Polymer Chain Statistics

4 POLYMER SHAPES IN SOLUTION 7.4.1 Polymer Chain Statistics 

This result follows from the fact that for large x, there is an equal likelihood of finding polymer repeat units oriented in opposite directions in an ensemble of polymer 

FIGURE 7.12 Equilibrium configuration of a single polymer chain in solution. 

Calculate the rms end-to-end distance of polyethylene with an average molecular weight of 140,000 g/mol. Compare your result with the fully extended length of this polymer. What is the physical significance of your result  

Solution Polyethylene is made up of methylene groups, CHj, of molecular weight 14 g/mol, separated by a bond distance of about 1.54A. Therefore, a = 1.54A, X = 10,000, and 

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The problem of investigation of polymer chain statistics without volume interactions entangled with an infinitely long string (in the 3D case) or with an obstacle (in the 2D case) was at first formulated by S.F. Edwards [10] and by S. Prager and H.L. Frisch [11] in 1967. These papers can be regarded as a comer... 

Like all mean-field theories, SCF theories replace the detailed, configuration-dependent interaction potentials with a mean potential averaged over the distribution of molecular configurations. Unlike other mean-field theories, SCF theory explicitly calculates the mean field by accounting for the polymer chain statistics. This field, in turn, controls the distribution of polymer configurations Hence the term self-consistent. ... 

AFe and AFC are, respectively, the electrostatic and chain free energy changes associated with die free ion — ion pair transition. An expression relating K to chain length is derived from its defining equation and polymer chain statistics... 

In its turn, the characteristic ratio C, which is polymer chain statistical flexibility characteristic [25], is determined according to the Eq. (97) [153] ... 

In Fig. 52 the dependence Q(C ) for the considered PUAr is adduced. As it was supposed, polymer chain statistical flexibility enhancement (C increasing) results to Q reduction. 

FIGURE 52 The dependence of conversion degree Q on polymer chain statistical flexibility, characterized by characteristic ratio C, for PUAr. 

In Fig. 54 the comparison of experimental and calculated according to the Eq. (27) Q dependences on DOPP contents c pp (at t=60 min) is adduced. Let us note several conclusions, following from the Fig. 54 plot. Firstly, a good correspondence of theory and experiment is observed. Since the theoretical values Q were calculated according to the known values, i.e., actually by PUAr chain flexibihty, then this confirms the proposed above treatment correctness, namely, the dependence of Q on polymer chain statistical flexibility. Secondly, the dependence Q(Cpopp) is located lower than the corresponding additive dependence, that is direct consequence of the dependence D c pp) character, adduced in Fig. 53. 

One more parameter—the characteristic ratio Coo — is characteristic of polymer chain statistical flexibility and determined according to the Eq. (98). In Fig. 57 the relation between and o- for the considered polyarylates is adduced. Since these parameters characterize the same polymer chain property, then between them the correlation is expected [25] that confirms the plot of Fig. 57. Certain scatter of data for two from the considered polyarylates can reflect different values of angles between bonds in chain [25]. 

We consider in this book the problem of polymer chain statistics in a disordered (say, porous) medium. If the porous medium is modelled by a percolating lattice [5], we can consider the following problem let the bonds (sites) of a lattice be randomly occupied with concentration p (> Pc, the percolation threshold) the SAWs are then allowed to have their steps only on the occupied bonds (through the occupied sites). We address the following questions [6,7] does the lattice irregularity (of the dilute lattice) affect the SAW statistics ... 

Analogous considerations were initially made and obeyed throughout the project for the accompanying theoretical methods ranging from computer simulations on the basis of simplified models up to demanding ab initio procedures. The combined theoretical and spectroscopic analysis provided particularly valuable information on the structure and dynamics of complex assemblies as well as on fundamental questions regarding polymer chain statistics. 

Then the value of characteristic ratio has been determined, which is an indicator of polymer chain statistical flexibility according to the equation [7] ... 

The first four chapters, making up the fundamental part, contain reviews of the latest knowledge on polymer chain statistics, their reactions, their solution properties, and the elasticity of cross-linked networks. Each chapter starts from the elementary concepts and properties with a description of the theoretical methods required to study them. Then, they move to an organized description of the more advanced studies, such as coil-helix transition, hydration, the lattice theory of semifiexible polymers, entropy catastrophe, gelation with multiple reaction, cascade theory, the volume phase transition of gels, etc. Most of them are difficult to find in the presently available textbooks on polymer physics. 

Let us consider the determination of molecular characteristics S and for butadiene-styrene rubber. As it is known [15], the value of macromolecule diameter square is equal to for polybutadiene - 20.7 and for polystyrene - 69.8 A. Calculating the macromolecule, simulated as cylinder, cross-sectional area for the indicated polymers according to the known geometrical formulae, let us obtain 16.2 A and 54.8 A, respectively. Further, accepting as S for butadiene-styrene rubber mean value of the cited above areas, let us obtain S=35.5 A. Further the characteristic ratio can be determined, which is a polymer chain statistical flexibility indicator [16], with the aid of the following empirical formula [14] ... 

The complete specification of molecular configuration would require a detailed knowledge of the dimensions and shapes of the monomeric units, local packing effects, and interactions with the solvent molecules. A valuable simplification can be made if one is willing to sacrifice consideration of short-range relationships and attendant behavior at the very highest frequencies. It utilizes the principle from polymer chain statistics that any two points on the chain backbone separated by perhaps 50 or more chain atoms will be related to each other in space in accordance with a Gaussian distribution of vectors, -20 provided the solvent is a 0-solvent. In... 

Polymer mechanical properties are one from the most important ones, since even for polymers of different special-purpose function a definite level of these properties always requires [20]. Besides, in Ref [48] it has been shown, that in epoxy polymers curing process formation of chemical network with its nodes different density results to final polymer molecular characteristics change, namely, characteristic ratio C, which is a polymer chain statistical flexibility indicator [23]. If such effect actually exists, then it should be reflected in the value of cross-linked epoxy polymers deformation-strength characteristics. Therefore, the authors of Ref [49] offered limiting properties (properties at fracture) prediction techniques, based on a methods of fractal analysis and cluster model of polymers amorphous state structure in reference to series of sulfur-containing epoxy polymers [50]. 

ELECTRONIC EXCITATION TRANSPORT AS A TOOL FOR THE STUDY OF POLYMER CHAIN STATISTICS... 

The parameters and characterize polymer chain statistical flexibility and molecular mobility level, respectively [2], The dimension can be determined with the aid of the following equation [2] ... 

Equation 2.7 gives two forms of analytical relation between d and fractal dimension depends not only on the order parameter value (p, but also on the polymer s molecular characteristics (C and 5), where is the characteristic ratio, which is an indicator of polymer chain statistical flexibility [42], S is a macromolecule cross-sectional area. This is the reflection of the above-mentioned rule, that for description of a polymer s condensed state structure, as a minimum two order parameters are required [12, 25]. 

For PCP = 0.585 nm and then the characteristic ratio C, which is a polymer chain statistical flexibility indicator [16], can be calculated according to Equation 1.8. The value for PCP is equal to 4.0, which is close enough to the value for cis-polyisoprene, which varies within the limits of 5.0-5.3 [10,17]. 

The value of characterises the polymer chain statistical flexibility and the degree of macromolecular coil compactness [25]. In Figure 5.5 the dependences CJcJ for modified EP are adduced. As follows from the data of Figure 5.5, more intense raising of is observed for EP-4 than for EP-3, which is due to the differences in the structures of these epoxy polymers. In the EP-4 case the adamantane fragment is built by two bonds into the network or partly forms a card polymer structure. For EP-3 the card polymer structure with one bond at the adamantane fragment is the most probable. Therefore, for macromolecular coil card elements of EP-4 chain structure are more effective in the role of steric hindrances. They do not allow to achieve the same degree of macromolecular coil compactness as do non-modifled EP-1 and EP-2 or EP-3. 

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