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

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1.6 Polymer Brushes 31 1.6.3 Diffusion Equation for a Parabolic Potential In Section 1.2, we established that the path-integral definition of the partition function is equivalent to the one based on the diffusion equation, just as Feynman’s path-integral formalism of quantum mechanics coincides with Schrödinger’s differential equation. As in quantum mechanics, the differential equation approach is generally the more practical method of calculating q(z,z 0 ,s). Here, we demonstrate that it indeed gives identical results to those obtained in the previous section, where the path-integral approach was used. Normally the diffusion Eq. (1.24) has to be solved numerically, but there already exists a known analytical solution for parabolic potentials (Merzbacher 1970). In fact, it was originally derived for the wavefunction of a simple harmonic oscillator; once again, we benefit from the analogy with quantum mechanics. Transforming the quantum mechanical wavefunction in Merzbacher (1970), according to Eqs. (1.50)-(1.56) for the parabolic potential in Eq. (1.113), gives ( q(z,z 0 ,s)= 3 4sin(πs/2) ) 1/2 ( exp − 3π[(z2 + z0 2)cos(πs/2) − 2zz ) 0] 4a 2 N sin(πs/2) (1.138) Although the derivation of this expression is complicated, it is relatively trivial to confirm that it is in fact the proper solution to Eq. (1.24) for the initial condition, q(z,z 0 , 0) = aN 1/2 δ(z − z 0 ). Substituting z =0and s =1into Eq. (1.138), we find that q(0,z 0 , 1) = √ 3/4, consistent with the path-integral calculation in Eq. (1.127). In terms of the partition function, the distribution of the sN’th segment is given by ρ(z; z 0 ; s) = q(0,z,1 − s)q(z,z 0,s) (1.139) q(0,z,1) The derivation is essentially the same as that for Eq. (1.31). Inserting Eq. (1.138) and simplifying with some basic trigonometric identities gives the identical expression to that in Eq. (1.134), as required by the equivalence of the two approaches. Given this, the remaining results of the preceding section follow. One slight difference is that we now obtain a specific value for the unknown constant in Eq. (1.137), because the initial condition for q(z,z 0 ,s), in effect, selects a proportionality constant for Eq. (1.127), but not one with any physical significance. Nevertheless, this provides us with a standard reference by which to compare calculations. 1.6.4 Self-Consistent Field Theory (SCFT) for a Brush The parabolic potential used up to now is only approximate. The argument for it assumes that the tension at the free chain-end is zero, but there is in fact a small entropic force that acts to broaden the end-segment distribution, g(z 0 ) (Matsen 2002b). Here, the self-consistent field theory (SCFT) is described for determining the field in a more rigorous manner. The theory starts with the partition function for the entire system, which is written as ∫ ( ) ∏ n Z ∝ ˜Dr α δ(z α (1) − ɛ) δ[1 − ˆφ] (1.140) α=1
32 1 Self-consistent field theory and its applications where the tildes, ˜Dr α ≡Dr α P [r α ], denote that the functional integrals are weighted by ( P [r α ]=exp − 3 ∫ 1 ) 2a 2 ds|r ′ N α(s)| 2 (1.141) 0 so as to account for the internal entropy of each coarse-grained segment. The functional integrals over each r α (s) are, in principle, restricted to the volume of the system, V = AL. To ensure that the chains are properly grafted to the substrate, there are Dirac delta functions constraining the s =1end of each chain to z = ɛ; we will eventually take the limit ɛ → 0. Furthermore, there is a Dirac delta functional that constrains the overall concentration, ˆφ(r), to be uniform over all r ∈V. Although the expression for Z is inherently simple, its evaluation is far from trivial. Progress is made by first replacing the delta functional by the integral representation, ∫ ( ∫ ) δ[1 − ˆφ] ρ0 ∝ DW exp dr W (r)[1 − N ˆφ(r)] (1.142) This expression is equivalent to Eq. (1.335) derived in the Appendix, but with k(x) substituted by −iρ 0 W (r)/N . The constants, ρ 0 and N, have no effect on the limits of integration, but the i implies that W (r) must be integrated in the complex plane along the imaginary axis. Inserting Eq. (1.142) into Eq. (1.140) and substituting ˆφ(r) by Eqs. (1.13) and (1.108) allows the integration over the chain trajectories to be performed. This results in the revised expression, ∫ Z ∝ ( ∫ DW (Q[W ]) n ρ0 exp N ) dr W (r) (1.143) where ∫ Q[W ] ∝ ( ˜Dr α exp − ∫ 1 0 ) ds W (r α (s)) δ(z α (1) − ɛ) (1.144) is the partition function of a single polymer in the external field, W (r). For convenience, the partition function of the system is reexpressed as ∫ ( Z ∝ DW exp − F [W ] ) (1.145) k B T where ( ) F [W ] Q[W ] nk B T ≡−ln − 1 ∫ AaN 1/2 V dr W (r) (1.146) There are a couple of subtle points that need to be mentioned. Firstly, the present calculation of Z allows the grafted ends to move freely in the z = ɛ plane, when actually they are grafted to particular spots on the substrate. As long as the chains are densely grafted, the only real consequence of treating the ends as a two-dimensional gas is that Q[W ] becomes proportional to A, the area available to each chain end. To correct for this, the Q[W ] in Eq.
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 31: 30 1 Self-consistent field theory a
Page 80 and 81: Bibliography Almdal, K., Rosedale,
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BIBLIOGRAPHY 81 Matsen, M. W. and G
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Index action 17 binodal point 46 bl
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Self-Consistent Field Theory and Its Applications by M. W. Matsen

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