From 45312651aecceb224f93d0e4fc6827a5e221b668 Mon Sep 17 00:00:00 2001 From: bursik Date: Thu, 19 Jan 2023 14:55:44 +0100 Subject: [PATCH 1/7] initial version, needs additional work --- docs/theory/dft/pDGT_theory_guide.md | 55 ++++++++++++++++++++++++++++ 1 file changed, 55 insertions(+) create mode 100755 docs/theory/dft/pDGT_theory_guide.md diff --git a/docs/theory/dft/pDGT_theory_guide.md b/docs/theory/dft/pDGT_theory_guide.md new file mode 100755 index 000000000..5f0e13d08 --- /dev/null +++ b/docs/theory/dft/pDGT_theory_guide.md @@ -0,0 +1,55 @@ +# Predictive Density Gradient Theory (pDGT) + +Predictive density gradient theory (pDGT) is an efficient approach for the prediction of surface tensions, which is derived from non-local DFT, see [Rehner, 2018](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.98.063312). A gradient expansion is applied to the weighted densities of the Helmholtz energy functional to second order as well as to the Helmholtz energy density to first order. + +Weighted densities (in non-local DFT) are determined from +$$ + n_\alpha(\mathbf{r})=\sum_in_\alpha^i(\mathbf{r})=\sum_i\int\rho_i(\mathbf{r}- \mathbf{r}')\omega_\alpha^i(\mathbf{r}')\diff\mathbf{r}'. +$$ +These convolutions are time-consuming calculations. Therefore, these equations are simplified by using a Taylor expansion around $\rb$ for the density of each component $\rho_i$ as +$$ + \rho_i(\mathbf{r}-\mathbf{r}')=\rho_i(\mathbf{r})-\nabla\rho_i(\mathbf{r})\cdot \mathbf{r}'+\frac{1}{2}\nabla\nabla\rho(\mathbf{r}):\mathbf{r}'\mathbf{r}'+\ldots +$$ + +In the convolution integrals, the integration over angles can now be performed analytically for the spherically symmetric weight functions $\omega_\alpha^i(\mathbf{r})=\omega_\alpha^i(r)$ +which provides +$$ + n_\alpha^i(\mathbf{r})=\rho_i(\mathbf{r})\underbrace{4\pi\int_0^\infty \omega_\alpha^i(r)r^2\diff r}_{\omega_\alpha^{i0}} + +\nabla^2\rho_i(\mathbf{r})\underbrace{\frac{2}{3}\pi\int_0^\infty\omega_\alpha^i(r)r^4\diff r}_{\omega_\alpha^{i2}}+\ldots + +$$ +with the weight constants $\omega_\alpha^{i0}$ and $\omega_\alpha^{i2}$. + +The resulting weighted densities can be split into a local part $n_\alpha^0$ and an excess part $\Delta n_\alpha$ as +$$ + n_\alpha(\mathbf{r})=\underbrace{\sum_i\rho_i(\mathbf{r}) \omega_\alpha^{i0}}_{n_\alpha^0} +\underbrace{\sum_i\nabla^2\rho_i(\mathbf{r})\omega_\alpha^{i2}+\ldots}_{\Delta n_\alpha}. +$$ + + +The second simplification is the expansion of the reduced residual +Helmholtz energy density $\phi(\{ n_\alpha\})$ around the local density approximation truncated after the second term +$$ + \Phi(\lbrace n_\alpha\rbrace) + =\Phi(\lbrace n_\alpha^0\rbrace) + +\sum_i\sum_\alpha\frac{\partial\Phi}{\partial n_\alpha}\omega_\alpha^{i2}\nabla^2\rho_i + \ldots + +$$ +The Helmholtz energy functional (which was introduced in the section about the \href{www.euler-lagrange-equation.de}{Euler-Lagrange equation}) then reads +$$ + F[\bm{\rho}(\mathbf{r})]=\int\left(f(\bm{\rho})+\sum_{ij}\frac{c_{ij}}{2}\nabla\rho_i\cdot\nabla\rho_j\right)\diff\mathbf{r} +$$ +with the density dependent influence parameter +$$ + \beta c_{ij}=-\sum_{\alpha\beta}\frac{\partial^2\Phi}{\partial n_\alpha\partial n_\beta}\left(\omega_\alpha^{i2}\omega_\beta^{j0}+ \omega_\alpha^{i0}\omega_\beta^{j2}\right). + +$$ +and the local Helmholtz energy density $f(\bm{\rho})$. + + + +For pure components the surface tension can be calculated as the surface excess grand potential per area, according to +$$ + \gamma=\frac{F-\mu N+pV}{A}=\int\limits_{-\infty}^\infty \left( \Phi^0 +\frac{c}{2} \rho^2 - \mu \rho + p \right) dz +$$ +Thus, no iterative solver is necessary to calculate the surface tension of pure components, which is a major advantage of pDGT. + From cbc1c5c1d31d9cdcd53a1c7984a87da26c2532d0 Mon Sep 17 00:00:00 2001 From: bursik Date: Thu, 19 Jan 2023 15:02:38 +0100 Subject: [PATCH 2/7] updated version --- docs/theory/dft/{pDGT_theory_guide.md => pdgt.md} | 0 1 file changed, 0 insertions(+), 0 deletions(-) rename docs/theory/dft/{pDGT_theory_guide.md => pdgt.md} (100%) diff --git a/docs/theory/dft/pDGT_theory_guide.md b/docs/theory/dft/pdgt.md similarity index 100% rename from docs/theory/dft/pDGT_theory_guide.md rename to docs/theory/dft/pdgt.md From a339c6207a10835556f18a69918432905fc82820 Mon Sep 17 00:00:00 2001 From: bursik Date: Thu, 19 Jan 2023 15:24:43 +0100 Subject: [PATCH 3/7] changes in index.md and adapt title --- docs/theory/dft/index.md | 1 + docs/theory/dft/pdgt.md | 16 ++++++++-------- 2 files changed, 9 insertions(+), 8 deletions(-) diff --git a/docs/theory/dft/index.md b/docs/theory/dft/index.md index a9fd61284..ab255620e 100644 --- a/docs/theory/dft/index.md +++ b/docs/theory/dft/index.md @@ -8,6 +8,7 @@ This section explains the implementation of the core expressions from classical euler_lagrange_equation functional_derivatives solver + pdgt ``` It is currently still under construction. You can help by [contributing](https://github.com/feos-org/feos/issues/70). \ No newline at end of file diff --git a/docs/theory/dft/pdgt.md b/docs/theory/dft/pdgt.md index 5f0e13d08..8fabb9d8a 100755 --- a/docs/theory/dft/pdgt.md +++ b/docs/theory/dft/pdgt.md @@ -1,12 +1,12 @@ -# Predictive Density Gradient Theory (pDGT) +# Predictive density gradient theory Predictive density gradient theory (pDGT) is an efficient approach for the prediction of surface tensions, which is derived from non-local DFT, see [Rehner, 2018](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.98.063312). A gradient expansion is applied to the weighted densities of the Helmholtz energy functional to second order as well as to the Helmholtz energy density to first order. Weighted densities (in non-local DFT) are determined from $$ - n_\alpha(\mathbf{r})=\sum_in_\alpha^i(\mathbf{r})=\sum_i\int\rho_i(\mathbf{r}- \mathbf{r}')\omega_\alpha^i(\mathbf{r}')\diff\mathbf{r}'. + n_\alpha(\mathbf{r})=\sum_in_\alpha^i(\mathbf{r})=\sum_i\int\rho_i(\mathbf{r}- \mathbf{r}')\omega_\alpha^i(\mathbf{r}')\mathrm{d}\mathbf{r}'. $$ -These convolutions are time-consuming calculations. Therefore, these equations are simplified by using a Taylor expansion around $\rb$ for the density of each component $\rho_i$ as +These convolutions are time-consuming calculations. Therefore, these equations are simplified by using a Taylor expansion around $\mathbf{r}$ for the density of each component $\rho_i$ as $$ \rho_i(\mathbf{r}-\mathbf{r}')=\rho_i(\mathbf{r})-\nabla\rho_i(\mathbf{r})\cdot \mathbf{r}'+\frac{1}{2}\nabla\nabla\rho(\mathbf{r}):\mathbf{r}'\mathbf{r}'+\ldots $$ @@ -14,8 +14,8 @@ $$ In the convolution integrals, the integration over angles can now be performed analytically for the spherically symmetric weight functions $\omega_\alpha^i(\mathbf{r})=\omega_\alpha^i(r)$ which provides $$ - n_\alpha^i(\mathbf{r})=\rho_i(\mathbf{r})\underbrace{4\pi\int_0^\infty \omega_\alpha^i(r)r^2\diff r}_{\omega_\alpha^{i0}} - +\nabla^2\rho_i(\mathbf{r})\underbrace{\frac{2}{3}\pi\int_0^\infty\omega_\alpha^i(r)r^4\diff r}_{\omega_\alpha^{i2}}+\ldots + n_\alpha^i(\mathbf{r})=\rho_i(\mathbf{r})\underbrace{4\pi\int_0^\infty \omega_\alpha^i(r)r^2\mathrm{d} r}_{\omega_\alpha^{i0}} + +\nabla^2\rho_i(\mathbf{r})\underbrace{\frac{2}{3}\pi\int_0^\infty\omega_\alpha^i(r)r^4\mathrm{d} r}_{\omega_\alpha^{i2}}+\ldots $$ with the weight constants $\omega_\alpha^{i0}$ and $\omega_\alpha^{i2}$. @@ -29,14 +29,14 @@ $$ The second simplification is the expansion of the reduced residual Helmholtz energy density $\phi(\{ n_\alpha\})$ around the local density approximation truncated after the second term $$ - \Phi(\lbrace n_\alpha\rbrace) - =\Phi(\lbrace n_\alpha^0\rbrace) + \Phi(\lbrace n_\alpha\mathbf{r}race) + =\Phi(\lbrace n_\alpha^0\mathbf{r}race) +\sum_i\sum_\alpha\frac{\partial\Phi}{\partial n_\alpha}\omega_\alpha^{i2}\nabla^2\rho_i + \ldots $$ The Helmholtz energy functional (which was introduced in the section about the \href{www.euler-lagrange-equation.de}{Euler-Lagrange equation}) then reads $$ - F[\bm{\rho}(\mathbf{r})]=\int\left(f(\bm{\rho})+\sum_{ij}\frac{c_{ij}}{2}\nabla\rho_i\cdot\nabla\rho_j\right)\diff\mathbf{r} + F[\bm{\rho}(\mathbf{r})]=\int\left(f(\bm{\rho})+\sum_{ij}\frac{c_{ij}}{2}\nabla\rho_i\cdot\nabla\rho_j\right)\mathrm{d}\mathbf{r} $$ with the density dependent influence parameter $$ From 7db16bfc438570ba2245dfa94a14920c11375e1d Mon Sep 17 00:00:00 2001 From: bursik Date: Fri, 20 Jan 2023 10:31:13 +0100 Subject: [PATCH 4/7] fixes and changes using sphinx --- docs/theory/dft/pdgt.md | 59 +++++++++++++++++++---------------------- 1 file changed, 27 insertions(+), 32 deletions(-) diff --git a/docs/theory/dft/pdgt.md b/docs/theory/dft/pdgt.md index 8fabb9d8a..b39fc3546 100755 --- a/docs/theory/dft/pdgt.md +++ b/docs/theory/dft/pdgt.md @@ -3,53 +3,48 @@ Predictive density gradient theory (pDGT) is an efficient approach for the prediction of surface tensions, which is derived from non-local DFT, see [Rehner, 2018](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.98.063312). A gradient expansion is applied to the weighted densities of the Helmholtz energy functional to second order as well as to the Helmholtz energy density to first order. Weighted densities (in non-local DFT) are determined from -$$ - n_\alpha(\mathbf{r})=\sum_in_\alpha^i(\mathbf{r})=\sum_i\int\rho_i(\mathbf{r}- \mathbf{r}')\omega_\alpha^i(\mathbf{r}')\mathrm{d}\mathbf{r}'. -$$ + +$$ n_\alpha(\mathbf{r})=\sum_in_\alpha^i(\mathbf{r})=\sum_i\int\rho_i(\mathbf{r}- \mathbf{r}')\omega_\alpha^i(\mathbf{r}')\mathrm{d}\mathbf{r}'.$$ + These convolutions are time-consuming calculations. Therefore, these equations are simplified by using a Taylor expansion around $\mathbf{r}$ for the density of each component $\rho_i$ as -$$ - \rho_i(\mathbf{r}-\mathbf{r}')=\rho_i(\mathbf{r})-\nabla\rho_i(\mathbf{r})\cdot \mathbf{r}'+\frac{1}{2}\nabla\nabla\rho(\mathbf{r}):\mathbf{r}'\mathbf{r}'+\ldots -$$ + +$$\rho_i(\mathbf{r}-\mathbf{r}')=\rho_i(\mathbf{r})-\nabla\rho_i(\mathbf{r})\cdot \mathbf{r}'+\frac{1}{2}\nabla\nabla\rho(\mathbf{r}):\mathbf{r}'\mathbf{r}'+\ldots$$ In the convolution integrals, the integration over angles can now be performed analytically for the spherically symmetric weight functions $\omega_\alpha^i(\mathbf{r})=\omega_\alpha^i(r)$ which provides -$$ - n_\alpha^i(\mathbf{r})=\rho_i(\mathbf{r})\underbrace{4\pi\int_0^\infty \omega_\alpha^i(r)r^2\mathrm{d} r}_{\omega_\alpha^{i0}} - +\nabla^2\rho_i(\mathbf{r})\underbrace{\frac{2}{3}\pi\int_0^\infty\omega_\alpha^i(r)r^4\mathrm{d} r}_{\omega_\alpha^{i2}}+\ldots - -$$ + +$$ n_\alpha^i(\mathbf{r})=\rho_i(\mathbf{r})\underbrace{4\pi\int_0^\infty \omega_\alpha^i(r)r^2\mathrm{d} r}_{\omega_\alpha^{i0}} + +\nabla^2\rho_i(\mathbf{r})\underbrace{\frac{2}{3}\pi\int_0^\infty\omega_\alpha^i(r)r^4\mathrm{d} r}_{\omega_\alpha^{i2}}+\ldots$$ + with the weight constants $\omega_\alpha^{i0}$ and $\omega_\alpha^{i2}$. The resulting weighted densities can be split into a local part $n_\alpha^0$ and an excess part $\Delta n_\alpha$ as -$$ - n_\alpha(\mathbf{r})=\underbrace{\sum_i\rho_i(\mathbf{r}) \omega_\alpha^{i0}}_{n_\alpha^0} +\underbrace{\sum_i\nabla^2\rho_i(\mathbf{r})\omega_\alpha^{i2}+\ldots}_{\Delta n_\alpha}. -$$ + +$$n_\alpha(\mathbf{r})=\underbrace{\sum_i\rho_i(\mathbf{r}) \omega_\alpha^{i0}}_{n_\alpha^0} +\underbrace{\sum_i\nabla^2\rho_i(\mathbf{r})\omega_\alpha^{i2}+\ldots}_{\Delta n_\alpha}.$$ The second simplification is the expansion of the reduced residual Helmholtz energy density $\phi(\{ n_\alpha\})$ around the local density approximation truncated after the second term -$$ - \Phi(\lbrace n_\alpha\mathbf{r}race) - =\Phi(\lbrace n_\alpha^0\mathbf{r}race) - +\sum_i\sum_\alpha\frac{\partial\Phi}{\partial n_\alpha}\omega_\alpha^{i2}\nabla^2\rho_i + \ldots - -$$ -The Helmholtz energy functional (which was introduced in the section about the \href{www.euler-lagrange-equation.de}{Euler-Lagrange equation}) then reads -$$ - F[\bm{\rho}(\mathbf{r})]=\int\left(f(\bm{\rho})+\sum_{ij}\frac{c_{ij}}{2}\nabla\rho_i\cdot\nabla\rho_j\right)\mathrm{d}\mathbf{r} -$$ + +$$ \Phi(\lbrace n_\alpha(\mathbf{r})\rbrace) + =\Phi(\lbrace n_\alpha^0(\mathbf{r})\rbrace) + +\sum_i\sum_\alpha\frac{\partial\Phi}{\partial n_\alpha}\omega_\alpha^{i2}\nabla^2\rho_i + \ldots $$ + +The Helmholtz energy functional (which was introduced in the section about the [Euler-Lagrange equation](euler_lagrange_equation.md)) then reads + +$$ F[\mathbf{\rho}(\mathbf{r})]=\int\left(f(\mathbf{\rho})+\sum_{ij}\frac{c_{ij}}{2}\nabla\rho_i\cdot\nabla\rho_j\right)\mathrm{d}\mathbf{r}$$ + with the density dependent influence parameter -$$ - \beta c_{ij}=-\sum_{\alpha\beta}\frac{\partial^2\Phi}{\partial n_\alpha\partial n_\beta}\left(\omega_\alpha^{i2}\omega_\beta^{j0}+ \omega_\alpha^{i0}\omega_\beta^{j2}\right). - -$$ -and the local Helmholtz energy density $f(\bm{\rho})$. + +$$ \beta c_{ij}=-\sum_{\alpha\beta}\frac{\partial^2\Phi}{\partial n_\alpha\partial n_\beta}\left(\omega_\alpha^{i2}\omega_\beta^{j0}+ \omega_\alpha^{i0}\omega_\beta^{j2}\right).$$ + +and the local Helmholtz energy density $f(\mathbf{\rho})$. For pure components the surface tension can be calculated as the surface excess grand potential per area, according to -$$ - \gamma=\frac{F-\mu N+pV}{A}=\int\limits_{-\infty}^\infty \left( \Phi^0 +\frac{c}{2} \rho^2 - \mu \rho + p \right) dz -$$ + +$$ \gamma=\frac{F-\mu N+pV}{A}=\int\limits_{-\infty}^\infty \left( \Phi^0 +\frac{c}{2} \rho^2 - \mu \rho + p \right) dz $$ + Thus, no iterative solver is necessary to calculate the surface tension of pure components, which is a major advantage of pDGT. From e00fab4032891301f1400f014213d69efb212f9e Mon Sep 17 00:00:00 2001 From: bursik Date: Mon, 23 Jan 2023 15:50:34 +0100 Subject: [PATCH 5/7] more consistency and other equation for gamma --- docs/theory/dft/pdgt.md | 23 +++++++++++++---------- 1 file changed, 13 insertions(+), 10 deletions(-) diff --git a/docs/theory/dft/pdgt.md b/docs/theory/dft/pdgt.md index b39fc3546..868e907ef 100755 --- a/docs/theory/dft/pdgt.md +++ b/docs/theory/dft/pdgt.md @@ -1,6 +1,6 @@ # Predictive density gradient theory -Predictive density gradient theory (pDGT) is an efficient approach for the prediction of surface tensions, which is derived from non-local DFT, see [Rehner, 2018](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.98.063312). A gradient expansion is applied to the weighted densities of the Helmholtz energy functional to second order as well as to the Helmholtz energy density to first order. +Predictive density gradient theory (pDGT) is an efficient approach for the prediction of surface tensions, which is derived from non-local DFT, see [Rehner et al. (2018)](https://journals.aps.org/pre/abstract/10.1103/PhysRevE.98.063312). A gradient expansion is applied to the weighted densities of the Helmholtz energy functional to second order as well as to the Helmholtz energy density to first order. Weighted densities (in non-local DFT) are determined from @@ -18,33 +18,36 @@ $$ n_\alpha^i(\mathbf{r})=\rho_i(\mathbf{r})\underbrace{4\pi\int_0^\infty \omega with the weight constants $\omega_\alpha^{i0}$ and $\omega_\alpha^{i2}$. -The resulting weighted densities can be split into a local part $n_\alpha^0$ and an excess part $\Delta n_\alpha$ as +The resulting weighted densities can be split into a local part $n_\alpha^0(\mathbf{r})$ and an excess part $\Delta n_\alpha(\mathbf{r})$ as $$n_\alpha(\mathbf{r})=\underbrace{\sum_i\rho_i(\mathbf{r}) \omega_\alpha^{i0}}_{n_\alpha^0} +\underbrace{\sum_i\nabla^2\rho_i(\mathbf{r})\omega_\alpha^{i2}+\ldots}_{\Delta n_\alpha}.$$ The second simplification is the expansion of the reduced residual -Helmholtz energy density $\phi(\{ n_\alpha\})$ around the local density approximation truncated after the second term +Helmholtz energy density $\Phi(\{ n_\alpha\})$ around the local density approximation truncated after the second term -$$ \Phi(\lbrace n_\alpha(\mathbf{r})\rbrace) - =\Phi(\lbrace n_\alpha^0(\mathbf{r})\rbrace) +$$ \Phi(\lbrace n_\alpha\rbrace) + =\Phi(\lbrace n_\alpha^0\rbrace) +\sum_i\sum_\alpha\frac{\partial\Phi}{\partial n_\alpha}\omega_\alpha^{i2}\nabla^2\rho_i + \ldots $$ The Helmholtz energy functional (which was introduced in the section about the [Euler-Lagrange equation](euler_lagrange_equation.md)) then reads -$$ F[\mathbf{\rho}(\mathbf{r})]=\int\left(f(\mathbf{\rho})+\sum_{ij}\frac{c_{ij}}{2}\nabla\rho_i\cdot\nabla\rho_j\right)\mathrm{d}\mathbf{r}$$ +$$ F[\mathbf{\rho}(\mathbf{r})]=\int\left(f(\mathbf{\rho})+\sum_{ij}\frac{c_{ij}(\rho)}{2}\nabla\rho_i\cdot\nabla\rho_j\right)\mathrm{d}\mathbf{r}$$ with the density dependent influence parameter -$$ \beta c_{ij}=-\sum_{\alpha\beta}\frac{\partial^2\Phi}{\partial n_\alpha\partial n_\beta}\left(\omega_\alpha^{i2}\omega_\beta^{j0}+ \omega_\alpha^{i0}\omega_\beta^{j2}\right).$$ +$$ \beta c_{ij}(\rho)=-\sum_{\alpha\beta}\frac{\partial^2\Phi}{\partial n_\alpha\partial n_\beta}\left(\omega_\alpha^{i2}\omega_\beta^{j0}+ \omega_\alpha^{i0}\omega_\beta^{j2}\right).$$ and the local Helmholtz energy density $f(\mathbf{\rho})$. -For pure components the surface tension can be calculated as the surface excess grand potential per area, according to +For pure components, as derived in the original publication, the surface tension can be calculated from the surface excess grand potential per area according to -$$ \gamma=\frac{F-\mu N+pV}{A}=\int\limits_{-\infty}^\infty \left( \Phi^0 +\frac{c}{2} \rho^2 - \mu \rho + p \right) dz $$ +$$ \gamma=\frac{F-\mu N+pV}{A}=\int_{\rho^\mathrm{V}}^{\rho^\mathrm{L}} \sqrt{\frac{c}{2} \left(f(\rho)-\rho\mu+p\right) } d\rho $$ -Thus, no iterative solver is necessary to calculate the surface tension of pure components, which is a major advantage of pDGT. + +Thus, no iterative solver is necessary to calculate the surface tension of pure components, which is a major advantage of pDGT. Finally, the density profile can be calculated from + +$$ z=\int_{\rho^\mathrm{V}}^{\rho^\mathrm{L}} \sqrt{\frac{c/2}{ f(\rho)-\rho\mu+p} } d\rho $$ From 37d4a6dd6102fb7f2ad689610bd27793345d621c Mon Sep 17 00:00:00 2001 From: bursik Date: Mon, 23 Jan 2023 16:30:31 +0100 Subject: [PATCH 6/7] rho fat and 2*c --- docs/theory/dft/pdgt.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/docs/theory/dft/pdgt.md b/docs/theory/dft/pdgt.md index 868e907ef..591018019 100755 --- a/docs/theory/dft/pdgt.md +++ b/docs/theory/dft/pdgt.md @@ -32,11 +32,11 @@ $$ \Phi(\lbrace n_\alpha\rbrace) The Helmholtz energy functional (which was introduced in the section about the [Euler-Lagrange equation](euler_lagrange_equation.md)) then reads -$$ F[\mathbf{\rho}(\mathbf{r})]=\int\left(f(\mathbf{\rho})+\sum_{ij}\frac{c_{ij}(\rho)}{2}\nabla\rho_i\cdot\nabla\rho_j\right)\mathrm{d}\mathbf{r}$$ +$$ F[\mathbf{\rho}(\mathbf{r})]=\int\left(f(\mathbf{\rho})+\sum_{ij}\frac{c_{ij}(\mathbf{\rho})}{2}\nabla\rho_i\cdot\nabla\rho_j\right)\mathrm{d}\mathbf{r}$$ with the density dependent influence parameter -$$ \beta c_{ij}(\rho)=-\sum_{\alpha\beta}\frac{\partial^2\Phi}{\partial n_\alpha\partial n_\beta}\left(\omega_\alpha^{i2}\omega_\beta^{j0}+ \omega_\alpha^{i0}\omega_\beta^{j2}\right).$$ +$$ \beta c_{ij}(\mathbf{\rho})=-\sum_{\alpha\beta}\frac{\partial^2\Phi}{\partial n_\alpha\partial n_\beta}\left(\omega_\alpha^{i2}\omega_\beta^{j0}+ \omega_\alpha^{i0}\omega_\beta^{j2}\right).$$ and the local Helmholtz energy density $f(\mathbf{\rho})$. @@ -44,7 +44,7 @@ and the local Helmholtz energy density $f(\mathbf{\rho})$. For pure components, as derived in the original publication, the surface tension can be calculated from the surface excess grand potential per area according to -$$ \gamma=\frac{F-\mu N+pV}{A}=\int_{\rho^\mathrm{V}}^{\rho^\mathrm{L}} \sqrt{\frac{c}{2} \left(f(\rho)-\rho\mu+p\right) } d\rho $$ +$$ \gamma=\frac{F-\mu N+pV}{A}=\int_{\rho^\mathrm{V}}^{\rho^\mathrm{L}} \sqrt{2c \left(f(\rho)-\rho\mu+p\right) } d\rho $$ Thus, no iterative solver is necessary to calculate the surface tension of pure components, which is a major advantage of pDGT. Finally, the density profile can be calculated from From bb5b4a4b9ba62b9c9fdb0620de5464d718e02dbd Mon Sep 17 00:00:00 2001 From: bursik Date: Mon, 23 Jan 2023 16:33:14 +0100 Subject: [PATCH 7/7] integral for z new bounds --- docs/theory/dft/pdgt.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/theory/dft/pdgt.md b/docs/theory/dft/pdgt.md index 591018019..0113234e9 100755 --- a/docs/theory/dft/pdgt.md +++ b/docs/theory/dft/pdgt.md @@ -49,5 +49,5 @@ $$ \gamma=\frac{F-\mu N+pV}{A}=\int_{\rho^\mathrm{V}}^{\rho^\mathrm{L}} \sqrt{ Thus, no iterative solver is necessary to calculate the surface tension of pure components, which is a major advantage of pDGT. Finally, the density profile can be calculated from -$$ z=\int_{\rho^\mathrm{V}}^{\rho^\mathrm{L}} \sqrt{\frac{c/2}{ f(\rho)-\rho\mu+p} } d\rho $$ +$$ z(\rho)=\int_{\rho^\mathrm{V}}^{\rho} \sqrt{\frac{c/2}{ f(\rho)-\rho\mu+p} } d\rho $$