# 双电层力

## 泊松——波尔兹曼模型

${\displaystyle {\mathcal {F}}=\int f\mathrm {d} z}$
${\displaystyle f={\frac {\epsilon }{8\pi }}(\psi '')^{2}+k_{B}T\left[n_{+}(z)\ln {\frac {n_{+}(z)}{n_{0}}}+n_{-}(z)\ln {\frac {n_{-}(z)}{n_{0}}}+n_{+}(z)+n_{-}(z)-2n_{0}\right]}$

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

${\displaystyle {\frac {\Pi }{k_{B}T}}=n_{+}(h/2)+n_{-}(h/2)-2n_{0}}$

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

${\displaystyle n_{\pm }({\mathbf {r} })=n_{\pm }^{0}e^{-z_{\pm }e\psi ({\mathbf {r} })/k_{B}T}}$

${\displaystyle -Vd\Pi +N_{+}d\mu _{+}+N_{-}d\mu _{-}=0}$

${\displaystyle \mu _{\pm }=\mu _{p}m^{(0)}+kT\ln n_{\pm }+z_{\pm }\psi }$

${\displaystyle d\Pi =k_{B}T(dn_{+}+dn_{-})+(z_{+}n_{+}+z_{-}n_{-})d\psi }$

${\displaystyle {\frac {\Pi }{k_{B}T}}=n_{+}(z)+n_{-}(z)-2n_{0}-{\frac {\epsilon }{2}}\left({\frac {d\psi }{dz}}\right)^{2}}$

### 无外加盐

${\displaystyle {\frac {\Pi }{k_{B}T}}={\frac {\epsilon k_{B}T}{2\pi e^{2}}}K^{2}={\frac {K^{2}}{2\pi l_{B}}}}$

${\displaystyle Kh\tanh(Kh)=-{\frac {2\pi e\sigma }{\epsilon k_{B}T}}h=h/b}$

#### 极限情况

${\displaystyle h/b\ll 1}$，带电表面为弱带电表面，渗透压可近似为：

${\displaystyle {\frac {\Pi }{k_{B}T}}\approx {\frac {1}{\pi l_{B}bh}}}$

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

${\displaystyle {\frac {\Pi }{k_{B}T}}\approx {\frac {\pi }{2l_{B}h^{2}}}}$

### 有外加盐

${\displaystyle {\frac {\Pi }{k_{B}T}}=2n_{0}(\cosh \psi _{m}-1)}$

${\displaystyle \psi _{s}=\psi _{m}+{\frac {2\lambda _{D}^{2}}{b^{2}}}}$

${\displaystyle {\frac {h}{2\lambda _{D}}}=\int _{\psi _{s}}^{\psi _{m}}{\frac {d\psi }{{\sqrt {(}}2\cosh \psi -2\cosh \psi _{m})}}}$

## 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. 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.