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A Markovian approach to electrodeposition
Ekctmctia Acta, Vd. 38, No. s, pp. 1147-l 148.1993 FVintcd in Great Britain
0
0013~4686/93 s&at + om 1993. Pmmon Prem Ltd
SHORT COMMUNICATION
A MARKOVIAN APPROACH TO ELECTRODEPOSITION THOMAS Z. FAHIDY Department of Chemical Engineering, University of Waterloo, Waterloo, ON, Canada N2L 3Gl (Received 16 November 1992; in revisedform
17 December
1992)
Abstract-The theory of Markov chains permits a relatively simple estimation of the long-term probabilities of finding a certain amount of deposit on an electrode surface, when some details of the electrode reaction mechanism are not known. Experimental observations may be used via likelihood functions to estimate the most plausible values of deposition probabilities. Key words: electrodeposition, markov chains, probability, likelihood.
probabilities[2]
NOMENCLATURE coefficient; the number of ways i objects can be chosen from a set of v objects L(q) likelihood function of single-event probability 4 N.4 Avogadro’s number conditional (Markov) probability Pij single-event probability of deposit redissolu4 tion (or decomposition); q, its numerical value that maximizes L(q) Markovian state at transition i S, length of unit observation time At L likelihood ratio mean of a Poisson distribution probability distribution z
iii)
binomial
Consider an electrode surface at which a particular reaction of unknown mechanism takes place under the influence of an electric or a magnetic field, solar or laser irradiation etc. involving electron transfer. The probability that n such transfers take place during equal active time intervals At is given by a certain probability distribution 4.; 4. may be theoretical or experimental. Let the electron transfer result in, eg discharge/deposition of an ionic species, and let the (single event) probability of redissolution (or decomposition) of this species be q. The probability of an arbitrary number of ions remaining on the electrode surface at sufficiently long times past the onset of the electron-transfer process may be estimated via Markov-chain theory [eg 1, 21. The set of such probabilities may be regarded as a measure of process efficiency. In a Markovian framework, the state S, (i = 0, 1, . . .) represents i number of ions deposited on the surface. During an active-time interval At a transition from state Si to state S, occurred if (i - v) ions, (v = 0, 1 . . . i), have redissolved or decomposed, and j - v ions, (j > v), remained deposited on the same surface. The transitional probabilities may be expressed in terms of individual Bernouilli trial
as
pij = vToC;(l - q)yqi-v4j_y.
i, j = 0, l...
(1)
Since the surface can contain, in principle, any number of discharged ions, all states are attainable. Assuming non-periodicity, the process may be regarded as an irreducible Markov chain. As shown recently by Fletcher et aI.[3, 43, various models can be applied to electrochemical nucleation to compute the expectation and the variance of the number of crystals, a binomial random variate, deposited on an initial number of active sites. For the sake of a simple illustration and following earlier studies of electrodeposition[5-81 growth nuclei kinetics has been considered as a random process with a characteristic Poisson distribution, and 4, = p” exp( - p)/n!
(2)
is taken, where p, the number of ions discharged on an average during time period At of the Poisson process, is of specific importance. Given equation (2) the long term (limiting) probabilities of finding j number of discharged ions on the surface: p? = exp( -&X&Y/j
!
(3)
are themselves Poissonian. As an illustration assume p = lo-‘N,, in an arbitrary deposition process (if the process were, eg a two-electron transfer for faradaic deposition, electrolysis over a period of one second at a current density of 2Adme2 and 98% current efficiency would produce approximately this value). For the sake of convenience, the amount of deposition is expressed in terms of lo-‘N,, units; then p = 1 is set and every integer j corresponds to an amount of lo-‘jN,. As shown in Table 1, containing long term probabilities of finding selected amounts of deposit on the surface at two distinct values of the (single-event) probability
1147
Short Communication
1148
Table 1. Long-term probabilities of finding lo-‘jN, amount of deposit on the electrode surface at q = 0.05 and q = 0.5 (equation 3), with p = 10-‘N, q=
i 10 15 18 19 20 21 25
q=
0.05 pl” 0.0058 0.0516 0.0844 0.0889 0.0888 0.0846 0.0446
0.5
i
pl”
0
0.136 0.271 0.271 0.180 0.0902 0.0361 0.0121
1
2 3 4 5 6
of redissolution, the pi” vs. j relationship possesses a shallow maximum. Within a relatively narrow jrange, arbitrary amounts of deposit share essentially the same probability. Consider now the reliability of a postulated value of q. Assuming that experimental observations yield lo-‘j. N, amount of deposit over time period At, the likelihood function[9]
p = lo-‘N,, if eg j = 25 were found experimentally, the value of Iz = 1.78 (based on q, = 0.04; L (q,,,) = 0.0795 and L (q) = 0.0446) would not discriminate between q,,, and q = 0.05. However, if j = 10 were the experimental finding, the value of Iz = 21.6 (q,,, = 0.1; L (q,,,) = 0.125; yq) = 0.00583) would indicate that the (assumed) value of 0.05 for the probability of deposit dissolution appears to be too low. In summary, the Markovian approach offers a relatively simple and straightforward interpretation of electrode processes in terms of probabilities, where an exact process mechanism is unknown. Further work to explore its full potential in this area is warranted.
REFERENCES 1. H. Stark and J. W. Woods, Probability, Random Processes and Estimation Theory for Engineers, Chapter 7. Prentice Hall, Englewood Cliffs, New Jersey (1986). 2. A. A. Sveshnikov, Problems in Probability Theory, Mathematical Statistics and Theory
(4i
3.
will be maximized at a certain q = q, , and a specific value ofj, and p. The likelihood ratio
4. 5. 6.
44) = exd - ~/&@%_L)
!
J.= 4qdl4q) between q, and q: if 1 2 10, a real difference in plausibility between q,,, and q is indicated, whereas I > 100 allows a strong preference for qm over q. Taking the illustration with is a means of discriminating
7.
of Random Functions,
Chapter VIII. Dover, New York (1968). R. L. Deutscher and S. Fletcher. J. electroanal. Chem. 164, 1 (1984). S. Fletcher, J. electroanal. Chem. 164, 11 (1984). M. Avrami, Cbem. Phys. 7,1103 (1939). S. K. Rangarajan, J. electroanal. Chem. Inter&. Chem. ss, 119 (1973). A. M. Bewick, M. Fleischmann and H. E. Thirsk, Trans.
Faraday Sot. !%, 2200 (1962). 8. S. Fletcher and T. Lwin, Electrochim. Acta 28, 237 (1983). 9. P. M. Reilly, Can. J. them. Engng. 48, 168 (1970).
0
0013~4686/93 s&at + om 1993. Pmmon Prem Ltd
SHORT COMMUNICATION
A MARKOVIAN APPROACH TO ELECTRODEPOSITION THOMAS Z. FAHIDY Department of Chemical Engineering, University of Waterloo, Waterloo, ON, Canada N2L 3Gl (Received 16 November 1992; in revisedform
17 December
1992)
Abstract-The theory of Markov chains permits a relatively simple estimation of the long-term probabilities of finding a certain amount of deposit on an electrode surface, when some details of the electrode reaction mechanism are not known. Experimental observations may be used via likelihood functions to estimate the most plausible values of deposition probabilities. Key words: electrodeposition, markov chains, probability, likelihood.
probabilities[2]
NOMENCLATURE coefficient; the number of ways i objects can be chosen from a set of v objects L(q) likelihood function of single-event probability 4 N.4 Avogadro’s number conditional (Markov) probability Pij single-event probability of deposit redissolu4 tion (or decomposition); q, its numerical value that maximizes L(q) Markovian state at transition i S, length of unit observation time At L likelihood ratio mean of a Poisson distribution probability distribution z
iii)
binomial
Consider an electrode surface at which a particular reaction of unknown mechanism takes place under the influence of an electric or a magnetic field, solar or laser irradiation etc. involving electron transfer. The probability that n such transfers take place during equal active time intervals At is given by a certain probability distribution 4.; 4. may be theoretical or experimental. Let the electron transfer result in, eg discharge/deposition of an ionic species, and let the (single event) probability of redissolution (or decomposition) of this species be q. The probability of an arbitrary number of ions remaining on the electrode surface at sufficiently long times past the onset of the electron-transfer process may be estimated via Markov-chain theory [eg 1, 21. The set of such probabilities may be regarded as a measure of process efficiency. In a Markovian framework, the state S, (i = 0, 1, . . .) represents i number of ions deposited on the surface. During an active-time interval At a transition from state Si to state S, occurred if (i - v) ions, (v = 0, 1 . . . i), have redissolved or decomposed, and j - v ions, (j > v), remained deposited on the same surface. The transitional probabilities may be expressed in terms of individual Bernouilli trial
as
pij = vToC;(l - q)yqi-v4j_y.
i, j = 0, l...
(1)
Since the surface can contain, in principle, any number of discharged ions, all states are attainable. Assuming non-periodicity, the process may be regarded as an irreducible Markov chain. As shown recently by Fletcher et aI.[3, 43, various models can be applied to electrochemical nucleation to compute the expectation and the variance of the number of crystals, a binomial random variate, deposited on an initial number of active sites. For the sake of a simple illustration and following earlier studies of electrodeposition[5-81 growth nuclei kinetics has been considered as a random process with a characteristic Poisson distribution, and 4, = p” exp( - p)/n!
(2)
is taken, where p, the number of ions discharged on an average during time period At of the Poisson process, is of specific importance. Given equation (2) the long term (limiting) probabilities of finding j number of discharged ions on the surface: p? = exp( -&X&Y/j
!
(3)
are themselves Poissonian. As an illustration assume p = lo-‘N,, in an arbitrary deposition process (if the process were, eg a two-electron transfer for faradaic deposition, electrolysis over a period of one second at a current density of 2Adme2 and 98% current efficiency would produce approximately this value). For the sake of convenience, the amount of deposition is expressed in terms of lo-‘N,, units; then p = 1 is set and every integer j corresponds to an amount of lo-‘jN,. As shown in Table 1, containing long term probabilities of finding selected amounts of deposit on the surface at two distinct values of the (single-event) probability
1147
Short Communication
1148
Table 1. Long-term probabilities of finding lo-‘jN, amount of deposit on the electrode surface at q = 0.05 and q = 0.5 (equation 3), with p = 10-‘N, q=
i 10 15 18 19 20 21 25
q=
0.05 pl” 0.0058 0.0516 0.0844 0.0889 0.0888 0.0846 0.0446
0.5
i
pl”
0
0.136 0.271 0.271 0.180 0.0902 0.0361 0.0121
1
2 3 4 5 6
of redissolution, the pi” vs. j relationship possesses a shallow maximum. Within a relatively narrow jrange, arbitrary amounts of deposit share essentially the same probability. Consider now the reliability of a postulated value of q. Assuming that experimental observations yield lo-‘j. N, amount of deposit over time period At, the likelihood function[9]
p = lo-‘N,, if eg j = 25 were found experimentally, the value of Iz = 1.78 (based on q, = 0.04; L (q,,,) = 0.0795 and L (q) = 0.0446) would not discriminate between q,,, and q = 0.05. However, if j = 10 were the experimental finding, the value of Iz = 21.6 (q,,, = 0.1; L (q,,,) = 0.125; yq) = 0.00583) would indicate that the (assumed) value of 0.05 for the probability of deposit dissolution appears to be too low. In summary, the Markovian approach offers a relatively simple and straightforward interpretation of electrode processes in terms of probabilities, where an exact process mechanism is unknown. Further work to explore its full potential in this area is warranted.
REFERENCES 1. H. Stark and J. W. Woods, Probability, Random Processes and Estimation Theory for Engineers, Chapter 7. Prentice Hall, Englewood Cliffs, New Jersey (1986). 2. A. A. Sveshnikov, Problems in Probability Theory, Mathematical Statistics and Theory
(4i
3.
will be maximized at a certain q = q, , and a specific value ofj, and p. The likelihood ratio
4. 5. 6.
44) = exd - ~/&@%_L)
!
J.= 4qdl4q) between q, and q: if 1 2 10, a real difference in plausibility between q,,, and q is indicated, whereas I > 100 allows a strong preference for qm over q. Taking the illustration with is a means of discriminating
7.
of Random Functions,
Chapter VIII. Dover, New York (1968). R. L. Deutscher and S. Fletcher. J. electroanal. Chem. 164, 1 (1984). S. Fletcher, J. electroanal. Chem. 164, 11 (1984). M. Avrami, Cbem. Phys. 7,1103 (1939). S. K. Rangarajan, J. electroanal. Chem. Inter&. Chem. ss, 119 (1973). A. M. Bewick, M. Fleischmann and H. E. Thirsk, Trans.
Faraday Sot. !%, 2200 (1962). 8. S. Fletcher and T. Lwin, Electrochim. Acta 28, 237 (1983). 9. P. M. Reilly, Can. J. them. Engng. 48, 168 (1970).