Size-Dependent Melting Behavior of Pb-17.5 At. Pct Sb-Free Biphasic Alloy Nanoparticles

Abstract

The present investigation reports the size-dependent melting behavior of bi-phasic Pb-17.5 at. pct Sb-free alloy nanoparticles, consisting of face-centered cubic (Pb) and rhombohedral (Sb) phases. These free nanoparticles (devoid of matrix) were synthesized via the solvothermal route, and subsequently different sizes were obtained by controlled heat treatment at 373 K under protective Ar atmosphere for the various durations. DSC and in situ TEM studies were used to probe the melting behavior. The experimental results reveal classical behavior, indicating the normalized eutectic temperature was inversely proportional to the nanoparticle radius. The in situ TEM investigation shows that melting initiates at the triple point among (Pb)/(Sb) and an outer surface and subsequently the melt front propagates into (Pb) phase first and then into (Sb) phase. Detailed thermodynamic modeling of the Pb-Sb system with size was carried out to decipher the size-dependent melting behavior of the biphasic nanoparticles. The experimental finding was corroborated with a theoretically predicted phase diagram as a function of size showing a reasonable match.

Appendix

1.1 Details of Thermodynamics Calculations

The objective of the appendix is to describe the details and procedure for obtaining the nanophase diagram using a thermodynamic database of molar volume and surface tensions of the respective pure metals. The method is adapted from the model developed by Weissmuller et al.[37]

The molar free energy, G^m, of a regular solution is commonly derived as:

$$G^{\text{m }} \left( {x,T} \right) = x_{\text{Pb}}^\circ G_{\text{Pb}} + \left( {1 - x_{\text{Pb}} } \right)^\circ G_{\text{sb}} + RT\left( {x_{\text{Pb}} { \ln }x_{\text{Pb}} + \left( {1 - x_{\text{Pb}} } \right){ \ln }\left( {1 - x_{\text{Pb}} } \right)} \right) + G_{mix}^{xs}$$

(A1)

where $x \; {\text{indicates }}$the mole fractions of components and $^\circ G_{\text{Pb}}$ and $^\circ G_{\text{sb}}$ are the standard Gibbs free energies of the pure components Pb and Sb, respectively. T is the temperature, R is the universal gas constant, and $G_{mix}^{xs}$ is the excess Gibbs free energy of mixing, which is extended by the Redlich-Kister formalism[48] as:

$$G_{mix}^{xs} = x_{\text{Pb}} \left( {1 - x_{\text{Pb}} } \right)\sum_{\vartheta } \varOmega_{\vartheta } \left( {2x - 1} \right)^{\vartheta }$$

(A2)

where $\varOmega_{\vartheta }$ is a non-ideal interaction parameter. $\vartheta$ is the order of expansion where for regular solution phases $\vartheta$ = 0 and $\vartheta$ = 1 for non-regular solution phases. Thermodynamic functions used during calculation are tabulated in Table IV.

Table IV Thermodynamic Functions Used in the Calculation of the Nano-Phase Diagram are Summarized in the Table

To evaluate the surface energies of solid phases ($\sigma^{\alpha}$ and $\sigma^{\beta }$) by using Butler’s equation, the molar surface area, surface energy and partial excess free energy of the pure component Pb or Sb need to be determined first. Therefore, the molar surface area can be calculated by using the following equation, $A_{i} = 1.091 N_{\text{A}}^{{\frac{1}{3}}} \left( {V_{i} } \right)^{{\frac{2}{3}}}$, where $N_{\text{A}}$ and $V_{i}$ are Avogadro’s number and the molar volume of component i. The molar volume$V$ is estimated by:

$$V = x_{\text{Pb}} V_{\text{Pb}} + (1 - x_{\text{Pb}} )V_{\text{sb}}$$

(A3)

$V_{\text{Pb}}$ and $V_{\text{sb}}$are the molar volumes of the pure components of Pb and Sb. The values of the molar volume of pure Pb and Sb in liquid state from the reported literature were used.[45,49] For evaluating the molar volume of pure components Pb and Sb in solid state, the following equation was used:

$$V_{x}^{\text{s}} = \frac{{V_{x}^{\text{L}} }}{{\left( {1 - \alpha_{x} } \right)}} \left( {x = {\text{Pb}}, {\text{Sb}}} \right)$$

(A4)

where $\alpha_{x}$ is the thermal expansion coefficient, which is a ratio of the change in volume of solid while melting, $\alpha_{x} = \frac{{V_{\text{x, m.p}}^{\text{L}} - V_{\text{x, m.p}}^{\text{s}} }}{{V_{\text{x, m.p}}^{\text{s}} }}$, where, $V_{\text{x, m.p}}^{\text{s}}$ and $V_{\text{x, m.p}}^{\text{L}}$ are the molar volumes of pure liquid and solid x (Pb or Sb) at melting temperature.

Furthermore, the surface energy values of pure Pb and Sb in liquid state were obtained directly from the literature.[44] Some of the reported references showed the surface energy value of the pure solid at the melting point is observed to be 25 pct larger than surface energy of the pure liquid.[49,50] Therefore, Eq. (A5) was used to approximate the surface energy of the pure solid component:

$$\sigma_{x}^{\text{s}} = 1.25 \sigma_{x,m.p}^{\text{L}} + \frac{{\partial \sigma_{x}^{\text{L}} }}{\partial T} \left( {{\text{T}} - {\text{T}}_{{{\text{x}},{\text{m}}.{\text{p}}}} } \right)\quad x = {\text{Pb or Sb}}$$

(A5)

where $\sigma_{x,m.p}^{\text{L}}$ is the surface energy of pure liquid x (Pb or Sb) at its equilibrium melting point T_x,m.p..

All the reported and calculated temperature-dependent surface energies and molar volumes of the pure components (Pb, Sb) in the solid as well as liquid state are summarized in Table III.

The partial excess Gibbs free energy of components Pb and Sb in the bulk can be evaluated by Reference 49:

$$\Delta G_{\text{Pb}}^{\text{b}} = G_{mix}^{xs} - \left( {1 - x_{\text{Pb}} } \right)\frac{{{\text{d}}\left( {G_{mix}^{xs} } \right)}}{{{\text{d}}x_{B} }}$$

(A6a)

$$\Delta G_{\text{sb}}^{\text{b}} = G_{mix}^{xs} + x_{\text{Pb}} \frac{{{\text{d}}\left( {G_{mix}^{xs} } \right)}}{{{\text{d}}x_{A} }}$$

(A6b)

For obtaining the partial excess Gibbs free energy of components Pb and Sb in the surface, Speiser, Yeum’s model has been used. According to Yeum’s model.[51,52]

$$\Delta G_{\text{Pb}}^{\text{s}} = \varOmega^{mix} \Delta G_{\text{Pb}}^{\text{b}}$$

(A7a)

$$\Delta G_{\text{sb}}^{\text{s}} = \varOmega^{mix} \Delta G_{\text{sb}}^{\text{b}}$$

(A7b)

$\varOmega^{mix} = 0.85$; liquid alloy and $\varOmega^{mix} = 0.84$; solid alloy

$\varOmega^{mix}$ represents the ratio of the coordination number in the surface to that in the bulk.[49]

The surface energies of solid phases ($\sigma^{\alpha }$ and $\sigma^{\beta }$) can be calculated by setting all the values in Butler’s Eqs. [A6a] and [A6b] computed by solving Eqs. [A3] through [A7a]. The equations are simultaneous equations with two unknowns: $x_{Pb,Sb}^{Surface}$ and$\gamma_{\alpha ,\beta }.$ Furthermore, these equations can be solved for those unknowns numerically.

The four possible states and corresponding free energy functions are given by$G^{{\text{total}},\alpha \beta },$$G^{{\text{total}},\alpha l},$$G^{{\text{total}},\beta l}$ and $G^{{\text{total}},l}.$ These four free energies were calculated as a function of the phase fraction at each temperature. Let us assume p₁ and p₂, the terminal mole fraction of the Pb- and Sb-rich phases, respectively. Here, p₁ and p₂ correspond to the intercepts of a common tangent with the free energy curve. To approximate the two minima of the Gibbs free energy curves, the following coupled equations are utilized:

$$\left. {\frac{{dG^{{\text{total}}} \left( x \right)}}{dx}} \right|_{{{\text{x }} = p_{1} }} = \left. {\frac{{dG^{{\text{total}}} \left( x \right)}}{dx}} \right|_{{{\text{x }} = p_{2} }}$$

(A8a)

$$\left. {\frac{{dG^{{\text{total}}} \left( x \right)}}{dx}} \right|_{{{\text{x }} = p_{1} }} *\left( {p_{1} - p_{2} } \right) = G^{{\text{total}}} \left( {p_{1} } \right) - G^{{\text{total}}} \left( {p_{2} } \right)$$

(A8b)

p₁ and p₂ are solved for ranges of temperature and plotted on a phase diagram as shown in Figure 8.