## Abstract

The complexity of biomineralized structures suggests the potential of organic constituents for controlling energetic factors during crystal synthesis. Atomic force microscopy was used to investigate the thermodynamic controls on carbonate growth and to measure the dependence of step speed on step length and the dependence of critical step length on supersaturation in precisely controlled solutions. These data were used to test the classic Gibbs-Thomson relationship and provided the step edge free energies and free energy barriers to one-dimension nucleation for calcite. Addition of aspartic acid, a common component in biomineralizing systems, dramatically affected growth morphology and altered the magnitude of the surface energy.

Many organisms mediate inorganic crystallization by selective application of organic compounds (1–3) to exert detailed control over the structure (3), orientation (1), growth kinetics (2), and nucleation sites (4) of inorganic crystals. An understanding of physical controls on biomineralization promises new routes to the controlled synthesis of complex crystalline structures for application across a broad spectrum of materials-based technologies. Because it occurs ubiquitously among biomineralizing systems (5) and it is easily crystallized, calcite—often in combination with aspartic acid–rich protein mixtures (1, 2, 5)**—**has emerged as a model crystal for developing this understanding (3, 6–8). Here we use in situ force microscopy (AFM) to quantify the fundamental energetic and kinetic parameters of calcite growth and the effect of organic molecules on those parameters in pure and aspartic acid–bearing solutions.

We examined calcite crystallization from solution onto a seed crystal according to established methods (9, 10). We carefully controlled solution chemistry with ionic strength fixed at 0.11 M, pH adjusted to 8.50, and the ratio of calcium to carbonate activity held at 1.04. The pCO_{2} was fixed by using a closed-system configuration. Calcium and carbonate activities were determined from the Davies equation (11). In situ AFM imaging at 25°C was performed in Contact Mode (Digital Instruments, Santa Barbara) by continuously flowing supersaturated solutions through the fluid cell at rates so that step velocity was independent of flow rate, ensuring that growth was not limited by mass transport to the surface. The birth of new steps at dislocations was monitored and the lengths of these steps were measured immediately before and after they began to advance. The critical length, *L _{c}
*, was taken as the average of these lengths (10). Step velocity was estimated by reference to a fixed point (the dislocation source) or by changes in the apparent step orientation (10).

The supersaturation, σ, is defined by (11)(1a)
(1b)or for CaCO_{3} in an aqueous solution(1c)where Δμ is change in chemical potential per molecule,*k _{b}
* is the Boltzmann constant,

*T*is absolute temperature,

*a*and

*a*are actual and equilibrium activity products,

_{e}*K*

_{sp}is the equilibrium solubility at zero ionic strength, and

*a*(

*i*) is the activity of the

*i*th species.

*K*

_{sp}(10

^{−8.54}) was calculated from the activity at which measured step speeds went to zero (12).

We present results for σ = 0.04 to 1.4. In this range, layer growth of calcite occurred on {101̅4} by advancement of 3.1 Å monomolecular steps generated at dislocations and separated by atomically flat terraces (Fig. 1, A and B). Step advancement led to the formation of polygonal growth hillocks with steps parallel to 〈4̅41〉 directions. Step geometry is shown in Fig. 1, C and D. The*c*-glide symmetry plane generated two distinct pairs of crystallographically identical steps denoted as the positive ([4̅41]_{+} and [481̅]_{+}) and negative ([4̅41]_{−} and [481̅]_{−}) directions (13).

The structure of the growth hillock depends on the thermodynamics of step advancement through terrace widths *T*
_{±}and step speeds *v*
_{±}, which are related by*T*
_{+}/*T*
_{−} =*v*
_{+}/*v*
_{−} . The growth rate of the crystal normal to the surface, *R*, is determined by these parameters through *R* = *h v*
_{±}/*T*
_{±}, where *h* is step height. The *v*
_{±} is determined by the attachment and detachment kinetics at kink sites and differs for the two step types because of the nonequivalence in kink site structure (9, 13). In contrast, *T*
_{±} should be determined by the equilibrium thermodynamics of the step edges through the Gibbs-Thomson effect (14). Consider the birth of a new spiral segment (Fig. 1, A and B). If it remains in equilibrium with the adjacent reservoir of growth units, then it will advance only when the change in free energy, Δ*g*, associated with the addition of a new row of growth units is negative. Taking into account the anisotropy in calcite step structure, one can show that Δ*g*for a straight step is(2a)
(2b)
(2c)where *L* is the length of the step,*b* is the 6.4 Å intermolecular distance along the step, *c* is the 3.2 Å distance between rows, γ_{+} and γ_{−} are the step edge free energies along the + and − steps, and γ_{++}, γ_{−−}, and γ_{+−} are contributions to the step edge free energy from the corner sites (Fig. 1, C and E). When step curvature is included, the second term on the right-hand side ofEq. 2a becomes an integral of 〈γ〉_{±} over the step segment as a function of step orientation. Setting Δ*g* to zero and substituting Eq. 1a into Eq. 2a shows that steps advance only if *L* exceeds a critical value, *L _{c}
*, given by(3)Alternatively, setting Δ

*g*= 0 for a step of arbitrary length gives the length-dependent equilibrium activity,

*a*(

_{e}*L*)(4a) (4b)where subscript ∞ refers to the infinitely long step and

*a*and

*a*

_{e,∞}are related by

*a*=

*i*

_{e,∞}exp(σ). Equations 4a and 4b are statements of the Gibbs-Thomson effect (14), which predicts that

*L*should scale inversely with σ as inEq. 3.

_{c}In agreement with Eq. 3, we observed that the measured*L _{c}
* varied inversely with σ for the two step directions on the calcite {101̅4} faces (Fig. 2). The slope of the relationship between

*L*and 1/σ for both types of steps gives values of 〈γ〉

_{c}_{+}= 2.75 and 〈γ〉

_{−}= 2.15 eV/nm. From Eqs. 2b and 2c, it follows that the difference between 〈γ〉

_{+}and 〈γ〉

_{−}associated with the ++ and −− corners is about 0.77 eV per molecule.

These measurements provide a direct test of the Gibbs-Thomson effect on crystal growth in any system and show that steps on calcite behave in accordance with equilibrium thermodynamics. This conclusion has implications for understanding the role of kinetics in controlling step advancement. Because the local atomic configurations at kink sites control the attachment and detachment kinetics, sites far from the corners should exhibit dynamics that are independent of step length. When σ > 1, then, even on a subcritical step, attachment to these sites will always be more rapid than detachment. Thus for the step to behave in accordance with Eq. 3, equilibrium with the surrounding thermodynamic reservoir must be maintained by the sites of high curvature near corners where detachment is preferred, and sites along the step must communicate with the near-corner sites on a time scale that is short compared with the time required to add on another row. These results imply that transport along the steps, whether by edge, surface, or bulk diffusion, is rapid enough to keep steps in equilibrium with the adjacent fluid.

The dependence of *L _{c}
*

_{±}versus 1/σ exhibits a nonzero intercept that marks the supersaturation σ

_{r}

_{±}, where

*L*

_{c}_{±}goes to zero (Fig. 2). For σ > σ

_{r±}, one-dimensional nucleation occurs along the step edges, and the steps become kinetically roughened. This supersaturation gives the approximate free energy barrier,

*g*

_{1D±}, to the formation of a stable dimer on step edges (15) through

*g*

_{1D±}=

*kT*σ

_{r}

_{±}. From the measured intercepts we obtain values for

*g*

_{1D±}of 0.14 and 0.10 eV, respectively.

When the steps are long (>500 nm), the dependence of *v* on saturation state in the absence of impurity effects is given by (9)(5)where ω is the volume per growth unit in the solid and β is the kinetic coefficient (16). Combining Eqs. 1, 4, and 5 leads to a length-dependent step speed(6)where *v*
_{±,∞} is given by Eq. 5 for *a _{e}
*(

*L*→ ∞). [When σ << 1, Eq. 6 reduces to a commonly used approximation:

*v*=

*v*

_{∞}(1 −

*L*/

_{c}*L*) (17).] The measured speed is independent of σ or step direction and rises much more rapidly than that predicted by Eq. 6 (Fig. 3).

Equation 6 was originally derived for spirals with isotropic step edge energies and kinetics and assumes a high kink site density so that the rate of attachment is not limited by the availability of kinks. Voronkov (18) proposed that, for highly polygonalized spirals, the distance between kink sites along a step is large so that the attachment rate is limited by their availability. He pointed out that the equilibrium shape of a step should be curved toward the corners and that the straight, central portion—which determines the speed—comprises only a small fraction of the step. (This prediction of step shape is verified in Fig. 1, A and B.) When the neighboring step advances, there is a rapid increase in the relative size of the straight portion and hence in the number of available kink sites. As a result, when *L* exceeds *L _{c}
*, the step speed rises rapidly to its limiting value as its length increases from

*L*to

_{c}*L*′, whose order of magnitude is given by

*L*+ 2ω/

_{c}*hc*σ. Substituting in the values of ω,

*h*, and

*c*for calcite, we obtain

*L*′/

*L*∼ 1.1, a prediction that is consistent with the results (Fig. 3).

_{c}This rapid rise of *v*(*L*) affects the magnitude of *T*
_{±} and hence *R*. Consider an isotropic square spiral. If *v* rose discontinuously from zero at *L* <*L _{c}
* to

*v*

_{∞}at

*L*>

*L*, then

_{c}*T*

_{±}would equal 4

*L*(17). However, because

_{c}*v*rises gradually from zero to

*v*

_{∞},

*T*

_{±}is larger by a factor that we refer to as the Gibbs factor,

*G*. The Gibbs-Thomson prediction (Eq. 6) leads to

*G*= 2.4 for small σ—giving

*T*

_{±}= 9.6

*L*

_{c±}(17). As σ increases,

*G*decreases to 1.0 in the limit of large σ.

To compare these predictions with our results for calcite, we must account for the anisotropy of the rhombohedral spiral on {101̅4} faces. The terrace widths for the two step directions become(7a)
(7b)where *B*is *v*
_{+}/*v*
_{−}, 〈*L _{c}
*〉 is the average value of

*L*for the two step directions, and θ is the angle between adjacent turns of the spiral. Figure 4 shows the measured dependence of

_{c}*T*

_{±}on σ as well as

*T*

_{±}= 2.4〈

*L*〉 obtained by using Eq. 6 at small σ. As predicted,

_{c}*T*

_{±}scales inversely with σ, but because of the anomalously rapid rise in

*v*(

*L*), the measured

*G*factor is close to one. Consequently, the growth rate,

*R*, of the calcite surface is about 2.5 times that predicted from classic theory (14, 17).

These results on pure calcite provide a firm basis for quantifying how biomolecules mediate mineralization. Addition of aspartic acid to growth solutions has a dramatic effect on spiral morphology (Fig. 5). While negative steps are still present, the glide plane no longer defines a line of symmetry. The hillock becomes asymmetric, exhibiting new step directions, terrace widths, and highly rounded step corners. Some of these morphological changes such as step rounding are due in part to kinetic factors, but each of these features demonstrates the importance of surface energetic controls. Most importantly, the presence of new equilibrium step directions shows that the low-energy facets as determined from the Wulff construction have changed. In addition, the much smaller terrace widths associated with these new steps demonstrate lower step edge free energy. We estimate the new step edge energy to be only ∼1 eV per molecule, smaller than for steps on pure calcite by a factor of about three.

The most plausible explanation for this behavior is that aspartic acid forms an ordered adsorption layer on the facets defined by the new step directions, thereby lowering the surface free energy through a change in the equilibrium activity of solutions contacting those facets. This behavior is documented for other systems such as NaCl, where a variety of compounds form ordered adsorption layers (19). These layers act as surfactants, lowering the surface free energy and stabilizing facets unexpressed in the pure system. The most notable of these compounds is glycine, which, like aspartic acid, possesses a terminal carboxyl group that binds to both Na and Ca and causes expression of {120} faces on NaCl (19). Thus it appears that the primary effect of aspartic acid on calcite crystallization is to alter the equilibrium thermodynamics of the growth surface.

↵* To whom correspondence should be addressed. E-mail: dove{at}eas.gatech.edu