Simulation of electrochemical growth of polycrystalline deposit under
galvanostatic conditions
E. Zinigrad, D. Aurbach
Bar-Ilan University, Ramat-Gan, Israel
ABSTRACT
The electrochemical growth of polycrystalline lithium deposits under
galvanostatic conditions was studied. An increase of the real surface area of the
polycrystalline deposits is theoretically described as a function of time. The
dependence of the number of simultaneously growing crystals on the current density
was obtained. The experimental data confirm the validity of the model simulation for
the growth of polycrystalline lithium deposits at constant current density in 1,3dioxolane (DN)/1 M LiAsF6 electrolyte solution.
INTRODUCTION
The electrochemical growth of single metals and alloys clusters under
galvanostatic and potentiostatic conditions was theoretically studied and
experimentally verified [1-3]. The relationship between the radius of the spherical
cluster and the growth current was obtained theoretically [1] using mass balance of
the deposited electroactive species. The theory that relates the number of crystals,
applied to electrochemical growth of metals, was developed by Schottky [4, 5] and
modified by Baraboshkin [6].
The electrochemical growth of polycrystalline lithium metal deposits under
galvanostatic conditions was studied in the present paper. An increase of the real
integral surface area of deposits was theoretically described. The verification of the
theoretical assumption was carried out by investigating galvanostatic deposition of
lithium in 1,3-dioxolane (DN)/1 M LiAsF6 electrolyte solution. The dependence of
simultaneously growing crystals’ number on current density was obtained.
THEORY
According to Milchev [1] the growth of single metal cluster during
galvanostatic deposition as a function of time is described by equation (1):
r3  r3 
0
3I
 (   0 ) ,
4 ( ) zF
(1)
where r is the cluster radius,
I - current for a single cluster, I = const,
v - molar volume of the metal deposited,
 - wetting angle of a single cluster growing on a foreign substrate under galvanostatic
conditions,
() = 1/2 – 3/4 (cos  + 1/4 cos3),
z - the electrochemical valence,
F - the Faraday constant.
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We neglect the initial mass of the critical nucleus and assume that the cluster has a
hemispherical shape. For a hemisphere, the wetting angle equals to 900, cos  = 0 and
hence () = 1/2. According to our assumption r0 = 0 at 0 = 0, then, for a
hemisphere, equation (1) takes the shape:
r3 
3I
 .
2  zF
(2)
The surface area of one hemispherical crystal equals:
s = 2r2.
(3)
Substituting (2) into (3), we obtain an equation that relates the surface area as a
function of time:
s ( )  2 (
3I 
 )2 / 3 .
2  zF
(4)
The deposits formation may be described in terms of the following
consecutive steps: formation of the nucleus, its growth up to the final size, formation
of new nucleus on top of its surface and so on. It’s impossible to measure the real
surface area of simultaneously growing crystals of polycrystalline deposit by a direct
microscopic observation as it is for a single crystal [1]. Therefore an indirect method
for surface area growth estimation may be applied. As an example, a method can be
developed for the studying of the electrochemical growth of a deposit, which
chemically reacts with the electrolyte solution components as in the case of lithium
deposition in 1,3-dioxolane (DN)/1 M LiAsF6 electrolyte solution. The rate of the
chemical interaction of the deposited metal with the electrolyte solution is
proportional to the real working surface.
It is known that during electrochemical deposition of lithium from aprotic
organic solution the deposited metal reacts with the electrolyte solution, namely – the
solvent, the anion and the contamination. We assume that the interaction occurs only
on the increscent surface area, i.e. on the surface of the crystals during the growth
time from its formation up to achieving the final size. In fact, the rate of the solvent
reaction on the bare Li surface is very fast and decreases quickly due to passivation.
Nevertheless, we roughly approximate average chemical interaction rate (V) as a
constant. The average rate represents the reaction of the solvent with the lithium
exposed to the solution during the cluster growth time (c) from its formation up to its
final size. The number of the moles of solvent reacted at time c with one cluster
equals:
c
M   V  s( )  d .
(5)
0
After the integration we obtain the number of moles of solvent that reacted with one
growing cluster as a function of its growth time:
6
3Iv
M  V   1 / 3 ( ) 2 / 3 c 5 / 3 .
5
zF
615
(6)
We express c using the final radius of the cluster from equation (2) and obtain the
number of the moles of solvent reacted with one growing cluster as a function of its
final radius:
M  V  0.8   2
zF 5
r .
I v
(7)
The total number of moles of solvent that reacted with the growing deposit of metal
over the whole experiment (Mtotal) can be calculated assuming that each of the clusters
has the same final size:
Mtotal = M .N,
(8)
where N is total number of the grains in the deposit that have grown over the time of
the constant current application.
It’s possible to calculate N from the experimental data:
N=Vtotal/Vsingle,
(9)
where Vtotal –volume of all formed grains and Vsingle=
2
   r 3 - volume of one
3
hemispherical crystal.
The volume of all grains, which is formed during the deposition process is
determined by the current density i and the test time τtotal:
Vtotal 
i   total 
.
zF
(10)
At this point the total number of the grains in the deposit equals:
N=
3  i   total
.
2 zF  r 3
(11)
By substituting (7) and (11) into (8), we obtain:
M total  Vtotal  1.2  n  r 2   total
,
(12)
where n = i / I, n is the number of the grains simultaneously growing on 1 cm2 of the
geometrical surface of an electrode.
EXPERIMENTAL
The growth of a lithium deposit was studied in 1,3-dioxolane solution
containing 1 M LiAsF6. The ready-made AA Li metal cell was used as two electrodes
electrochemical cell. AA cells are characterized as a system with standard conditions
and highly reproducible properties. AA cells were subjected to deposition-dissolution
cycles at different deposition current densities: 0.3, 0.75, 1.25, 1,75, 2.25 mA/cm2, up
to the end of life state, which was defined as cells’ capacity equals to zero. In all tests
the dissolution current density was the identical: 1.25 mA/cm2. The morphology of the
lithium deposits obtained at different current densities was studied by Scanning
Electron Microscope. To determine the mean grain size at the its final size, we treated
photomicrographs of the deposits using Scion Image software. At least 100 grains
were measured for each current density. All the tests were carried out at room
temperature.
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After 10 cycles
After 110 cycles
Fig. 1. SEM micrographs of lithium deposits formed at a deposition current
density 1.25 mA/cm2.
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RESULTS AND DISCUSSION
Mean grain diameter, micron
One of the several approximations that were assumed by us was the hemispherical
shape of the growing crystals. As seen in Fig. 1, in the dioxolane based electrolyte
solution, lithium crystals have a shape similar to the spherical one in the investigated
range of the current density. Figure 1 shows typical Li deposits obtained at a
deposition current density 1.25 mA/cm2 at different magnifications. As can be seen,
most of the crystals have indeed hemispherical shape. The morphology of the lithium
deposits was found to be practically independent on the cycle number. With the
increase of current density, the final grain size reduces (Fig. 2).
It is known [7] that the cathodes and the separators of exhausted batteries
were usually dry. In fact, the whole solvent disappeared due to its reduction by
growing lithium deposit. It is possible to calculate the mean rate of the interaction of
the lithium deposit with dioxolane (v) using the deposition time i. e. the total
accumulated time of the deposition processes in all cycles until the cell is exhausted.
It is important to note that the initial content of the dioxolane in AA cell is constant.
The mean rate of interaction between solvent and growing Li deposit can be obtained
as:
v = Mtotal /τtotal ,
(13)
35
30
25
20
15
10
5
0
0
0.5
1
1.5
2
2.5
Current density, mA/cm2
Fig. 2. Grain size variation as a function of current density.
where v is the mean interaction rate in mole/s; Mtotal – the total number of the moles
of dioxolane reacted over the whole period of time of accumulated lithium growth and
equals to the initial content of the dioxolane.
As demonstrated at Fig. 3, the mean interaction rate grows with the increase
the deposition current density. This phenomenon relates to the increase in the real
surface area of the growing deposit.
We assume that the number of the grains growing simultaneously on 1 cm2 of
the geometrical surface of electrode in the stationary state is constant for a given
current density. In accordance to (12), the real average interaction rate V calculated
for 1 cm2 of the real working surface area is:
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V
M total
.
1.2  n  r 2   total
(14)
The number of the hemispherical grains growing simultaneously on 1 cm2 of the
geometrical surface of the electrode in the stationary state can be obtained from (14):
n
M total
.
1.2  V  r 2   total
(15)
The value of the real average interaction rate is not expected to depend on the
current density at a constant temperature if the physico-chemical properties of the
lithium deposit do not change. The value Mtotal was constant in all of our tests.
Therefore, the variation of n with the change of the current density, depends on the
value 1/ r2. τtotal only.
In order to obtain an equation is suitable for interpretation of experimental
data we rewrite the formula (15) to the next form:
n
1.2  V
1
 2
.
M total
r   total
(16)
Interaction rate (v), mmole/s
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
Current density,
2
2.5
mA/cm2
Fig. 3. Interaction rate (v) dependence on current density.
Baraboshkin [6] obtained equations for calculating the number of crystals
formed on the cathode during electrochemical deposition at a constant current density.
In accordance to the theory of crystals’ number relevant to molten salts solution and
can be adopted for this problem as well, Z is proportional to i3/2. Such dependence
was obtained both for diffusion control:
Z
0.27 K 1 / 2i 3 / 2
3/ 2
zF 1 / 2C1 / 2 D 3 / 2 (c0  cS )3 / 2max
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,
(17)
where
K  W    2 / kT - the quantity, independent of overpotential ; W*
- work
of the formation of heterogeneous nucleus, η – overpotential, T – absolute
temperature, k – the Boltzman constant; C –full electrode capacity is assumed
independent of potential; D – diffusion coefficient; c0 and cs - concentrations of
lithium ions in the bulk of electrolyte solution and near surface of the growing cluster
correspondenly, the value of c0-cs is assumed constant during nucleus growth; max the highest phase overpotential at which nucleus is formed;
and for the migration control:
Z
84 z1 / 2 K 1 / 2i 3 / 2
3
 1 / 2C1 / 2  3 / 2max
,
(18)
where  is the spesific electroconductivity of electrolyte solution.
We assume that the n dependence on current density is correlated with Z
dependence on one. The value n is proportional to i3/2 as well as Z. The reason for
this correlation is the fact that n is as influenced by the nucleus formation rate as Z,
for the number of grains growing simultaneously on 1 cm2, will increase with the
growth of the nucleuses number, formed per time unit.
In confirmation of aforementioned theory, the dependence of 1/ r2. τtotal (16)
on the current density obtained in our tests, is shown in Fig. 4.
90
80
1/r2. ttotal, 1/cm2.s
70
60
50
40
30
20
10
0
0
1
2
i 3/2
3
4
(current density, mA/cm2)
Fig. 4. 1/ r   total vs. current density plot.
Dots are the experimental data. Solid line is a result
of calculation in accordance with equation (19).
2
The experimental data (dots) (Fig. 4) exhibit a good agreement with the
following linear equation:
1
r   total
2
 26i 3 / 2  6 .
620
(19)
1/ r2. τtotal equals n 
1.2V
(16), therefore, by substituning (19) into (16)
  M total
we obtain:
n
1.2V
 26i 3 / 2  6 .
  M total
(20)
This dependence (equation (20), Fig.4) confirms the validity of our model
simulation of the growth of polycrystalline lithium deposits at constant current density
in dioxolane solution.
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of single metal and alloy clusters – Part 1. Galvanostatic conditions’.
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clusters – Part 2. Potentiostatic conditions’. Electrochimica Acta 1995 40 (10)
1475-78.
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Electrochimica Acta 1996 41 (2) 329-35.
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Phys. Chem. 1962 31, 40-70.
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