Tex on nucleation theory fit, some notes and fixes

py/datafit.py

@@ -16,7 +16,7 @@ plt.rcParams.update({'font.size': 16,
 
 
 # As their layer was really thin, assume 2D theories
-dimensions = 2;
+dimensions = 2
 
 # Literature values for melting temperature, latent heat and interfacial energy
 # Taken from AIST
@@ -129,8 +129,8 @@ def nucleationratefit(plotrange=None, show=True):
                                           absolute_sigma=True)
 #                              p0=[b0,Literature.IE,Literature.L,Literature.Tm]
     if show:
-        print("Fitting J = b*dGv**2/(8*pi)/sqrt(IE**3*kB*T) * exp(-dGc/kB/T)")
-        print("        dGc = 16*pi*IE^3/(3*dGv)")
+        print("Fitting J = b*dGv^2/(8*pi)/sqrt(IE^3*kB*T) * exp(-dGc/kB/T)")
+        print("        dGc = 16*pi*IE^3/(3*dGv^2)")
         print("        dGv = L*(Tm-T)/Tm*2*T/(Tm+T)")
         print("b: ",popt[0],"+-",np.sqrt(pcov[0,0]))
         print("IE:",popt[1],"+-",np.sqrt(pcov[1,1]))

tex/experimentaldata.tex

@@ -89,7 +89,7 @@ materials as well.
   For late times however, as $f_a$ and $\Delta N$ get small,
   the fluctuations can get greater then the mean value,
   which makes it dificult to see if the rate is not constant.
-  It can be made easyer by integrating the rate again.
+  It can be made easier by integrating the rate again.
   This gives a number of nuclei corrected by the effect of the shrinking
   volume free to nucleation and is called $N_{\text{corrected}}$ here.
   
@@ -194,19 +194,20 @@ materials as well.
     \label{fig:expfit_jmak_comp}
   \end{figure}
 
-\section{Temperature Dependence of Groth Velocity}
+\section{Temperature Dependence of Growth Velocity}
   The estimated growth velocities for different temperatures can be described by
   one temperature dependent model.
   With only four data points available,
   an Arrhenius model can sufficiently describe the behaviour. % chisq/ndf of 3.0
   This can be seen in figure \ref{fig:expfit_v_arrhenius}.
   
-  Collectively, for the growth velocity
+  Collectively, the growth velocity
   \begin{align}
 %    v &= (1.9 \pm 3.0) \exp(-\frac{(21123 \pm 617)\unit{K}}{T}) \\
-   v  &= (1.9 \pm 3.0) \exp(-\frac{(1.82 \pm 0.05)\unit{eV}}{k_B T})
+   v  &= (1.9 \pm 3.0) \frac{\text{m}}{\text{s}} \cdot 
+           \exp\left(-\frac{(1.82 \pm 0.05)\unit{eV}}{k_B T}\right)
   \end{align}
-  should be useable for \sbte in this temperature range and vicinity.
+  should be useable for \sbte{} in this temperature range and vicinity.
   
   \begin{figure}[ht]
     \begin{subfigure}{.49\linewidth}
@@ -226,7 +227,41 @@ materials as well.
   Fitting an Arrhenius behaviour to the mobility gives less consistent results.
 
 \section{Classical Nucleation Theory}
-  %TODO
-  Estimation of $\gamma$, possibly more
+  In the data, the nucleation rates are not constant over time,
+  an effect not described by the introduced classical nucleation theory.
+  The time dependence of the rates are not similar for the different
+  measurements, which makes it especially complex to find a general model.
+  
+  To get a simple model providing values for the rate for other temperatures,
+  the mean rates of each measurement can be used.
+  The classical nucleation theory model was fitted to this data.
+  For $\dGv$, the introduced model was used
+  together with values from other experiments \citep{expdata,sb2teStructure}:
+  \begin{align}
+   T_m &\approx 837.5 \unit{K} \\
+   L &= 47.2 \, \frac{\text{J}}{\text{g}} \cdot 6.28 \, \frac{\text{g}}{\text{cm}^3} \\
+     &= 296.4 \cdot 10^6 \, \frac{\text{J}}{\text{m}^3}
+  \end{align}
 
+  This leaves the prefactor and the interfacial energy as free parameters.
+  With more than 4 measurements it should also be possible to fit with more than
+  two free parameters.
+  As the rates are not constant, this will probably not improve quality of the
+  results significant.
+  
+  \begin{figure}[h]
+    \centering
+    \includegraphics[width=.5\linewidth]{../img/expfit_J_T.png}
+    \caption{Classical nucleation theory fit onto experiment data on \sbte.
+        Here a rate per area is used, as the thickness was the same.}
+    \label{fig:expfit_cnt}
+  \end{figure}
   
+  The resulting values of
+  \begin{align}
+   \beta &= (8.3 \pm 34.0) \cdot 10^{80} \unit{s}^{-1} \\
+   \gamma &= (89.52 \pm 0.56) \frac{\text{mJ}}{\text{m}^2}
+  \end{align}
+  % correlation of 0.999986
+  have quite large uncertainties, but should be useable as an initial guess.
+  The fit is shown in figure \ref{fig:expfit_cnt}.

tex/simulationmodel.tex

@@ -90,7 +90,8 @@ and simple simulations to test the limits of the calculated behaviour.
     A ``branch to composition'' technique \citep{computerOrganizations} was used
     to avoid expensive branching operations.
     
-    Overall, the new implementation needs about $9\%$ the time of the old one.
+    Overall, the new implementation needs about $9\%$ the time of the old one
+    and should thus shorten the runtime by more than $20\%$.
   
   \subsection{Other Improvements and Performance Summary}
     Apart from the discussed changes, some minor improvements were made. \\

tex/sources.bib

@@ -336,10 +336,6 @@ abstract = { ▪ Abstract The phase-field method has recently emerged as a pow
   publisher={ACS Publications}
 }
 
-theory_of_mixtures-lecture-notes.pdf
-	Theory lectures how to come to phase field equations from a physical point,
-    not so ad-hoc like zhiyang
-
 On the performance lost from if statements (and also the technique I am using now)
 @article{computerOrganizations,
   title={Some computer organizations and their effectiveness},
@@ -351,3 +347,16 @@ On the performance lost from if statements (and also the technique I am using no
   year={1972},
   publisher={IEEE}
 }
+
+% Some properties of Sb2Te, not much, but at least a density
+% I read the given URL, which quotes this paper
+@article{sb2teStructure,
+    author = "Agafonov, V. and Rodier, N. and Ceolin, R. and Bellissent, R. and Bergman, C. and Gaspard, J.P.",
+    title = "Structure of Sb2Te",
+    journal = "Acta Crystallographica, Section C: Crystal Structure Communications",
+    year = "1991",
+    volume = "47",
+    pages = "1141-1143",
+    ASTM_id = "ACSCEE",
+    url = "https://materialsproject.org/materials/mp-6997/"
+}

tex/theory.tex

@@ -71,12 +71,16 @@ the phase field model.
     and a jump distance \citep{NucleationTheory}.
     Alternatively, it can be used as a free parameter when
     fitting this equation onto experimental data.
+%     As the actual rates differ from the calculated up to some orders of
+%     magnitude \citep{balluffi}, this might be a better approach.
     
     Alltogether, one gets
     \begin{align}
      J &= \beta \frac{Z}{V_m} e^{\frac{-\Delta G_c}{k_B T}} \\
        &= \frac{3 \beta}{4 \pi r_c^3} \sqrt{\frac{\Delta G_c}{3 \pi k_B T}}
           e^{\frac{-\Delta G_c}{k_B T}} \ ,
+%        &= \frac{\beta \dGv^2}{8 \pi} \sqrt{\frac{1}{\gamma^3 k_B T}}
+%           e^{\frac{-16 \pi \gamma^3}{3 \dGv^2 k_B T}}
      \label{eq:cnt_J}
     \end{align}
     where $V_m$ is the volume of a monomer
@@ -238,11 +242,22 @@ the phase field model.
     This equation has been used for phase change materials by several authors
     \citep{CrystallizationKinetics, SbTeArrhenius, AistExperiments, crystalGrowthInPcm}.
   
+  \subsection{Interafcial Energy}
+    At the interface between two phases, energetically higher structures
+    exist to fit the two structures together. %TODO citation
+    This energy per unit of interfacial area is called
+    interfacial energy $\gamma$.
+    
+    It plays a role in the classical nucleation theory as the energy barrier
+    for the formation of a new grain.
+    Its ratio to the bulk free energy defines the critical radius.
+    From a fit to the equation from this theory, a value can be estimated,
+    which is one way to determine this parameter \citep{SbTeArrhenius}.
+    
+    It is also responsible for the Gibbs-Thomson effect and can be determined
+    from measurements of this. %TODO citation, equation, ...
+  
   %TODO other physical quantities?
-%   \subsection{Interfacial Energy}
-%     ratio to bulk free energy defines critical radius of nucleation
-%     can be measured in Gibbs-Thomson effect
-%     or fitted from CNT
   
   \subsection{Crystallized Fraction}
     The main quantity of interest is the fraction $f_c$ of the material that