer species. We investigate a binary system with two different types of species α = A,B, with a volume fraction ϕ_α(r,t) that is a function of space r and time t. For a given species α, this volume fraction can be further divided into components of different molecular weight ϕ_α,Nα(r,t) such that and there is a molecular weight distribution w_α(N_α,r,t) =ϕ_α,Nα(r,t)/ϕ_α(r,t). We assume that the monomers of both species α are bi-functional and polymerize in linear step-growth fashion with a constant reaction rate k_α independent of molecular-weight.⁹⁵ Our numerical simulation scheme evolves ϕ_α,Nα(r,t) over time and space through diffusion and polymerization from an initial state featuring a homogeneous mixture of monomers A and B(ϕ_A,1(r,0) = ϕ_B,1(r,0) = 0.5). All simulations are performed on a two-dimensional grid.

2.1. Multi-component Cahn–Hilliard equation

The Cahn–Hilliard equation is used to predict the time-evolution of the component concentration fields, ϕ_α,Nα(r,t) = ϕ_i(r,t), in the context of phase separation for immiscible material blends.⁵⁷ Based on the previously-developed model,⁹⁴ we write the Cahn–Hilliard equation in a non-dimensional form:

This equation is rendered non-dimensional via a reference mobility M₀, a reference molecular weight N₀, a reference distance L₀, and a reference timescale t₀ = L₀²/M₀. In eqn (1), the indices i, j and k refer to both the species α and the specific degree of polymerization N_α for explicitly treating the polydispersity of each species. The Onsager coefficient M̃_ij is non-dimensionalized by the reference mobility M₀ = k_BT/6πaN₀ζ; k_BT, a and ζ represent thermal energy, effective hydrodynamic radius, and monomer friction coefficient, respectively. We treat as simply M̃_ij(r̃,t̃) = ϕ_i(r̃,t̃)M_iδ_ij/M₀ = ϕ_iM̃_iδ_ij under the Rouse dynamics assumption that mobility is inversely proportional to molecular weight (i.e. ∼ 1/N_i). Finally, we introduce a shorthand denoting the entire right-hand side of the first equation as [scr A, script letter A]_CH(r̃,t̃), which we later use in describing our numerical procedure combining the Cahn–Hilliard formalism with reaction kinetics.

In our approach to modeling the thermodynamics of polymers, in addition to a surface potential Ṽ(r) we will discuss later, we consider a Flory–Huggins contribution to the homogeneous free energy density f̃_h(r̃) = f̃_FH(r̃) + Ṽ(r̃) at position r̃:⁹⁴

This expression is normalized by the thermal energy k_BT. N_A,max and N_B,max represent the maximum degrees of polymerization that are chosen for practical considerations, and beyond which polymers are assumed to diffuse similarly slowly, χ is the Flory–Huggins interaction parameter, and ξ is a compressibility parameter. We choose ξ = 100 as a suitably high value to ensure that , while permitting small local fluctuations in concentration to enhance the numerical stability of our methodology. This deviation from traditional methods of solving the Cahn–Hilliard equation, which typically maintain strict mass conservation by evaluating n − 1 transport equations against a reference component,^{71–73,80,88,96} renders our approach more numerically manageable. This adaptation of f̃_FH incorporates different degrees of polymerization N_A or N_B, treating them as distinct species characterized by their respective density fields, ϕ_A,NA and ϕ_B,NB, which independently contribute to the mixing entropy. Concurrently, the sum over fields A and B (i.e.) impacts both the enthalpic χ term, and the incompressibility constraint.⁹⁷

The gradient energy coefficient k̃ is treated as a constant based on the Random Phase Approximation^98–101 such that κ = b²/(36〈ϕ_A〉〈ϕ_B〉) where b denotes the Kuhn length, and we use the overall volume fractions of the species 〈ϕ_A〉 and 〈ϕ_B〉 for simplicity. This form of the square gradient term is notably approximate and independent of chain length; however, more sophisticated treatments would account for N_A and N_B that may be non-negligible for short chains and of interest for future studies.¹⁰² We do not expect those to be important for this compressible scheme, where gradients in one species do not directly imply gradients in the other species, but this may also be of interest for future studies. To keep the length scale consistent with the previous work,⁹⁴ we keep k̃ = 10 and a reference molecular weight of N₀ = 10 to define the length scale L₀ = C × R_G,0 as a fraction C of the radius of gyration of the reference chain. Therefore, the use a reference chain length to render our equations dimensionless only sets the interfacial length scale versus the grid discretization, and not the relationship between chains of different lengths. Similarly, off-diagonal contributions to the square gradient term (i.e. relating the gradient of species A to species B) have been shown to be important for incompressible models.¹⁰³ Thus, C depends on the choice of the reference chain length N₀ such that the relationship C = [6〈ϕ_A〉〈ϕ_B〉k̃N₀]^−1/2 holds. The temporal and spatial dimensions of our model are constrained by the requirement for an adequately large κ to ensure numerical stability. Considering the typical length scales (i.e. a ∼ 0.5 nm) and based on prior definitions, we observe that phase separation occurs on the order of ca. 10 μs in our system.

For real systems with longer polymers, the relevant time scales would naturally be extended, but our reaction rate constant k̃ is considerably faster than those typically observed in experimental settings.⁹⁵ We anticipate that the physical arguments presented in this paper should scale directly to experimentally relevant time scales.

2.2. Surface

The presence of a surface significantly impacts the behavior of phase separation in polymer systems, playing a vital role in shaping the morphology and characteristics of the final material.^1,2,4–20 In our scheme, we implement the Neumann boundary condition at x̃ = 1 and x̃ = Z, where x̃ denotes the x-coordinate of r̃, and Z represents the number of discretized grids along both x- and y-directions, and is set to 64. This approach ensures no flux of polymers across the boundaries, effectively creating impenetrable surfaces at both ends of the x-axis. Conversely, we employ periodic boundary conditions along the y-direction.

The Neumann condition at the x-boundaries can be expressed as:

where μ is the chemical potential.

This combination of boundary conditions allows for the simulation of a confined system in the x-direction while maintaining periodicity in the y-direction, providing a robust framework for studying polymer dynamics under the presence of a surface.

2.3. Surface potential

Surfaces that exhibit energetically favorable interactions with only one species, polymer A, can induce different phase behaviors compared to those observed in the bulk. The characteristics of these interactions, whether attractive or repulsive, influence the adsorption, arrangement, and mobility of polymers on the surface, which may result in the preferential wetting of one component over another, thereby impacting the overall phase separation dynamics.^1–7,24 For our simulation, we apply the surface potential Ṽ for only one surface at x̃ = θ_s = 1, and it exponentially decays away from the surface in the x-direction as follows:

Ṽ(x̃) = Ṽ₀exp(−γ(x̃ − θ_s)) (4)

Here, the surface potential Ṽ is normalized by the thermal energy k_BT. Ṽ₀ is the initial surface potential at the surface (x̃ = θ_s), and γ is the decaying strength of the potential. In this method, we choose γ = 0.2, for which Ṽ₀ decays to less than 1 percent of its original value at x̃ = 25. In other words, the decaying potential spans roughly two-fifths of the simulation box along the x-direction.

2.4. Polymerization kinetics

In our framework, polymers grow through linear step-growth polymerization, adhering to the rate laws for each chemical species α:

The terms depicted on the right-hand side of the equation denote generation and consumption, respectively. We again introduce the shorthand [scr A, script letter A]_P for this term, to be used to describe our numerical calculation. In eqn (5), k̃ = kt₀/(ṽ₀L₀³) represents the reaction rate, which is normalized using the reference timescale and length scales, and encompasses the dimensionless monomer volume ṽ₀. The conversion from volume fraction to number densities is achieved by dividing volume fractions of corresponding polymers by N_α and N′_α. In our scheme, we impose a restriction on the upper limit of the molecular weight distribution, N_α,max, which has only minor quantitative effects on our results. For the linear step-growth polymerization, we adhere to the standard assumption that k̃ remains constant throughout the simulation, independent of molecular weight.⁹⁵

The polymer concentration field evolves over time via both diffusion and local extent of polymerization. To account for both of these process, we consider a straightforward combination of both eqn (1) and (5), that sequentially evaluates the change in polymer density at each time t̃ and location r̃. We write our time-evolution scheme as:

ϕ_α,Nα(r̃,t̃ + Δt̃) = ϕ_α,Nα(r̃,t̃) + Δt̃[[scr A, script letter A]_CH(r̃,t̃) + [scr A, script letter A]_P(r̃,t̃)] (6)

In this case, the change in density fields based on the Cahn–Hilliard equation and the polymerization rate laws are independently added to give the overall change in field densities.

In this work, we comprehensively examine the full range of molecular weights, enabling us to explore the entire mass distribution function w_A(N_A,t̃) as a function of t̃. Furthermore, our model incorporates an assessment of the reaction kinetics associated with step-growth polymerization by monitoring various metrics, including both the number-averaged 〈N_A〉_n and weight-averaged 〈N_A〉_w degree of polymerization of the A-component, as well as the entire distribution function w_A(N_A,r,t). Finally, we utilize equal reaction rates for both species (i.e. k̃_A = k̃_B, such that the system is symmetric) and data for each set of parameters reported represents an average over 30 independent trajectories.

3 Results

With our pCH model, we study polymerization-induced phase separation under the effect of surface potential which decays away from the surface. We consider symmetric blends where both A and B species exhibit the same polymerization rate constant k̃, and the only difference is how they are affected by the surface potential Ṽ. This complex process is quantified and characterized by using the degree of phase separation ψ(t̃), degree of anisotropy E(r̃,t̃), overarching order parameter ω(x̃,t̃), and ordering transition point x̃_ω=0(t̃). First, ψ(t̃) is defined as:

The angle brackets indicate a spatial average. This expression reflects the difference in local volume fractions between two different components, averaging and normalizing over the entire space to yield a number between ψ(t̃) = 0 and 1. A fully homogeneous system holds the relationship 〈ϕ_A(t̃)ϕ_B(t̃)〉 = 〈ϕ_A(t̃)〉〈ϕ_B(t̃)〉 such that ψ(t̃) = 0. In a fully phase-separated system, the values of ψ(t̃) is nearly 1 since ϕ_A(t̃) or ϕ_B(t̃) is close to zero, making the second term vanish in eqn (7). Thus, a phase separation process will proceed from the ψ(t̃) value of 0 toward 1 over time; the endpoint is determined by the phase separation intensity and the interface width between the two bulk phases. In addition, we similarly define ψ_x(x̃,t̃):

ψ _x (x̃,t̃) is averaged over only the y-direction to measure the degree of phase separation in one slice at each fixed x̃ value, unlike the overall degree of phase separation ψ(t̃) which is averaged over all space. Fig. 1a exhibits phase separation behavior along the x-direction from one surface (x̃ = 1) to the other (x̃ = 60); the time advances from red (t̃ = 0) to purple (t̃ = τ_p), where τ_p is the phase separation time for the system to reach ψ(τ_p) = 0.4. This arbitrary choice for the definition of τ_p is a heuristic that we find useful for quantifying the overall kinetics of phase separation, though is not specific to the surface-driven processes that we will consider in more detail later in this article. Fig. 1a plots ψ_x(x̃,t̃) versus x̃ for several time points t̃, and shows how the phase separation evolves. To be more specific, near x̃ = 1, ψ_x(x̃,t̃) rises quickly because the surface potential attracts the preferred species type A and induces phase separation, whereas in the “bulk” (x̃ = 40), it phase separates based on polymerization rather than surface potential due to its exponential decay. The strength of phase separation ψ_x(x̃,t̃) is weaker at x̃ = 40 than x̃ = 1, which we attribute to the decay of the surface potential. At x̃ = 40, the system effectively reproduces the bulk phase separation behavior unaffected by the surface potential. The smooth fluctuation in ψ_x(x̃,t̃ = τ_p) near the surface potential (x̃ < 20) is due to the presence of alternating interfaces between lamellar-like stripes, where eqn (8) outputs quantities close to 0 for ψ_x(x̃,t̃) at interfaces (e.g. 〈ϕ_A(r̃,t̃)〉_y = 0.5) and are not considered “phase separated” with this metric. Eqn (8) thus does not distinguish a locally well-mixed regime from the interface between phase separated domains, because their 〈ϕ_A(r̃,t̃)〉 values are the same. Although ψ_x(x̃,t̃) is a useful parameter to measure the degree of phase separation, it does not quantify the anisotropic ordering behavior arising from the surface potential.

Fig. 1 Plots illustrating a representative phase separation process, with V₀ = 10⁻², χ = 0.8, and k̃ = 10⁻². All figures share the same x̃-axis as the simulation box; (a) to (c) use color scale from red (t̃ = 0) to purple (t̃ = τ_p), and the color scale in (d) represents ϕ_A from 0 (purple) to 1 (red). (a) The degree of phase separation ψ_x(x̃,t̃), where 0 and 1 indicate well-mixed and fully phase-separated states, respectively. It represents earlier phase separation near the surface potential (x̃ = 1), and late occurrence of phase separation around x̃ = 40, which we define as the “bulk” regime. (b) The degree of anisotropy E_x(x̃,t̃), which scales from 1 to −1 for anisotropic and isotropic structures, respectively. It shows that an initially isotropic system develops an anisotropic structure near the surface potential (x̃ = 1). (c) Order parameter ω(x̃,t̃) shows directional phase separation; 1, −1 and 0 denote anisotropic, isotropic and no phase separation, respectively. Anisotropic phase separation takes place near the surface potential (x̃ = 1) and isotropic phase separation in the bulk (x̃ = 40). (d) Snapshot of the concentration profile showing surface-directed spinodal decomposition at t̃ = τ_p, where phase separation with directionality is shown.

To measure the extent to which phase separation driven by the surface potential produces an anisotropic concentration wave perpendicular to the surface and quantify this anisotropic ordering behavior, we introduce a value that quantifies the degree of anisotropy E_x(x̃,t̃):¹⁰⁴

Here, and where u_x and u_y are the unit vectors in the x- and y-direction, respectively. E_x(x̃,t̃) ranges from −1 to 1 for isotropic and anisotropic ordering structures, respectively, along the direction perpendicular to the potential surface; it is averaged over y-direction similar to ψ_x(x̃,t̃). Fig. 1b illustrates E_x(x̃,t̃) for multiple representative simulation times t̃, depicting the progressive ordering behavior extending from the surface potential (x̃ = 1) into the bulk towards the other surface (x̃ = 1) over time. We find that there is a big jump from the initial time in Fig. 1b reflecting the weak initial ordering behavior induced by the surface potential at x̃ = 1 whereas the regime near the surface on the other side (x̃ = 60) does not show anisotropic ordering behavior. However, Fig. 1b alone does not provide any information about the intensity of phase separation. In order to convey the details of both order parameters ψ_x(x̃,t̃) and E_x(x̃,t̃), and quantify directional phase separation caused by the surface potential, we introduce an overarching order parameter ω(x̃,t̃) as follows:

ω(x̃,t̃) =E_x(x̃,t̃) ×ψ_x(x̃,t̃). (10)

ω(x̃,t̃) is introduced to capture both the phase separation and the anisotropic structure; ω(x̃,t̃) at −1, 0 and 1 indicates isotropic full phase separation, no phase separation, and anisotropic full phase separation, respectively. Fig. 1c shows significant anisotropic phase separation near the surface potential at x̃ ≈ 5 as indicated by an increasing ω(x̃,t̃), while isotropic phase separation in the bulk is found near x̃ = 40 as indicated by the negative growth of ω(x̃,t̃). This is consistent with the corresponding concentration profiles exhibiting the behavior of surface-directed spinodal decomposition near the surface potential (x̃ = 1) shown in Fig. 1d. We note that a surface excess of A could also lead to a positive value of ω(x̃,t̃), though all χ ≠ 0 cases considered in this manuscript are at values of χN that quickly exceed the critical (χN)_crit = 2.0 where symmetric binary blends will spontaneously phase separate. This approach allows us to consider the interplay between surface potential and polymerization, which gets more complex as phase separation is competitively driven by both the surface potential and polymerization. ω(x̃,t̃) = 0 near the potential surface also allows us to characterize the ordering transition point, x̃_ω=0(t̃), at which bulk (isotropic) and lamellar-like (anisotropic) phase separation regimes intersect (x̃ ≈ 15) as shown in Fig. 1c and d. Measuring x̃_ω=0(t̃) provides insight into the effective potential range at which the concentration wave, induced by the potential surface, predominantly decays. This measurement also offers information about the isotropic–anisotropic transition in the system as shown in Fig. 1d.

The consideration of quantities ψ_x(x̃,t̃), E_x(x̃,t̃), ω(x̃,t̃), and x̃_ω=0(t̃) enables us to elucidate the interconnected roles of the Flory–Huggins interaction parameter, the reaction rate, and the surface potential in determining their impacts on the phase separation behavior under polymerization and selective surface.

3.1. Effect of surface potential strength Ṽ₀

We first consider how the surface potential strength affects the degree of phase separation ψ(t̃), and the effective potential range x̃_ω=0(t̃) over time. Here, we keep other parameters such as χ and k̃ the same to isolate the effect of the potential strength.

Starting with the effect of the potential strength on phase separation time, we plot the time-evolution of ψ(t̃) for various Ṽ₀ values including no potential case (Ṽ₀ = 0) in Fig. 2a. In the case of Ṽ₀ = 0, the phase separation is purely driven by the polymerization which increases the free energetic penalty for longer polymers to remain mixed. For sufficiently long polymer chains, a large value of χN leads to spontaneous phase separation. However, in a system with surface potential, not only the polymerization but also the surface potential plays a role as the driving force for phase separation, suggesting a competition between the two driving forces. In Fig. 2a, as Ṽ₀ increases, phase separation occurs at earlier times t̃, indicating the higher potential strength drives enhanced molecular transport towards the surface by preferentially attracting one species and prompting an earlier rise of ψ(t̃). We note that, in the absence of a driving force for phase separation (i.e. χN = 0.0), ψ ∼ [scr O, script letter O] (10⁻³) remains very small and there is only a slight increase in the A species at the surface.

Fig. 2 (a) ψ(t̃) over different Ṽ₀ at constant χ = 0.8 and k̃ = 10⁻³. Phase separation starts earlier as Ṽ₀ increases. (b) ω(x̃,t̃) at a specific time t̃ = 10 × 10⁵ from Fig. 2a. For Ṽ₀ = 0, only isotropic phase separation occurs whereas systems with surface potential produces anisotropic phase separation closer to surface potential (x̃ = 1). (c) Higher Ṽ₀ increases the anisotropic–isotropic transition front x̃_ω=0(t̃) (i.e. larger anisotropic regime near the surface potential), which gradually increases over time as the concentration wave propagates further into the bulk.

Then, we further investigate how anisotropic phase separation emerges in the x-direction at a specific time t̃ = 10 × 10⁵ denoted by the black dashed line in Fig. 2a. At t̃ = 10 × 10⁵, we plot ω(x̃) in Fig. 2b; ω(x̃) > 0 represents anisotropic lamellar-like phase separation in the direction perpendicular to the potential surface, and ω(x̃) < 0 indicates isotropic phase separation with no specific directionality. For the zero-potential case (Ṽ₀ = 0) in Fig. 2b, the system exhibits purely isotropic phase separation (ω(x̃) < 0), whereas systems with the potential Ṽ₀ > 0 show strong alignment (ω(x̃) > 0) near the potential surface (x̃ = 1). In this case, ω(x̃) gradually decreases below zero away from the potential surface, entering the “bulk” regime where the system phase separates isotropically. Moreover, the strength of the potential does not affect the intensity of ω(x̃) in the bulk regime (x̃ ∼ 40) as shown in Fig. 2b.

Lastly, we characterize the thickness of the anisotropic regime by the quantity x̃_ω=0(t̃) for different Ṽ₀ values to examine the extent of surface-guided phase separation over time; x̃_ω=0(t̃) describes when ω(x̃) = 0 (i.e. intersects with the black dashed line in Fig. 2b). Fig. 2c shows the time-evolution of x̃_ω=0(t̃), demonstrating that stronger Ṽ₀ initially produces a thicker anisotropic regime, and the subsequent growth with time is similar such that this initial offset persists regardless of the Ṽ₀ values.

3.2. Effect of polymer incompatibility χ

We now consider how the Flory–Huggins parameter χ governing the polymer incompatibility changes the degree of phase separation ψ(t̃), and possibly competes with surface potential as the driving force for phase separation.

To investigate the effect of polymer incompatibility on phase separation under the surface potential, we plot the time evolution of ψ(t̃) for three representative values of χ in Fig. 3. First, we show that larger χ values promote faster phase separation regardless of surface potential Ṽ₀ at a constant reaction rate k̃. In terms of Ṽ₀ at each χ, we observe a similar trend of ψ(t̃) with increasing Ṽ₀ relative to its corresponding zero-potential system (Ṽ₀ = 0). For a surface potential of Ṽ₀ = 10⁻³, the phase separation begins slightly earlier, as indicated by the earlier rise in ψ(t̃). As Ṽ₀ increases gradually from 0 to 10⁻¹, the influence of the surface potential becomes more pronounced. This enhances anisotropic phase separation (or wetting) behavior, leading to faster phase separation. Interestingly, the impact of Ṽ₀ is more significant for systems with lower χ. In these systems, weaker polymer incompatibility delays the phase separation because sufficient polymerization and the formation of larger chains are required to drive phase separation. This extended timescale allows the surface potential to attract more components, amplifying its effect on phase separation.

Fig. 3 The influence of the Flory–Huggins parameter, χ, which governs polymer immiscibility, on the degree of phase separation ψ(t̃) at a constant value of k̃ = 10⁻². Each value of χ is represented by a distinct color, while different line styles indicate various surface potential Ṽ₀. An increase in χ facilitates faster phase separation regardless of surface potential. For each value of χ, the effect of Ṽ₀ on the progression of phase separation is consistent; a higher surface potential leads to an earlier rise in ψ(t̃).

3.3. Effect of reaction rate k̃

Fig. 4 is the compilation of the order parameter ω(x̃,t̃) for wide ranges of k̃ and Ṽ₀ values at fixed χ of 1.0. In Fig. 4, each row and column shares the same Ṽ₀ and k̃, respectively. The rainbow color gradient represents the progression of time, transitioning from red (t̃ = 0) to purple (t̃ = τ_p), where τ_p is defined to characterize the phase separation time at ψ(t̃) = 0.4.

Fig. 4 The compilation of ω(x̃,t̃) under different conditions of Ṽ₀ and k̃ at constant χ = 1.0. Each row and column shares the same Ṽ₀ and k̃, respectively. For all 16 sub-Fig. 4a–p, the rainbow color gradient represents the progression of time, transitioning from red (t̃ = 0) to purple (t̃ = τ_p) denoted with an arrow in Fig. 4m. For the zero-potential systems shown in (a) to (d), systems only isotopically phase separate (ω(x̃,t̃) < 0) due to the absence of surface potential. With the surface potential shown in (e) to (p), systems develop positive values of ω(x̃,t̃) near the surface potential (x̃ = 1). For Ṽ₀ > 0, higher Ṽ₀ induces faster phase separation near the surface potential (x̃ = 1); the spatiotemporal evolution of ω(x̃,t̃) from red (t̃ = 0) to next lighter red appears more clearly regardless of k̃. At a reaction rate of k̃ = 10⁻⁵ in (e), (i) and (m), anisotropic phase separation dominantly propagates and induces ordering behavior (ω(x̃,t̃) > 0) throughout the system. Such enhanced penetration is attributed to the prevalent smaller molecules with higher mobility at lower reaction rate k̃ = 10⁻⁵. Furthermore, the difference in timescales between surface-directed and polymerization-induced phase separation plays a pivotal role; the system rapidly reaches τ_p due to surface-driven phase separation before polymerization-induced phase separation occurs. When increasing k̃, a weaker “wetting” behavior occurs near the surface potential. This phenomena is due to rapid emergence of larger chains with low mobility and rapid consumption of smaller molecules at higher k̃. In this regard, at lower k̃ values, the extended lifetime of smaller chains enhances polymer diffusion under the influence of surface potential.

First, in Fig. 4, we find that the evolution from the initial profile (t̃ = 0) to the subsequent one is higher for larger Ṽ₀. In other words, larger Ṽ₀ initially induces stronger accumulation of the preferred species, ϕ_A, at the potential surface x̃ = 1 (i.e. faster growth of ω(x̃,t̃) near x̃ = 1) regardless of the reaction rate. This stronger attraction of ϕ_A from a higher surface potential Ṽ₀ leads to faster transport and phase separation, and thus smaller τ_p as shown in Fig. 2a and 5, respectively.

Fig. 5 Phase separation time τ_p for various Ṽ₀ and k̃ at constant χ = 1.0. We find two different regimes for τ_p polymerization-limited and diffusion-limited regimes. As the reaction rate k̃ initially increases, it shortens phase separation time τ_p in a polymerization-limited regime (k̃ ≤ 10⁻⁴). In this regime, increasing Ṽ₀ noticeably reduces τ_p due to abundant short chains with high mobility. In a diffusion-limited regime (k̃ > 10⁻⁴), an increase in k̃ slows down the characteristic phase separation time τ_p while the effect of Ṽ₀ becomes negligible due to hindered diffusion from rapid accumulation of large polymers.

The analysis of varying reaction rates k̃ in Fig. 4 reveals several key trends. As k̃ decreases, the generation of larger chains capable of driving phase separation slows down. This results in reduced intensity for ω(x̃,t̃) < 0 near x̃ = 40 at t̃ = τ_p. At sufficiently low k̃ values, surface potential-driven phase separation becomes the dominant mechanism before t̃ = τ_p, as evidenced in Fig. 4e, i and m. Notably, systems with surface potential (Ṽ₀ > 0) at k̃ = 10⁻⁵ predominantly display ω(x̃,t̃) ≥ 0 for all x̃ and t̃, indicating no isotropic phase separation prior to τ_p. In contrast, increasing k̃ leads to rapid formation of larger polymers, resulting in a decrease in mobility and reduced polymer attraction to the surface potential as shown in Fig. 4. Moreover, Fig. 4k shows a unique behavior; the system at t̃ = τ_p (purple) shows contraction near the surface potential instead of further expansion toward the bulk (i.e. reduction of x̃_ω=0(t̃)). This indicates merging of two thin layers of the lamellar-like domain into one thicker layer.

In Fig. 5, to compare each τ_p which corresponds to the last purple curve of each sub-figure in Fig. 4, we plot τ_p for various combinations of k̃ and Ṽ₀ along with the zero-potential case (Ṽ₀ = 0). The observed trends across the parameter space are consistent for both diffusion-limited and reaction-limited regimes. For k̃ > 10⁻⁴, the system exhibits diffusion-limited behavior, as shown in Fig. 5. In this regime, increasing the reaction rate k̃ leads to a longer characteristic phase separation time, τ_p. We attribute this phenomena to the rapid accumulation of larger polymer chains with higher N_A, which significantly slows down molecular diffusion (∝ 1/N). Here, the influence of the surface potential Ṽ₀ remains consistent across different k̃ values, but the changes in τ_p are minimal. This is expected because, in a diffusion-limited regime, molecular diffusion driven by the surface potential is also restricted and cannot effectively attract long, less-mobile polymers. Conversely, in a reaction-limited regime (k̃ ⪅ 10⁻⁴), the phase separation time τ_p decreases with increasing k̃. In this scenario, the driving force from polymer incompatibility rapidly intensifies with higher k̃, enhancing the rate of phase separation. Unlike in the diffusion-limited regime, the surface potential Ṽ₀ significantly impacts the phase separation time τ_p in a reaction-limited regime, as illustrated in Fig. 5. This is because lower k̃ values extend the lifetime of smaller chains, making the surface potential more effective due to both the higher mobility and the prolonged lifetime of these smaller molecules. To be more specific, this is attributed to the potential-induced, selective attraction of ϕ_A to the surface, which allows for faster diffusive transport when the molecular weight is low (i.e. at the early stages of phase separation) and leads to an earlier onset of phase separation. As a result, the initial phase separation, driven by the surface potential, intensifies and grows away from the surface in layers. This continues until the average degree of polymerization increases sufficiently through polymerization to further drive phase separation. In essence, the surface potential compensates for the times over which phase separation does not occur due to slower polymerization, resulting in a more pronounced phase separation at smaller k̃.

To characterize the timescales between diffusion and reaction, the Damköhler number Da is defined:

Here, ζ₀ is the friction coefficient. At k̃ = 10⁻⁴, Da = 0.15, which denotes a reaction-limited regime where polymerization occurs slowly with respect to diffusion and phase separation. In the absence of the surface potential, this leads to longer τ_p as the rate k̃ decreases due to the slow polymerization of chains that must be sufficiently long before phase separation can occur. In Fig. 5, the surface potential drives phase separation for k̃ ≤ 10⁻⁴, thus the phase separation process is not limited by the extent of reaction; rather lower k̃ indirectly helps the process by not rapidly producing heavier polymers which will hinder the phase separation. Conversely, at k̃ = 10⁻³ or Da = 1.5, the system is in a diffusion-limited regime (k̃ > 10⁻⁴) as shown in Fig. 5. In this scenario, the surface potential slightly enhances the phase separation by preferentially attracting molecules but does not make a significant difference due to rapid accumulation of heavier chains from faster polymerization, leading to overall slower dynamics.

We also provide ψ(t̃) over different rates of reaction at Ṽ₀ = 10⁻³ in Fig. 6a with the corresponding molecular weight distribution w_A at t̃ = 2 × 10⁵ in Fig. 6b to examine the most abundant species with length N_A and its behavior at the onset of phase separation. Fig. 6a illustrates how the phase separation parameter ψ(t̃) varies with the reaction rate k̃, contrasting with the dependency on the Flory–Huggins parameter χ shown in Fig. 3. At a fixed surface potential Ṽ₀, two distinct trends emerge as previously identified in Fig. 5. Notably, for the system with k̃ = 10⁻⁵, indicated by the red color in Fig. 6a, phase separation is significantly delayed despite the presence of a surface potential. This delay is attributed to the competition between surface potential and the free energy of polymer mixing, where the surface potential strength (Ṽ₀ = 10⁻³) is insufficient to overcome the mixing free energy. Additionally, at this low reaction rate, polymer chains grow very slowly, maintaining miscibility for an extended period. This observation aligns with findings that systems with Ṽ₀ = 10⁻³ behave similarly to zero-potential systems regarding phase separation onset, as depicted in Fig. 3. It is further supported by the slow evolution of molecular weight distribution shown in Fig. 6b. Consistent with our observation, the molecular weight distribution w_A in Fig. 6b indicates that the smaller chains are abundant for lower k̃ and the systems accumulate larger chains more as k̃ increases; w_A shows which length of polymer is the primary participant in the phase separation process at early time (t̃ = 2 × 10⁵). Thus, we attribute an earlier rise of ψ(t̃) for lower k̃ to the surface potential which drives the diffusion of unreacted smaller molecules as shown in Fig. 6b, which results in enhanced phase separation as depicted in Fig. 2a, 3 and 5.

Fig. 6 (a) The effect of reaction rate k̃ on ψ(t̃) at constant Ṽ₀ = 10⁻³ and χ= 1.0. (a) shows consistency with previous observations in Fig. 5 that the initial increase in k̃ leads to faster phase separation but further increase slows down phase separation. (b) shows molecular weight distribution w_A at t̃ = 2 × 10⁵ in Fig. 6a. It describes the distribution of chain lengths participating in a phase separation process at the moment. This supports the findings that higher k̃ in a diffusion-hindered regime leads to a longer phase separation time because rapid accumulation of larger chains hinder diffusion and phase separation, and higher k̃ in a reaction-limited region leads to faster phase separation with increasing polymer incompatibility.

We analyze the phase separating regimes away from the surface potential to characterize the “bulk” phase separation regime at x̃ = 40, where isotropic phase separation (i.e. ω(x̃,t̃) < 0) occurs as shown in Fig. 4. First, we keep track of ω(x̃,t̃) values over time at x̃ = 40 as shown in Fig. 7 to represent the isotropic phase separation behaviors for the “bulk” regime. In Fig. 7a, we observe that ω(x̃ = 40,t̃) curves exhibit a near-collapse for identical k̃ > 10⁻⁴ (diffusion-limited regime), irrespective of Ṽ₀ values. This behavior is anticipated, given the sufficient distance from the anisotropic concentration wave induced by the surface potential in a diffusion-limited system. The trend closely resembles a system devoid of surface potential, where higher k̃ values necessitate extended time for phase separation due to the accelerated generation of slower-diffusing, heavier polymers.⁹⁴ Consistently, Fig. 7a demonstrates that increasing the reaction rate k̃ decelerates isotropic bulk phase separation at x̃ = 40, as illustrated in Fig. 5. Fig. 7b depicts ω(x̃ = 40,t̃) for lower reaction rates (k̃ ≤ 10⁻⁴) in the reaction-limited regime, revealing distinct behaviors compared to Fig. 7a. At k̃ = 10⁻⁴, the intensity diminishes significantly, except for Ṽ₀ = 0, relative to Fig. 7a. This reduction signifies the system's transition from a diffusion-limited to reaction-limited regime, where the surface potential exerts a more pronounced influence on phase separation, as evidenced in Fig. 5. Further reduction of k̃ to 10⁻⁵ yields positive values of ω(x̃ = 40,t̃), excluding the zero-potential system. This indicates a deeper penetration of the concentration wave from the surface potential and the development of anisotropic phase separation beyond the point (x̃ = 40) previously considered as “bulk”. In this scenario, surface-driven phase separation is sufficient and allows ample time for further propagation, expanding the anisotropic regime while suppressing isotropic phase separation, as demonstrated in Fig. 4e, i, and m.

Fig. 7 The bulk phase separation intensity ω(x̃ = 40,t̃) over k̃ and Ṽ₀ at constant χ = 1.0. (a) For varying Ṽ₀, the curves nearly collapse onto one another for the same k̃. This is expected for a diffusion-limited regime because the anisotropic concentration wave induced by the surface potential propagates less effectively, meaning that phase separation in the “bulk” depends on k̃. As the reaction rate k̃ increases, more time is required for isotropic phase separation in the bulk regime due to the rapid accumulation of longer chains. (b) In a reaction-limited regime, surface potential becomes more dominant due to the prevalent smaller molecules with faster diffusion, effectively altering the bulk phase separation behaviors. At k̃ = 10⁻⁵, ω(x̃ = 40,t̃) shows positive growth for Ṽ₀ > 0, indicating a deeper penetration of anisotropic concentration wave towards the bulk regime as shown in Fig. 4e, i and m.

To more thoroughly understand how different k̃ values affect the competition between surface-driven and bulk phase separation, we examine volume fractions of each individual chains with different degrees of polymerization N_A for species A. We do not plot species B, which shares the same large-x̃ behavior but exhibits oscillations that are opposite of those in A, and does not provide additional physical insight. We consider both slow (k̃ = 10⁻⁵) and fast (k̃ = 10⁻²) polymerization kinetics in Fig. 8 and 9, respectively. Fig. 8a provides the overall behavior of ω(x̃,t̃) at a smaller reaction rate (k̃ = 10⁻⁵) for a reaction-limited regime, which is the same as Fig. 4m. All of the same color in both Fig. 8a and c denotes the same time over the course of phase separation from red (t̃ = 0) to purple (t̃ = τ_p) as visualized in Fig. 8b.

Fig. 8 Plots demonstrating the dynamic evolution of the full molecular weight distribution for slow polymerization kinetics (i.e. k̃ of 10⁻⁵, reaction-limited regime). All figures share the same x-axis for a normalized simulation box, and color scale from red (t̃ = 0) to purple (t̃ = τ_p). (a) replicates ω(x̃,t̃) shown in Fig. 4m. (b) The visualization of one of the simulation trajectories at t̃ = 0 and τ_p. (c) The time-evolution of polymer concentration profiles along x̃. Phase separation near the surface potential gets more profound for short- or medium-sized N_A. Consumption and polymerization are shown with N_A = 1, 2 and 4 while generation and polymerization are found with N_A = 8, 16 and 32. In this slower reacting system, smaller molecules diffuse to the surface, accumulate, and polymerize locally, leading to local generation of heavier polymers at the concentrated regions such as near the surface potential.

Fig. 9 Plots demonstrating the dynamic evolution of the full molecular weight distribution for slow polymerization kinetics (i.e. k̃ of 10⁻², diffusion-limited regime). All figures share the same x-axis for the normalized simulation box, and color scale from red (t̃ = 0) to purple (t̃ = τ_p). (a) Replicated ω(x̃,t̃) shown in Fig. 4p. (b) Visualization of one of the simulation trajectories at t̃ = 0 and τ_p. (c) The time-evolution of polymer concentration profiles along x̃. Phase separation near the surface potential gets more profound for N_A = 16 or 32. No meaningful differences are found but to be completely consumed in N_A = 1,2,4 and 8 due to large k̃ of 10⁻². For N_A = 16, mostly consumption takes place while phase separating with small fluctuation near the surface. In N_A = 32, we observe that it accumulates first before it phase separates. This fast reacting system shows that polymerization is too fast for smaller molecules to diffuse, leading to accumulation of heavier polymers evenly throughout the space first, and then these larger molecules drive the phase separation.

In Fig. 8c, the concentration profiles of individual polymer chains with specific chain lengths (N_A) along the x-direction are presented for N_A = 1, 2, 4, 8, 16, and 32. This analysis examines the role of reaction rate under the influence of surface potential in relation to polydispersity. In the case of a slow reaction rate, monomers (N_A = 1) are weakly attracted to the surface potential over time and are continuously consumed through polymerization. For polymers with N_A = 2 and 4, an initial single peak (representing a polymer-rich or concentrated region) forms near the surface potential (x̃ = 1), followed by fluctuating concentrations extending toward x̃ = 60 as they are progressively consumed. This behavior persists for larger polymers (N_A = 8), with an additional observation of polymer generation during phase separation as the system evolves toward τ_p. The trend becomes more pronounced with increasing chain length. Heavier polymers (N_A = 16 and 32) exhibit simultaneous phase separation and generation, suggesting that larger molecules are formed based on the pre-established concentration fields of smaller chains. This eliminates the need for these larger molecules to diffuse and phase separate independently, as smaller molecules have already undergone phase separation over sufficient time at low reaction rates (k̃ = 10⁻⁵). Fig. 9 instead illustrates a diffusion-limited system with a relatively high polymerization rate constant k̃ = 10⁻², such that short chains are almost immediately consumed by the reaction. All other variables are kept constant as in Fig. 8 to specifically isolate the effect of the polymerization rate. Fig. 9a (which is also shown in Fig. 4p) shows two major differences in terms of ω(x̃,t̃). First, Fig. 9a exhibits ω(x̃,t̃) values below 0 (i.e. isotropic phase separation), which is not observed in Fig. 8a. This is attributed to the comparable timescales between surface-directed phase separation and polymerization-induced phase separation; for k̃ = 10⁻², the system quickly builds up sufficient incompatibility to initiate polymerization-induced phase separation before the surface-driven phase separation becomes dominant. Second, the system exhibits significantly less propagation of anisotropic concentration waves into the bulk, even under the same surface potential strength (Ṽ₀ = 10⁻¹). This is because smaller molecules are rapidly consumed, limiting their diffusive transport. This rapid consumption turns into the generation of larger molecules with reduced mobility, which hinders overall diffusion in the system even under the surface potential at k̃ = 10⁻².

Fig. 9c illustrates that for N_A = 1, 2, 4, and 8, the reaction (or consumption) rate is so high that molecules have insufficient time to diffuse near the potential surface. Only for larger chain lengths (N_A = 16) do we observe subtle simultaneous consumption and phase separation over time. At early times (indicated by red colors), the system exhibits minimal compositional variation and lacks evident phase separation. Gradual phase separation becomes apparent as the system approaches τ_p. For the largest chains considered (N_A = 32), the system initially shows a nearly uniform accumulation of polymers throughout space due to the rapid reaction rate. This is evident at early times when ϕ_A,32 averages around 0.07 for N_A = 32, as shown in Fig. 9c. A concentration wave emerges from the surface potential, propagating into the bulk and driving anisotropic phase separation. Additionally, bulk phase separation occurs at x̃ = 40 over longer timescales near τ_p, as depicted in Fig. 9a and b. However, this bulk phase separation does not result in significant changes in ϕ_A,32 at x̃ = 40 due to spatial averaging along the y-direction.

4 Conclusions

We have modified and developed a pCH model to study how phase separation in a polymerizing binary blend is affected by the presence of a preferential surface, and show that there is a complex interplay between polymer incompatibility, strength of the preferential surface interactions and polymerization kinetics on the dynamics and structure of the phase separating polymer.

We first demonstrate how the strength Ṽ₀ of an applied preferential surface potential affects phase separation (Fig. 2a), and use an order parameter ω(x̃,t̃) (Fig. 2b) and related phase ordering transition quantity x̃_ω=0(t̃) (Fig. 2c) to track the phase separation process over time. We found that phase separation at a preferential surface occurs before phase separation in the bulk, because larger values of Ṽ₀ locally concentrate one of the species near the surface potential; this increases both the local rate of polymerization and promotes the formation of the phase enriched in the preferred species. This leads to a layered, anisotropic structure at the surface, and we show that this structure grows monotonically with time by monitoring the anisotropic–isotropic transition point x̃_ω=0(t̃). This quantity reveals that stronger Ṽ₀ initially results in a thicker anisotropic regime, which repeats contraction upon merging of anisotropic domains and expansion upon further penetration of concentration wave over time.

We further investigate the influence of polymer incompatibility, characterized by χ, on the phase separation dynamics in the presence of a surface potential, Ṽ₀. As illustrated in Fig. 3, the behavior of ψ(t̃) aligns with expectations as χ varies; specifically, higher χ values result in accelerated phase separation. For each χ value, the system exhibits a monotonic dependence on Ṽ₀, where larger Ṽ₀ values lead to an earlier onset of the rise in ψ(t̃).

We define and employ the order parameter for directional phase separation, ω(x̃,t̃), across various combinations of the surface potential Ṽ₀ and reaction rate k̃, as illustrated in Fig. 4. Together with Fig. 5 and the Damköhler number Da analysis, we identify diffusion-limited and reaction-limited regimes. This framework enables us to systematically explore how phase behavior evolves as a function of k̃ and Ṽ₀ within each regime.

By explicitly accounting for polydispersity in our framework, we investigate the intricate interplay between surface potential and reaction rate, as illustrated by the molecular weight distribution w_A in Fig. 6b. This analysis provides a clear physical understanding of the dominant factors driving phase separation in both polymerization-limited and diffusion-limited regimes. Furthermore, we explore the spatiotemporal evolution of molecular weight under the influence of surface potential. Our findings reveal distinct phase separation pathways that arise from the competition between the timescales of surface-driven and polymerization-driven phase separation, governed by the polydispersity and molecular weight-dependent mobility.

Overall, we have demonstrated that phase separation under a preferential surface is governed by two distinct processes; surface-driven polymerization and bulk polymerization. The behavior of these two processes is governed by a competition between the polymerization rate and molecular diffusion, which has implications for practical materials systems such as surface-directed^105–108 or self-stratifying^108–110 coatings. This approach could also be further modified to account for other types of systems, such as coatings involving higher-functionality monomers that can undergo gelation,^111–113 or other types of polymerization reaction kinetics.^113–115 Similarly, it would be possible to modify this approach to account for other polymer dynamic behaviors, in particular directly including entangled polymer dynamics^64,113 through changes to the molecular-weight dependence of the diffusion constant.

Author contributions

Charles Sing and Hyeonmin Jeong designed the computational model and the theoretical framework, analyzed the data, carried out the implementation, and performed the calculations. Hyeonmin Jeong wrote the manuscript and designed the figures with the supervision of Charles Sing. Junsi Gu, Paul Mwasame, Kshitish Patankar, and Decai Yu provided guidance on the direction of the study. Paul Mwasame and Decai Yu helped connect Dows internal modeling resources. All authors provided critical feedback and helped shape the research and manuscript.

Data availability

Data for this article, including raw data (*.xlsx) for all figures and computational code (*.c) for running the polymerizing Cahn–Hilliard program, are available at https://github.com/Sing-group-ChBE-UIUC.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors acknowledge the Dow University Partnership Initiative (UPI Program) for financial support, and also the helpful discussions with Fabio Aguirre-Vargas (Dow), Michaeleen Pacholski (Dow), Harshad Shah (Dow), Guarav Singhal (UIUC), Lihong Lao (UIUC), Piyush Singh (UIUC), Paul Braun (UIUC), Charles Schroeder (UIUC), and Simon Rogers (UIUC). The authors also acknowledge helpful comments from the anonymous reviewers, who found a non-trivial error in our original method.

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