generic cluster dynamics derived from array nodal kernel. #434
generic cluster dynamics derived from array nodal kernel. #434lynnmunday wants to merge 8 commits into
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needs idaholab/moose#32230 merged |
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…ake it easier to run_tests from tests directory.
…es from a function
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| +Component 0 (monomer, $n=1$):+ | ||
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| !equation | ||
| \frac{dC_1}{dt} = G_1 - k_s C_1 - 2\beta_1 C_1^2 - \sum_{n=2}^{N} \beta_n C_1 C_n + 2\alpha_2 C_2 + \sum_{n=3}^{N} \alpha_n C_n |
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The terms G_1 and k_s represent the generation and sink rates for specific monomer species. G_1 and k_s primarily apply to point defects, such as vacancies and interstitials generated by irradiation, and are not applicable to chemical species. G_1 and k_s should be ignored for modeling solute clustering such as Cu precipitation. This has been confirmed by comparing against the paper by Bai etal (2017).
https://doi.org/10.1016/j.jnucmat.2017.08.042
| where the growth-in term is: | ||
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| !equation | ||
| \dot{C}_n^{\text{in}} = \begin{cases} \tfrac{1}{2}\beta_1 C_1^2 & n = 2 \\ \beta_{n-1} C_1 C_{n-1} & n > 2 \end{cases} |
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For n>2, the expression is correct, but I think the expression for n>2 also applies the same to n=2. I don't quite understand why there is a 1/2 for n=2, since c1 x c1 should be the opportunity to form a dimer multiplied by a rate constant. I'll need to figure this out. It seems the 1/2 is to avoid double counting of identical monomer–monomer collisions.
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I think you are right. I think each monomer forms a dimer so the divide by two gets rid of the extra dimer
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However this will violate mass conservation since we have - 2 * β1 * C_1^2 in equation one for monomer, that means the change in C_2 should be half the magnitude of 2 * β1 * C_1^2, which is β1 * C_1^2.
| Absorption and emission rate coefficients scale with the cluster surface area: | ||
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| !equation | ||
| \beta_n = \beta_0 n^{1/3}, \qquad \alpha_n = \alpha_0 n^{1/3} |
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We may need to rewrite this since the emission (alpha) and absorption (beta) coefficients are exponentially related by the binding energy of the cluster formation due to detail balance.
Absorption coefficient:
\beta_n = \frac{4\pi (r_1 + r_n) D_1}{V_{\mathrm{at}}},
where r_n is the radius of the cluster:
r_n = \left(\frac{3 n V_{\mathrm{at}}}{4\pi}\right)^{1/3}.
where V_at is the atomic volume (m^3/atom)
Detailed-balance relation between emission and absorption:
\alpha_{n+1} = \beta_n \exp!\left(-\frac{E_b(n+1)}{k_B T}\right).
where binding energy is defined as
E_b(n+1) = G_1 + G_n - G_{n+1}
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we can use the same expression in Bai's paper so the user can define the binding energy based on formation enthalpy and interface energy.
https://doi.org/10.1016/j.jnucmat.2017.08.042
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closes #429