Solving the Stefan problem for a solid phase growth on plane plate and spherical surfaces and testing of theoretical equations

Article abstract of DOI:10.1016/j.electacta.2005.11.011

Solutions of the Stefan problem in the 2D space considering a moving boundary of a solid deposit growing under mass transfer control on either plane plate or spherical solid substrates are reported. In the former case, the displacement of the growth front at the plane plate occurs perpendicularly to the substrate, whereas for the latter it shifts radially. For both substrates, in the absence of convection and surface roughness effects, the phase growth kinetics is determined by diffusion and advection, the latter being due to the linear displacement of the growth front with time. For both geometric arrangements the theory predicts two limiting kinetic situations, namely a diffusion control when the time and/or the radius of the substrate approach zero, and an advection control for the reverse conditions. For the spherical substrate, when its radius tends to infinity, the kinetics of the process approaches that found at the plane plate substrate. Theoretical potentiostatic current density transients are tested utilising growth pattern data for the formation of 2D silver dense branching electrodeposits on a plane plate cathode in a quasi-2D cell, and silver electrodeposits on spherical cathodes employing a high viscosity plating solutions.

Full text of DOI:10.1016/j.electacta.2005.11.011

Electrochimica Acta 51 (2006) 3969–3978
Solving the Stefan problem for a solid phase growth on plane plate and
spherical surfaces and testing of theoretical equations
M.A. Pasqualea,b,∗,1, S.L. Marchianoa,1, J.L. Vicentea, A.J. Arviaa,1
a Instituto de Investigaciones F´ısicoquimicas Teo´ricas y Aplicadas (INIFTA), Universidad Nacional de La Plata-Consejo Nacional de
Investigaciones Cient´ıficas y Te´cnicas, Sucursal 4, Casilla de Correo 16, (1900) La Plata, Argentina
b Facultad de Ciencias Fisicomatema´ticas e Ingenier´ıa, Pontificia Universidad Cato´lica Argentina, Buenos Aires, Argentina
Received 3 September 2005; received in revised form 29 October 2005; accepted 5 November 2005
Available online 15 December 2005
Abstract
Solutions of the Stefan problem in the 2D space considering a moving boundary of a solid deposit growing under mass transfer control on
either plane plate or spherical solid substrates are reported. In the former case, the displacement of the growth front at the plane plate occurs
perpendicularly to the substrate, whereas for the latter it shifts radially. For both substrates, in the absence of convection and surface roughness
effects, the phase growth kinetics is determined by diffusion and advection, the latter being due to the linear displacement of the growth front
with time. For both geometric arrangements the theory predicts two limiting kinetic situations, namely a diffusion control when the time and/or
the radius of the substrate approach zero, and an advection control for the reverse conditions. For the spherical substrate, when its radius tends to
infinity, the kinetics of the process approaches that found at the plane plate substrate. Theoretical potentiostatic current density transients are tested
utilising growth pattern data for the formation of 2D silver dense branching electrodeposits on a plane plate cathode in a quasi-2D cell, and silver
electrodeposits on spherical cathodes employing a high viscosity plating solutions.
© 2005 Elsevier Ltd. All rights reserved.
Keywords: Stefan problem; Diffusion–advection; Plane plate and spherical electrode; Silver electrodeposition
1. Introduction
cally denoted as Stefan problems, irrespective of the driving
force. They are often found in a number of processes occurring
in different areas of natural sciences and technologies.
Most rate equations that have been derived solving the
convective-diffusion differential equations for the growth of a
solid phase at a solid substrate have considered that the inter-
face remains at rest [1]. This implies a limitation to the extension
of those rate equations to processes in which the solid phase
growth front moves inward (forward) into the bulk of the reac-
tant supplier environment [2]. In this case, the solution of the
corresponding differential equations depends on the moving
boundary conditions [3].
Since about a century ago, solutions of moving boundary
problems have been proposed for heat transfer in iceberg dis-
placement [4–6], although their extension to mass transfer pro-
cesses, such as those involved in solid phase growth, has received
much less attention. These mass transfer problems are generi-
Inprinciple, bothFick’sandFourier’sequationscanbesolved
in each phase for different situations. (i) For a fixed boundary the
transfer equations are solved for the immobile interface assum-
ing constant spatial domains for each phase. (ii) For a moving
boundary the spatial domain of each phase changes as the inter-
face front moves according to a certain law that is known a priori.
(iii) For an implicit free boundary the spatial domain changes
and the moving boundary equations are unknown. They usually
depend on the physics of each problem and have to be found in
order to solve them [7].
This paper compares two solutions of the Stefan problem in
the 2D space considering the moving boundary of solid phase
growth [case (ii)] on either a plane plate or a spherical solid under
mass transfer control. In the former case, the displacement of
the growth front occurs perpendicularly to the substrate surface,
whereas for the latter it shifts radially. For this case, when the
radius tends to infinity, the limiting solution of the mass transfer
differential equation approaches the solution for the plane plate
∗
Corresponding author. Tel.: +54 221 4257439; fax: +54 221 4254642.
E-mail address: miguelp@inifta.unlp.edu.ar (M.A. Pasquale).
ISE member.
1
0013-4686/$ – see front matter © 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.electacta.2005.11.011

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M.A. Pasquale et al. / Electrochimica Acta 51 (2006) 3969–3978
substrate. Neither changes in the macroscopic roughness of the
moving boundary nor convective effects from density gradients
are considered in solving the transfer equations.
On the other hand, theoretical potentiostatic current density
transients are tested with experimental data related to the elec-
trochemical formation of silver dense branching patterns under
mass transfer control on plane plate and spherical cathodes. For
this purpose, to make the contribution of free convection negligi-
ble, in the former case a quasi-2D cell and conventional plating
solutions are utilised, whereas for the latter high viscosity plat-
ing solutions are employed. After correction for the roughness of
the electrodeposits, the agreement of theoretical and experimen-
tal data in the time range where the contribution of advection is
dominant is fairly good.
2. The solution of Fick’s equation with the Stefan
boundary condition
2.1. Plane plate substrate
Let us consider the deposition of species i on a plane plate
substrate of infinite dimensions by a diffusion process from a
fluid phase (plating solution) that occurs normally (y-direction)
to the plane plate. This process is expressed by Fick’s equation
in Cartesian coordinates
∂2ci
∂y2
∂ci
∂t
Di
=
(1)
Di and ci being the diffusion coefficient and the concentration
of species i (reactant) in the fluid phase and t is the deposition
time. The advance of the growth front occurs in the y-direction.
Eq. (1) is solved with the following boundary conditions:
Fig. 1. (a–c) Concentration profiles calculated for the plate plane with Eq. (3)
plotted as ci vs. y − βt at different values of β for t = 0.1, 1, 10 and 100 s. The
value of t increases as indicated by the arrows. The following values were used in
ci(y, 0) = ci0;
ci[s(t), t] = 0;
s(t) = β · t
y ≥ 0;
t > 0
s(t) = s(0)
(2a)
(2b)
(2c)
(2d)
the calculations: zi = 1; Di = 1.39 × 10−5 cm2 s−1; ci0 = 4.8 × 10−5 mol cm−3
.
These figures were taken from the electrodeposition of silver from aqueous
solution [9].
ci(y, t) = ci0;
y → ∞
Eq. (3) fulfills boundary conditions (2a) and (2b). Plots of ci
versus y − βt for various values of β and different values of t
are shown in Fig. 1. At constant y − βt the gradient of ci per-
pendicular to the substrate increases with β. This effect is more
remarkable at longer t.
Let us apply Eq. (3) to evaluate the potentiostatic current den-
sity transients, which represent the instantaneous rate of metal
electrodeposition from the plating solution on the plane plate.
The rate of this process under diffusion control is given in terms
of the cathodic current density jc(t)
Condition (2a) corresponds to the initially uniform reactant con-
centration in the plating solution, and s(t) is the front coordinate
at time t. The surface area of the growth front is equal to that
of the inert substrate upon which the growth of the solid phase
commences (t = 0); β denotes the advance velocity of the mov-
ing boundary. Condition (2b) indicates that the concentration
of i just at the growing front is null for t > 0, as expected from
mass transfer-controlled kinetics. Condition (2c), usually called
the Stefan condition, indicates the instantaneous location of the
growing front.
Solving Eq. (1) with boundary conditions ((2a)–(2d)), the
explicit expression for ci(y, t) results in
ꢁ
ꢄ
∂ci(y, t)
∂y
jc(t) = ziFDi
(4)
ꢀ
ꢁꢂ
ꢃ
ꢄ
ci0
β
y=βt
ci = ci0 −
exp
· (βt − y)
ꢃꢄ
2
Di
zi, ci and F being the electric charge per reactant species, the
reactant concentration in the plating solution and the Faraday
constant, respectively. Then, considering the differential expres-
ꢁ
ꢂ
ꢁ
ꢄꢅ
y − 2βt
y
√
√
× 1 − erf
+ 1 − erf
(3)
2 Dit
2 Dit

M.A. Pasquale et al. / Electrochimica Acta 51 (2006) 3969–3978
3971
t → ∞ is
lim [jc(t)]t→∞ = ziFci0β = (jc)adv
(7)
Eq. (7) corresponds to the advection limit to the mass transport
process. The greater the value of β, the faster the advection
regime is reached (Fig. 2b).
The linear dependence of jc on β from Eq. (7) contrasts with
the ꢁβꢂ1/2 dependence that would result, in general, from laminar
flow forced convection [1,11]. Therefore, the different hydro-
dynamic regimes that might be involved in the process can be
clearly distinguished (Fig. 2b).
On the other hand, Eq. (5) predicts a kinetic transition in the
electrodeposition process from a non-steady planar diffusion
Stefan process to a steady advection regime.
To further explore the behavior of Eq. (5), let us define the
dimensionless current ratio ρ(τ)
jc(τ)
ρ(τ) =
(8)
(jc)adv
and the dimensionless variable (τ)
β2t
τ =
(9)
4Di
Then, by replacing Eqs. (5) and (7) into Eq. (8), and further con-
sidering the dimensionless variable τ from Eq. (9), the following
dimensionless equation is obtained:
ꢁ
ꢄ
√
1
2
1
√
ρ(τ) =
exp(−τ) + erf( τ) + 1
(10)
Fig. 2. (a) Plots of jc vs. t calculated from Eq. (5) at different values of β, as
indicated. The height of the advection plateau increases with β. The same values
of zi, Di and ci0 indicated in Fig. 1 were used. (b) Plots of jc vs. β calculated
from Eq. (5) for different t, as indicated. Values of zi, Di and ci0 are the same as
those indicated in Fig. 1. The dashed trace (at the bottom) corresponds to the jc
vs. β1/2 plot resulting for laminar flow forced convection [1].
(πτ)
Obviously, due to the contribution of advection in Eq. (10), for
τ → ∞ it results in ρ(τ) → 1.
2.2. Spherical substrate
sion of Eq. (3), for jc(t) one obtains
Letusconsiderthesametypeofdiffusionprocessastheabove
that takes place on a spherical substrate in the radial direction
(r). This process is expressed by Fick’s equation in spherical
coordinates
ꢆ
ꢂ
ꢃ
ꢂ
ꢁꢂ
ꢃꢄ
1/2
Di
πt
β2t
4Di
ꢈ
jc(t) = ziFci0
exp
−
ꢇ
ꢇ
ꢃ
1/2
ꢂ
ꢃ
β
β
t
∂ci
∂t
∂2ci
∂r2
2 ∂ci
r ∂r
+
erf
+ 1
(5)
= Di
+
(11)
2
2
Di
The first RHS term in Eq. (5) is the product of the diffusion
term for a fixed plane plate and an exponential factor. The lat-
ter decreases the value of jc(t) faster than that predicted by the
solution of Fick’s equation for a fixed boundary [1]. The con-
tribution of the first term becomes more remarkable for t → 0.
The greater the value of β, the faster the decay of jc(t) (Fig. 2a).
The second β-dependent RHS term increases the value of jc(t)
even for t = 0. The limit of Eq. (5) for t → 0 is
In this case, the solid growth front advances in the r-direction.
Then, Eq. (11) is solved with the following boundary condi-
tions:
ci(r, 0) = ci0;
ci[s(t), t] = 0;
s(t) = r0 + βt
ci(r, t ) = ci0;
r ≥ r0
t > 0
(12a)
(12b)
(12c)
(12d)
ꢇ
ꢉ
ꢉ
Di
πt
β
Di
πt
(jc)adv
[jc(t)]tꢀ1 = ziFci0
+
= ziFci0
+
(6)
r → ∞
2
2
r0 beingtheinitialradiusofthesubstrate. Themeaningofbound-
ary conditions ((12a)–(12d)) is the same as that indicated above
for the plane plate substrate. The explicit solution of Eq. (11)
The second term for t → ∞ leads to a steady regime that cor-
responds to advection control [8,10]. The limit of Eq. (5) for

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M.A. Pasquale et al. / Electrochimica Acta 51 (2006) 3969–3978
with the conditions ((12a)–(12d)) is
Plots of ci versus ξ for r0 = 1 cm and different values of β are
shown in Fig. 3. The radial gradient of ci increases with β, the
effect being more remarkable at longer t.
For the electrodeposition of reactant i with the electric charge
zi the instantaneous rate equation, expressed as the cathodic cur-
rent density jc(t), is
ꢂ
ꢃ ꢁ
ꢄ
ꢊ
ci0
r
β
t
ci(r, t ) = ci0 −
exp
−
ξ
· r0ϕ(ξ, t) + β ϕ(ξ, t ) dt
2Di
0
(13)
where
ꢂ
ꢃ
ꢂ
ꢃ
∂ci
∂ξ
∂ci
∂r
ξ = r − r0 − βt
(14)
jc(t) = ziFDi
= ziFDi
(16)
(17)
ξ=0
r=r +βt
0
Then, the differential expression of Eq. (13) for ξ = 0 is
ꢉ
ꢁ
ꢄ
ꢀ
ꢂ
ꢃ
ꢂ
ꢃꢅ
∂c(r, t )
ci0
r
β
β
t
2
β2
√
=
1 +
[r0 + βt] + r0
erf
β
+ r0
exp
−
t
∂r
2Di
2Di
4Di
4Di
πDit
ξ=0
ꢌ
ꢆ
ꢊ
ꢋ
ꢍ
ꢈ
ꢂ
ꢃ
ci0
β
t
2
β2
t
√
+β
erf
β
+
exp
−
t
dt
r
2Di
4Di
4Di
πDit
0
as the differential expression of Eq. (15) for ξ = 0 is
and
ꢁ
ꢄ
ꢂ
ꢃ
ꢂ
ꢃ
βt1/2
√
1
β2t
ꢀ
ꢂ
ꢃ
ꢂ
ꢃ
∂ϕ(ξ, t)
∂ξ
β
1
β
ξ − βt
√
√
= −
erf
−
exp
−
ϕ(ξ, t) =
exp
−
ξ
erfc
2Di
4Di
4Di
πDit
ξ=0
2
2Di
4Dit
(18)
ꢂ
ꢃ
ꢂ
ꢃꢅ
β
ξ + βt
√
+ exp
ξ
erfc
(15)
To obtain the equation for jc(t) it is convenient to define the
following dimensionless variable:
2Di
4Dit
β2t
τ =
(19)
4Di
and
√
1
√
ψ(τ) = erf( τ) +
exp(−τ)
(20)
πτ
By introducing first Eq. (17) in Eq. (16), and then considering
Eqs. (19) and (20), the expression of jc(t) results in
ꢎ
ꢏ
1
r0
jc(τ) = ziFβc10 1 + ψ(τ)
2
r
ꢁ
ꢄ
ꢊ
ci0
τ
+ ziFDi
1 +
ψ(τ ) dτ
(21)
r
0
According to Eq. (21) (Fig. 4a) the fastest decay of jc(t) is
observed initially, whereas for t > 700 s, jc(t) tends to attain a
limiting value that increases with r0. By setting t at 1000 s, a
rapid increase in d[jc(t)]/dr0 with r0 is obtained. Likewise, the
decay of jc(t) exhibits a bump at about 100 s due to β. These
bumps are more noticeable as r0 is increased. Current density
transients for r0 = 1 cm also depend on β (Fig. 4b) and attain
a limiting value that increases linearly with β, as occurs for
the plane plate. In agreement with the advection contribution,
the greater the value of β, the sooner the appearance of cur-
rent density bumps (Fig. 4b, inset). For r → ∞ the value of jc(t)
approaches Eq. (6) for the plane plate substrate.
On the other hand, the limit of Eq. (21) for τ → ∞ and r → ∞
can be calculated considering the limit of Eq. (20) and the fact
that the numerical integration of the second term of Eq. (21) for
τ → ∞ approaches τ. Then, the limit of Eq. (21) results in
Fig. 3. (a–c) Plots of concentration profiles calculated for the sphere with Eq.
(13) at different values of β for t = 0.1, 1, 10 and 100 s, and r0 = 0.05 cm. The
value of t increases as indicated by the arrows. The same values of zi, Di and ci0
indicated in Fig. 1 were used.
3
lim [jc(τ)]t→∞
=
(jc)adv
(22)
4

M.A. Pasquale et al. / Electrochimica Acta 51 (2006) 3969–3978
3973
cell for the plane plate cathode, and glycerol to increase the vis-
cosity of the plating solution for the spherical cathodes, ensured
a negligible contribution of free convection to the process, as
required.
A quasi-2D-rectangular horizontal electrochemical cell was
made by a parallel edge arrangement of two 99.9% purity silver
sheets used as cathode and anode, respectively, with a cathode-
to-anode separation distance lc–a = 2 cm. To avoid border effects
and initially ensure a homogeneous primary current distribution
on the cathode, a Teflon® mask 0.025 cm thick was symmetri-
cally placed on both sides of the cathode. The top and the bottom
of the cells consisted of two parallel either Lucite® or glass flat
plates separated by the distance ls = 0.025 cm.
The cylindrical electrochemical cell 1 cm in height con-
sisted of a spherical cathode made by flame melting a 99.99%
purity, 0.0125 cm radius platinum wire to produce a sphere
with an initial radius r0 = 0.026 cm. The cathode was placed
at the centre of a silver ring anode suspended from the top
of the cell. The anode was made of 99.9% purity silver sheet
(ra = 1 cm). An optical quality plane glass at the bottom of the
cell allowed following the evolution of the silver electrodeposit.
Both cells were mounted on a suspended table to avoid the influ-
ence of spurious mechanical vibrations on the kinetics of the
process.
The following plating solutions were utilised. Solution I:
aqueous 0.024 M silver sulphate + 0.5 M sulphuric acid; solu-
tion II: aqueous 0.024 M silver sulphate + 1 M sulphuric acid;
solution III: aqueous 0.024 M silver sulphate + 1 M sulphuric
acid + 5 M glycerol. The excess of sulphuric acid behaved as
a supporting electrolyte to make the contribution of migration
negligible. Glycerol-containing solutions behave as Newtonian
fluids [15]. The viscosity (η) correction to the diffusion coeffi-
cient of the reacting ion (Di) was made using the Walden ratio
(Diη/T = 5.53 × 10−10 cm4 s−2 K−1 at 298 K). Solutions were
prepared from a.r. chemicals and MilliQ® water. Runs were
made at 25 1 ◦C.
Fig. 4. (a) Plots of jc vs. t calculated for the sphere with Eq. (21) for
β = 10 ␮m s−1 at different values of r0, as indicated. Values of zi, Di, and ci0 are
the same as those in Fig. 1. The current density bumps become more remarkable
as r0 is increased. (b) Plots of jc vs. t calculated with Eq. (21) for r0 = 1 cm at
different values of β, as indicated. Values of zi, Di and ci0 are the same as those in
Fig. 1. The inset shows the same plots for a shorter range of time to distinguish
the current density bumps.
3. Experimental
For each plating bath the cathodic polarization curves were
run at v = 0.30 V/s to determine the potential range of the mass
transport kinetic regime. Polarization curves were plotted as
cathode-to-anode voltage (ꢀEc–a) versus cathodic current (Ic).
The effective cathodic potential was ꢀEeff = ꢀEc–a − IcRo, Ro
being the ohmic resistance between the cathode and the anode
of the cell.
Silver patterns were grown at constant ꢀEc–a in the range
−0.90 ≥ ꢀEa–c ≥ −0.20 V. The pattern morphology and the
silver growth front velocity were followed from photographs
obtained with a stereoscopic microscope (Stemi 200 Zeiss) cou-
pled to a Canon Powershot G 5 camera. Simultaneously, the
cathodic current (Ic) and charge (Qc) were recorded using a
Radiometer 32 potentiostat.
Potentiostatic current density transient data from silver dense
branching pattern formation by electrodeposition from conven-
tional aqueous plating solutions were utilised to test the cor-
responding theoretical equations derived for plane plate and
spherical cathodes. The selected testing process offers a number
of advantages for our purpose. It rapidly reaches a diffusion-
controlled kinetics due to the high exchange current density
of the silver electrode in aqueous media [12–14]. The process
starts with the nucleation and growth of silver islands a few
monolayers in thickness that occur in a short time and involve a
negligible charge as compared to that of the diffusion-controlled
silver phase growth. Otherwise, the low, almost constant appar-
ent density of silver dense branching patterns leads to a large
almost constant growth front velocity closely approaching the
Stefan condition. The almost constant roughness of the growth
front and its small irregularities, most of them located within the
thickness of the diffusion layer, makes it feasible to introduce
a roughness correction factor in the evaluation of the cathodic
current density. Finally, the use of a quasi-2D electrochemical
4. Results and discussion
4.1. Plane plate substrate
The potentiostatic current transients (Fig. 5) exhibit a com-
plicated behavior in the range 5–150 s, i.e., in the interval where

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M.A. Pasquale et al. / Electrochimica Acta 51 (2006) 3969–3978
Fig. 5. Comparison of experimental (raw data) and theoretical plots of cathodic
current density transients for silver electrodeposition from a planar plate cath-
ode in aqueous 0.024 M silver sulphate + 1 M sulphuric acid. Experimental
data: (1) ꢀEc–a = −0.700 V and β = 26.2 ␮m s−1; (2) ꢀEc–a = −0.600 V and
β = 10.1 ␮m s−1; (3) ꢀEc–a = −0.475 V and β = 6.8 ␮m s−1. Theoretical data for
β = 26.2, 10.1 and 6.8 ␮m s−1 (solid traces).
the influence of the Stefan effect becomes remarkable and then
quasi-stationary advection sets in.
A sequence of photographs of quasi-2D silver electrodeposits
grown on the plane plate cathode (Fig. 6a) shows the stochas-
tic nature of silver electrodeposition at levels of the order of
10 ␮m. The apparent density and roughness of these electrode-
posits for t > 60 s (Fig. 6b) remain almost constant, as expected
for an isotropic growth. The roughness correction was estimated
for a minimal and maximal situation, namely, when there is
no interference of the surface regularities on the outer plane
of the diffusion layer the roughness factor approaches one,
whereas the maximal situation is given by the analysis of the
electrodeposit cross-section through the value of the root mean
square eight [9,16,17]. These roughness factors are comprised
between 1.8 and 3.0, i.e., they are greater than those obtained
from the ratio jc,exp/jc,the at time t. Therefore, to evaluate from
pattern profiles how large the influence of the front rough-
ness on the average diffusion layer thickness at the cathode is,
let us compare the average protrusion height and cavity depth
at the growth front, which are in the range 0.004–0.015 cm,
to the average diffusion layer thickness estimated from the
Fig. 6. (a) Sequence of silver growth patterns formed at the plane plate cathode
in aqueous 0.024 M silver sulphate + 1 M sulphuric acid. ꢀEc–a = −0.700 V. (b)
Apparent density of silver electrodeposits vs. t plots for ꢀEc–a = −0.700 and
−0.800 V. Error bars are indicated.
which is intermediate between that of the initial plane plate,
and the true roughness of the growth front. Correspondingly,
the average properties of the interface can be utilised to test the
continuous, deterministic model of advection. At the scale of
the quasi-stationary diffusion layer thickness the influence of
the fractal surface of the electrodeposit becomes unnoticeable
[18].
ratio ꢁδNꢂ = DAg /β, where DAg = 1.39 × 10−5 cm2 s−1 is the
silver ion diffusion coefficient in the aqueous solution, and
β is the average velocity of the growth front in the range
7 ≤ β ≤ 26 ␮m s−1. Accordingly, the value of ꢁδNꢂ is in the
range 0.005–0.020 cm. Then, the profile of the diffusion layer
should be to some extent modulated by the irregularities of
the front surface. This means that at the time scale of our
measurements the stochastic formation of protrusion and cav-
ities at the micro scale should actually produce a roughness,
Experimental data, after the above correction, fulfill reason-
ably well the linear dependence jc versus ꢁβꢂ plot predicted by
the advection term in Eq. (7) (Fig. 7). This dependence contrasts
the experimental ꢁβꢂ1/2 linear dependence that would result from
laminar flow forced convection perpendicularly on a plane plate
electrode [19,20].
There is a good correlation of data resulting from different
runs in terms of the dimensionless Eq. (10) (Fig. 8). The surpris-
ingly good prediction for the dimensionless correlation indicates
+
+

M.A. Pasquale et al. / Electrochimica Acta 51 (2006) 3969–3978
3975
Fig. 7. Plot of jc vs. β: (ꢀ) raw data and (ꢃ) data after roughness correction.
The straight line represents the advection term, and the lower trace corresponds
to laminar flow forced convection.
that the influence of roughness on the jc(τ)/(jc)adv ratio tends to
cancel.
4.2. The spherical substrate
The current transients (Fig. 9a) run on the spherical cathode
involve the contribution of the increase in the surface area, which
Fig. 9. (a) Potentiostatic current transient (raw data, dashed trace) for silver elec-
trodeposition on a spherical cathode (r0 = 0.026 cm). Aqueous 0.024 M silver
sulphate + 1 M sulphuric acid. (1) Glycerol-free and (2) 5 M glycerol-containing
solution. ꢀEc–a = −0.200 V. The full traces (3) and (4) are calculated with Eq.
(21), fortheabove-mentionedplatingsolutions, for β = 5and1.3 ␮m s−1, respec-
tively. (b) Plot of f vs. t for silver electrodeposits run at spherical cathodes with
the plating solutions indicated in (a), in the absence of glycerol (1), and 5 M
glycerol-containing solution (2). The dashed traces indicate the interval in which
a constant surface roughness electrodeposit is approached.
is accounted by the solution of Eq. (11), and a roughness mod-
ulation of the diffusion layer thickness similar to that described
above for the plane plate cathode. However, there is an important
difference between the plane plate and the spherical electrode
as the application of the theory to experimental data becomes
only possible for those electrodeposits approaching spherical
shapes and constant roughness. In fact, in the absence of free
convection silver dense branched electrodeposits tend to acquire
a spherical symmetry for t > 100–400 s for low viscosity plating
solutions (Fig. 10a), and for t > 400–600 s for high viscosity
plating solutions (Fig. 10b). For these values of t the average
growth pattern radius (ꢁrꢂ) increases linearly with t (Fig. 11).
Conversely, for t > 1500 s, an apparent increase in roughness and
a small decrease in the apparent density of growth patterns can
be noticed making these data useless for our purpose. These
changes can be due to a relevant contribution of free convec-
tion that turns the ball-shaped patterns into slightly conical ones
Fig. 8. Dimensionless plot of experimental potentiostatic cathodic current den-
sity transients for silver electrodeposition on the plane plate cathode. The full
trace represents Eq. (10). Symbols correspond to different values of β, as indi-
cated in the figure. (The permission of The American Chemical Society for
reproducing this figure from reference 10 is acknowledged.)

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M.A. Pasquale et al. / Electrochimica Acta 51 (2006) 3969–3978
Fig. 10. Silver growth pattern sequence at the spherical (r0 = 0.026 cm) cathode in aqueous 0.024 M silver sulphate + 1 M sulphuric acid. ꢀEc–a = −0.200 V. (a)
Glycerol-free and (b) 5 M glycerol.
[20,21]. The larger the initial radius of the cathode the higher
the value of jc and, correspondingly, the sooner the influence
of free convection due to density gradients in the bulk of the
solution.
The roughness factor (f), calculated from the ratio
f2D = jc,exp/jc,the, begins to increase after exceeding an induc-
growth patterns approach the spherical shape (Fig. 11). The
roughness factor correction, as applied to the plane plate
cathode can be extended to the spherical cathode only for
the range of t in which a sphere-shaped electrodeposit is
approached and its radius increases linearly with t. Under
these circumstances, the apparent density of the electrode-
posit becomes constant and the average roughness ꢁfꢂ in this
range of t (Fig. 9b) can be taken as the correction factor for
jc,exp. Accordingly, a fairly good agreement between the the-
oretical and experimental current density transients, irrespec-
tive of the viscosity of the plating solution can be obtained
(Fig. 12).
∼
∼
=
tion time tin 40 s for low viscosity solutions and t
200 s
=
in
for high viscosity solutions (Fig. 9b). These values of tin
could be assigned to the time required to achieve a cath-
ode completely silvered [22]. Later, the f versus t plots attain
an almost constant value within a certain range of t that
agrees with that in the ꢁrꢂ versus t plots where macroscopic

M.A. Pasquale et al. / Electrochimica Acta 51 (2006) 3969–3978
3977
Fig. 11. Plot of ꢁrꢂ vs. t from silver growth patterns produced at the spherical cathode of r0 = 0.026 cm. Aqueous 0.024 M silver sulphate + 1 M sulphuric acid.
ꢀEc–a = −0.200 V. (a) Glycerol-free and (b) 5 M glycerol. The linear portion of the r vs. t plots are indicated by the dashed traces. The horizontal trace corresponds
to the value of r0.
Fig. 12. Correlation of current density transient experimental data in the advection regime from the spherical cathode, after roughness correction, and the predictions
of Eq. (21). Silver electrodeposition from aqueous 0.024 M silver sulphate + 1 M sulphuric acid. ꢀEc–a = −0.200 V. (a) glycerol-free (β = 5 ␮m s−1) and (b) 5 M
glycerol (β = 1.3 ␮m s−1).
5. Conclusions
Acknowledgments
The solutions of the Stefan problem, as presented, are
of special interest to understand mass transport mechanisms
at moving boundary interfaces. The rate equations resulting
for a plane plate and a spherical geometry converge for the
limiting conditions derived for r → ∞. For both substrate
geometries, the solid growth under mass transfer kinetics and
linear displacement of the interfacial boundary with time, in the
absence of convection and surface roughness effects, involves
the simultaneous contribution of diffusion and advection. The
contribution of advection to the kinetics becomes more remark-
able when the duration of the process is longer and the radius
of the surface increases. For both geometries the solid growth
process exhibits a transition in kinetic control from diffusion to
advection.
This work was financially supported by the Consejo Nacional
´
´
de Investigaciones Cientıficas y Tecnicas (CONICET), the Uni-
versidad Nacional de La Plata, Argentina, and the Pontificia
´
Universidad Catolica Argentina, Buenos Aires, Argentina.
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