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. 2024 Feb 19;128(8):2018–2027. doi: 10.1021/acs.jpcb.3c07756

Dissipative Particle Dynamics Simulation for Reaction-Induced Phase Separation of Thermoset/Thermoplastic Blends

Yoshiaki Kawagoe †,*, Gota Kikugawa ‡, Keiichi Shirasu §, Yuuki Kinugawa †, Tomonaga Okabe †,∥,⊥
PMCID: PMC10911110  PMID: 38373192

Abstract

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Reaction-induced phase separation occurs during the curing reaction when a thermoplastic resin is dissolved in a thermoset resin, which enables toughening of the thermoset resin. As resin properties vary significantly depending on the morphology of the phase-separated structure, controlling the morphology formation is of critical importance. Reaction-induced phase separation is a phenomenon that ranges from the chemical reaction scale to the mesoscale dynamics of polymer molecules. In this study, we performed curing simulations using dissipative particle dynamics (DPD) coupled with a reaction model to reproduce reaction-induced phase separation. The curing reaction properties of the thermoset resin were determined by ab initio quantum chemical calculations, and the DPD parameters were determined by all-atom molecular dynamics simulations. This enabled mesoscopic simulations, including reactions that reflect the intrinsic material properties. The effects of the thermoplastic resin concentration, molecular weight, and curing conditions on the phase-separation morphology were evaluated, and the cure shrinkage and stiffness of each cured resin were confirmed to be consistent with the experimental trends. Furthermore, the local strain field under tensile deformation was visualized, and the inhomogeneous strain field caused by the phase-separated structures of two resins with different stiffnesses was revealed. These results can aid in understanding the toughening properties of thermoplastic additives at the molecular level.

Introduction

Thermoset resins are used extensively as matrix resins for composite materials and as adhesives. The chemical reaction between the base resin and curing agent forms a three-dimensional cross-linked structure, which results in resins having relatively high strength and stiffness despite being lightweight polymeric materials. However, several thermoset resins are brittle and exhibit poor toughness. Therefore, modifications have been made by introducing various tougheners.1−3 One major modification technique is reaction-induced phase separation. In this technique, a rubber component or thermoplastic resin is dissolved in a thermoset resin as a modifier, and a phase-separated structure is formed by demixing the modifier component during the curing process. In particular, thermoset/thermoplastic resin blends have been investigated extensively because they can provide improved toughness without significantly reducing the overall stiffness.1,4−13

In reaction-induced phase separation, the morphology of the phase separation changes depending on the curing conditions of the thermoset resin as well as the concentration and molecular weight of the toughener, and various properties can be expressed by controlling the morphology. Goossens et al.14 fabricated reaction-induced phase separations of diglycidyl ether of bisphenol A (DGEBA)/4,4′-diaminodiphenylsulfone (DDS)/polyoxymethylene (POM) blends under various curing conditions and investigated the morphological changes using small-angle laser light scattering, optical microscopy, and scanning electron microscopy (SEM). Rico et al.15 performed SEM observations on DGEBA/4,40-methylenebis (2,6-diethylaniline) (MDEA) with polystyrene (PS) at different molecular weights and concentrations. They clarified the correlation between the morphology, molecular weight, and concentration using thermodynamic interpretation based on the Flory–Huggins theory. As the cross-linking of thermoset resins is an exothermic reaction, the local heating rate may vary depending on the location. Odagiri et al.16 created phase-separated structures of DGEBA/diamino diphenylmethane (DDM)/poly(ether sulfone) (PES) blends at different heating rates and reported that the fracture toughness increased with the heating rate. Joy et al.17 investigated the morphology, glass transition temperature, impact strength, tensile strength, and Young’s modulus of the reaction-induced phase separation system of DGEBA/4,4′-DDS/methyl methacrylate acrylonitrile butadiene styrene (MABS). They reported that MABS concentrations below 10 phr resulted in a droplet-matrix structure and improved the ductility, whereas MABS concentrations above 15 phr resulted in a cocontinuous morphology and brittleness.

Numerical approaches have also been actively applied to understand various phase separation behaviors and experimentally observed properties. Numerical approaches must address phase separation over a wide range of temporal and spatial scales. In general, two approaches exist: field- and particle-based (or atomic-based). Field-based approaches, e.g., dynamic density functional theory (DFT)18−20 and dynamical self-consistent field theory,21,22 can reproduce various phase separation phenomena at a relatively low computational cost by determining the evolution of the compositional field. Oya et al.23 conducted a DFT simulation with the incorporation of a chemical reaction model and analyzed the multistep reaction-induced phase separation behavior in detail. However, the field-based method lacks information on the polymer chain conformation and the mechanical properties cannot be obtained.

All-atom molecular dynamics (MD) simulations have been used extensively in particle-based approaches to evaluate the resin properties.24−31 However, all-atom MD is not suitable for phase separation calculations because the computable spatial and time scales are too small, at ∼10 ns and ∼10 nm, respectively. Therefore, coarse-grained MD,32−34 multiparticle collision dynamics,35 and dissipative particle dynamics (DPD)36−38 have been used to model polymer molecules as coarse-grained chains and to track their behavior for reproducing phenomena on large time and spatial scales. These particle-based methods model chemical reactions by creating new bonds between particles within a distance threshold and reproducing reaction-induced phase separation. Li et al.32 used coarse-grained MD to reproduce the reaction-induced phase separation of bifunctional epoxy/diamine cross-linker/thermoplastic chains and reported that the domain growth follows a power law. Thomas et al.38 reproduced the reaction-induced phase separation of DGEBA/4,4′-DDS/PES using DPD. They introduced the Arrhenius-type reaction probability into the distance-based reaction model and evaluated the effect of the reaction parameters on the morphology formation. The reaction models and parameters in the above studies were determined parametrically and did not reflect the intrinsic material properties. In addition, these works focused on the phase separation formation and did not sufficiently discuss the material properties that were obtained from the phase separation structure.

Okabe et al.39 calculated the intrinsic reaction properties of resin species using quantum chemical calculations, proposed a new reaction model that incorporated these properties, and performed curing all-atom MD simulation for thermoset resins. This reaction model was applied to various resin types with chemical reactions.40−45 Kawagoe et al.46 extended this model to curing DPD simulation and showed that the reaction behavior could be produced with the same accuracy as that of all-atom MD simulation, with a significant reduction in the computational cost. In this study, curing DPD simulation was used to calculate the reaction-induced phase separation of thermoset/thermoplastic resin blends. DGEBA/DDM was used as the thermoset resin and PES was used as the thermoplastic resin. The reaction properties of the DGEBA/DDM were determined by ab initio quantum chemical calculations using global reaction route mapping (GRRM)47−52 to obtain the intrinsic material values. The DPD parameters were determined in a bottom-up manner using all-atom MD simulations. This enabled us to reproduce the behavior of the reaction and polymer molecules by incorporating the intrinsic resin properties and to reproduce the reaction-induced phase separation accurately, despite using a coarse-grained approach. In addition to the formation of the phase-separated structures, the curing properties, mechanical properties, and local strain distribution under deformation of the cured resin with phase-separated structures were evaluated.

Methodology

The chemical reaction properties of the thermoset resins were evaluated using GRRM, and the pure-component properties of the resins, which are necessary for identifying the parameters for DPD simulations, were determined from all-atom MD simulations. Curing DPD simulations were performed using the obtained reaction properties and DPD parameters, and the phase-separated structure formation and mechanical properties were evaluated. LAMMPS53a was used for all MD and DPD simulations, and an external python script was used to calculate cross-linking.

DPD Parameters

DGEBA and DDM were used as the base resin and curing agent, respectively, whereas PES was added to the thermoset resin as a toughener. Figure 1a depicts the coarse-graining of each molecule. DGEBA and DDM molecules were modeled as single beads, whereas a monomer unit was modeled as a single bead for the PES. Cross-linking between DGEBA and DDM beads occurs via chemical reactions: DGEBA beads can have a maximum of two bonds, whereas DDM beads can have a maximum of four bonds. PES has a linear structure that depends on its molecular weight. A schematic of the reaction in the DPD system is illustrated in Figure 1b. The dashed lines represent the bonds between DGEBA and DDM that are formed by the chemical reaction. The yellow beads in the figure represent 5-mer PES.

Figure 1.

Figure 1

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(a) Coarse-graining of DGEBA, DDM, and monomer of PES. (b) Reaction diagram of the DGEBA/DDM/PES blend. The dashed lines represent newly created bonds between the DGEBA and DDM beads. (c) Bonding diagram between DGEBA and DDM.

The time evolution of DPD beads is governed by Newton’s equation, and the force applied to the bead is expressed as

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Here, FCij is the conservative force between beads i and j, which is represented by the following soft repulsion

graphic file with name jp3c07756_m002.jpg 2
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where aij is the pairwise repulsion parameter; rc is the cutoff distance; and Inline graphic is the unit vector from beads j to i, where rij = ri – rj and rij = |rij|. The dissipative force FDij and random force FRij represent the viscous drag and thermal noise, respectively, and are expressed as

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Here, γ and σ are the amplitudes of the respective forces, which are related by σ2 = 2γkBT and γ = 4.5.38 The relative velocity vij = vi – vj, and ζij is the Gaussian random number with a zero mean and unit variance. The bond stretching force FBi was considered and calculated from the harmonic potential with a force constant of 4kBT/r2c and an equilibrium length of 0 for the intramolecular interactions. Therefore, the average bond length is determined from the bead radii. The previous study54 also attempted to estimate the bonding parameters from all-atom MD simulations. However, the estimated values were excessively large, and therefore, the same force constants were used for all bonds. The present study follows this approach.

The conventional parameterization scheme of DPD developed by Groot–Warren requires all beads to have the same density.55 In this study, we used the parameterization scheme proposed by Kacar et al. for beads with different local densities based on the molecular density.54,56 The DPD length scale was set such that the overall dimensionless DPD number density was approximately ρr3c = 3, which is a common value in DPD simulations. The bead volume ρ–1 was estimated from the average molecular volume, which is the weighted average of the pure species volumes, as follows

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In the above, Ni is the number of molecules of species i, ρpure,i is the pure component number density of species i, and PES1 represents the monomer of PES. The DPD parameters were determined from a system containing 20-mer PES at a concentration of 15 wt % (NDGEBA = 80,000, NDDM = 40,000, NPES1 = 1335 × 20 = 26,700), considering the various calculation conditions that are described later. The pure-component density ρpure,i was estimated from the respective pure-component all-atom MD simulations in the NPT ensemble (300 K and 1 atm), and the results are summarized in Table 1. In the MD simulation, the PES1 molecule was terminated with a hydrogen atom. The DREIDING57 force field was used, and the partial atomic charge was determined using QEq.58−60 According to eq 6, ρ–1 = 458.5 Å3; therefore, rc = 11.12 Å. Using the reference temperature T = 300 K and the mean molecular mass 281.98 g/mol, the reference time scale τ = 11.8 ps.

Table 1. Summary of Pure-Component MD Simulations.

species number of molecules mass density (g/cm3) molecular mass (g/mol) ρpure,i (Å–3)
DGEBA 200 1.0013 340.42 0.00177
DDM 340 0.9717 198.27 0.00295
PES1 380 1.2158 234.27 0.00313
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For like–like interactions; that is, interactions between the same species, the pairwise repulsion parameter can be obtained using the DPD equation of state

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where we assume that p = 23.8r–3ckBT and α = 0.101.56

For dislike interactions; that is, interactions between different species, the repulsion parameter can be determined by

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where χij is the Flory–Huggins interaction parameter, which is determined from the difference in the solubility parameter δi of each species.

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Table 2 summarizes the solubility parameter δi, the Flory–Huggins parameter χij, and the pairwise repulsion parameter aij. The solubility parameters were obtained from the literature,61 and these values were evaluated using Fedors’ method.62 The Flory–Huggins parameter indicates that DGEBA is a poor solvent for PES, whereas DDM is a good solvent. For the like–like parameter, a smaller ρpure,i; that is, a larger molecular volume, results in a larger aii and stronger repulsive interactions. The dislike parameter aij for good solvent pairs is smaller than that for poor solvent pairs.

Table 2. Summary of Solubility Parameter δi [(J/cm3)1/2], Flory–Huggins Parameter χij, and Pair-Wise Repulsion Parameter aij [kBT].

aij (χij) DGEBA δDGEBA = 20.5 DDM δDDM = 22.3 PES1 δPES = 23.1
DGEBA 35.63 (0) 21.95 (0.36) 22.27 (0.75)
DDM   11.86 (0) 11.53 (0.07)
PES1     10.45 (0)
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Curing Simulation

We used the curing DPD algorithm that was proposed in our previous study.46 This reaction model, which is an extension of the reaction model proposed for all-atom MD39 for DPD, has two criteria. The first criterion concerns the distance between the reactive beads; when the distance between the reactive beads is within the reaction cutoff, this bead pair becomes a reaction candidate. The reaction cutoff was set to rc,46 which is the same as the DPD cutoff radius, rc = 11.12 Å.

The second criterion is the reaction probability preact, which is calculated using the Arrhenius equation, as follows

graphic file with name jp3c07756_m011.jpg 10

where Ea is the activation energy, R is the gas constant, and T is the local temperature that is calculated from the corresponding pair, i.e., from two beads. The acceleration constant A, required to reproduce the curing reaction within a realistic computational time, is set to 1014. This value reproduced well the experimental curing behavior in the previous studies.44,46 A reaction occurs when preact is greater than a uniformly distributed random number between 0 and 1. Subsequently, the DPD simulation was performed for the cross-linking structure relaxation. Finally, the velocities of the reacted beads were scaled to be maintained at Kafter = Kbefore + βHf, where Kbefore and Kafter are the kinetic energies of the reacted bead pair before and after the reaction, respectively, and Hf is the heat of formation. The coefficient β is the adjustment factor that accounts for the reduction in degrees of freedom owing to coarse graining and is set to 0.069 (see ref (46) and Supporting Information). This local heating increases the probability preact of the next reaction judgment in eq 10.

Two types of reaction were considered for the DGEBA/DDM resin: epoxy/primary amine and epoxy/secondary amine reactions. The activation energies and heat of formation of these reactions were calculated using ab initio quantum calculations with GRRM at the B3LYP/6-31G(d) level of theory, as indicated in Table 3. The reaction probability preact varies with the value of the activation energy Ea. To distinguish between primary and secondary amines in the coarse-grained system, one of the two activation energies was selected probabilistically based on the number of bonds in the DDM bead: (1) no bonds: all reactions were epoxy/primary amine reactions, (2) one bond: Ea of epoxy/primary was selected with a probability of 2/3 and Ea of epoxy/secondary was selected with a probability of 1/3, because out of the three remaining unreacted sites in DDM, two correspond to the first reaction and one corresponds to the second reaction, (3) two bonds: each Ea was selected with a probability of 1/2, and (4) three bonds: all reactions were epoxy/secondary amine reactions. Using the selected activation energy and local temperature, the reaction probability was calculated with eq 10.

Table 3. Activation Energies Ea and Heat of Formation Hf of DGEBA/DDM Obtained in GRRM.

reaction type Ea (kcal/mol) Hf (kcal/mol)
epoxy/primary amine 55.09 15.32
epoxy/secondary amine 51.33 11.78
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In the curing simulation, the DPD relaxation and reaction judgment were performed alternately, and this cycle was continued until the desired cure conversion was achieved. In this case, the final conversion was set to 90% to allow for sufficient phase separation. The initial system size was determined such that the number density ρr3c = 3; for example, 35.73rc × 35.73rc × 35.73rc for a 10 wt % PES concentration. The DPD relaxation was performed for 10,000 steps with a time step of 0.005τ. In general, DPD simulations are calculated under a constant volume, whereas Lin et al.63 reported that DPD combined with a Berendsen barostat64 can reproduce phenomena under a constant pressure. In this study, the pressure was controlled using a Berendsen barostat at 23.8kBT/r3c to reproduce the cure shrinkage. The initial temperature was set to 1.0T (=300 K), and the temperature was increased by 0.002T per cycle. This is equivalent to a heating rate of 1.02 K/ns in real units. The temperature was maintained once it reached 1.51T (=453 K), which is the experimental curing temperature.16

Property Evaluation

Cure Shrinkage

The system shrank owing to pressure control in the DPD relaxation calculations for each curing cycle. The cure shrinkage was evaluated as the ratio of the final to uncured volumes V/V0 of each relaxation calculation. It was preliminarily confirmed that the aforementioned number of relaxation steps is sufficient to achieve volume convergence.

Structure Factor

We computed the partial radial distribution function gij(r)46,65 between PES beads (i = j = PES) to evaluate the structural properties of the phase-separated structure. The Faber–Ziman partial structure factor66 was calculated as follows

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where r is the length in the radial direction, q is the length of the scattering vector, and n0 is the PES number density.

Young’s Modulus

After curing, the system was cooled to 1T at a cooling rate of 2.04 × 10–3T/τ, followed by relaxation for 10,000 steps. Uniaxial tensile simulation was conducted at a strain rate of 1.0 × 10–5τ–1 (8.47 × 105 s–1). Orthogonal control was applied to maintain a constant system volume. The stress–strain responses were obtained, and the Young’s modulus was calculated by linear fitting below a 3% strain.

Results and Discussion

Reaction-Induced Phase Separation Behavior

The morphology transition during the curing reaction in the system of DGEBA/DDM containing 10 wt % 20-mer PES was investigated as a reference case. Figure 2 shows the morphology transitions during curing. Only PES beads are visualized in the figure, and the isosurface is drawn based on the particle density field. For clarity, the system is visualized by replicating it once in each direction. Figure 3 shows the evolution of the structure factor with curing; the downward arrows indicate the peak positions, which correspond to the largest periodic structure inside the system.

Figure 2.

Figure 2

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Evolution of the phase-separated structure with curing in the system of 20-mer PES at 10 wt %.

Figure 3.

Figure 3

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Evolution of the structure factor with curing in the system of 10 wt % 20-mer PES. The downward arrows indicate the peak positions.

The uncured DGEBA/DDM/PES blend existed in a single phase at the curing temperature. Below 20% cure conversion, no peaks appeared in the structure factor and the PES was still uniformly dispersed. The molecular weight of the thermoset resin increased as curing progressed. Therefore, the lower critical solution temperature (LCST) was expected to decrease and the mixture shifted to the two-phase region in the phase diagram.6 The peak of the structure factor appeared at 40% cure conversion, which indicates that phase separation started. The structure factor peak increased and shifted to the left with curing, indicating that the separation progressed and the domain grew. This is consistent with the light-scattering profile obtained from experimentally observed curing.6Figure 2 shows that PES could be observed in regions other than the large PES-rich domains below 50% cure conversion, whereas PES was ejected from the thermoset resin region and incorporated into the large PES-rich domains as the curing progressed. The cure conversion at the gelation point evaluated in the DPD simulation was 55%67 (cf. the theoretical value was 57.7%68). As the molecular mobility was much lower after the gelation point, the enlargement of the domains ceased and the peak shifted slightly to the right. As the curing progressed, the thermoset resin network structure and phase-separated structure were fixed. The phase separation was completed at 90% cure conversion, and spherical PES-rich phases with diameters of approximately 10–20rc were formed. The spherical thermoplastic-rich domain that was obtained in this simulation was the structure observed at low concentrations of thermoplastic resin, as reported in the experimental observations of previous studies.14−16

Effects of the Toughener Concentration

The toughener concentration is a factor that varies the phase-separation structure and contributes significantly to the mechanical properties of the toughened material. Kishi et al.7 observed a sea-island structure, with a PES-rich spherical phase/epoxy-rich matrix phase, at PES concentrations below 10 wt % using SEM in a DGEBA/DDM/PES blend. Furthermore, they reported a reversed epoxy-rich spherical phase/PES-rich matrix phase at concentrations above 20 wt % and an intermediate cocontinuous structure at 15 wt %. A similar concentration dependence was reported for a DGEBA/MDEA/PS blend by Rico et al.15 and a DGEBA/4,4′-DDS/POM blend by Goossens et al.14 We evaluated the phase-separated structure and properties at PES concentrations of 0, 5, 10, and 30 wt %.

Figure 4a presents the phase-separated structures that were obtained at different PES concentrations. A spherical phase-separated structure was formed at concentrations of 5 and 10 wt %, and the sphere size was larger at higher concentrations. This is consistent with previous experimental results.8,15 At 30 wt %, a cocontinuous structure was obtained, with the appearance of a mixture of lamellar and cylinder structures. Although the concentration conditions were quantitatively different, the experimental trends could be reproduced.7

Figure 4.

Figure 4

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Final phase-separated structure. (a) Effects of the PES concentration. (b) Effects of the PES molecular weight.

Figure 5 shows the degree of cure shrinkage as a function of the cure conversion for different PES concentrations. In the early stages of curing, thermal expansion due to heating is more dominant than cure shrinkage, and eventually cure shrinkage becomes dominant. The cure shrinkage was suppressed as the PES concentration increased. The degrees of the cure shrinkage of the only thermoset resin part at different concentrations were almost identical. The overall cure shrinkage decreased as the concentration of the nonshrinking PES increased. This is consistent with the experimental results obtained by Jose et al.9

Figure 5.

Figure 5

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Cure shrinkage with different PES concentrations.

Figure 6 depicts the stress–strain curves and Table 4 presents the obtained Young’s moduli. The stiffness of the thermoset/thermoplastic blend decreased with an increasing PES concentration because PES has a lower stiffness than thermoset resin with a cross-linked structure. Although the resin types differed, this trend is consistent with the experimental results of Francis et al.8 The relationship between the PES concentration and stiffness was almost linear, which suggests that the stiffness was determined by the concentration rather than by the phase-separated structure. As indicated in Figure 6, it was difficult to reproduce the nonlinear behavior of the resin because DPD simulates the molecules using soft bonds. Analysis using coarse-grained MD or all-atom MD with stiffer bond modeling may be necessary to evaluate the nonlinear behavior.

Figure 6.

Figure 6

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Stress–strain curve with different PES concentrations.

Table 4. Effects of the PES Concentration on Young’s Modulus and Cure Shrinkage.

concentration (wt %) molecular weight (mer) Young’s modulus (arb. units) cure shrinkage V/V0
0   3.41 0.901
5 20 3.18 0.904
10 20 3.00 0.906
30 20 2.45 0.919
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Effects of Toughener Molecular Weight

Subsequently, we compared systems with different molecular weights of PES at the same concentration. Curing simulations were performed using 5, 10, and 20-mer PES added at 10 wt %.

Figure 4b shows the phase-separated structure and Figure 7 shows the evolution of the structure factor with different PES molecular weights. Table 5 lists the Young’s modulus and cure shrinkage values. Although it was difficult to observe the characteristic structure of the 5-mer PES because the PES was still dispersed in the thermoset resin, Figure 7 indicates that it had a phase-separated structure. A larger molecular weight resulted in a larger domain size of the PES-rich phase. There was no significant difference in the cure shrinkage or Young’s modulus, which indicates that these properties were concentration dominated.

Figure 7.

Figure 7

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Comparison of structure factors at final conversion with different PES molecular weights.

Table 5. Effects of the PES Molecular Weight on Young’s Modulus and Cure Shrinkage.

concentration (wt %) molecular weight (mer) Young’s modulus (arb. units) cure shrinkage V/V0
0   3.41 0.901
10 5 2.98 0.910
10 10 2.96 0.907
10 20 3.00 0.906
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In general, a lower molecular weight of the additive resulted in a higher LCST. The increase in the molecular weight of the thermoset resin during curing caused the LCST to decrease and phase separation to begin. As a result, with a lower molecular weight of the PES, the cure conversion at which phase separation started occurred later. Figure 7 shows that the 20-mer PES had an obvious peak at 40% cure conversion, whereas the 10-mer and 5-mer PES had delayed peaks at 50 and 60% cure conversions, respectively. The structure was fixed before the phase separation was fully developed at low PES molecular weights, resulting in small domain sizes.

Effects of the Heating Rate

Although the relationship among the reaction frequency, diffusivity, and phase-separated structure has been discussed in previous numerical studies,32,34,37 it is difficult to link the experimental heating rate and phase-separated behavior quantitatively. In this study, we used a reaction model with activation energy and local temperature, which could directly reproduce the activation of the reaction and the increase in diffusivity with the increasing temperature. We evaluated the differences in the phase-separated structures for systems with different heating rates.

Heating rates of the 0.1T/cycle and the 0.002T/cycle were used, which correspond to 50.9 and 1.02 K/ns, respectively, in real units. Figure 8 shows the conversion curves with respect to the system temperature, and Figure 9 shows the phase-separated structure. The domain size of the PES-rich phase was larger at lower heating rates, which is consistent with the experimental results of Odagiri et al.16 The temperature rapidly reached the curing temperature (1.51T) at a high heating rate, resulting in the activation of the curing reaction from the early stage. As the curing reaction is superior to diffusion, the structure was fixed before the phase-separated structure was fully developed. In contrast, the curing reaction almost never occurred in the early stages at a low heating rate, and the reaction was gradually activated as the temperature increased. During the slow curing reaction, the structure was fixed after the phase-separated structure was fully developed via diffusion. Figure 10 shows the evolution of the structure factors at different heating rates. In both cases, the phase separation started at approximately 40% cure conversion, whereas high cure conversion was reached before the domain enlargement progressed (peak shift to the left) at high heating rates. No significant differences were observed in the cure shrinkage or Young’s modulus.

Figure 8.

Figure 8

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Conversion curves and temperature profiles at different heating rates.

Figure 9.

Figure 9

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Effect of heating rate on the final phase-separated structure.

Figure 10.

Figure 10

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Comparison of structure factor evolution at different heating rates.

Local Strain Distribution

Finally, we evaluated the local strain distribution in a resin with a phase-separated structure under uniaxial tensile deformation. When performing MD and DPD simulations, the direct conversion of the dynamics of molecular chains to continuum mechanics quantities, such as strain, is challenging. However, in the deformation of materials with heterogeneous structures, quantitative evaluation of the deformation field is important for understanding nonlinear response and fracture. In our previous study,69 we proposed a technique that combined all-atom MD and corrected smoothed particle hydrodynamics (SPH) to evaluate the local strain distribution with atomic resolution. The inhomogeneous strain and stiffness distributions of the thermoset resin near the solid under uniaxial deformation were revealed. A detailed description of this technique can be found in ref (69). In this study, we applied this technique to the phase-separated DPD simulations.

The strain distribution of the thermoset resin was visualized using DDM beads as SPH particles. The DGEBA beads moved together with the DDM beads; thus, the deformation could be adequately captured using only the DDM beads. The smoothing factor for SPH was set to 8rc. Conversely, the PES beads were highly fluid inside the agglomerates, making it difficult to evaluate the strain using this technique. Therefore, only the thermoset resin part is discussed here.

Three characteristic systems were used in this study. The first system was 5-mer PES at 10 wt %, in which the PES-rich phase was relatively well dispersed in the thermoset resin phase, as illustrated in Figure 4b. The second was 20-mer PES at 10 wt % with spherical PES-rich phases. The third was 20-mer PES at 30 wt %, which had a cocontinuous structure; that is, each phase was percolated.

Figure 11 presents a snapshot of the uniaxial tensile simulation. Figure 11a–c shows the PES-rich phase structure represented by a white surface and SPH particles (DDM beads) colored with the xx component of the Green–Lagrange strain ESPHxx calculated using this technique at applied strains of 3.5, 6.5, and 10%, respectively. The images were partially sliced to show the internal distribution. Figure 11d depicts only the PES-rich phase structure at an applied strain of 10%.

Figure 11.

Figure 11

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Snapshots of uniaxial tension of phase-separated resin. (a) Local strain ESPHxx distribution calculated from SPH at an applied strain of 3.5%. (b) ESPHxx distribution at an applied strain of 6.5%. (c) ESPHxx distribution at an applied strain of 10%. (d) PES-rich phase structures at an applied strain of 10%.

The difference in stiffness between the thermoset resin and PES resulted in an inhomogeneous strain field. For example, consider the system of 20-mer PES at 10 wt %. During early deformation (Figure 11a: 3.5% strain), the PES deformed selectively in response to macroscopic deformation owing to its softness. Consequently, the thermoset resin that was sandwiched between the PES-rich phases in the tensile direction was less deformed. However, the thermoset resin adjacent to the PES-rich phase in the transverse direction exhibited high strain because it was pulled by the greatly deformed PES-rich phase. Therefore, the PES-rich phase was expected to be significantly deformed in the polar regions, whereas the deformation in the inner region was relatively small. This trend was similar to that observed for other phase-separated structures. Under moderate deformation conditions (Figure 11b: 6.5% strain), the nonuniform strain distribution changed depending on the phase-separated structure. The system of 5-mer PES (10 wt %) had better dispersion and smaller PES-rich domains than the system of 20-mer PES (10 wt %); thus, the inhomogeneous strain field generated by the PES-rich phase was less pronounced. Consequently, the strain field was relatively uniform, and few low-strain regions appeared. As reported in the previous section, the Young’s moduli of these systems were nearly constant regardless of the molecular weight (Table 5), whereas the calculated local strain distribution was different. This result suggests that the failure behavior of the resin may differ depending on its molecular weight. In the systems of both 20-mer PES (10 wt %) and 20-mer PES (30 wt %), the low-strain regions between the PES-rich phases were more pronounced. In particular, small domains of thermoset resin surrounded by PES, such as those at the upper boundary of the figure for the 20-mer PES (30 wt %), exhibited almost no deformation. With further deformation (Figure 11c: 10% strain), the inhomogeneity became more pronounced in all phase-separated structures, and the high- and low-strain regions were distributed as layered structures. However, note that this amount of deformation can generally cause cracking or failure of the resin. Although it was difficult to reproduce the failure process in DPD because the molecules were simulated with very soft bonds, the evolution of an inhomogeneous strain field according to the phase-separated structure was revealed. Local large deformation can be the starting point of failure, and it helps to clarify the relationship between the phase-separated structure and toughening properties at the molecular scale.

Conclusions

In this study, coarse-grained curing simulations combined with ab initio quantum calculations and DPD simulations were used to accurately reproduce the reaction-induced phase separation of thermoset/thermoplastic blends. The DPD parameters were determined in a bottom-up manner from all-atom MD simulations and solubility parameters that reflected the properties of the resin species, rather than the idealized model parameters that have often been used. Furthermore, by incorporating a reaction model that consolidates the activation energy and heat of formation obtained from ab initio quantum chemistry calculations into DPD simulations, our analysis successfully combines polymer dynamics on a large space-time scale and precise chemical reactions. The effects of the PES concentration, PES molecular weight, and heating rate on the phase separation formation were investigated and the structure factor, cure shrinkage, and Young’s modulus were evaluated for each structure. These results were generally in agreement with the experimental results. Furthermore, the local strain distribution inside the phase-separated structure under uniaxial deformation was quantified. An inhomogeneous strain distribution that was dependent on the domain shape and size was observed in the phase-separated structure composed of soft thermoplastic and stiff thermoset resin. In particular, the thermoset that was adjacent to thermoplastic phases in the tensile perpendicular direction exhibited localized high strain, which could be the starting point of matrix failure. These results will aid in understanding the toughening properties of thermoplastic additives at the molecular level. It is difficult to capture the nonlinear and failure phenomena owing to the soft potentials used in DPD. In the future, we plan to partially reverse map the phase-separated structure obtained by DPD to an all-atom MD with realistic computational resources to investigate the detailed deformation and fracture behavior.

Acknowledgments

This research was supported by the Council for Science, Technology, and Innovation (CSTI) and the Cross-Ministerial Strategic Innovation Promotion Program (SIP), “materials integration for revolutionary design system of structural materials” (Funding agency: JST). This research was also supported by JSPS KAKENHI (grant nos. 21K14415 and 23H01291). Numerical simulations were performed on the “AFI-NITY” supercomputer system at the Advanced Fluid Information Research Center, Institute of Fluid Science, Tohoku University.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpcb.3c07756.

  • Phase-separated structure formation: concentration (MP4)

  • Phase-separated structure formation: molecular weight (MP4)

  • Phase-separated structure formation: heating rate (MP4)

  • Inhomogeneous strain field evolution (MP4)

  • Adjustment factor β for heat of formation and system size effect (PDF)

The authors declare no competing financial interest.

Footnotes

a

LAMMPS version: 23 June 2022.

Supplementary Material

jp3c07756_si_001.mp4 (24.7MB, mp4)
jp3c07756_si_002.mp4 (31MB, mp4)
jp3c07756_si_003.mp4 (14.3MB, mp4)
jp3c07756_si_004.mp4 (15.5MB, mp4)
jp3c07756_si_005.pdf (269.1KB, pdf)

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Supplementary Materials

jp3c07756_si_001.mp4 (24.7MB, mp4)
jp3c07756_si_002.mp4 (31MB, mp4)
jp3c07756_si_003.mp4 (14.3MB, mp4)
jp3c07756_si_004.mp4 (15.5MB, mp4)
jp3c07756_si_005.pdf (269.1KB, pdf)

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