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%3.020 Thermodynamics of Materials, Spring 2021
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% Title Information
\title{3.020 Lecture 25}
\author{Prof. Rafael Jaramillo}
\date{}

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\begin{document}

% Title Page
\begin{titlepage}
  \pagenumbering{gobble}
  \maketitle
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% Lectures
\pagenumbering{arabic}
% L25, B1
\section{Modeling the simplified Cr-Fe phase diagram}
\includegraphics[width=\textwidth]{figure1}
\hrule
% L25, B2
\begin{itemize}
    \item Start by building model for pure Fe
    \begin{itemize}
        \item needs to melt at $T_{M,Fe}$
        \item $G_{Fe}^{\alpha}$(T), $G_{L}^{\alpha}$(T) curves cross at $T_{M,Fe}$
        \item $ G = H - T S$
    \end{itemize}
    \includegraphics[width=0.4\textwidth]{figure2}\\
    $\Longrightarrow$ model temp dependence of $H$, $S$ and transformation quantities
    \[ \Delta H^{\alpha \rightarrow L}, \qquad \Delta S^{\alpha \rightarrow L}= \Delta H^{\alpha \rightarrow L} / T_M \]
\end{itemize}
\hrule
% L25, B3
\includegraphics[width=0.9\textwidth]{figure3}\\
enthalpy at standard state 25 $^o$C and 1 atm:
\begin{itemize}
    \item set to 0 for elements, by convention
    \item if pure component is a compound (e.g. \ce{SiO_2}), then use $\Delta H^o_{form,298}$
\end{itemize}
\hrule
% L25, B4
\includegraphics[width=0.9\textwidth]{figure4}\\
entropy at standard state ($S_{298}^o$)
\vspace{0.2cm}
\hrule
% L25, B5
\begin{itemize}
    \item [-] data needed for model of pure Fe
\end{itemize}
\begin{table}[t]
    \centering
    \begin{tabular}{c|c}
      standard state   & $S^0_{298}$, $\Delta H^0_{form,298}$ = 0 \\
      \hline
       heat capacity  & $C_P^{\alpha}(T)$, $C_P^{L}(T)$\\ 
       \hline
       transformation & $\Delta H^{\alpha \rightarrow L}(T_m)$, $\Delta S^{\alpha \rightarrow L}(T_m)$, $T_m$
    \end{tabular}
\end{table}
$C_P^{\alpha}(T)$, $C_P^{L}(T)$: each can be modeled as a polynominal \\
$\Delta H^{\alpha \rightarrow L}(T_m)$, $\Delta S^{\alpha \rightarrow L}(T_m)$, $T_m$ : a triple, two are independent \\
\begin{align*}
    C_P^{\alpha}(T) &= a^{\alpha} + b^{\alpha} T + C^{\alpha}/T^2 \\
    C_L^{\alpha}(T) &= a^{L} + b^{L} T + C^{L}/T^2 \\
    \Delta S^{\alpha \rightarrow L} (T_m) &= \Delta H^{\alpha \rightarrow L} (T_m)/T_M
\end{align*}
\hrule
% L25, B6
\begin{itemize}
    \item [-] temp-dependence of $H$
\end{itemize}
\begin{align*}
    H &= p_1 T^2 + p_2 T + p_3  \qquad \textrm{polynomial coefficients}\\
    \frac{\partial H}{\partial T} &= 2 p_1 T + p_2 \qquad \textrm{as in Matlab}\\
    &= a + bT \qquad \textrm{heat capacity model}
\end{align*}
\hrule
% L25, B7
\begin{itemize}
    \item Build model for pure Cr, as above
    \item Build solution models
    \begin{itemize}
        \item start with the spinodal phase $\alpha$ (BCC)
        \item model as simple regular $\longrightarrow$ only 1 adjustable parameter
    \end{itemize}
\end{itemize}
\begin{align*}
    \Delta H_{mix} &= a X_1 X_2 = a X_{Fe} X_{Cr} \\
    H &= H_{Fe}^o X_{Fe}+H_{Cr}^o X_{Cr}+a X_{Fe} X_{Cr}
\end{align*}
\hrule
% L25, B8
\begin{itemize}
    \item [-] estimate $a_0$ by eye by recording values at $X_{Cr} = 0.5$
\end{itemize}
\begin{minipage}{0.6\textwidth}
\begin{itemize}
    \item [-] $H^o_{Fe} \approx 25,750 ~ J/mol$
    \item [-] $H^o_{Cr} \approx 19,450 ~ J/mol$
    \item [-] $H(X_{Cr} = 0.5) \approx 29,000 ~ J/mol$
    \item [-] $a^{BCC} \approx 28,000 J/mol$
\end{itemize}
\end{minipage}
\begin{minipage}{0.4\textwidth}
\includegraphics[width=\textwidth]{figure5}
\end{minipage}
\begin{itemize}
    \item [-] alternatively could export data and fit
    \item [-] likewise build model fro liquid phase
\end{itemize}
\hrule
% L25, B9
\begin{itemize}
    \item At each temp, draw free energy-composition diagram and identify common tangents (if only)
\end{itemize}
\begin{align*}
    G^{\alpha} &= X_{Cr} \mu_{Cr}^{o,\alpha} + X_{Fe} \mu_{Fe}^{o,\alpha} + \Delta G^{\alpha}_{mix} \\
    G^{L} &= \underbrace{ X_{Cr} \mu_{Cr}^{o,L} + X_{Fe} \mu_{Fe}^{o,L}}_{\substack{\textbf{pure component models,}\\ \textbf{will contain reference } \\ \textbf{state changes}}} + \underbrace{\Delta G^{L}_{mix}}_{\textbf{Solution models}}
\end{align*}
\hrule
% L25, B10
\includegraphics[width=\textwidth]{figure6}
\end{document}