Abstract
We discuss a mesoscopic approach we recently devised to deal with the nucleation of a daughter phase in a metastable mother fluid, with application to bubble/droplet nucleation in metastable liquids/vapors. By numerical solution of the relevant stochastic system of partial differential equations the model is shown able to deal both with homogenous and heterogenous nucleation over surfaces of different wettability and geometry and to couple the nucleation process of the new phase with the large-scale flow dynamics. Indeed, the approach we discuss bridges the gap between the atomistic scale where nucleation takes place and the macroscopic dynamics, reaching an unprecedented range of length and time scales in a problem where thermal fluctuations are among the crucial ingredients. The review intends to drive the reader through the rich theory that, starting with the basic aspects of statistical mechanics and density functional theory, touches upon the subject of non-homogenous fluids and their dynamics. Details on the physical relevance of the proposed examples as well as many technical aspects, both concerning the theory and the numerical implementation, are purposely left out of consideration addressing the interested reader to the recently published literature.
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Notes
- 1.
Given a functional of the form \(\Phi \left[ h \right] = \int _\mathcal{D} \psi ( h(\mathbf{x}), \nabla h(\mathbf{x} )) d^nV(\mathbf{x})\), its first variation is \(\delta \Phi = \int _\mathcal{D} (\frac{\partial \psi }{\partial h} \delta h + \frac{\partial \psi }{\partial \nabla h} \cdot \delta \nabla h) d^nV(\mathbf{x}) = \int _\mathcal{D} (\frac{\partial \psi }{\partial h} - \nabla \cdot \frac{\partial \psi }{\partial \nabla h} ) \delta h d^nV(\mathbf{x}) + \int _{\partial \mathcal D} \mathbf{n} \cdot \frac{\psi }{\partial \nabla h} \delta h d^{n-1}S(\mathbf{x})\). Under many circumstances, depending on the boundary conditions, the last integral over the boundary of the integration domain vanishes altogether (i.e. when \(h_{|_{\partial \mathcal D}}\) is assigned, implying \(\delta h_{|_{\partial \mathcal D}} = 0\), or when \(\mathbf{n} \cdot \frac{\partial \psi }{\partial \nabla h}_{I_{\partial \mathcal D}} = 0\)). In these cases, the first variation is \(\delta \Phi = \int _\mathcal{D} \frac{\delta \Phi }{\delta h} \delta h d^nV(\mathbf{x})\), where the function \(\frac{\delta \Phi }{\delta h} = \frac{\partial \psi }{\partial h} - \nabla \cdot \frac{\partial \psi }{\partial \nabla h}\) is called the functional derivative of \(\Phi \) with respect to h. For general boundary conditions, the boundary terms arising from the integration by parts need to be retained.
- 2.
For a field v(x) with quadratic pdf \(\mathrm{p} = \exp \left( -1/2 \int v(x) k(x,y) v(y) dx dy \right) \) the correlation is such that \(\int C(x,y) k(y,z) dx = \delta (x-z)\). In general k(x, y) may involve operators but, in the simple case \(k(x,y) = k_* \delta (x,y)\), the equation for the correlation reduces to \(k_* \int C(x,y) \delta (y-z) dx = \delta (x,z)\) whose straightforward solution is \(C(x,y) = 1/k_* \delta (x-y)\).
- 3.
By repeated integration by parts, a functional of the form \(\exp \left( \int \left( A v(x)^2 - B |\nabla v(x)|^2\right) dx \right) \) can be also written as \(\exp \left( \int \left( A v(y) \delta (y-x) v(x) + B v(y) \delta (y-x) \nabla ^2 v(x) \right) dx dy \right) = \exp \left( \int v(y) \left( \left( A + B \nabla ^2 \right) \delta (y-x) \right) v(x) dx dy\right) \). The result then follows by the technique explained in the footnote 2.
- 4.
The easiest way to realize that Eq. (27) holds is to take the limit \(\Delta V \rightarrow 0\) of a discretized version of the path integral (27), \(\int \Pi _{k=1}^M d e_k d n_k \delta ({\hat{e}} - e_s) \delta ({\hat{n}} - n_s) \exp \left\{ \sum _{l=1}^M \left[ s(e_l,n_l) - s_0 - k_1 (e_l - e_0) -k_2 (n_l-n_0)\right] \Delta V \right\} /\mathcal{Z} = \exp \left\{ \left[ s({\hat{e}},{\hat{n}} ) - s_0 - k_1 ({\hat{e}} - e_0) -k_2 ({\hat{n}}-n_0)\right] \right\} /Z \).
- 5.
Given a quadratic entropy functional \(S[{\boldsymbol{\Delta }}]\), associated with the pdf \(\mathrm{p}[{\boldsymbol{\Delta }}] = 1/\mathcal{Z} \exp (S[{\boldsymbol{\Delta }}]/k_b)\), its functional derivative is \(\delta S/\delta {\boldsymbol{\Delta }} = - C^{-1} {\boldsymbol{\Delta }}\), where \({\mathbf {C}} = \langle {\boldsymbol{\Delta }} \otimes {\boldsymbol{\Delta }}\rangle \) is the correlation. A system of the form \(\partial {\boldsymbol{\Delta }}/\partial t = {\mathbf {L}} {\boldsymbol{\Delta }} + \mathbf{f}\), can then be rewritten as \(\partial {\boldsymbol{\Delta }}/\partial t = - {\mathbf {L}} {\mathbf {C}}\delta S/\delta {\boldsymbol{\Delta }} + \mathbf{f}\).
- 6.
Denoted by k the heat conduction coefficient and \(\mu \) the viscosity, the linearized capillary Navie-Stokes operator is
$$ {\mathbf {L}} = \begin{bmatrix} 0 &{} -\rho _m^* \nabla \cdot &{} 0 \\ &{} &{} \\ - {c_T^*}^2/\rho _m^* \nabla + \lambda \nabla \nabla ^2 &{} \mu /\rho ^*\left( \nabla ^2 - \frac{1}{3}\nabla \nabla \cdot \right) &{} - \left. 1/\rho _m \partial p/\partial \theta \right| _* \nabla \\ &{} &{} \\ 0 &{} \theta ^*/(\rho _m^* c_v) \left. 1/\rho _m \partial p/\partial \theta \right| _* \nabla \cdot &{} k/(\rho _m^* c_v \nabla ^2 ) \end{bmatrix}. $$ - 7.
The divergence operators appearing in Eq. (7) are obtained from the fluctuation-dissipation balance as part of the noise intensity operator \({\mathbf {K}}\).
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Gallo, M., Casciola, C.M. (2022). Theory and Simulation of Dynamic Nucleation in Metastable Fluids. In: Rega, G. (eds) 50+ Years of AIMETA. Springer, Cham. https://doi.org/10.1007/978-3-030-94195-6_23
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