{"cells":[{"cell_type":"markdown","metadata":{},"source":["# Droplets\n","\n","## Different morphologies of the droplet state\n","\n","Let us consider the equilibrium droplet state, _i.e._, $\\alpha<0$ and $-\\sqrt{\\frac{-\\alpha}{\\beta}}<\\phi_0<\\sqrt{\\frac{-\\alpha}{\\beta}}$, at $t\\rightarrow\\infty$.\n","For a fixed value of $\\alpha$, depending on the global average density $\\phi_0$, we can have 3 different types of droplet state.\n","For $\\phi_0<0$, we have a liquid droplet surrounded by vapour phase.\n","For $\\phi_0>0$, we have a vapour bubble surrounded by liquid phase.\n","Finally for $\\phi_0=0$, the liquid droplet forms a strip, as we can see from the figure below.\n","Note that periodic boundary condition is assumed in both $x$ and $y$-direction."]},{"cell_type":"code","execution_count":210,"metadata":{"tags":[],"trusted":false},"outputs":[{"data":{"image/png":"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","text/plain":["<IPython.core.display.Image object>"]},"execution_count":210,"metadata":{},"output_type":"execute_result"}],"source":["from IPython.display import Image\n","Image('https://raw.githubusercontent.com/elsentjhung/cahn-hilliard-droplet/master/figures/droplet-morphologies.png')"]},{"cell_type":"markdown","metadata":{},"source":["## Flat interface ($d=1$ dimension)\n","\n","Let us consider the strip geometry ($\\phi_0=0$).\n","As we can see from the middle picture above, the interface between the liquid and the gas phase is completely flat, reducing the problem to a one-dimensional problem $\\phi(x,t)$.\n","In the steady (or equilibrium) state $t\\rightarrow\\infty$, the density profile might look something like in the plot below:"]},{"cell_type":"code","execution_count":208,"metadata":{"trusted":false},"outputs":[{"data":{"image/png":"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","text/plain":["<IPython.core.display.Image object>"]},"execution_count":208,"metadata":{},"output_type":"execute_result"}],"source":["Image('https://raw.githubusercontent.com/elsentjhung/cahn-hilliard-droplet/master/figures/interfacial-profile.png')"]},{"cell_type":"markdown","metadata":{},"source":["The plot above shows the typical steady state density field $\\phi(x)$ in an infinite one-dimensional system.\n","At $x=-\\infty$, we have the gas phase, where $\\phi=-\\sqrt{\\frac{-\\alpha}{\\beta}}$ and at $x=+\\infty$, we have the liquid phase, where $\\phi=\\sqrt{\\frac{-\\alpha}{\\beta}}$.\n","We then have an interface, assumed to be located at the origin $x=0$, which separates the liquid from the gas phase.\n","As we can see, the interface is not sharp, but rather is spread across some interfacial width $\\xi$.\n","In real physical situations, the interfacial width is typically a few molecular lengths ($\\sim\\text{nm}$), however in simulations, we usually use a much wider interfacial width for numerical stability.\n","This is fine as long as the lengthscales of the problem we are interested in are much larger than $\\xi$.\n","\n","### Interfacial profile $\\phi(x)$ and interfacial width $\\xi$\n","\n","In this subsection, we will derive the interfacial profile $\\phi(x)$ and the interfacial width $\\xi$ for a flat interface, which is effectively a one-dimensional (1d) system.\n","\n","In 1d, the coarse-grained free energy can be written as:\n","\\begin{equation}\n","\\mathcal{F}[\\phi] = \n","A\\int_{-\\infty}^\\infty dx  \n","\\underbrace{\\bigg\\{ \\frac{\\alpha}{2}\\phi^2 + \\frac{\\beta}{4}\\phi^4 + \\frac{\\kappa}{2}|\\nabla\\phi|^2 \\bigg\\}}_{\\text{1d energy density}},\n","\\end{equation}\n","where $A$ is the area of the system along $y$ and $z$ direction.\n","(In 1d, the system is translationally invariant along $y$ and $z$.)\n","The 1d energy density consists of the local term: $\\frac{\\alpha}{2}\\phi^2 + \\frac{\\beta}{4}\\phi^4=f(\\phi)$, which we will call $f(\\phi)$, and the semi-local term $\\frac{\\kappa}{2}|\\nabla\\phi|^2$.\n","\n","The steady state density, is given by the solution to the equation:\n","\\begin{equation}\n","\\mu(x) = \\frac{\\delta\\mathcal{F}}{\\delta\\phi} = 0 \\quad\\Rightarrow\\quad\n","\\frac{df}{d\\phi} - \\kappa\\frac{d^2\\phi}{dx^2} = 0,\n","\\end{equation}\n","where $f(\\phi) = \\frac{\\alpha}{2}\\phi^2 + \\frac{\\beta}{4}\\phi^4$ is the local energy density.\n","To solve the above equation, we multiply both sides of the equation by $d\\phi/dx$ to get:\n","\\begin{align}\n","\\frac{df}{dx} - \\kappa\\frac{d^2\\phi}{dx^2}\\frac{d\\phi}{dx} = 0 \n","\\quad\\Rightarrow\\quad \n","\\frac{df}{dx} - \\frac{\\kappa}{2}\\frac{d}{dx}\\left(\\frac{d\\phi}{dx}\\right)^2 = 0\n","\\end{align}\n","Now we can integrate with respect to $x$ to get the __Noether equation__:\n","\\begin{equation}\n","f(\\phi) - \\frac{\\kappa}{2}\\left(\\frac{d\\phi}{dx}\\right)^2 = C, \n","\\end{equation}\n","where $C$ is the constant of integration. \n","The above equation can also be derived from Noether's theorem and the constant $C$ is a Noether current.\n","The constant of integration can be found by substituting $x=\\infty$ to get \n","$C=f\\left(\\sqrt{\\frac{\\alpha}{\\beta}}\\right)=-\\frac{\\alpha^2}{4\\beta}$.\n","After rearranging, we get:\n","\\begin{align}\n","\\frac{d\\phi}{\\sqrt{f(\\phi) + \\frac{\\alpha^2}{4\\beta}}} = \\sqrt{\\frac{2}{\\kappa}} dx,\n","\\end{align}\n","which we can integrate from $x=0$ to $x$.\n","\\begin{align}\n","\\int_{0}^{\\phi(x)} \\frac{d\\phi}{\\sqrt{f(\\phi) + \\frac{\\alpha^2}{4\\beta}}} \n","= \\sqrt{\\frac{2}{\\kappa}}x\n","\\end{align}\n","Note that we can factorize:\n","\\begin{align}\n","\\underbrace{\\frac{\\alpha}{2}\\phi^2 + \\frac{\\beta}{2}\\phi^4}_{f(\\phi)} + \\frac{\\alpha^2}{4\\beta} \n","= \\frac{1}{4\\beta}\\left(\\alpha+\\beta\\phi^2\\right)^2,\n","\\end{align}\n","so that the integral becomes:\n","\\begin{align}\n","\\int_{0}^{\\phi(x)} \\frac{d\\phi}{\\alpha + \\beta\\phi^2} = \\sqrt{\\frac{1}{2\\kappa\\beta}}x \\quad\\Rightarrow\\quad\n","\\frac{1}{\\alpha}\\int_{0}^{\\phi(x)} \\frac{d\\phi}{1 - \\frac{\\beta}{-\\alpha}\\phi^2} = \\sqrt{\\frac{1}{2\\kappa\\beta}}x\n","\\end{align}\n","Note that $\\alpha<0$. \n","Using $\\frac{d}{dx}\\tanh^{-1}(x)=\\frac{1}{1-x^2}$, we can then solve this integral to get:\n","\\begin{equation}\n","\\phi(x) = \\sqrt{\\frac{-\\alpha}{\\beta}}\\tanh\\left(\\frac{x}{\\xi}\\right), \\quad\\text{where}\\quad\n","\\xi = \\sqrt{\\frac{2\\kappa}{-\\alpha}}\n","\\end{equation}\n","is the interfacial width.\n","\n","### Surface tension $\\gamma$\n","\n","In this subsection, we will introduce the concept of surface tension $\\gamma$ and derive its expression for a flat interface.\n","First, let us consider the local free energy density $f(\\phi)+\\frac{\\kappa}{2}\\left(\\frac{d\\phi}{dx}\\right)^2$, which consists of the local and semi-local terms."]},{"cell_type":"code","execution_count":193,"metadata":{"trusted":false},"outputs":[{"data":{"image/png":"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","text/plain":["<IPython.core.display.Image object>"]},"execution_count":193,"metadata":{},"output_type":"execute_result"}],"source":["Image('https://raw.githubusercontent.com/elsentjhung/cahn-hilliard-droplet/master/figures/interfacial-energy.png')"]},{"cell_type":"markdown","metadata":{},"source":["The plot above shows the free energy density $f(\\phi(x))+\\frac{\\kappa}{2}\\left(\\frac{d\\phi}{dx}\\right)^2$ as a function of $x$ for the density profile $\\phi(x)$, shown in the previous plot.\n","In the bulk, the energy density is mostly flat and negative, which is equal to $f\\left(\\sqrt{\\frac{-\\alpha}{\\beta}}\\right)=-\\frac{\\alpha^2}{4\\beta}$.\n","Across the interface, the energy density goes up and then goes down, back to its bulk value.\n","The interfacial energy is defined to be excess free energy across this interface, _i.e._, the shaded region in the plot above.\n","More specifically, the interfacial energy is defined to be:\n","\\begin{equation}\n","\\mathcal{F}_{\\text{interface}} = A\\int_{-\\infty}^{\\infty}\n","\\left\\{f(\\phi(x)) + \\frac{\\kappa}{2}\\left(\\frac{d\\phi}{dx}\\right)^2 + \\frac{\\alpha^2}{4\\beta} \\right\\}dx.\n","\\end{equation}\n","Now we can refer to the __Noether equation__ which we derived above, _i.e._:\n","\\begin{equation}\n","f(\\phi) - \\frac{\\kappa}{2}\\left(\\frac{d\\phi}{dx}\\right)^2 = C = -\\frac{\\alpha^2}{4\\beta}\n","\\end{equation}\n","Substituting this to $\\mathcal{F}_{\\text{interface}}$, the interfacial energy becomes:\n","\\begin{equation}\n","\\mathcal{F}_{\\text{interface}} = A\\kappa\\int_{-\\infty}^{\\infty} \\left(\\frac{d\\phi}{dx}\\right)^2 dx.\n","\\end{equation}\n","Finally the surface tension $\\gamma$ is defined to be the interfacial energy per interfacial area. Thus we get:\n","\\begin{equation}\n","\\gamma = \\kappa\\int_{-\\infty}^{\\infty} \\left(\\frac{d\\phi}{dx}\\right)^2 dx = \\kappa\\int_{-\\sqrt{-\\alpha/\\beta}}^{\\sqrt{-\\alpha/\\beta}} \\frac{d\\phi}{dx} \\, d\\phi\n","\\end{equation}\n","Using Noether's equation, we can substitute $\\frac{d\\phi}{dx}$: \n","\\begin{equation}\n","\\frac{d\\phi}{dx} = \\sqrt{\\frac{2}{\\kappa}} \\sqrt{ f(\\phi) - f\\left(\\sqrt{\\frac{-\\alpha}{\\beta}}\\right) }\n","= \\sqrt{\\frac{2}{\\kappa}} \\sqrt{ f(\\phi) + \\frac{\\alpha^2}{4\\beta} },\n","\\end{equation}\n","so the surface tension becomes:\n","\\begin{equation}\n","\\gamma = \\sqrt{2\\kappa} \\int_{-\\sqrt{-\\alpha/\\beta}}^{\\sqrt{-\\alpha/\\beta}} \\sqrt{f(\\phi) + \\frac{\\alpha^2}{4\\beta}}\\,d\\phi\n","\\end{equation}\n","Substituting $f(\\phi)=\\frac{\\alpha}{2}\\phi^2 + \\frac{\\beta}{4}\\phi^4$, we can solve the above integral to get:\n","\\begin{equation}\n","\\gamma = \\sqrt{\\frac{-8\\kappa\\alpha^3}{9\\beta^2}}.\n","\\end{equation}\n","Note that $\\alpha<0$.\n","\n","## Curved interface ($d>1$ dimension)\n","\n","We will now go back to $d>1$ dimensional problems.\n","In general, the interface between the liquid and the gas phase is not always flat but often curved.\n","\n","### Microscopic picture of surface tension\n","\n","Let us consider a liquid water droplet. \n","Inside the bulk liquid, each water molecule has $z$ nearest neighbours. \n","However, the water molecules at the liquid/gas interface only have $z/2$ nearest neighbours. \n","That means to create a new interface, some bonds have to be broken.\n","The energy required to break these bonds (per unit interfacial area) is called the surface tension $\\gamma$. \n","\n","### Macroscopic picture of surface tension\n","\n","Let us consider a liquid droplet in three-dimension and far from the boundaries (such as solid walls).\n","The total free energy of the liquid droplet has two contributions:\n","\\begin{equation}\n","\\mathcal{F} = \\mathcal{F}_{\\text{bulk}} + \\mathcal{F}_{\\text{interface}}.\n","\\end{equation}\n","The first contribution comes from the bulk free energy $\\mathcal{F}_{\\text{bulk}}$, which is equal to the local free energy density $f(\\phi)$ multiplied by the volume of the droplet $V$:\n","\\begin{equation}\n","\\mathcal{F}_{\\text{bulk}} \\simeq f(\\phi_{l}) V = \\text{constant}.\n","\\end{equation}\n","Note that we have assumed the density of the liquid inside the droplet, $\\phi_{l}$, to be constant throughout the bulk of the droplet.\n","Consequently, the volume of the liquid droplet is also constant by conservation of mass.\n","Thus $\\mathcal{F}_{\\text{bulk}}$ is constant.\n","The second contribution to $\\mathcal{F}$ comes from the interfacial energy $\\mathcal{F}_{\\text{interface}}$, \n","which is equal to the surface tension $\\gamma$ multiplied by the surface area of the droplet $S$:\n","\\begin{equation}\n","\\mathcal{F}_{\\text{interface}} \\simeq \\gamma S.\n","\\end{equation}\n","Since $\\gamma>0$, the equilibrium shape of the liquid droplet, which minimizes the total free energy $\\mathcal{F}$, is therefore a sphere.\n","\n","The surface tension $\\gamma$ can also be thought as a force per unit length.\n","To illustrate this, let us consider a thin film of soap, as shown in the picture below.\n","Technically, a soap film is an air-liquid-air interface, so the surface tension would be doubled.\n","Let us imagine that this two-dimensional soap film is confined between three fixed rods (black colour in the picture below) and a movable rod which can move laterally along $x$ (red colour in the picture below).\n","The energy stored inside this soap film is then:\n","\\begin{equation}\n","\\mathcal{F}_{\\text{interface}} = \\gamma S = \\gamma \\ell x.\n","\\end{equation}\n","Therefore the force by the soap film on the red rod is equal to:\n","\\begin{equation}\n","F_{\\text{interface}} = -\\frac{d\\mathcal{F}_{\\text{interface}}}{dx} = -\\gamma \\ell.\n","\\end{equation}\n","$F_{\\text{interface}}$ is negative since the direction is to the left in the picture below.\n","Thus a surface tension $\\gamma$ is a force (per unit length) which tends to minimize the surface area of the interface."]},{"cell_type":"code","execution_count":192,"metadata":{"trusted":false},"outputs":[{"data":{"image/png":"iVBORw0KGgoAAAANSUhEUgAAASEAAADrCAYAAAAmGSB9AAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjQuMywgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/MnkTPAAAACXBIWXMAAAsTAAALEwEAmpwYAAAPqUlEQVR4nO3dfXRU9Z3H8c9NJmTCU0xICJBVERqEQJBKqA+0iHULy1KRIkJdRaBVUbYgItiiHBSOPHRJV4WqcFyOeLClGMUICBaP2MpuqWtyKCQSskBDErA8BIRASEkmc/ePwCg2gUkkfGcm79c5OTJzf3PvN3p8596E3HFc1xUAWImyHgBAy0aEAJgiQgBMESEApogQAFNECIApT2MWJyUluV27dm2mUQBEqry8vHLXdZPr29aoCHXt2lW5ubmXZyoALYbjOCUNbeNyDIApIgTAFBECYIoIATBFhACYIkIATBEhAKaIEABTRAiAKSIEwBQRAmCKCAEwRYQAmCJCAEwRIQCmiBAAU0QIgCkiBMAUEQJgiggBMEWEAJgiQgBMESEApogQAFNECIApIgTAFBECYIoIATBFhACYIkIATBEhAKaIEABTRAiAKSIEwBQRQljLzs5WbGysSkpKAs899thj6t69uw4fPmw4GYJFhBDWRo8erYyMDD333HOSpKysLK1evVrvv/++UlJSjKdDMDzWAwDfhOM4WrBggYYPH67u3btr/vz52rJli9LS0qxHQ5CIEMLekCFDNGDAAM2ePVvr16/XgAEDrEdCI3A5hpBUUVEh13WDWrtlyxbt2LFDrutyCRaGiBBC0qRJk3TTTTdp48aNF43Rjh07NGrUKC1dulQjR47UrFmzruCUuBycYL/aSFJmZqabm5vbjOPY8fl82rdvn/UYOMfv92vz5s166aWX5PV6NX36dE2YMOGCNSUlJbr11ls1adIkzZkzRwUFBerbt6+2bNmiwYMHm8yN+jmOk+e6bma924hQnY8++kiPPPKI9Rj4murqah08eFB+v19lZWXq3LmzJOn48eMaOHCgBg0apOXLlwfWjx07VqWlpdq2bZvVyKgHEULY2bt3r+bNm6dNmzZp6tSpmjp1quLj463HQhNdLEL8dAwhacmSJUpLS9PSpUuJT4QjQghJS5YssR4BVwg/HQNgiggBMEWEAJgiQgBMESEApogQAFMt/kf0d9xxhyorK63HAAI2btyoxMRE6zGumBYfodzcXFVUVFiPAQT4fD7rEa6oFh+h8ePH68CBY5o4cYr1KAjSP+X+UW6/Hqo+93tkkSYhIcF6hCsqIn93zOfz6dixY0HdWyYhIUEnTpxQUVGF2rZtdwWmwzfVOa2NnGnTpPnzrUdBkC72u2MR+Y3pyZMn6/rrr5ff77/k2vNrglmLEFFbW/eBiBBxEaqsrFR2drZqamq0du1a63EAXELEReiVV15Renq6MjIyNHfuXM5wgBAXURGqrKxUVlaWxowZo5SUFHm9Xs6GgBAXUREqKSnRuHHjdO2118pxHC1evFjFxcXWYwG4iIiKUHp6uhYvXhx4PHjwYM2cOdNwIgCXElERAhB+iBAAU0QIgCkiBMAUEQJgiggBMEWEAJgiQgBMESEApogQAFNECIApIgTAFBECYIoIATBFhACYIkIATBEhAKaIEABTRAiAKSIEwBQRAmCKCAEwRYQAmCJCAEx5rAfAldHhnu8r9k8f/cPzh/63VP7Uqw0mAuoQoRYi5rPtqvj5fJ358U8ueN7fsZPRREAdItQCRO/fp6iTJ1R9y21EByGH7wm1ADE78+RGR6umz7etRwH+ARFqAWLy8+TU1iqlb0d1SmurTmltlXx778D2Vv/zkdosy2rw9dFl++X9/btNOnb8jAeV/IN+avufc5v0ekQ+LsdagFY781Q1fLQqZi0MPOd64wJ/rh54u6oH3t7g62P/8HtFHS+Xht4V/EFra+XZu1ue0mId/eAvTRkbLQQRagFiCrbr1ONzVHvdt+rdnjj+TlX8YoHaL3pKNX2+rdiPP1D052U6vnKdnNOn1H7RLPmv6qC4Ddkqf/uPiio/ovhnH1fU0UNyvXH6Ynm2/B07KXH8narpka7YbX/QmdEPqN3SBZIcJf3wZpVv+LPi3vmt2qx4UU7VGdWmdNEXK96RG9daUQfLFD/7Z/IcLJVqqnX89Q2Sz1fvMRB5uByLcNGlxYo6cVw1GTc2uMazr0i+7tfLU1Sg2k6pKl+/TaemPCXvpndUfcttqunVV8dWb9bRD/4iN9arq37xiE4ufFnlm3JVdc94tVn5Ut1+igrk79BR5Rs+0ZkJ/66qkf+mk3NfUPmGP0uS/n7bUJVv+ERHP8xX7bXdFfvxB1J1tTo88K+qfHCajm7ervJ3/6TalC4NHgORhzOhCBezM0+u46imd796tzunT8n1euVUn5X8fp0ZN6nueX+t/AkdJNV9T6j26q6SJO/7OfLsLVTihBF1O6ipVtVdP5Zz+pScmmpVTpoe2Ldnd77O3PvTwOM2v1sh74ZsOdXViv5bmf4+9K660PX7TuBy0G0fL++7a+o9BiITEYpwMfl5qr0uTW7bdvVu9+wuUE2P3nX/vGHAV57PV9UPxyjq8wPyJ3eSHKduf7vzVTF7sapG3XfhcXK3qbr/rYF1kuTZv1e+69IkSXFvvi7P/+1S+dsfS3Fx6vjdHqpJS1ebN5arpt93LtxXA8dAZOJyLMKdmrVQR7YWNbg9pqhAvp4ZiikqUE36DV8+v2unfL0yFP15mWo7dQk8X9uxU91llOtKkjyF+YH91PTqG1jnnPhC/rbtpejowPbq/rdIcXFqvWq5oo4flT/1avmTU+QpKqh7kd8v5/ixBo+ByBS2ESovL9e4ceM0adIk61HCmmd3vmp69pFnd7586eci4rqKOnZE/uQU+XqkK/pAiZLvyJCnMF9VYyfKOXVSybf1UvIP+qn1mtcC+wm8XnVnM77r+wQen7l7nNq+tEgdRg1S9JG/qaZnRt3zYybIU/JXJd/eW8lDb1TM3sIGj4HI5LjnvtoEIzMz083NzW3GcYL36KOPKiEhQUuWLNHp06cv2JaTk6OVK1cqJyfnkvuJj49XRUWFCgtPqH37+GaaFpdT525eOdOmSYsWWY+CIDmOk+e6bmZ928LyTMjn8yk7O1vDhg1Tly5dLv0CACErLCO0bds29ejRQ3l5eRo4cKD1OAC+gbCM0NatW3XzzTdr3bp1GjVqlPU4aISqqiqtePV5Db2tl06dqrAeByEgLCOUm5urmJgY7dmzR8OGDbMeB0E4H5/v9e+i3F/OVmnJPlVWnr70CxHxwvLvCe3fv1+FhYWaPn26PJ6w/BRajKqqKv32jWV65fl5GlBdrQ1VZ3SjpM5xra1HQ4gIy/+DDx8+rKioKH48H8Iaig/wdWEZIY/Ho/nz56t1a76ahpqvx+e9qjOq7y5Grf21+v7Abyk6qvHfEYg+e1abDh1S/28+LkJA2EVow4YNKi0tVWxsrHbt2qXt27frvvv46/3Wgo3Peflnz+pME481RtLRM019NUJNWEWotrZWK1eu1BtvvKE5c+aoc+fOWrFihfVYkPST+/9Fn36yVTmuqyFBrG997qMpWjXxdQhNYRWh6OhovfXWW5LE2U+IyVqySr/OekZj3/2dHq6t1QxfjZKth0JYCMsf0SP0pKZeo4XPv6bNW4t0aNR96hHr1c89MTrawPrvxbVWh1hvkz4+lNQ6JuZKfnpoRmF1JoTQdz5GP5sxV7/OekY9Gjgz2ispe+On6tiEuyV27p+qDlfzXmmRggihWQQTo6uuSlRiYlKj9534lXsWIfxxOYZm1dBlWnVtrfVoCBFECFfE12NUJSk21ms9FkIAEcIVdT5GRXsrlZCQaD0OQgARgonoc7d9BYgQAFNECIApIgTAFBECYIoIATBFhACYIkIATBEhAKaIEABTROgKKyvbr9RURz6fr97tS5Ys0IwZDzZp31VVVRo//k717Bmvhx++R2vX/kb33vvlfQ5TUx0VF+9t0r6B5sKtPJrZTTd11eLF/6VBg/45qPVTpz7V5GO9995bKi8/rIKCY4G3Qho1ijtQIrRxJhRBDhwoUbduPXgvNoQVItSMpkwZp4MHSzVx4p1KS2url1/+j8C2tWt/owEDrlGfPkl68cX5ged/9atnNWXK/ZK+vHR7883X6137VVlZz+iFF+Zp3bo1Sktrq9WrV2jNmpUaOfK79a6fNm2CZs2arPvvH6a0tLa6666BOnLkkObMmab09AQNGtRTBQXbL+O/DaB+RKgZLV26Sqmp1+i119Zrz57Tmjz5ycC2Tz/9b338cZHWrPlQL7wwT3v2FDa4n2DWzpgxV1OmPKURI8Zqz57Tuvfen15yvvXr39STTz6n/PxytWoVqxEjblFGxo3Kzy/X8OGj9eyz05v2iQONQISMPP74M4qLi1Pv3jcoPf0G7dq147KsbYxhw36kvn37y+v1atiwHyk21qt77nlA0dHRGjFirD77jDMhND8iZOSrN3iPi2utysrTl2VtYyQlpQT+7PXGKTn5wseX6zjAxRChZuZwU3bgoohQM0tKSlFp6V+txwBCFhFqZlOmzNKLLz6nXr2u0rJlWdbjACHHcV036MWZmZlubm5uM45zeeTk5GjlypXKycm55Nr4+HhVVFSosPCE2rePb/7h8I117uaVM22atGiR9SgIkuM4ea7rZta3jTMhAKaIEABTRAiAKSIEwBQRAmCKCAEwRYQAmCJCAEwRIQCmiBAAU0QIgCkiBMAUEQJgiggBMEWEAJgiQgBMESEApogQAFMRFaHi4mItW7Ys8LigoECrVq0ynAjApURUhNq1a6enn35a5eXlkqSZM2fq9GneOwsIZR7rAS6npKQkPfTQQ3r77bdVWVmp/fv3B3WzewB2IipCkvTEE0+oW7du8ng8WrhwoWJjY61HAnAREXU5JknJyckaMmSIzp49q4kTJ1qPA+ASIu5MSJJeffVV7dy5k7MgIAxEZIQSExM1ePDgoNbefffdKi7+XK7rqqLiZPMOhssi9onZajV0kPwnI/O/V7t27RQVFXEXKQ2KyHdgbYzz78AKhIrDhw+rY8eO1mNcVhd7B9aIPBNqjPbt21uPAFzAcRzrEa6oFh+hsrIy6xGAFq3lXHgCCElECIApIgTAFBECYIoIATBFhACYIkIATBEhAKaIEABTRAiAKSIEwBQRAmCKCAEwRYQAmCJCAEwRIQCmiBAAU0QIgCkiBMAUEQJgiggBMEWEAJgiQgBMESEApogQAFNECIApIgTAFBECYIoIATBFhACYIkIATBEhAKaIEABTRAiAKSIEwBQRAmCKCAEwRYQAmCJCAEwRIQCmiBAAU0QIgCkiBMAUEQJgiggBMEWEAJgiQgBMESEApogQAFNECIApIgTAFBECYMpxXTf4xY5zVFJJ840DIEJd67pucn0bGhUhALjcuBwDYIoIATBFhACYIkIATBEhAKaIEABTRAiAKSIEwBQRAmDq/wF7dbTfY0mkhAAAAABJRU5ErkJggg==","text/plain":["<IPython.core.display.Image object>"]},"execution_count":192,"metadata":{},"output_type":"execute_result"}],"source":["Image('https://raw.githubusercontent.com/elsentjhung/cahn-hilliard-droplet/master/figures/work.png')"]},{"cell_type":"markdown","metadata":{},"source":["### Stress tensor \n","\n","Consider a smooth surface $S$, which can be open or closed.\n","For an open surface, we can arbitrarily define one side of the surface to be the _inner_ side and \n","the other side to be the _outer_ side.\n","For a closed surface, on the other hand, the inner side is always defined to be the volume which is enclosed by the surface.\n","At any point $P$ on the surface, we may then define the outward normal unit vector $\\hat{\\mathbf{n}}$ to be perpendicular to $S$ and point in the outer direction.\n","\n","Let us consider some surface element $d\\mathbf{S}=dS\\,\\hat{\\mathbf{n}}$ inside some fluid, \n","as depicted in the picture below.\n","The direction of $d\\mathbf{S}$ is defined to be the outward unit normal, which in this case is assumed to point to the right.\n","The differential force acting on the inner fluid by the outer fluid (or by the boundary if it is a solid wall on the other side) is then given by:\n","\\begin{equation}\n","dF_\\alpha = \\sigma_{\\alpha\\beta} dS_{\\beta},\n","\\end{equation}\n","where $\\sigma_{\\alpha\\beta}(\\mathbf{r})$ is the surface tensor, which is evaluated at a point on this surface element.\n","In the case of an isotropic stress, the stress tensor can be written as:\n","\\begin{equation}\n","\\sigma_{\\alpha\\beta}(\\mathbf{r}) = -p(\\mathbf{r})\\delta_{\\alpha\\beta},\n","\\end{equation}\n","where $p(\\mathbf{r})$ is called the pressure of the fluid at point $\\mathbf{r}$ in space.\n","Notice the negative sign. \n","If we have a fluid inside a cubic box of size $L\\times L\\times L$, \n","the fluid will push each side of the box, with force magnitude $pL^2$ (if $p$ is positive).\n","\n","Let us consider a fluid inside some volume $V$.\n","$V$ can be enclosed by other fluid or solid walls.\n","Let us denote $\\mathbf{f}(\\mathbf{r})$ to be the force density (force per unit volume) acting on the fluid at $\\mathbf{r}$.\n","The net force acting on the fluid (by the surrounding fluid or the solid boundaries) is then given by the integral of $\\mathbf{f}$ over the whole fluid volume:\n","\\begin{equation}\n","\\mathbf{F} = \\int_V \\mathbf{f} \\, dV.\n","\\end{equation}\n","However, from the definition of the stress tensor, the net force acting on the fluid by the boundary can also be written as:\n","\\begin{equation}\n","F_\\alpha = \\oint_{\\partial V} \\sigma_{\\alpha\\beta} \\, dS_\\beta.\n","\\end{equation}\n","Thus,\n","\\begin{align}\n","\\int_V f_\\alpha \\, dV = \\oint_{\\partial V} \\sigma_{\\alpha\\beta} \\, dS_\\beta = \\int_V \\partial_\\beta\\sigma_{\\alpha\\beta} \\, dV,\n","\\end{align}\n","and therefore,\n","\\begin{equation}\n","f_\\alpha = \\partial_\\beta\\sigma_{\\alpha\\beta}.\n","\\end{equation}"]},{"cell_type":"code","execution_count":191,"metadata":{"trusted":false},"outputs":[{"data":{"image/png":"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","text/plain":["<IPython.core.display.Image object>"]},"execution_count":191,"metadata":{},"output_type":"execute_result"}],"source":["Image('https://raw.githubusercontent.com/elsentjhung/cahn-hilliard-droplet/master/figures/stress-tensor.png')"]},{"cell_type":"markdown","metadata":{},"source":["Now we will derive the formula for the _elastic_ stress tensor for our free energy:\n","\\begin{equation}\n","\\mathcal{F}[\\phi] = \\int_V \\bigg\\{ \\underbrace{\\frac{\\alpha}{2}\\phi^2 + \\frac{\\beta}{2}\\phi^4 + \n","\\frac{\\kappa}{2}|\\nabla\\phi|^2}_{g(\\phi,\\partial_\\alpha\\phi)}   \\bigg\\}\\,dV.\n","\\end{equation}\n","Suppose that we have some fluid, confined to a trapezoidal box of volume $V$ in space, \n","as we can see in the picture below.\n","Now we can deform the fluid _affinely_ (_e.g._ by shearing the box) through some infinitesimal strain $\\delta\\mathbf{r}$.\n","In other words we displace every small patch of material at $\\mathbf{r}$ to $\\mathbf{r}+\\delta\\mathbf{r}$.\n","Under this _affine_ deformation, the fluid density $\\phi(\\mathbf{r})$ and the volume of the box $V$ transform as:\n","\\begin{align}\n","\\phi(\\mathbf{r}) \\rightarrow \\phi(\\mathbf{r}-\\delta\\mathbf{r})  \\quad\\text{and}\\quad\n","V &\\rightarrow V + \\delta V.\n","\\end{align}\n","In other words, affine deformation simply translates the value of $\\phi$ through space by $\\delta\\mathbf{r}$.\n","By Taylor expanding $\\phi(\\mathbf{r}-\\delta\\mathbf{r})$, we can find:\n","\\begin{equation}\n","\\phi(\\mathbf{r}-\\delta\\mathbf{r}) = \\phi(\\mathbf{r}) - \\underbrace{\\delta\\mathbf{r}\\cdot\\nabla\\phi}_{\\delta\\phi}.\n","\\end{equation}\n","Now we can calculate the change in the total free energy due to this affine deformation:\n","\\begin{align}\n","\\delta \\mathcal{F} &= \\mathcal{F}[\\phi+\\delta\\phi] - \\mathcal{F}[\\phi] \\\\\n","&= \\int_{V+\\delta V} g(\\phi+\\delta\\phi,\\partial_\\alpha\\phi+\\delta\\partial_\\alpha\\phi) \\, dV - \n","\\int_{V} g(\\phi,\\partial_\\alpha\\phi) \\, dV \\\\\n","&= \\int_{V} g(\\phi+\\delta\\phi,\\partial_\\alpha\\phi+\\delta\\partial_\\alpha\\phi) \\, dV + \n","\\int_{\\delta V} g(\\phi+\\delta\\phi,\\partial_\\alpha\\phi+\\delta\\partial_\\alpha\\phi) \\, dV - \n","\\int_{V} g(\\phi,\\partial_\\alpha\\phi) \\, dV\n","\\end{align}\n","Now we can Taylor expand:\n","\\begin{equation}\n","g(\\phi+\\delta\\phi,\\partial_\\alpha\\phi+\\delta\\partial_\\alpha\\phi) = \n","g(\\phi,\\partial_\\alpha\\phi) + \n","\\delta\\phi \\frac{\\partial g}{\\partial\\phi} + \n","(\\delta\\partial_\\alpha\\phi) \\frac{\\partial g}{\\partial(\\partial_\\alpha\\phi)}\n","\\end{equation}\n","In particular, the second integral becomes:\n","\\begin{equation}\n","\\int_{\\delta V} g(\\phi+\\delta\\phi,\\partial_\\alpha\\phi+\\delta\\partial_\\alpha\\phi) \\, dV\n","\\simeq \\int_{\\delta V} g(\\phi,\\partial_\\alpha\\phi) \\, dV,\n","\\end{equation}\n","where we have neglected order $\\sim\\delta^2$.\n","Thus the change in the total energy is\n","\\begin{align}\n","\\delta \\mathcal{F} &= \n","\\int_{V} \\bigg\\{ \\delta\\phi \\frac{\\partial g}{\\partial\\phi} + (\\partial_\\alpha\\delta\\phi) \\frac{\\partial g}{\\partial(\\partial_\\alpha\\phi)} \\bigg\\}\\,dV + \n","\\int_{\\delta V} g(\\phi,\\partial_\\alpha\\phi) \\, dV \\\\\n","&= \n","\\int_{V} \\bigg\\{ \\delta\\phi \\frac{\\partial g}{\\partial\\phi} + (\\partial_\\alpha\\delta\\phi) \\frac{\\partial g}{\\partial(\\partial_\\alpha\\phi)} \\bigg\\}\\,dV + \n","\\oint_{\\partial V} g(\\phi,\\partial_\\alpha\\phi) \\, \\delta\\mathbf{r}\\cdot d\\mathbf{S}.\n","\\end{align}\n","Note that when you displace a surface element $d\\mathbf{S}$ by $\\delta\\mathbf{r}$, the volume covered by this travelling surface element is $\\delta\\mathbf{r}\\cdot d\\mathbf{S}$.\n","Next we can use the integration by parts on the first term:\n","\\begin{align}\n","\\delta \\mathcal{F} &= \n","\\int_{V} \\bigg\\{ \n","     \\frac{\\partial g}{\\partial\\phi} -\n","     \\partial_\\alpha\\left(\\frac{\\partial g}{\\partial(\\partial_\\alpha\\phi)}\\right) \\bigg\\} \\delta\\phi \\,dV + \n","\\oint_{\\partial V} \\frac{\\partial g}{\\partial(\\partial_\\alpha\\phi)} \\delta\\phi \\, dS_\\alpha + \n","\\oint_{\\partial V} g \\delta r_\\alpha  \\,  dS_\\alpha \\\\\n","&= \n","-\\int_{V} \\mu \\delta r_\\alpha \\partial_\\alpha\\phi \\, dV - \n","\\oint_{\\partial V} \\frac{\\partial g}{\\partial(\\partial_\\alpha\\phi)} \\delta r_\\beta \\partial_\\beta\\phi \\, dS_\\alpha + \n","\\oint_{\\partial V} g \\delta r_\\alpha  \\,  dS_\\alpha \\\\\n","&= \n","\\int_{V}  \\phi \\partial_\\alpha(\\mu \\delta r_\\alpha) \\, dV - \n","\\oint_{\\partial V} \\phi \\mu  \\delta r_\\alpha \\, dS_\\alpha + \n","\\oint_{\\partial V} \\left\\{ g\\delta_{\\alpha\\beta} - \n","(\\partial_\\alpha\\phi)\\frac{\\partial g}{\\partial(\\partial_\\beta\\phi)}  \\right\\} \\delta r_\\alpha  \\, dS_\\beta \\\\\n","&= \\oint_{\\partial V} \\left\\{ (g - \\phi\\mu)\\delta_{\\alpha\\beta} - \n","(\\partial_\\alpha\\phi)\\frac{\\partial g}{\\partial(\\partial_\\beta\\phi)}  \\right\\} \\delta r_\\alpha  \\, dS_\\beta + \n","\\int_{V}  (\\phi \\partial_\\alpha\\mu) \\delta r_\\alpha \\, dV\n","\\end{align}\n","In the last line we assumed the fluid to be incompressible, _i.e._ $\\partial_\\alpha\\delta r_\\alpha=0$.\n","Using the first law of thermodynamics, the change in the free energy is also equal to:\n","\\begin{align}\n","\\delta \\mathcal{F} = \\delta E - T\\delta S = \\delta W - \\delta Q - T\\delta S,\n","\\end{align}\n","where $\\delta W$ is the work done on the system, $\\delta Q$ is the heat dissipated into the environment and $\\delta S$ is the increase in the system's entropy.\n","The heat dissipated into the environment causes the the entropy of the environment (or heat reservoir) $S_r$ to increase.\n","Assuming the reservoir to be frictionless, we can write $\\delta Q = T\\delta S_r$.\n","Thus, the change in the free energy is:\n","\\begin{align}\n","\\delta\\mathcal{F} = \\delta W - T \\underbrace{(\\delta S + \\delta S_r)}_{=0} = \\delta W.\n","\\end{align}\n","Note that $\\delta S + \\delta S_r$ is the change in the total entropy (_i.e._ entropy of the universe).\n","For all affine deformations, the process is time-reversible and thus should not contribute to the entropy of the universe.\n","Therefore, the change in the free energy is simply the work done on the system by the external forces: $\\delta F = \\delta W$.\n","\n","Now we will look at these external forces in more detail.\n","In order to apply the affine deformation to the system, we must exert:\n","1. some external surface force $\\mathbf{F}^{\\text{surface}}$ to the walls (using our hands), and/or\n","2. some external body force density $\\mathbf{f}^{\\text{body}}$ to the bulk of the fluid (such as gravity and electric field).\n","\n","Let us consider the external surface force first. \n","The force acting on the walls by the external force (such as our hands) is $\\mathbf{F}^{\\text{surface}}$.\n","The force acting on the walls by the fluid is $-\\oint_{\\partial V}\\sigma_{\\alpha\\beta}\\,dS_\\beta$.\n","Since we assume mechanical equilibrium throughout the deformation process, the net force on the walls has to be zero:\n","\\begin{equation}\n","F^{\\text{surface}}_\\alpha - \\oint_{\\partial V}\\sigma_{\\alpha\\beta}\\,dS_\\beta = 0, \\quad\\text{for } \\alpha=x,y,z.\n","\\end{equation}\n","Similar principle also applies for the external body force.\n","Let's consider some fluid element $dV$ inside the bulk of the fluid.\n","The force acting on this fluid element by the external force (such as gravity and electric field) is $\\mathbf{f}^{\\text{body}}\\,dV$.\n","The force acting on this fluid element by the surrounding fluid is $\\partial_\\beta\\sigma_{\\alpha\\beta}\\,dV$.\n","Since this fluid element is always in mechanical equilibrium, we also must have the force balance:\n","\\begin{equation}\n","f^{\\text{body}}_\\alpha + \\partial_\\beta\\sigma_{\\alpha\\beta} = 0, \\quad\\text{for } \\alpha=x,y,z.\n","\\end{equation}\n","Thus the work done by the external forces is $\\delta W = \\mathbf{F}\\cdot\\delta\\mathbf{r}$, or:\n","\\begin{equation}\n","\\delta W = \n","\\underbrace{\\oint_{\\partial V} \\sigma_{\\alpha\\beta} \\delta r_\\alpha \\, dS_\\beta}_{\\text{from external surface force}}  - \n","\\underbrace{\\int_{V} (\\partial_\\beta\\sigma_{\\alpha\\beta}) \\delta r_\\alpha \\, dV}_{\\text{from external body force}} \n","= \\int_V \\underbrace{\\sigma_{\\alpha\\beta}}_{\\text{stress}} \\underbrace{\\partial_\\beta \\delta r_\\alpha}_{\\text{strain}} \\, dV\n","\\end{equation}\n","Note that $\\partial_\\beta\\delta r_\\alpha$ is also called the strain tensor.\n","Now we can then equate $\\delta\\mathcal{F} = \\delta W$.\n","Comparing the surface term, we get the elatic stress tensor:\n","\\begin{equation}\n","\\sigma_{\\alpha\\beta} = \\underbrace{(g - \\phi\\mu)}_{-p}\\delta_{\\alpha\\beta}  -  (\\partial_\\alpha\\phi)\\frac{\\partial g}{\\partial(\\partial_\\beta\\phi)}, \n","\\end{equation}\n","where we have also identified the isotropic pressure to be $p=\\phi\\mu-g$.\n","Equating the volume term, we get the elastic force density:\n","\\begin{equation}\n","\\mathbf{f} = -\\phi\\nabla\\mu.\n","\\end{equation}\n","One can also verify that $f_\\alpha=\\partial_\\beta\\sigma_{\\alpha\\beta}$.\n","In the derivation above, we have assumed the fluid to be incompressible, _i.e._ $\\partial_\\alpha\\delta r_\\alpha=0$, for a compressible fluid, the results remain the same."]},{"cell_type":"code","execution_count":190,"metadata":{"trusted":false},"outputs":[{"data":{"image/png":"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","text/plain":["<IPython.core.display.Image object>"]},"execution_count":190,"metadata":{},"output_type":"execute_result"}],"source":["Image('https://raw.githubusercontent.com/elsentjhung/cahn-hilliard-droplet/master/figures/affine-deformation.png')"]},{"cell_type":"markdown","metadata":{},"source":["### Equation of state\n","\n","Let us consider the shifted Landau free energy for liquid/gas phase separation:\n","\\begin{equation}\n","\\mathcal{F}[n] = \\int dV \\bigg\\{ \n","\\underbrace{\\frac{\\alpha}{2}(n-n_c)^2 + \\frac{\\beta}{4}(n-n_c)^4}_{f(n)= \\text{bulk energy density}} + \n","\\frac{\\kappa}{2}|\\nabla  n|^2 \\bigg\\},\n","\\end{equation}\n","where $n_c,\\beta,\\kappa$ are positive and $\\alpha$ can be positive or negative.\n","Here $n(\\mathbf{r})>0$ represents the absolute number density of the fluid, rather than rescaled density $\\phi(\\mathbf{r})$, which can be positive or negative.\n","They are related through:\n","\\begin{equation}\n","\\phi(\\mathbf{r}) = \\frac{n(\\mathbf{r})-n_c}{n_c}.\n","\\end{equation}\n","For example, the number density of water molecules in liquid water at room temperature is $n\\simeq 33.4\\times 10^{27}\\,\\text{m}^{-3}$.\n","You can think of Landau free energy as a Taylor expansion of $\\mathcal{F}[n]$ around the critical point $(\\alpha=0,n=n_c)$ for small $n-n_c$.\n","In practice, one often uses Landau free energy even for large values of $n-n_c$, due to its simplicity.\n","\n","For $\\alpha<0$, the system favours phase separation into liquid and gas phase, whose number densities are given by the minima of the bulk energy density $f(n)$:\n","\\begin{align}\n","n_l = n_c + \\sqrt{\\frac{-\\alpha}{\\beta}} \\quad\\text{and}\\quad\n","n_g = n_c - \\sqrt{\\frac{-\\alpha}{\\beta}},\n","\\end{align}\n","respectively.\n","Note that the parameter $n_c$ has to be large enough such that $0<n_g<n_l$, since we are working with absolute number density rather than rescaled density.\n","\n","In the previous section, we have derived the isotropic pressure to be: \n","\\begin{equation}\n","P = n\\mu - g, \\text{ where }\n","\\mu = \\frac{\\delta\\mathcal{F}}{\\delta n} \\text{ and } \n","g = f(n)+\\frac{\\kappa}{2}|\\nabla n|^2.\n","\\end{equation}\n","In the bulk, the isotropic pressure is simply given by $P(n)=n f'(n)-f(n)$, where $f(n)$ is the bulk free energy density.\n","For the Landau free energy given above, the isotropic pressure in the bulk is given by:\n","\\begin{equation}\n","P(n) = \\alpha n(n - n_c) + \\beta n(n - n_c)^3 - \n","\\frac{\\alpha}{2}(n - n_c)^2 - \n","\\frac{\\beta}{4}(n - n_c)^4\n","\\end{equation}\n","The above equation is also called the equation of state, which relates the pressure $P$ to number density $n$ and temperature $T$ (the temperature is absorbed inside the parameter $\\alpha$).\n","Also note that we can add an arbitrary constant to $\\mathcal{F}[n]$ and hence to $P(n)$, which does not affect the overall dynamics.\n","Finally, let us define the molar volume to be $\\nu=1/n$, which is the average free volume occupied by a single molecule.\n","The equation of state in terms of the molar volume is:\n","\\begin{equation}\n","P(\\nu) = \\alpha \\nu^{-1}(\\nu^{-1} - n_c) + \\beta \\nu^{-1}(\\nu^{-1} - n_c)^3 - \n","\\frac{\\alpha}{2}(\\nu^{-1} - n_c)^2 - \n","\\frac{\\beta}{4}(\\nu^{-1} - n_c)^4.\n","\\end{equation}\n","The figure below shows the plot of $P(\\nu)$ as a function of $\\nu$ for $\\alpha<0$ (in some arbitrary units which we shouldn't worry about).\n","\n","The isothermal compressibility is defined to be:\n","\\begin{equation}\n","\\kappa_T = -\\frac{1}{\\nu}\\left(\\frac{\\partial \\nu}{\\partial P}\\right)_T.\n","\\end{equation}\n","$\\kappa_T$ is like the resistance of the fluid against compression.\n","As you can see in the $P(\\nu)$ plot below, for some range of $\\nu$, the isothermal compressibility $\\kappa_T$ is negative.\n","This signifies that the system is unstable for this range of $\\nu$.\n","Physically, the system will spontaneously phase-separate into liquid and gas phase.\n","The steady (or equilibrium) state will consist of two macroscopic phases: liquid and gas phase, separated by an interface.\n","In this section, we assume the interface to be mostly flat (in the next section we will generalize this to a curved interface such as droplets).\n","To find the densities of the bulk liquid and gas phase, we have to equate the chemical potential of the liquid and the gas phase (otherwise current will diffuse from one phase to the other):\n","\\begin{equation}\n","\\mu_l = \\mu_g \\quad\\Rightarrow\\quad f'(n_l) = f'(n_g).\n","\\end{equation}\n","The equation above alone is not enough to determine $n_l$ and $n_g$.\n","We also need to equate the pressure of the liquid phase to that of the gas phase:\n","\\begin{equation}\n","P_l = P_g \\quad\\Rightarrow\\quad n_l f'(n_l)-f(n_l) = n_g f'(n_g)-f(n_g).\n","\\end{equation}\n","If the pressures are not equal, the interface, separating the liquid from the gas phase, will be pushed in the direction from high to low pressure.\n","If we solve the two simultaneous equations above for our symmetric Landau free energy $\\mathcal{F}[n]$, we will find that the number densities of the liquid and the gas phase are given by the following expressions\n","\\begin{align}\n","n_l = n_c + \\sqrt{\\frac{-\\alpha}{\\beta}} \\quad\\text{and}\\quad\n","n_g = n_c - \\sqrt{\\frac{-\\alpha}{\\beta}},\n","\\end{align}\n","which also happen to be the minima of the free energy $\\mathcal{F}[n]$.\n","\n","Now with this information, we can verify Maxwell's construction, which states that the integral of $P(\\nu)-P_l$ from the liquid phase $\\nu_l$ to the gas phase $\\nu_g$ is zero.\n","In other words, the blue areas between the curve $P=P(\\nu)$ and the line $P=P_l$ below add up to zero.\n","To show this, we consider (recall $\\nu=1/n$):\n","\\begin{align}\n","\\int_{\\nu_l}^{\\nu_g} \\left[ P(\\nu) - P_l \\right] d\\nu &= \\int_{n_g}^{n_l} \\frac{P(n) - P_l}{n^2} dn \\\\\n","&= \\int_{n_g}^{n_l} \\frac{f'(n)}{n} dn - \\int_{n_g}^{n_l} \\frac{f(n)}{n^2} dn - P_l \\int_{n_g}^{n_l} \\frac{1}{n^2}dn \\\\\n","&= \\left[\\frac{f(n)}{n}\\right]_{n_g}^{n_l} + \\underbrace{\\int_{n_g}^{n_l} \\frac{f'(n)}{n^2} dn - \\int_{n_g}^{n_l} \\frac{f(n)}{n^2} dn}_{=0} + P_l \\left(\\frac{1}{n_l} - \\frac{1}{n_g} \\right) \\\\\n","&= \\frac{f(n_l)}{n_l} - \\frac{f(n_g)}{n_g} + \\left[n_l f'(n_l) - f(n_l)\\right] \\left(\\frac{1}{n_l} - \\frac{1}{n_g} \\right) \\\\\n","&= \\frac{n_gf(n_l) - n_lf(n_g) + n_ln_gf'(n_l) - n_gf(n_l) - n_l^2f'(n_l) + n_lf(n_l)}{n_ln_g} \\\\\n","&= \\frac{n_l\\left[n_gf'(n_g) - f(n_g)\\right] - n_l\\left[n_lf'(n_l) - f(n_l)\\right]}{n_ln_g},\n","\\end{align}\n","where we have used the fact that $f'(n_l)=f'(n_g)$. \n","Therefore the integral above becomes:\n","\\begin{align}\n","\\int_{\\nu_l}^{\\nu_g} \\left[ P(\\nu) - P_l \\right] d\\nu = \\frac{P_g - P_l}{n_g} = 0,\n","\\end{align}\n","since $P_l=P_g$.\n","Thus, we have verified Maxwell's construction.\n","Note that the derivation of the Maxwell's construction above remains valid for general double well potentials $f(n)$, which are not necessarily symmetric. "]},{"cell_type":"code","execution_count":189,"metadata":{"trusted":false},"outputs":[{"data":{"image/png":"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","text/plain":["<IPython.core.display.Image object>"]},"execution_count":189,"metadata":{},"output_type":"execute_result"}],"source":["Image('https://raw.githubusercontent.com/elsentjhung/cahn-hilliard-droplet/master/figures/Maxwell-construction.png')"]},{"cell_type":"markdown","metadata":{},"source":["### Convection vs diffusion\n","\n","Thus we have derived the formula for the elastic stress for the given free energy.\n","Under affine deformation, the fluid exerts some elastic stress to the walls, and also to each other inside the bulk.\n","However for a general deformation, this is not the only stress present in the fluid.\n","We also have a viscous stress, whose coefficient is called the viscosity, which is responsible for heat dissipation.\n","\n","One might also notice that the presence or absence of a mechanical stress does not affect the Cahn-Hiliard equation:\n","\\begin{equation}\n","\\frac{\\partial\\phi}{\\partial t} = M\\nabla^2\\frac{\\delta\\mathcal{F}}{\\delta\\phi} \\quad \\text{(Cahn-Hiliard equation)}\n","\\end{equation}\n","This is because Cahn-Hiliard equation only describes the diffusion process in the binary fluid.\n","In general, there are two ways a fluid can move inside the system: 1) diffusion and 2) convection.\n","\n","An example of diffusion process is when we put a drop of red ink inside a static and viscous solvent.\n","This ink will spread slowly until the whole fluid becomes homogenously red.\n","This diffusion process is due to the Brownian motion of the ink molecules.\n","Typically the higher the temperature is, the faster the diffusion process will be.\n","\n","An example of convection is when we put a beach ball on a flowing river. \n","This beach ball will move with the same velocity as the water.\n","If we put a drop of red ink on a flowing river, this blob of red ink will also move with the same velocity as the water, while at the same time diffuses around its centre of mass.\n","To describe the convection process in a binary fluid, we need to define another field, which is the velocity field $\\mathbf{u}(\\mathbf{r},t)$ (in addition to $\\phi(\\mathbf{r},t)$).\n","The time evolution of $\\mathbf{u}(\\mathbf{r},t)$ will be described by the Navier-Stokes equation:\n","\\begin{equation}\n","\\underbrace{\\rho}_{\\text{mass density}} \\underbrace{\\left(\\frac{\\partial}{\\partial t} + \\mathbf{u}\\cdot\\nabla\\right)u_\\alpha}_{\\text{acceleration}} = \n","\\underbrace{\\partial_\\beta\\sigma_{\\alpha\\beta}}_{\\text{force density}}.\\quad\\text{(Navier-Stokes equation)}\n","\\end{equation}\n","In essence, the Navier-Stokes equation is just a momentum conservation or Newton's second law.\n","Thus to take into account of both diffusion and convection process in a binary fluid properly, \n","we need to solve the coupled Cahn-Hiliard-Navier-Stokes equation:\n","\\begin{align}\n","\\frac{\\partial\\phi}{\\partial t} + (\\mathbf{u}\\cdot\\nabla)\\phi &= \n","M\\nabla^2\\frac{\\delta\\mathcal{F}}{\\delta\\phi} \\quad\\text{(Cahn-Hiliard equation)} \\\\\n","\\rho \\left(\\frac{\\partial}{\\partial t} + \\mathbf{u}\\cdot\\nabla\\right)u_\\alpha &= \n","\\partial_\\beta\\sigma_{\\alpha\\beta} \\quad\\text{(Navier-Stokes equation)} \\\\\n","\\nabla\\cdot\\mathbf{u} &= 0 \\quad\\text{(incompressibility condition)}. \n","\\end{align}\n","Notice that the first two equations are coupled through the stress tensor $\\sigma_{\\alpha\\beta}$, which depends on $\\phi$,\n","and the convection term $(\\mathbf{u}\\cdot\\nabla)\\phi$.\n","\n","In practice, we can also define the diffusion timescale $t_{\\text{diffusion}}$ and the convection timescale $t_{\\text{convection}}$ to be:\n","\\begin{align}\n","t_{\\text{diffusion}} = \\frac{L^2}{M\\alpha} \\quad\\text{and}\\quad\n","t_{\\text{convection}} = \\frac{L}{U},\n","\\end{align}\n","where $L$ is the system size and $U$ is the typical magnitude of the fluid velocity in the system (such the velocity of the flowing river under gravity).\n","If $t_{\\text{diffusion}}\\ll t_{\\text{convection}}$, then the diffusion process dominates over convection and the problem can be simply described by Cahn-Hiliard equation alone.\n","\n","### Laplace pressure\n","\n","Let us recall the Hamiltonian:\n","\\begin{equation}\n","\\mathcal{F}[\\phi] = \\int_V \\bigg\\{ \\underbrace{\\frac{\\alpha}{2}\\phi^2 + \\frac{\\beta}{4}\\phi^4}_{f(\\phi)} + \\frac{\\kappa}{2}|\\nabla\\phi|^2 \\bigg\\},\n","\\end{equation}\n","where $f(\\phi)$ is the bulk (or local) free energy density.\n","In equilibrium, and far from any boundary or curved interface, the density of the fluid is given by the minima of this bulk energy density.\n","We call this density the binodal density $\\pm\\phi_B$:\n","\\begin{equation}\n","\\frac{df}{d\\phi} = 0 \\quad\\Rightarrow\\quad \\phi=\\pm\\phi_B \\quad\\text{where}\\quad \\phi_B=\\sqrt{\\frac{-\\alpha}{\\beta}}.\n","\\end{equation}\n","As we have seen above, for a flat interface, the equilibrium density of the fluid on either side of the flat interface is given by the binodal densities $\\pm\\phi_B$.\n","On the other hand if the interface is curved, such as a liquid droplet surrounded by the gas phase, the density of the liquid and the gas are not exactly at the binodals (shown in the figure below).\n","They are in fact shifted away from the binodal values due to Laplace pressure.\n","Let us denote $\\phi_+$ to be the density of the liquid inside the droplet and $\\phi_-$ to be the density of the gas outside. \n","$\\phi_\\pm$ will be slightly shifted away from the binodal values, _i.e._:\n","\\begin{equation}\n","\\phi_\\pm = \\pm\\phi_B + \\epsilon_\\pm \\text{ (for curved interface)}.\n","\\end{equation}\n","This is because the pressure inside the droplet will be slightly higher than that outside.\n","The difference in this pressure is called the Laplace pressure.\n","\n","To illustrate this, let us consider a spherical liquid droplet of radius $R$ surrounded by gas phase, as shown in the figure below.\n","Let us denote $P_+$ and $P_-$ to be the pressure inside and outside the droplet respectively.\n","Now let us consider the upper hemispherical surface.\n","There are two forces acting on this hemispherical surface.\n","The first one is due to the pressure difference $\\Delta P = P_+ - P_-$ (Laplace pressure).\n","The force acting on the upper hemispherical surface is equal to $\\Delta P\\,\\pi R^2$ in the upwards direction.\n","The second force is due to the surface tension $\\gamma$: the upper hemispherical surface is pulled by the lower hemisphere right at the equator.\n","This force is equal to $\\gamma 2\\pi R$ in the downwards direction.\n","Thus balancing the two forces, we get:\n","\\begin{equation}\n","P_+ - P_- = \\frac{2\\gamma}{R}.\n","\\end{equation}\n","Note that for $d$-dimensional spherical droplet, the Laplace pressure is given by:\n","\\begin{equation}\n","P_+ - P_- = \\frac{\\gamma(d-1)}{R}.\n","\\end{equation}\n","\n","Apart from the force balance, the chemical potential inside the droplet also has to be the same as that outside the droplet.\n","If not, there will be a diffusive current from high chemical potential regions to low chemical potential regions.\n","Equating the two chemical potentials, we get:\n","\\begin{equation}\n","\\mu(\\phi_+) = \\mu(\\phi_-).\n","\\end{equation}\n","Now from the previous section, the chemical potential is given by the formula:\n","\\begin{equation}\n","\\mu = f'(\\phi) - \\kappa\\nabla^2\\phi.\n","\\end{equation}\n","However since we are only interested in the value of $\\mu$ in the bulk (where $\\phi$ is uniform), we can ignore the gradient in $\\phi$, and thus we get:\n","\\begin{equation}\n","f'(\\phi_+) = f'(\\phi_-) \\quad\\Rightarrow\\quad f'(\\phi_B + \\epsilon_+) = f'(-\\phi_B + \\epsilon_-).\n","\\end{equation}\n","Next, we can Taylor expand:\n","\\begin{align}\n","f'(\\phi_B + \\epsilon_+) &= f''(\\phi_B) \\epsilon_+ + \\mathcal{O}(\\epsilon^2) \\\\\n","f'(-\\phi_B + \\epsilon_-) &= f''(\\phi_B) \\epsilon_- + \\mathcal{O}(\\epsilon^2).\n","\\end{align}\n","Note that since $f(\\phi)$ is symmetric, $f''(\\phi_B)=f''(-\\phi_B)$.\n","Thus the equality of chemical potential implies that $\\epsilon_+=\\epsilon_-$, \n","_i.e._ the values of $\\phi_\\pm$ are shifted by the same amount from the binodals.\n","\n","Now let us go back to the Laplace pressure.\n","From the previous section, we have the formula for the pressure:\n","\\begin{align}\n","P = \\phi\\mu - g, \\text{ where } \n","g = f(\\phi) + \\frac{\\kappa}{2}|\\nabla\\phi|^2 \\text{ and }\n","\\mu = f'(\\phi) - \\kappa\\nabla^2\\phi.\n","\\end{align}\n","Since we are only interested in the value of $P$ in the bulk, we can ignore the gradient terms to get:\n","\\begin{equation}\n","P(\\phi) = \\phi f'(\\phi) - f(\\phi).\n","\\end{equation}\n","Now subsituting this to the Laplace pressure equation, we get:\n","\\begin{align}\n","\\phi_+ f'(\\phi_+) - \\phi_- f'(\\phi_-) - f(\\phi_+) + f(\\phi_-) &= \\frac{2\\gamma}{R} \\\\\n","(\\phi_B+\\epsilon) f'(\\phi_B+\\epsilon) - (-\\phi_B+\\epsilon) f'(-\\phi_B+\\epsilon) - f(\\phi_B+\\epsilon) + f(-\\phi_B+\\epsilon)  &= \\frac{2\\gamma}{R}\n","\\end{align}\n","We can then Taylor expand for small $\\epsilon$ and simplify to get\n","\\begin{align}\n","\\epsilon = \\frac{\\gamma}{R\\phi_B f''(\\phi_B)}.\n","\\end{align}"]},{"cell_type":"code","execution_count":186,"metadata":{"trusted":false},"outputs":[{"data":{"image/png":"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","text/plain":["<IPython.core.display.Image object>"]},"execution_count":186,"metadata":{},"output_type":"execute_result"}],"source":["Image('https://raw.githubusercontent.com/elsentjhung/cahn-hilliard-droplet/master/figures/Laplace-pressure.png')"]},{"cell_type":"markdown","metadata":{},"source":["### Droplet evaporation\n","\n","Now let us consider a single spherical liquid droplet of radius $R$, surrounded by the gas phase, as depicted in the figure below.\n","The density of the liquid inside the droplet and that of the gas just immediately outside the droplet will be shifted away from the binodal values due to the Laplace pressure as described in the previous section.\n","In particular, the density of the liquid inside the droplet and that of the gas on the surface of the droplet will be $\\phi_B+\\epsilon$ and $-\\phi_B+\\epsilon$ respectively, where\n","\\begin{equation}\n","\\phi_B = \\sqrt{\\frac{-\\alpha}{\\beta}}, \\quad\\text{and}\\quad \\epsilon = \\frac{\\gamma}{R\\phi_B f''(\\phi_B)}.\n","\\end{equation}\n","Very far away from the droplet, the density of the gas phase will go back to its equilibrium bulk value, _i.e._ $-\\phi_B$, as depicted in the figure below. \n","Thus, we expect a chemical potential gradient from $\\mu_R$ on the surface of the droplet to $0$ at infinity.\n","This chemical potential gradient will induce a current from the surface of the droplet to infinity, which then causes the droplet to shrink.\n","\n","Assuming the radius of the droplet $R(t)$ shrinks slowly, we can assume the chemical potential of the gas phase to be quasi-static: $\\nabla^2\\mu=0$.\n","Assuming spherical symmetry, $\\mu=\\mu(r)$, we then have:\n","\\begin{equation}\n","\\frac{1}{r^2}\\frac{d}{dr}\\left(r^2 \\frac{d\\mu}{dr}\\right) = 0 \\quad\\Rightarrow\\quad\n","\\mu(r) = \\frac{A}{r} + B.\n","\\end{equation}\n","Fixing the boundary conditions $\\mu(R)=\\mu_R$ and $\\mu(\\infty)=0$, we can solve the integration constants $A$ and $B$ to get $\\mu(r)$ and subsequently $\\mathbf{J}(r)$:\n","\\begin{equation}\n","\\mu(r) =\n","\\begin{cases}\n","\t\t\t\\mu_R & \\text{, } r<R \\\\\n","            \\frac{\\mu_R R}{r} & \\text{, } r\\ge R\n","\\end{cases} \\quad\\Rightarrow\\quad\n","\\mathbf{J}(r) = -M\\nabla\\mu = \n","\\begin{cases}\n","\t\t\t0 & \\text{, } r<R \\\\\n","            \\frac{M\\mu_R R}{r^2}\\hat{\\mathbf{r}} & \\text{, } r\\ge R\n","\\end{cases}.\n","\\end{equation}\n","The chemical potential inside/on the surface of the droplet is:\n","\\begin{equation}\n","\\mu_R = f'(\\phi_B+\\epsilon) \\simeq f''(\\phi_B)\\epsilon.\n","\\end{equation}\n","Using conservation of mass and divergence theorem, we have:\n","\\begin{equation}\n","\\frac{d}{dt}\\left[\\frac{4}{3}\\pi R^3 \\Delta\\phi \\right] = -\\oint_R \\mathbf{J}\\cdot d\\mathbf{S},\n","\\end{equation}\n","where $\\Delta\\phi=2\\phi_B$ is the density difference between the outside and the inside of the droplet.\n","We can solve the above equation to get the differential equation for the droplet radius $R(t)$:\n","\\begin{equation}\n","\\frac{dR}{dt} = -\\frac{M\\gamma}{2\\phi_B^2R^2} \\quad\\Rightarrow\\quad\n","R(t) = \\left[ R(0)^3 - \\frac{3M\\gamma}{2\\phi_B^2}t \\right]^{1/3}.\n","\\end{equation}\n","\n","Thus in an infinite sytem, a liquid droplet is always unstable and will evaporate completely (effectively the global density $\\phi_0$ is close to the binodal $-\\sqrt{\\frac{-\\alpha}{\\beta}}$). \n","However in computer simulations, the system size is finite and a liquid droplet can be made to be stable."]},{"cell_type":"code","execution_count":185,"metadata":{"trusted":false},"outputs":[{"data":{"image/png":"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","text/plain":["<IPython.core.display.Image object>"]},"execution_count":185,"metadata":{},"output_type":"execute_result"}],"source":["Image('https://raw.githubusercontent.com/elsentjhung/cahn-hilliard-droplet/master/figures/droplet-evaporation.png')"]},{"cell_type":"markdown","metadata":{},"source":["## Wetting\n","\n","In this section we will consider a liquid droplet on a flat solid surface, surrounded by the gas phase (see figure below).\n","In this case, we have three different interfaces:\n","- liquid-gas interface\n","- solid-gas interface\n","- solid-liquid interface,\n","\n","and three different surface tension: $\\gamma_{lg}$, $\\gamma_{sg}$, and $\\gamma_{sl}$ for each respective interface.\n","The line where all the three interfaces meet is called the contact line (indicated by the red dot in the figure below).\n","In our two-dimensional case, the contact line extends in the $z$-direction.\n","\n","The contact angle $\\theta$ is defined to be the angle between the tangent line to the liquid-gas interface and the surface at the contact line at equilibrium (see figure below). \n","If $\\theta<90^\\circ$, the surface is said to be hydrophilic (in the case of water), _i.e._, it attracts water.\n","If $\\theta>90^\\circ$, the surface is said to be hydrophobic, _i.e._, it repels water.\n","\n","The goal of this section is to derive the suitable boundary conditions for the density field $\\phi(\\mathbf{r},t)$ at the surface (which we assume to be at $y=0$ and $y=L$) for a given contact angle $\\theta$."]},{"cell_type":"code","execution_count":184,"metadata":{"tags":["hide_code"]},"outputs":[{"data":{"image/png":"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","text/plain":["<IPython.core.display.Image object>"]},"execution_count":184,"metadata":{},"output_type":"execute_result"}],"source":["Image('https://raw.githubusercontent.com/elsentjhung/cahn-hilliard-droplet/master/figures/contact-angle.png')"]},{"cell_type":"markdown","metadata":{},"source":["### Young's equation\n","\n","First we will express the contact angle $\\theta$ in terms of the three surface tension: $\\gamma_{lg}$, $\\gamma_{sg}$, and $\\gamma_{sl}$.\n","To do this let us consider one of the contact lines, indicated by the red dot in the figure below.\n","Note that the contact line extends infinitely in the $z$-direction (out of the screen).\n","The solid-liquid interface tends to pull this contact line to the right with magnitude $\\gamma_{sl}$ (per unit length of the contact line).\n","The gas-solid interface, on the other hand, tends to pull this contact line to the left with magnitude $\\gamma_{sg}$.\n","Finally the liquid-gas interface tends to pull the contact line upwards at an angle $\\theta$, relative to the $x$-axis, with magnitude $\\gamma_{lg}$.\n","These three interfacial forces acting on a single contact line are indicated by purple arrows in the figure below.\n","\n","In equilibrium, the contact line should not move, so we balance the net force in the $x$-direction to get the Young's equation for the contact angle:\n","\\begin{equation}\n","\\gamma_{sg} = \\gamma_{sl} + \\gamma_{lg}\\cos\\theta \\quad\\Rightarrow\\quad\n","\\cos\\theta = \\frac{\\gamma_{sg}-\\gamma_{sl}}{\\gamma_{lg}}.\n","\\end{equation}"]},{"cell_type":"code","execution_count":205,"metadata":{"tags":["hide_code"]},"outputs":[{"data":{"image/png":"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","text/plain":["<IPython.core.display.Image object>"]},"execution_count":205,"metadata":{},"output_type":"execute_result"}],"source":["Image('https://raw.githubusercontent.com/elsentjhung/cahn-hilliard-droplet/master/figures/Youngs-equation.png')"]},{"cell_type":"markdown","metadata":{},"source":["### Surface free energy\n","\n","So far we have written down the contact angle $\\theta$ in terms of the three surface tension: $\\gamma_{lg}$, $\\gamma_{sg}$, and $\\gamma_{sl}$, however, \n","we have not yet discussed how we are going to implement the contact angle $\\theta$ into our equation of motion, _i.e._, the Cahn-Hiliard equation. \n","This is actually done by modifying our free energy functional to include the contribution from the surface energy:\n","\\begin{equation}\n","\\mathcal{F}[\\phi] = \\int_V \\bigg[ \\underbrace{\\frac{\\alpha}{2}\\phi^2 + \\frac{\\beta}{4}\\phi^4}_{f(\\phi)} + \\frac{\\kappa}{2}|\\nabla\\phi|^2 \\bigg] dV - \\int_\\text{walls} h\\phi \\, dS,\n","\\end{equation}\n","where $h$ is some constant, which depends on the contact angle $\\theta$. \n","The last term above is a surface integral over the two wall surfaces at $y=0$ and $y=L$ (assuming they both have the same contact angle $\\theta$).\n","Thus, $h$ acts like an external field for $\\phi$ on the surface. \n","If $h>0$, it is more energetically favourable for the value of $\\phi$ to go up on the surface (compared to that in the bulk).\n","This corresponds to an hydrophilic surface or $\\theta<90^\\circ$.\n","On the other hand if $h<0$, it more energetically favourable for the value of $\\phi$ to go down on the surface.\n","This corresponds to an hydrophobic surface or $\\theta>90^\\circ$.\n","\n","The equilibrium state of the droplet corresponds to the minimum of the free energy $F[\\phi]$.\n","Let us now consider the variation in the free energy $F[\\phi]$, when we vary the density field $\\phi\\rightarrow\\phi+\\delta\\phi$:\n","\\begin{align}\n","\\delta F &= \\int_V \\bigg[ f'(\\phi)\\delta\\phi + \\kappa\\nabla\\phi\\cdot\\nabla\\delta\\phi \\bigg]dV - \n","\\int_\\text{walls} h\\delta\\phi \\, dS \\\\\n","&= \\int_V \\bigg[ f'(\\phi) - \\kappa\\nabla^2\\phi \\bigg]\\delta\\phi \\, dV + \n","   \\int_V \\kappa\\nabla\\cdot(\\delta\\phi\\nabla\\phi) \\, dV - \n","   \\int_\\text{walls} h\\delta\\phi \\, dS,\n","\\end{align}\n","where we have used integration by parts in the second line above.\n","Now using divergence theorem, the second term above becomes a surface integral over the whole system (see figure below):\n","\\begin{align}\n","\\int_V \\kappa\\nabla\\cdot(\\delta\\phi\\nabla\\phi) \\, dV &= \n","\\oint_{\\partial V} \\kappa \\delta\\phi\\nabla\\phi \\cdot \\hat{\\mathbf{n}}\\,dS \\\\\n","&= \\int_{y=0} \\kappa \\delta\\phi\\nabla\\phi\\cdot(-\\hat{\\mathbf{y}})\\,dS + \n","   \\int_{y=L} \\kappa \\delta\\phi\\nabla\\phi\\cdot\\hat{\\mathbf{y}}\\,dS + \n","\\underbrace{\n","    \\int_{x=0} \\kappa \\delta\\phi\\nabla\\phi\\cdot(-\\hat{\\mathbf{x}})\\,dS + \n","    \\int_{x=L_x} \\kappa \\delta\\phi\\nabla\\phi\\cdot\\hat{\\mathbf{x}}\\,dS \n","}_{=0}.\n","\\end{align}\n","Note that $\\hat{\\mathbf{n}}$ is the outward unit normal to the surface (see figure below). \n","Using periodic boundary condition, the integral over $x=0$ and $x=L_x$ surface vanish.\n","Thus we end up with just the surface integrals over $y=0$ and $y=L$.\n","Finally, the variation in the free energy is then given by:\n","\\begin{align}\n","\\delta F = \\int_V \\bigg[ f'(\\phi) - \\kappa\\nabla^2\\phi \\bigg]\\delta\\phi\\,dV - \n","           \\int_{y=0} \\bigg( \\kappa \\left.\\frac{\\partial\\phi}{\\partial y}\\right|_{y=0} + h \\bigg)\\delta\\phi\\,dS +\n","           \\int_{y=L} \\bigg( \\kappa \\left.\\frac{\\partial\\phi}{\\partial y}\\right|_{y=L} - h \\bigg)\\delta\\phi\\,dS.\n","\\end{align}\n","Now in equilibrium, the variation in the free energy (with respect to $\\delta\\phi$) is equal to zero, _i.e._ $\\delta F=0$.\n","Thus the volume term and the surface terms above should be equal to zero.\n","Equating the volume term to zero, we get:\n","\\begin{equation}\n","f'(\\phi) - \\kappa\\nabla^2\\phi = 0,\n","\\end{equation}\n","which is the same statement as the chemical potential being equal to zero: $\\mu=0$.\n","Now equating the surface terms to zero, we get a boundary condition for $\\phi$ at $y=0$ and $y=L$ respectively:\n","\\begin{equation}\n","\\left.\\frac{\\partial\\phi}{\\partial y}\\right|_{y=0} = -\\frac{h}{\\kappa}, \\quad\\text{and}\\quad\n","\\left.\\frac{\\partial\\phi}{\\partial y}\\right|_{y=L} = \\frac{h}{\\kappa}.\n","\\end{equation}"]},{"cell_type":"code","execution_count":211,"metadata":{},"outputs":[{"data":{"image/png":"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","text/plain":["<IPython.core.display.Image object>"]},"execution_count":211,"metadata":{},"output_type":"execute_result"}],"source":["Image('https://raw.githubusercontent.com/elsentjhung/cahn-hilliard-droplet/master/figures/surface-integral.png')"]},{"cell_type":"markdown","metadata":{},"source":["### Densities at the walls\n","\n","Now using the free energy functional given above, we will calculate the value of the density field $\\phi$ at the bottom wall $y=0$.\n","(The calculation for the top wall is similar.)\n","For simplicity, let us assume a homogenous liquid inside the system.\n","The density of the liquid is equal to $\\sqrt{\\frac{-\\alpha}{\\beta}}$ everywhere in the system \n","(except close to the walls).\n","Assuming $h>0$ for now, the density of the liquid at the walls, $\\phi_{\\text{wall}}$, will be slightly higher than that in the bulk, _i.e._ $\\phi_{\\text{wall}}>\\sqrt{\\frac{-\\alpha}{\\beta}}$ (see figure below).\n","Let us now calculate the value for $\\phi_\\text{wall}$.\n","From the previous section, we know that in equilibrium, the density field $\\phi(y)$ satisfies the equation:\n","\\begin{equation}\n","f'(\\phi) - \\kappa\\frac{d^2\\phi}{dy^2} = 0.\n","\\end{equation}\n","Note that we assume the system to be homogenous along $x$.\n","Now multiplying the above equation with $\\frac{d\\phi}{dy}$, and then integrating over $y$ from $y=0$ to $y=\\frac{L}{2}$, we get the __Noether's equation__:\n","\\begin{align}\n","\\int_0^{\\frac{L}{2}} \\frac{df}{dy} \\, dy - \n","\\frac{\\kappa}{2}\\int_0^{\\frac{L}{2}} \\frac{d}{dy}\\left(\\frac{d\\phi}{dy}\\right)^2 dy &= 0 \\\\\n","f\\left(\\phi\\left(\\frac{L}{2}\\right)\\right) - f(\\phi(y=0)) - \n","\\frac{\\kappa}{2} \n","\\left[ \n","    \\left(\\left.\\frac{d\\phi}{dy}\\right|_{y=\\frac{L}{2}}\\right)^2 - \\left(\\left.\\frac{d\\phi}{dy}\\right|_{y=0}\\right)^2 \n","\\right] &= 0.\n","\\end{align}\n","However, we know that the value of $\\phi(y)$ at $y=0$ is equal to $\\phi_\\text{wall}$ and the first derivative of $\\phi(y)$ at $y=0$ is equal to $-\\frac{h}{\\kappa}$.\n","Furthermore, $\\phi(y)$ at $y=\\frac{L}{2}$ takes the bulk value $\\sqrt{\\frac{-\\alpha}{\\beta}}$ and its derivative is zero since $\\phi$ is constant in the bulk (see figure below).\n","Therefore, the Noether's equation becomes:\n","\\begin{align}\n","f\\left(\\sqrt{\\frac{-\\alpha}{\\beta}}\\right) - f(\\phi_\\text{wall}) + \\frac{h^2}{2\\kappa} = 0.\n","\\end{align}\n","Now we can subsitute the $f(\\phi) = \\frac{\\alpha}{2}\\phi^2 + \\frac{\\beta}{4}\\phi^4$ to the equation above to get a quartic equation:\n","\\begin{align}\n","\\frac{\\alpha}{2}\\phi_\\text{wall}^2 + \\frac{\\beta}{4}\\phi_\\text{wall}^4 + \\frac{\\alpha^2}{4\\beta} - \\frac{h^2}{2\\kappa} = 0.\n","\\end{align}\n","We can then solve this quartic equation to get density of the fluid at the wall (_i.e._ at $y=0$ in this case):\n","\\begin{align}\n","\\phi_\\text{wall} = \\pm\\sqrt{\\frac{-\\alpha}{\\beta}}\\sqrt{1\\pm\\Omega}, \\quad\\text{where}\\quad\n","\\Omega = \\sqrt{\\frac{2\\beta}{\\kappa\\alpha^2}}h.\n","\\end{align}\n","In this case, since the density of the bulk is equal to $\\sqrt{\\frac{-\\alpha}{\\beta}}$ and $h>0$, \n","the density of the fluid at the wall has to be larger than $\\sqrt{\\frac{-\\alpha}{\\beta}}$.\n","Therefore, the appropriate solution to the quartic equation is:\n","\\begin{align}\n","\\phi_\\text{wall} = \\sqrt{\\frac{-\\alpha}{\\beta}}\\sqrt{1+\\Omega}.\n","\\end{align}\n","\n","We can easily generalize the result above to all four different cases: $h>0$, $h<0$, and liquid or gas bulk density.\n","The results are summarized in the table below:\n","\\begin{equation}\n","\\begin{array}{ |c|c|c| } \n","  \\hline\n","   & h>0 & h<0 \\\\ \n","  \\hline\n","  \\text{liquid} & \\phi_\\text{wall}=\\sqrt{\\frac{-\\alpha}{\\beta}}\\sqrt{1+\\Omega} & \\phi_\\text{wall}=\\sqrt{\\frac{-\\alpha}{\\beta}}\\sqrt{1+\\Omega} \\\\ \n","  \\hline\n","  \\text{gas} & \\phi_\\text{wall}=-\\sqrt{\\frac{-\\alpha}{\\beta}}\\sqrt{1-\\Omega} & \\phi_\\text{wall}=-\\sqrt{\\frac{-\\alpha}{\\beta}}\\sqrt{1-\\Omega} \\\\ \n","  \\hline\n","\\end{array},\n","\\end{equation}\n","where,\n","\\begin{equation}\n","\\Omega = \\sqrt{\\frac{2\\beta}{\\kappa\\alpha^2}}h \\quad\\Leftrightarrow\\quad\n","h = \\sqrt{\\frac{\\kappa\\alpha^2}{2\\beta}}\\Omega.\n","\\end{equation}\n"]},{"cell_type":"code","execution_count":217,"metadata":{},"outputs":[{"data":{"image/png":"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","text/plain":["<IPython.core.display.Image object>"]},"execution_count":217,"metadata":{},"output_type":"execute_result"}],"source":["Image('https://raw.githubusercontent.com/elsentjhung/cahn-hilliard-droplet/master/figures/wall-density.png')"]},{"cell_type":"markdown","metadata":{},"source":["### Three surface tensions\n","\n","After knowing the values of $\\phi$ at the walls for the four different cases, we can now compute the three surface tensions: $\\gamma_{lg}$, $\\gamma_{sg}$, and $\\gamma_{sl}$.\n","From the previous previous section, we have derived the liquid-gas surface tension:\n","\\begin{equation}\n","\\gamma_{lg} = \\sqrt{\\frac{-8\\kappa\\alpha^3}{9\\beta^2}}.\n","\\end{equation}\n","We are left two more surface tensions to compute.\n","\n","Recall that the surface tension is the excess free energy of the interface (per surface area of the interface).\n","Therefore, the solid-liquid surface tension is given by (we approximate the bulk $y=\\frac{L}{2}$ to be at infinity):\n","\\begin{equation}\n","\\gamma_{sl} = \\int_0^\\infty \\bigg[ f(\\phi) + \\frac{\\kappa}{2}\\left(\\frac{d\\phi}{dy}\\right)^2 \\bigg]dy - h\\phi_\\text{wall} - \\int_0^\\infty f\\left(\\sqrt{\\frac{-\\alpha}{\\beta}}\\right)\\,dy\n","\\end{equation} \n","We can then use Noether's equation again:\n","\\begin{equation}\n","f(\\phi) - \\frac{\\kappa}{2}\\left(\\frac{d\\phi}{dy}\\right)^2 = f\\left(\\sqrt{\\frac{-\\alpha}{\\beta}}\\right),\n","\\end{equation}\n","so we get:\n","\\begin{align}\n","\\gamma_{sl} &= \\kappa\\int_0^\\infty \\left(\\frac{d\\phi}{dy}\\right)^2 dy - h\\phi_\\text{wall} \\\\\n","&= \\kappa\\int_{\\phi_\\text{wall}}^{\\sqrt{-\\alpha/\\beta}} \\left(\\frac{d\\phi}{dy}\\right) d\\phi - h\\phi_\\text{wall}\n","\\end{align}\n","Using Noether's equation again, we get:\n","\\begin{align}\n","\\gamma_{sl} &= \\sqrt{2\\kappa}\\int_{\\phi_\\text{wall}}^{\\sqrt{-\\alpha/\\beta}} \\sqrt{f(\\phi)+\\frac{\\alpha^2}{4\\beta}}\\,d\\phi - h\\phi_\\text{wall}\n","\\end{align}\n","However, we know that $\\phi_\\text{wall}=\\sqrt{\\frac{-\\alpha}{\\beta}}\\sqrt{1+\\Omega}$, thus,\n","\\begin{align}\n","\\gamma_{sl} &= \\sqrt{2\\kappa}\\int_{\\sqrt{\\frac{-\\alpha}{\\beta}}\\sqrt{1+\\Omega}}^{\\sqrt{\\frac{-\\alpha}{\\beta}}} \\sqrt{f(\\phi)+\\frac{\\alpha^2}{4\\beta}}\\,d\\phi - \\sqrt{\\frac{\\kappa\\alpha^2}{2\\beta}}\\Omega\\sqrt{\\frac{-\\alpha}{\\beta}}\\sqrt{1+\\Omega}.\n","\\end{align}\n","Substituting $f(\\phi)=\\frac{\\alpha}{2}\\phi^2+\\frac{\\beta}{4}\\phi^4$, we can evaluate the integral:\n","\\begin{align}\n","\\gamma_{sl} &= \\sqrt{\\frac{-2\\kappa\\alpha^3}{9\\beta^2}}\\left[1-\\left(1-\\frac{\\Omega}{2}\\right)\\sqrt{1+\\Omega}\\right] -\n","\\sqrt{\\frac{-\\kappa\\alpha^3}{2\\beta^2}}\\Omega\\sqrt{1+\\Omega} \\\\\n","&= \\frac{\\gamma_{lg}}{2}\\left[1-\\left(1-\\frac{\\Omega}{2}\\right)\\sqrt{1+\\Omega}\\right] - \n","\\frac{3\\gamma_{lg}}{4}\\Omega\\sqrt{1+\\Omega} \\\\\n","&= \\frac{\\gamma_{lg}}{2} - \\frac{\\gamma_{lg}}{2}(1+\\Omega)^{3/2}.\n","\\end{align}\n","\n","Similarly, we can compute the solid-gas surface tension:\n","\\begin{align}\n","\\gamma_{sg} &= \\sqrt{2\\kappa}\\int_{-\\sqrt{\\frac{-\\alpha}{\\beta}}\\sqrt{1-\\Omega}}^{-\\sqrt{\\frac{-\\alpha}{\\beta}}} \\sqrt{f(\\phi)+\\frac{\\alpha^2}{4\\beta}}\\,d\\phi + \\sqrt{\\frac{\\kappa\\alpha^2}{2\\beta}}\\Omega\\sqrt{\\frac{-\\alpha}{\\beta}}\\sqrt{1-\\Omega} \\\\\n","&= \\sqrt{\\frac{-2\\kappa\\alpha^3}{9\\beta^2}}\\left[1 - \\left(1+\\frac{\\Omega}{2}\\right)\\sqrt{1-\\Omega}\\right] +\n","\\sqrt{\\frac{-\\kappa\\alpha^3}{2\\beta^2}}\\Omega\\sqrt{1-\\Omega} \\\\\n","&= \\frac{\\gamma_{lg}}{2} - \\frac{\\gamma_{lg}}{2}(1-\\Omega)^{3/2}.\n","\\end{align}\n","\n","Finally, applying the Young's equation, we get:\n","\\begin{align}\n","\\cos\\theta = \\frac{\\gamma_{sg}-\\gamma_{sl}}{\\gamma_{lg}} = \n","\\frac{1}{2}\\left[ (1+\\Omega)^{3/2} - (1-\\Omega)^{3/2} \\right].\n","\\end{align}\n","The equation above can be inverted (??) to get $\\Omega$ in terms of $\\theta$:\n","\\begin{equation}\n","\\Omega = 2 \\,\\text{sign}\\left(\\frac{\\pi}{2}-\\theta\\right)\n","\\sqrt{ \\cos\\left(\\frac{\\alpha}{3}\\right) \\left[ 1 - \\cos\\left(\\frac{\\alpha}{3}\\right) \\right]},\n","\\quad\\text{where}\\quad\n","\\alpha = \\cos^{-1}\\left(\\sin^2\\theta\\right).\n","\\end{equation}\n","Thus for a given contact angle $\\theta$, we can then find the appropriate value of $h$:\n","\\begin{equation}\n","h = \\sqrt{\\frac{2\\kappa\\alpha^2}{\\beta}} \\,\\text{sign}\\left(\\frac{\\pi}{2}-\\theta\\right)\n","\\sqrt{ \\cos\\left(\\frac{\\cos^{-1}\\left(\\sin^2\\theta\\right)}{3}\\right) \\left[ 1 - \\cos\\left(\\frac{\\cos^{-1}\\left(\\sin^2\\theta\\right)}{3}\\right) \\right]}\n","\\end{equation}"]},{"cell_type":"markdown","metadata":{},"source":["### Computer simulations\n","\n","Now we will put everything we have learnt above into practice.\n","Let us simulate a liquid droplet sitting on a flat solid surface with a given contact angle $\\theta$.\n","Our dynamical variable is the density field $\\phi(\\mathbf{r},t)$.\n","In computer simulations, space is discretized into: $x\\rightarrow i\\Delta x$ and $x\\rightarrow j\\Delta y$,\n","where $i=0,1,2,\\dots,N_x-1$ and $j=0,1,2,\\dots,N_y-1$.\n","$\\Delta x$ and $\\Delta y$ are the lattice spacing in the $x$- and $y$-direction respectively.\n","Ideally we require $\\Delta x$ and $\\Delta y$ to be much smaller than $1$.\n","The system size is then $L_x=N_x\\Delta x$ and $L_y=N_y\\Delta y$.\n","Similarly, time is also discretized into $t\\rightarrow n\\Delta t$, where $n=0,1,2,\\dots,N_t-1$, where $\\Delta t$ is the time step.\n","The total time of the simulation is then $N_t\\Delta t$.\n","Thus the density field becomes $\\phi(\\mathbf{r},t)\\rightarrow\\phi^n_{ij}$, which is represented as an array in Python:\n","\\begin{align}\n","\\phi_{ij} &=\n","\\begin{pmatrix}\n","\\phi_{00} & \\phi_{01} & \\ldots & \\phi_{0,N_y-1} \\\\\n","\\phi_{10} & \\phi_{11} &  & \\phi_{1,N_y-1} \\\\\n","\\vdots &  & \\ddots & \\vdots \\\\\n","\\phi_{N_x-1,0} & \\phi_{N_x-1,1} & \\ldots & \\phi_{N_x-1,N_y-1} \\\\\n","\\end{pmatrix} \\downarrow i\\text{-direction} \\\\\n","&\\quad\\quad\\quad\\quad\\quad\\longrightarrow j\\text{-direction}\n","\\end{align}\n","Note that the $x$- and $y$-axis are transposed.\n","The first and second derivatives are computed numerically as follow:\n","\\begin{align}\n","\\frac{\\partial\\phi}{\\partial x} &= \\frac{\\phi_{i+1,j}-\\phi_{i,j}}{\\Delta x} + \\mathcal{O}(\\Delta x) \\\\\n","\\frac{\\partial^2\\phi}{\\partial x^2} &= \\frac{\\phi_{i+1,j}-2\\phi_{i,j}+\\phi_{i-1,j}}{\\Delta x^2} + \\mathcal{O}(\\Delta x)\n","\\end{align}\n","The array $\\phi_{i+1,j}$ is then equivalent to shifting every elements inside the array $\\phi_{ij}$ upwards.\n","This is represented by the `np.roll` function in Python:\n","\n","`phi_i_plus_1 = np.roll(phi, -1, axis=0)`\n"]},{"cell_type":"code","execution_count":77,"metadata":{},"outputs":[],"source":["import numpy as np\n","import matplotlib.pyplot as plt\n","\n","class Wetting():\n","    # initialization\n","    def __init__(self, theta):\n","        # lattice parameters\n","        self.dx, self.dy = 1.0, 1.0 \n","        self.Nx, self.Ny = 64, 32\n","        self.dt = 0.01 \n","        self.Nt = 10000000\n","\n","        # parameters of the free energy\n","        self.M, self.alpha, self.beta, self.kappa = 1.0, -1.0, 1.0, 1.0 \n","\n","        # contact angle in radian\n","        self.theta = theta \n","        a = np.arccos(np.sin(self.theta)**2)\n","        self.h = np.sqrt(2.0*self.kappa*a**2/self.beta) \\\n","               * np.sign(0.5*np.pi-self.theta) \\\n","               * np.sqrt(np.cos(a/3.0)*(1.0-np.cos(a/3.0)))\n","\n","        # global variables\n","        self.phi = np.zeros((self.Nx, self.Ny))\n","        self.mu = np.zeros((self.Nx, self.Ny))\n","\n","        # array of cartesian coordinates (needed for plotting)\n","        self.x = np.arange(0, self.Nx)*self.dx\n","        self.y = np.arange(0, self.Ny)*self.dx\n","        self.y, self.x = np.meshgrid(self.y, self.x) \n","\n","        # initialize phi to be a half-spherical droplet\n","        radius = self.Nx*self.dx/4.0\n","        for i in range(0, self.Nx, 1):\n","            for j in range(0, self.Ny, 1):\n","                if (i*self.dx-0.5*self.Nx*self.dx)**2 + (j*self.dx)**2 < radius**2:\n","                    self.phi[i,j] = np.sqrt(-self.alpha/self.beta)\n","                else:\n","                    self.phi[i,j] = -np.sqrt(-self.alpha/self.beta)\n","\n","    # function to calculate second derivative\n","    def derivative_x_x(self, field):\n","        field_plus = np.roll(field, -1, axis=0)\n","        field_minus = np.roll(field, +1, axis=0)\n","        return (field_plus - 2*field + field_minus)/(self.dx**2)\n","\n","    def derivative_y_y(self, field):\n","        field_plus = np.roll(field, -1, axis=1)\n","        field_minus = np.roll(field, +1, axis=1)\n","\n","        # wall boundary condition\n","        if np.array_equiv(field, self.phi):\n","            field_minus[:,0] = field[:,0] + self.dx*self.h/self.kappa\n","            field_plus[:,self.Ny-1] = field[:,self.Ny-1]\n","        else:\n","            field_minus[:,0] = field[:,0]\n","            field_plus[:,self.Ny-1] = field[:,self.Ny-1]\n","\n","        return (field_plus - 2*field + field_minus)/(self.dy**2)\n","\n","    def laplacian(self, field):\n","        return self.derivative_x_x(field) + self.derivative_y_y(field)\n","\n","    # function to update phi\n","    def update(self):\n","        self.mu = self.alpha*self.phi + self.beta*self.phi*self.phi*self.phi - self.kappa*self.laplacian(self.phi)\n","        self.phi = self.phi + self.dt*self.M*self.laplacian(self.mu)\n","\n","    # simulation run\n","    def run(self):\n","        for nt in range(0, self.Nt, 1):\n","            self.update()\n","            if nt % 1000000 == 0:\n","                phi0 = np.sum(self.phi)/(self.Nx*self.dx*self.Ny*self.dy)\n","                print(f't = {nt*self.dt}, phi0 = {phi0}')"]},{"cell_type":"code","execution_count":78,"metadata":{},"outputs":[{"name":"stdout","output_type":"stream","text":["t = 0.0, phi0 = -0.59765625\n","t = 10000.0, phi0 = -0.5976562499999795\n","t = 20000.0, phi0 = -0.5976562499999944\n","t = 30000.0, phi0 = -0.5976562499949978\n","t = 40000.0, phi0 = -0.5976562499932141\n","t = 50000.0, phi0 = -0.5976562499823828\n","t = 60000.0, phi0 = -0.5976562499550506\n","t = 70000.0, phi0 = -0.597656249932575\n","t = 80000.0, phi0 = -0.5976562499109468\n","t = 90000.0, phi0 = -0.5976562498893808\n"]}],"source":["# hydrophilic case theta = 45 degrees\n","hydrophilic = Wetting(np.pi/4.0)\n","hydrophilic.run()"]},{"cell_type":"code","execution_count":91,"metadata":{},"outputs":[{"name":"stdout","output_type":"stream","text":["t = 0.0, phi0 = -0.59765625\n","t = 10000.0, phi0 = -0.5976562499999762\n","t = 20000.0, phi0 = -0.5976562499984726\n","t = 30000.0, phi0 = -0.5976562499917414\n","t = 40000.0, phi0 = -0.597656249988029\n","t = 50000.0, phi0 = -0.597656249988029\n","t = 60000.0, phi0 = -0.597656249988029\n","t = 70000.0, phi0 = -0.597656249988029\n","t = 80000.0, phi0 = -0.597656249988029\n","t = 90000.0, phi0 = -0.597656249988029\n"]}],"source":["# hydrophobic case theta = 135 degrees\n","hydrophobic = Wetting(3.0*np.pi/4.0)\n","hydrophobic.run()"]},{"cell_type":"code","execution_count":109,"metadata":{},"outputs":[{"data":{"image/png":"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","text/plain":["<Figure size 432x576 with 2 Axes>"]},"metadata":{"needs_background":"light"},"output_type":"display_data"}],"source":["# initialize figure and movie objects\n","fig, ax = plt.subplots(2, 1, figsize=(6,8)) \n","\n","for n in range(0, 2, 1):\n","    # set label\n","    ax[n].set_xlabel('$x$', fontsize=14)\n","    ax[n].set_ylabel('$y$', fontsize=14)\n","\n","    # set x range and y range\n","    ax[n].set_xlim(0, hydrophilic.Nx*hydrophilic.dx)\n","    ax[n].set_ylim(0, hydrophilic.Ny*hydrophilic.dy)\n","\n","    # set tick interval\n","    ax[n].tick_params(axis='both')\n","    ax[n].set_xticks(np.arange(0, hydrophilic.Nx, 10)*hydrophilic.dx)\n","    ax[n].set_yticks(np.arange(0, hydrophilic.Ny, 10)*hydrophilic.dy)\n","\n","    # set aspect ratio\n","    ax[n].set_aspect('equal')\n","\n","# create colormap of phi\n","ax[0].set_title('$\\\\theta=45^\\circ$')\n","colormap = ax[0].pcolormesh(hydrophilic.x, hydrophilic.y, hydrophilic.phi, shading='auto', vmin=-1.2, vmax=1.2)\n","ax[0].annotate('', xy=(5.5,0), xytext=(5.5+20,0+20), arrowprops=dict(edgecolor='red', arrowstyle='-', linestyle='--', linewidth=1))\n","ax[1].set_title('$\\\\theta=135^\\circ$')\n","colormap = ax[1].pcolormesh(hydrophilic.x, hydrophilic.y, hydrophobic.phi, shading='auto', vmin=-1.2, vmax=1.2)\n","ax[1].annotate('', xy=(24,0), xytext=(24-20,0+20), arrowprops=dict(edgecolor='red', arrowstyle='-', linestyle='--', linewidth=1))\n","\n","plt.show()"]},{"cell_type":"code","execution_count":null,"metadata":{},"outputs":[],"source":[]},{"cell_type":"markdown","metadata":{},"source":["References:\n","1. L. D. Landau and E. M. Lifshitz, _Theory of Elasticity, 3rd Edition_ (Elsevier, 1986).\n","2. M. E. Cates and E. Tjhung, Theory of binary fluid mixtures: from phase-separation kinetics to active emulsions, _J. Fluid Mech._ (2017).\n","3. T. Kruger, H. Kusumaatmaja, A. Kuzmin, O. Shardt, G. Silva, and E. M. Viggen, _The Lattice Boltzmann Method_ (Springer, 2017)"]}],"metadata":{"kernelspec":{"display_name":"Python 3","language":"python","name":"python3"},"language_info":{"codemirror_mode":{"name":"ipython","version":3},"file_extension":".py","mimetype":"text/x-python","name":"python","nbconvert_exporter":"python","pygments_lexer":"ipython3","version":"3.9.7"},"vscode":{"interpreter":{"hash":"cae271c71fdd3cc871ab10677ba94a2a4bc532d008bc675b95ab71698b6079a6"}}},"nbformat":4,"nbformat_minor":2}