Thermodynamically nonequilibrium solids

Yet another important property of fractals which distinguishes them from traditional Euclidean objects is that at least three dimensions have to be determined, namely, d, the dimension of the enveloping Euclidean space, df, the fractal (Hausdorff) dimension, and d the spectral (fraction) dimension, which characterises the object connectivity. [For Euclidean spaces, d = d = d this allows Euclidean objects to be regarded as a specific ( degenerate ) case of fractal objects. Below we shall repeatedly encounter this statement] [27]. This means that two fractal dimensions, d( and d are needed to describe the structure of a fractal object (for example, a polymer) even when the d value is fixed. This situation corresponds to the statement of non-equilibrium thermodynamics according to which at least two parameters of order are required to describe thermodynamically nonequilibrium solids (polymers), for which the Prigogine-Defay criterion is not met [28, 29]. 

Similar linear dependences for SP - OPD with various were obtained in Ref. [7] and they testify to molecular mobility level reduction at decrease and extrapolate to various (nonintegral) values at = 1.0. The comparison of these data with the Eq. (1.5) appreciation shows, that reduction is due to local order level enhancement and the condition = 1.0 is realized at values, differing from 2.0 (as it was supposed earlier in Ref [23]). This is defined by pol5miers sfructure quasiequilibrium state achievement, which can be described as follows [24]. Actually, tendency of thermodynamically nonequilibrium solid body, which is a glassy polymer, to equilibrium state is classified within the fimneworks of cluster model as local order level enhancement or (p j increase [24-26], However, this tendency is balanced by entropic essence straightening and tauting effect of polymeric medium macromolecules, that makes impossible the condition (p j= 1.0 attainment. At fully tauted macromolecular chains = 1.0)

polymer structure achieves its quasiequilibrium state at d various values depending on copolymer type, that is defined by their macromolecules different flexibility, characterized by parameter C. ... 

The important argument in favor of fractal approach application is the usage of two order parameter values, which are necessary for correct description of polymer mediums structure and properties features. As it is known, solid phase polymers are thermodynamically nonequilibrium mediums, for which Prigogine-Defay criterion is not fulfilled, and therefore, two order parameters are required, as a minimum, for their structure description. In its turn, one order parameter is required for Euclidean object characterization (its Euclidean dimension d). In general case three parameters (dimensions) are necessary for fractal object correct description dimension of Euclidean space d, fractal (Hausdorff) object dimension d and its spectral (fraction)... 

There are two main physical reasons, which define intercommunication of fractal essence and local order for solid-phase pol miers the thermodynamical nonequilibrium and dimensional periodicity of their structure. In Ref. [9], the simple relationship was obtained between thermod5mamical nonequilibrium characteristic - Gibbs function change at self-assembly (cluster structure formation of pol miers AG - and clusters relative fraction (p jin the form ... 

At solid body deformation the heat flow is formed, which is due to deformation. The thermodynamics first law establishes that the internal eneigy change in sample dU is equal to the sum of woik dW, carried out on a sample, and the heat flow dQ into sample (see the Eq. (4.31)). This relation is valid for any deformation, reversible or irreversible. There are two thermo-d5mamically irreversible cases, for which dQ = -dW, uniaxial deformation of Newtonian liquid and ideal elastoplastic deformation. For solid-phase polymers deformation has an essentially different character the ratio QIW is not equal to one and varies within the limits of 0.35 0.75, depending on testing conditions [37]. In other words, for these materials thermodynamically ideal plasticity is not realized. The cause of such effect is thermodynamically nonequilibrium nature or fractality of solid-phase polymers structure. Within the frameworks of fractal analysis it has been shown that this results to polymers yielding process realization not in the entire sample volume, but in its part only. 

The convective diffusion equations presented above have been used to model tablet dissolution in flowing fluids and the penetration of targeted macro-molecular drugs into solid tumors [5], In comparison with the nonequilibrium thermodynamics approach described below, the convective diffusion equations have the advantage of theoretical rigor. However, their mathematical complexity dictates a numerical solution in all but the simplest cases. 

In the pharmaceutical sciences, the nonequilibrium thermodynamics approach has been particularly important in the design of osmotic drug delivery devices, as discussed in Chapter 11. It has also been used to describe the convective transport of a binding antibody in an in vitro model of a solid tumor [8], As our appreciation of the roles of convection and osmosis in drug delivery increases, the nonequilibrium thermodynamics approach may find wider appeal. 

There is a corresponding paucity of experimental determinations of the surface tension of solids, probably because no direct experimental method has been developed. A review of the work on the surface tension of solid metals has been given by Shaler 27). These values were obtained, in most cases, near the melting point of the metals and thermodynamic equilibrium was achieved. These experiments are thus quite different from those where the nonequilibrium state persists, with incomplete relief of surface stress. As this review is mainly concerned with high surface area adsorbents in a state of considerable surface stress in vacuo at least), the above results with metals will not concern us further. 

Steinicke and Linke [17] refer to several microscopic and macroscopic states of mechanically stressed solids. Short time effects can be described by stochastic means or nonequilibrium thermodynamics. Long-lasting effects can be measured by calorimetry. The chemical potential and activity of the stressed solid can be measured depending on the induced defects. These defects include ... 

Early explanations about the effect of mechanical energy on the reactivity of solids are the hot-spot-model [23] and the magma-plasma-model [8]. The generation of hot-spof may be used to explain the initiation of a self-sustained reaction such as explosion, deflagration, or decomposition. Temperatures of over 1000 K on surfaces of about 1 pm2 for KM to 10-3 s can be created. These temperatures can also be found near the tip of a propagating crack [24]. Typically nonequilibrium thermodynamics are used to describe these phenomena. The magma-plasma-model allows for local nonequilibrium states on the solid surface during impact however, due to the very short time scale of 1(H s of these states only statistical thermodynamics can describe the behavior. 

We note that such problem does not appear in the nonequilibrium statistical thermodynamic approach (Sec. Ill), according to which micropores are considered together with their solid environment (micropore walls). Therefore, unlike the case of pyrolytic carbons, micropores in polymeric materials cannot be described in their own energy terms (chemical potential, etc.). 

The physical state of materials is often defined by their thermodynamic properties and equilibrium. Simple one-component systems may exist as crystalline solids, liquids or gases, and these equilibrium states are controlled by pressure and temperature. In most food and other biological systems, water content is high and the physieal state of water often defines whether the systems are frozen or liquid. In food materials science and characterization of food systems, it is essential to understand the physical state of food solids and their interactions with water. Equilibrium states are not typical of foods, and food systems need to be understood as nonequilibrium systems with time-dependent characteristics. 

For sodium palmitate, d-phase is the thermodynamically preferred, or equilibrium state, at room temperamre and up to 60°C b-phase contains a higher level of hydration and forms at higher temperatures and w-phase is an anhydrous crystal that forms at temperatures comparable with b-phase. Most soap in the solid state exists in one or a combination of these three phases. The phase diagram refers to equilibrium states. In practice, the drying routes and other mechanical manipulation used in the formation of solid soap can result in the formation of nonequilibrium phase structure. This point is important when dealing with the manufacmring of soap bars and their performance. 

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