Reaction-induced morphological transitions in a blend of diblock copolymers and reactive monomers: dissipative particle dynamics simulation

Abstract

The dissipative particle dynamics (DPD) method is applied to the morphological transitions of microphase-separated domains in a mixture of symmetric AB-diblock copolymers and reactive C-monomers, where polymerization and cross-linking reactions take place among C-monomers. The initial structure for the DPD simulation is an equilibrated cylindrical domain structure prepared by the density-biased Monte Carlo method with density profiles obtained from the self-consistent field theory. By introducing a cross-linking reaction among reactive C-monomers, we confirmed that the DPD simulation reproduces the morphological transitions observed in experiments, where the domain morphology changes due to segregation between A-blocks of diblock copolymers and cross-linking networks of C-monomers. When the cross-linking reaction of C-monomers is sufficiently fast compared to the deformation of the domains, the initial cylindrical domains are preserved, while the distance between the domains increases. On the other hand, when the formation of the cross-linking network is slow, the domains can deform and reconnect with each other in the developing cross-linking network. In this case, we observe morphological transitions from the initial domain morphology with a large-curvature interface to another domain morphology with a smaller-curvature interface, such as the transition from the cylindrical phase to the lamellar phase. We calculated the spatial correlations in the microphase-separated domains and found that such correlations are affected by the speed of the formation of the cross-linking network depending on whether the bridging between microphase-separated domains occurs in a nucleation and growth process or in a spinodal decomposition process.

---

1 Introduction

Controlling the properties of polymeric materials by blending polymers is of crucial importance in designing various composites. One of the successful examples is a thermosetting/thermoplastic polymer blend.^1–3 Epoxy resins have been widely used for industrial products including electronic and adhesive materials^4,5 and fiber-reinforced composites like CFRP (carbon-fiber-reinforced plastics)^6,7 because the epoxy resins have desirable mechanical and physical characteristics such as high thermal conductivity and insulation, except their toughness. It is well known that while epoxy resins possess large rigidity for many engineering applications, they allow cracks to grow and propagate easily, which leads to their undesirable brittleness. To overcome this brittleness of the epoxy resins, block copolymers have attracted great attention as a thermoplastic modifier to control the structure of nanoscale domains by their self-assembly.^8–12 The improvement of the fracture toughness has been observed in the experiments. Wu et al.⁸ reported that in the blend of epoxy and PBO–PEO (polybutylene oxide-b-ethylene oxide), the microphase-separated structure of branched worm-like micelles shows 19 times improved strain energy release rate, which characterizes the toughness, compared to the unmodified epoxy without PBO–PEO. Kishi et al.¹¹ also reported that in epoxy/acrylic block copolymers (PMMA-b-PnBA-b-PMMA, poly(methyl methacrylate)-b-poly(n-butyl acrylate)-b-poly(methyl methacrylate)), the nano-cylindrical structures exhibit a fracture toughness of 2530 J m⁻², which is twenty-fold compared to the unmodified epoxy thermosets. These observations undoubtedly indicate the importance of the precise prediction of the microphase-separated structures undergoing curing or cross-linking reactions to improve the functionalities of the epoxy resins.

To understand the mechanism of the microphase-separated structures of the epoxy/block copolymer blend, we focus on a pioneering experimental work reported by Lipic et al.^13,14 Their study employed a sample containing poly(ethylene oxide)-b-poly(ethyl ethylene-alt-propylene) (PEO–PEP) modifiers added to the diglycidyl ether of bisphenol-A (DGEBA) type epoxy thermosets. The authors reported that, depending on the volume fraction of modifiers and the temperature, the epoxy/block copolymer blend formed microphase-separated structures such as lamellar, gyroidal, hexagonal cylindrical, and spherical structures before the curing reaction. During the curing reaction, these microphase-separated structures transform into different morphologies, e.g., from a sphere to a cylinder. Their study demonstrates that “reaction-induced phase separation”^15,16 plays an important role in these morphological transitions.

As far as we know, few works have paid attention to reaction-induced morphological transitions for microphase-separated structures in view of the particle-based simulation, although several works studied reaction-induced “macrophase” separations.^17–20 In the present paper, by using dissipative particle dynamics (DPD) with a simple cross-linking reaction scheme, we simulate the epoxy/block copolymer blend in the mesoscopic scale. To simply mimic the target system studied by Lipic et al.,¹⁴ we investigate the morphological transitions in a blend of AB-diblock copolymers and reactive C-monomers, as explained in the next section. We first perform self-consistent field theory (SCFT) calculations to confirm the phase diagram obtained in the experiment. With the aid of the density profile of microphase-separated structures obtained from SCFT, we prepare well-ordered initial conditions for efficient DPD simulations. Starting from these initial conditions, we simulate the reaction-induced morphological transitions induced by the cross-linking reaction of reactive monomers. We also investigate the transient dynamics of the morphological transitions for various cross-linking conditions.

2 Model and simulation

To simulate our target system studied by Lipic et al.¹⁴ in the mesoscopic scale, we employ the standard DPD method^21,22 and we assume that our system consists of three types of DPD particles: A and B particles in the symmetric diblock copolymers (PEO–PEP) and reactive C particles (epoxy) as reactive solvent monomers as shown in Fig. 1. The reactive C particle forms a cross-linking network structure. We assume that the mixing of A particles in the diblock copolymer and the reactive C particles is athermal.

Fig. 1 Schematic picture of the model used in this study. Symmetric diblock copolymers are modeled by five A (blue) particles and five B (green) particles, which have a repulsive interaction with each other. Reactive C (red) particles are immiscible with B particles, and they can react to form cross-linking networks.

In the following section, we briefly review the DPD method used in the present article. This method has been applied to predict the kinetics of microphase separations of diblock copolymers,²³ and has recently been used for modeling polymer networks^24,25 and a thermoset resin curing simulation.²⁶ The motion of the DPD particles is described by Newton's equations of motion

where t is the time, m is the mass of a DPD particle, v_i is the velocity of the i-th particle, and F_i is the total force acting on the i-th particle. The total force F_i consists of four contributions:where F^C_ij is the conservative force from the j-th particle acting on the i-th particle, F^D_ij is the dissipative force, F^R_ij is the random force, and F^B_ij is the spring force which connects neighboring particles along the chain. The dissipative and random forces are given by

F^D_ij = −γω^D(|r_ij|)(r̂_ij·v_ij)r̂_ij (3)

and

F^R_ij = σω^R(|r_ij|)θ_ij(t)r̂_ij, (4)

where r_ij ≡ r_i − r_j, r̂_ij = r_ij/|r_ij|, and v_ij ≡ v_i − v_j. γ is the friction coefficient and σ is the noise strength which satisfies the fluctuation-dissipation theorem, σ² = 2γk_BT, and θ_ij is the Gaussian white noise characterized by

〈θ_ij(t)〉 = 0, (5)

and ω^D and ω^R are the weighting functions proposed by Español and Warren²³ as:where r_c is the cutoff radius.

As the conservative force, we adopt the soft repulsion force defined by

where a_ij is the repulsive interaction between i-th and j-th particles. In this paper, the values of a_ij are determined according to the study by Groot and Warren,²² where the repulsive interaction between the same types of particles a_ii is determined by

a_ii = 75k_BT/ρ, (9)

where ρ is the mean particle number density, and the interaction between the different types of particles a_ij is determined by

a_ij = a_ii + 3.27χ_ij (10)

for ρ = 3, where χ_ij is the Flory–Huggins χ-parameter. To construct bead-spring chains by the DPD particles, we introduce a linear spring between bonded particles,

F^B_ij = kr_ij, (11)

where k is the spring constant.

In addition to the standard DPD methodology explained above, we introduce a simple reaction algorithm for stochastic bond creation to reproduce the cross-linking process of reactive monomers. The reaction procedure is as follows:

1. For each reactive monomer, we list up surrounding reactive monomers within a given distance r_c.

2. We randomly choose one reactive monomer from the surrounding monomers listed up in step 1.

3. We create a permanent chemical bond between these monomers when p < p_c where p is the uniform random number in [0,1].

In this study, each reactive monomer can have up to 4 bonds. The above procedure is applied at every μ DPD time steps and for all monomers except for those with 4 bonds. Fig. 2 shows a schematic picture of the cross-linking reaction in this simulation explained above. A similar method for the cross-linking reaction was used in ref. 17 and 24.

Fig. 2 Schematic picture of the cross-linking reaction in this simulation. r_c is the cutoff radius of the monomer, within which the possible candidates for the cross-linking reactions are searched for. When a uniform random number p is smaller than a threshold value p_c, i.e., p < p_c, we assign a linear spring as a cross-linking bond between these monomers.

Before starting DPD simulations, we apply the SCFT for investigating the possible microphase-separated structures depending on the volume fraction of the A segment and the repulsive interaction between segments, and also apply the density-biased Monte Carlo method²⁷ for preparing well-ordered microphase-separated structures as initial conditions for DPD simulations. We use COGNAC and SUSHI in OCTA²⁸ as the DPD simulator and the SCFT simulator, respectively. About the theoretical details of SCFT and the density-biased Monte Carlo method, readers can refer to ref. 27 and 29.

3 Equilibrium microphase-separated structures before curing reactions

In this section, we discuss the equilibrium microphase-separated structures in a blend of symmetric AB-diblock copolymers and C-homopolymers before the cross-linking reaction takes place. It should be noted that in this section, we regard A and C monomers as the same species since their mixing is athermal. A theoretical phase diagram for such a blend of AB-diblock copolymers and A homopolymer system was obtained by Matsen^31,32 using the SCFT. Semenov and Likhtman^33,34 also studied a similar system in the strong segregation regime by using the strong segregation theory (SST) for N_d < N_h, where N_d and N_h are the molecular weights of the diblock copolymer and the homopolymer, respectively. Here we investigate the system similar to these studies but N_d = 10 and N_h = 1, and focus on the weak and intermediate segregation regimes. The diblock copolymer with N_d = 10 and reactive monomers with N_h = 1 correspond to the PEO–PEP and epoxy (BPA348), respectively, as mentioned in ref. 14. Since such a situation was not studied in ref. 32–35, we newly constructed a phase diagram of the blend of AB-diblock copolymers and homopolymers by using the SCFT, where the gyroid phase was not considered for simplicity as well as in our DPD simulations.

Fig. 3 shows the phase diagram predicted by our SCFT calculation. Our phase diagram is qualitatively similar to the experimental result obtained by Lipic et al.¹⁴ It should be noted that in the intermediate segregation regime with 20 < χN < 30, the location of the phase boundary between the lamellar and cylindrical phases and that between the cylindrical and spherical phases are independent of the incompatible parameter χN. Similar behavior is known to occur in the diblock copolymer melts (shown by the dashed curves in Fig. 3) but in a much stronger segregation region.^36,37 Moreover, the phase boundaries in the blend of the diblock copolymer and homopolymers locate at higher volume fraction regions than those in the diblock copolymer melts, because the homopolymers can fill the interstitial regions between the cylindrical domains or the spherical domains and relax the space-filling constraints imposed on the diblock copolymer melts.³³ We also note that the stable region of the spherical phase is much wider than the experimental result since the SCFT simulation does not include the thermal fluctuation effect which destabilizes the spherical phase. The density profile obtained by the SCFT simulation is used for efficiently generating the initial conditions of microscopic phase-separated structures for the DPD simulation using the density-biased Monte Carlo method mentioned in ref. 27.

Fig. 3 Phase diagram obtained using the SCFT. The horizontal axis represents the volume fraction of the A segment for AB-diblock copolymer melts, and the total volume fraction of A segments and C segments for a blend of AB diblock polymers and C homopolymers. The dotted regions colored by red, blue, green, and gray represent lamellar, cylindrical, spherical, and disordered phases of the blend, respectively. We do not consider the gyroid morphology for simplicity in this study. Coexistence regions are omitted as these regions are too narrow to be shown. The dashed curves show the phase boundaries of the diblock copolymer melts between L–G, G–C, C–S, and S–D, where L, G, C, S, and D represent lamellar, gyroid, cylindrical, spherical and disordered phases, respectively. These boundaries of the diblock copolymer melts are taken from ref. 30.

In the following sections, we show the results of the DPD simulation to investigate the effect of the cross-linking reaction on the initial domain morphologies. The total number of DPD particles are 139 986, which are placed in a simulation box with (L_x, L_y, L_z) = (44.0, 30.4, 34.9) for the cylindrical phase. A symmetric AB-diblock copolymer chain is composed of 5 A-particles and 5 B-particles, and thus a single block copolymer chain is composed of N_d = 10 DPD particles. The total number density of the DPD particles is fixed to ρ = 3, and the total number of block copolymer chains and reactive C monomers are (9797, 41 998) which correspond to the number fraction of the A particles and C particles as ϕ_A + ϕ_C = 0.65. The interaction parameters between AA, BB, CC, and AC-type pairs are set to a_AA = a_BB = a_CC = a_AC = 25, while the interaction parameters for AB and BC-type pairs are set to a_AB = a_BC = 46, which corresponds to χN = 64 using eqn (10). However, it should be noted that χN effectively reduces to (χN)_eff = 23–29 by considering the fluctuation effect due to a finite chain length according to the following relations:²³

orTherefore, the condition of our DPD simulations in the next section corresponds to the intermediate segregation regime. As the parameters for the spring constant of the harmonic bond and the friction, k = 4 and σ = 3.67 are used.²³ The time integration of the equations of motion is performed using a modified velocity-Verlet algorithm proposed by Groot and Warren²² with time step δt = 0.06. We monitored the temperature during the cross-linking reactions and confirmed that the temperature is controlled within the acceptable range,^22,23 even when we use δt = 0.06 in our reaction scheme. We performed three-dimensional DPD simulations starting from the initial conditions prepared in the above method. Fig. 4 shows the cylindrical phase as the uncured initial condition for the DPD simulations. To realize a true equilibrium state, we set the system size to satisfy the isotropic pressure condition, P_xx = P_yy = P_zz, where P is the pressure tensor. The same equilibrating procedure is used in ref. 38. It should be kept in mind that this condition is not useful for the spherical phase and the gyroid phase with the cubic symmetry because the isotropic pressure is achieved even when the system size is not optimized.

Fig. 4 The cylindrical phase as the initial condition for the DPD simulation obtained using the density-biased Monte Carlo method mentioned in ref. 27. Reactive C-monomers are not shown for visibility. We confirmed that these morphologies are stable in the simulation for 600 000 steps.

4 Cross-linking reaction

In the following subsections, we show the dynamics of domains triggered by the reaction-induced phase separations during the cross-linking reactions among reactive C-monomers in our DPD simulation. For the case of the rapid reaction where the cross-linking reaction is so fast that the initial domains cannot deform, we show that the domain morphology is fixed as it was in the initial structure. On the other hand, for the slow reaction case where morphological domains can deform and reconnect with each other, the initial microphase-separated structure transforms into other structures depending on the degree of convergence of the cross-linking reaction. These dynamics are then further studied in detail.

4.1 Domain growth dynamics

According to Lipic et al.,¹⁴ the reaction-induced morphological transitions were experimentally observed for a mixture of PEO–PEP and epoxy (BPA348), and the mechanism of these transitions is explained in their paper as follows: in the initial equilibrium state, the PEO sub-chain of the diblock copolymer and the epoxy are miscible. When the cross-linking reaction occurs, the molecular weight of the epoxy networks becomes larger, which results in a decrease in the translational entropy of the epoxy molecules. This decrease of the translational entropy leads to the segregation between the PEO block and the epoxy network. At the same time, the curvature of the interface of PEO–PEP reduces because the effective volume fractions of PEO blocks swollen by the epoxy decreases as a result of their segregation. The mechanism explained above is consistent with the morphological transition from the gyroid to the lamellar phase, and that from the spherical to the cylindrical phase during the curing process observed in the experiment.¹⁴

In the following discussion, we show that the above explanation is confirmed by our DPD simulations. First, we verify whether our simple cross-linking reaction scheme works. Fig. 5 shows the temporal change in the ratio of the number of unreacted C-monomers to the total number of C monomers in the system ρ_epoxy(N = 1) during the cross-linking reactions for the case of initial cylindrical phase. In our scheme, as the value of p_c decreases, the rate at which the number of unreacted monomers decreases slows down. In addition, when the product of μ⁻¹ and p_c is kept constant, ρ_epoxy(N = 1) decreased similarly. We use μ = 100 throughout our study.

Fig. 5 The ratio of the number of unreacted C-monomers to the total number of C-monomers in the system as a function of time for several reaction conditions from the cylindrical phase as the initial state. In our simple reaction scheme, the decay rate of the number of unreacted monomers in the system depends on μ and p_c only through the product μ⁻¹p_c.

To observe the reaction-induced morphological transitions during the cross-linking reaction, we prepare the cylindrical phase as the initial condition. We did not observe the morphological transition when we prepared the lamellar phase as the initial condition (not shown). In our study, three different cases are shown: (i) the moderate cross-linking reaction with p_c = 0.01, (ii) the slow cross-linking reaction with p_c = 0.0001, and (iii) the rapid cross-linking reaction with p_c = 1.0. This is justified by the following estimation: the relaxation time τ of the chain conformation with N = 10 in our system is roughly estimated as τ ∼ 100.²³ Now we define the reaction rate, i.e., the occurrence probability of the reaction during the unit time p_c,unit = p_c/(μδt) according to the relation shown in Fig. 5. We can regard p_c,unit as a reciprocal of the characteristic time for the growth of the cross-linking network, and then a dimensionless constant τp_c,unit gives 16, 0.16, and 0.0016 for p_c = 1.0, 0.01, and 0.0001, respectively. Therefore, if τp_c,unit ∼ 1, the growth of the cross-linking network is on the same order of the relaxation of the chain, which corresponds the case (i). For the case of τp_c,unit ≪1 (≫1), the network growth is much slower (faster) than the relaxation of the chain, which corresponds to cases (ii) ((iii)). We discuss case (iii) in the next subsection.

Fig. 6 shows the morphological transition in case (i). In this case, the cylindrical domains quickly transform into irregular lamellar structures. On the other hand, when the reaction probability is low as in case (ii), an ordered lamellar structure appears as shown in Fig. 7. We provide movies of temporal evolution of domain morphologies in each case as shown in the ESI.† In the provided movies, the reactive C-monomers are not shown for visibility of domains formed by diblock copolymers. It should be noted that these morphological transitions occur even if the mixing of the A-block of the diblock copolymer and the reactive C monomers is athermal. These morphological transitions can be purely caused by the decrease in the translational entropy of reactive monomers owing to the cross-linking reactions, even if there is no repulsive interaction among particles. In the case where the mixing of the A-block of the block copolymer and the reactive C monomer is not athermal, the partial compatibility may cause a nontrivial change in the result,³⁹ which is beyond the scope of this study.

Fig. 6 The time evolution of the domain morphology starting from the cylindrical phase under the cross-linking reaction with the moderate reaction probability p_c = 0.01. (left) t = 0, (right) t = 6000. In these figures, only particles are visualized without bonds. To view the temporal evolutions of domain morphologies, see the movie in the ESI.†

Fig. 7 Similar to Fig. 6 but with a low reaction probability p_c = 0.0001. (left) t = 26 400, (right) t = 48 000. To view the temporal evolutions of domain morphologies, see the movie in the ESI.†

Fig. 8 shows the instantaneous one-dimensional density profiles for each type of monomer. To calculate the continuous density distribution for each type of monomer from their coordinates, we use the extrapolation method, in which the density distributions on lattice points are calculated using the linear extrapolation function

where x_p is the position of the particle. Before the cross-linking reaction, A-blocks of diblock copolymers and reactive C-monomers do not segregate as shown in Fig. 8(a), since their mixing is athermal. After the cross-linking reaction, A-monomers and the cross-linking networks formed by the C-monomers segregate due to polymerization as shown in Fig. 8(b).

Fig. 8 Instantaneous one-dimensional density profile for each type of monomer along the x-axis at y = L_y/2 and z = L_z/2. (a) Initial condition t = 0 (Fig. 6(left)), and (b) t = 48 000 for the case of p_c = 0.0001 (Fig. 7(right)).

4.2 Domain spacing by cross-linking networks

In this subsection, we discuss the morphological behavior after the cross-linking reaction with p_c = 1.0, which we mentioned as case (iii) in the last subsection. When the formation of the cross-linking network is fast, we observe that the initial morphologies are preserved because the cross-linking network does not allow the cylindrical domains to deform and reconnect. Fig. 9 shows the cylindrical phase before and after the cross-linking reaction takes place. After the cross-linking reaction, the A-blocks of the diblock copolymer strongly segregate from the network matrix because the monomers lose their translational entropy due to the cross-linking reaction. In the experiment conducted by Lipic et al.,¹⁴ when the cross-linking reaction proceeds while the cylindrical domain morphology is preserved, a change in domain spacing (spacing between neighboring cylinders) is observed. To confirm the change in the domain spacing after the cross-linking reaction in DPD simulation, we estimate the most stable system size of the simulation box by measuring the anisotropy of the pressure. In order to measure anisotropy of the pressure, we first prepare an equilibrium system where P_xx = P_yy = P_zz is satisfied, and then we measure the pressure difference P_zz − (P_xx + P_yy)/2 after the cross-linking reaction by changing the system size in the parallel direction to the original cylindrical domains. After such preparation, we adjust the system sizes appropriately to find the isotropic equilibrium state.

Fig. 9 (left) A cylindrical phase as the initial domain morphology at t = 0. (right) The cylindrical domains in the cross-linking networks at t = 4800 under the cross-linking reaction with p_c = 1.0. The deformation of the cylindrical domains is hindered by the cross-linking network, and thus the cylindrical phase is preserved. It should be noted that both of particles and bonds are visualized here.

Fig. 10 shows the pressure difference after the cross-linking reaction. The simulations are performed by changing the system size L_x in the direction parallel to the cylindrical axis, where the total volume of the simulation box is kept constant. The horizontal axis shown in Fig. 10 represents the ratio of the simulated system size L_x,new to its original size before the cross-kinking reaction occurs, L_x. The vertical axis shows the anisotropy in the pressure tensor, whose zero value means that the system realizes an isotropic state with the deformed simulation box with L_x,new/L_x. When this ratio is unity, the anisotropy of the pressure tensor deviates from zero, which indicates that the structural anisotropy emerges due to the cross-linking reaction. To recover the isotropy of the pressure, we enlarge the system size L_x and L_y, and simultaneously reduce L_z to keep the system volume constant. Using this procedure, we obtain a state where an isotropic pressure is realized when L_x,new ≈ 1.055L_x. This suggests that the system expands in the direction parallel to xy-plane, that is, the plane perpendicular to the cylindrical axis, and shrinks along the direction parallel to the cylindrical axis (z-axis) during the cross-linking reaction. This behavior originates from the segregation between the B-block of the diblock copolymers and the monomers driven by the cross-linking reaction. When the cross-linking reaction occurs, the segregation becomes stronger because the monomers lose their translational entropy for mixing. This segregation causes an effective repulsive interaction between cylindrical domains. The enhancement of this immiscibility between the two phases inside and outside the cylindrical domains leads to an increase in the domain periodicity in the xy-plane. However, such a repulsion of the domains is compensated by the elastic force of the cross-linking network formed from the monomers. A balance between the repulsion force and the elastic attractive force determines the new equilibrium state. The similar behavior was observed in the experiment.¹³

Fig. 10 Anisotropy in the stress tensor due to the anisotropic domain morphology as a function of the ratio of the resized system size to the original system size. When the anisotropy is zero, the isotropic true equilibrium state is realized.

4.3 Morphological order for slow reactions

Here we discuss the dependence of the ordering kinetics on the reaction probability. In the previous subsections, we show the dynamics of morphological transition from the cylindrical structure to the lamellar structure during the cross-linking reaction, and we observe that the reaction probability p_c determines whether the system transforms into a regular (well-ordered) or irregular lamellar structure. To evaluate the dependence of ordering of the domain structure on the reaction probability p_c, we define the following correlation function:where 〈⋯〉 represents the ensemble average over 10 different simulations for a given reaction probability p_c. The three-dimensional density is defined as ρ(r) = ρ(x)ρ(y)ρ(z) by using eqn (14). The correlation function defined in eqn (15) represents the correlation or similarity of two-dimensional density distributions ρ(x, y; z) with the distance |z₁ − z₂|. Fig. 11 shows the correlation function according to eqn (15) for the case of each reaction probability, and Fig. 12 shows the examples of the two-dimensional cross-sections of the density distributions along the z-direction for each case.

Fig. 11 Correlation function C(z) of the density profiles in the planes normal to the cylindrical axis. For the case of p_c = 1, the initial microphase-separated structures are preserved and the correlation along the z-axis is kept high. For the case of p_c = 0.01, the correlation becomes much lower, which indicates that the lamellar structure is irregular along the z-axis. In the case of p_c = 0.0001, the correlation is higher than that in the case of p_c = 0.01, which indicates that the slower cross-linking reaction retains the regularity of the morphologies after the cross-linking reaction.

Fig. 12 Examples of two-dimensional cross-sections of the density distribution at some values of z-coordinates for different reaction probabilities. (a) In the case of the rapid reaction with p_c = 1.0, the preserved cylindrical domains are regularly arranged. (b) In the case of the reaction with the moderate rate p_c = 0.01, the cross-sections nearby (|Δz| ∼ 3) share a weak resemblance. (c) In the case of the slow reaction with p_c = 0.0001, the cross-sections at a large distance (|Δz| ∼ 6) are similar.

For the case of p_c = 1, which corresponds to the case of Fig. 9, the two-dimensional structure retains high correlation even for the two-dimensional cross-sections far from each other. The cross-linking network formed under the rapid reaction condition, as mentioned in the last subsection, does not allow the cylindrical domains to deform, and the cylindrical domains are preserved by the surrounding networks, as shown in Fig. 12(a). This leads to high correlation along the z-axis in the same manner as in the equilibrium state.

For the cases of p_c = 0.01 and 0.0001, the formation of cross-linking networks is rather slow and the domains can deform and reconnect with each other, leading to a reaction-induced cylinder-lamellar transition. As a result, the morphological transitions from cylinderical to lamellar structures are observed for both cases with p_c = 0.01 and p_c = 0.0001. However, the case with p_c = 0.0001 shows a correlation higher than the case with p_c = 0.01. This result suggests that ordering in the lamellar domains is promoted when the reaction is slow (p_c = 0.0001), while the lamellar domains are irregular when the reaction probability is moderate (p_c = 0.01), as shown in Fig. 12(b) and (c). These different behaviors are similar to those of reaction-induced viscoelastic phase separation.⁴⁰Fig. 13 depicts the intuitive explanation of the two processes of the connections of cylinders during cross-linking reactions. In the case of the moderate reaction rate, reactive monomers surrounding the cylindrical domains form a cross-linking network, whereby the translational entropy decreases rapidly and uniformly along the cylindrical domains, which causes an instability of the domain morphology similar to the spinodal decomposition (Fig. 13(a)). This leads to the random connections between cylindrical domains, where a given domain connects to some neighboring domains along the cylindrical axis. These random connections result in irregular lamellar structures in the case of moderate reaction probability. On the other hand, in the case of the slow reaction rate, the formation of the cross-linking networks proceeds gradually. When the cross-linking networks develop locally, a rung between neighboring cylinders is formed through an activation process due to thermal fluctuation. Once a rung is formed, it expands quickly along the cylindrical domain, leading to a formation of sheet-like domains. (Fig. 13(b)). As a result, a cylindrical domain is connected with fewer rungs to its neighboring cylinders in a regular manner compared to the case of the moderate reaction probability.

Fig. 13 Schematic pictures of the process of the connection between cylinders. (a) In the case of the moderate reaction rate, cylinders are randomly connected by bridging domains, as in the case of spinodal decomposition. (b) In the case of the slow reaction rate, the nucleation and growth process takes place, where a bridging domain between nearby cylinders is nucleated and grows along the cylinder swiftly.

5 Conclusions

In the present study, we perform DPD simulation to investigate the reaction-induced morphological transitions in a blend of symmetric diblock copolymers and reactive monomers. To prepare the true equilibrium state as an appropriate initial condition for the DPD simulation, we perform the SCFT calculation combined with the density-biased Monte Carlo method. Our SCFT simulation predicts the phase diagram that is consistent with the experimental results in the intermediate segregation regime with the incompatibility parameter χN. The experimentally observed morphological transitions during the cross-linking reaction from the microphase-separated structures with a high curvature to that with a lower curvature are well reproduced in our DPD simulation. Moreover, the experimental result of the increase in the domain space after the cross-linking reaction is successfully reproduced.

By simulating the cross-linking reactions with various reaction probabilities, we find that the morphological transitions undergo two different mechanisms, that is, nucleation and growth in the case of the slow reaction, and spinodal decomposition in the case of the moderate reaction. These mechanisms deeply relate to the reaction-induced instability and to the spatial arrangement and regularity of the microphase-separated domains after the cross-linking reaction. Our simulations offer a possible understanding of the dynamical pathway of the transition in domain morphologies, which contributes to applications in nanotechnology engineering.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The financial support from the Cross-ministerial Strategic Innovation Promotion Program (SIP) “Materials Integration for Revolutionary Design System of Structural Materials” (Funding agency: JST) is gratefully acknowledged.

References

1. A. F. Yee and R. A. Pearson, J. Mater. Sci., 1986, 21, 2462–2474.
2. R. A. Pearson and A. F. Yee, J. Mater. Sci., 1986, 21, 2475–2488.
3. R. A. Pearson and A. F. Yee, J. Mater. Sci., 1989, 24, 2571–2580.
4. A. Allaoui, Compos. Sci. Technol., 2002, 62, 1993–1998.
5. Z. Ahmadi, Prog. Org. Coat., 2019, 135, 449–453.
6. T. Okabe, T. Takehara, K. Inose, N. Hirano, M. Nishikawa and T. Uehara, Polymer, 2013, 54, 4660–4668.
7. B. A. Alshammari, M. S. Alsuhybani, A. M. Almushaikeh, B. M. Alotaibi, A. M. Alenad, N. B. Alqahtani and A. G. Alharbi, Polymers, 2021, 13, 2474.
8. J. Wu, Y. S. Thio and F. S. Bates, J. Polym. Sci., Part B: Polym. Phys., 2005, 43, 1950–1965.
9. E. Serrano, M. D. Martin, A. Tercjak, J. A. Pomposo, D. Mecerreyes and I. Mondragon, Macromol. Rapid Commun., 2005, 26, 982–985.
10. L. Ruiz-Pérez, G. J. Royston, J. P. A. Fairclough and A. J. Ryan, Polymer, 2008, 49, 4475–4488.
11. H. Kishi, Y. Kunimitsu, J. Imade, S. Oshita, Y. Morishita and M. Asada, Polymer, 2011, 52, 760–768.
12. H. Kishi, Y. Kunimitsu, Y. Nakashima, T. Abe, J. Imade, S. Oshita, Y. Morishita and M. Asada, EXPRESS Polym. Lett., 2015, 9, 23–35.
13. M. A. Hillmyer, P. M. Lipic, D. A. Hajduk, K. Almdal and F. S. Bates, J. Am. Chem. Soc., 1997, 119, 2749–2750.
14. P. M. Lipic, F. S. Bates and M. A. Hillmyer, J. Am. Chem. Soc., 1998, 120, 8963–8970.
15. K. Yamanaka and T. Inoue, Polymer, 1989, 30, 662–667.
16. K. Yamanaka, Y. Takagi and T. Inoue, Polymer, 1989, 30, 1839–1844.
17. H. Liu, H.-J. Qian, Y. Zhao and Z.-Y. Lu, J. Chem. Phys., 2007, 127, 144903.
18. A. A. Gavrilov, D. V. Guseva, Y. V. Kudryavtsev, P. G. Khalatur and A. V. Chertovich, Polym. Sci., Ser. A, 2011, 53, 1207–1216.
19. S. Thomas, M. Alberts, M. M. Henry, C. E. Estridge and E. Jankowski, J. Theor. Comput. Chem., 2018, 17, 1840005.
20. C. Li and A. Strachan, Polymer, 2018, 149, 30–38.
21. P. J. Hoogerbrugge and J. M. V. A. Koelman, Europhys. Lett., 1992, 19, 155–160.
22. R. D. Groot and P. B. Warren, J. Chem. Phys., 1997, 107, 4423–4435.
23. R. D. Groot and T. J. Madden, J. Chem. Phys., 1998, 108, 8713–8724.
24. G. Kacar, E. A. J. F. Peters and G. de With, Soft Matter, 2013, 9, 5785.
25. M. Langeloth, T. Sugii, M. C. Böhm and F. Müller-Plathe, J. Chem. Phys., 2015, 143, 243158.
26. Y. Kawagoe, G. Kikugawa, K. Shirasu and T. Okabe, Soft Matter, 2021, 17, 6707–6717.
27. T. Aoyagi, F. Sawa, T. Shoji, H. Fukunaga, J. Takimoto and M. Doi, Comput. Phys. Commun., 2002, 145, 267–279.
28. https://octa.jp/ .
29. Nanostructured Soft Matter, ed. A. V. Zvelindovsky, Springer, Netherlands, 2007.
30. M. W. Matsen, J. Chem. Phys., 2020, 152, 110901.
31. M. W. Matsen, Phys. Rev. Lett., 1995, 74, 4225–4228.
32. M. W. Matsen, Macromolecules, 1995, 28, 5765–5773.
33. A. N. Semenov, Macromolecules, 1993, 26, 2273–2281.
34. A. E. Likhtman and A. N. Semenov, Macromolecules, 1997, 30, 7273–7278.
35. M. W. Matsen, Macromolecules, 2003, 36, 9647–9657.
36. A. N. Semenov, Sov. Phys. JETP, 1985, 61, 733–742.
37. E. W. Cochran, C. J. Garcia-Cervera and G. H. Fredrickson, Macromolecules, 2006, 39, 2449–2451.
38. W. Liu, H.-J. Qian, Z.-Y. Lu, Z.-S. Li and C.-C. Sun, Phys. Rev. E, 2006, 74, 021802.
39. A. A. Gavrilov and A. V. Chertovich, Macromolecules, 2017, 50, 4677–4685.
40. Y. Oya, G. Kikugawa, T. Okabe and T. Kawakatsu, Adv. Theory Simul., 2022, 5, 2100385.

---

Footnote

† Electronic supplementary information (ESI) available: Movies of the morphological transitions under the cross-linking reaction discussed in Section 4.1. See DOI: https://doi.org/10.1039/d3sm00959a

---