Fun to (nano) imagine…

Richard Feynman once said that he was involved in Science for the fun of it, for the satisfaction of understanding nature. As a professor, it is a real pleasure to transmit my passion and love of science to summer students (that I make a point of hiring every summer!). Let’s have a quick peek at the project of one of them (Antoine Rincent).

The description of the energetic behavior of bulk phases has been extensively explored in the literature with great success. The popularity of the CALPHAD (Computer Coupling of Phase Diagrams and Thermochemistry from Computational Thermodynamics The Calphad Method by Hans Lukas, Suzana G. Fries and Bo Sundman) method partly lies in its simplicity and high flexibility.

But simplicity and flexibility both come to a price. Thermodynamic models of solutions derived from the (sub-)regular solution theory are embedding a lot of physical contributions (inter-atomic potentials and the influence of inter-distances as well as the chemical environment) in highly-empirical functions. For these reasons, the predictive ability of these models as a function of temperature, pressure, composition and especially size are limited.

In recent years, there has been a growing interest in the fascinating world of nano-systems:

  • nanoparticle catalysts (Acc. Chem. Res., 2013, 46 (8), pp 1671–1672)
  • nanoparticle precipitation in structural materials (CALPHAD, 32(1):164-170)
  • nanofabrication (ACS Nano20093 (5), pp 1049–1056)

One of the scientific challenges in the optimal design of nano-materials is the prediction of their thermo-physico-chemical behavior through physically sound models and representations. In this context, Antoine (a third year student in chemical engineering) is exploring the different theoretical approaches to study nano-systems.

As a first part of his project, Antoine reviewed the literature linked to nano-thermodynamics in search of equations and models to implement. The first accessible approach to describe the influence of particle size and chemistry on their melting temperature (liquidus/solidus) was found in the work of Cui et al. 2017.

Here are the nuts and bolts of their approach. Firstly, a little bit of thermodynamics. At equilibrium, each chemical potential \mu_i of the various species i  is the same in the different phases that define the equilibrium state.  Let’s take the Au-Cu system where a solid (s) and a liquid (l) solution are in equilibrium as an example.  In this case two independent equations emerge from the definition of the equilibrium state:

with
\label{eq1} \mu_{Au}^{s}=(\frac{\partial G_{s}}{\partial n_{Au}^{s}})_{T,P,n_{Cu}^{s}} \\ \mu_{Au}^{l}=(\frac{\partial G_{l}}{\partial n_{Au}^{l}})_{T,P,n_{Cu}^{l}}

G_{s}=n_{Au}^{s}g_{Au}^o+n_{Cu}^{s}g_{Cu}^o+RT[n_{Au}^{s}ln(\frac{n_{Au}^{s}}{n_{Au}^{s}+n_{Cu}^{s}})+n_{Cu}^{s}ln(\frac{n_{Cu}^{s}}{n_{Au}^{s}+n_{Cu}^{s}})]+\omega_{s}^{Au-Cu}(\frac{n_{Au}^{s}n_{Cu}^{s}}{n_{Au}^{s}+n_{Cu}^{s}}) \\ G_{l}=n_{Au}^{l}g_{Au}^o+n_{Cu}^{l}g_{Cu}^o+RT[n_{Au}^{l}ln(\frac{n_{Au}^{l}}{n_{Au}^{l}+n_{Cu}^{l}})+n_{Cu}^{l}ln(\frac{n_{Cu}^{l}}{n_{Au}^{l}+n_{Cu}^{l}})]+\omega_{l}^{Au-Cu}(\frac{n_{Au}^{l}n_{Cu}^{l}}{n_{Au}^{l}+n_{Cu}^{l}})

Introducing the molar enthalpy of fusion \Delta h^{fusion}_i of species i, we can ultimately arrive to the following set of equations to be solved for an imposed composition of the liquid phase, X_{Cu}^{l} in order to get T and X_{Cu}^{s}:

T=\frac{\Delta h^{fusion}_{Cu}-\omega_{s}^{Au-Cu}(1-X^{s}_{Cu})^{2}+\omega_{l}^{Au-Cu}(1-X^{l}_{Cu})^{2}}{R \cdot ln(\frac{X^{s}_{Cu}}{X^{l}_{Cu}})+\frac{\Delta h^{fusion}_{Cu}}{T^{fusion}_{Cu}}}                                          ( eq. 1)

T=\frac{\Delta h^{fusion}_{Au}-\omega_{s}^{Au-Cu}(X^{s}_{Cu})^{2}+\omega_{l}^{Au-Cu}(X^{l}_{Cu})^{2}}{R \cdot ln(\frac{1-X^{s}_{Cu}}{1-X^{l}_{Cu}})+\frac{\Delta h^{fusion}_{Au}}{T^{fusion}_{Au}}}                                                    (eq. 2)

A simple way for my student to use this approach was to solve this set of equations using the Maple software (without having to delve too deep in numerical methods, such as fixed point or Newton’s method).

Finally, in order to account for the effect of the particle diameter (D) on its melting temperature (for the Au-Cu system), we can estimate each component thermodynamic properties as a function of the system size using the following equation:

\frac{ \zeta(D,\lambda)}{\zeta} = (1-\frac{1}{\frac{12D}{D_{0}}-1}) \cdot e^{-\frac{2\lambda S_{0}}{3R} \cdot \frac{1}{\frac{12D}{D_{0}}-1}}

Where \zeta can be \Delta h^{fusion}_{i}, T^{fusion}_{i} and \omega^{i-j}D is the particle diameter, D_{0} its critical diameter and \lambda the particle shape factor. With these new values the last system of equation then gives the values of the nanometric system! To continue advancing on his project, he then programmed the system (and a resolution method through Newton’s method) in C++.

To obtain a better understanding of nanoparticles melting, Antoine is now working on the modeling of the surface tension of multicomponent liquids, as it is a key factor in this phase transition at the nanoscale.

Up to now his experience at the CRCT has granted him knowledge on programming, numerical methods and of course Thermodynamics and its applications. We are both looking forward to the next challenge!

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