Results and Discussion

The number of calcite crystallites that nucleated and grew on the PS substrates was proportional to the driving force, σ, of the reactant solution (Fig. 2 A–D, Insets). Rates were also PS-specific, with the highest and lowest number densities measured for heparin and chitosan, respectively (Fig. 2 A and D).

To quantify the thermodynamic basis for these differences, we evaluated the linear portion of the nuclei versus time data using classical nucleation theory (29, 30) (SI Text, section S1). The linear form of the general nucleation expression states that the flux of new nuclei onto a substrate is described bywhere J_o is the steady-state rate of heterogeneous nucleation (m⁻²⋅s⁻¹); A is dependent upon kinetic factors that include rates of ion desolvation, attachment, and detachment; and slope, B, is given bywhere F is a nucleus shape factor, ω is the molecular volume of calcite, k_B is the Boltzmann constant, T is temperature, and γ_net is the net surface free energy of the crystal–substrate–liquid system (mJ⋅m⁻²). Assuming that ω is relatively constant, the experimental variable within B is γ_net for a constant T. Because B is a proxy for the thermodynamic barrier to the nucleation (SI Text, section S1), its dependence upon γ_net represents the contribution of surface energies to the energy barrier. Eq. 2 gives a good fit to the rate data and shows that at the highest supersaturations substrates rich with carboxyl and sulfate functional groups (Fig. 2 A–C) promote the fastest nucleation rates, whereas the slowest rates occur on chitosan surfaces (Fig. 2D).

By evaluating the slopes of these trends, we find that values of B are PS-specific. The highest slopes, and thus the highest energy barriers to nucleation, are correlated with the charge-dense substrates [low-guluronic-acid (LG) alginate and heparin]. In contrast, nucleation onto chitosan substrates shows the lowest slope. Because B is directly proportional to surface energy cubed (Eq. 3), we can deduce that a low value of B is correlated with a low net surface energy. Our estimate of a low γ_net for chitosan is consistent with the surface energy values reported in a previous nucleation study (31). Comparisons of the two alginates show the form with high guluronic acid (HG) content has a smaller thermodynamic barrier to nucleation than LG alginate.

To decipher the physical basis for nucleation barriers estimated for each PS, we first examine the dependence of B on net charge. The expected charge per monosaccharide is estimated using reported acid dissociation constants of the functional groups for each PS and chemical characterizations by the vendor (SI Text, section S2 and Table S1). For our experimental conditions (final pH of the mixed solutions, 10.8 ± 0.1), carboxyl and sulfate functional groups are negatively charged, whereas hydroxyl, acetylamine, and amine groups have a neutral or slightly positive net charge. PSs with mixed functional group chemistry are assigned an average charge.

By correlating the average monosaccharide charge of each PS with B, we find a direct relationship (Fig. 3A). Heparin and the alginates, which derive a higher negative charge from carboxyl groups, are correlated with greater B values and, therefore, greater thermodynamic barriers to nucleation (Fig. 3A). In contrast, the small net positive charge of chitosan is associated with the lowest energy barrier to nucleation. The relationship in Fig. 3A suggests net charge of a PS influences nucleation through the surface free energy of the substrate–liquid–crystal system.

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Fig. 3.

Kinetic measurements provide insights to controls on thermodynamic barriers to nucleation. (A) Barriers are inversely correlated with charge density such that near-neutral PSs present the lowest barrier to calcite formation. (B) Measurements of surface free energy in air, a proxy for γ_SL, provide evidence for a competition between solvent and crystal interactions with the PSs that correlate with hydrophilicity. (C) Relationship between energy barrier and charge density can also explain the very high energy barrier measured on carboxylated SAMs. Offsets between SAMs and alginates of different chain lengths and stereochemistry suggest functional group structure and conformation also tune barrier height. SEs of B are derived from the linear fits of Eq. 2 to data in Fig. 2.

We can relate the observed trend to the underlying crystal– and solution–substrate interactions by evaluating the factors that influence the value of γ_net contained in B. The net surface free energy, γ_net, is a composite of three contributions:where γ_ij is the interfacial free energy of the liquid–crystal (LC), substrate–crystal (SC), and substrate–liquid (SL) interface, and h is a nucleus shape factor (for example, h = 1/2 for a hemisphere). For the conditions of these experiments, we assume that γ_LC is approximately constant. Thus, the higher values of B that we estimate for nucleation onto the most negatively charged PS are associated with large γ_SC, small γ_SL, or some combination of the two.

In principle, one can test the hypothesis that large negative charge is the basis for the large barrier to nucleation via smaller substrate–liquid energies by conducting independent measurements of γ_SL for each of the PS substrates. Unfortunately, this quantity is difficult to measure. Instead, we measured γ_S-air using established methods (32⇓–34) (Fig. S1, Table S2 and SI Text, section S3) to serve as a rough proxy for γ_SL. This value also provides insight on the hydrophilicity of the substrates. That is, surfaces with a large amount of charge prefer interaction with polar water molecules and therefore exhibit greater surface free energies while exposed to air and smaller surface free energies in water. Recall that smaller γ_SL, which corresponds to larger γ_S-air, contributes to a greater γ_net (Eq. 4) and thus a higher thermodynamic barrier to nucleation. Indeed we find PS substrates with high measured values of γ_S-air correlate with greater thermodynamic barriers to nucleation (Fig. 3B).

The trends in Fig. 3 A and B thus suggest the possibility of a broader relationship that could explain calcite nucleation behavior onto substrates ranging from neutral to highly charged. Measurements of energetic barriers to calcite nucleation onto carboxylated self-assembled monolayers (SAMs) provide insights into the influence of highly charged functional groups that are found in proteins and PSs in mineralizing systems (28). Due to the greater hydrophilicity of these substrates, this method of measuring γ_S-air cannot be used because the contact angles lie outside of the accessible range. Instead, we estimate the surface charge density from pK_a values of functional groups and packing density of the SAMs or PSs (SI Text, section S4). Fig. 3C shows that, indeed, charge density is correlated with the barrier height to calcite nucleation along a single trend that includes the SAM substrates.

The implication of Fig. 3C is that charge density through functional group chemistry has a primary control on the barrier to nucleation through differences in γ_SL. Small offsets in B between the two alginates and the carboxylated SAMs (Fig. 3C) suggest that differences in the structure of substrates with similar charge densities further regulate the barrier height. This conclusion is supported by recent simulations showing that variations in head group geometry that arise from SAM–monomer interactions cause subtle differences in substrate–crystal interfacial energies (35). Thus, nucleation barriers are largely controlled by substrate charge density through γ_SL and further regulated by small differences in γ_SC through subtle variations in structure.

Our results demonstrate the energy barrier to calcite nucleation is determined by a competition between the energetics of crystal nucleus attachment to a substrate and creating new interface (γ_SC) versus the energetics of eliminating a solvated substrate (γ_SL). By minimizing the change in free energy through these thermodynamic drivers, the systematic relationship between nucleation and the chemistry and structure of substrate environments can be understood. Hydrophilic substrates prefer to maintain the substrate–water interface, thus presenting a high barrier to making a new substrate–crystal interface. In contrast, neutral to less-hydrophilic surfaces present a lower thermodynamic barrier to displacing water to form a new substrate–crystal interface and thus to mineral nucleation. This interpretation of the experimental data expands upon the conclusions of earlier computational studies that explored the importance of surface ionization, as well as the competition between the energies of the substrate–crystal and liquid–crystal interfaces (36).

At first glance, the higher energy barriers we estimate for nucleation onto charged surfaces are counterintuitive given the long-standing perception that negatively charged functional groups promote nucleation by strong interactions with Ca²⁺ ions (14, 15, 37, 38). Moreover, a number of studies observe that calcite crystallites are preferentially formed on carboxyl-patterned regions, in sharp contrast to the surrounding hydrophobic methyl-functionalized areas (39⇓–41).

To understand why these seemingly disparate observations are consistent, recall that energy barriers are constant for a given substrate–liquid–crystal system and nucleation rates are dependent upon σ⁻². Thus, for any two substrates, there must be a value of σ above which, paradoxically, the one with the higher value of B gives faster nucleation rates (Fig. 2E). Because the studies described above were conducted at very high levels of supersaturation, the charge-dense macromolecules produced the most crystallites over short time intervals. Fig. 2E shows that as supersaturation is lowered these trends eventually flip and neutral substrates with lower energy barriers promote the fastest rates of nucleation (Eq. 2). This physical picture reiterates the point that whereas energy barriers to nucleation at a given supersaturation are constant and substrate-specific, the local supersaturation environment has the overarching control on relative rates of nucleation onto charged versus neutral substrates.

The findings suggest a simple mechanism-based framework for understanding nucleation as a continuum of substrate–liquid–crystal interactions that are governed by acidic and neutral chemical domains, irrespective of macromolecular class. The possibility of regulating the onset of nucleation by this construct suggests the need to revisit the long-standing assumptions regarding the roles of neutral and acidic macromolecules at sites of calcification. If near-neutral macromolecules present a lower energy barrier to nucleation by promoting the formation of new substrate–crystal interfaces, an important part of the mineralization picture may be currently missing.