Application of the significant structures theory to plastic crystals

Application of the significant structures theory to plastic crystals

Solid State Communications Vol. 4, pp. 219-222, 1966. Pergamon Press Ltd. Printed in Great Britain. APPLICATION OF THE SIGNIF CANT STRUCTURES THEORY ...

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Solid State Communications Vol. 4, pp. 219-222, 1966. Pergamon Press Ltd. Printed in Great Britain.

APPLICATION OF THE SIGNIF CANT STRUCTURES THEORY TO PLASTIC CRYSTALS M. E. Zandler and T. R. Thomson Department of Chemistry, University of Utah, Salt Lake City, U. S. A. (Received 26 March 1966 by F.R.N. Naborro)

The significant structures theory as developed for liquids by Eyring, Ree and collaborators is shown to be applicable also to the plastic crystal state. The properties of CBr4 (in the plastic crystal state) were calculated and compared with the experimental data. The calculations were performed on a digital computer using, without alteration, a program developed for the liquid state.

A NEW development in the theory of states of matter is the recognition of a state which is called by Timmermans ~ a “plastic crystal” (review given in Ref. 2). This state normally occurs between the melting point and transition point(s) of substances composed of molecules with a high degree of symmetry. Substances in which this state appears are characterized by their low entropy of fusion, their relatively high entropy of transition, and their high plasticity (softness) in the plastic crystal range.

solid lattice, It seems quite likely that the theory might also be applied to the plastic crystal state. In order to test this idea, the theory was applied to the plastic crystal state of CBr4, for which good experimental data is available. 5,8 The significant structure theory of pure iiquids is based on the recognition that the fdowing three structures give the most important contributions to the partition function of a liquid: (1) molecules with oscillattonal degrees of freedom, (2) the positional degeneracy of these oscillatory molecules and (3) molecules with translational degrees of freedom. A convincing argument is used to relate the relative proportions of molecules oscillational and translational degrees of with freedom to the excess volume of the liquid compared to the reference solid. More explicitly, the fraction of molecules having translational degrees of freedom is taken as (V-V V and V0 are thesolid molar volumes0)of/V, the where liquid and the reference respectively.

Many of the properties of this state suggest that the plastic crystal state is closely related to the liquid state. This viewpoint is in agreement with results of X-ray studies, optical pro3’4 perties,indicate which and dielectric that rotation constant and/or measurements movement of molecules takes place in the plastic crystal state to a degree comparable to the movement in liquids. Nevertheless, the motion can not be 6 suggests that the free rotation. ~ Ubbelohde plastic property of this state must be associated with a comparatively large concentration of crystal defects such as lattice vacancies. He guesses that “marked plasticity would be expected for concentrations of vacancies of the order 0. 1 to 1. 0 per cent”.

Based on these arguments the partitionfunction is then written as a product of the partition functions written for each of the structures, weighted by the fraction of molecules in each structure,

Since the significant structures theory as used by Eyring and collaborators7 is able to pre-

NV

7

N(V-V

I~ \

=

v

/ 1N(vv0) ~

dict quite accurately all the thermodynamic and many physical properties of a liquid using a model that assumes the volume expansion of a liquid to be due to the introduction of holes inthe

‘~iiq

“‘soP

‘deg’

‘gas /





V

I.

where the factorial term is due to the indistinguishability of the gas-like molecules (i. e. 219

220

SIGNIFICANT STRUCTURES THEORY TO PLASTIC CRYSTALS

molecules undergoing translational motion). Ustug standard expressions for f~ofand fga~, using Eyring’s expression for ~deg’ and applytag Stirlings approximation for the factortalterm the partition function for CBr4 becomes

Z1jq

r

E

4

pEIRT



.~,

______

L,_e/T~~

latory molecules (the sublimation energy Es, the molar volume V0 and the Einstein characteristic temperature e) as well as “a” are considered as parameters of the liquid state (or plastic crystal state). These are evaluated by fitting them to the following experimental data: the molar volume and vapor pressure of the liquid at the triple point (or of the plastic crystal at the transition temperature) and the vapor pressure at some other temperature. Knowing values for these parameters, and the molecular constants, the thermodynamic and mechanical properties may be calculated using the standard formulas of statistical mecharncs.

~

iI~ii”(I*IZ~~9

ev~N(V-V,)

12 .r”2(8fr2kT)~2(~.IbI~)”2~ff 9

L (2flmkT)~ ~

Vol. 4, No. 5

,,,

~-~i~i

,

where ~ = [aE V /(V-V )RT and the other notation is stan~ar~.The ~~roperties of the oscil-

TABLE 1 The Calculated and Observed Properties of Carbon Tetrabromide in the Plastic Crystal State T

P

°K

V

mm Hg cc/mole

S cal/ ° -mole

Cbs. Caic.

320. 00 (Ttr)

3. 14 (3. 14)

102. 91 (102. 91)

59.59

Cbs. Caic.

325.00

4.20 4.23

103.18 103.24

60. 11

Cbs. Calc.

330.00

5.62 5.63

103.50 103.57

60.62

Cbs. Calc.

335.00

7.33 7.42

103.80 103. 90

61. 13

Cbs. Caic.

340. 00

9.57 9.70

104. 14 104.23

61.62

Cbs. Calc.

345. 00

12.3 12.56

104.46 104.56

62. 11

Cbs. Caic.

350.00

15.9 16. 12

104. 77 104. 90

62.60

Cbs. Calc.

355. 00

20.3 20. 54

105. 10 105. 25

63. 07

Cbs. Caic.

360.00 25.7 25.95

105.60

Cbs. Caic.

363.25 30. 1 (Tt~) (30. 1)

105.83

‘Reference state

-

-

-

ideal gas at 0. 0°K

-

-

-

-

-

-

-

-

-

63.55 -

63. 85

11* cal/ mole -

-6941 -

-6772 -

-6605 -

-6438

L~Hub

11800 12295

6.78

46.4

25. 88 26. 05

33.96

11800 12238

6.10 6.43

38.0 45.3

25.81 26. 11

33.6 33.54

11800 12182

5. 77 6.32

38.9 45. 7

25. 78 26. 17

33. 11 33.41

11800 12127

5.82 6.30

46. 7

25.74 26.22

33.03 33.40

6. 02 6.35

48.2

25. 71 26.28

33. 09 33.45

25.68 26.33

33.25 33.54

25.65 26.38

33.38 33.65

25.62 26. 43

33. 56 33. 77

25.60 26.47

33.90 33. 90

26.50

33. 99

-

-

12071

-

-

-6103

12016

-

-

-5935

11961

-

Cp cal/ °-mole

x 10’ °K’

-

11905

-

-

-5598

11849

-

-

-5487

11813

-

-

6.42 -

6.51 -

6. 62 -

6. 73 -

6. 81

x 106 atm’

7 cal/

caf~ mole

-6271

-5767

C

-

-

-

-

49. 9 -

51. 8 -

53. 9 -

56.2

°

-

57. 7

-

-

Vol. 4, No. 5

SIGNIFICANT STRUCTURES ThEORY TO PLASTIC CRYSTALS

A computer program written in FORTRAN 9, language was developed by one of the authors not only to make these computations, but also to find the parameters in a simplified consistent manner. This progr;m was designed for calculations on the liquid state but could be used without alteration for the plastic crystal state. For purposes of the program the properties of the plastic crystal state were calculated as though it were a liquid; that is, the transition point of the plastic crystal is used as equivalent to the melting point of a normal liquid. This means that in a substance in which the plastic crystal state occurs, crystal form I and crystal form II correspond, respectively, to the liquid and solid states for a normal substance. The input data for CBr4 was as follows: molecular weight 1b ==331. =647; 1330the x lCr’°; moments theof inertia~° ‘a =

221

vibrational energies in units of cm’ and their degeneraciesU 269. 0(1), 122. 0(2), 183. 0(3), and 667. 0(3); Ttrans = 320. 00 °K; Ptr~ = 3. 14 mm; V(I~.~.s= 102. 905 cm3 Tt~ = 363.25 °K; and Ptp = 30. 1 mm. The values obtained for the parameters are: V 3, Es = 13648 cal/mole, a = 0.0 00015 = 101. and 08 cm 8 = 25. 17°K. The results of the calculations for the plastic crystal state of CBr 4 are given in Table I along with the observed properties. The calculated concentration of vacancies at the transition point is 1.81 percent agreeing well with the prediction made by Ubbelohde. It is seen that the accuracy of the calculated properties is quite good in most cases somewhat better than the accuracy expected for similar calculations of liquids. It must be concluded, therefore, that the Significant Structures Theory yields anstate crystal adequate as well description as the liquid of the state. plastic -

References 1.

TIMMERMANS J.,

J. Phys. Chem. Solids, 18, 1 (1961).

2.

STAVELEY L.A.K.,

3.

DUNNING W.J.,

4.

SMYTH C. P.,

5.

MARSHALL J.G.,

6.

UBBELOHDE A.R., J. Phys. Chem. Solids, 18,

7.

EYRING H., REE T. and HIRAI H., Proc. Nat. Acad. Sci. U.S., 44, 683 (1958). EYRING H. and REE T., ibid., 47, 526 (1961).

Ann. Rev. Phys. Chem., 13, 351 (1962).

J. Phys. Chem. Solids, 18, 21 (1961). J. Phys. Chem. Solids,

18, 40 (1961).

STAVELEY L.A.K. and HART K.R., Trans. Far. Soc., 52,

19 (1956).

90 (1961).

EYRING H., HENDERSON D. and REE T., Progress in International Research on Thermodynamic and Transport Properties, American Society of Mechanical Engineers, 88, New York (1962). EYRING H. and MARCHI R. P., J. Chem. Educ., 40, 562 (1963). EYRING H., REE T.S. and REE T., tnt. J. ~Engng. Sct., 3, 285 (1965). 8.

BRADLEY R. S. and DRURY T.,

9.

ZANDLER M. E., Ph. D. Thesis, Arizona State University (1965).

10. GELLES G. and PITZER K.S.,

Trans. Far. Soc.,

J. Am. Chem. Soc.,

55,

1844 (1959).

75, 5259 (1953).

11. HERTZBERG G., Molecular Spectra and Molecular Structures, I. Spectra of Diatomic Molecules, 2nd Ed., Van Nostrand, Princeton, N.J. (1950).

222

SIGNIFICANT STRUCTURES THEORY TO PLASTIC CRYSTALS

Vol. 4,

Die Theorie der kennzeichnenden Struktur wie von Eyring, Ree und ihren Mitarbeitern fffr Fltfssigkeiten entwickelt, beweist sich als anwendbar f~ir den plastisch kristallinen Zustand. Die EigenschaIten von CBr4 (im plastisch kristalllnen Zustand) wurden berechnet und mit den experimentellen Angaben verglichen. Die Berechnungen wurden mit Hilfe eines Rechenautomats durchgeftthrt, wobei ohne Ab~ndemnngem Programm gebraucht wurde, das fur den flifssigen Zustand entwickelt worden war.

No. 5