5464
J. Phys. Chem. 1996, 100, 5464-5469
Growth Anisotropy of N-Methylurea Crystals in Methanol Boris Yu. Shekunov*,† and Roger J. Latham‡ Departments of Pharmaceutical Sciences and Chemistry, School of Applied Sciences, De Montfort UniVersity, Leicester LE1 9BH, UK ReceiVed: October 17, 1995; In Final Form: January 2, 1996X
The present work investigates the relationship between the growth kinetics and the crystal structure of N-methylurea (NMU) crystals. Anisotropy of the step kinetic coefficient, step free energy, and normal growth rate of NMU crystals may be explained on the basis of a model which links the above quantities with the energy of kink formation in the growth steps. The proportionality condition between the solid-solid and solid-liquid bonds has been applied, and the coefficient of this interaction was obtained experimentally. The anisotropy of the crystal habit observed in these experiments is much greater than that predicted by the attachment energy model. We suggest that the method may be generally used to quantify the crystal growth anisotropy of molecular crystals.
1. Introduction The prediction of crystal growth behavior on the basis of crystal structure is one of the most important targets for the understanding of both the fundamentals of growth processes and applications in science and technology. Important crystallization problems such as growth rate anisotropy, morphological instability, and surface roughening are related to the kinetics of growth steps produced from dislocation defects on the crystal surfaces. This dislocation growth mechanism is typical for organic molecular crystals such as N-methylurea (NMU). The crystal habit is defined by both the intrinsic anisotropy of the crystal lattice and specific growth conditions such as dislocation structure, diffusion, solvent, and impurity interactions. It is also dependent on how far growth conditions are from the equilibrium state. One of the most important methods for the derivation of crystal form is based on a calculation of attachment energy according to the Hartman-Perdok theory.1 The normal growth rate is assumed to be proportional to the attachment energy, and the short range of van der Waals and hydrogen-bonding interaction within molecular crystals enables this energy to be calculated. An important development of the Hartman-Perdok method in terms of connected crystallographic nets2,3 also allows the roughening transition temperature to be predicted for different crystal faces. Thus, the attachment energy model is a good approximation for equilibrium crystal shape. In reality, however, the crystal form is not equilibrium; the structure of growth steps and the activity of dislocation source must be taken into account. In-situ laser interferometry4,6 offers a considerable advantage over measurements of crystal growth rate in that it allows the step velocity to be determined in different crystallographic directions. Whereas the growth rate is dependent on dislocation activity, the step velocity is not. The interferometric measurements also enable discrimination to be made between the intrinsic growth anisotropy and the anisotropy which is caused by diffusion or impurity effects. Detailed studies of growth kinetics and surface morphology of NMU crystals have been reported elsewhere.7,8 The main emphasis of the present investigation is to analyze the reasons †
Department of Pharmaceutical Sciences. ‡ Department of Chemistry. X Abstract published in AdVance ACS Abstracts, February 1, 1996.
0022-3654/96/20100-5464$12.00/0
for the growth anisotropy of NMU crystals. A simple computational analysis of the growth kinetics has been considered. 2. Experimental Results The surface kinetic measurements were carried out using an NMU solution in methanol as described in our previous works.7,8 The interferograms of the crystal faces corresponding to the crystal form in Figure 1d are shown in Figure 1 a-c. The space group symmetry of NMU is P212121,9 and there are three crystallographically independent faces, namely (011), (110), and (100). The distance between the interferometric fringes gives the steepness of the growth hillocks, p, whereas the movement of the fringes in the normal and tangential directions as a function of time allows the growth rate, R, and the growth step velocity, V, respectively, to be measured. The parameters p and V were obtained as functions of supersaturation σ for the following crystallographic directions: 1, [01h1], (011); 2, [100], (011); 3, [001], (110); 4, [1h10], (110), 5, [010], (100); 6, [001], (100). The normal growth rate, R, was also determined for the all three faces. In this work, the investigated interval of supersaturation was about 1% depending upon which crystal face was studied. At low supersaturation the growth kinetics are not limited by volume diffusion,7 and this enabled us to concentrate on the surface phenomena. Supersaturation was introduced by varying the temperature of the solution and was calculated by the equation 10
σ ) ln(c/c0)
(1)
where c and c0 are the actual and saturated concentrations of the solution in g L-1 reported elsewhere.7 The average temperature for the experiments was about 300 K. The velocity of growth steps, V, is the most important quantity in any description of the growth kinetics. Figure 2 shows the dependencies V(σ) for the crystallographic directions investigated. The dependencies are linear, proportional functions of supersaturation. This proportionality allows the step kinetic coefficient β to be determined according to the formula10,11
β ) (∆V/∆σ)/(F/c0)
(2)
where F is the density of NMU, which is equal 1.204 kg m-3. © 1996 American Chemical Society
N-Methylurea Crystals in Methanol
J. Phys. Chem., Vol. 100, No. 13, 1996 5465
Figure 1. Interferograms of the dislocation hillocks on (011) (a), (110) (b), and (100) (c) crystal faces of NMU (each photograph corresponds to approximately one-third of the whole face area) and the crystal habit. The anisotropy of the hillock form in the figures corresponds to the anisotropy of the growth velocity and hillock steepness on the crystal faces.
The calculated values of β are given in Table 1. The ratio between the kinetic coefficients in two perpendicular crystallographic directions gives the anisotropy of the dislocation hillock form (Figure 1a-c). The maximum step velocity in the [001] direction on the (100) face is about 25 times greater than the minimal step velocity in the [01h1] direction on the (011) face.
It is convenient to analyze dependencies p(σ) in the form of reciprocal coordinates 1/p (1/σ). According to the crystal growth theory, the dependence of 1/p (1/σ) should be linear for the same type of dislocation source.11 Such dependencies are given in Figure 3 for typical dislocation hillocks. The anisotropy of V for each particular face of NMU is not large. Assuming the proportionality between the free energy of growth step, R, and
5466 J. Phys. Chem., Vol. 100, No. 13, 1996
Shekunov and Latham
TABLE 1: Step Kinetic Coefficient, β, in Different Crystallographic Directions, the Ratio (r/m) between the Step Free Energy and Dislocation Activity on the Investigated Crystal Faces and the Anisotropy of the Growth Rate, AR, of NMU Crystals crystal face (011) (110) (100) a
β × 10-2 (m s-1) and step orientation
R/m, (kJ mol-1)
0.132, [01h1] 0.176, [100] 1.09, [001] 0.95, [1h10] 2.16, [010] 3.49, [001]
0.198 0.317 0.044 0.072 0.153 0.237
ARa 1 6.26 13.72
Measured at supersaturation σ ) 0.0025.
Figure 4. Normal growth rate, R, as a function of supersaturation σ for the (100) (a), (110) (b), and (011) (c) crystal faces.
the coefficient m. The ratio R/m, however, can be found from the slope of the lines in Figure 3, and the results are given in Table 1. Finally, the growth rate of the crystal faces, R, was measured as a function of supersaturation (Figure 4). The anisotropy of the NMU crystal form, AR, is the ratio between the growth rates of the different crystal faces and is shown in Table 1. The quantity AR was obtained at σ ) 0.0025. AR shows a weak dependence on supersaturation because the functions R(σ) ) pV are nonlinear and different for the different crystal faces. 3. Theoretical Analysis of the Step Anisotropy Figure 2. Dependence of the step velocity, V, on supersaturation, σ, corresponding to the following growth step orientations and crystal faces: a, [001], (100); b, [010], (100); c, [001], (110); d, [1h10], (110); e, [100], (011); f, [01h1], (011).
Figure 3. Dependence of the hillock steepness on supersaturation in the reciprocal coordinates (1/p)(1/σ) for the step orientations: a, [01h1], (011); b, [010], (100); c, [1h10], (110).
the kinetic coefficient, β,11 the following equation can be derived to quantify the dislocation activity on the crystal faces:
1/p ) F + (19Ra*/mhRgT)(1/σ)
(3)
where R is given in kJ mol-1, h is the height of the elemental growth layer (growth step) which is equal to the interplane distance, m is the number describing the length of the total Burger vector of dislocation source, a* is the mean distance between molecules along the growth step, Rg is the gas constant, and F is a constant which depends on the structure of the dislocation source and anisotropy of the step velocity. The absolute value of R cannot be calculated without knowledge of
The ultimate task for the prediction of growth step kinetics and growth step anisotropy is the determination of the density of step kinks (which are the active growth sites) and the free energy of growth steps.12 This can be done without taking into account solvent-solid and solvent-solvent interactions. Even for these conditions of vacuum morphology, a comprehensive analysis of the growth step configurations requires a computation of energetic and entropy components of the step free energy.13 Since the elemental cell of NMU and other molecular crystals consists of several molecules, the analytical computation of step configuration imposes some combinatorial problems, and the calculations become complicated. Modeling of the growth step can be significantly simplified by assuming that crystal consists of only one kind of molecule, and consequently, the strength of parallel intermolecular bonds is an average of individual bonds. This approach can be justified because the growth layer is packed by successive rows of molecules. As a result, the effect of step bonds on the step kinetics is averaged during the growth. According to this hypothesis, the analysis can be subdivided into the following steps: a. Calculation of the Intermolecular Bonds. The crystal structure of NMU in the two most representative projections (100) and (001) is shown in Figure 5. The solid-solid bond energies i were obtained by means of the program HABIT14 using 6-12 interatomic potentials reported by Lisson et al.15 The chosen set of potentials was specially derived for a variety of hydrogen-bonded amides that include the same structural groups as NMU. The strongest bonds between the nearest neighbors (Table 2) represent about 97% of the total lattice energy, and therefore, they define the major characteristics of the growth steps. A reasonable agreement was obtained between the calculated lattice energy El ) 83.06 kJ mol-1 and the enthalpy of sublimation measured as 87.24 kJ mol-1.16 b. Calculation of the Surface Energy and Specific Step Energy. This procedure determines the surface energy, W1, of
N-Methylurea Crystals in Methanol
J. Phys. Chem., Vol. 100, No. 13, 1996 5467
Figure 5. Crystal structure of NMU shown in two projection on the (100) (a) and (001) (b) planes. Broken lines and arrows show the most important crystal faces and growth layers; dotted lines show the most powerful hydrogen bonds.
TABLE 2: Intermolecular Bond Energies, Ei , for the NMU Crystal Structure Calculated between the Central Molecule and the Molecule Defined by [uWw], ja i
type
[uVw], j
d, Å
cos φ
i, kJ mol-1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
hydrogen hydrogen hydrogen hydrogen strong van der Waals strong van der Waals strong van der Waals strong van der Waals weak van der Waals weak van der Waals weak van der Waals weak van der Waals weak van der Waals weak van der Waals weak van der Waals weak van der Waals
1h11, 3 011, 3 021h, 2 020, 2 100, 4 11h0, 4 011h, 2 010, 2 1h00, 1 100, 1 001h, 1 001, 1 010, 3 1h10, 3 01h0, 4 000, 4
4.47 4.47 5.34 5.34 4.65 4.65 5.24 5.24 8.48 8.48 6.92 6.92 6.96 6.96 7.40 7.40
-0.95 0.95 -0.37 -0.37 0.49 0.49 -0.37 -0.37 -1.00 1.00 0.00 0.00 0.61 -0.61 -0.84 -0.84
-15.02 -15.02 -11.98 -11.98 -5.77 -5.77 -5.29 -5.29 -1.15 -1.15 -1.05 -1.05 -0.96 -0.96 -0.96 -0.96
u, V, w are the translation vectors of unit cell and j is the number of molecules in the unit cell. d is the intermolecular distance, and φ is the cosine of the angle between {100} and the bonding vector. a
the flat crystal faces and the specific energy of the step riser, W2, of smooth (T f 0) growth steps on the basis of the equation n
l
W1 (or W2) ) min[(1/n)∑∑ik]
(4)
i)1 k)1
where n is the number of symmetrically different molecules (i.e., not related by translation symmetry) which contribute to the energy of the flat surface (W1) or the step riser (W2), and l is the number of interrupted intermolecular bonds for an individual molecule. These quantities are defined as the minimal energy of crystallographic plane (W1) or step (W2) in a given direction. The magnitude of the surface and step energy is
strongly determined by the geometry of the bond nets. Thus, the most distinct difference can be seen between the {100} and {011} faces (Figure 5). The {100} faces have all the strongest bonds out of the face plane. The {011} faces contain a large number of the strongest bonds; in particular, five of six hydrogen bonds lie in these planes. Also, it should be noted that the condition of the minimal surface energy predicts the corrugated surface of the {110} faces (see Figure 5). c. Calculation of the Mean Energy of Formation of Growth Kinks, w. w is the additional surface energy which is created when a molecule is adsorbed into or evaporated from a smooth growth step: n
l
w ) (1/2n) ∑∑ik
(5)
i)1 k)1
In this case n means all possible molecular sites, and the coefficient 1/2 takes into account both positive and negative kinks created during adsorption (or evaporation). The geometry of the connected bond nets is very important for the definition of w. On the {100} faces the “smooth” step risers have a checked pattern (Figure 5) which is responsible for the small value of w on these faces. The direction of the strongest bonds on the {100} faces is diagonal to the step orientation, and when w is calculated, the strongest bonds do not contribute to the total magnitude of this energy. The values of W1, W2, and w are summarized in Table 3. This table also shows the important dimensional parameters a and h for the crystallographic directions investigated. d. Derivation of Anisotropy of the Kinetic Coefficient and Step Free Energy. When crystallization temperature is not very high compared to the roughening transition temperature, the growth kinetic coefficient β should be proportional to the density of growth kinks and the mean intermolecular distance, a, in the direction of the step velocity.10,12 This leads to the formula:
5468 J. Phys. Chem., Vol. 100, No. 13, 1996
β ∼ a exp(-Θw/RgT)
Shekunov and Latham
(6)
The coefficient Θ is introduced to take into account solventsolute interactions according to the proportionality condition.2 This means that the surface energy of crystal in solution is proportional to the strength of the solid-solid surface bonds (in vacuum). We also assume that other parameters which define the magnitude of β such as activation energy of solvatation or frequency of temperature oscillations are close to each other for the steps of different orientation because they are more related to the isotropic properties of solution. At relatively low crystallization temperatures, the proportionality condition leads to the approximation2
R ∼ ΘW2
(7)
4. Discussion. The Relationship between Crystal Structure and Growth Anisotropy According to eq 6, the dependence -ln(β/a) versus w should be linear. Figure 6 suggests that a linear relationship between these two values holds in general, and the magnitude of Θ found on this basis is 0.46 (squared correlation coefficient 0.86). The exponential dependence of the growth kinetic coefficient on the crystallographic orientation is a very important factor in determining the overall anisotropy of the crystal form. A comparison between Tables 1 and 3 also shows that the magnitude of the experimentally obtained quantity R/m is proportional to the calculated step energy W2. This is in agreement with eq 7. The much smaller magnitude of the quantity R/m compared to that of ΘW2 should be noted. This can be attributed, firstly, to the complicated structure of the dislocation source (m > 1) and, secondly, to the roughness of the growth step which decreases the step free energy.10,12,13 It is reasonable that both of these factors are relevant to an organic crystal structure such as NMU. Organic materials usually exhibit a high density of dislocations which means that the dislocation hillock is formed by a bunch of single dislocations. Relatively weak intermolecular interactions in organic crystals promote the formation of growth kinks with a consequent increase in the entropy and decrease in the free energy of growth steps. The anisotropy of the growth rate (the anisotropy of crystal form) may be derived on the basis of the formula11
R ) pV ∼ hβ/R
(8)
where β and R are taken for the same step orientation. The anisotropy of R, calculated from (8), is of the same magnitude as the experimentally observed anisotropy AR (Table 1), but it is somewhat higher than the experimental value. This difference can be explained using Figure 3 and eq 3: the lines (1/p)(1/σ) intersect the ordinate axis at different points above zero (F *
Figure 6. Experimental values of the step kinetic coefficient, β, in the six investigated crystallographic directions (Figure 2) versus calculated value of the kink energy, w, in the coordinates -ln(β/a)(w).
TABLE 3: Surface Energy, W1, the Specific Energy of Step Riser, W2, and the Energy of Kink Formation, w, Calculated for the Different Crystallographic Directions and Crystal Faces of NMUa crystal face
step orientation
W1, kJ mol-1
W2, kJ mol-1
w, kJ mol-1
a, Å
h, Å
(011) (011) (110) (110) (100) (100)
[01h1] [100] [001] [1h10] [010] [001]
14.01 14.01 23.02 23.02 23.86 23.86
12.00 22.17 4.26 8.18 10.32 10.64
16.17 8.64 7.60 5.04 1.00 < 0.83
4.92 4.24 6.92 3.66 6.98 6.92
4.92 4.92 5.39 5.39 8.48 8.48
a a is the mean intermolecular distance in the direction of growth step velocity, and h is the height of the growth step.
0 in (3)). This is caused by the complicated structure of the dislocation source. If the structure of the dislocation source was the same for the all crystal faces, the function R(σ) would be parabolic, and the anisotropy AR would then be independent of supersaturation. The coefficient Θ in formulas 6 and 7 has been determined from the kinetic dependencies in Figure 2. It is also possible to estimate Θ on the basis of thermodynamic data using the hypothesis of the equivalent wetting and proportionality conditions.2 These conditions imply that solid-fluid bonds on the surface are equal to the corresponding bonds in the bulk of the solution and also that solid-fluid and fluid-fluid bonds are proportional to the solid-solid bonds. The value of Θ could be obtained from2
Θ ) ∆dH/El
(9)
where ∆d H ) 14.67 kJ mol-1 is the enthalpy of dissolution determined from the van Hoff equation and the solubility data,7 and El is the lattice energy (referred to in section 3a). The value of 0.18 is considerably smaller than that calculated from the growth kinetics. This may indicate a breakdown of the equivalent wetting conditions because the solvent molecules near the crystal faces are ordered to a much greater extent than in the bulk solution. This conclusion correlates with the results of Bennema et al.2 where calculated roughening transition parameters (proportional to Θ) were usually found to be too low when compared with experimental data. In general, the proportionality conditions can be imposed when there is no special “tailor-made” resemblance between solvent and solute molecules. If this were the case, some crystal faces might have a strong specific bonding with solvent, and therefore, the coefficient Θ would be different for different faces. This could also lead to a strong adsorption on particular faces and, consequently, blocking of the growth steps. In the case of NMU crystals, the proportionality condition seems to be a good approximation. It can be seen from Figure 2 that the functions V(σ) are linear for the all crystal faces. This shows that the solvent effect on the crystal faces uniformly decreases the surface energy and that the solvent-surface interactions are relatively weak, thus creating a relatively minor barrier for the step movement. The strong, selective influence of solvent (or impurities) may lead to a form of nonlinear kinetics of the type that has been observed for some other crystals.17,18 The linear kinetics of NMU can be related to a high solubility of this material in methanol (462 g L-1) as, for example, compared with the solubility of 43 g L-1 for the NPP (N-(4-nitrophenyl)prolinol) crystals in methanol17 at the same temperature 300 K. The high bulk concentration of the solution results in a high surface concentration of solute molecules and better crystal growth. The growth rate anisotropy obtained in the present experiments and on the basis of calculated growth step anisotropy
N-Methylurea Crystals in Methanol
J. Phys. Chem., Vol. 100, No. 13, 1996 5469 Notation {hkl} (hkl) [hkl] a
Figure 7. Experimentally observed habit of NMU crystals (a) and habit derived on the basis of attachment energy model (b). The growth anisotropy differs by a factor of 10.
(section 3) is much greater than predicted by the attachment energy model.1 This model assumes proportionality between the growth rate and attachment energy. The attachment energy is approximately equal to the surface energy W1 (Table 3) in our consideration and that gives the growth anisotropy as 1.7 for the {011} and {100} crystal faces. As a result, the attachment energy model greatly underestimates the anisotropy of NMU crystal form (Figure 7). This shows a key difference between the present and the PBC model. According to eqs 6-8 the growth rate is mostly defined by the exponential function of energy w, and the resulting growth anisotropy exceeds by far that following from the linear relationship. The present simplified model gives a satisfactory explanation for the growth anisotropy. However, prediction of the exact values of the kink density and step free energy requires a more comprehensive analysis of step configurations. These computations are currently in progress. Furthermore, the coefficient Θ may be theoretically predicted for different crystal faces.19 The interferometry can experimentally determine the absolute value of the step free energy R and the roughening transition temperature when R f 0. These additional measurements would provide a complete picture of how the growth kinetics depends on the anisotropy of crystal bonds and solvent-surface interactions. 5. Conclusion The present work investigates the growth anisotropy of N-methylurea crystals growing in methanol. The growth anisotropy is a combined effect of the anisotropy of the step kinetic coefficient, β, step free energy, R, and the dislocation activity on the different crystal faces. These parameters were defined by means of the laser interferometric technique and were compared with the predicted anisotropy of β and R derived on the basis of the strongest intermolecular bonds. The proportionality condition between the solid-solid and solid-liquid bonds has been applied, and the coefficient of this interaction, Θ, has been experimentally obtained. The growth rate anisotropy observed in these experiments and derived on the basis of calculated step anisotropy is much greater than that predicted by the attachment energy model. We suggest that the method may be generally used to quantify the crystal growth anisotropy of molecular crystals. Additional experimental data and computational analysis are required to determine the absolute value of the step free energy and to obtain independently the magnitude of the coefficient Θ. Acknowledgment. We thank Dr. K. J. Roberts for his help with computer calculations. B.Y.S. gratefully acknowledges the financial support of this work by the Nuffield Foundation through a Research Grant to Newly Appointed Science Lecturers.
a* AR c c0 h m ∆dH El p R Rg T V w W1 W2 R β F i Θ σ
Miller indices of a crystal form Miller indices of a face crystallographic direction mean distance between molecules in the direction of step velocity mean distance between molecules along the growth step anisotropy of the normal growth rate (anisotropy of the crystal form) solution concentration equilibrium solution concentration height of the growth step total Burger vector of dislocation source given in h units enthalpy of dissolution lattice energy surface steepness normal growth rate of crystal face ()pV) gas constant absolute temperature of solution tangential velocity of growth (velocity of steps) energy of kink formation (in vacuum) surface energy of crystal face (in vacuum) specific energy of growth step riser (in vacuum) free energy of growth step kinetic coefficient of growth steps crystal density energy of solid-solid intermolecular interaction the coefficient between the surface energy in solution and in vacuum supersaturation of solution ()ln(c/c0))
References and Notes (1) Hartman, P. In Crystal Growth, An Introduction; Hartman, P., Ed.; North-Holland: Amsterdam, 1975; p 367. (2) Bennema, P.; Van der Eerden, J. P. In Morphology of Crystals; Sunagawa, I., Ed.; Terra Scientific Publishing: Tokyo, 1987; p 1. (3) Bennema, P.; Xiang Yang Liu; Lewtas, K.; Tack, R. D.; Rijpkema J. J. M.; Roberts, K. J. J. Cryst. Growth 1992, 121, 679. (4) Rashkovich, L. N.; Shekunov, B. Yu. J. Cryst. Growth 1990, 100, 133. (5) Rashkovich, L. N.; Shekunov, B. Yu. J. Cryst. Growth 1991, 112, 183. (6) Shekunov, B. Yu; Rashkovich, L. N.; Smolskii, I. L. J. Cryst. Growth 1992, 116, 340. (7) Ristic, R. I.; Shekunov, B. Yu.; Sherwood, J. N. Submitted to J. Cryst. Growth. (8) McEvan, A.; Shekunov, B. Yu.; Shepherd, E. E. A.; Sherwood, J. N. Submitted to J. Cryst. Growth. (9) Huiszoon, C.; Tiemerssen, G. W. M. Acta Crystallogr. 1976, B32, 1604. (10) Chernov, A. A. In Modern Crystallography III, Crystal Growth; Springer Series in Solid State Physics: Springer: Berlin, 1984. (11) Chernov, A. A.; Rashkovich, L. N.; Mkrtchan, A. A. J. Cryst. Growth 1986, 74, 101. (12) Barton, W. K.; Cabrera, N.; Frank, F. C. Philos. Trans. R. Soc. London 1951, 243A, 299. (13) Voronkov, V. V. SoV. Phys.sCrystallogr. (Eng. Transl.) 1966, 11, 259. (14) Clydesdale, G.; Docherty, R.; Roberts, K. J. Comput. Phys. Commun. 1991, 64, 311. (15) Lisson, S.; Hagler, A. T.; Dauber, P. J. Am. Chem. Soc. 1979, 101, 5111. (16) Ferro, D.; Barone, G.; Della Gatta, G.; Piacente, V. J. Chem. Thermodyn. 1987, 19, 915. (17) Shekunov, B. Yu.; Shepherd, E. E. A. Sherwood, J. N.; Simpson, G. S. J. Phys. Chem. 1995, 99, 7130. (18) Grant, D. J. W.; Ristic, R. I.; Shekunov, B. Yu.; Sherwood, J. N. Proceedings of the 1st International Particle Technology Forum; AIChE: New York, 1994, Vol. 1, p 293. (19) Xiang-Yang Liu; Bennema, P. J. Chem. Phys. 1993, 98, 145.
JP953075B
