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A Variable-Number Genetic Algorithm for Growth of 1-Dimensional Nanostructures into Their Global Minimum Configuration Under Radial Confinement

The versatility of genetic algorithms for determining the atomic structure of clusters has been well established starting with the seminal article of Deaven and Ho (Physical Review Letters 75, 288, 1995). The genetic algorithm approach has also been
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   PLEASE SCROLL DOWN FOR ARTICLE This article was downloaded by: [Indian Institute of Technology Kharagpur]  On: 11 March 2009  Access details: Access Details: [subscription number 901727275]  Publisher Taylor & Francis  Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK Materials and Manufacturing Processes Publication details, including instructions for authors and subscription information: A Variable-Number Genetic Algorithm for Growth of 1-DimensionalNanostructures into Their Global Minimum Configuration Under RadialConfinement T. E. B. Davies a ; D. P. Mehta b ; J. L. Rodríguez-López c ; G. H. Gilmer ad ; C. V. Ciobanu aa Division of Engineering, Colorado School of Mines, Golden, Colorado, USA b Department of Mathematicaland Computer Sciences, Colorado School of Mines, Golden, Colorado, USA c Advanced MaterialsDepartment, Instituto Potosino de Investigación Cientifica y Tecnológica, San Luis Potosí, San Luis Potosí,Mexico d Chemistry, Materials, and Life Sciences Directorate, Lawrence Livermore National Laboratory,Livermore, California, USAOnline Publication Date: 01 March 2009 To cite this Article Davies, T. E. B., Mehta, D. P., Rodríguez-López, J. L., Gilmer, G. H. and Ciobanu, C. V.(2009)'A Variable-NumberGenetic Algorithm for Growth of 1-Dimensional Nanostructures into Their Global Minimum Configuration Under RadialConfinement',Materials and Manufacturing Processes,24:3,265 — 273 To link to this Article: DOI: 10.1080/10426910802678172 URL: Full terms and conditions of use: article may be used for research, teaching and private study purposes. Any substantial orsystematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply ordistribution in any form to anyone is expressly forbidden.The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directlyor indirectly in connection with or arising out of the use of this material.   Materials and Manufacturing Processes , 24: 265–273, 2009Copyright © Taylor & Francis Group, LLCISSN: 1042-6914 print/1532-2475 onlineDOI: 10.1080/10426910802678172 A Variable-Number Genetic Algorithm for Growthof 1-Dimensional Nanostructures into TheirGlobal Minimum Configuration Under Radial Confinement T. E. B. Davies 1 , D. P. Mehta 2 , J. L. Rodríguez-López 3 , G. H. Gilmer 1  4 , and C. V. Ciobanu 1 1  Division of Engineering, Colorado School of Mines, Golden, Colorado, USA 2  Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado, USA 3  Advanced Materials Department, Instituto Potosino de Investigación Cientifica y Tecnológica,San Luis Potosí, San Luis Potosí, Mexico 4 Chemistry, Materials, and Life Sciences Directorate, Lawrence Livermore National Laboratory, Livermore, California, USA The versatility of genetic algorithms for determining the atomic structure of clusters has been well established starting with the seminalarticle of Deaven and Ho (Physical Review Letters 75, 288, 1995). The genetic algorithm approach has also been extended to spatially periodic structures where it has been recognized that the variation of the number of atoms enables, or at least facilitates, the convergence of the globalstructural search. Here, we present another application of genetic algorithms: we show that a real space, variable-number algorithm can be used toretrieve simultaneously the lowest-energy structure and the optimal number of atoms of 1-dimensional (1-D) nanostructures subjected to desiredconditions of radial confinement, starting from a single atom in the periodic unit cell. This algorithm is based on two-parent crossover operationsand zero-penalty “mutations,” the latter of which allowing for the algorithm to evolve even from a genetic pool made of identical structures.We show that a sufficiently rich set of crossover operations (attempted with equal probability) can make the procedure effective for finding theatomic structure very different 1-D nanomaterials. We test the algorithm for carbon nanotubes and for Lennard-Jones nanotubes using runs wherecrossovers are applied either individually or in combination. By analyzing the acceptance probabilities of the structures created in these runs, wediscuss the performance of the algorithm and possibilities for improvement. Keywords Carbon nanotubes; Genetic algorithms; Global optimization; Lennard-Jones; Nanowires. 1. Introduction One-dimensional (1-D) nanostructures presently showtremendous technological promise due to their noveland potentially useful material properties. The continuousminiaturization of electronics industry has achieved the limitin which the interconnection of devices in a reliableway is particularly challenging. Efforts are underwayfor synthesizing nanowires for specific molecular andnanoelectronics applications [1]. Such wires, possibly dopedor functionalized, can operate both as nanoscale devicesand as interconnects [2]. While remarkable progress hasbeen achieved in terms of preparation and characterizationof new 1-D materials [3–7], atomic-level knowledge of thestructure remains necessary for a complete understanding of usefulness of these nanowires for the device applications.Predictions of the structure of nanowires may be atpresent affected by the lack of robust methodologies(i.e., search algorithms coupled with model interactions)for searching the configuration space, and most studies todate rely on heuristically proposed structures as startingpoint for further stability studies at the ab-initio level[8–12]. Recently however, genetic algorithms coupled with Received July 27, 2008; Accepted November 18, 2008Address correspondence to C. V. Ciobanu, Division of Engineering,Colorado School of Mines, Golden, CO 80401, USA; empirical interactions potentials have made their way intothe field of predicting nanowire and nanotube structures[13–16].The application of genetic algorithms (GA) forstructural predictions of condensed-matter systems hasbeen introduced by Deaven and Ho, who showed thatthe lowest-energy structure of the C 60 cluster can beretrieved using a real-space genetic algorithm [17]. Most of the subsequent developments of genetic algorithms haveoccurred in area of structure optimization for clusters ;these developments include, for example, extended compactgenetic algorithms [22], differential evolution [23], andparticle swarm algorithms [23, 24]. Interestingly however,recent extensions of genetic algorithms to 2-D and 3-Dperiodic systems have proven very versatile for findingsurface reconstructions [19, 20] and predicting crystalstructures and polymorphs [21]. The key ingredient forthe efficient  use of GAs in the global optimization andstructure prediction for 2-D and 3-D systems is the provisionthat the number of atoms is allowed to vary. Such provision,which is necessarily absent in the case of clusters, facilitatesoptimization pathways that span systems with differentnumbers of atoms thus providing fast routes towards theglobal minimum of the appropriate fitness function. To ourknowledge, the use of  variable-number genetic algorithms for the global optimization of 1-D nanostructure has notbeen reported so far, as the applications of GAs for 1-D265  D o w nl o ad ed  B y : [ I ndi a n  I n s ti t u t e  of  T e ch n ol o g y  Kh a r a g p u r]  A t : 03 :00 11  M a r ch 2009  266 T. E. B. DAVIES ET AL. systems used constant numbers of atoms in the periodic cell[13–16].The purpose of this article is to present a novel applicationof genetic algorithms, namely, their use for finding thestructure of a 1-D nanotube via simulated growth . Becausethe simulation of growth is envisioned, we choose a variable-number  genetic algorithm which in the case of 2-D and 3-D periodic structures was shown to retrieve thecorrect global minimum of the relevant energetic quantity,i.e., surface energy in 2-D [19, 20], and cohesion energyper particle in 3-D [21]. The 1-D nanotubes investigatedhere are started with very few atoms in the periodic cell.The growth of the nanostructures takes off and proceeds solely through crossover operations, and stops when theoptimal structure (i.e., that with lowest energy per particle)for the given confinement conditions is found. As such,the growth is not a reflection of the kinetic processes thatoccur in actual synthesis experiments, but it is rather adifferent way to seek the optimal nanostructure that can besynthesized under the prescribed confinement conditions.The organization of the article is as follows. Section 2describes the details of the algorithm, in particular theselection criteria and the types of crossover operationsused in the genetic evolution. Our results for carbonnanotube (CNT) and Lennard-Jones (LJ) nanotube systemsare presented in Section 3, where we bring evidence thatthe determination of the optimal number of atoms andtheir locations during any given GA run are intrinsicallyrelated to one another. In Section 3 we also show that runsbased on a single crossover type are not always successful;we have found that a set of several crossovers attemptedwith equal probability make the algorithm more robust andapplicable for determining the optimal structure of verydifferent materials systems. We further discuss our resultsin Section 4, where we show how the acceptance probabilityof the operations in single-crossover GA runs varies duringthe evolution. Our results are summarized in Section 5,where we also give a brief account of possible extensionsof this work. 2. Description of the algorithm Nanotubes are simulated using periodic boundaryconditions along their axis [28] and spatial confinement inthe radial direction. The spatial period along the nanotubeaxis is kept constant, and the radial confinement is achievedby placing hard repulsive walls at two desired valuesof the radius [29]. 1 The nanotubes will be formed betweenthese two cylindrical walls, whose sole purpose is to keepthe system from expanding outside a desired cylindricalannulus. Our test systems are carbon atoms modeled viathe Tersoff potential [18], and LJ particles [30] with theparameters   = 1  5Å and  = 2  0eV. To increase the speedof the calculations, the range of the LJ potentials is truncatedat r  c = 2  7Å; this value has proved to be sufficiently large 1 The relative positions of the two repulsive walls are chosen such thatthe the interaction potential exerted by the wall does not significantlyaffect the energy of the nanotube created in between the two wall. Therepulsion potential of each wall has a form similar to that given in [29]. that the structures retrieved by our GA runs do not changeif the cutoff is increased further. Next, we give a descriptionof the algorithm that focuses on the genetic operationsused. For the systems considered here, “Generation Zero”(the starting set of structures) consists in a genetic pool of  p = 40 configurations, each member having a small numberof atoms placed at random locations inside a cylindricalannulus. Selection. The selection is based on a single fitnessfunction, namely, the potential energy per particle . A newstructure (i.e., the result of a crossover operation) is includedin the genetic pool if its energy is smaller than the highestenergy among the structures already present and  if it isno closer than  = 0  001eV to the energy of any one of the pool structures that has the same number of atoms.This means that a new structure is allowed in the pooldespite having an energy close to that of another structure  provided  that it has a different number of atoms. Allowingstructures with same energy but different number of atomsto enter the genetic pool helps to get the growth processstarted when the initial population members has one or veryfew atoms. Indeed, when particles are far apart and do notinteract, allowing a child with an larger number of atoms(that are still far enough that interactions are negligible)will increase the particle density and help the growth of the structure. The selection process that we use preventsthe duplication of members in the pool (whose size is keptat p = 40), duplication which often keeps the algorithmfrom retrieving the global minimum of the fitness function.For this GA procedure we impose no restriction on thenumber of atoms of any newly created structure, with theobvious exception that a structure with zero atoms is notallowed; the reason for leaving the number of atoms variableis that we want the optimal number of atoms to be found atthe same time as the optimal atomic structure. Admittedly,we do not know beforehand that the number of atoms wouldactually converge. The choice for a variable-number GA isonly justified a posteriori, by the results obtained. Crossovers. A crossover (which we often also call a move ,for simplicity) is an operation carried out by splicing tworandomly chosen (parent) structures from the genetic poolin order to create a new (child) structure [31]. The newstructure is considered for inclusion in the pool as explainedin the paragraph above. There are two types of crossoveroperations that we employ in this study. The first type,which we call sine crossovers requires cutting the parentsalong sinusoidal lines that are compatible with the periodicboundary conditions along the axis of the tube and withrespect to the angular coordinate. This procedure is adaptedfrom recent work in 3-D crystal structure prediction [21]which showed that, at least for 3-D periodic systems, thereal-space GA is more efficient when using cutting functionsthat obey the periodic boundary conditions.We have used three specific sine crossovers. In thesecrossovers, sine functions of randomly chosen frequency,amplitude, and initial phase were used to create two (e.g., S 1 or S 2 in Fig. 1) or four domains ( S 3 in Fig. 1) on each of thetwo parent structures. In the case of  S 1 and S 2 , we assemble  D o w nl o ad ed  B y : [ I ndi a n  I n s ti t u t e  of  T e ch n ol o g y  Kh a r a g p u r]  A t : 03 :00 11  M a r ch 2009  A VARIABLE-NUMBER GENETIC ALGORITHM 267 Figure 1. —(Color online) The crossovers S 1 , S 2 , S 3 , P  1 , and P  2 used in thegenetic algorithm optimization of 1-D tubular nanostructures. The differentcolors on the cylindrical surface indicate the domains that are taken from eachparent to create a new structure. one domain from each parent to create the child. For S 1 ,there are two cutting functions, each of which is a sinefunction that closes on itself and has the polar angle  asvariable. Similarly, in the case of  S 2 , there are two cuttingfunctions that depend on the axial coordinate z and obey theperiodic boundary condition along the z axis. The move S 3 uses four sine functions in order to create four domains oneach parent (Fig. 1). The child structure created with the S 3 move combines three domains on one of the parents withone domain of the other parent. Given that the contributionof one parent is larger in the final structure, for certainvalues of the amplitudes, phases or frequencies the singledomain that comes from the second parent can be viewedmerely as a perturbation on the configuration of the firstparent. Therefore, while we do not have explicit mutations in the algorithm, the S 3 move can act like a mutation of thefirst parent when the domain that comes from the secondparent (darker shade in the S 3 viewgraph of Fig. 1) is verysmall.The second type of crossovers that we used is based onplanar cuts as in the pioneering work of Deaven and Ho [17].We considered cuts with planes of arbitrary orientations,which are obviously not compatible with the periodicboundary conditions (refer to panel P  1 in Fig. 1, in which theupper and lower part belong to different parent structuresand thus create a parent-domain boundary and the z -boundsof the supercell). We also use, separately, crossovers basedon planes parallel to the symmetry axis of the system, whichare compatible with the boundary conditions ( P  2 in Fig. 1).The sine and planar crossover types were used individuallyin GA runs, as well as in combinations. The combinationwe describe here is the one in which all moves in Fig. 1 areattempted with equal probability at any given point of thegenetic evolution.  Zero-Penalty Moves. In order to increase the diversity of the children that given parents can help create, we introducezero-penalty moves for one of the parents that enter inany crossover operation. These moves do not alter theenergy of that parent structure (hence the term zero-penalty ),neither do they change its physical structure: a zero-penaltymove simply changes the relative positioning of one of theparents relative to the other before a crossover betweenthe two is performed. The specific moves that we useas zero-penalty “mutations” are: (i) rotations around theaxis by an arbitrarily chosen angle, (ii) axial displacementsof the structure by random z values through the periodicboundary conditions, and (iii) 180  rotations around anarbitrary axis that is perpendicular  to the z axis. To clearlyreveal the effect of zero-penalty moves in creating diversityin the genetic pool, let us consider the extreme case of a “Generation Zero” that is made up only of  identical structures. With such a starting point for the GA run, anycrossover is bound to create a child structure identical tothe already existing members in the pool. However, if aparent is to be, e.g., rotated about its axis before entering ina crossover operation, then that parent appears as distinctfrom the rest of the genetic pool, and the new structurecreated is different from any other structure in the pool.Therefore, zero-penalty moves create the diversity necessaryto a successful GA optimization even in cases where suchdiversity is completely lacking. When sufficient diversityis already present, a zero-penalty move does not hurt theperformance of the algorithm because its computationalcost is insignificant (the energy of the structure is nevercomputed after such move).With these descriptions of the ingredients of thealgorithm, the procedural steps in our typical GA evolutionare:(a) Two parents are randomly picked from the genetic pool;(b) One of the parents is subjected to a zero-penalty movechosen with equal probability from the set (i)–(iii)described above;(c) After the zero-penalty move on the first parent,a crossover operation is performed with the (unchanged)second parent;(d) The resulting child is subject to a conjugate-gradientrelaxation into the nearest local minimum of thepotential energy, and its “fitness” (potential energy perparticle) is computed;(e) The child is selected for inclusion in the pool, orrejected;(f) If the child does end up in the pool, then the membersof the pool are sorted from the lowest to the highestenergy per particle. One can easily implement and testvariations of this procedure, which, for example, mayuse zero-penalty moves on both parents before theyenter into a crossover. The cycle (a)–(f) is repeated fora prescribed number of crossovers or until at least thelowest energy member of the pool has converged to anumber of atoms and a value of the fitness function(energy per atom). The results from GA runs basedon single crossover types (individual moves) and thosebased on all crossovers are presented next. 3. Results for prototype nanotubes We have tested the genetic algorithm for two systems,carbon and LJ systems subjected to radial confinementconditions as described in the previous section. The purposeis to find out if nanotubes can evolve via genetic operationsfrom structures that have one or few atoms in the periodicunit cell. To this end, we start the GA runs for the CNTsystems from genetic pools in which each member hasone single carbon atom in the periodic cell. We follow theevolution of GA runs in which only one type of crossover isemployed from the set S 1 S 2 S 3 P  1 P  2  shown in Fig. 1,as well as one GA run where all these operations areattempted with equal probability .  D o w nl o ad ed  B y : [ I ndi a n  I n s ti t u t e  of  T e ch n ol o g y  Kh a r a g p u r]  A t : 03 :00 11  M a r ch 2009  268 T. E. B. DAVIES ET AL. Figure 2. —(Color online) Evolution of (a) the lowest energy in the pooland (b) the average energy across the pool for CNT systems in separate runsperformed with only one type of crossover, as well as in a run performedwith all crossovers attempted with equal probability. The horizontal axis of each plot shows the number of crossovers (moves) attempted. Note that theGA runs based solely on sine operations could not  find the global minimumstructure within 10,000 crossovers. The evolution of the lowest energy in the pool for eachof the six GA runs (i.e., five runs based on a single typeof crossover, and one run with all equiprobable crossovers)is plotted in Fig. 2(a), and the average energy across thepool is shown in Fig. 2(b) for each run. We observe thatonly the runs based on planar cuts ( P  1 and P  2 ) are ableto find the optimal defect free CNT structure in less than10 4 operations. The run based on all operations finds thecorrect structure in less than 5000 moves (crossovers).Since the individual sine crossovers are slow in findingthe global optimum structure, the success of the all-moveGA run is most likely due to the planar crossovers. TheGA runs that are based on sine crossovers evolve theirbest structures to defective CNTs. A question remains as towhether performing longer runs with single crossovers of the sine type would eventually yield the correct defect-freetubular structures.The structure of the best member in the genetic poolfor the all-crossovers GA run on CNT systems is shownat selected time-points in the evolution in Fig. 3. Asspecified, the run started with a single carbon atom forevery member of the pool, but we show two unit cellsfor each frame to help the visualization through periodicboundary conditions in the z direction. The best structuregrows from one atom to a string of atoms that spans thelength of the periodic cell (frame labeled 700 in Fig. 3).A rather large number of crossovers has to be attemptedin order for the string to grow wider (frames 2000, 2200),i.e., into a strip of sp 2 -hybridized carbon atoms (graphene)at frame 2200. Planar crossovers performed with parentstructures that consist of graphene strips will likely leadto two flat strips, as shown in frame 2300. The two stripssubsequently coalesce at an angle (frame 2700 in Fig. 3),acquire more atoms and start curving onto a cylindricalsurface due to the confining potential walls (frame 3300).Planar mating of parent structures such as that shown inframe 3300 results in closing the circumference, as shownin frame 3600; this closing occurs with defects along thecrossover planes, but such defects are systematically weededout later on (see frames 4100 and 4300).We have also tested the GA for growing LJ nanotubesunder radial confinement. To save time at the initial phaseof particle accumulation, we have started the LJ runs with10 atoms in each member of the GA pool located at randompositions within the periodic unit cell. The evolution of the best energy in the pool for LJ systems (again, withindividual crossovers and with all crossovers) is plotted inFig. 4(a), and the average energy across the genetic pool isshown in Fig. 4(b). We note that for the LJ system eachone of the GA operations is able to retrieve the correctoptimum of the LJ nanotube compatible to the boundary Figure 3. —Evolution of lowest-energy carbon structure during a GA run performed with all crossover types. The system starts with only 1 atom for eachmember of the genetic pool, and evolves towards a defect-free CNT as the lowest-energy pool member. The outer confining wall is shown by the white dashline, and the “time” (i.e., the index of the crossover operation) is indicated by the number shown atop each frame. The number of atoms is shown at the bottomof the frames; for clarity, two periodic lengths in the z direction are displayed.  D o w nl o ad ed  B y : [ I ndi a n  I n s ti t u t e  of  T e ch n ol o g y  Kh a r a g p u r]  A t : 03 :00 11  M a r ch 2009
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