Macromolecules 1993,26, 3344-3350
3344
Lamellar Ordering in Symmetric Diblock Copolymers R. A. Sones,’ E. M. Terentjev, and R. G . Petschek Department of Physics, Case Western Reserve University, Cleveland, Ohio 44106- 7079 Received November 12, 1992; Revised Manuscript Received March 1 , 1993
ABSTRACT We calculate the lamellar period L and interphase thickness a of an incompressible melt of symmetricdiblock copolymers from the onset of phase segregation (weak segregationlimit) to the limit where segregation is almost complete (strong segregation limit) by numerically solving a mean field lattice model, where the lattice spacing is taken small enough to approximate a continuum. Our results for L and a agree with previously derived theoretical formulas in both limits. Based on the first few terms of a Fourier series expansion of the density profile, we show analytically that L = O . ~ ~ ~ R ( X N )atO the . ~ ~weak ~ segregation limit, where R is the unperturbed molecular radius of gyration, x the Flory interaction parameter, and N the number of statistical segments per molecule; the compressibility of the melt-even in the incompressible limit-must be taken into consideration to get the correct dependence of L on xN, and the Fourier series approximation turns out to be accurate over a very small range of xN.
I. Introduction A polymer is called a diblock copolymer if it consists of two subchains (or blocks),one containing type Amonomers and the other type B. Typically A and B monomers have a positive free energy of mixing when the translation entropy-which is small for monomers constrained to be on a polymer-is ignored, and in a melt they tend to segregate into A-rich and B-rich domains. The linkage between A and B blocks precludes macroscopic phase separation and leads to interesting microscopic ordering on a scale of the radius of gyration of the polymer. A recent article by Bates1 reviews polymer-polymer phase behavior and references much of the literature. Theorists resort to various approximations when describing diblock copolymer melts. The molecules are typically modeled as ideal Gaussian chains with the same Kuhn length and volume per monomer for both blocks, and usually the melt is assumed incompressible. The statistical mechanics of this simplified model is then solved by the application of mean field theory or the random phase approximation. Within the frameworkof these approximations,analytic descriptions of microphase ordering have been proposed for two limiting cases. A melt of symmetric (same number of monomers in both blocks) diblock copolymers, with N monomers per molecule, unperturbed radius of gyration R, and Flory interaction parameter x (which is approximately inverselyproportional to temperature), is expected to exhibit a transition from a homogeneous mixture to an ordered structure when xN exceeds 1O.Ei2 In the weak segregation limit (xN = 10.5) the density distribution exhibits shallow sinusoidal modulation with predicted period 3.23R.2 As xN increases the modulation depth and period increase, until in the strong segregation limit (xN >> 10.5) the density distribution is lamellar with predicted period3
L = (192/7~~)~’~R(xiV)~’~ In this limit the predicted size of the transition region between A and B phases (the interphase width) is3y4
where the second term in brackets is a correction recently derived by S e m e n ~ v The . ~ interphase width is defined as the peak-to-peak modulation of the type A (or type B) density distribution divided by the magnitude of its slope 0024-929719312226-3344$04.00/0
at the center of the transition region. According to this definition, at the weak segregation limit where the density distribution is sinusoidal, the interphase width is 3.23RI a.
No analytic expressions exist for L or a for intermediate values of xN. In an early paper on block copolymers, Helfand6numerically calculated density distributions for xN = 20 and 74, from which values of L and a can be extracted. Recently, Melenkevitz and Muthukumar7and Shul18numerically calculated L for the entire range of xN from the weak to the strong segregation limits. Both find L R(xNW6 in the strong segregation limit, and L = 3.23R at the weak segregationlimit, but in the intermediate regime their results disagree. Shull discusses this disagreement and argues that his calculation involves fewer assumptions. Shull‘s results for L converge to eq 1in the strong segregation limit; this is reassuring, since his calculations are based on the same mean field theory for which eq 1was derived. Shull also numerically calculated a and compared his results to eq 2 without the correction term and found about 10% disagreement well into the strong segregation regime ( x N N 150); inclusion of the correction term eliminates this disagreement. The dependence of L on x N in the weak segregation limit is a topic of current interest. The belief is widespread that L R in this limit,7s9 but some measurementdo suggest L RIP3,one recent theory1’ predicts L RN0.6, and Shull8 deduces L RN0.6 from his numerical data. In this paper we calculate L and a for xN ranging from the weak to the strong segregation limits, by numerically solving a mean field model implemented on a lattice-an approach similar to that of Shull.* Then in section IILA we show analytically that our model yields L = 0.844R(~N)0.5~~ in the weak segregation limit.
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11. Mean Field Theory Our mean field model is similar to that of Helfand,4I6J2 but our notation and approximations are geared for direct numerical solution. Consider a system of volume V and temperature T (in energy units) containing Nc identical noninteracting diblock copolymer molecules. We regard each molecule as a freely-jointed chain with N segments and treat space as a simple cubic lattice with lattice constant 1, where P is the mean volume per segment, and require adjacent molecular segments to occupy nearestneighbor lattice sites.13 JV = VIP is the total number of lattice sites. As 1 0 the cubic lattice approaches a continuum and the freely-jointed chains become ideal
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1993 American Chemical Society
Macromolecules, Vol. 26, No. 13, 1993
Lamellar Ordering in Copolymers 3345
Gaussian chains; in our calculations we will take 1 small enough so that its size does not affect our results. (Though our interest in this paper is with symmetric diblock copolymers, the theory we present in this section does not assume both blocks have the same number of segments. However, we do assume the blocks have the same Kuhn length and volume per segment.) Let i = A, B be the block index, k = 1, 2, ...,N the segment index, and ik the value of the block index of the kth segment. When the chains are acted upon by a mean field p(i,r), which gives the energy per unit temperature of a type i segment at lattice site r, the system partition function (neglecting constant factors) is 2 = 2,NC
(3)
where 2, is the single-chain partition function (see, for example, eq 2.1 of ref 4) k
P(rl- r2)...P(rN-l- rN) (4) and the summations before the exponential are over all lattice sites. P(rk-1- r k ) is the probability that segment k is located at r k when segment k - 1 is at rk-1. For a random walk on a cubic lattice
Slightly different forms are used in other papers, but they all give identical results for polymer melts in the incompressible limit.14) We approximate the free energy (per chain per unit temperature) of a system of interacting chains as
where the first term on the right is the total free energy of a system of chains in the presence of the mean field but without segment-segment interactions (the reference state), the second is the free energy of the reference state due to interactions between segments and the mean field, and the third is the Flory-Huggins free energy evaluated in the reference state.lS The first and second terms give the entropy of the reference state, which makes it clear that we are simply using the entropy of the reference state (which we know how to calculate) as an approximation for the entropy of the interacting state. This procedure gives an upper bound to the free energy. The “best” such free energy is then given by functional minimization of eq 12 with respect to both 4 and p, resulting in $09 - 1 p(A,r) = + x+(B,r) K AB,r) = 7 - + x@(A,r)
where %, 9, and 2 are unit vectors. When N >> 1, this random walk results in a Gaussian distribution for the distance between chain ends with variance 2R2,where R2 = (N/6)12.13Equation 4 can be written
where
Z?k,r) = Z-(k,r) exp(-Aik,r))P(k,r) P(k,r) =
- r) exp(-p(ik,r’))Zf(kfl,r’)
(7)
where the Kronecker delta function restricts the summation to segments in block i. Equations 13 and 14 come from variational terms containing 64,while eq 15 comes from terms containing 6p. Variation of ln(2,) is facilitated by noting that
(8)
and at the chain ends
Z-(l,r) = Z+(N,r) = 1 (9) ZO(k,r) is the statistical weight (unnormalizedprobability) of finding segment k of a given chain at r, and Z*(k,r) are the statistical weights associated with the portions of the chain before (-) and after (+) segment k. Given p(i,r), the recursive structure of eq 8 allows one to calculate ZO(k,r) and 2, from eqs 5-9. Now consider interactions between segments. In the spirit of Flory-Huggins theory write the local free energy density
K is a dimensionless compressibility, x the Flory interaction parameter for A and B Segments, PO = NJV/ V the mean number density of segments, and 4(i,r) the number of type i segments per lattice site at r. Here $(i,r) is a reduced (dimensionless)partial density, while the total density is
Equations 6 and 15 imply the average value of $(r)is unity. Based on a mean field approximation and the FloryHuggins local free energy density, eqs 13-15 give a formal solution for a diblock copolymer system. The goal is to find distributions 4 and p which satisfy eqs 13-15 when ZO and 2, are calculated from eqs 5-9. 111. Solution Methods
A. Analytic Solution near the Weak Segregation Limit. In this section we investigate the behavior of our model for symmetric blocks near the weak segregation limit (xN 10.5) when N and x 0. Near the weak segregation limit the density profiles are one-dimensional sinusoids with infinitesimal amplitude,2 and we write
-
where
(Our f(r) is identical to that used by Helfand in ref 12.
1
$(A,z) = $1 $(&)
+ e COS(qZ) +
COS(2qZ))
(17)
= $1 - e cos(qz) + n2cos(2qz))
(18)
7e2
where q = 2?r/L,z is the position coordinate normal to the lamellae, e 0. The closer A is to zero, the closer 4 is to an exact solution. A is a scalar field, and one can use standard multidimensional minimization techniques (such as conjugate gradient descent"? to locate the solution point where A = 0. However, we use the following approach, which we have found to be simpler and more efficient. Guess an initial 4 and consider the line passing through 4 in the direction T4 - 4. This line may be written
&i,r;v) = $(i,r) + v(Td(i,r)- d(i,r)) (33) where 4is a point on the line located "distance" Y from 4. Every point on the line has a corresponding A given by eq 32. Use standard one-dimensional minimization technqiues (such as Brent's method17) to find the point on the line with minimum A (Le., minimize A with respect to v), then use this point as an improved new guess. This procedure can be iterated until A approaches zero. If at some step during iteration the minimum A occurs at Y = 0, convergence fails, and a different initial 4 should be chosen. The numerical solution strategy presented in this section is quite general, valid for three-dimensional distributions of asymmetric diblock copolymers. Specialization to the case of one-dimensional(lamellar)structure and symmetric blocks is discussed in Appendix B. IV. Results and Discussion Equations 1and 2 for L and a in the strong segregation limit depend only on xN and R, not the number of segments N. These equations were derived for N Q).Similarly, our numerical results for L and a depend only on xN and R when N >> 1,and we have taken N large enough so that the dependence on N is negligible (see Appendix B). A polymer melt is nearly incompressible ( K N 0). As K 0 our numerical results for L and a approach limiting values corresponding to an incompressible system (see Figure 1). In the followingnumerical calculationswe used K = 0.1 to simulate an incompressible system. Reducing
-
-
where the subscript zero denotes a value at equilibrium, the primes denote differentiation with respect to Q, and
*
Lamellar Ordering in Copolymers 3347
Macromolecules, Vol.26, No.13,1993
4*2
‘
3.8 0.01
I
0.1
I
I
I
1
10
100
,
.-
2.0
2.5
3.0
3.5
4.0
4.5
5.0
WxN)
K
Figure 2. Ratio of lamellar period to radius of gyration versus xN. Our numericalresults (TableI) are indicated by open circles, and the filled circle is Leibler’sa weak segregation limit. The dashed lines are analytic predictions for the weak (eq 31) and strong (eq 1) segregation limits. The calculations of Shull (extracted from Figure 2 of ref 8)are indicated by boxes, those of Helfande by triangles, and the measurements of ref 18 by diamonds. The solid line represents the solution of eq 26.
0.9
0.8
F 0.7
o.6
0.5
t
I
0.01
I
I
I
1
0.1
1
10
100
1
0
- -0.5 s-
Or,
v
-1.0
K
Figure 1. Dependence of (a,top) lamellar period and (b,bottom) interphase width on compressibility (xN = 21,N = 210). For K INf 2 116 if z = &I
(B2) 0 otherwise and seek solutions &(i,z)and p(i,z)periodic in z with period L. (Keep in mind that z is a discrete variable, and 4 and p are discrete distributions.) Because the blocks are symmetric, 4(A,t) = +(B,z+L/2)and p(A,z) = p(B,z+L/ 2), and knowledge of the solution over any half-period determines the entire solution. Specifically, let z = jl, where j = 1, 2, ..., JV and N = L/(2l), and impose the "reflectingwall" boundary conditions (requiredto evaluate the convolutions in eq 8) Ai,O) = p(i,l) 033) p ( i J P + h = p(iJ/2) 034) Ll(21)must be an integer, and the center of the interphase region is at L/4. The solution proceeds as follows: Given x, N, K , and JV, and given an initial guess for 4(i,z),iterate until A is within aspecified tolerance (typicaUyO.001) of zero, and calculate the free energy from eq 12. Then select a new value of JV, reinitialize 4(i,z), and repeat the procedure. Keep this up until the value of JV which minimizes the free energy is determined. The corresponding @(i,z) is the equilibrium density distribution, with period L = 2JVl and interphase
3360 Sones et al.
width
where 4’ represents the derivative with respect to z and is evaluated by fitting a natural cubic spline to +(A,z).In practice, we find the three even and three odd values of JV which bracket the minimum free energy, calculate the corresponding values of L and a, then use parabolic interpolation to estimate their values at the minimum free energy. The even and odd data sets are treated separately, and their results compared. If they disagree, then this means the lattice is too coarse to accurately describe the narrow interphase region. The solution is to increase the number of molecular segments N, which effectively decreases the segment length (and lattice constant) 1. But increasing N increases the computational burden. For each data point in Table I, N was made large enough so that the even and odd results agreed to within 0.7% for L and 3.3% for a. The initial density distribution we use is ~ ( A , Z ) = 1- ~ J ( B , z=) ~ ( L M - z ) (B6) where e(z)is the unit step function. However, for small K the iterative procedure fails (A does not go to zero). The solution is to startwith a relatively large value of K (typically lo), solve for r#~(i,z),then use this as a new initial distribution with a (typically 5 % ) smaller value of K. We proceed in this way until we reach the desired value K = 0.1.
Macromolecules, Vol. 26, No. 13, 1993
References and Notes (1) Bates, F. S. Science 1991, 251, 898. (2) Leibler, L. Macromolecules 1980,13, 1602. ( 3 ) Semenov, A. N. Sou. Phys. JETP 198S,61, 733. (4) Helfand, E.; Tagami, Y. J. Chem. Phys. 1972,56,3692. (5) Semenov,A. N. Theory of Block-CopolymerInterfacesin Strong
Segregation Limit. Submitted to Macromolecules.
(6) Helfand, E. Macromolecules 1975,8,652. Resulta for L and a were extracted from Figures 2 and 3, with an estimated uncertainty of *5 % (7) Melenkevitz, J.; Muthukumar, M. Macromolecules 1991, 24, 4199. (8) Shull, K. R. Macromolecules 1992,25, 2122. (9) Ohta, T.; Kawasaki, K. Macromolecules 1986,19, 2621. (10) Almdal, K.; Rosedale, J. H.; Bates, F. S.; Wignall, G. D.; Fredrickson, G. H. Phys. Rev. Lett. 1990,65, 1112. (11) Olvera de la Cruz, M. Phys. Rev. Lett. 1991,67,85. (12) Helfand, E. J. Chem. Phys. 1976,62,999. (13) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY,1979; pp 29-31. (14) Helfand, E.; Wawerman, Z. R. Macromolecules 1976, 9, 879. (15) Feynman, R. P. Statistical Mechanics; W. A. Benjamin, Inc.: Reading, MA, 1972; pp 67-70.
.
(16) Press, W. H.; Flannery, B. P.; Teukolsky, S. A,; Vetterling, W.
T. Numerical Recipes in C; Cambridge University Press: Cambridge, 1990; p 317. (17) Press, W. H.; Flannery, B. P.; Teukoleky, S. A.; Vetterling, W. T. Numerical Recipes in C; Cambridge University Press: Cambridge, 1990; p 299. (18) Anaatasiadis, S. H.; Russell, T. P.; Satija, S. K.; Majkrzak, C. F. J. Chem. Phys. 1990,92, 5617.
