Dimensions of the polymer

Muthukumar and Winter [42] investigated the behavior of monodisperse polymeric fractals following Rouse chain dynamics, i.e. Gaussian chains (excluded volume fully screened) with fully screened hydrodynamic interactions. They predicted that n and d (the fractal dimension of the polymer if the excluded volume effect is fully screened) are related by... 

The reptation model for polymer diffusion would predict that the thickness of the gel phase reflects the dynamics of disentanglement. The important factors here are chain length, solvent quality and temperature since they affect the dimensions of the polymer coils in the gel phase. The precursor phase, on the other hand, depends upon solvency and temperature only through the osmotic force it can generate in the system and the viscoelastic response of the system in the region of the front. These factors should be independent of the PMMA molecular weight. 

Flory s viscosity theory also furnishes confirmation of the w temperature as that in which a=V.2, and it permits the determination of the unperturbed dimensions of the Polymer chain. Even if a Q solvent is not available, several extrapolation techniques can be used for the estimating the unperturbed dimensions from viscosity data in good solvents. The simplest of these techniques seems to be that of Stockmayer. 

A scaling argument does not describe the full functional form of the dependence of the chain dimensions on concentration. We would expect the dimensions of the polymer to be a smoothly changing function between c and the concentration J where the coils reach their limiting radius equivalent to the 6 dimensions. However, to a first approximation we can suppose that Equation (5.73) applies up to the concentration ct ... 

Small angle X-ray scattering monitoring the distribution of electron density has been used to probe resin morphology [109]. More recently contrast matched small angle neutron scattering has also been employed [110, 111]. These techniques can also be applied to wet resins and tend to probe the very low dimensions of the polymer matrix structure. More studies are needed to identify the real value of these approaches. 

P(r) can be transformed into a distribution of the particle size as defined by the hydrodynamic radius Rh. But only for TDFRS, and not for PCS, a particle size distribution in terms of weight fractions can be obtained without any prior knowledge of the fractal dimension of the polymer molecule or colloid, which is expressed by the scaling relation of Eq. (39). This can be seen from the following simple arguments ... 

Due to the chain architecture and the large size of the macromolecules, the wetting behaviour of polymer liquids can be different from that of simple liquids. The effect becomes particularly strong when the dimension of the liquid phase, e.g. film thickness and droplet diameter, approaches the dimension of the polymer coil. In addition to the spreading coefficient and the surface pressure effects, entropic elasticity of the polymer chain provides a strong contribution to the free energy for a constant volume V0=Ad ... 

Closely connected with the conformational dimensions of the polymer coil, and therefore with the limiting viscosity number, are some other macroscopic quantities, viz. the limiting sedimentation coefficient and the limiting diffusivity. 

Fig. 1 shows proton conductivity for the perfluorosulfonic acid membranes, measured at 300 K in air by the DC resistance measurement, after gamma-ray irradiation at the several doses up to 414 kGy. The conductivity was calculated from the applied voltage and the measured current and dimension of the polymers. It can be seen in Fig.l that the conductivity increases with increasing the dose. The conductivities at 300 K in air atmosphere rapidly increased until about 50 kGy, and achieved to be higher by about three orders of magnitude than that of the unirradiated one. 

Usually one is interested in the geometry and thermodynamic properties of various models of polymer chains. In many cases, the effect of intramolecular long-range interactions on the shape and dimensions of the polymer chain are of interest. 

Many foods contain high-molecular weight polymers, such as proteins, pectins, and others. Often, they contribute significantly to the structure and viscosity of foods. In dilute solutions, the polymer chains are separate and the intrinsic viscosity, denoted as [ ], of a polymer in solution depends only on the dimensions of the polymer chain. Because [ ] indicates the hydrodynamic volume of the polymer molecule and is related to the molecular weight and to the radius of gyration, it reflects important molecular characteristics of a biopolymer. The concentrations of polymers used should be such that the relative viscosities of the dispersions are from about 1.2 to 2.0 to assure good accuracy and linearity of extrapolation to zero concentration (Morris and Ross-Murphy, 1981 da Silva and Rao, 1992). Intrinsic viscosity can be determined from dilute solution viscosity data as the zero concentration-limit of specific viscosity (ijsp) divided by concentration (c) ... 

A major goal in the physics of polymer melts and concentrated solutions is to relate measurable viscoelastic constants, such as the zero shear viscosity, to molecular parameters, such as the dimensions of the polymer coil and the intermolecular friction constant. The results of investigations to this end on the viscosity were reviewed in 1955 (5). This review wiU be principaUy concerned with advances made since in both empirical correlation (Section 2) and theory of melt flow (Section 3). We shall avoid data confined to shear rates so high that the zero shear viscosity cannot be reliably obtained. The shear dq endent behavior would require an extensive review in itself. 

These authors envisaged the critical step in the yield process as being the nucleation under stress of small disc-sheared regions (analogous to dislocation loops) that form with the aid of thermal fluctuations. The model explains quantitatively the variation of the yield stress with temperature, strain rate and hydrostatic pressure, using only two parameters, the shear modulus of the material and the Burgers vector of the shared region which is a constant related to the molecular dimensions of the polymer. 

The reciprocal of the fractal dimension of the polymer (see Section 1.4) is p. For an ideal linear chain p= j2 and the fractal dimension is l/i = 2. The Rouse time of such a fractal chain can be written as the product of... 

A characteristic feature of a dilute polymer solution is that its viscosity is considerably higher than that of either the pure solvent or similarly dilute solutions of small molecules. The magnitude of the viscosity increase is related to the dimensions of the polymer molecules and to the polymer-solvent interactions. Viscosity measurements thus provide a simple means of determining polymer molecular dimensions and thermodynamic parameters of interactions between polymer and solvents. These aspects will also be considered in a later part of this chapter. 

The second virial coefiicient A2, which is related to the Flory dilute solution parameters by Eq. (3.121), is a measure of solvent-polymer compatibility. Thus, a large positive value of A% indicates a good solvent for the polymer favoring expansion of its size, while a low value (sometimes even negative) shows that the solvent is relatively poor. The value of A2 will thus tell us whether or not the size of the polymer coil, which is dissolved in a particular solvent, will be perturbed or expanded over that of the unperturbed state, but the extent of this expansion is best estimated by calculating the expansion factor a. As defined by Eqs. (3.123) and (3.124), a represents the ratio of perturbed dimension of the polymer coil to its unperturbed dimension. 

If the universal constancy of is accepted, it is possible to calculate the average dimensions of polymer molecules in solution merely from knowledge of their intrinsic viscosities and molecular weights. More particularly, it is possible to calculate the natural, or unperturbed, dimensions of the polymer chain from the knowledge of intrinsic viscosity in a theta solvent [28,29]. 

Problem 3.25 For a fractionated sample of cis-1,4-polybutadiene of molecular weight 123x10 intrinsic viscosities were measured [30] in three different solvents at respective theta temperatures. From the results given below determine the variation of the unperturbed dimensions of the polymer molecule with temperature. 

During the final thermal tempering of PSiPI, about 40% weight loss of the polymer is expected and this is observed in a 40% reduction of the z (thickness) dimension of the polymer. Other... 

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 ... 

These conclusions are supported by the viscosity data plotted in Figure 1. The reduced viscosity of the hexyl copolymer is presented as a function of a in 0.2M solutions of TMACl, NaCl, and LiCl. The polymer concentration of 3.2 X 10"3 monomole/l was low enough to allow interpretation of the results in terms of the molecular dimensions of the polymer molecules. The findings demonstrate strikingly the differences in the effects of the TMA+ ion and the alkali metal ions. Whereas the polyacid showed an enormous expansion with increasing a in the presence of TMA+ ion, this expansion was suppressed almost completely by the alkali metal ions. The difference between the effects of... 

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