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

1.1 Gaussian Chain 7
structure involving an extensive amount of internal interface. Naturally, the properties of the
interface have a pronounced effect on the final material; in particular, a high interfacial tension
results in large domains, which is generally undesirable. One way of halting the tendency of
domains to coarsen is to chemically bond the unlike polymers together, forming block copolymer
molecules. The blocks still segregate into separate domains, but the connectivity of the
chain prevents the domains from becoming thicker than the typical size of a single molecule.
In fact, the domains tend to form highly-ordered periodic geometrics, of which the lamellar
phase of alternating thin layers pictured in Fig. 1.3(c) is the simplest. This ability to selfassemble
into ordered nanoscale morphologies of different geometries is a powerful feature
that researchers are now beginning to exploit for various high-tech applications.
1.1 Gaussian Chain
To start off, we consider linear homopolymer chains in a homogeneous environment, and disregard,
for the moment, all monomer-monomer interactions including the hard-core ones that
prevent the chains from overlapping. In the limit of high molecular weights, all such polymers
fall into the same universality class of non-avoiding random walks, and as a consequence they
exhibit a degree of common behavior. For instance, their typical size scales with their molecular
weight to the universal exponent, ν =1/2. To investigate this common behavior, there is
no need to consider realistic polymer chains involving segments such as those denoted in Fig.
1.2. The fact that the behavior is universal means that it is exhibited by all models, including
simple artificial ones.
For maximum simplicity, we select the freely-jointed chain, where each monomer has a
fixed length, b, and the joint between sequential monomers is completely flexible. Figure
1.4(a) shows a typical configuration of this model, projected onto the two-dimensional page.
It was produced by joining together M = 4000 vectors, r i , each generated from the random
distribution,
p 1 (r i )= δ(r i − b)
4πb 2 (1.1)
where r i ≡|r i |. The factor in the denominator ensures that the distribution is properly normalized
such that
∫
∫ ∞
dr p 1 (r) =4π dr r 2 p 1 (r) =1 (1.2)
0
The size of the particular configuration displayed in Fig. 1.4(a) can be characterized by
the length of its end-to-end vector,
R ≡
M∑
r i (1.3)
i=1
denoted by the arrow at the top of the figure. The length of this vector averaged over all
possible configurations, R 0 , can then be used as a measure for the typical size of the polymer.