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"# Problem set 6\n",
"\n",
"## Problem 6.1\n",
"\n",
"1. Salt ions disrupt the structure of the water molecules, and\n",
" stimulate the nucleation process. For the formation of a new phase\n",
" (here, a gas bubble) an energy barrier needs to be overcome. Small\n",
" particles or a dirty pan can lower the barrier of the nucleation\n",
" process locally. (this is actually a difficult question, and this\n",
" answer is not the full story)\n",
"\n",
"2. To prevent the formation of many different crystalline domains,\n",
" which gives an undesirable texture and look, and to make sure that\n",
" one ends up with the ’desirable’ crystalline phase (which forms\n",
" around 32$^\\mathrm{o}$C) which is the shiniest and makes the best\n",
" sound when it is broken.\n",
"\n",
"3. Important is to use a clean bottle, to cool rapidly, and to cool\n",
" deep enough, and not too long, to make sure that the nucleation\n",
" process does not take over.\n",
"\n",
"4. The formation and growth of ice crystals cuts organic material into\n",
" pieces at a microscopic scale. And if that is not harmful enough, it\n",
" also changes the osmotic pressure in the remaining fluid phase, such\n",
" that cells can dry out or burst.\n",
"\n",
"5. If one minimises $G$ one finds that\n",
"\n",
" $$\\frac{\\partial G}{\\partial R} = 0$$\n",
"\n",
" implies\n",
"\n",
" $$R_\\mathrm{crit} = \\frac{2 \\gamma}{\\Delta G}$$\n",
"\n",
" How the surface tension $\\gamma$ and change in free energy\n",
" $\\Delta G$ are calculated is a story for another time.\n",
"\n",
"## Problem 6.2\n",
"\n",
"1. For an ideal gas\n",
"\n",
" $$p = k_\\mathrm{B}T \\rho$$\n",
"\n",
" and a Van der Waals fluid\n",
"\n",
" $$p = k_\\mathrm{B}T \\frac{\\rho}{1-b\\rho} - a \\rho^2$$\n",
"\n",
"2. See plot at the bottom of this page.\n",
"\n",
"3. See lecture (slides)\n",
"\n",
"4. Anywhere where there are multiple densities with the same pressure,\n",
" one could expect a phase coexistence (draw a horizontal line in the\n",
" $p$ vs. $\\rho$ curve). However, the regime where the pressure\n",
" decreases with density should not be counted, because these\n",
" densities unstable (see lecture)\n",
"\n",
"5. For an ideal gas\n",
"\n",
" $$\\mu = k_\\mathrm{B}T\\ln{\\rho}$$\n",
"\n",
" and for a Van der Waals fluid\n",
"\n",
" $$\\mu = k_\\mathrm{B}T\\ln\\frac{\\rho}{1-b\\rho} + \\frac{k_\\mathrm{B}T}{1-b\\rho} - 2a\\rho$$\n",
"\n",
"6. See plot at the bottom of this page.\n",
"\n",
"7. Yes, if there are multiple densities with the same $\\mu$ one can\n",
" expect a phase coexistence. The regime where $\\mu$ decreases with\n",
" density should not be counted (see part d and lecture slides)\n",
"\n",
"8. This follows quickly after substituting $\\frac{N}{V}$ by $\\rho$.\n",
"\n",
"9. For the pressure (use the sum rule and the chain rule):\n",
"\n",
" $$p = - \\frac{\\partial}{\\partial V} Vf = - f - V \\frac{\\partial f}{\\partial V} = -f - V \\frac{\\partial \\rho}{\\partial V}\\frac{\\partial f}{\\partial \\rho} = \\rho \\frac{\\partial f}{\\partial \\rho} - f$$\n",
"\n",
" and for $\\mu$ similarly:\n",
"\n",
" $$\\mu = \\frac{\\partial}{\\partial N} Vf = V \\frac{\\partial f}{\\partial N} = V \\frac{\\partial \\rho}{\\partial N}\\frac{\\partial f}{\\partial \\rho} = \\frac{\\partial f}{\\partial \\rho}$$\n",
"\n",
"10. It may be obvious that the chemical potentials are the same, because\n",
" $\\mu = \\frac{\\partial f}{\\partial \\rho}$ is the slope of the curve,\n",
" and the two points obviously have the same slope:\n",
"\n",
" $$\\mu_1 = f'(\\rho_1) = f'(\\rho_2) = \\mu_2$$\n",
"\n",
" For $p_1 = p_2$ another condition has to be met. If that is the\n",
" case, then\n",
"\n",
" $$p_1 = \\rho_1 f'(\\rho_1) - f(\\rho_1) = \\rho_2 f'(\\rho_2) - f(\\rho_2) = p_2$$\n",
"\n",
" Swapping some terms, gives\n",
"\n",
" $$\\rho_1 f'(\\rho_1) - \\rho_2 f'(\\rho_2) = f(\\rho_1) - f(\\rho_2)$$\n",
"\n",
" but we know that $f'(\\rho_1)=f'(\\rho_2)$, so\n",
"\n",
" $$f'(\\rho_1) = f'(\\rho_2) = \\frac{f(\\rho_1) - f(\\rho_2)}{\\rho_1 - \\rho_2}$$\n",
"\n",
" This statement tells us that for the pressures to be the same, the\n",
" points need to have the same slope, and in addition to that, that if\n",
" the two points are connected with a straight line, that this line\n",
" has to have the same slope as well. This can be done very easily by\n",
" putting a ruler against the curve of $f(\\rho)$, as shown in\n",
" {numref}`fig:free_energy_density`."
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"import numpy as np\n",
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" return mu\n",
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"rho = np.linspace(0.01, 0.99, 200)\n",
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" line=dict(color=\"#ff5638\"),\n",
" mode='lines',\n",
" name='VdW fluid'), row=1, col=1)\n",
"\n",
" fig.add_trace( \n",
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" x=rho,\n",
" y=chempot(rho, a, 1),\n",
" line=dict(color=\"#ff5638\"),\n",
" mode='lines',\n",
" name='VdW fluid',\n",
" showlegend=False), row=1, col=2)\n",
"\n",
"traces_per_step = 2 # Number of traces per value of a\n",
"\n",
"# Show the traces for one value of a when loading the plot (initial setup)\n",
"active_a_index = 10\n",
"fig.data[active_a_index + base_traces].visible = True\n",
"fig.data[active_a_index + base_traces + 1].visible = True\n",
"\n",
"# Create and add slider\n",
"steps = []\n",
"for i in range(0, range_a.shape[0]):\n",
" visarray = [False] * len(fig.data)\n",
" visarray[0:base_traces] = [True] * base_traces\n",
" curr_idx = int(base_traces + i * traces_per_step)\n",
" next_idx = int(base_traces + (i+1) * traces_per_step)\n",
" visarray[curr_idx:next_idx] = [True] * traces_per_step\n",
" step = dict(\n",
" method=\"update\",\n",
" args=[{\"visible\": visarray}],\n",
" label=range_a[i]\n",
" )\n",
" steps.append(step)\n",
"\n",
"sliders = [dict(\n",
" active=active_a_index,\n",
" currentvalue={\"prefix\": \"a/b: \"},\n",
" steps=steps\n",
")]\n",
"\n",
"fig.update_layout(\n",
" sliders=sliders,\n",
" legend_title=\"Legend\",\n",
" height=400\n",
")\n",
"\n",
"fig.update_xaxes(title_text=r'$\\rho$', row=1, col=1)\n",
"fig.update_xaxes(title_text=r'$\\rho$', row=1, col=2)\n",
"\n",
"# Update yaxis properties\n",
"fig.update_yaxes(title_text=r'$p$', range=[0, 1], row=1, col=1)\n",
"fig.update_yaxes(title_text=r'$\\mu$', range=[-5.5, 0.25], row=1, col=2)\n",
"\n",
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