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Calculation of the elastic constants of phenanthrene crystals
239
Chemical Physics 115 (1987) 239-242 North-Holland, Amsterdam
CALCULATION OF THE ELASTIC CONSTANTS OF PHENANTHRENE CRYSTALS * R. KULVER,
K.-H. BROSE
and C.J. ECKHARDT
Department of Chemistry, University of Nebraska, Lincoln, NE 68588-0304, USA Received 16 March 1987
A lattice dynamical calculation in the harmonic approximation is employed to obtain the sound velocities in phenanthrene crystals at 0 K and with no stress. A Buckingham potential with Williams parameterization is used to determine the intermolecular potential energy. The potential obtained minimizes satisfactorily with respect to the known crystal structure. The elastic constants are obtained by using an iterative optimization procedure.
1. Introduction Little attention has been given to the lattice dynamical properties of molecular crystals, especially their phonon dynamics and elastic properties. For the greater proportion of these solids the available information, experimental or theoretical, is nonexistent. Nevertheless, these materials are of considerable interest since they represent a class of crystals where the types of binding forces are limited and, therefore, their properties should be easier to understand. In the case of the simple hydrocarbons, cohesion is due almost exclusively to van der Waals interactions. This usually provides for a clear separation in energy of the internal modes of the molecules from the external or lattice vibrations and, thus, surprisingly good results can be obtained from lattice dynamical calculations which treat the molecules as rigid. Our extension of piezomodulation spectroscopy to molecular crystals where a periodic elastic stress is applied to a crystal and a response, usually optical, is synchronously detected, has caused a much closer examination of the interaction of the excitations of the lattice with its mechanical properties [l]. Indeed, it has been demonstrated that
* Research supported by the Solid State Chemistry Program of the National Science Foundation.
the amount of mechanical energy involved with each excitation can be determined from the piezomodulated reflection spectra [l]. In the case of molecular crystals, an intimate connection between the van der Waals forces binding the lattice and the excitations of the system exists [2]. Calculations of elastic constants permits a test of the derived potential since experimental verification, such as Brillouin scattering, is possible. In addition, pressure- or stress-induced phase changes can be examined. This work presents the complete set of theoretical elastic constants for phenanthrene crystals. The molecule, a structural isomer of anthracene, possesses a small dipole moment estimated to be around 0.04 D [3] whereas its congener has none. The crystals of both are of the monoclinic class, but phenanthrene crystals have noticeably different physical properties. For example, the melting point of phenanthrene is 384 K while that for anthracene is 489 K. Although several calculations of the elastic constants for anthracene have been reported [4], there are none for the crystal of the isomer, phenanthrene. The stress applied to a crystal produces a macroscopic deformation, the external strain e. These are related through the compliance S: e=Sa. The elastic constants
0301-0104/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
which are elements of the
240
R Kulver et al. / Elastic constants of phenanthrene crystals
stiffness tensor form a fourth-rank tensor as do the compliances. This can lead to complications for the analysis of these elements because the low symmetries common to molecular solids generate a large number of finite tensor elements. For the monoclinic phenanthrene crystal one obtains Cl1
Cl2
Cl3
0
Cl5
0
Cl2
c22
c23
0
c25
0
Cl3
c23
c33
0
c35
0
0
0
0
CM
0
C&
Cl5
c25
c35
0
c55
0
0
0
0
c&j
0
c66
where the usual two-suffix notation is employed [5]. The determination of the stiffness tensor is not straightforward since sets of cubic equations in either thirteen (monoclinic) or twenty-one (triclinic) independent variables must be solved. This is nontrivial because of coupling through multiple products of elastic constants. In the next section the nature of the potential energy function and lattice dynamics are discussed. This is followed by the results of the calculation of the elastic constants.
2. Determination
of the potential energy function
The calculation of the elastic constants must begin with obtaining an appropriate lattice potential energy function from which the lattice frequencies and subsequently the elastic constants can be derived. It is crucial that the potential reasonably reproduces the experimentally observed crystal structure, the vibrational spectrum of the lattice and the binding energy of the crystal. This has been done for phenanthrene crystals. The lattice constants have been obtained by minimization of the lattice potential energy within the symmetry imposed by the experimentally observed P2, space group [6]. As a consequence of this symmetry, the potential energy surface is spanned by the a, b and c lattice periods, the angle fi and the three Euler angles describing the orientation of the molecule at the origin of the unit cell with respect to the Cartesian a&* coordinate system
and the center-of-mass coordinates of the molecules. Without the minimization, it is not certain that the stability of the crystal structure is assured and that a strain-free equilibrium configuration under the assumed potential energy function is obtained. Structure minimization was effected with parameters for the Buckingham potential (6-exp) obtained from the work of Williams [7]. A radius of summation of 28 molecules did not provide good values of the lattice constants and the molecular orientations. The summation radius of interactions had to be increased to 88 molecules to obtain good convergence of the energy. This was found to be in quite good agreement with the literature values of 21 to 22 kcal/mol [8]. Difficulty was still faced with the values of the lattice parameters. The work of MUM et al. indicated that the small but finite dipole of phenanthrene, 0.04 D, might be of importance in considerations of the polarizability [9]. Since other work in our laboratory on TCNQ indicated the importance of the role of electrostatic interactions [lo], it appeared reasonable to include these in the phenanthrene. Charges calculated using an MO SCF (3G STO) were incorporated following the approach of Govers [ll]. With this emendation to the calculation, a satisfactory agreement with the lattice parameters and molecular orientations was achieved. It is noteworthy that the inclusion of the coulombic interaction had no significant effect on
Table 1 Lattice minhization
a b ; x Y ; # 8 u latt
results a) Exp.
talc.
8.46 6.16 9.47 97.7 2.09 1.78 0.74 327.5 79.6 22.7 22
8.27 6.64 9.34 106.1 2.34 2.51 0.80 300.4 71.6 1.0 21.36
‘) a, b, c, x, y, z are in fmgstrijm; all angles in degree; lattice energy in kcal/mol.
R. Kulver et al. / Elastic constants ofphenanthrene
the lattice energy. These results are presented in table 1 together with the values obtained from the X-ray structure. The greatest disagreement is in the monoclinic angle. The intermolecular force constants must be used to calculate the elastic constants of the crystal. The force constants are the second-order terms in a Taylor series expansion of the lattice potential for infinitesimal displacements about molecular equilibrium positions and may thereby be obtained from the calculated lattice potential energy function. The dynamical matrices are formed from these. The normal Cartesian ubc* system was used and solutions to the secular equations obtained giving frequencies w(q, j) for magnitudes of the wavevector, q, extremely close to the center of the Brillouin zone. The slope of the appropriate acoustic phonon branches, j, yielded the sound velocities:
3. Calculation of elastic constants In the treatment of crystal acoustics, the sound waves which propagate in a solid yield an eigenvalue equation reminiscent of that obtained in the formulation of lattice dynamics. The crystal den-
Table 2 Calculated elastic phenanthrene stiffness (kbar) 101.01 74.15 19.76
stiffness
18.82 - 17.73 135.55
and
compliance
0.0
0.0 0.0 70.07
tensors
33.88 11.10 -60.24 0.0 25.72
for
0.0 0.0 0.0 16.72 0.0 8.07
compliance (lo-’
kbar-') 3.98 -30.33 26.31 267.00 -144.00 72.66
0.0 0.0
0.0
0.0
-413.56 0.0 0.0 197.72 0.0 28.23 0.0 -58.52 588.36 0.0 245.26
crysials
241
sity, and the velocity of the elastic (sound) wave, u(q) are related through Christoffel’s equation: IlIYq) - PV2111 = 0. Here r(q) is the macroscopic equivalent of the dynamical matrix and is a 3 X 3 matrix with elements of the form:
The cikjr are elastic constants and the q are the direction cosines of the wavevector to the k and I reference axes. To obtain the elastic constants we have adopted a procedure which differs from that normally employed but which has been applied to cubic systems [12]. The common approach solves for the elastic constants in a sequential process beginning with equations for wavevectors with the most elementary linear velocity-elastic constant relationships. Progressive solutions are obtained using previously calculated values. In this method the great drawback is that the errors propagate and accumulate such that the last determined off-diagonal elements of the tensor normally carry quite large uncertainties [13]. Additionally, the solution of the polynomial equations leads to ambiguities which require choices for intermediate results. An alternative approach requiring extreme effort is to set up a direct non-linear least-squares solution 1141. The calculations here employ an iterative optimization which is initiated with a complete but arbitrary set of elastic constants. In each iteration, the Christoffel determinants are solved for their eigenvalues, p 02,) for all wavevectors for those observed sound velocities considered. The error vector, e, is subsequently determined and its square formed:
This is minimized on successive steps of the iteration by systematically varying all elastic constants until there is no improvement. This procedure converges quickly for the systems investigated [15]. When applied to calculations reported in the literature, betters fits are obtained than found in the original work [15].
242
R. Kulver et al. / Elastic constants ofphenanthrene
Table 3 Sound wave velocities calculated from acoustic phonon frequencies and m-evaluated through Christoffel determinants q-vector
Initial (m/s)
Final (m/s)
100
1126.5 2283.9 3389.1
1102.5 1964.6 3413.2
010
480.4 776.0 2864.1
609.7 923.9 2513.0
001
1775.4 2954.3 4263.9
2016.9 2514.0 4165.0
110
741.7 1792.5 2957.7
843.9 1732.8 2718.7
101
807.1 2337.2 4894.9
946.2 2838.8 4879.9
011
1100.3 2098.8 2741.6
929.4 2052.4 3020.4
111
1361.7 2280.6 3549.9
1196.5 2353.3 3642.4
The stiffness and compliance tensors are given in table 2. Table 3 gives the sound velocities obtained from the final elastic constant tensor so that comparison with the original may be made. Other calculations of elastic constants performed in our laboratory indicate that the fit obtained for the sound velocities is better for a centric crystal such as TCNQ than it is for an acentric one such as phenanthrene. The translations in table 1 add three more variables to the energy minimization which may increase the error in a problem that is already prone to significant uncertainty [4] in the final result.
crystals
4. Conclusion The elastic constants of molecular crystals are important to the understanding of the mechanical properties of these materials. Of particular interest is their variance between crystals made up of isomers or between polymorphs. Comparison with the constants of the isomer, anthracene, are relevant. They are quite similar for the diagonal elements but vary significantly in their off-diagonal terms.
References [l] J. Merski and C.J. Eckhardt, J. Chem. Phys. 75 (1981) 3691; T. Luty and C.J. Eckhardt, J. Chem. Phys. 81 (1984) 520. [2] J. Merski and C.J. Eckhardt, J. Chem. Phys. 75 (1981) 3731. [3] A. Chablo, D.W.J. Cruickshank, A. Hinchchffe and R.W. Munn, Chem. Phys. Letters 78 (1981) 424. [4] H.B. Huntington, S.G. Gangoh and J.L. Mills, J. Chem. Phys. 50 (1969) 3844, G.S. Pawley, Phys. Stat. Sol. 20 (1967) 347. [5] G. Ventkataraman, L.A. Feldkamp and V.C. Sahni, Dynamics of perfect crystals (MIT Press, Cambridge, 1975). [6] J. Trotter, Acta Cryst. 16 (1963) 605. [7] T.L. Starr and D.E. Williams, Acta Cryst. A 33 (1977) 771. [8] Yu.A. Penshin, Physics and chemistry of organic compounds in the solid state (Nauka, Moscow, 1967). (91 R.W. Munn, Chem. Phys. 50 (1980) 119. [lo] K.-H. Brose and C.J. Eckhardt, Chem. Phys. Letters 125 (1986) 235. [ll] H.A.J. Govers, Acta Cryst. A 34 (1978) 960. (121 S.F. Ahmad, H. Kiefte, M.J. Couher and M.D. Whitmore, Phys. Rev. B 26 (1982) 4239. [13] C. Ecovilet and M. Sanquer, J. Chem. Phys. 72 (1980) 4145. [14] G.S. Pawley, Phys. Stat. Sol. 20 (1967) 347. [15] K.-H. Brose and C.J. Eckhardt, in preparation.
Chemical Physics 115 (1987) 239-242 North-Holland, Amsterdam
CALCULATION OF THE ELASTIC CONSTANTS OF PHENANTHRENE CRYSTALS * R. KULVER,
K.-H. BROSE
and C.J. ECKHARDT
Department of Chemistry, University of Nebraska, Lincoln, NE 68588-0304, USA Received 16 March 1987
A lattice dynamical calculation in the harmonic approximation is employed to obtain the sound velocities in phenanthrene crystals at 0 K and with no stress. A Buckingham potential with Williams parameterization is used to determine the intermolecular potential energy. The potential obtained minimizes satisfactorily with respect to the known crystal structure. The elastic constants are obtained by using an iterative optimization procedure.
1. Introduction Little attention has been given to the lattice dynamical properties of molecular crystals, especially their phonon dynamics and elastic properties. For the greater proportion of these solids the available information, experimental or theoretical, is nonexistent. Nevertheless, these materials are of considerable interest since they represent a class of crystals where the types of binding forces are limited and, therefore, their properties should be easier to understand. In the case of the simple hydrocarbons, cohesion is due almost exclusively to van der Waals interactions. This usually provides for a clear separation in energy of the internal modes of the molecules from the external or lattice vibrations and, thus, surprisingly good results can be obtained from lattice dynamical calculations which treat the molecules as rigid. Our extension of piezomodulation spectroscopy to molecular crystals where a periodic elastic stress is applied to a crystal and a response, usually optical, is synchronously detected, has caused a much closer examination of the interaction of the excitations of the lattice with its mechanical properties [l]. Indeed, it has been demonstrated that
* Research supported by the Solid State Chemistry Program of the National Science Foundation.
the amount of mechanical energy involved with each excitation can be determined from the piezomodulated reflection spectra [l]. In the case of molecular crystals, an intimate connection between the van der Waals forces binding the lattice and the excitations of the system exists [2]. Calculations of elastic constants permits a test of the derived potential since experimental verification, such as Brillouin scattering, is possible. In addition, pressure- or stress-induced phase changes can be examined. This work presents the complete set of theoretical elastic constants for phenanthrene crystals. The molecule, a structural isomer of anthracene, possesses a small dipole moment estimated to be around 0.04 D [3] whereas its congener has none. The crystals of both are of the monoclinic class, but phenanthrene crystals have noticeably different physical properties. For example, the melting point of phenanthrene is 384 K while that for anthracene is 489 K. Although several calculations of the elastic constants for anthracene have been reported [4], there are none for the crystal of the isomer, phenanthrene. The stress applied to a crystal produces a macroscopic deformation, the external strain e. These are related through the compliance S: e=Sa. The elastic constants
0301-0104/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
which are elements of the
240
R Kulver et al. / Elastic constants of phenanthrene crystals
stiffness tensor form a fourth-rank tensor as do the compliances. This can lead to complications for the analysis of these elements because the low symmetries common to molecular solids generate a large number of finite tensor elements. For the monoclinic phenanthrene crystal one obtains Cl1
Cl2
Cl3
0
Cl5
0
Cl2
c22
c23
0
c25
0
Cl3
c23
c33
0
c35
0
0
0
0
CM
0
C&
Cl5
c25
c35
0
c55
0
0
0
0
c&j
0
c66
where the usual two-suffix notation is employed [5]. The determination of the stiffness tensor is not straightforward since sets of cubic equations in either thirteen (monoclinic) or twenty-one (triclinic) independent variables must be solved. This is nontrivial because of coupling through multiple products of elastic constants. In the next section the nature of the potential energy function and lattice dynamics are discussed. This is followed by the results of the calculation of the elastic constants.
2. Determination
of the potential energy function
The calculation of the elastic constants must begin with obtaining an appropriate lattice potential energy function from which the lattice frequencies and subsequently the elastic constants can be derived. It is crucial that the potential reasonably reproduces the experimentally observed crystal structure, the vibrational spectrum of the lattice and the binding energy of the crystal. This has been done for phenanthrene crystals. The lattice constants have been obtained by minimization of the lattice potential energy within the symmetry imposed by the experimentally observed P2, space group [6]. As a consequence of this symmetry, the potential energy surface is spanned by the a, b and c lattice periods, the angle fi and the three Euler angles describing the orientation of the molecule at the origin of the unit cell with respect to the Cartesian a&* coordinate system
and the center-of-mass coordinates of the molecules. Without the minimization, it is not certain that the stability of the crystal structure is assured and that a strain-free equilibrium configuration under the assumed potential energy function is obtained. Structure minimization was effected with parameters for the Buckingham potential (6-exp) obtained from the work of Williams [7]. A radius of summation of 28 molecules did not provide good values of the lattice constants and the molecular orientations. The summation radius of interactions had to be increased to 88 molecules to obtain good convergence of the energy. This was found to be in quite good agreement with the literature values of 21 to 22 kcal/mol [8]. Difficulty was still faced with the values of the lattice parameters. The work of MUM et al. indicated that the small but finite dipole of phenanthrene, 0.04 D, might be of importance in considerations of the polarizability [9]. Since other work in our laboratory on TCNQ indicated the importance of the role of electrostatic interactions [lo], it appeared reasonable to include these in the phenanthrene. Charges calculated using an MO SCF (3G STO) were incorporated following the approach of Govers [ll]. With this emendation to the calculation, a satisfactory agreement with the lattice parameters and molecular orientations was achieved. It is noteworthy that the inclusion of the coulombic interaction had no significant effect on
Table 1 Lattice minhization
a b ; x Y ; # 8 u latt
results a) Exp.
talc.
8.46 6.16 9.47 97.7 2.09 1.78 0.74 327.5 79.6 22.7 22
8.27 6.64 9.34 106.1 2.34 2.51 0.80 300.4 71.6 1.0 21.36
‘) a, b, c, x, y, z are in fmgstrijm; all angles in degree; lattice energy in kcal/mol.
R. Kulver et al. / Elastic constants ofphenanthrene
the lattice energy. These results are presented in table 1 together with the values obtained from the X-ray structure. The greatest disagreement is in the monoclinic angle. The intermolecular force constants must be used to calculate the elastic constants of the crystal. The force constants are the second-order terms in a Taylor series expansion of the lattice potential for infinitesimal displacements about molecular equilibrium positions and may thereby be obtained from the calculated lattice potential energy function. The dynamical matrices are formed from these. The normal Cartesian ubc* system was used and solutions to the secular equations obtained giving frequencies w(q, j) for magnitudes of the wavevector, q, extremely close to the center of the Brillouin zone. The slope of the appropriate acoustic phonon branches, j, yielded the sound velocities:
3. Calculation of elastic constants In the treatment of crystal acoustics, the sound waves which propagate in a solid yield an eigenvalue equation reminiscent of that obtained in the formulation of lattice dynamics. The crystal den-
Table 2 Calculated elastic phenanthrene stiffness (kbar) 101.01 74.15 19.76
stiffness
18.82 - 17.73 135.55
and
compliance
0.0
0.0 0.0 70.07
tensors
33.88 11.10 -60.24 0.0 25.72
for
0.0 0.0 0.0 16.72 0.0 8.07
compliance (lo-’
kbar-') 3.98 -30.33 26.31 267.00 -144.00 72.66
0.0 0.0
0.0
0.0
-413.56 0.0 0.0 197.72 0.0 28.23 0.0 -58.52 588.36 0.0 245.26
crysials
241
sity, and the velocity of the elastic (sound) wave, u(q) are related through Christoffel’s equation: IlIYq) - PV2111 = 0. Here r(q) is the macroscopic equivalent of the dynamical matrix and is a 3 X 3 matrix with elements of the form:
The cikjr are elastic constants and the q are the direction cosines of the wavevector to the k and I reference axes. To obtain the elastic constants we have adopted a procedure which differs from that normally employed but which has been applied to cubic systems [12]. The common approach solves for the elastic constants in a sequential process beginning with equations for wavevectors with the most elementary linear velocity-elastic constant relationships. Progressive solutions are obtained using previously calculated values. In this method the great drawback is that the errors propagate and accumulate such that the last determined off-diagonal elements of the tensor normally carry quite large uncertainties [13]. Additionally, the solution of the polynomial equations leads to ambiguities which require choices for intermediate results. An alternative approach requiring extreme effort is to set up a direct non-linear least-squares solution 1141. The calculations here employ an iterative optimization which is initiated with a complete but arbitrary set of elastic constants. In each iteration, the Christoffel determinants are solved for their eigenvalues, p 02,) for all wavevectors for those observed sound velocities considered. The error vector, e, is subsequently determined and its square formed:
This is minimized on successive steps of the iteration by systematically varying all elastic constants until there is no improvement. This procedure converges quickly for the systems investigated [15]. When applied to calculations reported in the literature, betters fits are obtained than found in the original work [15].
242
R. Kulver et al. / Elastic constants ofphenanthrene
Table 3 Sound wave velocities calculated from acoustic phonon frequencies and m-evaluated through Christoffel determinants q-vector
Initial (m/s)
Final (m/s)
100
1126.5 2283.9 3389.1
1102.5 1964.6 3413.2
010
480.4 776.0 2864.1
609.7 923.9 2513.0
001
1775.4 2954.3 4263.9
2016.9 2514.0 4165.0
110
741.7 1792.5 2957.7
843.9 1732.8 2718.7
101
807.1 2337.2 4894.9
946.2 2838.8 4879.9
011
1100.3 2098.8 2741.6
929.4 2052.4 3020.4
111
1361.7 2280.6 3549.9
1196.5 2353.3 3642.4
The stiffness and compliance tensors are given in table 2. Table 3 gives the sound velocities obtained from the final elastic constant tensor so that comparison with the original may be made. Other calculations of elastic constants performed in our laboratory indicate that the fit obtained for the sound velocities is better for a centric crystal such as TCNQ than it is for an acentric one such as phenanthrene. The translations in table 1 add three more variables to the energy minimization which may increase the error in a problem that is already prone to significant uncertainty [4] in the final result.
crystals
4. Conclusion The elastic constants of molecular crystals are important to the understanding of the mechanical properties of these materials. Of particular interest is their variance between crystals made up of isomers or between polymorphs. Comparison with the constants of the isomer, anthracene, are relevant. They are quite similar for the diagonal elements but vary significantly in their off-diagonal terms.
References [l] J. Merski and C.J. Eckhardt, J. Chem. Phys. 75 (1981) 3691; T. Luty and C.J. Eckhardt, J. Chem. Phys. 81 (1984) 520. [2] J. Merski and C.J. Eckhardt, J. Chem. Phys. 75 (1981) 3731. [3] A. Chablo, D.W.J. Cruickshank, A. Hinchchffe and R.W. Munn, Chem. Phys. Letters 78 (1981) 424. [4] H.B. Huntington, S.G. Gangoh and J.L. Mills, J. Chem. Phys. 50 (1969) 3844, G.S. Pawley, Phys. Stat. Sol. 20 (1967) 347. [5] G. Ventkataraman, L.A. Feldkamp and V.C. Sahni, Dynamics of perfect crystals (MIT Press, Cambridge, 1975). [6] J. Trotter, Acta Cryst. 16 (1963) 605. [7] T.L. Starr and D.E. Williams, Acta Cryst. A 33 (1977) 771. [8] Yu.A. Penshin, Physics and chemistry of organic compounds in the solid state (Nauka, Moscow, 1967). (91 R.W. Munn, Chem. Phys. 50 (1980) 119. [lo] K.-H. Brose and C.J. Eckhardt, Chem. Phys. Letters 125 (1986) 235. [ll] H.A.J. Govers, Acta Cryst. A 34 (1978) 960. (121 S.F. Ahmad, H. Kiefte, M.J. Couher and M.D. Whitmore, Phys. Rev. B 26 (1982) 4239. [13] C. Ecovilet and M. Sanquer, J. Chem. Phys. 72 (1980) 4145. [14] G.S. Pawley, Phys. Stat. Sol. 20 (1967) 347. [15] K.-H. Brose and C.J. Eckhardt, in preparation.