Lamellar-in-lamellar self-assembled C–b–(B–b–A)_m–b–B–b–C multiblock copolymers: Alexander–de Gennes approach and dissipative particle dynamics simulations

Abstract

A simple theoretical analysis of the lamellar-in-lamellar self-assembled state of ternary C–b–(B–b–A)_m–b–B–b–C multiblock copolymer melts in the strong segregation limit is presented using the Alexander–de Gennes approximation. For a given value of m, the influence of the chain length of the various blocks and the value of the Flory–Huggins χ_AB and χ_BC interaction parameters on the number k of internal domains is discussed in detail. The theoretically predicted tendencies are corroborated by computer simulations using the dissipative particle dynamics technique.

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1. Introduction

During the past decades hierarchically ordered block copolymers and block copolymer based systems have been an attractive subject for experimental^1–15 and theoretical^16–22 investigations. The growing interest in this class of systems is due to the possibilities of creating periodic structures combining different periodicities. One of the first observations of a double-periodic behavior concerned the self-assembly in a melt of comb-shaped supramolecules consisting of polystyrene–block–poly(4-vinylpyridine) (PS–b–P4VP) diblock copolymers with hydrogen-bonded pentadecylphenol (PDP) side chains attached to the P4VP blocks.^1,2 The two different periods are connected to the two different intrinsic length scales: the long length scale period of the diblock copolymer and the short length scale period of the repeat motive of the comb-shaped P4VP(PDP) blocks. Depending on the relative volume fraction of the polystyrene block, lamellar-in-lamellar, lamellar-in-spheres, spheres-in-lamellar, etc. are formed.

Binary PS–b–(PI–b–PS)₄–b–PI–b–PS and ternary P2VP–b–(PI–b–PS)₄–b–PI–b–P2VP linear undecablock copolymers are the first examples of block copolymers with a linear architecture where double-periodic behavior has been observed experimentally.^4,5 Here P2VP, PI and PS denote poly(2-vinylpyridine), polyisoprene and polystyrene, respectively. For these multiblock copolymers all chemically different species are mutually immiscible. A 3-layered and a 5-layered lamellar-in-lamellar structure was observed for the binary, resp. ternary multiblock copolymer system. In these cases the long length scale period involves phase separation between the “long” PS resp. P2VP end-blocks and the middle multiblock, whereas the short length scale results from the microphase separation within the internal (PI–b–PS)₄–b–PI part of the multiblock containing much shorter blocks.

Both linear and graft-like multiblock copolymers have been extensively studied theoretically in the frameworks of self-consistent field calculations,^16,17 the weak segregation Landau approach^18–20 and strong segregation theory.^21,22 The copolymer composition, the architecture, the length of the different blocks and the values of the Flory–Huggins interaction parameters are the essential parameters. In our own theoretical work²¹ on ternary C–b–(B–b–A)_m–b–B–b–C multiblock copolymers the number k of internal A and B lamellar domains was addressed as a function of parameter m using self-consistent field considerations in the strong segregation limit. In ref. 22 we also analyzed and explained the essential difference between ternary C–b–(B–b–A)_m–b–B–b–C and binary A–b–(B–b–A)_m–b–B–b–A multiblock copolymers. The theoretical predictions obtained were in excellent agreement with the few experimental data available.^4,5

In this paper we reconsider the C–b–(B–b–A)_m–b–B–b–C multiblock copolymer melt using a much simpler theoretical approach based on the Alexander–de Gennes approximation.^23,24 Thus a very simple expression [eqn (12)] for the free energy is obtained which clearly demonstrates the influence of the number of internal layers k. We show that this simplified approach provides results that are virtually identical to the results obtained by the far more elaborated treatment. Using it we analyze and discuss the domain formation in ternary C–b–(B–b–A)_m–b–B–b–C multiblock copolymers in much more detail than before as a function of the chain length of the blocks and the Flory–Huggins χ_AB and χ_BC interaction parameters, limiting ourselves to the lamellar-in-lamellar self-assembled state. To corroborate the theoretically predicted tendencies, we performed computer simulations using the well-known dissipative particle dynamics (DPD) technique.^25–33

2. Theoretical model

Considering the ternary C–b–(B–b–A)_m–b–B–b–C multiblock copolymer melt in the Alexander–de Gennes approximation implies that we assume all 2m + 1 middle A- and B-blocks as well as the outer C-blocks to be stretched uniformly inside their respective layers. We assume that the 2m + 1 short middle blocks self-assemble into k internal layers confined between the outer C-layers.

In general, a global multiblock conformation can be either a bridge or a loop as illustrated in Fig. 1. Both global bridges and global loops consist in turn of local loops and bridges. In the spirit of the Alexander–De Gennes approximation,^23,24 we will not distinguish between global bridge and global loop conformations, assuming half of the global loop to be equivalent to half of the global bridge.

Fig. 1 A schematic representation of a global bridge (top) and a global loop (bottom) conformation for a C–b–(B–b–A)₄–b–B–b–C multiblock copolymer. A-, B- and C-blocks are denoted by red, yellow and green, respectively. In the illustration m = 4 and k = 5.

3. Theoretical analysis

Let n denote the degree of polymerization of the internal A- and B-blocks alike and N denote the degree of polymerization of the outer C-blocks, with N ≫ n. The statistical segment length and monomer volume are denoted as a and υ, respectively, and are assumed to be equal for all chemically different components. The thickness of the internal layers and the outer layer are denoted as h and H (Fig. 2). Furthermore Σ is used to denote the interfacial area per multiblock copolymer chain. The Flory–Huggins interaction parameters χ_AB, χ_BC and χ_AC are taken as positive, implying unfavourable interactions between all the chemically different species.

Fig. 2 A schematic representation of a global bridge conformation of a C–b–(B–b–A)_m–b–B–b–C copolymer for m = 6. h and H denote the thickness of the internal layers and half of the outer layers, respectively.

Incompressibility implies:

Nυ = HΣ (1)

(2m + 1)nυ = khΣ (2)

The total free energy per multiblock copolymer chain can be written as:

F = 2F_BC + (k−1)F_AB + mF_0A + (m + 1)F_0B + 2F_C + F_conf (3)

Here F_AB and F_BC are the interfacial free energies related to the interfacial tensions and the average interfacial area Σ per multiblock copolymer by:

F_AB = γ_ABΣ, F_BC = γ_BCΣ (4)

with interfacial tensions and

The elastic free energies of uniformly stretched short A- and B-blocks F_0A and F_0B are the same and in a scope of the Alexander–de Gennes approximation are:

Likewise, the elastic free energy F_0C for the outer C-blocks is given by:

The conformational term F_conf takes into account the global stretching of the middle multiblock chain. In our previous paper this conformational energy was approximated by F_conf = kln2.²¹ Here we introduce an improved expression obtained in the following way. The middle multiblock chain is divided into three parts, namely into two tail sections of type (B + A/2) containing one B-block and half of the A-block and a middle part consisting of (2m − 2) sections of type (A/2 + B/2) which contain half of an A- and half of a B-block. For the global bridge conformation the middle part of the multiblock chain starts at the centre of the second thin domain which is an A-domain and finishes at the centre of the (k − 1)th A-domain. Its conformation can be described by a random walk with drift on a periodic one-dimensional lattice with steps ending at the centres of the A- and B-domains. Assuming that the walk consists of k₊ steps in the positive direction and k₋ steps in the opposite direction (here we do not take into account the boundary conditions) so that k₊ + k₋ = 2m − 2 and k₊ − k₋ = k − 3, the conformational free energy (in k_BT energetic units) is given by:

A comparison with the very simple expression F_conf = kln2 used before,²¹ shows that the latter slightly overestimates the combinatorial contribution.

Taking into account eqn (1), (2), (4)–(7), the free energy expression (3) transforms into:

with:

γ* = 2γ_BC + (k − 1)γ_AB (9)

Note that eqn (8) is very similar to the corresponding equation for lamellar self-assembled diblocks. The main differences are the additional combinatorial terms and the obvious k-dependence of both the interfacial (γ*Σ) and the elastic stretching (Q/Σ²) contributions. Minimization of the free energy (8) with respect to Σ yields the equilibrium interface area:

which results in the final expression for the total free energy:

4. Results and discussions

We consider first the only system investigated experimentally so far, i.e., P2VP–b–(PI–b–PS)₄–b–PI–b–P2VP. It corresponds to χ_BC = 0.4, χ_AB = 0.1, χ_BCN = 340, χ_ABn = 17, m = 4 and n/N = 0.2. The free energy [eqn (12)] as a function of the number of internal layers k is presented in Fig. 3a. The minimum occurs for k = 5 precisely as found experimentally.⁴ The results for the values m = 3, 5 and 6 are presented in Fig. 3b, c and d. The values of k found are 5, 7 and 7. These results are in excellent agreement with the ones obtained from a much more elaborated mean-field calculation using the same set of parameters.²¹

Fig. 3 The free energy F of the lamellar-in-lamellar self-assembled C–b–(B–b–A)_m–b–B–b–C multiblock copolymer melt as a function of the number k of internal layers for χ_BCN = 340, χ_ABn = 17, n/N= 0.2. (a) m = 4, (b) m = 3, (c) m = 5, (d) m = 6.

In order to investigate the effect of the interfacial tension, numerical calculations were performed for different values of the Flory–Huggins χ_BC-parameter for m = 3, 4, 5 and 6, χ_AB = 0.1 and a fixed length of the internal blocks n = 200. Throughout the rest of this section the length N of the outer blocks is assumed to satisfy (2m + 1)n = 2N, thus assuring an equilibrium lamellar structure. The results are summarized in Table 1. Fig. 4a–d shows the free energy as function of k for m = 4 and Flory–Huggins parameter values χ_BC = 0.04, 0.4, 2.5 and 9.5, where the minima are found at k = 3, 5, 7 and 9, respectively.

Table 1 The number of internal domains k_opt as a function of χ_BC for χ_AB = 0.1 and n = 200

k opt	χ BC
	m = 3, N = 700	m = 4, N = 900	m = 5, N = 1100	m = 6, N = 1300	m = 7, N = 1500
3	≤0.16	≤0.08	≤0.06	≤0.045	≤0.042
5	0.17–2.6	0.09–0.64	0.07–0.3	0.05–0.18	0.043–0.13
7	≥2.7	0.66–6.7	0.31–1.76	0.19–0.83	0.13–0.48
9		≥6.8	1.77–13.2	0.84–3.75	0.49–1.82
11			≥13.3	3.77–22.4	1.83–6.85
13				≥22.5	6.9–34.5
15					≥35

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Fig. 4 Free energy F of lamellar-in-lamellar self-assembled C–b–(B–b–A)₄–b–B–b–C multiblock copolymer melt as a function of the number k of internal layers for n = 200, N = 900, χ_AB = 0.1 and (a) χ_BC = 0.04, (b) χ_BC = 0.4, (c) χ_BC = 2.5, (d) χ_BC = 9.5.

Larger χ_BC values force a reduction in the equilibrium BC interfacial area which in turn forces the internal short blocks to become more stretched. To relief this stretching the system starts to create more AB interfaces, i.e. larger values of k. Of course, in reality the values of the Flory–Huggins interaction parameters hardly ever exceed unity. The calculations for larger values are nevertheless useful to track and understand the tendencies in the layer formation in ternary C–b–(B–b–A)_m–b–B–b–C multiblock copolymers.

To see how these results depend on the elastic stretching of the blocks, the length of the internal blocks was decreased with a factor 2 to n = 100. Since we maintain the assumption (2m + 1)n = 2N, the length N of the outer blocks is likewise decreased by a factor of 2. We first consider fixed χ_AB = 0.1 and determine the equilibrium number of internal domains k_opt as a function of m and χ_BC. The results are presented in Table 2. A comparison with Table 1 shows that the decreased length of the blocks, implying “stiffer” springs, indeed requires larger values of χ_BC to obtain the same number of internal layers. As an explicit example we observe that for n = 100, resp. 200, 5 layers are found for 0.094 ≤ χ_BC ≤ 0.76 resp. 0.09 ≤ χ_BC ≤ 0.64.

Table 2 The number of internal domains k_opt as a function of χ_BC for n = 100 and χ_AB = 0.1

	χ BC
k opt	m = 3, N = 350	m = 4, N = 450	m = 5, N = 550	m = 6, N = 650	m = 7, N = 750
3	≤0.17	≤0.09	≤0.06	≤0.05	≤0.04
5	0.18–3.55	0.094–0.76	0.06–0.34	0.05–0.20	0.04–0.13
7	≥3.56	0.77–9.25	0.35–2.15	0.21–0.95	0.14–0.55
9		≥9.3	2.2–18.6	1.0–4.7	0.56–2.2
11			≥18.7	4.8–32.2	2.3–8.9
13				≥32.3	9.0–50.5
15					≥51

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Numerical calculations were also performed for a constant χ_BC = 0.1 as a function of χ_AB with internal block lengths of n = 200 and n = 100. The results are collected in Fig. 5 and 6. For the given value of χ_BC = 0.1, the minimal number of k = 3 internal layers is found for χ_AB to be sufficiently large, at smaller values a transition to a 5-layered and even a 7-layered structure (more A–B interface) is observed. Of course, 2nχ_AB has to be considerably larger than 10 to really have a strongly segregated lamellar-in-lamellar self-assembled state.

Fig. 5 The number of domains k_opt of lamellar-in-lamellar self-assembled C–b–(B–b–A)_m–b–B–b–C as a function of χ_AB and m for n = 200 and χ_BC = 0.1. Dashed lines delineate the range of χ_AB-values used.

Fig. 6 The number of domains k_opt of lamellar-in-lamellar self-assembled C–b–(B–b–A)_m–b–B–b–C as a function of χ_AB and m for n = 100 and χ_BC = 0.1. Dashed lines delineate the range of χ_AB-values used.

The interfacial contribution to the free energy is given by F_interface = 2F_BC + (k − 1)F_AB, which can be simply rewritten as where eqn (2) and (4) have been used. From this expression it follows straightforwardly that when χ_AB > 4χ_BC or equivalently γ_AB > 2γ_BC a 3-layered structure has a lower interfacial free energy than a 5-layered one. The results presented in the various tables, however, show that in reality a 3-layered structure is already formed at considerably smaller values of χ_AB, thus demonstrating in particular the importance of the conformational contribution [see eqn (8)].

The tendencies observed are corroborated by the results of computer simulations obtained by using dissipative particle dynamics simulation technique. The results are described in the next section, whereas the computational details are presented in the Appendix.

5. Dissipative particle dynamic simulations

In the dissipative particle dynamics simulation technique a large series of monomers are collected into a few bead-and-spring (DPD) particles in order to simulate the molecular behavior at longer time and length scales. The first situation simulated resembles the experimentally studied ternary P2VP–b–(PI–b–PS)₄–b–PI–b–P2VP linear undecablock copolymer system, i.e.m = 4.⁵Fig. 7 shows the corresponding self-assembled state observed for C₄–(B₁A₁)₄B₁–C₄ with the energy parameters representing the soft repulsion [see eqn (A2)] equal to a_BA = 85, a_BC = 320. Using eqn (A10) in the Appendix for the relation between these energy parameters and the familiar Flory–Huggins parameters, this corresponds to and . Fig. 7 demonstrates that a self-assembled lamellar state is formed with 5 “thin” internal layers as observed experimentally⁴ and calculated theoretically (Fig. 3a and ref. 21). The same result is obtained for internal blocks that are twice as long C₄–(B₂A₂)₄ B₂–C₄ (the subscripts of A, B and C denote the number of DPD particles taken for the calculations).

Fig. 7 An image of C₄–(B₁A₁)₄B₁–C₄ for a_BA = 85, a_BC = 320.

Subsequently, we address the issue of the dependence of the number of internal layers on the interfacial tension. For this purpose we take m = 5, as reasonable variations in the values of the Flory–Huggins parameters are theoretically predicted to induce transitions between a different number of internal layers. That this is also the case in the simulations is shown in Fig. 8, where 3 images of the same system C₄–(B₁A₁)₅ B₁–C₄ are presented for different energy parameters a_AB, a_BC. Fig. 8 demonstrates that when the A–B interaction becomes less unfavourable and the B–C interaction becomes more unfavourable, indeed transitions are observed from 3 to 5 to 7 internal layers. There is the obvious tendency to decrease the BC interface with a corresponding increase in the AB interface.

Fig. 8 Images of the self-assembled C₄–(B₁A₁)₅ B₁–C₄ multiblock copolymer melt for: (a) a_AB = 250, a_BC = 50; (b) a_AB = 75, a_BC = 120; (c) a_AB = 65, a_BC = 300.

That the number of internal layers also depends on the length of the blocks is illustrated by a simulation of C₃–(B₁A₁)₃ B₁–C₃ and C₃–(B₂A₂)₃ B₂–C₃ using the same energy parameter values. Fig. 9 shows that in the former case 3 internal layers are formed and 5 in the latter. Although this example can not be directly compared with the theoretical examples given, it confirms that shorter blocks, the rest being equal, push the system towards a smaller number of internal layers.

Fig. 9 Images of (a) C₃–(B₁A₁)₃ B₁–C₃ and (b) C₃–(B₂A₂)₃ B₂–C₃ for a_AB = 75 and a_BC = 120.

6. Summary

In this paper, we presented a simple theoretical analysis of the strongly segregated lamellar-in-lamellar self-assembled state of ternary C–b–(B–b–A)_m–b–B–b–C multiblock copolymers using the Alexander–de Gennes approach. This simplified description results in a very simple expression for the free energy clearly demonstrating the influence of the number of internal layers k on the various contributions. It allowed us to discuss in detail the influence of the pertinent parameters on the number of internal layers formed. The main observation concerns the sensitivity of k on the interfacial tension between the outer C-layers and the adjacent internal B-layers. The theoretically observed general tendencies were corroborated by the results of computer modeling using the dissipative particle dynamics technique. Experiments to verify these predictions are underway.

Computational Details

The dissipative particle dynamics (DPD) simulation method was introduced by Hoogerbrugge and Koelman²⁵ and first successfully applied by Groot and Madden²⁶ to AB diblock copolymer melts. The time evolution for a set of interacting particles is found by solving Newton's equations of motion. The force acting on the ith particle due to particle j is the sum of a conservative force , a dissipative force , and a random force is given by:where the sum is over all other particles within a certain cut-off radius r_C. Since r_C is the only length scale it is used as the unit of length and thus set equal to 1. The conservative force is a soft repulsive force given by:where a_ij is the repulsive interaction parameter between particles i and j, ,, . The dissipative force is a hydrodynamic drag force and is defined by:where γ is a friction parameter, ω^D is a r-dependent weight function. The random force describes thermal noise:where σ is the noise amplitude, ω^R is a weight function, and θ_ij is a random variable with normal distribution, Δt is a time-step. The dissipative force slows down the particles by removing the kinetic energy from them and this effect is balanced by the random force due to thermal fluctuations. Friction γ and noise σare related by:²⁷

σ² = 2γ k_BT (A5)

The associated weight functions satisfy the fluctuation–dissipation theorem if the following relation is satisfied:²⁸

ω^D = [ω^R(r)]² (A6)

The standard choice for ω^D is:

The spring force that acts on bead i due to its connection with beads j satisfies:

where C is a harmonic type spring constant, which is chosen to be equal to 4 (in terms of k_BT).²⁷

A modified version of the velocity–Verlet algorithm is used to solve Newton's equations of motion:²⁹

Groot and Warren²⁷ presented a detailed investigation of the effect of λ on the steady state temperature and showed that for a particle density ρ = 3 and noise σ = 3, the optimum value is given by λ = 0.65 for which the temperature control can be maintained even at large time-steps of Δt = 0.06. For our calculations we took accordingly λ = 0.65, Δt = 0.06, ρ = 3 and σ = 3.

The DPD simulations are performed in a cubic box of L³ grids with periodic boundary conditions. Since the particle density ρ is set equal to 3, the total number of simulated DPD beads equal 3L³. As reported in ref. 30–32, the morphology obtained by DPD simulations may depend on the finite size of the simulation box. In our simulations we have periodic structures with large periods and to exclude finite size effects we have to take the simulation box sufficiently large. The number of DPD beads per chain is in the range 9–17. The size of the simulation box volume used was taken in the range V = 1 × 10³–3 × 10³, in such a way that for each case considered L exceeded the length of the chains. All simulations were started from random positions.

Following the work of Groot and Warren,²⁷ the repulsive parameters between the same types of particles is taken as a_ii = 25. For different types of particles a_ij can be chosen from the relation between the energy parameter a_ij and the Flory–Huggins interaction parameter χ_ij

a_ij = a_ii + 3.497χ_ij (A10)

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