With the use of diffusionlimited aggregation modeling, we have investigated the effect of particle drift for dendritic growth. It is found that the morphology of dendritic growth is sensitive to the particle drift, i.e., the larger drift effect results in the denser growth of dendrite. From the analysis using the correlation function, we found the fractional dimension of each dendrite increases as the particles drift increases. Furthermore, we showed the height of dendrite significantly decrease for the slight change of particles drift. Finally, we discussed the strategy to reduce dendritic growth by modifying the transport properties of electrolytes.
1. Introduction
The study for morphology of growing surface has been an interesting topic in broad areas of biology such as morphogenesis of living organisms and physical sciences such as metallurgy, combustion, and electrochemistry
[1]
. The growing surfaces in solidphase are normally contacted with fluids including their ingredients in gas or liquid phase. Contrary to biological systems, inorganic systems are mainly governed by the condition of thermodynamic equilibrium. However, statistical fluctuations (timedependent deviation of physical quantities) often affect the morphology of surface significantly. When fluctuations are minor, a surface will grow in the crystalline structure, on the contrary, when fluctuations are dominant, the surface will grow in treelike shape, i.e. dendritic growth
[2
,
3]
.
One of representative theory describing dendritic growth is the diffusionlimited aggregation (DLA) which is proposed by Witten and Sander in 1981
[3]
. DLA is based on the random work of particles in the fluid, the socalled Brownian motion. In DLA model, when a diffusing particle is located on a site contacting with the part of the cluster (or seed), it becomes a part of cluster and does not move to other direction any more, i.e. growth. It has been known the aggregation generated in DLA model is followed by powerlaw correlations
[3

7]
, where the fractional power law behavior is related to the selfsimilarity as in fractals, which means there is no characteristic length scale to define the system.
In experiments, the dendrite has been observed in several areas such as an aggregation of silver atoms on a surface of platinum (epitaxy growth)
[8]
, the growth of succinonitrile dendrite (supercooling of the liquid)
[9]
, and the growth of zinc metal leaves (electrodeposition)
[10]
. Especially, the dendrite formation on the lithium metal anode in rechargeable lithium metal, lithiumsulfur, and lithiumair batteries has been observed
[11

13]
and becomes a serious problem, affecting life time and safety of battery cells due to shortcircuit
[14

17]
. To use a lithium metal anode for achieving high capacity in rechargeable lithium batteries, it is necessary to reduce the dendritic growth on the lithium metal surface. Therefore, the motivation of our study mainly originates from the question ‘How can the dendritic growth be controlled?’. Until now, several effects of adsorption
[4]
, drift
[5]
, concentration
[6]
, anisotropy
[18]
, and surface tension
[19]
have been suggested as main mechanisms. However, these studies were mainly performed in the radial geometry with a seed in the center as a starting point. Furthermore, even though the study of height of dendrite directly related to shortcircuit is important, the relation between the height and those effects are not clear yet.
Here, to study the effect of particle drift for dendritic growth, the DLA simulations have been performed in the twodimensional geometry with a linearshaped electrode. We have showed the morphology of dendrite becomes dense by increasing particles drift. We have also analyzed the fractional dimension of dendrites using a correlation function and showed a relationship between the degree of drift and dimensions. Finally, we conclude the height of dendrite, which is an important value for shortcircuit, can be changed significantly by increasing the probabilistic weight of particles drift slightly.
2. Computational Method
The DLA clustering was simulated in the twodimensional square lattice with (N
_{x}
, N
_{y}
) = (200,6000), where N
_{x}
and N
_{y}
are the number of grid in x and y directions, respectively. 5,000 particles were used for all simulations. Bottom lines in
Fig. 1
represent an initially flat anode surface. To realize the random work describing Brownian motion, random numbers were generated to decide the direction of particles movement each time. For the random work without drift, moving probabilities (P) for four directions, (P
_{x+}
, P
_{x}
, P
_{y+}
, P
_{y}
), were given by (0.25, 0.25, 0.25, 0.25). The four directions were shown in the inset of
Fig. 1a
. The degree of drift was changed by the change of moving probability in ‘y’ direction, in which other probabilities were given by equal amounts of (1P
_{y}
)/3. When diffusing particles meet the bottom line or the part of aggregated cluster, they do not move any more, i.e. attached to the cluster. The periodic boundary condition was also used in the boundaries of x = 1 and 200.
Morphologies of dendritic growth depending on the particle drift (P_{y}) of (a) 0.25, (b) 0.27, (c) 0.28, (d) 0.31, (e) 0.4, (f) 0.7, (g) 1.0; (insets) (a) moving directions of particles, (b)(g) enlargement of each dendritic growth.
3. Results and Discussion
Fig. 1(a)
shows a dendritic growth based on random work of particles with directional moving probability of (0.25, 0.25, 0.25, 0.25). The height of single big dendrite is 1,445 grids and the width is around 10 grids. This height is very large compared to 25 grids, calculated from 5,000 particles/200 sites, in closepacking of all particles. Therefore, in the random motion of particles, it can hardly keep the surface flat, then shortcircuit may occur in short time. This case may correspond to the movement of Li ion in electrolyte without drift. Hence the supply of Li ions to the surface of Li metal anode will be a very slow process. Therefore, particles have a large chance to contact with cluster before reaching the bottom electrode line and the morphology of aggregation becomes more dendritic as in
Fig. 1(a)
. By slightly increasing the probability of P
_{y}
from 0.25 to 0.27, i.e., from (P
_{x+}
, P
_{x}
, P
_{y+}
, P
_{y}
) = (0.25, 0.25, 0.25, 0.25) to (0.243, 0.243, 0.243, 0.27), the height of dendrite is significantly reduced as in
Fig. 1(b)
. Its height is 462 grids and width is around 25 grids. However, most particles are still attached in one large branch of dendrite. As seen in
Fig. 1(c)
and
1(d)
with P
_{y}
of 0.28 and 0.31, respectively, there is still one large dendrite distinguished from other dendrites. Even though the height of the large dendrite continually decreases with increasing P
_{y}
, its width does not increase any more. Instead, other smaller dendrites grow more.
In
Fig. 1(e)
with P
_{y}
of 0.4, the morphology of dendrite is drastically changed. The upper front of dendrites is flat compared with
Fig. 1(a)(d)
. From the inset in
Fig. 1(e)
, the dendrite with notable size is not found. The height is only 116 grids. By increasing Pymore, denser dendrites are obtained as in
Fig. 1(f)
with P
_{y}
of 0.7 and
Fig. 1(g)
with P
_{y}
of 1.0. The heights of dendrites become lower, 77 and 64 grids for P
_{y}
= 0.7 and 1.0, respectively. Especially, we can find an interesting point in the case with P
_{y}
= 1.0. Even though all 5,000 particles have moving possibility toward the electrode only, its height is not 25 grids as in closed packing but 64 grids. Why does this happen? All particles always move to the bottom electrode straightly, but initial positions of particles are randomly generated, therefore, some particles can exist in second layer before the first layer is completely filled. Normally, this prohibits particles from filling the first layer perfectly, since moving particles can be attached to the particles in second layer before reaching empty sites in bottom electrode. Without the consideration of thermodynamic diffusion of particles on the layers, it is impossible to make perfect packing of particles.
How can we measure the relation between the probabilistic weight of particles drift represented by P
_{y}
and the fractional dimension of dendrites? To quantify this relation, we now introduce the correlation function
[3]
,
where
r′
and
r
are the sites(positions) of particles and their neighboring particles, respectively.
ρ
(
r
) is the local density having a value of 0(empty) or 1(occupied) in site
r
and
N
is the number of particles. It has been known the correlation function shows a powerlaw behavior in DLA model
[1
,
3]
.
where
α
is an exponent of the correlation function. Furthermore, the
α
is also related to the fractional dimension of dendrite, which is given by
[1
,
3]
where
D
is the fractional dimension of dendrite and
d
is the dimension of space, specifically 2 in our case. We obtained the results of correlation functions for each P
_{y}
in
Fig. 2
. The correlation function is plotted in loglog scale, so the slope of each case corresponds to the exponent. In the analysis of correlation function, the slope of twodimensional case have zero (
α
= 0), since all neighboring sites of a specific particle are completely occupied which gives the relation of
N
(
R
) =
πR
^{2}
(
D
= 2), i.e. the area of circle. If the slope of calculated correlation function is close to zero, it means a fractional dimension is close to two dimension. In our case, P
_{y}
= 1.0 shows the smallest slope, so it is closest to the twodimensional closepacking among all cases of drift. This is already confirmed in the discussions regarding
Fig. 1
. The calculated
α
and
D
are listed in
Table 1
. As the P
_{y}
increases from 0.25 to 1.0, the
D
also increases from 1.53 to 1.88. It is reconfirmed the dimensionality of P
_{y}
= 1.0 is different from that of perfect closepacking (
D
= 2).
Analysis of the fractional dimension of dendritic growth for each number of particles drift (P_{y} = 0.251.0) by using correlation functions. The dashed lines show the slope of some correlation functions in loglog scale, which corresponds to the exponents (α) of correlation function in Equation (2).
Relationship between particles drifts (Py), exponents from correlation functions, dimensions and heights in dendritic growth.
Relationship between particles drifts (P_{y}), exponents from correlation functions, dimensions and heights in dendritic growth.
Finally, we have plotted the heights ratio of dendrites, which can be compared with experimental measurements, depending on dimensions for each P
_{y}
in
Fig. 3
. The heights ratio is obtained by dividing the height of flat dendrites by the maximum height of single large dendrite. We found the rapid increase of heights ratio for the slight increase of dimension of dendrite from P
_{y}
= 0.25 to 0.27. After that, the heights ratio increases linearly up to P
_{y}
= 0.4. The change of P
_{y}
from 0.25 to 0.4 results in dominant change for the heights ratio and dimension of dendrites. However, the range of P
_{y}
between 0.7 and 1.0 shows only small change of heights ratio and dimension of dendrites. The degree of drift given by P
_{y}
is proportional to the drift velocity of particles. The drift velocity is also proportional to the external force such as electric field and mobility of particles.
^{20)}
Therefore, in the same electric field, the higher mobility of particles may improve the drift velocity. From the StokesEinstein relation
[21]
, this mobility is proportional to the diffusion coefficient of particles and inversely proportional to the viscosity of the medium. Hence, the effect of particles drift can be enhanced by using electrolytes with lower viscosity which reduces the frictional drag of Li ions induced by surrounding solvent molecules or higher diffusion coefficient. Therefore, it can be a good strategy to modify electrolytes in the direction of lowering viscosity and raising diffusion coefficient, in order to mitigate dendritic growth.
Distribution of heights ratio of dendrites depending on fractional dimensions and particles drift (P_{y} = 0.25–1.0). The heights ratio is given by dividing the height obtained from flat dendrites (H_{flat}) by the maximum height obtained from single largest dendrite (H_{max}). Open circle at 2 in dimensionaxis represents the height ratio of close packing in twodimension.
4. Conclusions
We have studied the morphology, dimension, height of dendritic growth depending on the particles drift by using diffusionlimited aggregation method. The morphology of dendrite is intrinsically changed at P
_{y}
= 0.4, where the front of dendrite becomes almost flat. The fractional dimension of dendrite, analyzed by the correlation function, is almost saturated around 1.88 in the range of P
_{y}
= 0.7 and 1.0, i.e., it shows no significant change in morphology after the drift of P
_{y}
= 0.7. In addition, the height of dendrite significantly decreases by the slight change of drift and is saturated near P
_{y}
= 0.7. Based on these results, the P
_{y}
= 0.7 in drift effect may be a critical point showing the saturation of morphology, dimension, height in dendritic growth. Finally, it is concluded that the improved transport properties of lithium ions in electrolytes can be effective to enhance the cyclability of the lithium metal battery.
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