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The use of computer simulations to study the nature of an electrodeposit during the early stages of growth
Electrodeposition and Surface Treatment, 2 (1973174) 395 - 406 @ Elsevier Sequoia S.A., Lausanne - Printed in Switzerland
395
THE USE OF COMPUTER SIMULATIONS TO STUDY THE NATURE OF AN ELECTRODEPOSIT DURING THE EARLY STAGES OF GROWTH
J. W. OLDFIELD International Nickel Limited, European Research and Development Centre, Research and Technical Development Laboratory, Birmingham B16 0 AJ (Gt. Britain) (Received April 24, 1974)
Summary The early stages of growth of an electrodeposit have been studied using computer simulations. The result of this type of study is that predictions can be made regarding the effect of the electrodeposition parameters, e.g. nucleation rate, growth rate, etc., on the surface profile of the deposit. Introduction In recent years much attention has been given to nucleation as a ratedetermining step in electrodeposition [ 1 - 111. A number of authors have realised that nucleation is probabilistic, i.e. the exact location of nuclei formed at a given time cannot be predicted, and they have postulated various theories incorporating this fact [2 - 71. Others have studied nucleation and growth by computer simulation techniques [8 - 111. However, no attempt has so far been made to study the effect of nucleation and growth rates on the surface finish of a deposit. In this study computer simulation techniques are used to relate the individual rates of two-dimensional nucleation and growth to the surface profile of an electrodeposit formed at constant potential on a smooth underlying metal, In addition the requirements to produce particular types of surface profile are discussed in terms of parameters related to the electrodeposition process itself. This indicates the type of processes which are inherently suited to producing particular types of surface finish, e.g. bright. The overall aim of the work however is not only to relate nucleation and growth rates to surface finish for this simple situation but also to illustrate the techniques of computer simulation which could be used to study brightness and levelling in more complex electrodeposition processes, e.g. the formation of nodules over foreign particles, the mechanism of levelling, covering power, etc.
396
Theoretical
considerations
The formation of an electrodeposit of monomolecular height by twodimensional nucleation and growth of circular centres under potentiostatic conditions has been shown to be described by the following expression [S]
i=qnV2At2exp--
ITt3 V2A
(1)
3
where q is the charge involved in the formation of a layer (coulombs cme2), V is the rate of advance of an edge (cm set-I), A is the nucleation rate (cmP2 set-l ), i is the current density (amps cmP2) and t is time (seconds). The current/time relationship for electrodeposition involving the formation of successive layers by the same mechanism has not been derived but an expression relating the current due to the nth and (n + l)th layers has [l] , namely t 1,+1=
s
3/3(t-u)2 exp[ -/3(t-u)3]
i, dt
(2)
0
where u is the time at which the nth layer starts to be formed, and 0 is the constant 71PA 13. Since il is known from eqn. (1) it is possible to calculate the total current and the contributions from individual layers as a function of time and in fact this has been done [l] . However, inherent errors in the integration techniques limit the time axis to the formation of around the third monomolecular layer. It is not possible to solve eqn. (2) or the expression from the total current, namely m
I total = C n=l
L
thus simulation techniques time transients.
Simulation
remain as the sole method
of predicting
current
model
The theories mentioned above assume the growth of circular centres of monomolecular height and take into account the interaction between growth centres when they overlap, i.e. since the current is proportional to the total length of periphery when centres meet the current is reduced. To simulate the growth of circular islands is extremely difficult and the model chosen for this work was one involving the growth of squares such that the squares are always aligned parallel to one another. When sides of squares meet growth is halted. The area chosen was a grid, 100 X 100; to eliminate edge effects as far as possible the growth of centres off one edge was introduced back onto
Time
Fig. 1. Schematic representation of grid during simulation. diagonally shaded, first layer; shaded with squares, second corresponding to the position of the arrows. Fig. 2. Comparison of theory -___--eqn. (1); I simulation.
and simulation
for the growth
(5)
Unshaded areas, substrate; layer. A surface profile is shown of a single layer.
--
eqn. (4);
the grid on the opposite edge. Figure 1 shows a schematic simulated deposit. The simulation itself was carried out as follows. Deposition was split into discrete time intervals during which existing centres grow by an amount directly proportional to the growth constant V, (provided of course that they do not overlap within this distance in which case growth stops along this edge) after which a nucleation step occurs, the number of nuclei formed being directly proportional to A, the nucleation rate. The positions of the individual nuclei were determined using random numbers, generated by the computer. This process was repeated in each time interval. After each time interval the total growth that occurred within that interval was noted. These data were also split up into the growth for the individual layers. The total growth per unit time gave a measure of the deposition current. The exact shape of a current - time transient is a function of the locations of the nuclei and these are determined in a random manner; thus if a current - time transient is repeated its shape will differ somewhat from the original transient. In this work current - time-transients were repeated twenty times and the mean and standard deviation of these curves determined. In order to obtain a measure of the surface roughness the correlation function and the standard deviation of the thickness of the deposit were determined at various times during a transient. The correlation function indicated whether there was any correlation between the height of the deposit at positions a given distance apart, i.e. it gave a measure of the average spread of the surface undulations, while the standard deviation gave a measure of the average height of the undulation. Because the growth of a centre off one edge was introduced on the opposite edge the correlation function could
398
only be determined 100.
up to a distance apart of 50 points on the grid and not
Results Before using the simulation results to study the effect of nucleation and growth rates on the reproducibility and surface finish of the deposit we must first establish a theoretical expression to which the simulations correspond. The derivation of eqn. (1) assumes that circular nuclei are formed randomly over the electrode surface. If we assume that the nuclei are square rather than circular and that they are randomly oriented with respect to one another then the current time transient for the growth of a single layer is described by i=q4
V2At2exp----
4t3 V2A 3
.
(4)
In the model used for simulation the squares are not oriented at random with respect to one another but are aligned in parallel. To see whether this resulted in significant deviation from the theory the growth of a single layer was simulated and compared with the curves predicted by eqns. (1) and (4). In doing this and all other simulations the following assumptions were made as regards the grid size and the deposition process: 1. The formation of one molecule of the deposit involves a one electron change. 2. One molecule of the deposit is a 0.4 X 0.4 nm square. 3. The thickness of each layer is arbitrary and is termed a “step”. 4. One point on the grid represents a 10 X 10 nm square (i.e. one point is occupied by 625 molecules); the grid length is therefore lop4 cm, a realistic size. . 5. One discrete time interval is 1O-2 sec. This time is short enough to allow for only a small number of nuclei to be formed on the simulation area and therefore results in a relatively smooth current - time transient. The following values of the nucleation and growth constants were assumed for this comparison: V = 10e4 cm set-‘, A = lOlo nuclei crne2 sec~l, these values being typical for the formation of mercury salts on mercury [B, 12 - 141. Figure 2 shows the two theoretical curves; the simulation is represented by the vertical lines, these extending from the mean to plus and minus one standard deviation, being calculated from twenty repetitions of the growth transient. The simulation is in agreement with the theoretical curve for the growth of square centres and shows that eqn. (4) describes the mean of the simulated growth transients; i.e. the fact that the squares are aligned parallel to one another and not randomly oriented is of no consequence.
399
01
02
03 Time
0.4
(s)
Fig. 3. Current-time transient showing the mean (full line) and standard line) for V = 10e4cm set -’ and A = 9 X lOlo cm-‘set-l.
Time
deviation
(dotted
deviation
(dotted
(5)
Fig. 4. Current-time transient$owing line) for V= 3 X 10m4cm set and
the mean (full line) and standard see-l.
A = lOlo cm-’
From eqn. (4) we note that the mean current-time transient for the growth of a single layer (and therefore also the growth of the whole film) is a function of the combined rate constant V'A and not the individual values of V and A. This means that, provided that the product PA remains constant, both V and A can change without changing the shape of the average transient This presents a method of studying the effect of the individual nucleation and growth rates on the reproducibility (in terms of amount deposited in a given time) and the surface roughness of an electrodeposit. To illustrate this compare Figs. 3 and 4 corresponding to growth with V2A = 900;the individual values of V and A aregiven in Table 1. The curves are the same within the standard deviation but the standard deviation in Fig. 4 is far greater than that in Fig. 3. The magnitude of the standard deviation is a measure of the reproducibility of a current-time transient and therefore of the deposit itself. High
400 TABLE
1
Values of V and A studied
with V2A = 900
V(cm set-l )
A( crnm2 see-l )
1 x 1o-4
9 x 1o1O
3 x 1o-4
1 x 1o1O
irreproducibility implies large variations in the periphery length and therefore large variations in the average size and number of growing centres. To study in more detail this and other effects which are observed in simulated deposition experiments, simulations were carried out with V2A 1600 and 100 using the values of V and A given in Table 2. TABLE
=
2
Values of V and A studied -3
V2A(sec
)
V(cm
with V2A = 1600 and 100 set-l)
A(cmW2sec-‘)
1600
1 x 1o-4
1600
2 x lop4
4 x 1o1O
1600
4 x 1o-4
1 x 1o1O
100
1 x 1o-4
1 x 1o1O
Ttme
16 x 10”
(s)
Fig. 5. Mean current-time transients for the total current V= lop4 cm set-’ and A = 16 x lOlo cmp2 set-l.
and individual
layer currents
for
Figure 5 shows the mean current-time transient together with the mean current for the growth of individual layers for V = lop4cm set-’ and A = 16 X lOlo cmW2 set-I. Figure 6 shows the variation of the standard deviation as a percentage of the mean current with the growth constant V for V2A = 1600.This is linear and passes through the origin. Thus the variation of the
401
I
1
I 2 V 104km
3
4
500
sec.‘)
Fig. 6. Dependence of the standard constant V for V2A = 1600 sece3.
1000 V’A
deviation
as a percentage
1500
(se~-~)
of the mean current
on the
Fig. 7. Dependence of the standard deviation as a percentage of the mean current on the overall growth rate constant V2A for V = low4 cm set -’ and A = lOlo cmW2 see-‘.
standard deviation with l/A 1’2 for V2 A = 1600 is also linear and passes through the origin since if V2A is constant V is proportional to l/A1 ‘2. From this we can conclude that increasing V decreases the reproducibility of the deposit while increasing A increases its reproducibility. Figure 7 shows the dependence of the standard deviation as a percentage of the mean current on the overall growth rate, V2A and, as the latter increases at constant V the standard deviation decreases, while as it increases at constant A the standard deviation also increases. Clearly the reproducibility of the deposit is a marked function of the individual values of V and A rather than the combined constant V2A which governs the overall rate of growth. Now in order to relate these variations in V and A to surface roughness for the specific conditions considered the following parameters were determined: 1. Mean height of surface irregularities. To determine the mean height of surface irregularities the mean surface height was first determined. The standard deviation of the height of each individual row and column was then determined (100 rows and 100 columns) and the overall standard deviation was taken as the average of that of the rows and columns. The mean height of the surface irregularities from peak to trough is twice the average standard deviation. 2. Mean spread of surface irregularities. To determine this parameter the correlation function of the height of each row and column was determined and the average taken. The correlation
402
0
005
010 Dtstance
Fig. 8 Correlation 0.25 sec.
015 (104cm
function
for
n
025
050
1 V = lop4cm set -’ and A = 16
Time
X
075
10 0
(5)
lOlocm-’
set-1
after
Fig. 9. Dependence_;f distance of cF;rela>zon 02~ time. 0: V= 10P4cmsec ,A=16x 10 cm set ; X: V = 2 X 10m4 cm set-‘, A = 4 X lOlo cmd2 see-I; A: V = 4 X 10P4 cm set -l, A = lOlo cmP2 set-l.
function can take values of from +l to -1 and is a measure of the correlation between the height of points at varying distances apart. Values close to +1 and -1 indicate strong positive and negative correlation respectively; values close to 0 indicate little or no correlation. Both these parameters have been determined as a function of time for the combined growth rate constant V2A = 1600 with individual values of V and A asshown in Table 2. Figure 8 shows a typical plot of the correlation function against distance between two points; the 95% significance level is shown and we take this as denoting the distinction between correlation and no correlation. The distance at which this transition occurs corresponds to half the mean spread of the irregularities. This distance is shown as a function of time for the three combinations of A and V compatible with V2A = 1600 in Fig. 9. The fairly large scatter for the case with V = 4 X 1O-4 cm set-’ and A = 1 cme2 set-’ can be accounted for by the fact that the surface area represented by the grid is small for such a high value of V. In each case there is a characteristic time to reach a steady state and a characteristic distance of correlation (and therefore of mean spread). These data are shown in Table 3. Figure 10 shows the variation of the mean amplitude of the irregularities with time; again a characteristic time to reach the steady state mean amplitude is observed and in fact these times correspond to those in Fig. 9. The mean amplitudes are given in Table 3.
403 TABLE 3 Mean spread and amplitude of irregularities A(cmm2 set?)
V(cm set-l)
16 x lOlo 4 x 1o1O
1 x 1o-4 2 x 1o-4
1 x 1o1O
4 x 1o-4
Mean spread (Pm)
Mean amplitude (steps)
0.65 0.30
0.27 0.32
1.75 1.30
0.44
0.80
Time to steady state (set)
I 025
050 Time
075
1 O(
(5)
Fig. 10. Variation of the mean amplitude of irregularities with time. Symbols as in Fig. 9.
I 025
050 Distance
075
100
125
(104cm)
Fig. 11. Schematic diagram of the average appearance of the surface of an electrodeposit after growth for 1.0 sec. V = low4 cm set-‘, A = 16 X lOlo cm-’ set-‘; -____ V = 2 x 10W4 cm set-‘, A = 4 X lOlo cme2 set-‘; -.--.- V= 4 x 10v4 cm set-‘, A = lOlo cmP2 set-‘.
From Table 3, the surface structure of the deposit is markedly different depending on the values of V and A at constant V2A. Figure 11 illustrates
404
schematically the average appearance electrodeposits.
of the surface
for each of the three
Discussion From the results obtained we can draw the following conclusions with regard to the surface profile of a deposit and its reproducibility in terms of amount deposited in a fixed time under potentiostatic conditions when the overall growth rate, characterised by V2A, is constant. 1. Large nucleation rates result in a deposit the surface profile of which exhibits high amplitude, high frequency undulations. The amount deposited in a given time is reproducible. 2. High growth rates result in a deposit the surface profile of which exhibits low amplitude, low frequency undulations. The amount deposited in a given time is irreproducible. To explore in more detail the significance of these conclusions we must examine the parameters which determine V and A. The growth rate as a function of overpotential for an irreversible electrochemical reaction can be derived from the Volmer expression giving V= V, exp-
c~nFq RT
’ l-expI
anFq RT
cm set-l t
(5)
where V, is the growth rate at the reversible potential, 01 is the electrochemical transfer coefficient, q is the overpotential and n, F, R and T have their usual significance. It can be shown that the potential dependence of the nucleation rate, for the formation of layers of thickness h, is given by [12] cmP2 sec~l
(6)
where A, is a frequency factor, h is the layer thickness, (I is the surface free energy of the growing edge-solution int.erface, M is the molecular weight of the deposit, N is Avagadro’s number, p is the density of the deposit and e is the charge on an electron. In addition to this the nucleation rate will be increased by the number of dislocations and other surface defects present on the substrate. From an analysis of eqns. (5) and (6) Table 4 is constructed giving the requirements of various parameters such that a deposit falls into one or other of the above categories. This analysis can be used as a basis for practical investigations to improve the surface finish of a deposit in terms of brightness and levelling during the early stages of growth, e.g. if we wished to increase the amplitude and frequency of the surface undulations of a deposit formed by two-dimensional nucleation and growth we could do this by decreasing the film edge-
405 TABLE 4 Effect of various parameters on the nucleation and growth constants Requirements to increase amplitude and frequency of surface undulations
Requirements to decrease amplitude and frequency of surface undulations
Film edge-solution free energy (J)
Low
High
Molar volume of film (M/p)
Low
High
Film layer thickness (h)
Low
High
Electrochemical transfer coefficient (cl)
Low
High
Surface defects
High
Low
solution free energy (surface tension) by the addition of a suitable wetting agent (this would increase the nucleation rate). Alternatively if we wished to decrease the amplitude and frequency of surface undulations we could increase the molar volume of the film by adding inert agents which would be incorporated into the film (this would decrease the nucleation rate). Finally it should be pointed out that the relatively simple situation which has been dealt with in this work was deliberately chosen so as to validate the techniques used; this it has done since the conclusions in this case are obviously correct. However, the techniques can equally well be applied to the more complicated constant current situation. Also, in addition to nucleation and growth rates the effect of parameters such as diffusion layer thickness, addition of levelling agents, etc. on the surface profile of a deposit can be studied. Acknowledgements I should like to thank W. A. Davis who, besides writing all the necessary computer programs for this work, made many helpful suggestions regarding the actual simulation models. References 1 2 3 4 5 6 7
R. D. Armstrong and J. A. Harrison, J. Electrochem. Sot., 116 (1969) 328. S. Toschev and I. Markov, Electrochim. Acta, 12 (1967) 281. S. Toschev and I. Markov, J. Crystal Growth, 4 (1968) 436. I. Markov, Thin Solid Films, 8 (1971) 281. E. Budevski, J. Crystal Growth, 13/14 (1972) 93. S. Toschev, A. Milchev and S. Stoyanov, J. Crystal Growth, 13/14 (1972) 123. I. Markov and D. Kashchiev, J. Crystal Growth, 13/14 (1972) 131.
406 8 9 10 11 12 13
A. Bewick, M. Fleischmann and H. R. Thirsk, Trans. Faraday Sot., 58 (1962) 2200. U. Bertocci, Surface Sci., 15 (1969) 286. A. C. Adams and K. A. Jackson, J. Crystal Growth, 13/14 (1972) 144. G. H. Gilmer and P. Bennema, J. Crystal Growth, 13/14 (1972) 148. J. W. Oldfield, PhD Thesis, University of Newcastle upon Tyne, 1967. R. D. Armstrong, M. Fleischmann and J. W. Oldfield, J. Electroanal. Chem., 14 (1967) 235. 14 R. D. Armstrong and M. Fleischmann, Z. Physik. Chem., 52 (1967) 131.
395
THE USE OF COMPUTER SIMULATIONS TO STUDY THE NATURE OF AN ELECTRODEPOSIT DURING THE EARLY STAGES OF GROWTH
J. W. OLDFIELD International Nickel Limited, European Research and Development Centre, Research and Technical Development Laboratory, Birmingham B16 0 AJ (Gt. Britain) (Received April 24, 1974)
Summary The early stages of growth of an electrodeposit have been studied using computer simulations. The result of this type of study is that predictions can be made regarding the effect of the electrodeposition parameters, e.g. nucleation rate, growth rate, etc., on the surface profile of the deposit. Introduction In recent years much attention has been given to nucleation as a ratedetermining step in electrodeposition [ 1 - 111. A number of authors have realised that nucleation is probabilistic, i.e. the exact location of nuclei formed at a given time cannot be predicted, and they have postulated various theories incorporating this fact [2 - 71. Others have studied nucleation and growth by computer simulation techniques [8 - 111. However, no attempt has so far been made to study the effect of nucleation and growth rates on the surface finish of a deposit. In this study computer simulation techniques are used to relate the individual rates of two-dimensional nucleation and growth to the surface profile of an electrodeposit formed at constant potential on a smooth underlying metal, In addition the requirements to produce particular types of surface profile are discussed in terms of parameters related to the electrodeposition process itself. This indicates the type of processes which are inherently suited to producing particular types of surface finish, e.g. bright. The overall aim of the work however is not only to relate nucleation and growth rates to surface finish for this simple situation but also to illustrate the techniques of computer simulation which could be used to study brightness and levelling in more complex electrodeposition processes, e.g. the formation of nodules over foreign particles, the mechanism of levelling, covering power, etc.
396
Theoretical
considerations
The formation of an electrodeposit of monomolecular height by twodimensional nucleation and growth of circular centres under potentiostatic conditions has been shown to be described by the following expression [S]
i=qnV2At2exp--
ITt3 V2A
(1)
3
where q is the charge involved in the formation of a layer (coulombs cme2), V is the rate of advance of an edge (cm set-I), A is the nucleation rate (cmP2 set-l ), i is the current density (amps cmP2) and t is time (seconds). The current/time relationship for electrodeposition involving the formation of successive layers by the same mechanism has not been derived but an expression relating the current due to the nth and (n + l)th layers has [l] , namely t 1,+1=
s
3/3(t-u)2 exp[ -/3(t-u)3]
i, dt
(2)
0
where u is the time at which the nth layer starts to be formed, and 0 is the constant 71PA 13. Since il is known from eqn. (1) it is possible to calculate the total current and the contributions from individual layers as a function of time and in fact this has been done [l] . However, inherent errors in the integration techniques limit the time axis to the formation of around the third monomolecular layer. It is not possible to solve eqn. (2) or the expression from the total current, namely m
I total = C n=l
L
thus simulation techniques time transients.
Simulation
remain as the sole method
of predicting
current
model
The theories mentioned above assume the growth of circular centres of monomolecular height and take into account the interaction between growth centres when they overlap, i.e. since the current is proportional to the total length of periphery when centres meet the current is reduced. To simulate the growth of circular islands is extremely difficult and the model chosen for this work was one involving the growth of squares such that the squares are always aligned parallel to one another. When sides of squares meet growth is halted. The area chosen was a grid, 100 X 100; to eliminate edge effects as far as possible the growth of centres off one edge was introduced back onto
Time
Fig. 1. Schematic representation of grid during simulation. diagonally shaded, first layer; shaded with squares, second corresponding to the position of the arrows. Fig. 2. Comparison of theory -___--eqn. (1); I simulation.
and simulation
for the growth
(5)
Unshaded areas, substrate; layer. A surface profile is shown of a single layer.
--
eqn. (4);
the grid on the opposite edge. Figure 1 shows a schematic simulated deposit. The simulation itself was carried out as follows. Deposition was split into discrete time intervals during which existing centres grow by an amount directly proportional to the growth constant V, (provided of course that they do not overlap within this distance in which case growth stops along this edge) after which a nucleation step occurs, the number of nuclei formed being directly proportional to A, the nucleation rate. The positions of the individual nuclei were determined using random numbers, generated by the computer. This process was repeated in each time interval. After each time interval the total growth that occurred within that interval was noted. These data were also split up into the growth for the individual layers. The total growth per unit time gave a measure of the deposition current. The exact shape of a current - time transient is a function of the locations of the nuclei and these are determined in a random manner; thus if a current - time transient is repeated its shape will differ somewhat from the original transient. In this work current - time-transients were repeated twenty times and the mean and standard deviation of these curves determined. In order to obtain a measure of the surface roughness the correlation function and the standard deviation of the thickness of the deposit were determined at various times during a transient. The correlation function indicated whether there was any correlation between the height of the deposit at positions a given distance apart, i.e. it gave a measure of the average spread of the surface undulations, while the standard deviation gave a measure of the average height of the undulation. Because the growth of a centre off one edge was introduced on the opposite edge the correlation function could
398
only be determined 100.
up to a distance apart of 50 points on the grid and not
Results Before using the simulation results to study the effect of nucleation and growth rates on the reproducibility and surface finish of the deposit we must first establish a theoretical expression to which the simulations correspond. The derivation of eqn. (1) assumes that circular nuclei are formed randomly over the electrode surface. If we assume that the nuclei are square rather than circular and that they are randomly oriented with respect to one another then the current time transient for the growth of a single layer is described by i=q4
V2At2exp----
4t3 V2A 3
.
(4)
In the model used for simulation the squares are not oriented at random with respect to one another but are aligned in parallel. To see whether this resulted in significant deviation from the theory the growth of a single layer was simulated and compared with the curves predicted by eqns. (1) and (4). In doing this and all other simulations the following assumptions were made as regards the grid size and the deposition process: 1. The formation of one molecule of the deposit involves a one electron change. 2. One molecule of the deposit is a 0.4 X 0.4 nm square. 3. The thickness of each layer is arbitrary and is termed a “step”. 4. One point on the grid represents a 10 X 10 nm square (i.e. one point is occupied by 625 molecules); the grid length is therefore lop4 cm, a realistic size. . 5. One discrete time interval is 1O-2 sec. This time is short enough to allow for only a small number of nuclei to be formed on the simulation area and therefore results in a relatively smooth current - time transient. The following values of the nucleation and growth constants were assumed for this comparison: V = 10e4 cm set-‘, A = lOlo nuclei crne2 sec~l, these values being typical for the formation of mercury salts on mercury [B, 12 - 141. Figure 2 shows the two theoretical curves; the simulation is represented by the vertical lines, these extending from the mean to plus and minus one standard deviation, being calculated from twenty repetitions of the growth transient. The simulation is in agreement with the theoretical curve for the growth of square centres and shows that eqn. (4) describes the mean of the simulated growth transients; i.e. the fact that the squares are aligned parallel to one another and not randomly oriented is of no consequence.
399
01
02
03 Time
0.4
(s)
Fig. 3. Current-time transient showing the mean (full line) and standard line) for V = 10e4cm set -’ and A = 9 X lOlo cm-‘set-l.
Time
deviation
(dotted
deviation
(dotted
(5)
Fig. 4. Current-time transient$owing line) for V= 3 X 10m4cm set and
the mean (full line) and standard see-l.
A = lOlo cm-’
From eqn. (4) we note that the mean current-time transient for the growth of a single layer (and therefore also the growth of the whole film) is a function of the combined rate constant V'A and not the individual values of V and A. This means that, provided that the product PA remains constant, both V and A can change without changing the shape of the average transient This presents a method of studying the effect of the individual nucleation and growth rates on the reproducibility (in terms of amount deposited in a given time) and the surface roughness of an electrodeposit. To illustrate this compare Figs. 3 and 4 corresponding to growth with V2A = 900;the individual values of V and A aregiven in Table 1. The curves are the same within the standard deviation but the standard deviation in Fig. 4 is far greater than that in Fig. 3. The magnitude of the standard deviation is a measure of the reproducibility of a current-time transient and therefore of the deposit itself. High
400 TABLE
1
Values of V and A studied
with V2A = 900
V(cm set-l )
A( crnm2 see-l )
1 x 1o-4
9 x 1o1O
3 x 1o-4
1 x 1o1O
irreproducibility implies large variations in the periphery length and therefore large variations in the average size and number of growing centres. To study in more detail this and other effects which are observed in simulated deposition experiments, simulations were carried out with V2A 1600 and 100 using the values of V and A given in Table 2. TABLE
=
2
Values of V and A studied -3
V2A(sec
)
V(cm
with V2A = 1600 and 100 set-l)
A(cmW2sec-‘)
1600
1 x 1o-4
1600
2 x lop4
4 x 1o1O
1600
4 x 1o-4
1 x 1o1O
100
1 x 1o-4
1 x 1o1O
Ttme
16 x 10”
(s)
Fig. 5. Mean current-time transients for the total current V= lop4 cm set-’ and A = 16 x lOlo cmp2 set-l.
and individual
layer currents
for
Figure 5 shows the mean current-time transient together with the mean current for the growth of individual layers for V = lop4cm set-’ and A = 16 X lOlo cmW2 set-I. Figure 6 shows the variation of the standard deviation as a percentage of the mean current with the growth constant V for V2A = 1600.This is linear and passes through the origin. Thus the variation of the
401
I
1
I 2 V 104km
3
4
500
sec.‘)
Fig. 6. Dependence of the standard constant V for V2A = 1600 sece3.
1000 V’A
deviation
as a percentage
1500
(se~-~)
of the mean current
on the
Fig. 7. Dependence of the standard deviation as a percentage of the mean current on the overall growth rate constant V2A for V = low4 cm set -’ and A = lOlo cmW2 see-‘.
standard deviation with l/A 1’2 for V2 A = 1600 is also linear and passes through the origin since if V2A is constant V is proportional to l/A1 ‘2. From this we can conclude that increasing V decreases the reproducibility of the deposit while increasing A increases its reproducibility. Figure 7 shows the dependence of the standard deviation as a percentage of the mean current on the overall growth rate, V2A and, as the latter increases at constant V the standard deviation decreases, while as it increases at constant A the standard deviation also increases. Clearly the reproducibility of the deposit is a marked function of the individual values of V and A rather than the combined constant V2A which governs the overall rate of growth. Now in order to relate these variations in V and A to surface roughness for the specific conditions considered the following parameters were determined: 1. Mean height of surface irregularities. To determine the mean height of surface irregularities the mean surface height was first determined. The standard deviation of the height of each individual row and column was then determined (100 rows and 100 columns) and the overall standard deviation was taken as the average of that of the rows and columns. The mean height of the surface irregularities from peak to trough is twice the average standard deviation. 2. Mean spread of surface irregularities. To determine this parameter the correlation function of the height of each row and column was determined and the average taken. The correlation
402
0
005
010 Dtstance
Fig. 8 Correlation 0.25 sec.
015 (104cm
function
for
n
025
050
1 V = lop4cm set -’ and A = 16
Time
X
075
10 0
(5)
lOlocm-’
set-1
after
Fig. 9. Dependence_;f distance of cF;rela>zon 02~ time. 0: V= 10P4cmsec ,A=16x 10 cm set ; X: V = 2 X 10m4 cm set-‘, A = 4 X lOlo cmd2 see-I; A: V = 4 X 10P4 cm set -l, A = lOlo cmP2 set-l.
function can take values of from +l to -1 and is a measure of the correlation between the height of points at varying distances apart. Values close to +1 and -1 indicate strong positive and negative correlation respectively; values close to 0 indicate little or no correlation. Both these parameters have been determined as a function of time for the combined growth rate constant V2A = 1600 with individual values of V and A asshown in Table 2. Figure 8 shows a typical plot of the correlation function against distance between two points; the 95% significance level is shown and we take this as denoting the distinction between correlation and no correlation. The distance at which this transition occurs corresponds to half the mean spread of the irregularities. This distance is shown as a function of time for the three combinations of A and V compatible with V2A = 1600 in Fig. 9. The fairly large scatter for the case with V = 4 X 1O-4 cm set-’ and A = 1 cme2 set-’ can be accounted for by the fact that the surface area represented by the grid is small for such a high value of V. In each case there is a characteristic time to reach a steady state and a characteristic distance of correlation (and therefore of mean spread). These data are shown in Table 3. Figure 10 shows the variation of the mean amplitude of the irregularities with time; again a characteristic time to reach the steady state mean amplitude is observed and in fact these times correspond to those in Fig. 9. The mean amplitudes are given in Table 3.
403 TABLE 3 Mean spread and amplitude of irregularities A(cmm2 set?)
V(cm set-l)
16 x lOlo 4 x 1o1O
1 x 1o-4 2 x 1o-4
1 x 1o1O
4 x 1o-4
Mean spread (Pm)
Mean amplitude (steps)
0.65 0.30
0.27 0.32
1.75 1.30
0.44
0.80
Time to steady state (set)
I 025
050 Time
075
1 O(
(5)
Fig. 10. Variation of the mean amplitude of irregularities with time. Symbols as in Fig. 9.
I 025
050 Distance
075
100
125
(104cm)
Fig. 11. Schematic diagram of the average appearance of the surface of an electrodeposit after growth for 1.0 sec. V = low4 cm set-‘, A = 16 X lOlo cm-’ set-‘; -____ V = 2 x 10W4 cm set-‘, A = 4 X lOlo cme2 set-‘; -.--.- V= 4 x 10v4 cm set-‘, A = lOlo cmP2 set-‘.
From Table 3, the surface structure of the deposit is markedly different depending on the values of V and A at constant V2A. Figure 11 illustrates
404
schematically the average appearance electrodeposits.
of the surface
for each of the three
Discussion From the results obtained we can draw the following conclusions with regard to the surface profile of a deposit and its reproducibility in terms of amount deposited in a fixed time under potentiostatic conditions when the overall growth rate, characterised by V2A, is constant. 1. Large nucleation rates result in a deposit the surface profile of which exhibits high amplitude, high frequency undulations. The amount deposited in a given time is reproducible. 2. High growth rates result in a deposit the surface profile of which exhibits low amplitude, low frequency undulations. The amount deposited in a given time is irreproducible. To explore in more detail the significance of these conclusions we must examine the parameters which determine V and A. The growth rate as a function of overpotential for an irreversible electrochemical reaction can be derived from the Volmer expression giving V= V, exp-
c~nFq RT
’ l-expI
anFq RT
cm set-l t
(5)
where V, is the growth rate at the reversible potential, 01 is the electrochemical transfer coefficient, q is the overpotential and n, F, R and T have their usual significance. It can be shown that the potential dependence of the nucleation rate, for the formation of layers of thickness h, is given by [12] cmP2 sec~l
(6)
where A, is a frequency factor, h is the layer thickness, (I is the surface free energy of the growing edge-solution int.erface, M is the molecular weight of the deposit, N is Avagadro’s number, p is the density of the deposit and e is the charge on an electron. In addition to this the nucleation rate will be increased by the number of dislocations and other surface defects present on the substrate. From an analysis of eqns. (5) and (6) Table 4 is constructed giving the requirements of various parameters such that a deposit falls into one or other of the above categories. This analysis can be used as a basis for practical investigations to improve the surface finish of a deposit in terms of brightness and levelling during the early stages of growth, e.g. if we wished to increase the amplitude and frequency of the surface undulations of a deposit formed by two-dimensional nucleation and growth we could do this by decreasing the film edge-
405 TABLE 4 Effect of various parameters on the nucleation and growth constants Requirements to increase amplitude and frequency of surface undulations
Requirements to decrease amplitude and frequency of surface undulations
Film edge-solution free energy (J)
Low
High
Molar volume of film (M/p)
Low
High
Film layer thickness (h)
Low
High
Electrochemical transfer coefficient (cl)
Low
High
Surface defects
High
Low
solution free energy (surface tension) by the addition of a suitable wetting agent (this would increase the nucleation rate). Alternatively if we wished to decrease the amplitude and frequency of surface undulations we could increase the molar volume of the film by adding inert agents which would be incorporated into the film (this would decrease the nucleation rate). Finally it should be pointed out that the relatively simple situation which has been dealt with in this work was deliberately chosen so as to validate the techniques used; this it has done since the conclusions in this case are obviously correct. However, the techniques can equally well be applied to the more complicated constant current situation. Also, in addition to nucleation and growth rates the effect of parameters such as diffusion layer thickness, addition of levelling agents, etc. on the surface profile of a deposit can be studied. Acknowledgements I should like to thank W. A. Davis who, besides writing all the necessary computer programs for this work, made many helpful suggestions regarding the actual simulation models. References 1 2 3 4 5 6 7
R. D. Armstrong and J. A. Harrison, J. Electrochem. Sot., 116 (1969) 328. S. Toschev and I. Markov, Electrochim. Acta, 12 (1967) 281. S. Toschev and I. Markov, J. Crystal Growth, 4 (1968) 436. I. Markov, Thin Solid Films, 8 (1971) 281. E. Budevski, J. Crystal Growth, 13/14 (1972) 93. S. Toschev, A. Milchev and S. Stoyanov, J. Crystal Growth, 13/14 (1972) 123. I. Markov and D. Kashchiev, J. Crystal Growth, 13/14 (1972) 131.
406 8 9 10 11 12 13
A. Bewick, M. Fleischmann and H. R. Thirsk, Trans. Faraday Sot., 58 (1962) 2200. U. Bertocci, Surface Sci., 15 (1969) 286. A. C. Adams and K. A. Jackson, J. Crystal Growth, 13/14 (1972) 144. G. H. Gilmer and P. Bennema, J. Crystal Growth, 13/14 (1972) 148. J. W. Oldfield, PhD Thesis, University of Newcastle upon Tyne, 1967. R. D. Armstrong, M. Fleischmann and J. W. Oldfield, J. Electroanal. Chem., 14 (1967) 235. 14 R. D. Armstrong and M. Fleischmann, Z. Physik. Chem., 52 (1967) 131.