# Contact Melting of Metals Explained via the Theory of Quasi-Liquid Layer
## 2 Results and Discussion  
The statement of this problem has the following form (Figure 1):  
Figure 1: Mutual diffusion of atoms of two substances. Initial and boundary conditions for a homogeneous case (liquid or solid):  
\[c|_{x\rightarrow\infty}=1,\;\left.c|_{x\rightarrow-\infty}=0,\;\left.c|_{t=0}= \eta(x),\right.\right. \tag{2}\]  
where \(\eta(x)\) is a Heaviside step function. The solution of the equation, taking into account the initial and boundary conditions, has the form:  
\[c(x,t)=\frac{1}{2}\bigg{(}1+erf\bigg{(}\frac{x}{2\sqrt{Dt}}\bigg{)}\bigg{)} \tag{3}\]  
where  
\[erf(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}\exp\Bigl{(}-\xi^{2}\Bigr{)}d\xi \tag{4}\]  
The chemical potential of an atom in this case can be written as:  
\[\mu=\mu_{0}+kTInc, \tag{5}\]  
where \(\mu_{0}\) does not depend on the concentration of substances. In this formulation of the problem, we neglect the chemical reactions between the components, as well as various cross effects. However, until diffusion occurs, the metals cannot become liquid, because, being initially in a pure state, they are both solid. According to [PERSON]'s formula, the mean square displacement can be written as:  
\[\langle x^{2}\rangle^{1/2}=\sqrt{2 Dt} \tag{6}\]  
where \(D\) is the coefficient of mutual diffusion for two metals, and \(t\) is the time. For liquid metals, diffusion coefficients are approximately \(10^{-9}\) m\({}^{2}/\)s; however, for solid metals, diffusion coefficients are approximately \(10^{-20}\) m\({}^{2}/\)s. It is easy to see that for typical experimental times of the order of a minute, solids will not have time to diffuse, and the root mean square displacement will be smaller than the size of an atom. Thus, contact melting of metals should occur slowly, i.e., it should be determined by diffusion in a solid. Existing contact melting models (see, for example, refs. [8; 9; 10]) cannot explain this contradiction. To solve this problem, we will take into account that at temperatures close to the melting temperature a quasi-liquid layer (film) is located on the surface of a solid. Such quasi-liquid layers have been repeatedly recorded experimentally for different substances (see, for example, refs. [11; 12; 13; 14]). When calculating the volumetric thermodynamic characteristics of a substance, such a layer can be neglected; however, for some cases in which the surface condition is important, its presence is crucial. The temperature at which a quasi-liquid layer (premelting) is formed can be approximately estimated by taking into account the fact that an atom has a smaller number of neighbors on the surface of a solid compared with the atoms inside a sample (Figure 2). Let \(V(x)\) be the potential energy of the interaction of the selected atom with one of its nearest neighbors. The potential energy for the interaction with \(N\) atoms is equal to \(NV(x)\)  
Figure 2: Atoms on the surface of a metal. Each atom residing deep in the solid has six nearest neighbors, and an atom lying on the surface has five nearest neighbors. Therefore, the energies of this atom in such potential wells will be related by a 5/6 ratio. The energy required for melting should be proportional to the depth of the potential well; therefore, the temperature at which the quasi-liquid layer (premelting) appears in the first approximation is 5/6 of the melting temperature in the volume:  
\[T_{ql}\approx\frac{5}{6}T_{m}. \tag{7}\]  
In this regard, we consider contact melting experiments for some pairs of metals (Pb-Sn [6], Sn-In [7], Tl-Bi, Tl-Sn [2,7]), and show that when such metals contact each other, at least one of them has a quasi-liquid film on their surface. According to [2,7], contact melting was performed in a thermostat. The samples were cylinders with a diameter of \(\sim\)3 mm and a length of \(\sim\)2 cm. The contact surfaces of the samples were polished. From the moment of contact between the samples, the diffusion time begins. The growth of the liquid layer was observed visually using a microscope.