Microphase Separation in a Mixture of Block Copolymers in the Strong

1180

Macromolecules 1998, 31, 1180-1187

Microphase Separation in a Mixture of Block Copolymers in the Strong Segregation Regime Alexander L. Borovinskii and Alexei R. Khokhlov* Physics Department, Moscow State University, Moscow 117234, Russia Received May 5, 1997; Revised Manuscript Received October 13, 1997

ABSTRACT: Microphase separation in a blend of A-B and C-B block copolymers with strong A-B and C-B incompability is considered in the strong segregation regime. It is assumed that the length of the B block is always much larger than the lengths of A and C blocks, so the complex spherical micelles containing shorter A and longer C chains are formed. The inner structure of the complex A-C micelle is considered, and the distribution of volume fractions of A and C monomer units is calculated. It is shown that shorter A chains are distributed mainly in the periphery of the micelle, while C chains are coiled in the center of this micelle. Macroscopic separation in this blend is studied. It is shown that, in the mixtures with a small fraction of C-B block copolymer, a two-phase structure should be observed: one phase consists of practically pure A micelles, while the other phase the complex A-C micelles, with enriched contents of longer C chains, are formed. For chemically different A and C blocks, even slight A-C repulsion leads to phase separation between the phases with pure A and pure C micelles.

1. Introduction The formation of microdomain structures of different morphologies in melts and concentrated solutions of block copolymers is the topic of intensive experimental and theoretical studies.1-4 In particular, considerable attention is paid to the regime of strong segregation when immiscibility of the blocks is so high that they form practically pure one-component microdomains with narrow interphase regions.5-8 In the case of diblock copolymers consisting of two immiscible blocks A and B, with the numbers of monomer units NA and NB, the classical morphologies in the strong segregation regime are spherical A micelles in the B surrounding forming a body-centered cubic lattice (for the strongly asymmetrical case, NA , NB), cylindrical A micelles in the B surrounding forming a hexagonal lattice (for larger NA/ NB ratios), alternating A and B lamellae (for NA close to NB), and cylindrical and spherical B micelles in A surrounding (when the A component is dominant).1-4 Recently, a number of works were devoted to the microscopic separation in the melts of block copolymers with more complicated chain architecture10-15 and to the search for microdomain structures with more complicated morphologies (e.g., bicontinuous phases).16-18 However, even for simple diblocks or triblocks, many questions are far from being resolved. There is the problem of phase behavior and microstructures in the mixture of two block copolymers differing in the lengths and/or chemical nature of the blocks. A discussion of this problem was first presented by Halperin.19 Important results were obtained by Birshtein and Zhulina.20,21 Recent works22,23 represent detailed analysis and formulate a number of interesting theoretical predictions for the case of dilute solutions of diblock copolymers in selective solvents, where spherical micelles with insoluble core and soluble diffuse corona are formed. Here we will consider the opposite limiting case of the twocomponent melt of block copolymers with the components containing blocks of different length and chemical nature. More specifically, in this paper we will study the phase behavior and microdomain structure in the melt

mixture of two diblock copolymers, A-B and C-B, with NA , NB and NC , NB and strong immiscibility of the components A and B, C and B. Since the B component dominates, the spherical micelles should emerge. Depending on the parameters, these might be either mixed A-C micelles with A and C composition equal to the average one in the whole system, or it may be thermodynamically favorable to have a phase separation with two different A-C compositions in the micelles of the two phases. We will consider both the cases when the blocks A and C differ in length and in chemical nature. As to the B blocks in A-B and C-B block copolymers, we will assume that they are identical. The latter assumption means that, in the strong segregation regime, our analysis remains valid for the case of triblocks A-B-A (C-B-A) and C-B-C (with NA, NC , NB), and for corresponding multiblocks. Indeed, we will see that physically significant contributions to the free energy are localized near the spherical micelles, and for these contributions the fact of possible connection of A and C blocks by long B spacers is of no importance. In particular, our analysis can be applied to the mixture of two ionomers with identical backbone but different ionic groups (e.g., a mixture of polysterene (PS) chains containing a small fraction of Na-sulfonated PS with the PS chains containing Li-sulfonated PS). In refs 24-26, we have developed the approach to the theoretical consideration of the multiplet structure in ionomers which is based on the analogy of this structure with the microdomain structure in block copolymers in the limiting case when the length of one of the blocks tends to unity. It was shown24 that this analogy can be established for the so-called superstrong segregation regime, which is rarely encountered for realistic block copolymers. However, the relations for this regime can be easily obtained from the formulas for the strong segregation case, and after that addressing to the ionomer limit is not dificult. We will discuss below the conclusions for the ionomer case as well (in fact, consideration of this case was our initial motivation for the present study).

S0024-9297(97)00622-0 CCC: $15.00 © 1998 American Chemical Society Published on Web 02/07/1998

Macromolecules, Vol. 31, No. 4, 1998

Figure 1. Two block copolymer chains blended in the present system.

Microphase Separation of Block Copolymers 1181

tions from the isotropic micelles start at such large values of χAC that the system undergoes a phase separation, with the formation of pure A and pure C micelles. It is clear that, if the A and C blocks are significantly different in length, the shorter A blocks are located mainly at the periphery of the micelles, while within the central regions of the micelles only C units are present. This means the ends of shorter blocks A are distributed at distances r from the micellar surface in some interval 0 < r < r1. Correspondingly, the ends of C blocks are located within the sphere of the radius (R - r1), where R is the radius of the micelle. Following the usual arguments of refs 9 and 24, the free energy of the micelle can be written as a sum of four terms:

F ) F1 + F2 + F3 + F4

Figure 2. Structure of the mixed spherical A-C micelle.

The paper is organized as follows. In the next section we will consider the structure of the mixed A-C micelle and determine the main contributions to its free energy. Section 3 deals with the consideration of the phase equilibria in the system for the cases of the blocks A and C differing in length and in chemical nature. Comparison with experimental data and conclusions are formulated in section 4. 2. Structure and Free Energy of the Mixed Micelle We consider the melt formed by two diblock copolymers, A-B and C-B (Figure 1). The number of monomer units in the corresponding blocks obeys the inequalities NB . NA, NC, i.e., the B component is dominant in the system. To be definite, we will assume that NA e NC. The blocks A and B, C and B are strongly immiscible; as to the blocks A and C, the corresponding Flory-Huggins parameter χAC is either equal to zero (this corresponds to the case of chemically identical blocks differing only in length) or slightly positive or negative. Then, it is natural to expect the formation of mixed A-C micelles in the B surrounding (Figure 2). Let us consider the structure of such micelle and main contributions to its free energy. Let us denote the aggregation number of the micelle as Q and the fraction of the longer C blocks as f0. The micelle then includes Qf0 chains C and (1 - f0)Q chains A. In the strong segregation regime (between the blocks A and B, C and B), all the B blocks are outside the micelle, forming its corona. The junction points A-B and C-B are located within a narrow interphase layer near the micellar surface. The blocks A and C are elongated along the radial direction of the micelle. We assume here that the distribution of the concentration of A and C units is purely radial and does not depend on the angle. This assumption seems quite natural; moreover, using the results obtained below, it can be shown that the devia-

(1)

where F1 is the surface free energy, F2 is the free energy connected with the expansion of the B blocks in the corona, F3 describes the free energy of the expansion of A and C blocks in the core of the micelle, and F4 is the free energy of the interactions of A and C units in the micellar core. The contributions F1 and F2 can be written in the same way as for monodisperse melt of diblock copolymers:24

F1 ) 4πR2σ,

F2 3 aQ2 ) kBT 8π R

(2)

where σ is the surface tension coefficient associated with the micellar boundary and a is the linear size of the B monomer unit. The free energy of expansion of B blocks in eq 2 is calculated via summation of all the contributions of local extensions along the chains (cf. ref 24). In writing eq 2, we actually made two simplifying assumptions. First, we assumed that the value of σ does not depend on the composition of the A-C micelle. This is exactly the case if the immiscibility parameters χAB and χCB are equal. If χAB * χCB, the expression for F1 becomes more complicated; however, the final qualitative results of our analysis remain the same. Therefore, everywhere below we assume that σ ) const. Second, the expression for F2 can be written in the form of eq 2 for flexible B chains, with the Kuhn segment length of the order of the size of monomer unit. We will assume this in the subsequent analysis; moreover, we will suppose that a is the only microscopic length parameter for A, B, and C chains. This means that all these chains have the same Kuhn segment, which is of the order of the cross section of the chain. This assumption leads to significant simplification of the formulas and does not impose any significant restrictions on the generality of the consideration. The aggregation number of the micelle is connected with its radius by the simple geometrical relation

4πR3 ) Q((1 - f0)NA + NCf0)a3 ) QNAa3(1 + kf0) (3) where we have introduced the parameter characterizing the difference of NC and NA:

k≡

NC - NA NA

(4)

1182 Borovinskii and Khokhlov


Using eq 3, eq 2 can be written in the form

F1 3 2/3 a2σ ) 4π [QNA(1 + kf0)]2/3 kBT 4π kT

( )

(5)

F2 1 3 2/3 ) kBT 2 4π N

Q5/3 1/3 1/3 A (1 + kf0)

( )

(6)

As it was already mentioned, our main interest is in the case when the blocks A and C are neither chemically identical (i.e., they differ only in length), nor is the corresponding Flory-Huggins parameter χAC small. Therefore, in the calculation of F3 and F4, we will approximately assume that, even if χAC * 0, the structure of the micelle (i.e., the distribution of concentrations of A and C monomer units) is the same as for χAC ) 0. The significant deviations in the A and C concentration distribution from the case χAC ) 0 may occur at rather high positive values of χAC; however, this situation is never realized, because the macroscopic phase separation with the formation of practically pure A and C micelles occurs much earlier. The most complicated task is the calculation of the free energy F3 of expansion of A and C blocks in the interior of the complex A-C micelle. This calculation is presented in the Appendix. The free energy of the of the expansion of A and C blocks in the core of the micelle is given by eqs A29 and A27:

F3 )

Figure 3. Dependence of the penetration depth of the shorter A chains in the micelle, r1, on the micellar composition, f0, given by eqs A15 and A20 for different values of k.

Q5/3(1 + kf0)5/3 9π2 3 2/3 β(R,k) (7) 16 4π NA1/3(1 + k(1 - R)1/2)5

( )

where

(

)

1 5 + k(1 - R)1/2 1 - R + k(1 + k(1 15 3 2 R)1/2)R arcth[(1 - R)1/2] + k2(1 - R) 1 + R + 3 8 k2 2 - (1 - R)5/2 - k2(1 - R5/2) (8) k 5 3 15

β(R,k) )

(

)

(

)

and R is a dimensionless parameter, which is connected via eq A15 with r1, the maximum distance from the surface of the micelle reachable for shorter A blocks. The value of r1 can be obtained using eqs A13 and A20. The dependence of the depth of penetration of short A chains in the core of the micelle, r1 versus micellar composition, f0, for different values of k given by eqs A13 and A20 is shown in Figure 3. The local values of volume fractions of monomer units inside the micelle, ΦA(r/R) and ΦC(r/R), are defined by the eqs A24 and A14 of the Appendix. These functions are shown in Figure 4 for different values of k. To specify the free energy of the micelle, it remains to write down the contribution F4 which takes into account the interaction of A and C monomer units in the core of the miccelle. As it was already mentioned, in the expression for F4 we will assume that A and C concentration distributions are those which are calculated in the Appendix (see Figure 4); i.e., the fact that χAC is generally not equal to zero does not change significantly these distributions. We have

F4 ) χAC kBT

∫0r ΦA(r)ΦC(r)4π(R - r)2 dr 1

(9)

Figure 4. Distribution of the A and C volume fractions inside the complex A-C micelle, given by eq A24, for different values of k and for f0 ) 0.13.

Writing this integral in terms of variable x, we obtain

F4 1 + kf0 3 ) χACQNA kBT 2 (1 + k x1 - R)3

∫0RΦA(x)(1 -

(1 - x1/2 + k(1 - R)1/2)2

ΦA(x))

x1/2

dx (10)

The function ΦA(x) is defined by eq A24, and the integration in eq 10 can be performed numerically. Expressions 1, 5, 6, and 10 define completely the free energy of the mixed A-C micelle with the aggregation number Q and the composition f0. 3. Phase Equilibria in the Mixture of Block Copolymers Let us now turn to the consideration of the macroscopic system containing N block copolymer chains organized in N /Q micelles with the aggregation number Q. The free energy of the system has the form


F)

[ ( (


) )]

f0 N + [F1 + F2 + F3 + F4] + NkT f0 ln Q 1 + kf0 1 - f0 (11) (1 - f0) ln 1 + kf0

where the last term takes into account the entropy of mixing of short and long blocks in the micellar cores. By minimizing this expression with respect to Q, we find the equilibrium aggregation number for the complex A-C micelles with the composition f0:

Qeq )

1 + kf0 4πσ NA (12) 2 kBT 9π2 (1 + kf0) 1+ β(R,k) 8 (1 + kx1 - R)5

Equations 11 and 12 are valid when the system is homogeneous. However, for the mixture of two block copolymers, it is possible to have phase separation in the system with the formation of mixed A-C micelles with different composition in each phase. The free energy of the separated system can be written as follows:

F ) nIF(Q1,f1) + nIIF(Q2,f2) N

Figure 5. Phase diagrams for the mixture of chemically identical A-B and C-B block copolymers differing in the length of the blocks A and C for different values of NA and χAB. Phase separation regions are to the left of the curves presented.

(13)

where nI and nII are the fractions of the total number of N chains entering in phases I and II, respectively,

nI + nII ) 1

(14)

f1 and f2 are micellar compositions in the phases,

f1nI + f2nII ) f0

(15)

while Q1 and Q2 are the aggregation numbers of the micelles in the phases I and II; they depend on f1 and f2 via equations similar to eq 12. Minimization of the free energy (eq 13) with respect to nI and f1 gives the volumes and compositions of the coexisting phases. It turns out that the dependence of the free energy, F(Qeq,f0), on the composition always has two minima, one corresponding to f0 ) 0 (micelles formed only by short A chains). Therefore, when separation into two phases takes place, one of the coexisting phases always corresponds to the pure A micelles (f1 ) 0). Taking into account this fact, the minimization of the free energy (eq 13) with additional conditions (eqs 14 and 15) leads to the following equation:

dF(Q2,f2) )0 F(Q2,f2) - F(Q1,0) - f2 df2

(16)

for the determination of the composition of the complex A-C micelles in the second of the coexisting phases. The result of the solution of this equation for the case χAC ) 0 of block copolymers of identical chemical composition differing only in the length of blocks is presented in Figure 5. One can see that, for small fraction f0 of the block copolymer chains with long C blocks, phase separation with the formation of pure A micelles in one phase and complex A-C micelles with enhanced C content in the other phase is predicted. In other words, the existence of complex A-C micelles with small fraction of long C chains is thermodynamically unfavorable. The reason for this is the following. A small

Figure 6. Structure of the (thermodynamically unfavorable) complex A-C micelle with small fraction of long C chains.

number of long C chains in a predominantly A chain micelle should be essentially compressed with respect to the Gaussian dimensions (Figure 6). The gain in the surface free energy from the micelles of the type shown in Figure 6 cannot overcome significant enthropy losses from constrained conformations of C chains. Therefore, it is favorable to have a phase separation with the formation of the micelles with larger C content where the conformation of C chains is not so constrained. Some of the phase diagrams calculated for the general case χAC * 0 and for different values of k are shown in Figure 7. One can see that, for χAC < 0 (attraction between the A and C components), we still have the region of phase separation for the small content of long C chains. For higher values of f0, the homogeneous phase is always formed. On the other hand, for χAC > 0, it is enough to have a small repulsion between the A and C monomer units to obtain the phase separation into the practically pure A and pure C phases. The broadening of the phase separation region is very sharp, as it is possible to see from Figure 7. We can introduce the critical value of the immiscibility of A and C components (χAC)cr, which corresponds to the expansion of the right boundary of the phase separation region to ca. f0 ) 0.5. The dependence of (χAC)cr on k is illustrated in Figure 8. In the application of the results obtained above to the case of ionomers mentioned in the Introduction, we should consider the limit of NA f 1, NC f 1 and of


Figure 7. Phase diagrams for the mixture of two chemically different block copolymers for different values of k. Phase separation regions are to the left of the curves presented.


or in chemical structure between A and C blocks. Also, we analyzed the phase equilibria in this system and determined the conditions for macroscopic phase separation. We have shown that the mixtures with small fraction of C-B block copolymers with longer C blocks should separate with the formation of the phase with pure A micelles and complex A-C micelles with enriched C contents. For chemically different A and C blocks, even slight A-C repulsion leads to phase separation between the phases with pure A and pure C micelles. We are not aware of experimental verifications of the above results for the mixed blends of block copolymers. However, for the comicellization of block copolymer solutions in selective solvents, some of experimental results are available. This system should be somewhat similar to that considered in the present paper for the case of the mixture of two block copolymers with different lengths of insoluble blocks. Exactly this situation was considered recently in the literature.29,30 The main conclusion of ref 30 is the discovery of the regime with the coexistence of the two types of micelles: (i) pure A micelles (A being the shorter insoluble block) and (ii) complex A-C micelles with enhanced C content (see Figure 6 of ref 30). Altough the problem of the micelle formation in the solution is somewhat different from that considered in the present paper, most of the contributions to the micellar free energy (F1, F3, F4) are the same. The difference is only in the term F2 desribing the chain expansion in the corona of the micelle; however, the role of this term in the above analysis is not very significant. Therefore, the results of ref 30 can be regarded as the experimental confimation of one of the main conclusions of the present paper. On the other hand, we will analyze the micelle formation in the mixed solutions of block copolymers in selective solvents in a subsequent publication. Appendix

Figure 8. Dependence of the critical immiscibility parameter of A and C blocks, (χAC)cr, on k, for NA ) 50 and χAB ) 0.2.

strong interaction (repulsive or attractive) between A and C units. In general, our conclusion, then, should be the following: for the A-C attraction, one should expect the formation of mixed multiplets, while for repulsive interactions separate A and C multiplets will be performed. However, one should bear in mind that the ionomer multiplet structure corresponds to the superstrong segregation regime rather than to the strong segregation regime.24-26 In particular, this means a significant role of steric hindrances in the multiplet formation. For example, the difference of the self-volume and cross section between A and C monomer units should be taken into account, as was done in ref 24, and this may significantly renormalize the effective A-C affinity in the multiplet. Since here we have limited ourselves to the consideration of the theory with only one characteristic microscopic length a, we postpone the detailed analysis of the case of ionomeric multiplets to another paper. 4. Conclusions In the present paper, we have considered the structure of the complex spherical A-C micelles formed in the mixed blends of A-B and C-B block copolymers with a long B block and the difference in the length and/

According to the general method of the solution of similar problems first described in ref 9, let us characterize each chain by the trajectory r(n), where n is the monomer unit number along the chain counted from the surface of the micelle, r being dimensionless and expressed in the units of a. Then the local elongation of the chain is determined by the derivative dr/dn. Let us introduce the effective self-consistent field U(r), which induces the same elongation of the chains. It is assumed that the value of U is dimensionless, i.e., it is expressed in kT units. Physically, this self-consistent field appears as a result of the constant density condition, i.e., it is defined by the fact that monomer units have a self-volume. The potential U(r) should have such a form that a chain of N links is in equilibrium. When the mean-field approximation is valid, the free energy change F(Q + 1) - F(Q) to add one chain to the micelle containing Q chains is given by

e

-(F(Q+1)-F(Q))

)

∑ ∫dn

{ri(n)}

[( ) 1 dri

2 dn

2

]

- U({r(n)})

(A1)

where the first term in the integral corresponds to the stretching free energy and the second one accounts for excluded-volume free energy. The single-chain partition function in eq A1 is analogous to the path integral for one-particle quantum mechanics in a potential U(r). The integral in the right part of this expression corresponds



to the action with r(n) position of the particle at the time n. For the streched chain in the micelle, the partition function in eq A1 is dominated by the path which minimizes the action (cf. ref 9). By minimizing the single-chain action with respect to the path r(n), we obtain

d2r dU (r) )dr dn2

(A2)

Here it is convenient to further the mechanical analogy developed in refs 27 and 28. Let us assume that, in eq A2, the index n corresponds to the time of the motion of a particle, while the local chain elongation is analogous to the particle velocity. It is possible to say that eq A2 describes the mechanical motion of a particle in the field U(r). The free end of the chain located at some point r0 is not elongated,

dr (r ) ) 0 dn 0

(A4)

After the summation over all chains, we obtain

∫rRdr0g(r0) dn(r,r0) ) ∫r

dr0

g(r0) [2(U(r0) - U(r)]1/2

dn ∫0U dU dU 0

dr (U) ∫0U [2(U dU 1/2 dU - U)] 0

(

)

NC - NA 21/2 NA dr ) + dU π U1/2 (U - U )1/2 1

r(U) )

(A9)

23/2 (NAU1/2 + (NC - NA)(U - U1)1/2θ(U π U1)) (A10)

where θ(x) ) 1 at x > 0, and θ(x) ) 0 at x < 0. Upon the change to the variable U, the incompressibility condition eq A5 takes the form

(A6)

0

As is seen from eq A6, the chains of different lengths should terminate at the different intervals of the values of the potential. Returning now from general theory to the problem under consideration, we introduce the distance r1 from the surface of the micelle such that, at r1 < r < R, only

- U)]1/2

(A11)

where, by definition, g(U0) dU0 ) g(r0) dr0. The function g describes the distribution of the chain ends in the micelle: at U0 < U1, g(U0) dU0 defines the number of the ends of short A chains terminating in the interval dU0, while at U0 > U1, g(U0) dU0 gives the corresponding number for long C chains. By solving integral eq A11, we obtain

(R - r(U0)) dr dU0 1/2 dU 0 0 - U)

∫UA (U

g(U) ) 27/2

(A12)

Let us introduce the dimensionless parameters

R ) U1/A, x ) U/A

(A13)

Their connection with the spatial coordinate r follows from eq A10: 1/2

(A7)

g(U0)dU0

∫UA[2(U

0

(A5)

or, taking into account eq A4,

N(U0) )

while for U1 < U < A we have

4π(R - r(U))2 )

where g(r0) is the number of chains in the micelle whose ends lie at the distances between r0 and r0 + dr0 from the surface of the micelle. The potential U(r) is a priori unknown. However, we will not calculate U(r) explicitly, but use U as independent variable (instead of r); i.e., we will define the chain trajectory by the dependence of the number of monomer units n on U. Let us establish the connection between the new variable U and r. Let us assume that the end of the chain containing N monomer units and starting at the micellar surface corresponds to the potential U0. By definition, we adopt that, at the micellar surface, U ) 0. Then, after integration over the chain trajectory, we have

N(U0) )

(A8)

After the integration of dr/dU, we finally obtain

dn(r,r0) ) 2-1/2(U(r0) - U(r))-1/2

R

21/2NA dr ) dU πU1/2

(A3)

i.e., the fictitious particle starts at the point r0 with zero velocity. Following further this analogy, the number of monomer units in the narrow spherical layer of width dr at a distance r from micellar surface for a chain which end is fixed at the point r0 is

4π(R - r)2 )

the monomer units of long C chains are available in the micelle. Let us denote the corresponding value of the potential U1 ≡ U(r1). The ends of short A chains correspond to the interval 0 < U < U1, while the ends of the long C chains should be primarilly located in the interval U1 < U < A, where A ) U(R) is the value of the potential in the center of the micelle. After the substitution to the left side of eq A7 of the values of NA and NC, we obtain for 0 < U < U1

r x ) R

+ k(x - R)1/2θ(x - R) 1 + k(1 - R)1/2

r1 R1/2 ) R 1 + k(1 - R)1/2

(A14)

(A15)

It is to be reminded that k ≡ (NC - NA)/NA. Let us now substitute eqs A8, A9, and A10 into eq A12. Then, for 0 < x < R, we obtain


g(x) )

[

32NA2(2A)1/2 π2


ΦC(x) )

(1 + k(1 -

1 + (1 - x)1/2 - 2(1 - x)1/2 - 2k2((1 - x)1/2 R)1/2) ln 1 - (1 - x)1/2 (R - x)1/2) + k(1 + k(1 (1 - x)1/2 + (1 - R)1/2 R)1/2) ln (1 - x)1/2 - (1 - R)1/2 (2x0 - R) dx0 1 (A16) k R [x0(x0 - R)(x0 - x)]1/2

]

∫

while for R < x < 1, we have

g(x) )

π2

(1 + k(1 -

1 + (1 - x)1/2 - 2(1 + k2)(1 - x)1/2 + k(1 + R)1/2) ln 1/2 1 - (1 - x) (1 - x)1/2 + (1 - R)1/2 k(1 - R)1/2) ln (1 - R)1/2 - (1 - x)1/2 (2x0 - R) dx0 1 (A17) k R [x0(x0 - R)(x0 - x)]1/2

]

∫

Q)

∫

dxg(x) R

(A18)

∫0 dxg(x) 1

(A19)

f0(R) ) (1 + k(1 - R)1/2)((1 - R)1/2 - R arcth(1 -

)]

(

1 2R3/2 1 / R) ) - (2 + 5k2)(1 - R)3/2 - k - a + 3 3 3 1 2 2 - k (1 - R3/2) + k(1 + k(1 - R)1/2)((1 - R)1/2 + 3 3 k Rarcth(1 - R)1/2) - (1 - R)1/2(1 + 2R) (A20) 3 1/2

[

- x)1/2

0

≡ 1 - ΦA

In the center of the micelle (at R < x < 1), only monomer units C are present, therefore

ΦC(x) )

πNA2(1

1/2

-x

g(x0) dx0

∫x1(x

1 + k(1 - R)1/2)2

0

≡1 - x)1/2 (A23)

[ (

1 (1 + k(1 π(1 - x + k(1 - R)1/2)2 R-x R)1/2) π(1 - 2x1/2) - 2 arcsin 1 - 2 + 1-x

ΦA(x) )

1/2

(

1/2

1/2

]

Equations A13 and A20 establish the relations between the depth of penetration of short A chains in the micelle and its composition. This dependence for different values of k is shown in Figure 3. The knowledge of the function g(U) allows us also to find the local values of volume fractions of monomer units inside the micelle ΦA(U) and ΦC(U). For 0 < x < R,

1 ΦA(x) ) 1/2 2 πNA (1 - x + k(1 - R)1/2)2

R g(x0)

∫x (x

0

dx0

- x)1/2 (A21)

)

4(R - x) arcth[(1 - R) ] + x(1 - R) 1/2 - 2(1 + k2) [(1 - R)(R 4x arctan a-x

[

( ]) [ ]) ( ( )) (

x)]1/2 + (1 - x) arctan

R-x 1-R

1/2

+ πk2(R - x) + R-x 1-x

k2(1 - R) π - 2 arcsin 1 - 2

1

Dividing the first equation by the second, we obtain

[

-x

g(x0) dx0

∫R1(x

1 + k(1 - R)1/2)2

(A22)

1/2

The parameter R, which is unknown up to now, can be connected with the micellar composition f0. Indeed,

f0Q )

1/2

The latter equalities in eqs A22 and A23 can be explicitly checked upon the substitution of eqs A16 and A17 into these equations. As to the function ΦA(x) for 0 < x < R, the same substitution leads to the following explicit formula:

[

32NA2(2A)1/2

πNA2(1

+

1-x 2k2[(1 - R)(R - x)]1/2 ln - k -2(R R-x 1-x + 4(R - x)1/2(1 - R1/2) + x)1/2 ln R-x 1 + x1/2 R1/2 + x1/2 ln (A24) 2[x(R - x)]1/2 ln 1 - x1/2 R1/2 - x1/2

(

))]

The functions ΦA(r/R) and ΦC(r/R) given by eqs A24 and A14 are shown in Figure 4 for different values of k. Finally, let us calculate the free energy F3 of the expansion of A and C blocks in the interior of the complex A-C micelle with the agregation number Q. We have

F3 3 ) kT 2 3 2

dr (r,r0) ) ∫0Rdr0g(r0)∫0r drdn dr (U) ∫0AdU0g(U0)∫0U dU[2(U0 - U)]1/2dU 0

0

(A25)

Substituting here the expressions of eqs A8, A9, A16, and A17, we find

96 x2NA3A5/2 F3 ) β(R,k) kBT π2 where the function β(R,k) is defined as follows:

(A26)


(

)


1 5 + k(1 - R)1/2 1 - R + k(1 + k(1 15 3 2 R)1/2)R arcth[(1 - R)1/2] + k2(1 - R) 1 + R + 3 k2 2 8 k - (1 - R)5/2 - k2(1 - R5/2) (A27) 5 3 15

β(R,k) )

(

)

(

)

It is to be reminded that A is the potential in the center of the micelle (see eq A10):

23/2NAA1/2 (1 + k(1 - R)1/2) π

(7) (8) (9) (10) (11) (12) (13) (14) (15)

(A28)

(16) (17) (18)

Taking into account eq A28 and the expression of eq 3, we have finally

(19) (20)

R)

Q5/3(1 + kf0)5/3 9π2 3 2/3 β(R,k) F3 ) 16 4π N 1/3(1 + k(1 - R)1/2)5 A

( )

(A29)

Acknowledgment. The authors are grateful to Russian Foundation for Basic Research for financial support and to Dr. A. N. Semenov for valuable suggestions and comments. References and Notes (1) Bates, F. S. Science 1991, 251, 898. (2) Bates, F. S.; Fredrickson, J. H. Annu. Rev. Phys. Chem. 1990, 41, 525. (3) Erukhimovich, I. Ya.; Khokhlov, A. R. Vysokomol. Soed. 1993, 35, 1522. (4) Binder, K. Adv. Polym. Sci. 1994, 112, 181. (5) Meier, D. J. J. Polym. Sci., Part C 1968, No. 26, 81-98. (6) Helfand, E.; Wasserman, Z. R. Macromolecules 1976, 9, 879.

(21) (22) (23) (24) (25) (26) (27) (28) (29) (30)

Helfand, E.; Wasserman, Z. R. Macromolecules 1978, 11, 961. Helfand, E.; Wasserman, Z. R. Macromolecules 1980, 13, 994. Semenov, A. N. Sov. Phys. JETP 1985, 88, 1242. Erukhimovich, I. Ya. Polym. Sci. USSR 1982, 24, 2232. Olvera de la Cruz, M.; Sanchez, I. C. Macromolecules 1986, 19, 2501. Benoit, H.; Hadziioannou, G. Macromolecules 1988, 21, 1449. Mayes, A. M.; Olvera de la Cruz, M. J. Chem. Phys. 1989, 91, 7228. Mayes, A. M.; Olvera de la Cruz, M. J. Chem. Phys. 1991, 95, 4670. Dobrynin, A. V.; Erukhimovich, I. Ya. Macromolecules 1993, 26, 276. Matsen, M. W.; Schick, M. Macromolecules 1994, 27, 6761. Matsen, M. W.; Schick, M. Phys. Rev. Lett. 1994, 72, 1660. Floudas, G.; Pispas, S.; Hadjichristidis, N.; Pakula, T.; Erukhimovich, I. Ya. Macromolecules 1996, 29, 4142. Halperin, A. J. Phys. 1988, 49, 131. Birshtein, T. M.; Lyatskaya, V. Yu.; Zhulina, E. B. Polymer 1990, 31, 2185. Zhulina, E. B.; Birshtein, T. M. Polymer 1991, 32, 1299. Shim, D. F. K.; Marques, C.; Cates, M. E. Macromolecules 1991, 24, 5309. Sens, P.; Marques, C. M.; Joanny, J. F. Macromolecules 1996, 29, 4880. Nyrkova, I. A.; Khokhlov, A. R.; Doi, M. Macromolecules 1993, 26, 3601. Semenov, A. N.; Nyrkova, I. A.; Khokhlov, A. R. Macromolecules 1995, 28, 7491. Semenov, A. N.; Nyrkova, I. A.; Khokhlov, A. R. In Ionomers: Characterization, Theory and Applications; Schlick, S., Ed.; CRC Press: Boca Raton, FL, 1996; p 251. Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, 2610. Milner, S. T.; Witten, T. A.; Cates, M. E. Macromolecules 1989, 22, 853. Honda, C.; Hasegawa, Y.; Hirunuma, R.; Nose, T. Macromolecules 1994, 27, 7660. Honda, C.; Yamamoto, K.; Nose, T. Polymer 1996, 37, 1975.

MA970622D

Microphase Separation in a Mixture of Block Copolymers in the Strong

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