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@joyphys 2014-07-18T06:26:47.000000Z 字数 7927 阅读 1886

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=1nP!mnm=0k=1nPDRk(s)k=1nmDrmkk=1nCDrCkk=1nDDrDkDψexp(H)×rδ(jvjρ^j(r)1)rδ(f^(r)ρ^P(r)+g^(r)ρ^P(r)ρ^P(r))×δ(drρ^e(r))×k=1nCi=1nPNP0dsδ(rCkRi(s))k=1nDi=1nPNP0dsδ(rDkRi(s))

Hamiltonian

H=Hela+Hint+Hele+Hchem

Elastic term
Hela=k=1nPNP0ds32b2(Ri(s)s)2

Flory-Huggins term
Hint=drV0χρ^P(r)ρ^S(r)

Elactrostatic interaction
Hele=dr(ρ^e(r)ψ(r)18πlB|ψ(r)|2+duρ^D(r,u)puψ(r))

charge density operator
ρ^e(r)=f^(r)ρ^P(r)+mzmρ^m(r)

dipole density operator
ρ^D(r)=duρ^D(r,u)=g^(r)ρ^P(r)

Hchem=dr(μCf^(r)ρ^P(r)+μDg^(r)ρ^P(r)+μ+ρ^+(r)+μρ^(r)+μSρ^S(r))

microscopic density operator
ρ^P(r)=k=1nPNP0dsδ(rRi(s))

ρ^m(r)=k=1nmδ(rrmi)

charge fraction operator
f^(r)=nCk=1δ(rrCk)nPi=1NP0dsδ(rRi(s))

dipole operator
g^(r,u)=nDk=1δ(rrDk)δ(uuDk)nPi=1NP0dsδ(rRi(s))

Inserting identities

jDωj(r)Dρj(r)exp(dr)ωj(r)(ρj(r)ρ^j(r))=1

DγC(r)Df(r)exp(drγC(r)(f(r)ρP(r)f^(r)ρ^P(r)))=1

DγD(r,u)Dg(r,u)exp(drduγD(r,u)(g(r,u)ρP(r)g^(r,u)ρ^P(r)))=1

rδ(jvjρ^j(r)1)=Dη(r)exp(drη(r)(δ(jvjρ^j(r)1)))

rδ(f^(r)ρ^P(r)+g^(r)ρ^P(r)ρ^P(r))=Dα(r)exp(drα(r)(f^(r)ρ^P(r)+g^(r)ρ^P(r)ρ^P(r)))

δ(drρ^e(r))=dλexp(λdrρ^e(r))

Then the partition function

Z=jDωjDρjDψDηDαDfDgγCγDdλexp(F)

F=drjωj(r)ρj(r)+V0χρP(r)ρS(r)η(r)+η(r)jvjρj(r)(γC(r)f(r)+γD(r,u)g(r,u))ρP(r)+α(r)ρP(r)(f(r)+g(r)1)18πlB|ψ(r)|2+ψ(r)(zPf(r)ρP(r)+ionzionρion(r))+dug(r,u)ρP(r)puψ(r)nPln(QPnP)ionQionQSQCQD

species partition function
QP=DR(s)exp(NP0ds(32b2(R(s)s)2+ωP(R(s))))DR(s)exp(NP0ds(32b2(R(s)s)2))

QS=exp(μS)Vdrexp(ωS(r))

Qion=exp(μionzionλ)Vdrexp(ωion(r))

QC=exp(μC+λ)Vdrexp(γC(r))ρP(r)

QD=exp(μD)Vdrduexp(γD(r,u))ρP(r)

SCF equations:
ωP(r)=V0χρS(r)+vPη(r)α(r)

ωS(r)=V0χρP(r)+vSη(r)

ωion(r)=zionψ(r)

ρP(r)=nPQPNP0q(r,s)qdeg(r,s)

ρS(r)=exp(μS)Vexp(ωS(r))

ρion(r)=exp(μionzionλ)Vexp(ωion(r))

γC(r)=zPψ(r)+α(r)

γD(r,u)=puψ(r)+4πα(r)

f(r)=exp(μCzPλ)Vexp(γC(r))

g(r,u)=exp(μD)Vexp(γD(r,u))

2ψ(r)=4πlB[ionzionρion+P]

P=dug(r,u)ρP(r)pu

useful integral:

14πduexp(puψ(r))=sinh(p|ψ(r)|)p|ψ(r)|

14πduexp(puψ(r))pu=ψ(r))|ψ(r)|sinh(p|ψ(r)|)p|ψ(r)|L(p|ψ(r)|)

where L(x) is Langevin function
L(x)=coth(x)1/x

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