Numerical Simulation of Co-Continuous Morphologies in PEO/PS Polymer Blends

Abstract

This paper investigates co-continuous structures in immiscible polymer blends through three-dimensional (3D) computational calculations based on a multiphase phase-field equation for fluid flow. The mathematical model describes phase separation with the Cahn–Hilliard (CH) equation and fluid motion with the incompressible Navier–Stokes (NS) equations. Both polymers are treated as Newtonian viscous fluids, and the model includes surface tension, viscosity, and volume fraction effects. A semi-implicit finite difference method (FDM) solves the CH equation, and a projection method maintains the incompressibility of the flow field. Multigrid techniques solve the nonlinear systems efficiently. In addition, a connectivity-based detection algorithm determines whether a phase forms a connected structure that reaches all boundaries of the numerical domain. The numerical results show that the morphology changes from a droplet–matrix structure to a co-continuous structure as the volume fraction increases. The interfacial area per unit volume reaches a local maximum near the transition between these two regimes.

1. Introduction

Polymer–polymer blends are an important way to create materials that have combinations of properties that a single polymer cannot provide [1,2,3]. These materials are usually made by strongly mixing two immiscible polymers and then cooling the mixture to a temperature below the melting point of one component to obtain a solid material [1]. During the mixing process, different non-equilibrium heterogeneous microstructures (morphologies) can appear, and the cooling process can keep these structures in the solid material [4]. Common non-equilibrium morphologies include droplet–matrix structures, fibers, lamellae, and co-continuous sponge-like structures in which both phases form continuous networks [5]. Among these structures, co-continuous morphologies combine the properties of the two polymers more effectively. Each phase forms a continuous percolation network, and synergistic effects can be observed [6,7]. The formation and evolution of such complex microstructures can be analyzed or predicted using multi-physics modeling approaches, phase-field simulations, and advanced data-driven methods [8,9], which provide insights into the kinetics and morphology development under non-equilibrium conditions [10,11]. However, not all immiscible polymer blends can form co-continuous morphologies. The formation of such structures strongly depends on the rheological properties of the blend components, including viscosity ratio, viscoelasticity, and interfacial tension [12]. In particular, a balanced viscosity ratio between the two phases leads to the formation of co-continuous structures, whereas a large viscosity difference typically leads to droplet–matrix morphologies [13]. In addition, processing conditions such as shear rate and mixing time also influence morphology development during melt blending [14,15]. These effects can be observed in representatibe polymer blend systems [12]. Figure 1 shows sample SEM images of blends with co-continuous and droplet morphologies. The figure shows SEM images of polyethylene oxide (PEO)/polystyrene (PS) blends: a 50/50 blend after water extraction of PEO (left) and a 90/10 blend after toluene extraction of PS (right). The blends were obtained from PS ($M_w$ = 150,000 g/mol) and PEO ($M_w$ = 400,000 g/mol), where $M_w$ denotes the weight-average molecular weight, using melt mixing at 170 °C and subsequently quenching to preserve the microstructure.

Co-continuous microstructures have two phases that are both continuous and that pass through each other in three dimensions. Each phase supports itself and connects throughout the domain. The structure looks similar to a sponge. A clear mathematical definition that is suitable for our 3D numerical simulations and comparison with experiments is given in Section 2.2. To explain the difference between droplet–matrix and co-continuous morphologies, we show a simple illustration. Figure 2 shows the droplet–matrix morphology at 12% volume fraction. Figure 2a presents the structure before extraction of the matrix phase, whereas Figure 2b shows the structure after extraction. The remaining structure is not self-supporting. However, the interface shown in Figure 3a, which corresponds to a 50% volume fraction co-continuous phase, remains a self-supportive morphology in Figure 3b even after extraction of the other phase.

Polymer blends with a co-continuous structure have applications in various fields including conductive materials [17,18], electrostatic dissipative materials [19,20,21], and barrier film structures [22]. Many computational methods have been proposed to understand the formation mechanisms of such structures. Carolan et al. [23] studied the formation of droplet and co-continuous polymer microstructures using a finite volume-based numerical method of the Cahn–Hilliard (CH) equation. The mechanical response of the microstructures obtained was also analyzed under various stress states. The results showed that the morphology changes from a droplet structure to a co-continuous structure as the volume fraction increases, and that these morphological differences influence the elastic and plastic behavior of the composite. Zhu et al. [24] used a phase-field model based on the CH equation to generate the initial phase morphology and a two-phase moving-interface flow simulation to model phase coarsening. The co-continuous structure formed in polyether ether ketone (PEEK)/polyether sulfone (PES) immiscible polymer blends was then used to fabricate porous PEEK structures. Inguva et al. [25] presented a continuum-scale modeling method for polymer blends based on the CH equation and examined the effects of different thermodynamic models on phase separation and morphology formation. The authors also discussed that interconnected or co-continuous microstructures observed experimentally in applications such as organic solar cells and polymer membranes are closely related to the phase separation dynamics described by the CH model. Zhang et al. [26] developed a linear and unconditionally energy-stable method to simulate the CH equation-based copolymer dynamics in arbitrary complex domains. To further investigate the multiple copolymers on curved surfaces, Jia and Yang [27] recently developed a CH-type model and the associated efficient numerical scheme with closest-point method.

Despite these advances, it remains difficult to accurately identify and quantify co-continuous structures. Conventional experimental detection methods have limitations in that the procedures are complex and require significant time and cost. In general, the interfacial area increases as the number of minor components increases. However, when the structure becomes co-continuous, the total interfacial area decreases because elongated domains connect with each other. This behavior has been confirmed experimentally [16]. Current experimental detection methods are indirect and use measurements such as conductivity and surface area. In contrast, numerical simulations allow a direct assessment of whether a co-continuous structure exists. Therefore, numerical simulation-based approaches have become increasingly important. Among various numerical methods, meshless approaches such as the generalized FDM have been presented for the computation of complex physical problems and provide the treatment of complex geometries without mesh generation [28,29]. However, finite difference methods remain widely used due to their simplicity of implementation and computational efficiency, particularly for structured grid-based simulations. In this study, we present a numerical algorithm based on the FDM to simulate the coalescence process that leads to the formation of co-continuous morphologies; we also present a connectivity-based detection algorithm to evaluate the degree of co-continuity.

The remainder of this paper is organized as follows. Section 2.1 presents the governing equations for the coupled CH and incompressible Navier–Stokes (NS) system used to describe phase separation and fluid motion in polymer blends. Appendix A introduces the numerical discretization of the CH equation and the nonlinear multigrid solver used to efficiently compute the phase field, and also presents the discretization and solution procedure for the NS equations based on the projection method. Section 2.2 provides the definition and detection algorithm for identifying co-continuous morphologies in the three-dimensional (3D) computational domain. Section 3 presents numerical experiments that investigate the transition from droplet–matrix structures to co-continuous morphologies as the volume fraction varies and analyzes the corresponding changes in interfacial area and connectivity. Finally, Section 4 concludes the paper by summarizing the main numerical results of this study, highlighting the main contributions of the proposed method, and outlining possible directions for future research.

2. Methods

2.1. Governing Equations

Co-continuous morphology is a fully 3D phenomenon. In this study, we investigate the general mechanisms of co-continuous morphology formation in immiscible polymer blends. For simplicity, we treat the polymers as Newtonian viscous fluids. This assumption reduces computational complexity and allows us to investigate the physical mechanisms of phase separation and morphology evolution. Based on this assumption, the governing equations are as follows:

$$\begin{array}{ccc}\hfill {\mathbf{u}}_t(\mathbf{x},t)+\mathbf{u}(\mathbf{x},t)·\nabla \mathbf{u}(\mathbf{x},t)& =& -\nabla p(\mathbf{x},t)+{\displaystyle \frac{1}{Re}}\nabla ·[\eta (c(\mathbf{x},t))(\nabla \mathbf{u}(\mathbf{x},t)+\nabla {\mathbf{u}}^T(\mathbf{x},t))]\hfill \\ & & -{\displaystyle \frac{ϵ}{We}}(\Delta c(\mathbf{x},t)\nabla c(\mathbf{x},t)-{\displaystyle \frac{1}{2}}{\nabla |\nabla c(\mathbf{x},t)|}^2),\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill \nabla ·\mathbf{u}(\mathbf{x},t)& =& 0,\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill c_t(\mathbf{x},t)+\nabla ·(c(\mathbf{x},t)\mathbf{u}(\mathbf{x},t))& =& {\displaystyle \frac{1}{Pe}}\nabla ·(M(c(\mathbf{x},t))\nabla \mu (\mathbf{x},t)),\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill \mu & =& f(c(\mathbf{x},t))-{ϵ}^2\Delta c(\mathbf{x},t),\hfill \end{array}$$

(4)

where $\mathbf{u}=\mathbf{u}(\mathbf{x},t)=(u(\mathbf{x},t),v(\mathbf{x},t),w(\mathbf{x},t))$, $p=p(\mathbf{x},t)$, $c=c(\mathbf{x},t)$, and $\mu =\mu (\mathbf{x},t)$ are the velocity field, pressure, phase variable, and chemical potential, respectively. The spatial coordinate is denoted by $\mathbf{x}=(x,y,z)\in \Omega \subset {\mathbb{R}}^3$, and t is time. $\eta (c)$ is the viscosity, $M(c)=c(1-c)$ is the mobility function of the CH equation, and $f(c)=dF(c)/dc$, where $F(c)=0.25c^2{(1-c)}^2$ denotes the double-well potential. The dimensionless parameters are the Reynolds number, $Re={\rho }_{*}{V}_{*}{L}_{*}/{\eta }_{*}$, the Weber number, $W{e}_i={\rho }_{*}{L}_{*}{V}_{*}^2/{\sigma }_i$, and the diffusion Peclet number, $Pe={\rho }_{*}{L}_{*}{V}_{*}/(M_{*}{\mu }_{*})$. In this study, we focused on two-phase fluid systems to analyze their fundamental behavior and underlying mechanisms. Future research will extend this method to more complex systems, particularly ternary block copolymer systems, by using a ternary phase-field model [30]. This extension will allow us to capture richer interfacial dynamics and more intricate morphological evolution that cannot be described within the two-phase setting.

The Reynolds number represents the ratio of inertial to viscous forces, the Weber number describes the relative importance of inertial forces compared to interfacial tension, and the Peclet number indicates the ratio of convective to diffusive transport. Here, ${\rho }_{*}$, $V_{*}$, and $L_{*}$ denote the representative density, velocity, and length scales of the mixture, respectively, while ${\eta }_{*}$, ${\sigma }_{*}$ and $M_{*}$ are the reference viscosity, interfacial tension, and mobility. The numerical discretization and solution procedures are described in Appendix A.

2.2. Co-Continuity Detection Algorithm

We define a measure of co-continuity as follows: First, we determine n, the number of connected components in the flow. For each component that touches all six boundaries of the box, the degree of co-continuity is defined as $d_i=V_i/V$, where $V_i$ is the volume of the component and V is the total volume of the dispersed phase. The structure is fully co-continuous when $n=1$ and $V_i/V=1$. A blend is fully co-continuous only if one component can be completely extracted (100%), and the remaining structure is still self-supporting.

To determine the degree of co-continuity in the numerical simulations, we developed the following co-continuity detection algorithm. The algorithm begins at the point $(1,1,1)$ in the cubic computational domain $N_x\times N_y\times N_z$. For a given cubic array $A[][][]$, the objective is to identify the number of connected components and to compute the degree of co-continuity associated with each component. Algorithm 1 describes the procedure used to detect connected components and evaluate the degree of co-continuity in the three-dimensional array A. The algorithm scans the entire computational domain and identifies connected regions, where $A[i][j][k]>0.5$. For each detected region, a search procedure is performed to determine all connected grid points. The algorithm then checks whether the detected component touches all six boundaries of the domain. If this condition is satisfied, the co-continuity is calculated as the ratio of the number of grid points belonging to that component to the total number of grid points satisfying $A[i][j][k]>0.5$.

Algorithm 2 describes the detection subroutine used to identify neighboring grid points that belong to the same connected component. Starting from a given grid point $(m,n,l)$, the subroutine marks the point in the auxiliary array B and removes it from A to prevent repeated detection. It then examines all neighboring grid points in the $3\times 3\times 3$ neighborhood. If a neighboring point satisfies $A[i+m][j+n][k+l]>0.5$, the point is recorded in the register array and added to the current component. During the execution of Algorithms 1 and 2 is used, and array A is modified in this process. This modifies the input data permanently, which may affect subsequent analysis. To avoid this issue, a copy of A, denoted as $AA$, is created before Algorithm 1 is executed to preserve the original data.

Algorithm 1 Detection of Co-continuous Components.

1: components $\leftarrow 0$   2: co-continuity $\leftarrow 0$   3: total $\leftarrow {\sum }_{A[i][j][k]>0.5}1$   4: for $k=1$ to $N_z$ do   5:       for $j=1$ to $N_y$ do   6:             for $i=1$ to $N_x$ do   7:                  if $A[i][j][k]>0.5$ then   8:                        $m\leftarrow i$, $n\leftarrow j$, $l\leftarrow k$   9:                        $flag1\leftarrow 0$, $flag2\leftarrow 1$, $count\leftarrow 1$ 10:                        for $ii=1$ to $N_x$ do 11:                              for $jj=1$ to $N_y$ do 12:                                    for $kk=1$ to $N_z$ do 13:                                          $B[ii][jj][kk]\leftarrow 0$ 14:                                    end for 15:                              end for 16:                        end for 17:                        while $flag1<flag2$ do 18:                              DETECTION $(A,B,m,n,l,register)$ 19:                              $flag1\leftarrow flag1+1$ 20:                              $m\leftarrow register[i][1]$ 21:                              $n\leftarrow register[i][2]$ 22:                              $l\leftarrow register[i][3]$ 23:                              $flag2\leftarrow count$ 24:                        end while 25:                        components ← components $+1$ 26:                        Check whether B touches all six boundaries 27:                        if true then 28:                              co-continuity = ${\displaystyle \frac{{\sum }_{B[i][j][k]>0.5}1}{total}}$ 29:                        end if 30:                  end if 31:             end for 32:       end for 33: end for

Algorithm 2 Detection Subroutine.

1: procedure detection($A,B,m,n,l,register$)   2:       $B[m][n][l]\leftarrow 1$   3:       $A[m][n][l]\leftarrow 0$   4:       for $i=-1$ to 1 do   5:             for $j=-1$ to 1 do   6:                   for $k=-1$ to 1 do   7:                         if $0<m+i<N_x+1$ and $0<n+j<N_y+1$ and $0<l+k<N_z+1$ then   8:                               if $A[i+m][j+n][k+l]>0.5$ then   9:                                     $count\leftarrow count+1$ 10:                                     $register[count-1][1]\leftarrow i+m$ 11:                                     $register[count-1][2]\leftarrow j+n$ 12:                                     $register[count-1][3]\leftarrow k+l$ 13:                                     $B[i+m][j+n][k+l]\leftarrow A[i+m][j+n][k+l]$ 14:                                     $A[i+m][j+n][k+l]\leftarrow 0$ 15:                               end if 16:                         end if 17:                   end for 18:             end for 19:       end for 20: end procedure

3. Results and Discussion

The computational domain is $[0,1]\times [0,1]\times [0,1]$. Periodic boundary conditions are imposed on the side walls, and chaotic shear is applied at the walls $z=0$ and $z=1$, as shown in Figure 4. Thus,

$$\begin{array}{c}\hfill u(x,y,1)=-u(x,y,0)=\sin (\pi t/T),\\ \hfill v(x,y,1)=-v(x,y,0)=\cos (\pi t/T),\end{array}$$

where t is the time and T is a period [31].

The flow starts from rest. The initial concentration field consists of randomly distributed ellipsoids with specified volume fractions (Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10) or random perturbations of a uniform concentration with the same volume fraction (Figure 11). The simulation parameters are ${\eta }_1/{\eta }_2=1.0,{\rho }_1/{\rho }_2=1.0,Re=1.0,We=100.0,ϵ=0.01$, and $Pe=10.0/ϵ$.

At volume fractions of 35% or greater, the initial structures coalesce and form co-continuous microstructures, which agree with experimental observations [16]. Figure 5a shows the interface for a 20% volume fraction. Figure 5b–d show 2-D slices of the concentration field normal to the x-, y-, and z-axes, where the $c=0.5$ contour is filled. Figure 6a, Figure 7a, Figure 8a, Figure 9a and Figure 10a show the interfaces for 30%, 35%, 40%, 45%, and 50% volume fractions, respectively, where the surface $c=0.5$ represents the interface. Convex particles have no surrounded area, but particles with concave interfaces can surround the other phase. When the two phases strongly intertwine, each phase surrounds a significant portion of the other. This idea is used in the 2-D image analysis of [32] to detect co-continuous structures. The same effect appears in the 2-D slices shown here, but the full 3-D domain provides more information.

3.1. Interfacial Area per Unit Area

Experimentally, the interfacial area per unit volume shows a maximum near the boundary between droplet and co-continuous morphologies [16]. Figure 12a shows the average interface length per unit area of the micrographs as a function of blend composition for PEO/PS blends (PS, $M_w=150,000$ g/mol; PEO, $M_w=400,000$ g/mol). Two local maxima appear at $35\%$ and $65\%$ PEO. Figure 12b shows the corresponding numerical results for different volume fractions. In the simulations, peaks appear at $40\%$ and $60\%$. Similar to the experiments, the interface area remains nearly constant in the co-continuous region. The difference may result from experimental annealing and the non-Newtonian behavior of the fluids, while the simulations neglect viscoelastic effects. Figure 11 shows the evolution of initially random droplets ($30\%$, $35\%$, $45\%$, and $50\%$ volume fractions) under simple shear with shear rate $\lambda =1.0$. To describe the structure, we define a co-continuity measure. We first determine n, the number of connected components. Under periodic boundary conditions, opposite faces in each periodic direction are treated as connected, as shown in Figure 13. For each component that touches all non-periodic boundaries and at least one face in each periodic direction, the degree of co-continuity is defined as $d_i=V_i/V$, where $V_i$ is the volume of the phase and V is the total volume of the dispersed phase. The structure is fully co-continuous when $n=1$ and $V_i/V=1$. Values of co-continuity near zero indicate isolated particles, while larger values indicate concave or partially connected structures.

Table 1. Co-continuity and number of connected components for 30%, 35%, 45%, and 50% concentrations of the dispersed phase.

Case	Quantity	$\mathit{t}=0.3$	$1.35$	$1.8$	$1.95$	$4.2$	$6.9$	$10.05$
30%	Components	154	85	68	68	29	23	20
	Co-continuity	0.0	0.0	0.0	0.0	0.0	0.0	0.0
35%	Components	126	44	30	26	10	13	15
	Co-continuity	0.0	0.0	0.0	0.0	0.95	0.0	0.0
45%	Components	11	4	4	1	1	2	5
	Co-continuity	0.996	0.999	0.998	1.0	1.0	0.998	0.935
50%	Components	4	6	2	1	2	3	3
	Co-continuity	0.999	0.999	0.999	1.0	0.999	0.999	0.0

shows the values of $d_i$. In all cases, only one component touches all boundaries. When the volume fraction is $35\%$, a co-continuous structure forms at about $t=1.95$ and persists until about $t=6.75$. The structure forms through coalescence and later breaks under shear, which produces fiber-like tubes of the second phase.

Finally, Figure 14 shows the total interface area per unit area and the degree of co-continuity as functions of time for the simulations in Figure 11. The interface area decreases for all volume fractions. The $30\%$ case shows zero co-continuity until $t=6.0$ and reaches about $0.5$ between $t=6.0$ and $t=8.0$. The $35\%$ case is co-continuous between $t=2.0$ and $t=7.0$, while the $45\%$ and $50\%$ cases remain co-continuous throughout the simulation. This behavior is partly due to the initial conditions and requires further study.

3.2. Influence of Interfacial Tension

In Figure 15, two surface tensions are compared. The first row shows $\sigma =1.0$, and the second shows $\sigma =50.0$. The initial configuration and other parameters are the same as in Figure 5. For $\sigma =1.0$, little evolution is observed. For $\sigma =50$, the evolution is much faster, and breakup occurs due to Rayleigh instability, followed by coarsening through interface coalescence.

4. Conclusions

In this study, we performed 3D numerical simulations to investigate the formation and stability of co-continuous structures in immiscible polymer blends. The model combined the CH equation with the incompressible NS equations and assumed Newtonian viscous fluids. A semi-implicit FDM and a projection method were used, and the nonlinear systems were solved efficiently using a multigrid method. A connectivity-based algorithm was developed to detect co-continuity. The results are limited to the PEO/PS material system under the present assumptions. Based on these results, practical guidelines for achieving co-continuous morphologies during melt mixing are also discussed. The main findings are as follows:

• Morphology transition. The structure changed from droplet–matrix to co-continuous as the volume fraction increased, with co-continuity appearing at approximately $35\%$ or higher.
• Interfacial area. The interfacial area reached a maximum near the transition, which agrees with the experimental results.
• Shear effect. Co-continuous structures were formed via droplet merging and could later break under shear.
• Interfacial tension effect. Larger interfacial tension accelerated evolution and enhanced both coalescence and breakup.
• Practical implications (PEO/PS blends). Co-continuous structures can be achieved with the following:
  • Volume fraction around 35–50%.
  • Sufficient mixing and shear.
  • Appropriate interfacial tension.

The proposed method provides an effective tool for studying and controlling co-continuous morphologies in polymer blends. Future work should include more realistic models that account for non-Newtonian effects and complex processing conditions.