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

74 1 Self-consistent field theory and its applications
where φ A (r) is the concentration from n A noninteracting A-type polymers subjected to the
complex field, W A (r) =W + (r)+W − (r). In this case, φ A (r) is a complex quantity with no
physical significant. The same procedure can be adapted equally well to the partition functions
of other systems such as the diblock copolymer melts considered in Section 1.8.1.
Since the integrand of Z in Eq. (1.316) involves a Boltzmann factor, exp(−F/k B T ),
standard simulation techniques of statistical mechanics can be applied, where the coordinates
are now the fluctuating fields, W − (r) and W + (r). Ganesan and Fredrickson (2001) have used
Langevin dynamics while Düchs et al. (2003) have formulated a Monte Carlo algorithm for
generating a sequence 〉 of configurations with the appropriate Boltzmann weights. The ensemble
average,
〈ˆφA (r) , can then be approximated by averaging φ A (r) over the configurations
generated by the simulation. Even though φ A (r) is generally complex, the imaginary part will
average to zero. The need to consider two separate fluctuating fields is still computationally
demanding, but in practice the functional integral over the field, W + (r), can be performed
with the saddle-point approximation, which implies that as W − (r) fluctuates, W + (r) is continuously
adjusted so that φ A (r) +φ B (r) =1(Düchs et al. 2003). Even with the problem
reduced to a single fluctuating field, it is still an immense challenge to perform simulations
in full three-dimensional space. However, with the rapid improvement in computational performance
and the development of improved algorithms, it is just a matter of time before these
field-theoretic simulations become viable.
1.11 Appendix: The Calculus of Functionals
SCFT is a continuum field theory that relies heavily on the calculus of functionals. For those
not particularly familiar with this more obscure variant of calculus, this section reviews the
essential aspects required for a comprehensive understanding of SCFT. In short, the term
functional refers to a function of a function. A specific example of a functional is
F[f] ≡
∫ 1
0
dx exp(f(x)) (1.319)
This definition provides a well-defined rule for producing a scalar number from the input
function, f(x). For example, F[f] returns the value, e, for the input, f(x) =1, and it returns
(e − 1) for f(x) =x. Note that we follow a convention, where the arguments of a functional
are enclosed in square brackets, rather than the round brackets generally used for ordinary
functions.
The calculus of functionals is, in fact, very much like multi-variable calculus, where the
value of f(x) for each x acts as a separate variable. Take the case where 0 ≤ x ≤ 1,
and imagine that the x coordinate is divided into a discrete array of equally spaced points,
x m ≡ m/M for m =0, 1, 2, ..., M. Provided that M is large, the function, f(x), is well
represented by the set of values, {f 0 ,f 1 , ..., f M }, where f m ≡ f(x m ). In turn, the functional
in Eq. (1.319) is well approximated by the multi-variable function,
F({f m }) ≡ exp(f 0)
2M
+ 1 M
M−1
∑
m=1
exp(f m )+ exp(f M)
2M
(1.320)