# Implementing the Reactor Geometry in the Modeling of Chemical Bath Deposition of ZnO Nanowires

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{c}

^{stat}, in homeostatic conditions, from which the elongation of ZnO NWs, L

^{stat}, is deduced as follows:

_{0}is the concentration of Zn(II) ions (m

^{−3}), k

_{1}is the first-order reaction rate constant describing the crystallization process of ZnO (m·s

^{−1}), D is the diffusion coefficient of Zn(II) ions in aqueous solutions at a given temperature (m

^{2}·s

^{−1}), S is the c-plane top surface area ratio defined by the area of the top surfaces of all c-plane ZnO NWs divided by a given substrate surface area, ρ is the atomic density of wurtzite ZnO that is equal to 4.20 × 10

^{28}m

^{−3}, δ is the stagnant layer thickness subjected to diffusive transport (m), and t is the growth time (s). Cossuet et al. subsequently applied this expression to the growth of ZnO nanorods of different polarities selectively grown on ZnO single crystals, where the active area was sufficiently small compared to the volume of solution to consider static conditions [32]. They showed that Zn-polar ZnO nanorods had a higher growth rate compared to O-polar ZnO nanorods owing to their larger surface reaction rate constant k

_{1}. Later on, a novel theoretical model introducing dynamic conditions, and hence compatible with the use of a closed reactor subjected to no macroscopic convection, was established by solving the Fick’s diffusion equations [33]. The expressions of the time-dependent axial growth rate of ZnO NWs, R

_{c}

^{dyn}, and of their elongation, L

^{dyn}, are formulated as follows:

_{eq}is the equilibrium concentration of Zn(II) ions (m

^{−3}). Using a similar approach, Černohorský et al. employed finite element method calculations to solve the Fick’s diffusion equation in two dimensions and assessed the impact of the transverse diffusion of chemical reactants on the growth of small arrays of ZnO NWs [34]. However, these different studies have systematically considered the reactor as an infinite reservoir of chemical reactants, and the influence of the reactor size on the elongation process of ZnO NWs has basically not been investigated yet despite its critical importance.

## 2. Materials and Methods

#### 2.1. Deposition Techniques

^{2}Si (100) substrates (Si-Mat) cleaned with acetone and isopropyl alcohol in an ultrasonic bath. To favor ZnO nucleation from the substrate during the CBD process, a polycrystalline ZnO seed layer was deposited by dip coating from a solution containing 375 mM of zinc acetate dihydrate (Zn (CH

_{3}COO)

_{2}·2H

_{2}O, Sigma-Aldrich, St. Louis, MO, USA) and 375 mM of monoethanolamine (MEA, Sigma-Aldrich, St. Louis, MO, USA) in pure ethanol (Fisher Scientific, Merelbeke, Belgium, 99.8% purity). The substrates were dipped into the solution under a controlled atmosphere (air with <15% hygrometry) and subsequently annealed for 10 min at 300 °C to evaporate residual organic compounds for 1 h at 500 °C to crystallize the polycrystalline ZnO seed layer.

_{3})

_{2·}6H

_{2}O, Sigma-Aldrich, St. Louis, MO, USA) and hexamethylenetetramine (HMTA, C

_{6}H

_{12}N

_{4}, Sigma-Aldrich, St. Louis, MO, USA) in equimolar concentrations of 30 mM. To tune the reactor size by obtaining different heights of chemical bath ranging from 0.5 to 4.0 cm, the total volume of solution was varied from 4.6 to 37.4 mL. Each sample was maintained face down at the top surface of the chemical bath using Kapton, as depicted in Figure S1. No stirring was performed to favor diffusion processes during the CBD. The sealed reactors were placed in a regular oven heated at 90 °C for 41 h.

#### 2.2. Characterization Techniques

## 3. Results

#### 3.1. Description of the Theoretical Model under Dynamic Conditions

^{−3}) at a height z above the substrate and at a growth time t, C

_{eq}is the equilibrium concentration of Zn (II) ions (m

^{−3}), k

_{1}is the first-order reaction rate constant describing the crystallization process of ZnO (m·s

^{−1}), S is the c-plane top surface area ratio, and D is the diffusion coefficient of Zn (II) ions in aqueous solution at a temperature T (m

^{2}·s

^{−1}).

_{0}. Equation (6) accounts for the finite height of the reactor. It is derived from Fick’s first diffusion equation by assuming that the flux of Zn(II) ions at z = h equals 0. Equation (7) relates to the consumption of Zn(II) ions on the growth front located at the c-plane top facet of ZnO NWs. It is obtained by applying Fick’s first diffusion equation at z = 0, while considering Zn(II) ions as the limiting reactants. It is also worth noticing that the development of the growth front with time along the z axis is neglected and that S is considered as a constant parameter in a first approximation.

_{c}, we perform mass balance at the NW growth front, from which we can deduce the following relation:

^{28}m

^{−3}. The length of ZnO NWs, denoted as L, and its temporal dependence is, in turn, deduced by integrating Equation (11):

_{c}and L can be deduced only from the concentration profile of Zn(II) ions at z = 0. To simplify the problem, we implement the numerical approach of the inverse Laplace transform of Equation (10) instead of considering Equation (9). To compute this integral from discrete values of R

_{c}taken at different times, t, a linear interpolation is performed between two consecutive values of R

_{c}. Furthermore, the value of the equilibrium concentration of Zn(II) ions, C

_{eq}, depends on the CBD conditions and is generally not negligible [33]. However, it can be typically determined from thermodynamic simulations, as detailed in [33]. Additionally, the homogeneous formation of ZnO taking place in the bulk of the chemical bath can strongly impact the growth kinetics of ZnO NWs heterogeneously formed from the substrate, particularly in the case of CBD performed in standard pH conditions [22,33,38]. As the massive precipitation attributed to homogeneous growth is commonly observed during the first stages of CBD [33], we consider, in a first approximation, that it occurs instantaneously at t = 0. As a consequence, the effective value of the initial concentration, C

_{0}, is usually considered to be lower than the concentration of precursors introduced in the chemical bath and needs to be deduced from experimental data. A more detailed description of this theoretical model is available in the Supplementary Materials.

#### 3.2. Comparison of the Theoretical Model with the Case of a Semi-Infinite Reactor

_{0}, C

_{eq}, and k

_{1}, respectively, as determined in [33], whereas the Zn(II) ion diffusion coefficient D was deduced from [39] and set to 2.74 × 10

^{−9}m

^{2}/s. Finally, a typical value of the c-plane top surface area ratio, S, of 0.3 was considered [33]. We can notice that when h values increase from 0.5 to 2 cm, the evolutions of both the ZnO NW length and axial growth rate become increasingly similar to the evolutions observed in the case of a semi-infinite reactor, whose curves act as asymptotes when the reactor height tends to infinity. These results show the great consistency between the two models.

_{h}, the chemical bath is not significantly depleted in reactants and the elongation process of ZnO NWs follows similar kinetics as in the case of a semi-infinite reactor; (ii) when the effective growth time becomes larger than t

_{h}, the chemical bath depletes significantly faster than in the case of a semi-infinite reactor due to its limited height, which leads to a progressive decrease in the axial growth rate of ZnO NWs; and (iii) when the effective growth time is much larger than t

_{h}, the chemical bath is fully depleted and the NW growth is completely stopped.

_{h}, from which the reactor height, h, becomes significantly influential on the axial growth rate of ZnO NWs, we propose to define it more rigorously as the effective growth time needed to reach a relative difference, α, between the lengths of ZnO NWs grown in a reactor of semi-infinite height, L

_{∞}, and of finite height h, L

_{h}. Therefore, t

_{h}is reached when ∆L/L = (L

_{∞}—L

_{h})/L

_{∞}= α. With this criterion, t

_{h}was determined theoretically for different h values and for α values of 1%, 3%, and 5%, as revealed in Figure 3. The obtained values for α = 5% are also reported in Figure 2 (dashed lines). We observe that t

_{h}follows a parabolic evolution with h, as it rapidly increases for a long effective growth time—e.g., t

_{h}> 24 h, when h > 3 cm. Furthermore, the conditions in which the different NW growth regimes occur can readily be visualized from this graph: when t < t

_{h}, the reactor can be considered to be of semi-infinite height and the axial growth rate of ZnO NWs is maximized, whereas when t > t

_{h}, the reactor height must be taken into account in the determination of the ZnO NW length, as the NW growth becomes significantly slower. As a general rule, the value of α is typically chosen to reflect the maximum drop of axial growth rate that can be tolerated in a particular system.

_{c}< 50 nm/h) after 3.4, 9.3, 16.5, and 24.0 h for reactor heights of 0.5, 1.0, 1.5, and 2.0 cm, respectively. It is worth noticing that most of the reactors used experimentally have a height of a few centimeters—usually greater than 2 cm—and that effective growth times are only of a few hours—usually lower than 24 h. Therefore, t

_{h}is typically not reached, and the reactor can be approximated to a semi-infinite medium in the wide majority of cases in the present experimental conditions considered. However, the reactor height is expected to have a stronger influence in other non-standard, but important experimental conditions, such as when different chemical precursor concentrations [22,40,41], growth temperatures [41], or pHs [38] are used.

#### 3.3. Comparison of the Theoretical Model with Experimental Data

_{3})

_{2}and HMTA and with varying reactor heights, h, ranging from 0.5 to 4.0 cm. To be consistent with the geometry used in the theoretical model, the substrates were placed face down at the top of the chemical bath, as depicted in Figure S1. Furthermore, to obtain a noticeable variation of the length of ZnO NWs with the reactor height, a very long growth time of 41 h was chosen. The FESEM images collected on all the series of samples reveal a slight increase in the length of ZnO NWs as the reactor height is increased, as shown in Figure 4. The mean length of ZnO NWs for each reactor height was deduced from these FESEM images and is reported in Figure 5. The effect of the reactor height on the length of ZnO NWs was confirmed from this graph: the mean length of ZnO NWs increased from around 1.3 to 1.7 µm when the reactor height was increased from 0.5 to 4.0 cm.

_{0}, C

_{eq}, and k

_{1}were set to 19.1 mM, 15.2 mM, and 18.2 µm/s, respectively, as determined from [33], and D to 2.74 × 10

^{−9}m

^{2}/s, as deduced from [39]. One can notice that C

_{0}is considered to be lower than the concentration of chemical precursors introduced, which accounts for the initial consumption of reactants from homogeneous growth. Interestingly, when using the theoretical model with the effective growth time of 40 h and 25 min considered in the series of samples, the length of ZnO NWs was found to be greatly overestimated for reactor heights above 1 cm, as revealed in Figure 5 (black curve), whereas values reaching up to 4.1 µm were predicted for reactor heights above 3.5 cm. This apparent inconsistency between the theoretical model and the experimental data could be explained by the influence of the homogeneous growth of ZnO taking place in the bulk of the chemical bath, as this was not rigorously implemented in the present model. The homogeneous growth was indeed considered, in a first approximation, as an instantaneous phenomenon taking place at t = 0. While the massive precipitation attributed to homogeneous growth is mainly observed in the first stages of CBD in the present growth conditions [33], it is reasonable to assume that it can occur as well throughout the rest of the CBD until the chemical bath is fully depleted of reactants (i.e., when [Zn(II)] = C

_{eq}). This is supported by the fact that ZnO precipitates can typically be observed in the whole chemical bath even after several hours of growth [33]. As a consequence, homogeneous growth leads to a faster consumption of the chemical reactants where the complete depletion of the chemical bath occurs much sooner than the model predictions, thus significantly affecting the growth kinetics of ZnO NWs heterogeneously grown from the substrate. Additionally, it should be mentioned that spontaneous natural convection at the microscopic scale through eddy phenomena was not taken into account for the sake of simplicity, but may play a non-negligible role here as it is known to affect the diffusion of chemical species in the bulk of macroscopically still solutions [42,43].

#### 3.4. Using the Theoretical Model as a Predictive Tool

_{3})

_{2}and HMTA is shown in Figure 6a. For the sake of clarity, the theoretical data are also presented on 2D graphs at given reactor heights, h, in the range of 0.1 to 5 cm (Figure 6b) and at given effective growth times, t, in the range of 0.5 to 24 h (Figure 6c). We can notice that the evolution of the length of ZnO NWs with the reactor height reflects the growth regimes previously described: (i) in the first stages of the growth—i.e., when t < t

_{h}—the length of ZnO NWs is completely independent of the reactor height and the ZnO NW axial growth rate is maximized; (ii) in the later stages of the growth—i.e., when t > t

_{h}—the length of ZnO NWs becomes noticeably dependent of the reactor height and the ZnO NW axial growth rate correlatively decreases; and (iii) in the lasts stages of the growth—i.e., when t >> t

_{h}—the length of ZnO NWs becomes highly dependent upon the reactor height and the ZnO NW axial growth rate becomes negligible.

## 4. Conclusions

## Supplementary Materials

_{3})

_{2}and HMTA after 41 h at 90 °C with reactor heights, h, ranging from 0.5 to 4.0 cm. Figure S2: Top-view FESEM images of ZnO NWs grown by CBD at 90 °C with 30 mM of Zn(NO

_{3})

_{2}and HMTA for 41 h with reactor heights, h, ranging from 0.5 to 4.0 cm. The corresponding value of the c-plane top surface area ratio, S, which was measured from such images, is also indicated for each sample. The scale bar represents 500 nm.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic of the geometry used to establish the theoretical model describing the time-dependent axial growth rate and length of ZnO NWs by CBD under dynamic conditions, where a finite reactor height, h, is considered.

**Figure 2.**Theoretical evolutions of (

**a**) the length and (

**b**) the axial growth rate with the effective growth time of ZnO NWs for different reactor heights, h. For h values ranging from 0.5 to 2 cm (colored lines), the curves are determined by the numerical approach detailed in this work, whereas for an infinite h value (black lines), the curves are determined by Equations (2) and (3) (taken from [33]). The dashed lines represent the effective growth time t

_{h}(taken at α = ΔL/L = 5%, see text) for each reactor height, h, considered. The values taken for the different parameters correspond to typical growth at 90 °C with 30 mM of Zn (NO

_{3})

_{2}and HMTA (as determined in [33]), i.e., C

_{0}= 19.1 mM, C

_{eq}= 15.2 mM, k

_{1}= 18.2 µm/s, S = 0.3, D = 2.74 × 10

^{−9}m

^{2}/s, and ρ = 4.20 × 10

^{28}m

^{−3}.

**Figure 3.**Theoretical evolution of the effective growth time, t

_{h}, from which the ZnO NW length, L, becomes significantly lower than the case where the reactor is of semi-infinite height, i.e., when ΔL/L becomes greater than 1%, 3%, or 5%, as a function of h. If the effective growth time does not exceed t

_{h}, the problem can be simplified to the case where the reactor is of semi-infinite height. The values of the different parameters correspond to a typical growth at 90 °C with 30 mM of Zn (NO

_{3})

_{2}and HMTA (as determined in [33]), i.e., C

_{0}= 19.1 mM, C

_{eq}= 15.2 mM, k

_{1}= 18.2 µm/s, S = 0.3, D = 2.74 × 10

^{−9}m

^{2}/s, and ρ = 4.20 × 10

^{28}m

^{−3}.

**Figure 4.**Cross-sectional-view FESEM images of ZnO NWs grown by CBD at 90 °C with 30 mM of Zn(NO

_{3})

_{2}and HMTA for 41 h with reactor heights of (

**a**) 0.5, (

**b**) 0.9, (

**c**) 1.2, (

**d**) 1.9, (

**e**) 2.1, (

**f**) 2.7, and (

**g**) 4.0 cm, respectively. The scale bar represents 1 µm.

**Figure 5.**Length, L, vs. reactor height, h, of ZnO NWs grown by CBD at 90 °C with 30 mM of Zn (NO

_{3})

_{2}and HMTA for 41 h (corresponding to an effective growth time of 40 h and 25 min). The experimental data were fitted by the theoretical model, using C

_{0}= 19.1 mM, C

_{eq}= 15.2 mM, k

_{1}= 18.2 µm/s, S = 0.30, D = 2.74 ×10

^{−9}m

^{2}/s, and ρ = 4.20 × 10

^{28}m

^{−3}. The best adjustment is reached when an effective growth time of 7 h is considered, whereas the length of ZnO NWs is greatly overestimated if the expected effective growth time of 40 h and 25 min is instead considered.

**Figure 6.**(

**a**) 3D plot of the evolution of ZnO NW length vs. effective growth time, t, and reactor height, h, for a chemical bath kept at 90 °C initially containing 30 mM of Zn (NO

_{3})

_{2}and HMTA, as computed by the theoretical model using C

_{0}= 19.1 mM, C

_{eq}= 15.2 mM, k

_{1}= 18.2 µm/s, S = 0.30, D = 2.74 × 10

^{−9}m

^{2}/s, and ρ = 4.20 × 10

^{28}m

^{−3}. Two-dimensional plots of ZnO NW length vs. (

**b**) effective growth time, t, for specific h values and (

**c**) reactor height, h, vs. specific t values are also provided.

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**MDPI and ACS Style**

Lausecker, C.; Salem, B.; Baillin, X.; Consonni, V.
Implementing the Reactor Geometry in the Modeling of Chemical Bath Deposition of ZnO Nanowires. *Nanomaterials* **2022**, *12*, 1069.
https://doi.org/10.3390/nano12071069

**AMA Style**

Lausecker C, Salem B, Baillin X, Consonni V.
Implementing the Reactor Geometry in the Modeling of Chemical Bath Deposition of ZnO Nanowires. *Nanomaterials*. 2022; 12(7):1069.
https://doi.org/10.3390/nano12071069

**Chicago/Turabian Style**

Lausecker, Clément, Bassem Salem, Xavier Baillin, and Vincent Consonni.
2022. "Implementing the Reactor Geometry in the Modeling of Chemical Bath Deposition of ZnO Nanowires" *Nanomaterials* 12, no. 7: 1069.
https://doi.org/10.3390/nano12071069