VoIume 3. number 8 CHEMICAL PHYSICS LETTERS August 1969

ANOMALOUS HEAT CAPACITY OF SPHERICAL CLUSTERS OF ATOMS *

J. J. BURTON ** Department of Physics and the Materlols Research Laboratory.

Lkiucrsity of IZCinois. Urbana. Illinois 61801. USA

Received 11 July i969

The heat capacities of small spherical clusters of atoms arc calculated in the harmonic a for clusters of up to 55 atoms. The heat capacities do not take the expected form C( 2’) =A T B

proximation * BTS. The

heat capacity per atom is not monotonic in the number of atoms.

The surface contribution to tha vibrational density of states and heat capacity of solids has been the subject of theoretical interest recently Allen and de Wette [l] considered infinite slabs and showed that the surface contribution to the density of states is almost independent of the slab thickness for slabs more than ten layers thick. They also showed that the surface contri- butes a T2 term to the heat capacity of the infi- nite slab at low temperatures; this result is in accord with the results of approximate calcula- tions of the surface heat capacity [Z]. Dickey and Paskin [3] examined the vibrational states of fi- nite slabs and found both surface and edge mode contributions. Both Allen [l] and Dickey [3] looked at the surface as a perturbation on the bulk crystal.

Sufficiently smal1 crystals can not be regarded as basically solids with a surface; nearly all of their atoms are surface atoms. Therefore. we would not expect findings based on large crystals to be directly applicable to very smal1 crystals. This has led us to calculate explicitly the prop- erties of spherical clusters of atoms?.

The calculations were made for face-centered- cubic clusters of argon.atoms with the interac- tions between the atoms represented by a Lennard-Jones 6.12 potential [4]. Crystals hav- ing filled spherical shells of surface atoms were examined. The densities of states were obtained

* This work was supported in part by the U.S. Atomic Energy Commission under Contract AT(ll-l)-1198.

*= Present address: Henr.v Krumb School of hlines. Columbia University. New York. New York 10027, USA.

t The details of this work. incIuding the frequency distributions, will be published elsewhere.

in the harmonic approximation by diagonalizing the full force constant matrix considering all atomic interactions. The calculations were made on an IBM 360 computer. Limitations of the com- puter forced us to consider clusters having only 13. 19, 43, and 55 atoms. These cIusters have a central atom plus one, two. three, or four addi- tional shells of near neighbors.

The vibrational heat capacities of the clusters were calculated from the densities of states. In addition to the vibrational heat capacity, there is also a Lk heat capacity contribution arising from the rotations and translations of the cluster as a whole. We do not consider this term here though, as Jura and Pitzer [5] pointed out. it is larger than the vibrational heat capacity at sufficiently low temperature. The vibrational heat capacity per atcm of the 43 atom cluster is plotted in fig. 1. For comparison. the heat capacity of argon ca1culat.d using the Debye approximation and 6D = 93 K is also plotted. More sophisticated calculations have shown that the heat capacity of solid rare gases may be calculated quite well using the Lennard-Jones potential [a]. Certain basic features of the heat capacity are readily apparent. At high temperatures, the heat capacity per atom is less for the finite crystal than for the infinite crystal; the finite crystal has only 3N-6 normal modes to contribute to the heat capacity. At intermediate temperatures, the heat capacity for the finite crystals significantly exceeds that of the infinite crystal; the surface of the crystal contributes a low frequency peak to the density of states which leads to a larger low temperature heat capacity. Finally, at the very lOWeSt tern- peratures, the vibrational heat capacity of the smaller crystal goes to zero more rapidly than

594

Volume 3. number 3 CHENICAL PHYSICS LETTERS AuguSr: LYOY

30

2.5

20

$1.5

IO

.5

_ COAtoms

-.- 43

0; 0 10 20 30 40 50 60 70 80 90

TVK) Fig. 1. The vibrational heat capacity per atcm for a 43 atom cluster and for an infinite solid as a function of

temperature.

that of the infinite crystal; the small crystal has no very low frequency phonons as the possible wavelengths are limited by the size of the crys- tal.

At low temperatures the heat capacity of an infinite crystal [2] with a surface takes the form

C(T) =ATs i-3T2. (1)

C(T),‘T2 is plotted against T for all of the clus- ters in fig. 2. Again the Debye result for the in- finite crystal is plotted. Jf the heat capacity were represented by eq. (l), the curves in fig. 2 would be straight lines with non-zero intercepts at O°K. These curves are not straight lines because the

‘r _ CD Atoms 1

0 I 2 3 4 5 T (oKI

Fig. 2. The vibrational heat capacity per atom divided by T2 versus temperature for finite clusters and the

infinite solid.

small crystal is not like an infinite crystal with a surface.

C(T) is plotted against I,& for a number of different temperatures in fig. 3. At Low temper- aixres the vibrational heat capacity falls sharply for the smallest crystals. This effect is reIated to our earlier observation on fig. 1 - the heat capacity of the small cluster goes to zero faster than that of an infinite crystal. It is also appa- rent from fig. 3, that the low temperature heat capacity per atom is not monotonic in the number of atoms. This irregularity results from the com- petition of size and surface effects.

The properties of very smaiL clusters can not be obtained by extrapolation of the surface prop erties of large crystals. In fact, at low temper- atures, the vibrational heat capacity is not even monotonic in the number of atoms in the cluster. At high temperatures the entropy per atom. which is an integral of the heat capacity and so is af- fected by irregularities in the low temperature heat capacity, is also not monotonic in the num- ber of atoms. These observations have interest- ing implications for vapor phase nucleation and also cluster& in dense systems; these problems have been previously examined without proper consideration of the special properties of smaLL clusters. We are currently examining these pos- sibilities.

10 /~9’.__*

03 / 5”K\

01

d /-

i‘

3”

.03

01

z=

.003 2

\

.OOIb s, .2 A 6 0

+? Fig. 3. The heat capacitv per atom versus l/‘n where tz is the number of atoms in the cluster. The circles are calculated va!ues and the lines are smooth curves drawn through them. The temperatures are indicated on the curves. The heat capacity scale is logarithmic.

(0 55 43 ” 19 13 30 b - 93-K-e

25”K-., y -I S°K-_

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Volume 3. number 8 CHEMXCAL PHYSICS LETTERS August 1969

The au&or wishes to express his gratitude to G. Jura for suggesting consideration of this prob- lem, to R. Beyerlein and S. Fain for reading and commenting on the manuscript, and to D. Lazarus for helpful discussions and comments on the prob- lem.

REFERENCES

[l] R. E. Allen and F. W. de Wette. Phys. Rev. 179 (l969) 873.

[2J A. A. Maradudin. E. W. Montroll and G. H. Weiss. Solid State Phys. Suppl. (1963) 3.

(31 J. M. Dickey and A. Paskin. Phys. Rev. Letters 21 (1968) 1441.

[4] T. Kihara. J. Phys. Sot. Japan 3 (1948) 265. [5J G. Jura and K. S. Pitzer, J. Am. Chem. Sot. 74 (1952)

6030. 16) V. V. Goldman. G. K. Horton and M. L. Klein. Phys.

Rev. Letters 21 (1968) 1527.

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