) (1 θ P)( f ν/(1 f ) and taking ) θsν (surface coverage times the rate of removal from a kink site), we obtain θs ) f , which agrees with the Monte Carlo results. Figure 4b displays the sticking fraction for growth with typeII inhibitors as a function of f at ∆µ/kT ) 0.2 for different φ/kT. Similar results were obtained for the type-III inhibitors. As for not too high values of ∆µ/kT, S is proportional to ∆µ/kT, that is, S ) β∆µ/kT; the kinetic coefficient β can be deduced as a function of f and φ/kT. Excellent fits, shown in Figure 4b, were obtained for S using the empirical equation β ) β0 + β1 exp(-bf); for all φ/kT e 0.8, the R-square value of the fit is g0.99. Values of β0, β1, and b as a function of φ/kT are given in Figure 5. From this figure, it is evident that β0 is rather small; therefore, the function can be approximated as β ) β1 exp(-bf), where β1 is the kinetic coefficient in the absence of inhibitors and b is a blocking efficiency factor. This simplified expression can be derived if we assume that each adsorbed inhibitor blocks crystal growth on an average surface area b. The value of b does not necessarily correspond to the area of the growth unit that is covered by the additive; it can be larger as the inhibitor also partly “protects” the surface around. The surface area protected by m adsorbed inhibitors per unit surface area is s(m). Crystal growth can only take place
Figure 5. Fitting parameters, β0, β1, and b in S ) [β0 + β1 exp(-bf)] obtained from the S versus f curves in Figure 4b as a function of the bond strength φ/kT. (Type-II inhibitors, ∆µ/kT ) 0.2.)
on unprotected surface regions, u(m), the total surface area being s(m) + u(m) ) 1. Therefore, if inhibitors are adsorbed on the surface, the growth rate is S ) u(m)β1∆µ/kT, and the kinetic coefficient becomes β ) β1u(m). The change of protected surface area with the number of adsorbed inhibitors is given by
ds(m) ) b(1 - s(m)) dm
(4)
6384 J. Phys. Chem. C, Vol. 112, No. 16, 2008
van Enckevort and Los
Figure 6. Embedding of inhibitor units into the growing crystal surface via the kink (k) f step (st) f surface (su) positions.
where the second term in the right-hand part of this equation accounts for the overlap of protected areas. Substituting s(m) ) (1 - u(m)) gives
du(m) ) bu(m) dm
-
(5)
Solving this differential equation with the boundary condition u(m)0) ) 1 results in
u(m) ) exp(-bm)
(6)
and gives a kinetic constant of
β ) β1u(m) ) β1 exp(-bm)
(7)
Since in our case the surface area of an inhibitor unit is identical to that of a growth unit and taking the surface area of both units as the unit surface area, m is identical to the surface coverage θs, and b is the average number of growth units that are protected. Since for the rough faces θs = f , we finally come to
β ) β1 exp(-bf)
(8)
as was found by the Monte Carlo experiment. From the graph in Figure 5, it can be seen that for the lowest φ/kT ) 0.25, an adsorbed type-II inhibitor only “protects” two surface molecules, whereas for φ/kT ) 0.8, it “protects” the molecule below plus 11 neighbors. 4. Flat Faces 4.1. General. The {001} surfaces of the Kossel crystal grow layer-wise via steps if φ/kT g 0.8 and ∆µ/kT is less than the transition value for kinetic roughening.22,33 Then, the tailor-made inhibitors are not randomly incorporated into the crystal lattice but are preferentially incorporated via kink sites at the steps. These kink positions become step positions and finally surface positions upon further growth.34 This is shown in Figure 6. Once it has arrived at a surface position, with four horizontal and one vertical bond, the inhibitor is firmly embedded in the surface and is capable of retarding or pinning the step behind. Figure 7a shows the sticking fraction as a function of bond energy φ/kT for the three inhibitor types at ∆µ/kT ) 0.2 and fraction f ) 0.2. As compared to the clean case (without inhibitors), the growth rate decreases much faster for increasing φ/kT if inhibitors are added (note the logarithmic scale on the vertical axis). This implies that the effect of the tailor-made inhibitors increases rapidly with bond strength. Figure 7b gives the surface coverage, inhibitor ()vacancy) bulk concentration, and the sticking fraction ratio of the inhibited and clean surface as a function of φ/kT for the type-III inhibitors, again at ∆µ/kT
) 0.2 and f ) 0.2. From this figure, it can be inferred that in contrast to the sticking fraction ratio and inhibitor concentration, the surface coverage does not change much with φ/kT. In fact, θs ≈ f , similar as for the rough faces. Growth is retarded to a limited extent at lower values of φ/kT because there is sufficient time for the relatively weakly bonded inhibitors to release from their embedded surface positions. On the other hand, for larger φ/kT, the inhibitors are firmly attached to the surface, and especially for the type-II and -III inhibitors, step propagation gets almost completely blocked, and the crystal growth rate decreases several orders of magnitude. There is no sudden change in the dependence of S, θs, or θi ) θv on φ/kT upon going from rough to flat growth around φ/kT ≈ 0.8. Further, no evidence for step bunching was found during our simulations, as has been reported earlier for different kinds of impurities.35-38 For large φ/kT and lower driving force, the type-II and typeIII inhibitors behave similarly; they block crystal growth almost completely. However, at high ∆µ/kT, the sticking fraction does not attain a stationary value for the type-II case. Initially, growth proceeds rapidly, but in due time, it gradually slows down. Figure 8a shows the surface morphology after a growth run with type-II additives under such conditions (φ/kT ) 2.0; f ) 0.2; ∆µ/kT ) 2.5). The surface is very rough and is composed of many deep holes with an inhibitor molecule at the bottom. The impurity remains firmly anchored, and because of the SOS condition, it does not allow for deposition of growth units on top of it. This situation is not realistic. In practice, a vacancy will be formed above the inhibitor. This is modeled by the typeIII additives, which indeed give a constant growth rate after an initial period of relaxation. In addition, a realistic surface morphology (flat with steps) is obtained, as shown in Figure 8b. We therefore only consider type-III inhibitors in the remaining part of this section. 4.2. Dead Supersaturation Zone. Figure 9 shows the sticking fraction, surface coverage, and bulk fraction as a function of driving force for the type-III inhibitor. Figure 9a displays the results for φ/kT ) 1.2 and Figure 9b for φ/kT ) 2.0. In both cases, f ) 0.2. In contrast to the clean case as well as for the rough surfaces, no or very slow growth occurs for the lowest supersaturation values, that is, (dS/d(∆µ/kT))∆µf0 = 0, despite the presence of growth steps. Raising the supersaturation beyond this dead supersaturation zone of no growth leads to a gradual increase in sticking fraction, which eventually becomes linear with supersaturation, S ∝[∆µ/kT - (∆µ/kT)*]. We here define the width of the dead zone, (∆µ/kT)*, as that value of supersaturation at which the extrapolated linear growth curve intersects S ) 0 (see also Figure 9). The width of the no-growth region increases for increasing bond energies, being (∆µ/kT)* ) 0.64 for φ/kT ) 1.2 and (∆µ/kT)* = 1.3 for φ/kT ) 2. The width of the dead zone as a function of the fraction of inhibitor molecules in the fluid phase is summarized in Figure 10. Figure 10a displays several S versus ∆µ/kT curves for φ/kT ) 2, using f values ranging from 0 to 0.2, whereas Figure 10b shows the dependence of log[(∆µ/kT)*], abstracted from the graph in Figure 10a, on log(f ). The log-log plot is linear and corresponds to (∆µ/kT)* ) 3.14 × f 0.65. The occurrence of a dead supersaturation zone induced by impurities has been reported for many crystal growth systems,4 such as KH2PO4,39,40 paraffin,5,6 and K2Cr2O741 growing from solution. Cabrera and Vermilyea29 put forward an explanation for this phenomenon as early as 1958. They assumed that propagating steps are pinned by immobile impurities adsorbed
“Tailor-Made” Inhibitors in Crystal Growth
J. Phys. Chem. C, Vol. 112, No. 16, 2008 6385
Figure 7. Effects of tailor-made inhibitors as a function of bond strength φ/kT (∆µ/kT ) 0.2; f ) 0.2). (a) Sticking fraction for growth in the presence of the three types of inhibitors as compared to that for a clean surface; (b) sticking fraction ratio of the inhibited and clean surface, inhibitor ()vacancy) concentration, and surface coverage for growth in the presence of type-III inhibitors.
Figure 8. Surface morphologies obtained at a large bond strength and high supersaturation for (a) type-II and (b) type-III inhibitors. In both cases, φ/kT ) 2.0, ∆µ/kT ) 2.5, and f ) 0.2. Grey spheres are growth units; red spheres are inhibitor units.
Figure 9. Sticking fraction, surface coverage, and inhibitor fraction in the solid phase as a function of driving force for nonroughened faces growing in the presence of type-III inhibitors: (a) φ/kT ) 1.2, (b) φ/kT ) 2.0.
on the crystal surface. If the supersaturation is less than a critical value
(∆µ/kT )* ) CΩ2/3
γ 1/2 θ kT imp
(9)
then the steps are not able to pass the fence of adsorbed impurities, and crystal growth is blocked. In this equation, C is a constant, which is 2 for a square array of impurities29,42 and
1.51 for randomly distributed impurities.43 Further, Ω is the volume of one growth unit, and γ/kT is the dimensionless step free energy per unit step length, Ω1/3. For φ/kT ) 2, this step free energy equals 1.728, as follows from eq 5 in ref 16. The θimp is the surface concentration of the immobile impurities (i.e., the number of impurities per unit surface area, Ω2/3), which, in our case, are those inhibitor units that are embedded in the surface with four horizontal neighbors (“su” in Figure 6). For
6386 J. Phys. Chem. C, Vol. 112, No. 16, 2008
van Enckevort and Los
Figure 10. Width of the dead zone as a function of inhibitor concentration in the fluid phase for type-III inhibitors and bond strength φ/kT ) 2.0. (a) Sticking fraction against supersaturation for various inhibitor concentrations, f . The linear curve for f ) 0 represents the growth of a clean surface. (b) Log-log plot of the dependence of the critical supersaturation, (∆µ/kT)*, on inhibitor fraction, f . The two dashed lines are calculated values according to eq 10 using γ/kT ) 1.728, Ω ) 1, and a ) 1.0 and 0.65.
a bond strength of φ/kT ) 2, the total number of adsorbed inhibitors at (∆µ/kT)* is somewhat less than their fraction in the fluid phase, f . In addition, visual inspection of the grown surfaces showed that virtually all of the inhibitors are located at the above-mentioned immobile positions. Therefore, we can take θs ) a‚f , with the proportionality constant a somewhat less than one. Using eq 9, this gives an estimated theoretical relation
(∆µ/kT )* ) 1.5Ω2/3
γ 1/2 1/2 a f kT
(10)
As shown in Figure 10b, a reasonable though not perfect agreement of eq 10 with the Monte Carlo data is obtained by taking γ/kT ) 1.728, Ω ) 1, and a ) 1. The differences are likely due to the rather crude assumptions that are made in deriving the Cabrera and Vermilyea model and its application to the present case. A better fit is obtained for the smaller value of a ) 0.65. 4.3. Segregation. It is evident from Figure 9 that the concentration of type-III inhibitors incorporated into the grown crystal increases for increasing supersaturations up to a maximum value at (∆µ/kT)*. Beyond the dead supersaturation zone, that is, for ∆µ/kT > (∆µ/kT)*, θi ) θv decreases on the same footing as the surface coverage, or
- B(θ/s - θs) θi ) θv = θmax i
(11)
In this equation, θ/s is the surface fraction at (∆µ/kT)*, which is slightly less than the inhibitor concentration in the fluid phase, f , and B is a proportionality constant. The maximum inhibitor at the point of turnover at (∆µ/kT)* is concentration, θmax i approximately proportional to f , as depicted in Figure 11. The much smaller low inhibitor bulk fractions for ∆µ/kT than those for (∆µ/kT)* are explained by the fact that steps are completely blocked by the adsorbed inhibitors. A step can only propagate if the inhibitor molecules in front of it are released. Then, some growth can take place, but no or only a few inhibitor molecules will be incorporated. At higher supersaturation, the steps can pass the blocking centers and a large part of the adsorbed inhibitors are readily built into the crystal lattice. For the highest supersaturation, ∆µ/kT > (∆µ/kT)*, the surface and thus the bulk fraction of inhibitors decreases because the probability ratio of inhibitor attachment and growth unit
Figure 11. Maximal inhibitor ()vacancy) fraction in the solid phase, θmax ()θmax i v ), as a function of inhibitor fraction in the fluid, f , for type-III inhibitors at φ/kT ) 2.0.
attachment to the crystal surface decreases for increasing ∆µ/ + /P+ kT as Pi,ihb i ∝ exp(-∆µ/kT). 5. Etching On first sight, no dead undersaturation zone is expected for etching because inhibitor molecules embedded in the crystal surface are not expected to be able to retard or block step propagation significantly, as their rate of removal does not differ from the regular growth units. Looking at Figure 12a, one indeed expects that the etching velocity of a step with additives is very similar to that of a step on a clean surface. However, our simulations as well as experimental evidence reported in literature8,44,45 provide proof to the contrary. Figure 13 displays the measured negative sticking fractions, Sihb and Sno ihb of surfaces etched with and without additives as a function of driving force for different bond strengths φ/kT ) 0.7, 1.2, and 2.0 and f ) 0.2. In these simulations, a “clean” bulk crystal is etched. Type II-inhibitors are used. For etching, type-II inhibitors give the same results as type-III inhibitors, as no inhibitor molecules, and thus also no vacancies, are incorporated into the crystal lattice. In all cases, the sticking fraction ratio, RSF ) Sihb/Sno ihb, is less than 1 and is lowest for low |∆µ/kT|. For the rough surface (φ/kT ) 0.7; Figure 13a), the inhibition effect is limited, and no dead undersaturation zone exists, that is, [d|Sihb|/ d|∆µ/kT|]|∆µ/kT|f0 * 0. For the two flat faces (φ/kT ) 1.2 and 2.0; Figure 13b and c), however, [d|Sihb|/d|∆µ/kT|]|∆µ/kT|f0 tends
“Tailor-Made” Inhibitors in Crystal Growth
J. Phys. Chem. C, Vol. 112, No. 16, 2008 6387
Figure 12. Etching of a flat crystal surface with steps in the presence of tailor-made inhibitors. (a) The propagation of steps proceeds by stripping off growth and inhibitor units from kink positions. (b) Etch rate versus undersaturation of a one-dimensional crystal in the presence of inhibitor molecules; a dead undersaturation zone develops, extending up to |∆µ/kT| ) -ln[1 - 1/(1 - f )].
to zero, despite the presence of steps. Now, a dead undersaturation zone develops, which is wider for the largest φ/kT. Therefore, in fact, the situation is not very different from growth. In all three cases, RSF increases for increasing undersaturation |∆µ/kT|, going toward (1 - f /(1 - f )), which is 0.75 for the f ) 0.2 used in the simulations. Another difference from growth is the early decrease of the inhibitor surface coverage for increasing |∆µ/kT|, which already sets in at low undersaturations. For φ/kT ) 2.0, this decrease is very fast for |∆µ/kT| g 0.25, which is just beyond the dead undersaturation zone. To show analytically that retardation and blocking of step propagation also occurs for dissolution, we resort to a simple one-dimensional model. Consider a step with kinks as displayed in Figure 12a. Step propagation proceeds by stripping off the step by kink removal, as indicated by the arrow. However, growth and inhibitor units are not only removed but can also be added to the kink position during dissolution. The frequency of adding growth units to a kink site is P+ k ) ν exp(-|∆µ/kT|), ) f‚ν/(1 - f ). The frequency and that of the inhibitors is P+ k,ihb of removal of growth units depends on the probability, θk, that an inhibitor molecule occupies the kink site, Pk ) ν(1 - θk). The frequency of inhibitor removal from the kink is Pk,ihb ) ν‚θk. As for etching, no inhibitors are incorporated into or released from the bulk crystal, P+ k,ihb ) Pk,ihb or θk ) f /(1 - f ). Therefore, the kink propagation rate, Rk ) P+ k - Pk , is
Rk( f,|∆µ/kT|) ) ν[exp(-|∆µ/kT |) + f/(1 - f ) - 1] (12)
Figure 13. The effect of tailor-made inhibitors on crystal etching. Negative sticking fraction, |S|, surface coverage, θs, and sticking fraction ratio of the inhibited and clean surface, RSF, as a function of undersaturation, |∆µ/kT|, for type-II inhibitors at f ) 0.2. (a) φ/kT ) 0.7; (b) φ/kT ) 1.2; (c) φ/kT ) 2.0.
will be embedded in the surface, and the step coming from behind will be pinned. Therefore, growth will be blocked for this lower undersaturation regime. From this simple model, we expect a dead zone width of |∆µ/kT|* ) -ln[1 - f/(1 - f )]. For f ) 0.2, this corresponds with |∆µ/kT|* ) 0.29, which is not far from the observed value of 0.25 for φ/kT ) 2.0. Inspection of the surfaces etched below this critical undersaturation value indeed shows that almost all of the adsorbed inhibitor molecules are embedded in the surface with four horizontal neighbors and thus are capable of blocking step growth. For the highest |∆µ/kT |, the sticking fraction ratio
which is slower than the rate in absence of additives
Rk( f)0,|∆µ/kT |) ) ν[exp(-|∆µ/kT |) - 1]
(13)
Both stripping rates are displayed schematically as a function of undersaturation in Figure 12b. Growth is expected to occur if |∆µ/kT| < -ln[1 -f /(1 - f )], but then, inhibitor molecules
RSF )
Rk( f,|∆µ/kT|)/P+ k
) Rk( f)0,|∆µ/kT|)/P+ k exp(-|∆µ/kT|) - 1 + f /(1 - f ) (14) exp(-|∆µ/kT|) - 1
goes to (1 - f /(1 - f )), which agrees with our observations.
6388 J. Phys. Chem. C, Vol. 112, No. 16, 2008 It should be realized that our 1D model is oversimplified and only gives a qualitative understanding of the, rather unexpected, behavior during crystal etching in the presence of tailor-made inhibitors. 6. Applicability The number of possible mechanisms of the action of different types of additives and impurities on the growth of different kinds of crystal surfaces is almost endless. In this paper, we concentrated on the “classical” tailor-made additives as introduced by the research group at the Weizmann Institute of Science in the 1980s8-10 and applied this concept to the {001} surface of the simple cubic Kossel crystal. However, the conclusions and the analytical expressions derived in our study also hold for “real” additive molecules, provided that the strengths of the horizontal and downward bonds are similar to those of the growth units in the crystal face and the upward bond is weak or repelling. As only nearest-neighbor interactions and no surface diffusion is assumed, our results are most suited for the solution growth of organic, protein, and other macromolecular crystals as well as for some inorganic crystals. Except for F faces with large bond energies, φ/kT, the growth retarding effect of the tailor-made inhibitors used in this study is relatively mild. This agrees with the pioneering studies by Berkovitch et al.,9 who used relatively large amounts of additives, going from 1 to more than 10% w/w. On the other hand, literature reports a number of cases where only a few or a few tens of parts per million of impurity molecules are sufficient to block crystal growth completely at not too high supersaturation. Examples are the effects of Cr3+ and Fe3+ ions on the growth of {100} KH2PO4 crystals4,39,46 and of Fe3+ impurities on the growth and dissolution of K2SO4 crystals.44,45 In such cases, the strength of the downward bond and maybe also the horizontal bonds of the inhibitor molecules largely exceeds that of the growth units, and the inhibitor gets firmly attached to the growing crystal surface. Then, the inhibitor effect can be orders of magnitude stronger. Modeling this situation requires different simulations, using an extra variable (strength of the downward bond) and an essentially different analytical approach. Finally, our results must be considered with care for crystal growth from the vapor phase, which is generally dominated by surface diffusion. This is not included in our model. 7. Conclusions In this paper, we investigated the impact of “tailor-made” inhibitors on crystal growth by using kinetic Monte Carlo simulations. Tailor-made inhibitor molecules have horizontal and downward bonds that are very similar to those of the growth units in the crystal surface, but the strength of the upward bonds is weaker or even repelling. Adding such molecules leads to a controlled retardation of the growth or dissolution of specific crystal faces, which allows for “engineering” crystal morphologies. As a model system, we used the {001} surface of the Kossel crystal, growing or dissolving in the presence of three different types of inhibitor molecules. The effect of tailor-made inhibitors on the growth of thermally roughened faces is limited. The reduction of the growth rate is not very high, growth kinetics remains linear, and no dead supersaturation zone occurs. A simple model considering a blocking efficiency for each inhibitor molecule adsorbed at the crystal surface is used to describe the dependence of the growth rate on inhibitor concentration, supersaturation, and bond strength. The influence of the inhibitor molecules on the etching
van Enckevort and Los of rough faces is similar to that for growth; the effect on the sticking fraction is mild, and no dead undersaturation zone develops. Compared to the rough faces, the influence of the tailor-made inhibitors is large for the growth of flat faces because now the repelling inhibitor molecules are firmly embedded in the growing crystal surface. Growth kinetics is no longer linear, and a dead supersaturation zone develops, where crystal growth is blocked for supersaturations less than a critical value (∆µ/ kT)*. The width of the dead zone as a function of step free energy and inhibitor concentration in the mother phase agrees with Cabrera and Vermilyea’s theory of step pinning by adsorbed impurities. For strong bottom and side bonds and the most repelling top bonds, crystal growth is accompanied by the formation of vacancies on top of incorporated growth units. The concentration of the grown-in inhibitor molecules increases for increasing supersaturation up to (∆µ/kT)*, after which it decreases again. The maximum grown-in inhibitor fraction at (∆µ/kT)* is proportional with its concentration in the mother phase. Etching of flat faces again reveals nonlinear kinetics and the existence of a dead undersaturation zone, where dissolution is prohibited. This rather unexpected result is qualitatively explained by considering the process of etching as stripping off of surface steps by kink propagation in the presence of inhibitor molecules. References and Notes (1) Rome´ de l’Isle, J. B. L. Cristallographie; Didot jeune, Knapen and Delaguette: Paris, 1783; p 379. (2) Radenovic, N.; Kaminski, D.; van Enckevort, W. J. P.; Graswinckel, S.; Shah, I.; in ’t Veld, M.; Algra, R.; Vlieg, E. J. Chem. Phys. 2006, 124, 164706. (3) Buckley, H. E. Crystal Growth; John Wiley & Sons: New York, 1951. (4) Sangwal, K. AdditiVes and Crystallization Processes, from Fundamentals to Applications; John Wiley & Sons: Chichester, England, 2007. (5) Kubota, N.; Yokota, M.; Mullin, J. W. J. Cryst. Growth 1997, 182, 86-94. (6) Simon, B.; Grassi, A.; Boistelle, R. J. Cryst. Growth 1974, 26, 90-96. (7) Plomp, M.; McPherson, A.; Malkin, A. J. Proteins: Struct., Funct., Genet. 2003, 50 486-495. (8) Weissbuch, I.; Popovitz-Biro, R.; Lahav, M.; Leiserowitz, L. Acta Crystallogr., Sect. B 1995, 51, 115-148. (9) Berkovitch-Yellin, Z.; van Mil, J.; Addadi, L.; Idelson, M.; Lahav, M.; Leiserowitz, L. J. Am. Chem. Soc. 1985, 107, 3111-3122. (10) Weissbuch, I.; Lahav, M.; Leiserowitz, L. Cryst. Growth Des. 2003, 3, 125-150. (11) Davey, R. J.; Black, S. N. J. Cryst. Growth 1986, 79, 765-774. (12) Sarig, S. In Handbook of Crystal Growth; Hurle, D. T. J., Ed.; Elsevier: Amsterdam, The Netherlands, 1993; Volume 2b, Chapter 19, pp 1217-1269. (13) Gilmer, G. H.; Bennema, P. J. Appl. Phys. 1972, 43, 1347-1360. (14) Rak, M.; Izdebski, M.; Brozi, A. Comput. Phys. Comm. 2001, 138, 250-263. (15) Boerrigter, S. X. M.; Josten, G. P. H.; van de Streek, J.; Hollander, F. F. A.; Los, J.; Cuppen, H. M.; Bennema, P.; Meekes, H. J. Phys. Chem. A 2004, 108, 5894-5902. (16) Cuppen, H. M.; Meekes, H.; van Enckevort, W. J. P.; Vlieg, E.; Knops, H. J. F. Phys. ReV. B 2004, 69, 245404/1-245404/6. (17) Gilmer, G. H.; Huang, H.; Diaz de la Rubia, T.; Dalla Torre, J.; Baumann, F. Thin Solid Films 2000, 365, 189-200. (18) van Enckevort, W. J. P.; van der Berg, A. C. J. F. J. Cryst. Growth 1998, 183, 441-455. (19) Yoshioka, Y.; Matsui, T.; Kasuga M.; Irisawa, T. J. Cryst. Growth 1999, 198/199, 71-76. (20) Duffy, D. M.; Rodger, P. M. J. Phys. Chem. B 2002, 106, 1121011217. (21) Gilmer, G. H.; Bennema, P. J. Cryst. Growth 1972, 13/14, 148153. (22) Bennema, P. In Handbook of Crystal Growth; Hurle, D. T. J., Ed.; Elsevier: Amsterdam, The Netherlands, 1993; Vol. 1, Chapter 7, p 483.
“Tailor-Made” Inhibitors in Crystal Growth (23) Leamy, H. J.; Gilmer, G. H.; Jackson, K. A. In Surface Physics of Materials; Blackely, J. M., Ed.; Academic Press: New York, 1975; Vol. 1, Chapter 3, pp 121-188. (24) Hartman, P.; Perdok, W. G. Acta Crystallogr. 1955, 8, 49-52. (25) Hartman, P.; Perdok, W. G. Acta Crystallogr. 1955, 8, 521-524. (26) Hartman, P.; Perdok, W. G. Acta Crystallogr. 1955, 8, 525-529. (27) Bennema P.; van der Eerden, J. P. In Morphology of Crystals; Sunagawa, I., Ed.; Terrapub: Tokyo, 1987; Vol. A, Chapter 1, pp 1-75. (28) Leamy H. J.; Gilmer, G. H. J. Cryst. Growth 1974, 24/25, 499502. (29) Cabrera, N.; Vermilyea, D. In Growth and Perfection of Crystals; Doremus, R., Roberts, B. W., Turnbull, D., Eds.; Wiley: New York, 1958, pp 393-410. (30) Los, J. H.; van Enckevort, W. J. P.; Vlieg, E. J. Phys. Chem. B 2002, 106, 7321-7330. (31) Los, J. H.; van Enckevort, W. J. P.; Meekes, H.; Vlieg, E. J. Phys. Chem. B 2007, 111, 782-791. (32) Beatty, K. M.; Jackson, K. A. J. Cryst. Growth 1997, 174, 28. (33) van Veenendaal, E.; van Hoof, P. J. C. M.; van Suchtelen, J.; van Enckevort, W. J. P.; Bennema, P. Surf. Sci. 1998, 417, 121-138. (34) Cuppen, H. M.; Meekes, H.; van Veenendaal, E.; van Enckevort, W. J. P.; Bennema, P.; Reedijk, M. F.; Arsic, J.; Vlieg, E. Surf. Sci. 2002, 506, 183-195.
J. Phys. Chem. C, Vol. 112, No. 16, 2008 6389 (35) Kandel, D.; Weeks, J. D. Phys. ReV. B 1994, 49, 5554-5564. (36) Kandel, D.; Weeks, J. D. Phys. ReV. B 1995, 52, 2154-2164. (37) van der Eerden, J. P.; Mu¨ller-Krumbhaar, H. Phys. ReV. Lett. 1986, 57, 2431-2433. (38) de Theije, F. K.; Schermer, J. J.; van Enckevort, W. J. P. Diamond Relat. Mater. 2000, 9, 1439-1449. (39) Bredikhin, V. I.; Ershov, V. I.; Korolikhin, V. V.; Lizyakina, V. N.; Potapenko, S. Yu.; Khlyunev, N. V. J. Cryst. Growth 1987, 84, 509514. (40) Chernov, A. A.; Rashkovich, L. N. J. Cryst. Growth 1987, 84, 389393. (41) Derksen, A. J.; van Enckevort, W. J. P.; Couto, M. S. J. Phys. D: Appl. Phys. 1994, 27, 2580-2591. (42) van Enckevort, W. J. P. In Science and Technology of Crystal Growth; van der Eerden, J. P., Bruinsma, O. S. L., Eds.; Kluwer: Dordrecht, The Netherlands, 1995; pp 355-366. (43) Potapenko, Yu, S. J. Cryst. Growth 1993, 133, 141-146. (44) Kubota, N.; Fujisawa, Y.; Yokota, M.; Mullin, J. W. J. Cryst. Growth 1999, 197, 388-392. (45) Mullin, J. W. Crystallization; Butterworth and Heineman: Oxford, U.K., 2001; p 262. (46) Sangwal, K. J. Cryst. Growth 1999, 203, 197-212.
