CHEM E 455 Surface and Colloid Science Laboratory
Contents
Fluid Interface and Capillarity
Surface tension
| Description | Equations |
|---|---|
| Surface tension | $\sigma = \dfrac{dF}{dl}$ |
$T$ dependence of $\sigma$
| Description | Equations |
|---|---|
| Eotovs law | $\sigma v^{2/3} = k_E (T_c - T)$ |
| Guggenheim law | $\sigma = \sigma^* \left(1 - \dfrac{T}{T_c}\right)^{11/9}$ |
| Empirical linear equation ★ Modest T |
$\sigma =a-bT \newline b = \dfrac{d\sigma}{dT} = -0.1 \ \mathrm{mM/m \cdot K}$ |
$\sigma$ and intermolecular forces
| Description | Equations |
|---|---|
| van der Waals attraction | $\Phi_{\text{vdW}} = -\dfrac{B_{\text{vdW}}}{r^6}$ |
| Born repulsion | $\Phi_{\text{rep}} = \dfrac{B_{\text{rep}}}{r^{12}}$ |
| Lennard-Jones potential | $\Phi = 4\varepsilon \left[\left(\dfrac{\delta }{r}\right)^{12}-\left(\dfrac{\delta }{r}\right)^6\right]$ |
| Lennard-Jones attractive force | $F_{\text{attr}} = -\dfrac{24\varepsilon }{r}\left[2\left(\dfrac{\delta }{r}\right)^{12}-\left(\dfrac{\delta }{r}\right)^6\right]$ |
| Bakker’s equation | $\sigma = \displaystyle\int_{-\infty}^\infty (p - p_T) dz$ |
Components of $\sigma$
| Description | Equations |
|---|---|
| Components of surface tension | $\sigma = \sigma^{\text{disperison}} + \sigma^{\text{dipole}} + \sigma^{\text{induced dipole}} + \sigma^{\text{H-bond}} + \sigma^{\text{metallic bond}} + \cdots$ |
| Surface tension of liquid | $\sigma = \sigma^{\text{disperison}} + \sigma^{\text{acid-base}}$ |
| Surface tension of molten salt | $\sigma = \sigma^{\text{disperison}} + \sigma^{\text{metallic bond}}$ |
Combining rules of $\sigma$
| Description | Equations |
|---|---|
| Antanov | $\sigma_{AB} = \vert \sigma_{A(B)} - \sigma_{B(A)} \vert$ |
| Girifalco & Good | $\sigma_{AB} = \sigma_A + \sigma_B - 2\Phi \sqrt{\sigma_A \sigma_B}$ |
| Fowkes | $\sigma_{AB} = \sigma_A + \sigma_B - 2 \sqrt{\sigma_A^d \sigma_B^d}$ |
Young-Laplace equation
Curvature in 2D
| Description | Equations |
|---|---|
| Curvature of a plane curve | $\kappa = \dfrac{d\phi}{dS}$ |
| Curvature of a plane curve | $\kappa = \pm y'' [1+(y')^2]^{-3/2}$ |
| Curvature of a line | $\kappa = 0$ |
| Curvature of a circle | $\kappa = \dfrac{1}{R}$ |
Curvature in 3D
| Description | Equations |
|---|---|
| Curvature of a surface | $\kappa = \pm \left(\dfrac{1}{R_1}+\dfrac{1}{R_2}\right)=\pm \dfrac{2}{R_{\text{mean}}}$ |
| Curvature of a surface | $\kappa = \pm \dfrac{z_{xx}\left[1+\left(z_y\right)^2\right]-2z_xz_yz_{xy}+z_{yy}\left[1+\left(z_x\right)^2\right]}{\left[1+\left(z_x\right)^2+\left(z_y\right)^2\right]^{3/2}}$ |
| Curvature of a sphere | $\kappa = \dfrac{2}{R}$ |
| Curvature of a circular cylinder | $\kappa = \dfrac{1}{R}$ |
| Curvature of a cylindrical surface | $\kappa = \pm y'' [1+(y')^2]^{-3/2}$ |
| Curvature of an axially symmetric surface | $\kappa = $ |
Young-Laplace equation
| Description | Equations |
|---|---|
| Young-Laplace equation | $\Delta p = p'' - p' = \sigma\kappa$ |