J. Phys. Chem. 1991, 95, 1783-1786 singlet states by silyl groups has to be attributed solely to electronic effects. Note Added in Proof. A paper by Ho and M e l i ~ , where 3 ~ they present a modification of their empirical bond enthalpy correction factors, appeared during the reviewing of our manuscript. Their revised heats of formation which they cite for a few silicon hydrides are in good agreement with our values for systems like SiH3, SiH, and H2Si-SiH2 (3B), where a single determinant is sufficient as a reference wave function. Although their new values for the (33)
Ho,P.; Melius, C. F. J . Phys. Chem. 1990, 94, 5120.
1783
“multiply” bonded species like Si2H, Si2H3,and H2Si=SiH2 (IAg) are closer to ours than those cited in their previous work,’ significant differences still remain. This finding is, however, not very intriguing, since all their geometry optimizations were performed at the HF(UHF)/6-31G* level of theory and their modified correction factors too, in order to facilitate transition state predictions, depend in a complicated way on exponential bond length contributions and are, therefore, very sensible to calculated bond distances.
Acknowledgment. We are grateful for substantial suggestions given by Dr. G. Olbrich, Prof. P. Potzinger, and Prof. R. Walsh.
Elastic Constants of the Ethanol Adduct of Dianin’s Compound As Determined by Brillouin Spectroscopy Marek Zakrzewski,? Boguslaw Mr6z,**rHarry Kiefte,*.t Mary Anne White> and Maynard J. Cloutert Department of Chemistry, Dalhousie University, Halifax, Nova Scotia B3H 4J3. Canada, and Physics Department, Memorial University of Newfoundland, St. John’s, Newfoundland A1 B 3x7, Canada (Received: June 22, 1990; In Final Form: September 18, 1990)
Brillouin scattering measurements were carried out at room temperature on the ethanol adduct of 4-@-hydroxyphenyl)2,2,4-trimethylchroman (Dianin’s compo_und). This material forms clathrates with the mole ratio of ethanol guest to host molecule of 1:3. It crystallizes in the R3 space group (trigonal system). All seven independent elastic constants, for this symmetry, were determined. The results show that the system can be closely approximated by hexagonal symmetry.
Introduction Clathrates are a special class of molecular solids, especially since if the guest-host interactions are relatively weak they can offer the opportunity to see the macroscopic and microscopic properties associated with single-particle (guest) dynamics. Clathrates can have interesting chemical and physical properties, and these have attracted considerable attention recently.’ Our interests have been in thermodynamic and elastic properties of clathrates, especially those of clathrate One of our main aims is to understand the role of the guest species in the determination of these properties. To that end, we have i ~ ~ i t i a t ead ~series . ~ of investigations of a clathrate that can be prepared both as an inclusion compound and as an empty host lattice. 4-(p-Hydroxyphenyl)-2,2,4-trimethylchroman, 1, is an inclusion compound and was first reported
’0H 1
by Dianin in 1914.’ It forms very many different clathrates by crystallization from appropriate solvents/g~ests.*~~ The parent compound and its clathrates all appear to crystallize in the RJ space group.lO*iiThe general structure of the host lattice consists of hexamers of 1, linked by a network of hydrogen bonds involving the phenolic hydroxy groups, as shown in Figure 1. Monomeric units of the hexamers alternately point up and down along the
’Dalhousie University. 2 Memorial
University of Newfoundland. ‘On leave from the Institute of Physics, A. Mickiewicz University, Grunwaldzka 6, 60-780 Poznan, Poland.
c axis, and the binding force between hexamers is due to van der Waals interactions. The structure is composed of hour-glassshaped cages capable of accommodating even bulky molec~les?~~J’ The ethanol adduct of Dianin’s compound contains two ethanol guest molecules within each hexamer host cage, as shown by X-ray crystallographyi0*”and infrared difference spectroscopy.I2 From the latter methodI2 it is concluded that the two ethanol molecules form a hydrogen-bonded dimer. The position of each ethanol molecule is triply degenerate as deduced from the 3-fold rotational symmetry of the crystal. The large thermal parameters of the ethanol molecules in the X-ray diffraction study indicate disorder of the guest molecules and possibly occupational deficiencies. The purpose of this work was to determine the elastic constants and acoustic velocities and hence related thermodynamic quantities for the ethanol adduct of Dianin’s compound. We chose to examine this adduct because it is possible to grow crystals of sufficient size (several millimeters in each dimension) and quality to allow measurements by Brillouin spectroscopy. (These conditions are not always met and indeed precluded measurements of the empty host lattice.) Other than preliminary results for tetrahydrofuran clathrate h ~ d r a t ethis , ~ appears to be the first (1) Atwood, J. L.; Davics, J. E. D.; MacNicol, D. D. Inclusion Compounds; Academic Press: London, 1984; Vols. 1-3. (2) White, M. A.; MacLean, M. T. J . Phys. Chem. 1985, 89, 1380. (3) Kiefte, H.; Clouter, M. J.; Gagnon, R. E. J . Phys. Chem. 1985, 89, 3103. (4) Tse, J.; White, M. A. J . Phys. Chem. 1988, 92, 5006. (5) White, M. A.; Zakrzewski, M. J . Inclusion Phenom. Mol. Recognit. Chem. 1990, 8, 215-225. ( 6 ) Zakrzewski, M.; White, M. A.; Abriel, W. J . Phys. Chem. 1990, 94, 2203-2206. (7) Dianin, A. P. J . Russ. Phys. Chem. Soc. 1914, 46, 1310.
(8) Baker, W.; Floyd, A. J.; M a m i e , J. W. F.; Pope, G.;Weaving, A. S.; Wild, J. H. J . Chem. Soc. 1956, 2010. (9) Goldup, A.; Smith, G. W. Sep. Sci. 1971, 6, 791. (10) Flippen, J. L.; Karle, J.; Karle, I. L. J. Am. Chem. Soc. 1970, 92, 3749. ( I 1) MacNicol, D. D. Inclusion Compounds; Atwood, J. L., Davis, J. E. D., MacNicol, D. D., Eds.; Academic Press: London, 1984; Vol. 2, Chapter 1. (12) Davies, J. E. D. J . Inclusion Phenom. 1985, 3, 269.
0022-3654/91/2095-1783%02.50/00 1991 American Chemical Society
1784 The Journal of Physical Chemistry, Vol. 95, No. 4, 1991
Zakrzewski et al. 4
; I 1‘
I
L
~
1
Figure 1. Hexagonal unit cell of Dianin’s compound (phenolic protons are excluded for clarity). The hydrogen-bond network between phenolic oxygens is shown by dashed lines. The x and y axes shown illustrate the Cartesian reference frame used here. The crystallographic c axis is collinear with the Cartesian z axis and points out of the plane of the figure.
report of elastic constants for a clathrate. Brillouin spectroscopy is a very effective and sensitive method for investigation of the elastic properties of crystals. Phonon velocities, u, can be calculated directly from the Brillouin shifts, Av, by using the Brillouin e q u a t i ~ n : ’ ~ AY = vo(u/c)(t$
+ n$ - 2nin, cos
(1)
where v0 is the frequency of the laser radiation, c is the velocity of light in vacuum, ni and n, are the refractive indexes of the incident and scattered light respectively, and 0 is the scattering angle. The elastic constants can be determined from the solution to the Christoffel equation for different directions in the crystal lattice:I3 kikjflkql
- pu26i/l = 0
(2)
where q k and 9, are direction cosines, q k j / are the elastic constants written in full index notation, p is the crystal density, 6,/is the Kronecker 6. Brillouin scattering spectra for a single crystal consist of three values of frequency shifts, corresponding to the quasilongitudinal (L), and slow quasitransverse (TI) and fast quasitransverse (T2) acoustical modes. Experimental Section The ethanol adduct of Dianin’s compound was prepared from phenol and mesityl oxide according to the method of Baker et aL8 Single crystals were grown from ethanol by slow evaporation at room temperature. The measured density of the crystals at 23 OC, from determinations using a calibrated pycnometer matching the crystal density with aqueous KI solutions, was 1.2234 i 0.0002 the density g ~ m - which ~ , is in good agreement with 1.2252 g calculated from X-ray diffraction data.lOJ1The refractive indexes of the ethanol adduct of Dianin’s compound were determined by comparison with standard liquids (Cargille Laboratories) of known refractive indexes: n, = nv = 1.640 f 0.001 and n, = 1.641 f 0.001. The crystals were oriented by using a precession camera, which showed that the c axis was along well-formed edges of the crystals. Weissenberg photography showed that the a axis was perpendicular to two very well-formed walls. Since the elastic constant tensor is defined in the Cartesian frame of reference, a set of axes was assigned to the crystals by using Standards on Piezoelectric (13) Landau, L. D.;Lifshitz, E. M.Theory of Elasriciry; Addison-W&y: New York, 1959.
1
-
-10 1
1
1
0
1
+ 10
FREQUENCY SHIFT (GHd
Figure 2. Representative 2-h Brillouin scatteringspectrum for the single
crystal of the ethanol adduct of Dianin’s compound taken for the phonon direction [-0.35; -0.71; 0.611. One complete order is shown with quasilongitudinal (L) and quasitransverse (TI and T2) components up- and down-shifted from the central component (R). The free spectral range of the Fabry-Perot was 28.50 GHz. Cry~ta1s.I~The x , y, z reference axes are shown in Figure 1. Samples of five different orientations were cut out of the best crystals by using a wire saw manufactured by South Bay Technology Inc. and a wire blade of diameter 0.02 mm. Later the samples were polished by using silicon carbide grain of sizes from 600 to 1600. The measurements were taken on seven different crystals, of typical dimensions 5 mm X 5 mm X 5 mm. The Brillouin spectrometer used in these experiments has been described in detail e1se~here.l~The incident light source was provided by a single-mode argon laser (Spectra Physics 2020-03) operating a t h = 514.5 nm and filtered to a power of 30 or 50 mW. The scattered light was analyzed at 90“ by using a triple-pass piezoelectrically scanned Fabry-Perot interferometer (Burleigh RC 110) utilizing free spectral ranges (FSR) of 21.80, 26.95, and 31.66 GHz. The scattered light was detected by a cooled photomultiplier (ITT FW 130) connected to the multichannel analyzer of the data acquisition and stabilization system (Burleigh DAS-I). The spectral data were accumulated in the form of photon counts vs channel number (frequency). The DAS-1 automatically compensated for drifts in the laser frequency and separation of the Fabry-Perot reflectors and optimized the finesse for indefinitely long periods of time. A representative Brillouin spectrum of the ethanol adduct of Dianin’s compound with a central (Rayleigh) line and three Brillouin doublets (upand down-shifted) is shown in Figure 2. For these experiments, the errors in the determination of the frequency typically were of the order of 0.5-1%. Results and Discussion As discussed earlier, the experimental Brillouin shifts can be related directly to the elastic constants of a solid through the Christoffel equation (eq 2). The most common way to solve the Christoffel equation is to start with simple directions that give linear relationships between the elastic constants and phonon velocities; this is limited to the results of only a very few simple acoustic wave propagation directions. Subsequently results from more complex directions can be added to determine the remaining elastic constants. This method has some disadvantages, especially ( 14) Nye, J. F. Physical properties of crystals; Oxford Clarendon Press: New York, 1957. (IS) Ahmad, S.F.; Kiefte, H.; Clouter, M. J.; Whitmore, M . D. Phys. Rev. B 1975, 11, 1705.
The Journal of Physical Chemistry, Vol. 95, No. 4, 1991
Ethanol Adduct of Dianin's Compound
TABLE I: Brillouin Scattering Data for Single Crystals of the Ethanol Adduct of Dianin's Compound at
direction cosines X
1 0 0 0.7 I 0.7 1 0 0.50 -0.7 1 -0.7 I -0.50 0 0.7 1 -0.7 I 0.7 I -0.7 1 -0.35 0.26 -0.35 0.6 I -0.61 0.97 -0.50 -0.50
Y
0 1
0 0.7 1 0 0.7 1 -0.50 -0.5 1 -0.5 1 0.50 0.26 0.35 -0.35 -0.35 -0.6 I -0.7 1 0 0.71 0.7 1 0.7 1 0 -0.50 -0.50
T = 293 C
obsd freq, GHz Z
0 0 I 0 0.7 1 0.7 1 0.7 1 -0.5 1 -0.51 0.71 0.97 0.6 1 0.61 0.6 1 -0.35 0.61 0.97 0.6 1 0.35 -0.35 -0.26 -0.7 1 0.7 1
calcd freq, GHz
TI
T2
L
TI
T2
L
7.95 8.65 8.75 7.84 7.41 7.52
9.38 9.90 8.75 9.73
14.97 15.03 14.36 15.14 15.63 15.26 15.52 15.84 15.09 15.58 14.64 15.47 15.58 15.31 14.87 15.14 15.25 15.47 15.14 14.76 14.64 15.03 15.36
8.44 8.54 8.55 8.48 7.49 7.50 7.49 7.62 7.89 7.49 8.14 7.64 7.64 7.64 7.94 7.55 8.22 7.58 7.98 8.16 8.22 7.49 7.49
9.31 9.21 8.55 9.26 8.88 9.16 9.08 8.81 9.28 8.68 8.74 9.23 9.23 9.23 8.98 8.90 8.64 9.00 9.04 9.24 9.21 9.08 8.68
14.92 14.93 14.75 14.93 15.40 15.23 15.28 15.51 15.10 15.52 14.90 15.19 15.19 15.19 15.33 15.43 14.92 15.36 15.27 15.06 15.07 15.28 15.52
9.21 7.52 8.45 7.74 8.15 7.84 7.84 8.15 7.29 7.18
9.12 9.12 9.12 9.02 8.65 8.55 8.45
8.26 7.74 7.74 7.41
1785
8.93 9.02 8.05
T I , slow quasitransverse mode; T2, fast quasitransverse mode; L, quasilongitudinal mode.
the requirement of simple directions in the crystal lattice. Also, experimental errors in the determination of sound velocity accumulate in the calculation of the later elastic constants. Since Dianin's compound and its known adducts crystallize in such a low-symmetry space group, RJ in the trigonal the elastic constant tensor in the two-suffix matrix notation reduces to seven independent components.I4 For such a system, obtaining the elastic constants directly from the phonon velocities can lead to ambiguous results or even simultaneous equations with no unique solution. To solve the Christoffel equation for these results, a proven16J7 minimization proceduret8 was used. The algorithm consists of an iterative optimization procedure that fits all elastic constant tensor elements simultaneously to previously calculated sound velocities.I* In each iteration the Christoffel equations are solved for p u , , ~ . The weighted sum of the squares of the differences between these values and the corresponding puOb2 is calculated and then minimized by systematically varying the elastic constants until the fit is optimized. This approach is very similar to that used for cubic and hexagonal symmetry by the present authorsI5 and provides a superior method for determination of the complete set of elastic constants, especially since it does not rely on special directions and it does not propagate errors. In addition, the accuracy of the results improves with increasing the number of investigated directions. In the case of the R3 space group of the ethanol adduct of Dianin's compound, which gives no unique analytical solution to the Christoffel equation, this method was the only tractable approach to determine the elastic constants. This determination appears to be the first for all seven constants for this crystallographic symmetry. A total of 57 velocities were obtained by investigating 23 different directions in the crystal lattice. The experimental data and the best fits are shown in Table 1. The goodness of the fit can be judged by the closeness of the measured and calculated values of the shifts. The average absolute values of the differences between the observed and calculated values, expressed as a percentage of the observed values for the various modes, are as follows: TI, 3.7%; T2, 3.2%; L, 1.5%. In light of the quality of the crystals, in terms of relatively poor optical clarity and the existence of cracks, and the uncertainty (few degrees) in crystal orientation after cutting and polishing and visual ~
~~
(16) Brose, K.-H.; Eckhardt, C. J . Chem. Phys. Leii. 1986, 125, 235. (17) Dye, R. C.; Eckhardt, C. J . J . Chem. Phys. 1989, 90, 2090. (18) Brose, K.-H. Elcon. A Compuier Program for Fitting Elastic Con-
stanfs io Phonon Velocities: Department of Chemistry, Wayne State University: Detroit, 1989.
TABLE 11: Symmetrical Elastic Stiffness Tensor for the Ethanol Adduct of Dianin's Compound at 20 OC, As Determined by Brillouin SpectroscopyuSb cII
c12 cII
c13 c13 c33
c14
-25
-c14
?
0 0 0
0
C25
CU
c14
0 c44
(CII
- c12)/2
'ell = 1.34 (1.34). ~ 1 =2 0.32 (0.30). ~ 1 =3 0.65 (0.64). c I ~= 0.3 (0). cZ5 = 0 (0). cj3 = 1.31 (1.31). cU = 0.44 (0.43). bThe values given in brackets are for hexagonal symmetry. All numerical values of cu are in 1 O l o N d. TABLE III: Symmetrical Elastic Compliance Tensor for the Ethanol Adduct of Dianin's Compound at 20 OC, As Determined by Brillouin Spectroscopyu sll
s12
sI1
s14
-s25
s13
-s14
s25
s33
0
0
s44
0 34.4
0 0 2S25 2s14
2(Sll - s12) O s l l = 9.8 X IO-" m2 N-I. s12= -1.5 X mz N-I. s13= -4.9 X IO-" m2 N-I. s i 4 = 6.8 X m2 N-I. sZ5= 0 m2 N-I. s33 = 1.25 X m2 N-I. sU = 2.28 X m2 N-I.
alignment in the experimental setup, the fit is quite good. The Brillouin scattering results are confirmed by an independent determination of ultrasonic sound velocities along the c axis of a single crystal of the ethanol adduct of Dianin's compound. From the latter method the transverse mode velocity was determined to be 2.06 X lo3 m and the longitudinal mode velocity was 3.1 1 X lo3 m s-l; the corresponding values from the Brillouin experiment were 1.94 X IO3 and 3.18 X IO3 m s-l, The elastic stiffness constants and elastic compliance constants determined from the fit to the Brillouin scattering results are shown in Tables I1 and 111, respectively. By noting the changes in calculated velocities by varying cu, it is estimated that the relative uncertainties in the elastic constants are less than 0.02 X IOio N m-2. Some of the calculated frequency shifts for different phonon directions are virtually the same. This was caused by experimentally almost indistinguishable differences between Brillouin shifts for different directions. After several hundred iterations in the fit the cZ5elastic constant optimized at zero, causing the
J . Phys. Chem. 1991,95, 1786-1789
1786
R3 space group to look like the R3m space group, with respect to the elastic constants. The other elastic constant that is very close to 0 is ~ 1 4 .This suggests that the examined system could be closely approximated by hexagonal ~ y m m e t r y . ’ ~The , ~ elastic constant calculation within hexagonal symmetry was carried out and gave results (in brackets in Table 11) very similar to those for trigonal symmetry, verifying the close approximation of the symmetry here to hexagonal. I f hexagonal symmetry is assumed, the bulk modulus B,, given by
is 7.8 X IO9 N m-2 for the ethanol adduct of Dianin’s compound, and the anisotropy factor A, given by eq 4 is 0.83. A = 2 c 4 4 / ( c I I - c12)
(4)
The elastic constants for the ethanol adduct of Dianin’s compound are almost an order of magnitude larger than very soft hexagonal crystals such as 8-N2 and /3-C0.’s They are, nevertheless, much smaller (by about a factor of 5 ) than trigonal ferric (19) Holuj, P.; Drozdowski, M.; Czajkowski, M. Solid State Commun. 1985, 56, 1019.
( 2 0 ) Ngoepe, P. E.; Comins, J. D. Phys. Reu. Lett. 1988, 61, 978.
systems such as K3Na(Cr04)2,21than the trigonal layered compound Ca(OH)2,19or (by a factor of 10-50) than typical harder crystals such as A1203.22 The elastic constants measured in the present experiment are, in fact, very similar to those of (hexagonal) ice Ih,23the exception being c12,which is larger by a factor of 2 in hexagonal ice. In summary, the main result of the present work is that the complete elastic stiffness tensor of the ethanol adduct of Dianin’s compound has been determined. The crystals may be said to be very similar to ice close to the triple point. From the elastic point of view this material can reliably be approximated by hexagonal symmetry.
Acknowledgment. We thank Professor T. S. Cameron for help in determination of the crystal orientation, Professor K.-H. Brose for providing the Elcon program, Professor M. Jericho for assistance with the ultrasonic measurements, and B. Borecka for help with one figure. The Natural Sciences and Engineering Research Council of Canada supported this work through grants to H.K., M.A.W., and M.J.C. (21) MrBz, B.; Kiefte, H.; Clouter, M. J.; Tuszyfiski, J. A. Phys. Reu. B, in Dress. (22) Watchman, J. B.; Tefft, W. E.; Lam, Jr., D. G.; Stinchfield, R. P. J . Natl. Bur. Std. 1960, 64A, 213. (23) Gammon, P. H.; Kiefte, H.; Clouter, M. J. J . Phys. Chem. 1983,87, 4025.
Behavior of the Rotational Diffusion Tensor of Tetracene under Subslip Conditions M. J. Wirth* and S.-H. Chou Department of Chemistry and Biochemistry, University of Delaware, Newark, Delaware 19716 (Received: July 12, 1990; In Final Form: September 7 , 1990)
The fluorescence anisotropy decays of tetracene in 1-butanol, I-octanol,and I-dodecanol were measured at several temperatures by using frequency domain spectroscopy. Subslip rotational diffusion is observed. The double-exponential anisotropy decays were analyzed to determine the components, D,, Dy, and Dz, of the rotational diffusion tensor. The relative values of these componentsvary with temperature, indicating nonhydrodynamicrotational diffusion. Subslip behavior is found to be associated with a value of Dy/D that is significantlylarger than that predicted from hydrodynamics. The subslip phenomenon is interpreted as being a consequence of solvent structure.
Introduction The dynamics of solutes in microscopic media, such as micelles and surface monolayers, is very important to analytical chemistry because dynamics control the performance of liquid chromatography.’ Long-chain, n-alkyl functional groups are used for the most common chromatographic stationary phases and more recently for surfactant modifiers*” and pseudo stationary phases of electrokinetic ~hromatography.~For probing the dynamics of these media, rotational diffusion studies are valuable because fluorescence depolarization measurements can be made with the high sensitivity required for microscopic phases. However, an understanding of the rotational diffusion of solutes in bulk liquids is required in order to understand rotational diffusion of solutes in microscopic media. This understanding has not yet been reached for long-chain alkyl functionalities. The primary difficulty in understanding rotational diffusion for these systems is the description of the coupling between the ( I ) Dynamics of Chromatography; Giddings, J. C., Ed.; Dekker: New York, 1965. (2) Armstrong, D. W.; Henry, S. J. Anal. Chem. 1980, 3, 657. (3) Landy, J. S.;Dorsey, J. G. J . Chromatogr. Sci. 1984, 22, 68. (4) Terabe, S.; Otsuka, K.; Ichikawa, K.; Tsuchiya, A,; Ando, T. Anal. Chem. 1984, 56, 1 I 1 .
0022-3654/91/2095-1786%02.50/0
solute and its environment when the solute molecules are smaller than the solvent molecules. In hydrodynamic rotational diffusion, where the solvent is modeled as a continuum, the reorientation time, T ~ is~linear , in viscosity, 9: Tor
= (?V/kT)f,tickC
(1)
Vis the hydrodynamic volume of the Brownian ellipsoid,f,,ick is a parameter related to the geometric shape of the and Cis the coupling constant between the solute and solvent. In the stick boundary condition, C = 1. In the slip boundary condition, which is applied to nonpolar molecules because their interactions are weak, C < 1. The values for C in the slip boundary condition have been calculated for spheroidss and ellipsoid^.^ Nonpolar solutes in long-chain n-alkanes and n-alcohols reorient faster than that predicted by the slip boundary condition, Le., C < Cdi This subslip behavior was observed by Canonica, Schmid, and &Id for p-terphenyl and p-quaterphenyl in mixtures of pa(5) Polar Molecules, Debye, P., Ed.; Chemical Catalog Company: 1929. (6) Perrin, F. J . Phys. Radium 1936, 7, 1. (7) Deleted in proof. (8) Hu, C.-M.; Zwanzig, R. J. Chem. Phys. 1974, 60, 4354. (9) Youngren, G. K.; Acrivous, A. J . Chem. Phys. 1975, 63, 3846.
0 1991 American Chemical Society
