interaction_constants.ipynb

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    "Ion Interaction\n",
    "========\n",
    "Ion interaction effects are important for correcting the mobility of ions based on the ionic strength of solution. These effects are laid out in the *Irreversible Processes in Electrohoresis*, by Onsager and Fuoss. \n",
    "\n",
    "These effects have prefactors that need to be converted to appropriate units for use. Here, I'm going to resolve the units on these prefactors, and resolve the differences between the paper and the STEEP implimentation.\n"
   ]
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   "source": [
    "# imports\n",
    "from ionize import Aqueous\n",
    "from math import sqrt, pi\n",
    "import pint\n",
    "ur = pint.UnitRegistry()\n",
    "Q = ur.Quantity"
   ]
  },
  {
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   "execution_count": 2,
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   "source": [
    "# define values\n",
    "temperature = Q(25, 'degC')\n",
    "e = ur.elementary_charge\n",
    "kb = ur.boltzmann_constant\n",
    "dielectric = Aqueous.dielectric(temperature.magnitude)\n",
    "viscosity = Aqueous.viscosity(temperature.magnitude) * ur.pascal * ur.second\n",
    "Na = ur.avogadro_number\n",
    "eps = ur.vacuum_permittivity"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Robinson-Stokes\n",
    "---------------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'temperature' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-1-53761ae09cad>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[0;31m# STEEP: F*0.2297*z.*omega+31.410e-9\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0mcoefficient_1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m0.2297\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mtemperature\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mto\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'degK'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;36m.5\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mdielectric\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;36m3.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      3\u001b[0m \u001b[0mcoefficient_2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m31.410e-9\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mviscosity\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mtemperature\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mto\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'degK'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;36m.5\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mdielectric\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      4\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      5\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcoefficient_1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcoefficient_2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mNameError\u001b[0m: name 'temperature' is not defined"
     ]
    }
   ],
   "source": [
    "# STEEP: F*0.2297*z.*omega+31.410e-9\n",
    "coefficient_1 = 0.2297 * ((temperature.to('degK'))**.5*dielectric)**3.\n",
    "coefficient_2 = 31.410e-9 * viscosity * (temperature.to('degK')**.5*dielectric)\n",
    "\n",
    "print(coefficient_1, coefficient_2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### STEEP O-F\n",
    "     mob_new= F * omega - (F*0.78420*z.*factor.*omega* / dielectric^1.5 / temperature^1.5 + \n",
    "\n",
    "              31.410e-9 *  viscosity / dielectric^.5 / temperature^.5 .*\n",
    "              sqrt(IonicStr/2000)./(1+1.5*sqrt((yy(5)*yy(1))/(yy(6)*yy(2)))\n",
    "              *sqrt(IonicStr/2000));\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "#### STEEP RS\n",
    "    %Robinson-Stokes, just checking dont use this.\n",
    "    %mob_new=F*omega-(F*0.2297*z.*omega+31.410e-9).*sqrt(IonicStr/2000)./(1+1.5*sqrt(IonicStr/2000));\n",
    "    %Assemble matrix back"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Onsager-Fuoss\n",
    "------------\n",
    "The Onsager-Fuoss constants A and B."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[A] = [length] ** 1.5 / [substance] ** 0.5\n",
      "A = 0.5547184792860854 liter ** 0.5 / mole ** 0.5\n"
     ]
    }
   ],
   "source": [
    "# Bhaga Paper\n",
    "A = e**3 / 12 / pi \n",
    "A *= (Na / (dielectric * eps * kb * temperature.to('degK'))**3)**.5\n",
    "print('[A] =', A.dimensionality)\n",
    "print('A =', A.to('liter^.5/mol^.5'))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[B] = [current] * [length] ** 1.5 * [time] ** 2 / [mass] / [substance] ** 0.5\n",
      "B = 2.220870818024751e-08 liter ** 0.5 * meter ** 2 / mole ** 0.5 / second / volt\n"
     ]
    }
   ],
   "source": [
    "B = e**2 / 6 / pi / viscosity \n",
    "B *= (Na / (dielectric * eps * kb * temperature.to('degK')))**.5\n",
    "print('[B] =', B.dimensionality)\n",
    "print('B =', B.to('m^2/V/s* liter^.5/mol^.5'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Temperature-dependant values\n",
    "Below are values that only include temperature independant versions. \n",
    "\n",
    "A = Ap / dielectric^1.5 / temperature^1.5\n",
    "\n",
    "B = Bp x viscosity / dielectric^.5 / temperature^.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[length] ** 1.5 * [temperature] ** 1.5 / [substance] ** 0.5\n",
      "1980740.0746911524 kelvin ** 1.5 * liter ** 0.5 / mole ** 0.5\n"
     ]
    }
   ],
   "source": [
    "Ap = e**3 / 12 / pi \n",
    "Ap *= (Na / (eps * kb)**3)**.5\n",
    "print(Ap.dimensionality)\n",
    "print(Ap.to('liter^.5*degK^1.5/mol^.5'))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[current] * [length] ** 0.5 * [temperature] ** 0.5 * [time] / [substance] ** 0.5\n",
      "3.0225888785873703e-09 kelvin ** 0.5 * liter ** 0.5 * meter ** 2 * pascal / mole ** 0.5 / volt\n"
     ]
    }
   ],
   "source": [
    "Bp = e**2 / 6 / pi \n",
    "Bp *= (Na / (eps * kb))**.5\n",
    "print(Bp.dimensionality)\n",
    "print(Bp.to('m^2/V/s* liter^.5/mol^.5 * Pa * s * degK**.5'))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# TODO: check steep and spresso implementation\n",
    "# also check onsager-fuoss paper"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pitts\n",
    "-----"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[length] ** 0.5 / [substance] ** 0.5\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "0.9610960030607913 micrometer"
      ],
      "text/latex": [
       "$0.9610960030607913 micrometer$"
      ],
      "text/plain": [
       "<Quantity(0.9610960030607913, 'micrometer')>"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = (2 * e ** 2 * Na/ dielectric / eps / kb / temperature.to('degK'))**.5\n",
    "print(D.dimensionality)\n",
    "(D).to('m^.5/mol^.5')\n",
    "(1/D / Q('(1e-7 mol/L)^.5')).to('um')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[length] ** 0.5 * [temperature] ** 0.5 / [substance] ** 0.5\n"
     ]
    }
   ],
   "source": [
    "Dp = (2 * e ** 2 * Na / eps / kb)**.5\n",
    "print(Dp.dimensionality)\n",
    "Dp.to(())"
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
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