A monomolecular layer model of the surface phase of microdroplets was proposed, and the exact expression for Tolman length was derived in this paper. The Tolman lengths of water, n-pentane, and n-heptane were calculated by the expression, and the values are quite in agreement with the experimental values. By use of the Gibbs-Tolman-Kening-Buff equation, the exact relationship between the microdroplet surface tension and the radius is obtained, and the predicted values agree well with the simulated values. The results show that there is an obvious effect of the size of microdroplets (or nanoparticles) on the surface tension, and the surface tension decreases with decreasing droplet size. For the microdroplets of general liquid, only if their radius approaches or reaches 10(-9) m does the effect become significant.

Owing to their excellent adsorption properties compared with those of the corresponding bulk materials, nanoparticles have been widely applied in many fields. Their properties depend on the thermodynamics and kinetics of adsorption, which depend on the particle size. In this paper, we present universal theories of the thermodynamics and kinetics for nanoadsorption that have been developed over the past few years. Theoretically, we have derived relationships between the adsorption thermodynamic properties and the particle size, as well as those between the adsorption kinetic parameters and the particle size. Moreover, we discuss the regularities and mechanisms of influence of the particle size on the thermodynamics and kinetics of adsorption. Experimentally, taking the adsorption of methyl orange on nano-CeO in aqueous solution as a system, we have studied the size-dependent thermodynamics and kinetics of the system, and the size dependences were confirmed to be consistent with the theoretical relationships. The results indicate that particle size has a significant effect on the thermodynamic properties and kinetic parameters of adsorption: with decreasing particle size of nano-CeO, the adsorption equilibrium constant K and the adsorption rate constant k increase, while the molar Gibbs free energy of adsorption Δ G, the molar adsorption entropy Δ S, the molar adsorption enthalpy Δ H, the adsorption activation energy E, and the adsorption pre-exponential factor A all decrease. Indeed, ln K, Δ G, Δ S, Δ H, ln  k, E, and ln  A are each linearly related to the reciprocal of particle size. Furthermore, thermodynamically, Δ G and ln  K are influenced by the molar surface area and the difference in surface tensions, Δ S is influenced by the molar surface area and the difference in temperature coefficients of surface tension, and Δ H is influenced by the molar surface area, the difference in surface tensions, and the difference in temperature coefficients of surface tension. Kinetically, E is influenced by the partial molar surface enthalpy of the nanoadsorbent, ln  A is influenced by the partial molar surface entropy, and ln  k is influenced by the partial molar surface Gibbs energy. The theories can quantitatively describe adsorption behavior on nanoparticles, explain the regularities and mechanisms of influence of particle size, and provide guidance for the research and application of nanoadsorption.

Vanadium dioxide exhibits a reversible crystal transition from monoclinic phase to rutile phase, but the quantitative regularity of the effect of particle size on crystal transition thermodynamics still remains unclear. Herein, a new core−shell model was proposed and the universal equations for size-dependent crystal transition thermodynamics have been deduced. Experimentally, we researched the crystal transition behaviors of VO 2 (M) with different particle sizes. The results indicate that with the particle size of nano-VO 2 (M) decreasing, the temperature, enthalpy, and entropy of the crystal transition decrease, and these thermodynamic properties are all linearly related with the reciprocal of particle radius, which are consistent with the theoretical formulas. More importantly, by using the quantitative influence regularity of the particle size on the crystal transition temperature, we can obtain different crystal transition temperatures for different applications by purely controlling the particle sizes.

An equation for a phase transition in a dispersed system has been proposed, and the applications of the equation in various kinds of phase transitions have been discussed. The determinate relation between the interfacial tension and the radius of a droplet has been derived by the monolayer model. Applying the fusion transition equation and the interfacial tension relation, the melting temperatures of Au and Sn nanoparticles have been calculated, and the predicted melting temperatures are in good agreement with the available experimental data. The research results show that the phase transition equations can be applied to predict the temperatures of phase transitions of dispersed systems and to explain the phenomenon of metastable states; that the size of a dispersed phase has a remarkable effect on the phase transition temperatures, and the phase transition temperatures decrease with the radius of the dispersed phase decreasing; and that the depression of the melting temperature for a nanowire is half of that for a spherical nanoparticle with identical radius.

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