Transcript Document

Polyelectrolyte solutions
Polyelectrolyte solutions:
•
General properties
•
Manning’s approach
•
Cylindrical cell model – PB
equation
•
Comparison with experiment
Related, but not covered here:
•
Other cell geometries –
thermodynamics
•
Ions in external field – DF
theory
•
Interaction between electrical
double – layers
What are the polyelectrolytes ?
•
Polyelectrolytes are polymers consisting of
monomers having groups, which may ionize in a polar
solvent.
•
Physico-chemical properties of polyelectrolyte
solutions differ significantly from those of low –
molecular electrolytes as also from those of neutral
polymers. How?
1. How do these properties evolve by increasing the
charge and length of the polymer chain?
2. Can we explain experimental data in view of the
existent polyelectrolyte theories?
Why polyelectrolytes …
• Bio–polyelectrolytes (DNA, RNA); suspensions of lyophobic
colloids and surfactant micelles, polysaccharides.
•
Synthetic polyelectrolytes (co–polymers) have applications in
many areas of industry; especially in food industry, cosmetics,
and medicine, they are used for coating the surfaces, as superabsorbers in paper industry, and in waste–water management,
etc.
• They have very reach physical behavior, for example, weakly
charged PE may undergo conformational transition.
Polyions, counterions, co-ions
Polyelectrolytes come in various shapes: DNA is rod-like, synthetic
polyelectrolytes are flexible (chain-like) and some are globular as
fullerene derivatives or micelles and colloids.
Terminology:
counter-ions are small ions having the charge sign opposite to
polyions, and co-ions have the same charge sign as polyions.
An example - ionenes
CH3
|
CH3
|
– N+ –(CH2)x – N+ – (CH2)y –
|
|
CH3
CH3
x and y (numbers of
CH2 groups between the
nitrogen atoms) can be
3-3, 4-5, 6-6, and 6-9.
Cationic PE: N (blue), C (green) and H atoms (grey)
More examples …
a)
2.6
2.5
b)
c)
1.5
1.5
CH3
CH3
n
n
O
(a) Fulerene based weak
acid with 12 COOH groups.
6.0
S
ONa
O
O
S
ONa
O
O
CH3
(b) poly(styrene)sulfonic and
(c) poly(anethole)sulfonic acids
d)
(d) poly(p-phenylene) backbone
8.9
Properties of aqueous PE solutions
•
The activity and mobility of counterions are reduced well below their bulk
value (low osmotic pressure, high activity of water in solution).
•
When external field is applied to the solution, a fraction of counterions
travels as an integral part of a polyion.
•
In contrast to simple electrolytes the non-ideality increases upon dilution !
•
Thermodynamic properties are ion-specific !
•
Reduced viscosity increases upon dilution! Not true for uncharged
polymers.
•
Electrostatic theories are not always in agreement with theoretical results.
For heats of dilution often even the sign is not always predicted correctly!
Thermodynamics of aqueous solutions
0.5
0.4
0.3
j
a)
Osmotic coefficient as a function of polyelectrolyte
concentration: Lipar, Pohar, Vlachy, unpublished.
b)
0.2
NaPaSa
NaPSS
0.1
cp/ (mol/ L)
0
0.001
0.01
0.1
a) ‘normal’ aqueous solutions;
b) polyelectrolyte (HPAS –green and HPSS - red) solutions
1.
Osmotic coefficient of PE solutions
– divalent counterions
Φ = − n1/n2 ln a1
a1= P1/P●
Φ
- log m
Φ = P/Pideal is very low and, in contrast to low molecular
electrolytes, it decreases upon dilution!
Viscosity of PE solutions
Reduced viscosity of polyelectrolyte solutions:
A: In pure water,
B: with some additional salt,
C: in the excess of simple salt.
Indicates an extension of
polyelectrolyte upon dilution.
This behavior is in contrast to that
of the uncharged polymer
solutions, which behave as C.
Conformation and thermodynamic
properties are connected.
Levels of theoretical description
• Born-Oppenheimer level treats all the particles (ions and
water molecules) equivalently.
• McMillan-Mayer level: Only the solute particles are
treated explicitly and the properties of solvent are reflected
in the solute-solute interaction. The solution is treated as a
‘gas’ of solute particles in a solvent.
• The solvent-averaged potential (free energy) of solute
particles UN(r1,..., rN;T, P0) is determined at T and P0.
• M-M variables:
• B-O and experimental:
P0 + П, ca, cb, ..., T;
P0,
ma, mb, ...,T;
Aex
Gex
Modeling PE - preview (N=32)
The oligo-ions are freely jointed chains of charged hard spheres with
diameter 4 Å embedded in the continuous dielectric mimicking water.
Theories treat them as fully extended!
cm = 0.0195 monomol/L
cm = 0.195 monomol/L
Computer simulation of highly
asymmetric electrolyte
Polyelectrolytes
can be viewed as
electrolytes
asymmetric in
charge and size.
Hribar,Spohr, Vlachy, 1997
Manning’s ‘line charge’ approach
PE chain is replaced by an infinite line charge
with charge density parameter ξ = LB/b;
LB=e02/(4πkBTε0εr) (Bjerrum length for water
at T=298K is 7.14 Å );
β=zpe/b; L=Nb (N is the degree of
polymerization and b length of the monomer
unit);
zp is the valency of the charge on the polyion
and zi is the valency of counterions in solution.
Both entropy and energy are logarithmic functions of the distance r
from the polyion long axis; for ξ > 1 the ions do not dilute from
polyions (condense). For ξ ≤ 1 entropy prevails, as it always does
for spherical symmetry (approaching the ideal behavior).
Onsager’s observation
•
•
•
PE chain is replaced by an infinite line charge; ξ = LB/b.
The dielectric constant ε = 4πε0εr is taken as that of the pure solvent
For r=r0 , small enough distance from the line charge, the electrostatic
energy of the point charge is given by the unscreened potential:
uip  zi e(2β / 40 r ) ln r  2zi z p kBT ln r
Ai ( r0 ) 
r0
e
0
 uip ( r ) / kT
r0
2 r dr   r
(1 2 zi z p )
dr
0
If zizp<0, than phase integral diverges at the lower limit for all ξ such that
ξ ≥ |zi zp|-1 . For zpzi=−1 this means that it diverges for ξ = LB/b =
e02/(4πkBTε0εrb) ≥ 1. Sufficiently many counterions have to
‘condense’ on the polyion to reduce ξ to value just less than 1.
Counterion condensation: ξ<1
•
Counterions ‘condense’ on the polyion to lower the charge density parameter
ξ to 1 (actually to |zi zp|-1).
•
The uncondensed mobile ions are treated in the Debye-Hückel
approximation.
•
The potential ψ(r) at distance r from the line charge along the z-axis is given
by the superposition of the screened Coulomb potentials exp(-κr)/r from
infinitesimal segments of length dz (containing charge ezP/b dz).
 (r) 
( z p e / b)  exp(rt )
2 0 r
 (t
1
2
 1)
0.5
dt 
( 2 z p e / b)
4 0 r
K0 ( κr)
K0(κr) is the modified Bessel function of the zeroth order, which goes to
-ln (κr) for κr→0. In calculation, we used the substitution t2 = 1 + (z/r)2. For
two ionic species (counterions and co-ions): κ2=e2(n1 + n2)/ε0εr. Alternatively we
can obtain this result by solving the linearized PB eq in cylindrical symmetry.
The excess free energy: ξ<1
The function K(0) has the asymptotic behavior: K0 (κr)   lnκr ;
r  0
The term ln (r) is due to the line charge itself. So when r→0 the potential
of the ionic atmosphere ψ’(0) at the position of the line charge is:
 ' (0)  
2
4 0 r
ln(κ )  
z pe
2b 0 r
ln(κ )
The excess free energy fex due to interaction between mobile ions and the
segment dz of the line charge can be obtained by “charging” this segment up
from 0 to β = zpe/b.

f
ex
2
  ' (0)dβ  
ln(κ )
4 0 r
0
F ex / Vk BT   ne ln κ ;   1
The excess free energy for Np polyions in volume V (Fex =Np ∫ fex dz) is given
above. Note that N = b-1 ∫ dz and ne=N Np/V is the concentration expresed in
moles of the monomer units per volume (|zp|=1).
Osmotic coefficient: ξ<1
The osmotic coefficient φ = Π/Πid can be calculated from Fex as:
 F ex 
      
 k BT (j  1)(n1  n2 )

 V T , N P , N1 , N 2
ex
id
F ex   kBT Nm ln κ
1
  ln  


 V 
2V
Ni
 F ex 
1
1




k
TN

 k BTne  k BT (j  1)(n1  n2 )
B

2V
2
 V T , N P , N1 , N 2
1
2
N = neV is the number of monomer units.
With the use of X= ne/ns we finaly obtain:
And for X→∞ we have:
j  1 
1
2
j  1 
X
X 2
Osmotic coeficient for ξ>1
We distinguish the structural ξ = LB/b and ‘effective’ value of ξ. According
to the theory if ξ > 1 its effective value will be equal ξeff =1 since
condensed counterions will neutralize the fraction (ξ − 1) / ξ = 1 − ξ−1 of
the polyion charge (ξ−1 is the fraction of free counterions).


 φ(n1  n2 )  j (ne  2ns )  j (1,  1ne )  1ne  2ns
k BT

The effective concentration of ions is (ξ−1ne + ns); ns is the concentration of
added salt. φ(1,ξ−1 ) is the osmotic coefficient of a solution whose polyion
has effective ξ=1 and X= ne/ns replaced by ξ-1ne/ns..
1 X 1
1
j (1,  ne )  1 
2 X 1  2
And for X→∞ (salt-free
solutions) we have for ξ>1:
( 12 X 1  2)
j
; for   1
X 2
1
j
2
Comparison with experimental data
Apparent polyion charges defined
through Nernst-Einstein relation
as proportional to the ratio of
polyion electrophoretic mobility to
its self-diffusion coefficient were
measured. According to the
theory f = (ziξ)−1.
A:
B:
Results for chondroitin (ξ = 1.15)
are:
0.83 (theory 0.85) for Na+ ions
A: Osmotic coefficient in PEelectrolyte mixture (Na PMA + NaBr);
B: Mean activity coefficient of mobile
ions
0.42 (theory 0.44) for Ca++ ions
0.29 (theory 0.29) for La3+ ions
Cylindrical cell model
Each cell is an independent subsystem - other
cells merely provide the surroundings.
Poisson-Boltzmann equation theory
1 d  n d 
e
r

n


r dr  dr 
 0 r
 e   zi e0  i ( r ) 
i
 z e  ( R ) exp[ z e  / k T ]
i 0
i
i 0
B
i
For cylindrical systems n=1. The cell volume is related to the PE
concentration, which is measured in moles of the monomer units.
The boundary conditions are given by Gauss Law. The zero of
potential ψ(r) is chosen at r=R. Thermodynamic properties can be
obtained via the charging process (Aex) or by Eex and Sex separately.
Cell model vs. experiment
PB – dotted line
MC – broken line
PB
MC
Good agreement can be obtained for ξ / ξ0 ≈ 1.2 for 3,3 and
ξ/ξ0 ≈ 1.7 for 6,6 ionene Br; ξ0 is the structural value of LB/b.
Do not start simulations on full moon!
Which theory to use?
Theoretical approaches:
• Manning theory
• Poisson-Boltzman and MPB
• DFT – HNC/MSA approach
• Monte Carlo; MD
• Polymer RISM theory
• IE based on Wertheim’s O-Z
approach is able to treat flexible
polyions
Pearl-necklace (M-M) model
ei e j
uij (r) = u (r) +
εrij
*
ij
u*ij (r) = ,
u*ij (r) = 0,
for r  a,
for r  a.
Similar model(s): Kremer&Stevens,
and this year Yehtiraj et al.
Small ions are charged
hard spheres, while the
oligoions are modeled as a
flexible chain of charged
hard spheres.
Short-range function u*ij (r)
may include repulsive part
of the interaction, effect of
granularity of the solvent,
dielectric effect, etc.
Snapshots from actual simulations
Bizjak, Rescic, Kalyuzhnyi, Vlachy, JCP, December 2006
N = 8; cm=0.195 monomol/L
N=32
Activity coefficients and ΔH
Activity coeficients of counterions.
Enthalpies of dilution from c to 0.002
monomol/L are exothermic and become
less negative with the increasing N.
Eksperimental values are not available in this range of N !
Theory vs. MC simulation for N=16
c-c pdf
Osmotic coefficient
c-p pdf
Reduced density
Theory developed on ideas of M. Wertheim by M. Holovko, L.
Blum, Yu.V. Kalyuzhnyi, …., yields analytical (PMSA) solution.
Electrostatics is not everything …
1.0
Φ = − n1/n2 ln a1
Manning’s theory:
a1= P1/P●
Φ = 1 - ξ/2 for ξ <1
0.9
0.8
Φ = 1/(2ξ) for ξ >1
0.7

0.6
The agreement is
poor for ξ <1.
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.5
1.0
1.5
Charge density
2.0
ξ
2.5
3.0
Solutions exhibit
strong nonideality
even for ξ<0.5.
Other than Coulomb
forces are driving
water molecules out
of solution.
ΔH of ionene F are negative
Enthalpies of dilutions are
negative for 3,3 ionene
and agree very well with
the Manning LL and P-B
cell model theory.
The same ΔH<0 holds
true for Li (Na,Cs) salt of
PSSH at 298K, but NOT
for Cs salt at 273K!
Manning LL: ΔH/RT = (2ξ)-1 [ 1 + (d ln ε/ d ln T)] ln (c1/c2)
where c1 is the initial and c2 the final concentration; the term in brackets […] is
equal to −0.37 for water at 298K and P=1 bar.
ΔH of Cl and Br solutions > 0!
Cl
Br
1000
3,3
4,5
6,6
6,9
 HD / J mol
-1
800
600
400
200
0
-200
2,5
2,0
1,5
1,0
-3
- log (m / mol dm )
ΔH concentration dependence shows that Br counterions produce stronger
endothermic effect than Cl ions. The limiting slope is negative for 6,6 and 6,9
ionenes, but positive for the others. The differences between Br and Cl salt
are much larger for 3,3 and 4,5, than for 6,6 and 6,9 ionenes!
Conclusions
• Polyelectrolyte solutions are still
not understood sufficiently well.
Coworkers:
• The agreement between the
electrostatic theories and
experiment is at best semiquantitative.
A. Bizjak, J. Reščič, B. HribarLee, I. Lipar
• Solvent has to be included in
modeling to explain the ionspecific effects.
K.A. Dill (UCSF, USA)
• Effects of presence of uncharged
groups have to be included in
theory.
Yu.V. Kalyuzhnyi (ICMP,
Ukraine),
Sponsors:
ARD (Slovenia); NIH (USA);
and now also the Deutsche
Forschungsgemeinschaft
(Germany)
We hope at least a part of this will be done in Regensbug !