# Thermodynamically Consistent Models for Coupled Bulk and Surface Dynamics

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## Abstract

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## 1. Introduction

- Jing–Wang model [24]:We firstly define the free energy of the system as follows$$\begin{array}{c}\hfill \begin{array}{c}E\left[\varphi \right]={\int}_{\mathrm{\Omega}}{e}_{b}(\varphi ,{\varphi}_{t},\nabla \varphi ,\nabla \nabla \varphi )d\mathbf{x}+{\int}_{\partial \mathrm{\Omega}}\left[{e}_{s}(\varphi ,{\varphi}_{t},{\nabla}_{s}\varphi ,{\nabla}_{s}{\nabla}_{s}\varphi )\right]ds,\hfill \end{array}\end{array}$$$$\begin{array}{c}\hfill \begin{array}{c}{\varphi}_{t}=-({M}_{b}^{\left(1\right)}-\nabla \xb7{\mathbf{M}}_{b}^{\left(2\right)}\xb7\nabla ){\mu}_{b},\phantom{\rule{1.em}{0ex}}\mathbf{x}\in \mathrm{\Omega},\hfill \\ {\varphi}_{t}=-({M}_{s}+\frac{{\beta}^{2}}{\alpha})({\mu}_{s}+{\mu}_{c})+\frac{\beta}{\alpha}{\mu}_{b},\phantom{\rule{1.em}{0ex}}s\in \partial \mathrm{\Omega},\hfill \\ \alpha \mathbf{n}\xb7{\mathbf{M}}_{b}^{\left(2\right)}\xb7\nabla {\mu}_{b}=-{\mu}_{b}+\beta ({\mu}_{s}+{\mu}_{c}),\phantom{\rule{1.em}{0ex}}{\nabla}_{\mathbf{n}}{\varphi}_{t}=-{M}_{g}{\mu}_{g},\phantom{\rule{1.em}{0ex}}s\in \partial \mathrm{\Omega},\hfill \end{array}\end{array}$$
- Knopf–Lam model [28]:We define the free energy as follows$$\begin{array}{c}\hfill \begin{array}{c}E={\int}_{\mathrm{\Omega}}\frac{\u03f5}{2}{|\nabla \varphi |}^{2}+\frac{1}{\u03f5}F\left(\varphi \right)d\mathbf{x}+{\int}_{\partial \mathrm{\Omega}}\frac{\sigma}{2}{\left|{\nabla}_{s}\psi \right|}^{2}+\frac{1}{\sigma}G\left(\psi \right)+\frac{1}{2K}{(H\left(\psi \right)-\varphi )}^{2}ds,\hfill \end{array}\end{array}$$$$\begin{array}{c}\hfill \begin{array}{c}{\varphi}_{t}={\nabla}^{2}{\mu}_{b},\phantom{\rule{1.em}{0ex}}{\mu}_{b}=-\u03f5{\nabla}^{2}\varphi +\frac{1}{\u03f5}{F}^{\prime}\left(\varphi \right),\phantom{\rule{1.em}{0ex}}\mathbf{x}\in \mathrm{\Omega},\hfill \\ {\psi}_{t}={\nabla}_{s}^{2}{\mu}_{s},\phantom{\rule{1.em}{0ex}}{\mu}_{s}=-\sigma {\nabla}_{s}^{2}\psi +\frac{1}{\sigma}{G}^{\prime}\left(\psi \right)+\u03f5{H}^{\prime}\left(\psi \right)\mathbf{n}\xb7\nabla \varphi ,\phantom{\rule{1.em}{0ex}}s\in \partial \mathrm{\Omega},\hfill \\ \mathbf{n}\xb7\nabla {\mu}_{b}=0,\phantom{\rule{1.em}{0ex}}\u03f5\mathbf{n}\xb7\nabla \varphi =\frac{1}{K}(H\left(\psi \right)-\varphi ),\phantom{\rule{1.em}{0ex}}s\in \partial \mathrm{\Omega}.\hfill \end{array}\end{array}$$
- Liu–Wu model [27]:By setting $H\left(\psi \right)=\psi $ and $K\to 0$, $H\left(\psi \right)\to \varphi $ in the Knopf–Lam model, the Liu–Wu model is obtained which are given by$$\begin{array}{c}\hfill \begin{array}{c}{\varphi}_{t}={\nabla}^{2}{\mu}_{b},\phantom{\rule{1.em}{0ex}}{\mu}_{b}=-\u03f5{\nabla}^{2}\varphi +\frac{1}{\u03f5}{F}^{\prime}\left(\varphi \right),\phantom{\rule{1.em}{0ex}}\mathbf{x}\in \mathrm{\Omega},\hfill \\ {\varphi}_{t}={\nabla}_{s}^{2}({\mu}_{c}+{\mu}_{s}),\phantom{\rule{1.em}{0ex}}{\mu}_{c}+{\mu}_{s}=\u03f5\mathbf{n}\xb7\nabla \varphi -\sigma {\nabla}_{s}^{2}\varphi +\frac{1}{\sigma}{G}^{\prime}\left(\varphi \right),\phantom{\rule{1.em}{0ex}}\mathbf{n}\xb7\nabla {\mu}_{b}=0,\phantom{\rule{1.em}{0ex}}s\in \partial \mathrm{\Omega}.\hfill \end{array}\end{array}$$
- Knopf–Signori model [29]:Once the bulk and surface free energies are non-local as in Section 3.3 below, the non-local dynamics are given by$$\begin{array}{c}\hfill \begin{array}{c}{\varphi}_{t}={M}_{b}^{\left(2\right)}{\nabla}^{2}{\mu}_{b},\phantom{\rule{1.em}{0ex}}\mathbf{x}\in \mathrm{\Omega},\hfill \\ {\psi}_{t}=-(-{M}_{s}^{\left(2\right)}{\nabla}^{2}+\frac{{\beta}^{2}}{\alpha}){\mu}_{s}+\frac{\beta}{\alpha}{\mu}_{b},\phantom{\rule{1.em}{0ex}}s\in \partial \mathrm{\Omega},\hfill \\ \alpha {M}_{b}^{\left(2\right)}\mathbf{n}\xb7\nabla {\mu}_{b}=-{\mu}_{b}+\beta {\mu}_{s},\phantom{\rule{1.em}{0ex}}s\in \partial \mathrm{\Omega}.\hfill \end{array}\end{array}$$

## 2. Thermodynamically Consistent Phase Field Models with Consistent Dynamic Boundary Conditions

#### 2.1. Generalized Onsager Principle

#### 2.2. Models with the Free Energy up to Second Spatial Derivatives

**Remark**

**1.**

#### 2.2.1. Dynamics in the Bulk

#### 2.2.2. Dynamics on the Boundary

**Example**

**1**

**Example**

**2**

#### 2.3. Effect of Mobilities in the Bulk and on the Surface

## 3. Reduction to Limiting Cases

#### 3.1. The Jing–Wang Model [24]

#### 3.2. The Knopf–Lam Model [28] and the Liu–Wu Model [27]

#### 3.3. Non-Local Models including the Knopf–Signori Model [29]

#### 3.4. Reactive Transport Equation in a Binary Polymeric System

## 4. Numerical Results for a Binary Reactive System

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Time evolution of the spinodal decomposition of the binary polymer system with static (

**a**–

**c**) and dynamic boundary conditions (

**d**–

**i**), respectively. Snapshots of numerical solution $\varphi $ are taken at $T=0,4,8$, respectively. (

**a**–

**c**). The solution is obtained with homogeneous Neumann boundary conditions on ${\Gamma}_{i},i=1,2,3,4$. (

**d**–

**f**). The solution is obtained with a set of dynamic boundary conditions at ${k}_{s}=0,\eta =100,\alpha =1,\beta =1,\gamma =1$. Dynamic boundary effects on the bulk solution near ${\Gamma}_{1}$ are observed. (

**g**–

**i**). The solution is obtained with a set of dynamic boundary conditions at ${k}_{s}=5\times {10}^{-2},\eta =100,\alpha =1,\beta =1,\gamma =1$. The boundary reactive dynamics shown to impact the bulk solution more significantly near ${\Gamma}_{1}$. The other model parameter values used in the simulations are given as follows: $\beta =\gamma =n={Q}_{1}={Q}_{2}=1,{M}_{b}^{\left(1\right)}={M}_{c}^{\left(1\right)}=0,{M}_{b}^{\left(2\right)}={M}_{c}^{\left(2\right)}={M}_{s}^{\left(2\right)}=1\times {10}^{-4},\chi ={\chi}_{s}=4,{k}_{b}=0,b=0.02$.

**Figure 2.**Time evolution of the bulk volume fraction in (

**a**) and bulk free energy in (

**b**) with respect to the three simulations depicted in Figure 1. It is obvious that the bulk volume fraction with homogeneous Neumann boundary conditions keeps as a constant. However, no matter whether or not there is the reaction on the boundary, the bulk volume fraction increases with time when the boundary conditions are dynamic. Compared with the DBCs without reaction, the bulk volume fraction in the DBCs with reaction increases faster than that without reaction. The bulk free energy in each case decays in time as expected. Due to the existence of chemical reaction, the bulk free energy in the third simulation (with surface reaction) decreases faster than that in the second one (without surface reaction).

**Figure 3.**The effect of the boundary reaction rate and magnitude of the coupling free energy on bulk dynamics in the presence of inward/outward flux ($\alpha =\beta =\gamma =1$). Snapshots of the solution of $\varphi $ are taken $T=8$. (

**a**) Test 1: strong coupling without boundary reaction (${k}_{s}=0,\eta =1\times {10}^{2}$). (

**b**) Test 2: decoupling without boundary reaction (${k}_{s}=\eta =0$). (

**c**) Test 3: strong coupling with boundary reaction (${k}_{s}=5\times {10}^{-2},\eta =1\times {10}^{2}$). (

**d**) Test 4: decoupling with boundary reaction (${k}_{s}=5\times {10}^{-2},\eta =0$). The other model parameter values are the same as those listed in the caption in Figure 1. These four plots show that the non-trivial effect from both the coupling energy and the boundary reaction impacts the bulk dynamics nearby the boundary. In (

**a**,

**c**, and

**d**), there are large merging droplets or lamellar shown near the boundary.

**Figure 4.**Time evolution of the bulk volume fractions in (

**a**) and the total free energy in (

**b**) with respect to the four tests in Figure 2. We find the bulk volume fractions increase with time in tests 1, 3, and 4, which are related to the merging droplets or lamellar in Figure 3a,b,d. Since there is a leakage of bulk volume fraction in test 2, we do not find any large droplets or lamellar in Figure 3b. The total free energy in each test decays in time as expected.

**Figure 5.**The effect of the boundary reaction rate and magnitude of the coupling free energy on bulk dynamics in the presence of inward/outward flux ($\alpha =0.1,\beta =\gamma =10$). Snapshots of the solution of $\varphi $ are taken $T=8$. (

**a**) Test 5: strong coupling without boundary reaction (${k}_{s}=0,\eta =1\times {10}^{2}$). (

**b**) Test 6: decoupling without boundary reaction (${k}_{s}=\eta =0$). (

**c**) Test 7: strong coupling with boundary reaction (${k}_{s}=5\times {10}^{-2},\eta =1\times {10}^{2}$). (

**d**) Test 8: decoupling with boundary reaction (${k}_{s}=5\times {10}^{-2},\eta =0$). The other model parameter values are the same as those listed in the caption in Figure 1. These four plots show both the coupling energy and the boundary reaction impact the bulk dynamics near the boundary.

**Figure 6.**The effect of the boundary reaction rate and magnitude of the coupling free energy on bulk dynamics in the presence of inward/outward flux ($\alpha =10,\beta =\gamma =0.1$). Snapshots of the solution of $\varphi $ are taken $T=8$. (

**a**) Test 9: strong coupling without boundary reaction (${k}_{s}=0,\eta =1\times {10}^{2}$). (

**b**) Test 10: decoupling without boundary reaction (${k}_{s}=\eta =0$). (

**c**) Test 11: strong coupling with boundary reaction (${k}_{s}=5\times {10}^{-2},\eta =1\times {10}^{2}$). (

**d**) Test 12: decoupling with boundary reaction (${k}_{s}=5\times {10}^{-2},\eta =0$). The other model parameter values are the same as those listed in the caption in Figure 1. These four plots show that both the coupling energy and the boundary reaction impact the bulk dynamics near the boundary.

**Figure 7.**Time evolution of the bulk volume fractions in test 5–8 in (

**a**) and test 9–12 in (

**b**). We find the bulk volume fractions increase with time in tests 5–8, which are related to the merging droplets or lamellar in Figure 5. Since there are leakages of bulk volume fractions in tests 9–12, we find there are lamellar formed by the other phase in Figure 6.

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**MDPI and ACS Style**

Jing, X.; Wang, Q.
Thermodynamically Consistent Models for Coupled Bulk and Surface Dynamics. *Entropy* **2022**, *24*, 1683.
https://doi.org/10.3390/e24111683

**AMA Style**

Jing X, Wang Q.
Thermodynamically Consistent Models for Coupled Bulk and Surface Dynamics. *Entropy*. 2022; 24(11):1683.
https://doi.org/10.3390/e24111683

**Chicago/Turabian Style**

Jing, Xiaobo, and Qi Wang.
2022. "Thermodynamically Consistent Models for Coupled Bulk and Surface Dynamics" *Entropy* 24, no. 11: 1683.
https://doi.org/10.3390/e24111683