Multisite models to determine the distribution of kink sites adjacent to low-energy edges

Michael A. Lovette and Michael F. Doherty

Phys. Rev. E 85, 021604 – Published 15 February 2012

Abstract

Kink sites play a critical role in crystal growth. The incorporation of a growth unit into a kink site (1) maintains the free energy of the edge constant and (2) creates another site with the same properties. These properties allow growth through successive incorporation events to proceed in a self-sustaining manner such that the equilibrium spacing between kinks is maintained on average. Traditionally the distributions of kink sites have been determined using a single-site model whereby the probabilities of encountering a kink site adjacent to an edge and encountering a disturbance along an edge are assumed equivalent. In this paper, we develop multisite models that determine the probabilities of encountering kink sites; with the requirement that they obey both properties necessary for growth through successive self-sustaining incorporation events. The probabilities determined using the multisite models diverge significantly from the single-site model for edges with intermolecular interactions $≲ 6 k_{b} T$ between successive molecules. The implications of these findings for the development of predictive crystal shape models and experimental analysis are discussed.

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Received 22 July 2011

DOI:https://doi.org/10.1103/PhysRevE.85.021604

Authors & Affiliations

Michael A. Lovette and Michael F. Doherty^*

Department of Chemical Engineering, University of California, Santa Barbara, California 93106-5080, USA

^*mfd@engineering.ucsb.edu

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Vol. 85, Iss. 2 — February 2012

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Images

Figure 1
Schematic diagrams of typical Kossel edge segments with strong nearest-neighbor interactions in (a),(b) plan and (d) perspective view. For the segment in (b) and (d), a single disturbance results in a single kink site at the $(0, 1)$ position marked by the “x.” The plan-view geometries of positive and negative disturbances in this system are shown in (c), with dashed arrows demonstrating the change in position of the edge as a result of each disturbance. The blue lines in (d) indicate the interactions formed through incorporation events at the sites indicated by circles with dashed perimeters. Solid arrows indicate the outward normal directions of edges. The remaining symbols are described in Sec. 3.Reuse & Permissions
Figure 2
Periodic bond chain diagrams for all possible $F$ faces with two or three centrosymmetric PBCs. The unique edges for centrosymmetric systems are aligned with the PBC directions, which are indicated by lines. The general oblique lattice was drawn arbitrarily. Furthermore, diagonal edges were chosen to connect nodes from the “upper left” to “lower right;” the opposite case is equally plausible. White circles indicate the centers of mass for molecules contained within the faces, which are coincident with nodes in the different two-dimensional Bravais lattices.Reuse & Permissions
Figure 3
PBC diagrams showing slices of naphthalene (a),(b) and paracetamol (c),(d) aligned in the (20 $\bar{1}$ ) (a),(c) and (200) (b),(d) directions. White circles indicate the centers of mass for molecules contained within the slice. Cyan circles indicate the positions of nodes within the underlying Bravais lattices. Different colored lines indicate interactions of different strengths; with colors reused between the cases, e.g., interactions indicated by red lines have the same strength in (a) and (b), but not (a) and (c). The PBCs for naphthalene are centrosymmetric; as such the positions of molecules are nodes within oblique and rectangular two-dimensional Bravais lattices for the (20 $\bar{1}$ ) and (200) slices in (a) and (b), respectively. Although the same two-dimensional lattices exist in each slice, for the case of paracetamol a basis is required at each node to generate the center of mass positions.Reuse & Permissions
Figure 4
Possible states of the five-site edge segment which classify the unoccupied (0,1) position (marked by an “x”) as a thermodynamic kink. Only sites located within the slice containing the edge and contained either within the crystal or adjacent to the edge are shown. The outward normal directions of edges are aligned with the top of the page. These states are described in Table .Reuse & Permissions
Figure 5
Possible states of the five-site edge segment that do not result in the formation of a thermodynamic kink site at the unoccupied ( $0, 1$ ) position. Only sites located within the slice containing the edge and contained either within the crystal or adjacent to the edge are shown. The outward normal directions of edges are aligned with the top of the page. These states are described in Table .Reuse & Permissions
Figure 6
Probability of an unoccupied site adjacent to an edge existing within each group based on the five-site model for a Kossel crystal. The probabilities of encountering a disturbance along an edge, a thermodynamic kink, or a kink site adjacent to an edge diverge for $w ≲ 3 k_{b} T$ .Reuse & Permissions
Figure 7
Convergence of the native probabilities for a five-site segment to exist in the states shown in Figs. 1a and 1(b) with the native probabilities of encountering an edge adatom site and a disturbance along an edge, respectively.Reuse & Permissions
Figure 8
Schematic diagrams of typical edge segments with strong interactions on a (111) face of an fcc crystal: (a),(b) plan and (d) perspective view. A disturbance-free edge segment is shown in (a). For the segments in (b) and (d), a single disturbance results in a kink site at the (0,1) position designated by the “x.” The plan-view geometries of positive and negative disturbances in this system are shown in (c), with dashed arrows demonstrating the change in position of the edge as a result of each disturbance. Blue lines in (d) indicate the intermolecular interactions that are formed through the incorporation of a molecule into an unoccupied site. Arrows indicate the outward normal directions of edges.Reuse & Permissions
Figure 9
Possible states of the edge segment which classify the unoccupied ( $0, 1$ ) position, marked with an “x,” as a thermodynamic kink. Only sites located within the slice containing the edge and either within the crystal or adjacent to the edge are shown. Cyan sites indicate the positions within the terrace containing the edge that are a part of a disturbance but cannot be edge sites. The outward normal directions of edges are aligned with the top of the page. These states are described in Table . For brevity, only the sites pertaining to the classification of the segment are shown.Reuse & Permissions
Figure 10
Possible states of the edge segment that do not result in the formation of a thermodynamic kink site at the unoccupied ( $0, 1$ ) position. Only sites located within the slice containing the edge and either within the crystal or adjacent to the edge are shown. Cyan sites indicate the positions within the terrace containing the edge that are a part of a disturbance but cannot be edge sites. The outward normal directions of edges are aligned with the top of the page. These states are described in Table . For brevity, only the sites pertaining to the classification of the segment are shown.Reuse & Permissions
Figure 11
Probability of an unoccupied site adjacent to a native edge existing in a given group based on the multisite model for an fcc crystal. The probabilities of encountering disturbances, thermodynamic kinks, or kink sites diverge for $w ≲ 3 k_{b} T$ .Reuse & Permissions
Figure 12
Average number of molecules between kinks and thermodynamic kink sites determined using the multisite and single-site models are shown in (a). The error in using $λ / a$ to predict $\bar{n}$ is shown in (b). The error in measuring $\bar{n}$ and correlating it to $w$ using the single-site model is shown in (c) as a function of $λ / a$ . Each curve in (c) approaches a singularity at $\bar{n} (w = 0)$ .Reuse & Permissions
Figure 13
Successive incorporation events occurring at an edge (a)–(c), for which the average spacing between successive kink sites (a) and (b) and thermodynamic kinks (a)–(c) is constant. Incorporation events at the thermodynamic kink sites in (c) would result in the sealing of the edge and an increased average spacing between thermodynamic kinks.Reuse & Permissions
Figure 14
Hypothetical energy barrier for the attachment and detachment of molecules from thermodynamic kink sites is presented; where $q$ is the reaction coordinate for the attachment and detachment processes. Though not necessarily the case, $q$ is typically assumed to depend on the separation distance between the center of mass for the molecule attaching at the kink site (shown as a gray circle) and its position once incorporated.Reuse & Permissions
Figure 15
Molecular resolution AFM images of edges, and histograms of interkink distances (compiled from several images) for the (111) and (100) faces of apoferritin and insulin crystals, respectively. While disturbance densities along edges found on the (111) face of apoferritin appear beyond the applicable range of the single-site model, disturbances on the (100) face of insulin are consistent with traditional expectations. Images contained in the box marked apoferritin were adapted with permission from Yau et al. [26]. Images contained in the box marked insulin were adapted with permission from Georgiou and Vekilov [11]. Designations of (a)–(c) were retained from the original images and do not have correspondence within the figure.Reuse & Permissions
Figure 16
Molecules contained within and incorporation sites adjacent to the edge segments shown in Fig. 15 corresponding to apoferritin (a) and insulin (b). For the case of insulin, the red and yellow interactions are colored the same as corresponding interaction in Georgiou and Vekilov [11]. Arrows indicate the outward normal directions of edges.Reuse & Permissions

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