The elementary reactions for incorporation into crystals

Significance

Crystals are essential structural elements in living organisms and rocks and crucial constituents of the technologies that enable modern civilization. We unravel the mechanism of the chemical reaction between incoming molecules and the unique sites on a crystal surface that receive them, the kinks, which has remained elusive and subject to speculation for over 60 y. The presented paradigm of two-step incorporation may shed light on how minor solution components define the elaborate morphologies of natural crystal formations and guide the search for solvents and additives that stabilize the intermediate state to slow down the growth of, for instance, undesired polymorphs.

Abstract

The growth rates of crystals are largely dictated by the chemical reaction between solute and kinks, in which a solute molecule severs its bonds with the solvent and establishes new bonds with the kink. Details on this sequence of bond breaking and rebuilding remain poorly understood. To elucidate the reaction at the kinks we employ four solvents with distinct functionalities as reporters on the microscopic structures and their dynamics along the pathway into a kink. We combine time-resolved in situ atomic force microscopy and x-ray and optical methods with molecular dynamics simulations. We demonstrate that in all four solvents the solute, etioporphyrin I, molecules reach the steps directly from the solution; this finding identifies the measured rate constant for step growth as the rate constant of the reaction between a solute molecule and a kink. We show that the binding of a solute molecule to a kink divides into two elementary reactions. First, the incoming solute molecule sheds a fraction of its solvent shell and attaches to molecules from the kink by bonds distinct from those in its fully incorporated state. In the second step, the solute breaks these initial bonds and relocates to the kink. The strength of the preliminary bonds with the kink determines the free energy barrier for incorporation into a kink. The presence of an intermediate state, whose stability is controlled by solvents and additives, may illuminate how minor solution components guide the construction of elaborate crystal architectures in nature and the search for solution compositions that suppress undesirable or accelerate favored crystallization in industry.

Crystals endow natural and synthetic materials with essential structures and functions (1–8). The structural and functional utilities of crystals are, to a large extent, defined by their sizes and shapes (9, 10). In turn, the crystal sizes and shapes are determined by the crystal growth rate and its anisotropy related to the distinct structures of the crystal faces (8, 11–13). How fast crystal faces grow is largely dictated by the slow chemical reaction between solute molecules and specifically structured crystal surface sites, the kinks (11, 14–17), but the respective mechanism, i.e., the characteristic timescales and lengthscales, possible intermediates and their stabilities, remains elusive (18). The incorporation reaction rate is substantially slower than the rate of molecular supply to the kinks [referred to as the diffusion limit (19, 20)] and announces the presence of an activation barrier for incorporation. Since incorporation into kinks from a gas is nearly barrier-free, this barrier was ascribed (11, 17) to solvate shells that coat the kinks and envelope the solutes (7, 21–28). This model implies that the reaction for incorporation into kinks comprises breaking of several solute–solvent bonds and instituting new bonds with the molecules forming the kink. Fundamental mechanistic details on the pathway, along which bonds sever and rebuild, the associated energy relief, and potential long-lived intermediates, remain missing.

Indeed, this mechanism could not be probed before as molecular incorporation into kinks is often coupled to the complex kinetics of transport from the solution to the growth sites that may include adsorption on the terraces between steps, diffusion along the crystal surface, and desorption back to the solution (11, 17, 29–31). The molecular exchanges between solution, crystal surface, and kinks motivate multiparameter kinetic laws, in which the kinetics of incorporation into kinks convolve with the kinetics of the adjacent processes (32). In the absence of experimental data on the transport to the steps and reaction at the kinks, computer simulations of the growth of ionic crystals from aqueous solutions have suggested that the incorporation of ions into kinks may be a multistep process associated with step-wise dehydration of the ions (33, 34); interestingly, analogous simulations of the same system have yielded divergent intermediate steps (33, 35, 36). These computational studies shared an assumption of direct access of the incoming ions from the solution to the kinks, which was unsupported by experiment and may be unphysical (31). Still, if taken at face value, their results would concur with the classical expectation that the strength of the solute–solvent bonds governs the free energy barrier for incorporation into kinks (28, 37).

To elucidate a potential intermediate state that may be shifted by only fractions of a nanometer from the final molecular incorporation site, exist for times of order of nanoseconds, and thus be inaccessible to the current direct experimental methods, we employed as reporters four solvents with distinct structures and functions. To expose the reaction at the kinks we chose fully organic solute and solvent pairs, in which the solute molecules are expected to reach the steps directly from the solution (31, 38). This strategy is distinct from many solution crystallization systems studied to date, which employed fully or partially aqueous solvents (27, 39, 40) that favor the multistep surface diffusion pathway (31). We explored the growth of etioporphyrin I. Similarly to crystals of other porphyrins, etioporphyrin I crystals have low symmetry, which is conducive of appealing electronic properties (41, 42).

Results

How Etioporphyrin I Crystals Grow.

Etioporphyrin I comprises four pyrrole residues linked by methine groups into a flat porphyrin ring, decorated by alternating four ethyl and four methyl groups (Fig. 1A). In the known unsolvated crystal form with symmetry P1, the etioporphyrin I molecules arrange in columnar stacks supported by π–π bonds and oriented roughly along the [110] direction (Fig. 1B) (42). To illuminate how solute molecules incorporate into crystals, we compare the kinetics of crystallization of etioporphyrin I from three alcohols with increasing alkyl chain length, butyl, hexanyl, and octanyl, and dimethyl sulfoxide (DMSO), a smaller molecule with two short aliphatic residues. This comparison is substantiated by the identical P1 structures of the crystals that grow from the four solvents (SI Appendix, Figs. S1 and S2) (43). The UV–vis absorption spectra of etioporphyrin I in these solvents feature a prominent Soret band, at ca. 400 nm, and four Q bands in the wavelength range 475 to 630 nm (Fig. 1C) attributed to distinct transitions of the conjugated π electron systems of the constituent pyrroles (44–46). The Soret and the Q bands are similarly structured in the three alcohols and DMSO (Fig. 1C) and signify that the dominant solute species in the four solvents is a monomer (47–49).

Fig. 1.

Etioporphyrin I crystals and solutions. (A) The etioporphyrin I molecule. (B) Stacks of etioporphyrin I molecules in the crystal in P1 space group; Cambridge Structural Database REFCODE WOBVUF ref. 42; N is drawn in blue, C, in charcoal, and H, in silver. (C) UV–vis absorption spectra of etioporphyrin I dissolved in DMSO, butanol, hexanol, and octanol. (D) Scanning electron micrograph of an etioporphyrin I crystal; the prominent crystal faces are labeled. (E) In-situ atomic force microscopy (AFM) images of a (010) face of etioporphyrin I growing from solutions in DMSO, butanol, hexanol, and octanol.

(010), (011), and (101) faces are prominent in the habit of etioporphyrin I crystals grown from the three alcohols and DMSO (Fig. 1D). Typically, a (010) face orients parallel to the substrate (SI Appendix, Figs. S1 and S3) and affords a line of sight to atomic force microscopy (AFM). In situ AFM images (Fig. 1E) in supersaturated solutions, where the solute concentration $C$ exceeds the solubility $C_e$ (the concentration at which a crystal and the solution are at equilibrium), reveal that the (010) faces comprise unfinished layers wound into spirals that originate at the outcrops of screw dislocations, similarly to numerous other crystals growing from solution (50, 51).

The Mechanism of Molecular Ingress into Kinks.

The crystals grow as solute molecules associate to kinks, located along the edges of the unfinished layers, the steps (15, 52, 53). The crystal anisotropy enforces distinct velocities of the steps that propagate in different directions such that steps in the [100] or $\overrightarrow{a}$ direction grow fastest and reach furthest from the dislocation (Fig. 1E). In this way, the shape of the growth spiral reveals the ratios between the anisotropic step velocities; these ratios are similar in the four solvents (Fig. 1E and SI Appendix, Fig. S4) and independent of the supersaturation.

The rate of step growth on the crystal surface is a direct representation of the rate of the chemical reaction at the kinks. We used the step growth rate—or step velocity $v$—and its responses to thermodynamic and kinetic parameters to explore the mechanism of molecular incorporation into kinks. We measured $v$ by time-resolved in situ AFM (30, 52, 54–56). We tracked the locations of steps growing in the [100], or $\overrightarrow{a}$, direction (Fig. 2A). Step growth was steady in time (Fig. 2B and SI Appendix, Fig. S5), and the slopes of the time correlations of the step displacements present the step velocities $v$. In all solvents, v increased linearly with the solute concentration C (Fig. 2C), in common with numerous other solution-grown crystals (39). The linear $v\left(C\right)$ correlations indicate that the crystals grow by incorporation of the dominant solution species (30), etioporphyrin I monomers, in the four solvents. Mass preservation prescribes that $v$ equals the product of the rate of the chemical reaction of the solute with the kinks and the contribution of one molecule to step propagation, the solute molecular size $a$, $v=a{k}_a(C-C_e)$. We include in the rate constant $k_a$ the dimensionless kink density ${\overline{n_k}}^{-1}$, which is constant and close to the thermodynamic limit of ca. 0.3 (25, 52, 57, 58) on the rough steps of etioporphyrin I in all solvents (SI Appendix, Fig. S6). Subtracting the solubility $C_e$ accounts for the reversibility of molecular attachment. Alternatively, $v$ has been modeled as the product of the solute flux into a kink and the volume $\mathrm{\Omega }$ that a solute molecule contributes to the crystal, resulting in $v=\mathrm{\Omega }\beta (C-C_e)$ (11, 17), where $\beta$ is the step kinetic coefficient. Comparing the two relations informs that $a{k}_a=\mathrm{\Omega }\beta$. Dividing the slopes of the $v(C)$ correlations (Fig. 2C) by $a$ reveals the markedly different rate constants $k_a$ for incorporation into kinks from the four solvents (Fig. 2D).

Fig. 2.

The chemical reaction at the kinks. (A) Time resolved in situ AFM images of etioporphyrin I (010) face growing from butanol. Arrows trace the growth of a step in the [100] direction. (B) The evolutions of the step displacements during growth from butanol at printed values of $C-C_e$. Thirty independent measurements were averaged for each data point, and the error bars represent the respective SDs. (C) Step velocity $v$, determined from the slopes of the time correlations of the step displacement in (B) and SI Appendix, Fig. S5, as a function of the concentration $C$ in the four listed solvents. Error bars represent the SDs of the slopes in (B). Numbers represent the solubilities $C_e$ in the respective solvents in mM; see SI Appendix, Table S1 for the SDs. (D) Bimolecular rate constants $k_a$ for the reaction between incoming solute molecules and kinks evaluated from the $v(C)$ correlations in (C); error bars represent the SDs of the slopes in (C). (E) The viscosities $\eta$ of the four solvents at 28 °C, the temperature of the AFM measurements; error bars represent the SDs from SI Appendix, Table S2. (F) The product $k_a\eta$ for growth form the four solvents. Errors bars represent the propagation of the SDs for $k_a$ and $\eta$.

The rates of chemical reactions occurring in solution, in parallel with the reactants’ diffusivities (59), scale as the reciprocal viscosity of the solvent ${\eta }^{-1}$ refs. 60 and 61. To expose the molecular behaviors that govern solute incorporation into kinks, we account for the divergent viscosities of the solvents (Fig. 2E) and use the product $k_a\eta$ to quantify the reaction at a kink. The values of $k_a\eta$ are nearly identical for crystallization from solvents in the homologous series butanol–hexanol–octanol and lower by nearly half for crystallization from DMSO (Fig. 2F).

To further illuminate the mechanism of solute incorporation into kinks, we start by examining whether etioporphyrin molecules reach the steps directly from the solution (Fig. 3A) or, as numerous solution-grown crystals (30, 62–64), after adsorption on the terraces between steps followed by diffusion toward the steps (Fig. 3B). Two sets of observations affirm that during growth from the four solvents etioporphyrin I, in contrast to the studied water-based systems, prefers the direct incorporation pathway. First, in butanol, hexanol, and octanol, the velocities of steps as high as the $\overrightarrow{b}$ lattice parameter [$\left|\overrightarrow{b}\right|$ = 0.991 nm (42)], are close to those of steps of height $h=2|\overrightarrow{b}|$ (Fig. 3 C, D, and F and SI Appendix, Fig. S7). Second, $v$ is independent of the step separation $l$ for $l$s as short as a few nanometers (Fig. 3E). If solute reaches the steps via the crystal surface, the step supply field is constrained to two dimensions, which stunts the growth of closely spaced steps or steps of double height. Concurrently, analytical models of step growth mediated by surface diffusion predict that $v$ scales with $h^{-1}$ and sharply slows down at short l (31, 32). Experimental observations reveal that double-height steps that feed via the surface grow slower than single-height steps and may split into two single-height steps owing to unequal supply to the top and bottom layers (63, 65). By contrast, if the steps feed directly from the solution, the supply field is three-dimensional and abundant for closely spaced or twinned steps. Closed-form expressions for this growth mode predict negligible $v(h)$ and weak $v(l)$ correlations (11, 17, 31).

Fig. 3.

The pathway of a solute molecule into a kink. (A and B) Schematic representations of two pathways from solution to steps. (A) Solute molecules reach the steps directly from the solution. (B) Solute molecules adsorb on the crystal surface and diffuse toward the steps. (C) Time-resolved AFM images of the growth of single (silver arrows) and double (green arrows) height steps at C = 0.23 mM in hexanol. (D) The evolution of the surface profile along the dotted line in (C). The profiles at 21, 43, and 68 s are shifted vertically for clarity. Double-height steps (green arrows) advance over lengths similar to those of single-height steps (silver arrows). (E) The step velocity in the four solvents does not correlate with the step separation $l$. C = 0.23 mM in hexanol, 0.09 mM in butanol, 0.33 mM in octanol, and 0.25 mM in DMSO. The averages of 15 measurements for each $l$ interval, represented by horizontal bars, are shown. Vertical bars represent the SDs of the groups of measurements. (F) Comparison of the velocities $v$ of single, $h=|\overrightarrow{b|}$, and double, $h=2|\overrightarrow{b|}$, height steps in three solvents. The averages of 10 double and 10 single height steps are shown. Error bars represent the respective SDs. The values of C−C_e during these measurements are listed in the plots. No double-height steps were observed during growth from DMSO. (G) Schematic of the semi-spherical supply field for direct incorporation of solute into a kink. (H) Schematic of the free energy landscape along the classical one-step reaction pathway of incorporation of an etioporphyrin I molecule from the solution into a kink. The values for the standard free energy of crystallization $\mathrm{\Delta }{G}^o$ are from ref. 43. (I) Schematic of the free energy landscape along the two-step pathway of incorporation suggested by the free energy barriers for incorporation into kinks $\mathrm{\Delta }{G}^{‡}$ in the four solvents, evaluated from the ratios of the measured rate constants for incorporation into kinks $k_a$ (Fig. 2D) to their diffusion limits (SI Appendix, Table S4).

Factors That Determine the Rate Constant for Incorporation into Kinks.

We represent the bimolecular rate constant $k_a$ for the reaction between a kink and an incoming solute molecule as $k_a=k_0\exp \left(-\mathrm{\Delta }{G}^{‡}/k_{B}T\right)$, where $k_0$ is the rate constant of a reaction uninhibited by an activation barrier and proceeding at the diffusion limit, $\mathrm{\Delta }{G}^{‡}$ is the free energy barrier for incorporation of solute molecules into kinks, $k_B$ is the Boltzmann constant, and T is temperature (66). Molecules approach kinks only from half the space above the crystal (Fig. 3G), and the kinks can be assumed immobile and separated along the steps, on the average, by ${\overline{n}}_k$ molecules (29). These three constraints modify the Smoluchowski-type expression (67) for $k_0$ to $k_0=2\pi D{r}^{\ast }{N}_A/{\overline{n}}_k$ ($D$, etioporphyrin I diffusivity in the respective solvent; $r^{\ast }\cong$ 0.5 nm, reaction radius; and $N_A$, the Avogadro number). In turn, $D=k_{B}T/6\pi \eta a$. The constraints on a reaction at a kink and the elevated viscosities of the organic solvents depress the values of $k_0$ to order 10⁸ M⁻¹s⁻¹ (SI Appendix, Table S4), somewhat lower than the typical values of about 10¹⁰ M⁻¹s⁻¹ for diffusion-limited reactions of small molecules dissolved in water (66). The relations for $k_a$, $k_0$, and $D$ expectedly imply that $k_a$ scales as ${\eta }^{-1}$. Importantly, given that the kink densities in the four solvents are equal to ${\overline{n}}_k$’s thermodynamic limit (52, 58) (SI Appendix, Fig. S6), which is solvent-independent (29), the preexponential factor in the product $k_a\eta$ is solvent-independent and the barrier for solute incorporation into kinks $\mathrm{\Delta }{G}^{‡}$ solely regulates $k_a\eta$.

The found correlation between $\mathrm{\Delta }{G}^{‡}$ and $k_a\eta$ affords an opportunity to apply the $k_a\eta$ data to illuminate the mechanism that governs $\mathrm{\Delta }{G}^{‡}$. Classical crystal growth theories assign $\mathrm{\Delta }{G}^{‡}$ to a transition state, in which the crucial solute–solvent bonds are broken, but the solute has not yet bound to the kink (11, 17); as the solvent–solute bonds are broken, this complex is nearly identical in all solvents. Within this classical scenario, the activation free energy for incorporation would be instituted by the strength of the solute–solvent bonds and is greater in solvents that bind strongly with the solute (Fig. 3H).

To test the correlation between ΔG‡ and the strength of the solute-solvents bonds, we use that etioporphyrin I crystallizes as identical P1 crystals (SI Appendix, Figs. S1 and S2) from all four solvents (43). In consequence, the equilibrium free energies of crystallization $\mathrm{\Delta }{G}^o$, evaluated from the respective solubilities (43), characterize the relative strengths of the solute–solvent interactions in each solvent. A correlation between $\mathrm{\Delta }{G}^o$ and $\mathrm{\Delta }{G}^{‡}$ would exemplify the venerable Berthollet rule, according to which crystals grow faster from solvents, in which their solubility is lower (68). Etioporphyrin I, however, defies both the Berthollet rule and the classical understanding of activation barriers for solution crystallization. It binds weakly to butanol, and its interactions with DMSO, hexanol, and octanol are stronger and nearly equal (43). The classical principles concertedly predict lowest activation barrier and fastest growth from butanol solutions and comparable barriers and growth rates from DMSO, hexanol and octanol (Fig. 3H). The AFM results on the step velocities belie both parts of this prediction after accounting for the divergent viscosities of the solvents (Fig. 2F).

The discrepancy between the strength of the interactions of etioporphyrin I with the solvents, manifest as $\mathrm{\Delta }{G}^o$, and the activation barriers for crystallization ${\mathrm{\Delta }G}^{‡}$ advocates for two sequential elementary reactions of incorporation into kinks, separated by a metastable state that a solute molecule occupies in reaction one, before it invades the kink in reaction two. This intermediate complex locates along the direct access route of a molecule from the solution to a kink (Fig. 3A) and likely represents a state in which the solute molecule is bound, but only partially, to the molecules comprising the kink. The stability of the intermediate state, regulated by the strengths of the preliminary bonds with the kink and interactions with the solvent, dictates the height of the barrier that the solute molecule overcomes as it relocates to the kink (Fig. 3I). The values of ${\mathrm{\Delta }G}^{‡}$ in the four solvents imply that DMSO strongly stabilizes the intermediate state, and the contributions of the three alcohols are weaker and close (Fig. 3I).

A Microscopic View of the Reaction at the Kinks.

For a microscopic view of the pathway from the solution into a kink, we performed all-atom molecular dynamics (MD) simulations. We monitored the ingress of an etioporphyrin I molecule into a kink in contact with each of the four solvents (Fig. 4 A–C and Movie S1). Before an incoming molecule approaches a kink, it freely rotates. Further along the reaction pathway, the rotations are constrained by interactions with the molecules in the kink and not by the model assumptions (Fig. 4 A–C and Movie S1). We examined the normal distance of the molecule’s center of mass from the surface of the kink, $z_{\mathrm{COM}}$, and the number of solvent molecules in the kink cavity, $N_{\mathrm{solv}}$. The respective MD simulation trajectories reveal that incoming molecules stall at an intermediate state at ca. $z_{\mathrm{COM}}\approx 1.1\mathrm{n}\mathrm{m}$ (Fig. 4 B and D), where they associate with more solvent molecules than in the kink (Fig. 4E). In all solvents, the potential of mean force $F\left(N_{\mathrm{solv}},z_{\mathrm{COM}}\right)$ along the kink access route features two minima: a global minimum at $z_{\mathrm{COM}}\approx 0.7\mathrm{n}\mathrm{m}$, which corresponds to a molecule in the kink, and a metastable minimum at $z_{\mathrm{COM}}\approx 1.1\mathrm{n}\mathrm{m}$ (Fig. 4F and SI Appendix, Fig. S9), corresponding to the transient intermediate state observed in the MD trajectories (Fig. 4 B, D, and E).

Fig. 4.

Microscopic view of the ingress of an etioporphyrin I molecule into a kink. (A–C) Representative snapshots of an etioporphyrin I molecule in the solution near a kink (A), at the intermediate state (B), and at its final location in the kink (C). Classical MD simulation results. The green sphere in (B) defines our choice of the kink cavity, i.e., the volume where the dynamics of the solute molecule that partake in the reaction between a solute molecule and a kink are evaluated. Solvent molecules are omitted for clarity. The kink is viewed along the unfinished molecular row at the step edge. The direction of step growth is from Left to Right. Etioporphyrin I molecules in the crystal lattice are shown in gray. The lattice planes of etioporphyrin I molecules in front and behind the plane, which hosts the kink, are omitted. In the incoming etioporphyrin I molecule, C atoms are shown in teal, N in blue, and H in silver. (D and E) A representative MD simulation trajectory of an incoming etioporphyrin I molecule displayed in terms of the normal distance $z_{\mathrm{COM}}$ from the center of mass of an incoming molecule to that of the molecule at the bottom surface of a kink in (D) and the number of solvent molecules $N_{\mathrm{solv}}$ in the kink cavity [shown in green in (B)] in (E). (F) Two-dimensional potential of mean force profiles $F\left({N_{\mathrm{solv}},z}_{\mathrm{COM}}\right)$ for incorporation of a solute molecule into a kink from the four solvents. $F$ was computed from well-tempered metadynamics simulations. The locations of the three states shown in (A–C) are indicated in the DMSO plot.

The intermediate complex represents the stepping stone from which a molecule launches to surmount the activation barrier and relocate to the kink. The saddle point between the two minima hosts the transition state for transfer from the intermediate state into a kink and the energy difference between the intermediate state and the saddle point represents the activation barrier for molecular incorporation into kinks ${\mathrm{\Delta }G}^{‡}$ (Fig. 4F and SI Appendix, Fig. S9). In the four solvents, ${\mathrm{\Delta }G}^{‡}$ is between 11 and 14 kJ mol⁻¹ (Fig. 4F and SI Appendix, Fig. S9 and Table S5), close to the experimental estimates (Fig. 3I and SI Appendix, Table S5), although the statistical uncertainty in the computational values (ca. 2 kJ mol⁻¹, SI Appendix, Fig. S8 and Table S5) prevents precise quantitative comparisons.

How the Solvent Stabilizes the Intermediate State.

To rationalize the relative stabilities of the intermediate states in the four solvents, we turn to the dynamics of the solvent molecules occupying the ca. 0.4-nm gap between an incoming etioporphyrin I molecule and the lower molecule of the kink (Fig. 4B). Entropy maximization requires that this gap fills with solvent. This gap is populated by about two DMSO, one butanol, 2/3 hexanol, and 1/2 octanol molecules (Fig. 4F), which consistently comprise four -CH₃ or -CH₂- residues. The distributions of the centers of mass of the solvent molecules along the $z_{\mathrm{COM}}$ coordinate reveal that similar ranges of $z_{\mathrm{COM}}$ values are occupied, suggesting that the translational freedoms of the four solvents are similar. A substantial population of the smaller DMSO molecules retain significant orientational freedom (Fig. 5A), whereas the larger molecules of butanol, hexanol, and octanol mostly rest parallel to the lower kink surface (Fig. 5 A and B). We propose that the elevated orientational freedom and the greater number of DMSO molecules entropically stabilize the intermediate state in that solvent and also bolster the search for stronger bonds of the solvent with the etioporphyrin I molecules that frame the gap. The enhanced stability of the intermediate state during growth from DMSO increases ${\mathrm{\Delta }G}^{‡}$ and dictates the slower rate of growth from this solvent (Figs. 2F and 3I). The similar dynamics of the three alcohols in the gap (Fig. 5A) and their similar chemistries underlie the close stabilities of the intermediate complexes for incorporation from these solvents and the near identical values of the respective ${\mathrm{\Delta }G}^{‡}$ and $k_a\eta$ products (Fig. 2F).

Fig. 5.

Solvent dynamics in the intermediate state. (A) Distributions P(cos θ, z_COM) of the molecules of the four solvents trapped in the gap between an incoming solute and the base of a kink, Fig. 4B, along the vertical coordinate z_COM and the angle $\theta$ defined in (B). P(cos θ, z_COM) is normalized to unity far from the crystal surface. (B) Definition of the angle θ between the vertical axis z and a vector, shown in red, that characterizes the orientation of each solvent. $\cos \theta =0$ when $\theta =\pi /2$ and a solvent molecule rests parallel to the crystal surface. Etioporphyrin I crystal surface is shown in cyan. In the solvents, C is shown in gray, O, in red, S, in yellow, and H, in white.

Discussion

Crystallization of organic, inorganic, biomolecular, and colloid materials commonly involves the formation of several contacts between a solute and a kink. It is likely that two-step incorporation with an intermediate state will be a part of the respective mechanisms of access to kinks. For solutes that reach the kinks by the surface diffusion pathway, adsorption on the crystal surface leaves most of the solute valences saturated with solvent and allows the formation of a similar intermediate state as the solute interacts with a kink. It is feasible that for compounds with structures more complex than those of their respective solvents more than one intermediate state may exist, as suggested for the association of Ba²⁺ ions to steps on barite crystals (69). For other compounds, a long-living intermediate state may recruit more than one of the incoming solute molecules or ions (34). Rare instances of one-step solute incorporation into kinks are likely limited to high-symmetry solutes that are comparable in size to the solvent.

The breaking of the preliminary bonds between a growing crystal and an incoming molecule in the second stage of the reaction is necessary to accomplish incorporation into kinks but is highly counterintuitive. On the other hand, it is akin to backtracking, observed for amyloid fibril growth, in which an incoming peptide chain docks to the fibril tip by contacts that are distinct from those in the fibril bulk (70–76). The shared features of molecular incorporation of the relatively rigid (excepting the methyl and ethyl side groups) etioporphyrin I molecules and the flexible peptide chains suggest that backtracking may be a common feature of the mechanism of aggregation in solution.

The concept of an intermediate state for molecular incorporation may illuminate poorly understood examples of additive-driven faster crystal growth (7, 26, 77). Expectations, based on myriad prior studies, state that the additives would delay growth by binding to specific crystal surface sites (17). The acceleration of calcite growth by a line of polypeptides was attributed to reduced activation barriers due to “perturbation of the local water structure” by the peptides (7). The low concentration of polypeptides, from 10⁻⁷ to 10⁻⁵ M, needed to invoke growth acceleration, suggests that the peptides do not perturb the solvation of the calcium and carbonate ions in the solution, which would require additive amounts comparable to the solute concentrations of order of 10⁻⁴ M (50). On the other hand, the peptide effect on kinetics is consistent with destabilization of a potential intermediate state at the kinks, which exclusively localize at the crystal–solution interface at much lower concentrations. In another example, acetone, added to crystallization solutions of the peptide hormone insulin at concentrations up to 20 vol.%, increased multiple-fold the crystal solubility (78), suggesting that acetone weakens the crystal contacts. Instead of suppressing crystallization, however, acetone substantially boosted the rate of growth of the insulin crystals (26), another violation of the Berthollet rule (68). A possible explanation of the kinetic effect of acetone is that this additive weakens the contact that support an intermediate state for incorporation into kinks, which destabilizes it and lowers the activation barrier for association to the kinks.

Conclusions

The kinetics of etioporphyrin I crystal growth in four solvents disconnect from the respective crystallization thermodynamics and indicate that the incorporation of molecules into crystal growth sites is mediated by an intermediate state. MD simulations support the presence of an intermediate state, in which an incoming solute molecule forms preliminary contacts with the molecules from the kink. The breaking of the preliminary bonds of the solute with the solvent and the kink that stabilize this intermediate state constitutes the free energy barrier for incorporation into kinks.

The model of an intermediate state for molecular incorporation, in concert with the ubiquitous two-step crystal nucleation (79–83), suggests that two-step transitions may be common for phase transformations defined by multiple order parameters (84). The power of solvents and additives that uphold or undermine the intermediate state may illuminate how complex environments define the elaborate morphologies of natural crystal formations and guide the search for solvents and additives that suppress or accelerate crystallization during the synthesis of pharmaceuticals, fine chemicals, and advanced functional and structural materials.

Materials and Methods

Solution Preparation.

Etioporphyrin I was purchased from Santa Cruz Biotechnology Ltd. The solvents, octanol (anhydrous, ≥99%), butanol (anhydrous, ≥99%), hexanol (anhydrous, ≥99%), and DMSO (anhydrous, ≥99%) were bought from Sigma Aldrich. Deionized (DI) water for cleaning glass vials was produced by a Millipore reverse osmosis–ion exchange system (Rios-8 Proguard 2—MilliQ Q-guard). Solutions of etioporphyrin I in each solvent were prepared by dissolving etioporphyrin I in respective solvents at 70 °C in glass vials. The supersaturated solutions were filtered, and concentrations were measured using UV–vis spectroscopy. Solutions of desired concentration for bulk crystallization and in-situ AFM studies were made by subsequent dilution with respective solvents.

Viscosity.

The viscosity of organic solvents varies non-linearly with temperature. Solvent viscosities at 28 °C were determined by linear interpolation between two bracketing temperatures 25 °C and 30 °C or 35 °C (85–90). The solvent viscosities at 28 °C are reported in SI Appendix, Table S1.

Scanning Electron Microscopy.

For bulk crystal characterization with SEM, supersaturated solutions of etioporphyrin I (C = 0.5mM) in different solvents were passed through a 0.22-µm PTFE filter and kept in clean 20-mL capped borosilicate glass vials in an incubator. In addition, 15-µm diameter clean glass coverslips scratched at the center were kept at the bottom of the glass vials in contact with respective supersaturated solutions. Crystals of etioporphyrin I were made by slow cooling at the rate of 5 °C/15 min, upon reaching room temperature, the vials were kept undisturbed overnight. After 1 to 2 d, etioporphyrin I crystals nucleated on glass coverslips were taken out, rinsed with octanol to remove loose crystals, air dried, mounted on metal stubs with carbon tape and coated with 10 to 20 nm gold. SEM images (as in SI Appendix, Fig. S1) were collected at a system vacuum of 2 × 10⁻⁵ mbar pressure and EHT 3 kV.

UV–Vis Spectroscopy.

Weighted amounts of etioporphyrin I was completely dissolved in known volumes of solvents in borosilicate glass vials. The glass vials were left at 70 °C for several days with occasional stirring until complete dissolution. The stock solutions were filtered and diluted serially to achieve the desired concentrations. UV–vis absorbance spectra were recorded with DU Spectrophotometer 800 (Beckman) in the wavelength range 200 to 800 nm (Fig. 1C). For each porphyrin, distinct Soret and Q bands were clearly identifiable. The correlations between optical density at the wavelength of maximum absorbance in the respective Q bands and the concentration are linear (91). For experimental statistics, all measurements were carried out in triplicate, in independently prepared solutions. The extinction coefficients for each porphyrin–solvent pair were determined from the respective slopes of absorbance vs. concentration at a given wavelength (91).

Powder X-ray Diffraction.

Powder X-ray diffraction (PXRD) patterns (SI Appendix, Fig. S2) were obtained using BB Rigaku Samartlab X-ray diffractometer with primary monochromatic radiation CuKαI of wavelength 1.54056 Å. Powder patterns were collected by laying finely ground crystals on a quartz sample holder. Data were collected in the 2θ range 5 to 40°, with a 2θ step of 0.015˚ and 2s/step speed. A reference powder pattern for etioporphyrin I was computed using Mercury 3.6 (CCDC) software using single crystal data from the Cambridge Structural Database (42).

Crystal Face Indexing.

The crystal faces (SI Appendix, Fig. S3) were indexed on a Bruker D8 Venture diffractometer equipped with a charge-coupled device (CCD) detector using graphite-monochromated Cu Kα radiation (λ = 1.54056 Å) and an APEX-3 face indexing plug-in.

In Situ Monitoring of the Etioporphyrin I Crystal Evolution.

We used the multimode atomic force microscope (Nanoscope VIII and IV) from Bruker for all step velocity measurements. AFM images were collected in tapping mode using Olympus TR800PSA probes (Silicon nitride, Cr/Au coated 5/30, 0.15 N/m spring constant) with a tapping frequency of 32 kHz. Image sizes ranged from 2 μm to 20 μm. Due to the fast growth of etioporphyrin I layers on the (010) face, the scan rates were between 2.50 and 6.10 s⁻¹ depending on the scan size. Height, amplitude, and phase imaging modes were employed. The captured images contained 128 or 256 scan lines based on the scan aspect ratio, at angles depending on the orientation of the monitored crystal. The temperature in the fluid cell reached a steady value of 28 ± 0.1 °C within 15 min of imaging (92). This value was higher than room temperature (ca. 22 °C) owing to heating by the AFM scanner and laser.

Pure etioporphyrin I crystals were grown on glass disks as described above. The density of attached etioporphyrin I crystals was monitored under an optical microscope. We ensured similar crystal density for all samples to minimize potential depletion of solutes due to a high crystal number. The glass slides were mounted on AFM sample disks (Ted Pella Inc.), and the samples were placed on the AFM scanner. Etioporphyrin I solutions in respective solvents with desired concentrations were prepared less than 2 h in advance. This solution was loaded into the AFM liquid cell using 1-mL disposable polypropylene syringes (Henck Sass Wolf), tolerant of organic solvents. After loading, the system was left at rest for 10 min to equilibrate thermally. The crystal edges and surface morphology were used to identify the (010) crystal surface. Etioporphyrin I crystals are very dynamic hence the surface features changed rapidly in the presence of the growth solution. We set the scan direction approximately parallel to the [100] crystallographic direction and AFM images were collected for 30 min. The solution in the AFM fluid cell was refreshed every 10 min to maintain constant concentration. The evolution of the etioporphyrin I crystal surface was characterized by the velocity of growing steps v. To determine $v$, we monitored the displacements of 8 to 15 individual steps with a measured step height h = 1.17 ± 0.07 nm (Fig. 2A and SI Appendix, Fig. S4). Between 5 and 10 measurements were taken for each individual step with time. The step displacements were plotted as functions of time (Fig. 2B and SI Appendix, Fig. S5). The slope of these plots was used to determine the step velocity v (Fig. 2C) at a given supersaturation.

Observation of Kinks along the Steps.

The kink densities on the (010) face of etioporphyrin I were determined using high-resolution images obtained by Cypher ES environmental AFM (Asylum Research), with Bruker SNL 10 AFM tips. In situ monitoring of small step segments on (010) face of etioporphyrin I in the four solvents at their respective solubilities show abundant kink densities in all the four solvents chosen for the study. The lateral resolution of these images, ca. 2 nm, is insufficient to resolve individual kinks (50, 52, 57, 93, 94). The lack of straight step segments, however, suggests that the average separation between kinks is comparable to the resolution limit, ca. 3 molecules, equivalent to kink density ${\overline{n}}_k^{-1}$ close to the thermodynamic limit of 0.3 (SI Appendix, Fig. S6).

Determinations of Activation Free Energies.

The activation energy of solute molecule incorporation into the growth site was determined from the step velocities. The rate constant for solute incorporation may be presented as $k_a=k_o\mathrm{e}\mathrm{x}\mathrm{p}(\mathrm{\Delta }{G}^{‡}/k_{B}T)$, where $k_0=2\pi D{r}^{\ast }{N}_A/{\overline{n}}_k$ combines the rate constant for reaction between a kink and a solute molecule only limited by diffusion with the constant kink density ${\overline{n}}_k^{-1}$. The $k_a$ values were determined from the slopes of the step velocity correlation with solute concentration. The $k_0$, values are determined from the known values of solvent viscosity η, kink density ${\overline{n}}_k^{-1},$ and assuming $r^{\ast }$ = 0.5 nm. The resulting values of $k_0$ are listed in SI Appendix, Table S4. The values of activation free energy $\mathrm{\Delta }{G}^{‡}$ are listed in SI Appendix, Table S5.

Molecular Dynamics Simulations.

To investigate the attachment/detachment of etioporphyrin I molecules to etioporphyrin I crystals in DMSO, octanol, hexanol, and butanol, we carried out all-atom molecular dynamics (MD) simulations. We employed GROMACS 5.1.5 (95). Etioporphyrin I and the solvent species were modeled with the generalized AMBER force field (96, 97) (GAFF), using the ACPYPE package (98) to generate topology files for GROMACS. Partial charges were derived from B3LYP/6-31G(d) density functional theory calculations using the restrained electrostatic potential (RESP) method implemented in the PyRED program (99). The equations of motion were propagated using the leapfrog integration algorithm with a 2-fs timestep. All the bond lengths were constrained using the LINCS (100) algorithm to remove fast degrees of freedom associated with bond vibrations and facilitate the use of a large time step. Van der Waals interactions were truncated using a 1-nm cutoff, and periodic boundary conditions were applied in all three dimensions. Coulomb interactions were handled with the particle mesh Ewald method (101), using a cutoff of 1 nm for the real-space contributions and an error tolerance of 10⁻⁵.

Unit cell information for etioporphyrin I was obtained from the Cambridge Structural Database REFCODE WOBVUF (42). The unit cell parameters predicted by the GAFF model were evaluated by performing a 10-ns MD simulation of a 4 × 4 × 4 supercell of etioporphyrin I and computing averages over configurations sampled every 10 ps over the last 5 ns. During the MD simulation, the temperature was maintained at 300 K and isostress conditions of 1 bar were imposed using a Bussi–Donadio–Parrinello thermostat (102) (0.1-ps time constant) and an anisotropic Parrinello–Rahman barostat (103) (2-ps time constant), respectively. Unit cell parameters from the GAFF model were found to deviate from the literature data by less than 3% (SI Appendix, Table S6).

The kink site on the (010) face of etioporphyrin I was modeled by replicating the unit cell to create a $7\times 2\times 5$ supercell, which was then rotated to align the $(010)$ face with the z-axis of the simulation box, preserving the triclinic periodicity of the crystal. Next, the z-dimension of the simulation box was enlarged to create a crystal slab with top and bottom (010) faces exposed to vacuum; the z-dimension of the simulation box was chosen to be at least 3 times that of the crystal slab thickness to minimize finite size effects in the simulations. A portion of the top layer of the crystal slab was then cleaved to expose an obtuse kink in the $[100]$ direction on a [001] facing step. The final kink model consisted of 52 etioporphyrin molecules in the crystal and one tagged etioporphyrin occupying the kink site. Last, the space above and below the slab was filled with molecules of the desired solvent (SI Appendix, Table S7).

Energy minimization of the solvated kink models was performed using the steepest descent algorithm with a stopping criterion of 1,000 kJ mol^−1nm−1 for the maximum force on an atom. Each model system was then equilibrated by performing a 20-ns MD simulation in the canonical ($\mathit{NVT}$) ensemble at 300 K using a Bussi–Donadio–Parrinello thermostat (102) with a 0.1-ps time constant. The $\mathit{NVT}$ simulation was followed by an additional 20 ns of equilibration in the isothermal–isostress ($N{P}_{z}T)$ ensemble at 300 K and 1 bar, coupling the z-dimension of the simulation box normal to the surface to a Parrinello–Rahman barostat (103) with a 2-ps time constant while fixing the x- and y-dimensions.

The equilibrated systems were used to initialize well-tempered metadynamics (WTmetaD) (104) simulations to study the attachment/detachment of a tagged etioporphyrin molecule at the kink site. The WTmetaD simulations were performed in the $N{P}_{z}T$ ensemble at 300 K and 1 bar using the PLUMED 2.4.3 free energy plug-in for GROMACS. The choice of collective variables (CVs) was motivated by a study, which showed that both slow solvent and solute degrees of freedom must be sampled to properly converge potential of mean force (PMF) calculations for the detachment/attachment of Cl⁻ and Na⁺ at kink sites on NaCl crystals (36). Accordingly, we used two CVs: the normal distance between the kink site and the center of mass (COM) of the tagged etioporphyrin molecule, $z_{\mathrm{C}\mathrm{O}\mathrm{M}}$, and the number of solvent molecules in the kink cavity, ($N_{\mathrm{s}\mathrm{o}\mathrm{l}}$). The first CV was defined as the normal distance from a reference carbon in the etioporphyrin I crystal located on the bottom surface of the kink. The second CV was defined using a stretched and shifted rational switching function to obtain a CV suitable for biasing in WTmetaD:

$$\begin{array}{c}{N}_{\mathrm{solv}}=\frac{1}{n_h}\sum _j\frac{s\left(r_{\mathit{ij}}\right)-s\left(d_{\max }\right)}{s\left(0\right)-s\left(d_{\max }\right)}H\left(d_{\max }-r_{\mathit{ij}}\right),\hfill \end{array}$$

where

$$\begin{array}{c}s\left(r\right)=\left\{\begin{array}{ll}1,& \left(\frac{r-d_0}{r_0}\right)\le 0\\ \frac{1-{\left(\frac{r-d_0}{r_0}\right)}^m}{1-{\left(\frac{r-d_0}{r_0}\right)}^{2m}},& \left(\frac{r-d_0}{r_0}\right)<0\end{array}\right.,\hfill \end{array}$$

$r_{\mathit{ij}}$ is the distance between solvent heavy atom j and the reference carbon atom on the kink’s surface, $n_{\mathrm{h}}$ is the number of heavy atoms in a single solvent molecule, $H$ is the Heaviside function, $r_0=0.9$ nm, $d_0=0.85$ nm, $d_{\mathrm{m}\mathrm{a}\mathrm{x}}=0.90$ nm, and $m=12$. This CV was implemented using the DENSITY and INSPHERE functions in PLUMED and provides a continuous approximation to the number of solvent molecules within a 0.9-nm radius of the kink site.

The WTmetaD simulations were performed using a bias factor of 15 and a 1-ps interval between the deposition of repulsive Gaussian potentials. The initial height of the Gaussian potentials was 1 kJ/mol, and Gaussian widths of 0.02 nm and 0.15 molecules were chosen for $z_{\mathrm{C}\mathrm{O}\mathrm{M}}$ and $N_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{v}}$, respectively. To limit the explored CV region, upper quadratic walls were placed at $z_{\mathrm{C}\mathrm{O}\mathrm{M}}=1.7$ nm and $N_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{v}}=11$ molecules using spring constants of 5,000 kJ/(mol⁻¹ nm⁻²) and 5,000 kJ mol^−1nm−2, respectively. To prevent lateral drift away from the kink site, an auxiliary upper quadratic wall was also placed on the in-plane (x-y) distance between the reference carbon atom on the kink’s surface and the center of mass of the tagged etioporphyrin molecule at $0.4$ nm using a spring constant of 2,000 kJ mol^−1nm−2. For each solvent, three independent WTmetaD simulations were run for 5.5 to 9.5 µs (SI Appendix, Table S6) to obtain converged estimates of the PMF characterized by the two CVs. The PMF from each WTmetaD simulation was calculated from the generated time series data using the reweighting method proposed in ref. 105. The final PMF reported for each solvent was obtained by averaging over the results from the three independent simulations, using the SD to estimate statistical uncertainties.

Supplementary Material

Appendix 01 (PDF)

Dataset S01 (XLSX)

Movie S1.

The ingress of an etioporphyrin I molecules into a kink. Successive snapshots of an etioporphyrin I molecule in the solution near a kink (ass seen in Fig. 4a), at the intermediate state (as in Fig. 4b), and at its final location in the kink (as in Fig. 4c). Classical MD simulation results. Solvent molecules are omitted for clarity. The kink is viewed along the unfinished molecular row at the step edge. The direction of step growth is from left to right. Etioporphyrin I molecules in the crystal lattice are shown in grey. The lattice planes of etioporphyrin I molecules in front and behind the plane, which hosts the kink, are omitted. In the incoming etioporphyrin I molecule, C atoms are shown in teal, N in blue, and H in silver.

Data, Materials, and Software Availability

The datasets generated and analyzed during the current study are provided in SI Appendix. No custom-made computer code was used in this work. All other data are included in the manuscript and/or supporting information.