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Molecular dynamics simulation of vacancy properties in plastic adamantane
Physica 131B (1985) 249-255 North-Holland, Amsterdam
M O L E C U L A R DYNAMICS SIMULATION OF VACANCY P R O P E R T I E S IN PLASTIC ADAMANTANE M. M E Y E R * S.R.M.P. CEN Saclay, 91191 Gif sur Yvette Cedex, France
C. M A R H I C L.P.M. CNRS 1 Place A. Briand 92195, Meudon-Principal Cedex, France
Molecular dynamics simulation techniques are used to calculate the formation energy of a single vacancy in plastic adamantane. Two different temperatures are investigated, 310 K and 400 K. The formation energies calculated for these two temperatures do not differ significantly. There is no translational relaxation around the vacancy, but there is a noticeable increase of orientational disorder in the neighbourhood of the point defect.
1. Introduction
Some polyatomic molecular crystals undergo a transition to a high temperature phase associated with a rather important transition heat. The characteristics of these high temperature phases which are generally referred to as 'plastic crystals' differ significantly from those of other molecular crystals: the entropy of fusion is smaller, the crystalline structure is cubic or hexagonal closepacked with a high degree of orientational disorder, the mechanical deformation is easy and the diffusion coefficients are high [1, 3]. These properties are valid for all 'plastic crystals' and it is important to point out that the orientational disorder is dynamical. But it is also interesting to notice that some of their characteristics, such as the amount of disorder, may vary significantly when crystals with a very low entropy of fusion are compared to those with a relatively high entropy of fusion. From the point of view of the diffusion mechanism, the point defects are vacancy type defects, but in order to explain some experimental results, obtained in highly disordered crystals, it is necessary to suggest the existence of relaxed defects and divacancies [2, 4]. The relaxation process is slightly more * On leave of absence from L.P.M. CNRS 1 PI.A Briand 92195, Meudon-Principal Cedex, France.
complicated in a polyatomic molecular crystal than in a monatomic one since it may be translational and/or rotational. In highly disordered crystals this rotational part may be appreciable since it is already known that the coupling between rotational and translational motion is important to explain some dynamical properties [1]. Molecular dynamics simulation is a suitable method to study some defect properties as a function of the temperature [5, 6] and it is particularly suitable when dynamical properties such as orientational disorder are involved. Recent advances in computer simulation techniques make it possible to explore the properties of polyatomic molecular crystals at finite temperature. For instance the computation time is reduced when a new version of the constraints method is used to study rigid molecules [7]. Adamantane or tricyclo(3,3,1,1)decane (C10H16) may be considered as an example of a 'plastic crystal' with a limited amount of disorder. The plastic phase is stable between 208.6 and 5 4 3 K [8]. The structure is face centered cubic with a lattice parameter a = 9.45/k at room temperature [9, 10]. The orientational disorder is due to the fact that the molecules are equally and randomly distributed between two distinct equilibrium orientations [11]. Diffusion experiments [12] and positron annihilation techniques [13] show that point defects are mostly
0378-4363/85/$03.30 O Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
250
M. Meyer and C. Marhic / Vacancy properties in plastic adamantane
single vacancies with a formation energy Ef = 70+ 10kJmo1-1. The structural properties and the rotational motion have already been studied in plastic adamantane by molecular dynamics simulation [14]. The good agreement between the results of these calculations and the experimental data, show that the model of rigid molecules interacting via a pairwise additive potential is suitable to calculate properties of the plastic phase of adamantane. It seems reasonable to calculate, with the same simulation technique, the formation energy of a single vacancy as a function of temperature. This is the object of the present work which also contains a study of the orientational disorder introduced by the point defect. Computational details are given in section 2; results are discussed in section 3. The paper ends with concluding remarks.
Table I A t o m - a t o m potential B ~ e x p ( - C~#r)
[15].
V~B = - A ~ r +
a fl
A,,~ o (kJ m o l - l A 6)
B~ (kJ mo1-1)
C~ (A-l)
C~7 C-H H-H
2376 523 114
349908 36677 11104
3.6 3.67 3.74
ditions were used to simulate an infinite crystal. Due to the nearest image convention, the potential energies functions were truncated at r~ = 14 A. The adamantane molecule has 26 atoms (C~0H~6: fig. la) and the a t o m - a t o m model requires a lot of computation time since the energy of a pair of molecules involves 676 interactions. In order to perform the simulation more efficiently, the algorithm used to evaluate the forces has been
2. Computational details The formation energy of a single vacancy and the disorder introduced by this point defect were evaluated by molecular dynamics simulation. The calculations were performed in plastic adamantane by comparing for each of the two investigated temperatures (310, 400 K) a crystal without defect and a crystal containing one single vacancy. The formation energy Ef was obtained by calculating the potential energy difference between the two crystals. The orientational disorder was investigated by comparing the orientational probability density function for the molecules in the defect-free system and for the nearest neighbours of the vacancy.
parameters
carbon
atoms
• hydrogen ....
z
T I
2.1. Molecular dynamics simulation Molecular dynamics calculations were performed on two systems, the first one containing 108 molecules (defect free) and the second one 107 molecules and one vacancy. Forces and energies were calculated with empirical pairwise additive a t o m - a t o m potentials. The parameters fitted by Williams (set IV) [15] were used in (exp-6) functions to calculate C-C, C-H, H - H interactions (see table I). Periodic boundary con-
/
b
Fig. 1. a) Diagram of the a d a m a n t a n e molecule C10H16 with four carbon atoms bonded to one hydrogen atom (tertiary carbons) and six carbon atoms bonded to two hydrogen atoms (secondary carbons); b) Regular tetrahedron build on the four tertiary carbons, these atoms are not bonded together, the lines between the atoms are drawn to show the tetrahedral shape. The dashed lines indicate the molecular axes.
M. Meyer and C. Marhic / Vacancy properties in plastic adamantane
vectorised and the calculations were performed on a C R A Y 1S computer. As the molecule is assumed to be rigid, a method of constraints in Cartesian coordinates is used [7] and the number of equations of motion which have to be integrated explicitly is reduced to those of four non-coplanar atoms per molecule. The atoms selected to determine the motion of each molecule are the four tertiary carbons (fig. lb). The equations of motion were integrated with Verlet's algorithm [16] and a time step At = 0.01 ps. 1500 steps were allowed for equilibration and static properties were obtained by averaging over a further 2500 steps. Selected values of the static properties are listed in table II (mean potential energy U, pressure P, mean square displacements of the centre of mass of the molecules
(u2,)).
2.2. Orientational order The tetrahedral molecular symmetry is well depicted by t h e regular tetrahedron built on the four tertiary carbons (fig. lb). In order to discuss the orientations of the adamantane molecules in the cubic Fm3m lattice, it is interesting to describe two relevant orientations of a regular tetrahedron in a cubic field. i) T o orientation: the four threefold axes of
251
the tetrahedron coincide with four of the eight threefold axes of the cube (fig. 2). There are two distinct T o orientations. ii) C3v orientation: one threefold axis and 3 mirror planes containing this axis are common to the tetrahedron and to the cube (fig. 2). This orientation is obtained from the previous one by a 7r/3 rotation of the tetrahedron about a threefold axis. There are eight distinct Csv orientations. The experimental results show that the most probable positions of the molecules in plastic adamantane are the two T d orientations [11]. In this paper, the orientational order is described by the orientational probability density function P(O, ~) where 0 and ~0 are the polar angles of a vector joining the centre of mass of a molecule to a tertiary carbon. P ( 0 , ~ ) is expanded in terms of symmetry adapted functions which are the surface harmonics relative to the cubic point groups [20] or cubic harmonics, P(O,
~) =(1/4~)(1 + •
(KI)KI(O , ~)) .
I
',
i Table II Molecular dynamics results
Td
T (K)
P (MPa)
U (kJ tool -I)
N
(u 2) (~2)
307.5 312 306 309
19.5 13.5 9.9 14.5
60.9 61 60.5 60.4
108 108 107 107
0.032
401.2 398.7 401.4
3.5 4.1 4.8
55.6 55.1 55.1
108 107 107
0.052
(u~)n,, (,~2)
~ 1/ / 3
, //
a
0.038
C3v 0.066
* The subscript nn refers to the nearest neighbours of the vacancy. T h e experimental values of the latent heat of sublimation in the temperature range 298-366 K vary between 59.3 and 53.5 kJ tool -1 [17, 19]. T h e experimental value of the m e a n square displacement (u 2) is equal to 0.046 A at T = 3 0 0 K [10].
t:~.
3
Fig. 2. Two relevant orientations of a regular tetrahedron in a cubic field. Td orientations: the threefold axes (3) of the tetrahedron coincide with four of the eight threefold axes of the cubic site. C3v orientation: one threefold axis (3) of the tetrahedron coincide with one of the cube.
252
M. Meyer and C. Marhic / Vacancy properties in plastic adamantane
The cubic harmonics are linear combinations of spherical harmonics [21] and have been tabulated up to l = 12 [20]. The symmetry properties of the orientational probability density function depend on the molecular and lattice symmetries [22]. For a d a m a n t a n e a complete basis of Kt is obtained for 1 values equal to 4, 6, 8, 9, 10, 12. The coefficients (Kt) were determined by averaging on the 2500 time steps the values of Kt(O,~o) calculated for the tertiary carbons. P(O,~) converges rapidly enough to be calculated with good accuracy in spite of the limitation of the expansion to / = 12.
10.0
5.0
0.0
-5.0
-10.0
3. Results
3.1. Structural properties The centre of mass coordinates of the molecules surrounding the defect were calculated, and the average positions do not vary in comparison with the defect-free lattice. The projected trajectories of the centre show that there is no distortion of the lattice around the vacancy (fig. 3). The mean square displacement of nearest neighbour molecules to the vacancy has been calculated at 310 and 4 0 0 K ; it is 20% higher than for the molecules in the defect-free crystal (table II). T h e r e is no translational relaxation around the defect but the presence of a vacancy induces an increase in the vibrational amplitude of the nearest neighbour molecules. Contour line diagrams were plotted with the values of P(O, q~) calculated at T -- 310 and 400 K in a defect-free lattice and for the 12 nearest neighbours of the vacancy. An example is given in fig. 4, where the orientationai probability density fucntion obtained at T = 4 0 0 K in a defect-free lattice is plotted with 0 and q~ varying from 0 to ~. These diagrams show clearly that the most probable molecular orientations are the T d ones; the m a x i m u m value of the probability density function decreases when the t e m p e r a t u r e increases. There are also local maxima corresponding to C3v orientations; their probability increases with the temperature. The orientational probability density function of the 12
TJ -10.0
\TJ -5.0
0.0
5.0
10.0
Fig. 3. Projected trajectories of the centre of mass of the molecules calculated at 400 K in a crystal containing one vacancy. These trajectories correspond to the molecules belonging to the 3 (100) planes containing the vacancy. T h e outlines of the molecules are drawn for an ideal fcc lattice (straight lines) containing one vacancy (dotted outline).
nearest neighbours of the vacancy is also at a m a x i m u m for the T d orientations but C3v orientations are more probable than in the defect-free crystal (fig. 5). The orientational probability of C3v is not negligible since there are 8C3v orientations and only 2T d. The values of the maxima and of the minima of P(O, q~) show that the orientational order decreases around the vacancy and the relative probabilities of the T d and C3~ orientations are also modified by the defect.
3.2. Energy of formation of a vacancy The potential energies of the two systems were evaluated for temperatures close to 310 K and 400K. The equilibrium values reached by the t e m p e r a t u r e and the pressure are not exactly the same for the crystal without defect and for the crystal containing one vacancy. The main preblem is due to the evaluation of the pressure which varies significantly from one system to the other (table II). The values of t e m p e r a t u r e and pressure are closer for the systems simulated around 400 K. The values of the potential energy differences are summarized in table III together
M. Meyer and C. Marhic / Vacancy properties in plastic adamantane
253
Table III Formation energy of a single vacancy
with the values of pressure and the relative variations of the temperature. Since there is a wide spread in the equilibrium values of the pressure obtained at 310K the corresponding values of the formation energy of a single vacancy are scattered. If one neglects at 310 K (table III), the first two values for which the pressure differences between the two systems are very high, the formation energies of a single vacancy at 310 and 4 0 0 K are respectively (60.5-+2.5) and ( 5 9 +- 1) kJmol -i. Within the limits of accuracy of this calculation can conclude that, for adamantane, there is no variation of the formation energy of a single vacancy with the temperature.
Et
T
AT/T
(kJ mol -I)
(K)
(%)
P(108) (MPa)
59.5 58
400 400
0 0.6
3.5 3.5
48 53 58 63
310 310 310 310
0.5 0.5 1.0 2.0
19.5 19.5 13.5 13.5
P(107) (MPa) 4.8 4.1 9.9 14.5 9.9 14.5
P(108) pressure of the crystal without defect; P(107) pressure of the crystal containing one vacancy; Et formation energy of the vacancy (potential energy difference); A T/T relative variation of the temperature.
"ff
a
4>
'!'1"
'
1"I 2
o
,.-., "1"1"
7
I
0
Fig. 4. Contour lines plotted with the orientational probability density function calculated at 400 K for the tertiary carbons. 0 and ~0 vary from 0 to lr. Four main maximums correspond to the two possible Td orientations (0.5), 16 secondary maximums (4 of them coincide with the Td orientations) correspond to the C3~orientations (0.055). This diagram has been plotted for a crystal without defect. (a) P(O, ~p) indexed contour lines; (b) P(O, ~) Td and C~ orientation are indicated.
254
b
1T 2
0 O "IT
©
17
O
~®
3vl° ie
0
Fig. 4. (continued).
b
a
.02
• 04
.15
O~
- 20
.03 --~ .02
.30
4,
f 0
0
rr
0
0
_rr 2
Fig. 5. Comparison of the orientational probability density function relative to tertiary carbons calculated at 400 K in a crystal without defect (a) and for the 12 molecules nearest neighbour of the vacancy (b). The contour lines are drawn for 0 and ~ varying between 0 and ~/2 (1/4 of the orientational space investigated in fig. 4), the Td and C3v orientations are indicated with the same symbols as in fig. 4b).
M. Meyer and C. Marhic / Vacancy properties in plastic adamantane
4. Conclusion The study by molecular dynamics simulation at 310 and 400 K of a vacancy in plastic adamantane show that there is no translational relaxation around the defect. The translational and orientational disorder is higher for the molecules surrounding the vacancy. The mean square displacement of their centre of mass is increased; the preferential orientation of these molecules is the same as in the perfect crystal but with a lesser probability. The occurrence of another orientation is increased for the molecules around the defect, this may be related to a modification of the rotational jumps of the molecules reorienting between equilibrium positions. More information could be obtained by studying the frequencies and the types of rotational jumps of the molecules surrounding the vacancy. The formation energy Ef of a single vacancy is close to the experimental value and there is no significant variation of Ef with the temperature.
Acknowledgement This work has been supported by the Scientific Council of CCVR who supplied the computer time on a CRAY 1S.
References [1] The Plastically Crystalline State, J.N. Sherwood, ed. (Wiley, New York, 1979).
255
[2] A.V. Chadwick, Mass Transport in Solids, F. Beniere and C.R.A. Catlow, eds., NATO ASI series B Physics 97 (1983) 285. [3] J.M. Chezeau and J.H. Strange, Phys. Rep. 53 (1979) 1. [4] M. Brissaud, Ann. Chim. Fr. 4 (1979) 534. [5] C.H. Bennett, Diffusion in Solids, A.S. Nowick and J.J. Burton, eds. (Academic Press, New York, 1975) p. 73. [6] G. Jacucci, Mass Transport in Solids, F. Beniere and C.R.A. Catlow, eds., NATO ASI series B Physics 97 (1983) 131. [7] G. Ciccotti, M. Ferrarrio and J.P. Ryckaert, Mol. Phys. 47 (1982) 1253. [8] S.S. Chang and E.F. Westrum, J. Phys. Chem. 64 (1960) 1547. [9] C.E. Nordman and D.L. Schmittkons, Acta Cryst. 18 (1965) 764. [10] J.P. Amoureux, M. Bee and J.C. Damien, Acta Cryst. B36 (1980) 2633. [11] J.P. Amoureux and M. Bee, Acta Cryst. B36 (1980) 2636. [12] J. Bleay, P.W. Salthouse and J.N. Sherwood, Phil. Mag. 36 (1977) 885. [13] D. Lightbody, J.N. Sherwood and M. Eldrup, Chem. Phys. Lett. 70 (1983) 487. [14] M. Meyer and G. Cicotti, Submitted to Mol. Phys. [15] D.E. Williams, J. Chem. Phys. 47 (1967) 4680. [16] L. Verlet, Phys. Rev. 159 (1967) 98. [17] P.J. Wu, L. Hsu and D.A. Dows, J. Chem. Phys. 54 (1971) 2714. [18] W.K. Bratton and I. Szilard, J. Org. Chem. 32 (1967) 2019. [19] M. Mansson, N. Rapport and E.F. Westrum, J. Am. Chem. Soc. 92 (1970) 7296. [20] C.J. Bradley and A.F. Cracknell, The Mathematical Theory of Symmetry in Solids (Clarendon, Oxford, 1972) p. 51. [21] F.M. Mueller and M.G. Priestley, Phys. Rev. 148 (1966) 638. [22] M. Yvinec and R. Pick, J. Physique 41 (1980) 1045.
M O L E C U L A R DYNAMICS SIMULATION OF VACANCY P R O P E R T I E S IN PLASTIC ADAMANTANE M. M E Y E R * S.R.M.P. CEN Saclay, 91191 Gif sur Yvette Cedex, France
C. M A R H I C L.P.M. CNRS 1 Place A. Briand 92195, Meudon-Principal Cedex, France
Molecular dynamics simulation techniques are used to calculate the formation energy of a single vacancy in plastic adamantane. Two different temperatures are investigated, 310 K and 400 K. The formation energies calculated for these two temperatures do not differ significantly. There is no translational relaxation around the vacancy, but there is a noticeable increase of orientational disorder in the neighbourhood of the point defect.
1. Introduction
Some polyatomic molecular crystals undergo a transition to a high temperature phase associated with a rather important transition heat. The characteristics of these high temperature phases which are generally referred to as 'plastic crystals' differ significantly from those of other molecular crystals: the entropy of fusion is smaller, the crystalline structure is cubic or hexagonal closepacked with a high degree of orientational disorder, the mechanical deformation is easy and the diffusion coefficients are high [1, 3]. These properties are valid for all 'plastic crystals' and it is important to point out that the orientational disorder is dynamical. But it is also interesting to notice that some of their characteristics, such as the amount of disorder, may vary significantly when crystals with a very low entropy of fusion are compared to those with a relatively high entropy of fusion. From the point of view of the diffusion mechanism, the point defects are vacancy type defects, but in order to explain some experimental results, obtained in highly disordered crystals, it is necessary to suggest the existence of relaxed defects and divacancies [2, 4]. The relaxation process is slightly more * On leave of absence from L.P.M. CNRS 1 PI.A Briand 92195, Meudon-Principal Cedex, France.
complicated in a polyatomic molecular crystal than in a monatomic one since it may be translational and/or rotational. In highly disordered crystals this rotational part may be appreciable since it is already known that the coupling between rotational and translational motion is important to explain some dynamical properties [1]. Molecular dynamics simulation is a suitable method to study some defect properties as a function of the temperature [5, 6] and it is particularly suitable when dynamical properties such as orientational disorder are involved. Recent advances in computer simulation techniques make it possible to explore the properties of polyatomic molecular crystals at finite temperature. For instance the computation time is reduced when a new version of the constraints method is used to study rigid molecules [7]. Adamantane or tricyclo(3,3,1,1)decane (C10H16) may be considered as an example of a 'plastic crystal' with a limited amount of disorder. The plastic phase is stable between 208.6 and 5 4 3 K [8]. The structure is face centered cubic with a lattice parameter a = 9.45/k at room temperature [9, 10]. The orientational disorder is due to the fact that the molecules are equally and randomly distributed between two distinct equilibrium orientations [11]. Diffusion experiments [12] and positron annihilation techniques [13] show that point defects are mostly
0378-4363/85/$03.30 O Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
250
M. Meyer and C. Marhic / Vacancy properties in plastic adamantane
single vacancies with a formation energy Ef = 70+ 10kJmo1-1. The structural properties and the rotational motion have already been studied in plastic adamantane by molecular dynamics simulation [14]. The good agreement between the results of these calculations and the experimental data, show that the model of rigid molecules interacting via a pairwise additive potential is suitable to calculate properties of the plastic phase of adamantane. It seems reasonable to calculate, with the same simulation technique, the formation energy of a single vacancy as a function of temperature. This is the object of the present work which also contains a study of the orientational disorder introduced by the point defect. Computational details are given in section 2; results are discussed in section 3. The paper ends with concluding remarks.
Table I A t o m - a t o m potential B ~ e x p ( - C~#r)
[15].
V~B = - A ~ r +
a fl
A,,~ o (kJ m o l - l A 6)
B~ (kJ mo1-1)
C~ (A-l)
C~7 C-H H-H
2376 523 114
349908 36677 11104
3.6 3.67 3.74
ditions were used to simulate an infinite crystal. Due to the nearest image convention, the potential energies functions were truncated at r~ = 14 A. The adamantane molecule has 26 atoms (C~0H~6: fig. la) and the a t o m - a t o m model requires a lot of computation time since the energy of a pair of molecules involves 676 interactions. In order to perform the simulation more efficiently, the algorithm used to evaluate the forces has been
2. Computational details The formation energy of a single vacancy and the disorder introduced by this point defect were evaluated by molecular dynamics simulation. The calculations were performed in plastic adamantane by comparing for each of the two investigated temperatures (310, 400 K) a crystal without defect and a crystal containing one single vacancy. The formation energy Ef was obtained by calculating the potential energy difference between the two crystals. The orientational disorder was investigated by comparing the orientational probability density function for the molecules in the defect-free system and for the nearest neighbours of the vacancy.
parameters
carbon
atoms
• hydrogen ....
z
T I
2.1. Molecular dynamics simulation Molecular dynamics calculations were performed on two systems, the first one containing 108 molecules (defect free) and the second one 107 molecules and one vacancy. Forces and energies were calculated with empirical pairwise additive a t o m - a t o m potentials. The parameters fitted by Williams (set IV) [15] were used in (exp-6) functions to calculate C-C, C-H, H - H interactions (see table I). Periodic boundary con-
/
b
Fig. 1. a) Diagram of the a d a m a n t a n e molecule C10H16 with four carbon atoms bonded to one hydrogen atom (tertiary carbons) and six carbon atoms bonded to two hydrogen atoms (secondary carbons); b) Regular tetrahedron build on the four tertiary carbons, these atoms are not bonded together, the lines between the atoms are drawn to show the tetrahedral shape. The dashed lines indicate the molecular axes.
M. Meyer and C. Marhic / Vacancy properties in plastic adamantane
vectorised and the calculations were performed on a C R A Y 1S computer. As the molecule is assumed to be rigid, a method of constraints in Cartesian coordinates is used [7] and the number of equations of motion which have to be integrated explicitly is reduced to those of four non-coplanar atoms per molecule. The atoms selected to determine the motion of each molecule are the four tertiary carbons (fig. lb). The equations of motion were integrated with Verlet's algorithm [16] and a time step At = 0.01 ps. 1500 steps were allowed for equilibration and static properties were obtained by averaging over a further 2500 steps. Selected values of the static properties are listed in table II (mean potential energy U, pressure P, mean square displacements of the centre of mass of the molecules
(u2,)).
2.2. Orientational order The tetrahedral molecular symmetry is well depicted by t h e regular tetrahedron built on the four tertiary carbons (fig. lb). In order to discuss the orientations of the adamantane molecules in the cubic Fm3m lattice, it is interesting to describe two relevant orientations of a regular tetrahedron in a cubic field. i) T o orientation: the four threefold axes of
251
the tetrahedron coincide with four of the eight threefold axes of the cube (fig. 2). There are two distinct T o orientations. ii) C3v orientation: one threefold axis and 3 mirror planes containing this axis are common to the tetrahedron and to the cube (fig. 2). This orientation is obtained from the previous one by a 7r/3 rotation of the tetrahedron about a threefold axis. There are eight distinct Csv orientations. The experimental results show that the most probable positions of the molecules in plastic adamantane are the two T d orientations [11]. In this paper, the orientational order is described by the orientational probability density function P(O, ~) where 0 and ~0 are the polar angles of a vector joining the centre of mass of a molecule to a tertiary carbon. P ( 0 , ~ ) is expanded in terms of symmetry adapted functions which are the surface harmonics relative to the cubic point groups [20] or cubic harmonics, P(O,
~) =(1/4~)(1 + •
(KI)KI(O , ~)) .
I
',
i Table II Molecular dynamics results
Td
T (K)
P (MPa)
U (kJ tool -I)
N
(u 2) (~2)
307.5 312 306 309
19.5 13.5 9.9 14.5
60.9 61 60.5 60.4
108 108 107 107
0.032
401.2 398.7 401.4
3.5 4.1 4.8
55.6 55.1 55.1
108 107 107
0.052
(u~)n,, (,~2)
~ 1/ / 3
, //
a
0.038
C3v 0.066
* The subscript nn refers to the nearest neighbours of the vacancy. T h e experimental values of the latent heat of sublimation in the temperature range 298-366 K vary between 59.3 and 53.5 kJ tool -1 [17, 19]. T h e experimental value of the m e a n square displacement (u 2) is equal to 0.046 A at T = 3 0 0 K [10].
t:~.
3
Fig. 2. Two relevant orientations of a regular tetrahedron in a cubic field. Td orientations: the threefold axes (3) of the tetrahedron coincide with four of the eight threefold axes of the cubic site. C3v orientation: one threefold axis (3) of the tetrahedron coincide with one of the cube.
252
M. Meyer and C. Marhic / Vacancy properties in plastic adamantane
The cubic harmonics are linear combinations of spherical harmonics [21] and have been tabulated up to l = 12 [20]. The symmetry properties of the orientational probability density function depend on the molecular and lattice symmetries [22]. For a d a m a n t a n e a complete basis of Kt is obtained for 1 values equal to 4, 6, 8, 9, 10, 12. The coefficients (Kt) were determined by averaging on the 2500 time steps the values of Kt(O,~o) calculated for the tertiary carbons. P(O,~) converges rapidly enough to be calculated with good accuracy in spite of the limitation of the expansion to / = 12.
10.0
5.0
0.0
-5.0
-10.0
3. Results
3.1. Structural properties The centre of mass coordinates of the molecules surrounding the defect were calculated, and the average positions do not vary in comparison with the defect-free lattice. The projected trajectories of the centre show that there is no distortion of the lattice around the vacancy (fig. 3). The mean square displacement of nearest neighbour molecules to the vacancy has been calculated at 310 and 4 0 0 K ; it is 20% higher than for the molecules in the defect-free crystal (table II). T h e r e is no translational relaxation around the defect but the presence of a vacancy induces an increase in the vibrational amplitude of the nearest neighbour molecules. Contour line diagrams were plotted with the values of P(O, q~) calculated at T -- 310 and 400 K in a defect-free lattice and for the 12 nearest neighbours of the vacancy. An example is given in fig. 4, where the orientationai probability density fucntion obtained at T = 4 0 0 K in a defect-free lattice is plotted with 0 and q~ varying from 0 to ~. These diagrams show clearly that the most probable molecular orientations are the T d ones; the m a x i m u m value of the probability density function decreases when the t e m p e r a t u r e increases. There are also local maxima corresponding to C3v orientations; their probability increases with the temperature. The orientational probability density function of the 12
TJ -10.0
\TJ -5.0
0.0
5.0
10.0
Fig. 3. Projected trajectories of the centre of mass of the molecules calculated at 400 K in a crystal containing one vacancy. These trajectories correspond to the molecules belonging to the 3 (100) planes containing the vacancy. T h e outlines of the molecules are drawn for an ideal fcc lattice (straight lines) containing one vacancy (dotted outline).
nearest neighbours of the vacancy is also at a m a x i m u m for the T d orientations but C3v orientations are more probable than in the defect-free crystal (fig. 5). The orientational probability of C3v is not negligible since there are 8C3v orientations and only 2T d. The values of the maxima and of the minima of P(O, q~) show that the orientational order decreases around the vacancy and the relative probabilities of the T d and C3~ orientations are also modified by the defect.
3.2. Energy of formation of a vacancy The potential energies of the two systems were evaluated for temperatures close to 310 K and 400K. The equilibrium values reached by the t e m p e r a t u r e and the pressure are not exactly the same for the crystal without defect and for the crystal containing one vacancy. The main preblem is due to the evaluation of the pressure which varies significantly from one system to the other (table II). The values of t e m p e r a t u r e and pressure are closer for the systems simulated around 400 K. The values of the potential energy differences are summarized in table III together
M. Meyer and C. Marhic / Vacancy properties in plastic adamantane
253
Table III Formation energy of a single vacancy
with the values of pressure and the relative variations of the temperature. Since there is a wide spread in the equilibrium values of the pressure obtained at 310K the corresponding values of the formation energy of a single vacancy are scattered. If one neglects at 310 K (table III), the first two values for which the pressure differences between the two systems are very high, the formation energies of a single vacancy at 310 and 4 0 0 K are respectively (60.5-+2.5) and ( 5 9 +- 1) kJmol -i. Within the limits of accuracy of this calculation can conclude that, for adamantane, there is no variation of the formation energy of a single vacancy with the temperature.
Et
T
AT/T
(kJ mol -I)
(K)
(%)
P(108) (MPa)
59.5 58
400 400
0 0.6
3.5 3.5
48 53 58 63
310 310 310 310
0.5 0.5 1.0 2.0
19.5 19.5 13.5 13.5
P(107) (MPa) 4.8 4.1 9.9 14.5 9.9 14.5
P(108) pressure of the crystal without defect; P(107) pressure of the crystal containing one vacancy; Et formation energy of the vacancy (potential energy difference); A T/T relative variation of the temperature.
"ff
a
4>
'!'1"
'
1"I 2
o
,.-., "1"1"
7
I
0
Fig. 4. Contour lines plotted with the orientational probability density function calculated at 400 K for the tertiary carbons. 0 and ~0 vary from 0 to lr. Four main maximums correspond to the two possible Td orientations (0.5), 16 secondary maximums (4 of them coincide with the Td orientations) correspond to the C3~orientations (0.055). This diagram has been plotted for a crystal without defect. (a) P(O, ~p) indexed contour lines; (b) P(O, ~) Td and C~ orientation are indicated.
254
b
1T 2
0 O "IT
©
17
O
~®
3vl° ie
0
Fig. 4. (continued).
b
a
.02
• 04
.15
O~
- 20
.03 --~ .02
.30
4,
f 0
0
rr
0
0
_rr 2
Fig. 5. Comparison of the orientational probability density function relative to tertiary carbons calculated at 400 K in a crystal without defect (a) and for the 12 molecules nearest neighbour of the vacancy (b). The contour lines are drawn for 0 and ~ varying between 0 and ~/2 (1/4 of the orientational space investigated in fig. 4), the Td and C3v orientations are indicated with the same symbols as in fig. 4b).
M. Meyer and C. Marhic / Vacancy properties in plastic adamantane
4. Conclusion The study by molecular dynamics simulation at 310 and 400 K of a vacancy in plastic adamantane show that there is no translational relaxation around the defect. The translational and orientational disorder is higher for the molecules surrounding the vacancy. The mean square displacement of their centre of mass is increased; the preferential orientation of these molecules is the same as in the perfect crystal but with a lesser probability. The occurrence of another orientation is increased for the molecules around the defect, this may be related to a modification of the rotational jumps of the molecules reorienting between equilibrium positions. More information could be obtained by studying the frequencies and the types of rotational jumps of the molecules surrounding the vacancy. The formation energy Ef of a single vacancy is close to the experimental value and there is no significant variation of Ef with the temperature.
Acknowledgement This work has been supported by the Scientific Council of CCVR who supplied the computer time on a CRAY 1S.
References [1] The Plastically Crystalline State, J.N. Sherwood, ed. (Wiley, New York, 1979).
255
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