Calculation of solid–liquid interfacial free energy of silicon based on classical nucleation theory

Abstract

The solid–liquid interfacial free energy of silicon was calculated by the method based on classical nucleation theory (CNT), where the molecular dynamic (MD) simulations were carried out, and a series of cylindrical solid nuclei were equilibrated with undercooled liquid phase to create an ideal model of a homogeneous nucleation. The interfacial free energy was extracted from the relationship between the critical nuclei radii and their corresponding equilibrium temperatures. The influence of the interfacial curvature on its free energy was for the first time considered in our work, the influence can be measured by a Tolman length which was introduced to modify the traditional CNT; therefore, more accurate results were obtained. The averaged melting point and Tolman length extracted from simulations were 1678.27 K and 2.82 Å, respectively, which are consistent with the expected results. The averaged interfacial free energy is 401.92 mJ/m², which is in good agreement with the results from experiments.

APPENDIX

A. The determination of melting point

The equilibrium melting point T_M is a crucial reference point for determining the interfacial free energy γ_SL. In our work, we obtained T_M from solid–liquid coexistence simulations.28,29 As shown in Fig. 7, a box filled with silicon atoms (in orientation of [100] × [010] × [001]) was designed to form a sandwich structure. The cross-section size of box was 106.82 × 106.82 Å; the size of two blue sections in x direction was 54.307 Å, while the size of red section was changeable; to keep the symmetry of the system, the size of red section in the middle is twice as larger as two sides and periodical boundaries were used in three directions. The atoms in red region were heated to 2400 K to obtain liquid phase while atoms in blue section were kept fixed. Then the liquid was subsequently quenched to an estimated melting temperature and relaxed the whole system by 1000 ps. Equilibrium temperature T was obtained when the solid–liquid interfaces became stable.

FIG. 7

Sandwich structure for determining the melting temperature. Atoms in red section were used to obtain liquid phase, while atoms in blue remained solid phase.

Figure 8 shows the relation between equilibrium temperature and the distance of two solid–liquid interfaces. If the distance between interfaces increased, the equilibrium temperature decreased. We attribute the reason of this phenomenon to the interaction between two interfaces. In fact, Broughton and Gilmer30 noticed similar phenomenon in the research of grain boundaries premelting. In their work, the interaction between two interfaces was expressed as follows:

$$E({{\lambda}}) = ({{{\gamma}}_{{\rm{GB}}}} - 2{{{\gamma}}_{{\rm{SL}}}})g({{\lambda}})\quad ,$$

(11)

where λ is the distance between two interfaces and γ_GB is the grain boundaries energy which arises when λ = 0. The term g(λ) accounts for the transition from crystalline order at small λ, where g(0) = 1, to liquid at larger λ, with g(∞) = 0. Hoyt et al.31 have studied the form of g(λ), and gave an exponential expression. Broughton and Gilmer studied the situation that

$${{{\gamma}}_{{\rm{GB}}}} - 2{{{\gamma}}_{{\rm{SL}}}}>0,$$

(12)

the interaction E(λ) is positive, and two interfaces are repulsive32 to keep the liquid layer from collapsing below the melting point. However, in our case, the grain boundaries energy γ_GB = 0, because of the same crystalline orientation on both sides of liquid phase, there is the following:

$${{{\gamma}}_{{\rm{GB}}}} - 2{{{\gamma}}_{{\rm{SL}}}} = - 2{{{\gamma}}_{{\rm{SL}}}}{\rm{<}}\;0\quad .$$

(13)

Therefore, the interaction E(λ) is negative, and two interfaces attract in our simulations. To keep the liquid phase from solidification, the temperature must be higher than actual melting point. When λ is larger enough, this attraction can be neglected, and the equilibrium temperature T can be used as approximated the melting point T_M. As shown in Fig. 8, when the distance λ was larger than 20 Å, the equilibrium temperature became stable, and a value of 1678 K was obtained at λ = 109 Å.

FIG. 8

Dependence of equilibrium temperature on the distance between two solid–liquid interfaces. As the distance increased, the equilibrium temperature decreased.

B. The latent heat of fusion

Before determining the latent heat of fusion L_V, some bulk properties of solid and liquid phases must be obtained. Firstly, we heated a single-phase solid sample that consists 8000 atoms from 500 to 2500 K in 1000 ps. Simulations were carried out by NPT, the pressure was set as zero and periodic boundary conditions are applied. The sample became liquid at the upper superheating temperature about 2250 K. In this process, the density and enthalpy per atom of solid phase were measured and fitted by third order polynomial, provided:

$$\eqalign{{{{\rho}}_{\rm{S}}}\left(T \right) = 0.04998 - 7.678 \times {10^{\, - 7}}T + 2.36511 \;\;\;\; \quad \quad \quad \,\, \times {10^{\, - 10}}{T^{\,2}} - 4.26653 \times {10^{\, - 14}}{T^{\,3}}\,({\mathop {\rm{A}}\limits^ \circ{^{- 3}})\quad ,}} $$

(14)

$$\eqalign{& {H_{\rm{S}}}\left(T \right) = - 4.3377 + 2.61343 \times {10^{\,4}}T + 2.17603 \;\;\;\; \quad \quad \quad \,\,\, \times {10^{\, - 9}}{T^{\,2}} + 2.2285 \times {10^{\, - 12}}{T^{\,3}}({\rm{eV}}). \cr} $$

(15)

Then, the system was cooled back to 500 K in another 1000 ps, the density and enthalpy per atom of liquid phase were measured and fitted by the following:

$$\eqalign{& {{{\rho}}_{\rm{L}}}\left(T \right) = 0.05123 + 4.06181 \times {10^{\, - 6}}T - 2.31245 \;\;\;\; \quad \quad \quad \,\, \times {10^{\, - 9}}{T^{\,2}} + 3.0682 \times {10^{\, - 13}}{T^{\,3}}\left({{{\mathop {\rm{A}}\limits^ \circ}^{\,3}}} \right), \cr} $$

(16)

$$\eqalign{& {H_{\rm{L}}}\left(T \right) = - 4.26721 + 5.5461 \times {10^{\, - 4}}T - 1.01167 \;\;\;\; \quad \quad \quad \,\,\,\, \times {10^{\, - 7}}{T^{\,2}} + 1.3192 \times {10^{\, - 11}}{T^{\,3}}\left({{\rm{eV}}} \right). \cr} $$

(17)

Figure 9 shows the hysteretic loops of the heating and cooling processes of bulk system, from which the latent heat of fusion was calculated by the following:

$${L_{\,{\rm{V}}}} = \left[{{H_{\,{\rm{L}}}}\left(T \right) - {H_{\,{\rm{S}}}}\left(T \right)} \right] \times {{{\rho}}_{\,{\rm{S}}}}\left(T \right){|_{\,T = {T_{\,M}}}}.$$

(18)

FIG. 9

The hysteretic loops of the heating and cooling processes of a homogeneous bulk system. (a) The density loop. (b) The enthalpy loop. The polynomial fits to the data are given in Eqs. 14–17.

The melting point T_M = 1678 K was substituted into Eq. (18) and obtained L_V = 2544.47 mJ/m².