The Gaussian Chain

The foregoing model gives a good physical picture of the flexible, randomly oriented molecule in the liquid or glass, but it has two weaknesses. First, there is some ambiguity concerning n how many segments should be adopted in the model Second, it doesn t lead to further analysis. 

The Gaussian chain (or model) does not suffer from these weaknesses the model assumes that the end-to-end separation of a macromolecule follows Gaussian statistics. It encompasses the freely jointed chain as a special case. 

Consider a representative chain OA (see Fig. 2.16) with a coordinate system attached at one end. Let the end-to-end vector of the chain be r and extend from the origin to the point (x, y, z), i.e. 

The chain OA can take up an enormous number of different conformations, each characterized by a value of r. By conformation is meant a particular shape of the chain. The chain can be thought of in a simple manner, as a flexible string. When its ends are held at fixed r it can take up a certain number of conformations. Each value of r will have a specific probability the greater the number of conformations for a particular r, the greater the probability of occurrence of that value of r. What is the probability that the chain displacement vector reaches from the origin to the point r and lies within the volume element dV = dxdy dz  

A function which models closely this behaviour in the region of interest is the Gaussian function 

we consider a chain where the length of each segment has the Gaussian distribution, i.e. 

Then the configurational distribution function of the chain is given by 

except for the spring potential, there is no interactional potential among the beads. The total potential of all the springs in the chain is expressed as 

In equilibrium, the Boltzmann distribution of such a bead-spring model is exactly Eq. (1.45). 

The Gaussian chain has an important property the probability distribution of the vector R — R between any two beads n and m is a Gaussian function, i.e. 

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A basic theoretical model for flexible polymers is the Gaussian chain which assumes N ideal beads with intramolecular distance between them following a Gaussian distribution, so that the mean quadratic distance between two beads separated by n-1 ideal and not correlated bonds is given by [ 15,20]... 

The Gaussian chain model yields a spring constant even for a single bond k=3k T/ , where k is the Boltzmann constant. From Eq. 3.3 the chain extension between arbitrary points along the chain may be computed to R n) -R(m)y)= n - m ... 

In addition to the above experimental point, one can raise a theoretical objection against the way in which Volkenstein et al. introduce the effect which the structure in a network has on its elastic behaviour. In their theory the Gaussian chain statistics are left unchanged in spite of the fact that the chain molecules run through bundles. Such a decoupling of chain statistics and bundles is unwarranted. In Fig. 29 c a schematic representation of the approach of Volkenstein et al. to a structured network is given. Only a two chain network is drawn, although it should, of course, be remembered that in reality a bundle structure will comprise parts of many molecules. 

In this case the mapping just expresses the relation among segment size and segment number which holds for the Gaussian chain. 

Thus all seems perfect. We have constructed an RG mapping, wliich indeed shows a fixed point. However, the expression (8.32) for / is not satisfactory. It must be independent of A, otherwise dilatation by A2 does not lead to the same result as repeated dilatation by A. Now Eq. (8.32) is only approximate since in Eq. (8,31) we omitted terms O 0 2. This is justified only if 0 is small. We thus need a parameter which allows us to make. If arbitrarily small, irrespective of A. Only e — 4 — d can take this role. In all our results the dimension of the system occurs oidy in the form of explicit factors of d or It thus can be used formally as a continuous parameter. To make our expansion a consistent theory, we have to introduce the formal trick of expanding in powers of e — 4 — d. 3 vanishes for = 0, consistent with the observation (see Chap, fi) that the excluded volume is negligible above d = 4, not changing the Gaussian chain behavior qualitatively. For e > 0 Eq. (8.32) to first order in yields... 

Variations of the Equilibrium Distance r0 between Crosslinks. Thermodynamical calculations start from the hypothesis that the (gaussian) chain is unperturbed and lead to... 

An important property of the Gaussian chain is that the distribution of the distance between any two particles of the chain is Gaussian and is similar to function (1.5). So, the mean values of the functions of the vector r — ra = e 7 r° r7 where a and 7 are the labels of the particles of the chain, can be calculated with the help of the distribution function... 

One can expect that the parameter of hydrodynamic interaction (2.11) behaves universally for subsequent division of the chain. One can reasonably guess that the quantity (2.11) does not depend on the length of the macromolecule and on the number of subchains. In this case, the hydrodynamic radius of the particle for the Gaussian chain... 

With regard to the molecular origin of these hindrances, it should be mentioned that the linear dimension of Aerosil (300 m g ) particles (about 7 nm) is comparable with the mean average distance between primarily filler particles in the PDMS matrix. Since the Gaussian chain statistics might be applied for PDMS chains in the filled rubbers [18], it is easy to show that the chain portions between the primarily filler particles should contain about 30-80 elementary units. This value is in the same range as the apparent number of elementary chain units between topological hindrances as measured by NMR... 

Figure 15 shows the dependence of the chain anisotropy (with respect to the anisotropy of the Kuhn segment fiA) on its relative length x = L/a according to Eqs. (39) and (40). At x—> 0 the curves have the slope 0.5 and rapidly attain an a mptotic limit corresponding to the anisotropy of the Gaussian chain. 

Curve I describes the dependence of y on x obtained for kinetically rigid worm-like chains disregarding the dependence of p on x. The asymptotic limit of Curve I is 0.833. .. (instead of y = 1 for Curve 4) because in Reference the anisotropy of the Gaussian chains is taken to be (3A/2 rather than 3/3A/5 used for plotting Curves 1 -4. 

Fig. 67a—c. Various modes of intramolecular motion in the Gaussian chain a first mode = rotation of a single-segment chain as a whole b second mode = rotation of parts of a two-segment chain c third mode = rotation of parts of a three-s ment chain... 

Fig. 4. Scaled end-to-end distance distribution for an excluded volume chain (solid line), Eq. (28), and the Gaussian chain (shaded area)...

In Fig. 4, the conformational behavior of the excluded volume chain (d = 3) represented by Eq. (28) is drawn as a function of the rescaled end-to-end distance to be compared with that of the Gaussian chain. As one can see, the selfavoiding walk does not very much differ from the Gaussian statistics. In the vicinity of r = 1, the difference is only 10%. Thus, replacing real chains (d = 3) with the Gaussian chain is never a poor approximation. However, what is important is that the difference revealed in Fig. 4 between the excluded volume chain and the Gaussian chain does not disappear even in the limit of N— °°. In this real world (d = 3), one can never fit real chains to the Gaussian chain by properly... 

The study of the simple case where the chain is made of a finite number N of discrete links will be used as a basis for a general study of the Gaussian chains. In this case, the probability law of the chain is given by... 

In particular, it appears that the probability distribution of the vector joining the chain ends, for a chain with excluded volume, strongly differs from a Gaussian function, and this fact greatly diminishes the value of the Gaussian chains. 

The Gaussian chain is obviously an oversimplification of the real polymer chain. The description of the shape of a polymer chain can in fact vary greatly according to the size scale with which we examine it. 

At high extensions, departure from the Gaussian chain approximation becomes significant and has led to the development of a more general but semiempirical theory based on experimental observations. This is expressed in the Mooney, Rivlin, and Saimders (MRS) equation... 

Fig. 3.7 The Gaussian chain with one end coincident with the origin.

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