Gaussian coils

The coil demention for polyacrylamide may be obtained from the relation, which is applicable for non-Gaussian coils ... 

A similar scaling transition has been proposed to account for the response of an isolated coil to tensile stress [24,25]. If a force is applied to a Gaussian coil Equation (8) can be used to calculate the response of the coil since at thermal equilibrium the applied force F dE/dR so,... 

Tables of this sort are valid for Gaussian coils only. In thermodynamically good solvents the Gaussian behaviour of chain molecules is perturbed by what is called the excluded volume effect 30. The P/(0) function depends on the distribution of mass within the particle and this, in turn, is changed if the volume effect is operative.

Fig. 32. Molecular weights obtainable in principle from the angular dependence of LS by linear Gaussian coils...

It is well known that for Gaussian coils the mean square dimensions P2 and h- are related bv the ratio h2/ 2 S. It was of interest to check the validity of this ratio for cross-linked coils. It can be seen from table for the long chains /N =100, 150 J the ratio h. /P2 6 is practically constant at different cross-linkages number, while for the short chains /Ns 30, 50 7 this ratio is decreased with the degree of cross-linking. 

For random coils, is directly proportional to the contour length. If n is the number of main chain atoms in the chain, = an. The parameter a is relatively insensitive to environment (21), and has been calculated for a number of polymers from strictly intramolecular considerations using the rotational isomeric model (22). The root-mean-square distance of segments from the center of gravity of the coil is called the radius of gyration S. The quantity S3 is an approximate measure of the pervaded volume of the coil. For Gaussian coils,... 

A restriction will be made to polymer chains without strong and specific intersegmental forces, such as may exist in proteins and many other macromolecules. The reason is that the elasticity of networks composed of such chains cannot even be approached from the Gaussian coil point of view, since the chains are helical or at least partly so. At the moment no good theoretical treatment of these "proteinlike systems is available. 

Fig. 11. Density of segments belonging to a Gaussian coil (full line) and average segment density in the system (dotted line) as a function of the distance from the center of mass of the selected coil. A dilute solution B concentrated solution or...

DuSek, K. The heterogeneity and extent of intramolecular crosslinking in a system of Gaussian coils. Collection Czech. Chem. Commun. 33, 1100 (1968). 

The peculiarity of this expression, however, is that it does not make sense for dilute solutions of Gaussian coil molecules. In fact, the free-draining case is characterized by the limit of infinetely smaE friction coefficient . For this case, the contributions of the chain molecules to the viscosity of the solution becomes zero. 

In ref. 63 a lower numerical factor than that of eq. (5.3) is given. This is due to the fact that the original Flory-Fox parameter, viz. 2.1 X 1023, has been used there (98). The present numerical factor is preferred as it is consistent with the concept of Gaussian coils in the absence of excluded volume. A comparison between experimental results of ref. (69)... 

The second eq. (5.4) is automatically fulfilled. It is also seen that x = 2P/V It is noticed that this statistics is rather suitable for the description of short chains which do not form Gaussian coils. 

The following observations seem to lead to a key for the understanding of the quasistatic treatment of the stress-optical behaviour (777). With the aid of third eq. (5.4) and eq. (5.8), the ratio of Maxwell constant and intrinsic viscosity, as valid for Gaussian coil molecules in a matching solvent, can be formulated as follows ... 

Kuhn for statistically coiled molecules. The two dotted lines denoted by F and N stand for the free-draining and the non-draining case of Zimm s theoty for Gaussian coils. The hatched area indicates the area where the experimental points obtained on solutions of anionic polystyrenes are located (See Fig. 3.1). 

A universal calibration curve (log [> ] Mw vs. elution vol.) was prepared for the gel permeation chromatograph using samples of anionically prepared polystyrene (Fig. 7). The figure includes samples of polyf4-methyl-1 -pentene). It has been found that polymers for which the relationship between log[ ] Mw and the G.P.C. elution volume is linear are generally Gaussian coils in solution. 

Fig. 2.1. A random walk chain of 50 monomers of fixed length (thin polygon), leaking the position of every fifth monomer as segment coordinates, we find our model chain (fal polygon), which is approximated by a Gaussian coil...

Theoretical estimates of the quantity zv — 1 are in the range from 0.5 (non-draining Gaussian coil), to 1.11 (draining coil with excluded-volume interaction). A compilation of empirical values of K and of the power exponents for different polymers and different solvents may be found in the literature (Flory 1969, Tsvetkov et al. 1964). The empirical values of the exponent zv — 1 do not exceed 0.9, which indicates significant impermeability of the macro-molecular coil in a flow. We may note that once a relation of type (6.24)... 

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