受限壁附近高分子链的构象统计

柔性链

无限大平面附件的柔性高分子链

如上图所示，高分子链一端位于距离平面(l)处，另一端坐标为(z)，设高分子链链节大小(b)，高分子链链长为(N_c)，则高分子链构象配分函数为

egin{equation} egin{split} G(l,z)=&sqrt{frac{3}{2pi N_cb^2}}left (expleft [-frac{3(z-l)^2}{2N_cb^2} ight ] - expleft [-frac{3(z+l)^2}{2N_cb^2} ight ] ight )\ =&frac{1}{2sqrt{pi }R_g}left (expleft [-frac{(z-l)^2}{4R_g^2} ight ] - expleft [-frac{(z+l)^2}{4R_g^2} ight ] ight ) end{split} label{Gcoil} end{equation}

其中，(R_g=sqrt{frac{N_c}{6}}b)，为柔性链的回转半径。

配分函数为

egin{equation} egin{split} mathcal Z=&int_0^infty G(l,z)mathrm dz\ =&frac{1}{2sqrt{pi }R_g}int_0^infty expleft [-frac{(z-l)^2}{4R_g^2} ight ] mathrm dz\ &-frac{1}{2sqrt{pi }R_g}int_0^infty expleft [-frac{(z+l)^2}{4R_g^2} ight ] mathrm dz \ =&frac{1}{sqrt{pi }}int_{-l/2R_g}^infty expleft (-t^2 ight ) mathrm dt quad left (mathrm{where} quad t=frac{z-l}{2R_g} ight )\ &-frac{1}{sqrt{pi }}int_{l/2R_g}^infty expleft (-t^2 ight ) mathrm dt quad left (mathrm{where} quad t=frac{z+l}{2R_g} ight )\ =&frac{1}{sqrt{pi }}int_{-l/2R_g}^0 expleft (-t^2 ight ) mathrm dt + frac{1}{sqrt{pi }}int_0^infty expleft (-t^2 ight ) mathrm dt \ &- left ( frac{1}{sqrt{pi }}int_0^infty expleft (-t^2 ight ) mathrm dt -frac{1}{sqrt{pi }}int_0^{l/2R_g} expleft (-t^2 ight ) mathrm dt ight )\ =&frac{1}{sqrt{pi }}int_{-l/2R_g}^0 expleft (-t^2 ight ) mathrm dt + frac{1}{sqrt{pi }}int_0^{l/2R_g} expleft (-t^2 ight ) mathrm dt\ =&frac{2}{sqrt{pi }}int_0^{l/2R_g} expleft (-t^2 ight ) mathrm dt\ =&mathrm {erf}left (frac{l}{2R_g} ight ) end{split} label{Zcoil} end{equation}

上式用到了误差函数：

egin{equation} mathrm {erf}(x) = frac{2}{sqrt {pi}} int_0^x exp(-t^2) mathrm dt = frac{2}{sqrt {pi}} int_{-x}^0 exp(-t^2) mathrm dt label{error} end{equation}

自由能为

egin{equation} frac{F_c(l,N_c)}{k_BT} = -ln mathcal{Z} = -ln mathrm {erf}left (frac{l}{2R_g} ight )= -ln mathrm {erf}left (sqrt{frac{3}{2N_c}}frac{l}{b} ight ) label{Fcoil} end{equation}

硬棒——柔线双嵌段高分子链

无限大平面附近的柔线——硬棒双嵌段高分子链

如上图所示，硬棒——柔线（rod-coil）柔线一端位于距离受限壁(l)处，硬棒可以自由旋转，并忽略硬棒与柔线的相互作用。设硬棒和柔线的链长分别为(N_d)和(N_c)，两嵌段库恩长度相等，均为(b)。构象分布为：

egin{equation} egin{split} G(l,z, heta)=&sqrt{frac{3}{2pi N_cb^2}}left (expleft [-frac{3(z-l)^2}{2N_cb^2} ight ] - expleft [-frac{3(z+l)^2}{2N_cb^2} ight ] ight )\ &frac{1}{2}expleft [ -U(z, heta) ight ]\ =&frac{1}{2}expleft [ -U(z, heta) ight ]G(l,z) end{split} label{Grodcoil} end{equation}

其中( heta)为硬棒与(z)轴夹角，硬棒与受限壁相互作用为

egin{equation} U(z, heta)= egin{cases} 0,& heta lt pi-arccos{frac{z}{N_db}}, or quad z ge N_d b \\ infty, & heta ge pi-arccos{frac{z}{N_db}} ,and quad z le N_d b end{cases} label{U} end{equation}

配分函数为：

egin{equation} egin{split} mathcal Z&=int_0^infty mathrm dz G(l,z, heta)int_0^pi mathrm d heta frac{1}{2}expleft [ -U(z, heta) ight ]sin heta \\ &=int_0^{N_db} mathrm dz G(l,z, heta)int_0^pi mathrm d heta frac{1}{2}expleft [ -U(z, heta) ight ]sin heta + int_{N_db}^infty mathrm dz G(l,z)\\ &=int_0^{N_db} mathrm dz G(l,z) frac{N_db+z}{2N_db} + int_{N_db}^infty mathrm dz G(l,z)\\ &=int_0^infty mathrm dz G(l,z)-frac{1}{2}int_0^{N_db} mathrm dz G(l,z) +frac{1}{2N_db}int_0^{N_db} z G(l,z) mathrm dz \\ &=mathrm {erf}left (sqrt{frac{3}{2N_cb^2}}l ight )-frac{1}{2}I_1+frac{1}{2N_db}I_2 end{split} label{Zrodcoil} end{equation}

下面计算(I_1)：

egin{equation} egin{split} int_0^{N_db} mathrm dz G(l,z)= &frac{1}{2sqrt{pi }R_g}int_0^{N_db} mathrm dz expleft [-frac{(z-l)^2}{4R_g^2} ight ]\ &-frac{1}{2sqrt{pi }R_g}int_0^{N_db} mathrm dz expleft [-frac{(z+l)^2}{4R_g^2} ight ] \ =& frac{1}{sqrt{pi}}int_{-l/2R_g}^{(N_db-l)/2R_g} exp(-t^2)mathrm dt \ &-frac{1}{sqrt{pi }}int_{l/2R_g}^{(N_db+l)/2R_g} exp(-t^2)mathrm dt\ =& frac{1}{sqrt{pi}}int_{-l/2R_g}^0 e^{-t^2}mathrm dt +frac{1}{sqrt{pi}}int_0^{(N_db-l)/2R_g} e^{-t^2}mathrm dt \ &-frac{1}{sqrt{pi }}int_0^{(N_db+l)/2R_g} e^{-t^2}mathrm dt + frac{1}{sqrt{pi }}int_0^{l/2R_g} e^{-t^2}mathrm dt\ =&mathrm {erf}left (frac{l}{2R_g} ight )+frac{1}{2}mathrm {erf}left (frac{N_db-l}{2R_g} ight )-frac{1}{2}mathrm {erf}left (frac{N_db+l}{2R_g} ight ) end{split} label{I1} end{equation}

下面计算(I_2)：

egin{equation} egin{split} int_0^{N_db} zG(l,z) mathrm dz = &frac{1}{2sqrt{pi }R_g}int_0^{N_db} zexpleft [-frac{(z-l)^2}{4R_g^2} ight ] mathrm dz\ &-frac{1}{2sqrt{pi }R_g}int_0^{N_db} zexpleft [-frac{(z+l)^2}{4R_g^2} ight ] mathrm dz\ =& frac{1}{sqrt{pi }}int_{-l/2R_g}^{(N_db-l)/2R_g} (2R_gt+l)exp(-t^2)mathrm dt \ &-frac{1}{sqrt{pi }}int_{l/2R_g}^{(N_db+l)/2R_g} (2R_gt-l)exp(-t^2)mathrm dt\ =&frac{R_g}{sqrt{pi }}left {expleft [-left ( frac{l}{2R_g} ight )^2 ight ] - expleft [-left ( frac{N_db-l}{2R_g} ight )^2 ight ] ight }\ &+frac{l}{sqrt{pi }} int_{-l/2R_g}^{(N_db-l)/2R_g} exp(-t^2)mathrm dt \ &-frac{R_g}{sqrt{pi }}left {expleft [-left ( frac{l}{2R_g} ight )^2 ight ] - expleft [-left ( frac{N_db+l}{2R_g} ight )^2 ight ] ight }\ &+frac{l}{sqrt{pi }} int_{l/2R_g}^{(N_db+l)/2R_g} exp(-t^2)mathrm dt\ =& frac{R_g}{sqrt{pi }}left {expleft [-left ( frac{N_db+l}{2R_g} ight )^2 ight ]-expleft [-left ( frac{N_db-l}{2R_g} ight )^2 ight ] ight }\ &+frac{l}{sqrt{pi }} int_{-l/2R_g}^{(N_db-l)/2R_g} exp(-t^2)mathrm dt\ &+ frac{l}{sqrt{pi }} int_{l/2R_g}^{(N_db+l)/2R_g} exp(-t^2)mathrm dt\ =& frac{R_g}{sqrt{pi }}left {expleft [-left ( frac{N_db+l}{2R_g} ight )^2 ight ]-expleft [-left ( frac{N_db-l}{2R_g} ight )^2 ight ] ight } \ &+frac{l}{2}mathrm {erf}left (frac{N_db-l}{2R_g} ight )+frac{l}{2}mathrm {erf}left (frac{N_db+l}{2R_g} ight ) end{split} label{I2} end{equation}

自由能为：

egin{equation} frac{F_{d-c}(l,N_d,N_c)}{k_BT} = -ln left [ mathrm {erf}left (sqrt{frac{3}{2N_cb^2}}l ight )-frac{1}{2}I_1+frac{1}{2N_db}I_2 ight ] label{Frodcoil} end{equation}