# Contact Angle Relaxation during Droplet Spreading

Contact Angle Relaxation during Droplet Spreading

In this Demonstration, the macroscopic behavior of the contact line during spontaneous spreading of a nonvolatile droplet on a substrate is studied. A droplet is placed on a substrate and initially makes contact with the substrate at some nonequilibrium contact angle . The drop then begins to spread until it reaches its equilibrium contact angle . The dynamics of this process are simulated in this Demonstration.

θ

0

θ

eq

The molecular-kinetic theory of viscous spreading [1] is used to describe the dependence of the contact angle on the speed of the wetting line, where is the base radius of the drop:

θ

W=

dR

dt

R

W=2λsinhcos-cosθ=asinhbcos-cosθ

0

K

γ

LV

θ

eq

2nT

k

B

θ

eq

In this model, is the frequency of molecular displacements at the wetting line, the average length of displacements, is the surface tension of the liquid, is the number of adsorption sites per unit area, is Boltzmann's constant, and is the temperature. In this Demonstration, these parameters are consolidated into two parameters called and . The equilibrium contact angle is denoted by .

0

K

λ

γ

LV

n

k

B

T

a

b

θ

eq

V=π(2+cosθ)sinθ3

3

R

2

(1+cos(θ))

Noting that , the expression for the time rate of change of the contact angle can be derived:

W=

dR

dt

dθ/dτ=-sinhbcos-cosθ

1/3

(π/3)

θ

eq

4/3

(2-3cosθ+θ)

3

cos

2

(1-cosθ)

where is dimensionless time.

τ=ta/

3

V

Using the sliders, you can vary the initial contact angle when the drop is placed on the substrate, the dimensionless wetting parameter , the maximum time for spreading , and the equilibrium contact angle . The Demonstration shows how and =R/ vary with time and the dependence of on the speed of the contact line =W/a. You can see the 3D shape of the spreading drop at different spreading times by clicking the spherical drop tab in the drop-down menu and then varying the time slider .

θ

0

b

τ

max

θ

eq

θ

R

1/3

V

θ

W

τ

τ