Network statistical thermodynamics

It is not particularly difficult to introduce thermodynamic concepts into a discussion of elasticity. We shall not explore all of the implications of this development, but shall proceed only to the point of establishing the connection between elasticity and entropy. Then we shall go from phenomenological thermodynamics to statistical thermodynamics in pursuit of a molecular model to describe the elastic response of cross-linked networks. 

Statistical thermodynamics of swollen polymer networks. J. Polymer Sci. 59, 191 (1962). 

Flory PJ (1953) Principles of Polymer Chemistry. Cornell University Press, Ithaca NY Flory PJ (1976) Statistical thermodynamics of random networks. Proc Royal Soc London A 351 (1666) 351-380... 

Pore systems of solids may vary substantially both in size and shape. Therefore, it is somewhat difficult to determine the pore width and, more precisely, the pore size distribution of a solid. Most methods for obtaining pore size distributions make the assumption that the pores are nonintersecting cylinders or slit-Uke pores, while often porous solids actually contain networks of interconnected pores. To determine pore size distributions, several methods are available, based on thermodynamics (34), geometrical considerations (35-37), or statistical thermodynamic approaches (34,38,39). For cylindrical pores, one of the most commonly applied methods is the one described in 1951 by Barrett, Joyner, and Halenda (the BJH model Reference 40), adapted from... 

Statistical Thermodynamics of a Cross-Linked Polymer Network... 

An expression for the force as a function of strain can be established by statistical thermodynamic analysis of the chain, and then of a network of chains. 

According to the statistical thermodynamic approach to be developed below, each conformation that a network chain segment may take is equally probable. The number of such conformations depends on the end-to-end distance, r, of the chain, reaching a rather sharp maximum at tq. The retractive force of an elastomer is developed by the thermal motions of the chains, statistically driven toward their most probable end-to-end distance, Tq. 

It is shown that the problem of formulating the statistical thermodynamics of networks can be resolved by the use of ideas from quantum statistical mechanics by appropriate generalization. Examples are given in terms of gaussian and liquid crystal polymer networks. 

Evidently, the entropic forces must originate from the polymer chains which set up the network. It is easy to see the physical basis of the retraction mechanism When chains are extended on stretching the network, the number of available rotational isomeric states and thus the entropy decreases, and this produces a retractive force. Statistical thermodynamics can describe this effect in more detail, employing model considerations. 

Rory PJ. Statistical thermodynamics of random networks. Proc R Soc Ixind A 1976 351 351-80. 

Sato H, Fujikake H, Lino Y, Kawakita M, Kikuchi H (2002) Flexible grayscale ferroelectric liquid crystal device containing polymer walls and networks. Jpn J Appl Phys 41 5302-5306 Sato H, Fujikake H, Kikuchi H, Kurita T (2003) Rollable polymer stabilized ferroelectric liquid crystal device using thin plastic substrates. Opt Rev 10(5) 352-356 Schrader DM, Jean YC (1988) Positron and positronium chemistry. Elsevier, Amsterdam Shinkawa K, Takahashi H, Fume H (2008) Ferroelectric liquid crystal cell with phase separated composite organic film. Ferroelectrics 364 107-112 Simha R, Somcynsky T (1969) On the statistical thermodynamics of spherical and chain molecule fluids. Macromolecules 2 342-350... 

The properties of elastomeric materials are controlled by their molecular structure which has been discussed earlier (Section 4.5). They are basically all amorphous polymers above their glass transition and normally crosslinked. Their unique deformation behaviour has fascinated scientists for many years and there are even reports of investigations into the deformation of natural rubber from the beginning of the nineteeth century. Elastomer deformation is particularly amenable to analysis using thermodynamics, as an elastomer behaves essentially as an entropy spring . It is even possible to derive the form of the basic stress-strain relationship from first principles by considering the statistical thermodynamic behaviour of the molecular network. 

Actually, we can apply this to stracture elements and ions in all the models of solutions available in thermodynamics and in particular those deduced Irom statistical thermodynamics such as the model of strictly regular solutioa The basic assumptions of these models apply without reservation to stracture elements and even, in this case, the assumption of a pseudo-network, which it is necessary to admit in the case of liquid phases, does not obviously present arty difficulty for crystallized solids. 

Polymer networks are conveniently characterized in the elastomeric state, which is exhibited at temperatures above the glass-to-rubber transition temperature T. In this state, the large ensemble of configurations accessible to flexible chain molecules by Brownian motion is very amenable to statistical mechanical analysis. Polymers with relatively high values of such as polystyrene or elastin are generally studied in the swollen state to lower their values of to below the temperature of investigation. It is also advantageous to study network behavior in the swollen state since this facilitates the approach to elastic equilibrium, which is required for application of rubber elasticity theories based on statistical thermodynamics. ... 

Self-consistent approaches in molecular modeling have to strike a balance of appropriate representation of the primary polymer chemistry, adequate treatment of molecular interactions, sufficient system size, and sufficient statistical sampling of structural configurations or elementary transport processes. They should account for nanoscale confinement and random network morphology and they should allow calculating thermodynamic properties and transport parameters. 

There are models for water in the literature which aim at building up an effective Hamiltonian from a description of structure V. These models must give the same results for long-time dynamics as those that will emerge straightforwardly from our statistical model, based on H-bond dynamics. This must be expected in the temperature range at which the structural features of the H-bond network are dominant, lliese literature models are limited to descaibing the short-time dynamics and some thermodynamic properties of water. 

Many of the thermodynamic and transport properties of liquid water can be qualitatively understood if attention is focused on the statistical properties of the hydrogen bond network [9]. As an example, let us observe the temperature dependence of density and entropy. As temperature decreases, the number of intact bonds increases and the coordination number is closer to the ideal value 4. Because of the large free volume available the temperature decrease is associated with an increase of the local molecular volume. This effect superimposes of course to the classical anharmonic effects which dominate at high temperature, when the number of intact bonds is smaller. The consequence of both effects is a maximum on the temperature dependence of the liquid density. This maximum is actually at 4°C for normal water and 11 °C for heavy water. Such a large isotopic effect can also be understood because the larger mass of the deuterium makes the hydrogen bonds more stable. 

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