add basic tutorial and update helpers

phase_diagram/LiAl_poscar

+POSCAR file written by OVITO
+1
+19.0693264 0.0 0.0
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+Li Al 
+216 216 
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phase_diagram/export.csv

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+0,0,finished,,x0_sol,/x0_sol,lial/lial_thermodynamics_composition,2022-05-27 00:15:29.733653,2022-05-27 00:21:34.589072,364.0,pyiron@cmleo26#4,Calphy,0.4,,
+1,1,finished,,x0_lqd,/x0_lqd,lial/lial_thermodynamics_composition,2022-05-27 00:21:34.911022,2022-05-27 00:26:39.285689,304.0,pyiron@cmleo26#4,Calphy,0.4,,
+2,2,finished,,x0_alli,/x0_alli,lial/lial_thermodynamics_composition,2022-05-27 00:26:39.609501,2022-05-27 00:29:36.121032,176.0,pyiron@cmleo26#4,Calphy,0.4,,
+3,3,finished,,x1_sol,/x1_sol,lial/lial_thermodynamics_composition,2022-05-27 00:29:36.439161,2022-05-27 00:34:08.102581,271.0,pyiron@cmleo26#4,Calphy,0.4,,
+4,4,finished,,x1_lqd,/x1_lqd,lial/lial_thermodynamics_composition,2022-05-27 00:34:08.393575,2022-05-27 00:39:53.247222,344.0,pyiron@cmleo26#4,Calphy,0.4,,
+5,5,finished,,x1_alli,/x1_alli,lial/lial_thermodynamics_composition,2022-05-27 00:39:53.742922,2022-05-27 00:44:11.875476,258.0,pyiron@cmleo26#4,Calphy,0.4,,
+6,6,finished,,x2_sol,/x2_sol,lial/lial_thermodynamics_composition,2022-05-27 00:44:12.178900,2022-05-27 00:50:18.355354,366.0,pyiron@cmleo26#4,Calphy,0.4,,
+7,7,finished,,x2_lqd,/x2_lqd,lial/lial_thermodynamics_composition,2022-05-27 00:50:18.664603,2022-05-27 00:54:28.207543,249.0,pyiron@cmleo26#4,Calphy,0.4,,
+8,8,finished,,x2_alli,/x2_alli,lial/lial_thermodynamics_composition,2022-05-27 00:54:28.696677,2022-05-27 01:01:56.467409,447.0,pyiron@cmleo26#4,Calphy,0.4,,
+9,9,finished,,x3_sol,/x3_sol,lial/lial_thermodynamics_composition,2022-05-27 01:01:56.729048,2022-05-27 01:07:48.331324,351.0,pyiron@cmleo26#4,Calphy,0.4,,
+10,10,finished,,x3_lqd,/x3_lqd,lial/lial_thermodynamics_composition,2022-05-27 01:07:48.548404,2022-05-27 01:13:14.236686,325.0,pyiron@cmleo26#4,Calphy,0.4,,
+11,11,finished,,x3_alli,/x3_alli,lial/lial_thermodynamics_composition,2022-05-27 01:13:14.712876,2022-05-27 01:18:11.052385,296.0,pyiron@cmleo26#4,Calphy,0.4,,
+12,12,finished,,x4_sol,/x4_sol,lial/lial_thermodynamics_composition,2022-05-27 01:18:11.514232,2022-05-27 01:24:25.968831,374.0,pyiron@cmleo26#4,Calphy,0.4,,
+13,13,finished,,x4_lqd,/x4_lqd,lial/lial_thermodynamics_composition,2022-05-27 01:24:26.355495,2022-05-27 01:29:27.170347,300.0,pyiron@cmleo26#4,Calphy,0.4,,
+14,14,finished,,x4_alli,/x4_alli,lial/lial_thermodynamics_composition,2022-05-27 01:29:27.552810,2022-05-27 01:59:49.532453,1821.0,pyiron@cmleo26#4,Calphy,0.4,,
+15,15,finished,,xp_sol,/xp_sol,lial/lial_thermodynamics_composition,2022-05-27 06:49:08.217437,2022-05-27 06:51:56.469011,168.0,pyiron@cmleo26#4,Calphy,0.4,,
+16,16,finished,,xp_lqd,/xp_lqd,lial/lial_thermodynamics_composition,2022-05-27 06:51:56.705974,2022-05-27 06:54:42.587704,165.0,pyiron@cmleo26#4,Calphy,0.4,,
+17,17,finished,,xp_alli,/xp_alli,lial/lial_thermodynamics_composition,2022-05-27 07:20:21.591321,2022-05-27 07:22:29.775974,128.0,pyiron@cmleo26#4,Calphy,0.4,,

phase_diagram/helpers.py

 import pandas as pd
 import os
+import numpy as np
+from calphy.integrators import kb
+from scipy.optimize import fsolve
 
 file_location = "../potentials/AlLi.eam.fs"
 pot_al = pd.DataFrame({
@@ -23,5 +26,93 @@ pot_alli = pd.DataFrame({
     'Species': [['Al', 'Li']],
     'Config': [['pair_style eam/fs\n', 'pair_coeff * * AlLi.eam.fs Al Li\n']]
 })
+pot_lial = pd.DataFrame({
+    'Name': ['LiAl_eam'],
+    'Filename': [[os.path.abspath(file_location)]],
+    'Model': ["EAM"],
+    'Species': [['Li', 'Al']],
+    'Config': [['pair_style eam/fs\n', 'pair_coeff * * AlLi.eam.fs Li Al\n']]
+})
+potential_list = [pot_al, pot_li, pot_alli, pot_lial]
+
+def fe_at(p, temp, threshold=1E-1):
+    """
+    Get the free energy at a given temperature
+    
+    Parameters
+    ----------
+    p: pyiron Job
+        Pyiron job with calculated free energy and temperature
+        
+    temp: float
+        Required temperature
+        
+    threshold: optional, default 1E-1
+        Minimum difference needed between required temperature and temperature found in pyiron job
+        
+    Returns
+    -------
+    float: free energy value at required temperature
+    """
+    arg = np.argsort(np.abs(p.output.temperature-temp))[0]
+    th = np.abs(p.output.temperature-temp)[arg] 
+    if th > threshold:
+        raise ValueError("not a close match, threshold %f"%th)
+    return p.output.energy_free[arg]
+
+def normalise_fe(fe_arr, conc_arr):
+    """
+    Get the enthalpy of mixing by fitting and subtracting a straight line connecting the end points.
+    
+    Parameters
+    ----------
+    fe_arr: list of floats
+        array of free energy values as function of composition
+        
+    conc_arr: list of floats
+        array of composition values
+    
+    Returns
+    -------
+    norm: list of floats
+        normalised free energy
+    
+    m: float
+        slope of the fitted line
+    
+    c: float
+        intercept of the fitted line
+    """
+    m = (fe_arr[-1]-fe_arr[0])/(conc_arr[-1]-conc_arr[0])
+    c = fe_arr[-1]-m*(conc_arr[-1]-conc_arr[0])
+    norm = fe_arr-(m*conc_arr+c)
+    return norm, m, c
 
-potential_list = [pot_al, pot_li, pot_alli]
+def find_common_tangent(fe1, fe2, guess_range):
\ No newline at end of file
+    """
+    Do a common tangent construction between two free energy curves.
+    
+    Parameters
+    ----------
+    fe1: numpy array
+        first free energy curve
+    
+    fe2: numpy array
+        second free energy curve
+    
+    guess_range: list of floats length 2
+        The guess range to find end points of the common tangent
+    
+    Returns
+    -------
+    res: list of floats length 2
+        The end points of the common tangent
+    """
+    def _ct(x, p1, p2):
+        p1der = np.polyder(p1)
+        p2der = np.polyder(p2)
+        term1 = np.polyval(p1der, x[0])-np.polyval(p2der, x[1])
+        term2 = (np.polyval(p1, x[0]) - np.polyval(p2, x[1]))/(x[0]-x[1]) - np.polyval(p1der, x[0])
+        return [term1, term2]
+    res = fsolve(_ct, guess_range, args=(fe1, fe2))
+    return res
\ No newline at end of file

phase_diagram/tutorial.ipynb

+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "10444b16-f852-48eb-ab46-b5ce43e962b3",
+   "metadata": {},
+   "source": [
+    "<table border=\"0\">\n",
+    " <tr>\n",
+    "    <td style=\"width:30%\"><img src=\"img/potentials_logo.png\" width=\"100%\" align=\"justify\"></td>\n",
+    "    <td style=\"width:70%\"> <p style=\"width:100%;color:#B71C1C;font-size:24px;text-align:justify\"> From electrons to phase diagrams </p> <p style=\"width:100%,font-size:16px\">Day 03 Tutorial</td>\n",
+    " </tr>\n",
+    "</table>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ad615a36-1703-4c42-822f-062109a99e3b",
+   "metadata": {},
+   "source": [
+    "In this notebook, we will use the potentials fitted in the previous days for the calculation of thermodynamic properties such as Helmholtz and Gibbs free energies, which in turn can be used for the calculation of phase diagrams. We will discuss calphy, the tool for automated calculation of free energies, and the methology involved.\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "9b8152d6-5edb-4be2-9d7b-2b80017f70bb",
+   "metadata": {},
+   "source": [
+    "---"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bfdfcbbf-cc13-4b43-8b6e-31bc25cd4212",
+   "metadata": {},
+   "source": [
+    "### <font style=\"color:#B71C1C\" face=\"Helvetica\" > Phase diagrams and how to calculate them </font>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c53d510c-cd30-4e38-9a42-1d84132c0e19",
+   "metadata": {},
+   "source": [
+    "<table border=\"0\">\n",
+    " <tr>\n",
+    "    <td style=\"width:40%\"><img src=\"img/phase_dia_1.png\" width=\"100%\" align=\"justify\"></td>\n",
+    "    <td style=\"width:60%\"> <p style=\"font-size:14px\">Phase diagrams provide a wealth of information such as: coexisting lines, melting temperature, phase stability, nucleation mechanism. </p></td>\n",
+    " </tr>\n",
+    "</table>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "61ecb35b-a285-45d8-a8e5-a6a1f62d6459",
+   "metadata": {},
+   "source": [
+    "### <font style=\"color:#B71C1C\" face=\"Helvetica\" > Calculation of phase diagrams: the essential ingredients</font>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7d819917-9a56-4589-9627-baff1e4296f3",
+   "metadata": {},
+   "source": [
+    "<table border=\"0\">\n",
+    " <tr>\n",
+    "    <td style=\"width:50%\"><img src=\"img/cimg4.png\" width=\"100%\" align=\"justify\"></td>\n",
+    "    <td style=\"width:50%\"> <p style=\"font-size:14px\">Phase diagrams can be evaluated from free energy diagrams. <br> <br>\n",
+    "        The required input are: <br> <br>\n",
+    "        &#9679; $G(P, T)$ for unary systems <br>\n",
+    "        &#9679; $G(x, T)$ for binary systems </p></td>\n",
+    " </tr>\n",
+    "</table>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6198fc41-3bc2-4436-a272-1a168e44307a",
+   "metadata": {},
+   "source": [
+    "### <font style=\"color:#B71C1C\" face=\"Helvetica\" > Calculation of free energies: Thermodynamic integration </font>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "607f3c8d-4e49-4fb6-be8d-39999eba5f7b",
+   "metadata": {},
+   "source": [
+    "<img src=\"img/fig1.png\" width=\"1000\">"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "8bab199b-a1a0-42eb-a595-cd545d23873a",
+   "metadata": {},
+   "source": [
+    "- free energy of reference system are known: Einstein crystal, [Uhlenbeck-Ford model](https://doi.org/10.1063/1.4967775)\n",
+    "- the two systems are coupled by \n",
+    "$$\n",
+    "H(\\lambda) = \\lambda H_f + (1-\\lambda)\\lambda H_i\n",
+    "$$\n",
+    "- Run calculations for each $\\lambda$ and integrate \n",
+    "$$\n",
+    "G_f = G_i + \\int_{\\lambda=0}^1 d\\lambda \\bigg \\langle  \\frac{\\partial H(\\lambda)}{\\partial \\lambda } \\bigg \\rangle\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "4756bb61-6a01-48f8-98b1-9d076b7dccc0",
+   "metadata": {},
+   "source": [
+    "To calculate $F$,\n",
+    "\n",
+    "- for each phase\n",
+    "    - for each pressure\n",
+    "        - for each temperature\n",
+    "            - for each $\\lambda$\n",
+    "\n",
+    "If we choose 100 different $\\lambda$ values; 100 calculations are needed for each temperature and pressure! \n",
+    "\n",
+    "**Dimensionality: (phase, $P$, $T$, $\\lambda$)**\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "6ab61ee6-2273-408b-aad5-329601bf79a3",
+   "metadata": {},
+   "source": [
+    "### <font style=\"color:#B71C1C\" face=\"Helvetica\" > Speeding things up: Non-equilibrium calculations </font>"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "c4a61c05-3f8c-47ff-b01a-8ae2c3672998",
+   "metadata": {},
+   "source": [
+    "##### Non-Equilibrium Hamiltonian Interpolation\n",
+    "\n",
+    "<img src=\"img/cimg5.png\" width=\"600\">"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "bb228e05-ae48-4150-a033-9e0004f09db6",
+   "metadata": {},
+   "source": [
+    "In this method:\n",
+    "\n",
+    "- Discrete $\\lambda$ parameter is replaced by a time dependent $\\lambda(t)$\n",
+    "- Instead of running calculations at each $\\lambda$, run a single, short, non-equilibrium calculation"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "7968c6d8-f410-499f-9341-5abb6243f0a2",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "pyiron-atom-dev",
+   "language": "python",
+   "name": "pyiron-atom-dev"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.10.4"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}