Single-chain partition function

As an example of the first type we consider the single chain partition function, Equations (4.19), (11.10) yield... 

To calculate the single chain partition function Q/, it is useful to define the end segment distribution, qj(r, t), which describes the probability of finding the end of a chain fragment—containing the segments [0 f] and being exposed to the fields Wa(oi = A, B) —at position r [9] ... 

The special situation of a homogeneous saddle point, corresponding to a homogeneous disordered phase, is particularly interesting. In that case, ex-phdt analytical relations between the single chain partition function and the fields can be obtained, and one recovers the well-known random phase approximation (RPA). To illustrate this approach, we shall now derive the RPA structure factor for the case of a symmetric binary homopolymer blend. 

More recently, stress and strain have been incorporated as a conjugated pair of slow dynamic variables to extend the model of the SCF theory. This allows us to capture some effects of viscoelasticity [76]. Similar to the evaluation of the single chain partition function by enumeration of explicit chain conformations, one can simulate an ensemble of mutually non-interacting chains exposed to the effective, self-consistent fields, U and W, in order to... 

The single chain partition function <2i (/, T) under the action of the tension / for the random flight model chain with a bond length a takes the asymptotic form... 

In the case of general chain architectures, however, the mean fleld problem of a single chain in an external fleld cannot be cast in the form of a modified diflusion equation, and the density that a single chain creates in the external field and the concomitant single-chain partition function have to be estimated by partial enumeration [30-34], This methodology has been successfully applied to study the packing of short hydrocarbon chains in the hydrophobic interior of lipid bilayers [31,32,34] and polymer brushes [33] and to quantitatively compare the results of Monte Carlo simulations to the predictions of the mean field theory without adjustable parameters [30], The latter application is illustrated in Figure 5.2. 

Monte Carlo Sampling of the Single-Chain Partition Function and Self-Consistent Brownian Dynamics... 

While this strategy mitigates the sampling problem for the density, it does not solve the problem for the estimation of the single-chain partition function,... 

In Eq. (6.86), the Q s are the partition functions of individual components in the presence of a field. Explicitly, for the polyelectrolyte chain with smeared charge distribution along the backbone, single chain partition function is given by... 

Then, we can calculate the segment density 4>p and the single chain partition function Q/>[153, 160, 161, 162] ... 

The quantity Qc uJa ) in the above expression is the single-chain partition function in the external field... 

Because the single-chain partition function Qc( uj ) depends on the field tUo,(r) in an implicit form, it is useful to express 0c( tu ) in terms of a series of w (r), leading to an explicit expression. This can be achieved by writing the fields tUa(r) in the form. 

Using the perturbation solutions of the propagators, Qa r,s r ), the single-chain partition function can be obtained as a series expansion of the terms,... 

Using these expressions, the single-chain partition function are given by... 

Prove that the single chain partition function of a freely jointed polymer chain of size M is... 

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