Does Impurity-Induced Step-Bunching Invalidate Key Assumptions of

Does Impurity-Induced Step-Bunching Invalidate Key Assumptions of the Cabrera-Vermilyea Model? Rile I. Ristic,*,† James J. DeYoreo,*,‡ and Chun M. Chew† Chemical Engineering Department, UniVersity of Sheffield, Mappin Street, Sheffield S1 3JD, UK, and Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720

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ReceiVed October 23, 2007; ReVised Manuscript ReceiVed January 29, 2008

ABSTRACT: We show that the growth recovery mechanism of the {110} faces on crystals of the pharmaceutical paracetamol in the presence of its intrinsic impurity acetanilide occurs in the same way as in the growth of inorganic KH2PO4 (KDP) crystals with Fe(III) and Al(III) impurities, by initial movement of macrosteps while elementary steps remain pinned. This suggests that the mechanism of recovery by activation of elementary step motion assumed by Cabrera and Vermilyea (C-V) is not applicable to a diverse set of common and technologically important crystal systems. Recognizing that impurity-driven macrostep formation depends on an imbalance in the concentration ahead and behind the step, we propose a general condition that must be met for a crystal-impurity system to behave according to the C-V predictions. A well-known classical theory proposed by Cabrera and Vermilyea (C-V)1 has been widely used to analyze growth inhibition by pinning of elementary steps due to their interaction with adsorbates.2–4 According to this theory, the pinning effect on elementary steps spreading across the growing crystal face leads to a “dead zone” (Figure 1) in which the steps are stopped from advancing if the supersaturation σ is below a critical supersaturation σ*, since the critical radius of step curvature (rc ) γΩ/kTσ) is greater than the impurity spacing (where γ is the step-edge free energy, Ω is the effective molecular volume, k is Boltzman’s constant, and T is temperature of crystallization). If the supersaturation increases above the critical value, σ*, the C-V model assumes that the elementary steps initially pinned by the impurity molecules start to break through the “impurity fence”. As a result, the train of elementary steps on a given surface becomes mobile again. However, it was shown through in-situ atomic force microscopy (AFM) that growth on the {100} face of inorganic KH2PO4 (KDP) in the presence of Fe(III),5,3 Cr(III),3 and Al(III)3 recovers from impurity poisoning by propagation of macrosteps instead of elementary steps as supposed in the C-V model. Here we report that this discrepancy between the C-V model and observations of growth recovery is also exhibited by a very different crystal system: organic paracetamol (C8H9NO2) crystals. On the basis of these results and observations of other crystals, we propose a condition on the relative time scales for impurity adsorption and terrace lifetime for any crystal system that must be met before the standard C-V model can be applied. Paracetamol powder of purity >99%, purchased from Merck, was recrystallized and subjected to high performance liquid chromatography to quantify the presence of unintentionally added (intrinsic) impurities. This method revealed the presence of 0.0055% w/w of acetanilide (C8H9NO), an intrinsic impurity incorporated in the system as a result of the industrial synthesis. Other impurities were not detectable. Tangential velocity, V, of growth steps spreading across the {110} paracetamol faces was measured as a function of supersaturation σ ) ln(C/Ce), where C and Ce are the actual and equilibrium concentrations, using in-situ laser interferometry,6 at constant temperature T ) 296 K, and a flow rate of 30 cm/s. This dependence (Figure 1) shows that there is indeed a pronounced “dead zone” for 0 < σ < 10.5% in which the growth was not detectable by this technique. Although this is one of the most powerful conventional * E-mail: [email protected] (R.I.R.); [email protected] (J.J.D.). † University of Sheffield. ‡ Lawrence Berkeley National Laboratory.

Figure 1. Tangential velocity V ) f(σ) for the {110} faces. σ* ≈ 10.5% is the critical supersaturation at which growth resurrection is observable by this technique (after 8). Inset: Schematic showing the region below σ* where the step speed is small but nonzero.

tools7 for studying the kinetics of a single crystal face, its resolution is too low to reveal detailed characteristics of elementary steps such as height, bunching, and in particular, extremely slow growth (“zero”) kinetics. Consequently, in-situ AFM imaging was performed over this range of supersaturation. This technique enabled us to probe any crystal growth activity in the dead zone seen by laser interferometry from nanometer to micrometer length scales and check the validity of the postulated C-V scenario for growth resurrection. The crystallization temperature was kept constant, T ) 296 K for all σ. Other experimental details are described elsewhere.8 Figure 2a-c shows three images of the same area (10 × 10 µm2) of the {110} face for three different values of σ: (a) 3%, (b) 8%, and (c) 12%. Under these conditions, the step generation at hillocks H1 and H2, as well as the movement of elementary steps generated by them, was “frozen”. The identical surface topographs, with markedly roughened step morphology, confirm that acetanalide pins the steps and creates a dead zone for elementary steps over this range of σ. This dead zone seen by AFM might initially seem a satisfactory answer to the “zero” growth rate of the {110} faces observed in the interferometric dead zone (Figure 1). However, Figure 2c shows that the elementary steps are not only immobile in the interferometric dead-zone, but also slightly above it, at σ ) 12%, a supersaturation at which growth recovery of the {110} faces was previously observable by the interferometry (Figure 1). But to go above this value of σ and try to restart growth of elementary steps was a challenging task for three reasons. The first

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Figure 2. A series of AFM deflection images collected in contact mode illustrating the effect of elementary step pinning and the existence of a dead-zone on the {110} faces of paracetamol containing 0.0055% w/w of acetanilide as an intrinsic impurity. The steps remain nearly immobile even with increasing supersaturation from (a) 3%, (b) 8% to (c) 12%. Elementary step height ≈ 7.5 Å.

two are related to principal weaknesesses of AFM. One is concerned with the relatively small area of study, less than 20 × 20 µm, that is required to observe elementary steps on inorganic crystals.9 This implies a lack of conclusive evidence that events over such a small area reflect the growth behavior of the surface as a whole. For example, bunching and macrostep movement may already take place on the rest of the surface. Second, the time needed for capturing an AFM image is on the order of a minute, whereas velocities of growth steps on paracetamol are frequently on the order of 1 µm · s-1, which means they cross a 10 µm image in only 10 s. The third reason lies in the time lapse of 5–10 min needed between

Communications successive changes of supersaturation before a new sequence of AFM scans could be taken. As a result, our attempts at resurrecting elementary steps by increasing supersaturation above 12%, appreciably above the dead zone measured interferometrically, led to rapid coverage of the area by macrosteps before the target suppersaturation was established. Nonetheless, in spite of these difficulties, the sudden movement of macrosteps made it obvious that growth recovery of the {110} faces did not occur by the movement of elementary steps. To further explore this phenomenon, we examined the simultaneous behavior of both elementary and macrosteps at σ ) 3%, an appreciably smaller value than the critical one (σ* ≈ 10.5%) seen interferometrically. In the downward AFM scan of Figure 3a, the surface is dominated by elementary steps that are immobile or nearly so due to pinning by the intrinsic impurities. Near the completion of the first scan, a train of macrosteps enters at the lower left corner, marking the start of growth ressurection by propagation of macrosteps, instead of the elementary steps. In subsequent scans (Figure 3b,c), the train of macrosteps advances further and overruns the elementary steps, while the elementary steps remain almost immobile. This result forces us to modify Figure 1 to include a region of low but nonzero growth between σ* and some value σd, which is less than 3%, as illustrated by the inset to Figure 1. This same behavior has been documented for growth of KDP in the presence of both Fe(III) and Cr(III) where extensive measurements by both interferometry2 and AFM3,5 have demonstrated the failure of the C-V model to predict both the dependence of step speed on supersaturation and the variation in dead zone width with impurity conccentration. It follows from all of these studies, that growth resurrection can occur by activation of macrosteps, rather than elementary steps as postulated in the C-V model. Given the rather stark differences between KDP, a simple ionic salt, and paracetamol, an organic compound with extensive hydrogen bonding, these results suggest that the C-V postulate may be in disagreement with the actual behavior of a wide variety of crystallizing systems. In contrast, recent results on growth of calcium oxalate monohydrate (COM) in the presence of citrate, a well known inhibitor of COM growth, are in excellent agreement with the predictions of C-V-type models.4 This is true not only for the dependence of step speed on supersaturation, but even for the variation in dead zone width on citrate concentration. So one must ask the question, “What is different between these systems that leads to such stark differences in step dynamics?”. The most obvious difference between these systems is the solubility, which is 100-1000 times larger for both paracetamol and KDP than for COM. Step speeds in these systems all follow the typical linear rate law, V ) Ωβ(C-Ce) where β is the kinetic coefficient. In all of them, the value of β is nearly the same, about 10-1 cm · s-1. Over the range of supersaturation used in the studies on KDP and paracetamol (σ < ∼20%), σ ) ln(C/Ce) is well approximated by C/Ce -1, or C -Ce = σCe so V = βσCe. Thus, at equivalent values of σ, step speed scales with solubility Ce. Indeed, in AFM studies on COM, typical step speeds are measured in tens of nm · s-1, while in paracetamol and KDP, they are measured in µm · s-1 and tens of µm · s-1, respectively. This difference can have serious implications for the tendency towards step bunching. In the two classical models of impurityinduced step bunching,10,11 both of which start with the C-V model as the basis for predicting the speed of any individual step, an essential element is that surface impurity concentration is greater on larger terraces than on smaller ones. Thus, if a step slows down so that the leading terrace grows and the trailing terrace shrinks, that step sees a greater than average impurity level while the trailing step sees a lower than average level, reinforcing the imbalance in step speed and causing the steps to bunch. Van der Eerden and Muller-Krumhaar12 later recognized that the surface impurity concentration should be a function of terrace exposure time rather than terrace width. By assuming that step speed followed a C-V dependence on surface impurity concentration and

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Figure 4. Schematic showing Θe(1-e-t/τ) dependence of surface impurity coverage Θ on time t, where the characteristic adsorption time is τ and the equilibrium coverage is Θe.

Figure 3. A series of AFM deflection images collected in contact mode illustrating regeneration of the {110} face of paracetamol at σ ) 3% by the movement of macrosteps following poisoning near equilibrium by 0.0055% w/w of acetanilide, an intrinsic impurity. Contrary to the C-V assumption, elementary steps remain nearly immobile while growth occurs on the macrosteps. Elementary step height ≈ 7.5 Å.

that this concentration had a saturable time dependence, increasing as 1 - e-t/τ where τ is a characteristic adsorption time, they obtained a pattern of progressive and regular step bunching. Thomas et al.5,13 found that, for KDP (100) faces, progressive doubling of step bunches was observed and well described by this model with a τ of about 0.1 s, although the identity of the impurity was unknown. Their analysis of KDP (100) growth recovery from Fe3+ poisoning gave a τ of 1–10 s.2 The predictions of bunching from classical step growth models arise from differences in impurity coverage on leading and trailing

terraces. To establish such differences requires that the impurity coverage depends strongly on the width of the terrace, w. However, under dynamic growth conditions, this dependency only arises if the terrace lifetime, w/V, where V is the step velocity, is comparable to or smaller than the characteristic time associated with terrace adsorption, τ. This idea is illustrated schematically in Figure 4, where dimensionless impurity surface coverage Θ/Θe, where Θ and Θe are the actual and equilibrium fractional impurity coverage, is plotted as a function of dimensionless time, t/τ. Surface coverage varies strongly when t/τ e 1 but changes only very slowly at longer times. Hence, a strong dependence of surface coverage on terrace width will occur in systems where w/V e τ or when the ratio of time scales of terrace lifetime to surface adsorption is order unity or less, w/(Vτ) e 1. The value of this dimensionless group w/(Vτ) depends only upon physical properties of the growth system and thus measures the inherent propensity for any given system to be susceptible to step bunching instabilities driven by the C-V mechanism. In the case of KDP (100), terrace widths are on the order of 0.5 microns, step speeds are on the order of 5 microns/s, and τ is 0.1 s or greater, giving w/(Vτ) ≈ 1 or less, which satisfies the criterion needed for step bunching. For paracetamol, terrace widths are about a micron and step speeds are on the order of 1 micron/s. For a characteristic terrace absorption time comparable to that of KDP, paracetamol exhibits w/(Vτ) ≈ 1, and the bunching criterion is still satisfied. In contrast, on COM, terrace exposure times are more like w/V ) 10–50 s. For a value of τ comparable to that estimated from the KDP studies, w/(Vτ ) ≈ 100–500 and step bunching would not be expected to occur. From this analysis of the models, we can postulate an inherent source for the difference in behavior between a low-solubility crystal such as COM and highly soluble crystals such as KDP and paracetamol. The 100–1000× difference in step speeds places their respective terrace lifetimes on opposite sides of τ. In further support of this hypothesis, we note that despite extensive AFM studies on calcite in the presence of a wide variety of inorganic and organic impurities,14–16 no step bunching has ever been observed. Coincidentally, the solubility of calcite is about 10× smaller than even that of COM. Certainly this analysis is only approximate as different impurities will differ in their characteristic adsorption times, and one should not be surprised to find a more slowly adsorbing impurity that induces step bunching even on a crystal like COM. Moreover, other factors inherent to a crystal growth system will influence the onset of step bunching. Recently, numerical simulations that can take into account transient phenomena, changing terrace widths, and other effects that are not easily represented in crude analytical models of step growth have been used to analyze step bunching driven by solution flow as well as step stabilization by step-induced convection.17 Such analyses even indicate that the time scale for solute adsorption to the terrace may be relevant for step bunching, even without the presence of impurities (J. J. Derby, personal

1122 Crystal Growth & Design, Vol. 8, No. 4, 2008 communication). Analyses like these are needed to develop a clearer picture of the role of competing time scales in impurity-driven bunching as well as its consequence for face stability, but the general trend with increasing solubility will still be towards step bunching and a failure of the C-V model to predict recovery from impurity poisoning. In conclusion, AFM studies show that the C–V model fails to account for the observed mode of recovery from impurity poisoning for diverse materials. Apart from the need for reviewing the generally accepted key assumption of the C-V model, these studies also highlight other important issues that need to be resolved. For example, the initial formation of macrosteps even at very low supersaturations and often near crystal edges8 and gross structural defects, the physicochemical properties and dynamics of macrosteps, and a mechanism of overruning of elementary steps by macrosteps are all poorly understood. Perhaps most importantly, virtually no direct measurements of impurity adsorption times are available for any crystals, yet that time plays a crucial role in determing the tendency towards step bunching. Understanding of these issues will likely be a pivotal prerequsite for controlling structural and compositional uniformity during crystallization of a wide range of materials such as nonlinear optical crystals, pharmaceuticals, and proteins.

Acknowledgment. We gratefully acknowledge the valuable suggestions of Prof. Jeffrey Derby as well as those of an anonymous reviewer. We acknowledge the support from the National Institutes of Health (DK61673) and the U.S. Department of Energy (DOE), Office of Basic Energy Science (BES), Division of Chemical, Biological and Geological Sciences, This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC5207NA27344.

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References (1) Cabrerra, N.; Vermilea, D. A. In Growth and Perfection of Crystals,Proceedings of the International Conference, Cooperstown, NY, August 1958; Wiley: New York, 1958; pp 393410. (2) Rashkovich, L. N.; Kronsky, N. V. J. Cryst. Growth 1997, 182, 434. (3) Thomas, T. N.; Land, T. A.; DeYoreo, J. J.; Casey, W. H. Langmuir 2004, 20, 7643. (4) Weaver, M.; Qiu, S. R.; Hoyer, J. H.; Nancollas, G. H.; Casey, W. H.; De Yoreo, J. J. J. Cryst. Growth 2007, 306, 135. (5) Land, T. A.; Martin, T. L; Potapenko, S.; Palmore, G. T.; DeYoreo, J. J. Nature 1999, 399, 442. (6) Rashkovich, L. N.; Shustin, O. A. SoV. Phys. Usp. 1987, 30, 280. (7) Booth, N. A.; Stanojev, B.; Chernov, A. A.; Vekilov, P. G. ReV. Sci. Instrum. 2002, 73, 3540. (8) Ristic, R. I.; Finnie, S.; Sheen, D. B.; Sherwood, J. N. J. Phys. Chem., B 2001, 105, 9057. (9) DeYoreo, J. J.; Orme, C. A.; Land, T. A. In AdVances in Crystal Growth Research; Sato, K.; Furukawa, Y.; Nakajima, K., Eds.; Elsevier Publishing Company: Tokyo, 2001; p 361. (10) Cabrerra, N.; Vermilea, D. A. In Growth and Perfection of Crystals, Proceedings of the International Conference, Cooperstown, NY, August 1958; Wiley: New York, 1958; p 411. (11) Chernov, A. A. SoV. Phys. Usp. 1961, 4, 116. (12) Van der Eerden, J. P.; Muller-Krumbhaar, H. Electrochim. Acta 1986, 31, 1007. (13) Thomas, T. N.; Land, T. A.; Casey, W. H.; DeYoreo, J. J. Phys. ReV. Lett. 2004, 92, 216103-1. (14) De Yoreo, J. J.; Dove, P. M. Science 2004, 306, 1301. (15) De Yoreo, J. J.; Dove, P. M.; Wierzbicki, A. CrystEngComm 2007, 9, 1144. (16) Wasylenki, L. E.; Dove, P. M.; Wilson, D. S.; De Yoreo, J. J. Geochim. Cosmochim. Acta 2005, 69, 3017. (17) Kwon, Y. I.; Dai, B.; Derby, J. J. Prog. Cryst. Growth Charact. Mater. 2007, 53, 167.

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