Heterogeneous nucleation capability of conical microstructures for water droplets

Abstract

The presence of microstructures on a substrate has a great effect on the heterogeneous nucleation of water droplets. A circular conical apex and a cavity are adopted as the physical model to represent the typical defects which exist widely on substrates, and classic nucleation theory is used to quantitatively analyze the nucleation capability of different microstructures at different condensation conditions. The results indicate that conical cavities with narrower cone angles can reduce the nucleation free energy barrier as compared with apexes and a planar substrate, yielding a relatively higher nucleation capability. With the vapor pressure and supersaturation increasing, the nucleation rate increases rapidly, and some of the cavities that are originally not preferred for nucleation gradually translate into active nucleation sites. Consequently, the activated nucleation sites are finite for practical substrates under certain nucleation conditions, and the nucleation sites number density can be affected by the condensation conditions and the distribution of micro cavities on the substrate. The analysis also indicated that it is possible to realize spatial control of nucleation sites by the construction of micro cavities, and the nucleation sites number density can be intensified by increasing the amount of micro cavities on the substrate.

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1. Introduction

It can be observed that, when dew drops form, although they may be positioned randomly on flat leaves, they often tend to accumulate in the direction of the leaf tip as time continues. The mechanism of this behavior has been reported by Shanahan¹ in terms of surface free energy minimization. Another question that naturally arises is: why exactly do the initial dew drop doses not preferably form at the leaf tip in the first place?

The formation of the initial dew drop is a typical heterogeneous nucleation process of liquid droplets from a bulk vapor phase, which is a typical process that exists in nature and industrial applications, such as the formation of rain drops and hailstones, crystal formation,^2–5 chemical vapor deposition,^6–8 and nucleation of initial droplets in dropwise condensation. As far as the nucleation process is concerned, the thermodynamic model was usually adopted and referred to as the Classic Nucleation Theory (CNT).^9,10 Subsequently, the CNT model has been modified to obtain more accurate results,^11–14 and the effect of a planar substrate and the heterogeneous nucleation model were considered.^15–24 Experimental results revealed that the nucleation process can be affected by artificially distributed hydrophilic areas to realize the so called “spatial control” of nucleation sites.²⁵ On the other hand, the relationship between the substrate structure and the nucleation process,^26–33 and the effect of micro particles on the nucleation rate, were also widely investigated.^34,35 The results suggest that the initial droplets tend to appear on the substrate defects and deposited heterogeneous particles,^26,27,33 and the nucleation sites number density increases with the surface roughness.^28,36 These results all indicated that the substrate properties, such as the wettability of the substrate material, and the distribution of microstructures, have a great effect on the nucleation of the initial nucleus. However, most of the works concentrated on the physico-chemical properties that are obtained from statistics, such as the apparent contact angle and surface roughness, and the underlying mechanism of how the substrate structures affect the nucleation process is still not very clear. Hence, a better understanding of the behavior of initial dew drop formation on different structures is very significant to explain natural phenomenon and predict the effect of substrate structures on nucleation processes.

In this paper, the behavior of heterogeneous nucleation of water droplets on conical microstructures will be explored. A circular conical apex and a cavity are adopted as the physical model to represent the typical defects widely existing on the substrates, and the CNT model is adopted to quantitatively analyze the nucleation capability of the microstructures under different condensation conditions.

2. Physical model

A circular conical apex and a cavity are shown in Fig. 1 to represent the defects which exist in actual substrates, and a planar substrate is also considered for comparison. The structures are characterized by the intrinsic wetting angle α of the substrate material and the cone angle β. As the initial nucleus is relatively small, gravitational force can be neglected as compared with the surface tension effect. According to the classic capillary approximation¹⁶ and molecular dynamics simulation results,³⁷ the nucleus will be deposited on three different substrate structures in the form of a spherical cap as shown in Fig. 1, with the local contact angle equal to the intrinsic wetting angle of the substrate material. It has to be pointed out that, for structures that fulfill the condition of α + β/2 < 90°, the meniscus within micro cavities will be concave instead of that described in Fig. 1c. The additional pressure provided by the concave meniscus is favorable for the stability of embryos of any sizes and hence is favorable for the formation of the nucleus. Here, the term of embryo is adopted to denote a molecular cluster that has not reach up to the critical size, in order to differentiate from nucleus. Based on this fact, the structures fulfilling the α + β/2 < 90° condition will not be considered in this paper. The sizes of the microstructures are comparable to that of the initial nucleus, which is on the nanoscale under typical nucleation conditions of water droplets.

Fig. 1 Schematic representation of an axisymmetric nucleus on three microstructures. (a) Circular conical apex, (b) planar substrate, (c) circular conical cavity.

In the following thermodynamic model section, it can be found that the nucleation process is greatly affected by the volume and liquid–vapor and liquid–solid interfacial areas of the initial nucleus. For a nucleus deposited on an apex as described by Fig. 1a, the related parameters can be calculated from the geometry:

R = r* × cos(α − β/2) (1)

h_cap = r* × [1 + sin(α − β/2)] (2)

h_cone = R × cot(β/2) (3)

where r* is the critical curvature radius of the initial nucleus, R is the bottom radius of the cone, and h_cap and h_cone are the heights of the spherical cap and the cone, respectively.

The volume of the nucleus V_drop, the liquid–vapor and liquid–solid interfacial areas of S_lv and S_ls can be obtained as follows:

S_lv = 2πr*²[1 + sin(α − β/2)] (5)

S_ls = πr*² cos²(α − β/2)/sin(β/2) (6)

In the case of the planar substrate, β can be simply set as 180°. For a cavity structure, the cone angle β* can be converted into β by the formula β = 360 − β*, and the formulas above are also applicable. Consequently, eqn (1)–(6) can be used to calculate the parameters related to all three physical structures.

3. Thermodynamic model

Generally, the nucleation capability at certain conditions can be evaluated either by the nucleation free energy barrier ΔG(r*) or the nucleation rate J. Here, ΔG(r*) is defined as the Gibbs free energy barrier that has to be overcome to form a nucleus of critical size r*; J is defined as the number of initial nuclei formed within a unit time period and a unit volume or area for homogeneous or heterogeneous nucleation processes. Based on classic nucleation theory, the nucleation rate J can be expressed as follows:^19,25where J₀ is a kinetic pre-factor that is directly connected with vapor conditions and nucleus configurations (spherical shape for homogeneous nucleation, and spherical cap for heterogeneous nucleation on a planar substrate).^19,38 k_B is the Boltzmann constant and T is the temperature.

Using the CNT model, the capillary approximation^16,19,21 and Young’s equation, the nucleation free energy barrier of critical nucleus can be expressed as:

where ρ, σ_lv and M are the density, surface tension and molecular mass of the condensate liquid, respectively. According to the capillary assumption,^16,19,21 ρ and σ_lv are obtained from the physico-chemical properties of the bulk liquid. N_A is Avogadro’s number, and S is the supersaturation defined as the ratio between vapor pressure P_v and the equilibrium pressure at nucleation temperature.

The critical radius of the nucleus can be obtained as:²⁵

where n_l is the number of molecules per unit volume of condensate liquid.

By substituting eqn (9) into eqn (4)–(6) to calculate V_drop, S_lv and S_ls, and then substituting them into eqn (8), a general formula of ΔG(r*) can be obtained as follows with respect to the microstructure configurations:

with F(α,β), the form factor that is connected with substrate structure parameters, given by the following expression:

For nucleation processes of liquid droplets from the bulk vapor phase, the nucleation rate J can be deduced from the following simplified general formula:²¹

where N(n) is the number density of embryos containing n molecules per unit area for heterogeneous nucleation, j_e(n) is the attachment rate of vapor molecules onto an embryo of size n, and A(n) is the vapor–liquid interfacial area of an embryo.

The integral formula in eqn (12) means that the nuclei are developed from embryos with various sizes. The number distribution of these embryos N(n) can be obtained from the classic cluster size distribution model:^11,21,24

where ρ_N,v is the number of vapor molecules per unit volume of the bulk vapor. The pre-factor of the above formula is either ρ_N,v or (ρ_N,v)^2/3 for homogeneous or heterogeneous nucleation.²¹ In the present study, the nucleation occurs on the substrate, and the pre-factor of (ρ_N,v)^2/3 is used. The analysis is conducted under steady state conditions, and there are no depletion effects on the finite number of microstructures. ΔG(r) is the free energy change for the formation of embryo of radius r. The difference between ΔG(r) and ΔG(r*) is whether the critical radius is achieved.

The radius of the embryo r is related to the number of molecules n in the embryo by:

where m is the mass of one molecule, and v_l is the specific volume of the condensate, which is also adopted from the bulk liquid.^16,19,21

ΔG(r) can be obtained by expanding the right hand side term of eqn (10) in a Taylor power series in terms of r − r* about the equilibrium radius r*:²¹

j_e(n) can be deduced from the kinetic theory of gases:

With the consideration of microstructure configurations, the interfacial area of A(n) can be expressed as:

A(n) = 2πr²[1 + sin(α − β/2)] (17)

Substituting eqn (13), (16) and (17) into eqn (12), the nucleation rate J is written:

Meanwhile, the relationship between cluster size n and r can be described by eqn (14). Substituting eqn (14) into the above formula, J can be organized as:

Substituting eqn (15) into the integration term of the above formula, we can get:

Setting intermediate variable B as:

eqn (20) then can be simplified as follows:

Finally, substituting eqn (22) into eqn (19), a general form of J can be obtained as:

If β is taken to be 180°, eqn (23) becomes identical to the expression for heterogeneous nucleation on a planar substrate.

Meanwhile, the kinetic pre-factor J₀ in eqn (7) also can be obtained as:

As indicated by eqn (10) and (23), the nucleation rate J is a function of nucleation conditions (P_v and S) and substrate structure parameters (α,β).

4. Results and discussion

4.1 Effect of β on the nucleation rate

According to eqn (23), the nucleation rate of the heterogeneous nucleation process is determined by J₀ and ΔG(r*). The calculated results of J₀ and ΔG(r*) under various structure parameters are shown in Fig. 2. As β increases, the substrate structures translate from apexes to cavities, with J₀ and ΔG(r*) decreasing rapidly in the same manner. As indicated by the schematic diagram of the physical model, a relatively larger liquid–vapor interfacial area is expected for an initial nucleus deposited on the top of an apex, providing a relatively higher probability for vapor molecular attachment. As a result, J₀ for an apex is larger than that of planar substrate or cavities. For instance, the calculated results of J₀ for an apex with β = 60° is 1.37 times that of a planar substrate (β = 180°) and 3.73 times that of a cavity with β = 300° (β* = 60°). On the other hand, as β increases, the volume and interfacial area of the initial nucleus decrease accordingly due to the space-confining effect of cavities. Considering that the main part of the nucleation free energy barrier is caused by the formation of new interfaces, the decrease of the interfacial area is thus favorable for the decrease of ΔG(r*). For instance, ΔG(r*) for an apex with β = 60° is 1.87 times that of a planar substrate and 13.93 times that of a cavity with β = 300°.

Fig. 2 ΔG(r*), J₀ and J under various structure parameters (P_v = 100 kPa, S = 1.5, α = 90°).

It is necessary to point out that, as the nucleation rate is an exponential function of ΔG(r*), the nucleation free energy barrier is thus decisive to the heterogeneous nucleation process. According to eqn (23), the nucleation rate for a cavity with β* = 60° is 9.0 × 10³² times that of a planar substrate and 1.6 × 10⁶⁶ times that of an apex with β = 60° under the considered nucleation condition. As the microstructures translate from apexes to cavities, comparably lower nucleation free energy barriers are required to form the initial nucleus due to the space-confining effect of cavities, and the nucleation rates are thus increased. Based on the discussions above, the order of the nucleation capability of the three structure configurations is cavities, planar substrate and apexes. The presence of cavities with narrower cone angles is favorable for nucleation processes.

The nucleation rate J calculated from eqn (23) is shown in Fig. 3, under the nucleation condition of P_v = 100 kPa and S = 1.5. Apexes are not considered as they are not preferred for nucleation compared with a planar substrate and cavities. The structure parameter β is translated into β* using the relation of β* = 360 − β, and a smaller β* denotes a narrower cavity. The intrinsic wetting angle α is restricted within 60–110° with the consideration of most practical substrate materials.²¹ As expected, the nucleation rate decreases with α sharply, indicating that the nucleation rate on a hydrophilic surface was higher than a hydrophobic one for any substrate structures. Meanwhile, the nucleation rates for narrower cavities are obviously greater than the planar substrate for the same α, indicating relatively higher nucleation capabilities for micro cavities. It has been reported by Varanasi²⁵ that the nucleation rate on a hydrophilic surface with α ∼ 25° is about 10¹²⁹ times higher than that on a hydrophobic surface with α ∼ 110°, and the nucleation sites can be artificially controlled by a hydrophilic–hydrophobic hybrid surface. According to the model analysis above, so called spatial control of nucleation sites also can be realized by the appropriate substrate structure constructions, except for the regulation of surface wettability.

Fig. 3 Nucleation rate as a function of wetting angle α and structure parameter β* (P_v = 100 kPa, S = 1.5).

It also can be found that the effect of structure parameter is so great that the nucleation rates under certain conditions are extremely low and almost no nuclei can be formed under these circumstances. This behavior actually provides a threshold of α and β for heterogeneous nucleation processes. In the contour map of Fig. 3, a threshold of 1 m⁻² s⁻¹ is chosen following Carey’s analysis.²¹ As a result, only those micro cavities that fulfill the threshold can be activated to form initial nuclei (see the lower-right part of the contour map with color fill).

For water vapor condensation, the kinetic pre-factor J₀ is usually in the range of 10²³–10²⁶ m⁻² s⁻¹, which means that the exponential part of eqn (23) has to be greater than 10⁻²⁶–10⁻²³ to fulfill the threshold. The threshold proposed here actually provides an upper limit for the nucleation free energy barrier. To ensure effective nucleation, ΔG(r*) has to be low enough, and the upper limit of ΔG(r*) can be calculated from eqn (23) at different condensation conditions. In principle, a relatively lower wetting angle and narrower cavity are favorable for nucleation processes. This also explains why randomly distributed cavities, grooves, scratches (smaller β*) and heterogeneous particles (smaller α) can act as nucleation sites.^26,27,39

One of the important conclusions obtained from the above analysis is that the number of active nucleation sites will be finite on a practical condensation substrate with randomly distributed micro cavities. The number of nucleation sites per unit surface area is usually defined as the nucleation sites number density (N_s), an important parameter in dropwise condensation heat transfer theory.⁴⁰ Considering that the actual condensation substrate is composed of randomly distributed micro cavities, grooves and apexes of different structure parameters, the nucleation capabilities of different areas are inherently different depending on whether the above thresholds are well fulfilled. During the initial condensation stage, the micro cavities with higher nucleation capabilities will be rapidly occupied by initial nuclei. As condensation continues, more cavities that fulfill the threshold will be gradually activated, and the number of initial nuclei increases accordingly until all of the possible nucleation sites are occupied, yielding a maximum value of nuclei numbers. After that, the subsequent condensation process will be realized by the growth of pre-existing nuclei from critical size to micro droplets, while no nuclei could form on the blank surface between adjacent droplets. The blank area is inherently not preferred for nucleation basically due to the failure to fulfill the threshold.

4.2 Nucleation capability of micro cavities at various condensation conditions

As indicated by eqn (9) and eqn (23), the nucleation rate is also a function of vapor pressure and supersaturation. The calculated results of nucleation rate at various condensation conditions are presented in Fig. 4 and 5, with S varying from 1.2 to 1.6 under the vapor pressure of 100 kPa, and P_v varying from 10 kPa to 100 kPa under the supersaturation of 1.5, respectively. As described earlier, an arbitrary threshold of J = 1 m⁻² s⁻¹ is chosen to determine whether the micro cavity can be activated as an nucleation site, and the combination of α and β that fulfils the threshold is shown in Fig. 4 and 5 by color fill.

Fig. 4 Nucleation capability of micro cavities under various supersaturations.

Fig. 5 Nucleation capability of micro cavities under various vapor pressures.

As S increases from 1.2 to 1.6, the nucleation rate for the same structure increases accordingly. According to eqn (9), the critical size of the nucleus decreases sharply as S increases, yielding a comparably lower nucleation free energy barrier as indicated by eqn (10). Consequently, the nucleation rate is greatly increased for the same micro cavity. Meanwhile, some of the micro cavities that are originally not preferred for nucleation under low supersaturations can translate into effective nucleation sites when S is increased to a certain degree.

On the other hand, the nucleation rate also increases with P_v under a constant supersaturation, as shown in Fig. 5. According to eqn (9) and (10), as vapor pressure increases, the critical radius of the nucleus decreases slightly, yielding a relatively lower nucleation free energy barrier that is preferred for nucleation. Meanwhile, as indicated by eqn (24), the kinetic pre-factor J₀ also increases with P_v. As a result, J increases rapidly with P_v for the same S and α ∼ β. Similar to the behavior observed in Fig. 4, some of the micro cavities that are originally not preferred for nucleation under low vapor pressures can translate into effective nucleation sites when P_v is increased to a certain degree, as shown in Fig. 5. As P_v and S increase, more micro cavities with wider cone angles can translate into active nucleation sites, suggesting that N_s may increase with P_v and S accordingly. The relationship between N_s and the condensation conditions has been noticed by different researchers,^41,42 and the results also suggest that N_s increases with P_v and S.

5. Conclusions

A circular conical apex and a cavity are proposed as physical models to represent the typical defects which widely exist on substrates, and classic nucleation theory is adopted to quantitatively analyze the nucleation capability of different microstructures at different water vapor nucleation conditions.

The results indicate that the kinetic pre-factor and nucleation free energy barrier all decrease when the substrate structures translate from apexes to cavities, and the nucleation rate of narrower cavities is higher than for a planar substrate and apexes. The cavities that are distributed on a substrate can act as nucleation sites, and the activated nucleation sites are finite for practical substrates due to the different nucleation capabilities of different surface areas. The spatial control of nucleation sites and the intensification of the nucleation sites number density can be realized by substrates with relatively lower wetting angle and the presence of narrower cavities.

The nucleation capability can also be affected by condensation conditions. As vapor pressure and supersaturation increase, the nucleation rate increases rapidly for the same structures, and some of the cavities that are originally not preferred for nucleation gradually translate into active nucleation sites, suggesting that the nucleation sites number density may increase with vapor pressure and supersaturation.

Acknowledgements

The authors greatly appreciate the financial support from the National Natural Science Foundation of China (no. 51236002, no. 51476018).

Notes and references

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