Ticket #1234: Jeffery Orbits.ipynb

File Jeffery Orbits.ipynb, 109.1 KB (added by pkienzle, 18 months ago)

corrected filename

Line 
1{
2 "cells": [
3  {
4   "cell_type": "markdown",
5   "metadata": {
6    "colab_type": "text",
7    "id": "eNi--SgpRsBA"
8   },
9   "source": [
10    "## Jeffery Orbits\n",
11    "\n",
12    "### Introduction\n",
13    "\n",
14    "Demonstration of angle for ellipsoids tumbling in shear from using equations from Stover and Cohen (1990).  \n",
15    "\n",
16    "\n",
17    "<table>\n",
18    "    <!-- \n",
19    "    <tr><td>\n",
20    "    <img src=\"Stover1990_Fig1.png\" width=\"320\"/>\n",
21    "    </td><td>\n",
22    "    <img src=\"Stover1990_Fig2.png\" width=\"300\"/>\n",
23    "    </td></tr> \n",
24    "    -->\n",
25    "    <tr><td>\n",
26    "    <img src=\"https://docs.google.com/uc?export=download&id=1KdqU_9-pH5pCShPPZ6-rJJAeZaOFF_EX\" width=\"320\"/>\n",
27    "    </td><td>\n",
28    "    <img src=\"https://docs.google.com/uc?export=download&id=17MVbd4FUyr3VPgDOVhO52TMlpnkQgiSa\" width=\"300\"/>\n",
29    "    </td></tr>\n",
30    "    <tr><td colspan=\"2\">Fig. 1 and Fig. 2 from Stover and Cohen (1990) showing the definition of the coodinate system and Jeffery orbits for different values of $C$</td></tr>\n",
31    "</table>\n",
32    "\n",
33    "Tumble period $T$ depends on effective aspect ratio $r_e$ and shear rate $\\dot\\gamma$ as\n",
34    "\n",
35    "$$\n",
36    "T = \\frac{2\\pi}{\\dot\\gamma}\\left(r_e + \\frac{1}{r_e}\\right)\n",
37    "$$\n",
38    "\n",
39    "For ellipsoids, $r_e$ is length/diameter.  For other particles it can be determined experimentally or by simulations, such as those in Ingber (1994).\n",
40    "\n",
41    "From tumble rate we can get angle as a function of time as\n",
42    "\n",
43    "$$\n",
44    "\\tan \\phi = r_e \\tan\\left(\\frac{2\\pi t}{T} + \\kappa\\right)\n",
45    "$$\n",
46    "\n",
47    "and\n",
48    "\n",
49    "$$\n",
50    "\\tan \\theta = \\frac{C r_e}{\\sqrt{r_e^2 \\cos^2 \\phi + \\sin^2\\phi}}\n",
51    "$$\n",
52    "\n",
53    "where $\\kappa$ is the phase angle determined from the initial orientation of the particle.\n",
54    "\n",
55    "$C$ is the orbit constant which governs the evolution of $\\theta$ and $\\phi$ over time (Fig. 2 above).  From Stover and Cohen (1990), \"If Jeffery's assumptions are not violated, the value of $C$ is constant and the particle follows one of these trajectories indefinitely. $C$ is zero when the particle is permanently aligned with the vorticity axis and infinity when the particle lies in the $x$-$y$ plane.\"\n",
56    "\n",
57    "A full treatment requires acknowledging wall effects, particle inertia, medium viscosity, particle interactions, *etc.*, but in the idealized world of the dilute limit we will ignore these.\n",
58    "\n",
59    "\n",
60    "### References\n",
61    "\n",
62    "[Stover1990]: Stover, C.A., Cohen, C., 1990. *The motion of rodlike particles in the pressure-driven flow between two flat plates.* Rheologica Acta **29**, 192–203. https://doi.org/10.1007/BF01331355\n",
63    "\n",
64    "[Ingber1994]: Ingber, M.S., Mondy, L.A., 1994. *A numerical study of three‐dimensional Jeffery orbits in shear flow.* Journal of Rheology **38**, 1829–1843. https://doi.org/10.1122/1.550604\n",
65    "\n",
66    "[Jeffery1922]: Jeffery, G.B., 1922. *The Motion of Ellipsoidal Particles Immersed in a Viscous Fluid.* Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences **102**, 161–179. https://doi.org/10.1098/rspa.1922.0078"
67   ]
68  },
69  {
70   "cell_type": "code",
71   "execution_count": 0,
72   "metadata": {
73    "colab": {},
74    "colab_type": "code",
75    "id": "FIufCJ8sRsBC"
76   },
77   "outputs": [],
78   "source": [
79    "import matplotlib\n",
80    "import numpy as np\n",
81    "from numpy import pi, sin, cos, tan, arctan, arctan2, sqrt, unwrap, degrees, radians\n",
82    "import matplotlib.pyplot as plt\n",
83    "%matplotlib inline"
84   ]
85  },
86  {
87   "cell_type": "code",
88   "execution_count": 0,
89   "metadata": {
90    "colab": {},
91    "colab_type": "code",
92    "id": "ox6F4_WERsBG"
93   },
94   "outputs": [],
95   "source": [
96    "def jeffery(t, re, shear, C=1, phase=0, t_is_norm=True):\n",
97    "    T = 2*pi/shear*(re + 1./re)\n",
98    "    if t_is_norm:\n",
99    "        t_norm, t = t, t*T\n",
100    "    else:\n",
101    "        t_norm, t = t/T, t\n",
102    "    x = 2*pi*t_norm + phase\n",
103    "    phi = arctan(re*tan(x))\n",
104    "    theta = arctan(C*re/sqrt(re**2*cos(phi)**2 + sin(phi)**2))\n",
105    "    #theta = arctan(C*sqrt(re**2*sin(x)**2 + cos(x)**2))\n",
106    "    #phi, theta = degrees(unwrap(phi)), degrees(unwrap(theta))\n",
107    "    return t, phi, theta"
108   ]
109  },
110  {
111   "cell_type": "markdown",
112   "metadata": {
113    "colab_type": "text",
114    "id": "ttmcNVTXRsBI"
115   },
116   "source": [
117    "### Evolution of Jeffrey orbits\n",
118    "\n",
119    "Twiddle orbit constant ```C``` = $C$, ```phase``` = $\\kappa$, aspect ratio ```re``` = $r_e$ and ```shear``` = $\\dot\\gamma$ below.  Using normalized time, ```t_over_T```, we can specify the number of periods to plot.  If using just time then it is harder to see that line shape depends only on aspect ratio."
120   ]
121  },
122  {
123   "cell_type": "code",
124   "execution_count": 3,
125   "metadata": {
126    "colab": {
127     "base_uri": "https://localhost:8080/",
128     "height": 313
129    },
130    "colab_type": "code",
131    "id": "wMHQuUQ3RsBM",
132    "outputId": "b4d4c5b8-1f52-423b-cc5a-a7c3087295e7"
133   },
134   "outputs": [
135    {
136     "data": {
137      "text/plain": [
138       "<matplotlib.legend.Legend at 0x7f29785a2438>"
139      ]
140     },
141     "execution_count": 3,
142     "metadata": {
143      "tags": []
144     },
145     "output_type": "execute_result"
146    },
147    {
148     "data": {
149      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYcAAAEXCAYAAABGeIg9AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nIVSIziGEEEKF/wE6ziBGuPWgjAAAAABJRU5ErkJggg==\n",
150      "text/plain": [
151       "<Figure size 432x288 with 1 Axes>"
152      ]
153     },
154     "metadata": {
155      "tags": []
156     },
157     "output_type": "display_data"
158    }
159   ],
160   "source": [
161    "t_over_T = np.linspace(0, 1, 400)\n",
162    "C = 0.5\n",
163    "phase = 0.\n",
164    "re = 8.4\n",
165    "shear = 30\n",
166    "t_s, phi, theta = jeffery(t_over_T, re, shear, C=C, phase=radians(phase))\n",
167    "t = t_over_T if 1 else t_s    \n",
168    "plt.plot(t, degrees(phi), label=r'$\\phi$')\n",
169    "plt.plot(t, degrees(theta), label=r'$\\theta$')\n",
170    "if t is t_over_T:\n",
171    "    plt.xlabel('normalized time (t/T)')\n",
172    "else:\n",
173    "    plt.xlabel(r'time (s)')\n",
174    "plt.ylabel(r'angle ($ ^\\circ$)')\n",
175    "plt.title(r'$r_e = %g, C = %g$'%(re, C))\n",
176    "plt.legend()\n"
177   ]
178  },
179  {
180   "cell_type": "markdown",
181   "metadata": {
182    "colab_type": "text",
183    "id": "jo-0vhK2VBMg"
184   },
185   "source": [
186    "### Simulation of $\\theta_d$ and $\\phi_d$ distribution\n",
187    "\n",
188    "Generate an initial population of $\\theta$ and $\\phi$ with equal probability in each direction.  In practice, this means that small values of $\\theta$ will be rare because the line of latitude near the pole is short.  Use $\\arccos(U[-1,1])$ where $U[-1,1]$ is the uniform distribution over $[-1, 1]$.  This does covers a half circle corresponding to lines of latitude.\n",
189    "$\\phi$ is uniform over the lines of latitude.\n",
190    "\n",
191    "Next find $C$ and $\\kappa$ from $\\theta$ and $\\phi$ assuming we are at time $t=0$.  This is a simple inversion of the $\\tan \\phi$ and $\\tan \\theta$ expressions above.  Note that $C$ will be negative for $\\theta < 0$.\n",
192    "\n",
193    "Now forward propagate the system to random time $t$, unifomly covering some multiple of the period.   Can't use a fixed time $t$ since $t$ reverts to its original scrambled state every period.  In practice, there will be random phase shifts over time (and switching to neighbouring $C$ orbits) so the time average should be a reasonable representation.   Besides, except for extremely long periods our measurement should be the time average of the system, and so it doesn't matter where each particle is along its orbit, just the distribution of orbits.\n",
194    "\n",
195    "Histogram the population of the forward propagated $\\theta$ and $\\phi$ showing both correlation and independent histograms.  As we see below, regardless of what $r_e$ we select we still get $\\theta$ and $\\phi$ acting independently.\n",
196    "\n",
197    "Note that the histograms from the simulation represent the true weights of $\\theta$ and $\\phi$.  When using this within *sasmodels*, which does a correction for latitude in the distribution, need to set $\\theta_d = (90-\\theta)$ and modify the weight of the $\\Delta\\theta_d$ distribution by $1/\\cos(\\theta_d)$ compared to whatever analytic form the distributions below exhibit.\n",
198    "\n",
199    "The plots below use $\\phi$ instead of $\\phi_d$ to show that there is a $90^\\circ$ shift when the long axis switches between polar and equatorial radius.  They use $\\theta_d$ instead of $\\theta$ because $\\theta = 0^\\circ$ is perpendicular to the flow in this coordinate system, and it makes the plots confusing having $P(\\theta) = 0$ at the center.  The preferred orientation for theta is along the flow regardless of aspect ratio, so there is no advantage to displaying $\\theta$."
200   ]
201  },
202  {
203   "cell_type": "code",
204   "execution_count": 20,
205   "metadata": {
206    "colab": {
207     "base_uri": "https://localhost:8080/",
208     "height": 283
209    },
210    "colab_type": "code",
211    "id": "CoN2Puw8Zm1c",
212    "outputId": "a1bb05d3-cb18-4eb9-ed7e-b1ccd74bb4f3"
213   },
214   "outputs": [
215    {
216     "data": {
217      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAY8AAAEKCAYAAADq59mMAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nwIEDBw7uG07wcODAgQMH943/H/o0n1q0UbNTAAAAAElFTkSuQmCC\n",
218      "text/plain": [
219       "<Figure size 432x288 with 3 Axes>"
220      ]
221     },
222     "metadata": {
223      "tags": []
224     },
225     "output_type": "display_data"
226    }
227   ],
228   "source": [
229    "re = 3\n",
230    "n_pop = 1000000\n",
231    "theta_pop = np.arccos(np.random.rand(n_pop)*2-1)\n",
232    "phi_pop = np.random.rand(n_pop)*2*pi\n",
233    "C_pop = tan(theta_pop)*sqrt(cos(phi_pop)**2 + sin(phi_pop)**2/re**2)\n",
234    "k_pop = arctan(tan(phi_pop)/re)\n",
235    "t_pop = np.random.rand(n_pop)\n",
236    "#_ = plt.hist(np.log10(C_pop), bins=200, density=True)\n",
237    "#_ = plt.hist(k_pop, bins=200, density=True)\n",
238    "t, phi_next, theta_next = jeffery(t_pop, re, shear=1, C=C_pop, phase=k_pop)\n",
239    "if re >= 1: phi_next[phi_next<0]+=pi # phase unwrap\n",
240    "#if re >= 1: phi_next -= pi/2  # phi => phi_d\n",
241    "phi_lim = [0, 180] if re>=1 else [-90, 90]\n",
242    "theta_next = pi/2 - theta_next  # theta => theta_d\n",
243    "theta_next[theta_next >= pi/2] -= pi # phase unwrap\n",
244    "theta_lim = [-90, 90]\n",
245    "weights = np.ones_like(theta_next)#/cos(theta_next)\n",
246    "phi_ticks = np.arange(phi_lim[0], phi_lim[1]+1, 30)\n",
247    "theta_ticks = np.arange(theta_lim[0], theta_lim[1]+1, 30)\n",
248    "plt.subplot(221)\n",
249    "plt.hist(degrees(phi_next), bins=200, density=True, weights=weights)\n",
250    "plt.xticks([])\n",
251    "plt.ylabel(r'$P(\\phi)$')\n",
252    "plt.xlim(phi_lim)\n",
253    "plt.subplot(224)\n",
254    "plt.hist(degrees(theta_next), bins=200, density=True, weights=weights, orientation='horizontal')\n",
255    "plt.yticks([])\n",
256    "plt.xlabel(r'$P(\\theta_d)$')\n",
257    "plt.ylim(theta_lim)\n",
258    "plt.subplot(223)\n",
259    "# Use log density to show more detail\n",
260    "from matplotlib import colors\n",
261    "plt.hist2d(degrees(phi_next), degrees(theta_next), bins=200, weights=weights, norm=colors.LogNorm())\n",
262    "#plt.xlabel(r'$\\phi_d$ for $\\phi$ = %d ($^\\circ$)'%(90 if re > 1.0 else 0))\n",
263    "plt.xlabel(r'$\\phi$ ($^\\circ$)')\n",
264    "plt.ylabel(r'$\\theta_d$ ($^\\circ$)')\n",
265    "plt.xlim(phi_lim)\n",
266    "plt.xticks(phi_ticks)\n",
267    "plt.ylim(theta_lim)\n",
268    "plt.yticks(theta_ticks)\n",
269    "_ = None"
270   ]
271  },
272  {
273   "cell_type": "markdown",
274   "metadata": {
275    "colab_type": "text",
276    "id": "shokLKwv1g9m"
277   },
278   "source": [
279    "\n",
280    "### Analytic forms for $\\Delta\\theta_d$ and $\\Delta\\phi_d$\n",
281    "\n",
282    "**Work in Progress: This section is incomplete and incorrect.**  \n",
283    "\n",
284    "See simulation cell below for the empirical forms of the distribution.\n",
285    "\n",
286    "In order to develop a orientation distribution for use with *sasmodels*, assume we start with a set of particles in random orientation. The scattering pattern is integrating over the population of particles over time, so we need a distribution that captures the proportion of time spent at each orientation.  From Fig 2 in Stover and Cohen (1990), we can assume that each initial orientation will determine a unique $C$ and $\\kappa$ pair, with $C$ remaining constant and $\\kappa$ moving along the orbit as the system evolves.\n",
287    "\n",
288    "\n",
289    "We can ignore the initial phase $\\kappa$ since it only serves to shift the curves uniformly, and will vanish in the integration. So we need to determine the distribution of $C$ values from the initial orientation and for each $C$, determine the joint distribution of $\\theta$ and $\\phi$, then marginalize.  That is, find the form of $P(\\theta, \\phi, C)$ then integrate to get $P(\\theta) = \\int_{\\phi, C} P(\\theta | \\phi, C) d\\theta dC$ and $P(\\phi) = \\int_{\\theta, C} P(\\phi | \\theta, C) d\\phi dC$.  Should cross check by random sampling of evolved systems, and verify that $\\int_C P(\\theta, \\phi | C) dC$ is separable.\n",
290    "\n",
291    "Using\n",
292    "\n",
293    "$$ \\frac{d}{dy} f^{-1}(y)\\Big|_{y=y_o} = \\left(\\frac{d}{dx}f(x)\\right)^{-1}\\Big|_{x=f^{-1}(y_o)}$$\n",
294    "\n",
295    "density should be inversely related to the slope, so for $\\phi$ with $x = \\pi\\,t/T \\in [0, 1]$ covering $[-\\pi/2, \\pi/2]$ with\n",
296    "\n",
297    "$$ \\phi(x) = \\tan^{-1}(r_e\\tan(x))$$\n",
298    "\n",
299    "then\n",
300    "\n",
301    "$$\n",
302    "\\frac{d\\phi}{dx} = (r_e \\sin^2(x) + (1/r_e)\\cos^2(x))^{-1}\n",
303    "$$\n",
304    "\n",
305    "and so\n",
306    "\n",
307    "$$\n",
308    "\\frac{dx}{d\\phi} = r_e \\sin^2(\\phi) + (1/r_e)\\cos^2(\\phi)\n",
309    "$$\n",
310    "\n",
311    "Normalize by the integral to make it into a probability\n",
312    "\n",
313    "$$\n",
314    "\\int_{-\\pi/2}^{\\pi/2} \\frac{dx}{d\\phi}\\,d\\phi = \\tfrac{\\pi}{2} (r_e + 1/r_e)\n",
315    "$$\n",
316    "\n",
317    "Some more trig:\n",
318    "\n",
319    "\\begin{align}\n",
320    "\\cos(\\arctan(x)) &= 1\\big/ \\sqrt{1 + x^2} \\\\\n",
321    "\\sin(\\arctan(x)) &= x \\big/ \\sqrt{1 + x^2}\n",
322    "\\end{align}\n",
323    "\n",
324    "so\n",
325    "\n",
326    "\\begin{align}\n",
327    "\\tan \\theta &= \\frac{C r_e}{\\sqrt{r_e^2 \\cos^2 \\phi + \\sin^2\\phi}} \\\\\n",
328    "                   &= \\frac{C r_e}{\\sqrt{r_e^2 (1 + \\tan^2 x)/(1 + r_e^2\\tan^2 x)}} \\\\\n",
329    "                   &= C\\sqrt\\frac{1 + r_e^2\\tan^2 x}{1 + \\tan^2 x} \\\\\n",
330    "                   &= C\\sqrt{r_e^2\\sin^2 x + \\cos^2 x}\n",
331    "{}\\end{align}\n",
332    "\n",
333    "and\n",
334    "\n",
335    "$$\n",
336    "\\frac{d\\theta}{dx} = \\frac{C\\,(r_e^2 - 1)\\,\\sin x \\cos x}{\\sqrt{r_e^2\\sin^2 x + \\cos^2 x}\\, \n",
337    "(C^2 (r_e^2 \\sin^2 x + \\cos^2 x) + 1)}\n",
338    "$$\n"
339   ]
340  },
341  {
342   "cell_type": "code",
343   "execution_count": 0,
344   "metadata": {
345    "colab": {
346     "base_uri": "https://localhost:8080/",
347     "height": 300
348    },
349    "colab_type": "code",
350    "id": "2bTDi7VcYmhC",
351    "outputId": "78868af1-44cf-4b85-cdcf-58e20aca513d"
352   },
353   "outputs": [
354    {
355     "data": {
356      "text/plain": [
357       "(0, 0.010955673858649729)"
358      ]
359     },
360     "execution_count": 5,
361     "metadata": {
362      "tags": []
363     },
364     "output_type": "execute_result"
365    },
366    {
367     "data": {
368      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAY8AAAEKCAYAAADq59mMAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nx5cAAAAASUVORK5CYII=\n",
369      "text/plain": [
370       "<Figure size 432x288 with 1 Axes>"
371      ]
372     },
373     "metadata": {
374      "tags": []
375     },
376     "output_type": "display_data"
377    }
378   ],
379   "source": [
380    "re=8.4\n",
381    "#re=1e10\n",
382    "phi = 180*t_over_T - 90\n",
383    "dx_dphi = re*sin(pi*t_over_T)**2 + cos(pi*t_over_T)**2/re\n",
384    "dx_dphi_norm = 90*(re+1/re)\n",
385    "p_phi = dx_dphi / dx_dphi_norm\n",
386    "plt.plot(phi, p_phi)\n",
387    "plt.xlabel(r'$\\phi ( ^\\circ)$')\n",
388    "plt.ylabel(r'$P(\\phi)$')\n",
389    "plt.ylim([0, max(p_phi)])"
390   ]
391  }
392 ],
393 "metadata": {
394  "colab": {
395   "collapsed_sections": [],
396   "name": "Jeffery Orbits.ipynb",
397   "provenance": [],
398   "version": "0.3.2"
399  },
400  "kernelspec": {
401   "display_name": "Python 3",
402   "language": "python",
403   "name": "python3"
404  },
405  "language_info": {
406   "codemirror_mode": {
407    "name": "ipython",
408    "version": 3
409   },
410   "file_extension": ".py",
411   "mimetype": "text/x-python",
412   "name": "python",
413   "nbconvert_exporter": "python",
414   "pygments_lexer": "ipython3",
415   "version": "3.6.7"
416  }
417 },
418 "nbformat": 4,
419 "nbformat_minor": 1
420}