混合溶剂中的高分子凝胶理论推导

参考资料：THE JOURNAL OF CHEMICAL PHYSICS 137, 024902 (2012)

高分子凝胶的自由能(F)有两项，混合项(F_{mix})和弹性项(F_{el})：

egin{equation} F=F_{mix}+F_{el} label{Fen} end{equation}

两种混合溶剂中的高分子凝胶，混合自由能为

egin{equation} F_{mix}=frac{k_BT}{v_0}left[ V_g f(phi_{1g},phi_{2g},phi_3)+V_s f(phi_{1s},phi_{2s},0) ight ] label{Fmix} end{equation}

其中，(V_g) 和 (V_s) 分别为凝胶和凝胶外部溶液的体积，(v_0)为单体和溶剂分子的体积，(f(phi_{1},phi_{2},phi_3))为Flory-Huggins混合自由能：

egin{equation} f(phi_{1},phi_{2},phi_3)=phi_{1}lnphi_{1}+phi_{2}lnphi_{2}+sum_{ilt j}chi_{ij}phi_iphi_j label{FH} end{equation}

其中，(phi_{1})和(phi_{2})分别为溶剂分子的体积分数，下标(phi_{ig})和(phi_{is})分别表示第(i)（(i=1,2)）种组分凝胶内外的体积分数，(phi_{3})表示高分子网络的体积分数，(chi_{ij})为(i)和(j)两种组分的相互作用参数。

高分子网络的熵弹性能为

egin{equation} F_{el}=frac{1}{2}k_BT u V_{g0}left[ 3left ( frac{phi_{30}}{phi_3} ight )^{2/3}-2Blnleft ( frac{phi_{30}}{phi_3} ight ) ight ] label{Fel} end{equation}

其中，(V_{g0}) 为处于参考态的凝胶的体积，( u) 为交联点密度，(B) 为非线性弹性系数，(phi_{30}) 为处于参考态的凝胶的体积分数。

凝胶内外满足不可压缩性条件：

egin{equation} phi_{1g}+phi_{2g}+phi_3=1 label{Incom1} end{equation}

egin{equation} phi_{1s}+phi_{2s}=1 label{Incom2} end{equation}

可定义如下巨势：

egin{equation} Omega=F_{mix}+F_{el}-mu_2(V_gphi_{2g}+V_sphi_{2s})+kappa (V_g+V_s) label{Grand} end{equation}

其中，(mu_2)和(kappa) 分别为保证第二种组分和总体积不变的拉格朗日乘子。对巨势求极小可得平衡态：

egin{equation} frac{partial Omega}{partial phi_{2g}}=frac{partial Omega}{partial phi_{2s}}=0 label{Mini1} end{equation}

egin{equation} frac{partial Omega}{partial V_g}=frac{partial Omega}{partial V_s}=0 label{Mini2} end{equation}

由

egin{equation*} frac{partial Omega}{partial phi_{2g}}=frac{partial F_{mix}}{partial phi_{2g}}-mu_2V_g=frac{k_BT}{v_0}V_gfrac{partial f}{partial phi_{2g}}-mu_2V_g=0 end{equation*}

于是得

egin{equation*} ilde{mu}(phi_{2g},phi_3)=frac{partial f}{partial phi_{2g}}=frac{v_0mu_2 }{k_BT} end{equation*}

同理，由(frac{partial Omega}{partial phi_{2s}}=0)可得

egin{equation*} ilde{mu}(phi_{2s},0)=frac{partial f}{partial phi_{2s}}=frac{v_0mu_2 }{k_BT} end{equation*}

由方程eqref{Incom1}和eqref{Incom2}，可将eqref{FH}化为：

egin{equation*} egin{split} f(phi_{2},phi_3)=&(1-phi_{2}-phi_3)ln(1-phi_{2}-phi_3)+phi_{2}lnphi_{2}\ &+chi_{12}(1-phi_{2}-phi_3)phi_2 +chi_{13}(1-phi_{2}-phi_3)phi_3+chi_{23}phi_2phi_3 end{split} end{equation*}

于是得

egin{equation} ilde{mu}(phi_{2},phi_{3})=lnfrac{phi_2}{1-phi_2-phi_3}+chi_{12}(1-2phi_2-phi_3)+(chi_{23}-chi_{13})phi_3 label{ChemPot} end{equation}

下面再看方程eqref{Mini2}，

egin{equation} frac{partial Omega}{partial V_g}=frac{partial F_{mix}}{partial V_g}+frac{partial F_{el}}{partial V_g}+kappa=0 label{POV} end{equation}

方程eqref{POV}中第一项

egin{equation} frac{partial F_{mix}}{partial V_g}=frac{k_BT}{v_0}left [f(phi_{2g},phi_3)+V_gfrac{partial f(phi_{2g},phi_3)}{partial V_g} ight ] label{PFmVg1} end{equation}

其中，

egin{equation*} egin{split} V_gfrac{partial f(phi_{2g},phi_3)}{partial V_g}=& -phi_{2g}frac{partial f(phi_{2g},phi_3)}{partial phi_{2g}}-phi_{3}frac{partial f(phi_{2g},phi_3)}{partial phi_{3}} \ =&-(phi_{2g}+phi_3)ln(1-phi_{2g}-phi_3)-phi_{2g}lnphi_{2g}\ &-chi_{12}(1-phi_{2g}-phi_3)phi_{2g}+chi_{12}phi_{2g}^2+chi_{13}phi_{2g}phi_3\ &-chi_{23}phi_{2g}phi_3+phi_3+chi_{12}phi_{2g}phi_3+chi_{13}phi_3^2\ &-chi_{13}(1-phi_{2g}-phi_3)phi_3-chi_{23}phi_2phi_3 end{split} end{equation*}

代入方程eqref{PFmVg1}，得

egin{equation} egin{split} frac{partial F_{mix}}{partial V_g}left (frac{k_BT}{v_0} ight )^{-1}=& ln(1-phi_2-phi_3)+phi_3\ &+chi_{12}phi_{2g}^2+chi_{13}phi_{3}^2\ &-(chi_{23}-chi_{12}-chi_{13})phi_{2g}phi_3 end{split} label{PFmVg2} end{equation}

方程eqref{POV}中第二项

egin{equation} egin{split} frac{partial F_{el}}{partial V_g}=&frac{partial F_{el}}{partial phi_3}frac{partial phi_3}{partial V_g }=-frac{phi_3}{V_g}frac{partial F_{el}}{partial phi_3}\ =&-k_BT u phi_3 frac{V_{g0}}{V_g}left [-left (frac{phi_{30}}{phi_3} ight )^{2/3}frac{1}{phi_3}+frac{B}{phi_3} ight ]\ =&k_BT u left [left (frac{phi_{3}}{phi_{30}} ight )^{1/3}-Bfrac{phi_{3}}{phi_{30}} ight ] end{split} label{PFelVg} end{equation}

方程eqref{Mini2}中第二个偏导的结果为

egin{equation} frac{partial Omega}{partial V_s}=frac{partial F_{mix}}{partial V_s}+kappa=0 label{POVs} end{equation}

其中

egin{equation} egin{split} frac{partial F_{mix}}{partial V_s}=&frac{k_BT}{v_0}left [f(phi_{2s},0)+V_sfrac{partial f(phi_{2s},0)}{partial V_s} ight ]\ =&frac{k_BT}{v_0}left [ln(1-phi_{2s})+chi_{12}phi_{2s}^2 ight] end{split} label{PFmVs} end{equation}

综上，凝胶平衡态结构由如下方程给出：

egin{equation} egin{split} &lnfrac{phi_{2g}}{1-phi_{2g}-phi_3}+chi_{12}(1-2phi_{2g}-phi_3)+(chi_{23}-chi_{13})phi_3=\ &lnfrac{phi_{2s}}{1-phi_{2s}}+chi_{12}(1-2phi_{2s}) end{split} label{Struc1} end{equation}

egin{equation} egin{split} &ln(1-phi_{2s})-ln(1-phi_{2g}-phi_3)-phi_3\ &-chi_{12}(phi_{2s}^2-phi_{2g}^2)-chi_{13}phi_3^2+(chi_{23}-chi_{12}-chi_{13})phi_{2g}phi_3\ &- u v_0 left [left (frac{phi_{3}}{phi_{30}} ight )^{1/3}-Bfrac{phi_{3}}{phi_{30}} ight ]=0 end{split} label{Struc2} end{equation}