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**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

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**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? 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? HOW – they differ from polymers and how from simple electrolytes!!!!!! Mention this.

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**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 super-absorbers in paper industry, and in waste–water management, etc. They have very reach physical behavior, for example, weakly charged PE may undergo conformational transition. They have extremely reach physical behavor which extendes over many length scales – from few angstrems to long-range Coulomb interaction

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**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. PE are complex in nature. They came in various shapes, DNA is rodlike and the synthetic PE are flexible (chain-like) as shown in this figure. PE can be globular (spherical shape) micelles, proteins, fullerene derivatives. But they are electrolytes so essentially the same theoretical methods apply. Terminology: counter-ions are small ions having the charge sign opposite to polyions, and co-ions have the same charge sign as polyions.

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An example - ionenes CH CH3 | | – N+ –(CH2)x – N+ – (CH2)y – x and y (numbers of CH2 groups between the nitrogen atoms) can be 3-3, 4-5, 6-6, and 6-9. WE performed MC simulation to the pearl neclase model – freely jointed charged hard spheres. Cationic PE: N (blue), C (green) and H atoms (grey)

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**More examples … (b) poly(styrene)sulfonic and a) b) c)**

(a) Fulerene based weak acid with 12 COOH groups. WE performed MC simulation to the pearl neclase model – freely jointed charged hard spheres. (b) poly(styrene)sulfonic and (c) poly(anethole)sulfonic acids (d) poly(p-phenylene) backbone d)

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**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! How do PE differ from low-weigth symmetric electrolytes? They are electrostatically bound to polyion and therefore their activity and mobility is low! In an external field bound counterions travel with the polyion in an opposite direction as they actually should.

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**Thermodynamics of aqueous solutions**

Osmotic coefficient as a function of polyelectrolyte concentration: Lipar, Pohar, Vlachy, unpublished. a) 0.1 0.2 0.3 0.4 0.5 0.001 0.01 1. cp/ (mol/ L) j NaPaSa NaPSS b) ‘normal’ aqueous solutions; b) polyelectrolyte (HPAS –green and HPSS - red) solutions

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**Osmotic coefficient of PE solutions – divalent counterions**

Φ - log m Φ = P/Pideal is very low and, in contrast to low molecular electrolytes, it decreases upon dilution!

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**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!

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**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.

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**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: P0 + П, ca, cb, ..., T; Aex B-O and experimental: P0, ma, mb, ...,T; Gex MM theory is esentially mimicking the osmotic experiment; one one side there is pure water and on the other side of the semi-permeable membrane there is an electrolyte solution. The chem poten of water as the only permeable species must be equal on both sides when the equilibrium is reached.

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**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! Pearl-neclase model – we see very non-uniform distribution of particles in dilute solution. Polyions are like ‘islands’ of irregular shape surrounded by counterions – lots of volume is free!!! This makes PE very difficult to model. cm = monomol/L cm = monomol/L

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**Computer simulation of highly asymmetric electrolyte**

Polyelectrolytes can be viewed as electrolytes asymmetric in charge and size. Hribar,Spohr, Vlachy, 1997

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**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).

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**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: 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.

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**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). 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.

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**The excess free energy: ξ<1**

The function K(0) has the asymptotic behavior: 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: 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. Note the logarithmic dependence on kapa and for simple electrolyte we have kapa**1.5 !!!! 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).

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**Osmotic coefficient: ξ<1**

The osmotic coefficient φ = Π/Πid can be calculated from Fex as: N = neV is the number of monomer units. With the use of X= ne/ns we finaly obtain: And for X→∞ we have:

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**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). 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.. And for X→∞ (salt-free solutions) we have for ξ>1:

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**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. Results for chondroitin (ξ = 1.15) are: 0.83 (theory 0.85) for Na+ ions 0.42 (theory 0.44) for Ca++ ions 0.29 (theory 0.29) for La3+ ions A: B: A: Osmotic coefficient in PE-electrolyte mixture (Na PMA + NaBr); B: Mean activity coefficient of mobile ions

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**Cylindrical cell model**

Each cell is an independent subsystem - other cells merely provide the surroundings.

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**Poisson-Boltzmann equation theory**

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.

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**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.

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**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

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**Pearl-necklace (M-M) model**

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. Similar model(s): Kremer&Stevens, and this year Yehtiraj et al.

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**Snapshots from actual simulations**

Bizjak, Rescic, Kalyuzhnyi, Vlachy, JCP, December 2006 N=32 N = 8; cm= monomol/L

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**Activity coefficients and ΔH**

Activity coeficients of counterions. Enthalpies of dilution from c to monomol/L are exothermic and become less negative with the increasing N. Eksperimental values are not available in this range of N !

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**Theory vs. MC simulation for N=16**

Osmotic coefficient c-c pdf c-p pdf The idea is to make aneclase – how to do this. We coul glue somehow beads together by make sticky spots on both side. But we cannot go to very long chains – so we make a circle – some properties not all will be ell reprodueced. The theory is analytical in the MSA approximation. Reduced density Theory developed on ideas of M. Wertheim by M. Holovko, L. Blum, Yu.V. Kalyuzhnyi, …., yields analytical (PMSA) solution.

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**Electrostatics is not everything …**

Manning’s theory: Φ = 1 - ξ/2 for ξ <1 Φ = 1/(2ξ) for ξ >1 The agreement is poor for ξ <1. Solutions exhibit strong nonideality even for ξ<0.5. Other than Coulomb forces are driving water molecules out of solution. Φ = − n1/n2 ln a1 a1= P1/P● ξ

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**Enthalpies of dilution: salts of PSSH**

Vesnaver, Skerjanc, Pohar, JPC 1988

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**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.

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**ΔH of Cl and Br solutions > 0!**

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!

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Conclusions Polyelectrolyte solutions are still not understood sufficiently well. The agreement between the electrostatic theories and experiment is at best semi-quantitative. Solvent has to be included in modeling to explain the ion-specific effects. Effects of presence of uncharged groups have to be included in theory. Coworkers: A. Bizjak, J. Reščič, B. Hribar-Lee, I. Lipar Yu.V. Kalyuzhnyi (ICMP, Ukraine), K.A. Dill (UCSF, USA) Sponsors: ARD (Slovenia); NIH (USA); and now also the Deutsche Forschungsgemeinschaft (Germany) Take home lesson – important for interpretation of experimental data !!!!! We hope at least a part of this will be done in Regensbug !

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