A chain polymer molecule consists of a sequence of repeating units (monomers) of the same or variable type. In the most simple statistical model, one assumes a coarse grained representation of the chain in terms of a sequence of beads connected by bonds such that there is no correlation between bonds connecting adjacent beads of the chain and that all bond directions have the same probability. Clearly, all interactions between chain beads are then neglected. This model of a freely jointed chain is equivalent to an uncorrelated random walk with fixed step length.
In the simplest case, one can consider lattice chains equivalent to a random walk on a regular lattice with the coordination numberz [Doi96]. Let us assume that the polymer chain is made up ofN successive bond vectors~rn(n= 1, ..., N) with|~rn|=bandb being the nearest distance between two neighboring sites of the lattice. One can treat the length of the end-to-end vector R~ connecting two ends of the chain as a measure of the extent to which the chain spreads out
R~ =
N
X
n=1
~rn. (1.1)
The average of R~ is zero because the system is isotropic. Therefore, we define the size of the than the total length of the chain L =bN . In other words, the equilibrium state of the chain due to its flexibility corresponds to a randomly shaped coil.
Let us assume that one end of the chain is fixed at the origin and calculate the probability distribution functionP(R, N~ ) that the end-to-end vector of the chain reaches the valueR~ after N successive steps. As the chain beads can only occupy the sites of the regular lattice with the coordination number z, the bond vector of the chain can take with probability 1/z a set of possible values~bi (i= 1, ..., z). Therefore, to reach the positionR~ afterN steps, the end-to-end vector at the (N−1)th step must coincide with probability 1/z with one of the positionsR~−~bi
so that
we obtain finally the following equation for P(R, N~ ) from (1.4),
∂P
1.1 Simple Polymer Chain Models
where the prefactor (2πN b3 2)3/2 is found from the normalization condition R
d3RP(R, N~ ) = 1. The first moment of the distribution (1.9) coincides with (1.3)
More complicated off-lattice models of the ideal chain explicitly account for flexibility mech-anisms. They take into account “short-range interactions” which occur between neighboring segments along the chain but ignore “long-range interactions” between segments belonging to widely separated parts. However, there is a common general feature of these models that the orientational correlations between the bonds exponentially decay along the chain and become negligible over the persistence length ˜l of the chain
h~rn~rmi ≃b2exp
−s/˜l
, (1.10)
where s is the distance between the segments n and m along the chain and h...i stands for averaging over all possible configurations of the chain. A section of the polymer chain whose length is smaller than ˜l is practically stiff, whereas two sections separated by a distance larger than ˜l are independent of each other. Because the persistence length is hard to measure in the experiment, one usually defines the Kuhn segment length l from the mean squared end-to-end vector hR~2i = (L/l)l2 =Ll which is of the same order of magnitude as the persistence length. Considering the whole polymer as an effective freely jointed chain ofL/lpractically stiff segments, one can again obtain the Gaussian distribution for the end-to-end vector. Evidently, the vector connecting two points of the chain which lie at a distanceslarger thanl also satisfies the Gaussian distribution.
A model that is made up of uncorrelated bond vectors is called the Gaussian model of the polymer chains. Let us write the position of then-th bead of the chain asR~n,wheren = 0, ..., N so that bond vectors become
~rn =R~n−R~n−1, (1.11)
A state of the Gaussian chain is defined by the set of the bead positionsn R~n
o
= (R~0, ~R1, ..., ~RN) which is realized with the probability
P n
where ˜b is the mean length of the bond vector.
One can think of (1.12) as the probability distribution in the canonical ensemble of N har-monic springs linked together. The energy of the chain can then be written as
U = 1
with the spring constantk = 3kBT /b2of theentropicorigin .The Gaussian model is often called the bead-spring model; it is very useful because of its simple mathematical formulation.
A more convenient measure of the polymer size which is readily accessible in experiments based on light scattering is the radius of gyration defined by
R2G = 1
where R~cm is the center of mass of the chain
The radius of gyration for the ideal chain in the limit N → ∞ reads R2G= 1
6N˜b2. (1.16)
The Fourier transform of the pair correlation function of the chain g(~r) = 1 can be easily calculated for the Gaussian chain; the final result reads
g(~q) = Z
d3rg(~r)ei~q~r=N 2
x4 exp(−x2)−1 +x2
, (1.18)
wherex=q RGand the function in the brackets with the prefactor 2/x4is the Debye scattering function. It is clearly seen from (1.18) that the scattered intensity from the chain which is proportional to g(~q) involves only one microscopic length scale, namely RG. In the limit qRG ≫1 one can find
g(~q)≃ 2N
q2RG2 , (1.19)
and for qRG ≪1 one has g(~q)≃N .
More complicated models of the polymer chain take into account excluded volume effects, which produce repulsive interactions between beads separated by an arbitrary distance along the chain. In order to avoid unfavorable contacts between the beads, the chain favors swollen states as opposite to the ideal chain. A number of ideal chain configurations are disallowed due to excluded volume effects and the average size of such chains become larger than that of the ideal chain. The radius of gyration of the excluded volume chain depends stronger on the number of beads,
RG ∼Nν (1.20)
with ν >1/2. The exponent ν obtained in computer simulations on self-avoiding chains with large lengths in the three-dimensional space was found to be approximately 0.588, see [Doi96].
A simple mean field argument due to Flory [deGen] leads to ν = 3/5.