Transient variation in contact angle

Chapter 5. Evaporation of sessile binary droplet on a heated spherical particle

5.2.5. Transient variation in contact angle

In the pinning mode of evaporation, contact angle decreases with time following a linearly decaying trend. Although, an inverse relationship of contact angle and surface tension at gas- liquid interface can be seen from Young’s equation which could explain the decreasing con- tact angle with increasing temperature, however, the present observation shows a linearly de- caying trend in contact angle even when the droplet temperature (hence surface tension)

Chapter 5 156 reaches an steady stage after the initial heating period (Figure 5.11). It is worth to notice that, contact angle hysteresis were obtained for only water droplet but not for water-glycerol drop- lets due to the relatively lower surface tension of water-glycerol which found really difficult to stay stable on the particle’s apex with a large volume.

Temporal variation in contact angle of an evaporating droplet can be derived from the volumetric evaporation rate based on the spherical cap assumption, which is given as (see derivations in Appendix E):

( )2

3

1 cos( )t

t

c

d V

dt r

θ θ

π

= +  (5.3)

where θtis the contact angle including the particle segment as shown in Figure 5.4a, the contact angle of droplet on particleθ is given asθ θ θ= t − p (θp →0for flat surface): rc is half of the chord length cap previously shown in Figure 5.4, Vis the volumetric evaporation rate of the spherical particle-droplet cap (V =Vcap L, +Vcap P, ) which is described as follows;

( T ) Hu&Larson

V = f Ma ×V (5.4)

In Eq. (5.4), VHu&Larson =(πD ri c /ρL)(1−RH C) v(0.27θ2 +1.3)is the evaporation rate pro- posed by the well-known study of Hu and Larson (2002) at room temperature operating con- ditions which was used as the base case in this study; where RH is relative humidity, Di is binary diffusion coefficient in the vapour phase; Cv is a temperature dependent saturated va- pour concentration. Change in droplet temperature due to heat received from the heated sub- strate via conduction and loss of heat at interface due to latent heat of vaporisation can be computed using the following heat balance as dT dt=(AS L− kL*∆T h/ *L +LVρLVCap L, )

(VCap L, ρLCp L, )where AS L− is the wetted area, k*Lis effective thermal conductivity of the liquid

Chapter 5 157 takes into account the internal convections (Abramzon & Sirignano, 1989) within the droplet and hL*is the droplet effective thickness through which heat conduction occurs.

The Hu and Larson model was found to under-predict the evaporation rate compared with experimental data in the present study which was also reported in the recent work of P. Chen et al. (2017). The effect of thermal Marangoni flows under the conditions of heated surface is expected to contribute to the enhancement in evaporation (see section 5.2.6) for an analysis of the internal convections). This effect in Eq. (5.4) was included through an empirical factor

( T)

f Ma in the present study which was obtained by matching the baseline evaporation rate predicted by Hu and Larson (2002) model with the actual evaporation rate i.e. the slope of the droplet volume reduction with time for three different surface temperatures and is given as:

( T) 2.7910 4 1.00

f Ma = − Ma+ (5.5)

whereMaTis thermal Marangoni number driven by interfacial temperature difference [Eq.

(5.6)]. Equation (5.5) gives f Ma( T)=1.00when MaT =0 presenting the base evaporation model at room temperature condition at which thermal Marangoni effect is assumed negligi- ble.

A 4th order Runge-Kutta method implemented in Matlab (version 2015b) was used to solve Eq. (5.3) and the predicted reduction in the contact angle in the pinning mode is presented in Figure 5.17 for the “gly10” case (90 % water – 10 % glycerol). Reasonable agreement be- tween model predictions [Eq.(5.3)] with experimental data was found for three different solid temperatures with root mean square error 0.53 K, 1.48 K and 2.55 K for solid surface temper- ature of 323 K, 343 K and 358 K, respectively).

Chapter 5 158 Figure 5.17 Measured contact angle and spreading diameter reduction with time, at three dif-

ferent solid temperatures. Operating conditions: 90 % water - 10 % glycerol droplet. d0 = 2.75 mm, dp = 10 mm. TL,0 = 299.5 K. TS = 323-358 K.

Also shown in Figure 5.17 the maximum contact angle, θmax which was obtained after excluding first second data to exclude the effect of the initial oscillations as discussed earlier.

This contact angle was found smaller at a lower solid surface temperature which was ~ 70o, 79o and 80o for surface temperature at 323, 343 and 358 K, respectively. Initial contact angle was determined after the initial spreading stage which occurs approximately within a second (under the present experimental conditions). During this initial spreading stage, a higher heat supply at the solid-liquid interface leads to a higher speed of the contact line movement against the spreading (while the spreading speed of a gently deposited droplet remains low) which in turn results in a higher contact angle. Early experiment of Di Marzo and Evans (1986) depicted a decreasing trend in contact angle of water droplet on aluminium surface

Chapter 5 159 with increasing substrate temperature. Experimental data of Chandra and Avedisian (1991b) showed a slight increase in the contact angle (from ~ 32 to 40o) for a heptane droplet deposit- ed on stainless steel substrate when surface temperature was increased from 24 to 98 oC.

Bernardin et al. (1997) however reported a negligible change in the contact angle for water droplet (~ 90o) deposited on a polished aluminium surface. Crafton and Black (2004) investi- gated the evaporation behaviour of water and heptane droplet on aluminium surface heated under saturation temperature but did not mention any direct relationship between initial con- tact angle and solid temperature. Vadgama and Harris (2007) measured the contact angle of R134a on aluminium and copper surface at temperature up to 80 oC. Their results showed that contact angle decreased with surface temperature on aluminium but increased with tempera- ture on copper surface. The experiment of Misyura (2017), for the same surface properties (copper, structured wavy shape), initial contact angle of a water droplet increased from ~ 67 to ~ 74o when diameter growths from ~ 0.7 to ~ 3.5 mm. Experimental data of Sefiane and Tadrist (2006) presented this behaviour for water droplet on an aluminium surface heated from 20 to 100 oC (Fig. 4a in Sefiane and Tadrist, 2006) however the dependency of contact angle on substrate temperature was not discussed in the study. Different wall temperatures at 75 and 95 oC resulted in a slight difference in contact angle of ~ 1.4o under these conditions.

Different views have been reported against the effect of surface temperature on the initial contact angle therefore extra studies are necessary.

Effect of liquid concentration on the contact angle is presented in Figure 5.18. The contact angle reaches a maximum value after the deposition occurring within the first one second and then decreases continually with time as the evaporation progresses. For solid temperature at 323 K and glycerol concentration varying in the range of 0 to 35 %, maximum contact angle remains in the range of ~ 66 to 84o. A larger concentration of glycerol in the droplet could be seen leading to a lower maximum contact angle which could possibly be at-

Chapter 5 160 tributed to the reduction in surface tension with a higher percentage of glycerol in the mixture as previously explained.

Figure 5.18 Contact angles of water-glycerol droplets at different concentration of glycerol from 0 to 35 %. Operating conditions: TS = 323 K, dp =10 mm, Ta = 296 K, relative humidity

RH = ~ 50 %. The contact angle decrease rate found at 0.16 o/s, 0.14 o/s, 0.12 o/s and 0.10 o/s for gly00, gly10, gly25 and gly35 respectively.

All the contact angle profiles exhibit a similar linearly decaying trend until the more volatile component (water) is nearly depleted from the system. The present experimental finding on continuous reduction in contact angle with time during a major time of the evapo- ration is consistent with the previous measurement of Chandra et al. (1996) and Crafton and Black (2004), in which continual decrease in contact angles were noted for a majority time of evaporation. The effect of composition on the maximum contact angle (θmax) is also noticea- ble here which decreases from ~76 to 65 o with the increasing glycerol concentration from 0

Chapter 5 161 to 35 % which is attributed to the decreasing droplet volume (d0 = 2.61 - 2.91 mm) with in- creasing glycerol concentration as explained earlier. The linear relationship of contact angle with time breaks down towards the end which occurs at ~ 150 s for 35 % glycerol; ~ 215 s for 25 % glycerol; and ~ 250 s for 10 % glycerol concentration, respectively. The contact angle decrease rate within the linear stage was found to vary slightly from ~ 0.16 to 0.10 o/s when glycerol concentration was increased from 0 to 35 %.

Figure 5.19 depicts the temporal change in contact angle of the more volatile system – water-IPA droplets, which shows a large range in initial contact angle for IPA concentration from 0 to 15 %. While the transition stage is clear for the widely different boiling point wa- ter-glycerol droplets at the end of the process, indefinite transition can be noticed at the early stage of evaporation for water-IPA droplet

Figure 5.19 Contact angles of water-IPA droplet at different IPA concentration from 0 to 15 %. Operating conditions: TS = 343 K, dp =10 mm, Ta = 296 K, relative humidity RH =

~50 %

Chapter 5 162 Figure 5.20 Transient change in contact angles of “Butanol05” droplet. Operating conditions:

TS = 353 K, dp =10 mm, Ta = 296 K, relative humidity RH = ~ 50 %.

Change in contact angle of the water-butanol droplet at 5 % butanol shown in Figure 5.20 appears different from the earlier results for water-glycerol and water-IPA systems. The three stages of evaporation for this particular mixture was previously discussed in Figure 5.16 Figure 5.21 shows the correlation between the spreading ratios (βpin) and the maxi- mum measured contact angles (θmax). A linearly decreasing profile was obtained through lin- ear regression given asφmax =θmax /θS,0 =−0.86βp ni +2.23 (R2 ~ 0.88) where the static con- tact angle for water droplet on brass spherical surface measured at room temperature as

o

,0 74

θS = . The obtained slope of ~ 0.86 in the present study is for water-glycerol system at maximum concentration of glycerol at 35 % while it is 0.73 for water-IPA system at maxi-

I II III

Stage I: CCR

Stage II: decrease CR, increase CA Stage III: CCR

Chapter 5 163 mum concentration of glycerol at 15 %. The smaller slope of water-glycerol system is thought due to its higher surface tension, which resulted in higher contact angle and therefore smaller wetted area (or wetted diameter) compared with water-IPA system. The slope ob- tained for water on aluminium flat surface in the study of Di Marzo and Evans (1986) is of ~ 1.01 with the consideration of a static contact angle for water droplet on polished aluminium surface of ~ 90o (Bernardin et al., 1997).

Figure 5.21 Experimental relationship between the pinned spreading diameter and contact angle at solid temperatures from 323 - 358 K and glycerol concentrations from 0 – 35 %. Op-

erating conditions: dp = 10 mm, Ta = 296 K, relative humidity RH = ~ 50 %;