Temperature dependence of the surface free energy

The equilibrium shape of Pb crystallites of micrometer size on a graphite substrate has been analyzed by scanning electron microscopy to derive the polar diagram of the surface free energy. From shape analyses performed at different temperatures, the temperature dependence of the surface free energy anisotropy, expressed as ratio y(uj) /yim) has been determined. The result is presented in Fig. 10. The surface free... 

Fig. 10. Temperature dependence of the surface free energy anisotropy of Pb. Equilibrium shapes of Pb crystallites on graphite have been analysed at 473 K, 523 K, 548 K and 573 K. A decrease of the anisotropy with increasing temperature is apparent from the negative slopes of the curves [83Hey].

Experiments on the equiUbrium shape of small crystallites and on the groove angle at grain boundaries indicate an orientation dependence of the surface free energy. The value the surface free energy varies by several percent for different surface orientations at high temperature. This anisotropy is e.g. responsible for the faceted shape of the crystallites shown in Fig. 4. 

Fig. 4. Scanning electron microscopy image of the anisotropic faceted morphology of a lead crystallite. This equihhrium shape has been analysed to derive the orientation dependence of the surface free energy at various temperatures, see chapter 4.4.7.3.1 [83Hey],...

Before turning to the surface enthalpy we would like to derive an important relationship between the surface entropy and the temperature dependence of the surface tension. The Helmholtz interfacial free energy is a state function. Therefore we can use the Maxwell relations and obtain directly an important equation for the surface entropy ... 

Equation (2.25) shows that the temperature dependence of the surface energy is determined by the surface entropy, which should generally be positive. Thus, the surface energy decreases as the temperature is increased. From the definition of the Helmholtz free energy. Equation (1.28), the surface Helmholtz firee energy, internal energy and entropy are related by... 

Temperature dependence of the anisotropy of the surface free energy... 

The actual surface termination of oxides not only is a result of relaxation or reconstruction but also depends critically on the environment. Preparation of oxide surfaces for UHV studies often requires high-temperature or high-pressure oxygen treatment. Such factors can lead to dynamic adsorption/desorption processes, which determine the surface structure under these conditions. At given experimental parameters, the surface structure with the lowest surface free energy will be the most stable one. A formahsm has been developed that allows the calculation of the surface free energy from ah initio methods as a function of pressure and... 

More recently, the temperature dependence of the above vibrational component was studied13. Still for the 100 face of NaCl, the contribution of this component to the total surface energy, see Section II.3., was approximately —2 erg/cm2 and to the free surface energy near - 16 erg/cm2, both at 273 °K. No correction for the polarizability of the ions was employed. 

Since surface pressure is a free energy term, the energies and entropies of first-order phase transitions in the monolayer state may be calculated from the temperature dependence of the ir-A curve using the two-dimensional analog of the Clausius-Clapeyron equation (59), where AH is the molar enthalpy change at temperature T and AA is the net change in molar area ... 

The YBG equation is a two point boundary value problem requiring the equilibrium liquid and vapor densities which in the canonical ensemble are uniquely defined by the number of atoms, N, volume, V, and temperature, T. If we accept the applicability of macroscopic thermodynamics to droplets of molecular dimensions, then these densities are dependent upon the interfacial contribution to the free energy, through the condition of mechanical stability, and consequently, the droplet size dependence of the surface tension must be obtained. 

This relationship identifies the surface energy as the increment of the Gibbs free energy per unit change in area at constant temperature, pressure, and number of moles. The path-dependent variable dWs in Eq. (2.60) has been replaced by a state variable, namely, the Gibbs free energy. The energy interpretation of y has been carried to the point where it has been identified with a specific thermodynamic function. As a result, many of the relationships that apply to G also apply to y ... 

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