illusory spirals and loops in crystal growth

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Illusory spirals and loops in crystal growth Alexander G. Shtukenberg 1 , Zina Zhu, Zhihua An, Misha Bhandari, Pengcheng Song, Bart Kahr, and Michael D. Ward 1 Molecular Design Institute and Department of Chemistry, New York University, New York, NY 10003 Edited by Michael L. Klein, Temple University, Philadelphia, PA, and approved September 10, 2013 (received for review June 18, 2013) The theory of dislocation-controlled crystal growth identies a continuous spiral step with an emergent lattice displacement on a crystal surface; a mechanistic corollary is that closely spaced, oppositely winding spirals merge to form concentric loops. In situ atomic force microscopy of step propagation on pathological L-cystine crystals did indeed show spirals and islands with step heights of one lattice displacement. We show by analysis of the rates of growth of smaller steps only one molecule high that the major morphological spirals and loops are actually consequences of the bunching of the smaller steps. The morphology of the bunched steps actually inverts the predictions of the theory: Spi- rals arise from pairs of dislocations, loops from single dislocations. Only through numerical simulation of the growth is it revealed how normal growth of anisotropic layers of molecules within the highly symmetrical crystals can conspire to create features in apparent violation of the classic theory. B urton, Cabrera, and Frank (BCF) (13) launched the mod- ern era of crystal growth (4) with the idea that screw dis- locations on a crystal surface continually extrude steps to which molecules can attach. The living endsof these emanating spirals resolved the paradox of fast growth from solutions at low supersaturation. BCF theory, conceived originally for simple cen- trosymmetric cubic lattices, anticipated that one screw disloca- tion would generate a spiral whereas a closely spaced pair of dislocations spiraling in opposite directions would annihilate one another to form closed loops (the so-called FrankRead source mechanism) (5). Soon thereafter, both mechanisms were dem- onstrated (6, 7), forever sensitizing investigators of crystal growth to the coexistence and dichotomy of spirals and loops. Later it was discovered that crystals containing screw axes normal to the growth face could exhibit interlacing step patterns that, despite having morphologies that were more complex than crystals with proper symmetry axes, were easily understood (810). Hexagonal L-cystine crystals studied here by real-time in situ atomic force microscopy (AFM) exhibit extremely puzzling patterns: Single dislocations apparently form closed loops whereas pairs of dis- locations generate spirals. Although careful analysis has resolved this apparent contradiction and conrmed the correctness of BCF theory, these observations vividly illustrate how crystals lacking proper rotation axes and containing several translationally nonequivalent growth units can produce unusual and deceptive morphological features that can lead to incorrect conclusions about growth mechanisms. L-cystine crystals can form in the kidneys, leading to cystinuria, a painful and chronic condition. Our laboratory has used in situ AFM measurements of step growth rates on well-developed (0001) faces of hexagonal L-cystine (noncentrosymmetric space group P6 1 22, ref. 11) in the presence of additives to identify po- tent growth inhibitors (12). Remedies of this sort, however, must build on a complete understanding of spiral growth in the absence of additives, which motivated the investigation described herein. Real-time in situ AFM of L-cystine crystals grown from su- persaturated aqueous solution (equilibrium solubility C eq = 0.7 mM) revealed hexagonal growth hillocks that resemble stacks of islands as if formed from closed loops. Each island is 5.6 nm high, corresponding to the c-unit cell length (c = 5.6275 nm). The 6 1 screw axis inherent to the space group sym- metry is reected by a pinwheel of minor steps, successively ro- tated clockwise around the c axis by 60°, which spin out from each island, intersecting the edges of the island below (Fig. 1 A and B; Fig. S1). The height of these minor steps is ca. 1 nm, equivalent to c/6, the thickness of one of six elementary layers in the crystal structure that stack around the c axis and equivalent to the height of one L-cystine molecule. Occasionally, the 5.6-nm-high major steps form a macrospiral rather than closed loops (Fig. 1 C and D). In this case, the minor steps connect successive turns of the major step. The smallest Burgers vector––the direction and magnitude of the lattice dislocation––normal to the (0001) face is equal to the translation vector c (Fig. 2). The elementary c/6 layers are asy- mmetric, each bounded by six inequivalent edges aligned with the h10 10i directions. These edges dene six minor step planes with unique surface energies that anticipate different step lengths and growth velocities. Low-energy steps sever the fewest strong bonds, hydrogen bonds in this case (1315). Minor steps that fulll these conditions are depicted by slices A, B, and C in Fig. 2B. Step A would truncate the fewest hydrogen bonds, leading to the smallest step surface energy and velocity. Minor step A can advance in either the forward or reverse direction, denoted A(+) and A(). Greater velocities would be expected for directions B(+) and B() as well as C(+) and C(), which truncate more H bonds. This anisotropy is masked distant from the core, where the minor steps of the pinwheel are equivalent (because of the 6 1 screw axis) and correspond to the slowest moving step. Near the dislocation core, however, the six ineq- uivalent steps of a single elementary layer can be discerned by AFM. The white trace in Fig. 3 depicts the six steps of a single elementary layer emanating from the core (also Fig. S4). Direct AFM measurements revealed different step velocities for all six f10 10g steps, remarkably spanning an order of magnitude, the slowest step denoted arbitrarily as A(+). For example, the step velocities (nm/s) at [L-cystine] = 1.2 mM were measured as A(+) = 1.8, B(+) = 13, C(+) = 9.5, A() = 5.1, B() =12.7, C() = 10.9 (also Table S1, Fig. S5). Related features have been postulated for symmetry-induced interlacingof elementary growth layers Signicance Molecular mechanisms of crystal growth from solution remain ill-dened. Scanning probe microscopies have begun to illus- trate what was before insightful theory. The in situ observa- tions described here for hexagonal L-cystine crystals, which are known to form kidney stones, demonstrate that crystals with certain symmetries can exhibit unusual structural and growth behaviors that produce unexpected and deceptive morpho- logical features. Such features can appear to violate a classic theory of crystal growth enshrined more than 60 y ago and could lead to incorrect conclusions about growth mechanisms. Author contributions: A.G.S., Z.Z., and M.D.W. designed research; A.G.S., Z.Z., and M.B. performed research; A.G.S., Z.Z., Z.A., and P.S. analyzed data; and A.G.S., B.K., and M.D.W. wrote the paper. The authors declare no conict of interest. This article is a PNAS Direct Submission. 1 To whom correspondence may be addressed. E-mail: [email protected] or mdw3@ nyu.edu. This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1311637110/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1311637110 PNAS | October 22, 2013 | vol. 110 | no. 43 | 1719517198 CHEMISTRY

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Page 1: Illusory spirals and loops in crystal growth

Illusory spirals and loops in crystal growthAlexander G. Shtukenberg1, Zina Zhu, Zhihua An, Misha Bhandari, Pengcheng Song, Bart Kahr, and Michael D. Ward1

Molecular Design Institute and Department of Chemistry, New York University, New York, NY 10003

Edited by Michael L. Klein, Temple University, Philadelphia, PA, and approved September 10, 2013 (received for review June 18, 2013)

The theory of dislocation-controlled crystal growth identifies acontinuous spiral step with an emergent lattice displacement ona crystal surface; a mechanistic corollary is that closely spaced,oppositely winding spirals merge to form concentric loops. In situatomic force microscopy of step propagation on pathologicalL-cystine crystals did indeed show spirals and islands with stepheights of one lattice displacement. We show by analysis of therates of growth of smaller steps only one molecule high that themajor morphological spirals and loops are actually consequencesof the bunching of the smaller steps. The morphology of thebunched steps actually inverts the predictions of the theory: Spi-rals arise from pairs of dislocations, loops from single dislocations.Only through numerical simulation of the growth is it revealedhow normal growth of anisotropic layers of molecules withinthe highly symmetrical crystals can conspire to create features inapparent violation of the classic theory.

Burton, Cabrera, and Frank (BCF) (1–3) launched the mod-ern era of crystal growth (4) with the idea that screw dis-

locations on a crystal surface continually extrude steps to whichmolecules can attach. The “living ends” of these emanatingspirals resolved the paradox of fast growth from solutions at lowsupersaturation. BCF theory, conceived originally for simple cen-trosymmetric cubic lattices, anticipated that one screw disloca-tion would generate a spiral whereas a closely spaced pair ofdislocations spiraling in opposite directions would annihilate oneanother to form closed loops (the so-called Frank–Read sourcemechanism) (5). Soon thereafter, both mechanisms were dem-onstrated (6, 7), forever sensitizing investigators of crystal growthto the coexistence and dichotomy of spirals and loops. Later itwas discovered that crystals containing screw axes normal to thegrowth face could exhibit interlacing step patterns that, despitehaving morphologies that were more complex than crystals withproper symmetry axes, were easily understood (8–10). HexagonalL-cystine crystals studied here by real-time in situ atomic forcemicroscopy (AFM) exhibit extremely puzzling patterns: Singledislocations apparently form closed loops whereas pairs of dis-locations generate spirals. Although careful analysis has resolvedthis apparent contradiction and confirmed the correctness ofBCF theory, these observations vividly illustrate how crystalslacking proper rotation axes and containing several translationallynonequivalent growth units can produce unusual and deceptivemorphological features that can lead to incorrect conclusionsabout growth mechanisms.

L-cystine crystals can form in the kidneys, leading to cystinuria,a painful and chronic condition. Our laboratory has used in situAFM measurements of step growth rates on well-developed(0001) faces of hexagonal L-cystine (noncentrosymmetric spacegroup P6122, ref. 11) in the presence of additives to identify po-tent growth inhibitors (12). Remedies of this sort, however, mustbuild on a complete understanding of spiral growth in the absenceof additives, which motivated the investigation described herein.Real-time in situ AFM of L-cystine crystals grown from su-

persaturated aqueous solution (equilibrium solubility Ceq =0.7 mM) revealed hexagonal growth hillocks that resemblestacks of islands as if formed from closed loops. Each islandis ∼5.6 nm high, corresponding to the c-unit cell length (c =5.6275 nm). The 61 screw axis inherent to the space group sym-metry is reflected by a pinwheel of minor steps, successively ro-

tated clockwise around the c axis by 60°, which spin out from eachisland, intersecting the edges of the island below (Fig. 1 A and B;Fig. S1). The height of these minor steps is ca. 1 nm, equivalent toc/6, the thickness of one of six elementary layers in the crystalstructure that stack around the c axis and equivalent to the heightof one L-cystine molecule. Occasionally, the 5.6-nm-high majorsteps form a macrospiral rather than closed loops (Fig. 1 C andD).In this case, the minor steps connect successive turns of the majorstep.The smallest Burgers vector––the direction and magnitude of

the lattice dislocation––normal to the (0001) face is equal to thetranslation vector c (Fig. 2). The elementary c/6 layers are asy-mmetric, each bounded by six inequivalent edges aligned withthe h1010i directions. These edges define six minor step planeswith unique surface energies that anticipate different steplengths and growth velocities. Low-energy steps sever the feweststrong bonds, hydrogen bonds in this case (13–15). Minor stepsthat fulfill these conditions are depicted by slices A, B, and Cin Fig. 2B. Step A would truncate the fewest hydrogen bonds,leading to the smallest step surface energy and velocity. Minorstep A can advance in either the forward or reverse direction,denoted A(+) and A(−). Greater velocities would be expectedfor directions B(+) and B(−) as well as C(+) and C(−), whichtruncate more H bonds. This anisotropy is masked distant fromthe core, where the minor steps of the pinwheel are equivalent(because of the 61 screw axis) and correspond to the slowestmoving step. Near the dislocation core, however, the six ineq-uivalent steps of a single elementary layer can be discerned byAFM. The white trace in Fig. 3 depicts the six steps of a singleelementary layer emanating from the core (also Fig. S4). DirectAFM measurements revealed different step velocities for all sixf1010g steps, remarkably spanning an order of magnitude, theslowest step denoted arbitrarily as A(+). For example, the stepvelocities (nm/s) at [L-cystine] = 1.2 mM were measured as A(+) =1.8, B(+) = 13, C(+) = 9.5, A(−) = 5.1, B(−) =12.7, C(−) = 10.9(also Table S1, Fig. S5). Related features have been postulatedfor “symmetry-induced interlacing” of elementary growth layers

Significance

Molecular mechanisms of crystal growth from solution remainill-defined. Scanning probe microscopies have begun to illus-trate what was before insightful theory. The in situ observa-tions described here for hexagonal L-cystine crystals, which areknown to form kidney stones, demonstrate that crystals withcertain symmetries can exhibit unusual structural and growthbehaviors that produce unexpected and deceptive morpho-logical features. Such features can appear to violate a classictheory of crystal growth enshrined more than 60 y ago andcould lead to incorrect conclusions about growth mechanisms.

Author contributions: A.G.S., Z.Z., and M.D.W. designed research; A.G.S., Z.Z., and M.B.performed research; A.G.S., Z.Z., Z.A., and P.S. analyzed data; and A.G.S., B.K., and M.D.W.wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence may be addressed. E-mail: [email protected] or [email protected].

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1311637110/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1311637110 PNAS | October 22, 2013 | vol. 110 | no. 43 | 17195–17198

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Page 2: Illusory spirals and loops in crystal growth

sharing a screw axis (8), including 61, and observed for lower-orderscrew axes (9, 10, 16, 17). The L-cystine spirals are distinguished bywell-defined steps corresponding to each elementary layer as wellas the six steps belonging to a single elementary layer, permittingdirect measurement of the step velocity anisotropy that is re-sponsible for the deceptive microscopic morphologies.As a group of six minor steps advances from the core along

the six h1010i directions, the slowest step ultimately will limit theentire group, leading to bunching of the minor steps and for-mation of the 5.6-nm major steps flanking the hexagonal islandsand macrospirals in Fig. 1 B and D, respectively. Fig. 1A depictssix steps radiating from a single dislocation, each related to thenext by a 60° rotation and an elevation of c/6. Consequently, eachstep belongs to a different elementary layer. Fig. 4A reveals sixequivalent steps emanating from a dislocation core, with eachelementary layer denoted by a different color. As a particularminor step denoted at A(+) in each elementary layer advances ata velocity VA(+), minor step B(+) (from the same elementarylayer) is pulled from the core (depicted here as 33% of a com-plete angular cycle). Minor step B(+) then advances until itexceeds a critical length LB(+), at which time C(+) is born. OnceC(+) achieves LC(+) (65%), A(−) is born, and so on. BecauseVB(+) > VA(+), the distance between A(+) and B(+) in adjacentlayers decreases until B(+) merges with A(+) (71%), thusforming a permanent bunch. A bunch of six has a riser of 5.6 nm.The process of bunching in all directions normal to step edges atan equal distance from the core produces a hexagonal island(100%). The process then repeats with the next A(+) elementarystep (133%). An animated version of this simulation is providedin Movie S3.The Frank–Read source in Fig. 1C is the product of two

closely spaced dislocations of opposite sense. The total Burgersvector of this pair of dislocations is zero. The generation of new

steps occurs as for single dislocations (Fig. 4B). As an elementarylayer is extruded through the dislocation pair, its minor stepsconnect the opposing dislocation cores. Eventually, the six minorsteps of an elementary layer form an asymmetric six-sided poly-gon (also known as a loop) but with concavities expected (alsoknown as reentrant angle) during loop formation. The velocitiesof the two segments forming the reentrant angle are doubled inthe simulation because of the well-known acceleration of thestep motion at reentrant angles (18). The cycle is complete whensix loops corresponding to six elementary layers are formed,bunching along the directions of the slowest advancing steps(100%). By the time the first generation bunches (indicated bythe red arrow at 130%) it has met a second-generation bunch(indicated by the blue arrow). Here, the illusion of an overallmacrospiral first becomes evident as the asymmetric loops stackand the slowest steps of successive loops along the stack arerelated by a 60° rotation. In other words, the continually in-creasing distance between the step bunches and the core, com-bined with growth rate anisotropy, creates an apparent spiral(Fig. 4B; Movies S4 and S5).The well-accepted idea that spiral morphologies arise from a

single dislocation whereas a stack of islands is a telltale sign of aFrank–Read dislocation source (19, 20) is apparently turned upsidedown in the case of L-cystine. The splitting of a major step andsubsequent bunching reverses the morphological consequences

Fig. 1. Real-time in situ AFM of growth hillocks. Hillocks formed by (A)a single dislocation and (C) by a pair of dislocations, a so-called Frank–Readsource. Corresponding models B and D are illustrated, respectively. Six dif-ferent colors are used to depict the minor steps that define the six differentc/6 elementary layers. [L-cystine] = 2 mM. An AFM movie of growth from theFrank–Read source is provided in Movie S1. A movie of features like that in Acan be found in ref. 9. Step sources consisting of two or more dislocations ofthe same sense also are observed occasionally (Fig. S2).

Fig. 2. Crystal structure of L-cystine. (A) Three adjacent helices of L-cystinemolecules, viewed normal to the ð1010Þ plane, each winding about the 61screw axis that coincides with the c axis. Six L-cystine molecules span the 5.6-nm c axis along the flank of the six f1010g planes. (B) One of the c/6 ele-mentary layers as viewed normal to the (0001) plane. The A, B, and C stepsmarked by red, green, and blue lines, respectively, define six growth direc-tions (steps may advance from either side of each line). Line A slices thefewest number of hydrogen bonds and is considered to be the step of lowestsurface energy. Atom color code: carbon (gray), oxygen (red), nitrogen(blue), sulfur (yellow), hydrogen (white). (C) Schematic illustration of a hex-agonal L-cystine crystal with Miller planes denoted. A cut-and-fold papermodel for construction of a 3D hexagonal hillock of L-cystine is provided forthe convenience of the reader (Fig. S3).

17196 | www.pnas.org/cgi/doi/10.1073/pnas.1311637110 Shtukenberg et al.

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of dislocation-actuated growth: the spirals form closed loopswhereas proximal dislocations form macrospirals. In both cases,the deceptive hillock features are a consequence of bunching ofsymmetry-related elementary layers governed by the slowest ad-

vancing step. These features are special consequence of the inter-action of two screw axes, the sixfold screw with a translationalcomponent of c/6, and that created only at the dislocation core witha translational component of c. The BCF theory was originallydeveloped for so-called Kossel crystals, simple centrosymmetriccubic lattices with one growth unit per unit cell. Recent efforts havefocused on adjustments to BCF theory to account for low sym-metries and more than one growth unit (21–23). The observationsdescribed here for L-cystine indicate that such non-Kossel crystalscontaining high-order screw axes can exhibit micromorphologiesthat appear to be inconsistent with BCF theory, requiring verycareful analysis to unwrap their true identities.

Materials and MethodsPreparation of Hexagonal L-Cystine Crystals. All crystallization experimentswere performed at pH = 6.3. The hexagonal form of L-cystine was crystallizedfrom a supersaturated L-cystine solution prepared by adding 70 mg ofL-cystine (Sigma-Aldrich) to 100 mL of deionized water (3 mM), followedby heating under reflux at 100 °C for ∼30 min with stirring until the L-cystinewas completely dissolved. The resulting solution corresponds to a relativesupersaturation C/Ceq ∼ 4.3, based on the reported solubility (Ceq = 0.7 mMat pH 7, 25 °C) (24–26). The solution was then allowed to cool slowly withstirring for 75 min, after which 30-mL aliquots were transferred to separateglass containers that were subsequently sealed to prevent evaporation andexposure to airborne particulates. The solutions were then stored at roomtemperature in a quiescent environment for 72 h, after which single crystalswere collected by vacuum filtration (Whatman Grade 1 filters, 11-μm pores)and air dried.

Fig. 3. Single frame captured during real-time in situ AFM imaging ofa step source. Numbers correspond to each unique c/6 elementary layer,letters to the six unique directions within an elementary layer. The whitelines trace the steps of an elementary layer, arbitrarily assigned here as layer 1.The edges 1C(−), 1B(−), 1A(−), and 1(B)+ are obscured by layers above[L-cystine] = 1.5 mM. AFM Movie S2 illustrates the growth of the elementarylayers near the dislocation core.

Fig. 4. (A) Apparent island formation from a single dislocation: In the first panel (0%) the different colors correspond to identical steps rotated by +60° andsuccessively elevated by c/6. These steps are assigned to the slowest-advancing step A(+). The percentage of angular period is denoted in each image. Forexample, minor step B(+) reaches its critical length at 33% and 133% of the spiral revolution period. (B) Macrospiral formation from a pair of dislocations(Frank–Read source). The yellow arrows in the first panel of B (0%) depict the positions of the dislocation cores. At 130% of the spiral revolution period thefirst- and second-generation bunches, denoted by the red and blue arrows, respectively, meet to begin the illusion of the macrospiral. The percentage ofangular period required for a new turn of the macrospiral is denoted in each image. See Movies S3 and S4.

Shtukenberg et al. PNAS | October 22, 2013 | vol. 110 | no. 43 | 17197

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Crystal Growth Measurements. L-cystine crystals, prepared by the proceduredescribed above, were mounted on an AFM specimen disk coated withNorland optical adhesive (type 81) that had been cured partially by exposureto UV light (model EA-106, Spectroline, Spectronics Corporation, λ = 365 nm)for 1 min. The crystals were mounted by gently pressing the coated diskagainst hexagonal platelets on the filter paper, which naturally aligned theflat (0001) faces parallel with the specimen disk. The optical adhesive wasthen cured completely by additional UV exposure (∼15 min), which firmlybonded the crystals to the AFM specimen disk. The mounted L-cystine crys-tals were etched slightly by immersion in deionized water for 30 s at 65 °C toremove any amorphous deposits or impurities present on their surfaces. Insitu AFM was performed with a Digital Instruments Nanoscope IIIa Multi-mode system to collect sequential images of growing crystals, which thenwere compiled to produce real-time movies of crystal growth. All meas-urements were performed in a fluid cell in contact mode using Veeco NP-BSi3N4 tips on silicon nitride cantilevers with a spring constant of 0.12 N/m(triangular, 196 μm length, 41 μmwidth). Step velocities were determined bymeasuring the step displacement from a reference point (usually froma hillock center) in consecutive deflection images acquired at periodicintervals that ranged from 5 to 14 s (Fig. S5). Measurements were acquiredunder continuous flow by injection of fresh supersaturated L-cystine sol-utions into the AFM cell with a syringe pump, at a rate of 10 mL/h. Thisensured constant supersaturation and additive concentration, which en-sured constancy of step velocities for a particular set of conditions.

Simulations. The simulations of L-cystine hillock growth, portrayed in Fig. 4and Movies S3 and S4, were constructed using Java and configured as aplugin for ImageJ, which was used to output the animations. In the case of aspiral (Fig. 4A), the initial configuration is defined by the edges of six

crystallographically identical minor steps (color-coded to denote the suc-cessive rotations and elevations of 60° and ±c/6) radiating from the dislo-cation core. A step velocity V (perpendicular to the edge) and critical lengthare assigned to each edge. In these simulations, the step velocities for the sixminor steps within each individual anisotropic sublayer were assigned basedon values measured experimentally. The critical lengths were assumed to beidentical for all edges. The evolution of the spiral is processed frame byframe with a specified time interval between frames, here set as t = 0.01 s.Once the length of an edge exceeds the critical length, the simulation allowsthe edge to advance a distance equal to Vt concomitant with the creation ofan adjoining edge. The position of an edge is not allowed to move beyondthe edge in the underlying layer. This is repeated for all edges during a givenframe, and the process is repeated for a specified number of frames, therebygenerating arms of a spiral. The images of each frame are then combined tocreate the entire simulation. The simulation of the Frank–Read source iscreated in a similar manner, but the initial configuration is constructed fromsix inequivalent minor steps, each with a unique velocity, which connect twoproximal dislocation cores, separated by a distance equivalent to two criticallengths. When a reentrant angle appears in the simulation, the step veloc-ities of adjacent edges are doubled. Java script files for the spiral and Frank–Read source are included as Datasets S1 and S2.

ACKNOWLEDGMENTS. This work was supported by the Materials ResearchScience and Engineering Center Program of the National Science Foundation(NSF) under Award DMR-0820341. B.K. and M.D.W. also acknowledge sup-port from the NSF (CHE-0845526, DMR-1105000, and DMR-1206337) andNew York University. Z.Z. is grateful to New York University for a Hori-zon Fellowship.

1. Burton WK, Cabrera N, Frank FC (1949) Role of dislocations in crystal growth. Nature163(4141):398–399.

2. Frank FC (1949) The influence of dislocations on crystal growth. Discuss Faraday Soc 5:48–54.

3. Burton WK, Cabrera N, Frank FC (1951) The growth of crystals and the equilibriumstructure of their surfaces. Phil Trans Roy Soc London A 243(866):299–358.

4. Chernov AA (1989) Formation of crystals in solutions. Contemp Phys 30(4):251–276.5. Frank FC, Read WT (1950) Multiplication processes for slow moving dislocations. Phys

Rev 79(4):722–723.6. Verma AR (1951) Spiral growth on carborundum crystal faces. Nature 167(4258):939.7. Amelinckx S (1951) Spiral growth on carborundum crystal faces. Nature 167(4258):

939––940.8. van Enckevort WJP, Bennema P (2004) Interlacing of growth steps on crystal surfaces

as a consequence of crystallographic symmetry. Acta Crystallogr A 60(6):532–541.9. Pina CM, Becker U, Risthaus P, Bosbach D, Putnis A (1998) Molecular-scale mechanisms

of crystal growth in barite. Nature 395(6701):483–486.10. Stoica C, Van Enckevort WJP, Meekes H, Vlieg E (2007) Interlaced spiral growth and

step splitting on a steroid crystal. J Cryst Growth 299(2):322–329.11. Oughton BM, Harrison PM (1959) The crystal structure of hexagonal L-cystine. Acta

Crystallogr 12(5):396–404.12. Rimer JD, et al. (2010) Crystal growth inhibitors for the prevention of L-cystine kidney

stones through molecular design. Science 330(6002):337–341.13. Hartman P, Perdok WG (1955) On the relations between structure and morphology of

crystals. I. Acta Crystallogr 8(1):49–52.14. Hartman P, Perdok WG (1955) On the relations between structure and morphology of

crystals. II. Acta Crystallogr 8(9):521–524.

15. Hartman P, Perdok WG (1955) On the relations between structure and morphologyof crystals. III. Acta Crystallogr 8(9):525–529.

16. Qiu SR, et al. (2005) Modulation of calcium oxalate monohydrate crystallization bycitrate through selective binding to atomic steps. J Am Chem Soc 127(25):9036–9044.

17. Petrova EV (2011) AFM observation of a retgersite crystal surface growing in a water–ethanol solution. Moscow Univ Phys Bull 66(5):437–442.

18. Rashkovich LN, De Yoreo JJ, Orme CA, Chernov AA (2006) In situ atomic force mi-croscopy of layer-by-layer crystal growth and key growth concepts. Crystallogr Rep51(6):1063–1074.

19. Forty AJ (1954) Direct observations of dislocations in crystals. Philos Mag 3(9):1–25.20. Rashkovich LN, Petrova EV, Shustin OA, Chernevich TG (2003) Formation of a dislo-

cation spiral on the (010) face of a potassium hydrogen phthalate crystal. Phys SolidState 45(2):400–407.

21. Chernov A, Rashkovich L, Vekilov P (2005) Steps in solution growth: Dynamics ofkinks, bunching and turbulence. J Cryst Growth 275(1-2):1–18.

22. Rashkovich L, Petrova E, Chernevich T, Shustin O, Chernov A (2005) Non-Kossel crys-tals: Calcium and magnesium oxalates. Crystallogr Rep 50(S1):S78–S81.

23. Cuppen H, Meekes H, Van Enckevort WJP, Vlieg E (2004) Kink incorporation and steppropagation in a non-Kossel model. Surf Sci 571(1-3):41–62.

24. Carta R, Tola G (1996) Solubilities of L-cystine, L-tyrosine, L-leucine, and glycinein aqueous solutions at various pHs and NaCl concentrations. J Chem Eng Data 41(3):414–417.

25. Kallistratos G, Malorny G (1972) Experimentelle Untersuchungen zur Frage derchemischen Auflösung von Cystinsteinen. Arzneimittelforschung 22(9):1434–1444.

26. Königsberger E, Wang ZH, Königsberger LC (2000) Solubility of L-cystine in NaCl andartificial urine solutions. Monatsh Chem 131(1):39–45.

17198 | www.pnas.org/cgi/doi/10.1073/pnas.1311637110 Shtukenberg et al.