root/trunk/matml/transport/resources/evaprat/evaprat.tex

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\title{Multicomponent Evaporation Kinetics}
\date{June 19, 2005}
\author{Adam Powell}
\maketitle

This short description of multicomponent evaporation begins with a
thermodynamic description of vapor pressure and the Langmuir equation of pure
element evaporation, then derives an ``evaporation ratio'' which relates the
ratio of components in the vapor to the ratio in the evaporating liquid or
solid.  For a dilute solute, this evaporation ratio is related to the activity
coefficient.

\paragraph{The Clausius-Clapeyron Equation}

The Clausius-Clapeyron equation gives the vapor pressure of a liquid or solid
as a function of temperature.  For two phases of the same chemical compound in
equilibrium, changes in free energy along a coexistence curve are equal:
\begin{equation}
  \label{eq:dgsame}
  dG_1 = dG_2 \Rightarrow
  V_1dP_1 - S_1dT_1 = V_2dP_2 - S_2dP_2.
\end{equation}
Since the pressures and temperatures of the two phases are the same, this can
be expressed in terms of changes in volume and entropy, to give an expression
for $dP/dT$:
\begin{equation}
  \label{eq:dpdt}
  \Delta V dP = \Delta S dT \Rightarrow \frac{dP}{dT} = \frac{\Delta S}{\Delta
    V}.
\end{equation}
This condition, called the Clapeyron equation, applies across liquid-solid,
liquid-gas and solid-gas transitions in pressure-temperature space.  For
example, in most substances, the volume change during melting $\Delta V_m$ is
positive, and since melting increases entropy, $\Delta S_m/\Delta V_m$ is
positive, as is the slope of the pressure-temperature coexistence curve.  For
the few such as water and silicon where the volume change during melting is
negative, that slope is negative.

\begin{figure}[htbp]
  \centering \pdfimageresolution 330
  $\ $\pdfximage{SLVmet.png}\pdfrefximage\pdflastximage$\ $
  $\ $\pdfximage{SLVwater.png}\pdfrefximage\pdflastximage$\ $
  \caption{Schematic phase diagrams for an element/compound: typical metal
    (left), water or silicon (right).}
  \label{fig:PTphases}
\end{figure}In evaporation, the volume of the gas is so much larger than that
of the liquid that $\Delta V_v$ is approximately the gas volume, which on a
molar basis is $RT/P$ according to the ideal gas law; thus:
\begin{equation}
  \label{eq:almostclausclap}
  \frac{dP_v}{dT} = \frac{P_v}{RT}\Delta S_v.
\end{equation}
Since the free energy change at the coexistence curve is zero, $\Delta
G_v=0=\Delta H_v-T\Delta S_v$, and the entropy change is simply $\Delta
S_v=\Delta H_v/T$.  This substitution and some rearrangement gives an equation
which on integration yields the Clausius-Clapeyron equation:
\begin{equation}
  \label{eq:clausclap1}
  \frac{dP_v}{P_v} = \frac{dT}{RT}\frac{\Delta H_v}{T} =
  \frac{\Delta H_v}{RT^2}dT,
\end{equation}
\begin{equation}
  \label{eq:clauslap}
  \ln P_v = -\frac{\Delta H_v}{RT} + B,
\end{equation}
where $B$ is the integration constant.  If $\Delta H_v$ depends linearly on
temperature, a third term $C\ln T$ is added, and in some cases a fourth term
$DT$ is also added to improve fit to data.  In tables\footnote{For example,
  {\em Smithells Metals Handbook} (E. Brandes ed., Butterworth, 1983) has these
  constants for numerous metals.}, this is summarized:
\begin{equation}
  \label{eq:finalclausclap}
  \log P_v = -\frac{A}{T} + B + C\log T + DT.
\end{equation}

For a multi-component liquid or solid, the vapor pressure of each component is
its activity $a$ times the vapor pressure of the pure substance.  From this
point on, $\bar{P}_v$ will denote the vapor pressure of the pure substance and
$P_v=a\bar{P}_v$ its partial pressure in solution.

\paragraph{Evaporation Into a Vacuum: the Langmuir Equation}

For a gas with energies following a Boltzmann distribution, the number of atoms
(moles) striking a surface per unit area per unit time is given by:
\begin{equation}
  \label{eq:langmuir}
  J = \frac{P}{\sqrt{2\pi MRT}}.
\end{equation}
At equilibrium, the flux of evaporating atoms leaving the surface equals that
arriving at the surface, so replacing $P$ with $P_v$ in equation
\ref{eq:langmuir} gives its evaporation flux.  Take away the gas, and this
gives the net evaporation rate into a vacuum.  The vapor pressures and
evaporation fluxes of several metallic elements are given in figure
\ref{fig:metalvap} below.  Three lessons to draw from that figure are:
\begin{itemize}
\item Vapor pressures and evaporation rates are {\em very strong} functions of
  temperature, much stronger than convective heat transfer coefficients or
  evaporation rates.
\item Vapor pressure decreases with increasing melting point.
\item $\bar{P}_v$ and $J$ curves do not change their slopes or shapes
  dramatically across melting points.
\end{itemize}
\begin{figure}[htbp]
  %% Redo these figures with my own data...
  \centering \pdfimageresolution 230
  $\ $\pdfximage{elem-pv.png}\pdfrefximage\pdflastximage$\ $
  $\ $\pdfximage{elem-J.png}\pdfrefximage\pdflastximage$\ $
  \caption{Vapor pressures $\bar{P}_v$ (left) and evaporation rates $J$ (right)
    of pure elements.  Small circles represent melting points.  From Schiller,
    Heisig and Panzer, {\em Electron Beam Technology}, Wiley, 1982.}
  \label{fig:metalvap}
\end{figure}

\paragraph{Reaction Rate Coefficients}

For an element or compound preferentially evaporating from a solution into a
vacuum, a first order heterogeneous rate coefficient $k''$ can describe its
removal from the surface:
\begin{equation}
  \label{eq:heteratecoeff}
  J_B = k''_B C_B,
\end{equation}
where $J_B$ is the molar flux of species $B$ from the surface (the ``reaction
rate'') and $C_B$ is its molar concentration at the surface of the solution.
The units of $k''$ are length/time, and this can be thought of as a diffusive
average velocity of the solute atoms through the solution near the surface.

Note that if the surface concentration is not known, one can often estimate a
mass transfer coefficient $h_D$ relating flux to difference between the surface
and bulk concentrations.  The sum of the two resistances due to diffusion
through the surface boundary layer $1/h_D$ and evaporation $1/k''$ then
describes the total resistance to transport from the bulk to the gas phase.

Because a solute's partial vapor pressure $P_{vB}$ is its activity $a_B$
times its pure vapor pressure $\bar{P}_{vB}$, its evaporation flux $J_B$ is the
product of activity and evaporation rate of the pure substance:
\begin{equation}
  \label{eq:partialvap}
  J_B = \frac{P_{vB}}{\sqrt{2\pi M_BRT}} =
  \frac{a_B\bar{P}_{vB}}{\sqrt{2\pi M_BRT}}.
\end{equation}
For a dilute solution of solute $B$ in solvent $A$ in the Henry's Law
r\'{e}gime, the activity is proportional to mole fraction $X_B$, and the
proportionality constant defines the activity coefficient $\gamma_B$:
$a_B=\gamma_BX_B$.  The mole fraction, in turn, is the quotient of solute
concentration $C_B$ and the sum of all concentrations, which in dilute solution
is approximately the solvent molar density:
\begin{equation}
  \label{eq:molefrac}
  X_B = \frac{C_B}{\sum C_i} \simeq \frac{C_B}{C_A} = \frac{C_B M_A}{\rho},
\end{equation}
where $\rho$ is the solution density.  Substituting equation \ref{eq:molefrac}
and the definition of the activity coefficient into equation
\ref{eq:partialvap} yields an expression of solute flux $J_B$ which is
proportional to concentration $C_B$, and an expression for reaction rate
coefficient $k''_B$:
\begin{equation}
  \label{eq:finalrate}
  J_B = \frac{\gamma_BC_BM_A}{\rho}\frac{\bar{P}_{vB}}{\sqrt{2\pi M_BRT}} =
  k''_B C_B,
\end{equation}
\begin{equation}
  \label{eq:finalratecoeff}
  k''_B = \frac{\gamma_BM_A}{\rho}\frac{\bar{P}_{vB}}{\sqrt{2\pi M_BRT}}.
\end{equation}
This rate coefficient has many uses, {\em e.g.} in models of batch and
continuous flow reactors, this helps to assess the effectiveness of vacuum
distillation processes for solute removal.

\paragraph{Evaporation Ratio}

[Adapted from A. Powell, J. Van Den Avyle, B. Damkroger, J. Szekely and U. Pal
``Analysis of Multicomponent Evaporation in Electron Beam Melting and Refining
of Titanium Alloys,'' {\em Metall. Mater. Trans.} {\bf 38B}, 1227-1239 (1997).]
For a dilute solution (liquid or solid) of solute $B$ in solvent $A$
evaporating into a vacuum, the Evaporation Ratio $ER_B$ is defined as:
\begin{equation}
  \label{eq:evratdef}
  ER_{\rm B} = \frac{\rm wt\%B_{vapor}/wt\%A_{vapor}}
  {\rm wt\%B_{solution}/wt\%A_{solution}}.
\end{equation}
This ratio will be equal to the equivalent ratio of mole fractions, and the
ratio of mole fractions in the vapor is in turn equal to the ratio of Langmuir
evaporation rates (equation \ref{eq:langmuir}), so the evaporation ratio can be
rewritten as
\begin{equation}
  \label{eq:evrat2}
  ER_{\rm B} = \frac{J_{\rm B}}{X_{\rm B}}\frac{X_{\rm A}}{J_{\rm A}} =
  \frac{P_{v{\rm B}}}{X_{\rm B}\sqrt{M_{\rm B}}}
  \frac{X_{\rm A}\sqrt{M_{\rm A}}}{P_{v{\rm A}}},
\end{equation}
where $X_i$ represents the mole fraction of species $i$ in the liquid or solid,
$P_{vi}$ its vapor pressure, and $M_i$ its molecular weight.  Assuming titanium
activity roughly follows Raoult's law, its vapor pressure is the product of the
vapor pressure in its pure state and mole fraction in the liquid or solid, so
we rewrite the evaporation ratio again as
\begin{equation}
  \label{eq:evrat3}
  ER_{\rm B} = \frac{P_{v{\rm B}}}{X_{\rm B}\bar{P}_{v{\rm A}}}
  \sqrt{\frac{M_{\rm A}}{M_{\rm B}}},
\end{equation}
where $\bar{P}_{vi}$ represents the vapor pressure of pure species $i$.  We
then assume Henrian behavior and use the definition of the activity coefficient
$P_{v{\rm B}}=\gamma_{\rm B}\bar{p}_{v{\rm B}}X_{\rm B}$ to arrive at
\begin{equation}
  \label{erlast}
  ER_{\rm B}=\gamma_{\rm B}\frac{\bar{P}_{v{\rm B}}}{\bar{P}_{v{\rm A}}}
  \sqrt{\frac{M_{\rm A}}{M_{\rm B}}}.
\end{equation}

Thus the evaporation ratio, which relates compositions in the vapor and
condensed phases, is a simple function of the activity coefficient, pure
element vapor pressures and molecular weights.
\end{document}
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