Polyelectrolyte
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Ngày 23/10/2018 |
51
Chia sẻ tài liệu: Polyelectrolyte thuộc Bài giảng khác
Nội dung tài liệu:
31
Polyelectrolyte Chains at Finite Concentrations
Counterion Condensation
N=187, f=1/3, eLJ =1.5, u=3
cs3 = 1.5 10-4
cs3 = 1.5 10-2
32
Counterion Condensation
The electrostatic attraction between polyelectrolyte chain and counterions in solutions
Can results in condensation of counterions on polyelectrolyte chain. The counterion condensation appears to be due to a fine interplay between the electrostatic attraction and the loss of the translational entropy by counterions due to their localization in the vicinity of polymer chain.
Polymer
chain
counterions
33
Tutorial: Electrochemical Potential
Equilibrium distribution of charge density in external electric fields
Consider distribution of a charged particles with charge eq in nonhomogeneous external electric field E(x)
E(x)eqc(x)DxDy
Force balance on element with area DxDy
At equilibrium the sum of all forces acting on the element DxDy is equal to zero.
p(x,y+Dy)Dx
34
Counterion Condensation
Two-State Model (Oosawa-Manning )
Counterion in solution are divided into:
State 1: Counterion localized inside potential valleys of radius r0 along polymer backbones;
State 2: Counterions freely moving outside the region occupied by polyelectrolyte chains.
The total solution volume V is divided into two regions.
One with volume Npv (Np number of chains in a system) –
localization volume and another outer region (state 2) with
volume V- Npv.
35
Counterion Condensation
Two-State Model
Relationship between fraction of condensed counterions, linear charge density and polymer volume fraction is given by the following equation
Dependence of the inverse reduced effective linear charge density on the Oosawa-Manning condensation parameter g0
36
Counterion Condensation
Two-State Model for Flexible Polyelectrolyte Chains
Size of a polyelectrolyte chain depends on fraction of free counterions (1-b)f
Thus, the counterion condensation parameter is equal to
In this case the relationship between fraction of condensed counterions, linear charge density and polymer volume fraction is written as
37
Counterion Condensation
Tutorial: Poisson-Boltzmann Equation
Two-zone Model
Cylindrical zone
Spherical zone
38
Two-Zone Model of Dilute Polyelectrolyte Solutions
Poisson-Boltzmann equation for the cylindrical cell
Boundary condition at the surface of the rod
Boundary condition at the surface of the outer
cylinder
, where
, where
Two-parameter solution for counterion density
, where
39
Diagram of Phases
Counterions leave
cylindrical zone upon
dilution increasing gR
I. Polyion charge density
g0 is too low to localize
counterions.
II. Polyion charge density
g0 is high.
Counterions condense
to reduce electrostatic
energy.
III. Entropy wins at very low concentrations and
counterions leave polyions (even those with high g0).
There should be phase transitions between different regimes.
weakly
charged
g0
gR
Deshkovski et al ‘01
40
I. Weakly Charged Polyions
Counterions distribution in the cylindrical zone c(r) ~ r-2g0 is dominated
by the outer regions (g0<1).
Changes of gR only change
the prefactor of c(r) ~ r-2g0
MD of 16 rod-like polymers
with N=97 and lB=3s
41
II. Saturated Condensation
Counterions are distributed self-similarly throughout the whole
cylindrical zone.
Effective charge density of a polyion
together with condensed counterions is constant.
g0=3
Two-zone model
predicts c(r) ~ r-2
42
III. Unsaturated Condensation
At gR = 1 there is a transition to the unsaturated condensation
regime with polyion starving for more counterions.
g0=1.5
Two-zone model
predicts c(r) ~ r-2gR
43
Osmotic Coefficient
Osmotic coefficient (ratio of osmotic
pressure to that of an ideal gas
of counterions).
f = p /(kBTc)
Osmotic coefficient increases
towards unity with decreasing
concentration of dilute solutions.
Prediction of 2-zone model
44
Diagram of Phases for Flexible
Chains
unsaturated
condensation
saturated
condensation
weakly
charged polyions
Two-zone model for flexible chains
For flexible chains both parameters
g0 and gR are functions of polymer
concentration.
g0
45
Osmotic Coefficient of Flexible Chains
Osmotic coefficient of
flexible chains
Prediction of two-zone model for
flexible chains
46
Ion-binding and Ion Localization
Models of Counterion Condensation
These models also separate counterions into two different classes: bound and free counterions
This leads to nonlinear equation for the fraction of bound counterions
At equilibrium the chemical potentials of counterions in two states are equal
to each other mbound=mfree
Q-solvent
poor solvent
Continuous change with f
Avalanche-like condensation
with change of f
47
Dependence of a Chain Size on
Polymer Concentration
Counterion condensation leads to reduction of a chain size with
increasing polymer concentration.
Poor solvent
q-solvent
Dilute Solution
Regime
Dilute Solution
Regime
48
Electrostatic Persistence Length
In salt solutions the electrostatic interactions are screened at distances larger than the Debye screening length k.
Polyelectrolyte chain in salt solutions behaves as semiflexible
polymer with salt dependent persistence length.
49
Electrostatic Persistence Length
Odijk-Skolnick-Fixman approach
50
Electrostatic Persistence Length
Flory-like Calculations
51
Electrostatic Persistence Length
Flory-like Calculations
q
Polyelectrolyte Chains at Finite Concentrations
Counterion Condensation
N=187, f=1/3, eLJ =1.5, u=3
cs3 = 1.5 10-4
cs3 = 1.5 10-2
32
Counterion Condensation
The electrostatic attraction between polyelectrolyte chain and counterions in solutions
Can results in condensation of counterions on polyelectrolyte chain. The counterion condensation appears to be due to a fine interplay between the electrostatic attraction and the loss of the translational entropy by counterions due to their localization in the vicinity of polymer chain.
Polymer
chain
counterions
33
Tutorial: Electrochemical Potential
Equilibrium distribution of charge density in external electric fields
Consider distribution of a charged particles with charge eq in nonhomogeneous external electric field E(x)
E(x)eqc(x)DxDy
Force balance on element with area DxDy
At equilibrium the sum of all forces acting on the element DxDy is equal to zero.
p(x,y+Dy)Dx
34
Counterion Condensation
Two-State Model (Oosawa-Manning )
Counterion in solution are divided into:
State 1: Counterion localized inside potential valleys of radius r0 along polymer backbones;
State 2: Counterions freely moving outside the region occupied by polyelectrolyte chains.
The total solution volume V is divided into two regions.
One with volume Npv (Np number of chains in a system) –
localization volume and another outer region (state 2) with
volume V- Npv.
35
Counterion Condensation
Two-State Model
Relationship between fraction of condensed counterions, linear charge density and polymer volume fraction is given by the following equation
Dependence of the inverse reduced effective linear charge density on the Oosawa-Manning condensation parameter g0
36
Counterion Condensation
Two-State Model for Flexible Polyelectrolyte Chains
Size of a polyelectrolyte chain depends on fraction of free counterions (1-b)f
Thus, the counterion condensation parameter is equal to
In this case the relationship between fraction of condensed counterions, linear charge density and polymer volume fraction is written as
37
Counterion Condensation
Tutorial: Poisson-Boltzmann Equation
Two-zone Model
Cylindrical zone
Spherical zone
38
Two-Zone Model of Dilute Polyelectrolyte Solutions
Poisson-Boltzmann equation for the cylindrical cell
Boundary condition at the surface of the rod
Boundary condition at the surface of the outer
cylinder
, where
, where
Two-parameter solution for counterion density
, where
39
Diagram of Phases
Counterions leave
cylindrical zone upon
dilution increasing gR
I. Polyion charge density
g0 is too low to localize
counterions.
II. Polyion charge density
g0 is high.
Counterions condense
to reduce electrostatic
energy.
III. Entropy wins at very low concentrations and
counterions leave polyions (even those with high g0).
There should be phase transitions between different regimes.
weakly
charged
g0
gR
Deshkovski et al ‘01
40
I. Weakly Charged Polyions
Counterions distribution in the cylindrical zone c(r) ~ r-2g0 is dominated
by the outer regions (g0<1).
Changes of gR only change
the prefactor of c(r) ~ r-2g0
MD of 16 rod-like polymers
with N=97 and lB=3s
41
II. Saturated Condensation
Counterions are distributed self-similarly throughout the whole
cylindrical zone.
Effective charge density of a polyion
together with condensed counterions is constant.
g0=3
Two-zone model
predicts c(r) ~ r-2
42
III. Unsaturated Condensation
At gR = 1 there is a transition to the unsaturated condensation
regime with polyion starving for more counterions.
g0=1.5
Two-zone model
predicts c(r) ~ r-2gR
43
Osmotic Coefficient
Osmotic coefficient (ratio of osmotic
pressure to that of an ideal gas
of counterions).
f = p /(kBTc)
Osmotic coefficient increases
towards unity with decreasing
concentration of dilute solutions.
Prediction of 2-zone model
44
Diagram of Phases for Flexible
Chains
unsaturated
condensation
saturated
condensation
weakly
charged polyions
Two-zone model for flexible chains
For flexible chains both parameters
g0 and gR are functions of polymer
concentration.
g0
45
Osmotic Coefficient of Flexible Chains
Osmotic coefficient of
flexible chains
Prediction of two-zone model for
flexible chains
46
Ion-binding and Ion Localization
Models of Counterion Condensation
These models also separate counterions into two different classes: bound and free counterions
This leads to nonlinear equation for the fraction of bound counterions
At equilibrium the chemical potentials of counterions in two states are equal
to each other mbound=mfree
Q-solvent
poor solvent
Continuous change with f
Avalanche-like condensation
with change of f
47
Dependence of a Chain Size on
Polymer Concentration
Counterion condensation leads to reduction of a chain size with
increasing polymer concentration.
Poor solvent
q-solvent
Dilute Solution
Regime
Dilute Solution
Regime
48
Electrostatic Persistence Length
In salt solutions the electrostatic interactions are screened at distances larger than the Debye screening length k.
Polyelectrolyte chain in salt solutions behaves as semiflexible
polymer with salt dependent persistence length.
49
Electrostatic Persistence Length
Odijk-Skolnick-Fixman approach
50
Electrostatic Persistence Length
Flory-like Calculations
51
Electrostatic Persistence Length
Flory-like Calculations
q
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