J. Phys. Chem. 1995, 99, 4757-4762
4757
Investigation of Solidification of Benzophenone in the Supercooled Liquid State Daniel J. Graham,* Peter Magdolinos,t and Michelle Tosi Department of Chemistry, Loyola University of Chicago, Chicago, Illinois 60626 Received: April 13, 1994; In Final Form: November 18, 1994@
A study of the solidification of supercooled liquid benzophenone (BZP) is presented. As is well-known, liquid BZP can be extensively supercooled without crystallization or glass formation. In the experiments, solid phase transition is activated by either seeding or momentary freezing (using liquid the BZP liquid nitrogen) of a portion of liquid. Measurements of the position of the liquidsolid interface as a function of time are reported. In the case of seeding, the temperature dependence of the interface propagation is examined over a range exceeding 120 K. BZP solidification is found to occur via two modes, each demonstrating its own distinct kinetics and morphology. For purposes of comparison, results of experiments with 4,4’dimethylbenzophenone are also presented. A cellular model for BZP solidification and liquid metastability is offered.
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I. Introduction In 1951 Hildebrand and co-workers focused on the unusual properties of liquid phosphorous, citing related work published as early as 1882.1-4 The properties included a persistent metastability demonstrated by liquid P4 at temperatures more than 115 K below the freezing point. Among other things, these researchers measured the velocity of the liquidsolid interface derived from momentary freezing of a small portion of liquid. Unusually high velocities of 23-210 c d s dependent on temperature were reported. Work such as Hildebrand’s is significant for the questions it raised: What underlies the persistent metastability of liquids such as phosphorous? What molecular rearrangements activate the liquid solid phase transition? How are the structures of an undercooled liquid and its solid phase related? The more recent literature attests to the progress and continued intrigue regarding glass-forming system^.^,^ The present study is aimed at this diverse research field, having drawn initial inspiration from Hildebrand’s work. The properties of amorphous and crystalline benzophenone (BZP) have been detailed in many place^.^ From early experiments, this glass-forming compound is known to demonstrate at least two crystalline morphologies, a and p, with respective freezing points of 323 and 299 K.8 The a crystal is orthorhombic with space group P212121 and appears to be more extensively investigated than the p form.g The latter appears as a low-melting crystal having a monoclinic structure.’O Interestingly, pure BZP liquid can be cooled more than 120 K below the a freezing temperature with no evidence of either crystal or glass formation. A glass is readily obtained for BZP liquid immersed in liquid nitrogen.” In recent experiments, the authors have examined these characteristics, especially as they relate to the process of solidification. The present study focuses on the kinetics of the liquid solid phase transition as well as the effects of select molecular substituents. Experiments which connect with the distant and recent alike have proven interesting. Among other things, BZP supercooled liquid demonstrates two modes of solidification leading to the a and p structures. Each mode displays its
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* To whom all correspondence should be addressed. Present address: Department of Chemistry, Northern Illinois University, DeKalb, Illinois. @Abstractpublished in Advance ACS Abstracts, March 1, 1995.
own distinct kinetics and solid material distribution. The experiments have suggested a model for the phase transition kinetics, which is posed in section IV. For purposes of comparison, results of experiments with 4,4’-dimethylbenzophenone (DMBZP) are also presented. This paper is obviously geared to the peculiarities of one aromatic ketone and a derivative compound. The ideas and analysis, however, may be relevant to other systems, as is noted at the end of section V.
11. Procedure Early experiments of this project were performed using BZP which had been extensively zone-refined. It was later found that Aldrich Gold Label material (melting temperature 322324 K) produced identical results without additional purification. The procedure involved crushing BZP to a fine dry powder which could then be added to dry,cylindrical glass tubes. Using a heat gun, one can easily melt the powder so as to form a transparent viscous liquid. The resistance of BZP liquid to crystallization was noted in section I, not to mention the early literature.8 In the authors’ experience, BZP liquid persisted indefinitely, several months and longer at room temperature under moderately dust-free conditions. The dimensions and composition of the container tubes had no bearing on these results. The data contained in this paper are derived from BZP contained in borosilicate tubes of dimensions 3 and 2 mm for the outer and inner diameter, respectively. Different methods of initiating the solidification were examined, and two proved to be of particular interest. The first method involved addition of granular a-BZP (typical edge lengths 1 mm, mp 322-324 K) to liquid held at a fixed temperature. The second involved momentary (5-10 s) freezing of a 1-2 mm portion of liquid using a boiling nitrogen (77 K) immersion bath. Both methods gave rise to a liquidsolid interface which propagated along the container axis. Records were made of the interface position as a function of time and temperature. This was a straightforward task given that the velocities were on the order of 0.1 c d s or less; one required only a stopwatch and ruler to make the measurements. It was sufficient to regulate the sample temperature using assorted aqueous electrolyte and dry ice baths. Microscopic examination of crystals was carried out using American Optical and Bausch & Lomb instruments plus linear polarizers.
0022-3654/95/2099-4757$09.00/00 1995 American Chemical Society
4758 J. Phys. Chem., Vol. 99, No. 13, 1995
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Figure 1. Results of grain-initiated solidification experiments with liquid BZP. Upper frame illustrates the position (cm) of the liquid/ solid interface as a function of time (s), relative to placement of the activating grain. Interface velocities are the slopes of the best-fit straight lines. InsetJumbers Iefer to the sample temperature (K). Lower frame illustrates U-versus-T, the reduced variables being defined in text.
Experiments were also performed with DMBZP obtained from Eastman Chemical and extensively zone-refined. The freezing point of this material at 368 K is higher than that of BZP.l2 The physical and chemical properties of DMBZP and BZP are otherwise very similar. Orthorhombic DMBZP single crystals can be prepared under the same solution conditions (hexane or ethanol solvent) which favor a-BZP growth. The DMBZP unit cell structure is found to be isomorphous with that of a-BZP.I3 Only one crystal morphology appears to have been reported for DMBZP.14 111. Results
Figure 1 contains results of grain-initiated solidification experiments with liquid BZP. The upper frame illustrates position-versus-time for the liquidsolid interface at three different temperatures. A linear time dependence is observed in these and all other cases. Interface velocities v are given by the slopes of best-fit lines allied with these data. The experiments demonstrated velocities at ca. 0.1 c d s to be virtually independent of temperature less than 300 K. By contrast, the velocity decreased sharply over the range bounded by ca. 300 K and the a melting point. The lower frame illustrates the temperature dependence, employing reduced variables = v/v,, (vma = 0.093 c d s ) , T = T/Ta,where T, is the a melting temperature, 323 K). In all experiments, grain-initiated solidification resulted in a white powdery material distributed uniformly throughout the container. The a morphology was readily verified on the basis of the melting temperature. The size and shape of the activating grain had no bearing on the velocities and temperature dependence. The interface velocities proved independent of cooling schedules and container orientation in the gravitational field. The a phase of BZP (space group P212121) is ~ h i r a l .Even ~ so, the authors found it exceedingly difficult to pinpoint the handedness of tiny single crystals of these experiments. However, microscopic examination of sundry grains revealed multiple single crystals bonded together. When viewing these crystal aggregates using a microscope with
Figure 2. Results of freezing-initiated solidification experiments with liquid BZP at 293 K. Upper frame illustrates the position (cm) of the solidification front as a function of time (min), relative to location of initially frozen surface. The trace drawing contained in the lower frame derives from a Xerox image of a sample tube after several hours. The /3 crystallites coexistent with metastable liquid are indicated by blackening. In using this imaging method, one must be careful not to
allow copy machine heat to melt the crystals. crossed polarizers, various “dark” regions were observed. It would then appear that optical rotatation caused by one type of enantiomer in the complex is canceled by its neighbors. This suggests that both enantiomers are produced in the experiments in roughly equal numbers. Momentary freezing of a portion of BZP liquid gave rise to a behavior which was quite unusual. The upper frame of Figure 2 illustrates the position of the BZP solidification front as a function of time at 293 K. By “solidification front”, one refers to the solid material located farthest from the initially frozen (BZP glass) portion. For this particular experiment, one observes a discontinuity in the position-versus-time dependence at ca. 150 min. While the occurrence and time of this discontinuity varied from experiment to experiment, the front position always scaled with time according to a fractional exponent between 0.5 and 0.7. For example, log-log plots of the Figure 2 data show the front position to scale as time f.66*0.05 at r < 150 min and as f.53*o.05 for 150 < t < 500 min. One notes the front “velocity” to be on the order of 0.01 c d m i n . This is more than 2 orders of magnitude less than that observed for typical grain-initiated solidification. Surprisingly, solid material in the freezing-initiated experiments was not powdery and uniformly distributed. The morphology was instead transparent and single-crystal-like. Sundry crystallites had edge lengths of 1-2 mm and appeared as thin sheets dispersed throughout the container. The lower frame of Figure 2 contains a trace drawing derived from one experiment, with solid distinguished from liquid by blackening. One observes the crystallites to form a network which threads regions of persistently metastable liquid. Upon microscopic examination of samples, only a single plane of optical symmetry was apparent. The morphology of individual crystals was thus assigned tentatively as monoclinic.l5 The observed melting point readily identified the crystals as the ,8 form of BZP. Results such as in Figure 2 proved very sensitive to temperature. Growth of ,8 material was sustainable only at temperatures within a few degrees of room temperature. Outside this range, ,8 solidification transformed after a few minutes into
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Solidification of Benzophenone the more rapid a-type process. Further, growth of p-BZP was quite sensitive to the manner of freezing. In the authors’ experience, the process would only take place if the initial freezing was uniform for a cross section of liquid perpendicular to the container axis. In other words, p crystallites could only be educed from a single smooth surface of BZP glass. The preponderance of p crystallites evolved with growth axes and planes perpendicular to the container axis. In experiments with DMBZP, virtually no resistance to crystallization was demonstrated. Pure liquid DMBZP solidified spontaneously only a few degrees below the melting temperature. The time dependence of the liquidkolid interface position was found to be linear in all cases. Velocities near 0.1 c d s were recorded irrespective of nucleation details. These kinetics were nearly identical to those observed for a-BZP growth at temperatures I300 K.
IV. A Cellular Model for BZP Solidification One imagines a lattice occupied by various BZP cells or elements, one per site; each cell contains the same number of molecules. Let the process of transforming liquid elements L to either a or p solid BZP elements proceed according to two distinct sets of reactions. In the first set, an L can convert to a only if an adjacent lattice site is occupied by an a. The a and L exert no effect on other a and L, respectively, nearest neighbor or otherwise. The reactions of the first set are written as follows: La-aa
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One notes (1)-(3) to describe an autocatalytic process. A system composed entirely by L would persist indefinitely given (2). Further, an a produced in (1) may catalyze an L transformation in a later step. The presence of a single a in the midst of L’s leads to 100% “solidification”, complete occupation of the lattice by a elements. The second reaction set includes (2); (4)-(9) are the remaining steps in converting L to /?on a model lattice. By analogy with (3), solid @) elements exert no mutual effects:
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In the second reaction set, solid elements exert no catalytic activity per se:
Instead, a catalytic intermediate I participates in several reactions, only one of which yields a solid element: LI-1p
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Note that in the “solidification” step (6), nearest neighbor L and I are transformed to I and ,l3, respectively. In a competing step, nearest neighbor L and I are transformed to respective I and L: LI-IL
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In another competing reaction, nearest neighbor L,? and I are transformed to I and L, respectively: PI-IL
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In the final possible reaction, nearest neighbor I and /3 are transformed, respectively, to /3 and I elements:
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The second reaction set also specifies an autocatalytic process: Element I is required for all liquid solid transitions; the catalyst is a product of all reactions in which it is involved. Unlike the first reaction set, however, the second provides for reversibility in the liquid solid transition. The model considers only the effect of a single catalytic I on a lattice occupied by L. Thus, the reaction I I ... does not enter the discussion. Elementary reactions specify the model. However, another perspective is offered using “reaction profiles” (Figure 3) envisaged (cartooned) for liquid and crystalline phase space of BZP. For the profile (upper frame) allied with (1)-(3), L lacks sufficient free energy to cross a barrier without catalytic intervention. An adjacent a possesses catalytic activity by virtue of excess free energy. Reaction (1) can be viewed as catalytic a jumping in the direction of crystalline phase space and simultaneously transferring free energy to an adjacent L. In so doing, L crosses the barrier (is transformed into an a ) and acquires catalytic activity. Note that the number of catalysts is not strictly conserved and that all motion occurs in the direction of a crystalline phase space. Solid elements surrounded entirely by other a are taken to lie on the far rhs of the barrier. The former elements demonstrate no catalytic activity according to (3). For the second reaction set (bottom frame), liquid and solid elements again lie on opposite sides of a free energy barrier; the catalyst is positioned at the apex. Reaction 6 can be viewed as I taking a jump to 3’/ phase space and transferring free energy to a nearest neighbor L. Reaction 7 involves I jumping to liquid phase space and transfemng energy to a nearest neighbor L. Reaction 8 involves I jumping to liquid phase space and transferring energy to a nearest neighbor p. In reaction 9, I jumps to crystalline phase space and transfers energy to a nearest neighbor ,!?. Note that in the second reaction set the number of catalysts is conserved and that motion is bidirectional along the reaction coordinate. The model ignores any number of factors: heat of fusion, free volume and density changes, material transport, handedness of a , etc. Even so, computer constructions of the model readily mimic the experimental results. The simpler constructions take the lattice to be 2D and rectangular. In the authors’ experience, the precise lattice dimensions had little bearing on the model’s correlation with experimental results. The model results presented here, however, are derived from lattices of 10 x 60 sites, upon which various BZP configurations were constructed and evolved according to the two reaction sets. In such constructions, an interior cell (a, p, L, or I) has eight nearest neighbors, while a border cell has a maximum of five. All model constructions were carried out using a Gateway PC with 486 DX processor. In models of a-BZP growth, one or more border sites of the lattice are charged with a (initial catalysts), the remaining sites occupied by L. This arrangement mimics the experiment whereby an a-BZP grain is added to one end of a tube containing supercooled liquid. A fundamental time unit C is defined by the random selection of an element and nearest neighbor on the lattice followed by execution of the appropriate reaction, ( l ) , (2), or (3). By analogy with experiment, the model tracks the position of the L-a interface along the axial lattice dimension X as a function of C. Concerning temperature effects, the occurrence of reaction 1 is weighted by the factor exp[Ta/D(Ta - r)] where D is a constant and Ta is the a melting temperature.
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4760 J. Phys. Chem., Vol. 99, No. 13, 1995
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Figure 3. Reaction profile view of BZP solidification. For the profile contained in the upper frame, liquid and a crystalline phase space are separated by a free energy barrier. Reaction 1 occurs when catalytic a takes a step in the direction of crystalline phase space and transfers free energy to a nearest neighbor liquid element. One or both reaction products can have catalytic activity, depending on the configuration of neighboring a and L. The lower frame illustrates the reaction profile pertaining to the formation of B-BZP. Inset numbers denote reactions specified in text. 1
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The upper frame of Figure 4 illustrates the position of the model L-a interface as a function of time unit C. This plot derives from an initial configuration of three a located at the border of an L-filled lattice. A linear time dependence is
Figure 5. Results of a computer model of B solid formation using a 10 x 60 lattice. The upper frame contains a plot of position X of solidification front versus time C. Note abscissa labels denote 100 OOOs of time units, Included is a plot of the function AP.6,where A is a constant. The lower frame illustrates a portion of the lattice after 3 x lo6time units have elapsed. Sites occupied by B elements are indicated by hatching, while clear cells denote the liquid elements. The catalytic element is indicated by a small solid square and (at the time of the snapshot) occupies a cell near the lower center of the illustration.
observed in these and all other cases. As in the experiments, the interface “velocities” of the model proved independent of the number and configuration of initial catalytic a positioned near the lattice border. The lower frame of Figure 4 illustrates model u-versus-?. for D = 100. Note the similarity between this graph and the experimental data contained in Figure 1. Figure 5 illustrates the results of model solidification govemed by the second reaction set. The upper frame plots the position X of the p-L front as a function of time unit C, following placement of a single I at the middle of the lattice border. The lower frame contains a snapshot of the lattice after 3 x lo6 time units have elapsed. One observes aggregates of ,!? (hatched cells) to coexist with L (clear cells). Further, the “solidification front” advances as a series of spurts; the velocity is more than 2 orders of magnitude less than that observed in constructions of the a growth. While the precise scaling of the front position with time is unclear, the behavior is nominally described by a fractional exponent. Figure 5 includes a plot of the function where A is a constant. A temperature dependence was not built into the model given the experimental inability to activate p-BZP growth over a significant temperature range.
V. Discussion Solidification phenomena are autocatalytic in nature. In the present case, a solid BZP surface (a-or ,!?-like) provides the free energy requisite to a barrier crossing which takes adjacent liquid to crystalline phase space. A surface produced in one solidification step is able to catalyze a later step. The experiments highlight several features peculiar to BZP. Most notably, BZP liquid demonstrates a persistent metastability at temperatures well below the freezing point, in addition to two distinct modes (a,p) of solidification. Each mode displays its own kinetic and morphological characteristics which are determined by the activation conditions. In the faster mode, the entirety
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Solidification of Benzophenone of liquid is converted to a crystallites, irrespective of the size, shape, etc., of the activating a grain. The interface position scales linearly with time. By contrast, the slower mode can be triggered only by a surface of BZP glass. Only a fraction of liquid converts to ,!? material, with the front position scaling with time according to a fractional exponent. At the macroscopic level, BZP solidification is governed by the interplay of surface energy, the heat of fusion, and material transport. Theory developed over the years provides for a detailed analysis at this level.16 In addition, a layered-growth macroscopic model has been applied to BZP crystallization by previous researchers.17 The experimental results, however, emphasize the importance of small-scale autocatalytic details. One gathers this from the fact that the container walls demonstrated no catalytic activity and a grains gave rise only to a crystallites. Further, angstrom-scale perturbations imposed by methyl groups obliterated the metastability of BZP liquid and propensity for two modes of crystallization. It can then be argued that the liquid solid transition in BZP entails a type of small-scale or molecular recognition. This process can be obstructed, triggered, and/or steered toward different regions of phase space, depending on small-scale structural details. l8 Reactions 1-3 and model constructions capture the essentials of a-BZP growth. An analogous version applies to DMBZP solidification equally well. The model demonstrates the same scaling behavior of the liquid-solid interface as recorded in experiments. The model further shows how the process kinetics can be insensitive to details of the initial a catalyst distribution placed near the sample border. Concerning temperature dependence, one notes the similarity of model factor exp[-Tu/ D(Tu - T)] to the Vogel-Tamman expression.lg Similarity of the lower frames of Figures 1 and 4 is then not surprising; the Vogel-Tamman expression is of general utility in describing the non-Arrhenius kinetics of amorphous phases. The second reaction set and model constructions mimic the formation of ,!?-BZP in supercooled liquid. This process is markedly different from a growth in terms of scaling behavior, material distribution, and crystallization velocity. Notably, the /3 solidification appears reversible to an extent, with the free energy of supercooled liquid and crystalline states being nearly equal. Several issues remain to be resolved, such as the solid front discontinuities, disparity of scaling exponents, and temperature sensitivity. Nonetheless, one simple way to visualize the ,!? growth is via the reversible catalytic activity of an element which lies at the edge of liquid and @) crystalline phase space. This type of solidification, while unusual, should not be unique to BZP. Materials having a solid phase with free energy nearly equal to that of the liquid should demonstrate analogous (diffusive-like) growth patterns. The solidification behavior of supercooled gallium is offered as one possibility.20 A basis for the persistent metastability of BZP liquid is offered as follows. One considers a mixed array of a and ,!?elements in equilibrium with the liquid phase, for example,
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One then notes that the above system will never solidify in the sense of converting to a- or ,!?-BZP given reactions 1-9. The authors conjecture that supercooled BZP may be viewed alternatively as a mixture of a and /3 clusters having different molecular-scale structures and reactivities. If the nucleation efficacies of a and /3 are competitive, then different reaction
specificities may indeed obstruct solidification. Emphasis may be placed on the word “conjecture”, as direct experimental support of this view is lacking. However, one also recalls the results of experiments with DMBZP. This compound demonstrated only one (a-type)crystalline phase12-14 and no resistance to crystallization. It may be that only liquids such as BZP, P4, and gallium which couple readily to more than one crystalline phase can be extensively supercooled. The persistent metastability of certain liquids may derive from angstrom-scale cluster mixtures constrained by elementary reaction steps. Clusters, free energy barriers, and reaction coordinates allied with supercooled liquids are subjects of considerable recent attention.21,22In this lab, the solidification properties of a variety of organic glass formers and derivatives continue to be investigated. Acknowledgment. The authors are grateful to Professor Stephen Pavkovic for helpful discussions and the use of crystal optics equipment. The detailed constructive comments of two anonymous referees are appreciated. References and Notes (1) Hildebrand, J. H.; Rotariu, G. J. J. Am. Chem. SOC.1951,73,2524. Powell, R. E.; Gilman, T. S.; Hildebrand, J. H. J. Am. Chem. SOC. 1951, 73, 2525. (2) Bridgman, P. W. Phys. Rev. 1914, 3, 186. (3) Thomas, C. D.; Gingrich, N. S. J. Chem. Phys. 1938, 6, 659. (4) Gemez, D. C. R. Hebd. Seances Acad. Sei. 1882, 95, 1278. (5) See reviews of glass-forming systems: Angell, C. A,; Clark, J. H. R.; Woodcock, L. V. Adv. Chem. Phys. 1981, 48, 397. Frenkel, D.; McTague, J. Ann. Rev. Phys. Chem. 1980, 31, 491. (6) Representative solidification references are listed as follows: Christian, J. W. The Theory of Transfomtions in Metals and Alloys, 2nd ed.; Pergamon: Oxford, 1975. Andres, R. P. In Nucleation; Zettlemoyer, A. C., Ed.; Dekker: New York, 1969; p 80. Wu, D. T. J. Chem. Phys. 1992, 97, 2644. Oxtoby, D. W. Nature 1990, 347, 725. Oxtoby, D. W. Adv. Chem. Phys. 1988, 70, 263. Chemov, A. A. In Crystal Growth; Springer: Berlin, 1984; p 122, Langer, J. S. Rev. Mod. Phys. 1980, 52, 1. Chalmers, B. Principles of Solidification; Wiley: New York, 1964. (7) Beilstein Handbook, Second Supplement to Volume 7, p 349. Hochstrasser, R. M.; Lin, T. S. J. Chem. Phys. 1968, 49, 4929. Anderson, R. J. M.; Kohler, B. E. J. Chem. Phys. 1976, 65, 1976. Schonherr, G.; Bassler, H.; Silver, M. Philos. Mag. 1981, 47, 369. Peter, L. M.; Vaubel, G. Phys. Status Solidi B 1973, 58, 593. (8) Representative early studies of supercooled and crystalline BZP can be found in the following: Livingston, J.; Morgan, R.; Stone, E. C. J. Am. Chem. SOC.1913, 35, 1505. Schaum, K. Z. Phys. Chem. 1898, 25, 722. Richards, W. T.; Kirkpatrick, E. C.; Hutz, C. E. J. Am. Chem. SOC. 1936,58,2243. Meyer, J.; Pfaff, W. Z. Anorg. Allgem. Chem. 1934, 217, 257. Tammann, G. Z. Phys. Chem. 1899, 29, 51. Hasselblatt, M. Z. Anorg. Allgem. Chem. 1922, 119, 325. Moms, J. B.; Strickland-Constable, R. F. Trans. Faraday SOC.1954, 50, 1378. (9) The structure and growth of a-BZP crystal has been investigated by the following: Banerjee, K.; Haque, A. Indian J. Phys. 1938, 12, 87. Fleischer, E. B.; Sung, N.; Hawkinson, S. J. J. Phys. Chem. 1968, 72,4311. Vul, E. B.; Lubanova, G. M. Sov. Phys. Crystallogr. 1967, 12, 355. Ovsienko, D. E.; Alfintsev, G. A.; Chemerinskii, G. P. J. Cryst. Growth 1982, 107, 60. Pech, S.; Vignes-Adler, M. J. Cryst. Growth 1978.43, 123. Bleay, J.; Hooper, R. M.; Narang, R. S.; Shenvood, J. N. J. Cryst. Growth 1978, 43, 589. (10) Groth, P. Chemische Krystallographie;Engelmann: Leipzig, 1919; Vol. V, p 89. (1 1) Hunter, T. F.; McAlpine, R. D.; Hochstrasser, R. M. J. Chem. Phys. 1968, 50, 1140. (12) Reference 10, p 131. Beilstein Handbook, Second Supplement to Volume 7, p 387. (13) Toussaint, J. Mem. SOC.R. Sei. Liege 1952, 12, 1. Wilson, A. J. C., Ed. Structure Reports 1952, 16, 516. (14) The crystal structure of DMBZP has been reported most recently by the following: Kojic-Prodic, B.; Bresciani-Pahor, N.; Horvatic, D. Acta Crystallogr. 1990, C46, 430. Ito, Y.; Matsuura, T.; Tabata, K.; Meng, J.B.; Fukuyama, K.; Sasaki, M.; Okada, S . Tetrahedron 1987, 43, 1307. (15) Temstra. P.: Codd. L. W. Crvstallometrv: Academic: New York, 1961; Chapter IV. (16) See. for examde: Katz J. L.; Donohue, M. D. Adv. Chem. Phys. 1979,40,49. Weeks, .f D.; Gilmer, G. H. Adv. Chem. Phys. 1979,40, 157. (17) See Ovsienko, et al., of ref 9.
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4762 J. Phys. Chem., Vol. 99, No. 13, 1995 (18) Molecular-level aspects of crystallization have been addressed by the following: Kahr, B.; McBride, J. M. Angew. Chem., In?. Ed. Engl. 1992, 31, 1. McBride, J. M.; Bertman, S. B. Angew. Chem., Znt. Ed. Engl. 1989, 28,330. McBride, J. M.Angew. Chem., In?. Ed. Engl. 1989,28,377. Vaida, M.; Shimon, L. J. W.; Weisinger-Lewin, Y.; Frolow, F.; Lahav, M.; Leiserowitz, L.; McMullan, R. K. Science 1988, 241, 1475. WeisingerLewin, Y.; Frolow, F.; McMullan, R. K.; Koetzle, T. F.; Lahav, M.; Leiserowitz, L. J. Am. Chem. SOC.1989, 111, 1035.Whitesell, J. K.; Davis, R. E.; Wong, M. S.; Chang, N. L. J. Am. Chem. SOC.1994, 116, 523.
(19) Angell, C. A. J. Non-Cryst. Solids 1991, 13, 131. (20) Defrain, A. J. Chim. Phys. 1977, 74, 851. (21) See, for example: Variyar, J. E.; Kivelson, D.; Lynden-Bell, R. M. J. Chem. Phys. 1992, 97, 8549 and cited references. (22) See,for example: van Duijneveldt, J. S.; Frenkel, D. J. Chem. Phys. 1992, 96, 4655 and cited references.
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