Figure 8.1: (a) Schematic of the coarse-grained discrete Gaussian chain model with N+l
segments and N springs. The segment positions are denoted by rɑ, rɪ, ..., rjy, and the bond
vectors are denoted by bɪ, b2, ..., bjy. (b) Schematic of the continuous Gaussian chain
model. The configurations of the polymer chain are described by the r (s), where s ∈ [0, N]
is a contour parameter.
The partition function of a single chain can be expressed as
Z0 = / drw+1exp [-∕3f∕0(r'v+1)] , (8.1)
where r'v+1 = (r0, rɪ, ..., r^) denotes the set of N + 1 segment positions and Uo(rw+1)
is the potential energy associated with the particular configuration of the polymer
chain. An alternative representation is the set of N bond vectors, bv = (bɪ, b2, ...,
bjv), where bi = ri - ri-ι.
Z0=Vl dbN exp [—∕5l7o(byv)] ,
209
(8.2)
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