A thin film of the ferroelectric metal oxide lead titanate $(PbTi{O}_3)$ is deposited on a

single crystal platinum substrate that is$,$ to a first order approximation, completely inert

with respect to reaction with oxygen atoms and in which oxygen atoms are insoluble.

The resulting grain structure of the $PbTi{O}_3$ is such that the entire $50nm$ film thickness

is spanned by single $PbTi{O}_3$ grains and the lateral grain size is much larger than $50$

$nm.$ You are interested in studying oxygen diffusion in $PbTi{O}_3$ and the possible effects

of oxygen vacancies $($empty sites on the oxygen ion sublattice$)$ on the ferroelectric

properties of $PbTi{O}_3.$ To do this, you decide to anneal the $PbTi{O}_3$ film, after its

deposition on the Pt substrate, in an atmosphere enriched' in the $"$IS $O$ isotope of oxygen,

which will act as tracer for probing oxygen ion motion. You plan to place the $PbTi{O}_{3}Pt$

sample in a sealed chamber having valves and gas handling capabilities that allow

exchange of the atmosphere in the chamber. The sample will be initially annealed in

non$-$isotope$-$enriched oxygen at $1$atm pressure and at $500C$ to equilibrate the sample

prior to quickly pumping out the gas and replacing it with $97\%{?}^{11}{O}_2($balance ${?}^{16}{O}_2)$ at

$1$ atm total pressure and continuing the anneal. You plan to characterize the fractional

concentration of ${?}^{18}O$ as a function of depth, $z,$ into the $PbTi{O}_3$ thin film,

$C(z,t)-=\frac{c_{{18}_0}(z,t)}{c_{{16}_0}(z,t)+c_{{18}_0}(z,t)}$

using secondary ion mass spectrometry $($SIMS$)$ depth profiling.

Before conducting experiments, you decide to first perform a simple diffusion

calculation to get an idea of the best annealing conditions to use. You perform the

calculation for an annealing temperature of $500C.$ Based on prior reports on oxygen

diffusion in similar materials, you estimate that the diffusivity of oxygen ions in $PbTi{O}_3$

at this temperature will be $D_0$~~$2\mathrm{.}5{10}^{-15}{m}^2{s}^{-1}.$ You assume that the ${?}^{18}\frac{O_2}{{?}^{16}}{O}_2$ gas

mixture in the sealed chamber is sufficiently well mixed that it's reasonable to ignore

the existence of isotope concentration gradients in the gas phase.

The boundary condition for the fractional ${?}^{18}O$ concentration at the top surface of the

$PbTi{O}_3$ film is dietated by the kinetics of the isotope surface exchange reaction:

${?}^{18}{O}_2(g)+2^{16}{O}_0^x=2^{18}{O}_0^x+{?}^{16}{O}_2(g),$

where $O_0^x$ denotes an oxygen ion sitting on an oxygen sublattice site in $PbTi{O}_3.$

Assuming $1^{tt}$ order oxygen exchange kinetics, the flux of ${?}^{18}O$ into the film surface

$0$ will be given by

$J_0(0,t)=K[C_G-C(0,t)]$

where $K$ is the isotope exchange rate constant, and $C_0=0\mathrm{.}97,$ the isotope concentration

in the solid that is in equilibrium with the ${?}^{11}{O}_2$ enriched gas phase composition. A report

${?}^1$ The natural abundance $($concentratica$)$ of ${?}^{In}{O}_2$ in exyzen gas is $0\mathrm{.}2$mol$\%,$ with the balance being ${?}^{14}{O}_2$

in the literature suggests that $K$~~$5{10}^{-12}\frac{m}{s}$ is a reasonable value to assume for the

oxygen tracer surface exchange rate on a $PbTi{O}_3$ surface at $500C.$

a$)$ Write down the oxygen tracer diffusion equation, and the initial and boundary

conditions for this calculation in terms of $C,$ the fractional concentration of ${?}^{18}O$

isotope in the film.

b$)$ What are the initial $(t=0)$ and "long time" $(t)$ values of the flux of $"$O from

the gas phase into the film surface? Explain, briefly.

e$)$ Use a caleulation to determine the minimum value of the fractional ${?}^{18}O$ isotope

concentration within the PbTi