Effects of concentration and chain length of the sequence copolymer on interfacial properties of homopolymers/sequence copolymers ternary blends: A DPD simulation study

Dongmei Liu; Huifeng Bo; Yongchao Jin; Deyang Li; Zhanxin Zhang; Kai Gong; Ye Lin; Sijia Li

doi:10.1371/journal.pone.0270094

Abstract

The effect of the concentration and chain length of the copolymer AB with sequence length τ = 8 on the interfacial properties of the ternary mixtures A₁₀/AB/B₁₀ are investigated by the dissipative particle dynamics (DPD) simulations. It is found that: i) As the copolymer concentration varies from 0.05 to 0.15, increasing the copolymer enrichment at the center of the interface enlarges the interface width ω and reduces the interfacial tension. However, as the concentration of the sequence copolymers further increases to 0.2, because the interface has formed micelles and the micellization could lower the efficiency of copolymers as a compatibilizer, the interfacial tension exhibits a slightly increase; ii) elevating the copolymer chain length, the copolymer volumes vary from a cylinder shape to a pancake shape. The blends of the copolymer with chain length N_cp = 24 exhibit a wider interfacial width w and a lower interfacial tension γ, which indicates that the sequenced copolymer N_cp = 24 exhibits a better performance as the compatibilizers. This study illustrates the correlations between the reduction in interfacial tension produced by the sequence copolymers and their molecular parameters, which guide a rational design of an efficient compatibilizer.

Citation: Liu D, Bo H, Jin Y, Li D, Zhang Z, Gong K, et al. (2022) Effects of concentration and chain length of the sequence copolymer on interfacial properties of homopolymers/sequence copolymers ternary blends: A DPD simulation study. PLoS ONE 17(7): e0270094. https://doi.org/10.1371/journal.pone.0270094

Editor: Rafael Verduzco, Rice University, UNITED STATES

Received: April 4, 2022; Accepted: June 4, 2022; Published: July 26, 2022

Copyright: © 2022 Liu et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the paper and its Supporting information files.

Funding: This work is financially supported by the Natural Science Foundation of Hebei Province (Grant B2020507044), the Science and Technology Project of Hebei Education Department (grant QN2022058).

Competing interests: The authors have declared that no competing interests exist.

Introduction

The interfacial behavior of copolymers AB between phase-separated homopolymers A and B has long been the focus of research [1]. It was shown in previous studies that block copolymers tend to aggregate at the interface [1–8]. The aggregated block copolymers can compatibilized the immiscible homopolymers and improve the mechanical strength of the composite materials. Though considerable attention has been focused on the simplest diblock copolymer compatibilizers during the last three decades, some research found that the compatibility of a copolymer strongly correlates to the architecture and sequence distribution of copolymers and, thereby, impacts the ability of a copolymer to strengthen the interface [9].

Both theoretical [10, 11] and experimental [3, 9] studies have proven that as the sequence distribution of a copolymer varies from alternating copolymer to diblock, the ability of the copolymer to modify a phase separation interface will also change. When the diblock copolymers are used to compatibilized the immiscible homopolymers, they will form a cylindrical or dumbbell shape. As the sequence length of the copolymer decreases under the fixed copolymer chain length, the time of the copolymer crossing the interface will increase. As the sequence length of the copolymer further decreases and becomes alternating, the copolymer will cover a substantial area of the interface and attain a pancake-type structure. The number of times the alternating copolymer crosses the interface and the degree of penetration into the homopolymer phase are far less than that of the diblock copolymers [9].

Computer simulations provide another way to probe the structural and interfacial properties of the ternary mixtures comprised of sequence copolymers. Ko and Jo [12] found that the copolymer of different architectures exhibited different conformation by a Monte Carlo simulation. That is, the diblock copolymer chains spanned across the interface once and each block extended to the corresponding homopolymer phases, the random copolymers weaved back and forth across the interface [12], whereas the alternating copolymers only lay on the interface and hardly extended to the homopolymer phases. Dadmun [12–15] employed Monte Carlo simulations to explore the structure and miscibility of a ternary blend with different copolymer architectures. They found that though both block and alternating copolymers can modify the interface, the purely random copolymer had the weakest effect on interfacial strengthening. Genzer and coworkers [16] compared the effectiveness of block, alternating, random copolymers and protein-like copolymers (PLCs) at low copolymer concentration (0.66%) using discontinuous molecular dynamics (DMD) computer simulations. It was found that as efficient interfacial compatibilizers, PLCs with high-molecular-weight are likely to outperform the random or alternating copolymers. Subsequently, they explored the dynamics of the phase separation in the blends of A/PLC/B by kinetic Monte Carlo simulation, which showed that the PLC not only could effectively slow down the process of phase separation but also minimize the unfavorable components contacts, thereby reducing the interfacial tension [17]. In a recent paper [18] an investigation was presented on the interfacial and structural properties for ternary mixtures of copolymers with different sequence distributions by dissipative particle dynamics (DPD) simulations. The simulations showed that the interfacial tension decreased by increasing the sequence length of the copolymer from τ = 2 (the alternating structure) to 8, whereas it increased by further increasing the sequence length of the copolymer from τ = 8 to 32 (the diblock), thus the copolymer with sequence length τ = 8 was more effective in reducing the interfacial tension. We defined τ = 2 to correspond to an alternating structure and τ = 32 to a diblock referred to the studies of Genzer et al. [16] and Chang et al. [19]. Therefore, it would be necessary to further systematically study the effects of other molecular parameters (e.g. the concentration and the chain length of the copolymer) on the interfacial and structural properties of the interfaces for the ternary mixture composed of copolymer with sequence length τ = 8.

In the present paper, the earlier work is extended by investigating the effect of the concentration and length of copolymer with a sequence length τ = 8. DPD simulations are employed to study the interfacial and structural properties of ternary blends composed of copolymers with sequence length τ = 8. Below, we first describe the DPD methods and the simulation details of this work. Then, in the following section, we show the simulation results, and systemically analyze and discuss the results. We highlight the important role of the sequence copolymer concentration in reducing interfacial tension. Further, we elucidate the fundamental mechanism that the copolymers AB(τ = 8) with the chain length N_cp = 24 for strengthing the interface is more prominent. Finally, a brief summarize and some concluding remarks are offered.

Method

Dissipative particle dynamics (DPD) is a coarse-grained method presenting two important features [20]. First, in DPD simulation a bead usually represents a whole molecule or a molecule fragment. Second, the interaction between different beads occurs through soft potential, which results in the beads can overlap considerably. These features allow studying the structural and dynamic properties of polymer systems in a much larger length and time scale, which hardly be accessed by the all-atom molecule dynamics simulations [21]. These features also allow DPD to efficiently study the highly dispersed polymer mixtures. The model of this paper was constructed based on the previous studies of dispersed polymeric mixtures [18, 20, 22–37], which were briefly introduced as follows.

Model

The motion of all DPD beads abides by Newton’s second law [38], (1) where r_i, v_i, m_i represent the position vector, velocity vector, and mass of bead i, respectively. For the convenience of calculation, the mass m_i is taken as 1. The total external force F_i exerting on bead i can be divided into four component forces, which are the conservative force , the dissipative force , the random force and the spring force . The total force acting on ith bead by other beads and the component forces are given by: (2)(3)(4)(5)(6) where α_AB is the repulsive parameter. r_ij = r_i − r_j, r_ij = |r_ij|, e_ij = r_ij/r_ij is the unit vector of r_ij. r_ij represents the distance between beads i and j; γ is the friction coefficient of dissipative force; σ is the nose amplitude, which determines the intensity of the random; ξ_ij is a Gaussian random number. To ensure that the simulation obeys a canonical ensemble, the weight functions ω^D(r_ij) and ω^R(r_ij) for the dissipative and random processes, as well as the friction coefficient γ and the nose amplitude σ followed the fluctuation-dissipation theorem [38]. (7)(8) where k_BT = 1, k_B and T represents the Boltzmann constant and temperature.

The repulsive parameter α_AB can be calculated from the Flory-Huggins parameter χ_AB according to expression [38]. (9)

The repulsive parameters between the same and the different types of beads are α_AA = α_BB = 25 and α_AB = 40, respectively [28].

Simulation details

Materials Studio was used as the simulator of the present work. All simulations were carried out in a 30 × 30 × 30 cubic box in DPD reduced units with the number density ρ = 3, thus the system contains approximately 81000 beads. Periodic boundary conditions are applied in all three directions. The interaction radius and the friction coefficient are set as r_c = 1 and γ = 4.5 in DPD reduced units respectively. To ensure that the simulations reach the equilibration state, 2.0 × 10⁵ time steps are first performed with the time step Δt = 0.05 in DPD reduced units [39]. After that, 5.0 × 10⁴ time steps are performed. To achieve good statistics, we averaged the data of 10³ − 10⁴ independent samples from 5 parallel simulation runs.

In this paper, we focus on the mixtures composed of homopolymer A₁₀, B₁₀, and copolymer AB with sequence length τ = 8. The sequence length τ = 8 is defined by its periodicity, e.g. if a copolymer has an architecture that 4 A beads followed by 4 B beads, the sequence length of the copolymer is τ = 8. The content of homopolymers A₁₀ and B₁₀ are equal. In order to investigate the effect of the sequence copolymer concentration on the interfacial properties, we vary copolymer concentration from c_cp = 0.01 to 0.2 (c_cp = the number of copolymer beads / the total number of beads) while the chain length and the sequence length of the copolymer were fixed as of N_cp = 32 and τ = 8 respectively. The concentration here represents the volume fraction of the sequence copolymer AB in the simulation system. Fig 1 shows the schematic of AB copolymers of τ = 8, N_cp = 32. To explore the effect of sequence copolymer AB(τ = 8) chain length on the interfacial properties, the chain length of the copolymers are set as N_cp = 8, 16, 24, 32, 48, 64 with the copolymer concentration c_cp = 0.15.

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Fig 1. Schematic for copolymer AB of the sequence length τ = 8 and chain length N_cp = 32, where the blue and green spheres denote beads A and B of copolymer respectively.

https://doi.org/10.1371/journal.pone.0270094.g001

Interfacial tension is an important parameter to measure the properties of the polymeric blend system. In the conditions of the interface is perpendicular to the x-axis, the interfacial tension can be calculated following the Irving-Kirkwood equation [40] (10) where P_xx, P_yy, P_zz represent the pressure tensor of the x, y, and z-direction respectively, <> denotes the ensemble average.

To explore the microscopic configurations of the copolymers, we calculate the mean-square radius of gyration and its three components , and , and the orientation parameter q of the sequence copolymers. is the component normal to the interface, and are the components parallel to the interface. The orientation parameter q is obtained by [28] (11)

Additionally, we calculate the interfacial width w by fitting the function tanh ((x + d)/w) to the (ρ^A(x) − ρ^B(x))/ρ(x) across the interface, where d is the shift of the interface center along the x-axis [29].

Results and discussion

Effects of the sequence copolymer AB(τ = 8) concentration

In this section, the influence of the concentration of the sequence copolymer AB on the interfacial properties is discussed. To accelerate the formation of interfaces, the homopolymers A₁₀, B₁₀, and copolymers AB(τ = 8) were initially placed in distinct positions of the x-direction in the box. The copolymer concentration is set as c_cp = 0.05, 0.1, 0.15 and 0.2 while the chain length and the sequence length of the copolymers were N_cp = 32 and τ = 8 respectively.

Figs 2 and 3 show the morphology snapshots and density profiles for the blends of A₁₀/AB(τ = 8)/ B₁₀ of different copolymer concentrations, respectively. It is found that as the copolymer concentration increases from c_cp = 0.05 to 0.2, the enrichment of the copolymers at the interfaces exhibits significant increases [Figs 2 and 3(a)], which results in a decayed correlation between the A₁₀ and B₁₀ homopolymers, as shown in Fig 3(b). Specifically, as the concentration of the sequence copolymer AB(τ = 8) increases from c_cp = 0.05 to 0.1, the density of the beads A and B of the copolymers at the interfaces increases [the black squares and red dots in Fig 3(a)], whereas as c_cp of the sequence copolymer AB(τ = 8) further increases to 0.15 and 0.2, the density of the beads A and B of the copolymers at the center of the interfaces almost unchanged, the distribution widths of beads A and B of copolymers broaden. It is noticeable that as c_cp = 0.15 and 0.2, the interfaces have reached saturated: i) at c_cp = 0.15, the interfaces curved, ii) at c_cp = 0.2, the copolymers formed micelles at the interface. To investigate the effect of initial positions, we simulated the evolution of the blends’ morphology with all the chain positions that were initially randomized, which is shown in S1 Fig. S1 Fig shows that as the system did not reach equilibrium t = 50000, some copolymers dissolved in the bulk, whereas as the system reached equilibrium t = 200000, the superfluous copolymers also form micelles at the interface instead of dissolving in the bulk. We infer that this result is related to the energy change in the system.

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Fig 2. Morphology snapshots of systems at different sequence copolymer concentrations.

(a)A₁₀/AB(τ = 8)/B₁₀ ternary blends and (b) AB(τ = 8) copolymers at N_cp = 32. The red and yellow spheres denote bead A and bead B of homopolymers A₁₀ and B₁₀, and the green and blue spheres represent beads A and B of the AB copolymers.

https://doi.org/10.1371/journal.pone.0270094.g002

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Fig 3. Density profiles of beads A and B of the (a) AB(τ = 8) copolymers and (b) A₁₀ and B₁₀ homopolymers in the x-direction as a function of the copolymer AB(τ = 8) concentration at N_cp = 32.

https://doi.org/10.1371/journal.pone.0270094.g003

We calculate the mean-square radius of gyration and its three components , and , the orientation parameter q of the sequence copolymers AB(τ = 8) at different copolymer concentrations, as shown in Fig 4(a) and 4(b). It is found that as the sequence copolymers AB(τ = 8) concentration increases from c_cp = 0.05 to 0.2, exhibits an obvious increase, whereas and decrease slightly, which results in an increase of the orientation parameter q [Fig 4(b)]. Because of that and are greater than , the orientation parameter q < 0, which indicates that the copolymer AB with the sequence length τ = 8 is mainly oriented along with the directions of parallel to the interfaces.

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Fig 4. Mean-square radii of gyration

and the three components

,

and

(a), and orientation parameter q (b) of copolymers as a function of the copolymer AB(τ = 8) concentration at N_cp = 32.

https://doi.org/10.1371/journal.pone.0270094.g004

The variations of the interfacial width ω and the interfacial tension γ in the ternary blends with increasing the sequence copolymers AB(τ = 8) concentration c_cp are shown in Fig 5(a) and 5(b). As expected, the interfacial width ω increases monotonically with increasing the sequence copolymers AB(τ = 8) concentration c_cp, which is caused by the increase of the enrichment of the copolymers at the interfaces. However, the interfacial tension γ decreases fast first and then increases slightly with increasing the sequence copolymers AB(τ = 8) concentration c_cp. Specifically, with increasing the sequence copolymers AB(τ = 8) concentration c_cp from 0.05 to 0.15, a remarkable drop in the interfacial tension is observed, this is because as c_cp increases from 0.05 to 0.15, the density of beads A and B of the sequence copolymers AB(τ = 8) at the interface increases [as illustrated in Figs 2 and 3(a)], which results in the decayed correlations between the A₁₀ and B₁₀ homopolymer, thus the interfacial tension γ decreases. Further, as c_cp increases from 0.15 to 0.2, the interfacial tension γ increases slightly. A possible explanation for the elevatory interfacial tension at the concentration c_cp = 0.2 may be the result of the formation of the micelles [Fig 2 c_cp = 0.2]. The micellization can lower the efficiency of the copolymer as compatibilizers [41], therefore, the interfacial tension γ increases slightly.

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Fig 5. Interfacial width ω (a), and the interfacial tension γ (b) of the blends as a function of the copolymer AB(τ = 8) concentration at N_cp = 32.

https://doi.org/10.1371/journal.pone.0270094.g005

Effects of the sequence copolymer AB (τ = 8) chain length

To investigate how the chain length of the copolymer AB (τ = 8) influences the compatibilization of the phase-separated, we vary the chain length of the copolymers AB (τ = 8) from N_cp = 8 to N_cp = 64, at c_cp = 0.15.

Fig 6 shows the relative density profiles of the copolymers AB (τ = 8) and the homopolymers A₁₀ and B₁₀ with copolymers chain length N_cp = 8, 24, 64. It is found that very little difference in the density profiles of the copolymers at various copolymer AB chain lengths is observed, as N_cp = 24, the densities of A+B beads of the copolymers AB (τ = 8) at the center of the interface are lower slightly [the red dots of Fig 6(a)], and as N_cp = 64, the density of A+B beads of the copolymers AB (τ = 8) at the center of the interface is higher slightly [the blue up triangles of Fig 6(a)]. However, little to no difference is observed in the density profiles of the A₁₀ and B₁₀ homopolymers at different copolymer chain lengths, as shown in Fig 6(b). This is because, despite the fact that the chain length of the AB copolymers is increased, the sequence length τ = 8 of copolymers remains unchanged. Hence, the density profiles of homopolymers at different copolymer chain lengths almost coincide.

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Fig 6. Density profiles of beads A and B of the (a) copolymers AB(τ = 8) and (b) homopolymers A₁₀ and B₁₀ in the x-direction as a function of the copolymer AB(τ = 8) chain length at c_cp = 0.15.

https://doi.org/10.1371/journal.pone.0270094.g006

Fig 7(a) and 7(b) show dependence of the mean-square radius of gyration and its three components , , , and the orientation parameter q of copolymers on the copolymer AB(τ = 8) chain lengths. We found that as the chain length of the copolymers AB(τ = 8) increases from N_cp = 8 to 64, almost remains unchanged, and increase monotonically, thus orientation parameter q decreases [Fig 7(b)]. Moreover, it is obvious that as N_cp = 8, is larger than and , the copolymer volume is shaped like a cylinder [16], as N_cp > 16, = = , whereas N_cp > 16, is smaller than and , the copolymer volumes are shaped like a pancake [16]. This means that when the sequence length of the copolymer τ = 8, with increasing the copolymer chain length, the volumes for the copolymer distributed at the interfaces vary from a cylinder shape to a pancake shape.

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Fig 7. (a) Mean-squared radius gyration

and the three components

,

, (b) orientation parameter of copolymer as a function of the copolymer AB(τ = 8) chain length at c_cp = 0.15.

(N_cp = 8, 16, 24, 32, 48, 64).

https://doi.org/10.1371/journal.pone.0270094.g007

Fig 8(a) and 8(b) depict the dependence of the interfacial width ω and the interfacial tension γ on the chain length N_cp of the copolymers AB(τ = 8). We found that although the density profiles of the copolymers and homopolymers at different copolymers chain lengths little change, the interfacial width ω and interfacial tension γ vary significantly. Specifically, as the copolymers’ chain length increases from N_cp = 8 to 24, the interfacial width w increases [Fig 8(a)], and the interfacial tension γ decreases [Fig 8(b)]. This result is inconsistent with the results of diblock and triblock copolymers, in a blended system of diblock or triblock copolymers, the shorter the chain length of the diblock and the triblock copolymer the lower the interfacial tension [24, 28]. As the chain length of the sequenced copolymer further increases from N_cp = 24 to 64, the interfacial width w decreases the interfacial tension increases. These results indicate that the blends composed of the copolymers AB(τ = 8) with the chain length N_cp = 24 displays a wider interfacial width and a lower interfacial tension than those of other chain lengths. Therefore, we conclude that the copolymers AB(τ = 8) with the chain length N_cp = 24 serve as more effective compatibilizers than the short diblock and the long sequence copolymer for strengthening the interfaces. S2 Fig shows the morphology snapshots for the blends of A₁₀/AB(τ = 8)/B₁₀ of different copolymer chain lengths, respectively. We found that as the chain lengths of copolymers increase from N_cp = 8 to 64, the interfaces undergo the following changes: i) flat interfaces of not saturation (N_cp = 8 and 16), ii) interface saturation with change in the interfacial geometry (N_cp = 24 and 32), iii) formation of micelles (N_cp = 48 and 64). Therefore, we inferred that the reason of the nonmonotonic behavior of the interfacial tension and the interfacial width in Fig 8 might be the no saturation for N_cp < 24 and the over saturation for N_cp > 24.

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Fig 8. Interfacial width ω (a), and the interfacial tension γ (b) of the blends as a function of the copolymer AB(τ = 8) chain length at c_cp = 0.15.

(N_cp = 8, 16, 24, 32, 48, 64).

https://doi.org/10.1371/journal.pone.0270094.g008

Conclusion

In this paper, the DPD simulation method is used to explore the effects of the concentration and the chain length of sequence copolymer with sequence length τ = 8 on the interfacial properties of A₁₀/AB(τ = 8)/B₁₀ ternary blends.

Our simulations show that as the concentration of the copolymer AB(τ = 8) (N_cp = 32) increases from c_cp = 0.05 to 0.15, the density of beads A and B of copolymer AB(τ = 8) at the interface significantly increases, which results in the reduction of the interfacial tension. However, as c_cp of the AB(τ = 8) copolymers further increases from c_cp = 0.15 to 0.2, due to the copolymers AB(τ = 8) at the interface has formed micelles at c_cp = 0.2, and the micellization could lower the efficiency of the copolymers as a compatibilizer, therefore, the interfacial tension increases slightly. By elevating the chain length of the sequence copolymer AB(τ = 8) from N_cp = 8 to 64 at c_cp = 0.15, the copolymer volumes vary from a cylinder shape to a pancake shape. As the chain length of the copolymer AB(τ = 8) N_cp = 24, the blend exhibits a wider interfacial width ω and a lower interfacial tension, which indicates the copolymers AB(τ = 8) of the chain length N_cp = 24 for strengthing the interface is more prominent.

The obtained results indicate that the structural and mechanical properties of the blends composed of the sequenced copolymer are strongly correlated to the concentration and the chain length of the copolymers. In this context, we deduce it would be necessary to further investigate the dependence of the interfacial and phase behaviors of the ternary blend composed of sequence copolymer on the other molecular parameters.

Supporting information

S1 Fig. Representative morphology snapshots for A₁₀/AB(τ = 8)/B₁₀ ternary at different simulation times.

Compositions are A₁₀/AB(τ = 8)/B₁₀ ternary blends for (a1, a2), and AB (τ = 8) copolymers for (b1, b2). Chain length and concentration of the copolymer are fixed as N_cp = 32, c_cp = 0.2. Red and yellow spheres represent bead A and bead B of homopolymers, and green and blue spheres represent beads A and B of the copolymers.

https://doi.org/10.1371/journal.pone.0270094.s001

(PDF)

S2 Fig. Morphology snapshots of systems at different copolymer chain length.

(a)A₁₀/AB(τ = 8)/B₁₀ ternary blends and (b) AB (B₁₀) copolymers at c_cp = 0.15. The red and yellow spheres denote bead A and bead B of homopolymers A₁₀ and B₁₀, and the green and blue spheres represent beads A and B of the AB copolymers.

https://doi.org/10.1371/journal.pone.0270094.s002

(PDF)

Acknowledgments

We are grateful for the essential supports of Hebei Key Laboratory of Data Science and Application and School of Materials of Sun Yat-sen University.

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Figures

Abstract

Introduction

Method

Model

Simulation details

Results and discussion

Effects of the sequence copolymer AB(τ = 8) concentration

Effects of the sequence copolymer AB (τ = 8) chain length

Conclusion

Supporting information

S1 Fig. Representative morphology snapshots for A10/AB(τ = 8)/B10 ternary at different simulation times.

S2 Fig. Morphology snapshots of systems at different copolymer chain length.

Acknowledgments

References

S1 Fig. Representative morphology snapshots for A₁₀/AB(τ = 8)/B₁₀ ternary at different simulation times.