Etioporphyrin I crystal growth phenomenology
Etioporphyrin I crystals are rod-like and their habit is dominated by elongated lateral {010} and {001} faces and small axial {101} faces (Fig. 1b). The rates of growth of the individual faces R
characterize the rate of conversion of the solute into crystalline matter and thus constitute crucial variables of crystal growth. We measured the growth rates from the shape evolution of crystals growing in a microfluidics channel (Figs. S1, S2) where we maintain constant solute concentration C by continuously supplying fresh growth solution in octanol 57. We used concentrations that equal or exceed the solubility Ce = 0.25 mM, i.e., the concentration at which crystals are in equilibrium with the solution and neither grow nor dissolve; Ce was measured in independent experiments (Fig. S3). The intense maroon color of the crystals affords easy microscopic determination of the growth rate R of the axial {101} faces from the increase of crystal length (Fig. 1c, d). The length-to-width ratio of the crystals, ca. 10:1, persists after growth at various concentrations, indicating that the R(C) correlation for the {101} faces is similar to those of the lateral faces.
Etioporphyrin I crystal faces are flat and intersect at sharp edges (Fig. 1b). Flat crystal faces are expected to grow by spreading of layers wherein molecules incorporate at the terminal molecular rows of unfinished layers—the steps 16,58—which then grow with velocity v. Mass preservation dictates that R should equal the product \(h{l}^{-1}v\), where h is the step height and l is the average step separation 22,23. As steps advance through a bimolecular reaction between solute molecules and kinks 59 and since the kink density is often independent of C 22,23, v is expected to scale linearly with C 60–63. If new layers are generated by dislocation outcropping on the face, as is typical at low supersaturations, \({l}^{-1}\) increases linearly or sublinearly with C 27,29 (Fig. S4), whereas h = 0.991 nm 49 is a constant determined by the crystal structure. Therefore, the product \(h{l}^{-1}v\) = R should scale no more sharply than as a quadratic function of C 22,23.
The measured R(C) of the {101} faces defy these expectations (Fig. 1d). The crystals grow at any C > Ce and R exponentially reaches ca. 20-fold greater values at C = 0.35 mM, at which the relative supersaturation C/Ce – 1 = 0.4 (Fig. 1d, inset). Within the range 0.35 < C < 0.47 mM (at the latter C/Ce – 1 = 0.88), R(C) is linear (Fig. 1d). The rate of increasing R appears to slow down at C > 0.47 mM. The slower R(C) increase in the highest concentration range can be attributed to insufficient supply of solute to the faster growing crystal. The measured combination of exponential and linear increases, however, is hard to reconcile with classical models 22,23,29,30.
Four scenarios could potentially underlie the observed R(C) correlation (Fig. 1e – h). First, the steep R(C) increase could be due to a lack of dislocation outcrops on the measured faces, forcing the generation of new crystal layers by two-dimensional nucleation (Fig. 1e) 64,65. The quasi-exponential increase of the rate of two-dimensional nucleation with higher solute concentration 66,67 would then motivate a steep increase of the step density \({l}^{-1}\) and a quasi-exponential R(C) correlation 58. Monitoring the (010) face with atomic force microscopy (AFM), however, reveals new crystal layers wound into a spiral (Fig. 1i) that originates at the point where a screw dislocation outcrops on the growing face 22,23,68–70. The dislocation-generated steps grow at velocity v which is steady (Fig. 1j) and a linear function of C at C/Ce – 1 as high as 0.6 (Fig. 1k). The spiral attains an elliptical shape elongated along the [100] or \(\overrightarrow{a}\) direction (Fig. 1i) driven by the crystal anisotropy, which enforces fastest v in that direction 27,29. Dislocations are linear defects that do not constrain their outcrop points to a single face and move among faces as the crystal grows 65,71,72. Thus, dislocations are likely to outcrop and drive growth on the {101} face.
Second, the dislocation-generated steps may, at low supersaturations, exhibit a kink density lower than the thermodynamic limit of ca. 0.3 per molecule at the step edge (Fig. 1f) 24,26,73,74. Elevated supersaturations may induce increased kink density owing to either one-dimensional nucleation of new lattice rows 75 or attachment of two-dimensional clusters 26. The steep kink density increase would manifest in a quasi-exponential R(C) correlation 26. AFM imaging of the step edge on a (010) etioporphyrin I face at Ce reveals, however, a kink density close to the thermodynamic limit (Figs. 1l, S5). Since kink density is regulated by the average bond strength in the crystal lattice 22, steps on the {101} face are likely also kink-rich.
Third, the steep R(C) increase may manifest the action of foreign molecules that adsorb on terraces between steps and force advancing steps to bend as they pass between them, a mechanism often called “step pinning” (Fig. 1g) 58,76. The imposed curvature raises the chemical potential of the molecules in the curved step segments and lowers the supersaturation 76. Step pinning arrests growth within a rather broad range of low supersaturations, sometimes referred to as “the dead zone”, and invokes fast v ascent at intermediate supersaturations 70,76–79. In contrast to the predictions of this mechanism, R(C) exhibits no dead zone (Fig. 1d).
Excluding these three scenarios that could induce quasi-exponential R(C) leaves one potential mechanism, that the fast increase of R reflects the association of precursors to the growing faces, wherein the precursors preform in the solution at numbers that sensitively increase at higher solute concentrations (Fig. 1h).
Growth by spreading of multilayer stacks empowered by organic solvents
The above hypothesis of precursor-mediated growth suggests that precursors may be more common at elevated supersaturations. Indeed, time-resolved in situ AFM imaging of a (010) face growing at supersaturation C/Ce – 1 = 0.56 (Fig. 2a) reveals precursors that land on the crystal surface. The precursors shape as circular domes with diameters ca. 400 nm and height ca. 6–12 nm, corresponding to volumes of order 105 – 106 nm3. The precursors’ growth indicates that they are condensates of etioporphyrin I, the only compound (other than the solvent) available in substantial amounts. Initially they grow both in height and width, as expected for nonfaceted condensates. In time (ca. 5 min, Fig. 2b, c) the precursors develop a flat top parallel to the underlying (010) face, their vertical growth ceases, and they transform into stacks of 25 to 50 layers (Fig. 2a – c) that spread along the surface. The precursor-generated stacks prolongate in the same \(\overrightarrow{a}\) direction as the crystal’s own dislocation-generated single layers (Fig. 1g), indicating that they are crystalline and their lattice aligns with the lattice of the underlying crystal. The alignment between substrate lattices and those generated from precursors indicates that the precursors are initially liquid and allow the periodic force field of the substrate to guide the formation of their crystal structure. As a result, there is a disorder-to-order transition wherein the precursors integrate into the underlying crystal and substantially contribute to its growth.
Probing the hypothesis that the precursors may be crystallites that orient to align with the crystal lattice, we note that a 100 nm crystallite would have translational and rotational diffusivities \({D}_{trans}={k}_{B}T{\left(6\pi \eta {R}_{p}\right)}^{-1}\) ≈ 5×10–13 m2s-1 and \({D}_{rot}={k}_{B}T{\left(8\pi \eta {R}_{p}^{3}\right)}^{-1}\) ≈ 180 s-1 (kB, Boltzmann constant; T, temperature; h = 7.36 mPa s, solvent—1-octanol—viscosity 80; Rp, particle radius). The interactions with the crystal surface that may align the crystallite, according to the oriented attachment scenario 42,45–47, extend to less than 2 nm 81 59. According to the Einstein relation, the particle would take about 1 µs to diffuse over this distance. During this time, it would probe angles of ca. 0.02 rad, substantially less than the 4p minimum needed for oriented attachment of its lattice to another lattice. This analysis suggests that oriented attachment may be limited to precursors with sizes on the order of 1 nm—as in most of the documented cases 42,45–47—which rotate five to six orders of magnitude faster while translating only 100-fold faster.
At longer times the stacks of layers originating from individual precursors merge (Fig. 2d). No gaps remain between the merging layers, in further support of alignment of the lattices of the precursor-generated stacks and the underlying lattice and the precursor fluidity. Dislocations, revealed by the emergence of spiral steps, appear on their flat tops (Fig. 2d). Dislocations represent plastic deformations of the crystal lattice enforced by the accumulation of lattice strain. To gauge the strain of the precursor-generated latices, we employ the growth velocity v of crystal layers. This v is expected to scale as \(\text{exp}\left[({\mu }_{s}-{\mu }_{c})/{k}_{B} T-1\right]\cong C/{C}_{e} -1\), where µs and µc are the chemical potentials of the solute in the solution and in the crystal, respectively 78,82. This relationship, which assumes that the activity coefficients of the solute are close to unity at both C and Ce, is reasonable at C < 1 mM in a solution with no charged species and it motivates the observed linear \(v\left(C\right)\) correlation (Fig. 1h). Lattice strain would increase µc, lower the driving force for growth, and thereby retard v. Time-resolved in situ AFM measurements demonstrate that steps on the flat tops of precursor-generated stacks grow 20 to 40% slower than steps on the underlying crystal, with steps on surfaces of thicker stacks, monitored later in the stack evolution, growing faster. The layers within a stack grow slower by about half (Fig. 2e). The elevated strain levels may be induced by imperfect alignment between the newly formed crystal layers and the underlying lattice. This imperfect fit, in turn, may be caused by enhanced viscosity of the condensate that comprises the precursors, consistent with observations of mesoscopic solute rich-clusters in lysozyme aqueous solutions that are fluid and readily host crystal nucleation when freshly-formed, but gel, solidify, and deactivate as they age 37.
Besides strain, the edges of the stacked layers, separated by about 10 nm, may grow slower owing to overlapping of their supply fields 22,23,69,83. Monitoring the growth of steps as close as a few nanometers and steps of double height, however, reveals that their velocities are similar to that of single-height steps far from their neighbors (Fig. S6). The l-independence of v suggests that the solute molecules reach the steps directly from the solution and not after adsorption on the terraces and diffusion towards the steps 84 and that the precursor-generated stacks grow slower owing to accumulation of strain and not to supply field overlap.
A far-reaching consequence of the lack of step-field overlap is that the stacks of layers spread laterally with considerable velocity (Fig. 2e) and provide a major contribution to crystal growth at high supersaturations. This contribution would be greatly reduced if closely-spaced steps slowed down owing to competition for supply from the crystal surface (Fig. S6b). Documented examples reveal complete growth cessation of steps closer than ca. 50 nm 62,69,83. Thus, competition for scarce solute supply via the crystal surface would transform a stack of ca. 30 steps spaced at ca. 10 nm into a major stalled macrostep 85–87. The surface diffusion pathway, however, may be constrained to crystallization from water-containing solvents, in which hydrogen bonds between solvent and solute lower the adsorption enthalpy and the activation barrier for surface diffusion and thus enable this supply route 84. This line of argument suggests that the nonclassical growth mode, whereby liquid precursors generate stacks of crystal layers that spread laterally, may be unique to organic solute-solvent pairs, in which the weak solute-solvent interactions empower the direct solute supply route to the steps.
Mesoscopic etioporphyrin-I-rich clusters enabled by weak solute-solute interactions
To understand the properties and elucidate the formation mechanisms of the precursors revealed by AFM monitoring of etioporphyrin I crystal growth—their relatively uniform ca. 100 nm diameter, fluidity, and solute concentration-dependent numbers—we characterized etioporphyrin I solutions with oblique illumination microscopy (OIM, Fig. 3a-c) 80,88–94. OIM reveals aggregates in etioporphyrin I solutions, which exhibit a narrow size distribution and an average radius Rc = 75 ± 12 nm (Fig. 3d, e). Such aggregates would contain ca. 50,000 moderately packed etioporphyrin I molecules with an effective diameter of ca. 3 nm and solvent occupying the voids. Importantly, the volume of an aggregate is ca. 1×106 nm3. The similarity of this volume to the volumes of the crystal growth precursors seen by AFM (Fig. 2b) suggests that the growth precursor may originate as aggregates. The flattening of the aggregates upon their association to the crystal surface is another evidence of their fluidity. Values of Rc are steady for at least six hours (Fig. 3d). The narrow size distribution and steady size stand in contrast to expectations for crystals or other solid or liquid aggregates that result from first-order phase transitions 33,95,96, for which growth persists and R increases in time 97. In further challenge to aggregates’ crystallinity, the highest concentration at which aggregates were monitored, 0.06 mM (Fig. 3f, g), is about four-fold lower than the crystal solubility, Ce = 0.25 mM.
The aggregates exist in dynamic equilibrium with the host solution. Indeed, the number of aggregates N decreases from 1.2×108 cm-3 to 0.02×108 cm-3, ca. six-fold, in response to a three-fold reduction of concentration, from 0.06 to 0.02 mM, whereas Rc is consistently ca. 75 nm within this concentration range (Fig. 3f, g). The response of N to reduced concentration is incompatible with irreversibly disordered agglomerates, whose concentration would be diluted in parallel with that of the solute. The concentration of the solution in equilibrium with the aggregates Cf is approximately equal to the initial C0 (Fig. 3h), demonstrating that the aggregates capture a minor fraction of the solute. Consistently, the fraction of the solution volume they occupy, \({\varphi }_{2}\approx \frac{4}{3}\pi {R}_{c}^{3}N\) ≈ 10− 7 at the highest solute concentration examined, 0.06 mM (cluster characterization at higher concentrations was hampered by the absorption of the illuminating OIM wavelength by etioporphyrin I 59), is small and even smaller at lower etioporphyrin I concentrations. Surprisingly, Cf is not constant, but instead increases with C0 (Fig. 3h). The finding of variable equilibrium concentration defies expectations for phase equilibria between solutions and crystals, amorphous aggregates, and liquids, which equilibrate with solutions of concentration that is constant and independent of the initial concentration of the solution in which they form 33,98–100.
These behaviors of etioporphyrin I aggregates are consistent with observations of mesoscopic solute-rich clusters of proteins 41,101–103 and organic molecules 68,69,104,105. According to recent models, the mesoscopic clusters form due to accumulation of transient oligomers (Fig. 3i, where the transient oligomers are tentatively represented as dimers) 93,94,103,106,107. Accounting for concurrent chemical and phase equilibria predicts a strong correlation between the final Cf and the initial C0 solution concentrations 93,94. The mesoscopic aggregates of etioporphyrin I appear to comply with the predictions of this model and we conclude that the aggregates, which act as crystal growth precursors, are mesoscopic etioporphyrin-I-rich clusters.
A crucial part of the cluster mechanism is the formation and decay of transient oligomers. These dynamics require a propensity to form relatively weak bonds, of several kBT units, between solute molecules 103,108–110. Etioporphyrin I can partake in several such bonds: p-stacking 68; quadrupole; and dispersion 9. All-atom molecular simulations have revealed that etioporphyrin I readily forms a line of weakly bound dimers, in which the constituent monomers are mostly parallel (Fig. S8) 63. Absorption spectroscopy supported by density functional theory modeling confirmed the presence, at a low concentration, of stacked etioporphyrin dimers, consistent with dimer configuration and weak binding, predicted by the simulations 63. Importantly, the potential minimum corresponding to the dimer (Fig. S8) falls within the range of several kBT units required for formation of mesoscopic solute-rich clusters 109. The structural complexity of many organic molecules brings about a rich variety of bonds in aqueous, mixed, and purely organic solvents, some of which may be of the strength needed for cluster formation. The bonds that support mesoscopic clusters differ from those that bind crystals only by strength and not by range or directionality, in contrast to earlier theoretical scenarios 44,111. These considerations underlie the prediction that nonclassical and dual classical-nonclassical crystal growth modes may be common in organic crystallization.
Incorporation of amorphous particles and lattice strain.
Not all particles that land on the crystal surface integrate with the crystal lattice and generate crystal layers. For instance, we observed particles that land on the surface and within ca. 10–15 min shrink in both height and width (Fig. 4a, b). We measured the particle heights with respect to the surrounding crystal surface, which continued to grow during the observation. The addition of new layers within the period of observation (Fig. 4a) contributed ca. 5 nm to the base from which the particle size is measured, with the net negative change in the particle size attributable to its dissolution.
Dissolution of particles in a supersaturated solution, in which steps in the immediate vicinity of the particles on the crystal surface grow, is highly counterintuitive. For nonclassical pathways involving crystallization by particle attachment (CPA), such behavior has been reported for mis-oriented attachment when particles are crystalline and there is a mismatch with the underlying crystal lattice 42,44. Given, however, that the precursors of etioporphyrin I are liquid, our
observation seems to indicate the dissolving particles consist of a distinct phase, which is substantially less stable, i.e., carries higher \({\Delta }{G}^{o}\), than the crystal. The P1 crystals that we study are the only crystal form of etioporphyrin I. We conclude that the particles represent a solid amorphous phase, known to have higher free energy than crystals owing to looser molecular packing supported by fewer intermolecular bonds. We submit that the mesoscopic etioporphyrin-I-rich clusters transform to amorphous particles as they age in the solution, where the lack of crystal templates prevents their disorder-to-order transformation (Fig. 2). The proposed evolution of the mesoscopic clusters is similar to previous observations, in aqueous solutions, with mesoscopic clusters and dense liquids of lysozyme and olanzapine 31,37,69,112. Importantly, these observations indicate that amorphous particles are poor precursors for crystal growth in organic media, unlike observations for minerals, e.g., zeolites, that form in aqueous environments via CPA involving disorder-to-order transition of amorphous particles 52,113. Besides the failure of etioporphyrin-I particles to integrate in the crystal lattice and contribute to growth, they may become embedded within it as gross defects and diminish the crystal quality.
Fast crystal growth carries substantial technological advantages 114. The exponential increase of the growth rate of etioporphyrin I crystals with solute concentration makes fast growth readily accessible at moderate supersaturations, at which spurious nucleation of new crystals is still minimal. The transition to mostly nonclassical growth, however, with clusters associating to growing surfaces and generating stacks of layers that merge and integrate with the lattice may potentially have deleterious effects on the crystal quality.
In view of the dominant application of etioporphyrin I crystals in optical elements we examine their optical properties using complete transmission polarimetry 115, namely Mueller matrix imaging microscopy 116. The polarization states of incoming light and that transmitted through the sample and microscope optics are modulated by continuously rotating waveplates at different frequencies. The Mueller matrix M, the linear operator that relates the input polarization or Stokes state (S), into a new polarization state for the outgoing light, characterized by the output vector (S') can be constructed from the characteristic frequencies of the time dependent transmitted intensity signal I(t) 117–119. The S vectors are a four-element vectors that describe the complete polarization characteristics of light, whereas M is a 4×4 operator that establishes the relationship between S and S', fully encoding the polarization-altering properties of the sample: \(\overrightarrow{S{\prime }}=M\overrightarrow{S.}\) The Mueller matrix changes as polarization evolves during the propagation of light through the crystal. The characteristic properties related to the linear retardance (LR) and the linear extinction (LE) that arise in the anisotropy of the refraction and absorption can be evaluated from the logarithm of the M 120–122.
Etioporphyrin I crystals are strongly absorbing across the visible part of the electromagnetic spectrum. We compared the LR at wavelength 500 nm of etioporphyrin crystals grown by the two mechanisms at C = 0.3 mM (where classical dislocation growth dominates) or 0.5 mM (at which the crystals mostly grow nonclassically, by the spreading of precursor-generated layer stacks). Here the absorbance is large (ca. 1) but unchanging (flat) and the linear extinction is near zero (Fig. 4d). The average values of linear retardance for the two crystals were 0.019 and 0.023, respectively, which is essentially the same given the accuracy to which we know the pathlengths. The linear retardance |LR| is an extrinsic phase shift |LR| = 2pDnL/l, that depends on the optical pathlength (L) and the intrinsic birefringence (Dn) and l is the wavelength in nm. Thus, the crystals grown classically and nonclassically were not distinguishable polarimetrically. The optical birefringence measurements demonstrate that growing crystals fast, at high supersaturations, where the nonclassical mode dominates, does not compromise their quality.