Macromolecules 1986,19, 2501-2508
value for PEMO corresponds with the lowest enthalpy and entropy of melting. This last quantity is usually of paramount importance in determining the melting temperature, and a low value is often interpreted as due to restricted flexibility of the macromolecule in the molten state.13 This is not the case in those polyoxetanes that are characterized by a high degree of flexibility. The most flexible of these polyoxides, in the equilibrium sense of having the smallest characteristic ratio, is poly(trimethylene oxide) -(O(CH2)&.14 Its low melting temperature has been associated with a high degree of conformational disorder in the molten state corresponding a large value of Sm.15The conformationalrandomness is only slightly diminished by the restrictions that the dialkyl substitution at the 3-position adds to the rotations of the 0-C bonds. The steric encumbrance is scarcely modified on passing from PDMO to PEMO and PDEO and consequently the lower entropy of fusion of PEMO in comparison with the other two polymers c a n be related to a disordered structure in the crystalline state. The departure from the regularity will alter the entropy of this state, leading to a decreased entropy of fusion. In conclusion, the synthesis of PEMO, carried out under experimental conditions where a random placement of the substituents along the chain is expected, gives a polymer that is crystallizable from the melt state. Its thermal properties are well described by a two-phase model, amorphous and crystalline.
2501
Acknowledgment. Financial support from the Comisidn Asesora de Investigacidn Cientifica y TBcnica is gratefully acknowledged. Registry No. PEMO (SRU), 103349-60-6; PEMO (homopolymer), 103349-59-3; triethyloxonium hexafluoroantimonate, 19554-80-4.
References and Notes (1) Penczek, S.; Kubisa, P.; Matyjaszewski, K. Adu. Polym. Sci. 1980, 37.
(2) PBrez, E.; Gbmez, M. A.; Bello, A,; Fatou, J. G. Colloid Polym. Sci. 1983, 261, 571. (3) Takahashi, Y.; Osaki, Y.; Tadokoro, H. J. Polym. Sci., Polym. Phys. Ed. 1980,18, 1863. (4) Schmoyer, L. F.; Case, L. C. Nature (London)1960,187,592. (5) Bello, A,; PBrez, E.; Fatou, J. G. Makromol. Chem. 1984, 185, 249. (6) PBrez, E.; Gbmez, M. A.; Bello, A.; Fatou, J. G. J. Appl. Polym. Sci. 1982, 27, 3721. (7) Haldon, R. A,; Schell, W. J.; Simha, R. J. Macromol. Sci. ( B ) 1967, I, 759. (8) Wunderlich, W. Macromol. Phys. 1973, 1. (9) Lupinacci, D.; Winter, W. T. J.Polym. Sci., Polym. Phys. Ed. 1982, 20, 1013. (10) Lovinger, A. J.; Cais, R. E. Macromolecules 1984, 17, 1939. (11) Ivin, K. J. J. Polym. Sci., Polym. Symp. 1978, No. 62, 89. (12) Kops, J.; Hvilsted, S.; Spanggaard, H. Macromolecules 1980, 13, 1058. (13) Mandelkern, L. Crystallization of Polymers; McGraw-Hill: New York, 1964. (14) Takahashi, Y.; Mark, J. E. J.Am. Chem. SOC.1963,98,3756. (15) PBrez, E.; Fatou, J. G.; Bello, A. Eur. Polym. J.,in press.
Theory of Microphase Separation in Graft and Star Copolymers Monica Olvera de la Cruz*t and Isaac C. Sanchez* Institute for Materials Science and Engineering, National Bureau of Standards, Gaithersburg, Maryland 20899. Received March 13, 1986
ABSTRACT: Phase stability criteria and static structure factors have been calculated for simple AB graft copolymers, for star copolymers with equal numbers of A and B arms, and for n-arm star diblock copolymers. The A-B interactions are characterized by the usual x parameter. The fraction of A monomer in the graft copolymer is denoted as f and the fractional position along the A chain backbone a t which the B graft is chemically linked is denoted as T. When T = 0 or 1the graft copolymer degenerates to a simple diblock copolymer. Leibler previously calculated that the critical value, (xN),, a t which an AB diblock copolymer containing N monomer units undergoes microphase separation is 10.5. This critical value occurs at f = 0.5 and is the only composition for which the transition is second order. According to the present theory, a graft copolymer (0 < T < 1) does not have a critical point for any f; i.e., all transitions are first order. For a given T , the spinodal values, (xN),,always reach a minimum value at f = 0.5; for T = f = 0.5, (xN), = 13.5. However, star copolymers with equal numbers ( n )of A and B arms each containing N / 2 monomers (f = 0.5) have a critical point at (xN), = 10.5 for all values of n. Like the graft copolymers, the n-arm star diblock copolymers (each arm is a diblock copolymer of composition f containing N monomer units) do not have a critical point. At f = 0.5, ( x N ) , equals 8.86, 7.07, 5.32, and 4.33 for n = 2, 4, 10, and 30, respectively. At a spinodal point the static structure factor S(q) diverges at a finite wave vector q*. Near a critical point q * / 2 ~determines the periodicity of the lowest symmetry-ordered structure (mesophase) and is expressed in units of the copolymer's radius of gyration R.
Introduction It is well-known from Flory-Huggins theory that a binary mixture of A and B homopolymers phase separates at a critical value of the interaction parameter x given by (XN), = 4
(1)
when each of the homopolymers has N/2 monomer units. Leibler' was the first to point out that a simple 5050 NBS Guest Scientist from the University of Massachusetts.
* Current address: Alcoa Laboratories, Alcoa Center, P A
15069.
0024-9297/86/2219-2501$01.50/0
diblock AB copolymer containing N monomer units has a larger critical x value:
-
(xru), = 10.5
(2)
Since classically x 1/T (T is temperature), this implies that phase separation is more difficult (requires a lower temperature) in diblock copolymers than in analogous homopolymer systems. Chemically joining two homopolymers of the same size to form a diblock copolymer reduces the critical temperature for phase separation by a factor of 2.6. In this paper the effects of other molecular architectures on the phase separation and critical behavior 0 1986 American Chemical Society
2502 de la Cruz and Sanchez
Macromolecules, Vol. 19, No. 10, 1986 AB diblock copolymer
Simple AB graft copolymer
/‘\-.’
A282 copolymer star
\
of long polymers interacts with many other chains, making the correlation length very long. The longer the correlation length, the larger the region of the phase diagram where mean field theory is applicable. In polymer blends, for example, the critical region at which mean field theory breaks down is reduced by 1 / N . So, in practice, blends of long polymers phase separate, in agreement with mean field t h e ~ r y . ~ In what follows, the system will be characterized in terms of an order parameter ApA(r) (or more generally its Fourier components), which will now be defined. Consider an incompressible copolymer melt of composition f and total mean monomer density p E PA + p B . Incompressibility means that locally at the point r PAW
c
I
+ PB(d
=P
(3)
The canonical averages of these local densities are (AB), copolymer star
Figure 1. Schematic representation of four types of copolymers.
(PA(r)) = P A = f p
(44
(PB(r)) = PB = (1 - f ) P
(4b)
and In general, there are density (concentration) fluctuations and PA(r)
of copolymers have been investigated. Specifically, three types of copolymers have been considered: a simple graft copolymer and two types of star copolymers denoted as A,B, and (AB),. All three copolymers are illustrated in Figure 1. Iff is the fraction of A monomers in the copolymer, then a simple graft copolymer is formed by joining a B chain of (1 - f ) N monomers to an A chain of f N monomers at an arbitrary fractional position 7 along the A chain (0 I 7 d 1). When 7 = 0 or 1, the graft copolymer degenerates to a simple diblock copolymer. An A,B, star copolymer has 2n arms, n equal arms of fNo A monomers and n equal arms of ( 1 - f)NoB monomers. The total number of monomers in the copolymer is N = nN,. An (AB), star copolymer has n equal arms, but each arm is a diblock chain containing f N oA monomers and (1- f)No B monomers. Note that n = 1 corresponds to a diblock copolymer and n = 2 corresponds to a triblock copolymer. Notice also that as the number of arms ( n )increases, the core of the star will naturally become richer in A (or B) monomers and the monomers deep in the core will be effectively “screened” from interacting with B monomers. This self-segregation or self-micellarization makes the phase transition behavior of (AB), stars qualitatively different from those of A,B, stars. (AB),, stars have recently been synthesized and characterized.2 In a mixture of two homopolymers, equilibrium phase separation occurs on a macroscopic size scale. In block copolymers or in copolymers such as those shown in Figure 1, phase separation can only occur on a microscopic level; the constraint that the A and B chains be chemically connected forces the phase separation to occur on size scales of the order of the radius of gyration (R) of the copolymer. Although each phase (one rich in A and the other rich in B) is disordered, the phases themselves are arranged in a periodic ordered structure (a mesophase). Three structures have been predicted:’ body-centered cubic, hexagonal, and lamellar. The ordered structure that is formed is the one that minimizes the free energy and depends on the copolymer composition and temperature. In this paper the transitions are assumed to be properly described by mean field theory. A single chain in a melt
- PA E
(5)
APA(r) # 0
Since the system is incompressible,ApA(r) = -ApB(r), only one order parameter is needed to specify the free energy of the system (see below). The static structure factor S(q)is defined as the Fourier transform of the monomer density-density correlation function: 1 s(q) E 7~ [ e x P ( k r )( A l P A ( ~ ) A P A ( ~dr )) (6) VP where V is the system volume and the factor Vpzhas been introduced to make S(q)dimensionless. In the disordered phase S(q)will only depend on q, the magnitude of q. S(q) can be determined by scattering experiments [q = (4a/X) sin ( 8 / 2 ) ,where X is the wavelength of the radiation and 8 the scattering angle] and is a direct measure of the concentration fluctuations in the system. A primary objective in this paper is to calculate S(q) in the melt for the copolymers in Figure 1; S(q) is the essential link between theory and experiment. Herein functional (path) integral methods are used to calculate S(q) which are equivalent to the random phase approximation (RPA) method used by Leib1er.l With the Fourier transform of the order parameter ApA(r) denoted by ApAq, it is well-known from linear response theory that the contribution to the free energy from fluctuations is
F / k T = Fo/kT + (1/2VP)C (APA&*Aq)/s(q)
+
920
(higher order terms in ApAq) (7) where p*Aq is the complex conjugate of pAq. Notice that the Fourier components of the concentration fluctuations are weighted by S-’(q) and that the fluctuations tend to increase the free energy above Fo. The most probable fluctuations are those for which S(q) is a maximum. In a binary mixture of two homopolymers, S(q) would have its maximum at q = 0, which corresponds to fluctuations of infinite wavelength. At large q, S(q) would fall off as q-2, which is characteristic of scattering from a single chain.4 Although the melt of a Figure 1 copolymer is a pseudo-two-component system, S(0) = 0 at all temperatures because concentration fluctuations on macroscopic size scales are impossible (the chemical connection between
Microphase Separation in Star Copolymers 2503
Macromolecules, Vol. 19, No. 10, 1986
A and B chains prevents the possibility of such fluctuations). However, as before, S(q) approaches zero as q-2 for large q. In the melt the most probable concentration fluctuations will be of order R, and S(q) will, in general, pass through a maximum at q* where q*R is of order unity; q* is independent of temperature but S(q*) increases as the temperature is lowered. Finally, it diverges at the spinodal temperature or at a characteristic value of xN denoted as (XI?),. Hereafter, ( X I ?is) defied , as that value of xN where S(q*) m. The disorder-order transition occurs at with (xNt2 (XI?), (both (xWt and (XI?), depend on f cc PAID). A critical value of (XI?)occurs when (xWt = (XW,and is denoted as (XI?),. At ( X N the , transition is second order whereas it is first order everywhere else. At the critical point the third-order term in the free energy expansion in powers of the order parameter (eq 7) vanishes. So at the critical point the free energy is invariant to the interchange of A ~ toA-APA~. ~ As expected by the symmetry of the molecule an A,B, star copolymer exhibits a critical point at f = 0.5 and (XI?), = (XI?),. On the other hand, a graph with 0 < 7 < 1and an (AB), star with n 1 2 have no critical point: the third-order term does not vanish for any value off. In this paper we only calculate the spinodal, so we restrict our analysis to only the second-order term in the free energy expansion, eq 7. Herein S ( q ) is calculated in the disordered phase (melt) by expanding the free energy in the order parameter. For all of the copolymers considered, S ( q ) has the following functional form:
-
where D[r(s)] = NGr(s) and N is the appropriate normalization constant. To illustrate the method, consider the simple graft copolymer shown in Figure 1. Each of the nm chains has f N monomer units of A and (1- f ) N monomer units of B. The B chain is attached at a fractional position 7 (0 IT I1)along the A chain; Le., the attachment is at ?LA. The partition function for a system of ~ z A Bidentical graft copolymer chains is given by
where U is the system potential energy and 6 l/kT. Notice that the 6 function accounts for the chemical connectivity between the A and B chains. It is assumed throughout that 1A = 1B = 1. To evaluate the effect of concentration fluctuations on the partition function or free energy, it is necessary to specify the microscopic densities:
where i = A or B and LA = fNl and LB = (1- ONl. The mean densities are given by Pi E
( l / V l p i ( r ) dr = nABLi/Vl
The Fourier components piq of the microscopic densities are defined by pip =
where the function Q depends specifically on the copolymer type. Thus, S ( q ) has a maximum at q*, and q* is determined by the equation aQ/aq = 0. As x x,, S(q*) 03 and xs is given by 2xs = Q(q*)/N,. Notice that S(q*) diverges as (x,- x ) - I , which is analogous to how S(0) (or osmotic compressibility) diverges classically for a binary mixture. There is another analogy with critical phenomena of ordinary two-component mixtures. Expand Q ( q ) around q* to second order; then S ( q ) can be rewritten in a modified Ornstein-Zernike form:
-
-
(9)
where the correlation length 5 is given by
t2= (1/4)Q"(q*)/(x, - X ) N O
(10)
(14)
leiqrpi(r)dr
t1 5 4
or by using eq 13
(Factors of (2a)-' that arise from the definition of a Dirac 6 function are suppressed throughout.) Equation 15b is the fundamental equation that relates the amplitudes of the density fluctuations ( p C ) to the path variables ri(s). Note that as defined, the p i p are dimensionless and (16) It is assumed that the pair interactions between monomers i and j at r and r' are short-ranged and contribute a term oij6(r- r') to the system potential energy (aijhas units of energy-volume). The total potential energy U of the copolymer melt may then be expressed as
Thus, 5 diverges as (x,- x)-1/2as does the classical correlation length in an ordinary two-component mixture.
Theoretical Approach The theoretical formalism used here will closely follow the path integral methods developed by E d ~ a r d s Using .~ this approach, Edwards has shown6 that the excluded volume effects in polymer melts are fully screened Le., the chains behave nearly ideally and satisfy Gaussian statistics. Throughout our calculation the chains are assumed to be Gaussian. In a continuous model of a Gaussian chain of N monomers of step length 1 and contour length L = N1, the probability that a chain configuration lies between the continuous space curves r(s) and r(s) + 6r(s) is
or by using eq 16, one can express U in terms of the density Fourier components:
u = UO+ ('/VI C C wi,PiqP*jq iJ=A,B q # O
(17b)
where Uocontains the q = 0 contribution. In Fourier space the incompressibility condition, eq 3, becomes PAq + PBq = P 6 ( q ) (18) Using eq 18 in eq 17 yields
u = UO - (kT/VP) x
PAqP*Aq
(17~)
q+o
where x is the usual interaction parameter:
x P@[aAB- 1/Z(@AA + WBB)] In terms of x, Vo is given by
t 19)
Macromolecules, Vol. 19, No. 10, 1986
2504 de la Cruz and Sanchez
N J O = PVf(1 - f)x +
(unimportant constants and terms linear in f ) (20)
In the Appendix, the expansion of the partition function in terms of pAq for a simple graft copolymer is outlined; it is found that
where D,(x) = cu2D(ax);
cy
= f or 1 - f
x = Nq212/6
(22)
(23)
and D(p) is the well-known Debye function: DG) = (~/P')[/I+
- 11
(24)
and where
F,,(x) = [2 - e - a T X F,(x)
- e-a(l-T)X1 /x
FaT(7 = 0 or 1) = [l - e-.']/x
(25) (26)
When 7 = 0 or 1, S(q) reduces to the result obtained by Leibler' for a diblock copolymer. An approach similar to that used to calculate S ( q ) for a graft copolymer was used to calculate S ( q ) for the two star copolymers in Figure 1. The difference is the chemical constraint that defines the copolymer used in the partition function:
I
B(rA(TLA) - r B ( 0 ) )
= (xn?,. At this point the transition is second order and the free energy is dominated by fluctuations of order q*. In the neighborhood of the critical point (xWtis close to (xN), and the third- and fourth-order terms may be evaluated around q*. This then allows the determination of (XN),. For the graft and (AB), star copolymers, the third-order terms are nonzero at f = 1/2 and all other values off. On the other hand, A,B, star copolymers are symmetric around f = 'I2 and the third-order term vanishes at this composition. Since the graft and (AB), copolymers have no critical point, there is no justification for assuming that (xNtmay be evaluated a t q*. In this paper no attempt was made to determine (xNt(see Concluding Remarks). The divergence of S(q*) at (xN), is on a size scale of the order of the radius of gyration ( R ) of the copolymer; i.e., the product q*R will be of order unity. As is well-known for a linear chain of N monomers (or segments), R2 = N12/6. In what follows it is convenient to relate R of the copolymers to the radius of gyration of the linear diblock copolymer containing No monomers. Thus R2 = gRo2
Ro2 = N012/6 where g = 1 - 6f27(1 - 7)(1 - f ) I1
1 + 2(n - 1)[1- 3f(l - f)]/n 5 3 - 6f(l - f )
graft
'43"
Note that the constraint for (AB), stars places the A units in the core of the star, but this is arbitrary. The result of the S ( q ) calculation for the A,B, star copolymer is S(q)-' = (Df + D i + + 2nFfFl-f + ( n - 1)(Ff2+ Fl-f2))/(No[D@l-f + ( n - l)(DfF,-f2 + Dl-fFf2) - (2%- - 1)(FfF1-f)2]) - 2~ (27)
and for an (AB), star copolymer S(q)-' = {Df+ D1-f+ ( n - 1) X [Ff2 + Fl-f2exp(-2fxo)] 2FfF,-f[l + ( n - 1) X ~ ~ P ( - ~ ~ o ) I ) / I N o+~ D ( n@ - U[Dl-fFf2 ~-~ + DfF1-t X exp(-2fxo)I - (FfFl-f)'[l + 2(n - 1) exp(-fx~)lIl- 2% (28)
+
where xo = N0q212/6
(29)
and all of the D, and F, functions in eq 27 and 28 are functions of x,; the total number of monomers in the star Copolymer N = nNo. Note that when n = 1both eq 27 and 28 reduce to the Leibler result for a diblock copolymer.
Results and Discussion As mentioned in the Introduction, S(q) passes through a maximum at q*. The position of this maximum is independent of temperature (x), but the magnitude of S(q*) diverges at the spinodal temperature or at (xN),. In general, the disorder-to-order transition occurs prior to reaching (xN),; Le., (xWt 5 (xN),. A t the critical point, second- and third-order terms vanish and (XN), = (xN),
graft
g=
g =1
chemical constraint II;=,B[r,'(LA) on each copolymer = rBi(o)]n~i~8[rAL(LB) rA'"(LA)]
(30)
+ 2(n - l ) / n
I3
A,B, (31)
(AB),
The above g relations are all easily obtained from general formulas given in Yamakawa's book.' In the previous section xo = (qR$ and this variable appears naturally in the formulae for S ( q ) . Graft Copolymers. For small and large values of x,, eq 21 can be expanded with the result (here N = No and x = x,)
S(x> VI. The net entropy change per (AB), star molecule is approximately given by ASo - [(n- l ) / n ]In [ V / f ( V - VI)] N ASo [ ( n- U / n l In U / f ) (35) Thus, even at f = 1 / 2 it is expected that (xNo),for (AB), stars will be smaller than for an AB diblock. As can be seen in Figure 7, the calculated values of (xNo),are in agreement with this qualitative prediction. Notice also that the (xNo),curves are asymmetric in the way predicted by eq 35.
01 0.0
I
I
I
0.2
0.4
0.6 f
Figure 7. Variation of (xN0),with composition and arm number for (AB), star copolymers. 5*0/ I
i
n=30/
I
1
I
\ I\
I
/
2.5 0.0
I
0.2
I
0.4
/
I
1
0.6
0.8
J 1.0
f
Figure 8. Variation of (q*Ro)zwith composition and arm number for (AB), star copolymer.
Like the A,B, stars, the leading term in the expansion of S ( q ) for small or large q is independent of n. Thus, to determine the dependence of q* on n, a close examination of what is happening on size scales of the order Ro is required. The formation of an (AB), star (a triblock copolymer) from two diblocks doubles the length of the copolymer and R2 = 2R:. This increase in the dimensions of the copolymer should lower q*; as n becomes large, R2 3R: (see eq 31) so this effect of lowering q* with increasing n rapidly diminishes. As n gets large, the (AB), star begins to develop a “core and shell” type structure. The core will be rich in A monomer and the shell will be rich in B monomer even in the disordered state. This self-segregation or self-micellarization tends to create significant concentration fluctuations at the core-shell interface, which is at size scales of the order f I 2 R . This latter effect will become more significant as n increases and will tend to increase q*. These arguments suggest that q* will pass through a minimum as n increases. This qualitative prediction is in fact observed as is illustrated in Figure 8. The radius of gyration of the core of the star is R t = fR2. If (q*Rf)2is plotted against f instead of (q*R#, the curves look very systematic as is illustrated in Figure
-
9.
Concluding Remarks To our knowledge, the predictions presented in this paper have largely been untested. Although the meso-
Macromolecules, Vol. 19, No. 10, 1986
Microphase Separation in Star Copolymers 2507
I
J4
01 0.0
I
0.2
I
I
0.6
0.4
I
0.8
1.0
f
Figure 9. Variation of (q*R )2 with composition and arm number for (AB), star copolymers. is the radius of gyration of the core of the star; Rf2 = fR2, where R is the radius of gyration for the (AB), star.
if
phases of (AB), stars have been studied,, scattering in the one-phase region and the effect of arm number on transition temperatures have not been examined. The predicted effects of “molecular architecture” on transition temperatures are especially dramatic. For example, if the molecular weight of a diblock was increased by a factor of 2, keeping its composition constant at f = l/,, the critical temperature would increase by a factor of 2. (Transition temperatures for all other compositions would all also increase by about a factor of 2.) However, if the molecular weight was increased by a factor of 2 by forming an (AB), star [2(AB) (AB),], Le., a triblock copolymer, the transition temperature at f = ‘1, would only increase by a factor of about 10.5/8.86 N 1.2. For an AzB2star [2(AB) AzBz]the transition temperature at f = l/z would not increase at all! The predictions of this paper are based on the assumptions of free interpenetration of A and B units in the disordered state and of Gaussian statistics. In stars with large n the A and B units do not overlap much at the core of the stars even a t x = 0. Also, the chains are expected to be stretched at the core. A simple density argument p = nN/ V ,if the chains are assumed Gaussian (V MI2), constrains n to be less than MI2. Nevertheless we expect the predictions in this paper to break down for n even smaller than N12. Predicting the symmetry of a mesophase or its periodicity is burdened with difficulties, some of which are noted below: 1. Leibler’s predictions1 of mesophase symmetries (lamellar, hexagonal, or BCC) are based on the theory of Alexander and McTaque (AM).9 The AM theory was developed for the liquid-solid transition involving atoms of the same size and assumes that only one size scale (q*) is important. When f # l/z the block lengths in a diblock are unequal and it has not been proved (or disproved) that one size scale is sufficient to describe mesophase symmetries in diblock copolymers. 2. A new symmetry, a bicontinuous ordered structure called the double-diamond,has been observed in (AB), star copolymers.2 Recently, this structure has also been observed in simple diblock copolymers.lOJ1This suggests that to predict the ordered structures, a more detailed analysis is required. 3. It is assumed in AM and Leibler’s theory that the periodicity of the ordered structure is 2a/q*. In Leibler’s
-
theory, q* is calculated in the one-phase (disordered)region where the chains are Gaussian. It is now well documented, both e~perimentallyl~-’~ and t h e ~ r e t i c a l l y , ~that ~ ” in the mesophase the chains are “stretched” and deviate substantially from their usual Gaussian dimensions. Periodicities vary as PI3rather than “Iz. At best, a correction of order N1J6must be applied to q* to correlate predicted spacings with those observed experimentally, when the transition is away from the critical point. The above reservations notwithstanding, it is interesting to note that the periodicities observed in (AB), stars with hexagonal symmetry2 (23000 arm molecular weight, 70% core material) appear to exhibit a maximum at about n = 8. This observation is in qualitative agreement with the predictions shown in Figure 8 for the variation of q* with arm number. Acknowledgment. M. Olvera de la Cruz thanks Prof. Frank Karasz for supporting this work and Dr. Jeff Marqusee for helpful discussions. Appendix In order to evaluate the effect of fluctuations on the free energy, the path integrations required in eq 12 must be carried out subject to the constraint that each of the piq satisfy eq 15b. This constraint is handled in the usual way by introducing 6 functions
and can be parameterized in terms of the auxiliary fields hiq:
-
-
(A.2) Introduction of these 6 functions is a unit Jacobian for a change of integration variables from the path variables ri(s) to the density amplitudes pip. The total contribution to the free energy from density fluctuations requires an integration over the piCl. However, for simplicity, these integrations are not explicitly displayed in what follows because our primary interest is not the total free energy but the contribution of the fluctuations to the free energy. The chemical constraint 6[rA(TLA) - rg(O)] is also parameterized in terms of an e field:
where P[ri(s)] = e ~ p [ - ~ ” ~ [ 3 r ~ (+ s )iAi] ~ / 2ds/l] = Po[ri(s)]exp[-iJL’Ai ds/l] (A.5) and where r(s) 9 ar(s)/as and Ai is an effective potential or field with Fourier components Xi,: Ai
=
Xiqeiwrc(s)
qzo
(A.6)
2508 de la Cruz and Sanchez
Macromolecules, Vol. 19, No. 10, 1986
Notice that all of the q = 0 contributions have been extracted in eq A.4; they contribute to the partition function, where Uois defined by eq 20. If Po and P are integrated over all paths but with the chain ends fixed at r and r’, the following subsidiary (Green’s) functions are defined: Go(r,r’;L) =
and the D, and Fa functions are defined by eq 22,25, and 26. Note that VnABis the ideal entropy contribution to the partition function. Substituting (A.15) into (A.ll) and integrating over dX r’(L) Po[r(s)]D[r(s)]= l d k eik.(r-r’)-kZL1/6 yields r(0) Z = 2, exp[-(1/2) p(A-’ + W)pT] (A.17) (A.7)
1
q+o
and Zo = P m exp[-PUo - (1/2)
C In
(Det A)]
(A.18)
P+O
G(r,r’;L) = Go - iG1 - y2G2+ ...
G1(r,r’;L) =
(A.8)
1
ds / 1 dr, Go(r,r,;s)A(rs)Go(r,,r’;L-s) 64.9)
G2(r,r’;L) =
Go(r,r,;s)A(r,)Go(r,,r,~;sCs)A(r,~)GO(rs~,r’;L-s’) (A.lO) Higher order Gi are defined similarly. For the present, (A.8) will be truncated at G2. The partition function can now be written in the following compact form: Z = e-@uoJdX ( l d e l H Ad r d r ’ l H B d r dr’lnABX exp C [ipX - y2 pWpT] (A.11) qfO
where (A.12)
and
HB = e-if*rG(r,r’;LB)
(A.13)
In eq A.12 and A.13 each Green’s function is expanded as in eq A.18. Also p = (pAq,pBq),X = (XAq,XBq), and W is the matrix (A.14)
With the definition of A given in (A.6), the required integrations over r, r‘, r,, c, and s can now be carried out, with the result
(A.15)
where
lexp[iX.p - f/,XAXT]dX = 2n(Det A)-II2 exp[-1/2PA-’pT] (A.19) Finally, the incompressibility condition, eq 18, is invoked, i.e., pAq = -pBq for q # 0, to obtain (note for q # 0, ApAq
dr, dr,’
H A = lG(r,r,;sLA)ei“.G(r~,r’;(l-7)LA) dr,
In the derivation of (A.17), use was made of the general formula
=
PAq)
z= 20exP(-’/,pV
(PAqP*Aq)/s(q))
(A.20)
P+O
where S(q) is given by eq 21 or S(q)-l= pV[1 A lT/(Det A)] - 2x
(A.21)
where 1 = (1, 1). As previously mentioned, the total free energy of the system still requires an integration over pAq followed by a summation (integration) over q. Since our primary objective was to calculate S(q), these subsequent integrations have not been carried out. References and Notes (1) Leibler, L. Macromolecules 1980, 13, 1602. (2) Alward, D. B.; Kinning, D. J.; Thomas, E. L.; Fetters, L. J. Macromolecules 1986,19, 215. (3) de Gennes, P . 4 . J. Phys.,-Lett. 1977, 38, L-441. (4) de Gennes, P.-G. Scaling Concepts in Polymer Physics; Corne11 University: Ithaca, NY, 1979; pp 62-68. (5) Edwards, S . F. In Path Integrals; Papadopoulos, G. J., Ed.; Plenum: New York, 1977; pp 285-313. (6) Edwards, S. F. J. Phys. A 1975, 9, 1670. (7) Yamakawa, H. Modern Theory of Polymer Solutions; Harper and Row: New York, 1971; pp 47-52. (8) DiMarzio, E. A. J. Chem. Phys. 1967, 47, 3451. (9) Alexander, S.; McTague, J. Phys. Reu. Lett. 1978, 41, 702. (10) Thomas, E. L., private communication. (11) Hashimoto, T., private communication. (12) Hasegawa, H.; Hashimoto, T.; Kawai, H.; Lodge, T. P.; Amis, E. J.; Glinka, C. J.; Han, C. C. Macromolecules 1985, 18, 67. (13) Hadziioannou, G.; Picot, C.; Skoulios, A.; Ionescu, M.-L.; Mathis, A,; Duplessix, R.; Gallot, Y.; Lingelser, J.-P. Macromolecules 1982, 15, 263. (14) Hashimoto, T.; Shibayama, M.; Kawai, H. Macromolecules 1980,13, 1237. (15) DiMarzio, E. A.; Sanchez, I. C. Bull. Am. Phys. SOC.1984,29, 281.
(16) Helfand, E.; Wasserman, Z. R. Macromolecules 1976, 9, 879. (17) Meier, D. J. Prep. Polym. Colloq., SOC.Polym. Sei., Jpn. 1977, 83.
