Edwards continuous chain

The Edwards Hamiltonian is an appealing but most formal object. To mention a simple fact, shrinking to zero the segment size of the discrete chain model as done in the continuous chain limit, we in general get a continuous but not differentiable space curve. Strictly speaking the first part, of Vj, does not exist. Further serious mathematical problems are connected to the (5-function interaction. Hie question in which sense Ve is a mathematically well defined object beyond its formal perturbation expansion is ari interesting problem of mathematical physics. 

In two panmieter theory we ignore the microscopic length Lscale i. It is thus intimately related to the continuous chain limit > 0, where the discrete sequence of segments is replaced by a continuous space curve. On the level of the configurational probability JF exp [—17] this leads to the Edwards... 

A continuous chain can also be derived by making all springs in a spring-bead chain vanishingly short, with its total contour length held fixed. The resulting chain has no stiffness but is completely fiexible. The Edwards chain discussed later in this chapter is an example. 

A variety of approximations have been proposed to calculate eq 1.3. All the analytical calculations so far reported have actually dealt not with discrete chains but with continuous ones in the sense that the sums in eq 1.3 were replaced by appropriate integrals. The Edwards continuous chain version of eq 1.3 is shown to become... 

As mentioned in Section 3.2 of Chapter 1, the Hamiltonian H of the Edwards continuous chain depends on L, b, /3c, and A. Hence, any equilibrium property of this chain in dilute solutions ought to be expressed as a function of these four parameters. Interestingly, eq 1.9 indicates that actually (i ) is governed by three combined parameters Lb, z, and X/L formed from the four (for continuous chains (R )o = Lb according to eq 1-3.15). Hence, if X/L < 1, essentially depends on Lb and (or or depends only on z). We can expect that the same holds for any equilibrium behavior of the Edwards continuous chain in dilute solutions. Any theory of dilute polymer solutions fitting to this expectation is called the two-parameter theory. We may say that it is a theory of the Edwards continuous chain subject to the condition A/L 1. The use of the Edwards Hamiltonicui combined with this condition is hereafter referred to as the two-parameter approximation. 

The excluded-volume variable introduced in eq 1.11 has been defined for the Edwards continuous chain with length L and excluded-volume strength V. Since all the theoretical expressions presented above for or and as arc concerned with such chains, the variable appearing in them is the one so defined. However, though identical notation was used, the z that appeared in... 

Since global polymer behavior is not affected by local details of the polymer chain, its formulation may be made by use of a suitably coarse-grained polymer chain. We may choose the random flight chain, the spring-bead chain, the Edwards continuous chain, and others, depending on the global property concerned, the method of formulation, and the accuracy of the results to be derived. In what follows, we are concerned with the theories developed with the Edwards continuous chain. 

As explained in Section 3 of Chapter 1, the Hamiltonian H of the Edwards continuous chain contains three parameters chain length L,... 

We denote any renormalizable quantity of the Edwards continuous chain by Q. Then there exist an infinite number of coarse-grained Edwards chains, each having different A but the same value of Q. Such chains are called equivalent with respect to Q and their ensemble is called the renormalization group. 

To start with, we note that, as mentioned in Section 3.2 of Chapter 1, the Hamiltonian of a d-dimensional Edwards continuous chain is given by eq 1-3.26 with c(s) and v defined as follows ... 

Here, omitting the mathematical details, we sketch a relatively new dieoty due to Muthukumar and Edwards [43]. The M-E theory considers a uniform solution consisting of a pure solvent and the Edwards continuous chains with a Kuhn segment length b. It is formulated on the assumption that the Hamiltonian H of the solution is given by... 

The analytical solution of the problem has lead to the fact that at n = 0, the Laplace transform of the Green function G j is the probability to find the end of a polymer chain with the length h = t (where t is time in the problem of the general type) in the point rj, if the first segment is located at point fy. As a matter of fact, it is in agreement with the formulation of the continuous-chain model by Edwards. 

Des Cloizeaux (1981) has offered another approach using the direct renormalization method on the basis of the continuous chain model in continuous [Pg.656]

Edwards continuous chain with its corresponding Hamiltonian IIq is obviously the most suitable, in every respect, model of a polymer chain. However, this model involves plenty of fine details of the conformational structure, which actually have no influence on the experimentally measured quantities, eg. the mean-srjuare end-to-end distance. The theoreticians (Freed, des Cloizeaux, Oono, Ohta, Duplanticr, Schaffer, et el.) have found such renormalization group procedures of the source Hamiltonian Ho to drive it to the fixed point Hamiltonian //, which allow access, by the conventional methods of statistical physics, to characteristic quantities close to their experimental values. 

Curtiss and Bird introduce reptation in a maimer which does not involve the tube concept, at least not in an explicit way. Their model leads to a constitutive equation in which the stress is the sum of two contributions. On contribution is exactly 1/3 the expression obtained by Doi and Edwards when those workers invoke the independent alignment approximation, i.e., that contribution is a special case of the BKZ relative strain equation. The othCT contribution depends on strain rate and is proportional to a link tension coefficent c (0 < e < 1) which diaracterizes the forces along the chain arising from the continued displacements of chain relative to surroundings. 

In Edwards s theory (1966),12 the chains are represented by continuous curves. The interaction between chains is transformed so as to be replaced by an external random potential whose effect is mathematically equivalent. The chain behaviour in this external potential is studied in an approximate manner, and subsequently, the result is averaged over the set of potentials. 

---