Categories: Fluid mechanics, Fluid statics, Physics, Surface tension.

Young-Dupré relation

In fluid mechanics, the Young-Dupré relation relates the contact angle of a droplet at rest on a surface to the surface tensions of the interfaces. Let αgl\alpha_{gl}, αsl\alpha_{sl} and αsg\alpha_{sg} respectively be the energy costs of the liquid-gas, solid-liquid and solid-gas interfaces:

αsg−αsl=αglcos⁡θ\begin{aligned} \boxed{ \alpha_{sg} - \alpha_{sl} = \alpha_{gl} \cos\theta } \end{aligned}

The derivation is simple: this is the only expression that maintains the droplet’s boundaries when you account for the surface tension force pulling along each interface.

A more general derivation is possible by using the calculus of variations. In 2D, the upper surface of the droplet is denoted by y(x)y(x). Consider the following Lagrangian L\mathcal{L}, with the two first terms respectively being the energy costs of the top and bottom surfaces:

L=αgl1+(y′)2+(αsl−αsg)+λy\begin{aligned} \mathcal{L} = \alpha_{gl} \sqrt{1 + (y')^2} + (\alpha_{sl} - \alpha_{sg}) + \lambda y \end{aligned}

And the last term comes from the constraint that the volume VV of the droplet must be constant:

V=∫0Lydx\begin{aligned} V = \int_0^L y \dd{x} \end{aligned}

The total energy to be minimized is thus given by the following functional, where the endpoints of the droplet are x=0x = 0 and x=Lx = L:

E[y(x)]=∫0L(αgl1+(y′)2+(αsl−αsg)+λy)dx\begin{aligned} E[y(x)] = \int_0^L \Big( \alpha_{gl} \sqrt{1 + (y')^2} + (\alpha_{sl} - \alpha_{sg}) + \lambda y \Big) \dd{x} \end{aligned}

In this optimization problem, the endpoint LL is a free parameter, i.e. the LL-value of the optimum is unknown and must be found. In such cases, the optimum y(x)y(x) needs to satisfy the so-called transversality condition at the variable endpoint, in this case x=Lx = L:

0=(L−y′∂L∂y′)x=L=(αgl1+(y′)2+(αsl−αsg)+λy−(y′)21+(y′)2)x=L=(αgl11+(y′)2+(αsl−αsg)+λy)x=L\begin{aligned} 0 &= \Big( \mathcal{L} - y' \pdv{\mathcal{L}}{y'} \Big)_{x = L} \\ &= \bigg( \alpha_{gl} \sqrt{1 + (y')^2} + (\alpha_{sl} - \alpha_{sg}) + \lambda y - \frac{(y')^2}{\sqrt{1 + (y')^2}} \bigg)_{x = L} \\ &= \bigg( \alpha_{gl} \frac{1}{\sqrt{1 + (y')^2}} + (\alpha_{sl} - \alpha_{sg}) + \lambda y \bigg)_{x = L} \end{aligned}

Due to the droplet’s shape, we have the boundary condition y(L)=0y(L) = 0, so the last term vanishes. We are thus left with the following equation:

αgl11+(y′(L))2=αsg−αsl\begin{aligned} \alpha_{gl} \frac{1}{\sqrt{1 + (y'(L))^2}} = \alpha_{sg} - \alpha_{sl} \end{aligned}

At the edge of the droplet, imagine a small right-angled triangle with one side dx\dd{x} on the xx-axis, the hypotenuse on y(x)y(x) having length dx1+(y′)2\dd{x} \sqrt{1 + (y')^2}, and the corner between them being the contact point with angle θ\theta. Then, from the definition of the cosine:

cos⁡θ=dxdx1+(y′(L))2=11+(y′(L))2\begin{aligned} \cos\theta = \frac{\dd{x}}{\dd{x} \sqrt{1 + (y'(L))^2}} = \frac{1}{\sqrt{1 + (y'(L))^2}} \end{aligned}

When inserted into the above transversality condition, this yields the Young-Dupré relation.

References

  1. B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.