C HAPTER 3
3.3 Results and discussion
f2(κ) = Rcosh 2κ
2κ2 −Rsinh 2κ 2κ3 − R
2κ2, and f3(κ) = 2 tanhκ
Rρlg dσ
dT
−2σtanhκ Rρl2g
dρl dT
.
Equation (3.9) is a first-order non-linear ordinary differential equation in terms ofκ(T). In this paper, this equation is solved by the fourth-order Runge-Kutta method (RK4) for different constant temperatures starting from a reference temperature,Teq(at whichκ(Teq)is a known quantity). The assumption is that temperature variation ofρl(T),ρa(T),σ(T)are known quantities, and so subsequently temperature derivatives are known. These temperature derivative values of ρl(T), ρa(T) and σ(T), are obtained from empirical relationships or fitting relationships to the published data (e.g. Fig. 3.2). Within the small range of temperatures (T ∈[Tlc,Tuc])or metastable states at adiabatic condition, it is assumed that Marangoni effects are negligible.
The solution of Eq. (3.9) providesκ(T), which straightway is used to calculate SCA (θe).
θe(T) =
cot−1(sinh(κ(T))); θe(T)∈ 0,π
2 i
∀T ∈[Tlc,Tuc].
π+cot−1(sinh(κ(T))); θe(T)∈hπ 2,π
∀T ∈[Tlc,Tuc].
(3.10)
Due to CLP, meniscus line will be contracted or stretched depending on the dominance of upward or downward forces respectively within the temperature rangeT ∈[Tlc,Tuc]. Range of temperatures i.e.Tlc andTuccan be evaluated from a fitting relationship ofθe-vs.-T plot, if the CAH is known a priori. Assuming the CAH is known to be (α1+α2)i.e. (θe+α1) and (θe−α2), then fromθe-vs.-T plot, a polynomial relationship can be obtained as T =
∑ni=0 biθei
= ft(θe)(nis a positive integer;bn,bn−1,. . .,b0are the curve-fitting parameters andbn6=0). Therefore, the range of temperatures will be: Tuc= ft θe(Teq)−sgn(bn)α1
and Tlc= ft θe(Teq) +sgn(bn)α2
, where sgn(x)is the sign function ofx. The above formulations can be extended to multicomponent system as well as to dynamic situation as discussed briefly in Appendix-A3.1-A3.2.
or down (translation). The radius and the length of the capillary tube are taken as 50 µm and 0.50 m, respectively for the numerical simulations. Initial SCAs for different liquids on solids (Table3.1) are considered at 298.15 K i.e.θe(Teq=298.15 K). The temperature,Teq= 298.15 K is the reference value, and the parameter,κ at this initial or reference temperature is κ(Teq) =κeq. Equation (3.9) is, thereafter, numerically solved using κeq as the initial value to obtain κ(T) for the intended range of temperature. In addition, the temperature- dependent values of liquid densitiesρl(T), air densityρa(T), and surface tensionsσ(T)are collected from various literature sources (as shown in Fig.3.2) and are fitted with polynomial relationships. From the initial conditions likeκeqandθe(Teq)(see Table3.1), the parameter κ(T) is calculated for different constant temperatures (say, T > or < 298.15 K) at the adiabatic condition. Therefore, SCAs (θe(T)) for these constant temperatures are calculated from Eq. (3.10). Figure3.3shows the variation of SCA for the temperature range ofTeq±20 K. Even if the figure shows the continuous line, in theory,θe(T)values are distinct at constant temperatures withinTeq±20 K. This range of temperatures is selected arbitrarily just to test the proposed model.
100 200 300 400 500 600
100 101 102 103 104 105
T (K) ρ (kg/m3 )
100 200 300 400 500
10−2 10−1 100
T (K)
σ (N/m)
Water n−Hexadecane Glycerol Mercury Ethanol Air
Water n−Hexadecane Glycerol Mercury Ethanol
Figure 3.2: Temperature-dependent densities of liquid, air and surface tension i.e. ρl(T),ρa(T)andσ(T) respectively. Data sources of densities: water (Dean,1999),n-hexadecane (Onken et al.,1989), glycerol (Glycerine Producers’ Association,1963), mercury (Dean,1999), ethanol (Maggi and Alonso-Marroquin,2013), air (Golfman,2012). Data sources of surface tension: water (Vargaftik et al.,1983),n-hexadecane (Neumann et al.,1971), glycerol (Glycerine Producers’ Association,1963), mercury (Dean,1999), ethanol (Maggi and Alonso-Marroquin,2013).
The SCA decreases withT in the case ofθe<90◦(i.e. wetting fluids, e.g. water,n-hexadecane and glycerol), however, it increases with temperature in the event ofθe>90◦(i.e. non-wetting fluids, e.g. mercury) (Fig. 3.3). Non-dimensionalθe with non-dimensionalT is, therefore, plotted for all the liquids mentioned above for comparison purposes (Fig. 3.4). This trend of
Table 3.1:Initial SCAs for different liquids
Liquid on Solid θe (Teq=298.15 K) Reference
Water on Glass 50◦ Liechti et al.(1997)
n-Hexadecane on Polytetrafluoroethylene 46◦ Neumann et al.(1971)
Glycerol on Glass 48.5◦ Liechti et al.(1997)
Mercury on Glass 133◦ Ellison et al.(1967)
SCA of wetting and non-wetting fluids is qualitatively analogous to Wenzel state, as surface becomes more hydrophilic and more hydrophobic in the presence of surface roughness, which is defined by Wenzel (1936) as the ratio of real surface to the projected one (Das et al., 2012, and see references therein). In this analysis, this trend is, however, happening due to temperature and other CAH-inducing surface parameters like roughness.
270 280 290 300 310 320 46
48 50 52
(a)
T (K)
θ
e( ° )
270 280 290 300 310 320 42
44 46 48 50
(b)
270 280 290 300 310 320 47
48 49 50
(c)
270 280 290 300 310 320 132.5
132.75 133 133.25 133.5
(d)
Figure 3.3: Temperature-dependent SCAs for (a) water, (b)n-hexadecane, (c) glycerol and (d) mercury for temperature ranges of 298.15 K±20 K.
CAH (CH) can be construed as the slope of the graph between SCA and temperature at Teq i.e. for a small and fixed ∆T =max(|Tuc−Teq|,|Tlc−Teq|)†
, CH = |θmax−θmin|
≈ dθe/dT|T=Teq
|∆T|. The temperature variation of θe is susceptible to density, as it is
†max(r,s) =maximum value amongrands=12(r+s+|r−s|)(seeKalman’s (1984) paper).
0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 0.92
0.94 0.96 0.98 1 1.02 1.04 1.06 1.08
T / T
eq
θ e (T) / θ e (T eq)
Water n−Hexadecane Glycerol Mercury
Figure 3.4:Non-dimensional SCAs versus non-dimensional temperatures with respect to reference temperature Teq=298.15 K for water,n-hexadecane, glycerol and mercury.
0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08
0.98 0.985 0.99 0.995 1 1.005 1.01 1.015 1.02 1.025
T / T eq θe (T) / θe (Teq)
R = 50 µm R = 150 µm R = 250 µm R = 350 µm R = 450 µm
Figure 3.5:Non-dimensional SCAs-vs.-non-dimensional temperatures with respect to reference temperature Teq=298.15 K for water, assuming differentRand keeping other parameters constant.
observed that the lighter fluid like n-hexadecane has more slope than other heavier fluids like water or glycerol or mercury (Figs. 3.5-3.6). The thermal variation of SCA is more for the lighter liquids. Magnitudes of slope, |(dθe/dT)|(≡CH) in decreasing order are: (1) n-hexadecane: 0.14461◦K−1andρl(Teq) =771.44 kg m−3, (2) water: 0.088505◦K−1and ρl(Teq) =996.66 kg m−3, (3) glycerol: 0.043374◦K−1andρl(Teq) =1258.18 kg m−3and (4) mercury: 0.016932◦K−1andρl(Teq) =13533.13 kg m−3. This shows a dependency of densities of liquids on CAH. Thermal variation ofθedepends only onθe(Teq)and the radius (R) of the capillary tube and does not depend onL. It is evident from Eq. (3.9) thatCH(T) is directly proportional toRfor wetting fluids (see Fig.3.5) and vice versa for non-wetting fluids (see Fig. 3.6) (if all other parameters remain constant). Asθe(Teq)reduces in the case of wetting liquid (e.g. water from 89◦to 10◦) or increases in the case of non-wetting liquid (e.g. mercury from 91◦to 170◦),|(dθe/dT)|varies from 0.002◦K−1to 0.514◦K−1for water and 0.0002◦K−1to 0.145◦K−1for mercury (Fig. 3.7).
Another interesting observation is thatCH(T)tends to zero, or the curve ofθe(T)becomes asymptote (Fig. 3.8) asθe(T)tends to 0◦or 180◦(from Eq. (3.10)). Hitherto, if the initial SCA (θe(Teq)) is closer to the value of 0◦ or 180◦, then the chances of θe(T) reaching asymptotic value are high, within small temperature ranges (especially for lighter liquids as they are much susceptible to temperature).
0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08
0.9995 0.9996 0.9997 0.9998 0.9999 1 1.0001 1.0002 1.0003 1.0004 1.0005
T / T eq θe (T) / θe (Teq)
R = 50 µm R = 150 µm R = 250 µm R = 350 µm R = 450 µm
Figure 3.6:Non-dimensional SCAs-vs.-non-dimensional temperatures with respect to reference temperature Teq=298.15 K for mercury, assuming differentRand keeping other parameters constant.
0.9 0.95 1 1.05 1.1 0.7
0.8 0.9 1 1.1 1.2 1.3 1.4
T/Teq
θ e (T)/θ e (T eq)
(a)
θe (Teq) = 10°
θe (T
eq) = 45°
θe (T
eq) = 70°
θe (T
eq) = 89°
0.9 0.95 1 1.05 1.1
0.996 0.997 0.998 0.999 1 1.001 1.002 1.003 1.004 1.005
T/Teq
θ e (T)/θ e (T eq)
(b) θe (T
eq) = 91°
θe (T
eq) = 110°
θe (Teq) = 135°
θe (Teq) = 170°
Figure 3.7:Non-dimensional SCAs versus non-dimensional temperatures with respect to reference temperature Teq=298.15 K for (a) water and (b) mercury, assuming differentθe(Teq)and keeping other parameters constant.
0 2 4 6
T (K) θ e (T) (°)
275 280 285 290 295 300 305 310 315 320177 178 179 180
Mercury (θe (T
eq) = 179°) Water (θ
e (T
eq) = 1°)
Figure 3.8:Temperature-dependent SCAs for mercury and water for temperature ranges of 298.15 K±20 K withθe(Teq) =179◦and 1◦(i.e. closer values to stable angles) respectively.
3.3.1 Validation against published results
We have compared our results with the experimental values ofNeumann et al.(1971) and with Adamson’s potential distortion model (PDM) (Adamson,1973) (see Fig. 3.9). We calculate our results for a particular liquid with knownθe(Teq),σ(T)andρl(T)(see Table3.2). Our
model matches with experimental data quite well. Large range of temperatures shows that contact angle changes not exactly in linear fashion, also predicted by PDM (Adamson,1973).
This large range is just to show the non-linearity of θe-vs.-T relation. Adamson (1973) showed that contact angle will be zero at a critical temperature (Tc) . This is not true in our case as our model predicts that at stable angles (i.e. 0◦or 180◦),θe(T)becomes asymptotic with increasing temperature (see Fig. 3.8). This may be because we have not included interfacial energies of solid-air or solid-liquid that could be one of the reasons to reduce the asymptotic behavior near stable angles. Our model is specifically targeted to CAH due to CLP phenomenon. We assume in deriving the Eq. (3.9) that CLP happens within small range ofT. Within small range ofT, our model matches quantitatively with PDM and experimental result ofNeumann et al.(1971) satisfactorily.
Table 3.2:Temperature-dependent properties required for verification purpose
Liquid on teflon σ(T)(1) ρl(T)(2) θe(Teq=288.15 K)
n-octane 0.04949−0.00009509T 703.1224−0.8099T−0.0003471T2 29.91◦(3) n-decane 0.050792−0.00009197T 730.1536−0.7678T−0.001962T2 38.13◦(4) n-undecane 0.05107−0.0000901T 740.7080−0.7064T−0.0002179T2 41.70◦(4) Data sources: (1): Jasper and Kring (1955), (2): Egloff(1939), (3): Sutula et al. (1967),
(4):Neumann et al.(1971)
2800 300 320 340 360 380 400 420 440 460 480 500
5 10 15 20 25 30 35 40 45
T (K) θe(T) (°)
n−undecane n−decane
n−octane
Figure 3.9:SCAs-vs.-T on teflon for liquids:n-octane (R=800µm),n-decane (R=400µm) andn-undecane (R=40µm) withLbeing 0.5 m andρa(T)being same as in Fig.3.2. Here symbols - circle (◦), dash-dot (−·), and solid line (−) represent the results of experiment, PDM, and our model respectively. Experimental data ofn-octane are taken fromSutula et al.(1967) (throughAdamson,1973), andn-decane andn-undecane from Neumann et al.(1971).