# 双电层力

## 泊松——波尔兹曼模型

$\mathcal F = \int f \mathrm d z$
$f = \frac{\epsilon}{8\pi}(\psi'')^2 + k_BT \left [n_+(z)\ln\frac{n_+(z)}{n_0}+n_-(z)\ln\frac{n_-(z)}{n_0}+n_+(z)+n_-(z)-2n_0 \right ]$

$\Pi = - \frac{\delta \mathcal F}{\delta h}$

$\frac{\Pi}{k_BT} = n_+(h/2)+n_-(h/2)-2n_0$

$\nabla^2\psi(\bold r) = - \frac{4\pi e}{\epsilon}[z_+n_+(\bold r)+z_-n_-(\bold r)]$

$n_{\pm }(\bold r) =n_{\pm}^0 e^{-z_{\pm }e\psi(\bold r)/k_BT}$

$-V d\Pi + N_+ d\mu_+ + N_- d\mu_- = 0$

$\mu_\pm = \mu_pm^{(0)} + kT \ln n_\pm +z_\pm \psi$

$d\Pi = k_BT(dn_+ + dn_-) + (z_+n_+ + z_-n_-) d \psi$

$\frac{\Pi}{k_BT} = n_+(z)+n_-(z)-2n_0 - \frac{\epsilon}{2}\left(\frac{d\psi}{dz}\right)^2$

### 无外加盐

$\frac{\Pi}{k_BT} = \frac{\epsilon k_BT}{2\pi e^2} K^2 = \frac{K^2 }{2\pi l_B}$

$Kh\tanh(Kh) = -\frac{2\pi e \sigma}{\epsilon k_BT}h = h/b$

#### 极限情况

$h/b \ll 1$，带电表面为弱带电表面，渗透压可近似为：

$\frac{\Pi}{k_BT} \approx \frac{1}{\pi l_B b h}$

$h/b \gg 1$，且$Kh \rightarrow \pi$，带电表面为强带电表面，渗透压可近似为：

$\frac{\Pi}{k_BT} \approx \frac{\pi}{2 l_B h^2}$

### 有外加盐

$\frac{\Pi}{k_BT} = 2n_0 (\cosh\psi_m -1)$

$\psi_s = \psi_m + \frac{2\lambda_D^2}{b^2}$

$\frac{h}{2 \lambda_D} = \int_{\psi_s}^{\psi_m}\frac{d\psi}{\sqrt(2\cosh \psi - 2 \cosh \psi_m)}$

## 参考文献

1. ^ W. B. Russel, D. A. Saville, W. R. Schowalter, Colloidal Dispersions. Cambridge University Press: Cambridge, 1989.
2. ^ W C K Poon, D Andelman. Soft Condensed Matter Physics in Molecular and Cell Biology. Taylor & Francis Group. 2006: 107. ISBN 0-7503-1023-5.
3. ^ 3.0 3.1 J. Israelachvili, Intermolecular and Surface Forces. Academic Press: London, 1992.
4. ^ W C K Poon, D Andelman. Soft Condensed Matter Physics in Molecular and Cell Biology. Taylor & Francis Group. 2006: 107. ISBN 0-7503-1023-5.
5. ^ D. ANDELMAN. Handbook of Biological Physics. Elsevier Science. 1995: 617.