Simulation Studies in Electrocrystallisation JEFFREY AND SARRIKHAI K. RANGARAJAN A. HARRISON School of Chemistry University of Newcastle upon Tyne Newcastle upon Tyne NE1 7RU Received 10th August 1977 The properties of a simple “chess board” type computer simulation of electrocrystallization are investigated. The importance of the “discrete effect” and the “ingestion effect” is evaluated for the formation of a single layer. A quantitative investigation of the Avrami equation for square nuclei is carried out. Some comments are made about multilayer formation. The results suggest that this method can be used to make quantitative predictions in cases where analytical mathematical solutions are difficult. The importance of simulation studies in electrocrystallisation lies in bridging the gap between the naive idealisation of theoretical models and the complex response of an actual experimental ~ystem.~ A similar philosophy is also used in other prob- lems where information is sought on a molecular level for example by (molecular dynamics) computer experiments in studies on radial correlation functions computer studies of nucleation and crystal growth etc.In real problems of electrocrystallisa-tion there will be interactions which can only be understood by setting up simulation models. The purpose of the present paper is to present some of the elementary pro- perties of a simulation and to suggest their relation to real events. The parameters of the simulation are no doubt related to molecular processes for example lattice incorporation and electron transfer but it is not necessary that the scales of distance and time involved are those of the elemental processes.More specifically consider the activation-growth model. The familiar Avrami equations are expected to follow but should they? After all the kernel of this approach is the introduction of the concept of an extended area S and the relation of this to the true area accounting for overlap by S = 1 -exp (-Sx). (1) S itself is then evaluated by simple addition or convolution according to conditions of instantaneous or progressive nucleation. Several questions can be asked (a) Does eqn (1) correctly evaluate the overlap effects especially for a geometry other than a circular one e.g. square or hexagonal? (b) Is it possible that eqn (1) has to be replaced by a more general law S =f(S,)? (c) Is eqn (1) reliable for both instantaneous and progressive nucleation conditions.This is not so obvious because there are two types of randomness involved namely spatial and temporal. (d)Even if eqn (1) holds does eqn (2) hold and if so what are the values of p and n? S = 1 -exp (-PI”). (2) 7 Permanent address :Department of Inorganic and Physical Chemistry Indian Institute of Science Bangalore 560012 India. JEFFREY A. HARRISON AND SARRIKHAI K. RANGARAJAN Obviously there must be limitations to eqn (1) and it will be interesting to obtain indications of these by simulation studies. An even more specific question which this paper can answer is how important is the ingestion effect.Under conditions of progressive nucleation the rate of successful nucleus activation is governed not only by the rate of nucleation but also by the availability of free surface for nucleation. This introduces into the calculation of S a new factor e-Sx and consequently results in a non-linear differential equation for S,. It will be interesting to know the significance of this as no new parameter is involved in the model. A simulation study can naturally incorporate this phenomenon and hence provide an interesting ground for discussion of this aspect. Problems become even more intriguing when these ideas are extended to multi- layer growth. The problem can be treated as a cascade process and visualises a linear relation between the n + 1 layer and the layer n namely wheref is the rate of monolayer formation on the nth layer.Eqn (3) and (1) constitute the simple but fundamental relations in most theories. Any relaxation of simplicity e.g.,fn+l #fn can make the solutions very difficult to derive but simulation can incorporate such realistic features more conveniently. THE SIMULATION A square matrix L x L (nominally 40 x 40 or 100 x 100)was chosen to represent the ~urface.~,~ Nucleation sites N per time step were chosen by random number generator. If a coordinate chosen for monolayer growth fell on an already occupied portion of the surface this was ignored. Each nucleus was then grown by advancing the periphery by G length units per time step. The nuclei were squares.The bound- aries were cyclical in that nuclei moving over a boundary were reintroduced at the 240 --190-0 -0 X -+ 150 E -.--_w 97 -+ -X0 48 - - +9 0 I I 1 I I I I I I 1 1.o 2.8 4.6 6.4 8.2 10 "1 FIG.1.-Example of a simulation result N = 5 G = 1 L = 39 showing successive averages (+)second;(O)fourth;( x)fifth. SIMULATION STUDIES IN ELECTROCRYSTALLISATION opposite edge. This procedure generates cylindrical boundaries in the two directions. The program was organised as a series of subroutines for setting the nuclei growing the nuclei and dealing with the boundaries. Monolayer or multilayer growth could be carried out. In the case of multilayers the same subroutine for nucleation was used except that nuclei falling on an occupied surface adopted the identity of the next layer.The scan subroutines scanned for periphery (isim)or for area covered (Ssim). The scans for isimor Ssim as a function of nT* were accumulated in separate runs and the average values plotted by computer so that the standard deviation at each value of nT could be reported. However averaging was very effective in all the runs reported here and the final result which was then independent of further averaging was reached in 3 or 4 runs. A typical simulation result is shown in fig. 1. GROWTH PARAMETERS Since simulation experiments only involve numbers it is worthwhile showing how to make contact with " real situations ". For example the L x L grid with N square centres introduced in a stepwise fashion can map a real situation as follows.If a unit spacing in the grid corresponds to a real distance E cm say and the unit of time is then At s table 1 shows the correspondence rules in terms of the thus-far unknown E and At. TABLE 1.-RELATION BETWEEN REAL AND SIMULATION PARAMETERS real system simulation correspondence number of nuclei nucleation rate growth rate N" dN m AN* $= zFyk dt No N G N* = Nol(L2e2) AN* =I-N/(At * L2e2) zFyk = GEfAt time t nT t = nT * At In the case of instantaneous nucleation the extended area S becomes S = (4NoG2nT2/L2) (4) a non-dimensional form independent of E and At. If the nucleation is progressive again independent of E and At. As an example if N* = 1OI2 cm-2 in the real system and No = 100 and L = 100 in the simulation it follows that E -lo-' cm i.e.,if the values of No and L are used in the simulation for No* -10l2cm'2 each unit in the grid then corresponds to lo-' cm.In addition if the progressive nucleation rate A in the real system is known to be -10-1 s and N = 10 in the simulation the unit of time is then fixed and At -1 s. With these values if G is assumed to be 5 then it follows that the characteristic time constant for the growth in the real system becomes accessible in -5 time steps in the simulation Le. when nT = 5 the coverage is nearly unity. * See table 1 for definition. JEFFREY A. HARRISON AND SARRIKHAI K. RANGARAJAN EQUATIONS FOR S AND i On the basis of the last section and the Avrami eqn (I) then the appropriate equa- tions for S and i in real parameters are' S = 1 -exp (-PIt2) (6) l=qm-2PIt * exp (7) where P = 4N*(zFyk)*,(see table I) for instantaneous nucleation and S = I -exp (-ppt3) (8) i = 3ql,,ppt exp ( -ppt3) (9) where Pp= 4/3AN*(~Fyk)~ for progressive nucleation.The corresponding simulation equations are Ssim= L2[l -exp (-Ti)] isim= 4LNo*GT exp (-Ti) where Ti = 4NoGnT2/L2 for instantaneous nucleation and isim= 4(3/4)2/3L2( $2)'Tp2e-Tp3 NG2 where Tp3= (413)-n-r3 L2 for progressive nucleation. RESULTS MONOLAYER SIMULATION The first question is whether the Avrami eqn (1) is valid for the 2D instantaneous nucleation model. In this model there is only spatial randomness and no further complications are expected.The test which has been applied to the simulation is two-fold (1) to confirm that S is in fact given by 1 -exp (S,) and (2) to confirm that the extended area S is the sum of the individual centres and = 4No(ykt)2. In the above formulae the geometrical factor x appropriate for circular centres' has been replaced by 4. Since there is no continuum in the simulation studies S is written as 4NOG2nT2/L2.Fig. 2 in which the non-dimensional current iND in the form In (isim/4LGNo*TI) is plotted against the non-dimensional time T clearly demonstrates how satisfactory both Avrami's equation and the expression for S are. The case of progressive nucleation is probed in a similar way in fig. 3 and 4; here the non-dimensional current iND in the form In (isi,/[4(~)3L2(NG2/L2)~~~2)~ is plotted against the non-dimensional time Tp3.In graphs 2-4 the circles give the Avrami result i.e. a 45" line. Any deviation in view of the agreement between simulation and the Avrami equation for instantaneous nucleation could be attributed to S,. Possible sources of deviation lie in the stochastic nature of the activation process the discrete nature of time imposed by the simulation and the ingestion effect. SIMULATION STUDIES IN ELECTROCRYSTALLISATION -0202 W -0.95 - m a ;f -1.9 .,= cl 5 -2.8- 0 + 0 -4- - 3.7 - 0 + 0 f -4.7 1 L-LU FIG.2.-Ln (4L&'& for instantaneous nucleation of a monolayer, plotted against 'v No = 8 G = 1 L = 39 after 8 averages.In order to eliminate the stochastic effects in this paper all the figures refer to averaged data so that only an averaged activation process is considered. This leaves only time discretisation or ingestion to be considered. Fig. 3 and 4 show how deviations from the Avrami result can be seen in simula-tions. However it is somewhat more convenient to consider the behaviour of Ssim and this will be used in the subsequent discussion. + 0 i 0 -3.1 t -4.0) I I I I + 0.0033 0.67 1.3 20 2.7 3.3 FIG.3.-Ln (4(~)2,3L2(Nc2,L2>',3Tpzplotted against F2 nT3for progressive nucleation of a mono-) layer (+)N = 4 G = 1 L = 39 after 8 averages. JEFFREY A. HARRISON AND SARRIKHAI K.RANGARAJAN If the extended area Sx,in the case of progressive nucleation is evaluated taking into account the discrete nature of the time steps then sx(nT) becomes 0 4--2.6 c 0 T; FIG.4.-Plot as fig. 3 for N = 1 G = 1 L = 39 after 6 averages. Note that when nT = 0 Sx(nT) = 0. Also when nT -co,s behaves as predicted by continuum theory. It seems likely that this effect must play a significant role. Fig. 5(a) and (b) verify this by plotting for the various cases Ssim against Tp3and In (1 -Ssim/L2) Because the graphs are from a computer printout a against (Tp3)disc. square represents where two or more points cross. The agreement is satisfactory and shows a strong correlation with the discrete effect. Thus the Avrami equation and the parameter S are being employed correctly.Even a geometric " fudge " factor in addition to the factor 4 is not necessary. However it is still necessary to evaluate the extent of the ingestion effect. This can be evaluated in the continuum case by solving the non-linear differential equation for S S = T2IoTexp [-S,(z)]dz z exp [-S,(z)]dz -I-loT z2 exp [ -Sx(z2)]dz where T3/3= Tp3.Note that as T-0 S +T3/3,as expected in the classical version. The fact that progressive rates of nucleation become proportional to the available " free " area is responsible for the coefficient e-sx in the analysis. An inter- esting limit is predicted as T+co,namely S cccT2,but the effect per se does not seem SIMULATION STUDIES IN ELECTROCRYSTALLISATION H X + X f .:I 0.60 0 0 .LoL m H B OS2Ot+ 0.00088 m I I I I 0.00088 0.18 0.35 0.53 0.70 0.88 u -0-0'3" -1.0 i3 4- -2.0 - 0 I -3.0 - C -4.1 - 0 + -5.1 1 I I I d -4.1 51 ( Tg disc (6) FIG.5.-(a) Ss1,/L2against Tp3,N = 5 G = 1 L = 39 4 averages (+).( x ) is the discrete result, (0) N Avrami and (a) due to ingestion.(b)Ln (1 -Ssi,/L2) against (Tp3)disc= 5 G = 1 L = 39 4 averages (+). (0) is a 45"line. JEFFREY A. HARRISON AND SARRIKHAI K. RANGARAJAN + I- X 0.60L 0.40 0.0035 I 1 1 1 1 0.0035 0.70 1.4 2.1 2. a 3.5 0 + 0 -0.1 1 I I 1 1 9 0.021 1.6 3.3 4.9 6.5 8.1 (~p3 disc (b) FIG.6.4~) Ssim/L2 against Tp3,N =2 G =2 L =39; 5 averages (t).(x)is the discrete result Avrami and (0)due to ingestion.(b)Ln (1 -Ssim/L2)against (TP3)disc, (0) N =2 G =2 L = 39 5 averages (+). (0) is a 45" line. SIMULATION STUDIES IN ELECTROCRYSTALLISATION 0.98-+ + 8 ? X f X 0.78-+8 X +$ E 0.59 -k 8 (4 .GI4 0.39-5 8 N f 8 0.20-D a $ -0.74- ? 0 -1.5-*.:I- t 0 + 0 + -2.2 - 0 c- + 0 -3.0- + 0 -3.7 I I I I + JEFFREY A. HARRISON AND SARRIKHAI K. RANGARAJAN 1.0- + + v m w X . 0.80- + 8 X 0.61 - 8 + -x 0.41 il 0.21 -0 0 0 0 0 161 I I I I 9 0.042 3.3 6.5 9.7 13.0 16.0 FIG.8.-(a) Ssi,/L2 against Tp3,N = 1 G =4,L = 39 3 averages (-I-).(x ) is the discrete result (0)Avrami (a)due to ingestion. (6) Ln (1 -Ssi,,,/L2)against (Tp3)disc N = 1 G =4 L = 39 3 averages (+). 80 SIMULATION STUDIES IN ELECTROCRYSTALLISATION to be important except at larger times where the coverage is large. The equation for S has been solved in this work by both numerical and perturbation methods. Fig. 6(a)is a typical curve which demonstrates the magnitude of the ingestion effect com- pared with the discrepancy between the simulation and classical Avrami theory. It seems likely that a discrete version of the ingestion effect however would make the agreement between the simulation and the discrete version of the Avrami equation even better. Fig. 7(a) reflects the same analysis for a different choice of N and G.MULTILAYER SIMULATION Similar investigations have been made to verify eqn (3) used in multilayer analysis. Only one aspect will be reported here uiz. the dependence of the steady state multi- layer current i on the growth parameters. It has been shown7 that an exact solution of this problem leads to i =qm/imexp (-S,)dz in particular for 2D progressive nucleation Approximate versions of the above result are also available by numerical lointegration of eqn (3) and by simulation.ll Fig. 9 indicates that the simulation results support the dependence on pP1l3, where -I 8 -h .--.+ -1r 0 -!! lo-’ I I I I I Ill1 I I I Ill Ill I I I I I Ill log-NG L2 NG2 FIG.9.-Log (isim)m/L2) against log 7for multilayer growth each point is the average of approxi-L mately 10 runs.a simulation equivalent of Pp i.e. NG2/Lis actually plotted. The coefficient (-1.82) seems to differ from the theoretically predicted one (1.12) (fig. 10). An analysis for ingestion effect on multilayer form has been made by evaluating the integral in eqn (17) but the correction is not sufficient to explain the difference in coefficient. Pre-liminary studies by using eqn (14) to evaluate the integral in eqn (17) however indicate that the discrete effect is probably largely responsible. JEFFREY A. HARRISON AND SARRIKHAI K. RANGARAJAN / OeSr ~:~~ 0 .1 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 against (f 9)' FIG. lO.-(is,&/L2 for inultilayer growth.The full Iine has slope 1.82. CONCLUSION This paper has considered some of the features of simulation which must be talcen into account if these are to be used to investigate more realistic models of nucleation and growth processes. We believe that the discrete and ingestion efkcts also have their physical counterparts in real systems. Unfortunately not many experiments are available in simple and well-characterised systems. However the form of some recent experiments12on "dislocation " free Ag surfaces indicate that some of the effects discussed in this paper may operate. On the other hand experiments on real metal5 systems are also available. In order to make substantial progress in understanding these processes in necessarily more complex models it will be essential to set up models and identify the key parameters.Simulation is a convenient way of achieving this and we are embarking on a programme to investigate some of the obvious effects. In the field of monolayer formation we are investigating the effect on the kinetics of growth of the geometry and arrangement of centres the nature of the growth process fluctua- tions in space and the effect of interactions among centres also the influence of the ingestion effect for more complicated growth profiles (other than for example deter- mined by constant potential conditions). For multilayer growth we are using simula- tion to investigate the transients the effect of different growth and nucleation rates for successive layers and the influence of fluctuations and correlations in the activation process.We hope to present some of these results shortly. M. Fleischmann and H. R. Thirsk in Advances in Electrochemistry and Electrochemical Eng-ineering ed. P. Delahay (Interscience New York 1963) vol. 3 p. 123. J. A. Harrison and H. R. Thirsk in Electroanalytical Chemistry ed. A. J. Bard (Marcel Dekker New York 1971) vol. 5 p. 67. E. Budevski in Progress in Surface and Membrane Science ed. D. A. Cadenhead and J. F. Danielli (Academic Press New York 1976) vol. 11 p. 71. SIMULATION STUDIES IN ELECTROCRYSTALLISATION J. A. Harrison S. K. Rangarajan and H. R. Thirsk J. Electrocherit. SOC.,1966 113 1120. W. Davisoii J. A. Harrison and J. Thompson Faraday Disc. Chem. SOC.,1973 56 171. M. Avrami J. Chem.Phys. 1939,7,1103; 1940,8,212; 1941,9,177. S. K. Rangarajan J. Electroarmlyt. Chenz. 1973 46 119 125. M. M. Clark J. A. Harrison and H. R. Thirsk 2.phys. CIzem. N.F. 1975 98 153. J. A. Harrison and W. J. Lorenz J. Electroanalyt. Chenz. 1977 76,375. lo R. D. Armstrong and J. A. Harrison J. Electrochem. Sac. 1969 116 328. l1 U. Bertocci Surface Sci. 1969 15 286. l2 E. Budevski personal communication.