Phase transformations of nanoparticles exposed to hydrostatic pressure

Abstract

A thermodynamic analysis of phase transformations of nanoparticles under hydrostatic pressure has revealed important differences between phase transformations under isothermal or adiabatic conditions. This presuppositionless analysis fully explains a hysteresis with respect to the phase fraction and the pressure observed experimentally. It is important to mention that the results of this analysis may be transferred to the role of any external volumetric field acting on phase transforming nanoparticles. Typical examples are phase transformations of ferromagnetic intermetallics subjected to the influence of magnetic fields.

Appendices

Appendix 1: Heat transfer of a particle to its environment

Let us consider a spherical particle with radius R, the thermal conductivity λ and the dimension-free thermal diffusivity $\tilde{\kappa }\left( {\tilde{\kappa } = \lambda /\left( {\rho_{p} c_{{{F}, {p} }} R^{2} } \right),\;\rho_{p} \;{\text{density,}}\;c_{{F, {p} }} \;{\text{specific}}\;{\text{heat}}} \right).$ The sphere has an initial, relative temperature T ₁. The actual temperature T at a position ρ = r/a with r being the radial distance, 0 ≤ ρ ≤ 1, is denoted as $T_{1} \tilde{T}\;{\text{with}}\;\tilde{T}$ being a dimension-free temperature.

As first case, we study the heat transfer to the environment, staying at a temperature T = 0, performed by convection with a heat flux $\lambda \partial \tilde{T}/\partial \rho = - Rh\tilde{T}$ at the surface of the particle; h is the heat transfer coefficient. The solution of this problem can be found in Carslaw and Jaeger (1976), with a dimension-free time $\tau = t\tilde{\kappa }$ with t being the actual time. The average temperature $\tilde{T}_{\text{av}}$ in the spherical particle can be reported as

$$ \tilde{T}_{\text{av}} = 6\left( {{\frac{Rh}{\lambda }}} \right)^{2} \mathop \sum \limits_{n = 1}^{\infty } \exp \left( { - \alpha_{n}^{2} \tau } \right) \cdot {\frac{1}{{\alpha_{n}^{2} \left[ {\alpha_{n}^{2} - \left( {1 - Rh/\lambda } \right)Rh/\lambda } \right]}}}. $$

(17)

The quantity α _n is the n-th root of the equation $\alpha_{n} {\text{cotan }}\left( {\alpha_{n} } \right) + \left( {Rh/\lambda - 1} \right) = 0.$ For Rh/λ ≪ 1 a series development of the a.m. transcendent equation yields $\alpha_{1} \sim \sqrt {3Rh/\lambda }$. For the roots α _i , i ≥ 2, an estimation α _i  ~ 4.49 + (i − 2)π exists. Therefore, the multiplier of $\exp \left( { - \alpha_{n}^{2} \tau } \right)$ decreases in the order of 1/n ⁴, which yields a quick convergence of the series. In many practical cases it is, therefore, sufficient to use as an estimation of $\tilde{T}_{\text{av}}$ the first term of the series in (17), which follows from a binomial series developments and some analysis as

$$ \tilde{T}_{\text{av}} \sim \exp \left( { - 3Rh/\lambda \cdot \tau } \right)\left( {1 - {\frac{3Rh}{2\lambda }}} \right)\quad{\text{for}}\;\,\sqrt {3Rh/\lambda } \ll 1 $$

(18)

Taylor series development of the first term in (18) yields

$$ \tilde{T}_{\text{av}} \sim 1 - {\frac{3Rh}{\lambda }}\tau = 1 - {\frac{3h}{{R\rho_{p} c_{{F,_{p} }} }}}\,t. $$

(19)

As second case, we study the heat transfer to a matrix, in which the particle is embedded, with the same thermal properties as the particle. An exact analytical solution for this problem can be found in Frank and Mises (1961) as

$$ {\tilde{{T}}} = {\frac{ 1}{ 2}}\left[ {{\text{erf}}\left( {{\frac{1 + \rho }{2\sqrt \tau }}} \right) + {\text{erf}}\left( {{\frac{1 - \rho }{2\sqrt \tau }}} \right)} \right] + {\frac{1}{\rho }}\sqrt {{\frac{\tau }{\pi }}} \left[ {\exp \left( {{\frac{{ - \left( {1 + \rho } \right)^{2} }}{4\tau }}} \right) - \exp \left( { - {\frac{{\left( {1 - \rho } \right)^{2} }}{4\tau }}} \right)} \right]. $$

(20)

The function erf(x) is the error function of the argument x. The average temperature $\tilde{T}_{\text{av}}$ in the spherical particle can be found by integration as

$$ \tilde{T}_{\text{av}} = 1 - {\frac{1}{\sqrt \pi }}\left[ {\sqrt \tau \left( {3 - 2\tau } \right) + \exp \left( { - {\frac{1}{\tau }}} \right)\left( {1 - \sqrt \tau + 2\tau \sqrt \tau } \right)} \right]. $$

(21)

For a short time period τ the average temperature can be approximated as

$$ \tilde{T}_{\text{av}} = 1 - {\frac{3}{\sqrt \pi }}\sqrt \tau = 1 - {\frac{3}{\sqrt \pi }}\left( {{\frac{\lambda }{{R^{2} \rho_{p} c_{{F,_{p} }} }}}\,t} \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} . $$

(22)

Appendix 2

Thermodynamic data used for the model calculations

Due to a lack of a consistent set of thermodynamic data for nanomaterials in the open literature, an artificial data set was developed that allows the demonstration of the properties of phase transformations in different enclosures. The following values are selected:

• Material data:

  Density: ρ = 5,000 kg m⁻³

  Density ratio: ρ₂/ρ₁ = 0.91

  Molecular weight: M = 0.05 kg mol⁻¹

  Surface energy phase 1: γ₁ = 1.0 J m⁻²

  Surface energy phase 2: γ₁ = 1.1 J m⁻²

• Thermodynamic data:

  Enthalpy phase 1: u ₁ = 0 J mol⁻¹ (Reference state)

  Enthalpy phase 2: u ₂ = 6,000 J mol⁻¹

  Entropy phase 1: s ₁ = 34 J T⁻¹ mol⁻¹

  Entropy phase 2: s ₂ = 40 J T⁻¹ mol⁻¹

  Heat capacity: $c_{p} = 25[{{{\left( {D_{1} - 2 \times 10^{ - 9} } \right)^{3} + 1.2\left( {D_{1}^{3} - \left( {D_{1} - 2 \times 10^{ - 9} } \right)^{3} } \right)} /{D_{1}^{3} }}}]$

This formula meets approximately the influence of the surface on the heat capacity, assuming a surface layer of 1 nm thickness with increased heat capacity; D ₁ is the diameter of the particle in phase 1 in meter.