混合溶剂中的高分子凝胶中的渗透压的一般计算

在博文混合溶剂中的高分子凝胶理论推导中，渗透压计算是按照各组分均匀分布来计算的。在一般情况下，各组分不是均匀分布，那该如何计算渗透压呢？

对于混合溶剂中的高分子凝胶，由于体积不可压缩性约束条件，三种组分不是独立的，我们把独立组分选为凝胶和一种溶剂。自由能密度为

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*}

渗透压为

egin{equation*} Pi=phi_2frac{partial f}{partial phi_2}+phi_3frac{partial f}{partial phi_3}-f(phi_{2},phi_3) end{equation*}

第一项

egin{equation*} egin{split} phi_2frac{partial f}{partial phi_2}=&phi_2left [-ln(1-phi_{2}-phi_3)+lnphi_{2}+chi_{12}(1-phi_{2}-phi_3)-chi_{12} phi_{2} \ -chi_{13}phi_3+chi_{23}phi_3 ight ]\ =&-phi_2ln(1-phi_{2}-phi_3)+phi_2lnphi_{2}+chi_{12}(1-phi_{2}-phi_3)phi_2 \ &-chi_{12} phi_{2}^2-chi_{13}phi_2phi_3+chi_{23}phi_2phi_3 end{split} end{equation*}

第二项

egin{equation*} egin{split} phi_3frac{partial f}{partial phi_3}=&phi_3left [-ln(1-phi_{2}-phi_3)-1-chi_{12}phi_{2}-chi_{13}phi_3+\ chi_{13}(1-phi_{2}-phi_3)+chi_{23}phi_2 ight ]\ =&-phi_3ln(1-phi_{2}-phi_3)-phi_3-chi_{12}phi_{2}phi_3-chi_{13}phi_3^2+\ &chi_{13}(1-phi_{2}-phi_3)phi_3+chi_{23}phi_2phi_3 end{split} end{equation*}

于是渗透压为

egin{equation*} egin{split} Pi=&phi_2frac{partial f}{partial phi_2}+phi_3frac{partial f}{partial phi_3}-f(phi_{2},phi_3)\ =&-ln(1-phi_{2}-phi_3)-phi_3-chi_{12}phi_{2}phi_3-chi_{12}phi_{2}^2-chi_{13}phi_2phi_3 -\ &chi_{13}phi_3^2+chi_{23}phi_2phi_3 \ =&-ln(1-phi_{2}-phi_3)-phi_3-chi_{13}phi_3^2-chi_{12}phi_{2}^2+Gphi_2phi_3 end{split} end{equation*}

其中

egin{equation*} G=chi_{23}-chi_{12}-chi_{13} end{equation*}

本体溶液自自由能密度

egin{equation*} f_s(phi_{2s})=(1-phi_{2s})ln(1-phi_{2s})+phi_{2s}lnphi_{2s}+chi_{12}(1-phi_{2s})phi_{2s} end{equation*}

渗透压为

egin{equation*} egin{split} Pi_s=&phi_{2s}frac{partial f_s}{partial phi_{2s}}-f_s(phi_{2s})\ =&-ln(1-phi_{2s})-chi_{12}phi_{2s}^2 end{split} end{equation*}

其他体系的渗透压计算与此类似。