Size, Pressure, and Temperature Dependence

According to Debye approximation, temperature elevation weakens the bond energy AEp in the form of, 

Debye approximation Cv(7/ ), of constant volume and approaches a constant value of 3/ R is the ideal gas constant) at high temperature. At T the integration of the specific heat or the specific internal energy depends linearly on T. At low temperatures, on the other hand, the integration shows nonlinearity with respect to the temperature in a 7 manner. 

Using the core-shell configmation, the bond nature (m), solid shape (t), solid size K), pressure (P), and temperature (7) dependence of the relative change of the energy density can be obtained by summing contributions toward the outermost three atomic layers. The relative change of the Y that is proportional to the energy density (Puen) of a solid to the values at measured Tq — OK can be expressed as. 

The first part represents purely the effect of size, and the second part the joint effect of other stimuli. The effect of multi-filed coupling proceeds only in the surface up to skin depth, as seen from the second term in the second part. The undercoordinated atoms in the surface skin dictate the relative change of the elasticity and extensibility of the entire nanosolid whereas atoms in the core interior retain their bulk feamres. 

For the bulk materials (C, = 1 and Aj = 0), the skin effect can be omitted. The Y then mms to be P and T dependent only. The thermally driven softening and mechanically stiffening exhibit the bulk feature, and no summation over the surface layers is necessary. Apart from the effect of thermal expansion, the present form agrees with Anderson s model of T -dependent Y with further identification of 

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Table 27.5 Parameters derived from theoretical reproduction of the size, pressure, and temperature dependence of the bulk modulus and the Raman shift for TiOa [151]...

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