Early Onset of Kinetic Roughening due to a Finite Step Width in Hematin Crystallization

The structure of the interface of a growing crystal with its nutrient phase largely determines the growth dynamics. We demonstrate that hematin crystals, crucial for the survival of malaria parasites, transition from faceted to rough growth interfaces at increasing thermodynamic supersaturation $\mathrm{\Delta }\mu$. Contrary to theoretical predictions and previous observations, this transition occurs at moderate values of $\mathrm{\Delta }\mu$. Moreover, surface roughness varies nonmonotonically with $\mathrm{\Delta }\mu$, and the rate constant for rough growth is slower than that resulting from nucleation and spreading of layers. We attribute these unexpected behaviors to the dynamics of step growth dominated by surface diffusion and the loss of identity of nuclei separated by less than the step width $w$. We put forth a general criterion for the onset of kinetic roughening using $w$ as a critical length scale.

Figure 1

Rough and smooth crystal interfaces, illustrated in (a) and (b), respectively. A kink is highlighted in red and a step edge in blue. (c)–(f) Steady-state morphology of (100) hematin faces, imaged in phase mode, at four hematin concentrations $c_H$ and corresponding supersaturations $\mathrm{\Delta }\mu /k_{B}T$. At low $c_H$ and $\mathrm{\Delta }\mu$, in (c), hematin crystals grow by the generation and spreading of layers. At higher $c_H$ and $\mathrm{\Delta }\mu$ in (d)–(f), the surface is rough.

Figure 2

Hematin surface roughness. (a),(b) Schematics of kinetic roughening at increasing supersaturation $\mathrm{\Delta }\mu$ by (a) denser 2D nucleation and (b) higher step density in a step train. The radius of a 2D island $R$ is illustrated in (a) and a step width $w$, interstep separation $\mathbf{\ell }$, and correlation length $\xi$ in (b). (c)–(f) The distribution of the mean squared roughness $R_q$ for $100\times 100{\mathrm{nm}}^2$ areas at four $\mathrm{\Delta }\mu$ values, corresponding to the images in Figs. 1. (g) Average $R_q$ computed from the data in (c)–(f) as a function of $\mathrm{\Delta }\mu$. The range of rough growth is shaded in gray.

Figure 3

The critical 2D nucleus radius $R_c$ and the step width $w$. (a)–(f) Illustration of the determination of $R_c$ at $c_H=0.21\mathrm{mM}$ and $\mathrm{\Delta }\mu =0.27k_{B}T$ from the size and shape fluctuations of a newly nucleated island, indicated by the arrow in (b). (g) The dependence of $R_c$ on the supersaturation $\mathrm{\Delta }\mu$ [24]. The solid line denotes the predicted $R_c=\gamma \mathrm{\Omega }/\mathrm{\Delta }\mu$ dependence. The shaded area denotes the region of rough growth. A horizontal dashed line marks the value of $R_c$ at the roughening transition. (h) A high-resolution image of a step edge at $c_H=0.28\mathrm{mM}$, highlighted with a white contour; $w$ is indicated.

Figure 4

The normal growth rate $V$ and the overlapping of step supply fields. (a)–(e) A sequence of in situ AFM height mode images of a (100) face, recorded at $c_H=0.23\mathrm{mM}$, displaying the growth of two steps (left and right) in opposing directions. (f) $V$ of (100) faces determined as discussed in Supplemental Material [26]. The face kinetic coefficient ${\beta }_{\text{face}}$ is defined as the slope of the $V(c_H-c_e)$ correlation.