Dynamics of Biomineral Formation at the Near-Molecular Level

4784

Chem. Rev. 2008, 108, 4784–4822

Dynamics of Biomineral Formation at the Near-Molecular Level S. Roger Qiu and Christine A. Orme* Physical and Life Sciences Directorate, Lawrence Livermore National Laboratory, 7000 East Avenue, Mailstop L-367, Livermore, California 94550 Received April 24, 2008

Contents 1. Introduction 2. Solution Speciation and Crystal Growth Parameters 3. Monitoring Crystal Growth using Scanning Probe Microscopy 3.1. Instrumentation and Measurement Considerations 3.2. Step Motion 3.2.1. Derivation of the Relationship between Step Velocity and Supersaturation 3.2.2. Kinetic Coefficient 3.3. Minimum Step Lengths for Step Motion 3.3.1. Critical Lengths for Step Motion 3.3.2. Step Movement Length Based on Kinetic Arguments 3.4. Geometric Effects on Kinetics 3.4.1. Dependence of Step Kinetics on Step Length 3.4.2. The Dependence of Step Kinetics on Inside Corners 3.5. Hillock Geometry (Step Density) 3.6. Impurity Interactions 3.6.1. Effects of Additives on Step Kinetics 3.6.2. Step Pinning Models 3.6.3. Effects of Additives on Shape 4. Pathological Crystallization 5. Calcium Oxalate Monohydrate 5.1. Structure and Growth in Pure Solution 5.2. Metal Ions 5.3. Citrate 5.4. Naturally Occurring Macromolecules 5.5. Synthetic Macromolecules 5.5.1. Synthetic 27-Residue Peptides 5.5.2. Synthetic PolyD, PolyE, and PolyAA 6. Brushite 6.1. Growth and Dissolution of Brushite in Solutions without Impurities 6.1.1. Atomic Structure of Brushite 6.1.2. Crystal Habit 6.1.3. Steps Structure and Relative Kinetic Coefficients 6.2. Size-Dependent Phenomena during Dissolution 6.2.1. Bulk Dissolution Behavior 6.2.2. Etch Pit Observations 6.3. Additives with Carboxyl Moieties 6.3.1. Interaction of Citrate with Brushite

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6.3.2. Interplay between Brushite and COM 6.3.3. Interplay between Brushite and Apatite 6.3.4. The Chelation of Calcium 7. Hydroxylapatite 7.1. Growth on the Prism {10-10} Planes 7.2. Growth on the Basal {0001} Plane 8. Outlook 9. Acknowledgments 10. References

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1. Introduction Biomineral formation is critical to biological functions of living organisms. For this reason, the synthesis of complex organic-inorganic composites that typify mineralized tissue such as bones and teeth and also the prevention of pathological mineralization that manifests as kidney stones, arthritis, plaque formation, and caries are important challenges for the biomineralization community. One piece of this challenge is a more complete understanding of the biological environment and the regulatory processes that lead to the creation, inhibition, and dissolution of mineralized tissues. This begins with the tissue itselfsits composition and structure, its material properties, and its evolution during formation or dissolution. Equally important is the biological milieu that modifies the crystallization process, including the solution state, the proteins or macromolecules, and the matrix materials that guide nucleation and growth. And the third piece is the regulatory pathway and cellular processes that choreograph the initiation of mineralization and the evolution of the crystallizing environment. Downstream from the biological process, another area of the biomineralization field concentrates primarily on the question of mineral formation/dissolution and typically utilizes model environments inspired from, but not faithful to, the biological milieu. Questions of mineral modulation are broken down into the more tractable questions of synthesisshow does one control the shape, size, orientation, growth rate, or polymorph of a mineral using templates, macromolecules, or environments adapted from those found in biosystems but separated from questions of biological regulation. The goal of these simplified experiments is to develop guiding principles that can be used to better understand biomineralization within the organism as well as to enable the synthesis of new materials. Within the biomineralization literature, there are several mechanistically distinct paradigms (Figure 1) that are thought to guide synthesis in biomineral systems: these include the use of templates to direct growth, the use of molds to shape amorphous phases, the use of directed aggregation to assemble nanoparticles, and the use of additives to modify

10.1021/cr800322u CCC: $71.00  2008 American Chemical Society Published on Web 11/12/2008

Biomineral Formation at the Near-Molecular Level

S. Roger Qiu is a physicist within the Physical and Life Science Directorate of Lawrence Livermore National Laboratory. His research interests include biomineralization, biomolecular imaging, dynamics at the solid/liquid, solid/ vapor interface, and crystal growth. He received his Ph.D. degree in physics from the University of California, Riverside in 2000, with a National Science Foundation IGERT Fellowship, where he discovered a new mechanism for self-limiting film growth on a transition metal surface employing molecular precursors and developed a novel and safer means to remove radioactive pollutants from environments. He was a senior physicist at Semiconductor Solutions, Schlumberger Technologies, San Jose, CA, before taking a position at the Lawrence Livermore National Laboratory in 2002, where he studied the biological control over crystallization. He is a member of the American Association for Crystal Growth and the Materials Research Society.

Christine A. Orme is a physicist within the Physical and Life Science Directorate of Lawrence Livermore National Laboratory. She studied physics at the University of Michigan, receiving her Ph.D. in 1995 in the area of surface evolution during vapor deposition. In 1996 she joined Lawrence Livermore National Laboratory, where she has served as a project leader, group leader, and institute director. Orme’s laboratory explores molecular processes at interfaces and uses in situ tools to address how these modify the way that materials assemble and disassemble. She is particularly interested in biomineralization, biomimetic approaches to material assembly, corrosion, andsunderlying all of thesesthe fundamental physics of growth and dissolution. She currently studies metal-organic interactions to understanding how ligands direct shape control during synthesis. Orme is active in several professional organizations, including the Materials Research Society, where she served as the Spring 2008 Meeting cochair, and the American Association for Crystal Growers, where she serves on the Executive Board. She is the recipient of a Science and Technology Award from Lawrence Livermore National Laboratory (2001), an Office of Science Early Career Scientist and Engineer Award (2002), and a Presidential Early Career Award in Science and Engineering (2002) for her work in biomineralization.

crystal growth. In practice, biomineralization (or bioinspired synthesis, more broadly) may draw from aspects of each of these mechanisms.

Chemical Reviews, 2008, Vol. 108, No. 11 4785

The first of these ideas is the use of templates to control the nucleation of minerals. Work in this area has demonstrated that substrates functionalized with organic molecules can be used to dictate the location, density, and orientation of biominerals.1 It is thought that control is achieved through a combination of surface charge, surface energy, and surface structure. However, there is ongoing discussion as to the relative weight of these terms. Current research explores when the concept of epitaxial fit does and does not apply2-4 as well as the extent to which both the mineral and the organic template rearrange to accommodate the interface between them.5,6 To independently test the influence of charge, structure, and surface energy, most studies make use of artificial templates such as alkanethiol self-assembled monolayers (SAMs) and Langmuir-Blodgett films, which can be systematically altered. Both systems have been recently reviewed.4,7 The second paradigm explores the use of two and threedimensional molds that shape materials when they are in an amorphous phase. The amorphous materials may then transform to a crystalline material retaining, to a large extent, their molded shape. As amorphous phases are known for both calcium phosphates8,9 (ACPs) and calcium carbonates10,11 (ACCs) (as well as several other phosphate minerals12,13), this is an attractive route for creating more complex shapes. In organisms, the amorphous phases of calcium carbonates and phosphates do not typically persist through maturation, and minerals are more commonly found in a crystalline phase in the adult. It is currently not believed that ACP is found in nascent bone14,15 although transient ACP phases have been observed in chiton teeth.16 ACC has been found as a precursor to crystalline calcite in sea urchin larvae17,18 as well as a precursor to aragonite in sea urchin spines.19 In recent years, progress has been made in describing the physics and chemistry that describe the crystallization process and in elaborating on the potential such an approach might have as a synthetic route.20-22 A description of biomineralization stemming from amorphous precursors explores concepts in liquid-liquid phase separation, phase transitions, hydration, and dehydration. A review of the role of ACC in biomineralization can be found in ref 23. The third paradigm considers the aggregation of nanoscale to micron-scale building blocks, akin to colloidal assembly. The building blocks may be amorphous or crystalline and attach either through alignment of specific facets24 or via mediating additives such as surfactants.25,26 The underlying physics and chemistry draw from aspects of nanoparticle synthesis and surface functionalization along with the characterization of these nanomaterials. Assembly relies on surface properties such as surface charge and surface energy and the diffusional properties of these building blocks. A recent review can be found in ref 27. The fourth paradigm, and the focus of this review, considers how additives and crystal growth parameters influence crystal morphology and kinetics. Crystal growth starts from the point of view that the solution environment alters the free energy landscape of the mineral surface and, in doing so, changes the activation barriers associated with adsorption and desorption of molecules to and from the surface. Additives, which vary in complexity from small ions such as Mg2+ to large protein aggregates, interact with the surface in several mechanistically distinct ways and can alter either the energetics of the surface or the kinetics of incorporation (or both). This area has its roots in the physical

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Figure 1. Schematic representation of four strategies used to engineer crystalline materials. Pictures in the lower panels are examples for each category. All examples are of calcite. (a) Calcite crystals nucleated onto a patterned alkanethiol substrate. Reproduced with permission from ref 1. Copyright 2004 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim. (b) Molds enable complex three-dimensional shapes. Schematic reproduced with permission from ref 1. Copyright 2004 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim. Example of a single crystal precipitated from an amorphous phase within polymer replicas of sea urchin skeletal plates. Reproduced with permission from ref 22. Copyright 2002 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim. (c) Aggregation of amorphous phases or aggregation of crystalline building blocks represents another growth mode. Reproduced with permission from refs 25 and 26. Copyright 2003 and 2004 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim. (d) Additives modify the free energy landscape (schematically depicted) by altering the kinetic barriers associated with moving molecules between the solution phase and the crystal. Reprinted with permission from ref 40. Copyright 2001 Nature Publishing Group. Table 1. Chemical Formula and Solubility Products (Ksp) for Calcium Containing Biominerals Discussed within the Text chemical name 32,33

calcium carbonate calcium hydrogen phosphate dihydrate (DCPD)d34,35 hydroxylapatite (HAP)e38,39 calcium oxalate monohydrate (COM)36,37

mineral name

crystalline product

reactants

solubility –log Ksp

na

calcite brushite

Ca + CO3 Ca2+ + HPO42- + 2H2O

CaCO3 CaHPO4•2H2O

8.48 6.63c

2 2

structure R3jc Cc

apatite whewellite

5Ca2+ + OH- + 3PO43Ca2+ + C2O42- + H2O

Ca5(OH)1(PO4)3 CaC2O4•H2O

58.65c 8.78c

9 2

P63/m P21/n

2+

2-

b

a The number of growth units per molecule, assuming the reactants shown and omitting waters. b 25 °C. c 37 °C. d DCPD is an abbreviation for dicalcium phosphate dihydrate (which is nonstandard chemical nomenclature). e The reaction for one molecule (half of a unit cell is shown). Often called hydroxyapatite rather than the more chemically correct hydroxylapatite.

chemistry of solutions, mineral crystallography, and the models of crystal growth developed in the 1950s by Burton, Cabrera, and Frank.28 Nevertheless, it is somewhat misleading to call this paradigm traditional crystal growth because the models needed to adequately describe biomineral growth have not been established yet. Scanning probe microscopy29 (SPM) is the most recent of a set of observational tools, such as constant composition methods30,31 (CC), interferometry, and in situ scanning xray diffraction (SXRD), that have helped transform the study of aqueous crystal growth by emphasizing real-time quantification of kinetics. While these observations are in no way limited to biominerals, many of the most critical studies have been applied to biomineral surfaces, and for this reason, the study of biomineralization is inextricably interwoven with advances in crystal growth concepts. This review will discuss SPM studies of calcium-based biominerals (Table 1). The minerals include calcium hydrogen phosphate dihydrate (frequently denoted as DCPD),32,33 calcium oxalate monohydrate (COM),34,35 and hydroxylapatite (HAP).36,37 All of these minerals are considered sparingly soluble, spanning from the most soluble brushite32 to the least soluble HAP.36 The primary focus will be minerals found in the body under pathological or healthy settings (COM, DCPD, HAP). The review will also discuss examples of calcium carbonate in the polymorph calcite,38,39

where they serve to illustrate aspects of crystal growth but will not fully cover the literature. To address differences in specialization of the diverse biomineralization community, this review includes background material in the area of solution speciation and crystal growth. The review will only superficially cover the topics of nucleation and the advances in crystal growth that have resulted from SPM studies of organic crystals. There are several recurring themes that appear throughout the discussion. First, interactions at steps play a defining role in the morphology and kinetics of biominerals, and in many cases, step edges (as opposed to facets) serve as the molecular docking sites for growth modifiers.40,41 The second is the sheer diversity of mechanisms by which additives change the rate of crystal growth. Results from calcium oxalate, calcium carbonate, and calcium phosphate mineral suggest that biominerals control rate and shape using a combination of kinetic and thermodynamics controls. These include the following: • altering the kink or step density;42 • changing the kinetic barriers associated with moving ions between the solution and the solid;43-45 • pinning or blocking step motion;46 • chelating solution ions, thereby reducing the supersaturation;47-49 or



Table 2. Composition Ranges of Inorganic Ions (in mM) in Human Fluids with Corresponding Mineral Saturation States of Phosphatesa,c Na+ ClCa2+ Pi HCO3K+ Mg2+ FSO42NH4+ Growth Parameters pH {Ca}/{Pi} I (mM) σgu DCPD σgu HAP

serum

saliva55,56

130-150 99-110 2.1-2.9 0.74-1.5 8.2-9.3 3.6-5.6 0.74-1.5 0.01-0.02 0.08-0.12

10 23 0.40-2.1 2.9-11 2.1-25 23 0.21 0.005

enamel fluid57

plaque fluid58

140 150 0.5 3.9 10 21 0.8 0.005

4 7.4 1.31-2.54 152-153 -0.9-0.5 2.3-2.7

5.5-7.5 0.02-0.69 39-46 -1.6-1.0 -0.2-3.8

7.2-7.3 0.145b 165b -0.50b 2.22b

urine59

6.8-50 14-52 0.8-8.6 7.8-29

50-250 64-380 0.81-7.8 7.2-45

38-90 1.3-5.5 0.0013-0.018

20-96 0.70-7.8

19-64

0.50-50 10-56

5.69-7.08 0.01-1.40 152-155 -1.1-1.7 0.2-4.2

4.8-8.0 0.05-1.89 258-274 -2.1-1.6 -1.3-4.8

a σgu is defined by eq 2. Species in the top portion of the table are put into a speciation program (with many more complexes than shown). The results of the speciation are summarized by the growth parameters in the lower portion of the table. Reproduced with permission from ref 54. Copyright 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim. b Average values.cMany of the ions (particularly Ca) complex with proteins and other organic molecules also found in these fluids. The “free” rather than total values are given in the table.

• incorporating into the solid, thereby shifting the mineral solubility.50 The crystal shape is modified when any one of these rate controlling variables alter specific steps or facets. For shape modification, it is important to consider why certain steps or facets are engaged while others are not, and this specificity often involves stereochemistry.51 With the quantification made available by in situ tools such as SPM, the community is at a point where we can move beyond terminology that describes the end result alone, such as “shape modifier” and “inhibitor”, to address the mechanism by which these effects occur.

2. Solution Speciation and Crystal Growth Parameters In situ approaches, such as constant composition methods, interferometry, and SPM, have the opportunity to measure dynamics with sufficient accuracy that control and quantification of environmental parameters have become increasingly important. Accordingly, solution speciation modeling,52 which can be used to translate the solution composition into parameters that control crystal growth, has become an essential part of quantitative SPM experiments. Speciation programs, made widely available by the geochemical community, calculate the concentrations and activities of all solution ions and complexes from a list of initial reactants and a database of potential reactions with their respective association constants (Ka) and solubility products (Ksp).53 Parameters that drive crystal growth, such as the supersaturation, pH, ionic strength, and ratio of cations to anions, are calculated from the species activities. Table 2 shows the speciation and resultant crystal growth parameters54 for several common body fluids.55-59 While the solution properties (such as supersaturation, pH, ionic strength, and ratio of cations to anions) determine the thermodynamic relationship between the solution and the crystal, these parameters can also affect kinetics. In general, these effects cannot be calculated a priori, as they alter activation barriers or surface speciation, where little is known about the exact nature of the states. This is, in fact, one of the areas that SPM experiments address by measuring step kinetics as a function of these parameters. The qualitative effect of each of these variables is summarized in Table 3.

Table 3. Crystal Growth Parameters parameter (symbol)

effect near crystal surface

net flux to surface; determines growth mode (island nucleation vs step flow) net charge of surface; hydrated state of surface ratio of growth kinetics of incorporation when activation units (R) barriers for different ions are not the same ionic strength (I) Debye length of the double layer temperature kinetics of adsorption, desorption, (T) diffusion additive concentration various actions: step-pinning, surfactant, (Ci) blocking layer, incorporation, chelator, etc. supersaturation (σ) pH

The most important crystallization parameter is the thermodynamic driving force or the supersaturation. The supersaturation, σ ) ∆µ/kBT, is a unitless number proportional to the chemical potential difference associated with molecules transfering from the bulk solution to the bulk solid phase and can be determined from speciation calculations. Three related representations for the driving force are found in the literature: the supersaturation, the supersaturation ratio, and the relatiVe supersaturation. The supersaturation ratio can be computed by using the solution speciation results to calculate the ion activity products (IAP) for minerals of interest. The supersaturation ratio, S, is then given by

∏i aim

i

S)

Ksp

,

Sbrushite )

{Ca2+}{HPO42-} Ksp,brushite

(1)

where the product multiplies the activity of each species, ai, raised to the power of the number of each species in the chemical formula, mi. The supersaturation ratio for the calcium phosphate mineral brushite is shown explicitly. For S ) 1, the mineral and solution are in equilibrium, for S < 1, the solution is undersaturated and the mineral will dissolve, and for S > 1, the solution is supersaturated and the mineral will grow. In this example, one molecule of brushite has two independent ions (Ca2+ and HPO42-) that play a role in crystallization. These are termed growth units. Traditionally, the supersaturation is written per molecule rather than per growth unit and is related to S through σ ) ∆µ/kBT )


Qiu and Orme

ln S. To allow comparisons between materials with different numbers of growth units, it is useful to define a driving force that is normalized for the number of growth units (n),

∆µgu σ 1 ) ln S1/n ) ln S ) (2) kT n n where “gu” denotes growth unit. The assumed growth units of crystals discussed in this review are given in Table 1. The third definition of supersaturation is used to describe the kinetics of steps. The description of step kinetics stems from the laws of first order rate equations. As shown explicitly in section 3.2.1, step velocity (v) is proportional to e∆µ/kBT - 1, where ∆µ is the difference in chemical potential between the solution and activated state, kB is Boltzmann’s constant, and T is absolute temperature. This leads to a step velocity that is proportional to the relative supersaturation, defined as σgu )

σrel ) S1/n - 1

(3)

When only one growth unit is involved (n ) 1), this can be written as the difference between the actual activity (a) and the equilibrium activity (ae) to give the commonly used expression, σrel ) S - 1 ) (a - ae)/ae. Notice that near saturation (S ∼ 1) the logarithm in the definition of σgu can be expanded (ln(x) ≈ 1 - x for x ≈ 1) such that σgu ≈ σrel. In the literature, σgu and σrel are often used interchangeably. This is only reasonable near saturation. Similarly, σrel ) S - 1 is only correct for single component crystals with n ) 1. Positive values of σ, σgu, and σrel favor crystal growth. Most solutions have a metastable region, where the solution is supersaturated but not enough to overcome the energy barrier that prevents crystals from spontaneously precipitating from the solution phase (on reasonable time scales). SPM experiments typically operate in this regime where growth occurs on existing crystal surfaces without nucleating new crystals. Because the supersaturation also determines the step velocity (through σrel) and the step density (as shown in section 3.4), this parameter is used to tune the solution state such that dynamics are within the measurement limitations of SPM. The second parameter, pH, affects both the solution as well as the mineral surface. In mineral systems, a shift to lower pH will typically lower the saturation state. For example, in phosphate systems, this occurs by shifting the balance of phosphate species from PO43- to HPO42- to H2PO4- as the pH is lowered, and similarly in carbonate systems, lower pH shifts species from CO32- to HCO3- to H2CO3. At the mineral surface, the pH can shift the surface charge by changing the distribution of proton and hydroxyl groups hydrating the interface. In principle, these may change the activation barriers associated with crystallization, as the hydrating ions must be removed to allow a crystallizing ion to join the crystal surface. The third parameter, ionic strength plays a role in screening both ion-ion electrostatic interactions in solution (which is accounted for by the activity coefficient) and electrostatic interactions between ions in solution and the surface. The ionic strength of a solution, I, is defined as

I ) 1/2∑ [i]zi2

(4)

i

where [i] is the concentration and zi the charge of each ionic species, i. Most biomineralization studies operate at relatively high ionic strengths of 0.15-0.5 M to simulate body fluids

or seawater environments and thus necessarily use Davies60,61 or B-dot62 activity coefficients that extend the utility of the Debye-Huckel activity coefficients to higher ionic strength values. The cation to anion activity ratio (for example of calcium to phosphate or calcium to carbonate) acknowledges that the activation barriers for the cation and anion may not be the same. In these cases, the growth rates will depend on the concentration of the rate limiting ion rather than simply the supersaturation.63-67 The growth of a multispecies crystal relies on the relative rates of adsorption and desorption of the various ions or growth units that make up the unit cell. A simple salt such as NaCl has two growth units (n ) 2 in eqs 2 and 3), namely Na+ and Cl- ions. However, in general, the growth units represent the pathway with the lowest activation barrier that allows an ion to move from the solution state to the solid state and vice versa. In a binary ionic compound such as brushite (CaHPO4 · 2H2O), it is tempting to think of the growth units as Ca2+ and HPO42-, but it is possible that in the process of shedding waters of hydration and incorporating into the solid that the activation barrier is lower for a multistep process wherein one of the other phosphate complexes (e.g., H2PO4- or PO43-) adsorbs and then adds or sheds a hydrogen. Several studies have shown that ion ratios play a role in kinetics,68-73 and progress has been made at modeling these effects;63-67,74 however, most crystal growth models assume a single species, and more work is needed to fully describe multicomponent crystals. In this review, we assume the growth units are those shown as the reactants in Table 1.

3. Monitoring Crystal Growth using Scanning Probe Microscopy This section discusses the use of scanning probe microscopy (SPM) to investigate solid-solution interfaces. We begin with instrumental considerations and in particular the influence of instrument resolution and environmental fidelity on measurement. We then discuss the types of kinetic measurements made by SPM and put these in the context of the classical arguments of Gibbs-Thomson75-77 and Burton, Cabrera, and Frank.28 The high precision of SPM has opened up the possibility of critically testing these theoretical models, and in this section we show several examples where they fail. One of the underlying assumptions in these models is that steps have a high kink density due to thermal fluctuations. It is not yet clear whether this assumption breaks down in biominerals, but it is an area under active investigation and is one of the principle areas where the current spatial and temporal resolution limits of the instrument prevent a clear picture. In the last section, the effects of impurities are discussed.

3.1. Instrumentation and Measurement Considerations There are three elements requiring control for successful imaging of crystal growth: (1) the scanning probe microscope; (2) the crystal substrate; and (3) the solution environment. The solution-crystal system determines the type and speed of various interfacial processes (such as attachment, detachment, and diffusion) and thus sets the time and length scale of the events of interest. Whereas the instrument, with its associated spatial and temporal limits, defines which


surface processes are observable, together these determine the spatial and temporal scales of the surface processes that are imaged. Scanning probe microscopy (SPM) measures surface morphology by rastering a sharp probe over a surface. By tracking the probe’s vertical movement as it crosses over surface features, the microscope is able to create topographic maps of the surface with subangstrom resolution. What makes scanning probe microscopy invaluable for studying crystal growth is its ability to image atomic steps in fluid environments. Since its invention in 1986,29 many books have been written that discuss operating modes, instrumentation, imaging artifacts, and interfacial forces; the reader is referred to these for most details on the technique itself.78,79 Because the lateral resolution is limited by the probe used to scan the surface (typical probe diameters are 10 nm), the individual growth units (such as Ca2+, HPO42-, or PO43-) of a growing crystal are not reliably resolvable. While this is changing as instrumentation and tips improve, this currently limits imaging of features smaller than ∼5 nm such as kinks, adatoms, small clusters, and short steps. The exception is crystals composed of large building blocks such as proteins where kink density, kink motion, and kink attachment/detachment rates can be explicitly measured.80,81 Dynamic measurements are limited by both the lateral and the temporal resolution of the instrument. The spatial resolution has contributions from the finite size of the SPM tip and, at higher scan rates, the finite response time of the instrument. The temporal resolution is a function of the entire system and, thus, is a function of the cantilever, the photodiode detector, the feedback circuit, and the piezotransducer. However, for most commercial systems in contact mode, the height image has a response time limited by the piezotransducer and the feedback circuit. This is typically 1 kHz but is dependent on the instrument and gain settings. The error signal (deflection mode) has a faster response time than the closed loop height image and is limited by the cantilever resonance (∼10-15 kHz for most contact mode tips). The faster response time of the error image allows greater scan speeds but is eventually limited by the stability of the piezoscanner, which resonates due to the impulse associated with changing direction at the beginning and end of the scan. Also, only relatively flat and hard surfaces can accommodate the variable force that results from higher scan speed and gains insufficient to fully track the surface. However, widely spaced atomic steps on mineral surfaces are good candidates for this partially open loop mode of operation, and for this reason most dynamic biomineral images are displayed in error mode where the step edges are highlighted. The instrument limits give rise to minimum pixel dimensions ∼3-5 nm (set by the probe) and maximum scan speeds less than approximately 15 Hz (set by the instrument feedback and mechanics). As discussed by Higgins and Hu, these resolution limits have implications for simultaneously measuring step lengths and step kinetics.82 To move beyond these limitations, new instruments and tips are needed that improve spatial and temporal resolution. Although true atomic resolution was demonstrated in the 1990s,83,84 it was not routine. Over time, a new generation of instruments has been developed and demonstrated in both vacuum85 and fluid environments.86,87 Although these instruments are not designed to have the high temporal resolution needed for many kinetic measurements, they have the potential to greatly improve our ability to understand


Figure 2. A faceted crystal surface is composed of steps, terraces, and adatoms. Reprinted with permission from ref 81. Copyright 2007 American Chemical Society.

adsorbates on steps and terraces and near equilibrium phenomena. At the same time, several approaches have been used to improve temporal resolution.88 These include the development of smaller cantilevers with concomitant higher frequencies,89,90 more rapid vertical transducers,91,92 improved feedback electronics,93 and faster scanning transducers.94,95 These improvements are interrelated and must be addressed simultaneously to realize the full potential of fastscanning instruments. Although some crystal growth studies have been performed on these faster scanning instruments,96 most studies are constrained by the limitations of commercially available instruments. However, these limits are constantly evolving and the future should see more emphasis placed on kinks and kink dynamics rather than the step and step dynamics that are the focus of this review. The topography of a crystal surface is composed of steps, islands, kinks, and adatoms (Figure 2). At modest supersaturations, crystals grow by attachment to existing steps, usually created by dislocations. Above a critical supersaturation, growth will proceed via two-dimensional nucleation of islands. And at yet higher supersaturation, threedimensional crystals will precipitate in the solution phase. All three regimes are important for biomineralization; however, only step-flow growth and two-dimensional nucleation can be visualized via SPM, and of these, most studies to-date have focused on step-flow. The solution saturation state is the primary “knob” that experimentalists use to ensure that step speeds, critical lengths, and step densities fall within a range that can be measured by SPM. The control of environmental conditions (such as flow, pH, and temperature) is an essential component of quantitative crystal growth.97,98 Figure 3 illustrates a typical arrangement where temperature- and pH-controlled solutions are pumped through a fluid cell as the crystal is being imaged. Ideally, the hydrodynamics are such that diffusive and convective transport within the fluid cell compensates the growth or dissolution at the crystal surface, making the ion concentrations at the crystal surface identical to that of the reservoir. In reality, the solution near the crystal interface is influenced by the reactions at the crystal surface. In some in situ studies, custom fluid cells have been designed to explicitly test and model flow and concentration within the cell.99,100 This approach allows quantitative assessment of kinetics even when concentrations are locally depleted or enhanced. More commonly, the fluid flow is increased until the surface kinetics are independent of flow rate.101 This helps to reduce the limitations of mass transport within the cell but does not ensure that conditions at the crystal are identical to those at the reservoir. To address this issue, finite element modeling has been used to estimate the concentration profile directly under the AFM tip. For the case of a calcium oxalate monohydrate crystal growing in a commercial fluid cell,102 it was found that the solute is depleted by ∼3% due


Qiu and Orme

that step mobility is limited by the rate of nucleating kinks. The study of step fluctuations has been used to great advantage in metal and semiconductor evolution to estimate kink densities.103 Most of these studies use scanning tunneling microscopy to image spatial and temporal fluctuations in the equilibrium step configuration. This type of study is not yet part of the biomineral literature in large part due to the difficulty in achieving atomic resolution.

3.2. Step Motion

Figure 3. (a) A schematic drawing that shows a common fluid cell setup where a peristaltic pump is used. This particular fluid cell is made of glass and uses a silicone o-ring. In this configuration, the distance between the bottom of the fluid cell and the sample and the o-ring diameter defines the internal volume, which is ∼50 µL. (b) Calculated supersaturation variation within the fluid cell. Reprinted with permission from ref 102. Copyright 2006 American Chemical Society. Table 4. Summary of Measurable Quantities Common in SPM Experiments step measurable

determines

step velocity critical step length morphology step-edge fluctuations

kinetic coefficient (β) step-edge free energy (γedge) stable step orientations; pinning kink attachment/detachment rates; kink density

to solute consumption and obstructed flow by the cantilever. This was found for a typical growth solution under a wide range of flow rates or Reynolds numbers. Results are expected to be similar for other crystals with similar or slower growth rates under flow conditions typical of quantitative measurements. (For reference purposes, a typical aqueous flow rate of 1 mL/min ∼ 0.02 cm3/s corresponds to a Reynolds number of 26 for the commercial cell shown). Although this supersaturation change is small for calcium oxalate monohydrate (and by analogy hydroxylapatite and brushite growing under similar conditions), it will be more significant for crystals composed of slowly diffusing molecules or for crystals with faster growth rates. In these cases, custom cells99 or modeling102 may be needed to interpret kinetic measurements. Dynamic measurements include the velocity of atomic steps, the critical length for step motion, step morphology, and step fluctuations. What these measurables reveal about crystal growth is summarized in Table 4. As will be shown, the velocity of a step depends upon its step length. In the following sections, we first discuss step motion for the two extremes of step lengths, namely the motion of an infinitely long step and the lack of motion for steps shorter than the critical length. Both of these extremes appeal to thermodynamic arguments and in particular assume that steps are populated with an equilibrium distribution of kinks. We then discuss step motion between these extremes, which depends upon the kink density. At high kink density, steps are not kinetically limited from moving, and at low kink density, they can be (depending on the length of the step relative to the average distance between kinks). In several examples, we show evidence of step motion that does not follow the dynamics predicted by the high kink density limit, suggesting

Step velocities, when measured as a function of solution parameters, provide a wealth of information related to the activation barriers that set the rate at which ions or molecules transfer between the solution and the solid phase. The sensitivity of step kinetics to most solution parameters makes it a responsive probe of solute-solid interactions. But this sensitivity also makes it challenging to interpret data when more than one variable is changed (as is often the case). For example, one expects the step velocity to vary with pH, supersaturation, temperature, impurity concentrations, and, in multicomponent systems, the ratios of species. Thus, interpretation of velocity data relies on detailed knowledge of the solution state, motivating the previous discussion defining solution parameters. There are three standard ways to measure step kinetics: (1) measurement with respect to a fixed object; (2) measurement in a disabled mode; and (3) measurement using an apparent step angle. In all cases, it is advantageous to image at the site of a screw dislocation because, at this position, steps are continuously sourced and all step directions can be imaged simultaneously. These methods are discussed in the literature.82,97,104 Quantitative measurements of velocity versus supersaturation are found in the literature for a wide variety of solution grown crystals including minerals such as calcite,105,106 barite,107,108 hydroxylapatite;109 calcium oxalate monohydrate,41,110 brushite,97,111 optical crystals such as ammonium dihydrogen phosphate (ADP),112 and potassium dihydrogen phosphate (KDP);113 several proteins,81,114-117 and other organic crystals such as viruses,118 hydrogen bonded tapes,119 and uric acid.120 The study of velocity versus other solution variables such as ionic strength,121 pH,100,104,120 the cation to anion ratio,68,74 and temperature107,112,122,123 has also been quantified.

3.2.1. Derivation of the Relationship between Step Velocity and Supersaturation The relationship between the step kinetics and solution parameters is not simple, and details remain a matter of current investigation. However, the most straightforward derivation64,81,124,125 is worth detailing because it is the most commonly used and because many authors use variations that make systematic comparisons difficult. The simplest case describes a single-component crystal (A(aq) f A(s)) known as a Kossel crystal in which all kinks are the same. In the limit of infinitely long steps with growth occurring via attachment and detachment at kink sites and under the further assumption that nucleation of kink sites is not rate limiting, the step velocity (Vs) can be related to the motion of a single kink (Vk) by Vs ) b⊥FkVk, where b⊥ is the molecular distance in the direction of the step motion (or perpendicular to the step edge) and Fk is the kink density. In general, both Fk and Vk are functions of supersaturation (and other solution



parameters); however, in what follows, we will assume the high kink density limit and take Fk to be a constant independent of solution state. The motion of a single kink on a step is related to the difference between the rates of attachments (j+) and detachments (j-) in units of the number of molecules per site per unit time. The kink velocity, Vk is given by

Vk/b|| ) j+ - j-

(5)

where b| is the molecular distance parallel to the step. In equilibrium, the kink velocity is zero, implying that j+ ) j-. This will be used to relate the attachment and detachment rates to the solubility product. To rewrite in terms of known experimental values such as the supersaturation ratio (S) and the solubility product (Ksp), the rates of attachment and detachment must be expressed in terms of chemical potentials, allowing us to separate out the explicit concentration dependence from the other terms. The attachment and detachment rates are related to the chemical potentials of the starting (µ) and activated (µ*) states, by j( ) ν(e(µ-µ*)/kBT with µ ) µ0 + kBT ln a, where µ0 represents the standard state, ν( is a frequency factor commonly thought of as an attempt frequency but can also be thought of as diffusion over a repulsive barrier,81,126 and the activity is unitless, defined as a ) γc/c0.127 The choice of standard reference state (µ0) sets the definition of both the activity coefficient (γ) and the standard concentration (c0). For aqueous species, the standard state is typically chosen as the chemical potential of the hydrated ion in a hypothetical 1 molal solution and c0 is then 1 molal. (If molar or molal chemical potentials are used, then kBT f RT in the above expressions.) In the end, the units of the standard state will need to be converted to number of molecules per site per unit time, as needed for the definition of j(. During attachment, the starting state is the molecule within the aqueous phase (subscript aq). This leads to j+ ) ν+e(µ0,aq-µ*)/kBTa, which is usually simplified to j+ ) k+a. The concentration independent on-rate, k+, is often called the first order rate constant for attachment. During detachment, the starting state is the molecule within the crystal (subscript xt), where the activities are defined to be 1, leading to j- ) ν-e(µ0,xt-µ0*)/kBT ) k-. The kink velocity can then be written in terms of the chemical potentials as Vk/b| ) υ-e(µxt-µ*)/kBT[(υ+/υ-)e(µaq-µxt)/kBT - 1]. In equilibrium, the chemical potentials of the solution and solid state are defined to be equal, µext ) µeaq. Additionally, in equilibrium, the kink velocity is zero, which necessitates that the frequency factors be equal, υe+ ) υe-. This constrains the functional form of the frequency factor128 such that υ+ ) υ f(µaq/kBT) and υ- ) υ f(µxt/kBT), where f(µ/kBT) is, in general, an unknown function. To arrive at the most commonly utilized expression for the velocity, the frequency factors are equated such that υ+ ) υ- ) υ (or f(µ/kBT) ) 1). The kink velocity can then be written as

Vk * ) υe(µxt-µ )⁄kBT(e(µaq-µxt)⁄kBT - 1) b|

(6)

In several places in the text, we refer to this result, typical of first order chemical reactions,129 as V ∝ (e∆µ/kBT - 1), where ∆µ represents the change in chemical potential between the initial and final states. To connect with the chemistry literature and speciation databases, it is more convenient to rewrite in terms of the

first order rate constants (k(), the supersaturation ratio (S), and the solubility product (Ksp). Equating the attachment and detachment rates in equilibrium, j+ ) j-, and noting that the equilibrium activity is related to the solubility product leads to ae ) Ksp ) k-/k+. Using the definition of the supersaturation ratio S ) a/ae, the kink velocity then becomes Vk/b|| ) k+Ksp(S - 1), demonstrating that the kink mobility depends on the relative supersaturation (σrel ) S - 1). The step velocity is then Vs ) b||b⊥Fkk+Ksp(S - 1), where both S ) (γc)/(γece) and Ksp ) (γece)/(c0) are unitless but Ksp is given in terms of the standard state units. To re-express j( in terms of the number of molecules per site per unit time, the standard state units are converted to number of molecules per volume times the volume per site. When the standard m state is in molar units, this becomes Ksp [mol/L] * 10-3 3 3 # [L/cm ] Na [#/mol] Ω [cm /site] ) Ksp [#/cm3] Ω [cm3/site], where Na is Avogadro’s number and Ω is the molecular volume of the kink site. The kink density is itself a complex function that can depend on supersaturation, geometry, temperature, and, for multiple component crystals, the ratio of species.125 However, in the rough step limit, where kink-nucleation and supersaturation dependent effects are negligible compared to kinks produced by thermal fluctuations, the kink density may be expressed as a constant Fk ) 1/nkb|| with nk, the number of sites between kinks, estimated to be between 1 and 10.63,67,130 This leads to the common expression, first articulated by Chernov,124 # Vs ) βsΩKsp (S - 1)

(7)

where β is termed the kinetic coefficient with units of velocity and K#sp must be expressed as a number density such # that ΩKsp is unitless. From this expression, one can see that the step velocity is proportional to the relative supersaturation and to the mineral solubility. All orientation specific properties are contained within the kinetic coefficient. A similar derivation can be followed for crystals with more than one growth unit,64,74 resulting in expressions where both the kink velocity and kink density are functions of not only the ion activity product (through the supersaturation) but also the ion activity ratios of the multiple components. However, in the limit where the activities and the attachment and detachment rates are equivalent, then for a twocomponent crystal (Aaq + Baq T ABs), the step velocity has the form of a Kossel (single component) crystal except that the solubility product, the molecular volume, and the supersaturation ratio are adjusted for the number of growth units. For a two-component crystal such as calcite, brushite, # 1/2 or calcium oxalate, the step velocity is Vs ) βs(Ksp ) (ΩAB/ 1/2 2)(S - 1), where ΩAB is the molecular volume.64 More generally, for an n-component mineral,

Ωn 1⁄n (S - 1) (8) n For biominerals, (Ksp)1/n is typically expressed in molar units 1/n and requires the conversion (K#sp)1/n ) (Km [mol/L] * 10-3 sp) 3 [L/cm ] Na [#/mol]. The step kinetics depend directly on the solubility product (normalized per growth unit). All things being equal, crystals with lower solubility grow more slowly. This explains why the kinetics associated with sparingly soluble biominerals are amenable to SPM imaging whereas highly soluble crystals such as NaCl have step mobilities that are too fast to image except very close to equilibrium. # Vs ) βs √Ksp n


Qiu and Orme

Table 5. Comparison of Factors Influencing Step Kinetics for Several Calcium Mineralsa system DCPD dissolution [10-1]Cc DCPD growth [10-1]Cc HAP growth calcite COM [101] various proteins

Ωgu (×10-23 cm3)b

ae (#/cm3)c

6.16 6.16 2.95 3.06 5.47

3.87 × 10 3.87 × 1017 1.62 × 1014 3.47 × 1016 2.45 × 1016

βd (cm/s)

17

0.020 0.26-0.3 0.0033 0.26 0.1-0.26 0.2-420 × 10-4

citation 111 97 141 106 110 81

a Step velocities and relative supersaturation values typical of SPM measurements. b We distinguish between Ωu, the unit cell volume; Ω, the molecular volume, and Ωgu, the average volume per growth unit. They are related by Ω ) Ωu/Z and Ωgu ) Ω/n, where Z is the number of molecules per unit cell and n is the number of growth units per molecule (Table 1). c The solubility product converted to number density units, ae [#/cm3] ) 10-3[L/cm3] Na [#/mol] Kspm [mol/L]. d The values shown in the table were recalculated from literature values to reflect the definition of β shown in eq 8. Common variations found in the literature are the use of Ce rather than Ksp, the use of unit cell or molecular volume rather than the average volume of a growth unit, and the use of S rather than S1/n.

The ability of SPM to more quantitatively assess surface processes has led to data that tests and challenges step kinetic models. Several examples of this will be shown in upcoming sections. One of the open questions is whether step motion can be treated under the paradigm set forth by Burton, Cabrera, and Frank, where it is assumed that steps have a high concentration of kinks. The consequence of this assumption for step kinetics is that steps are treated as sinks for adatoms and kink nucleation is not a rate-limiting step. The validity of this approach depends on the distance between kinks compared to the step length. As was pointed out by Frenkel in 1945,131 steps in equilibrium above T ) 0 will necessarily contain kinks due to thermal fluctuations. Thus, from purely thermodynamic considerations, kinks are part of the step structure. The equilibrium distance between kinks, Lk, is given by Lk ) 1 /2b|(eε/kBT + 2), where b| is the intermolecular distance along the step motion and ε is the kink formation energy.28 Kink energies can vary over several orders of magnitude with reported values of ε/kT between 0.1 and 10 for metals and semiconductors.103 Reliable values are not known for the biominerals under consideration in this review, but the range of interest can be estimated from interfacial energy measurements obtained from nucleation experiments. These give ε/kT ) 2.4 and 2.5 for calcite and brushite, respectively.132 For ε/kT equal to 1, 3, and 5, the distance between kinks is 2, 11, and 75 atomic spacings, respectively. Thus, for typical atomic spacings of 0.5 nm, Lk is between 1 and 37 nm over this range. For step lengths less than or near Lk, the assumption of high kink density certainly fails and the nucleation of kinks from the flux (rather than through thermal fluctuations in the step) must be considered. The expression for the kink density then becomes a function of the supersaturation, Fk ) 2S1/2ne-ε/kBT rather than a constant, as used in the derivation of eq 8.125 These considerations also impact the minimum length for step motion as discussed in section 3.3.

3.2.2. Kinetic Coefficient There are many exceptions to the simple form shown in eq8,includingvariationsasafunctionofsteplength,82,107,133-135 variation due to low kink density,67,125,134,136,137 variation due to multiple kink types,64-66,74 and variation due to impurity interactions.138,139 Remarkably, given the number of caveats associated with the derivation, the linear relationship between step velocity and relative supersaturation holds well for many diverse systems, allowing the kinetic coefficient to be measured from the slope. The kinetic coefficient,

β)

b⊥ + b⊥ + (µ0,aq-µ*)⁄kBT k ) ν e nk nk

(9)

is an important quantity that contains (and hides) all of the physics associated with the desolvation, adsorption, diffusion, and incorporation. It is also a function of the number of molecules between kinks (nk), the attempt frequency (ν+), and the activation barrier, none of which are typically known. Land and De Yoreo tabulate kinetic coefficients for several inorganic and organic systems, demonstrating that they vary over several orders of magnitude.115 Vekilov expanded the list to compare over a dozen proteins.81 Interestingly, he found that the kinetic coefficient did not correlate with the mass or symmetry of the macromolecule. In a comparison between ferritin and apoferritin, two proteins that have identical outer shells that differ only in that ferritin has a ferrite core and thus a greater mass, Vekilov and co-workers found that the two proteins had identical kinetic coefficients.81 From this lack of mass dependence, they argue that the attempt frequency should not be thought of as the vibration frequency proposed by Eyring129 for gases, which scales as m-1/2, but rather a Kramer’s-style kinetic mechanism,140 which can be thought of as diffusion over a repulsive barrier. In a Kramer’s-type kinetic law, the prefactor corresponding to the “attempt frequency”, ν, is given by b⊥ν ) D/λ, where D is the diffusion coefficient and λ is a length scale associated with the barrier (and is typically ∼1 nm). As both apoferritin and ferritin have the same diameter and shell chemistry, they have the same diffusion coefficient, resulting in the same kinetic coefficient. Vekilov also found that the kinetic coefficient did not trend with the symmetry of the protein, as one would expect if rotational entropy played a significant role in the kinetic coefficient.81 Based on the mass and symmetry independence of the kinetic coefficient, he concluded that the kinetic coefficient is dominated by restructuring of water molecules due to the approach of the crystallizing molecule to the surface and that the activated state is better thought of as a higher energy state due to the partial rearrangement of the water shell rather than a state where the molecule is partially bound to the substrate but with stretched bonds (as is the case for an Eyring model). This is backed up by experiments on calcite that showed that changes to the kinetic coefficients scaled with the hydrophilicity of the interacting peptide.45 A comparison of several calcium biominerals (Table 5) shows that the kinetic coefficients for growth97,106,110,141 are within a factor of 3 of one another, with the exception of HAP, which is 2 orders of magnitude smaller. The small value of the kinetic coefficient for HAP is used to argue that



3.3.1. Critical Lengths for Step Motion

Figure 4. The relationship between critical length and step-edge free energies depends on the geometry of the crystal. Schematic of the addition of a row of atoms (in red) to an existing step for two geometries: (a) rhombohedral (such as calcite) and (b) triangular (such as brushite). Dark lines indicate the newly created edge length, each with its own step-edge free energy, γedge (J/m). The three corner types of calcitesobtuse-obtuse (oo), obtuse-acute (oa), and acute-acute (aa)sand two atom typessCa2+ (blue) versus CO32-(gray)sare indicated in part a. For brushite (b), only one atom type is shown.

The critical length needed for step motion is an outcome of the Gibbs-Thomson relation76,77 and is an important measure that provides a link between stable island or step size and the step-edge free energy. This is of particular importance to the area of biomineralization, as one of the ways that growth modifiers alter crystal growth is by changing surface and step energies.42 Although the idea of a critical length is an integral element of classical crystal growth models proposed in the 1950s,28 in situ approaches such as SPM were necessary for direct measurements, and thus, the critical length was first measured for spiral growth almost 50 years after it was first proposed.96,133 The critical length results from the competition between the chemical potential and the step-edge energy. When a step grows from a supersaturated solution, molecules move from the solution phase to the solid phase, reducing the energy of the system; however, step edges are created in the process, which increases the energy of the system. The critical length is the length at which these two opposite influences balance. The classical arguments are typically shown for the creation of an island or pit on a planar substrate. Here we show the argument for the incremental addition of a row of atoms to an existing island. For an island with a rhombohedral symmetry (Figure 4a), the incremental change in Gibbs free energy is

∆G ) -m∆µ + hb2γriser ∆x2

Figure 5. (a-d) Sequential AFM images illustrating the measurement of the critical length, Lc. The progression of the step indicated by an arrow is being monitored. In parts a and b, the step does not propagate because its length is less than the critical length, whereas in part c it begins to move, as indicated by the new edge that forms on the step labeled UR. The upper steps are acute; the lower steps are obtuse. A scale bar indicates 20 nm; capture time 1.2 s per image. Reprinted with permission from ref 96. Copyright 1998 American Institute of Physics. (e) For calcite the critical lengths were used to determine the step-edge free energy of the acute and obtuse steps. Reprinted with permission from ref 133. Copyright 1998 American Association for the Advancement of Science.

the growth unit in these systems is a cluster of molecules rather than the single ions.109

3.3. Minimum Step Lengths for Step Motion Steps must achieve a certain length before they are stable and will propagate forward. In this section we introduce two lengths: (1) the critical length Lc, which is based on thermodynamic arguments and assumes a sufficiently high kink density that equilibrium can be achieved, and (2) the kinetic length, Lkinetic, which is based on kinetic arguments and assumes low kink density. Steps shorter than the critical length cannot move, steps between Lc and Lkinetic may move with sufficient kink density, and steps larger than Lkinetic will move.

(10)

where m is the number of molecules that add to the step, ∆µ is the change in chemical potential per molecule, 2b2/2 is the change in step-edge length, h is the height of the step, and γriser ()γedge/h) is the step-edge free energy in units of energy per area. (To write the simplest expression, we are assuming the step-edge free energy is isotropic, such that γriserh ) γedge ) γο ) γa). Note that the length in the direction of step motion does not count as a new edge, as it existed before the row of molecules was added. The step length, L, is related to the number of molecules added by L ) mb1, where b1 is the dimension of a molecule parallel to the step (assuming an AB crystal such as calcite). Setting ∆G/∆x2 to zero and solving for the critical length, one obtains

Lc )

b1b2hγriser Ωγriser ) ∆µ ∆µ

(11)

where ∆µ ) kT ln S, as defined previously in section 2, and Ω is the molecular volume. The important feature is that the critical length is proportional to the step-edge free energy and inversely proportional to the supersaturation. Step lengths smaller than Lc are thermodynamically unstable and do not propagate, whereas steps greater than this length propagate forward (Figure 5). Experimentally, one can determine whether the step emerging from the dislocation source is stationary (subcritical) or moving (supercritical) by comparing its location relative to the dislocation source. After measuring the length of this initial step over numerous sequential images, the critical length is bounded by determining the longest subcritical step and the shortest supercritical step. The critical step length for a dislocation hillock was measured directly for the first time using SPM by Teng et al., demonstrating the inverse scaling with supersaturation133 (Figure 5e). Paloczi et al. demonstrated the virtue of high speed imaging for these measurements (Figure 5a-d).96


Figure 6. Examples demonstrating geometric effects on the step kinetics. (a) The step velocity depends on step length. Graph of normalized velocity as a function of step length normalized to the critical step length. The velocity rises to its infinite step value more quickly than is predicted by eq 14. Reprinted with permission from ref 133. Copyright 1998 American Association for the Advancement of Science. (b) SPM image (6 µm × 6 µm) of multiple merging hillocks on a growing brushite surface. The velocity of the [-101] step direction (noted with arrows in the schematic) depends upon the corner type and roughly doubles when it terminates at an inside rather than an outside corner. In scanned images, the observed step angles are a function of both the crystallographic orientation and the step velocity. To ascertain that the velocity rather than the orientation was altered, images were compared scanning both down and up. Reprinted with permission from ref 67. Copyright 2004 the Materials Research Society.

These authors used short cantilevers to achieve a frame time of 1.2 s per image, an improvement in temporal resolution of ∼10 compared to conventional SPM. Their work showed that the critical lengths differed for all four step directions, which cannot be explained from the discussion given above. Unfortunately, imaging was only performed on one hillock and thus may be a result of the defect structure of that particular hillock rather than a more general result. However, it is clear that high-speed imaging extends the supersaturation range and improves the quality of critical length measurements. Critical length measurements for calcite were revisited by Fan et al.101 SPM has also been used to estimate the critical island size for two-dimensional nucleation.118 As for spirals, the critical island size is determined by measuring the size for which islands first become stable. However, unlike step growth from dislocation sources, subcritical islands shrink rather than remain stationary. In general, the critical length will vary depending on the step orientation and geometry of the crystal. For the case of brushite (Figure 4b), the critical length of the ith step is Lc,i ) bih(γ1b1 + γ2b2 + γ3b3)/∆µ, and thus, the ratio of Lc1/ Lc2/Lc3 is expected to be b1/b2/b3. Similar arguments for calcite would predict that the critical length is independent of orientation, Lc,i ) b1b2h(γo + γa)/∆µ. Surprisingly, orientation dependence is observed for obtuse versus acute steps.101,133 This difference has been attributed to corner sites and is related to subtle differences between islands and spirals. When a step on a spiral begins to propagate, it must create a new corner site, which is not the case for islands where only edge length is added. The energy cost associated with each step direction then depends on these corner

Qiu and Orme

energies. Teng et al.133 argue that this is the case for calcite, which has three corner types: obtuse-obtuse (oo), obtuseacute (oa), and acute-acute (aa). If E denotes the energy associated with creating a new corner site, one obtuse step is related to Eoa and the other to Eoo. The corner associated with each step will depend on whether the spiral is rotating clockwise or counterclockwise, but on average, the energy associated with the obtuse step depends on (Eoa + Eoo)/2; similarly, for an acute step, it is (Eoa + Eaa)/2. If the energies are sufficiently large, then this can explain the orientation dependence observed in Figure 5e. Thus far, only average values have been reported and neither the third critical length nor the dependence on rotation direction have been noted. The corner energies are expected to be quite small, differing from the step by only one bond. Teng et al.133 point out that the corner energies represent the step free energy associated with the curved corner sites, not the literal singular corner site depicted in a diagram such as Figure 4. Nevertheless, the magnitude of corner energies and corner curvature remains a matter of debate in the crystal growth.

3.3.2. Step Movement Length Based on Kinetic Arguments The critical length is a necessary but insufficient criterion for step motion. Steps shorter than the critical length cannot propagate; steps longer than the critical length will propagate if they are in equilibrium but may not due to kinetics. Implicit within the Gibbs-Thomson relation is the assumption that steps are populated with kinks and that kink-nucleation is not a rate-limiting step for propagation. When this is not the case, a length analogous to the critical length exists but is based on kinetic arguments. In the low-kink limit, step propagation relies on the nucleation of one-dimensional islands along the step. While there is an energetic cost associate with creating the first adsorbed atom on a straight step, the addition of subsequent atoms does not change the chemical potential of the system and thus there is no “critical” size at which the onedimensional nucleus becomes stable. (This is in contrast to two- and three-dimensional nucleation, in which there is a well-known critical size at which the nucleus becomes stable and results in supersaturation dependent thresholds where these phenomena occur.) Islands that have a greater probability of increasing their length compared to dissolving back will contribute to the growth of the step. The competing factors are the forward motion of the kink, which is given by the kink velocity, and the backward motion of the kink, which is due to a random walk back and forth. The net backward motion in a given time t can be estimated as136,142

l ) b|(nback - nforward) = 2√Dkt - Vkt

(12)

where n is the number of steps, Dk is the diffusion constant of the kink, Vk is the kink velocity, and t is the time. From this, it can be seen that the backward motion of the step goes through a maximum that corresponds to a maximum length, Lkinetic ) D/Vk. Steps smaller than this are more likely to dissolve back due to the random walk of the kink. Steps larger than this are likely to persist due to the steady forward motion of the kink stemming from the supersaturated state. The diffusion constant Dk = (b|)2k-, and Vk ) b|k-(S1/n 1), leading to


Lkinetic )

b| (S - 1) 1⁄n


(13)

In the kinetic regime, the length at which a step begins to propagate then reflects the probability that nucleated kinks grow to a sufficient size that they are unlikely to dissolve back. Ideas such as these can also account for orientation specific “critical lengths” because, in general, the nucleation rate and the kink propagation rate are sensitive to the step orientation. In the case of calcite, acute and obtuse steps would be expected to have different nucleation rates due to the difference between γo and γa, and the kink velocities on the two steps are known to be different. To address the question of whether the length at which a step begins to move is associated with kinetics or thermodynamics, it will become important to couple critical length measurements with independent surface energy measurements and to develop analytic methods that explicitly account for kink-nucleation dynamics. As temporal and spatial resolutions improve, one would hope to image kink nucleation dynamics directly, as can be done for larger, slower moving molecules such as proteins.126,143

3.4. Geometric Effects on Kinetics Geometry also plays a role in step kinetics. We will discuss two cases: the first, which has already been alluded to, is the effect of step length on velocity, and the second is the effect of merging hillocks, where two joining steps can act as sources of kinks.

3.4.1. Dependence of Step Kinetics on Step Length As discussed in section 3.2.1, infinitely long steps propagate according to eq 8 (under the assumptions presented), and steps shorter than the critical length (Lc) do not propagate based on thermodynamic arguments (section 3.3). If we again assume that the step motion is not limited by kinetics, then the Gibbs-Thomson relation can be used to determine the velocity as a function of step length. This problem has been treated in the literature46,133 and (for one growth unit n ) 1) results in

V(L) ) V∞

(S - SLc⁄L) S-1

(14)

where S is the supersaturation ratio, Lc the critical length, L the length of the step, and V∞ the velocity of a step with infinite length. This equation is obtained by noting that the length dependent equilibrium activity (which takes into account the increase in energy due to the step-edge free Ωγriser/L energy) is given by astep . Substituting this e (L) ) Kspe into the expression for the step velocity, V(L) ) Ωβ(a astep e (L)) and using the definition of the critical length (eq 11) results in eq 14. Near equilibrium, in the limit S ) 1 + ε with ε , 1, the supersaturation ratio in eq 14 can be expanded as (1 + ε)q ) 1 + qε + Ο(ε2), where Ο(ε2) represents terms of order ε2 and higher. This results in

(

)

Lc for S ≈ 1 (15) L This approximation is often found in the literature and is the starting point for most step-pinning models, as discussed in section 3.6.2. V(L) ≈ V∞ 1 -

Closer examination of this relationship in calcite growth (Figure 6a) has shown that the velocity rises from zero to its value at infinite length faster than is predicted by eq 14 and is one piece of evidence suggesting that the assumption of high kink density may not be valid for sparingly soluble minerals with straight steps.133 A similar fast rise was observed for potassium hydrogen phthalate.135 By contrast, the step velocity of barite107 rises more slowly than predicted by eq 14. Equation 14 is based on the Gibbs-Thomson relation and assumes high kink density. On the other hand, in the low kink density limit, the functional form of the velocity is more complex. Voronkov was the first to discuss this problem.134 A functional form was reported by Rashkovich that was attributed to Voronkov but not derived in the literature.135 For this reason it is difficult to assess. The form is

L -1)A Lc

V ⁄ V∞

√1 - (V ⁄ V∞)

2

(

( ))

2 V 1 - ArcSin π V∞

with A )

πkBTb 2Ωγriser

(16)

where b is a molecular spacing, Ω the molecular volume, kB Boltzmann’s constant, T the absolute temperature, and γriser the step energy per area (as described in section 3.3.1). When the velocity is close to zero (V/V∞ ) ε, ε ≈ 0), eq 16 can be expanded to give V ) (V∞/A)(L/Lc - 1) + Ο(ε2). Thus, when L ≈ Lc, the velocity increases linearly with L. The velocity then rises rapidly to V/V∞ ) 1. This behavior is independent of the supersaturation. This is consistent with observations of calcite and potassium hydrogen phthalate discussed above.

3.4.2. The Dependence of Step Kinetics on Inside Corners There is also evidence that step kinetics depend on the presence of inside corners. Consider two singular steps that have different crystallographic orientations and propagate to meet one another. When they meet, they form an inside angle where there is no potential barrier for kink generation. Therefore, this angle can serve as a kink source that can accelerate step propagation when motion is limited by kinknucleation. Indeed, as Figure 6b shows, on brushite (CaHPO4 · 2H2O), when the steps [101j]Cc and [1j00]Cc touch one another, the [101j]Cc step is accelerated, as indicated by the change in apparent slope of the step.97,144 A single image cannot distinguish between a change in orientation and a change in step speed. However, successive images scanned down and up can be used to measure both step speed and step orientation. In this example, only the step speed is changing, not the step orientation. This step acceleration implies that there is a higher kink generation rate due to the presence of the re-entrant corner and that the step kinetics are otherwise limited by kink nucleation rate. Other examples of step acceleration due to the merging of inside corners have also been reported in the literature.145 During calcite dissolution studies, it was shown that the dissolution rate was slow until two pits merged. Once the steps merged, a new fast moving step direction was created. This effect is also observed during brushite dissolution (section 6.2). We note that this effect differs from that described above because both orientation and step speed are altered.


Qiu and Orme

3.6. Impurity Interactions

Figure 7. The change in hillock density and geometry when the critical length and velocity are changed. The first column shows the effect of changing one direction whereas the second column displays the effect when all step directions are changed by the same amount simultaneously. Reprinted with permission from ref 97. Copyright 2007 American Institute of Physics.

3.5. Hillock Geometry (Step Density) The length at which a step begins to move (Li) and the step velocity (Vi) for step direction i both affect the step density (or terrace widths, wi) near the apex of a hillock. (For the purposes of this discussion, it does not matter whether the step movement length stems from thermodynamic (Lc) or kinetic underpinnings (Lkinetic)). The terrace width in a particular direction depends on the time for a full revolution of the spiral (T) and the velocity of the step, wi ) ViT. Thus, the relative terrace widths reflect the relative velocities, w1/w2/w3 S V1/V2/V3, and once the velocity is known in one orientation, the others can be determined from the step densities.146 The time for a spiral revolution is a function of the velocities and the critical lengths in all directions. If one makes the simplifying assumption that the velocities are zero for lengths less than Li and V∞ for all other lengths, then for a triangular hillock the time is given by the sum of the times needed to grow each critical length:

T)

|L

1

| |

| |

Sinθ1 L2 Sinθ2 L3 Sinθ3 + + V2 V3 V1

| (17)

Figure 7 illustrates the effect of changing the critical length and the velocity on hillock geometry. Starting from a hillock with isotropic velocities and critical lengths, the first column demonstrates the effect of changing only one step direction whereas the second column demonstrates the effect of changing all step directions simultaneously. In particular, changes to the critical length (whether isotropic or anisotropic) change the overall density of steps whereas only anisotropic changes to the step velocity affect the step densities. These effects explain the change in hillock geometry caused by increasing the supersaturation. When the supersaturation is increased, the velocity of all of the steps increases in approximately the same way (V ∝ (e∆µ/kBT-1)). Although the steps move faster, the hillock geometry does not change (Figure 7, bottom row). However, the critical length (eq 11) and the kinetic length (eq 13) also change with supersaturation, which leads to a higher density of steps. This effect is observed in Figure 27b.

Additives (or impurities) play a major role in modifying the kinetics, morphology, and phase of biominerals. In fact, it might be said that it is the impurities that transform minerals into functional materials. The utility of additives is one of the distinctions between traditional manufacturing of materials and bioinspired approaches. SPM has been particularly useful in advancing the science of impurity interactions because it has the ability to monitor both the morphology and kinetics of atomic step motion as a function of impurity concentration. Unlike bulk studies, results are not averaged over different step directions or different facets that may all interact with the impurity in unique ways. There are several generic ways that adsorbates can affect growth. They can incorporate into the crystal, they can change kinetic coefficients, they can pin steps, and they can act as surfactants. Each of these alters the step kinetics in characteristic ways that allow the differing mechanisms to be distinguished.

3.6.1. Effects of Additives on Step Kinetics For the simplest case, a one-component, Kossel crystal growing by step flow, the step kinetics obey V ) βΩ(a ae). Accordingly, additives can alter step kinetics by changing the kinetic coefficient (β), the solution activity (a), or the equilibrium solubility (ae). Because the facet growth rate depends on the step density as well as the step kinetics, additives that alter step density also affect growth rate. For spiral growth, the step density is related to the critical length (which in turn is a function of the step free energy). For this reason, additives that alter the step free energy also alter crystal growth rates. Examples of additives that change a, ae, β, and Lc can all be found within the biomineralization literature. To deconvolve these mechanistically distinct effects, step kinetics can be measured as a function of supersaturation at several impurity concentrations (Figure 8). Additives that change the solubility constant (ae) (for example, by incorporating within the mineral and causing strain)50,110,147 will cause a shift in the velocity versus supersaturation curve (Figure 8a). The point at which the step velocity goes to zero gives the new equilibrium activity, ae. A similar effect will occur if the ion activity product is altered due to complexation within the solution but not accounted for in calculating the supersaturation. In either case, these shifts reflect changes to the thermodynamic databases used to calculate the solution speciation. Additives that alter the kinetic coefficient, β, change the slope of the velocity versus relative supersaturation curves (Figure 8b). Although curves such as Figure 8b make it clear when the kinetic coefficient is altered, moving deeper into the root cause requires considerably more work. As discussed in section 3.2.2, the kinetic coefficient is composed of several difficult factors: the kink density, the attempt frequency or prefactor, and the activation barrier. Devising methods to independently measure the impact of impurities on these factors is an area for future development. In the absence of other information, one typically assumes that the activation barrier is altered. Additives can also act like surfactants (Figure 8d). In this case, the molecule attaches to growing steps, altering their shape without necessarily altering their kinetics. The mol-



Figure 8. Schematic showing how different mechanisms of impurity interactions change the step velocity. As the impurity concentration (Ci) is increased, the velocity versus relative supersaturation (a-c) or velocity versus orientation (d) change in characteristically different ways. (a) Strain caused by the substitution of impurity ions changes the solubility, which is the concentration where the step velocity goes to zero. This causes the velocity curves to shift over but not change shape. (b) Impurities can either increase or decrease the kinetic coefficient. This causes the slope to change. (c) Impurities that adsorb to steps can prevent the steps from moving due to the high local curvature of the step between the blocked points. This causes a “dead zone” where the velocity of the steps is slow compared to the clean solution (solid line). (d) Surfactants change the step free energy. Two-dimensional slices of pseudo Wulff plots of calcite before and after the addition of aspartic acid, showing that the step-edge free energy, γ(θ), changes due to adsorbed molecules at the steps.

ecule does not incorporate within the mineral but rather rides along the surface.

3.6.2. Step Pinning Models In many cases, impurity interactions sufficiently alter growth that the step velocity is no longer linear with supersaturation. The Cabrera-Vermilyea138 (C-V) model and more recent modifications139 accounts for impurities that block (or pin) the motion of the step. As described briefly below, the step velocity then depends explicitly on the critical length as well as the concentration, lifetime, and distribution of impurities on the surface. The C-V model was the first to propose the slowing of step kinetics due to a pinning mechanism based on impurity adsorption at surfaces, steps, or kinks.138,148-150 The C-V model builds from the Gibbs-Thomson concept of a critical length and the length dependent velocity discussed by BCF and described in section 3.4.1. Within the C-V model, impurities act as blockers at the sites where they adsorb, preventing the crystal step from propagating locally and thus causing a straight step to become scalloped. As steps advance past the blocked sites, they curve and effectively become a collection of smaller step segments pinned at the sites where impurities persist. As the blocker density increases, the step segments approach the critical length, and their velocity slows, as described in section 3.4.1 until eventually they are stopped when their radius of curvature reaches the critical radius (as defined in section 3.3). The basic form of the velocity stems from eq 14 and is given by

(S - SLc⁄L) S-1 with β ) f(Ci) and L ) f(Ci)

V(L) ) Ωβ(a - Ksp)

(18)

where the length of steps between blockers, L, and the kinetic coefficient β can be functions of the impurity concentration. The two competing length scales are the average distance between blockers, which is a function of the impurity concentration in solution, Ci, and the critical length, which is a function of the mineral supersaturation. For this reason, the degree of inhibition depends on the supersaturation and the “blocker” concentration (Ci) (shown schematically in Figure 8c). Higher concentrations of adsorbate cause a greater reduction in velocity; these effects are more pronounced at lower supersaturations. This behavior leads to a region of no growth (the “dead zone”) at low supersaturations that then transitions to the growth rate of the impurity-free solution at higher supersaturation

The three major elements of the C-V model are the critical length, the length dependent velocity, and the average distance between blocking sites. Most variations on the original C-V description develop better models for the effective distance between blocking sites. The original model considers adsorption of impurities to terraces using a Langmuir isotherm to connect solution concentrations with surface coverage. Later models consider binding to steps149 and kinks, the relative merits of different adsorption isotherms, and the finite lifetime of impuritites.139 Sangwal reviews the evolution of impurity models with a focus on understanding “supersaturation barriers”, the critical supersaturation needed for step advancement, also known as the dead zone.150 In an application to biomineral growth, Weaver et al. have adjusted the C-V model to simultaneously include the effects of impurities on the kinetic coefficient, β, as well as pinning.46 This effectively combines the effects shown in Figure 8b and 8c, resulting in eq 19, described in section 5.2.

3.6.3. Effects of Additives on Shape In equilibrium, crystal shape is dictated by minimization of the surface free energy per unit volume, as first described by Wulff.151 Thus, crystal shape can be related to plots of the total surface free energy as a function of the angular orientation of plane normal vectors n(θ). Such plots are called Wulff plots or γ plots. Herring152 showed that cusps in the surface energy plots corresponded to facets, and Taylor153 later proved that the relationship between shape and surface free energy was unique. The shape can be constructed geometrically from the γ-plot by drawing a line from the origin to γ(θ) at each angle θ. The set of lines perpendicular to these radial vectors inscribe the equilibrium crystal shape. Figure 9 shows an example of a two-dimensional slice of a Wulff plot (solid line) with the corresponding crystal shape (dashed line) demonstrating the correspondence between γ and shape and in particular how cusps with the smallest surface free energy dominate the crystal shape. When crystals are not in equilibrium, a similar formalism can be used to relate facet growth rate as a function of facet orientation to the crystal growth shape.124,154,155 The velocity maps V(θ), known as idiomorphs or kinetic Wulff plots, similarly dictate that the slowest growing facets dominate the growth shape just as the smallest surface free energy dictates the equilibrium shape. For a simple case of a cubic crystal with velocity minima in the {100} and {111} facet directions, the ratio of the {100} to {111} growth rates (R


Figure 9. Two-dimensional slice of a Wulff plot with surface energy as a function of angular orientation (solid line) and corresponding equilibrium shape (dotted line). Reprinted from ref 285, Copyright 2004, with permission from Acta Materialia Inc., Published by Elsevier Ltd.

Figure 10. The growth shape is dominated by the slowest growing facets. The shape changes from cubic to octahedron as the ratio (R) of growth rates of {100} to {111} facets increases from 0.58 to 1.73. Reprinted with permission from ref 286. Copyright 2000 American Chemical Society.

Figure 11. Schematic of shape transition from (a) to (c) when impurities block the growth of B-facets (b). Reprinted with permission from ref 51. Copyright 1985 the National Academy of Sciences of the U.S.A.

) V{100}/V{111}) determines the shape. The shape shifts from cubic to octahedron as the ratio increases (Figure 10). Additives can strongly influence the shape by anisotropically altering either the surface energy or the facet growth rate. The standard paradigm for shape modification used in the biomineralization literature (when molds are not used) considers the effects of additives on the velocity map. In this scenario, additives bind preferentially to a particular facet and in doing so block growth sites, causing the facet velocity to slow. The modified crystal habit expresses these slowgrowing facets as the faster growing facets grow out and are eliminated from the shape (Figure 11). In a typical assay, crystals are grown in the presence and absence of additives. New facets found in the presence of additives are identified as those to which the additive binds. In the context of this model, Addadi and Weiner51 discussed the importance of stereochemistry (spatial arrangements of atoms within molecules) in describing the ability of acidic proteins to alter calcium dicarboxylate salts. Their work demonstrated that newly expressed facets had carboxylate groups approximately perpendicular to surface. They proposed that these facets were selected because the orientation and spacing of the carboxylate groups in the

Qiu and Orme

Figure 12. SPM (top row) and SEM (bottom row) of calcite crystals in the (a) absence of impurities and the (b) presence of L-aspartic acid and the (c) presence of D-aspartic acid. Reprinted with permission from ref 40. Copyright 2001 the Nature Publishing Group.

crystal matched the shape and orientation of binding carboxyl groups from the aspartic acid rich proteins. Such conclusions were based on the observation of the macroscopic habit of the crystal in the absence and presence of additives. More recently, SPM has also been used to investigate the effect of shape modifiers during growth. SPM has the advantage of being an in situ technique allowing the shape transition to be monitored in real time. From these experiments, a more detailed picture emerges based on additives binding to steps rather than facets. Using SPM, atomic steps growing from a dislocation hillock are imaged before and after the addition of an additive. The exact location is imaged with and without the additive, making the correlation direct and simultaneously providing both morphological and kinetic changes. This technique was used to assess the impact of chiral aspartic acid molecules on calcite growth (Figure 12).40 Calcite, whose habit is a rhombohedra bounded by {104} facets in the absence of additives, instead transforms to a cylindrical habit elongated along the [001] direction with pyramidal {104} caps on each end when grown in the presence of aspartic acid (and other molecules with carboxylate groups156). The traditional explanation for this behavior is that the additive stabilizes the family of facets that compose the cylinder by binding to these faces and blocking their growth.51 However, by directly imaging the transition, SPM revealed that this interpretation was not correct; the additive instead interacted with the growing atomic steps on the {104} pyramidal faces. Additionally, the additive acted like a surfactant to change the step-edge free energy of the growth steps. This is akin to changing the orientation dependence of the γ-plot and differs mechanistically from altering the velocity map. One immediate implication of this result is that the molecular docking site is at a step edge on a pyramidal face not on a facet of the cylinder. Nevertheless, energetically favorable binding sites on steps were found in which the carboxyl groups of the aspartic acid molecule mimicked the carbonate groups of the solid, supporting the importance of stereochemistry in creating specificity. To critically test the ideas of stereochemistry, D- and L-aspartic acid were both imaged on the same growth hillock to demonstrate that the binding site reflected the symmetry of the molecule.



Table 6. Pathological Calcium Containing Minerals Found in the Body tissue

minerala,b

disease

model solution

ref

loops of henle teeth salivary glands joint

CaOx, HAP, DCPD DCPD, TCP, OCP DCPD CPPD, DCPD, HAP, CaOx

kidney stone formation calculus/caries sialolith rheumatoid arthritis, osteoarthritis, and chondrocalcinosis

urine plaque/saliva saliva synovial fluid

160 165, 166 164 161-163

a Calcium oxalate (CaOx), hydroxylapatite (HAP), brushite (DCPD), tricalcium phosphate (TCP), octacalcium phosphate (OCP), calcium pyrophosphate dihydrate (CPPD). b The minerals are typically substituted forms of the phases indicated. This is well documented for the apatites.9 The degree of substitution of the other phases is less well-known.

Another outcome for this particular case is that the mechanism of interaction is more like a surfactant than a blocker (Figure 8d). However, as will be seen through examples, when discussing COM, shape modification can occur for any of the reasons discussed in section 3.6 (pinning, incorporation, changes to the kinetic coefficient, or changes to the interfacial energy), provided they occur anisotropically. In general, the more detailed view provided by imaging atomic steps versus macroscopic habit has added to our knowledge of binding sites. And, the ability of SPM to probe kinetics and the use of kinetics to distinguish mechanisms of impurity interaction have led to a richer dialogue to describe the stabilization of facets.

4. Pathological Crystallization The calcium phosphate phases of interest in healthy human biology include amorphous calcium phosphate, as well as biogenic apatites: bone, enamel, and dentine. These materials have been thoroughly reviewed by several authors.9,157 Biogenic apatites have the structure of hydroyxlapatite but are highly substituted with other cationic and anionic species. There are several classic examples of pathological crystallization in humans that involve apatitic as well as other calcium-based minerals (Table 6). These include crystallization within the urinary track, producing kidney stones;158-160 crystallization within the joints, which causes various forms of arthritis;161-163 plaque formation within teeth or salivary ducts, resulting in caries and other dental disorders;164-166 and plaque formation within the vascular system, which results in arthrosclerosis.167 Kidney stones (renal lithiasis) afflict approximately 10% of the population. In the kidney, mineral deposits contain various phases of calcium salts such as calcium phosphate and calcium oxalate. Under some conditions, calcium phosphate minerals are thought to be the initiator to stone formation, as they are often observed in the core of kidney stones.168 Most kidney stones consist of calcium oxalate (CaOx) in either the monohydrate (COM, whewellite) or dihydrate (COD, wheddellite) forms. However, the majority of the stones are primarily composed of COM crystals.169 Formation of stones within the urinary tract is a complex process influenced by multiple factors. Although normal urine is frequently supersaturated with respect to calcium oxalate, most humans do not form stones. Typically, any crystals formed are rapidly passed before achieving a size sufficient for retention. Increased quantities of calcium and/ or oxalate are excreted by many COM stone formers.170,171 However, greater supersaturation alone does not account for the incidence of stone disease.172 The lack of stone formation in most humans is better explained by the presence of inhibitors that decrease the formation, growth, and aggregation of COM crystals and their binding to renal cells.173-176 In vitro assays studies have shown that urinary proteins and other naturally occurring small molecules and macromol-

ecules are responsible for the inhibitory effect. Although thought to be protective, the underlying principles by which these inhibitors control COM crystallization are still not well defined. Crystallization within the joints causes arthritis. Crystals are typically 1-10 µm, composed of calcium pyrophosphate dihydrate, calcium oxalate, and calcium phosphate that form within joints, causing inflammation and arthritic conditions. SPM has been used to image calcium pyrophosphate dihydrate and OCP crystals within the synovial joint fluid of arthritic patients,177,178 but no kinetic studies have been performed to date. In addition, there are several examples of crystallization of proteins or organic molecules that cause disease. These include the crystallization of a mutated form of hemoglobin to cause sickle cell anemia, the crystallization of monosodium urate monohydrate to cause arthritic gout, the crystallization of cholesterol, which plays a role in atherosclerosis, and the crystallization of uric acid, commonly found in kidney stones. While SPM has been used to investigate the kinetics of uric acid,179 cholesterol monohydrate,180,181 and hemoglobin,182-184 in this review we concentrate on the calcium based minerals and in particular COM, brushite, and HAP.

5. Calcium Oxalate Monohydrate Recently, researchers have utilized SPM based techniques to study the growth and dissolution of COM74,185,186 with and without additives and to reveal the fundamental mechanism by which small molecules41,46,186,187 and biological molecules185,188-193 control the formation of COM. While most of the SPM studies reveal that COM follows spiral growth on dislocation hillocks, some have observed that it also grows via 2D nucleation.187-189 Growth kinetics and crystal shape are prone to be modified when small molecules and macromolecules are present in the growth medium. Touryan et al.187 found that rare earth trivalent ions such as Eu3+ and Tb3+ modified growth shape by inducing an ionic switch in the orientation of the COM unit cell, which resulted in a pillar-like structure in one particular face. They also found that the divalent Mg2+ ions modified the crystal habit by pinning surface steps and retarded the spiral kinetics. On the other hand, Gvozdev et al.186 found that Al and Fe ions greatly increased the COM dissolution rate. For biological molecules, Qiu et al.41,192 showed that step-specific interactions are responsible for the primary effects of citrate and osteopontin (OPN) being exerted on different faces of COM crystals. Similarly, Guo et al.185 observed that anionic molecules and polypeptides with significant anionic functionalityssuch as that presented by carboxylatessexhibit face and step specific interactions with COM. In a related study, Sheng et al.194-196 used AFM chemical force microscopy to investigate the binding of various functional groups to COM surfaces and found strong binding by carboxyl groups to COM surfaces, with the strongest adhesion force occurring


Qiu and Orme

Table 7 Deganello and Piro Tazzoli and Domeneghetti

symmetry

a (Å)

b (Å)

c (Å)

β (deg)

P21/n P21/c

9.976 6.290

14.588 14.580

6.291 10.116

107.05 109.46

on the (-101) face. They also found that carboxylate functionalized SPM tips exhibited reduced adhesion to COM crystals when the solution contained polyaspartic acid. In this article, the focus is on work aimed at revealing the shape and kinetic modification of COM growth under the influence of representative additives, including metallic ions, citrate, naturally occurring urinary proteins such as osteopontin (OPN), and relevant synthetic macromolecules such as polypeptides and polymers. The review will be given in the order of increasing molecular size as well as the complexity of structures. We start the section by reviewing COM growth in pure solution.

5.1. Structure and Growth in Pure Solution COM is a monoclinic crystal, and its unit cell structure has been described by several conventions. Two commonly adopted conventions in the literature are the Deganello and Piro convention35 and the Tazzolli and Domeneghetti convention.197 In this article, we use the Deganello notation convention to describe the unit cell as well as the faces. The unit cell parameters in both conventions are given in Table 7. Moreover, the list of the index of selected faces that are discussed in this paper is included in Table 8, which shows the correspondence between the indices under the two different conventions. The equilibrium shape and size of COM crystals depend on growth conditions such as growth medium, ionic strength, pH, temperature, and other local environmental factors.186,189,198-204 Most of the synthetic COM crystals exhibit habits with three distinct facessnamely the {-101}, {010}, and the {120} facesswith aspect ratios similar to those shown in Figure 13a and b, where the crystal shown in Figure 13b was grown under high ionic strength and nearly neutral pH.203 However, crystal habits with different faces and aspect ratios have also been reported under other growth conditions.189,199 For example, Jung et al.189 developed a method to utilize a double jet precipitation at high temperature and low ionic strength to produce COM crystals with an enlarged (010) face. They also developed a method by slowly evaporating saturated calcium oxalate aqueous solutions under acidic conditions to fabricate crystals with a shape like a xiphoid. This method produced crystals with parallelogramshaped (-101) faces instead of the familiar hexagonal faces. Examples of the crystal habits grown from these methods are shown in Figure 14a-c. In order to better interpret the SPM measurement results on the growth of COM in pure solution, the molecular structures of the commonly investigated faces are discussed. Representative molecular structures of the (-101) and the (010) faces are shown in Figure 13c and d. For ease of viewing, the molecular coordinates are slightly shifted with respect to their correct positions on both faces, and on the (-101) face, oxalate groups at the edge are only partially displayed. In both planes, the surface layers contain two sublayers of oxalate groups: one is parallel and the other is perpendicular to the planes, respectively. While the perpendicular oxalate groups on the (-101) plane lie below the

Figure 13. (a) Schematic of COM crystal habits with three commonly expressed planes (-101), (010), and {120}. (b) COM equilibrium habit grown under high ionic strength at room temperature. (c) Molecular structure and stacking layers of the (-101) face. (d) Molecular structure of the (010) face. Green, Ca2+; red, O; dark gray, C; off-white, H. (a, c, and d) Reprinted with permission from ref 41. Copyright 2005 American Chemical Society. (b) Reprinted with permission from ref 203. Copyright 1993 the Journal of Urology.

Figure 14. SEM images of COM crystals grown under different conditions: (a) acidic conditions; (b and c) under high temperature and low ionic strength. AFM images showing the growth hillocks on the (-101) face (d), the (010) face (e), and the (021) face (f). High resolution SPM images (raw data, upper portion; filtered data, lower portion) showing the lattice of the (-101) face (g), the (010) face (h), and the (021) face (i). (a-c and e-i) Reprinted with permission from ref 158. Copyright 2007 the Mineralogical Society of America. (d) Reprinted with permission from ref 41. Copyright 2005 American Chemical Society.

topmost surface layer, those on the (010) plane are exposed and extend beyond the plane. These structural differences make the (-101) plane richer in Ca2+ than the (010) plane


and dictate the strength of their interactions with solution additives to which both faces are exposed during crystal growth. On the (-101) plane, the molecular rows stack in repeating sequences of AA′BB′AA′ (Figure 13c). This stacking sequence leads to step bunching and step interlacing on the (010) surface. If the water molecules in the COM structural models are replaced by oxygen atomssa common simplificationssignificant steric features associated with the water molecules vanish. This omission gives rise to an AABBAA stacking sequence of molecular rows on the (-101) plane. Because the molecular stacking sequence has implications for both step dynamics and hillock structure, this clear difference between the models with and without oxygen atoms shows the importance of including the entire water molecule in structural models in order to correctly interpret SPM data. SPM results reveal that COM crystals grow on complex screw dislocation hillocks74,189,192 comparable to those of many other solution grown crystals such as KDP205 and calcite.206 The growth hillock density is much higher on the (-101) face than on the (010) face. In general, crystals grow mainly on a few dominant hillocks, with other minor sources emerging on the flanks of these hillocks. Representative examples of growth hillocks on the (010), (-101), and (021) faces are shown in Figure 14d-f.41,189 As shown in Figure 14d, the growth hillocks on the (-101) face exhibit a triangular-shaped morphology that is quite different from its equilibrium parallelogram or hexagonal habit. Steps propagating toward the {010} faces and the (1-20) and (120) were not present. Instead, steps moving toward [101] were observed that truncate the angle formed by the [-1-20] and [-120] directed steps. The implication is that steps directed toward these three unrepresented directions have considerably higher speeds than the others. This means that there is a large anisotropy in step speed or attachment/detachment kinetics between the [1-20]/[120] and [-1-20]/[-120] directions, despite the fact that the only significant difference is in the tilt angle of the oxalates (Figure 13c). The highly anisotropic step kinetics gives rise to closely spaced steps along the two slower directions, as shown in Figure 14d. Although double steps are observed at some hillocks, the majority of the hillocks have only single steps with a height of 6 Å, implying that either the growth unit contains two molecular sublayers, as described earlier, or the addition of one type of unit occurs on a time scale which is short compared to the time required for growth units of the other type. The growth hillock on the (010) face is shown in Figure 14e, which exhibits a four-sided spiral dislocation. The planes that form the dislocations are the {021} and {120} planes. The geometry of the growth hillock is rather complicated and contains multiple screw dislocations, often of the Frank-Reed type.207 For the first several turns of the growth spiral, the step heights range from one to three H, where H is the height of an elementary step, about 3.98 Å. However, at a sufficient distance away from the dislocation source, all steps are quadrupled with heights of ca. 16 Å. This quadrupling of steps is caused by, so-called, “step interlacing”,208,209 which is most apparent at the hillock corners, as shown in Figure 14e. Both the step interlacing and step bunching have their origin in the AA′BB′ packing sequence described above. This packing structure creates a screw-axis symmetry element perpendicular to the (010) face and leads


to asymmetries in step speed from layer to layer with a fourlayer periodicity. As shown in Figure 13c, the AA′BB′AA′ stacking sequence of the molecular rows in the (-101) plane comprises the elementary steps that define sequential terraces of hillocks on the (010) face. There are two types of water molecules and oxalate groups within these rows: type 1 lying flat and type 2 perpendicular to the (-101) plane. The orientations of the type 2 oxalate groups and calcium pairs in layers A and A′ are the same and are mirrored by their counterparts in layers B and B′. The water molecules on the other hand are different in all four molecular layers, although they are related by a symmetry operation. On the (010) face, this structural switch results in small differences in step speeds among the layers. Because the faster steps eventually catch up to slower steps (Figure 14e), after only a few turns of the dislocation spiral, step bunches displaying the periodicity of the AA′BB′ stacking are generated. Thus, the unique stacking structure along with the change in orientation about the screw-axis at the A/B′ interface leads to growth of the (010) face on quadruple height steps. On the (021) face, as reported by Jung et al.,189 although growth hillocks are occasionally observed, 2D islands were the main sources of growth. The 2D islands have growth shapes mimicked by the xiphoid habit of the bulk crystal,189 as shown in Figure 14f. This result suggests that the (021) face has a much higher surface energy than that of the (-101) and (010) faces. High resolution SPM images of the three different faces were also collected by Jung et al.189 to reveal the lattice structure of each individual face. The near-atomic resolution images of the (-101), (010), and (021) faces are shown in Figure 14g-i, respectively. While the upper portion of the image shows the original data, the lower portion gives the Fourier-filtered view. For the (-101) face, it was difficult to obtain the two-dimensional periodicity from the raw SPM image; however, the filtered image revealed one-dimensional features oriented along the [201] direction (Figure 14g). On the other hand, for both the (010) and the (021) faces, twodimensional periodicity was clearly discernible (Figure 14h and i), and within the experimental error, the measured unit cell parameters (except one) were in good agreement with their corresponding lattice constants on each plane. The slightly smaller measured lattice parameter along one direction on the (021) face suggested a minor surface reconstruction in comparison to that of the bulk structure. Moreover, since the measured step height on the (021) face was identical to the vertical lattice parameter perpendicular to the surface, the surface reconstruction only occurred laterally. This observation may also support the argument that the (021) face has a higher surface energy than the other faces. Based on the discussion in section 3.2.2, the kinetic coefficient, β, of COM along specific step directions can be estimated from the measured step speed. Qiu et al. estimated the kinetic coefficient along the [101], [-1-20], and [100] steps to be 0.2, 0.02, 0.03 cm · s-1, respectively. These were calculated by assuming a linear dependence of step speed on solute concentration.210,211 In a follow-up study, Weaver et al.110 introduced background electrolytes to the solution and found that the kinetic coefficient along the [101] direction strongly depended on the background electrolyte. For all the alkali-metal ions examined, the largest β was found to be associated with K+ at 0.52 cm · s-1 while it was only 0.19 cm · s-1 with Na+. (It should be noted that these reported


Figure 15. Representative SPM images showing the morphology of the (-101) face on a COM crystal under various growth conditions. (a) Surface morphology under a pure supersaturated solution showing a growth spiral and flat terraces. (b) Vertical nanostructures on the (-101) face after Eu3+ ions were introduced to the solution. (c) Growth hillocks after a pure solution was reintroduced to the liquid cell. Image horizontal dimension: (a) 1 µm; (b–c) 1.5 µm. Reprinted with permission from ref 187. Copyright 2004 Nature Publishing Group.

values are a factor of 2 larger than the ones obtained from eq 8, as n was chosen to be 1 in refs 41 and 110. Nevertheless, these values are similar to those found for other inorganic salts.210,211 As for the discrepancy and the influence in modifying the magnitude of the kinetic coefficient for different background electrolytes, no satisfying explanation was found. However, it was believed that the effect must lie in the electrostatic influence of the ions on the growth step and that the metal ions either block kink sites or raise the attachment energy for calcium at the step edges.

5.2. Metal Ions Metallic ions such as Mg2+, Sr2+, and Fe3+ are often found in Ca-based biominerals, which includes calcium phosphate, calcium carbonate, and calcium oxalate.9,212-214 In addition, rare earth ions such as Eu3+ and Te3+ are also found in the aforementioned biominerals that reside in human kidney stones.64 It has been reported that Mg2+ changes the growth of calcite by enhancing its solubility.50 The exact role of these multivalent ions in regulating COM formation is still unclear, however, and is of interest to the medical field for developing effective stone therapy as well as to the materials science community for inspired inorganic synthesis. In a recent study, using COM as a model system, Touryan et al.187 discovered that the trivalent europium ion can switch the surface morphology of COM back and forth from a flat crystalline sheet to nanostructures that are oriented perpendicular to the surface. In this study, SPM was used to monitor the evolution of the COM (-101) face under the influence of Eu3+ ions. When growing in pure solution, the surface exhibited similar flat terraces and spiral hillocks to those shown in Figure 14d. A representative image from this work is shown in Figure 15a. However, when a small amount of Eu3+ ions were added to the supersaturated solution, the (-101) face was then covered with vertically oriented nanostructures (Figure 15b) that were perpendicular to the surface. Spectroscopy results showed that these nanostructures were still crystalline COM. Interestingly, these nanostructures would eventually disappear if a fresh Eu3+-free supersaturated calcium oxalate solution was subsequently infused to the liquid cell and, instead, the spiral terraces were recovered on the (-101) face (Figure 15c). The same effect was observed for another rare earth ion, Tb3+. However, when divalent ions such as Mg2+ and citrate were added to the solution, no nanostructures were observed. The surface was still growing on the spiral hillocks but at a reduced rate. This observation was consistent with other reports on the effect of divalent ions on biomineral growth.215,216

Qiu and Orme

The results shown by Touryan et al.187 indicated that the switching ability over the COM morphology was unique to trivalent ions. It was proposed that the addition of the trivalent ions to COM had led to a partial substitution of the divalent calcium ions. In the lattice where the substitution occurred, an extra charge was created. Although the incorporated trivalent ion would coordinate well with the neighboring oxalates ions in the COM unit cell, similar to that of Ca2+, the extra charge from the Eu3+ would need to be balanced. Such a requirement was satisfied by binding an additional oxalate ion from solution. It was believed that because the oxalate ion was elongated with an aspect ratio of 1.2:1, the addition of the extra oxalate group caused the COM unit cell to switch orientation, which led to the appearance of the nanoscale structures on the (-101) face. Alternatively, there may be other possible explanations for the observed structural switching. When trivalent ions are incorporated into the lattice, they introduce local strain. Strain can cause a shift in the solubility of the surface overlayers. Subsequently, the supersaturation at the sites where the trivalent ions resided increased. The enhancement in supersaturation could drive a fast 2D nucleation with a preferred growth along the direction that was perpendicular to the (-101) face. Elongated nanostructures would then be formed like those shown in Figure 15b. Such effects in calcite have been recently reviewed.147 The fact that Eu3+ ions can switch the growth mode to produce different structures on the COM surface suggests a potential strategy for engineering hierarchical materials by sequentially adding or depleting metal ions in growth medium. The valences of the metal ions can be selective depending on the host materials. Furthermore, this new finding may also provide clues to the formation of various shapes of kidney stones158 in pathological mineralization.

5.3. Citrate Many clinical studies have shown that urinary citrate plays an important role in regulating kidney stone disease.217 For example, citrate deficiency in the urinary tract is often found in patients with calcium oxalate stone disease and, on the other hand, oral potassium citrate is effective in prevention of recurrent stone diseases.217 By forming ion pairs or other solution complexes, citrate can decrease urinary supersaturation with respect to calcium salts. In the meantime, citrate has direct effects on crystallization that include inhibition of the nucleation, growth, and aggregation of COM crystals.218,219 Additionally, citrate may also enhance the effectiveness of protein inhibitors of crystallization. For example, inhibition of aggregation of COM crystals by Tamm-Horsfall protein is increased by citrate.175 Therefore, a fundamental understanding of how citrate affects the growth of COM can provide insights into the role that citrate plays in renal stone pathogenesis. Furthermore, a physiochemical understanding of such a role may uncover new avenues for more effective therapy of renal stone disease.217 The molecular scale view of COM modulation by citrate has been revealed from the investigation by combining in situ SPM with molecular modeling.41,192 Citrate altered COM growth morphology as well as growth kinetics by selective binding to atomic steps on the existing crystal faces with the dominate effect on the (-101) face. Both the magnitude of the effect and the rate at which the full effect was observed increased with citrate concentrations. Step speeds were


Figure 16. SPM images showing the morphological changes of growth hillocks on the (-101) faces after the effect of citrate is fully realized: (a) in pure solution and (b) in the presence of citrate. Image horizontal dimension: (a) 10 µm; (b) 4 µm. Reprinted with permission from ref 41. Copyright 2005 American Chemical Society. Table 8. Index of Selected Crystal Faces in Two Notation Conventions199 Deganello and Piro

Tazzoli and Domeneghetti

{-101} {010} {120} {021}

{100} {010} {021} {12-1}

reduced and step edges became roughened and the changes were direction dependent. The drastic morphological changes of the step at the dislocation hillocks on the (-101) face induced by citrate are displayed in Figure 16. Step edges along all three directions were roughened, with the most dramatic changes occurring at the [101] step. When the full effect was realized, the [101] step lost its stability and presented a highly ramified step front, while the [-120] and [-1-20] steps evolved to a single half-circle-shaped step with its center coinciding with the dislocation origin. Accompanying the modification in step and growth hillock morphology were changes in step speeds. When the full effect was realized, the step speed reduction was quite anisotropic with the reduction along the [101] step more severe than that along the other directions. The net result of these changes in step speed and shape was to alter the morphology of the growth hillocks from a triangular to a nearly circular shape, as shown in Figure 16b. The final morphology of the growth hillocks depended on the citrate concentration in solution. Moreover, there was a strong correlation between the shape change at the elementary step level and the macroscopic growth habit. Comparison between the SPM images41 and the bulk crystal growth experiments202,203 demonstrates that the evolution of crystal habit with increasing citrate concentration mimics the growth hillock geometry on the (-101) face. The final shape of both was directly related to the citrate concentration. Similar phenomena were also observed in calcite growth under the influences of amino acids or abalone nacre proteins where the bulk shapes bore resemblances to the morphology of the microscopic growth hillocks.40,43 The close correspondence of the morphological features in all these systems from both the macro- and microscales suggested that the overall crystal shape was being controlled by the elementary step kinetics and not by considerations of minimum surface energy. This conclusion does not, however, rule out the possibility that the step-edge and interfacial energies are modified in a similar fashion. The effect of citrate on the (010) face was minimal, and no significant changes in either step morphology or step


kinetics were observed.41,192 Thus, the interaction between citrate and the (010) face was much weaker than that with the (-101) face. Since the shape change at the elementary step due to the addition of citrate was reflected in the macroscopic growth habit,41,202,203 the modification in COM morphology and kinetics was mainly due to step-specific pinning on the (-101) face. The underlying principles of the asymmetric and highly site-specific interactions between citrate and COM were revealed by molecular modeling with energy minimization to calculate the binding energies of citrate docking to steps and faces of COM for various possible configurations.41 The calculated binding energies are summarized in Table 9. As shown in the table, the binding energies on the surfaces were much smaller than that on the steps for both faces.202 Moreover, the binding to the steps on the (-101) surface was more favorable, as suggested by the fact that the binding energies were much higher than those on the (010) surface. These binding energy calculations were consistent with the SPM observations. The molecular modeling work revealed the important geometric factors that dictated the stereochemical recognition in citrate binding. The first was the orientation of the oxalate groups within the planes that form the step. The dicarboxylic acids at both ends of oxalate molecules presented domains that repelled the carboxylic acids of the citrate molecule. Thus, steps and terraces that exposed flat oxalate configurations were favored. The second factor was the configuration of Ca sites on the crystal surface. The conformation of the nonplanar citrate molecule was relatively rigid so that its three carboxylic acids could not be easily rotated. Thus, a Ca configuration that mimics the carboxylate geometry maximized the binding energy. The acute geometry of the [101] step on the (-101) face provided a favorable steric configuration for citrate to bind with minimal strain because its geometry optimized both factors. Specifically, the flat orientation of the oxalate groups on the basal plane avoided electrostatic repulsion of carboxylates, while the configuration of Ca sites on both the basal and riser planes accommodated all three carboxylic acids, as shown in Figure 17a. In contrast, on the (010) face, dicarboxylic acids of the oxalates were exposed and extend beyond the (010) plane, as demonstrated in Figure 17b, making it difficult for citrate molecules to bind either to the steps or to the face due to the electrostatic repulsion. Moreover, the 90° angle between the basal plane and the step riser resulted in a poor geometric match between the Ca ions and the citrate carboxylates, a steric condition that could only be accommodated by strain associated with distortion of the citrate molecule. Third, the chemical details within the citrate molecule also contributed to the strong binding to the [101] step on the (-101) face. As seen in Figure 17a, the hydrogen atom from the hydroxyl group of the citrate molecule formed a hydrogen bond with one of the oxygen atoms within oxalate ions in the basal plan. Besides enhancing the binding strength, due to its flexibility, the hydrogen bond also behaved as a pivot to hold the citrate molecule in place and thus to stabilized the attachment when it was docked to the acute step. This could also be part of the reason that the citrate molecule was slightly skewed from the [010] direction. Thus, the strong interaction between citrate and the steps on the (-101) face resulted in pinning of all the steps on the surface. The difference in the binding energies among the steps on both faces led to the anisotropic reduction in step kinetics, which ultimately resulted in morphological changes over COM growth.


Qiu and Orme

Figure 17. Binding configurations at minimum energy of citrate to the steps on the (-101) face and the (010) face. Chemical bonds between the citrate molecules and calcium ions and between the hydroxyl groups and the oxalate ions are displayed by green lines. (a) To the acute [101] step on the (-101) face viewed approximately perpendicular to the step. (b) To the [100] step on the (010) face. Blue, Ca2+; red, O; dark gray, C; off-white, H. Reprinted with permission from ref 41. Copyright 2005 American Chemical Society.

Citrate modifies both the morphology and kinetics of COM growth by pinning step motion. The is supported by the observation of the serrated step morphology for the newly expressed step edges,41 the existence of a dead zone in step speed, and the reduction in kinetic coefficient as determined from in situ SPM studies.46,110 The step kinetics in the presence of citrate can be best described by a modified Cabrera-Vermilyea model of step pinning.46,110 In this reformulated model, the step speed is written in terms of the curvature through the Gibbs-Thomson relationship (eq 18).133 The kinetic coefficient is modified due to kink blocking, making it an explicit function of impurity concentration. Moreover, the adsorption of impurities to the terraces is described by Langmuir adsorption kinetics. This also makes the step segment length an explicit function of impurity concentration. If Vi represents the step speed in the presence of citrate and V0 is the step speed in pure solution, a generalized expression for C-V-type step velocity can be written as

{ [

A2Ci Vi ⁄ V0 ) 1 - A3 1 + A2Ci

] }{ 0.5

1-

eA1

[

A2Ci 1 + A2Ci σ

]

0.5

e -1

-1

}

(19)

In this formula, A1 ) [2γriser(Ωh)1/2]/(B1kBT), A2 ) kA/kD, and A3 ) [γriser(Ωh)1/2]/nk,0. B1 is the product of the following proportionality constants:57 the fraction of adsorbed surface impurities that stick to a step and pin it, the geometric factor relating linear spacing to aerial density, and the percolation threshold for a step to move through a field of blockers.220 γriser is the step-edge free energy per unit step height, Ω is the molecular volume in solid, h is the step height, kB is the Boltzmann constant, T is the temperature, nk,0 is the kink site density in pure solution, and kA and kD are the attachment and detachment rate coefficients of the impurities, respectively. It is apparent that the fitting parameters in the generalized formula for step speed Vi are controlled by three

Figure 18. Dependence of normalized velocity, V/V0, on citrate concentration for each supersaturation. The dotted line is fit to the original C-V model. Dashed lines are fits to the data according to eq 19. Reprinted from ref 110, Copyright 2007, with permission from Elsevier B.V.

fundamental parameters of the system: the step-edge free energy, the kink density, and the ratio of the impurity adsorption/desorption rate coefficients. The measured step velocity at the presence of citrate is shown in Figure 18. The dotted curve is the best fit of the original C-V model, which does not accurately reflect the experimental data. However, the dashed lines, which are fits based on the reformulated step speed expression, as given in eq 19, show excellent match to the experimental results. The agreement validates that the reformulated C-V model is best to describe the step kinetics of COM under the influence of citrate molecules. The predictions of the C-V model had also been linked to studies of citrate adsorption in macroscopic crystallization of COM.221 The macroscopic crystallization kinetics of COM in the presence of citrate at 37 °C and an ionic strength of 0.15 mol L-1 was investigated using the CC method, which has been described in detail elsewhere.30,222 The dependence of growth rate on both supersaturation and citrate concentration is shown in Figure 19a, where Ri and R0 are the macroscopic growth rates of COM with and without the presence of citrate, respectively. Based on earlier discussion of the AFM observation, one can assume that citrate inhibits growth of COM only by pinning steps on the (-101) face; the relative growth rate Ri/Ro can be written in the following general form as define by the classical C-V model,220

Ri 1 ) 1 - GBLc√Ci R0 4

(20)

Here, G is a number of order 1 that depends on the geometry of the crystal lattice and is approximately 2 for a square lattice, B is a proportionality coefficient that reflects a combination of geometric factors, sticking probability, and impurity lifetime on the surface, and Lc is the critical radius. It is apparent that the theoretical curves (dashed lines in Figure 19a derived from eq 20 do not agree well with the experimental data. First, the measured relative growth rate falls faster and flattens more suddenly than predicted. Moreover, the minimum rate is far below that predicted for pinning of a single type of face. However, if the kinetic coefficient that relates step speed to supersaturation is assumed to decrease with increasing citrate levels on all faces, and step pinning only applies to the (-101) face (as indicated by the SPM investigation), then



including contributions from all COM faces, the best fit to all data sets for a relative growth rate is of the form

Ri (1 - A(-101)√Ci)(1 - B(-101)√Ci) + ) R0 4 (1 - B(010)√Ci) (1 - B(120)√Ci) + (21) 4 2 which 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 thus is given by Ri ) 1 - 52.6√Ci + 734.7Ci R0

(22)

It should be noted that the best fitting results reported here may not be unique, as the fitting coefficients may co-vary. Because the uncertainties of reported values were not given in ref 221, it is hard to assess the accuracy of the fitting results. Nevertheless, the formula shows the correct dependence of velocity on impurity concentration. The dashed curve in Figure 19b shows that this model gives an excellent fit to the data. From these results, we conclude that the COM-citrate system is reasonably described by the C-V model, with the modification that citrate also impacts the kinetic coefficient of step motion on all faces.

5.4. Naturally Occurring Macromolecules OPN is a naturally occurring protein in humans and is a single-chain protein with a peptide molecular weight of ∼33 kDa. OPN has an abundance of sequence domains rich in aspartic acid,223 and in Vitro evidence implicates OPN as one of several macromolecular inhibitors of urinary crystallization with potentially important actions at several stages of COM crystal nucleation, growth, or aggregation.172,224 Normal human urine contains levels of OPN (>100 nM) that markedly inhibit several aspects of COM crystallization.172 Since the acidic residues of organic molecules have been implicated in the control of mineralization in a wide range of organisms and mineral systems,13,225 analysis of modulation of COM by OPN at a molecular level may also offer general insights concerning the molecular control of biomineral formation. In striking contrast to the case of citrate, in situ SPM investigation shows that OPN modifies the morphology and inhibits the growth kinetics strongly on the (010) face of COM. It has very little effect on the (-101) face.192 These changes are due to step-specific OPN interactions leading to pinning. Qualitatively similar effects were seen at all OPN levels (1-25 nM) but were more rapid and quantitatively greater at higher OPN levels. The changes in step morphology of the growth hillocks on the (010) face are shown in Figure 20a and b. When 5 nM OPN was introduced to the growth solution, both the [100] and [021] steps became strongly pinned and lost lateral stability. In addition, step speed along both directions was also dropped by an order of magnitude. However, OPN did not alter either step speeds or morphology on the (-101) face (Figure 20c and d). Although interactions of OPN with steps on this face were apparently weak, discrete adsorbates appeared on the (-101) terraces at all OPN concentrations investigated (1-25 nM). The number of adsorbates increased over time without changes in their dimensions, showing that they were not simply 2D

nuclei. While the lateral dimensions among these adsorbates had a wide distribution, their heights were fairly constant at about 10 Å. These results suggest that OPN molecules interact with the terraces strongly enough to form adsorbates, but these bound OPN molecules and those in solution interact weakly with the steps on this face and thus cause no changes in step kinetics or morphology. In fact, the steps propagated freely without interference from the adsorbates on the terraces.41 The adsorbates in these figures remained intact following the passage of growing steps, which apparently move beneath them. As with citrate, divergent effects of OPN upon the (-101) and (010) faces must reflect differences in geometric relationships of functional domains with crystal steps and terraces. The dicarboxylic acid residues of aspartic acid-rich proteins are known to be responsible for their strong interaction with crystal faces.51 Phosphorylation of OPN has also been linked to inhibition of COM growth.226 However, protein binding to heterogeneous interfaces reflects both the charge density of functional groups and the spacing of these groups with relation to the local geometry of the mineral face. Although OPN molecules are highly flexible in 10 mM phosphate solution,227 a fixed conformation may be induced by Ca2+ ions228 (or by binding to a mineral surface), as suggested by the apparently inert nature of the OPN adsorbates on (-101) terraces. Among the local step characteristics that should impact OPN binding is the height. On the (-101) face, step heights are only 6.0 Å. However, the step height is 16 Å on the (010) surface due to step bunching as described in the previous section.192,208,229 Thus, the much greater height of the quadruple steps on the (010) face facilitates binding of a large number of carboxylic acid and phosphate groups of OPN to the step riser and the basal plane, leading to a strong OPN-step interaction that pins the steps. On the (-101) face, where the step-risers may not be tall enough to satisfy the steric requirement for the binding of OPN anionic groups, a weak binding of OPN molecules to the planar terrace may well predominate because of the larger availability of binding sites. Similar behavior has been observed for the Tamm-Horsfall proteins (THP) on the crystallization of COM in a recent study.230 During growth of the (-101) face, THP forms filament-like deposits that cover the surface without altering the geometry of the growth hillocks. In fact, despite this deposition, THP accelerates step kinetics on this face. In contrast, no visible THP deposits appear on the (010) face. Still, although the initial shape of the growth hillocks is preserved, they become poorly resolved as if a thin film of THP is coating the surface. The effect of THP on the growth speeds of the two types of steps on the (010) face is bifunctional. That is, at lower supersaturation THP inhibits growth, while at higher supersaturation it promotes growth. In light of these complex results, further work is required to reveal the underlying principles of how proteins control the crystallization of COM and other biominerals in general.

5.5. Synthetic Macromolecules 5.5.1. Synthetic 27-Residue Peptides In the effort to mimic the putative action region of osteopontin and other aspartic acid-rich proteins, a pair of 27-residue linear aspartic acid-rich peptides of (DDDS)6DDD (short by DDDS) and (DDDG)6DDD (short by DDDG) were designed, where D ) aspartic acid, S ) serine, and G )


Qiu and Orme

Figure 19. (a) Relative growth (Ri/R0) rates collected under various supersaturation and citrate concentrations (symbols and solid lines). The dashed lines are examples of theoretical curves calculated with eq 20. (b) Fit of experimental data to the modified C-V model that takes into account decreasing β on all faces with increasing citrate level. Reprinted with permission from ref 221. Copyright 2006 American Chemical Society.

Figure 20. Temporal AFM images showing changes of COM growth hillocks in the presence of OPN on the (010) face (a and b) and on the (-101) face (c and d): (a) pure; (b) 5 nM; (c) pure; (d) 1 nM. Image horizontal dimension; (a and b) 1.7 µm; (c) 6.0 µm; (d) 4.5 µm. Reprinted with permission from ref 192. Copyright 2004 the National Academy of Sciences of U.S.A.

Figure 21. AFM images show the evolution of growth hillocks on the (010) face (a-c) and the (-101) face (d and e) during the growth in a supersaturated (σ ) 0.82) solution containing the synthetic 27-residue peptides: (a) pure; (b) in the presence of 7 nM DDDS; (c) in the presence of 8 nM DDDG; (d) pure; (e) in the presence of 1.4 nM DDDS. Image horizontal dimension: (a–c) 2 µm; (d and e) 5 µm. Reprinted with permission from ref 190. Copyright 2006 American Chemical Society.

glycine.190 In situ SPM observations show that the 27-residue (27-mer) peptides modify both the morphology and kinetics of step propagation on the growing COM crystal faces, with

DDDS peptide being more potent. This is true on both the (010) and (-101) faces. On the (010) face, the DDDS peptide causes much more severe changes to the step morphology than that with DDDG. As shown in Figure 21b, in the presence of 6.67 nM DDDS, step edges along both the 〈021〉 and 〈100〉 directions were heavily roughened, which indicates that these steps were highly pinned. Morphological changes in the 〈100〉 direction were more prominent than those in the 〈021〉 direction, particularly in a relative reduction of terrace width. When the full effect was realized, the rectangular shaped hillocks on the (010) face were transformed into a series of elongated step segments along the 〈021〉 direction. Similarly, DDDS also markedly altered step kinetics in both directions. The measurable reduction in step speed was 50% after only 20 min of exposure to 6.67 nM of DDDS. All growth had ceased after ∼1 h, and no additional changes in surface morphology were detected in images obtained at later times. The modification to both morphology and kinetics on the (010) face by DDDG is much less significant. Figure 21c shows the hillock shape in the presence of 8 nM DDDG. Unlike the morphology shown in Figure 21b, the step edges on the dislocation hillocks remained clearly delineated throughout the experiment. Moreover, no strong pinning of step edges in either direction was observed, although the overall shape of the hillocks was slightly changed. The major alteration of the shape of dislocation hillocks was an elongation along the crystallographically equivalent corners between the [-100] and [021] directions. This was due to a relative slowing of the step speed in the direction of the other two corners of the hillock, which led to a flattening of the hillock in that direction. Because the difference between these two sets of corners is related to differences in the structure of left-facing and right-facing kink sites along the steps, this result indicates a stronger interaction with those kinks facing the slow corners. Overall, there was less effect on step speed than observed with the DDDS peptide, with only a 40% reduction in step speed after 1 h of exposure to the DDDG peptide, and no further reduction at later times. On the (-101) face, both peptides caused changes to the step morphology and kinetics. Similar to the case on the (010) face, the effects exerted by the DDDS peptide were much greater than those exerted by the DDDG peptide. The evolution of surface features of the (-101) face in the presence of DDDS at 1.34 nM is shown in Figure 21d and e. Both the [101] step and the 〈120〉 steps were strongly



Figure 23. SPM images showing the morphological changes of growth hillocks on the (010) face due to the presence of polyD: (a) pure; (b) after 2.5 µg/mL of polyD was added in solution. Reprinted with permission from ref 189. Copyright 2004 American Chemical Society. Figure 22. Inhibition of COM growth rate by a series of concentrations of DDDS (solid circles) and DDDG (open triangles). The concentration of DDDG needed to achieve either 50% inhibition or 90% inhibition of COM growth is more than 30 times larger than that for DDDS. Reprinted with permission from ref 190. Copyright 2006 American Chemical Society.

affected. The step edges along both directions were pinned and eventually became undefined. In addition, a new step is expressed along the [120] direction.190 The emergence of the [120] step suggests that this step is slowed by the presence of DDDS, since it was not detected during its rapid growth in pure solution. Step speeds along the [101] and the 〈120〉 directions were also significantly reduced. After 90 min of exposure to a low level (1.34 nM) of DDDS, growth in all step directions ceased. In contrast, effects of the same level of DDDG on the step kinetics and morphology were very minimal.190 First, the presence of [120] steps was not detected. Next, DDDG reduced step speed by only 20%, and finally, the maximal morphologic change was much less drastic, as shown in Figure 21e. SPM analysis clearly demonstrated that DDDS is a much stronger growth modifier than DDDG. This observation is consistent with bulk kinetics analysis on the growth of COM under the influence of these two peptides using the constant composition method.190 Figure 22 shows the percent inhibition of COM growth in the presence of both DDDS and DDDG. The results undoubtedly show that, for all peptide levels investigated, nearly 30 times more DDDG is required in solution to achieve the same magnitude of inhibition in COM growth by DDDS. Thus, the potency of growth inhibition by DDDS is much stronger than that by DDDG. Although it is suggested by many other studies that an increasing the number of aspartic acid residues in the sequence is responsible for the increased growth inhibition,185,226 the divergence between DDDG and DDDS, which have the identical number of aspartic acids, states otherwise. Thus, the identity of the spacer used in the linear 27-mers suggests that the sequence detail is a crucial determinant of the interaction with COM crystals. The flanking spacers may change the local charge density that is presented to the steps of COM crystals. They may also change the peptide structures at the interface. For DDDS, the existence of the OH group in the side chain of the serine spacer may increase the binding strength to the local steps, either by forming a hydrogen bond with the oxygen atom or participating in binding with calcium ions in either the basal or riser planes of the steps.41 Hydrogen bond formation can play an important role in stabilizing molecular structures bound to

steps229 and has been observed to enhance the strength of the interaction between citrate and COM steps.41,192 Moreover, since all amino acids in DDDS are hydrophilic, DDDS is most likely present itself as a linear chain in solution. This enables all the carboxylic groups to cooperatively interact with the crystals when they reach the step edges. The hydrophobic nature of glycine may cause DDDG to be flexible and unstable in solution. Because glycine is not hydrophilic, the presence of glycine within the sequence may increase the tendency of DDDG to fold, making the structure more compact. The consequence of compaction is that the number of available carboxylic groups for interaction with steps will be greatly reduced. If there is no folding, DDDG is probably highly mobile and flexible.231 By making the peptide structure less stable, the entropic contribution to the binding strength will be increased. However, since the entropy contribution is negative to the binding events,232 the interaction between DDDG and the steps is weakened. Thus, both features conferred by glycine would be expected to reduce inhibition of COM growth.

5.5.2. Synthetic PolyD, PolyE, and PolyAA Since their first investigation on the effect of poly aspartic acid (polyD) and poly glutamic acid (polyE) on COM growth by monitoring the filling rate over the pits on the (-101) surface using SPM,185 Jung et al. performed a more comprehensive examination of the effect of macromolecules on COM growth.189 First, the investigation was extended to examine three faces (-101), (12-1), and (010) (via Deganello and Piro notation) and steps were observed in all three faces. Second, besides polyD and polyE, poly(acrylic acid) (polyAA) was included in the additive list. Because the steps were imaged on all three surfaces, the control in growth was revealed by measuring the changes in step speed and growth morphologies of all relevant atomic step directions. Similar to data reported for citrate and protein,41,192,233 the synthetic polyD, polyE, and polyAA modify COM growth by selectively interacting with steps on the existing faces The three synthetic polypeptides showed strong control over COM growth morphology. For example, the growth hillock on the (010) face completely lost its identity and stability when polyD was present in the growth solution. As shown in Figure 23, for an addition of polyD at as low as 2.5 µg/mL, both the 〈021〉 step and the 〈100〉 step were strongly modified. As a result, the growth hillocks changed from a rectangular-shaped spiral to a band of serrated stripes. The serrated feature of the steps suggests that polypeptides


Qiu and Orme

Figure 24. Dependence of step speed on the concentration of polyAA, polyD, and polyE: (a and b) steps on the (010) face; (c) step on the (-101) face. The notation for the step speed direction is converted to the Deganello and Piro35 convention. Reprinted with permission from ref 189. Copyright 2004 American Chemical Society.

control the growth by pinning the steps. The modified shape shows a great resemblance to that of the DDDS and OPN. The polypeptides also showed strong inhibition to the COM growth kinetics. On the (010) face, the reduction of the step speed along both the [021] step and the [100] step (Figure 24a and b) exhibited similar characteristics for each impurity, respectively. As clearly shown in those figures, the step speed decreased substantially at the low impurity concentration and then leveled off as the peptide concentration increased, followed by a rapid drop at high concentrations. Similar characteristic effects were observed over the steps on the (-101) surface for the same polypeptides (see Figure 24c). Judging from the kinetics curve in both figures, it was apparent that polyAA was the strongest inhibitor on both faces. In contrast to polyD and polyE, only small amounts of polyAA were required to completely suppress the step growth on both faces. This effect was more prominent on the (-101) surface, with the advancement of steps stopped at the polypeptide concentration as low as 0.05 µg/mL. The inhibitory potency, however, for polyD and polyE strongly depended on the surface on which the interaction took place. On the (010) surface, polyE was more effective than polyD, but the role was reversed on the (-101) surface. The stronger inhibition of the (-101) face in general may be related to the atomic scale structure of steps on the surface. As discussed previously, the steric configuration of steps on the (-101) surface, along with the high content of Ca ions in both the riser and basal planes, makes it ideal for strong binding with negatively charged carboxylate groups. This is consistent with the kinetic results for polyD but not with that for polyE. The difference between polyD and polyE is the number of methylene groups in the anionic side chain. The results suggest that, with a short side chain, the peptides bind more strongly to the steps on the (-101) surface. This is consistent with the observation for the polyAA, which also bears a short side chain. Thus, the stereochemical match between the steric steps and the side chain conformation determines the interaction. There are two interesting points that should be noted from Figure 24c. First, there appears to be a slight acceleration to the [101] step advancement in the presence of polyE in the medium concentration range. Next, there is a sharp drop in the step speed at higher concentration for both polyD and polyE. Although one could argue that, within the error bar of the experimental data, this small acceleration in step speed may not be true, such behavior has been reported in other crystal growth systems under the influence of polypeptides.45

In that study, the role of polypeptides in modifying calcite growth was credited to the hydrophilicity, which reduced the desolvation energy barrier of the ions of growth units in solution or ions at the kink site. Since the polypeptides also pin the steps, the competition between pinning steps and reducing the activation energy barrier depends on the polypeptide concentration in solution. When the concentration is high, the ability for the polypeptides to pin the steps dominates and, consequently, the propagation of the step will be drastically slowed down and eventually ceased. A similar rationale can be applied to explain the results shown in Figure 24c, especially for polyD and polyE. The formation of the finger-like step bunches shown in Figure 23b189 is very similar to that shown in Figure 20b for OPN and Figure 21b for the DDDS 27-mers. In fact, the similar morphology of step bunches was also observed on the crystals of KH2PO4 grown under the influence of Fe3+ ions.216,234 It was believed that the modification was intrinsically related to the dynamic competition between the pinning of steps by Fe3+ adsorption and step-step interactions within the step bunches. Similarly, the inherent tendency of steps on the (010) face to bunch into quadruple-height steps is likely to be a source of the same behavior in the COM system.41,189,192 A recent study shows that, for long chain macromolecules, the change in kinetics and morphology may relate to the overlap between the time scale for peptide adsorption and the lifetime of the interstep terraces.235,236 The competition between the step terrace lifetime and time for modifier to reach equilibrium on the terraces is critical in controlling the modification of COM growth by large size macromolecules.

6. Brushite As noted in Table 6, the mineral brushite (CaHPO4 · 2H2O) is found under pathological conditions in kidney stones, some forms of arthritis, and caries. It is also found under some conditions within the salivary glands and exists in both young and (to lesser extent) old calculus. In general, brushite is undersaturated in common body fluids such as saliva, urine, or serum (see Table 2) but will form at sufficiently low pH. It is known that brushite has faster nucleation kinetics than the other calcium phosphate phases due to its higher solubility.237,238 Accordingly, it has been proposed that brushite is a transient precursor for phases such as octacalcium phosphate (OCP) and HAP168 and may form even when it is not the most stable phase. It has also been suggested that macromolecules can slow the transition to other phases



Table 9. Binding Energies (kJ mol-1) of Citrate Bound to Various Steps and Faces of COM Crystal41 (-101) face (-101) plane [101] acute step [101] obtuse step [-1-20] obtuse step [120] acute step (010) plane [100] straight step [100] obtuse step [021] obtuse step [021] straight step

(010) face

-65.4 -166.5 -110.0 -133.8 -162.4 -48.9 -73.2 -92.8 -56.9 -102.4

by inhibiting the dissolution or hydrolysis reactions necessary for transformation to the more stable phases. For example, brushite has been shown to remain stable in extracted salivary solutions (pH > 7) despite the fact that OCP and HAP are more thermodynamically stable phases in this pH range.239 On the other hand, brushite transforms to OCP under the same solutions once filtered, suggesting that macromolecules inhibited the transformation in the unfiltered solution. Taken together, kinetics may cause brushite to form more readily and to remain within the body for longer durations than equilibrium solution speciation of body fluids might predict. One pathological condition of particular interest is the role of brushite in the formation of kidney stones. As discussed more fully in section 4, calcium oxalate monohydrate (COM), the major component of kidney stones, is supersaturated in urine and along parts of the urinary track, but in healthy patients it typically remains within the metastable region and thus is unlikely to undergo homogeneous nucleation on reasonable time scales. For this reason, it has been suggested that other materials may act as nucleation centers. As brushite is the phase that precipitates most readily in urine environments at pH less than 6.9, it has been postulated that brushite enables the nucleation of COM.240 The following section reviews brushite dissolution and growth dynamics as measured by SPM (Table 9) and examines brushite in the presence of four molecules with carboxyl moieties: citrate, oxalate, osteocalcin, and poly(sodium)aspartate.

6.1. Growth and Dissolution of Brushite in Solutions without Impurities The growth and dissolution of brushite is highly anisotropic with three distinctly different steps that vary in their kinetics, kink density, and bonding environment. Scudiero et al.,241 who were the first to use SPM to study the surface kinetics, examined these differences in bonding environments by applying a mechanical load using the SPM tip. Later, Tang et al.242-245 examined dissolution kinetics, comparing them to bulk measurements, and Kanzaki et al.111 measured step dissolution rates versus degree of undersaturation to obtain a kinetic coefficient for dissolution. Similarly, Orme and Giocondi97 obtained a kinetic coefficient for growth. The atomic structure, which dictates the stable steps and the relative rates of dissolution or growth, is reviewed below.

6.1.1. Atomic Structure of Brushite Brushite is a noncentrosymmetric monoclinic crystal with four CaHPO4 · 2H2O molecules per unit cell. The structure is composed of corrugated rows of calcium cations (light blue spheres) intertwined with HPO42- anions (gray tetra-

Figure 25. Overview of the brushite atomic structure oriented in the Cc crystal class with Ca in light blue, O from HPO4 in red, O from water in dark blue, P displayed as gray tetrahedrons coordinated with four oxygen atoms, and H in white. (a) One 7.6 Å bilayer looking down on the (010) face with cuts along the three primary step directions observed by SPM. (b and c) Calcium and phosphate clusters, respectively, in the same orientation as in part a. The right cluster is higher than the left cluster, as best viewed in the side view provided in part d.

hedrons) such that adjacent calcium (or phosphate) clusters are at alternate heights, forming a corrugated bilayer structure (Figure 25). Water molecules bound to the calcium cations point outward at the top and bottom edges of these bilayers, resulting in layers of water between the Ca2+- and HPO42-containing sheets (Figure 25d). The weaker bonding of the water molecules to one another creates a cleavage plane between the two water layers perpendicular to the b-axis. For this reason, the {010} faces are fully hydrated even within the bulk structure. The {010} faces dominate the macroscopic habit,246 leading to a platelike morphology. The earliest X-ray analysis performed by Maclennan and Beevers247,248 showed brushite to be similar to its sulfate analogue, gypsum (CaSO4 · 2H2O), and reported the main crystallographic parameters. The structure was later revised to the Ia crystal class,33 where (010) and (0-10) are mirror images of one another. Curry and Jones33 used neutron scattering to analyze the positions of the hydrogen atoms


Qiu and Orme

possible that are less frequently observed experimentally, making it clear that there are limitations to the successes of these simple models, which are based only on in-plane bonding and do not include interactions with the solvent.

6.1.3. Steps Structure and Relative Kinetic Coefficients

Figure 26. A common brushite macroscopic habit looking down onto the (010) face. The facets are given in the Cc crystal classification and differ from Legeros’ work,246 which did not account for brushite’s lack of inversion symmetry. The growth step directions observed by SPM are shown as a black triangle.

and determined that the acidic hydrogen within the HPO42moiety occupies a unique site on the longest P-O bond. The current description of brushite’s structure is complicated by the fact that four different crystallographic classes are used in the literature. These are Ia/2,246,248 which as noted above was later corrected to Ia,33 Aa,249,250 and Cc.251 While most work is presented in class Ia, the Ia classification is not recognized in the standard tables, and thus, more recent papers instead use Aa and Cc. Of these, Cc is recommended to describe crystals with a unique b-axis (such as brushite), and thus, we use it here but provide a table to help unify the various descriptions, particularly because most experimental data is given in Ia while most modeling (and crystallographic software) is presented in the other two systems.

6.1.2. Crystal Habit Brushite crystals can be readily grown using several methods, including precipitation directly from supersaturated solutions,246,252 slow diffusion in silica gels,246,253 and electrochemical deposition on conducting substrates.254 The {010} cleavage planes dominate the brushite habit, leading to thin platelike crystals. However, aside from the large flat {010} faces, the crystal habit has considerable variationsfrom triangles to trapezoids to more complex shapes. The shape depends sensitively on impurities and growth conditions, suggesting that brushite surfaces interact readily with other ions to allow the system to transition between different morphologies.246,255 While this makes the system interesting, these variations also make identification of the side facet orientations difficult using only the macroscopic habit as the guide. The original indexing was performed by Legeros;246 however, at the time, the lack of inversion symmetry was not realized and the no distinction was made between (010) and (0-10) facets. For this reason, the indexing was later revisited by Abbona.249 The habit was further investigated using periodic bond chain methods256 to show that the most commonly observed facets have “flat” forms (denoted F in Tables 10 and 11), which have two or more chains of chemical bonds within the plane and thus are less reactive with their environment, leading to slower kinetics and concomitantly larger area. The exception is (111)Cc, which is a “stepped” form (denoted S in Table 10) with only one chemical chain within the plane. Stepped forms are expected to be more reactive, and indeed, as will be discussed in later sections, this step direction has the greatest velocity of the three that compose the growth triangles. However, several other F-forms are theoretically

All SPM imaging to date has been performed on the (010) or (0,-1,0) face. This is due to the tablet geometry and convenience of using the cleavage plane to expose a clean, flat surface at the start of an experiment. As discussed by Scudiero et al.,241 the asymmetry of the surface oxygen atoms visualized from atomic resolution images can be used to distinguish the (010) from the (0-10) face. Similarly, the shapes of the triangular dissolution pits are mirror images of one another on the two faces and also serve as unique identifiers once the correlation is known. The same holds for growth hillocks (keeping in mind that growth hillocks are mirror images of dissolution pits). There are several features of the atomic structure that play a role in the crystallization dynamics. Within the crystal, each calcium ion is bonded to eight oxygen atoms (Figure 25b): six from neighboring phosphates (in red) and two from water molecules (in dark blue). Thus, at a step edge, where oxygen atoms are not available from neighboring phosphates, it is likely that the calcium ion will complete its coordination by binding water or OH- groups from the solution. As a reminder that unfulfilled oxygen bonds exist on these edges, the step edges displayed in Figure 25a and 25d are cut such that the CaO8 coordination (Figure 25b) remains intact, although the exact form of the hydrated step edge is unknown. As the crystal grows, the oxygen atoms from the solution will need to be removed (or rearranged) to accommodate the adsorbing HPO42- ion, and thus, dehydration is expected to be an important part of the activation barrier for growth.52 It is also interesting that the {010} faces are fully hydrated as part of the bulk crystal structure and thus the removal of tightly bound water at an {010} surface is not a part of the activation barrier on this facet. In other words, the large surface area of this facet is due to low surface energy rather than to kinetic barriers associated with dehydration. And, in fact, brushite has a relatively low interfacial energy of 4.5 mJ/m2 42 compared to those of other biominerals, such as 13.1 mJ/m2 for COM.257 SXRD studies show that this water layer is crystalline, but not icelike, and does not impart order of water molecules into the solution, as might be expected from ice.258 The fully hydrated surface also suggests that hydrophobic proteins are unlikely to bind to these surfaces. The stable step directions were measured originally from observations of etch pit morphologies using SEM259 and more recently from SPM observations of both etch pits111,241-244 and growth hillocks.42,54 Brushite atomic steps measure 0.7-0.8 nm tall, reflecting one layer bounded by waters on each side. The steps typically have a triangular morphology (Figure 25), although under some conditions other steps can be observed. The three step directions within the (010) face are [1,0,-1]Cc, [1,0,1]Cc, and [-100]Cc. The step riser planes associated with each step direction cannot be measured directly with SPM or SEM; thus, the riser indices are inferred by comparing with observed macroscopic facets or calculated low energy faces (for example, as determined by PBC) that cut the (010) surface in the given step direction. The comparison between observed atomic step directions and



Table 10. Kinetic SPM of Calcium Hydrogen Phosphate Dihydrate calcium phosphate

additive

DCPD

none

DCPD DCPD DCPD

none none poly (sodium) aspartate, poly (L+)lysin, BSA

DCPD

citrate

DCPD

oxalate

DCPD

osteocalcin

primary finding

citations

mechanical stresses enhance chemical dissolution rates kinetic coefficients dissolution, size-dependent effects polymers exert kinetic controls on dissolution by chelating calcium and changing the interfacial energies citrate reduces step density via changes to the step free energy oxalate chelates calcium, initiating the transformation to COM osteocalcin chelates calcium, initiating the transformation to more stable HAP

241 97, 111 242-245 47 42 49, 54 48

Table 11. Correlation between the Primary Facets and Step Directions in the Three Crystallographic Classes Used in the Literaturea Ia

Cc

Aa

faceb (hkl)

stepc [UVW]

face (hkl)

step [UVW]

face (hkl)

facet family

PBC formd

(010) (-121) (11-2) (110)

[101] [-20-1] [00-1]

(010) (02-1) (111) (-111)

[-100] [10-1] [101]

(0-10) (1-20) (-1-1-1) (-1-11)

{010} {120} {111} {-111}

F F S F

Opposite signs are needed for both facets and steps to describe the (01j0) face. b The facets are given in the direction of step motion for hillocks growing on a (010) facet and are assumed to create an angle that is obtuse with respect to the underlying plane, as is suggested by the macroscopic crystal habit. c The step direction is defined as the cross product between the (010) face and the riser facet and, thus, is a vector lying within the (010) plane parallel to the step (rather than perpendicular to it). The direction of the step (advancing versus retreating) is made unique by choosing the (hkl) of the riser to point in the direction of the step motion. d Classification according to periodic bond chain analysis. F denotes “flat” and S “stepped” forms. a

macroscopic habit is shown in Figure 26. The inferred step risers are those found commonly in macroscopic crystals. Kanzaki et al.111 measured the kinetic coefficient during dissolution (Figure 27a) and reported values for the two fastest dissolving step directions of β[10-1]Cc ) 0.007 cm/s and β[-100]’Cc ) 0.014 cm/s. When these are adjusted to reflect eq 8, these become β[10-1]Cc ) 0.02 cm/s and β[-100]′Cc ) 0.038 cm/s. The authors note that the fastest growing step [-100]′Cc is oriented 5-7° off of the expected [-100]Cc direction. From the images available in the paper, these fast steps always have an inside corner. It is likely that this step is not a new orientation but instead moving faster than the other [-100] steps due to the availability of kinks at the inside angle and thus appears to have a different angle in a scanned image. This effect is discussed for growth in section 3.4.2. The kinetic coefficient reported for growth97 (Figure 27b) is an order of magnitude larger than the dissolution value with β[10-1]Cc ) 0.26-0.3 cm/s. Additionally, the value depended on whether the supersaturation was adjusting by adding more Ca2+ ions or by adding more HPO42- ions, suggesting that the ratio of Ca2+ to HPO42- played a role in the kinetics. By monitoring the step velocity under conditions of constant supersaturation and varying the Ca2+ to HPO42ratio, the kinetic coefficient was found to change by a factor of ∼2 (Figure 27c), with the growth rate limited by HPO42incorporation.97 Part of the difference between the growth and dissolution kinetic coefficients is likely due to differences in the [Ca2+]/[ HPO42-] ratios between the experiments, as the dissolution experiments were performed under conditions of excess calcium (serum-like solution) whereas the growth experiments were under conditions of excess phosphate (urine-like solution). The anisotropic nature of the steps makes it interesting to correlate stable step structure and relative kinetic coefficients with the underlying crystal structure. Both growth and

dissolution data report that the [-100]Cc step has the slowest kinetics.42,111,241 There are several features that make this step unique compared to the other two. First, along this step direction, the chains of calcium and hydrogen phosphate are bonded at the same level rather than in a corrugated manner, as they are in the other two directions. This means that this direction has tight ionic bonding within the step, giving it low step specific energy. Scudiero et al. point out that the calcium ions within this step have five nearest neighbor bonds compared with four nearest neighbor bonds for the other steps. Another feature that may play a role in the dynamics241,249 of this step is that the acidic hydrogen atom points into solution at the step edge (Figure 25, in black). It has been suggested that the OH- molecules hydrogen bond with water in solution, which must then be removed before the next crystallizing molecule can be adsorbed, leading to higher activation barriers for this step. The complementary [100]Cc step, which is chemically similar within the plane of the step but does not have an OH- group extending into solution, is not observed under normal conditions, supporting this idea.

6.2. Size-Dependent Phenomena during Dissolution In an interesting series of experiments that began simply enough as a bulk measurement of dissolution rates, Tang et al. found that brushite crystals did not dissolve fully despite a constant degree of undersaturation.242-244,260 This finding is contrary to conventional concepts of dissolution, which maintain that dissolution will proceed until all materials are consumed by the dissolution process. To gain a clearer understanding of what prevents materials from dissolving in undersaturated solutions, they employed SPM to directly monitor the surface. The results, discussed below, enunciate the importance of surface energy during dissolution (just as


Figure 27. Kinetic coefficients for brushite. Step velocities versus relative supersaturation during (a) dissolution and (b) growth with corresponding examples of surface morphology. (a) Orientations given in the Ia space group. One-micron scale bar shown. (b) Images correspond to a supersaturation ratio, S, of 1.63, 1.13, and 1.09 from top to bottom. All images are 6 µm × 3 µm. (c) Step velocity (normalized to high phosphate starting solution) versus free [Ca2+]/[ HPO42-] ratio showing higher growth rates with excess HPO42-. (a) Reprinted from ref 111, Copyright 2002, with permission from Elsevier B.V. (b and c) Reprinted with permission from ref 97. Copyright 2007 American Institute Physics.

it is for growth) and may have broad implications for the stability of nanoscale biominerals.

6.2.1. Bulk Dissolution Behavior In bulk measurements using constant composition techniques,30 Tang et al. found that dissolution rates of brushite crystals had an unusual pattern where the dissolution rate slowed almost to a halt before all of the starting materials were consumed despite a constant degree of undersaturation. Periodically, dissolution would resume in a stepwise manner, suggestive of a burst of dissolution that would then slow or stop.244 They found that the proportion of starting seed crystals dissolved was a function of both the size of the starting crystals and the degree of undersaturation. This behavior brings to mind some of the ideas originally suggested by the Ostwald-Freundlich description of solubility, which expresses solubility as a function of interfacial tension and particle size. However, they predict that the solubility increases as particle size is reduced, which is contrary to the findings of this study. More commonly, the solubility is equated with the solubility product Ksp and considered independent of surface effects.

Qiu and Orme

Figure 28. SPM image showing the dissolution of brushite at (a and b) σ ) -0.06 and (c and d) σ ) -0.172. Both sets of images are from the same sample in approximately the same location with the orientation shown. The temperature is held constant at 37 °C, and the solution is flowed at 0.5 mL/min throughout imaging to ensure a constant undersaturation value. The surface is allowed to equilibrate under flowing solution for at least 30 min at each undersaturation value. (a and b) Height image of 5 µm × 5 µm area, taken 6 min apart. Lighter colors represent topographically higher areas; thus, islands are light whereas pits are dark. The islands (white arrows) have dissolved noticeably, as have larger pits (gray arrows) whereas smaller pits are mostly unchanged. As expected at lower undersaturation levels (c and d), the surface dissolves faster; hence, 15 µm × 15 µm areas are shown 2 min apart. Dotted outlines in part d mark the original step locations to facilitate comparisons. Again, island geometries (Figure 29 case a) dissolve faster than compact pits. In these images, the rate of dissolution (over this area) increased when the three leftmost pits merged, thereby creating a geometry like Figure 29 case a. These images indicate that surface energy terms play a role in the dissolution rate. Reprinted with permission from ref 243. Copyright 2003 American Chemical Society.

6.2.2. Etch Pit Observations To investigate the role of the surface in these events, SPM was used to monitor step motion and morphology under solution conditions where dissolution was observed to slow or stop. Because the object was to follow pit evolution, a cleaved brushite crystal was first rinsed with water, creating a roughened surface populated with shallow pits. Two time points are compared (Figure 28) with undersaturation values of σrel ) (S1/2 - 1) ) -0.06 and -0.172. Further experimental details are provided in the figure caption and cited papers. The images illustrate the dissolution of both pit and island morphologies. Within the field of view, no new pits are formed due to the modest undersaturation levels.



Figure 29. Schematic of surface morphologies during dissolution. Gray scale coloring indicates topography, with lighter colors indicating higher regions than darker colors. Parts a and d are in side view, whereas parts b and c are in plane view.

What is immediately clear from these sequences is that not all steps are freely dissolving despite the undersaturated conditions. Rather, morphology, size, and orientation play a role. At both undersaturation values, the images show that islands dissolve readily but that pits dissolve comparatively slowly and in some instances not at all (or immeasurably slowly). For example, the smallest pits in Figure 32a have evolved very slowly whereas regions of islands (marked with white arrows) have a comparatively rapid dissolution rate. It is also clear that the step orientation plays a role in dissolution, with certain step directions dissolving much more slowly (notably, the [101]Cc) than others. In addition, the pit shape changes at the two saturation levels. From an energetic perspective, dissolution is guided by the incremental change in Gibbs free energy due to the removal of a row of atoms from a step. This involves both the chemical potential and surface energy terms, just as was discussed for growth (section 3.3.1). The degree to which step-free energies limit dissolution depends on the geometry of the surface, as depicted by the four cases shown in Figure 29. A nonsingular surface that begins with some roughness due to islands, hillocks, and/or steps (case a) will undergo spontaneous dissolution because both the chemical energy term and the step free energy term are negative due the simultaneous removal of atoms from the solid and the reduction of step length. In this case, there is no critical size; all steps dissolve independent of their length. Existing pits and pits emanating from dislocations, cases b and c, respectively, may or may not dissolve spontaneously, depending on the step lengths. As the step dissolves, its length increases. Because these step edges increase the energy of the system, this is an energetically unfavorable event until the step lengths are sufficiently large that the energy decrease associated with dissolution overcomes the energy increase associated with creating more edge length. This is completely analogous to the arguments presented in section 3.3 for the presence of a critical length for step motion during growth. In these cases, the incremental change in the Gibbs free energy (for triangular symmetry such as brushite) is ∆Gi/∆xi ) -(L/bi)|∆µ| + h∑iγibi, where L is the length of the dissolving step and bi and γi are the crystal lattice spacing and step free energy, respectively, in the ith direction. The step free energy term is constant; thus, once the step length is sufficiently large or the solution is sufficiently undersaturated, the negative chemical potential term wins and dissolution becomes energetically favorable. Strictly speaking, the distinction between cases a and c has less to do with the fact that case a is an island and case c a pit than whether the step length increases or decreases due to dissolution. For example, it is possible for noncompact islands to increase the step edge during dissolution; in which case, steps will have a critical length. Similarly, noncompact pits will dissolve freely until they reach a compact shape. At the farthest extreme, on a defect free flat surface (case d), a pit must be nucleated, which also involves the creation

Figure 30. Schematic diagram of the incremental changes in Gibbs free energy of a step due to the removal of a layer of atoms (step dissolution). The critical length is defined at ∆G /∆n ) 0.

of step edges. In this instance, the removal of one molecule must overcome the free energy term, ∆G/∆n ) -|∆µ| + h∑iγibi. Once the solution is sufficiently undersaturated that new pits are initiated with high probability (case d), then all other dissolution processes (a-c) are energetically favored and occur spontaneously, independent of the size of the feature. Thus, to explore the idea of pit evolution under conditions where new pits are not formed, one must first precondition the surface by exposing it to a sufficiently undersaturated solution that pits form and then moving closer to equilibrium. Most conventional theories of dissolution do not consider step or surface free energies. One reason for this is that critical phenomena such as described above can only be observed sufficiently near equilibrium. How close one needs to be depends on the step free energies. The widths of the regimes are summarized in Figure 30, which schematically plots the incremental change in Gibbs energy (black dashed lines) for pit nucleation versus pit evolution. In both cases, the step energy term (solid gray line) is a positive constant and the chemical potential terms (dashed gray lines) scale with ∆µ; but the slope of the chemical potential term differs according to the number of atoms removed (one for nucleation and n* ) L*/b for dissolution of steps of length L*). The points where the resultant change in Gibbs energy (dotted black line) is zero demark the regions of interest. In region III, pits form spontaneously, and in regions I and II, dissolution depends on geometry and step length. For any given length of step L*, lengths less than this will dissolve in region II but not in region I. When the surface energy terms are small, the point where pitting begins moves closer to equilibrium, as is easy to see by shifting the Gibbs free energy lines down, making size dependent phenomena difficult to measure. However, sparingly soluble materials, which encompass most biominerals, typically have correspondingly high surface energies.261 And these high surface and step energies create a larger range of undersaturations where geometry and length play a role in dissolution. The reasoning behind why surface energies are typically not considered in conventional dissolution theories is also a matter of the pathway taken. If an entire surface plane dissolves simultaneously, the size of the crystal is reduced and, hence, both the chemical potential term and the surface energy terms are negative. In this instance, analogous to case a, size does not play a role. However, if the path to dissolution involves the removal of surface steps, rather than an entire plane, then the length of the steps can be important. Note that the progression of this argument can move down in scale one more level. In the above discussion, we have


Qiu and Orme

considered an entire step dissolving simultaneously, which assumes that the steps have a high kink density. If the path to dissolution involves kink nucleation, then the details of the discussion will change, although one still expects steplength dependent effects (as discussed in section 3.3.2). The implications of this study are that, near equilibrium, materials with high surface energies or low solubility will have dissolution rates that are moderated by their surface geometry and size. The effects become more important as the particle size approaches the critical lengths, as more of the surface is likely to be composed of steps shorter than the critical length. In particular, an effect such as this may proffer dissolution resistance to nanoparticulate HAP, which may have implications for bones and teeth.244,245,262 Such effects have also been discussed for calcite.101

6.3. Additives with Carboxyl Moieties 6.3.1. Interaction of Citrate with Brushite One class of renal kidney stones is composed of brushite crystals that form within the inner medullary collecting ducts, eventually plugging them and causing damage to the surrounding cells.171 Brushite kidney stones are much less common than COM stones (2.8% compared with ∼75%); however, they are not as amenable to removal via extracorporeal shock wave lithotripsy (in which sound waves are used to break crystals into smaller pieces), and thus, prevention is highly desirable. It has also been proposed that brushite, due to its lower interfacial energy and higher solubility and, hence, faster nucleation kinetics, may act as a transient seed that initiates the formation of the more common calcium oxalate or mixed calcium oxalate and apatite stones.240 For both these reasons, it is of interest to understand whether there is potential therapeutic merit in citrate (as discussed more fully in section 5.3) in reducing the crystallization of brushite. In constant composition experiments, it was found that citrate, C3H5O(COO)33-, inhibits the bulk growth rate of brushite seeds.42 A concentration of 2.1 µM citrate reduced the growth rate by 50%, and at 10 µM citrate, the growth rate was 95% inhibited. In these experiments, the [Ca2+]/ [citrate] ratio was >80; thus, the supersaturation was not significantly altered due to citrate-calcium complexation in the solution. To investigate the cause of the inhibition, SPM was used to monitor the step kinetics and morphology. In these experiments, the same dislocation hillock was imaged in a succession of environments alternating between pure and citrate bearing (1-20 µM) solutions. Surprisingly, even at the highest citrate concentrations, brushite steps neither slowed nor significantly changed morphology (Figure 31b-d), although these same solutions inhibited the bulk growth rate (Figure 31a). Instead, the primary effect suggested by SPM images was a reduction in step density. Because the bulk growth rate is a function of both the step velocity and the step density, the SPM and bulk observations could be reconciled. However, the mechanism of inhibition is unlike the classical behavior, as, for example, was described for the case of COM, where citrate was observed to reduce the step velocity via step pinning. The step density on spiral growth hillocks is related to the critical length for step propagation (section 3.5), and in particular, a larger terrace width (as was observed in the presence of citrate) corresponds to a larger critical length.

Figure 31. Interaction of citrate with brushite. (a) Titrant volume (mL) versus time for constant composition experiments demonstrating the effect of citrate (1-10 µM). The slopes (normalized per crystal surface area) give growth rates. The rate is reduced by 95% at 10 µM citrate. (b) Corresponding SPM measurements of step velocities show constant values for citrate concentrations over this full range. SPM images in the (c) absence and (d) presence of citrate demonstrating that citrate decreased step density in this range of concentrations. Reproduced with permission from ref 42. Copyright 2005 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

With the supersaturation held constant, the critical length (Lc ∝ γ/∆µ) is proportional to the step free energy. Thus, the larger terrace widths can be explained by an increased step free energy. The authors further verified this result by using a thin layer wicking method to measure the surface energies of brushite powders in the absence and presence of citrate, finding that the interfacial energy increased by 95% at 10 µM citrate. The higher surface interfacial energy also increases the barrier to nucleation and thus increases the induction time, consistent with experimental results. The implications for macroscopic crystallization are that brushite crystals are less likely to nucleate in the presence of citrate (even at the same supersaturation value). This effectively expands the metastable regime, delaying the precipitation of crystals. Also, brushite crystallites that do form (or already existed) have significantly slower growth rates in the presence of citrate.



In solutions where the calcium and citrate concentrations are more comparable, the Ca-citrate complexes that form within the solution can significantly decrease the free Ca2+, resulting in a lower supersaturation. While this slows the step velocity, this type of “inhibitory” effect is in the solution, not at the crystal surface, and can be adjusted for once the solution speciation is properly taken into account.

6.3.2. Interplay between Brushite and COM When two minerals compete for a common ion, both the kinetics of formation and the relative thermodynamic stability of the two solids play roles in the temporal evolution of the solid-solution mixture. This is nicely illustrated in the case of brushite and COM, where the former has faster formation kinetics and the latter has greater stability. COM is the more thermodynamically stable phase, as suggested by the solubility product of 2.47 × 10-9 versus 2.36 × 10-5 (Table 1). Accordingly, as long as oxalate is available, COM will out compete brushite for calcium. However, from classical nucleation theory, the activation barrier associated with homogeneous nucleation from solution has a magnitude that depends sensitively on the interfacial energy,

∆gnuc ∝ γsl3/∆µ2

(23)

That is, at the same driving force, the solid with the lower interfacial energy, γsl, will have the lower nucleation barrier. From the kinetic perspective, brushite crystals will precipitate faster from solution due to their lower interfacial energy of 4 versus 13.1 mJ/m2 for COM.49 This effect has been noted in both urine samples240 as well as model solutions without crystallization inhibitors.49 In evaluating urine samples from patients with and without nephroliasis, Pak et al.240 showed that brushite was the first phase to nucleate out of solution in samples at pH less than 6.9 for both groups. From this evidence, the authors suggested that brushite was likely to play a regulatory role in stone formation by creating a transient phase that would initiate the nucleation of other calcium phosphate or oxalate phases. In idealized solutions without the full complexity of urine samples, Tang et al.49 tested this hypothesis using dual constant composition31 (DCC) experiments to measure the induction time associated with nucleating brushite and COM. The induction time (t) is the time needed for a supersaturated solution to precipitate crystals and is related to the activation barrier associated with nucleation,

ln τ ∝ C1 + C2

γsl3 kB3T3(ln S)2

(24)

Tang and co-workers showed that brushite crystals consistently nucleate before COM and related the lag time to the difference in surface energies between these materials. Interestingly, the authors also found that the presence of brushite seeds reduced the subsequent induction time for COM nucleation from over 2 days to approximately 1 h, suggesting that brushite surfaces in some way promote the formation of COM. To better understand this process, SPM was used to investigate how brushite might aid in nucleating COM. A brushite surface was imaged in solutions similar to the DCC experiments, starting with a brushite growth solution and then monitoring the surface response as oxalate was titrated into

solution (Figure 32). The initial surface shows a triangular hillock representative of normal growth. At low oxalate concentrations, the steps show some evidence of pinning and new facet forms, indicating specific interactions as the surface. However, once the oxalate concentration passed a threshold sufficient to precipitate COM crystals, the most apparent effect of oxalate was to chelate calcium first from solution and then cause rapid dissolution of the brushite seed. No islands of COM were observed to form directly on the brushite surface; instead, the transformation from brushite to COM occurred via a dissolution-reprecipitation process. At the end of the experiment, randomly oriented COM crystals could be observed on the brushite seed. The SPM results demonstrated that the brushite surface does not act as a heterogeneous nucleation site for COM in the classical sense. Typically, one expects a nucleation promoter to have a higher interfacial energy that becomes covered by a lower surface energy material or to act like a template providing an epitaxial match between the minerals. Instead, brushite seeds act as a reservoir of relatively concentrated calcium (typically 2-10 times more so than in the solution) and thus represent a region of higher supersaturation that can aid in the precipitation and rapid growth of COM. Additionally, there is evidence that COM crystals that form in this manner deposit on the brushite surface and are more likely to be held together as aggregates rather than dispersed in the solution.49

6.3.3. Interplay between Brushite and Apatite The transformation of brushite to apatite has also been observed by SPM. Flade et al.48 showed that brushite transformed to apatite-like hexagonal plates via a dissolution reprecipitation process in the presence of osteocalcin.263 Osteocalcin is a small 46-50 amino acid protein produced by osteoblasts and implicated in the formation of bone. Hauschka et al.264 proposed that osteocalcin preferentially binds to the (0001) surface of HAP due to the near epitaxial match between the γ-carboxyglutamate (Gla) residues of the protein and the Ca2+ ions in the (0001) HAP planes. Flade et al.48 chose SPM to investigate the early stages of nucleation to ensure sufficient resolution that the protein could be directly imaged. The authors used ex situ imaging to monitor the progression of a brushite seed crystal in an undersaturated, buffered solution containing 250 µg/mL (∼0.04 mM) osteocalcin. After 5 min, the authors saw protein binding to brushite steps but not the (010) facet, and after 1 h, thin apatite-like hexagonal plates coated with protein were observed covering the brushite surface (Figure 33). The authors propose a model in which osteocalcin binds to brushite steps where it reacts with the calcium and phosphate dissolving from the brushite surface. They then suggest that osteocalcin acts as a template for HAP nucleation due to the ordering of Gla residues. It is not clear from these experiments whether HAP nucleates in solution or on the brushite surface. And while it is clear that osteocalcin promotes HAP nucleation, it is not proven that osteocalcin acts as a template, although it is compelling. The authors convincingly demonstrate the strong affinity of osteocalcin for the (0001) face of apatite, in agreement with the model proposed by Hauschka et al.264 In both COM and apatite reprecipitation, it is not fully resolved whether brushite’s role is to act as a reservoir of calcium or whether the surface additionally organizes or concentrates the additive to accelerate nucleation of the new


Qiu and Orme

Figure 32. Interaction of oxalate with brushite. Each image is 20 µm × 20 µm: (a) in a pure growth solution (S ) 1.1, [Ca]/[P] ) 0.01); (b) with 1.1 mM oxalate ([Ca]/[Ox] ) 0.4); (c) with 1.5 mM oxalate ([Ca]/[Ox] ) 0.4). In part c, the solution is undersaturated and pitting occurs. Reprinted with permission from ref 49. Copyright 2006 the International Society of Nephrology.

Figure 33. Interaction of brushite with osteocalcin. In undersaturated solutions, brushite dissolves and then reprecipitates as apatite. (a) SPM image of a 2 µm × 2 µm dissolving brushite surface with osteocalcin bound to steps but not terraces. (b) Apatite-like hexagonal structure precipitated on a brushite surface. Osteocalcin coats the apatite surface. 1.5 µm × 1.5 µm. Reprinted with permission from ref 48. Copyright 2001 American Chemical Society.

phase. It would appear that the presence of the brushite surface serves to aggregate the newly precipitated crystals. The explanation for this is unclear, as the new phases do not have an epitaxial relationship to the substrate (they are randomly oriented) and they have higher interfacial energy, which would not favor coating the surface. Thus, it may be assumed that the additives have some role as a binder.

6.3.4. The Chelation of Calcium It is interesting to compare citrate, oxalate osteocalcin, and aspartic acid containing polymers, as each of these interacts with calcium via carboxyl groups in different geometries. Revisiting the case of citrate, which has three carboxyl groups, at low concentrations of citrate, brushite growth was inhibited by altering the interfacial energy. At higher concentrations, however, the chelation of calcium sufficiently reduces the supersaturation that this becomes the dominant effect. And, as observed for oxalate (Figure 32), citrate can cause brushite seeds to dissolve. However, in the case of citrate, the Ca-citrate complex is an aqueous species, which has two positive ramifications. First, it forms quickly, as it does not have a nucleation barrier associated with creating a critical size. And most importantly, it chelates calcium in an aqueous species and does not contribute to stones. Similar etching effects are observed for other carboxylate containing polymers. This is not surprising, as acidic polymers such as these are used commercially as antiscalants265,266 where this is their function. Peytcheva and Antonietti47 observed an increased dissolution rate in the presence of poly(sodium)aspartate (MW 18000 g/mol-1, ∼135 aspartic acids per molecule). The authors attribute the transition from shallow, slow growing pits in water to faster dissolving and deeper pits in poly(sodium)aspartate to

Figure 34. Poly(sodium)apartate etches brushite surfaces by chelating calcium. (a) SPM image showing etch pit; scale bar one micron. (b) Optical image displaying deep triangular pits (crystal length 0.5 mm). Reproduced with permission from ref 47. Copyright 2001 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.

chelation of calcium and changes in the interfacial energy of the surface in the presence of the polymers (Figure 34). Lower molecular weight poly(sodium)aspartate (MW 10000 g/mol-1) had an effect more similar to that of water, and thus, the authors argue that the enhanced pitting does not scale strictly with chelating power and, therefore, suggest specific interactions with the surface that rely on the polymer size. The authors do not observe direct adsorption of polymers on the surfaces and thus propose that the proximity of the polymer lowers the crystal interfacial energy. It should be noted that the pit observations are not at the same degree of undersaturation (due to calcium chelation), which complicates the interpretation of the images, as some changes may be expected simply due to changes in undersaturation independent of the effects of surfactants that alter surface energy (as is observed in section 6.2). And in



measurements of step kinetics.268 The hydrothermal process allows the synthesis of hexagonal needles bounded by {10-10} faces up to 0.2 mm in diameter and several millimeters in length.270 Using psuedophysiological solutions with Ca and P concentrations similar to those of serum (Table 2), they observed that growth proceeded by both step flow and island nucleation on the dominant a-face but only island nucleation on the smaller basal planes.

7.1. Growth on the Prism {10-10} Planes

Figure 35. Crystal structure of hydroxlapatite: (a) looking down on the hexagonal basal {0001} surface; (b) looking down on a prism {10-10}surface. Ca, light blue; OH, dark blue; O, red; PO3, gray tetrahedron.

particular, a higher degree of undersaturation is expected to form deeper etch pits, as the mode of dissolution transitions from step flow dissolution to etch pit nucleation. The authors also investigate the etching effects of poly(L+)lysine, suggesting that it binds to phosphate rather than calcium and thus interacts with different steps. They are not able to obtain direct SPM evidence of this, however, due to the adsorption of the polymer on the SPM tip, which prevents quality imaging. In summary, the interaction of carboxyl groups with brushite surfaces has three distinct effects: alteration of step free energies; specific interactions that stabilize facets or pin steps; and the chelation of calcium. The chelation of calcium by carboxyl groups lowers the supersaturation, causing Brushite to dissolve. The aqueous calcium complexes are used as therapeutics for kidney stones or as antiscalants more generally. Solid complexes consume brushite via dissolution reprecipitation reactions with time scales dependent on the supersaturation and interfacial energy. A more complete view of how additives alter mineralization would be obtained by systematic investigating the supersaturation versus the additive to calcium ratio, as different effects dominate in different regions.

7. Hydroxylapatite Hydroxylapatite (HAP) is a hexagonal crystal with two (Ca5(OH)1(PO4)3) molecules per unit cell. HAP is in the P63/m space group with unit cell parameters a ) b ) 9.432 Å, c ) 6.881 Å, and γ ) 120°.37 The crystals typically grow as rods elongated along the c-axis, and in teeth, the rods are bundled together and oriented such that the (0001) prism face forms the outer surface exposed to the fluid environment of the mouth. Looking down on the (0001) plane, the hydroxyl groups are aligned along the c-axis, resulting in hexagonal channels. Figure 35 shows the surface looking down on a basal {0001} face and a prism {10-10} face. Simulations of hydrated HAP surfaces have calculated surface energies of 420 and 660 mJ/m2 for the (0001) and (10-10) faces, respectively.267 The expected equilibrium crystal morphology based on these surface energies would be a hexagonal plate dominated by the (0001) face rather than the rods found in practice. HAP is difficult to image due to its typically small (

Dynamics of Biomineral Formation at the Near-Molecular Level

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