Dissipative particle dynamics simulations of the morphologies and dynamics of linear ABC triblock copolymers in solutions

Abstract

Due to the great potential in the field of multifunctional nanoreactors and carriers, several previous works have shown the interesting morphologies of multicompartment micelles from simple linear ABC triblock copolymers in dilute solutions. In this work, for linear ABC terpolymers, their concentration-induced morphologies and morphological transitions are investigated with dissipative particle dynamics simulations. Firstly, several novel morphologies beyond those already known for dilute solutions, including the spherical core–shell–corona (CSC) micelle containing a small reverse CSC inside, the hexagonal packed cylinders in a lamella, the disk CSC micelle with a ring core, the multi-segment with a ring shell, and so on, were observed by varying the terpolymer concentration and the ratio of the three blocks. Secondly, the increase of concentration generally resulted in the morphological transition from three- or two-dimensional (3D or 2D) to one-dimensional (1D) structures. Finally, the dynamic pathway of morphological formation is similar to that of ABC star miktoarm terpolymers, which has three steps, i.e. nucleation, coalescence and growth. Moreover, the qualitative analysis showed that interfacial tension plays a definite role in the formation of the final morphologies. This work enriches the molecular-level knowledge of the morphology of multicompartment micelles from the concentration-induced self-assembly of simple linear ABC terpolymers and reveals their formation pathways, which will be useful for the future preparation and application of novel micelles.

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1 Introduction

Exquisitely ordered nanostructures from the self-assembly of block copolymers has attracted much attention in the field of drug delivery, microelectronic materials, advanced plastics, and so on.^1–4 Precise synthetic methods developed over the past decade have afforded access to a broad portfolio of multiblock copolymers,^5–7 which offers unparalleled opportunities for designing new nanostructures by the fascinating bottom-up self-assemble strategy. At the same time, it has also opened Pandora’s box, as the myriad of molecular design possibilities poses a daunting challenge for experiments, even for theories and simulations.⁸ Up to now, the simplest AB diblock copolymers have been most extensively investigated by various means, leading to an in-depth understanding on their self-assembly behaviours.¹ It is well known that, by only adding a third block, the ABC triblock copolymer can dramatically expand the spectrum of accessible morphologies and, therefore, these are rapidly been given more attention. This kind of terpolymer mainly generates a variety of well-controlled multicompartment micelles with nanosized structural units, as originally proposed by Helmut Ringsdorf.⁹ Accordingly, in order to know what valuable ABC terpolymers should be synthesized for any given application, understanding the fundamental principles that govern their morphologies at the molecular level is of fundamental importance.

Early studies regarded concentric (core–shell–corona, CSC) onion morphologies as the default structures adopted by linear ABC (l-ABC) triblock copolymers.¹⁰ After the pioneering work by Lodge et al.,¹¹ most investigations have been focused on the self-assembly of ABC miktoarm star copolymers (μ-ABC). The main reason is that the miktoarm star copolymer can effectively suppress the concentric onion structure, leading to a fascinating array of new morphologies, such as bowl, worm, hamburger, raspberry, and so on.^12,13 Especially, a series of simulation and theoretical studies have also been performed to give a molecular-level understanding of the self-assembly of μ-ABC.^14–19 Recently, the works of Laschewsky and co-workers revealed an inspiring result where l-ABC terpolymers with different block sequences can also form non-concentric complex micelles.^20–24 Additionally, depending on the mixtures of THF, water and organic multiacids, Müller and co-workers also found several novel micelles (vesicle, toroid and undulated ribbon) from the self-assembly of l-ABC terpolymers.²⁵ These significant experiments richly prove that the potential of the simple l-ABC terpolymer is grossly underestimated. Subsequently, we utilized DPD simulations to explore the phase diagram of l-ABC in a dilute solution. Rich complex morphologies beyond the traditional understanding, such as raspberry-onion, helix-on-sphere, cage, ring, bowl, were revealed and two interesting dynamic evolution mechanisms, “contacting and fusing” and “folding and fusing”, were also shown.^26,27 Based on DPD simulations, there are other important investigations of the morphologies of linear ABC triblock copolymers in dilute solutions and blends.^15,16,28,29 In addition, the Monte Carlo technique and the self-consistent field theory have been successfully used to explore the complex self-assemble behaviours of l-ABC in different conditions.^30–32 The continual studies using experiments, simulations and theories have enriched our knowledge on the self-assembly behaviours of linear and star ABC triblock copolymers, especially in dilute solutions. However, for linear ABC terpolymers, simulation works (experimental or theoretical) focusing on concentration-induced morphologies and morphological transitions are still rare. Experimentally the EISA (solvent evaporation induced self-assembly) method has been used to fabricate the template from ABC triblock copolymers.³³ Moreover, our previous DPD simulations also showed that, for the rod–coil–rod (ABA) triblock copolymer, the increase of concentration can effectively induce many novel morphologies.³⁴ Thus, in this work, we again use the DPD technique to predict the concentration induced morphologies of l-ABC terpolymers with different block sequences and block lengths. Furthermore, the dynamic evolution pathways of several new morphologies are also investigated.

2 Method and model details

The DPD method, originally developed by Hoogerbrugge and Koelman,³⁵ is a coarse-grained particle-based dynamics simulation technique, which is similar to MD but allows the simulation of the hydrodynamic behaviour in much larger, complex systems up to the microsecond range. Like MD, DPD particles obey Newton’s equation of motion, and the total forces on a particle i include a conservative force F^C, a dissipative force F^D and a random force F^R, i.e.

The three forces for non-bonded beads are given by

where α_ij is the repulsion parameter between bead i and j, which reflects the chemical characteristics of interacting beads; and r_ij = r_i − r_j, r_ij = |r_ij|, e_ij = r_ij/r_ij and v_ij = v_i − v_j. ζ_ij is a Gaussian random number with zero mean and unit variance. γ is the friction constant and σ characterizes the noise strength. To ensure the system satisfies the fluctuation–dissipation theorem and corresponds to the Gibbs canonical ensemble, only one of the two weight functions, w^D and w^R, can be chosen arbitrarily and this choice fixes the other one.³⁶ There is also a relation between the amplitudes (σ and γ) and k_BT. It is w^D =(w^R)² and σ² = 2γk_BT, where k_B is the Boltzmann constant and T is the temperature. Generally, based on the work of Groot and Warren,³⁷ the simple form for w^D = (w^R)² = (1 − r_ij)² and σ = 3 (γ = 4.5) are used, and Newton equations for all beads are integrated by a modified version of the velocity-Verlet algorithm with λ = 0.65. The particles connected by the spring force can be used to represent a polymer, and the interaction force F^S for bonded beads iswhere l_ij is the bond length between the two beads i and j. Here, the spring coefficient k_S = 4 and the balance bond length l₀ = 0 are chosen. For easy numerical handling, the cutoff radius r_c, the bead mass m, and the temperature k_BT are chosen as the units of the simulated system.

Fig. 1 shows our coarse-grained models for l-ABC (x-y-z), l-BAC (y-x-z) and l-ACB (x-z-y), which consist of the solvophilic block (A), the weakly solvophobic block (B) and the strongly solvophobic block (C), x, y and z are the number of beads of A, B and C, respectively. The solvent is represented by an individual bead S. The interaction parameters chosen are listed in Table 1. In fact, these appropriate DPD parameters were first defined by Zhong et al. to describe the multicompartment micelles of the famous μ-EOF miktoarm terpolymer.¹⁴ The following DPD simulations also used the same parameters and successfully explored the morphological diversity of the ABC terpolymer systems.¹⁵ However, Sheng et al. think that the interactions between a poly(perfluoropropylene oxide) (F) block and another two blocks are very strong and provide a suite of gentle parameters.¹⁶ Here, our main aim is to obtain the universal rule of governing concentration-induced morphologies for l-ABC in solutions. Therefore, we still adopt these present parameters for a better comparison with our previous works.

Fig. 1 The coarse-grained model of the linear ABC triblock copolymer with different block sequences.

Table 1 Repulsion parameters (DPD unit) in this work

	A	B	C	S
A	25	45	90	27
B	45	25	75	50
C	90	75	25	120
S	27	50	120	25

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To avoid the finite size effect, our simulations are performed in a larger cubic box of size (30r_c)³ under periodic boundary, containing 81 000 DPD beads with a random distribution at ρ = 3. The concentrations (φ, represented by the volume fraction of the terpolymers) of l-ABC in solution are chosen at the range of 0.05–0.3 to check the efficacy of varied concentrations. The time step Δt is 0.05 and a total of 1–2 × 10⁶ DPD time steps are carried out to guarantee the equilibration for each system.

3 Results and discussion

In general, the morphology of ABC triblock copolymers in a solvent is influenced by the concentration, the block length (or ratio), the interaction between different compositions, and so on. This is a large parameter space. It is difficult to consider all the factors influencing the morphology. In the present work, we fixed all the interaction parameters listed in Table 1 and focused on the effects of a varied concentration (φ) and the block length (by varying x, y and z) on the morphologies and their dynamics evolutions. For each one of the three sequences (l-ABC, l-BAC and l-ACB) there are 7 kinds of linear terpolymers with different block lengths (2-2-2, 2-2-8, 2-8-2, 8-2-2, 2-8-8, 8-2-8, 8-8-2), 6 kinds of concentrations (φ = 0.05, 0.1, 0.15, 0.2, 0.25, 0.3) and a total of 42 combinations. Considering all three sequences, a total of 126 equilibrium micellar morphologies (see the ESI†) are provided by our simulations. Several novel morphologies, not found by self-assembly in dilute solutions (generally φ = 0.1) are provided first. Next, we will introduce these new morphologies and their dynamics evolutions according to the order of l-BAC, l-ABC and l-ACB in detail.

For l-BAC, the middle solvophilic block (A) links two solvophobic blocks (B and C) as a bridge. The terpolymers with this kind of block sequence can bring more novel morphologies than the other two block sequences (introduced in the following text). Fig. 2 gives the phase diagram and detailed morphologies of l-BAC as a function of block length (y-x-z) and concentration (φ), respectively. As shown in Fig. 2 (top), the common core–shell–corona micellar structures (CSC_S and CSC_D) rarely appear in the whole phase space. A new CSC morphology, CSC², is revealed for l-BAC (2-2-8) at φ = 0.15 and 0.2. Our previous simulations^26,27 missed this interesting micellar morphology because of the low concentration (φ = 0.1). To carefully examine the internal structure of CSC², we give its density profile and the cutting image in Fig. 3 (top). The density profile shows that, under ρ ≈ 1.2, blocks A and B form a shell surrounding block C. However, beyond ρ ≈ 1.2, blocks A and B hide in block C. Based on the sunken degree of ρ of bead C, we can deduce the formation of a small inner core. Due to the spherical symmetry, the other two density profiles in the x and y directions have a similar result, and we do not show and discuss them again. Combining the cutting image of CSC², we can find that the outer-portion is a natural CSC structure, i.e. B is the shell and C is the core. However, the inner is a reversed CSC, i.e. C is the shell and B is the core (A is always located at the interface of the B and C domains). Only considering the thermodynamic factor, it is obvious that the interaction between the block and the solvent drive the formation of the outer-portion, and the interaction among the blocks bring the inner one. For an in-depth understanding, we also investigate the dynamics evolution of the CSC² micelle and the result is given in Fig. 3 (down). It is easy to see that the shorter solvophilic B block makes the system unstable and several droplets with the common CSC structure are formed after a short time (t = 3 × 10⁴). After that, the neighbouring droplets coalescence and become larger with the increase of time. At the latter stage, once a larger micelle contacts another one (t = 1 × 10⁵), both blocks A and B gather at the contacted interface (see the arrows labelled in Fig. 3) and then are embedded in the interior by the fusing of block C. The result indicates that the formation of the structure follows four steps, which are nucleation, coalescence, growth and embedding. If the latter stage (after t = 1 × 10⁵) is broken by the dynamic method, the common CSC, not the novel CSC² structure, would be obtained. Therefore, we know that controlling the dynamic evolution pathway is also a good alternative to tune the different morphologies for l-BAC terpolymers in solutions. Experimentally, Cui and co-workers used divalent organic counter ions and solvent mixtures to drive the organization of block copolymers down the specific dynamic pathway into complex one-dimensional structures.³⁸

Fig. 2 Top, the morphological phase diagram of l-BAC terpolymers in solution. The numbers listed along the y-axis are yxz, which represents the length of block B, A and C, respectively. φ on the x-axis is the concentration of l-BAC. Down, simulation morphologies represented by symbols in the phase diagram. Red, blue and green represent the A-, B- and C-rich domains, respectively. For clarity, several morphologies use the isodensity surface. CSC_S and CSC_D, spherical and disk core–shell–corona (CSC) micelle, respectively. CSC², a big CSC micelle containing a small CSC inside. RAS, raspberry micelle. MAS, multi-arm segment. CiS, cylinder in strip. CiL, cylinder in lamella. PML, perforated multi-layer. ML, multi-layer. PL_C, perforated lamella with a middle of block C. L_C, lamella with a middle of block C. SiG, sphere in gel. MC, multicore.

Fig. 3 Top, the density profile along the z direction for the CSC² micellar morphology and the inset image is the cutting image. Down, the dynamic evolution of the CSC² micelle. Red, blue and green represent the A-, B- and C-rich domains, respectively.

Fig. 4 gives the dynamic evolution of other three interesting morphologies, which are ML, SiG and CiL, respectively. As for the ML (representing the multi-layer structure) of l-BAC (2-2-2) at φ = 0.3, alternate solvophobic blocks B and C (layers) linked by the solvophilic block A form a one-piece lamella with a thickness of about 15 DPD unit and the thickness of single layer (B or C) is about a third of the lamella. The evolution of the ML structure in Fig. 4a shows that discrete blocks B and C-rich domains come forth first, and then individually fuse into the continuous rich domains. Finally, the whole evolves into a lamella. When the concentration decreases to φ = 0.25, the perforated multi-layer (PML) can be formed. SiG (sphere in gel) consists of blocks B and C assembled into the unattached spheres distributed in the gels of block A. l-BAC terpolymers with a longer solvophilic block A prefer to form this kind of structure and the different ratio of blocks B and C would result in different spherical sizes. For example, the ratio of B to C in l-BAC (8-8-2) is 4, which brings the bigger blue spheres of block B and the smaller green spheres of block C in the red gels of block A (see the ESI†). As shown in Fig. 4b, the formation of the SiG morphology has only two steps, nucleation and growth. Despite the simple dynamic formation pathway, the gelation behaviour of ABC terpolymers³⁹ and their application in drug release⁴⁰ have been given more and more attention. Then, the other interesting CiL (cylinder in lamella) structure from l-BAC (8-2-2) at high concentration φ = 0.3 consists of hexagonal packed cylinders of block C set in the lamella of block B, and the two domains are linked by block A. However, at low concentrations φ = 0.05 and 0.1, the l-BAC (8-2-2) triblock copolymer forms the raspberry-like (RAS) morphology. By comparing we can find the obvious effect of concentration on the morphology again. Fig. 4c gives the formation pathway of the CiL morphology. The result shows that block C first nucleate in the gel of block B at early stages (t = 3 × 10³), then coalesce and grow. At the same time, the shape of the whole develops into a lamella (t = 5.1 × 10⁴). When the lamella is stable, block C maintains the evolution trend (coalescence and growth) and begins to have a cylindrical shape (t = 1.5 × 10⁵), up to the formation of hexagonal packing (t = 1 × 10⁶). In fact, the dynamic evolution of RAS morphology is very different with the above one due to the different concentration, which follows the mechanism of “contacting and fusing” from the small raspberry nucleation.²⁷ Therefore, the concentration increase of the ABC terpolymers in solution can not only influence the final morphology but also the dynamic pathway.

Fig. 4 Dynamic evolution of ML (a), SiG (b) and CiL (c) micelles, respectively. Red, blue and green represent the A-, B- and C-rich domains, respectively. For clarity, the B and C-rich domains for the SiG and the C-rich domains for the CiL morphologies use the isodensity surface, respectively.

For l-ABC, the solvophilic block A links the weakly solvophobic block B and the strongly solvophobic block C. The terpolymers with this sequence generally form large numbers of concentric CSC morphologies, as shown in Fig. 5. In the left phase diagram, it is mainly occupied by the typical spherical and disk CSC micelles, which have a core of block C, a shell of block B and a corona of block A, and are represented by green spheres, especially at low concentrations. Here, we don’t distinguish the spherical CSC morphologies as our previous work based on their different shells, which can adopt several interesting morphologies, such as ring, cage and helix.²⁶ In addition, several new morphologies (C_RCS, Cyl_C, FT_C and DL_C) driven by the high concentration are still found, which also testify the effect of concentration on morphology. However, when the solvophilic block A is long enough, such as l-ABC terpolymers (8-2-2, 8-2-8 and 8-8-2), they prefer to form the absolute small CSC micelles and assemble like a gel (the symbol is the three spheres in Fig. 5), which is different from the SiG morphology from l-BAC (2-8-2, 2-8-8 and 8-8-2). The obvious difference is that we can draw out an integrated CSC micelle from the above gels and not obtain anything from the SiG gels.

Fig. 5 Left, the morphological phase diagram of l-ABC in solutions. xyz along the y-axis represent the length of A, B and C, respectively. Right, simulation morphologies represented by the symbols in the phase diagram. Cyl_C, cylindrical CSC micelle with a core of block C. C_RSC, disk CSC micelle with a ring core of block C. FT_C, flat tube CSC structure with a core of block C. DL_C, distorted lamella with a middle of block C. The three green spheres represent the congeries of several single CSC micelles. Red, blue and green represent the A-, B- and C-rich domains, respectively. For clarity, the B-rich domains in partial snapshots use the isodensity surface.

In addition, it is interesting that the new C_RSC (disk CSC micelle with a ring core of block C) micelle is detected first for l-ABC (2-8-2) at φ = 0.15. Fig. 6 (top) shows the internal structure of the C_RSC micelle based on its density profile and the section image along the longest diameter. The ρ data of bead C has an obvious decrease in the structural middle. Combining the cutting image, we can clearly know that the ring structure of block C results in the valley of ρ data. The raised peak of ρ data for bead A and B indicates that a number of beads A and B concentrate in the central hole of the C ring. Additionally, the cutting image shows a small hole in the centre of the whole micelle, which demonstrates that the whole C_RSC is also a ring structure. To explore the formation of the C_RSC micelle, we also simulate its dynamics evolution pathway, as shown in Fig. 6 (down). Before t = 1.5 × 10⁵, the l-ABC (2-8-2) terpolymer has the same evolution, i.e. nucleation of block C, coalescence and growth, as several l-ABC terpolymers with the disk-like CSC morphology (CSC_D). However, the transformation from CSC_D to C_RSC is a direct process without a transition state. It is a pity that we do not find out the direct reason for this process. An apparent explanation is that the ring structure of C_RSC is to ensure the biggest contacting area between the solvent S and the solvophilic block A. Of course, from the shape variation from an integrated disk to a ring we can speculate a qualitative change of interfacial tension for the micelle. Though the interfacial tension for planar interfaces can be calculated by the pressure tensor with the definition of Irving and Kirkwood,^42–44 for curving interfaces it is still not simulated exactly. To date, therefore, we can only give a qualitative estimation that the interfacial tension is important to drive the dynamic evolution of the micelle from the self-assembly of ABC terpolymers.

Fig. 6 Top, the density profile for the C_RSC micellar morphology and the inset image is the cutting image. Down, the dynamic evolution of the C_RSC micelle. Red, blue and green represent the A-, B- and C-rich domains, respectively. For clarity, the C-rich domains use the isodensity surface.

From the phase diagrams, we can see that, at high concentrations, linear ABC triblock copolymers are generally inclined to form the lamella morphology, including PL_C (perforated lamella with a middle of block C), L_C (lamella with a middle of block C) and DL_C (distorted lamella with a middle of block C), which are very different from the above PML and ML morphologies. They are the normal three-layer lamella, i.e. the strong solvophobic block C in the middle, the weakly solvophobic block B in the two sides and the solvophilic block A on the outside. This kind of array is easy to understand from the thermodynamic angle, therefore, we further focus on their dynamic pathway and the result is given in Fig. 7a. Taking PL_C from l-ABC (2-2-2) at φ = 0.2 as an example, it is interesting to see a similar formation pathway with C_RSC from l-ABC (2-8-2) at φ = 0.15. They all first form the CSC_D structure by the process of nucleation, coalescence and growth of block C before a given time (here t = 2.2 × 10⁵ for PL_C). For PL_C, in the following process, the larger CSC_D micelle turns into a lamella structure through the edge fracture of the integrated disk core (block C) and shell (block B). Obviously, the last morphological transformation should also be ascribed to the influence of interfacial tension. In addition, we also check the dynamics evolution of the congeries of CSC micelles in Fig. 7b. The simulation result shows that there are three common steps during the formation of congeries no matter what the concentration, which are nucleation, coalescence and growth. When the single CSC micelle arrives at a given size, the growth is over. The reason is that the longer solvophilic A has enough ability to better screen the influence of the solvent on solvophobic blocks B and C, which contributes to the stable gel-like systems.

Fig. 7 Dynamic evolution of PL_C (a) and the congeries of CSC micelles (b), respectively. Red, blue and green represent the A-, B- and C-rich domains, respectively. For clarity, the C-rich domains use the isodensity surface.

Compared with l-ABC, l-ACB merely exchanges the sequence of the weakly solvophobic block B and the strongly solvophobic block C. Therefore, like l-ABC, the phase diagram (Fig. 8, top) of l-ACB is dominated by the typical spherical and disk CSC micelle (blue), especially at low concentrations. The only difference is that the cores here are composed of block B (blue sphere). It is clear that l-ACB (8-2-2, 8-2-8 and 8-8-2) with longer solvophilic A still prefers to form the congeries of CSC micelles (labelled by the three sphere in Fig. 8) the same as l-ABC terpolymers. At high concentrations, l-ACB terpolymers are also inclined to the normal three-layer lamella, i.e. PL_B, L_B and DL_B (block B in the middle) similar to PL_C, L_C and DL_C (block C in the middle) in appearance. We have already shown the characteristic and dynamic evolution pathways for these morphologies, and compared them with those from l-BAC. Here we will not do so again. In fact, the little alternation in the block sequence still brings several different morphologies. For example, l-ACB (2-8-2) with a longer strong solvophobic block C forms a series of spotted CSC micelles, and the increase of concentration decides the different shape transition of the micelles (from sphere, ellipsoid, cylinder to strip, see Fig. 8, down). Among them, the spherical C²SC micelle is apparently similar to the above spherical CSC² micelle. However, by careful comparison, we can see that they have an entirely different kernel. The core of C²SC only comprises block B, and that of CSC² is a reversed CSC containing the three blocks. In addition, the l-ACB (2-8-8) at φ = 0.1–0.2 forms an interesting MSR morphology where block C builds a round segment surrounding the segment of block B. Interestingly, it is in accord with the morphology of laterally structured vesicles with a core from ABC miktoarm star terpolymers found by Wang et al.⁴¹ They have also given its formation pathway including three steps (i.e. nucleation, coalescence and growth).

Fig. 8 Top, the morphological phase diagram of l-ACB in solution. xzy along the y-axis represent the length of A, C and B, respectively. Down, simulation morphologies represented by the symbols in the phase diagram. C²SC, spotted CSC micelle with a core of block B. SE_B, spotted ellipsoid with a core of block B. SSR_B, spotted strip with a ring core of block B. SC_B, spotted cylinder with a core of block B. MS, multi-segment. HAM, hamburger. MSR, MS with a ring segment. PL_B, perforated lamella with a middle of block B. DL_B, distorted lamella with a middle of block B. FT_B, flat tube CSC structure with a core of block B. L_A, lamella with a middle of block B. Red, blue and green represent the A-, B- and C-rich domains, respectively.

In order to give an insight into these special morphologies, their formation pathways are investigated and given in Fig. 9. Firstly, the SE_B (spotted ellipsoid with a core of block B) micelle from l-ACB (2-8-2) at φ = 0.1 is taken as an example to represent the dynamic evolution of the series of spotted CSC micelles. Fig. 9a clearly shows that the l-ACB (2-8-2) system quickly forms the small CSC micelles in a short time (t = 5 × 10³), which have the core of block C and the spotted shell of block B. In the following process, the coalescence of CSC micelles makes several separate shells of block B into the core interior of block C and becomes the central deep-seated core. The same dynamic pathway is also found for the raspberry-onion-like micelle of l-ABC (2-12-2) at φ = 0.1 in our previous work.²⁷ Finally, the spherical CSC micelles gradually develop into the larger elliptic SE_B micelle by coalescence. As a whole, micelles with an inner core can be obtained from all linear ABC triblock copolymers with three sequences by the appropriate parameters. It is noteworthy that the tuning of dynamic evolution pathways is one of these important factors. Fig. 9b gives the dynamic evolution of the multi-segment (MS) worm-like micelle, whose morphology is close to the MC micelle from l-BAC (2-2-2) at φ = 0.05 and 0.1. However, there is a little discrepancy between these two morphologies. The alternate B and C disks in MC are separated and also linked by block A, however, those in MS are directly linked together and block A distributes the surface of block C as a corona. We have revealed the interesting dynamic pathway of “folding and fusing” for the MC micelle. Here, the dynamic evolution pathway for MS micelles shows a difference, it experiences the nucleation of solvophobic blocks, the coalescence by the “head-to-head” and “shoulder-to-shoulder” fashions, and the final multi-segment (MS) worm-like micelles. Like l-ABC (2-2-8) at φ = 0.25, all l-ACB (2-2-2) at φ = 0.15, 0.2 and (2-2-8) at φ = 0.2 form the flat tube (FT) CSC structure. The only difference is that the former has a core of block C (FT_C) and the latter has a core of block B (FT_B). The formation pathway of FT_B from l-ACB (2-2-8) at φ = 0.2 is shown in Fig. 9c. From its appearance, we generally think that this kind of flat tube morphology should be from the cylindrical CSC structures, only by simple extrusion. Interestingly, before the final flat tube structure, it also forms the disk-like CSC structure (t = 1 × 10⁵) the as same as C_RSC and PL_C. Obviously, the interfacial tension plays a crucial function during the last transformation of these morphologies. In Fig. 9d, we can also see the significant effect of interfacial tension on the formation of the hamburger (HAM) micelle from l-ACB (2-8-8) at φ = 0.25. The former steps, nucleation and coalescence, don’t show this particularity. It should be observed that a flat tube CSC structure appears at t = 1.1 × 10⁵, the difference is that the tube has several holes. With an increase of simulation time, the thin sides of the flat tube disappear and the shape of a long hamburger appears gradually. After a longer equilibrium time, HAM micelles are achieved at the final stage. In a word, during the dynamic evolution process of the concentration induced self-assembly for linear ABC triblock copolymers, the interfacial tension plays a vital role to decide the final morphology.

Fig. 9 Dynamic evolution of SE_B (a), MS (b), FT_B (c) and HAM (d) morphologies, respectively. Red, blue and green represent the A-, B- and C-rich domains, respectively. For clarity, the B and C-rich domains in several snapshots use the isodensity surface.

4 Conclusions

In summary, based on our previous work, we further utilize the DPD simulation method to provide an insight into the concentration induced self-assembled morphologies of linear ABC triblock copolymers and their dynamic evolution pathways, where A is a solvophilic block, B and C are weakly solvophobic and strong solvophobic blocks, respectively. The concentration (described by the volume fraction, φ) has a remarkable influence on the morphologies and dynamic evolutions of the micelles. The detailed phase diagrams show that, except for the conventional concentric core–shell–corona and other interesting micelles found in the previous works, several novel morphologies beyond our previous knowledge in the dilute solution are found, such as CSC², CiL, ML (or PML), SiG, C_RSC, FT_{C or B}, SSR_B, MSR, L_{B or C}, and so on. In general, with an increase of concentration, the morphologies change from spherical (three-dimensional, 3D) and multi-layer worm-like, cylindrical or flat tube-like (two-dimensional, 2D), to lamella (one-dimensional, 1D) structures. Furthermore, the formation pathways of several novel structures are analyzed and a universal mechanism with three steps, nucleation, coalescence and growth, can be found. In detail, for three typical C_RSC (3D to 2D), FT_B (2D) and PL_C (1D) morphologies, a kind of similar concentric disk-like CSC structure is found in their dynamic stage of growth. Based on the evolution of the whole shape, we can estimate that the interfacial tension plays a significant function in the final formation of the morphologies. These results are helpful to understand the formation mechanism of complex multicompartment micelles by the concentration induced self-assembly of linear ABC triblock copolymers and assisting experiments to find new morphologies.

Acknowledgements

All the authors appreciate very much the financial support from the Foundation of CAEP (no. 2014B0302040, 2014-1-075) and the National Nature Sciences Foundation of China (no. 11402241).

Notes and references

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Footnote

† Electronic supplementary information (ESI) available: All equilibrium micelle morphologies. See DOI: 10.1039/c5ra09661h

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