The elementary reactions for incorporation into crystals

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.

Keywords: activation barrier; crystal growth; molecular mechanisms; organic materials; solvent interactions.

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. ; 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.

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 $\eta$.

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. . (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).

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.

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.