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In Situ Atomic Force Microscopy Investigation of the {100} Face of KH2PO4 in the Presence of Fe(III), Al(III), and Cr(III) Tiffany N. Thomas,*,†,‡ Terry A. Land,‡ Jim J. DeYoreo,‡ and William H. Casey†,§ Department of Land, Air, and Water Resources, Chemistry Graduate Group, University of California at Davis, Davis, California 95616, Chemistry and Materials Science Division, Lawrence Livermore National Laboratory, Livermore, California 94550, and Department of Geology, University of California at Davis, Davis, California 95616 Received February 20, 2004. In Final Form: May 17, 2004 Here we report the effects of dissolved metal complexes of Fe(III), Al(III), and Cr(III) on the step velocities of the {100} face of KH2PO4 (KDP) as observed with atomic force microscopy. The dependence of step velocity on supersaturation (σ) exhibits a dead zone that scales with adsorbate concentration. The observed dependence varies with the metal complex. From these data, we derive values for the characteristic adsorption time (τ) for the Al(III) and Cr(III) step-pinning adsorbates as being on the order of several hundred microseconds as compared to 10-100 s for the corresponding Fe(III) step-pinning adsorbates. The values of τ are strikingly different than rates of ligand exchange but are associated with the adsorbate-induced morphology of the surface, including elementary steps that bunch into macrosteps and supersteps. The stoichiometry of the adsorbate species is assumed to be M(HxPO4)x, where M ) Fe(III), Al(III), or Cr(III). KDP crystals grown in the presence of the dissolved metals were analyzed using laser ablation inductively coupled plasma mass spectroscopy. The data revealed sectoral zoning on the {100} face, with the concentrations of the incorporated adsorbates in the sector with slower moving elementary steps being 1.7-2.0 times greater than those measured on the sector with fast moving elementary steps.
Introduction Detailed knowledge of the growth of crystals in the presence of impurities has far-reaching applications in fields as diverse as pharmaceutical production, biomineralization, optics production, and geochemistry. Impurity adsorbates can induce drastic changes in the growth behavior of crystal surfaces1-4 including reduction of growth rate,1-3 changes in the crystal habit,2 and changes in the step morphology.2,3 Moreover, impurity incorporation often has a severe impact on physical properties5 of crystals. To better understand the complexity of the interactions between adsorbates and surfaces, it is necessary to develop a model for interactions in wellcharacterized systems where some of the key variables can be isolated. Because of the extensive history of work on KH2PO4 (KDP) crystal growth,6 KDP provides an excellent experimental model for the investigation of * To whom correspondence may be addressed. Tiffany N. Thomas: One Shields Ave., 1110 PES bldg., Davis, CA 95616; phone (530) 752-2107; e-mail
[email protected]. Terry A. Land: Lawrence Livermore National Laboratory, L-491, 7000 East Ave., Livermore, CA, 94550; phone (925) 423-5836; fax (925) 422-5781; e-mail
[email protected]. † Department of Land, Air, and Water Resources, Chemistry Graduate Group, University of California at Davis. ‡ Chemistry and Materials Science Division, Lawrence Livermore National Laboratory. § Department of Geology, University of California at Davis. (1) Rashkovich, L. N.; Kronsky, N. V. J. Cryst. Growth 1997, 182, 434-441. (2) Land, T. A.; Martin, T. L.; Potapenko, S.; Palmore, G. T.; DeYoreo, J. J. Nature 1999, 399, 442. (3) Thomas, T. N.; Land, T. A.; Casey, W. H.; DeYoreo, J. J. Phys. Rev. Lett., in press. (4) Verdaguer, S. V.; Clemente, R. R. J. Cryst. Growth 1986, 79, 198. (5) DeYoreo, J. J.; Burnham, A. K.; Whitman, P. K. Int. Mater. Rev. 2002, 47, 113 and references therein. (6) Rashkovich, L. N. KDP Family of Crystals; Adam-Hilger: New York, 1991 and references therein.
adsorbate-surface interactions. More specifically, salts of trivalent metals are known to affect the rates and morphologies of growth of the {100} face of KDP.1-4 Iron, Al(III), and Cr(III) all form stable coordination complexes in aqueous solutions and are therefore appropriate choices with which to further study this phenomenon. Previous research on the effects of these impurities on KDP growth using interferometry1 revealed the presence of a “dead zone”sa region of positive supersaturation where no growth occurssin the dependence of step velocity on supersaturation, presumably caused by adsorption of aqueous complexes of Al(III), Fe(III), and Cr(III).1,2 Many aspects of this dead zone were found to be consistent with the classic Cabrera-Vermilyea (C-V) model of step pinning.7 In this model, the normal flow of elementary steps across the growing crystal face is disrupted by a field of adsorbate “stoppers”. These stoppers prevent the step from advancing when the distance between the adsorbate stoppers, Li, is approximately less than the critical radius of step curvature, rc, given by
rc ) Rω/kTσ
(1)
where R is the step-edge free energy per unit step height, ω is the molecular volume, k is Boltzmann’s constant, T is temperature, and σ is supersaturation defined as
σ ) ln(C/Ce)
(2)
where C and Ce are the actual and equilibrium molarity of solute, respectively. A detailed analysis using percolation theory8 shows that the criterion for step motion through the field of stoppers is Li > Arc, where A is a number of order 1 that depends (7) Cabrera, N.; Vermilyea, D. A. Growth and Perfection of Crystals; Chapman and Hall: London, 1958; p 393. (8) Potapenko, S. Y. J. Cryst. Growth 1993, 133, 147.
10.1021/la049546f CCC: $27.50 © 2004 American Chemical Society Published on Web 07/27/2004
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on the geometry of the lattice and is of order 2 for a square lattice,8 a reasonable approximation for the {100} face, leading to an expression for the critical supersaturation, σ*, in terms of Li, above which the steps begin to move and which is given by
σ* ) 2ωR/kTLi
(3)
Recognizing that Li ) (1/ni)0.5, where ni is the surface concentration of adsorbates, we have that
σ* ∼ ni0.5
(4)
Moreover, for the low impurity levels that are required to stop growth of KDP (∼1 × 10-5 to 1 × 10-4 mol of adsorbate/ mol of KDP), one can reasonably assume that ni ∼ Ci where Ci is the bulk solution concentration of the adsorbing species. According to the C-V model,7 at σ*, growth of the face recovers by the motion of elementary steps that advance with a speed, V, equal to
(
V ) V0 1 -
σ* σ
1/2
)
(5)
where V0 is the speed in the pure system (or in the impure system for σ . σ*). The interferometric studies mentioned earlier,1 which provided much of the existing quantitative knowledge about the dead zone, revealed features of the dead zone that were not predicted by the C-V model7 and that could not be explained purely on the basis of that data. Specifically, on surfaces doped with Fe(III), the dependence of V on σ deviates significantly from that of eq 5. Moreover, in the case of Al(III), the dependence of σ* on Ci does not obey the square root law of eq 4. These discrepancies call into question the degree to which the physics and chemistry of impurity pinning is understood. Because atomic force microscopy (AFM) studies provide both measurements of V vs σ and detailed morphological information, they can help to explain the observed anomalous behavior. Indeed, our previous AFM investigations show that elementary steps are not responsible for recovery of growth in these systems.2,3 Instead, step bunches of various heights begin to move first at a supersaturation σd < σ*, the supersaturation at which the elementary steps regain motion. More specifically, these previous AFM studies showed that the growth of the {100} face of KDP in the presence of aqueous solutes of Fe(III), Al(III), and Cr(III) occurs via three distinct step classes: elementary steps, which are singular monolayers that typically emanate from a dislocation; macrosteps, which are bunches of 2-50 elementary steps; supersteps, which are bunches of 50 to several hundreds of elementary steps.3,9 Data from these AFM experiments were used to derive fundamental kinetic parameters for KDP growth, as well as to evaluate theoretical models for step bunching due to the adsorption of trace impurities.9 Investigation of KDP growth using AFM in the presence of these impurities revealed a correlation between the morphological changes and four characteristic regions of the velocity vs supersaturation (σ) curve: (i) the dead zone; (ii) at the supersaturation when steps first begin to move (σd); (iii) the region of rapid increase of velocity with supersaturation; (iv) the region where step velocities again equal those of an undoped system. (9) Thomas, T. N.; Land, T. A.; Martin, T. L.; Casey, W. H.; DeYoreo, J. J. J Cryst. Growth 2004, 260, 566.
Land et al.2 presented a model for growth in the FeIIIKDP system that took into account the finite rate of impurity adsorption. When the time scale for achieving the equilibrium coverage of an adsorbate is long compared to the terrace lifetime, morphological changes arise that depend on the details of both the adsorption kinetics and the stereochemical requirements for impurity pinning of both elementary steps and step bunches. Consequently, the specific behavior of the surface in response to each adsorbate, as well as the shape of the velocity curve during the recovery from the dead zone, is unique for each adsorbate. The model of Land et al.2 predicts that the shape of the velocity curve is determined by the characteristic time, τ, for adsorbates to reach the equilibrium concentration on the terraces, which is given by
τ)
( )( )
1 LE σ* 2 VM σd
2
(6)
where LE is the terrace width between two elementary steps, VM is the velocity of the macrosteps, and σd is the supersaturation at which macrosteps are mobile, but elementary steps remain pinned. According to eq 6, by measuring the values of LE, VM, σ*, and σd for each of the metals studied, we can calculate the value of τ for each metal. In doing so, we obtain quantitative information about the impurity adsorption kinetics that had not otherwise been measured. A complete derivation, including all assumptions and approximations, of eq 6 would be lengthy to present here but is included in ref 10. Fe(III) and Al(III) are labile cations in solution, meaning that ligand exchanges around the hydrated monomer metal ion, (M(H2O)6)3+, occur rapidly, e.g., more than once per second.11 While Cr(H2O)6)3+ is not labile in strongly acidic solutions, at pH ) 4.2, the native pH of the KDP solution, this hydrated metal interconverts rapidly with Cr(OH)(H2O)52+, which is labile. In the KDP solution, speciation calculations reveal that all three metal ions exist as various phosphate species.4,12,13 Undoubtedly, the rates of ligand exchange of bound waters in these phosphato complexes are more rapid than the corresponding hydrolysis complexes. Here we look in detail at the effects of the adsorbate complexes of Fe(III), Al(III), and Cr(III) on the growth of the {100} face of KDP using in situ AFM. We correlate step morphology with σ, derive τ for each of these metals, and relate this value to the behavior of the growth surface as it is resurrected from the dead zone. Experimental Section The details of the experimental procedure are discussed in depth elsewhere2,3,9 and will be reviewed briefly here. Filtered KDP solutions of known concentration were doped with either Fe(NO3)3, Al(NO3)3, or CrK(SO4)2 to reach a desired concentration. Because Al(III) and Fe(III) are labile, the counterion in solution (NO3- or Cl- in previous studies) should not affect the speciation and pinning activity. We performed preliminary experiments on solutions doped with Cl- and NO3- and confirmed that the counterions do not impact growth. Therefore the choice of counterion is arbitrary. The equilibrium value, Ce, was calculated based on the solubility curves of Frost et al.,14 measured on salts (10) Thomas, T. N. Archerite (KH2PO4) Growth Morphology and Kinetics in the Presence of Surface Adsorbates. Ph.D. Dissertation, University of California at Davis, 2004. (11) Richens, D. T. The Chemistry of Aqua Ions; Wiley: New York, 1997. (12) Holroyd, A.; Salmon, J. E. J. Chem. Soc. 1956, 269. (13) Jameson, R. F.; Salmon, J. E. J. Chem. Soc. 1954, 4013. (14) Frost, T.; Britten, J. A.; DeYoreo, J. J. Unpublished.
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from the same source used in this study, and is given by
Ce ) 1.049 + (2.30 × 10-2)T + (9.72 × 10-5)T2 mol/L
(7)
where T is in units of degrees centigrade. Typical dopant concentrations were 7.5 × 10-6 mol and 15 × 10-6 mol of dopant/mol of KDP, which equate to approximately 1.5 and 3.0 ppm Al(III) by weight, or approximately 9.75 × 10-6 and 19.5 × 10-6 M. These units of concentration (mol of dopant/ mol of KDP) were chosen to simplify the comparison of the three dopants since the units of parts per million can lead to confusion when discussing several dopants of differing atomic weights. Dopant concentrations were confirmed via elemental analysis of solution samples collected after the experiments so that any impurities added into the solution by the apparatus (solution vessels, tubing, fluid cell, etc.) were accounted. These solutions were allowed to equilibrate at 80 °C for several hours before experimentation so that the solute speciation did not change during an experiment. Upon addition of the concentrated Al(NO3)3 solution to the KDP solution, a solid milky white floc forms immediately. However, the solid disappears after several hours of equilibration at 80 °C, as was verified by the absence of a Tyndall effect (no scattering of transmitted laser light). At this pH, any aluminum phosphate solid that forms due to a local supersaturation created when the concentrated solution is initially added quickly redissolves into solution. The details of the AFM technique are described by Thomas et al.9 and references therein. The term “in situ” refers to investigations performed during growth on a seed crystal in the Digital Instruments AFM fluid cell while solution was constantly flowed (flow rate of 6 mL/min) at a controlled temperature. The term “ex situ” refers to observations of crystals grown from solution in small (100-500 mL) Teflon or polymethylpentene vessels at a constant temperature for a specific duration, and preserved by pulling from solution through a stream of hexane.9 For in situ AFM measurements, crystals (2 mm × 2 mm × 1 mm) were glued to glass cover slips for use as seeds. A Digital Nanoscope III force microscope was used in contact mode with standard Si3N4 cantilevers having a nominal force constant of 0.1 N‚m-1. The KDP solution was heated throughout the experiment to prevent spontaneous crystallization and continuously stirred with a Teflon-coated magnetic stir bar to ensure a homogeneous solution. Only Tygon tubing was used to flow the solution from the reservoir to the fluid cell and out from the fluid cell to the collection vessel (the solution was not recirculated). The solution only briefly contacted the glass cover slip, which is not a source of impurities. The temperature of the fluid cell was controlled to (0.1 °C using a Peltier heater/cooler connected to a commercial temperature controller.15 A coated type-T thermocouple was inserted through the tubing just inside the outlet port of the fluid cell in order to determine the temperature of the solution in the cell. Sequential up-and-down scans were collected, and the velocities were calculated by methods discussed in Land et al.16
Results Solution Speciation. To more thoroughly understand the interactions of adsorbate complexes of trivalent metals with the step edge, it is important to understand what complexes might be present and at what concentrations. Verdaguer and Clemente,4 using formation constants for each protonation state of the metal-phosphate complexes, calculated that the dominant species of these three metals in a KDP solution would be a monoprotonated monophosphate species (i.e., M(HPO4)+). A range of other stoichiometries of Al-phosphate complexes may also be present, including Al(H2PO4)2+, Al(H2PO4)2+, (15) DeYoreo, J. J.; Orme, C. A.; Land, T. A. In Advances in crystal growth research; Sato, K., Nakajima, K., Furukawa, Y., Eds.; Elsevier Science: Amsterdam, 2001. (16) Land, T. A.; DeYoreo, J. J. J. Cryst. Growth 2000, 208, 623-637.
Figure 1. Schematic of dependence of V on σ for Al(III) and Fe(III) the dead zone (region a). For Fe(III)-doped surfaces, the velocity curve has a roughly linear region of slow growth that is not predicted by the C-V model (region b), followed by a rapid rise in velocity (region c) after which the velocity rejoins that of the undoped system (region d). The shapes of the velocity curves for surfaces grown in the presence of Al(III) have a shape that is at least qualitatively similar to that predicted by the C-V model but in fact, as we will show later, do not quantitatively fit the C-V prediction. The curves for both Al(III) and Cr(III) exhibit a dead zone at low supersaturations (region a), a critical supersaturation, σ* where growth suddenly begins (point e), and a rapid rise in velocity (region f) after which the velocity curve rejoins that of the clean curve (region d). Cr(III) curves are the same shape as the Al(III) curves. The finely dotted line is a typical growth curve from an undoped solution.
Al(HPO4)2-, Al(HPO4)32-, Al(PO4), Al2(PO4)3+, Al(PO4)23-, Al(PO4)23-, Al2(PO4)2, Al2(OH)2(PO4)+, Al2(OH)3(PO4), and Al3H-q(H3PO4)r9-q where (q,r) can be (6,1), (5,2), (6,3), (8,3), (6,4), and (8,5).12,13,17,18 However, in these conditions the dominant Al species are [Al(PO4)] > [Al(HPO4)+] > [Al(H2PO4)2+] > [Al2(PO4)(OH)2], with all other species negligibly present.19,20 Speciation calculations21 of solutions containing Fe(III) and Cr(III) also reveal that the dominant species in the KDP solution are [Fe(HPO4)2-] . [FeHPO4+] and [CrHPO4+] . [CrH2PO42+], respectively. It should be noted that only Al(III) is present in small concentrations as a dinuclear complex. Although M(HxPO4)x is the dominant species, it is not necessarily the complex responsible for step pinning. The surface area of a typical seed crystal in our experiments is roughly 10-6 m2. Taking σ* ) 0.05 and using eq 3, we calculate that Li is on the order of 200 Å. If we assume that this distance represents the average distance between pinning sites, it would require only about 10-14 mol of adsorbate to completely pin the surface (calculated by dividing the total surface area by Li and dividing by Avogadro’s number). Estimating the total volume of the fluid cell to be 0.001 L, even if we require the velocity to go to zero upon the addition of one fluid cell of growth solution containing the adsorbate complex, for a strongly adsorbing species the adsorbate concentration in the growth solution would only have to be a maximum 10-11 M. The actual concentration of the added impurity is 106 times greater. Therefore, even if we take into account the (17) Wilson, M. A.; Collin, P. J. Anal. Chem. 1989, 61, 1253. (18) Ciavatta, L.; Iuliano, M. Ann. Chim. 1996, 86, 1 and references therein. (19) Dayde, S.; Filella, M.; Berthon, G. J. Inorg. Biochem. 1990, 38, 241. (20) Smith, R. M.; Martell, A. E. NIST Critically Selected Stability Constants of Metal Complexes Database Version 1, 1993, U.S. Department of Commerce, Technology Admin., NIST, Gaithersburg, MD 20899. (21) WINSGW, based on SOLGASWATER: Erickson, G. Anal. Chim. Acta 1979, 112, 375.
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Figure 2. Dependence of step speed on σ for the undoped system (a), for solutions containing Fe(III) (b), for solutions containing Al(III) (c), and for solutions containing Cr(III) (d). For all plots, the open symbols are for the fast direction and the closed symbols are for the slow direction. Triangles and circles represent 7.5 × 10-6 mol and 15 × 10-6 mol of metal/mol of KDP, respectively. In b-d, the curves for the pure system are shown as dashed and solid lines for the fast and slow sides, respectively. All scatter in the curve for the doped systems lies within the 95% confidence limits in the pure system indicated by error bars in (a). The red lines in (b-d) are guides to the eye.
finite desorption rate of all adsorbate species, species that represent very small fractions of the total impurity content can still be responsible for step pinning. In light of this discussion, we will simply consider the pinning complex to be some form of a yet-unidentified metal-phosphate species with simple stoichiometry. What remains unclear is the geometrical structure of the complex and, more specifically, whether the phosphate bonds to the metal are monodentate or bidentate, as has been previously suggested.12 While these questions are difficult to address considering the labile nature of the metals in this study, we will revisit them in future publications. Effect on Step Velocity and Morphology. The locations of σ* and σd are shown in Figure 1 as a basis of comparison for following figures. As shown schematically in Figure 1, the dependence of step speed on supersaturation in the presence of the adsorbates is specific to each adsorbate. The step speed vs supersaturation curves for pure KDP and for solutions doped with Fe(III), Al(III), and Cr(III) are given in panels a-d of Figure 2, respectively. Due to the amount of data collected for the pure KDP system, we present these data in Figure 2a as a curve indicating mean velocities with error bars corresponding to the 95% confidence interval. Because the {100} face has two distinct growth sectors with different step speeds,9 curves for the “fast” and “slow” directions are presented. These curves are also included in subsequent figures (Figure 2b-d). As
required by the geometry of a dislocation growth hillock, the ratio of the step speeds for the two step directions is equal to the inverse ratio of the hillock slopes so that the fast step direction is the on the shallow slope of the hillock. (On a (100) face, the slow directions are 〈001〉 and 〈001 h 〉, and the fast directions are 〈010〉 and 〈01 h 0〉.) A comparison of the experimental velocity curves for the Al(III)- and Cr(III)-doped surfaces versus the C-V model is also included (Figure 3). The morphology of the growth surface also changes dramatically with supersaturation. Images of the growth surface in each of the four regions are given in Figures 4a-d, 5a-d, and 6a-d for Fe(III)-, Al(III)-, and Cr(III)doped solutions, respectively. Because high concentrations of Fe(III) produce a surface morphology that is unlike that of lower Fe(III) concentrations, the morphology of the highly doped Fe(III) surfaces is illustrated separately in Figure 7. For all concentrations of Al(III) and Cr(III) and high concentrations of Fe(III), supersteps appear and resurrect growth at σ*. Surfaces grown in the presence of low concentrations of Fe(III) express only elementary steps and macrosteps. A detailed description of morphological changes with σ in each of these systems follows. A more detailed description of supersteps can be found elsewhere.3 The relationship between σ* and concentration of Al(III) and Cr(III) dopants is given in Figure 8 and compared to the true root dependence given in eq 4.
Crystal Growth in the Presence of Impurities
Figure 3. Comparison of 7.5 × 10-6 mol of Cr(III) (closed circles, dotted line) and 1.5 × 10-5 mol of Al(III) (open circles, dashed and dotted line) per mol of KDP with the undoped growth curve (dashed line) and the C-V model prediction (solid line) for the velocity curve at the same adsorbate concentration.
Discussion General Effect of Trivalent Metal Adsorbates on Growth. Although the specific behaviors of each adsorbate are varied, there are several common characteristics of all adsorbate complexes. All three adsorbates induce a velocity “dead-zone” at low supersaturations. At a critical supersaturation (σ*) the velocity begins to recover and, over a small range of σ, rejoins the velocity curve for surfaces grown in the absence of adsorbates. At low σ, the surface consists of immobile elementary steps and small step bunches. At σ* steps begin to bunch into groups of various heights and regain motion. However, each adsorbate produces slightly different morphology at high values of σ. For example, Fe(III) induces only the formation of macrosteps, while Al(III) induces the formation of supersteps at all σ > σ*. Chromium(III) induces the formation of supersteps only during the rapid rise in velocity at σ* but only macrosteps at σ > σ*. Never-theless, given the strong similarities between all three adsorbates, a comprehensive model for growth behavior in the presence of adsorbates can be derived. The following subsections discuss the specific behavior of the growth surface in the presence of each of the three adsorbate ions as well as a comprehensive model for growth behavior in the presence of adsorbates. Effect of Fe(III) Adsorbates on Growth. Ferric iron is the only dopant that causes two distinct regimes in the velocity curve during the resurrection from the dead zone (Figures 2b and 4a-d). The effect of low concentrations of Fe(III) in solution has been discussed earlier (and elsewhere2) but will be summarized here. For all σ < σd, the surface is immobile, with all elementary and macrosteps pinned by a field of adsorbate “stoppers” (Figure 4a). At σd, macrosteps begin to move across the surface, resurrecting the step velocity from the dead zone as supersaturation increases beyond σd. Simultaneously, individual elementary steps remain pinned and motionless on the surface (Figure 4b). Elementary steps are constantly incorporated into the bottom of an advancing macrostep as the macrostep overruns them and are simultaneously stripped from the top of the macrostep, so that a quasi-steady-state exists in a key supersaturation range, σd < σ < σ*. Highly doped Fe(III) solutions induce different behavior as the surface recovers from the dead zone, which is unlike
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that seen in the presence of low concentrations of Fe(III). The surface morphology expressed at high Fe(III) concentrations (∼3 × 10-5 mol of Fe(III)/mol of KDP) is given in Figure 7. At exactly σ*, supersteps form and move across the surface, apparently unimpeded by adsorbate impurities. These supersteps consist of hundreds of elementary steps that are tightly bunched together as a single advancing step. These supersteps can be up to 500 nm tall and quickly bury any existing morphological features on the surface as they move. All macrosteps and elementary steps previously visible on the surface are quickly covered by the advancing superstep. Supersteps can appear anywhere on the growth surface, including the top of the dislocation hillock, as shown in Figure 7. The initial motion of supersteps across the face is erratic. Supersteps do not appear at regular intervals and are not consistently the same height. The supersteps rapidly increase in velocity over a small range of σ, returning the step velocity to that of the undoped system at the same supersaturation. A more detailed discussion of supersteps and their general behavior can be found elsewhere.3 Effect of Al(III) Adsorbates on Growth. Solutions doped with Al(III) (shown in Figure 2c and Figure 5a-d) affect the surface very differently than solutions doped with Fe(III). Unlike Fe(III), Al(III)-doped experiments show no slow increase in velocity in the range of σd < σ < σ*, so that σd ) σ*. For all σ < σ*, the surface is comprised of pinned elementary steps and macrosteps (Figure 5a). At σ*, macrosteps only briefly recover motion while the elementary steps remain pinned (Figure 5b). Just as the macrosteps begin to move, supersteps appear and quickly cover the existing surface. The step bunches left behind by the superstep begin to move, but are again covered by the next advancing superstep. This behavior continues over a very small transition region of approximately 0.1% σ (Figure 5c). Above σ*, supersteps dominate the growth surface and, within a range of 0.2% σ, velocities of the Al(III)-doped surface recover to that of an undoped KDP surface (Figure 5d). For the remainder of the supersaturation range investigated, supersteps are primarily responsible for growth, with macrosteps occasionally visible on the surface. The velocity curve presented in Figure 2c, shown as the velocity contributions of each step type, is presented in Figure 9. Analysis of the resulting velocity curve using eq 6 based on measurements of terrace widths from crystals grown ex situ following growth under identical conditions yields a value of τ on the order of several hundred microseconds. (Since the elementary steps that comprise a superstep are very close together, we were unable to measure individual terrace widths directly. Instead, we measured the step riser angle and determined terrace widths to be on the order of 10-9 m. An exact value of τ is difficult to assign due to the range in terrace widths that result from the dependence of the step face angle on height.9 However, taking this range into consideration, we estimate τAl is between 2 × 10-4 and 6 × 10-4 s.) Additionally, we note that Al(III) was added as Al(NO3)3 to all growth solutions since the choice of counterion should not affect the pinning behavior. Although previous studies noted a difference in the dependence of step velocity on supersaturation between nitrate and chloride species,1 preliminary experiments using our starting materials indicate that the addition of NO3- or Cl- alone does not impact growth. Effect of Cr(III) Adsorbates on Growth. Cr(III)doped experiments exhibit some similarities to the Al(III)doped experiments. Like Al(III) adsorbates, Cr(III) adsorbates induce a dead zone in which the surface consists
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Figure 4. Changes in growth morphology with supersaturation in the 15 × 10-6 mol of Fe(III)/mol of KDP system at various points along the growth curve: (a) in the dead zone, σ < 2%; (b) near σd, σ ) 3.6%; (c) near σ*, σ ) 4%; (d) after the velocity has fully recovered, σ ) 6%. These images were taken from Land et al.2
of pinned elementary and macrosteps (Figures 2d and 6a-d). Recovery from the dead zone exhibits only one rise in velocity so that σd is nearly equal to σ*. Trains of supersteps quickly move across the surface during the velocity resurrection from the dead zone. However, unlike Al(III), once the velocity of the growth surface rejoins that of the undoped system, macrosteps once again dominate the morphology of the surface. Supersteps are not evident on the growth surface at higher σ. As with Al(III), with Cr(III) the calculated value of τCr lies between 2 × 10-4 and 6 × 10-4 s. The velocity curve presented in Figure 2d, shown as the velocity contributions of each step type, is presented in Figure 10. Comparison to C-V Model. We compared the shape of the step velocity vs supersaturation curve for Cr(III)doped surfaces to that predicted by the C-V model as given in eq 5. Using the relationships between rc and Li and σ and σ*, respectively, eq 5 becomes
V ) VO(1 - 2rC/〈Li〉)1/2
(8)
Fitting the experimental data presented in Figure 3, we calculate an average distance between adsorbates of ∼2.9 × 10-8 m or ∼290 Å on a surface exposed to a growth solution containing 7.5 × 10-6 mol of Cr(III)/mol of KDP, and ∼2.1 × 10-8 m or ∼210 Å for 1.5 × 10-5 mol of Cr(III)/ mol of KDP. For Al(III)-doped surfaces, we find that the distances between adsorbates in the presence of 7.5 × 10-6 and 1.5 × 10-5 mol of Al(III)/mol of KDP are ∼4.8 × 10-8 m or ∼480 Å and ∼2.4 × 10-8 m or ∼240 Å,
respectively. Using these distances and the analysis discussed in the results section, we compared the concentration of adsorbates in solution to the concentrations of the dominant species in solution and find what percentage of the various species must adsorb to pin the surface. Taking the typical crystal size and an average impurity spacing of 250 Å, the concentration of adsorbates in solution required to pin steps assuming 100% adsorption, is on the order of 10-11 M, meaning that only 0.0001% of the original dopant concentration needs to adsorb to the surface to arrest growth. This result reinforces the uncertainty of the stoichiometry of the step-pinning adsorbate, since most species present in solution are at sufficiently high concentrations to completely pin the surface, even with substantial desorption rates. Additionally, the dependence of V on σ undergoes a rapid rise at σ*, which is also not predicted by the C-V model. This anomalous behavior is most likely the result of the unexpected formation of macrosteps and supersteps. However, while no adsorbate investigated thus far fits the predictions of the C-V model entirely, the C-V model is, none the less, an excellent foundation toward the understanding of adsorbate interactions with growing crystal surfaces. Adsorbate Concentration and Dead Zone. As discussed above, the Cabrera-Vermilyea model predicts that the width of the dead zone should be proportional to n1/2, where n is the concentration of adsorbates,7 assuming that the velocity is resurrected via the motion of elementary steps. Since we now know that elementary steps are
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Figure 5. Changes in growth morphology with supersaturation in the 15 × 10-6 mol of Al(III)/mol of KDP system at various points along the growth curve: (a) in the dead zone, σ ) 4.1%; (b) near σ*, σ ) 6.1%; (c) during resurrection, σ ) 6.2%, where a superstep is seen moving into the bottom of the image; (d) after the velocity has recovered to that of an undoped system, σ ) 8%. (Note: for all σ > σ*, the surface grows on supersteps.)
not primarily responsible for the surface recovery from the dead zone, we do not necessarily expect a root dependence of σ* on n. Indeed, the data presented in Figure 8 illustrate that σ* does not always have a square-root dependence on solution concentration of adsorbates. In fact, σ* is linearly dependent on Al(III) concentration. This linear relation could be due to either the formation and movement of step bunches (macrosteps and supersteps) or the adsorption of a multinuclear adsorbate complex. With the latter in mind, the dependence of σ* on Al(III) concentration (Figure 8) might shed some light on the stoichiometry of the pinning complex. It is possible that the stoichiometry of the Al(III) adsorbate complex is not mononuclear, but rather binuclear. The equilibrium concentration of the binuclear complexes depends on the concentration of Al(III) squared (i.e., [Al2(PO4)x] ) [Al3+]2). Therefore, the width of the dead zonesi.e., σ*sbecomes a linear function of the total Al(III) concentration via the dependence of σ* on n (i.e., σ* ∼ ([Al3+]2)1/2 ) [Al3+]). Equilibrium speciation calculations of Al(III) in solution reveal a low concentration of Al2PO4(OH)2+ (0.6% of the total Al species), which could be sufficient to cause the observed step pinning. However, we must note that for all the trivalent metals we have studied (Al(III), Fe(III), and Cr(III)), only Al(III) exhibits this linear dependence and appears to be the anomaly, rather than the norm. Adsorption Time and Ligand Exchange Rates. We correlated the value of τ for each metal adsorbate to the
ligand-exchange rate of the metal. The values of τ decrease in the following order: Fe(III) > Al(III) ∼ Cr(III), while the ligand exchange rates of the aqueous metal ions trend in the reverse order. Decreased pH and complexation to phosphate could increase the rates of ligand exchange. However, it is unlikely that aqueous complexes of both Al(III) and Cr(III) would reach similar rates at pH ) 4.2, given their wide difference in inherent reactivities. Due to the labile nature of the metal ions, we cannot draw conclusions about their speciation either in solution or at the surface. Speciation and ligand exchange rates will certainly impact the value of τ. However, the time scales of τ do not match reasonable rates of ligand exchange for the ions. Ferric iron complexes typically exchange waters of hydration, an accepted estimate of ligand-exchange rates, in time scales of ∼10-3 s or less, yet we calculate an adsorption time of 10 s. Aqueous complexes of Al(III) and Cr(OH)2+ typically exchange hydration waters over time scales of seconds but have absorption times on the order of ∼10-4 s. One explanation is that, at the surface, the Al(III) and Cr(III) cannot quickly dissociate from the inner-sphere phosphate as it is incorporated into the surface. It follows that the Al(III) and Cr(III) concentrations would build up more quickly at the surface. Conversely, Fe(III) takes longer to build up an equilibrium concentration on the surface because it quickly dissociates from the innersphere phosphate as the phosphate incorporates into the surface growth site. Hence, the formerly bound phosphate
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Figure 6. Changes in growth morphology with supersaturation in the 15 × 10-6 mol of Cr(III)/mol of KDP system at various points along the growth curve: (a) in the dead zone, σ ) 4%; (b) at σ*, σ ) 7.1%; (c) during resurrection, σ ) 7.3%, where a superstep can clearly be seen by the large black line from the upper right to the lower center of the image; (d) after the velocity has rejoined that of an undoped system, σ ) 10.5%, where all growth occurs as macrosteps and supersteps are no longer visible.
Figure 7. Morphology for σ > σ* in a system containing a high concentration of Fe(III) (∼3 × 10-5 mol of Fe(III)/mol of KDP). Growth occurs on supersteps that emerge from the dislocation hillock. (a-c) Sequential images at the top of a dislocation hillock; the supersteps are 1500 nm in height.
group is left at the surface while the Fe(III) itself diffuses away into solution. A Comprehensive Model for Step Bunching. Superstep velocities are surprisingly unaffected by adsorbate accumulation on the elementary step terraces that comprise the superstep, but an explanation can be constructed from the values of τ. Our previous investigations3 showed that, in the case of Al(III), the angle between the terrace and the superstep riser increases with step height until it reaches a limiting value of 11.8°. This angle corresponds to an average spacing of 18 Å between the elementary steps that comprise the superstep. First, we
note that this is similar in magnitude to the calculated spacing between pinning sites for Al(III) and Cr(III) at σ*. Moreover, for supersteps moving at a velocity of ∼3 µm/s, the terrace lifetime is on the order of a few hundred microseconds.3 This is the same order of magnitude as the adsorption time estimated from the small interval of supersaturation between σd and σ*. These observations suggest the following comprehensive physical picture for the appearance and evolution of all three families of steps. Step trains of all three families of steps consist of one or more individual elementary steps separated by terraces that fluctuate in size but have some
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Figure 8. Plot of σ* vs Ci for Al(III) (closed circles) and Cr(III) (open circles). The relationship between σ* and Ci for Al(III) is best described by a linear dependence, where y ) 0.413 ( 0.000 x. In the case of Cr(III), the data fits a root dependence on Ci given by y ) 1.85 ( 0.02x0.5. Because of the anomalous early rise in velocity on surfaces doped with Fe(III), we did not include the Fe(III) data in this plot. The error bars for each point are smaller than the size of the symbol.
Figure 9. Plot of velocity vs σ for growth from a solution doped with 1.5 × 10-5 mol of Al(III)/mol of KDP, shown as the specific velocity of each step type as a function of σ. At low σ, elementary steps and macrosteps have the same velocity (the points overlie one another). Note that as σ is increased, elementary steps and eventually macrosteps are no longer observed on the surface. At high σ (above σ*) all growth occurs via the motion of supersteps.
average width and lifetime. When the average terrace lifetime is much longer than the impurity adsorption time, the individual elementary steps rapidly become immobilized by impurity pinning.7,22,23 For well-separated elementary steps, this occurs at σ* or the supersaturation that corresponds to the onset of growth.7 The formation of step bunches occurs even in undoped growth solutions and has been compared to classic models of step bunch formation.9 The predictions of both the “shock wave” model24 and the impurity-induced step doubling model25 are consistent with our measurements of step bunching (22) Land, T. A.; DeYoreo, J. J.; Lee, J. D.; Furguson, J. R. Mater. Res. Soc. Symp. Proc. 1995, 355, 45. (23) van Enckevort, W. J. P.; van den Berg, A. C. J. F. J. Cryst. Growth 1998, 183, 441. (24) Chernov, A. A. Sov. Phys. Usp. 1961, 116. (25) van der Eerden, J. P.; Muller-Krumbhaar, H. Electrochim. Acta 1986, 31, 1007.
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Figure 10. Plot of velocity vs σ for growth from a solution doped with 1.5 × 10-5 mol of Cr(III)/mol of KDP, shown as the specific velocity of each step type as a function of σ. At low σ, elementary steps and macrosteps have the same velocity (the points overlie one another). At σ*, supersteps alone have nonzero velocity. Above σ*, macrosteps are the only step type observed on the surface.
in undoped solutions, and therefore our data do not allow us to differentiate between the two.9 Growth can still occur at supersaturations below σ*, provided that the steps bunch sufficiently to ensure that the terrace lifetime approaches the adsorption time. Then the average adsorbate spacing falls and the steps can remain mobile at lower supersaturations. None-the-less, terraces fluctuate in width and impurities can still adsorb, pinning steps locally. This local pinning eventually leads to the immobilization of the step. In growth solutions doped with Fe(III), pinned elementary steps are remobilized when a second step catches up from behind. However, the structural relationship between the adsorbate and the KDP lattice that leads to this fact is unknown. Future research will model the molecular-scale interactions between the adsorbate molecule and the step edge. Where σd e σ e σ*, step bunches can only continue to advance as long as the rate at which the step bunch “collects” steps at its base is greater than or equal to the rate at which they are removed from the top.2 At sufficiently low supersaturations (eσd), this condition fails to hold. The step bunches become disorganized, decay, and are pinned. For adsorbates with large τ, the separation between σd and σ* is large. Conversely, for adsorbates with small τ, σd and σ* become indistinguishable. Now we consider the affect of this competition between adsorption time and terrace lifetime on step-bunch stability. Because smaller terraces collect fewer adsorbates with larger distances between adsorbates, and therefore faster step speeds, steps will continue to bunch. This bunching will continue to reduce the average terrace width and increase the step speed. Thus, a step bunch will catch up to other step bunches and become higher and steeper. Eventually the step bunch reaches a state where the terrace lifetime is far enough below the adsorption time and the step speed becomes indistinguishable from that of elementary steps in undoped solutions. These are supersteps. The faster impurities adsorb to the terraces, the stronger the driving force for superstep formation. In other words, the collapse of σd onto σ* and the tendency toward superstep formation go hand in hand. For Fe(III) adsorbates, which build up slowly on the terrace, smaller terraces with shorter lifetimes (i.e., macrosteps) are less affected by the adsorbates than are
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elementary steps that are separated by large terrace widths. When σd < σ < σ*, the Fe(III)-adsorbate concentration on the terraces is less than the equilibrium amount. The lower concentration of adsorbates allows motion of step bunches (and elementary steps with narrow terraces). As σ increases, the lifetimes of the terraces become shorter, allowing fewer adsorbates to adhere. Finally, at σ*, the velocity undergoes a rapid rise to rejoin that of the undoped system (Figure 2b-d) where even elementary steps are again mobile. For Fe(III), the adsorption time is slow so that σd and σ* are well separated and supersteps only form at very high concentrations. For Al(III) and Cr(III), the species that pins steps adsorbs so rapidly that σ* and σd are indistinguishable and supersteps readily form at much lower adsorbate concentrations. Conclusion While interferometric data has also seen a step velocity dead zone on surfaces doped with adsorbate complexes, the lack of morphological data limits the interpretation of the results. However, the use of AFM to simultaneously record step velocity and morphology changes with supersaturation yields a more comprehensive picture of the interaction between the adsorbate complex and the growing surface. We have found that labile phosphato complexes of Fe(III), Al(III), and Cr(III) cause step velocities to decrease and form a velocity dead zone on the {100} face of KDP at low supersaturations. The characteristic time for adsorption of the complex can be derived from the relationship between velocity and supersaturation and is related to the morphology of the surface during the growth process. Fe(III)-doped surfaces show a unique dead zone that is not predicted by the C-V model and results from the long time scale needed for Fe(III) to reach equilibrium adsorbate concentrations on terraces (τFe ∼ 10-100 s). The pinning of elementary steps causes macrosteps to form that are responsible for growth of the surface. Only in the presence of high concentrations of Fe(III) do supersteps form.
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The behavior of the Al(III) adsorbate on the surface and the rapid adsorbate build-up on the terrace (2 × 10-4 < τAl < 6 × 10-4 s) leads to the formation of macrosteps and supersteps which are ultimately responsible for the resurrection of the surface. This observation is contrary to the C-V model, which states that the surface would be resurrected by elementary steps. The step velocities on the Al(III)-doped surface resurrect solely by the propagation of supersteps that dominate growth for all σ > σ*. Cr(III)-doped surfaces behave much in the same way as do the Al(III)-doped surfaces, including the formation and motion of supersteps during the recovery from the dead zone. Cr(III) adsorbates also have similar characteristic adsorption times (∼2 × 10-4 < τCr < 6 × 10-4 s). The primary difference between the Al(III)- and Cr(III)-doped surfaces is the morphology of the growth surface after velocities resurrect from the dead zone. Cr(III) adsorbates induce the expression of supersteps only during the resurrection from the dead zone. However, after the velocity recovers, the surface is once again dominated by the motion of macrosteps. The relationship between σ* and n is best described by a root dependence for Cr(III) and a linear dependence for Al(III)-doped solution and could be a result of the stoichiometry of the adsorbate complex. Future work will address the specific speciation and geometry of the pinning complexes using inert metals as model complexes (i.e., Rh(III)). Acknowledgment. We thank Dave Wruck for the elemental analysis of our samples and US DOE Grant DE-FG03-02ER15325 and US NSF Grant NSF-EAR0207709 for support. This work was performed under the auspices of the US Department of Energy, National Nuclear Security Administration by the University of California, Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48 (UCRL-ADS202287). LA049546F
