@joyphys 2014-07-18T06:26:47.000000Z 字数 7927 阅读 2106

偶极聚电解质刷

research

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 = 1 n P ! \prod m \sum n m = 0 \infty \prod k = 1 n P \int D R k (s) \prod k = 1 n m \int D r m k \prod k = 1 n C \int D r C k \prod k = 1 n D \int D r D k \int D ψ exp (- H) \times \prod r δ (\sum j v j ρ^j (r) - 1) \prod r δ (f^(r) ρ^P (r) + g^(r) ρ^P (r) - ρ^P (r)) \times δ (\int d r ρ^e (r)) \times \prod k = 1 n C \sum i = 1 n P \int N P 0 d s δ (r C k - R i (s)) \prod k = 1 n D \sum i = 1 n P \int N P 0 d s δ (r D k - R i (s))

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

Hamiltonian

H = H e l a + H i n t + H e l e + H c h e m

$H=H_{ela}+H_{int}+H_{ele}+H_{chem}$
Elastic term

H e l a = \sum k = 1 n P \int N P 0 d s 3 2 b 2 (\partial R i ( s ) \partial s) 2

$H_{ela}=\sum_{k=1}^{n_P}\int_0^{N_P}\mathrm ds\frac{3}{2b^2}\left (\frac{\partial \mathbf R_i(s)}{\partial s} \right )^2$
Flory-Huggins term

H i n t = \int d r V 0 χ ρ^P (r) ρ^S (r)

$H_{int} = \int \mathrm d\mathbf r V_0\chi \hat\rho_P(\mathbf r) \hat\rho_S(\mathbf r)$
Elactrostatic interaction

H e l e = \int d r (ρ^e (r) ψ (r) - 1 8 π l B | \nabla ψ (r) | 2 + \int d u ρ^D (r, u) p u \cdot \nabla ψ (r))

$H_{ele}=\int \mathrm d\mathbf r \left ( \hat\rho_e(\mathbf r)\psi(\mathbf r)-\frac{1}{8\pi l_B}|\nabla\psi(\mathbf r)|^2 + \int \mathrm d\mathbf u\hat \rho_D(\mathbf r, \mathbf u)p\mathbf u \cdot \nabla\psi(\mathbf r) \right )$
charge density operator

ρ^e (r) = f^(r) ρ^P (r) + \sum m z m ρ^m (r)

$\hat\rho_e(\mathbf r) = \hat f(\mathbf r)\hat\rho_P(\mathbf r) + \sum_mz_m\hat \rho_m(\mathbf r)$
dipole density operator

ρ^D (r) = \int d u ρ^D (r, u) = g^(r) ρ^P (r)

$\hat \rho_D(\mathbf r)=\int \mathrm d\mathbf u \hat \rho_D(\mathbf r, \mathbf u)=\hat g(\mathbf r)\hat \rho_P(\mathbf r)$

H c h e m = \int d r (μ C f^(r) ρ^P (r) + μ D g^(r) ρ^P (r) + μ + ρ^+ (r) + μ - ρ^- (r) + μ S ρ^S (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

ρ^P (r) = \sum k = 1 n P \int N P 0 d s δ (r - R i (s))

$\hat \rho_P(\mathbf r) = \sum_{k=1}^{n_P}\int_0^{N_P} \mathrm ds \delta(\mathbf r -\mathbf R_i(s))$

ρ^m (r) = \sum k = 1 n m δ (r - r m i)

$\hat \rho_m(\mathbf r) = \sum_{k=1}^{n_m} \delta(\mathbf r -\mathbf r_{mi})$
charge fraction operator

f^(r) = \sum n C k = 1 δ ( r - r C k ) \sum n P i = 1 \int N P 0 d s δ ( r - R i ( s ) )

$\hat f(\mathbf r) =\frac{\sum_{k=1}^{n_C}\delta(\mathbf r - \mathbf r_{Ck})}{\sum_{i=1}^{n_P}\int_0^{N_P}\mathrm ds\delta(\mathbf r - \mathbf R_i(s))}$
dipole operator

g^(r, u) = \sum n D k = 1 δ ( r - r D k ) δ ( u - u D k ) \sum n P i = 1 \int N P 0 d s δ ( r - R i ( s ) )

$\hat g(\mathbf r, \mathbf u) =\frac{\sum_{k=1}^{n_D}\delta(\mathbf r - \mathbf r_{Dk})\delta(\mathbf u - \mathbf u_{Dk})} {\sum_{i=1}^{n_P}\int_0^{N_P}\mathrm ds\delta(\mathbf r - \mathbf R_i(s))}$

Inserting identities

\prod j \int D ω j (r) \int D ρ j (r) exp (\int d r) ω j (r) (ρ j (r) - ρ^j (r)) = 1

$\prod_j\int D\omega_j(\mathbf r)\int D\rho_j(\mathbf r) \exp(\int \mathrm d\mathbf r )\omega_j(\mathbf r)(\rho_j(\mathbf r)-\hat\rho_j(\mathbf r))=1$

\int D γ C (r) \int D f (r) exp (\int d r γ C (r) (f (r) ρ P (r) - f^(r) ρ^P (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 γ D (r, u) \int D g (r, u) exp (\int d r d u γ D (r, u) (g (r, u) ρ P (r) - g^(r, u) ρ^P (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 r δ (\sum j v j ρ^j (r) - 1) = \int D η (r) exp (- \int d r η (r) (δ (\sum j v j ρ^j (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 r δ (f^(r) ρ^P (r) + g^(r) ρ^P (r) - ρ^P (r)) = \int D α (r) exp (- \int d r α (r) (f^(r) ρ^P (r) + g^(r) ρ^P (r) - ρ^P (r)))

$\prod_{\mathbf r}\delta(\hat f(\mathbf r)\hat\rho_P(\mathbf r) +\hat g(\mathbf r)\hat\rho_P(\mathbf r)-\hat\rho_P(\mathbf r)) = \int D\alpha(\mathbf r)\exp(-\int \mathrm d\mathbf r \alpha(\mathbf r)(\hat f(\mathbf r)\hat\rho_P(\mathbf r) +\hat g(\mathbf r)\hat\rho_P(\mathbf r)-\hat\rho_P(\mathbf r)) )$

δ (\int d r ρ^e (r)) = \int d λ exp (- λ \int d r ρ^e (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 ω j \int D ρ j \int D ψ \int D η \int D α \int D f \int D g \int γ C \int γ D \int d λ exp (- F)

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

F = \int d r ⎡ ⎣ - \sum j ω j (r) ρ j (r) + V 0 χ ρ P (r) ρ S (r) - η (r) + η (r) \sum j v j ρ j (r) - (γ C (r) f (r) + γ D (r, u) g (r, u)) ρ P (r) + α (r) ρ P (r) (f (r) + g (r) - 1) - 1 8 π l B | \nabla ψ (r) | 2 + ψ (r) (z P f (r) ρ P (r) + \sum i o n z i o n ρ i o n (r)) + \int d u g (r, u) ρ P (r) p u \cdot \nabla ψ (r) ⎤ ⎦ - n P ln (Q P n P) - \sum i o n Q i o n - Q S - Q C - Q D

$F=\int \mathrm d\mathbf r\left [ -\sum_j\omega_j(\mathbf r)\rho_j(\mathbf r) +V_0\chi \rho_P(\mathbf r)\rho_S(\mathbf r) -\eta(\mathbf r)+\eta(\mathbf r)\sum_jv_j\rho_j(\mathbf r)\\ - (\gamma_C(\mathbf r)f(\mathbf r)+\gamma_D(\mathbf r,\mathbf u) g(\mathbf r,\mathbf u))\rho_P(\mathbf r)+\alpha(\mathbf r)\rho_P(\mathbf r)(f(\mathbf r)+g(\mathbf r)-1)\\ -\frac{1}{8\pi l_B}|\nabla\psi(\mathbf r)|^2+\psi(\mathbf r)(z_Pf(\mathbf r)\rho_P(\mathbf r)+\sum_{ion}z_{ion}\rho_{ion}(\mathbf r))\\ +\int \mathrm d \mathbf u g(\mathbf r,\mathbf u)\rho_P(\mathbf r)p\mathbf u\cdot \nabla \psi(\mathbf r) \right ]\\ -n_P\ln(\frac{Q_P}{n_P})-\sum_{ion}Q_{ion}-Q_S-Q_C-Q_D$
species partition function

Q P = \int D R ( s ) exp ( - \int N P 0 d s ( 3 2 b 2 ( \partial R ( s ) \partial s ) 2 + ω P ( R ( s ) ) ) ) \int D R ( s ) exp ( - \int N P 0 d s ( 3 2 b 2 ( \partial R ( s ) \partial s ) 2 ) )

$Q_P=\frac{\int D\mathbf R(s)\exp(-\int_0^{N_P}\mathrm ds(\frac{3}{2b^2}(\frac{\partial \mathbf R(s)}{\partial s})^2+\omega_P(\mathbf R(s))))}{\int D\mathbf R(s)\exp(-\int_0^{N_P}\mathrm ds(\frac{3}{2b^2}(\frac{\partial \mathbf R(s)}{\partial s})^2))}$

Q S = exp ( - μ S ) V \int d r exp (- ω S (r))

$Q_S=\frac{\exp(-\mu_S)}{V}\int \mathrm d\mathbf r\exp(-\omega_S(\mathbf r))$

Q i o n = exp ( - μ i o n - z i o n λ ) V \int d r exp (- ω i o n (r))

$Q_{ion}=\frac{\exp(-\mu_{ion}-z_{ion}\lambda)}{V}\int \mathrm d\mathbf r\exp(-\omega_{ion}(\mathbf r))$

Q C = exp ( - μ C + λ ) V \int d r exp (- γ C (r)) ρ P (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 = exp ( - μ D ) V \int d r d u exp (- γ D (r, u)) ρ P (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:

ω P (r) = V 0 χ ρ S (r) + v P η (r) - α (r)

$\omega_P(\mathbf r)=V_0\chi \rho_S(\mathbf r)+v_P\eta(\mathbf r) -\alpha(\mathbf r)$

ω S (r) = V 0 χ ρ P (r) + v S η (r)

$\omega_S(\mathbf r)=V_0\chi \rho_P(\mathbf r)+v_S\eta(\mathbf r)$

ω i o n (r) = z i o n ψ (r)

$\omega_{ion}(\mathbf r)=z_{ion}\psi(\mathbf r)$

ρ P (r) = n P Q P \int N P 0 q (r, s) q d e g (r, s)

$\rho_P(\mathbf r)=\frac{n_P}{Q_P}\int_0^{N_P}q(\mathbf r,s) q^{deg}(\mathbf r,s)$

ρ S (r) = exp ( μ S ) V exp (- ω S (r))

$\rho_S(\mathbf r)=\frac{\exp(\mu_S)}{V}\exp(-\omega_S(\mathbf r))$

ρ i o n (r) = exp ( μ i o n - z i o n λ ) V exp (- ω i o n (r))

$\rho_{ion}(\mathbf r) = \frac{\exp(\mu_{ion}-z_{ion}\lambda)}{V}\exp(-\omega_{ion}(\mathbf r))$

γ C (r) = z P ψ (r) + α (r)

$\gamma_C(\mathbf r)=z_P\psi(\mathbf r)+\alpha(\mathbf r)$

γ D (r, u) = p u \cdot \nabla ψ (r) + 4 π α (r)

$\gamma_D(\mathbf r,\mathbf u)=p\mathbf u\cdot \nabla \psi(\mathbf r) +4\pi \alpha(\mathbf r)$

f (r) = exp ( - μ C - z P λ ) V exp (- γ C (r))

$f(\mathbf r)=\frac{\exp(-\mu_C-z_P\lambda)}{V}\exp(-\gamma_C(\mathbf r))$

g (r, u) = exp ( - μ D ) V exp (- γ D (r, u))

$g(\mathbf r,\mathbf u)=\frac{\exp(-\mu_D)}{V}\exp(-\gamma_D(\mathbf r,\mathbf u))$

\nabla 2 ψ (r) = - 4 π l B [\sum i o n z i o n ρ i o n + \nabla \cdot P]

$\nabla^2\psi(\mathbf r)=-4\pi l_B\left [ \sum_{ion}z_{ion}\rho_{ion}+\nabla \cdot \mathbf P \right ]$

P = \int d u g (r, u) ρ P (r) p u

$\mathbf P = \int \mathrm d \mathbf u g(\mathbf r,\mathbf u)\rho_P(\mathbf r)p\mathbf u$

useful integral:

1 4 π \int d u exp (- p u \cdot \nabla ψ (r)) = sinh ( p | \nabla ψ ( r ) | ) p | \nabla ψ ( r ) |

$\frac{1}{4\pi}\int \mathrm d\mathbf u\exp(-p\mathbf u\cdot \nabla\psi (\mathbf r))=\frac{\sinh(p|\nabla\psi(\mathbf r)|)}{p|\nabla\psi(\mathbf r)|}$

1 4 π \int d u exp (- p u \cdot \nabla ψ (r)) p u = ψ ( r ) ) | \nabla ψ ( r ) | sinh ( p | \nabla ψ ( r ) | ) p | \nabla ψ ( r ) | L (p | \nabla ψ (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) $L(x)$ is Langevin function

L (x) = coth (x) - 1 / x

$L(x)=\coth(x)-1/x$

偶极聚电解质刷

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