偶极聚电解质刷

research

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Counterion adsorption on tethered polyelectrolyte chain
References:
THE JOURNAL OF CHEMICAL PHYSICS 2007, 126, 244902
J. Phys. Chem. B 2009, 113, 11076–11084
THE JOURNAL OF CHEMICAL PHYSICS 2012, 136, 234901

The derivation follows ''J. Phys. Chem. B'' 2009, 113, 11076–11084
Partition function

$$Z=\frac{1}{n_P!}\prod_m\sum_{n_m=0}^{\infty}\prod_{k=1}^{n_P} \int D \mathbf R_k(s)\prod_{k=1}^{n_m}\int D \mathbf r_{mk}\\ \prod_{k=1}^{n_C}\int D \mathbf r_{Ck} \prod_{k=1}^{n_D}\int D \mathbf r_{Dk}\int D\psi\exp(-H) \times \\ \prod_{\mathbf r}\delta(\sum_jv_j\hat\rho_j(\mathbf r)-1) \prod_{\mathbf r}\delta(\hat f(\mathbf r) \hat\rho_P(\mathbf r)+\hat g(\mathbf r) \hat\rho_P(\mathbf r)-\hat\rho_P(\mathbf r))\times \\ \delta(\int \mathrm d \mathbf r \hat\rho_e(\mathbf r)) \times \\ \prod_{k=1}^{n_C}\sum_{i=1}^{n_P}\int_0^{N_P}\mathrm ds\delta(\mathbf r_{Ck}-\mathbf R_i(s)) \prod_{k=1}^{n_D}\sum_{i=1}^{n_P}\int_0^{N_P}\mathrm ds\delta(\mathbf r_{Dk}-\mathbf R_i(s))$$

Hamiltonian

$$H=H_{ela}+H_{int}+H_{ele}+H_{chem}$$
Elastic term
$$H_{ela}=\sum_{k=1}^{n_P}\int_0^{N_P}\mathrm ds\frac{3}{2b^2}\left (\frac{\partial \mathbf R_i(s)}{\partial s} \right )^2$$
Flory-Huggins term
$$H_{int} = \int \mathrm d\mathbf r V_0\chi \hat\rho_P(\mathbf r) \hat\rho_S(\mathbf r)$$
Elactrostatic interaction
$$H_{ele}=\int \mathrm d\mathbf r \left ( \hat\rho_e(\mathbf r)\psi(\mathbf r)-\frac{1}{8\pi l_B}|\nabla\psi(\mathbf r)|^2 + \int \mathrm d\mathbf u\hat \rho_D(\mathbf r, \mathbf u)p\mathbf u \cdot \nabla\psi(\mathbf r) \right )$$
charge density operator
$$\hat\rho_e(\mathbf r) = \hat f(\mathbf r)\hat\rho_P(\mathbf r) + \sum_mz_m\hat \rho_m(\mathbf r)$$
dipole density operator
$$\hat \rho_D(\mathbf r)=\int \mathrm d\mathbf u \hat \rho_D(\mathbf r, \mathbf u)=\hat g(\mathbf r)\hat \rho_P(\mathbf r)$$
$$H_{chem}=\int \mathrm d\mathbf r\left ( \mu_C\hat f(\mathbf r)\hat\rho_P(\mathbf r)+\mu_D\hat g(\mathbf r)\hat \rho_P(\mathbf r) + \mu_+\hat\rho_+(\mathbf r)+ \mu_-\hat\rho_-(\mathbf r)+ \mu_S\hat\rho_S(\mathbf r)\right )$$
microscopic density operator
$$\hat \rho_P(\mathbf r) = \sum_{k=1}^{n_P}\int_0^{N_P} \mathrm ds \delta(\mathbf r -\mathbf R_i(s))$$
$$\hat \rho_m(\mathbf r) = \sum_{k=1}^{n_m} \delta(\mathbf r -\mathbf r_{mi})$$
charge fraction operator
$$\hat f(\mathbf r) =\frac{\sum_{k=1}^{n_C}\delta(\mathbf r - \mathbf r_{Ck})}{\sum_{i=1}^{n_P}\int_0^{N_P}\mathrm ds\delta(\mathbf r - \mathbf R_i(s))}$$
dipole operator
$$\hat g(\mathbf r, \mathbf u) =\frac{\sum_{k=1}^{n_D}\delta(\mathbf r - \mathbf r_{Dk})\delta(\mathbf u - \mathbf u_{Dk})} {\sum_{i=1}^{n_P}\int_0^{N_P}\mathrm ds\delta(\mathbf r - \mathbf R_i(s))}$$

Inserting identities

$$\prod_j\int D\omega_j(\mathbf r)\int D\rho_j(\mathbf r) \exp(\int \mathrm d\mathbf r )\omega_j(\mathbf r)(\rho_j(\mathbf r)-\hat\rho_j(\mathbf r))=1$$
$$\int D\gamma_C(\mathbf r)\int Df(\mathbf r)\exp(\int \mathrm d \mathbf r\gamma_C(\mathbf r)(f(\mathbf r)\rho_P(\mathbf r)-\hat f(\mathbf r)\hat\rho_P(\mathbf r)))=1$$
$$\int D\gamma_D(\mathbf r,\mathbf u)\int Dg(\mathbf r,\mathbf u)\exp(\int \mathrm d \mathbf r \mathrm d \mathbf u \gamma_D(\mathbf r,\mathbf u)(g(\mathbf r,\mathbf u) \rho_P(\mathbf r)-\hat g(\mathbf r,\mathbf u)\hat\rho_P(\mathbf r)))=1$$
$$\prod_{\mathbf r}\delta(\sum_jv_j\hat\rho_j(\mathbf r)-1)=\int D \eta(\mathbf r)\exp(-\int \mathrm d\mathbf r\eta(\mathbf r)(\delta(\sum_jv_j\hat\rho_j(\mathbf r)-1)))$$
$$\prod_{\mathbf r}\delta(\hat f(\mathbf r)\hat\rho_P(\mathbf r) +\hat g(\mathbf r)\hat\rho_P(\mathbf r)-\hat\rho_P(\mathbf r)) = \int D\alpha(\mathbf r)\exp(-\int \mathrm d\mathbf r \alpha(\mathbf r)(\hat f(\mathbf r)\hat\rho_P(\mathbf r) +\hat g(\mathbf r)\hat\rho_P(\mathbf r)-\hat\rho_P(\mathbf r)) )$$
$$\delta(\int \mathrm d\mathbf r \hat\rho_e(\mathbf r))=\int \mathrm d\lambda\exp(-\lambda\int \mathrm d \mathbf r \hat\rho_e(\mathbf r))$$

Then the partition function

$$Z=\prod_j\int D\omega_j\int D\rho_j \int D\psi \int D\eta \int D\alpha \\ \int Df \int Dg\int \gamma_C\int \gamma_D \int \mathrm d\lambda \exp(-F)$$

$$F=\int \mathrm d\mathbf r\left [ -\sum_j\omega_j(\mathbf r)\rho_j(\mathbf r) +V_0\chi \rho_P(\mathbf r)\rho_S(\mathbf r) -\eta(\mathbf r)+\eta(\mathbf r)\sum_jv_j\rho_j(\mathbf r)\\ - (\gamma_C(\mathbf r)f(\mathbf r)+\gamma_D(\mathbf r,\mathbf u) g(\mathbf r,\mathbf u))\rho_P(\mathbf r)+\alpha(\mathbf r)\rho_P(\mathbf r)(f(\mathbf r)+g(\mathbf r)-1)\\ -\frac{1}{8\pi l_B}|\nabla\psi(\mathbf r)|^2+\psi(\mathbf r)(z_Pf(\mathbf r)\rho_P(\mathbf r)+\sum_{ion}z_{ion}\rho_{ion}(\mathbf r))\\ +\int \mathrm d \mathbf u g(\mathbf r,\mathbf u)\rho_P(\mathbf r)p\mathbf u\cdot \nabla \psi(\mathbf r) \right ]\\ -n_P\ln(\frac{Q_P}{n_P})-\sum_{ion}Q_{ion}-Q_S-Q_C-Q_D$$
species partition function
$$Q_P=\frac{\int D\mathbf R(s)\exp(-\int_0^{N_P}\mathrm ds(\frac{3}{2b^2}(\frac{\partial \mathbf R(s)}{\partial s})^2+\omega_P(\mathbf R(s))))}{\int D\mathbf R(s)\exp(-\int_0^{N_P}\mathrm ds(\frac{3}{2b^2}(\frac{\partial \mathbf R(s)}{\partial s})^2))}$$
$$Q_S=\frac{\exp(-\mu_S)}{V}\int \mathrm d\mathbf r\exp(-\omega_S(\mathbf r))$$
$$Q_{ion}=\frac{\exp(-\mu_{ion}-z_{ion}\lambda)}{V}\int \mathrm d\mathbf r\exp(-\omega_{ion}(\mathbf r))$$
$$Q_C=\frac{\exp(-\mu_{C}+\lambda)}{V}\int \mathrm d\mathbf r \exp(-\gamma_{C}(\mathbf r))\rho_P(\mathbf r)$$
$$Q_D=\frac{\exp(-\mu_{D})}{V}\int \mathrm d\mathbf r \mathrm d\mathbf u \exp(-\gamma_{D}(\mathbf r,\mathbf u))\rho_P(\mathbf r)$$
SCF equations:
$$\omega_P(\mathbf r)=V_0\chi \rho_S(\mathbf r)+v_P\eta(\mathbf r) -\alpha(\mathbf r)$$
$$\omega_S(\mathbf r)=V_0\chi \rho_P(\mathbf r)+v_S\eta(\mathbf r)$$
$$\omega_{ion}(\mathbf r)=z_{ion}\psi(\mathbf r)$$
$$\rho_P(\mathbf r)=\frac{n_P}{Q_P}\int_0^{N_P}q(\mathbf r,s) q^{deg}(\mathbf r,s)$$
$$\rho_S(\mathbf r)=\frac{\exp(\mu_S)}{V}\exp(-\omega_S(\mathbf r))$$
$$\rho_{ion}(\mathbf r) = \frac{\exp(\mu_{ion}-z_{ion}\lambda)}{V}\exp(-\omega_{ion}(\mathbf r))$$
$$\gamma_C(\mathbf r)=z_P\psi(\mathbf r)+\alpha(\mathbf r)$$
$$\gamma_D(\mathbf r,\mathbf u)=p\mathbf u\cdot \nabla \psi(\mathbf r) +4\pi \alpha(\mathbf r)$$
$$f(\mathbf r)=\frac{\exp(-\mu_C-z_P\lambda)}{V}\exp(-\gamma_C(\mathbf r))$$
$$g(\mathbf r,\mathbf u)=\frac{\exp(-\mu_D)}{V}\exp(-\gamma_D(\mathbf r,\mathbf u))$$
$$\nabla^2\psi(\mathbf r)=-4\pi l_B\left [ \sum_{ion}z_{ion}\rho_{ion}+\nabla \cdot \mathbf P \right ]$$
$$\mathbf P = \int \mathrm d \mathbf u g(\mathbf r,\mathbf u)\rho_P(\mathbf r)p\mathbf u$$

useful integral:

$$\frac{1}{4\pi}\int \mathrm d\mathbf u\exp(-p\mathbf u\cdot \nabla\psi (\mathbf r))=\frac{\sinh(p|\nabla\psi(\mathbf r)|)}{p|\nabla\psi(\mathbf r)|}$$
$$\frac{1}{4\pi}\int \mathrm d\mathbf u\exp(-p\mathbf u\cdot \nabla\psi (\mathbf r))p\mathbf u = \frac{\psi(\mathbf r))}{|\nabla\psi(\mathbf r)|} \frac{\sinh(p|\nabla\psi(\mathbf r)|)}{p|\nabla\psi(\mathbf r)|}L(p|\nabla\psi(\mathbf r)|)$$
where $L(x)$ is Langevin function
$$L(x)=\coth(x)-1/x$$