纳米粒子添加物对高分子凝胶体积转变的影响



符号列表:

  • $\phi$:凝胶体积分数
  • $\phi_0$:凝胶参考态体积分数
  • $N$:凝胶内支链长度
  • $\psi$:纳米粒子体积分数
  • $\chi_{ij}$:组分$i$与$j$之间的弗洛里-哈金斯参数

凝胶自由能密度:

\begin{equation} \begin{split} f_{\mathrm {gel}}=&\frac{3\phi}{2N}\left[\left(\frac{\phi_0}{\phi} \right)^{2/3}-\frac{4}{3f}\ln\frac{\phi_0}{\phi} \right]+ \\ & \frac{\psi}{P}\ln\psi + (1-\psi-\phi)\ln(1-\psi-\phi)+ \\ & \chi_{\mathrm {NP}}\phi\psi + \chi_{\mathrm {NS}}\phi(1-\psi-\phi) + \chi_{\mathrm {PS}}\psi(1-\psi-\phi) \end{split} \label{fgel} \end{equation}

本体溶液自由能密度:

\begin{equation} f_{\mathrm {bulk}}=\frac{\Psi}{P}\ln\Psi + (1-\Psi)\ln(1-\Psi)+\chi_{\mathrm {PS}}\Psi(1-\Psi) \label{fbulk} \end{equation}

凝胶内外化学势相等:

\begin{equation} \begin{split} &\frac{\partial f_{\mathrm {gel}}}{\partial \psi}=\frac{\partial f_{\mathrm {bulk}}}{\partial \Psi}\Longrightarrow \\ &\frac{1}{P}\ln\frac{\psi}{\Psi}-\ln\frac{1-\psi-\phi}{1-\Psi}+2\chi_{\mathrm {PS}}(\Psi-\psi)+(\chi _{\text{NP}}-\chi _{\text{NS}}-\chi _{\text{PS}})\phi = 0 \end{split} \label{muP} \end{equation}

凝胶内外渗透压相等:

\begin{equation} \begin{split} &\frac{\partial f_{\mathrm {gel}}}{\partial \phi}\phi+\frac{\partial f_{\mathrm {gel}}}{\partial \psi}\psi-f_{\mathrm {gel}}=\frac{\partial f_{\mathrm {bulk}}}{\partial \Psi}\Psi-f_{\mathrm {bulk}}\Longrightarrow \\ &\frac{\phi_0^{2/3}}{N}\phi^{1/3}+\phi(1-\frac{2}{fN})+\\ &\ln\frac{1-\psi-\phi}{1-\Psi}+(1-\frac{1}{P})(\psi-\Psi)+\\ &\chi _{\text{NS}}\phi^{2}+\chi_{\mathrm {PS}}(\psi^2-\Psi^2)+(\chi _{\text{NS}}+\chi _{\text{PS}}-\chi _{\text{NP}})\phi\psi = 0 \end{split} \label{osmotic} \end{equation}

标签: 凝胶

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