Polymers root mean square dimension

The root-mean-square distance Vr separating the ends of the polymer chain is a convenient measure of its linear dimensions. The dissymmetry coefficient will be unity for (VrV 0< l and will increase as this ratio increases. 

Apart from their utility in determining the correction factor 1/P( ), light-scattering dissymmetry measurements afford a measure of the dimensions of the randomly coiled polymer molecule in dilute solution. Thus the above analysis of measurements made at different angles yields the important ratio from which the root-mean-square... 

Thus the ratio of the size of the particle to the root mean square end to end distance of the polymer is an important factor. With small particles, diameter ca.50 run, bridging occurs easily and since they are small, or similar in dimensions, to the polymers flocculation occurs easily moreover, the number concentration of particles is high and hence a large number of bridges are formed. With large particles, diameter > 1 ym, the polymer molecule is much smaller than the particle and hence it forms an adsorbed layer which often confers a degree of steric stability (48). 

For a given polymer-solvent system, the intrinsic viscosity varies with the molecular weight of the polymer. According to Flory, the intrinsic viscosity is directly proportional to the hydrodynamic volume occupied by the random coil of the polymer molecule in a solution. In addition, the hydrodynamic volume is related to the cube of the typical linear dimension of the random coil (root mean square end-to-end distance). The intrinsic viscosity is expressed as ... 

In spite of all the recent success which can be traced to the three techniques, SANS, SAXS, SALS, there are serious limitations in their scope Q). For instance only average values, such as root-mean-square end-to-end distance and radius of gyration of a polymer chain can be extracted from these techniques. In other words, these techniques cannot provide information on the actual conformation or dimension of the individual polymer chain. Also since all three techniques are indirect, the analysis of the raw data is not straightforward and requires various manipulations and faith in existing theories to draw conclusions. Finally with regard to SANS, it must be noted that only a limited number of facilities in the U.S.have this capability, thus restricting its use. 

One parameter that is commonly used to specify the dimension of a linear polymer molecule is the root-mean-square (rms) end-to-end length. The simplest, and also the most primitive, model for a polymer molecule is the random flight chain, also termed the freely jointed chain. In this model, the bonds are represented by volumeless lines in space, and there are no restrictions on the valency angles or on the rotations about the bonds. The rms end-to-end length, can be... 

Basically, all techniques involve taking into account the effect of long range interactions which alter the dimensions of the polymer by a factor a50). As a result, the root-mean-square end-to-end distance, good solvent may be expressed by ... 

One of the characteristic dimensions of polymer coils is the root-mean-square end-to-end distance ( r ) ), which for a linear chain of n bonds is calculated by considering the backbone bonds as vectors (6 ) [49]. 

Any physical property of a polymer molecule that depends on its conformation can ordinarily be expressed as a function of some sort of average dimension. The polymer dimension that is most often used to describe its spatial character is the displacement length, which is the distance from one end of the molecule to the other. For the fully extended chain, this quantity is referred to as the contour length. Given the extremely large number of possible conformations and number of chains, a statistical average, such as the root-mean-square end-to-end distance, (rj), is required to appropriately express this quantity. 

Solution K is a constant that is essentially independent of the molecular weight of the polymer and the character of the solvent medium, as amply demonstrated by the above data. However, K shows a decrease with increasing temperature. From Equation 12.72, K is proportional to the factor t JM. We recall that the unperturbed root-mean-square end-to-end distance (r ) is expanded invariably to greater dimensions relative to completely free rotation as a result of the effects of hindrances to free rotation. As the temperature is increased, the tendency to completely free rotation is enhanced as the effects of these hindrances are diminished. Consequently, K also decreases. 

The fractal dimension of purely statistical models, i.e., models without the effect of excluded volume, can be determined accurately [see Equation (11.9a)]. For linear polymers, this model corresponds to phantom random-walk. In the case of branched statistical fractals, the corresponding model is a statistical branched cluster, whose branching obeys the random-walk statistics. Since the root-mean-square distance between the random-walk ends is proportional to the number of walk steps N, then D = 2 irrespective of the space dimension. These types of structures have been studied [61, 75-77]. The value D = 4 irrespective of d was obtained for a branched fractal. Unlike ideal statistical models, models with excluded volume, i.e., those involving correlations, cannot be accurately solved in the general case. The Df values for these systems are usually found either using numerical methods such as the Monte Carlo method or taking into account the spatial position of a renormalisation group. 

The average conformation of the polymer is called a statistical coil. The dimensions of such a coil are commonly expressed by the radius of gyration and by the root mean square end-to-end distance h, as shown in Figure 12.4. The radius of gyration is a kind of average radius of the coil, defined as... 

First attempts to quantitatively estimate stabilizing capabilities of polymers date back to Faraday s time." Steric stabilization becomes probable because spatial dimensions of at least comparatively low-molecular compounds are commensurable with the range of London s forces of attraction or even exceed them. If the diameter of a macromolecule of a linear polymer coincides with the root-mean-square (rms) distance between its ends then the relationship between the mean geometric radius of the particle (r versus polymer molecular mass M can be expressed by the following relationship ... 

The expansion factor, a, is used to characterize the size of polymer coils when they are not under unperturbed conditions—that is, to measure the effects of the polymer s environment on its overall dimensions. To do so, a is defined as the ratio of the root mean squared end-to-end vector relative to its value in the unperturbed state according to the following equation ... 

As seen in the above examples, confinement lowers the number of dimensions available to a polymer chain. In the Gaussian chain, on the one hand, the confinement changes the confined components only. The root mean square end-to-end distance changes only by a numerical coefficient without changing the dependence of Rp on N. In the real chain, on the other hand, the decrease in the dimensionality changes qualitatively the relationship between N and R from that in the fiee solution. The confinement manifests the excluded volume effect more prominently. 

In a theta solvent (A2 = 0), polymer-solvent interactions are just balanced by polymer-polymer and solvent-solvent interactions. Long-range interactions disappear and the polymer chain assumes its so-called unperturbed dimensions which manifest themselves for linear chains by a dependence of the root-mean square radii of gyration s, on the square root of molar masses ... 

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