高分子刷的解析平均场理论有两种表述方式。一个是MWC理论(Macromolecules 1988, 21, 2610-2619),另外一个就是Zhulina和Birshtein这两位俄罗斯老太太的理论(Macromolecules 1991, 24, 140-149),后者在物理上更直接,我重新整理一下,是为此文。
高分子刷的(平均一根链的)自由能(Delta F)为链的熵弹性(Delta F_{el})与排除体积作用能(Delta F_{conc})之和:
egin{equation} Delta F=Delta F_{el}+Delta F_{conc} label{eq:F} end{equation}
排除体积作用能
egin{equation} Delta F_{conc}=frac{sigma}{a^3}int f[varphi(x)] mathrm dx label{eq:Fconc} end{equation}
其中(sigma)为平均一根链在接枝面上所占据的面积,(varphi(x))为高分子体积分数,(f[varphi(x)]/a^3)为相互自由能密度。
接枝链的熵弹性:
egin{equation*} egin{split} Delta F_{el}(x')&=frac{3}{2a^2}int_0^Nleft (frac{mathrm dx}{mathrm dn} ight )^2mathrm dn=frac{3}{2a^2}int_0^Nfrac{mathrm dx}{mathrm dn}frac{mathrm dx}{mathrm dn}mathrm dn\ &=frac{3}{2a^2}int_0^{x'}frac{mathrm dx}{mathrm dn}mathrm dx =frac{3}{2a^2}int_0^{x'}frac{mathrm dx}{mathrm dn}mathrm dx\ &=frac{3}{2a^2}int_0^{x'}E(x,x')mathrm dx end{split} end{equation*}
其中(x')为高分子链的末端所在位置,(H)为刷的高度,(E(x,x')=frac{mathrm dx}{mathrm dn}),并满足:
egin{equation}int_0^{x'} frac{1}{E(x,x')}mathrm dx=N label{eq:Econs}end{equation}
接枝链的末端的分布为(g'(x')),(g'(x')mathrm dx')为(x')处(Amathrm dx')体积范围内接枝链末端的数目,满足
$$Aint_0^H g'(x')mathrm dx'=n_P$$
其中(A)为接枝表面的总面积,(n_P)为接枝链的总数目。
平均一条链的熵弹性能为:
egin{equation} egin{split} Delta F_{el}&=frac{A}{n_P}int_0^H Delta F_{el}(x')g'(x')mathrm dx'\ &=frac{3}{2a^2}int_0^H g(x')mathrm dx'int_0^{x'}E(x,x')mathrm dx end{split}label{eq:Fel}end{equation}
其中,(g(x')=frac{A}{n_P}g'(x')),为(x')处(mathrm dx')厚度范围内接枝链末端的数目,满足(int_0^H g(x')mathrm dx'=1)。
高分子体积分数(varphi(x))满足:
egin{equation} egin{split} varphi(x)&=frac{a^3}{sigma}int_0^Hfrac{mathrm dn}{mathrm dx} g(x')mathrm dx'\ &=frac{a^3}{sigma}int_0^Hfrac{g(x')}{E(x,x')} mathrm dx' end{split}label{eq:varphi}end{equation}
egin{equation}sigmaint_0^{H} varphi(x)mathrm dx=Na^3label{eq:varphicons}end{equation}
要得到刷的结构,需要对如下泛函求变分:
egin{equation}F'=Delta F+lambda_1 int_0^{H} varphi(x)mathrm dx +int_0^H lambda_2(x')mathrm dx'int_0^{x'} frac{1}{E(x,x')}mathrm dxlabel{eq:Fp}end{equation}
其中(lambda_1)和(lambda_2(x'))分别为拉格朗日乘子。
对(F')变分有:
egin{equation} egin{split} delta F'=&delta Delta F_{el}+delta Delta F_{conc} + lambda_1 int_0^{H}delta varphi(x)mathrm dx\ &-int_0^H lambda_2(x')mathrm dx'int_0^{x'} frac{delta E(x,x')}{E^2(x,x')}mathrm dx\ =& frac{3}{2a^2}int_0^H mathrm dx'int_0^{x'}left [g(x')delta E(x,x') + E(x,x') delta g(x') ight ]mathrm dx \ &+frac{sigma}{a^3}int frac{delta f[varphi(x)]}{delta varphi(x)} delta varphi(x) mathrm dx + lambda_1 int_0^{H}delta varphi(x)mathrm dx\ &-int_0^H lambda_2(x')mathrm dx'int_0^{x'} frac{delta E(x,x')}{E^2(x,x')}mathrm dx end{split}label{eq:var} end{equation}
根据方程eqref{eq:varphi},有:
egin{equation} delta varphi(x)=frac{a^3}{sigma}int_0^Hleft [frac{delta g(x')}{E(x,x')}-frac{g(x')}{E^2(x,x')}delta E(x,x') ight ] mathrm dx' label{eq:varvarphi} end{equation}
将方程eqref{eq:varvarphi}带入方程eqref{eq:var},得
egin{equation} egin{split} delta F'= &int_0^H mathrm dx' int_0^{x'} mathrm dx delta E(x,x')\ &left [frac{3g(x')}{2a^2}-frac{lambda_2(x')}{E^2(x,x')}-left (lambda_1+frac{delta f[varphi(x)]}{delta varphi(x)} ight )frac{g(x')}{E^2(x,x')} ight ]\ & int_0^H delta g(x') mathrm dx' int_0^{x'} mathrm dx \ &left [frac{3E(x,x')}{2a^2}+frac{1}{E(x,x')}left (lambda_1+frac{delta f[varphi(x)]}{delta varphi(x)} ight ) ight ] end{split}label{eq:varesult} end{equation}
相应地我们可得如下两个变分方程:
egin{equation} frac{3g(x')}{2a^2}-frac{lambda_2(x')}{E^2(x,x')}-left (lambda_1+frac{delta f[varphi(x)]}{delta varphi(x)} ight )frac{g(x')}{E^2(x,x')} =0 label{eq:var1} end{equation}
egin{equation}
frac{3E(x,x')}{2a^2}+frac{1}{E(x,x')}left (lambda_1+frac{delta f[varphi(x)]}{delta varphi(x)}
ight )=0
label{eq:var2}
end{equation}
由方程eqref{eq:var1},
egin{equation} E^2(x,x')=U_1(x')-U_2(x) label{eq:EU12} end{equation}
其中,
egin{equation} U_1(x')=frac{2a^2lambda_2(x')}{3g(x')} label{eq:U1} end{equation} egin{equation} U_2(x)=-frac{2a^2}{3}left ( lambda_1+frac{delta f[varphi(x)]}{delta varphi(x)} ight ) label{eq:U2} end{equation}
链的末端不受拉伸,则(E(x,x)=0),于是 (U_1=U_2),我们有
egin{equation} E(x,x')=sqrt{U(x')-U(x)} label{eq:EU} end{equation}
(U(x))仍是未知函数,将方程eqref{eq:EU}代入方程eqref{eq:Econs},得
egin{equation} U(x)=frac{pi^2 x^2}{4N^2} label{eq:Ux} end{equation}
将方程eqref{eq:Ux}代入方程eqref{eq:EU}得
egin{equation} E(x,x')=frac{pi }{2N}sqrt{x'^2-x^2}label{eq:Ex} end{equation}
将方程eqref{eq:Ux}代入方程eqref{eq:U2}得
egin{equation}lambda_1+frac{delta f[varphi(x)]}{delta varphi(x)}=-frac{3pi^2x^2}{8N^2}label{eq:varphix} end{equation}
将方程eqref{eq:Ux}代入方程eqref{eq:varphicons},得如下积分方程:
egin{equation} varphi(x)=frac{2Na^3}{pisigma}int_0^{x'}frac{g(x')}{sqrt{x'^2-x^2}}mathrm dx label{eq:inteq}end{equation}
从方程eqref{eq:varphix}到高分子体积分数(varphi(x)),解积分方程eqref{eq:inteq}就可得高分子链末端的分布。积分方程的解可从积分方程手册中查到,在pp21。