During the evaporation process, non-uniform temperature distribution along the droplet interface generates a Marangoni convection inside the evaporating droplet. Marangoni flow significantly affects the flow field and the particle deposition pattern. The temperature distribution within the droplet depends on both the conduction of heat within the droplet and the conduction of heat to the droplet from the underlying substrate. If the thermal conductivity of the substrate (ks) is higher than the thermal conductivity of the liquid drop (kL), then the droplet base in contact with the substrate will be isothermal. On the other hand, if ks is small then the liquid will be nonisothermal along the base of the droplet (Larson, 2014). A reduced value of substrate thermal conductivity will lower the overall temperature in the droplet as well as temperature gradient along the droplet interface (Ristempart et al., 2007; Xu et al., 2010). Ritsenpart et al. (2007) using
1.2 Review of the literature asymptotic analyses have shown that the direction of circulation depends on both the contact angle and ratio of the thermal conductivities of the substrate and liquid (KR). They have shown that for KR > 2 the Marangoni flow is directed radially outward along the substrate whereas for 1.45 < KR < 2, the flow direction depends on the value of contact angle, and for KR < 1.45 the circulation direction is radially inward along the substrate as shown in Figure 1.15. This theory is formulated only for substrate with infinite thickness.
For the case of finite substrate thickness, where the bottom of the substrate temperature can be assumed to be at ambient temperature, a dimensionless number (RN) can be defined that control the temperature distribution along the droplet free surface. RN can be defined as the ratio of thermal resistance of the substrate to the thermal resistance of the liquid droplet. The total temperature difference between the top of the droplet and the bottom of the substrate is estimated as ΔTtot = ΔT + ΔTs where ΔT is the temperature drop across the height of the droplet and ΔTs is the drop across the finite thickness of the substrate. The overall temperature different can be written as (Larson 2014),
∆𝑇𝑡𝑜𝑡 = 2 ∆𝐻𝑣𝑎𝑝 𝐷𝑣𝑎𝑝 𝜌𝑣𝑎𝑝 ℎ𝑜
𝑘𝐿 𝑅 (1 +𝑅𝑁)
(1.29)
where, Hvap is the heat of vaporization per unit of mass (kJ/kg), Dvap is the diffusivity of water vapor (m2/s), ρvap is the mass density of vapor (kg/m3), ho is the height of the droplet (cm), kL is the thermal conductivity of the liquid, R is the radius of the droplet, hs is the thickness of the substrate (cm) and 𝑅𝑁 = 𝑘𝐿ℎ𝑠
𝑘𝑠ℎ𝑜. If 𝑅𝑁 ≫ 1, then the eqn. (1.29) becomes
∆𝑇𝑡𝑜𝑡= 2 ∆𝐻𝑣𝑎𝑝 𝐷𝑣𝑎𝑝 𝜌𝑣𝑎𝑝 ℎ𝑜
𝑘𝐿 𝑅
(1.30)
When the thermal resistance in the substrate is greater than that in the droplet, it
1.2 Review of the literature the contact line, where evaporation is fastest, rather than at the top of the droplet, where the droplet height is maximum, but the evaporation rate is the lowest. Xu et al. (2010) predicted that the temperature distribution not only depends on the relative thermal conductivity of the substrate and liquid but also the ratio of the substrate thickness to the contact angle of the droplet. They have neglected heat convection and proposed the following condition for lowest temperature to be at the contact line: RN = [sin2θ − 4λ tan(θ/2)]/ (4λ + 2sin2θ). For small contact angles, this condition is well approximated by RN > 1. For large contact angles, greater than 45o, increasing the contact angle further decreases the value of the RN needed to drive the coldest temperature to the contact line, and thereby reverse the Marangoni flow.
The criterion for direction of surface temperature can be more understood from the Figure 1.15b. David et al. (2007) have shown experimentally that uniform evaporation rate along the droplet free surface increases with the thermal conductivity of the substrate.
Recently, Zhang et al. (2014) have shown through numerical simulation that uniform surface temperature distribution for all the conditions of evaporation is not reasonable and a non-monotonic surface temperature profile may exist during the transition period i.e.
when the contact angle is near to the critical value as shown in Figure 1.16. They reported that the surface temperature distribution can be defined by two factors i.e. non-uniform evaporation rate along the droplet free surface and another one is non-uniform conduction path from the substrate to the droplet free surface. The surface temperature due to conduction path may be predominant over the evaporative cooling when the contact line of the droplet is large, whereas the surface temperature due to evaporation cooling is dominant when the contact angle reduces and the influence of conduction path and evaporative cooling are comparable during the transition period of the evaporating droplet.
1.2 Review of the literature
(a) Surface temperature gradient near the contact line of an evaporating droplet. (From Ristenpart et al. 2007, Phys. Rev. Lett. 99, 234502. Copyright 2007 by the American Physical Society). (b) The temperature distribution relative to thermal conductivity as well as the contact angle of the droplet.
(From Xu et al. 2010, Langmuir 26, 1918-1922. Copyright 2013 by the American Chemical Society).
Phase diagram of the temperature distribution along the interface of an evaporating droplet. (From Zhang et al. 2014, Phys. Rev. E. 89, 032404.
1.2 Review of the literature