{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Mott-Schottky model for calculating the space charge potential\n",
    "\n",
    "**Running the code in this notebook (under Mott-Schottky conditions for the 6 defined temperatures) takes approximately 3 minutes (iMac with 4 Ghz i7 processor).**\n",
    "\n",
    "Analysis of grain boundary effects on ionic conductivity typically considers the space-charge potential as a characteristic parameter. However $r_{GB}$ can be obtained directly from experimental impedance spectroscopy, whereas the space charge potential cannot and is therefore approximated using the Mott-Schottky model. \n",
    "\n",
    "$$ r_{GB} = \\frac{\\rho_{i,x}}{\\rho_{i,\\infty}} = \\frac{\\mathrm{exp}(z\\Phi_0 / k_BT)}{2z \\Phi_0 / k_BT } $$\n",
    "\n",
    "The conventional Mott-Schottky analysis assumes that the grain boundart region is negligably thin and that oxygen vacancies are fully depleted in the space-charge region. This gives an analytical description of the space-charge behaviour which can be completely characterised from the space charge potential.\n",
    "\n",
    "The function to calculate the Mott-Schottky aprroximation for the space charge potential takes one argument, and that is the charge of the mobile defect.\n",
    "```\n",
    "c_o.solve_MS_approx_for_phi( valence[0] )\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from pyscses.defect_species import DefectSpecies\n",
    "from pyscses.set_of_sites import SetOfSites\n",
    "from pyscses.constants import boltzmann_eV\n",
    "from pyscses.calculation import Calculation, calculate_activation_energies\n",
    "from pyscses.set_up_calculation import calculate_grid_offsets\n",
    "from pyscses.grid import Grid\n",
    "\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "boundary_conditions = 'periodic'\n",
    "site_charges = False\n",
    "systems = 'mott-schottky'\n",
    "core_models = False\n",
    "site_models = 'site_explicit'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "alpha = 0.0005\n",
    "\n",
    "conv = 1e-8\n",
    "grid_x_min = -2.0e-8\n",
    "grid_x_max = +2.0e-8\n",
    "bulk_x_min = -2.0e-8\n",
    "bulk_x_max = -1.0e-8\n",
    "\n",
    "dielectric = 1\n",
    "\n",
    "index = 111\n",
    "\n",
    "b = 5e-9\n",
    "c = 5e-9\n",
    "\n",
    "temp = [ 773.15, 873.15, 973.15, 1073.15, 1173.15, 1273.15 ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "valence = [ +1.0, -1.0 ]\n",
    "site_labels = ['site_1', 'site_2']\n",
    "defect_labels = ['defect_1', 'defect_2']\n",
    "mole_fractions = [ [ 0.2, 0.2 ] ]\n",
    "initial_guess = [ [ 0.2, 0.2 ] ]\n",
    "mobilities = [ 1.0, 1.0 ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = '../input_data/example_data_2_one_seg_energies.txt'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "--- 184.58980011940002 seconds ---\n"
     ]
    }
   ],
   "source": [
    "space_charge_potential_list = []\n",
    "MS_space_charge_potential_list =[]\n",
    "\n",
    "limits, laplacian_limits = calculate_grid_offsets( data, grid_x_min, grid_x_max, 'single' )\n",
    "\n",
    "for m in mole_fractions:\n",
    "    for t in temp:\n",
    "    \n",
    "        defect_species = { l : DefectSpecies( l, v, m, mob ) for l, v, m, mob in zip( defect_labels, valence, m, mobilities ) }\n",
    "\n",
    "        all_sites = SetOfSites.set_of_sites_from_input_data( data, [grid_x_min, grid_x_max], defect_species, site_charges, core_models, t )\n",
    "        if site_models == 'continuum':\n",
    "            all_sites, limits = SetOfSites.form_continuum_sites( all_sites, grid_x_min, grid_x_max, 1000, b, c, defect_species, laplacian_limits, site_labels, defect_labels )\n",
    "        if systems == 'mott-schottky':\n",
    "            for site in all_sites.subset( 'site_2' ):\n",
    "                site.defect_with_label('defect_2').fixed = True\n",
    "        if systems == 'gouy-chapman':\n",
    "            for site in all_sites.subset( 'site_2' ):\n",
    "                site.defect_with_label('defect_2').fixed = False\n",
    "        grid = Grid.grid_from_set_of_sites( all_sites, limits, laplacian_limits, b, c )\n",
    "        \n",
    "        c_o = Calculation( grid, bulk_x_min, bulk_x_max, alpha, conv, dielectric, t, boundary_conditions )\n",
    "        c_o.form_subgrids( site_labels )\n",
    "        if systems == 'gouy-chapman':\n",
    "            c_o.mole_fraction_correction( m, systems, initial_guess )\n",
    "        c_o.solve(systems)\n",
    "        c_o.mole_fractions()\n",
    "        c_o.calculate_resistivity_ratio( 'positive', 2e-2 )\n",
    "        c_o.solve_MS_approx_for_phi( valence[0] )\n",
    "        \n",
    "        space_charge_potential_list.append(max(c_o.phi))\n",
    "        MS_space_charge_potential_list.append(c_o.ms_phi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The space-charge potential calculated using the Mott-Schottky model can be compared to the space-charge potential calculated using the Poisson-Boltzmann solver as demonstrated below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11849d2b0>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot( temp, space_charge_potential_list, label = '$\\Phi_0^{PB}$' )\n",
    "plt.plot( temp, MS_space_charge_potential_list, label = ' $\\Phi_0^{MS}$' )\n",
    "plt.xlabel( 'Temperature (K)' )\n",
    "plt.ylabel( 'Potential (eV)' )\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.6.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}