- Email: [email protected]
Crystallization simulation in macromolecular crystals
Journal of Crystal Growth 220 (2000) 130}134
Crystallization simulation in macromolecular crystals C. FalcoH n RodrmH guez *, F. FalcoH n RodrmH guez, S. Aguilera Morales, R. Mulet Genicio Departamento de Teorn& a de Funciones, Facultad de Matema& ticas y Computacio& n, Universidad de La Habana, La Habana, Cuba Laboratorio de Crecimiento de Cristales, Instituto de Materiales y Reactivos, Universidad de La Habana, La Habana, Cuba Departamento de Fn& sica, Universidad Cato& lica del Norte, Antofagasta, Chile Laboratorio de Superconductividad, Instituto de Materiales y Reactivos } Facultad de Fn& sica, Universidad de La Habana, La Habana, Cuba Received 12 January 2000; accepted 12 July 2000 Communicated by A. McPherson
Abstract In order to avoid the di$culties encountered in the crystallization computer simulation of macromolecular crystals, we introduce the so-called `growth cella that is, the simple geometric volume occupied by a structural growth unit (i.e., monomer, dimer, etc.) in the crystal. The number and orientation of bonds that are satis"ed by the surrounding ones are accounted in the surfaces of these growth cells. This construction allows us to compare the attachment probabilities of growth units in di!erent crystallographic orientations on macromolecular crystals surfaces. The application of our method to the recent experimental observation of the formation of steps in the 10 1 02 direction on the (1 0 1) concanavalin A crystal surface is presented. 2000 Published by Elsevier Science B.V. Keywords: Crystallization; Macromolecular; Simulation
1. Introduction Usually, Monte Carlo (MC) simulation of crystallization processes only take into account the binding energy per bond and the number of bonds to calculate the attachment, detachment and migration probabilities of the growth unit [1,2]. Due to this lack of crystallographic speci"city of MC and because the development of potentials for even simple systems is a time-consuming and usually * Corresponding author. E-mail addresses: [email protected] (C. FalcoH n RodrmH guez), [email protected] (F. FalcoH n RodrmH guez), saguiler@ socompa.ucn.cl (S. Aguilera Morales), mulet@!.oc.uh.cu (R. Mulet Genicio).
empirical task in molecular dynamics (MD), both methods cannot be used to address molecular-level e!ects that are too speci"c. In this sense, `ab initioa calculations constitute a complementary approach which can improve our understanding of processes on the microscopic level, although applications to dynamical processes are still very limited [1,3]. On the other hand, the real images observed by means of SEM, AFM and other techniques [4,5] evidences a `growth near equilibriuma process, where surface di!usion cause the redistribution and smoothing of solid}liquid interface, far away from the simple `solid-on-solida (SOS) model, generally used in these simulations. In addition, the measurements of the step current, combined with the estimate values for the di!usion length, strongly
0022-0248/00/$ - see front matter 2000 Published by Elsevier Science B.V. PII: S 0 0 2 2 - 0 2 4 8 ( 0 0 ) 0 0 6 1 8 - 7
C. Falco& n Rodrn& guez et al. / Journal of Crystal Growth 220 (2000) 130}134
131
indicate that surface di!usion rather than bulk di!usion is the controlling mechanism of solute transport to the steps in these systems [4]. Undoubtedly, the big di!erences commonly found between the simulated surface structures obtained by MC and MD and the real images observed by SEM, AFM and other techniques, prove that these calculations must be improved in the search of a better agreement of the simulation results with the experimental "ndings. This improvement of the simulation may include the consideration of a large number of parameters, among these:
surface crystallization kinetics becomes di$cult. In order to avoid these di$culties, we introduce the so-called `growth cella that is, the simple geometric volume occupied by a structural growth unit (i.e., monomer, dimer, etc.) in the crystal. The number and approximate location of bonds that are satis"ed by surrounding units are accounted for in the surfaces of these growth cells. This construction allows us to compare the attachment probabilities of growth units in di!erent crystallographic orientations in a group of macromolecular crystal surfaces.
(a) the electrostatic attraction on the growth unit in solution exerted by the di!erent concentrations and orientations of unsatis"ed bonds at crystal surface, (b) the solid angle of the solution on which the previously cited electrostatic attraction exerts in#uence, determining in this way the available number of growth units for attachment, (c) the correspondence in number and distribution of unsatis"ed bonds between the growth unit and the attachment site, (d) the steric impedance for attachment of the growth unit at the selected site, (e) the terrace width, when the selected sites are found in the border of a step.
3. Formation of steps on concanavalin A (0 0 1) crystal surface
To elaborate an algorithm considering all the above-mentioned parameters is a very di$cult task and the computation time involved in the calculation would probably be prohibitive. Thus, one should disregard, based on physical intuition, those parameters with minor in#uence in a given problem. This approach in systems where the binding energies per bond are nearly the same makes it possible to substitute usual Monte Carlo simulations by simpler Celular Automata Models.
2. Growth cell In the case of macromolecular crystals, due to the structural complexity of the growth units, the analysis of the in#uence of each parameter on the
Kashimoto et al. [6] have proposed the steric hindrance for attachment in the 10 1 02 direction as the cause of formation of steps on the (1 0 1) crystal surface. Following these authors, we assume the dimer as the growth unit but due to the high number of inter-dimer bonds (32 bonds), we also considered that the formation of tetramers occurs immediately after the attachment of a dimer in the crystal surface. In Fig. 1(a) is shown the growth cell (dimer) of a concanavalin A crystal prevailing in the experimental growth conditions studied by Kashimoto et al. and in Fig. 1(b) the cell (tetramer) created by the immediate union of two dimers. In Fig. 2(a) is shown the distribution of unsatis"ed bonds per dimer in the #at (1 0 1) crystal surface and in Fig. 2(b) we can observe the correct orientation that must take the dimers in the solution near the liquid}solid interface in order to attach to this surface. In the subsequent analysis, we consider that the free dimers in solution ful"ll a succesion of actions in order to attach to the surface. They are: (i) approximation to the crystal, due to the attraction exerted between the unsatis"ed bonds on the surface and on the free dimer, (ii) rotation of the free dimer, in order to accomplish the maximal correspondence between its and surface unsatis"ed bonds, and (iii) migration to a more energetically favorable site.
132
C. Falco& n Rodrn& guez et al. / Journal of Crystal Growth 220 (2000) 130}134
Fig. 1. Growth cells of a concanavalin A crystal } (a): dimer; (b): tetramer, created by the union of two dimers.
Fig. 3. Attached tetramer on the #at (1 0 1) crystal surface. Fig. 2. (a): Flat (1 0 1) surface of concanavalin A crystal; (b): dimer correctly oriented to attach to this surface.
Considering that the initial attraction on a dimer in solution exerted by the (1 0 1) surface must occur preferentially between the six unsatis"ed bonds localized near the edges of the normal surfaces to 10 1 02 of the free dimer and the six unsatis"ed bonds localized near the convex edge between (1 1 0), and (1 1 0) surfaces of a dimer step in the #at (1 0 1) crystal surface, we conclude that the position with highest attachment probability must be the I-denoted, less probably appear the II-denoted position, and with minimal attachment probability appear the III-denoted.
Immediately after the attachment of a dimer in the I-position the attachment of a second dimer must occur in order to complete a tetramer as shown in Fig. 3. This represents the "rst stage in the formation of a 2-d nucleus, in accordance with the birth-and-spread growth model. This tetramer, on the #at (1 0 1) surface, appears as a triangular pyramid bounded shaped by (1 1 0), (1 1 0) and (0 0 1) planes, with height"51.5 As . We may analyze the attachment probabilities on each of the eight di!erent neighbors of the 2-d nucleus and, as a consequence, the most probable attachment order. Due to their greater altitude, the most active must be the groups of six and four unsatis"ed
C. Falco& n Rodrn& guez et al. / Journal of Crystal Growth 220 (2000) 130}134
bonds near the pyramidal vertex. In addition, taking into account the bond correspondence criterion, the dimers attracted by the high bond density edge formed by (1 1 0) and (1 1 0) 2-d nucleus surfaces must preferently adsorb to the (1 1 0), to the detriment of the (1 1 0) surface, and later attach to the most energetically favorable IV-denoted site. Less probable becomes attachment to the I-denoted site because of the larger migration jump that it implies, and still less the attachment to the II-denoted site, for which a very large migration jump is necessary without rotation to accomplish inter-dimer binding. After the occupation of the IVdenoted site, the I-denoted becomes the most probable site for attachment of a third tetramer. The second stage in this analysis corresponds to the structure shown in Fig. 4, after the attachment of a third tetramer in the most probable I-position. In this case, we can observe the appearance of a high density of unsatis"ed bonds at the interface of both tetramers. Again taking into account the necessary correspondence of bond distributions between the attachment site at the surface and the free dimer in solution, we conclude that the most probable neighbor position for attachment is the II-denoted, to the detriment of the III-denoted. On the other hand, in accordance with Land et al. [4], the attachment probability of position V in Fig. 3 is enhanced due to the duplication of the terrace width in the 11 0 02 direction. In accordance with Kashimoto et al. [6], the attachment probabilities of VI and VII positions
Fig. 4. First stage in the formation of a 2-d nucleus.
133
diminishes due to the steric hindrance caused by the restricted solid angle and the appropriate orientation the dimers must observe in order to remain "xed in both sites. Finally, the attachment probability of VIII-denoted sites, taking into account their relatively poor interaction with the large attractive convex edge between the (1 1 0) and (1 1 0) surfaces of the 2d-nucleus, must remain approximately the same as for the #at surface. As a consequence of this process a narrow terrace array in the 10 1 02 direction must arise that, takes into account the evidence for di!usion-"eld overlap on narrow terraces (j<=, where j is the surface di!usion length, = the terrace size). This implies that the step velocity diminution and homogenization [4] gives rise to stable step arrays on the (1 0 1) surface that may transform this surface to a new (n 1 n) surface where, accordingly with Kashimoto et al. [6], the value of n ranges from 7 to 11. From the previous discussion we developed, as a "rst approximation for concanavalin step formation, cellular automation in a bidimensional lattice. The lattice is initially empty (∀i, j h [i, j]"0), then a site (i, j) is randomly chosen and set to 1 (h [i, j]"1). This site mimics the dimer deposited in position I of Fig. 2. Then the attachment of the next dimer is simulated by the following rules: Rule 1: If h [i, j]"1 then h [i!1, j#1]"1 L L> and h [i#1, j#1]"1. L> Rule 2: If h [i, j]"1 and h [i!1, j#1]"1 then L L h [i!1, j]"1. L> Rule 3: If h [i, j]"1 and h [i!1, j]"1 then L L h [i!1, j!1]"1 L> Rule 1 simulates the deposition processes in positions IV and V of Fig. 3; Rule 2 represents the deposition of a tetramer in site I of Fig. 3, while Rule 3 represents the deposition of a tetramer in site II of Fig. 4. In Fig. 5 are shown the results of the application of this algorithm. It represents a view of the (1 0 1) surface along the 10 0 12 direction. The di!erent numbers represent the sequence order of attachment of tetramers at the corresponding sites. Advance directions of I and II terraces actually depend on the attachment probability relations for
134
C. Falco& n Rodrn& guez et al. / Journal of Crystal Growth 220 (2000) 130}134
4. Conclusions Surface kinetics analysis using the `growth cella concept allows a comparation of attachment probabilities along all crystallographic directions and, as a consequence, the resulting crystallization surface microtopography. The results shown in Fig. 5 demonstrate that this analysis reproduces the birth and spread of steps in the 10 1 02 direction on the (1 0 1) surface of concanavalin A crystals, in accordance with the assumption of Kashimoto et al.
Fig. 5. View of (1 0 1) surface along the 10 0 12 direction. The numbers represent the sequence order of attachment of tetramers to the 2-d nucleus.
IV, V, VI and I, II, IV sites in Fig. 3, respectively. In Fig. 5 is shown the speci"c case where all attachment probabilities have the same values. The terrace in the 10 1 02 direction exhibits a minimal advance velocity due to the low attachment probabilities for III, VII and VIII sites in Fig. 3. The terrace translation in this case must occur preferentially by means of classical attachment at one site-completion of the terrace step mechanism.
References [1] M. Kotrla, Computer Physics Communications 97 (1996) 82. [2] S.C. Ke, L.J. De Lucas, J.G. Harrison, Phys. D: Appl. Phys. 31 (1998) 1064. [3] D.D. Vvedensky, in: H.W.M. Salemink, M.D. Pashley (Eds.), Semiconductor Interfaces at the Sub-nanometer Scale, Kluwer Academic Publishers, Netherlands, 1993, pp. 45}55. [4] T.A. Land, J.J. De Yoreo, J.D. Lee, Surface Sci. 384 (1997) 136. [5] A.J. Malkin, Yu.G. Kuznetsov, A. Mc Pherson, Surface Sci. 393 (1997) 95. [6] E. Kashimoto, G. Sazaki, K. Hasegawa, T. Nakada, S. Miyashita, H. Komatsu, K. Sato, Y. Matsuura, H. Tanaka, J. Crystal Growth 186 (1998) 461.
Crystallization simulation in macromolecular crystals C. FalcoH n RodrmH guez *, F. FalcoH n RodrmH guez, S. Aguilera Morales, R. Mulet Genicio Departamento de Teorn& a de Funciones, Facultad de Matema& ticas y Computacio& n, Universidad de La Habana, La Habana, Cuba Laboratorio de Crecimiento de Cristales, Instituto de Materiales y Reactivos, Universidad de La Habana, La Habana, Cuba Departamento de Fn& sica, Universidad Cato& lica del Norte, Antofagasta, Chile Laboratorio de Superconductividad, Instituto de Materiales y Reactivos } Facultad de Fn& sica, Universidad de La Habana, La Habana, Cuba Received 12 January 2000; accepted 12 July 2000 Communicated by A. McPherson
Abstract In order to avoid the di$culties encountered in the crystallization computer simulation of macromolecular crystals, we introduce the so-called `growth cella that is, the simple geometric volume occupied by a structural growth unit (i.e., monomer, dimer, etc.) in the crystal. The number and orientation of bonds that are satis"ed by the surrounding ones are accounted in the surfaces of these growth cells. This construction allows us to compare the attachment probabilities of growth units in di!erent crystallographic orientations on macromolecular crystals surfaces. The application of our method to the recent experimental observation of the formation of steps in the 10 1 02 direction on the (1 0 1) concanavalin A crystal surface is presented. 2000 Published by Elsevier Science B.V. Keywords: Crystallization; Macromolecular; Simulation
1. Introduction Usually, Monte Carlo (MC) simulation of crystallization processes only take into account the binding energy per bond and the number of bonds to calculate the attachment, detachment and migration probabilities of the growth unit [1,2]. Due to this lack of crystallographic speci"city of MC and because the development of potentials for even simple systems is a time-consuming and usually * Corresponding author. E-mail addresses: [email protected] (C. FalcoH n RodrmH guez), [email protected] (F. FalcoH n RodrmH guez), saguiler@ socompa.ucn.cl (S. Aguilera Morales), mulet@!.oc.uh.cu (R. Mulet Genicio).
empirical task in molecular dynamics (MD), both methods cannot be used to address molecular-level e!ects that are too speci"c. In this sense, `ab initioa calculations constitute a complementary approach which can improve our understanding of processes on the microscopic level, although applications to dynamical processes are still very limited [1,3]. On the other hand, the real images observed by means of SEM, AFM and other techniques [4,5] evidences a `growth near equilibriuma process, where surface di!usion cause the redistribution and smoothing of solid}liquid interface, far away from the simple `solid-on-solida (SOS) model, generally used in these simulations. In addition, the measurements of the step current, combined with the estimate values for the di!usion length, strongly
0022-0248/00/$ - see front matter 2000 Published by Elsevier Science B.V. PII: S 0 0 2 2 - 0 2 4 8 ( 0 0 ) 0 0 6 1 8 - 7
C. Falco& n Rodrn& guez et al. / Journal of Crystal Growth 220 (2000) 130}134
131
indicate that surface di!usion rather than bulk di!usion is the controlling mechanism of solute transport to the steps in these systems [4]. Undoubtedly, the big di!erences commonly found between the simulated surface structures obtained by MC and MD and the real images observed by SEM, AFM and other techniques, prove that these calculations must be improved in the search of a better agreement of the simulation results with the experimental "ndings. This improvement of the simulation may include the consideration of a large number of parameters, among these:
surface crystallization kinetics becomes di$cult. In order to avoid these di$culties, we introduce the so-called `growth cella that is, the simple geometric volume occupied by a structural growth unit (i.e., monomer, dimer, etc.) in the crystal. The number and approximate location of bonds that are satis"ed by surrounding units are accounted for in the surfaces of these growth cells. This construction allows us to compare the attachment probabilities of growth units in di!erent crystallographic orientations in a group of macromolecular crystal surfaces.
(a) the electrostatic attraction on the growth unit in solution exerted by the di!erent concentrations and orientations of unsatis"ed bonds at crystal surface, (b) the solid angle of the solution on which the previously cited electrostatic attraction exerts in#uence, determining in this way the available number of growth units for attachment, (c) the correspondence in number and distribution of unsatis"ed bonds between the growth unit and the attachment site, (d) the steric impedance for attachment of the growth unit at the selected site, (e) the terrace width, when the selected sites are found in the border of a step.
3. Formation of steps on concanavalin A (0 0 1) crystal surface
To elaborate an algorithm considering all the above-mentioned parameters is a very di$cult task and the computation time involved in the calculation would probably be prohibitive. Thus, one should disregard, based on physical intuition, those parameters with minor in#uence in a given problem. This approach in systems where the binding energies per bond are nearly the same makes it possible to substitute usual Monte Carlo simulations by simpler Celular Automata Models.
2. Growth cell In the case of macromolecular crystals, due to the structural complexity of the growth units, the analysis of the in#uence of each parameter on the
Kashimoto et al. [6] have proposed the steric hindrance for attachment in the 10 1 02 direction as the cause of formation of steps on the (1 0 1) crystal surface. Following these authors, we assume the dimer as the growth unit but due to the high number of inter-dimer bonds (32 bonds), we also considered that the formation of tetramers occurs immediately after the attachment of a dimer in the crystal surface. In Fig. 1(a) is shown the growth cell (dimer) of a concanavalin A crystal prevailing in the experimental growth conditions studied by Kashimoto et al. and in Fig. 1(b) the cell (tetramer) created by the immediate union of two dimers. In Fig. 2(a) is shown the distribution of unsatis"ed bonds per dimer in the #at (1 0 1) crystal surface and in Fig. 2(b) we can observe the correct orientation that must take the dimers in the solution near the liquid}solid interface in order to attach to this surface. In the subsequent analysis, we consider that the free dimers in solution ful"ll a succesion of actions in order to attach to the surface. They are: (i) approximation to the crystal, due to the attraction exerted between the unsatis"ed bonds on the surface and on the free dimer, (ii) rotation of the free dimer, in order to accomplish the maximal correspondence between its and surface unsatis"ed bonds, and (iii) migration to a more energetically favorable site.
132
C. Falco& n Rodrn& guez et al. / Journal of Crystal Growth 220 (2000) 130}134
Fig. 1. Growth cells of a concanavalin A crystal } (a): dimer; (b): tetramer, created by the union of two dimers.
Fig. 3. Attached tetramer on the #at (1 0 1) crystal surface. Fig. 2. (a): Flat (1 0 1) surface of concanavalin A crystal; (b): dimer correctly oriented to attach to this surface.
Considering that the initial attraction on a dimer in solution exerted by the (1 0 1) surface must occur preferentially between the six unsatis"ed bonds localized near the edges of the normal surfaces to 10 1 02 of the free dimer and the six unsatis"ed bonds localized near the convex edge between (1 1 0), and (1 1 0) surfaces of a dimer step in the #at (1 0 1) crystal surface, we conclude that the position with highest attachment probability must be the I-denoted, less probably appear the II-denoted position, and with minimal attachment probability appear the III-denoted.
Immediately after the attachment of a dimer in the I-position the attachment of a second dimer must occur in order to complete a tetramer as shown in Fig. 3. This represents the "rst stage in the formation of a 2-d nucleus, in accordance with the birth-and-spread growth model. This tetramer, on the #at (1 0 1) surface, appears as a triangular pyramid bounded shaped by (1 1 0), (1 1 0) and (0 0 1) planes, with height"51.5 As . We may analyze the attachment probabilities on each of the eight di!erent neighbors of the 2-d nucleus and, as a consequence, the most probable attachment order. Due to their greater altitude, the most active must be the groups of six and four unsatis"ed
C. Falco& n Rodrn& guez et al. / Journal of Crystal Growth 220 (2000) 130}134
bonds near the pyramidal vertex. In addition, taking into account the bond correspondence criterion, the dimers attracted by the high bond density edge formed by (1 1 0) and (1 1 0) 2-d nucleus surfaces must preferently adsorb to the (1 1 0), to the detriment of the (1 1 0) surface, and later attach to the most energetically favorable IV-denoted site. Less probable becomes attachment to the I-denoted site because of the larger migration jump that it implies, and still less the attachment to the II-denoted site, for which a very large migration jump is necessary without rotation to accomplish inter-dimer binding. After the occupation of the IVdenoted site, the I-denoted becomes the most probable site for attachment of a third tetramer. The second stage in this analysis corresponds to the structure shown in Fig. 4, after the attachment of a third tetramer in the most probable I-position. In this case, we can observe the appearance of a high density of unsatis"ed bonds at the interface of both tetramers. Again taking into account the necessary correspondence of bond distributions between the attachment site at the surface and the free dimer in solution, we conclude that the most probable neighbor position for attachment is the II-denoted, to the detriment of the III-denoted. On the other hand, in accordance with Land et al. [4], the attachment probability of position V in Fig. 3 is enhanced due to the duplication of the terrace width in the 11 0 02 direction. In accordance with Kashimoto et al. [6], the attachment probabilities of VI and VII positions
Fig. 4. First stage in the formation of a 2-d nucleus.
133
diminishes due to the steric hindrance caused by the restricted solid angle and the appropriate orientation the dimers must observe in order to remain "xed in both sites. Finally, the attachment probability of VIII-denoted sites, taking into account their relatively poor interaction with the large attractive convex edge between the (1 1 0) and (1 1 0) surfaces of the 2d-nucleus, must remain approximately the same as for the #at surface. As a consequence of this process a narrow terrace array in the 10 1 02 direction must arise that, takes into account the evidence for di!usion-"eld overlap on narrow terraces (j<=, where j is the surface di!usion length, = the terrace size). This implies that the step velocity diminution and homogenization [4] gives rise to stable step arrays on the (1 0 1) surface that may transform this surface to a new (n 1 n) surface where, accordingly with Kashimoto et al. [6], the value of n ranges from 7 to 11. From the previous discussion we developed, as a "rst approximation for concanavalin step formation, cellular automation in a bidimensional lattice. The lattice is initially empty (∀i, j h [i, j]"0), then a site (i, j) is randomly chosen and set to 1 (h [i, j]"1). This site mimics the dimer deposited in position I of Fig. 2. Then the attachment of the next dimer is simulated by the following rules: Rule 1: If h [i, j]"1 then h [i!1, j#1]"1 L L> and h [i#1, j#1]"1. L> Rule 2: If h [i, j]"1 and h [i!1, j#1]"1 then L L h [i!1, j]"1. L> Rule 3: If h [i, j]"1 and h [i!1, j]"1 then L L h [i!1, j!1]"1 L> Rule 1 simulates the deposition processes in positions IV and V of Fig. 3; Rule 2 represents the deposition of a tetramer in site I of Fig. 3, while Rule 3 represents the deposition of a tetramer in site II of Fig. 4. In Fig. 5 are shown the results of the application of this algorithm. It represents a view of the (1 0 1) surface along the 10 0 12 direction. The di!erent numbers represent the sequence order of attachment of tetramers at the corresponding sites. Advance directions of I and II terraces actually depend on the attachment probability relations for
134
C. Falco& n Rodrn& guez et al. / Journal of Crystal Growth 220 (2000) 130}134
4. Conclusions Surface kinetics analysis using the `growth cella concept allows a comparation of attachment probabilities along all crystallographic directions and, as a consequence, the resulting crystallization surface microtopography. The results shown in Fig. 5 demonstrate that this analysis reproduces the birth and spread of steps in the 10 1 02 direction on the (1 0 1) surface of concanavalin A crystals, in accordance with the assumption of Kashimoto et al.
Fig. 5. View of (1 0 1) surface along the 10 0 12 direction. The numbers represent the sequence order of attachment of tetramers to the 2-d nucleus.
IV, V, VI and I, II, IV sites in Fig. 3, respectively. In Fig. 5 is shown the speci"c case where all attachment probabilities have the same values. The terrace in the 10 1 02 direction exhibits a minimal advance velocity due to the low attachment probabilities for III, VII and VIII sites in Fig. 3. The terrace translation in this case must occur preferentially by means of classical attachment at one site-completion of the terrace step mechanism.
References [1] M. Kotrla, Computer Physics Communications 97 (1996) 82. [2] S.C. Ke, L.J. De Lucas, J.G. Harrison, Phys. D: Appl. Phys. 31 (1998) 1064. [3] D.D. Vvedensky, in: H.W.M. Salemink, M.D. Pashley (Eds.), Semiconductor Interfaces at the Sub-nanometer Scale, Kluwer Academic Publishers, Netherlands, 1993, pp. 45}55. [4] T.A. Land, J.J. De Yoreo, J.D. Lee, Surface Sci. 384 (1997) 136. [5] A.J. Malkin, Yu.G. Kuznetsov, A. Mc Pherson, Surface Sci. 393 (1997) 95. [6] E. Kashimoto, G. Sazaki, K. Hasegawa, T. Nakada, S. Miyashita, H. Komatsu, K. Sato, Y. Matsuura, H. Tanaka, J. Crystal Growth 186 (1998) 461.