Chemical Physics 369 (2010) 122–125

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Chemical Physics

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Change in optimum genetic algorithm solution with changing banddiscontinuities and band widths of electrically conducting copolymers

Avneet Kaur, A.K. Bakhshi *

Department of Chemistry, University of Delhi, Delhi 110 007, India

a r t i c l e i n f o

Article history:Received 9 January 2010In final form 13 March 2010Available online 17 March 2010

Keywords:CopolymersElectronic propertiesGenetic algorithmTheoretical chemistryArtificial IntelligenceConducting polymers

0301-0104/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.chemphys.2010.03.015

* Corresponding author. Tel.: +91 9818352820; faxE-mail address: akbakhshi2000@yahoo.com (A.K. B

a b s t r a c t

The interest in copolymers stems from the fact that they present interesting electronic and optical prop-erties leading to a variety of technological applications. In order to get a suitable copolymer for a specificapplication, genetic algorithm (GA) along with negative factor counting (NFC) method has recently beenused. In this paper, we study the effect of change in the ratio of conduction band discontinuity to valenceband discontinuity (DEc/DEv) on the optimum solution obtained from GA for model binary copolymers.The effect of varying bandwidths on the optimum GA solution is also investigated. The obtained resultsshow that the optimum solution changes with varying parameters like band discontinuity and bandwidth of constituent homopolymers. As the ratio DEc/DEv increases, band gap of optimum solutiondecreases. With increasing band widths of constituent homopolymers, the optimum solution tends tobe dependent on the component with higher band gap.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

The band gap of a polymer determines its electrical and opticalproperties, therefore design and synthesis of low band-gap poly-mers have been an area of intense interest in recent years. Thesematerials are finding several technological applications includingelectromagnetic shielding, electrostatic charge dissipation, opto-electronic displays and sensors. Several ways have been developedto synthesize low band-gap polymers [1]; one of the most success-ful approaches is copolymerization [2–5]. There are many conduct-ing polymers, such as polyaniline, polyacetylene, polythiopheneand polypyrrole. Of all known conducting polymers, choosingtwo polymers with apt concentrations to make a copolymer withspecific properties is a daunting task [6]. To overcome this prob-lem, genetic algorithm (GA) has been used recently [7,8]. Geneticalgorithm uses evolution operations (crossover and mutation)[9,10] to transform a population of data objects, each with individ-ual fitness values, into a new population with higher average fit-ness values thereby generating a solution (optimized) with goodfitness. GA has been shown to be a general optimization techniquewith applications in engineering design [11–15]. In the presentstudy, GA generates automatic solutions of the optimum relativeconcentration of constituent homopolymers in binary, ternaryand quaternary copolymers presenting some pre-specified proper-ties. GA is combined with NFC and DOS analysis to predict elec-tronic structure and conduction properties of a copolymer.

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: +91 11 27662504.akhshi).

The two fundamental electronic properties, the conductionband discontinuity (DEc) and the valence band discontinuity(DEv), and their ratio DEc/DEv are known to control the electronicstructures and conduction properties of copolymers. The conduc-tion band discontinuity (DEc) is defined as the difference in themagnitude of electronic affinities of the two components whilethe valence band discontinuity (DEv) is equal to the correspondingdifference in the ionization potentials. In this paper, we investigatethe effect of the change in DEc/DEv ratio on the GA solution. Thecopolymers, on the basis of the band alignment of constituenthomopolymers may belong to any one of the four different classesof quasi-one-dimensional super-lattices viz; Type-I, Type-II stag-gered, Type-II misaligned and Type-III [16]. For the purpose ofthe present study six different model systems of copolymers wereseen, all of which differ from one another in respect of both valenceand conduction band discontinuities and hence the ratio DEc/DEv

differs, but the class of copolymers to which these systems belongi.e. Type-II staggered, remains the same. The model systems stud-ied are very much related to the real systems given as referencesabove. The advantage of studying model systems lies in the factthat they help in predicting the trend of various properties therebyproviding an insight to the parameters to be taken care of whilesynthesizing novel polymers.

Band width of a polymer is known to be a measure of the mobil-ity of the charge carriers produced as a result of p-doping or n-dop-ing. Herein, we investigate the effect of change of valence andconduction bandwidths of the homopolymers on the optimumsolution (obtained from GA) of model copolymers AxB100�x belong-ing to the class of Type-II staggered. The systems investigated are

A. Kaur, A.K. Bakhshi / Chemical Physics 369 (2010) 122–125 123

different from one another in respect of band widths of both va-lence and conduction bands of constituent homopolymers (A)x

and (B)x. The trends observed in the electronic properties of thecopolymers are reported as a function of bandwidth of constituenthomopolymers.

2. Methodology

In the GA technique, one finds optimum relative concentrationfor binary copolymers presenting minimum gap value and maxi-mum electronic delocalization. To describe the electronic structureof our polymeric chains we are using a Linear Combination ofAtomic Orbital (LCAO) approximation:

wpj¼Xn

r¼1

cjrur ð1Þ

where wpj is the jth molecular orbital, ur is the atomic orbital for therth atom, and Cjr is the coefficient of the rth atomic orbital in the jthmolecular orbital, with r varying from one to number of atomicorbitals (n).

The electronic density of states is obtained by solving the Hüc-kel determinant (tight binding) of the copolymer chain consistingof N units:

HðkÞ ¼

a1 � k b12 0 . . . 0b21 a2 � k b23 . . . 00 b32 a3 � k b34 00 0 b43 . . . bNðN�1Þ

0 0 0 . . . aN � k

������������

������������

¼ 0 ð2Þ

where a’s and b’s are the usual Coulomb and hopping integrals andk is the eigenvalue. Now our main task is to calculate the eigen-values (or the energy values) so that we can calculate the bandgap of the copolymer. Negative eigenvalue theorem [17] is an effi-cient method of computing the distribution of eigenvalues of a realsymmetric matrix. Once obtained the eigenvalues, the eigenvectorsof interest can be obtained (one by one) through the use of inverseiteration method (IIM), and consequently the Inverse ParticipationNumber (IPN) [18]. The IPN (Ij for a particular orbital) is a measureof the level of delocalization of a molecular orbital:

Generate population of five chromosomes

Generate the polymeric chain for each chromosome

Generate the Hückel matrix

Calculate band gap and IPN

Fig. 1. Overview of steps involved in GA

Ij ¼Pn

r¼1jCjr j4Pn

r¼1jCjr j2� �2 ð3Þ

The gap value of a copolymer is obtained from the energy differencebetween the lowest unoccupied molecular orbital (LUMO) andhighest occupied molecular orbital (HOMO) and IPN is obtainedfrom expansion coefficients of HOMO. Once the gap and IPN valuesare obtained the fitness of a particular combination of homopoly-mers is calculated using the fitness function [f (x)]:

f ðxÞ ¼ 1ð1=qÞgapþ IPN

ð4Þ

where q is the energy difference between the higher LUMO andlower HOMO out of the constituent homopolymers. The steps in-volved in GA and evaluation of fitness function are shown in Fig. 1.

3. Results and discussion

In the present calculation of the density of electronic states, wehave consistently used a chain length of 300 units and an energygrid size of 0.001 eV. Various parameters used in GA and NFCmethod are given in our recent paper [18,19]. It is evident fromFig. 2 that both the homopolymers, (A)x and (B)x, constituting thecopolymers have the same values of band gap (Eg) in all the six sys-tems. Also the values of conduction and valence band width ofboth the homopolymers (A)x and (B)x remain the same in all thesix systems of copolymers. However, these systems differ fromone another with respect to the band alignments of both valenceand conduction band of homopolymer (A)x. Consequently, the val-ues of both the conduction band discontinuity (DEc) and the va-lence band discontinuity (DEv) and hence their ratio DEc/DEv isdifferent in each of these six systems of copolymers. For example,the magnitude of the ratio DEc/DEv in system 2a is 0.20 eV, how-ever for system 2b its value is 0.33 eV and becomes 0.50 eV in sys-tem 2c and so on. It, therefore, follows that as we go from system2a to system 2f, both the valence and conduction band discontinu-ity and also their ratio DEc/DEv increase. The results (Table 1) showthat the optimal solution obtained from GA varies with banddiscontinuities in case of Type-II staggered system. The solutionfrom GA for discontinuity ratio of 0.20 and 0.33 is A20B80 whereas

Evaluate fitness

Test Convergence

STOP

Selection +

Reproduction

YES

NO

and evaluation of fitness function.

(a) (b)

(c) (d)

(e) (f)

(B)x

-2.0

-8.0

-2.5

(A)x

-9.5

(B)x

-2.0

-8.0

-4.25

(A)x

-11.25

(B)x

-2.0

-8.0

-4.0

(A)x

-11.0

-2.0

-8.0

-3.5

(A)x

-10.5

(B)x (B)x

-2.0

-8.0

-3.0

(A)x

-10.0

-2.0

-8.0

-2.25

(A)x

-9.25

(B)x

Fig. 2. Valence and conduction band alignments of model constituent homopoly-mers in six different systems of copolymers belonging to the category of Type-IIstaggered. All values in eV.

Table 1Band gap and IPN values for optimal solutions obtained from GA for systems 2a–2fwith varying band discontinuities.

System DEc/DEv GA solution IPN Band gap (eV)

2a 0.20 A20B80 0.007876 5.81072b 0.33 A20B80 0.007879 5.58972c 0.50 A13B87 0.006431 5.11682d 0.60 A13B87 0.006431 4.62942e 0.67 A13B87 0.006431 4.13702f 0.69 A13B87 0.006432 3.8890

124 A. Kaur, A.K. Bakhshi / Chemical Physics 369 (2010) 122–125

for ratios of 0.50, 0.60, 0.67 and 0.69 the GA solution comes out tobe A13B87.

3.1. Discussion of DOS curves of systems 2a–2f (Fig. 3)

It is important to note that the energy positions of the peakscorresponding to the two homopolymers shifts regularly with in-crease in DEc/DEv ratio from system 2a to system 2f. This resultsdue to change in a (diagonal Hückel parameters) values withchange in band alignments of valence and conduction bands ofthe homopolymers from system 2a to system 2f. As a result theelectronic properties of these copolymers change with DEc/DEv ra-tio. The DOS distribution of these systems consists of relativelybroader regions of allowed energy states. The result is that theband gap decreases from system 2a to system 2f .

Table 2 gives the variation in optimum solution with varyingbandwidths. For the systems discussed in Table 2, the bandwidthsof the constituent homopolymers – their ionization potential (IP)(corresponding to the top of valence band), the electronic affinity(EA) (corresponding to the bottom of the conduction band) andthe band-gap (Eg) remains the same. The effect of changing bandwidths on optimal solution is studied for systems belonging tothree categories. Systems (Table 2a) belonging to first category

have the bandwidths varying for both components A and B by anequal amount. For systems in Table 2a, the band width values ofboth the valence and conduction bands of both the homopolymers(A)x and (B)x are increased by 0.5 eV simultaneously. For secondcategory of systems (Table 2b) the bandwidths of component Avary but that for component B remains the same. In the third cat-egory of systems (Table 2c) the bandwidths of component B varywhereas for component A remain the same. In the first category,when the bandwidths vary from 1.0 to 3.5 eV for valence band ofcomponent A, the GA solution values does not vary .i.e. the opti-mum solution remains the same as A13B87. For the second categorythe optimum solution varies from A13B87 to A34B66, while for thethird category again the optimal solution does not vary. This im-plies that changing the band width of the component with higherband gap (in this case A) changes the optimal solution. Whenbandwidth for both the components increase by equal amounts,the optimal solution is not affected as the effect due to increasein band width of A is compensated by increase in band width ofB. The gap and IPN values for the optimal solutions however showa trend. For the first category, with the increase in band widths, theband gap values of optimal solution of copolymer increases and theIPN almost remains constant. In the second category of systems,the band gap of the copolymers initially increases with the increasein band widths of component A. When the band width of VB andCB of A becomes 4.5 and 4.0 eV, respectively, A constitutes largerpercentage of the optimal solution. The band gap of the copolymerdecreases but the IPN increases drastically. For the systems belong-ing to third category, the band gap values of copolymers decreaseswith the increase in band width of component B. Type-I copolymerwas also studied and it was found that the GA solution does notvary with varying band discontinuities and bandwidths for thistype of copolymer.

4. Conclusion

The obtained results show that the optimum solution (obtainedfrom GA) changes with varying parameters like band discontinuityand band width of constituent homopolymers. As the ratio of con-duction band discontinuity to valence band discontinuity (DEc/DEv) increases, the band gap of optimum solution decreases. Thisis due to decrease in the energy difference between the lowerLUMO and higher HOMO out of the constituent homopolymers.The optimum solution contains lesser percentage of A withincreasing discontinuity ratio. This results in increasing both theintrinsic conductivity and dopantphilicity of the copolymer. Sys-tems 2a and 2b have almost same value of IPN and systems 2c–2f have similar values. This implies that IPN is not generally af-fected by changing discontinuity. With increasing band widths ofconstituent homopolymers, the optimum solution tends to bedependent on the component with higher band gap.

The trends obtained in the GA solution of the copolymers as afunction of the band discontinuity and band width are very usefulfor designing copolymers with desired electronic and conductionproperties.

Recently, Ant Algorithm has also been used to get optimized re-sults for the copolymers [19]. Our group has done a comparativestudy of Genetic and Ant algorithms applied to polymeric super-lattices [20]. The obtained results show that both the algorithmspresented are able to give an optimum solution to the problemof designing conducting polymers. In GA, very good solutions aregenerated by probing less than 3% of the solution space as on anaverage 30 generations consisting of five individuals each are re-quired to find an optimum solution. The percentage of solutionspace probed in AA is more than GA but both give same resultsfor Type-I systems. Hence, extensive studies are required to judge

0

1

2

3

4

5

6

7

012345678

-15 -10 -5 0 5

E (eV)-15 -10 -5 0 5

E (eV)

-15 -10 -5 0 5

E (eV)-15 -10 -5 0 5

E (eV)

-15 -10 -5 0 5

E (eV)-15 -10 -5 0 5

E (eV)

0

2

4

6

8

10

0

2

4

6

8

10

NN

N

NN

N

0

2

4

6

8

10

0

2

4

6

8

10

Syst

em

2aSy

stem

2b

Syst

em

2c

Syst

em

2dSy

stem

2e

Syst

em

2f

Fig. 3. DOS curves of GA solution of systems 2a–2f. Energy in eV and number of states N in relative units.

Table 2Band gap and IPN values for optimal solutions obtained from GA for systems withvarying band widths of (a) both A and B, (b) component A only, (c) component B only.

System Band widths GAsolution

IPN Band gap(eV)

VB CB

A B A B

(a)1 1.0 1.5 0.5 1.0 A13B87 0.006434 5.02882 1.5 2.0 1.0 1.5 A13B87 0.006433 5.05483 2.0 2.5 1.5 2.0 A13B87 0.006431 5.07784 2.5 3.0 2.0 2.5 A13B87 0.006430 5.09885 3.0 3.5 2.5 3.0 A13B87 0.006428 5.1168

(b)1 3.0 3.5 2.5 3.0 A13B87 0.006428 5.11682 3.5 3.5 3.0 3.0 A13B87 0.006427 5.13933 4.0 3.5 3.5 3.0 A13B87 0.006425 5.16134 4.5 3.5 4.0 3.0 A34B66 0.015888 5.0954

(c)1 3.00 3.00 2.50 2.50 A13B87 0.006428 5.12282 3.00 3.50 2.50 3.00 A13B87 0.006428 5.11683 3.00 4.00 2.50 3.50 A13B87 0.006428 5.11084 3.00 4.75 2.50 4.25 A13B87 0.006427 5.0988

A. Kaur, A.K. Bakhshi / Chemical Physics 369 (2010) 122–125 125

very firmly which algorithm is better. For this purpose, we arestudying different types of polymeric systems like Type-II stag-gered and Type-II misaligned.

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