Electrowetting of Ionic Liquids in Solid–Liquid–Liquid Systems - Drops and Bubbles in Contact with Solid Surfaces - page 246

Figure 2. Hydrodynamic and molecular-kinetic approaches to the dynamics of wetting.

states, the dynamic contact angle, θ , changes in time and is a function of the speed,
u , of the moving contact line.
The velocity dependence of the contact angle is most often described in terms
of a hydrodynamic model or a molecular-kinetic model [52] (Fig. 2). In the hy-
drodynamic approach, a small zone in the vicinity of the contact line (characteristic
size l ) is effectively excluded from consideration as the no-slip condition is violated
[53-55].
An elaborated hydrodynamic model has been provided by Cox [54] but a simpli-
fied version, derived by Voinov, performs well for dynamic contact angles smaller
than 130 ◦ . Voinov's equation is [53]

9 μu
γ

ln L θ 3 θ 0 + =

l .

(6)

In this model the main dissipation mode is viscous friction and the viscosity of the
liquid, μ , naturally appears in equation (6). The influence of the macroscopic length
scale, L , and the microscopic length scale, l , is weak because of the logarithmic
dependence. The viscous dissipation, D HD , has been estimated for a liquid wedge
contained between L (e.g., droplet size) and l (e.g., slip length) assuming a film
flow under a fixed pressure gradient [55]

μ ∂u y
∂x

2

const μu 2
θ

ln L

d x d y d z =

D HD =

l .

(7)

The molecular-kinetic approach describes the macroscopic movement of the contact
line as a series of molecular jumps. Each molecular jump is described as a rate-
activated process and arrives at the following equation [56-58]

2 k B T
γλ 2

u
2 k 0 λ ,

sinh − 1 cos θ = cos θ 0 − (8)

where λ is the average size of the molecular jump, k 0 is the jump frequency at a
static contact line ( u 0), k B is the Boltzmann constant, and T is the absolute tem-

perature. The dissipation mode in this case is molecular friction at the contact line,

=