Self-Consistent Field Theory and Its Applications by M. W. Matsen

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1.6 Polymer Brushes 37 With the normalization constant set to √ 4 √ λ i C i = 2 √ λ i +sin ( 2 √ ) (1.168) λ i the basis functions obey the orthonormal condition in Eq. (1.60). Once the basis functions are determined, there are a couple of useful quantities to calculate. The first is the symmetric tensor, Γ ijk ≡ 1 ∫ dzf i (z)f j (z)f k (z) (1.169) L and the second is the set of coefficients for the identity function, I i ≡ 1 ∫ L sin (√ ) λ i dzf i (z) =C i √ (1.170) L 0 λi defined such that ∑ i I if i (z) =1. With the preamble complete, we are now ready to solve for the partial partition function. The procedure is slightly more involved than in Section 1.5.1, because the field is now omitted from Eq. (1.165), but that is the price we pay to have simple basis functions. Nevertheless, the procedure is much the same. The expansions, q(z,s) = ∑ ∑ i q i(s)f i (z) and w(z) = k w kf k (z), are first inserted into the diffusion Eq. (1.24), giving ∑ dq i (s) ds f i(z) =− a2 N ∑ 6L 2 λ i q i (s)f i (z) − ∑ w k q i (s)f i (z)f k (z) (1.171) i i ik By multiplying through by f j (z), integrating over r, and using the orthonormal condition, this can be rewritten as a system of first-order linear ordinary differential equations, d ds q i(s) = ∑ j A ij q j (s) (1.172) subject to the initial conditions, q i (0) = I i . The coefficients of the matrix, A, are given by A ij ≡− λ ia 2 N 6L 2 δ ij − ∑ k w k Γ ijk (1.173) Note that the boundary conditions on q(z,s) are already accounted for by those on f i (z). The solution to Eq. (1.172) is simply q i (s) = ∑ j T ij (s)q j (0) (1.174) where T(s) ≡ exp(As) is referred to as a transfer matrix. To evaluate it, we perform a matrix diagonalization A = UDU −1 (1.175)
38 1 Self-consistent field theory and its applications where the elements of the diagonal matrix, d k ≡ D kk , are the eigenvalues of A and the respective columns of U are the corresponding normalized eigenvectors. Since A is symmetric, all the eigenvalues will be real and U −1 = U T . We can then express T(s) =U exp(Ds)U T , from which it follows that T ij (s) = ∑ k exp(sd k )U ik U jk (1.176) Similarly, the solution for the complementary function, q † (z,s) ≡ ∑ i q† i (s)f i(z),isgivenby q † i (s) =∑ j T ij (1 − s)q † j (1) (1.177) where q † i (1) = 1 L ∫ L 0 (√ ) aN dzq † 1/2 (z,1)f i (z) =C i cos λi L (1.178) This result is obtained by first integrating with q † (z,1) = aN 1/2 δ(z − ɛ), and then taking the limit, ɛ → 0. Once the partial partition functions are evaluated, the partition function for the entire chain is obtained by Q[w] V = ∑ i q i (s)q † i (s) =∑ i exp(d i )¯q i (0)¯q † i (1) (1.179) where we have made the convenient definitions, ¯q i (0) ≡ ∑ j q j(0)U ji and ¯q † i (1) ≡ ∑ j q† j (1)U ji. 2 w(z) a2 N/L 2 1 0 -1 -2 2 4 L/aN 1/2 = 1 -3 0.0 0.2 0.4 0.6 0.8 1.0 z/L Figure 1.7: Self-consistent field, w(z), plotted for several brush thicknesses, L. The dashed curve denotes the SST prediction in Eq. (1.113), shifted vertically such that its average is zero.
Page 1 and 2: Self-Consistent Field Theory and It
Page 3 and 4: 2 Contents Index 83
Page 5 and 6: 4 1 Self-consistent field theory an
Page 11 and 12: 10 1 Self-consistent field theory a
Page 37: 36 1 Self-consistent field theory a
Page 80 and 81: Bibliography Almdal, K., Rosedale,
Page 82 and 83: BIBLIOGRAPHY 81 Matsen, M. W. and G
Page 84 and 85: Index action 17 binodal point 46 bl

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