Maximum spreading factor

Chapter 1 Introduction

5.3 Results and Discussion

5.3.4 Maximum spreading factor

Figure 5.10. Log-log plot ofβ as a function ofW eof the droplet. βincreases with the increase of W e all impact regimes. Each square marker represents the mean value of multiple data points and error bars indicate the standard deviations from the mean.

W e ≈1.7 toW e ≈76.7. Empirically, the maximum spreading factor in this impact regimes can be approximated by a scaling of β ∼ W e^1/5. A significant transition occurs right before W e₂, such that β increases at a faster rate in the W e ≥ W e₂ impact regime.

In the W e≥W e₂ impact regime, the maximum spreading factor can be approxi- mated by a scaling of β ∼W e^1/2. Such scaling, is in fact, the generalized scaling law derived from a straightforward energy conservation approach. Using this approach, the scaling of maximum spreading factor is obtained by calculating the radius of the droplet when the initial kinetic energy of the droplet is converted completely into surface energy (Chandra and Avedisian, 1991; Bennett and Poulikakos, 1993).

The agreement between this scaling and the experimental data presented here sug- gests that a straightforward energy conservation approach may adequately predict the droplet spreading behavior upon impact on a non-wetted surface in the medium and high Weber number regime.

As expected, the scaling of β ∼ W e^1/4, as suggested by Clanet et al. (2004), Bartolo et al. (2005), and Biance et al. (2006), fails to capture the droplet spreading behavior in all impact regimes. Note that this scaling law was proposed to counter the straightforward energy conservation approach because of the presence of an internal flow in the droplet during the impact. Such internal flow suggests that the initial kinetic energy is not completely converted into surface energy even when the droplet reaches its maximum spreading diameter, and thus β should be determined by the balance between gravity and surface forces. However, as mentioned by Eggers et al.

(2010), these scaling laws are actually similar to each other in such a small range of W e, such that they cannot be reliably discriminated. In the low Oh and Ca regime, where the effect of viscous and capillary forces is minimal, the maximum spreading radius should not be dictated by a balance between gravity and surface tension (Eggers et al., 2010).

In the largeW eandReregimes, two different mechanisms to predict the maximum spreading diameter have been assumed (Eggers et al., 2010). The first assumption is that the effect of viscosity can be neglected such that the maximum spreading

Figure 5.11. Log-log plot of β as a function of Re of the droplet. β increases with the increase of Reall impact regimes.

diameter only depends on the balance between kinetic energy and surface tension of the droplet. The second assumption is that the viscous dissipation dominates surface tension and the maximum spreading diameter is determined by the balance between inertia and viscosity. Using the first assumption, the maximum spreading diameter should scale as β ∼ W e^1/2, while using the second assumption, it should scale as β ∼Re^1/5 (Clanet et al., 2004; Bartolo et al., 2005). It has been demonstrated above that β ∼W e^1/2 in theW e≥W e₂ impact regime, confirming the validity of the first assumption.

The second assumption, which is derived from an idea that the spreading behavior is limited the effect of viscosity, may only be satisfied in the high Oh regime. Since this current study was done in the very low Oh regime, the obtained experimental data could not be used to confirm the validity of the second assumption. For the sake of argument, β is plotted versus Re to show if the scaling of β ∼ Re^1/5 is satisfied (Figure 5.11). Although β does increase with the increase of Re, the scaling of β ∼Re^1/5 is not satisfied. In fact, empirical approximation of the experimental data shows that the maximum spreading factor scales as β ∼Re^1/2. Such poor prediction

of β by the second assumption infers that the loss of kinetic energy of the droplet during impact due to viscous dissipation is indeed minimal. Therefore, the scaling of β ∼ Re^1/5 may not be relevant for use in predicting the droplet spreading behavior on a non-wetted surface in the small viscosity regime.

As stated earlier, many previous studies have been conducted in the past (Chandra and Avedisian, 1991; Pasandideh-Fard et al., 1996; Kim and Chun, 2001; Ukiwe and Kwok, 2004; Vadillo et al., 2009; Li et al., 2010; Eggers et al., 2010; Lee and Lee, 2011) to find a mathematical model that could reliably describes β. These studies show that an accurate model can be made based on the energy conservation approach and by taking the effect of viscous dissipation and surface wettability into account.

Based on the energy conservation approach, a balance of energy between phases of drop before, during, and after the impact can be written as follows:

KE₁+SE₁ =KE₂+SE₂+W₁₂ (5.13a)

KE2+SE2−W23=KE3+SE3 (5.13b) whereKE1 andSE1 are the droplet’s kinetic energy and surface energy, respectively, during the free fall phase right before the impact; KE₂ and SE₂ are the droplet’s kinetic energy and surface energy, respectively, at the instant of maximum spread during the impact phase; KE₃ and SE₃ are the droplet’s kinetic energy and surface energy, respectively, during the rebound phase right after the impact. The loss of en- ergy due to viscous dissipation when the droplet spreads and retracts on the surface during the impact are denoted by W12 and W23 respectively. At the instant of maxi- mum spread, the droplet’s kinetic energy can be assumed to be negligible (KE₂ = 0) because the velocity of the droplet is basically zero at that exact moment. Equa- tion (5.13a) represents the energy balance between the total of the droplet’s energy during the free fall phase right before it impacts the surface and the sum of droplet’s energy when it reaches its maximum spreading diameter and the energy consumed to spread the droplet on the surface. Equation (5.13b) represents the energy balance between the sum of the droplet’s energy when it reaches its maximum spreading di-

ameter and the energy consumed to retract the droplet on the surface and the total of the droplet’s energy during the rebound phase right after it lifts off from the surface.

Since it can be assumed that the loss of energy due to viscous dissipation during the spreading and retraction phase is equal, W₁₂ =W₂₃ = W, Equation (5.13a) and Equation (5.13b) can be written in a single equation as following:

KE1+SE1 =KE3+SE3+ 2W (5.14)

where the kinetic energy of the droplet during the free fall and rebound phases is given by

KE₁ =πρd³_iv_i²/12 (5.15a) KE₃ =πρd³_rv_r²/12 (5.15b) and the surface energy of the droplet during the free fall and rebound phases is given by

SE₁ =πd²_iσ (5.16a)

SE₃ =πd²_rσ (5.16b)

Equation (5.14) can be simplified by substituting v_r and d_r with v_i and d_i using the relation given by the volume ratio ξ = (d_r/d_i)³ and the coefficient of restitution ε=v_r/v_i. Hence, it can be written as:

πρd³_iv²_i

12 +πdi²σ= πρξd³_iε²v_i²

12 +πξ^2/3di²σ+ 2W (5.17) By substituting Equation (5.15a) and Equation (5.16a) into Equation (5.17), W can be expressed as:

W = 1

2[(1−ε²ξ)KE₁+ (1−ξ^2/3)SE₁] (5.18) where the volume ratio (ξ) is described in Equation (5.1) and the coefficient of resti- tution (ε) is described in Equation (5.12). According to Ukiwe and Kwok (2004), the droplet’s surface energy when it reaches its maximum spreading diameter can be

expressed as:

SE₂ =πσd_sl_s+π

4σ(d_s−l_s)²(1−cosθ_s) (5.19) where l_s and θ_s are the thickness and the equilibrium contact angle of the droplet when it reaches its maximum spreading diameter. Experimental data show that the thickness of the droplet when it reaches its maximum diameter decreases rapidly with the increase of Weber number. At a small Weber number, W e < W e₁, the droplet’s thickness is still comparable to the maximum diameter of the droplet. However, at a moderate Weber number, W e₁ ≤ W e < W e₂, the droplet’s thickness becomes an order of magnitude smaller than its maximum diameter. At an even larger Weber number, W e ≥ W e₂, the droplet’s thickness is found to be l d_s, and can no longer be observed. Hence, as suggested by Pasandideh-Fard et al. (1996), Vadillo et al. (2009), and Lee and Lee (2011), it is reasonable to neglect the first term of Equation (5.19). As mentioned earlier, the static contact angle of the droplet on the superhydrophobic carbon nanotube array is about 171^◦ and the contact angle hysteresis is about 3^◦, which suggests that the equilibrium contact angle of the droplet during the impact is about 171^◦(±3^◦). Consequently, the last term of Equation (5.19) can be approximated as πσd²_s/2. The energy balance between phases of the drop before and during the impact can then be written as:

ρdiv_i²

12σ + 1 = d²_s 2d²_i + 1

24[(1−ε²ξ)π

12ρd³_iv_i²+1

2(1−ξ^2/3) (5.20) After simplification, β can be expressed as:

β = [1

12(1 +ε²ξ)W e+ 1

2(1 +ξ^2/3)]^1/2 (5.21) where ξ is described in Equation (5.1) and ε is described in Equation (5.12).