Flexible molecular chain models

There have been two basic approaches along these lines  

The free volume theory is modified so that the changes in free volume with temperature relate to a discontinuity that occurs at rather than Tg. 

It is considered that Ti represents a true thermod3mamic transition temperature. Adam and Gibbs [19] have developed a modified transition state theory in which the frequency of molecular jumps relates to the cooperative movement of a group of segments of the chain. The number of segments acting cooperatively is then calculated from statistical thermodynamic considerations. 

Despite the successes mentioned above there have been objections to free volume theories [13,20] based on observations that a few polymers behave similarly under both constant pressure and constant volume conditions, although in the latter case free volume must decrease with increasing temperature. 

Condensed matter physicists calculate many properties of eiystalline solids in terms of a model, due initially to Debye, in which massive point atoms are connected by linear elastic springs. The dynamies of molecular chains can be considered from this starting point. The theories diseussed below, although initially derived for polymer solutions, can be used to predict relaxation spectra and time temperature equivalence for amorphous solid polymers. As a full treatment involves quite advanced mathematics, we shall discuss the theories only in outline. 

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It is somewhat difficult conceptually to explain the recoverable high elasticity of these materials in terms of flexible polymer chains cross-linked into an open network structure as commonly envisaged for conventionally vulcanised rubbers. It is probably better to consider the deformation behaviour on a macro, rather than molecular, scale. One such model would envisage a three-dimensional mesh of polypropylene with elastomeric domains embedded within. On application of a stress both the open network of the hard phase and the elastomeric domains will be capable of deformation. On release of the stress, the cross-linked rubbery domains will try to recover their original shape and hence result in recovery from deformation of the blended object. 

The system used in the simulations usually consists of solid walls and lubricant molecules, but the specific arrangement of the system depends on the problem under investigation. In early studies, hard spherical molecules, interacting with each other through the Lennard-Jones (L-J) potential, were adopted to model the lubricant [27], but recently we tend to take more realistic models for describing the lubricant molecules. The alkane molecules with flexible linear chains [28,29] and bead-spring chains [7,30] are the examples for the most commonly used molecular architectures. The inter- and intra-molecular potentials, as well as the interactions between the lubricant molecule and solid wall, have to be properly defined in order to get reliable results. Readers who intend to learn more about the specific techniques of the simulations are referred to Refs. [27-29]. 

Simulation methods have been proved to be useful in the study of many different molecular systems, in particular in the case of flexible polymers chains [ 14]. According to the variety of structures and the theoretical difficulties inherent to branched structures, simulation work is a very powerful tool in the study of this type of polymer, and can be applied to the general problems outHned above. Sometimes, this utility is manifested even for behaviors which can be explained with simple theoretical treatments in the case of linear chains. Thus, the description of the theta state of a star chain cannot be performed through the use of the simple Gaussian model. The adequate simulation model and method depend strongly on the particular problem investigated. Some cases require a realistic representation of the atoms in the molecular models [10]. Other cases, however, only require simplified coarse-grained models, where some real mon-... 

The physical significance of 2 in Equation (73) is somewhat harder to define. At first glance it appears to be the length of the repeating unit, about 0.25 nm for a vinyl polymer. We must remember, however, that the derivation of Equation (73) assumed that the coil was connected by completely flexible joints. Molecular segments are attached at definite bond angles, however, so an actual molecule has less flexibility than the model assumes. Any restriction on the flexibility of a joint will lead to an increase in the dimensions of the coil. The effect of fixed bond angles on the dimensions of the chain may be incorporated into the model as follows. 

The flow-strength criteria stated in equation (10.2) has been examined experimentally by Fuller and Leal [72] and Dunlap and Leal [149] using four- and two-roll mills, respectively. These devices allow one to systematically vary the flow type (the relative amount of pure extension to pure rotation). The birefringence was measured for dilute and semidilute polymer solutions as a function of both the magnitude and type of the flow. Simple molecular models of flexible polymer chains suggest that such data, when plotted as a... 

We present here the results of such a systematic investigation on the dependence of the self-diffusion coefficient of flexible polymer chains as a function of P and N, conducted on polydimethylsiloxane (PDMS). This model polymer is well above its glass temperature at room temperature (Ta = - 120°C), so that one can expect that spurious effects associated with the variation of the free volume and of the local monomer-monomer friction coefficient with the molecular weights of the chains are minimised. 

The relationship between chemical structure and viscoelastic behaviour is established through molecular models considering that polymers relax or diffuse in the same way they are considered as flexible statistical chains trapped between the topological constraints created by the surrounding chains. 

Integral equation ideas on the structure of monatomic liquids were first modified and applied to molecular liquids by Chandler and Andersen, Their classic work is now referred to as the reference interaction site model (RISM) of molecular liquids. Polymer RISM (PRISM) is essentially an extension of RISM theory that successfully describes the structure of flexible polymer chains in the liquid state. 

In the diaryl-substituted polysilane, the global dimensions are approximated by a wormlike chain model with a persistence length of 100 A. Thus, for the highest molecular weight studied, LIq — 100, and the chain dimensions are similar to those of a flexible-model Kuhn chain with bond lengths IkOf —200 A, that is,... 

The freely jointed chain model is most appropriate for synthetic polymers, such as polyethylene and polystyrene. For other molecules, such as DNA and polypeptides, the molecular flexibility is better described by the worm-like chain model (described in Section 2.2.4), whose force law can be approximated by a simple expression due to Marko and Siggia (1995), namely. 

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