pubs.acs.org/jpcc Sharp Fickian Behavior of Electrogenerate Chemiluminescence Waves in Conjugate Polymer Films Davi A. Ewars* Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716-553, Unite States ABSTRACT: One technique to stuy electrochemical oxiation phenomena in thin polymer films is the generation of electrogenerate chemiluminescence (ECL waves. Such waves are sharp an easy to image, an recent experiments have shown both constant-spee an Fickian-style wave behavior. One way to moel such waves mathematically is to track the concentration of the ion clusters in the polymer film. One such moel has a stanar Fickian form but with a highly nonlinear iffusion coefficient. This moel is analyze numerically, an the results are compare with previous asymptotic analysis. The results emonstrate that ECL waves corresponing to this moel are inee sharp an move in a Fickian way. Hence more complicate effects must be inclue in the moel if constant-spee behavior is to be observe. 1. INTRODUCTION Electrochemical conjugate polymer films show great promise for avances in many technologies. These films can be use in electrochromic smart winows that control the level of transparency as voltage is applie. 1 Such films are use in many types of sensors, incluing those use for etecting chemicals, etermining humiity an ph, an mimicking human senses. They form the basis for organic light-emitting ioes (OLEDs, 3 which are use to create screens for small electronics an now even televisions. 4 Because oxiation an reuction can change the volume of polymer films, one can exploit this process to create electrochemical actuators 5 or even artificial muscles. 6 A robust knowlege of the behavior of the unerlying kinetics is crucial to controlling an optimizing such evices. One useful approach to stuying these evices is through electrogenerate chemiluminescence (ECL, 7 which allows optical stuy of the reuction an oxiation processes. A key feature of these processes is the formation of a sharp front in the concentration of ionic clusters, which causes bright patterns in ECL experiments. 8 This behavior has alreay been exploite inustrially for use in fingerprinting evices. 9 Given a source of clusters, this front propagates with time; hence these patterns are sometimes referre to as ECL waves or ECL solitons. 10,11 One key quantity to etermine is the spee at which these waves propagate. For many years, experiments have shown sharp fronts in such systems; 1 however, some experiments show the front moving proportional to t, as one woul expect from a purely iffusive process, 13 whereas others claim to see the front moving with constant spee, 14,15 as one might expect from incluing the effects of swelling an other changes in the polymer matrix. 16 18 Still others see the front moving in both ways epening on experimental parameters. 19 Most current moels for ECL systems inclue only iffusive effects, 0 although several have highly nonlinear iffusion coefficients. 1 Because Fickian moels allow similarity solutions epening on x/ t, it is unlikely that such moels can capture constant-spee behavior. 1 In Guo et al., 8 the authors stuy the oxiation of a thin poly(9,9-ioctylfluorene-co-benzothiaiazole film. The film is embee with an array of Au electroe posts that provie nucleation sites for ionic clusters that sprea through the film, causing the ECL waves. The authors claim that their numerical simulations show sharp fronts moving with constant spee; however, asymptotic analysis of the same moel shows that the sharp fronts move like t. We shall use a numerical scheme to verify those asymptotic results an explain the iscrepancy between the two groups.. RESULTS AND DISCUSSION.1. Governing Equations. The full escription of the experiment uner consieration is provie in Guo et al.; 8 below we summarize the etails relevant for analyzing the resulting system. The ECL wave is moele through the concentration C of ionic clusters in a thin ( 50 nm film mae of poly(9,9-ioctylfluorene-co-benzothiaiazole. The governing (imensionless equation is the following: = t DC ( C Dw DC ( = + D + κ C CT 1 e ( (1a (1b where the subscripts w an refer to wet an ry. Here x measures istance along the film an C T is a saturation Receive: January 15, 013 Revise: March 7, 013 Publishe: March 7, 013 013 American Chemical Society 6747 x.oi.org/10.101/jp400476m J. Phys. Chem. C 013, 117, 6747 6751
concentration, so we consier the region where C > C T to be wet. Initially, there are no clusters in the polymer, so Cx (, 0 = 0 ( In the experiments by Guo et al., 8 Au posts at x = ± 1 provie (the same constant concentration of clusters, which is higher than the saturation concentration. These bounary conitions are even in x, as are eqs 1a, 1b, an. Hence we may exploit this symmetry to take the interval as x [0,1] with (0, t = 0 C(1, t = 1, C < 1 T (3a (3b Note that the interval [0,1] is generic. If the posts were space L units apart, then we coul efine a new variable y = x/l an obtain our previous system with a scale value of D(C. With a iffusion coefficient of the form in eq 1b, making κ large causes a great change in the size of the iffusion coefficient as the polymer changes from ry to wet. Hence we woul like to examine the solution of eqs 1a an 1b for large κ. Moreover, D is usually quite small; in particular, Guo et al. inicate that D can be up to six orers of magnitue smaller than D w. 8 To reuce the number of perturbation parameters uner consieration, we take D D = * Dw κ (4 where D * is consiere to be O(1. Substituting eq 4 into eq 1b, we obtain 1 DC ( = D κ 1 + e w ( C C D + * T κ (5 which varies only with κ. A graph of D(C on a logarithmic scale is shown in Figure 1. The parameters chosen are similar to those in Guo et al. 8 to replicate the sharp ECL waves seen experimentally. 13 15,19 In particular, we choose C T = 0.95, D w = 0.1, D * = 0.01 (6 Note that with κ = 0, we have D constant. Note also that as κ increases the curve gets steeper an D gets smaller. In fact, for very large κ, one can think of D(C as being piecewise constant (at least to leaing orer... Asymptotic Solution. In Ewars, the author constructs the solution of this system for large κ; we summarize the results here. First, introuce the similarity variable 1 x ζ = Dt w (7 Note that if the solution can be shown to epen only on ζ (which we o below, then the isoclines of C (which correspon to ECL waves will move like t, as consistent with stanar Fickian behavior. In particular, we efine ζ = ζ s to be the isocline corresponing to the saturation value at leaing orer: C( ζ s = C T (8 In this formulation, there are five separate regions of interest, as shown in Figure. Note that three of these regions are very Figure. Schematic of various regions with ifferent behavior in the iffusion coefficient. Figure 1. D(C versus C on a logarithmic scale. Here C T = 0.95, D w = 0.1, an D * = 0.01. Dotte line: D(C given by eq 1b with κ = 0. Soli curves: D(C given by eq 5 with κ = 1, 10, an 100. Note that as κ increases, the curve steepens an the iffusion coefficient in the ry region ecreases. thin, epen on the size of κ, an exist aroun C = C T, which is where the iffusion coefficient changes rapily in Figure 1. Hence each region correspons to ifferent behaviors in eq 5: In Region I, the concentration is far enough above C T that the exponential term in the numerator of the first brackete term goes to zero. Hence the brackete term is 1, D(C D w, an the evolution procees as if the iffusion coefficient was constant. In Region II, the concentration is near enough to C T that the nonlinear term in eq 5 must be consiere but not the D * term. In Region III, both terms in eq 5 are of the same size. In Region IV, C is far enough below C T that D(C D, an the evolution procees as if the iffusion coefficient is constant. In particular, there is a sharp transition in the concentration between the saturation value an zero. In Region V, C is exponentially small, which means that the polymer can be consiere to be essentially ry. 6748 x.oi.org/10.101/jp400476m J. Phys. Chem. C 013, 117, 6747 6751
Because of the structure iscusse above, the similarityvariable solution is a goo approximation only when region V (with its exponentially small flux contains the bounary x = 0, where eq 3a is applie. Hence the similarity transformation will work only until t max =(4ζ s D w 1 ; after that the wet solution occupies the entire omain. Because of the ifferent regions, we must break the solution into two parts. In the ry region where ζ > ζ s, the solution is given by ζκ ζ ζ ζκ ζ ζ s ( s s ( s C( ζ = CT exp + D * D * log κ uc( ζ, ζ > ζs κ κ (9a where we must restrict the omain on ζ because the exponential term in eq 9a will iverge for ζ > ζ s. u c may be efine implicitly through u ζ κ ζ ζ * c Duc e = sct ( s + A (9b or written explicitly in terms of the Lambert W function. 3 Here A is a constant yet to be etermine. In the wet region, the solution is given by ζ κζ ζ ζ C( ζ = 1 (1 C erf sct( s 1 1 e T + log erf ζs κ κζsc T( ζs ζ log κ uc( ζ, ζ < ζs κ κ (10 The two solutions 9a an 10 match at ζ = ζ s, with a smooth erivative, as require. ζ s itself is given by the following equation, which arises from balancing the flux at ζ = ζ s ζ T T s s s (1 C e = ( C ζ erf ζ π (11 Note that eq 11 is inepenent of D w (because that epenence is scale away in the efinition of ζ..3. Numerical Simulations. We implement a numerical solution to check the asymptotic approach. First, we note that if we efine then Dw Fx t = DCx t + + κ [ Cxt (, (, (, log(1 e CT] κ F = D + an eq 1a becomes Dw κe κ 1 + e κ[ Cxt (, CT] κ[ Cxt (, CT] = DC ( C (1 (13 = F t (14 which provies a natural way to iscretize the problem using a conservative scheme. Because we will be applying spatial ifferences to F, we nee bounary conitions for it: F (0, t = 0 Dw F t = D + + κ (1 (1, log(1 e CT κ where we have use eqs 3a an 3b. (15a (15b We iscretize by letting ( i 1 Fij, = F,( j 1 Δ t, i = 1,..., N + 1, N j 1 (16 an similarly for C. From the x iscretization in eq 16 we have foun that Δx = N 1. Because of the nonlinearity in F, any implicit metho woul require a nonlinear solver like Newton s metho. Instea, we choose the explicit metho Cij, + 1 Cij, Fi 1, j Fij, + Fi+ 1, j = Δt ( Δx (17 For a iffusion equation with constant D, the following constraint must be satisfie to guarantee convergence (cf. Sauer, 4 Theorem 8.: Δ t < ( Δ x D (18 For our problem, the most conservative estimate for eq 18 uses the largest value taken on by D(C (namely, D w + D, so to ensure convergence we take Δt =(3D w N 1. By using the stanar approach of introucing a phantom point at Δx to take care of the Neumann conition 15a, we see that the algorithm 17 may be written in matrix-vector notation as C = C + MF; C, F ; M j+ 1 j j j j 0 0 0 0 0 1 1 0 0 0 0 1 M = 3Dw 0 0 0 0 1 1 0 0 0 0 0 0 0 n+ 1 ( n+ 1 ( n+ 1 (19a (19b Here the last row of M is all zeroes because C n+1,j = 1 for all t from the bounary conition 3b. To valiate the algorithm, we began by simulating the case where κ = 0. In this case, the iffusion coefficient is constant an an exact solution can be foun using separation of variables. Numerical simulations showe agreement with error proportional to N, as preicte by the theory. 4 Next, we check the accuracy of the asymptotic solution for the full problem against the numerical simulations with N = 100. In particular, with the parameters given in eq 6, we have foun that ζ s = 0.1608, tmax = 96.655 (0a For κ, motivate by Guo et al., 8 we select a value large enough to reprouce the sharp ECL waves seen experimentally: 13 15,19 κ = 100 D = 10 5 (0b where we have use eqs 4 an 6. The final parameter to be ientifie is A. From stanar asymptotic matching practice, A woul be etermine by the next term in the ry solution [which is O(κ 1 ]. When comparing with numerics, an equivalent approach is to require that the numerical an asymptotic solutions match at the ry bounary. Hence we matche the asymptotic value of C(0,Δt to the numerical value, yieling A = 0.01461. 6749 x.oi.org/10.101/jp400476m J. Phys. Chem. C 013, 117, 6747 6751
Clearly a key question is whether the ECL wave moves with constant spee, as claime in Guo et al., 8 or like t, consistent with the Fickian theory. In Figure 3, we plot the evolution of Figure 4. Concentration profiles from the numeric (symbol an asymptotic (line solutions for various times. Note the sharp nature of the front. The time intervals are chosen so that the front moves a fixe istance between snapshots. Figure 3. Comparison of asymptotic an numerical solutions (for various values of N for the isocline s(t, where C(s(t,t = C T, emonstrating that it moves proportional to t, consistent with Fickian behavior. 1 s(t is plotte because the front moves from right to left. the isocline where C = C T. Because we are now consiering the full asymptotic solution as given by eqs 9a, 9b, an 10, we track the isocline irectly, rather than using the leaing-orer approximation 8. The Figure presents the results from numeric an asymptotic cases; for each, linear interpolation between gri points was use to estimate the position of s(t, where C(s(t,t =C T. By efinition, the isocline for the asymptotic solution behaves like t, as shown by the soli curve in Figure 3. Note the very close agreement with the numerical solution, showing that the front oes inee behave like t, at least until the entire polymer saturates (rightmost ata in Figure 3. Note that the numerically compute front values are the same for each N, as one woul expect from a convergent numerical scheme. Although this t epenence contraicts the conclusion of Guo et al., 8 it is actually compatible with their results. In Figure S of appenix I in the supporting information to their work, the authors show that the front spee ecreases inversely with the step size Δx. Suppose that the front s(t avances Δs = mδx over a time step Δt. Then, their ata imply that Δs Δ = m Δ x Δ 1 Δ Δ s t t x Δt 1 Δt as expecte from Fickian ynamics. Next, we compare the actual solution profiles from the asymptotic an numerical solutions to see if they show the type of sharp front profile seen experimentally. 13 15,19 In Figure 4, we plot a series of snapshots in time comparing the numerical an asymptotic solutions. The time intervals are chosen so that the front moves a fixe istance between snapshots. Note the close agreement between the numerical an asymptotic solutions, with significant ifferences visible only for the last time snapshot. Although it is ifficult to iscern ue to the narrow with of the front, the solutions also exhibit a shouler-like profile, as a very sharp ben in the wet polymer transitions to a much wier profile in the ry region, which is reminiscent of experimental ata for these systems. 13 15,19 3. CONCLUSIONS One way for scientists to better unerstan the oxiation processes in polymer films is through ECL. Although experimental ata consistently show sharp ECL waves in such systems, there is ebate as to whether those waves which correspon to concentration isoclines of ionic clusters move like t or t. 14,15,19 Certainly iffusion in polymers can exhibit sharp fronts moving with constant spee, as foun by Thomas an Winle, 18 although typically the mathematical moels for such behavior must inclue other effects besie nonlinear Fickian iffusion. 16,17 In Guo et al., 8 the authors propose a nonlinear Fickian-type moel for ECL waves an claim that their simulations show the moel allows for constant front spee. In this work, we examine the moel numerically, compare these results with the asymptotic solutions in Ewars, an emonstrate that although the nonlinear iffusion coefficient oes prouce sharp fronts, those fronts still move proportional to t. In the limit of large κ, the omain can be ivie into several separate regions, where ifferent processes ominate. Far from the ionic source, a nearly ry precursor region permits the use of a similarity-variable solution, which forces the front to behave like t. The nearly piecewise-constant behavior of D(C for large κ forces a sharp front near the saturation value. In the bulk of the wet region, the iffusion coefficient is a constant D w, an the ynamics are again stanar Fickian. In this work, we implemente a conservatively ifference explicit numerical metho to solve eqs 1a an 1b subject to eqs, 3a, an 3b to compare with the theoretical work in Ewars. The results from the asymptotic an numeric approaches agree, showing a sharp front moving proportional to t, profiles that agree well with experimental results. 13 15,19 Although the analysis in this article focuse on the moel from Guo et al., 8 the conclusions are quite general. Consier any moel of the form 1a subject to eqs an 3a an 3b. As long as the iffusion coefficient in the ry region is negligible, an x/ t similarity variable can be use, no matter the exact functional form of D(C. Hence to capture constant front spee in ELC waves, one has to incorporate aitional effects that 6750 x.oi.org/10.101/jp400476m J. Phys. Chem. C 013, 117, 6747 6751
arise ue to the nature of the polymer film, such as viscoelastic stresses in the polymer. 16,17 AUTHOR INFORMATION Corresponing Author *E-mail: ewars@math.uel.eu. Notes The authors eclare no competing financial interest. ACKNOWLEDGMENTS The author thanks the reviewers for their many insightful suggestions. REFERENCES (1 Rauh, R. Electrochromic Winows: An Overview. Electrochim. Acta 1999, 44, 3165 3176. 3r International Meeting on Electrochromics (IME-3, Imp. Coll., Lonon, Sept 7 9, 1998. ( Ahikari, B.; Majumar, S. Polymers in Sensor Applications. Prog. Polym. Sci. 004, 9, 699 766. (3 Armstrong, N.; Wightman, R.; Gross, E. Light-Emitting Electrochemical Processes. Annu. Rev. Phys. Chem. 001, 5, 391 4. (4 van er Vaart, N.; Lifka, H.; Buzelaar, F.; Rubingh, J.; Hoppenbrouwers, J.; Dijksman, J.; Verbeek, R.; van Wouenberg, R.; Vossen, F.; Hiink, M.; Rosink, J.; Bernars, T.; Giralo, A.; Young, N.; Fish, D.; Chils, M.; Steer, W.; Lee, D.; George, D. Towars Large-Area Full-Color Active-Matrix Printe Polymer OLED Television. J. Soc. Inf. Disp. 005, 13, 9 16. International Symposium of the Society for Information Display (SID 004, Seattle, WA, May 5 7, 004. (5 Jager, E.; Smela, E.; Inganas, O. Microfabricating Conjugate Polymer Actuators. Science 000, 90, 1540 1545. (6 Smela, E. Conjugate Polymer Actuators for Biomeical Applications. Av. Mater. 003, 15, 481 494. (7 Richter, M. Electrochemiluminescence (ECL. Chem. Rev. 004, 104, 3003 3036. (8 Guo, S.; Fabian, O.; Chang, Y.-L.; Chen, J.-T.; Lackowski, W. M.; Barbara, P. F. Electrogenerate Chemiluminescence of Conjugate Polymer Films from Patterne Electroes. J. Am. Chem. Soc. 011, 133, 11994 1000. (9 Xu, L.; Li, Y.; Wu, S.; Liu, X.; Su, B. Imaging Latent Fingerprints by Electrochemiluminescence. Angew. Chem., Int. E. 01, 51, 8068 807. (10 Chang, Y.-L.; Palacios, R. E.; Chen, J.-T.; Stevenson, K. J.; Guo, S.; Lackowski, W. M.; Barbara, P. F. Electrogenerate Chemiluminescence of Soliton Waves in Conjugate Polymers. J. Am. Chem. Soc. 009, 131, 14166 14167. (11 Chen, J.-T.; Chang, Y.-L.; Guo, S.; Fabian, O.; Lackowski, W. M.; Barbara, P. F. Electrogenerate Chemiluminescence of Pure Polymer Films an Polymer Blens. Macromol. Rapi Commun. 011, 3, 598 603. (1 Aoki, K.; Aramoto, T.; Hoshino, Y. Photographic Measurements of Propagation Spees of the Conucting Zone in Polyaniline Films uring Electrochemical Switching. J. Electroanal. Chem. 199, 340, 17 135. (13 Tezuka, Y.; Aoki, K.; Ishii, A. Alternation of Conucting Zone from Propagation-Control to Diffusion-Control at Polythiophene Films by Solvent Substitution. Electrochim. Acta 1999, 44 1871 1877. Workshop on the Electrochemistry of Electroactive Polymer Films (WEEPF97, Douran, France, Sept 4, 1997. (14 Tezuka, Y.; Ohyama, S.; Ishii, T.; Aoki, K. Observation of Propagation Spee of Conuctive Front in Electrochemical Doping Process of Polypyrrole Films. Bull. Chem. Soc. Jpn. 1991, 64, 045 051. (15 Wang, X.; Shapiro, B.; Smela, E. Visualizing Ion Currents in Conjugate Polymers. Av. Mater. 004, 16, 1605 1609. (16 Ewars, D. A. Constant Front Spee in Weakly Diffusive Non- Fickian Systems. SIAM J. Appl. Math. 1995, 55, 1039 1058. 6751 (17 Durning, C. J.; Ewars, D. A.; Cohen, D. S. Perturbation Analysis of Thomas an Winle s Moel of Case II Transport. AIChE J. 1996, 4, 05 035. (18 Thomas, N.; Winle, A. A Theory of Case II Diffusion. Polymer 198, 3, 59 54. (19 Wang, X.; Smela, E. Experimental Stuies of Ion Transport in PPy(DBS. J. Phys. Chem. C 009, 113, 369 381. (0 Lacroix, J.; Fraoua, K.; Lacaze, P. Moving Front Phenomena in the Switching of Conuctive Polymers. J. Electroanal. Chem. 1998, 444, 83 93. (1 Wang, X.; Shapiro, B.; Smela, E. Development of a Moel for Charge Transport in Conjugate Polymers. J. Phys. Chem. C 009, 113, 38 401. ( Ewars, D. A. Sharp Fickian Fronts in Conjugate Polymer Films. IMA J. Appl. Math., submitte. (3 Corless, R.; Gonnet, G.; Hare, D.; Jeffrey, D.; Knuth, D. On the Lambert W Function. Av. Comput. Math. 1996, 5, 39 359. (4 Sauer, T. Numerical Analysis, n e.; Pearson: New York, 01. x.oi.org/10.101/jp400476m J. Phys. Chem. C 013, 117, 6747 6751