Constant Composition Studies Verify the Utility of the Cabrera-Vermilyea (C-V) Model in Explaining Mechanisms of Calcium Oxalate Monohydrate Crystallization Lijun Wang,† James J. De Yoreo,‡ Xiangying Guan,† S. Roger Qiu,‡ John R. Hoyer,§ and George H. Nancollas*,†
CRYSTAL GROWTH & DESIGN 2006 VOL. 6, NO. 8 1769-1775
Department of Chemistry, Natural Sciences Complex, UniVersity at Buffalo, State UniVersity of New York at Buffalo, Amherst, New York 14260, Department of Chemistry and Materials Science, Lawrence LiVermore National Laboratory, LiVermore, California 94551, and The Children’s Hospital of Philadelphia, School of Medicine, UniVersity of PennsylVania, Philadelphia, PennsylVania 19104 ReceiVed December 29, 2005; ReVised Manuscript ReceiVed June 13, 2006
ABSTRACT: The classic theory of Cabrera and Vermilyea (C-V) postulates that inhibition of crystallization by impurities is the result of pinning of step motion. Although generally accepted, the predictions of the C-V model have not been previously linked to studies of impurity adsorption in macroscopic crystallization systems. Since calcium oxalate monohydrate (COM) is the primary constituent of most human kidney stones, effects of impurities on COM crystallization are biologically relevant. Recent in situ atomic force microscopy (AFM) studies suggest that citrate molecules adsorb to COM step edges on the (-101) face, thereby pinning step motion and decreasing the velocity of the steps. We have now investigated the crystallization kinetics of COM in the presence of citrate at 37 °C and I ) 0.15 mol L-1 using the constant composition (CC) method. The dependence of growth rate on both supersaturation and citrate concentration were measured, and the behavior of the COM-citrate system was found to be in reasonable agreement with the predictions of the C-V model, provided that, in addition to pinning of step motion on the (-101) face through the Gibbs-Thomson effect, the kinetic coefficient that relates step speed to supersaturation is assumed to decrease with increasing citrate levels on all faces. The dependence of macroscopic crystal aspect ratio on citrate level in these CC studies also fits this model. Thus, we find that the results provide a mechanistic link between the microscopic AFM measurements of step kinetics and the macroscopic CC data on crystal growth rates and crystal habit. Introduction Kidney stones are aggregates of microcrystals and most commonly contain calcium oxalate monohydrate (COM) as the primary constituent.1-3 Numerous studies have investigated the mechanisms of in vitro formation and inhibition of COM crystal growth that contribute to the abundance of COM in stones.4 Investigations of COM crystallization have included seed crystal based growth studies using various conditions of fluid dynamics5-14 and evaluation of spontaneous precipitation of COM in highly supersaturated solutions.15,16 In contrast, the CC system17 of bulk-crystallization studies more effectively simulates the in vivo biologically stabilized conditions for COM crystallization. Thus, precise measurements of rates of formation of COM determined by CC analysis more accurately reflect the in vivo kinetics of crystallization.18 In addition to its own role in stone formation and inhibition, citric acid, HOOC(CH2)2CH(COOH)(CH2)2COOH, has also served as a model for the more complicated naturally occurring proteins that are also rich in the carboxylates of acidic amino acids. AFM measurements of adhesion forces between molecules immobilized on AFM probes and crystals have shown binding of carboxylates to the COM crystal surface.19 Citrate is normally abundant in the urinary milieu and inhibits COM growth and aggregation in vitro at concentrations similar to those found in urine (1-2 mM).20 Citrate also blocks adhesion of COM crystals to renal epithelial cells.21 Dual actions of citrate in urine oppose crystal formation by both thermodynamic and * To whom correspondence should be addressed. Tel: +1-716-645-6800 ext. 2210. Fax: +1-716-645-6947. E-mail:
[email protected]. † State University of New York at Buffalo. ‡ Lawrence Livermore National Laboratory. § University of Pennsylvania.
kinetic mechanisms.22,23 Deficiencies of urinary citrate predispose individuals toward renal stone formation.24 At present, citrate is the only natural inhibitor that is used in medical treatment and routinely monitored by its levels in urine.25 Although the majority of studies have been concerned with the overall kinetics of bulk crystallization, recent investigations have been performed at a molecular scale.26-28 AFM has been used to measure the effect on COM crystal growth of various additives, including citrate and osteopontin.13,14 These studies and related investigations15,29 have revealed highly specific binding of molecular species to crystal planes or steps.30 The results of the AFM studies suggest that addition of citrate markedly alters the morphology and kinetics on the (-101) face but has little or no effect on the (010) face.13,14 Citrate molecules adsorb to step edges or accumulate on terraces ahead of migrating steps, thereby pinning step motion and decreasing the velocity of the steps.13 The classic model of inhibition proposed by Cabrera and Vermilyea31 analyzes the effect of step pinning by comparing the critical curvature of a step to the average impurity spacing. Because the critical curvature decreases as the supersaturation is increased, this model predicts that (1) at sufficiently high inhibitor concentration, growth ceases because steps can no longer pass between the adsorbed inhibitor molecules, creating a so-called “dead zone” of supersaturation, and (2) when a threshold supersaturation is met, the steps break through the chain of adsorbed impurities and rapidly achieve the step velocity characteristic of the pure system. Despite the general acceptance of this theory, no macroscopic observations of growth rates have ever been directly linked to this microscopic impurity step-pinning model. The objective of this work is to examine data from new CC crystallization kinetics studies of COM in the context of the C-V impurity model. These studies show that the traditional C-V model can reasonably account
10.1021/cg050679o CCC: $33.50 © 2006 American Chemical Society Published on Web 07/11/2006
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Table 1. Experimental Conditions of COM Crystallizations at I ) 0.15 mol L-1 and pH 7.0
σ)
supersaturation σ
WCaCl2 ) WK2C2O4/ 10-4 mol L-1
VK2C2O2/ 10-3 mol L-1
VCaCl2 ) WNaCl/ mol L-1
VNaCl/ mol L-1
0.095 0.144 0.266 0.388
2.150 2.250 2.500 2.750
5.430 5.450 5.500 5.550
0.149 0.149 0.149 0.149
0.288 0.288 0.288 0.288
for the inhibition of COM crystallization by citrate that is experimentally observed. Experimental Section COM Seed Preparation. COM crystals were prepared by the dropwise addition of 500 mL aliquots of 0.40 mol L-1 of calcium chloride and potassium oxalate into 1 L of triply distilled deionized water (TDW) at 70.0 ( 0.1 °C. The suspension was stirred continuously at this temperature for 6 h to allow for complete conversion of the precipitate to calcium oxalate monohydrate. The solid was then filtered and washed with TDW until free from residual chloride ion, suspended in TDW, and aged at 37.0 °C for at least 1 month prior to use. It has been shown that COM crystallites may be stored for long periods of time, without conversion of the monohydrate to other hydrates.32 Specific Surface Area of COM Seed. The specific surface area, 3.3 m2 g-1, was determined by a single-point Brunauer-Emmitt-Teller method (Quantasorb Sorption System, Quantachrome Corp.) using a 20/80 nitrogen/helium gas mixture. Constant Composition of COM Growth. Supersaturated solutions (ionic strength I ) 0.15 mol L-1, pH 7.0) were prepared by slowly mixing filtered (0.22 µm Millipore filter) calcium chloride, potassium oxalate, and sodium chloride. Nitrogen, saturated with water vapor at 37 °C, was purged through the reaction vessels to exclude carbon dioxide. Crystallization experiments, initiated by the introduction of known amounts of COM crystals, were conducted in magnetically stirred (450 rpm) double-jacketed vessels thermostated at 37.0 ( 0.1 °C. In the CC method, a titrant solution containing the lattice ions was simultaneously added to the reaction solutions to compensate for changes due to crystal growth. When growth was initiated, changes in electrode potential triggered the addition of two titrant solutions, which were designed to maintain a constant activity of all ionic species in the reaction solutions. The titrants were prepared having COM stoichiometries. For titrant buret 1
VCaCl2 ) 2WCaCl2 + Ceff
(1)
VNaCl ) 2WNaCl - 2Ceff
(2)
VK2C2O4 ) 2WK2C2O4 + Ceff
(3)
and for titrant buret 2
W and V are the total concentrations in the reaction solutions and titrants, respectively, and Ceff is the effective titrant concentration (5.00 × 10-3 mol L-1 in this study). The experimental conditions are summarized in Table 1. For the impurity-doped experiments, citrate concentrations ranged from 1.0 × 10-5 to 1.0 × 10-3 mol L-1. Titrant addition was triggered by a potentiometer (Orion 720A) incorporating a calcium ion selective (Orion 93-20) and a reference electrode (Orion 900100). During the crystallization, the output of the potentiometer was constantly compared with a preset value and the difference, or error signal, activated motor-driven titrant burets, thereby maintaining a constant thermodynamic driving force. Chemical analysis of solution samples periodically withdrawn and filtered (0.22 µm Millpore filter) showed that the total calcium (atomic absorption) concentration remained constant to within (1.5% during the experiments. The driving force for the crystal growth of COM in supersaturated solutions is due to the change in Gibbs free energy from the supersaturated solutions to equilibrium. The relative supersaturation, σ, is defined by
[
]
(Ca2+)(C2O42-) Ksp
1/2
-1
(4)
where (Ca2+) and (C2O42-) are the ionic activities, calculated using the Davies extended form of the Debye-Hu¨ckel equation with mass balance expressions for total calcium and total oxalate with appropriate equilibrium constants by successive approximation for the ionic strength.33 The solubility activity product of COM at 37.0 °C, Ksp, was 2.20 × 10-9 mol2 L-2.34 Scanning Electron Microscopy (SEM). Crystallites for SEM investigation (Hitachi S-4000 FESEM) were separated from the solution by filtration (0.22 µm Millipore filters) at the end of the experiments and dried at room temperature. After the crystallites were coated with 20 nm of evaporated carbon, images were collected using a primary beam voltage at 20 keV.
Results and Discussion Growth Rate. The crystal growth rates at any instant were determined from plots of titrant volume added as a function of time (the first 15 min of reaction). The overall growth rate, R, is defined by eq 5, where dV/dt is the titrant curve gradient, SA
R)
Ceff dV SA‚ms dt
(5)
is the specific surface area of the seed crystals, and ms is the initial seed mass. Figure 1 shows typical CC curves for COM growth at different supersaturations (the relative supersaturation with respect to COM, σ, ranges from 0.095 to 0.388; the pH and ionic strength are 7.0 and 0.15 mol L-1, respectively). The COM growth rates in pure solutions, R, at σ values of 0.095, 0.144, 0.266, and 0.388, respectively, are (4.77 ( 0.09) × 10-6 (n ) 7), (9.42 ( 0.05) × 10-6 (n ) 7), (1.43 ( 0.05) × 10-5 (n ) 7), and (1.97 ( 0.10) × 10-5 mol m-2 min-1 (n ) 7), respectively, and they are almost unchanged at citrate concentrations below 1.00 × 10-5 mol L-1. It can be seen that the rates gradually decreased with an increase in the citrate concentration, about 50% inhibition of growth in the presence of 1.00 × 10-4 mol L-1 citrate, eventually approaching a maximum degree of inhibition of about 93% at a citrate concentration of 1.00 × 10-3 mol L-1 (Figure 2). It is important to note that changes in the calcium concentration as a result of the formation of calcium citrate complexes can be ruled out, since speciation calculations were made to maintain free ionic calcium concentrations in the presence of citrate. Thus, citrate is an effective COM crystal growth retardant. C-V Model of Step Pinning by Citrate. Citrate adsorption can dramatically change aspects of crystal growth such as step morphology and kinetics. Combining force microscopy with molecular modeling, Qiu et al. showed that citrate controls growth habit and kinetics by pinning step motion on the (-101) face through specific interactions in which the stereochemical relationship between the citrate molecule and the step structure determine the effectiveness.13,14 To interpret these results, we consider the impact on CC measurements of step pinning on one distinct set of faces, as was observed in AFM experiments. The appearance of serrated step edges and slowing of the step speed suggest that the effect of citrate may be described by the classic C-V model of step pinning.31 In this model, the effects of impurities on crystal growth are caused by impurity adsorption at kinks, edges, and terraces of a growing surface, thus reducing the growth rate by reducing or hindering the movement of growth steps.35-39 Impurities adsorbed at step edges or on the terraces ahead of migrating steps create a field of “impurity stoppers” that act to
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Crystal Growth & Design, Vol. 6, No. 8, 2006 1771
rc )
Rω kTσ
(7)
Here R is the step-edge free energy per unit step height, ω is the molecular volume, k is Boltzmann’s constant, and T is the absolute temperature. The criterion for step motion through the field of inhibitors is then Li > Grc. The expression for the critical supersaturation, σ*, in terms of Li, above which the steps begin to move, is given by
Rω σ* ) G kTLi
(8)
Combining eqs 6-8 leads to an expression for Vi/V0 in terms of σ*, for σ g Gσ*:
Vi ) V0(1 - Gσ*/σ) Figure 1. CC growth curves of COM at different supersaturations (the curves have been normalized to the same seed mass of 10.0 mg). The growth rates are calculated from the gradient of these CC curves.
(9)
Recognizing that Li ) (1/ni)0.5,36 where ni is the surface concentration of adsorbates, we have that
σ* ≈ ni0.5
(10)
Under the assumption that the surface coverage of impurities scales with the bulk solution concentration (i.e., it just scales with flux to the surface),31 then ni ) B2Ci, where Ci is the bulk solution concentration of the adsorbing species and B is a proportionality coefficient that reflects a combination of geometric factors, sticking probability, and impurity lifetime on the surface. (In the limit of infinite lifetime, B is the rate coefficient in a Langmuir adsorption model.) Then
Rω 0.5 C kT i
σ* ) GB
Figure 2. COM crystal growth rates in the presence of various concentrations of citrate. Data points represent the average of seven measurements. The standard deviations for experiments in the presence of citrate were not larger than those for the experiments in pure solution that are reported in the text.
block the motion of elementary steps, thereby decreasing their velocity. When the average spacing between impurity molecules (Li) is less than a critical distance, whose magnitude is approximately given by the Gibbs-Thomson critical diameter (2rc), the steps are unable to advance. As a consequence, the C-V model predicts the presence of a “dead zone”s a region of positive supersaturation (σd) where no growth occurs. As the supersaturation is increased beyond σd and the critical diameter becomes smaller than Li, the steps begin to squeeze through the “fence” of impurities and the step speed (Vi) rises rapidly, eventually reaching that of the pure system (V0).31,40,41 The C-V prediction of the dependence of step speed on impurity spacing, for Li g Grc, is given by41
Vi ) V0(1 - Grc/Li)
(6)
where G is a number of order 1 that depends on the geometry of the crystal lattice and is of order 2 for a square lattice41 and the critical radius is given by
(11)
To apply eq 9 to the macroscopic CC data, we need to relate the step speed to the growth rate for each face. For a crystal growing isotropically, we note that that there is a special relationship between the faces; for the jth and kth faces, ajRj ) akRk, where aj is the area of the jth face. However, when the relative growth rates of the faces change suddenly, this relationship no longer holds. For a COM crystal, the total growth rate R0AT ) ∑ajRj (AT is the total crystal area; R0 is the initial average normal growth rate in the absence of impurity) is given by
R0AT ) 2a(-101)R(-101) + 2a(010)R(010) + 4a(120)R(120)
(12)
For the impurity-free case, applying the relationship between the face growth rates and normalizing all growth rates to that on the (010) face gives
R0AT ) 8a(010)R(010)
(13)
Finally, we can relate the face growth rate Rj to the elementary step growth rate Vj, by using the expression Rj ) pjVj, where pj is the average slope of the face, i.e., the step height times step density. Then the impurity-free growth rate of a COM crystal is
R0AT ) 8a(010)p(010)V(010)
(14)
When impurities are added that pin steps on a single face, we have two regimes. The first occurs for σ < σ*. In this regime, the growth rate is given by the sum of the growth rates of the unaffected faces and is independent of Ci. However, above σ*, the growth rate of the affected face is controlled by eq 9 and
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Figure 3. (a) Relative growth rates of COM crystals in the presence of various concentrations of citrate at different supersaturations. R0 and Ri are the rates of reaction in the absence and presence of citrate, respectively (pH 7.0, 37 °C, and I ) 0.15 mol L-1). Dashed lines are examples of the C-V curves. (b) C-V curves for four different cases.
thus varies with Ci through eq 11. Consider the effect of citrate on the (-101) face:
RiAT ) 2a(-101)p(-101)V(-101)(1 - GBrcCi0.5) + 2a(010)p(010)V(010) + 4a(120)p(120)V(120) (15) Using the relationship between face growth rates at the instant citrate is added, for GBrcCi0.5 < 1
RiAT ) a(010)p(010)V(010)[6 + 2(1 - GBrcCi0.5)] 1 ) 8a(010)p(010)V(010) 1 - GBrcCi0.5 4
(
)
(16a)
and for GBrcCi0.5 g 1
RiAT ) 6a(010)p(010)V(010)
(16b)
D is defined as GB/4 with G and B defined in the text following eqs 6 and 11, respectively. We obtain the following relationships for the relative change in growth rate normalized to the initial surface area, which is directly comparable to CC data:
Ri ) 1 - DrcCi0.5 R0
(17)
R 0 - Ri ) DrcCi0.5 R0
(18)
In fact, no matter what faces are affected, we will get a result of the general form, for DrcCi0.5 e 1:
Ri ) 1 - DrcCi0.5 R0
(19)
As Figure 3a shows, the CC data for citrate are qualitatively in agreement with this C-V dependence in that, independent of supersaturation, Ri/R0 decreases rapidly and gradually reaches a minimum growth rate. However, comparison with the theoretical curves (heavy dashed lines in Figure 3a) show that there are two significant differences. The first is that the rate falls faster and flattens more suddenly than predicted. The second is that the minimum rate is far below that predicted for pinning of a single type of face. At first consideration, the results then appear to be in contradiction to the published AFM results,
in that the latter show only pinning on the (-101) faces. Most importantly, because (-101) faces account for only two out of eight crystal faces, this result would imply that the minimum growth rate should be 75% of the growth rate in pure solution. However, the CC data have been collected to considerably higher citrate levels than the AFM data, in which the highest citrate level was less than 2 × 10-5 M. The CC data presented here have been collected at citrate levels up nearly 100 times that level. If we look at the relative growth rate measured in the CC measurements at the much lower citrate levels of the AFM experiments, in fact, Ri/R0 is about 75%. Nonetheless, even though the two sets of results are not in contradiction, the CC results show that, at sufficiently high citrate levels, (1) both of the other types of facess(010) and (120)sare affected by citrate and (2) the curves shown in Figure 3a must be composite curves representing inhibition characterized by significantly different values of D for the various faces. While one could derive the three different values of D by fitting the data to an equation of the form given in eq 19, where
D)
D(-101) 4
+
D(010) 4
+
D(120) 2
the reliability of the derived coefficients is suspect for two reasons. The first is that, given the scatter in the data, a threeparameter fit will give a wide range of acceptable values. However, the second is both more subtle and significant. Recent AFM measurements on the (-101) face show that, in the pure system, the step speed rises linearly with calcium concentration.42 The slope of the V vs C plot gives the kinetic coefficient, β, for step motion, which reflects factors such as the attempt frequency for attachment to the step, the kink density, the sticking coefficient, and the detachment probability. Those results also show that, in addition to step pinning, the addition of citrate reduces the kinetic coefficient. That is to say, once the supersaturation is well in excess of the σ* value, like the unimpeded step in pure solution, the step speed rises linearly with calcium concentration, but it has a lower value and rises with a slower rate than the unimpeded step. As a result, eq 19 no longer holds. Instead, one must modify eq 15 to explicitly call out the dependence of step speed on citrate concentration for each face. The result is a minimum six-parameter equation which, without independent constraints, cannot be used to reasonably determine either Di or βi. Nonetheless, we can look at the expected behavior for some limiting cases. Figure 3b
COM Crystallization Kinetics
Crystal Growth & Design, Vol. 6, No. 8, 2006 1773
Figure 4. (a) Plots of relative growth rates, Ri/R0, against σ for COM crystallization kinetics in the presence of various concentrations of citrate at pH 7.0, 37 °C, and I ) 0.15 mol L-1, showing that, while there is a small amount of curvature at low σ, over most of the range of σ used here Ri/R0 is constant and is independent of supersaturation for σ < σ*. (b) Predications of the behavior of eq 20 for a range of impurity levels in the limit of no pinning (G ) 0 in eq 20) and the limit of constant β (E ) 0 in eq 20), showing that the decrease in β is the dominant effect of citrate on the (010) and (120) faces.
shows four cases: the first is one in which only the (-101) face is affected, by both step pinning and reduction in β. The growth rate is expected to drop rapidly with increasing citrate content but becomes a constant at Ri/R0 ) 0.75. The second case is one in which, in addition to step pinning on the (-101) face, the other two types of faces have a kinetic coefficient that decreases as Ci0.5. That is, β scales with surface impurity coverage, which implies a square-root dependence on bulk concentration, just as in the case of step pinning (see eq 9 and preceding discussion.). The third case is the same as the the second, but now the other two types of faces also exhibit step pinning to a lesser degree than the (-101) face. This is equivalent to saying that the dead zone is much narrower for a given citrate level. The fourth and final case is the simple C-V model in which all faces are treated the same and have the weak degree of step pinning assumed in the previous case for the (010) and (120) steps but no change in β. This corresponds to the dashed curves in Figure 3a. As Figure 3b shows, an important consequence of including the decrease in β is that the deviation in the dependence of Ri/R0 on citrate level from the C-V prediction as seen in Figure 3a can now be understood. The strong pinning of the (-101) face leads to the rapid drop at low citrate levels, and the decrease in kinetic coefficients combined with some degree of pinning on the other two faces leads to the less rapid decrease at higher citrate levels. Finally, we can use the fact that step pinning gives a nonlinear dependence of V on σ to assess the degree to which high citrate levels leads to step pinning vs a reduction in β on the (010) and (120) faces. For any face, we can write
Ri σ* ) 1 - G (1 - ECi0.5) R0 σ
(
)
(20)
where the first term reflects the effect of step pinning and the second gives the decrease in β. Here E is a proportionality constant that, like B, reflects a combination of geometric factors, impurity lifetimes, and sticking probabilities. We know that growth on the (-101) face ceases at low citrate levels ( σ*, otherwise there would be no growth at all. Then, according to eq 20, V vs σ should be an increasing
Figure 5. Fit of experimental data to the C-V model, modified to take into account decreasing β on all faces with increasing citrate. The fitted dashed line is calculated using eq 22.
nonlinear function of σ. As Figure 4a shows, while there is a small amount of curvature at low σ, over most of the range of σ used here, Ri/R0 is constant. Figure 4b emphasizes this point by showing the behavior of eq 20 for a range of impurity levels in (1) the limit of no pinning (G ) 0 in eq 20) and (2) the limit of constant β (E ) 0 in eq 20). These results show that the decrease in β is the dominant effect of citrate on the (010) and (120) faces. The most straightforward interpretation of this result is that citrate poisons kinks but does not create permanent pinning sites43 and that the coefficient E gives the relative reduction in available kink sites with increasing citrate concentration. The results in Figures 3a and 4a now allow us to constrain the number of parameters needed to fit the data to a C-V model with decreasing kinetic coefficient. The best fit to all data sets assuming the form
Ri 1 ) (1 - A(-101)Ci0.5)(1 - B(-101)Ci0.5) + R0 4 1 1 (1 - B(010)Ci0.5) + (1 - B(120)Ci0.5) (21) 4 2
1774 Crystal Growth & Design, Vol. 6, No. 8, 2006
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Figure 6. SEMs of COM crystals grown in (a) pure supersaturated solution and in citrate-containing solutions at citrate concentrations of (b) 2.5 × 10-4 mol L-1 and (c) 10-3 mol L-1. Scale bar: 600 nm.
gives A(-101) ) 165.1, B(-101) ) 17.8, B(010) ) 13.58, and B(120) ) 7.0. The final form of the modified C-V is given by
Ri ) 1 - 52.62Ci0.5 + 734.7Ci R0
(22)
The dashed curve in Figure 5 shows that this model gives an excellent fit to the data. From these results we conclude that the COM-citrate system is reasonably well described by the C-V model, with the modification that citrate also impacts the kinetic coefficient of step motion on all faces. Influence of Citrate on COM Morphology Through Selective Binding to Atomic Steps. The presence of citrate in the supersaturated solutions affects not only the kinetics of crystal growth but crystal morphology as well. In the absence of citrate, the crystals developed a prismatic habit with (-101), (010), and (120) as the principal faces and were elongated along the [101] direction (Figure 6a), as also observed previously by others.27,44 In the presence of citrate the crystals were less elongated, and the effect increased with increasing citrate concentrations (Figure 6b,c). To quantify this effect, the ratio D{120}/D{010} was measured in more than 20 crystals for each citrate concentration. Bouropoulos et al.45 used this ratio and showed the effects of soluble macromolecules extracted from plant leaf crystals on the morphologies of COM crystals. D{120} and D{010} are the respective perpendicular distances between the edges of the (120) and (010) faces measured on the (-101) face (Figure 7a), and their ratio is a direct measure of the relative growth rates normal to the two faces, (pV)(120):(pV)(010). The results in all cases are shown in Figure 7b, the values were significantly different from the control (p < 0.05), and the effect is concentration dependent, indicating possibly specific inhibition of crystal growth. The decrease in the (010) plane surface area shows that the (010) face is becoming the fastest growing, suggesting that its growth is the least inhibited by the citrate. The shape of the curve in Figure 7b is reminiscent of the C-V curves for Ri/R0 (Figure 5), suggesting that, as in the case of pinning, the reduction in kinetic coefficient reflects the relationship between bulk citrate concentration and surface coverage. These observations of the bulk crystal habit are consistent with our previous AFM results.13 In that study, after citrate was introduced, the step speed along all directions on the (-101) face was decreased. At a citrate level of ∼1.00 × 10-5 M, the speed along [101] was reduced by a factor of 25 while the speeds along [-120] and [-1-20] were only reduced by about a factor of 2. As a result, the growth hillock shape on this face in citratecontaining solutions evolved from the hexagonal-like habit to that of a round disk, similar to the case for the face itself.14 To understand the stereospecifity of citrate binding to COM crystals, Qiu et al.13 used molecular modeling with energy minimization to calculate binding energies of citrate at steps and surfaces of COM crystals for several possible configurations and showed that the strength of the citrate-COM interaction is
Figure 7. (a) Schematic representation of the morphology of a COM crystal showing the expressed faces. D{120} and D{010} are the respective perpendicular distances between the edges of the {120} and {010} faces measured on the (-101) face. (b) Effect of citrate on the morphologies of COM crystals as expressed by the D{120}/D{010} ratio. The lower values of the ratio compared with the controls indicate inhibition of the (-101) face and/or (-120) face.
much greater at steps than on terraces and is highly step specific. The maximum binding energy, -166.5 kJ mol-1, occurs for the [101] step on the (-101) face. The binding energies for steps of the (010) face are much smaller, reaching as little as -56.9 kJ mol-1 for the [120] step. All other step binding energies lie between these extremes.14 This high selectivity leads to preferential binding of citrate to the acute [101] step on the (-101) face. This specificity of the citrate interaction with the acute [101] step is achieved due to the excellent matching between the separation of carboxylic groups of citrate and the calcium ion distribution in the step.14 Moreover, the acute geometry allows multiple accommodations of carboxylate groups with the least amount of distortion, as the distance between two adjacent calcium ions along the [010] direction is 7.3 Å, which is close to the 7.0 Å distance between the two end carboxylic groups of citrate.14 Meanwhile, the hydrogen atom from the hydroxyl group forms a hydrogen bond with one of the oxygen atoms
COM Crystallization Kinetics
within the oxalate ions in the basal plane. This steric recognition behaves as a pivot for the citrate molecule and also stabilizes the attachment when it is docked to this step.14 Strong citrate-step interactions on the (-101) face leads to pinning of all steps, while the weak interactions with the steps on the (010) face produces little or no step pinning. These resultant changes in the anisotropy of the step kinetics are, in turn, responsible for changes in the shape of macroscopic COM crystals. Thus, the molecular predictions are fully consistent with the experimental observations.14 Both macro- and microscale results show that overall crystallization kinetics and crystal shape are being controlled by the elementary step kinetics and track the predictions of the model presented here. Summary and Conclusions The crystallization kinetics of COM was investigated using the CC method both in the absence and in the presence of citrate at 37 °C and I ) 0.15 mol L-1. The normalized COM growth rates gradually decreased with the increase of citrate concentrations. For citrate concentrations of 1 × 10-5, 5 × 10-5, 1 × 10-4, 2.5 × 10-4, 5.0 × 10-4, and 1 × 10-3 mol L-1, Ri/R0 ) constant ) 0.82 ( 0.04, 0.72 ( 0.03, 0.50 ( 0.02, 0.35 ( 0.03, 0.20 ( 0.02, and 0.07 ( 0.01, respectively, regardless of supersaturations. The CC crystallization kinetics data are well fit by the C-V model, provided a decrease in kinetic coefficient with increasing citrate level for all faces is included. These results are in good agreement with AFM measurements on the (-101) and (010) faces. The macroscopic habit of the COM crystals changed to a disklike shape with increasing citrate concentration. The dependence of aspect ratio on citrate level is consistent with the model fit to the CC data. Moreover, it agrees with the previous AFM results as well as the molecular-scale analyses that show highly selective COM-citrate interactions with preferential binding to steps of the (-101) face and weak interactions with the other faces. We conclude that the behavior of the COMcitrate system is in excellent agreement with the predictions of the C-V model, with the modification that, in addition to causing step pinning through the Gibbs-Thomson effect, citrate also impacts the kinetic coefficient of step motion on all faces. A simple kink poisoning model is consistent with the latter, but no independent proof for this hypothesis exists. Acknowledgment. This work was supported by research grants from the National Institutes of Health (Nos. DE03223, DK33501, and DK61673). This work was performed, in part, under the auspices of the U.S. Department of Energy by the University of California, Lawrence Livermore National Laboratory, under Contract No. W-7405-Eng-48. References (1) (2) (3) (4) (5) (6) (7) (8)
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