Sieving effect of agarose on quasi-2D silver pattern electroformation: A pinning-depinning-like transition resulting from gelled silver plating solutions

Article abstract of DOI:10.1021/jp037835r

New features of silver patterns grown in quasi-2D electrochemical cells of parallel plate cathode/anode design utilizing sol and gel aqueous silver plating solutions are reported. Morphology transitions in the silver growth patterns that depend on the composition of plating solutions are observed. These transitions are explained by both a sieving effect due to the presence of agarose and the change in the relative contribution of diffusion and advection to the mass-transport-controlled electrochemical process. Characteristic scaling lengths from growth patterns are related to both the gel structure and the geometry of electrodeposits. Gels consist of a percolated network of gel and randomly distributed colloidal particles, their size and velocity being represented by hyperbolic distribution functions. For silver plating gels a pinning-depinning transition in growth patterns is also observed. From the dynamic scaling analysis of growth pattern 2D profiles, the critical growth and roughness exponent as well as the characteristic lengths of the environment were evaluated. Values of these exponents approach those predicted either by the Kardar, Parisi, and Zhang (KPZ) deterministic equation, or by the cellular automata lattice model that has been proposed for the dynamics of a driven interface in a medium with random pinning forces.

Full text of DOI:10.1021/jp037835r

9720
J. Phys. Chem. B 2004, 108, 9720-9727
Sieving Effect of Agarose on Quasi-2D Silver Pattern Electroformation: A
Pinning-Depinning-like Transition Resulting from Gelled Silver Plating Solutions
M. A. Pasquale, S. L. Marchiano, and A. J. Arvia*
Instituto de InVestigaciones F ´ı sicoquimicas Te o´ ricas y Aplicadas (INIFTA), UniVersidad Nacional de La
Plata-Consejo Nacional de InVestigaciones Cient ´ı ficas y T e´ cnicas, Sucursal 4, Casilla de Correo 16,
(1900) La Plata, Argentina
ReceiVed: December 12, 2003; In Final Form: April 12, 2004
New features of silver patterns grown in quasi-2D electrochemical cells of parallel plate cathode/anode design
utilizing sol and gel aqueous silver plating solutions are reported. Morphology transitions in the silver growth
patterns that depend on the composition of plating solutions are observed. These transitions are explained by
both a sieving effect due to the presence of agarose and the change in the relative contribution of diffusion
and advection to the mass-transport-controlled electrochemical process. Characteristic scaling lengths from
growth patterns are related to both the gel structure and the geometry of electrodeposits. Gels consist of a
percolated network of gel and randomly distributed colloidal particles, their size and velocity being represented
by hyperbolic distribution functions. For silver plating gels a pinning-depinning transition in growth patterns
is also observed. From the dynamic scaling analysis of growth pattern 2D profiles, the critical growth and
roughness exponent as well as the characteristic lengths of the environment were evaluated. Values of these
exponents approach those predicted either by the Kardar, Parisi, and Zhang (KPZ) deterministic equation, or
by the cellular automata lattice model that has been proposed for the dynamics of a driven interface in a
medium with random pinning forces.
1
. Introduction
ion intermediates in the electrodeposition pathway. Additionally,
the presence of colloidal additives in the plating solution
introduces some kind of obstacles for the growing front
displacement, producing a sort of sieving effect that may
influence the growth mode of the solid phase.
The growth mode of solid phases is influenced by the
characteristics of the substrate, the environment, and the
operation routine. Accordingly, new solid phases with specific
bulk and surface properties, such as crystallographic structure
and roughness, and particle shape and size distribution for
dispersed materials can be obtained by adjusting the working
variables. These problems are of utmost importance in fast-
developing areas of applied science related to materials science
and engineering, for instance, in the conformal electrodeposition
In contrast to abundant data available for metal electrodepo-
sition utilizing conventional aqueous plating baths, data for the
electrochemical growth of solids in colloidal systems are very
scarce. In fact, data is available only for copper electrodeposition
from carboxi-methyl-cellulose-containing aqueous plating baths
9
employing either three-dimensional, 3D, or quasi-2D electro-
1
of metals and the design of new materials with preestablished
properties. To have a reliable fundamental support to these
10
chemical cells. In these cases a remarkable influence from
2
the colloidal environment on the growth mode of metal
electrodeposits was observed.
types of processes a key point is to understand the kinetics and
mechanism of solid phase formation under a variety of
environmental conditions.
To study the kinetics of a solid-phase growth such as the
electrocrystallization of metals, a useful approach consists of
following the growth pattern utilizing quasi-2D cells. Data from
this type of experiments permit to distinguish changes in the
kinetics of solid-phase growth at different length and time
scales. The growth rate of the growing front depends on the
electrochemical perturbation routine, the composition of the
environment and its rheological properties, and the presence of
impurities. The growth mode and transitions in the morphology
of electrodeposits are also influenced by the dominant mass
transport mechanism.6-8
Additives in metal plating solutions play a key role in the
characteristics of the electrodeposits. They are involved in
different physical and chemical mechanisms, such as additive
adsorption at the growing front, and the formation of complex
On the other hand, for non-Newtonian fluids, such as agarose-
containing media, the apparent viscosity of sols increases with
agarose concentration to reach a value in which gels are
produced. The high apparent viscosity of gels is able to
3
11-13
overcome buoyancy forces acting at the metal growth front.
Under these conditions one would expect that the effect of
sieving on the growth front will be comparable to that of a
random pinning force quenching to some extent the thermal
noise of depositing metal ions. Accordingly, pinning forces offer
a resistance against the growth front displacement.
In this work, novel features of silver growth patterns produced
by quasi-2D electrochemical cells of parallel-plate cathode-
anode design, utilizing different plating baths under the form
of agarose sols and gels, are reported.
Silver branched patterns produced in supporting electrolyte-
containing plating baths show a decrease in the number of
branches, accompanied by an increase in their average thickness.
These changes in the pattern morphology can be explained by
the complex sieving effect due to the presence of agarose that
4
,5
*
Corresponding author. E-mail address: ajarvia@inifta.unlp.edu.ar
1
0.1021/jp037835r CCC: $27.50 © 2004 American Chemical Society
Published on Web 06/19/2004

Sieving Effect of Agarose
J. Phys. Chem. B, Vol. 108, No. 28, 2004 9721
added slowly. For gels (caga > caga*) two procedures were
followed. For the in situ procedure (hereafter denoted as “gel
in situ”), as the plating solution attained about 50 °C, it was
poured into the cell and allowed to reach the gel-forming
temperature there. For the ex situ procedure (hereafter called
“
gel ex situ”), gel formation was made outside the cell, and the
electrochemical cell was filled up by pasting and pressing the
gel between its plates to ensure good electrical contacts at each
electrode/environment interface.
-1
Cathodic polarization curves at 0.30 V s , utilizing different
plating solutions, allowed us to establish the potential range
where the rate of the electrochemical reaction was under mass-
transport control. Polarization curves were plotted as cathodic
current (Ic) versus cathode-to-anode voltage (∆Ec-a). Values of
Figure 1. Top and lateral view of the quasi-2D and rectangular
electrochemical cells. WE ) working electrode; CE ) counter
∆
Ec-a are referred to the anode since in our case the anodic
overvoltage for silver electrodissolution is negligible. The
cathodic overpotential (ηc) was corrected for the ohmic drop,
i.e., ηc ) ∆Ec-a - IcRc, Rc being the ohmic resistance between
the cathode and the anode. Because of the high value of the
electrode; l ) thickness of the solution layer.
s
modifies the relative contribution of diffusion, advection, and/
or free convection in the mass-transport-controlled process. The
characteristic scale lengths of silver patterns are related to the
structural parameters of gels. The structure of agarose gels can
be described as a percolated network of colloidal material with
randomly distributed colloidal agglomerates, their size distribu-
+
20
exchange current density of the Ag (aq)/Ag electrode, for
η c e -0.20 V, ηc becomes practically equal to the concentration
overpotential of the silver electrode.
Silver growth patterns were produced at constant ∆Ec-a in
the range -1.20 e ∆Ec-a e -0.38 V, for different electrodepo-
sition times (t). Simultaneously, the cathodic current (Ic) and
charge (Qc) transients were recorded using a Radiometer 32
potentiostat. Runs were performed at 298 K.
Silver pattern morphology, root-mean-square roughness (Wrms),
and velocity of the growing front were determined from the
profile of pictures obtained with a stereoscopic microscope
tion following a hyperbolic function and their velocity decreas-
ing with the agglomerate size.14-17
A scheme of the growing front is advanced. It consists of an
immobile percolated network, and small mobile agglomerates,
although their mobility is smaller than that of ionic species in
the “agarose-free” domains of the plating bath. When the driven
force for silver electrodeposits overcomes pinning, the abrupt
change in the velocity of the growing front is consistent with a
pinning-depinning-type transition.
The dynamic scaling analysis of 2D profiles from growth
patterns allowed us to evaluate the growth and roughness
exponents, as well as the characteristic lengths of the environ-
ment. Values of these exponents approach the prediction of the
Kardar, Parisi, and Zhang (KPZ) deterministic equation includ-
ing a random noise term as well as that of a cellular automata
lattice model that has been proposed for the dynamics of a driven
interface in a medium with random pinning forces.18,19 To clear
this ambiguity further work is in progress.
(Stemi 200 Zeiss) coupled to a camera (CCD Hitachi 220),
connected to a video screen and a computer equipped with a
frame grabber and an image analyzer (Contron Electronics
KS300). Occasionally, a sequence of photographs was taken
using a Contax camera 167 MT coupled to the microscope.
To estimate the size distribution of the colloidal agglomerates,
Rhodamin B-stained gel samples were observed by confocal
laser scanning microscopy (CLSM) using an Olympus FV 300
microscope.
3
. Results
.1. Characteristics of Agarose-Containing Sols and Gels.
From the rheological standpoint, agarose-containing aqueous
.01 M sulfuric acid is sufficiently stable to be used for a few
hours without a noticeable decrease in the flow time (tf)
3
2
. Experimental Section
0
Quasi-2D rectangular cells with a vertical plate electrode
arrangement were utilized to follow the evolution of growth
patterns resulting from silver electrodeposition. Each rectangular
cell consisted of top and bottom parallel flat plates made of
either Lucite or glass separated by the distance ls ) 0.025 cm
and a parallel-edge silver (99.9% purity) cathode-anode ar-
rangement (width lw ) 5 cm, cathode-to-anode separation
distance lc-a ) 2 cm; Figure 1).
1
1,13
determined with a viscometer.
Agarose sols, either freshly
prepared or after aging for a certain time, behave as non-
newtonian fluids, whereas gel-like structures obey Bingham
1
2
plastic flow. It should be noted that for caga g 0.02% w/v, the
increase in tf can be related to a significant increase in the
percent of gel structure. It has been shown that the relative
percent of free agglomerates (XL) depends on caga according to
Silver growth patterns were obtained from x M silver sulfate
1
2
equation
(0.008 e x e 0.024) + caga % w/v agarose (0 e caga e 0.6)
aqueous solution, with addition of 0.5 M sodium sulfate + 0.01
M sulfuric acid as supporting electrolyte. Plating solutions, either
as sols or gels, were prepared at the appropriate range of caga,
considering that the critical micellar concentration of agarose
1
/2
XL ) 1 - 0.21c
aga
(1)
which indicates the occurrence of at least two distinguishable
structural domains in both sols and gels. Thus, for caga ) 0.05%
w/v (sol), eq 1 yields XL ) 0.95, and for caga ) 0.6% w/v (gel),
XL ) 0.85.
Small magnification pictures from “gel in situ” and “gel ex
situ” (Figure 2) show a structure dominated by agglomerates
12
(MW ≈ 120,000) is caga* ) 0.12% w/v. Analytical reagent
grade chemicals and Milli-Q water were used for plating solution
preparation. These solutions were used after nitrogen saturation.
Agarose-containing solutions (sol, caga < caga*) were made
by first dissolving silver sulfate and sodium sulfate in water
under heating followed by agarose addition under stirring. As
the system was cooling, drops of concentrated sulfuric acid were
2
and islands greater than 2000 µm , the largest islands being
observed for “gels in situ”. Conversely, due to the mechanical

9722 J. Phys. Chem. B, Vol. 108, No. 28, 2004
Pasquale et al.
Figure 4. Cathodic polarization curves (0.03 mV s-1) after ohmic drop
correction; 0.024 M silver sulfate + 0.5 M sodium sulfate + 0.01 M
sulfuric acid (dotted trace); 0.024 M silver sulfate + 0.5 M sodium
sulfate + 0.01 M sulfuric acid +0.6 % w/v agarose gel (solid trace).
Quasi-2D rectangular cell; 298 K.
Figure 2. Photographs of 0.024 M silver sulfate + 0.5 M sodium
sulfate + 0.01 M sulfuric acid + 0.6 % w/v agarose gel layers from ex
situ (a) and in situ (b) procedures.
In our case, the Korcak empirical law is fulfilled for C ) 2.5
and B ) 1.3. A similar decaying distribution function has been
observed for gels formed from other carageen derivatives.22
From eq 2, as the fractal dimension is DF ) 2B, it results in DF
=
2.5, as expected for a percolation process under chemical
2
1
bonding control.
For caga > 0.2% w/v, the agglomerate size distribution (Figure
) starts from an agglomerate average diameter of about 1 µm,
3
the minimum size that can be discriminated and associated with
agarose agglomerates from CLSM photographs. Accordingly,
assuming a spherical agglomerate of agarose (MW ) 120,000)
-
4
of radius 0.5 × 10
agglomerate is 5.2 × 10
apparent gel density is close to 1 and the average weight of a
cm, the average volume of each
cm . Further, considering that the
-
13
3
-
19
single agarose molecule is 2 × 10
g, the average numbers
of agarose and water molecules in 1 µm diameter agglomerates
6
9
are 2.6 × 10 and 1.0 × 10 , respectively. The size range of
these agglomerates agrees fairly well with those that have been
1
5,22
reported elsewhere.
3.2. Electrochemical Data. Cathodic polarization curves from
Figure 3. Plot of P(A > a), the probability of appearance of
agglomerates of area A greater than the minimum area a, versus a. (9)
Experimental data. The curve represents the Korcak empirical law.
either agarose-free or agarose-containing plating solution with
an excess of supporting electrolyte show well-defined cathodic
limiting currents (Ic,lim; Figure 4). For the agarose-free solution
-2
Ic,lim ) 22.7 mA cm , and for the agarose-containing bath Ic,lim
manipulation involved in the preparation method, “gels ex situ”
show an agglomerate size distribution wider than that of “gels
in situ”.
Images of both gels (Figure 2a,b) confirm the appearance of
randomly distributed percolated islands forming channels that
are occupied by a large number of randomly distributed small
agglomerates. Therefore, agarose gels can be described as a
percolated, almost immobile structure with a large number of
small agglomerates.
Data from both conventional images and CLSM photographs
were plotted as a size distribution function of agglomerates with
area (A) larger than a value (a) versus a (Figure 3). According
to the Korcak empirical law,21 the probability P(A>a) of an
agglomerate area A exceeding a minimum area a is given by
the hyperbolic function
-2
)
19.7 mA cm . This difference in Ic,lim is consistent with a
change in the mechanism of silver electrodeposition caused by
the suppression of macroscopic flux in gelled media and by a
slight change in the solvodynamic radius of ionic species in
solution that may result from hydrogen bonding interactions with
agarose molecules at the hydration sheath of ions.
Data from potentiostatic current transients for gels with
different cAg SO were displayed as Qc/Ic versus t plots (Figures
2
4
5
and 6). Despite the current fluctuations due to stochastic
disorder in the colloidal environment, data from potentiostatic
current transients for “gels in situ” (Figure 5) tend to approach
a straight line with slope 1 for t < 100 s. The same slope results
from data of “gels ex situ” for t < 15 s (Figure 6). In the former
case, the fastest displacement of the growing front in the
direction perpendicular to the electrode surface is observed, as
expected for the more open structure of the environment.
Conversely, data resulting from “gels in situ” (Figure 5) for t
-B
P(A>a) ) Ca
(2)

Sieving Effect of Agarose
J. Phys. Chem. B, Vol. 108, No. 28, 2004 9723
Figure 6. Q
cell; ∆Ec-a ) -1.2 V, x M silver sulfate + 0.5 M sodium sulfate +
.01 M sulfuric acid + 0.6 % w/v agarose “gel ex situ”. (a) x ) 0.008;
b) x ) 0.012; (c) x ) 0.018; 298 K.
c c
/I versus t plots obtained with the quasi-2D rectangular
Figure 5. Q
cell; ∆Ec-a ) -1.2 V, x M silver sulfate + 0.5 M sodium sulfate +
.01 M sulfuric acid + 0.6 % w/v agarose “gel in situ”, (a) x ) 0.016;
b) x ) 0.018; (c) x ) 0.024; 298 K.
c c
/I versus t plots obtained with the quasi-2D rectangular
0
(
0
(
>
100 s tend to approach the straight line with slope 2, and a
similar slope results from “gels ex situ” for 15 e t e 100 s.
For t > 100 s, both gels approach again the slope 1. This change
in the slope of the Qc/Ic versus t plots can be related to a change
in the dominant mass-transport mechanism from diffusion to
advection, influenced to some extent by the change in surface
area of the cathode.23
3
.3. Silver Pattern Morphology. Growth patterns produced
at ∆Ec-a ) -1.20 V from agarose-free 0.024 M silver sulfate
0.5 M sodium sulfate + 0.01 M sulfuric acid baths (Figure
a) involve an initial first quasi-compact silver layer about 0.05
+
7
mm thick (not observable at the scale of the photograph) that
appears for 0 e t e 15 s. Subsequently, a second branched
layer about 0.20 mm thick is formed for 15 e t e 30 s. The
electrodeposit morphology consists of protrusions that turn into
branches growing in the direction of the electric field. For t >
3
0 s, the growing pattern consists of separated dense trees
growing in the same preferred direction, partially screening the
growth of smaller neighbor trees and columns.
On the other hand, growth patterns from 0.024 M silver
sulfate + 0.5 M sodium sulfate + 0.01 M sulfuric acid + 0.6%
w/v agarose (“gel in situ”) (Figure 7c) for t < 15 s consist of
a first a quasi-compact silver layer, whereas for t > 15 s,
protrusions and branching are observed. In this case, the average
interbranch distance is shorter than that observed for “gels ex
situ”. Additionally, for 45 e t e 1275 s, a pinning effect on
the growing front is evident. Finally, for t > 1300 s, only a
few branches continue growing, displaying a dense fan-like
branched pattern.
Figure 7. Sequences of silver patterns obtained in the quasi-2D
rectangular cell: ∆Ec-a ) -1.2 V, (a) 0.024 M silver sulfate + 0.5 M
sodium sulfate + 0.01 M sulfuric acid; (b) idem + 0.6 % w/v agarose
“
gel ex situ”; (c) idem + 0.6 % w/v agarose “gel in situ”. (d) Details
of the growth sequence from (c). 298 K. For t ) 90 s, channels can be
clearly distinguished in (d). The influence of gel structure on the
electrodeposit morphology can be appreciated.
Similar experiments run in “gel ex situ” at ∆Ec-a ) -1.20
V (Figure 7 b) show, for t < 60 s, the formation of a large
number of protrusions merging from the initial quasi-compact
silver layer. The number density of these protrusions decreases
rapidly with t due to coalescence of a number of neighboring
protrusions and hindrance of further growth for others. Subse-

9724 J. Phys. Chem. B, Vol. 108, No. 28, 2004
Pasquale et al.
Figure 8. Images of silver quasi-2D growth patterns formed from 0.024
M silver sulfate + 0.5 sodium sulfate + 0.01 M sulfuric acid + 0.6%
w/v agarose gel at ∆Ec-a ) -1.20 V and 298 K, and their corresponding
histograms. A reference box was drawn at the border of a spot of
methylene blue-stained gel opposite to the cathode. (a) Grayscale
histogram obtained either in the absence of a silver electrodeposit or
when the latter was still far from the reference box edge. (b) Histogram
obtained when the moving front was just ready to touch the reference
box edge. In this case, the gray density (darker pixels) on the left-hand
side of the reference box has notoriously increased. Note the significant
change in both the half width and symmetry of the histogram, as well
as the shift in its maximum value.
Figure 9. Plots of 〈V〉 and V
M
versus cAg SO from x M silver sulfate
2
4
+
0.5 M sodium sulfate + 0.01 M sulfuric acid at ∆Ec-a ) -1.2 V,
+) agarose-free solution; (∆) 0.6 % w/v agarose “gel ex situ”; (O)
.6 % w/v agarose “gel in situ”; 298 K.
(
0
number of darker pixels in the reference box (Figure 8), as one
should expect from the displacement of smaller agglomerates
pushed ahead by the growing front. From these results, it is
reasonable to consider that small nonpercolated moving ag-
glomerates interfere with the displacement of the growing front,
i.e., the local rate of silver electrodeposition.
quently, for t > 200 s, only a few columns dominate the growth
pattern, and for t > 500 s, the average velocity of the growing
front perpendicular (〈V〉) to the initial electrode surface dimin-
ishes, yielding wider column tips.
3.5. Influence of Silver Ion Concentration on the Velocity
of the Growing Front. In our experiments, the driven force of
the electrochemical process is the concentration gradient of silver
ions at the cathode/solution interface. Correspondingly, to
investigate the influence of the driven force on the pattern
morphology, runs setting ∆Ec-a ) -1.2 V and caga in the range
0.008 e cAg SO e 0.024 M (the solubility of silver sulfate at
The correlation between pattern morphology and gel structure
(Figure 7d) confirms, in these cases, the key role of the presence
of agarose-containing environment in determining the growth
mode of silver electrodeposits (Figures 5 and 6). The average
electrodeposition rate slows down considerably at channels
and voids with a high density of small agglomerates, in contrast
to free channels where fast branching is accomplished. The
pinning effect is reflected throughout the change in the apparent
density of silver electrodeposits, particularly in gelled plating
solutions.
2
4
298 K) were made.
The plots of both the average velocity of the growing front
perpendicular to the initial electrode surface 〈V〉, and the
maximum growing front velocity in the same direction VM,
versus cAg SO for agarose-free solution (Figure 9a,b) show a
2
4
linear increase in both 〈V〉 and VM with cAg SO . Conversely, for
2
4
-
2
3
.4. Mobility of Small Agarose Particles. The agarose
both gels “in-situ” and “ex-situ” and cAg SO < 1.7 × 10 M,
2 4 2 4
2
4
agglomerate motion hypothesis in gels was demonstrated by
running the following experiment in which the distribution of
preset blue methylene-stained spots was disturbed by the
growing front itself. For this purpose, “gel ex situ” containing
methylene blue-stained spots was located approximately 0.25
cm away from the working electrode. The comparison of
pictures before and after silver electrodeposition allowed us to
detect changes in the distribution of stained spots caused by
the motion of the growing front. We selected a 700 × 700 µm
rectangular region of the image (open drawn down rectangle in
Figure 8) so that one-half of the stained spots were allocated
inside the selected region and the rest were placed between this
region and the working electrode surface. A first grayscale
histogram (blank) was constructed for pixels inside the box,
providing information about how stained agglomerates (darker
ones) were distributed in the selected region. Then, silver
electrodeposition was started, and as the growing front reached
spots just outside the selected region, another grayscale histo-
gram was made. The latter showed the appearance of a larger
the VM versus cAg SO and 〈V〉 versus cAg SO plots show a slow
increase in both VM and 〈V〉 with cAg SO , whereas for cAg SO >
2
4
2
4
-
2
1.7 × 10 M, both VM and 〈V〉 increase rapidly with cAg SO .
2
4
Correspondingly, there is a critical concentration of silver sulfate,
-
2
c*Ag SO ≈ 1.7 × 10 M, that is associated with a dramatic
2
4
increase in the velocity of the growing front that approaches
the value for agarose-free plating solution. This sudden change
in the displacement of the growing front correlates with the
morphology transition of silver electrodeposits. This transition
starts from the slow growth of branched columns initially formed
on the first compact thin silver layer, to the fast growth of a
fan-like morphology of only a few large branched columns
(Figure 7). Then, the morphology transition that appears for
cAg SO > c*Ag SO indicates that the driving force of the
2
4
2
4
electrochemical reaction becomes sufficiently large to overcome
the resistance imposed by the sieving effect due to the presence
of agarose. Hence, the overall effect resembles that of a
pinning-depinning transition that has been observed in the
2
4
growth of solid phase in porous media.

Sieving Effect of Agarose
. Discussion
.1. Fundamental Aspects of Agarose Sols and Gels.
J. Phys. Chem. B, Vol. 108, No. 28, 2004 9725
4
4
Electrochemically produced silver patterns depend on both the
sol and gel structure and composition of the plating solution.
For aqueous agarose sols, domains with different structures
(Figure 2) can be explained via specific inter- and intramolecular
interactions of OH groups from agarose molecules and water.
Agarose is an alternating polysaccharide copolymer of alga
origin, its backbone containing (1f4) and (1f3) linked 3,6-
anhydro-R-L-galactose.2
5-27
The 4-linked residues give rise to
an extended ribbon, whereas the 3-linked units generate a double
helix consisting of parallel 3-fold chains with a left-hand
direction and a pitch of 1.90 nm. A large part of the agar
backbone can be replaced by neutral and charged groups. This
structure can persist under hydrated conditions in sol and gel
three-dimensional networks. The formation of gels involves the
association of chains through double helices to develop ran-
domly coiled agglomerates in a three-dimensional network
structure.2
5,26
In agarose the repeating sequence structure is
28
interrupted by the occurrence of other residues. Such inter-
ruptions can produce a remarkable modification in bulk proper-
ties by terminating intermolecular association through structur-
ally and hysterically regular junction zones, leading to a 3D
aggregate formation. The same effect is also produced by
departure from the regular primary sequence. Then, the inter-
chain association through double helical junction zones is
terminated by galactose residues in the normal, unbridged ring
form in place of the fused anhydride ring structure. This
Figure 10. Scheme for silver electrodeposition in agarose gels.
Colloidal particles are characterized by wide size and velocity distribu-
tion functions. 〈V〉 is the average moving front velocity; V
p
is the
velocity of electrodepositing cations; V denotes the velocity of agarose
L
agglomerates. The dashed trace represents the average moving front
line.
c c c
TABLE 1: Values of K Estimated from the Q /I versus t
Plots for Gelled Silver Plating Solutions under Different
Transport Mechanism on an Initially Planar Surface
introduces a backbone kink that cannot be accommodated within
the ordered structure.2
5,28
The formation of this molecular
mechanism
area
K
organization, in particular intermolecular networks, has been
1
2
pure diffusion
pure diffusion
advection
S(t) ) constant
S(t) ) m t
2
2/3
1
proved by rheological studies of agarose sols and gels.
The sieving effect caused by the presence of agarose in the
plating bath depends on the disordered structure of sols and
gels (Figures 2 and 3), which in turn depends on the preparation
procedure. Data from both in situ and ex situ gels indicate that
the sieving effect is equivalent to a pinning effect on the average
rate of displacement of the growing front. The fractal structure
of gels is consistent with an anisotropic rate of displacement of
the growing front imposed by the pinning effect.
S(t) ) constant
V ) constant
f
advection
S(t) ) m t
) constant
0.5
V
f
and enhancing advection in going from agarose-free to gelled
plating solution as concluded from potentiostatic current
transient data plotted as Qc/Ic versus t plots (Figures 5 and 6).
For each regime, the relationship Qc/Ic ) Kt can be obtained to
evaluate the proportionality constant K. For mass-transport-
controlled electrochemical processes, K becomes overpotential-
independent but changes with the dominant mass-transport
The pinning effect can be related to characteristic lengths
for the growing phase in the direction perpendicular and parallel
to the electrode surface, respectively. These lengths are related
to the size of aggregates and channels in the environment. Then,
the value of c*Ag SO related to the pinning-depinning transition
4
mechanism and evolution of the cathode area. Thus, for solid
2
4
metal electrodeposition the ratio Qc/Ic is obtained from the
expression
in the growth mode of silver electrodeposits should be related
to the critical lengths derived from the gel structure.
The gel structure can be better described as randomly
distributed large quasi-frozen percolated structures and small
mobile agglomerates, as depicted in Figure 10. The ratio
between the average rate of displacement of small agglomerates
and the average silver electrodeposit front rate becomes a
relevant parameter that determines the global morphology of
the electrodeposits and the kinetics of the process. In our case,
the electrodeposition rate depends on the local concentration
gradient of silver ions and can be modified by properly adjusting
the concentration of silver ions in the plating bath. Then, the
environment-dependent kinetics of silver pattern growth can be
understood in terms of a random size distribution of pinning
agglomerates (random walkers) covering a wide range of
mobility that increases with the reciprocal of the average
agglomerate radius.
∫
jcS dt
Qc
)
(3)
Ic
j S
c
In eq 3, S is the effective area of the cathode surface, j ) I /S;
j ) j (t) and S ) S(t). The expression j ) j (t) depends on
c
c
c
c
c
c
whether the data correspond to the range of time in which
diffusion, stationary free convection, or advection dominates
mass transport, and S(t) accounts for the evolution of S during
silver electrodeposition.
The stationary cathodic limiting current values of K resulting
from the preceding situations are assembled in Table 1. Their
comparison to experimental data (Figures 5 and 6) provided an
idea of the dominant mass-transport mechanisms during silver
electrodeposition.
The presence of agarose also alters the mass transport
properties in the environment by suppressing free convection
The dynamic scaling theory was applied to obtain relevant
information about the characteristics of roughening, the domi-

9726 J. Phys. Chem. B, Vol. 108, No. 28, 2004
Pasquale et al.
nant molecular mechanism determining silver pattern shape, and
the occurrence of possible transitions in the growth mode.
4.2. Dynamic Scaling Analysis. The dynamic scaling theory
predicts that the interface width of a growing front (profile),
i.e., the mean square surface height fluctuation, W(L, t) is defined
by
1 L
2
Wrms ) W(L,t) )
[h(i,t) - hh (t)]
(4)
∑
x
L
i)1
with h(i,t) being the height of the i column, and hh (i,t) is the
mean height
L
1
hh (t) )
h(i,t)
(5)
∑
L
i)1
For length scale L and growing time t, W(L, t) scales as24,29,30
W(L,t) ∝ L f(t/LR/â)
R
(6)
(7)
(8)
Equation 6 for t . LR/â becomes
W(L,t) ∝ LR
Figure 11. Log Wrms versus log t and log Wrms versus log L plots from
silver electrodeposit profiles: ∆Ec-a ) -1.20 V; (a) 0.024 M silver
sulfate + 0.5 M sodium sulfate + 0.01 M sulfuric acid + 0.6 % w/v
agarose “gel ex situ”; (b) idem + 0.6 % w/v agarose “gel in situ”, 298
K. Dashed traces represent the slopes referred to in the text.
and for t , LR/â
W(L,t) ∝ tâ
In equations 6-8, R and â are the roughness and growth
exponent, respectively. The value of R is related to the surface
texture of the deposit, and correspondingly, to the fractal surface
dimension DF of the self-affine surface by DF ) 3-R. Thus, for
R f 1, DF f 2, the surface tends to be Euclidean (ordered),
whereas when R f 0, DF f 3, the surface exhibits an increasing
degree of disorder. As exponents R and â are interdependent,
there is a simple way to collapse the temporal and spatial
randomly distributed quenched noise, and the size and velocity
distribution of quenchers. For gels, irrespective of their prepara-
tion, the values R ) 1.25 ( 0.2 and â ) 0.95 ( 0.15 are very
different from R ) 0.63 and â ) 0.63 resulting from either the
simple directed percolation depinning (DPD) model or the DPD
3
1-33
with some modifications.
Despite data scattering and the limited extension of the linear
relationship usually obtained from experimental data, scaling
exponents measured for silver electrodeposition from both gels
approach those predicted by either the Kardar, Parisi, and Zhang
R
R/â
W(L, t) data onto a single curve by plotting W/L versus t/L
.
Let us first consider silver growth patterns in supporting
electrolyte-containing “gel in situ” (caga ) 0.6% w/v). The
logW(L,t) versus log t plot (Figure 11a) shows three ranges of
t where linear functionalities are fulfilled. For the range 1.15
e log t e 1.8, the slope of the best linear function is â ) 0.88
(
KPZ) deterministic equation or by a cellular automata lattice
1
9,34
model
that has been proposed for the dynamics of a driven
interface in a medium with random pinning forces. The
scattering of data may be partly attributed to the irregular
structure of gels. Additional data are, however, required to have
the possibility to discriminate between these two models. The
main drawback lies in the fact that the complexity of gel systems
exceeds that of the models that have been proposed, particularly
the occurrence of static and dynamic domains in the environment
producing a combined pinning effect. This fact encourages
further experimental work about these systems that will provide
data to be used to advance in the system modeling.
(
>
0.15; for 1.8 < log t e 2.2, â ) 0.33 ( 0.15, and for log t
2.2, â ) 0.88 ( 0.2. This plot shows two crossovers for log
t* ) 1.85 (t* ) 70.8 s), and for log t* ) 2.25 (t* ≈ 180 s). On
the other hand, data for log t > 2.2 were plotted as log W versus
log L (Figure 11a) to evaluate R. This plot exhibits two linear
ranges of log L with different slopes, namely, for the range 2.25
e log L e 3.0, R ) 1.25 ( 0.15; for log L > 3.13, the value
of W remains almost constant within 0.1 log unit, i.e., R ) 0 (
0
.1. This plot defines a crossover for log L* ≈ 2.8 (L* ≈ 790
µm).
Similar plots from data obtained with the same “gel ex situ”
caga ) 0.6% w/v) led to a log W versus log t plot (Figure 11b)
with two linear portions with different slopes, namely, â ) 1
0.2 for 1.9 e log t e 2.5 and â ) 0.33 ( 0.05 for log t >
.5, and a crossover for log t* ) 2.5 (t* ) 316s). Otherwise,
5. Conclusions
(
By increasing agarose concentration in the plating solutions,
the contribution of natural convection to the mass-transport-
controlled silver electrodeposition is progressively diminished.
For gels, a pure diffusion control is reached, as could be
concluded from the potentiostatic current transients and from
Qc/Ic versus t plots for a constant cathode area.
(
2
the log W versus log L plot (Figure 11b) also exhibits two linear
portions, a first one for log L < 2.95 approaching R ) 1.25 (
0
.05, and a second one for log L > 2.95 with R ) 0 ( 0.1, and
a crossover at log L* ) 3 (L* ) 1000 µm).
.3. Scaling Analysis Data and Proposed Models. The
Gels consist of islands and channels that affect mass-transport
properties of depositing ions and produce a sieving effect that
is reflected in the electrodeposit growth pattern. Gels produce
a randomly distributed quenched noise at the moving interface
of the electrodeposit. A pinning-depinning transition appears
4
dynamic scaling analysis of growth patterns raises the possibility
of advancing in the interpretation of the main mechanism
associated with silver electrodeposition in an environment with

Sieving Effect of Agarose
J. Phys. Chem. B, Vol. 108, No. 28, 2004 9727
by changing the driving force of the moving interface. There is
a critical silver salt concentration associated with this transition.
Generally, silver-electroformed pattern morphology in a gelled
environment is characterized by a relatively small number of
wide branches with tip splitting, greatly influenced by the island-
channel structure of gels, which depends on the preparation
method, i.e., ex-situ or in-situ. The complicated cathode area
versus t function is related to the gel structure.
From the dynamic scaling analysis of growth pattern 2D
profiles, the critical growth and roughness exponents, as well
as the characteristic lengths of the environment, were evaluated.
These exponents approach the predictions of both the Kardar,
Parisi, and Zhang (KPZ) deterministic equation, and a cellular
automata lattice model that has been proposed for the dynamics
of a driven interface in a medium with random pinning forces.
Results reported in this work are interesting in two main
aspects. They provide, for the first time, quantitative information
for the general problem of growth patterns of a solid phase under
diffusion control. For this purpose, data derived from silver
growing patterns, using silver electrodeposition as test process,
offer the possibility of a first comparison to model data.
(8) Barkey, D. P.; Watt, D.; Liu, Z.; Raber, S. J. Electrochem. Soc.
994, 141, 1206.
1
1
3
(9) Arvia, A. J.; Baz a´ n, J. C.; Carrozza, J. S. W. Electrochim. Acta
968, 13, 81.
(10) Wang, M.; Ming Nai-ben; Bennema, P. Phys. ReV. E 1993, 48,
825.
(
11) Pasquale, M. A.; Marchiano, L. S.; Bolz a´ n, A. E.; Arvia A. J. J.
Appl. Electrochem. 2003, 33, 431.
12) . Gavira Ruiz, J. M.; Novella, M. L.; Moreno, R.; Gavira, J. A. J.
Cryst. Growth 2001, 232, 165.
13) . Carro, P.; Marchiano, S. L.; Hern a´ ndez Creus, A.; Gonz a´ lez, S.;
Salvarezza, R. C.; Arvia, A. J. Phys. ReV. E 1993, 84, R2374.
14) Starchev, K.; Sturm, J.; Weill, G.; Brogen, C.-H. J. Phys. Chem. B
1997, 101, 5659.
15) Attwood, T. K.; Nelmes, B. J.; Sellen, D. B. Biopolymers 1988,
7, 201.
(
(
(
(
2
(16) Waki, S.; Harvey, J. D.; Bellamy, A. R. Biopolymers 1982, 21,
1
909.
(17) Stellwagen, J.; Stellwagen, N. C. Biopolymers 1994, 34, 187.
(
(
(
18) Leschhorn, H. Physica A 1993, 195, 324.
19) Leschhorn, H., Ph.D. Thesis, Ruhr University, Bochum, 1994.
20) Gerischer, H.; Tischer, R. P. Z. Electrochem. 1957, 61, 1159.
(21) Avnir, D. The Fractal Approach to Heterogeneous Chemistry, J.
Wiley: New York, 1989; Ch. 5.
22) Reid, D. S.; Bryce, T. A.; Clark, A. H.; Rees, D. A. Discuss.
Faraday Soc. 1974, 57, 275.
23) Pasquale, M. A.; Marchiano, S. L.; Vicente, J. L.; Arvia, A. J., (to
(
(
be submitted).
Acknowledgment. This work was financially supported by
the Consejo Nacional de Investigaciones Cient ´ı ficas y T e´ cnicas
CONICET) and PICT 98 No. 06-03251 from Agencia Nacional
de Promoci o´ n Cient ´ı fica y Tecnol o´ gica of Argentina.
(24) Barab a´ si, A.-L.; Stanley, H. E. Fractal Concepts in Surface Growth,
Cambridge University Press: New York, 1995.
(
25) Rees, D. A.; Morris, E. D.; Thon, D.; Madden, J. K., In The
(
Polysaccharides; Aspinall, G. O., Ed.; Academic Press: New York, 1982;
Vol I, Ch. 5, p 195.
(
(
26) Arnott, S.; Fulmer, A.; Scott, W. E. J. Mol. Biol. 1974, 90, 269.
27) Rees, D. A.; Welsh, E. J. Angew. Chem., Int. Ed. Engl. 1977, 16,
References and Notes
2
14.
(
1) Moffat, T. P.; Bonevich, J. E.; Huber, W. H.; Stanisheky, A.;
(28) Perlin, A. S.; Casu, B. In The Polysaccharides; Aspinall, G. O.,
Stafford, D. R.; Josell, D. J. Electrochem. Soc. 2000, 147, 4524.
Ed.; Academic Press: New York, 1982; Vol. I, Ch. 4, p 182.
(29) Godr e´ che, C. Solids Far From Equilibrium; Cambridge University
Press: Cambridge, England, 1990; Ch. 6.
(30) Family, F.; Vicsek, T. Dynamics of Fractal Surfaces; World
Scientific: Singapore, 1991.
(31) Tang, L. H.; Leschorn, H. Phys. ReV. A 1992, 45, R8309.
(32) Buldyrev, S. V.; Barab a´ si, A.-L.; Caserta, F.; Havlin, S.; Stanley,
H. E.; Vicsek, T. Phys. ReV. A 1992, 45, R8313.
(33) Leschhorn, H.; Tang, L. H. Phys. ReV. Lett. 1994, 49, 1238.
(34) Sol e´ , R. V.; Manrubia, S. C., Orden y Caos en Sistemas Complejos,
Universitat Polit e´ cnica de Catalunya, Spain, 1996.
(
2) Fleury, V.; Barkey, D. Europhys. Lett. 1996, 36, 253.
(3) Matsushita, M.; Sano, M.; Hayakawa, Y.; Hongo, H.; Sawada, Y.
Phys. ReV. Lett. 1984, 5, 286.
4) . Pasquale, M. A.; Marchiano, S. L.; Arvia, A. J. J. Electroanal.
Chem. 2002, 532, 255.
5) . Pasquale, M. A.; Marchiano, S. L.; Arvia, A. J. J. Electrochim.
Acta 2002, 48, 153.
6) . Fleury, V.; Chazalviel, J.-N.; Rosso, M.; Sapoval, B. J. Electroa-
nal. Chem. 1990, 290, 249.
7) Kuhn, A.; Argoul, F. J. Electroanal. Chem. 1995, 397, 93.
(
(
(
(