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

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1.6 Polymer Brushes 35 partition functions would be solved under the boundary conditions, q(0,s)=q † (0,s)=0, which prevents the polymers from crossing the z =0plane and ensures that φ s (z) → 0 as z → 0. For discussion purposes, we will assume that the profile is not so sharp as to invalidate our coarse-grained model. It will certainly be sharp enough for the ground-state-dominance approximation to hold, which implies that the surface field is given by w s (z) = a2 N 6 ∇ 2√ φ s (z) √ φs (z) (1.156) However, one further relation between w s (z) and φ s (z) is required in order to solve for the surface profile. This is where the details of the molecular interactions must enter. Provided that we are not concerned with knowing the details of the surface, such as φ s (z) and the resulting surface tension, it is possible to proceed without a full treatment of the surface profile. This involves implementing the alternative boundary conditions, ∂ ∂z q(0,s)= ∂ ∂z q† (0,s)=0 (1.157) and solving the SCFT for w(z) ignoring the deviation from φ(z) =1at the surface. The full solution would only differ in the narrow region next to the substrate where φ s (z) < 1. More specifically, if we were given φ s (z), the proper solution would be obtained by making the substitutions, q(z,s) ⇒ q(z,s) √ φ s (z) (1.158) q † (z,s) ⇒ q † (z,s) √ φ s (z) (1.159) w(z) ⇒ w(z)+w s (z) (1.160) The added factor of √ φ s (z) would restore the proper boundary condition for the partial partition functions, and would transform the concentration profile from φ(z) =1to φ(z) =φ s (z). One can also confirm that the transformation in the field is exactly as required for the transformed partition functions to satisfy the diffusion Eqs. (1.24) and (1.28); the proof of this requires that the pretransformed partition functions satisfy [ ][ ] [ ][ ] ∂ ∂ ∂z q(z,s) √ ∂ ∂ √ φs (z) = ∂z ∂z q† (z,s) φs (z) =0 (1.161) ∂z Indeed, it is safe to assume that the boundary conditions in Eq. (1.157) extend over the narrow region where φ s (z) has a nonzero gradient. For most problems such as the brush, we simply assume that there is no variation in the profile, φ s (z), of each surface, rendering them as inert features that conveniently decouple from the behavior of the remaining system. Thus, we can continue to ignore the details of the molecular interactions and solve the field subject to the incompressibility condition, φ(z) =1, with the partial partition functions satisfying reflecting boundary conditions, Eq. (1.157), at z =0and z = L. For the case of the polymer brush, there is still one more issue to deal with at the z =0 substrate. The grafting of the chain ends in a perfect plane, which for the moment is at z = ɛ, produces a delta function in the field (Likhtman and Semenov 2000). Our intention is to
36 1 Self-consistent field theory and its applications incorporate this singularity into the boundary condition, restoring w(z) to a continuous, finite function. We must, however, begin with the true field, w t (z) = βa2 N δ(z − ɛ)+w(z) (1.162) 6 containing the singularity at a small infinitesimal distance, ɛ, from the substrate. The diffusion Eq. (1.24) requires the delta function in w t (z) to be balanced by an infinity in ∇ 2 q(ɛ, s). In other words, the delta function produces a sharp kink in ∂ ∂zq(z,s) at z = ɛ, and similarly in ∂ ∂z q† (z,s). The strength of the kink is determined by integrating the diffusion equation from z =0to 2ɛ, which gives a 2 N 6 [ ∂ ∂z q(2ɛ, s) − ∂ ∂z q(0,s) ] − βa2 N q(ɛ, s) ≈ 0 (1.163) 6 ∂ ∂z q z(0,s)=0. Dropping this term Our original reflecting boundary condition implies that from Eq. (1.163) and then taking the limit ɛ → 0 yields the new boundary condition, ∂ q(0,s)=βq(0,s) (1.164) ∂z Naturally, the complementary partition function, q † (z,s), satisfies the same boundary condition at z =0. The constant β is determined by the requirement that w(z) does not develop a delta function at z =0. 1.6.6 Spectral Solution to SCFT There are two main ways to solve the SCFT; one is a direct real-space method (Matsen 2004), where the diffusion equations are solved using finite differences for the derivatives (the Crank- Nicolson algorithm), and the other involves a spectral method (Matsen and Schick 1994a) much like that discussed in Section 1.5.1. The spectral method is particularly efficient for lower grafting densities, while the real-space one is better suited to thick brushes. Note that there are also semi-spectral algorithms (Rasmussen and Kalosakas 2002) that combine the advantages of the two methods. Here, we demonstrate the spectral method. Since the field is unknown, the functional expansions will be performed in terms of Laplacian eigenfunctions, f i (z), defined by ∇ 2 f i (z) =− λ i L 2 f i(z) (1.165) subject to the boundary conditions, f i ′(0) = βf i(0) and f i ′(L) =0. Note that the L2 in Eq. (1.165) has been included so that λ i becomes dimensionless. The solution to this eigenvalue problem is (√ ) f i (z) =C i cos λi (1 − z/L) (1.166) where the boundary conditions require √ λi tan (√ λi ) = βL (1.167)
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 35: 34 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

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

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