Deformation entropy

Other types of deformation may be handled similarly. Shear, for example, may be treated as a homogeneous strain involving an increase in one coordinate x) while another (z) remains constant, the volume being constant also. Thus, we may let aa = a, oLy = a, and az=. On substitution of these conditions in Eq. (41), the deformation entropy per unit volume becomes... 

M in g/mol or kg/kmol). a can then be obtained according to Eq. (38), with X taken from Eqs. (36) or (37). If the molar deformation entropy VM ASdef = — R/2 is considered in both cases, X follows from the transcendent equations... 

Table 7. Linear and isotropic deformation ratios X = 1/1 yielding the molar deformation entropy —R/2 J mol-1 K I at the given n contacts of the sorbed polystyrene coil with the linear polystyrene gel (e.g. X = 1.65 means 1 — lo is equal to 65 % of lo)...

Table 8. Young s modulus E, shear modulus G and tensile stress a as calculated for the deformation entropy —R/2 according to Table 7, of sorbed polystyrene coils of given degree of polymerization P which form n contacts with the linear polystyrene gel at the temperatures 283 and 293 K...

Because a gel can be regarded mechanically as a solid and thermodynamically as a solution, the thermodynamics of gels can be formulated as an expansion of that of polymer solutions however, there is a macroscopic deformation of networks that does not exist in solutions and calculation of its deformation entropy is somewhat challenging. Here this calculation will be briefly discussed and readers are referred to the original literature for more details. 

Fig. 1 Conceptual diagram for the calculation of deformation entropy of networks.

Damping ratio, 332, 334 DD-MAS, 283, 285, 287, 288, 290 Debye-Buiche equation, 195-6 Debye-Waller factor, 214, 217 Deformation entropy of networks, 70 Degree of polymerization, 99,125, 128, 130, 133 distribution, 37-41 number average, 31-5 total probability, 32 weight average, 31-5, 36 Deoxyribonucleic acid (DNA), 356 Dielectric constant, 363-4, 390 Dielectric properties, 363-79... 

Dislocation theory as a portion of the subject of solid-state physics is somewhat beyond the scope of this book, but it is desirable to examine the subject briefly in terms of its implications in surface chemistry. Perhaps the most elementary type of defect is that of an extra or interstitial atom—Frenkel defect [110]—or a missing atom or vacancy—Schottky defect [111]. Such point defects play an important role in the treatment of diffusion and electrical conductivities in solids and the solubility of a salt in the host lattice of another or different valence type [112]. Point defects have a thermodynamic basis for their existence in terms of the energy and entropy of their formation, the situation is similar to the formation of isolated holes and erratic atoms on a surface. Dislocations, on the other hand, may be viewed as an organized concentration of point defects they are lattice defects and play an important role in the mechanism of the plastic deformation of solids. Lattice defects or dislocations are not thermodynamic in the sense of the point defects their formation is intimately connected with the mechanism of nucleation and crystal growth (see Section IX-4), and they constitute an important source of surface imperfection. 

The fact that shock waves continue to steepen until dissipative mechanisms take over means that entropy is generated by the conversion of mechanical energy to heat, so the process is irreversible. By contrast, in a fluid, rarefactions do not usually involve significant energy dissipation, so they can be regarded as reversible, or isentropic, processes. There are circumstances, however, such as in materials with elastic-plastic response, in which plastic deformation during the release process dissipates energy in an irreversible fashion, and the expansion wave is therefore not isentropic. 

The transition obtained under stress can be in some cases reversible, as found, for instance, for PBT. In that case, careful studies of the stress and strain dependence of the molar fractions of the two forms have been reported [83]. The observed stress-strain curves (Fig. 16) have been interpreted as due to the elastic deformation of the a form, followed by a plateau region corresponding to the a toward [t transition and then followed by the elastic deformation of the P form. On the basis of the changes with the temperature of the critical stresses (associated to the plateau region) also the enthalpy and the entropy of the transition have been evaluated [83]. 

In order to arrive ultimately at the entropy change accompanying deformation, we now proceed to calculate the configurational entropy change involved in the formation of a network structure in its deformed state as defined by a, ay, and (We shall avoid for the present the stipulation that the volume be constant, i.e., that axayag=l.) Then by subtracting the entropy of network formation when the sample is undeformed (ax = ay = az=l)j we shall have the desired entropy of deformation. As is obvious, explicit expressions will be required only for those terms in the entropy of network formation which are altered by deformation. 

For ordinary deformations of rubberlike substances (excluding swelling phenomena to be discussed in the following chapter) it is permissible to assume constant volume, i.e., axayaz = l. The logarithmic term in Eq. (41) then disappears. In the particular case of elongation at constant volume, = = giving for the entropy of deforma-... 

Conventional implementations of MaxEnt method for charge density studies do not allow easy access to deformation maps a possible approach involves running a MaxEnt calculation on a set of data computed from a superposition of spherical atoms, and subtracting this map from qME [44], Recourse to a two-channel formalism, that redistributes positive- and negative-density scatterers, fitting a set of difference Fourier coefficients, has also been made [18], but there is no consensus on what the definition of entropy should be in a two-channel situation [18, 36,41] moreover, the shapes and number of positive and negative scatterers may need to differ in a way which is difficult to specify. 

The reversible recovery of a deformed elastomer to its original (undeformed) state is due to an entropic driving force. The entropy of polymer chains is minimum in the extended conformation and maximum in the random coil conformation. Cross-linking of an elastomer to form a network structure (IX) is... 

Let in the deformated m-ball in a moment of break the part of the residual intertwining chains is equal to a. Then the entropy of mixing will be equal to... 

Each submolecule will experience a frictional drag with the solvent represented by the frictional coefficient /0. This drag is related to the frictional coefficient of the monomer unit (0- If there are x monomer units per link then the frictional coefficient of a link is x(0- If we aPply a step strain to the polymer chain it will deform and its entropy will fall. In order to attain its equilibrium conformation and maximum entropy the chain will rearrange itself by diffusion. The instantaneous elastic response can be thought of as being due to an entropic spring . The drag on each submolecule can be treated in terms of the motion of the N+ 1 ends of the submolecules. We can think of these as beads linked... 

The first ingredient in any theory for the rheology of a complex fluid is the expression for the stress in terms of the microscopic structure variables. We derive an expression for the stress-tensor here from the principle of virtual work. In the case of flexible polymers the total stress arises to a good approximation from the entropy of the chain paths. At equilibrium the polymer paths are random walks - of maximal entropy. A deformation induces preferred orientation of the steps of the walks, which are therefore no longer random - the entropy has decreased and the free energy density/increased. So... 

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