``Madelung model'' prediction for dependence of lattice parameter on nanocrystal size.
Vasili Perebeinos\corauthref{cor}
The competition between the long range Coulomb attractive and the short range repulsive interaction in ionic nanocrystals creates an effective negative pressure, which causes the lattice parameter $a$ to increase with decreasing nanoparticle size $d$. A simple ``Madelung model'' is used to predict the dependence of the lattice parameter for CeO$_2$ and BaTiO$_3$ on $d$, $\delta a/a=\alpha/d$, for $\alpha=$0.22 \AA \ and 0.18 \AA \ respectively. The model predictions are compared with experimental results.
paper.tex
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\begin{frontmatter}

\title{``Madelung model'' prediction for dependence of 
lattice parameter on nanocrystal size.}

\author{Vasili Perebeinos\corauthref{cor}} 
\ead{vasili@bnl.gov}
\ead[url]{http://cmth.phy.bnl.gov/$\sim$perebein}
\corauth[cor]{Corresponding author.}
\address{Department of Physics, Brookhaven National Laboratory, Upton,
New York 1973-5000}


\author{Siu-Wai Chan} and
\author{Feng Zhang}
\address{Department of Applied Physics and Applied Mathematics,
Columbia University, New York, NY 10027}

%\author{Vasili Perebeinos\address[BNL]{
%{Department of Physics, Brookhaven National Laboratory, Upton, 
%New York 1973-5000}}
%\author{Vasili Perebeinos\corauthref{cor}}
%\author{label2]{Siu-Wai Chan}
%\author{label2]{Feng Zhang}
%\address{Department of Physics, Brookhaven National Laboratory, Upton, 
%New York 1973-5000}
%\address[label2]{Department of Applied Physics and Applied Mathematics, 
%Columbia University, New York, NY 10027}

\begin{abstract}
The competition between the long range Coulomb attractive 
and the short range repulsive interaction in ionic nanocrystals 
creates an effective negative pressure, which causes the
lattice parameter $a$ to increase with decreasing nanoparticle size $d$. 
A simple ``Madelung model'' is used to predict the dependence of the 
lattice parameter for CeO$_2$ and BaTiO$_3$ on $d$, $\delta a/a=\alpha/d$, 
for  $\alpha=$0.22 \AA \ and 0.18 \AA \ respectively. 
The model predictions are compared with experimental results. 
\end{abstract}

\begin{keyword} A. Nanostructures \sep A. Surfaces and interfaces  

\PACS 61.50.Ah \sep 61.82.Rx \sep 68.35.Md \sep 79.60.Jv 
\end{keyword}
\end{frontmatter}


How properties of ionic crystals change
when their surface to volume ratio 
becomes large is an active area of 
research \cite{Zhang,Spanier,Tsunekawa2,Tsunekawa1,Tsunekawa3,Sayle,Cordatos}. 
The ceria nanoparticles CeO$_2$ have been studied experimentally using 
Raman, X-ray diffraction, photoabsorption, high resolution transmission 
electron microscopy (HRTEM) \cite{Zhang,Spanier},
and transmission electron microscopy \cite{Tsunekawa2,Tsunekawa1,Tsunekawa3}.
The fascinating property of ionic nanoparticles is the lattice parameter  
increase with nanoparticle decreasing size \cite{Zhang,Tsunekawa1,Tsunekawa3}. 
Intuitively one would expect the lattice parameter  to decrease as a result 
of surface stress as was observed in gold nanoparticles \cite{Mays}. 

%Intuitively one would expect 
%the lattice parameter  to shrink as a result of surface stress \cite{Mays}.  

Nanocrystal properties are significantly 
different from their bulk counterparts. Previous theoretical treatments 
\cite{Sayle,Cordatos} used realistic potentials for small Ce$_n$O$_{2n}$ 
cluster calculations. It was shown that flourite structure for ceria 
was not achieved for $n$ smaller than $50$. In the present work we investigate
lattice parameter dependence on a cluster size for $d=4-20$ nm or 
$n=10^3-10^5$. 
Charge neutral octahedra nanocrystals are surface terminated with oxygen 
anion layers along the eight $\{111\}$ faces. 
The balance between the long range Coulomb and the short range atomic 
interactions leads to the optimal lattice parameter  $a_0$ dependent on 
the size of the cluster. 
The volume energy per formula unit in the bulk material is usually modeled as
%
\begin{eqnarray}
\varepsilon_{\rm v}=\frac{BV_0}{2}\left(\frac{V}{V_0}-1\right)^2,
\label{Ev}
\end{eqnarray}
%
where $B$ is the bulk modulus and  $E_{\rm v}=N\varepsilon_{\rm v}$ is the 
volume energy of $N$ formula units of CeO$_2$, each filling volume space 
$V_0=a_0^3/4$. 
In the infinite crystal Eq. (\ref{Ev}) is the total free energy which 
is minimum at the bulk lattice parameter  $a_0$. 
In the nanocrystal the balance between the Coulomb and short range 
forces is altered due to the surface. 
The surface Coulomb energy  
creates a negative pressure which leads to the lattice parameter  expansion 
in the nanocrystal in accord with observation \cite{Zhang}. 

To calculate the Coulomb energy in ionic nanoparticles we use the same 
procedure as Madelung energy calculation in bulk material \cite{Kittel}. 
In the fluorite (CaF$_2$) structure the Ce$^{4+}$ ions occupy 
a fcc sublattice and  O$^{2-}$ ions occupy the tetrahedral 
interstitial sites forming a simple cubic sublattice of length $a_0/2$. 
This gives 
a Madelung energy per formula unit 
$\varepsilon_{\rm Mad}=\beta_{\rm bulk}e^2/a$, 
where $\beta_{\rm bulk}=-46.55$.
The value of $\beta=Mz_iz_j/r_0$ 
is related to the Madelung constant $M=2.52$ for the 
CaF$_2$ crystal structure \cite{Lide}, where ionic charges are
$z_i=2$, $z_j=4$ and the nearest neighbor distance is $r_0=\sqrt{3}/4$ 
in units of $a$.
To calculate the surface Madelung energy we use a finite size cluster of 
Ce$^{4+}$ and O$^{2-}$ ions terminated by the oxygen layers along the 
eight $(111)$ planes. 
The surface energy is proportional to the area of  
nanoparticle $\sim N^{2/3}$. 
The diameter $d=(3N/2\pi)^{1/3}a_0$ is 
related to the number of Ce atoms $N$ such that the sphere of equal diameter 
contains an equal volume $NV_0$. 
The Madelung energy 
of a cluster (with a uniform lattice parameter) 
can be calculated for different diameters. The results are
shown on Fig. \ref{fig1}(a), which are well fitted by
%
\begin{eqnarray}
\frac{{\rm E}_{\rm Mad}}{N}=\left(-\beta_{\rm bulk}+
\frac{\beta_{\rm sur}}{d}\right)\frac{e^2}{a},
\label{Emad}
\end{eqnarray}
%
with two parameters $\beta_{\rm bulk}=46.55$ and $\beta_{\rm sur}=4.13$ nm.



%\vspace{0.0pc}
%
\begin{figure}
\psfig{figure=fig1.eps,height=2.39in,width=2.97in,angle=0}
\caption{Closed circles are model calculations of the Madelung energy 
on ionic (a) CeO$_2$ and (b) BaTiO$_3$ nanocrystals. 
The CeO$_2$ nanocrystal was terminated by 
eighth (111) planes and BaTiO$_3$ by six (100) planes. 
The solid lines are the best fits using Eq. (\protect{\ref{Emad}}).}
\label{fig1}
\end{figure}
%





The  uniform lattice parameter  expansion reduces the surface Madelung energy
creating the stress in the nanocrystal estimated with Eq. (\ref{Ev}). 
The total energy per Ce atom on the nanoparticle is
%
\begin{eqnarray}
\frac{{\rm E}_{\rm tot}}{N}=\frac{BV_0}{2}\left(\frac{a^3}{a_0^3}-1\right)^2
+\frac{\beta_{\rm sur}}{d}\frac{e^2}{a}.
\label{Emad2}
\end{eqnarray}
%
Here the first term is the volume contribution, which includes the 
Madelung energy of the infinite crystal, and the second term is the 
surface Coulomb energy from Eq. \ref{Emad}. 
Minimization of the total energy with respect 
to the lattice parameter  for fixed $N$ gives
%
\begin{eqnarray}
\frac{\delta a}{a_0}=\frac{1}{9BV_0}\frac{e^2}{a_0}\frac{\beta_{\rm sur}}{d}
=\frac{\alpha}{d}.
\label{res}
\end{eqnarray}
%

The bulk modulus $B=230\pm10$ GPA has been measured by high-pressure 
x-ray diffraction \cite{Duclos}. 
Taking $a_0=5.409$ \AA \ we obtain the relation $\delta a/a_0=0.22$ \AA$/d$
for the lattice parameter expansion shown on 
Fig. \ref{fig2}. The experimental lattice parameter  
(also shown on Fig. \ref{fig2}) expands with a larger slope $\alpha=0.37$ \AA.
Our model predicts 60\% of the measured value of $\alpha$. 
The missing 40\% of the lattice parameter  expansion 
might be assigned to the formation of point defects.
These defects are oxygen vacancies and simultaneous reduction of two 
ions of Ce$^{4+}$ to Ce$^{3+}$ for each oxygen vacancy formed,
maintaining charge neutrality. 
Electron spectroscopy \cite{Mullins} and 
X-ray photoelectron spectroscopy \cite{Tsunekawa2} show the presence of a
Ce$^{3+}$ component in nanocrystals.
Since Ce$^{3+}$ has a larger radius compared to Ce$^{4+}$ there may
be a lattice expansion accompanying the 
replacement of (Ce$^{4+}$)$_2$O$^{2-}$ by Ce$^{3+}V$ where $V$ is 
a neutral oxygen vacancy.
An upper limit in the Ce$^{3+}$ contribution to expansion 
was estimated \cite{Zhang} by interpolation of published 
results of the variation in ceria lattice parameter  
with RE$^{3+}$ radii from doping with different rare earth oxides 
(RE$_2$O$_3$) \cite{rev,McBride}. 
Recent neutron diffraction measurements on nano-scale powder of CeO$_2$ 
\cite{Egami} suggest associated oxygen defect formation 
(involving oxygen vacancies and self-interstitials) in
the fluorite structure. 
Such defects are not included in our model and may alter 
the surface Madelung energy estimation. 

\vspace{1.0pc}

%
\begin{figure}
\psfig{figure=fig2.eps,height=2.10in,width=2.97in,angle=0}
\caption{Filled circles are experimental lattice parameters versus inverse 
diameter of CeO$_2$ nanoparticle.
Dashed line is the best fit for the experimental data 
\protect\cite{Zhang}. Solid line is a Madelung model prediction 
$\delta a/a=0.22$ \AA $/d$.}
\label{fig2}
\end{figure}
%


The Madelung model is valid for any ionic nanocrystal.
We apply it for the BaTiO$_3$ nanoparticles to calculate
the competition between the long range Coulomb and the short range 
atomic interactions. The results are shown on Fig.\ref{fig1}(b). 
Ionic charges Ba$^{2+}$, Ti$^{4+}$ and O$^{2-}$ were used to calculate 
the surface energy of the nanoparticle. 
To preserve charge neutrality the cubic nanocrystal with size $d$ 
was terminated by the three BaO and three TiO$_2$ $\{100\}$ planes. 
The calculated value of $\beta_{\rm sur}=3.6$ nm along with
the experimental lattice parameter  $a_0=3.996$ \AA \ and bulk modulus
$B=196$ GPa \cite{Fischer} predicts $\delta a/a_0=0.18$ \AA $/d$. 
Existing data \cite{Tsunekawa3} on BaTiO$_3$ nanoparticles 
are not sufficient to verify the relationship 
Eq. (\ref{res}) and further experiments are needed. 




In conclusion we propose a mechanism for lattice 
parameter  expansion of ionic nanocrystals 
due to the effective negative Madelung pressure. 
Using the experimental bulk modulus we quantitatively explain a
lattice parameter  expansion in ionic materials. 
A discrepancy between the model prediction and observed effect in CeO$_2$ 
can be attributed to the Ce$^{3+}$ component \cite{Tsunekawa2,Tsunekawa1}
and other defects not included in our model. 
The Ce$^{3+}$ concentration deduced from magnetic measurements 
\cite{Tsunekawa1} suggests a much smaller value than expected from 
lattice expansion measurements in accord with our theory.
A non-uniform lattice parameter  expansion 
and structural defect formation \cite{Egami} would 
change the quantitative answer, but not the conclusion that 
an effective negative Madelung pressure is significant in ionic 
nanocrystals. 



{\bf Acknowledgments}

We are grateful to Philip B. Allen for critical reading of the manuscript, 
Michael Weinert for valuable discussions, and Stephen O'Brien for 
taking our attention on BaTiO$_3$ nanocrystals.
This work was supported in part by DOE  Grant No.\ DE-AC-02-98CH10886.
S.-W.C. and F.Z. acknowledge support by the MRSEC program of the National 
Science Foundation under DMR-9809687.

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%\bibitem{Lee} T. Lee and I. A. Aksay, Crystal Growth and Design, 
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%\bibitem{Tsukada} M. Tsukada, C. Satoko, and H. Adachi, 
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%\bibitem{Cordatos} H. Cordatos, D. Ford, and R. J. Gorte, J. Phys. Chem. 
%{\bf 100}, 18128 (1996).

\end{thebibliography}

\end{document}

cond-mat0208048.extracted_bib
\begin{thebibliography}{00}


\bibitem{Zhang} F. Zhang, S.-W. Chan, J. E. Spanier, E. Apak, Q. Jin, 
R. D. Robinson, and I. P. Herman, Appl. Phys. Lett. {\bf 80}, 127 (2002).


\bibitem{Spanier} J. Spanier, R. D. Robinson, F. Zhang, S.-W. Chan, and 
I. P. Herman, Phys. Rev. B {\bf 64}, 245407 (2001). 

\bibitem{Tsunekawa2} S. Tsunekawa, T. Fukuda, and A. Kasuya, Surf. Sci. Lett. 
{\bf 457}, L437 (2000). 

\bibitem{Tsunekawa1} S. Tsunekawa, R. Sivamohan, S. Ito, A. Kasuya, and 
T. Fukuda, NanoStructured Materials, {\bf 11}, 141 (1999).

\bibitem{Tsunekawa3} S. Tsunekawa, K. Ishikawa, Z.-Q. Li, Y. Kawazoe, and 
A. Kasuya, Phys. Rev. Lett {\bf 85}, 3440 (2000).

\bibitem{Sayle} T. X. T. Sayle, S. C. Parker, and C. R. A. Catlow, 
Surf. Sci. {\bf 316}, 329 (1994). 

\bibitem{Cordatos} H. Cordatos, D. Ford, and R. J. Gorte, 
J. Phys. Chem. {\bf 100}, 18128 (1996).

\bibitem{Mays} C. W. Mays, J. S. Vermaak, and D. Kuhlmann-Wilsdorf, 
Surf. Sci. {\bf 12}, 134 (1968). 

\bibitem{Kittel} C. Kittel, in {\it Introduction to Solid State Physics}, 
(J. Wileys \& Sons Inc, New York, 1996) p. 634. 

\bibitem{Lide} D. R. Lide, in {\it CRC Handbook of Chemistry and Physics}, 
(CRC Press, Boca Raton, 82 Ed. 2001-2002) p. 12-36.

\bibitem{Duclos} S. J. Duclos, Y. K. Vohra, A. L. Ruoff, A. Jayaraman, and 
G. P. Espinosa, Phys. Rev. B {\bf 38}, 7755 (1988).

\bibitem{Mullins} D. R. Mullins, S. H. Overbury, and D. R. Huntley, 
Surf. Sci. {\bf 409}, 307 (1998).

\bibitem{rev} L. Eyring, in {\it Handbook on the Physics and Chemistry of Rare 
Earths}, edited by K. A. Gschneider and L. Eyring (North-Holland, Amsterdam, 
1979), Vol. 3, Chap. 27. 

\bibitem{McBride} J. R. McBride, K. C. Hass, B. D. Poindexter, and 
                 W. H. Weber, J. Appl. Phys. 76, 2435 (1994).

\bibitem{Egami} E. Mamontov and T. Egami, J. of Phys. Chem. Sol. 
{\bf 61}, 1345 (2000).

\bibitem{Fischer} G. J. Fischer, Z. Wang, and S. Karato, Phys. Chem. Minerals 
{\bf 20}, 97 (1993). 
 
%\bibitem{Lee} T. Lee and I. A. Aksay, Crystal Growth and Design, 
%{\bf 1}, 401 (2001).

%\bibitem{Tsukada} M. Tsukada, C. Satoko, and H. Adachi, 
%J. Phys. Soc. of Japan {\bf 48}, 200 (1980).


%\bibitem{Cordatos} H. Cordatos, D. Ford, and R. J. Gorte, J. Phys. Chem. 
%{\bf 100}, 18128 (1996).

\end{thebibliography}