Numerical solution of SCF equations

That is, to arrive at z in j steps, a chain must have been at one of the adjoining layers on the previous step. The factors 1/6 and 2/3 are for a simple cubic lattice. If one imposes appropriate boundary conditions on V (l,z), then one can iteratively solve eq. (9.11) for a given F(z). For chains end-grafted to a planar wall, F(0) = oo and V (l,z) = 6, 1. 

The monomer density p z) for a given ip s, z, z ) is the normalized weight for a chain to travel from its origin z — 0 through the point z in j steps to some other endpoint z, whieh it must reach in IV — 5 steps. For a one component brush in a solvent, the origin is the grafting surface and as such 

The self-consistency comes in because depends on p(z) through V(z). This set of equations can then be solved by a relaxation technique. For diblock copolymers and blends, additional distribution functions are necessary, but the general form of the solution is similar. Scheutjens and Fleer were the first to exploit the numerical solution of the SCF equations on a lattice for polymers in solution using eq. (9.11). It is possible now to generate an extremely large number of conformations and study N as large as 100000. Their method has been widely applied to study adsorption of poly- 

The SCF equations were originally written in terms of a diffusion equation for tj s,r,r ). If one expands exp(-F(r)/kaT) = 1/(1 -h V v)/kBT), which is exact in the large N limit, then it is easy to see that eq. (9.11) is simply the discrete representation of the modified diffusion equation of Edwards,  

The polymer chain is then treated as a continuous curve and space is dis- 

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