Influence of Aggregation and Dialysis on Conductivity of Salt-Free Polyelectrolyte Solution in the Dilute/Semidilute Region

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Article

Influence of Aggregation and Dialysis on Conductivity of

Salt-Free Polyelectrolyte Solution in the Dilute/Semidilute Region

King-Fu Lin, Szu-Nang Yang, Hon-Long Cheng, and Yu-Hui Cheng

Macromolecules, 1999, 32 (14), 4602-4607 • DOI: 10.1021/ma9817976 Downloaded from http://pubs.acs.org on November 30, 2008

Influence of Aggregation and Dialysis on Conductivity of Salt-Free

Polyelectrolyte Solution in the Dilute/Semidilute Region

King-Fu Lin,* Szu-Nang Yang, Hon-Long Cheng, and Yu-Hui Cheng Institute of Materials Science and Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China

Received November 18, 1998; Revised Manuscript Received May 4, 1999

ABSTRACT: The conductivity of sodium poly(styrenesulfonate) solution was found to increase during dialysis against water as indicated by the in-situ conductivity measurements at a concentration of c ) 0.1 g/L, which is in the dilute/semidilute region. To interpret this unexpected phenomenon, the domains of polyion chains and their aggregates are assumed to have a uniformly charged density. Although the aggregation resulted from the higher attractive forces rather than the repulsion between the polyion chains, their size was determined by the counterbalance of ionic osmotic forces due to the fact that the interior of aggregated domains has more counterions than the exterior. When the polyelectrolyte solution was dialyzed against water, extra osmotic forces acted on the solution due to Donnan’s effects. On the basis of the mean field theory and above arguments, the increase of conductivity during dialysis was interpreted in terms of the aggregation of polyions in the aqueous solution.

Introduction

The aggregation of polyelectrolyte chains in the aqueous solutions has raised great research interest over the past two decades. In the presence of a high concentration of low molecular weight salt that screens off charges from polyelectrolytes, the polyions may be homogeneously dispersed. Odijk1_{has pointed out that}

the theoretical predication on the solution behavior of polyelectrolytes with added salt was straightforward, whereas no satisfactory theory in the absence of salt was available. The polyelectrolyte solutions with no or low added salt have been often regarded as a hetero-geneous solution in which the polyions formed a mul-tichain cluster (or domain).2-11_{For example, Ise et al.}2

observed a single broad peak in the small-angle X-ray scattering of polyelectrolyte dilute solution and sug-gested the presence of quasi-lattice-like ordering do-mains. In the dynamic light scattering of polyelectrolyte solutions with no or low added salt, a double-exponential correlation function interpreted as consisting of fast and slow modes was frequently reported.4-11_{The fast mode}

was interpreted as a coupled diffusion of polyions and counterions. The slow mode was interpreted as the diffusion of multichain domains and regarded as evi-dence of the existence of polyions’ aggregation.10_{It was}

also demonstrated that the slow mode appeared along with a slight increase of apparent equivalent conductiv-ity of poly(L-lysine-HBr) when the external NaBr salt concentration is decreased below a certain critical value.12

The conformation of polyelectrolyte chains in the aqueous solution is not yet clear. Theoretical efforts on interpretation of polyion expansion focused on two models, i.e., the chain model and spherical model.13,14

The first chain model was made by Kuhn et al.15 _in

1948. They assumed that in dilute solutions the coun-terions escape from the polyion domain so that they do not influence the mutual repulsion of the fixed charges carried by polyions. This simplified chain model leads to a rod limit of polyions in the extremely dilute solution. The model has been shown to be oversimplified because

a significant fraction of counterions remained in the polyion domain.14_{Odijk et al.}1,16_{had modified the chain}

model by describing the expansion of polyelectrolytes according to the wormlike chain model,17 _counterion

condensation theory,18_{electrostatic persistence length,}19

and polymer scaling law.20 _{However, the model could}

only apply to the semidilute region of polyelectrolyte solution with added salt. On the other hand, the spherical model was first proposed by Hermans and Overbeek,21_{who assumed that the polyelectrolyte chains}

formed as a sphere with a uniformly charged density in aqueous solution with added salt. The expansion coefficient of polyelectrolyte chains was calculated ac-cordingly by using the Poisson-Boltzmann equation under high salt concentration. Flory22,23_{also treated the}

swelling of polyion in the presence of salt as a uniformly charged sphere. He proposed a Donnan assumption that the major driving forces of the swelling process was the osmotic forces, which are mainly due to an excess of mobile ions in the interior of the polyion domain and partly (but negligibly) due to the mixing of solvent and polymer segments. However, the spherical model is unrealistic since the polyion chains become increasingly asymmetric as they expand. Nevertheless, in the ab-sence of salt, due to the heterogeneous nature of polyelectrolyte solution, no existing theory so far is satisfactory to interpret its solution properties.

Conductometry is an important tool to investigate the solution properties of polyelectrolytes. In this study, when the sodium poly(styrenesulfonate) (NaPSS) aque-ous solution at a concentration of 0.1 g/L was dialyzed against water, the conductivity was found to increase with time. The increase of conductivity during dialysis of polyelectrolyte solution was never reported in the literature before. So, the experiment was repeated several times and came out with a consistent result. To interpret this phenomenon, we have assumed that the polyions may aggregate and form a uniformly charged domain. Thus, on the basis of the mean field theory, the increase of conductivity during dialysis was interpreted satisfactorily.

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Experimental Section

Sodium poly(styrenesulfonate) purchased from Aldrich has a number-average molecular weight of 226 000 and a Mhw/Mhn ratio of 2.57, measured by gel permeation chromatography (Testhigh Series III RI detector with H2O/CH3OH/NaNO3/ NaH2PO4buffer solution as a mobile phase). The calibration curve was obtained from the monodispersed poly(ethylene oxide) standard sample. The purification of NaPSS followed the procedure provided by ref 11 with some modifications.

In general, 1 g of NaPSS was dissolved in 0.2 M HCl (250 cm3_{). The solution was then dialyzed in a Spectra/Pro} regener-ated cellulose membrane tube (molecular weight cutoff at 3500) against water (ultrapure, 18.3 MΩ), which was purified by the Barnstead Easypure RF system. The dialysis tube was cleaned by heating in a detergent solution, followed by extensive rinsing with ultrapure water. The solution was dialyzed until the conductivity of the repeatedly refreshed water surrounding the dialysis tube closed to 1µS/cm. Then it was converted to the NaPSS solution by potentiometric titration with 0.1 M NaOH and dialyzed again to remove the excess NaOH. The solution was then freeze-dried to a solid product, which retained up to 10 wt % H2O as measured by thermogravimatric analyzer (Du Pont model 9900-954 TGA). To prepare the 0.1 g/L NaPSS aqueous solution samples for the in-situ conductivity measurements during dialysis, the freeze-dried NaPSS sample with corrected weight (subtracting the water content) was dissolved in the ultrapure water. For comparison, a 0.1 g/L unpurified NaPSS aqueous solution was also prepared. Two batches for dialysis of unpurified and purified NaPSS individually were prepared as follows: 100 mL of NaPSS solution was loaded into a cleaned dialysis tube, which was then immersed in a beaker containing 1.5 L of ultrapure water. During the dialysis, the beakers were ther-mostated at 30 °C with a deviation of 0.01 °C. One of the beakers with purified NaPSS sample was performed conduc-tivity measurement under a nitrogen atmosphere by using a conductivity cell, with platinized electrodes and a cell constant of 3.64 cm-1, immersed inside the dialysis tube. The other beaker with unpurified sample was performed conductivity measurement in open air. The signal was recorded on a Radiometer Copenhagen CDC-230 conductivity meter, and the data were automatically stored by the personal computer during dialysis.

The purified NaPSS was also used to measure the conduc-tivity at various concentrations. A concentration of 0.8 g/L, the highest concentration in this study, was prepared first. The other concentrations were prepared by the successive dilution with ultrapure water. The concentrations of NaPSS after prepared were rechecked by the spectrophotometric method at a wavelength of 261.5 nm.11 _{The conductivity} measurements were conducted in a beaker thermostated at 30 °C with a deviation of 0.01 °C.

Results

Figure 1 shows the specific conductivity of 0.1 g/L unpurified NaPSS solution as a function of the dialysis time. In the beginning, the conductivity decreased from 33 to 29 µS/cm rapidly and then turned to increase. After 50 h, the increasing rate was higher, and the conductivity versus time follows the S-pattern until 275 h and then stayed at 38.1µS/cm. The increase ratio by dialysis from the lowest to highest conductivity was 31.4%. Because the increased conductivity by dialysis was an unexpected result, the experiment was repeated several times, and the results were quite consistent. To ensure that the increase of conductivity was not caused by the condensing CO2and impurities, the next

experi-ment was conducted in a nitrogen atmosphere with the purified NaPSS.

Figure 2 shows the specific conductivity of 0.1 g/L purified NaPSS solution versus the dialysis time. In the

initial stage, the conductivity was decreased slightly from 14.9 to 14.7 µS/cm but picked up to 15.1 µS/cm before 25 h. In comparison with the unpurified one, the purification process has decreased the conductivity from 33µS/cm (Λ ) 68 S cm2_{/monomol) to 14.9}_{µS/cm (30.7}

S cm2_{/monomol), which was considered to be due to the}

removal of low molecular weight species of NaPSS and impurities. Moreover, the initial decrease of conductivity for purified NaPSS resulted from the diffusion of water into the solution across the dialysis tube, because the purified NaPSS had the molecular weight too high to diffuse out in a short time. After dialysis for 25 h, the conductivity was increased from 15.1 to about 20µS/ cm. Similar to the unpurified NaPSS, the increase of conductivity followed an S-pattern, and the total pro-cessing time to reach the maximum conductivity was 275 h. The increase ratio of specific conductivity by

Figure 1. Specific conductivity of 0.1 g/L unpurified NaPSS aqueous solution as a function of the dialysis time at 30 °C.

Figure 2. Specific conductivity of 0.1 g/L purified NaPSS aqueous solution as a function of the dialysis time at 30 °C. Macromolecules, Vol. 32, No. 14, 1999 Conductivity of Salt-Free Polyelectrolyte 4603

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dialysis is 36%, which was apparently caused by the osmotic pressure induced by dialysis.

Figure 3 shows the equivalent conductivity of purified NaPSS as a function of concentration at 30 °C. It may be divided into three regions. At c > 0.14 g/L, the equivalent conductivity barely changed. In the region 0.14 g/L > c > 0.02 g/L, the equivalent conductivity increased with decreasing concentration at a slope of ∼-45.7 (S cm2_{/monomol) (L/g). At c < 0.02 g/L, the}

equivalent conductivity was abruptly increased at a slope of∼-6990 (S cm2_{/monomol) (L/g).}

Discussion

According to de Gennes et al.,24_{the onset of}

concen-tration, cG*, that the fully stretched chain cannot orient

freely any more, may be defined by

where M is molecular weight of the polyelectrolyte, NA

is the Avogadro number, and l is the contour length of polyions. For NaPSS, l (Å) may be estimated by the following equation18

where M and Mmare molecular weights of the

polyelec-trolyte and of the monomer unit, respectively. From eqs 1 and 2, cG* of NaPSS in our system was estimated as

0.019 g/L. On the other hand, Odijk1 _{has defined a}

dilute/semidilute region in c* > c > cG*, where c* may

be estimated by the following equation

where Q ) e2_/_{RT is the Bjerrum length (e is the}

electronic charge and  is the dielectric constant of

water) and b is the charge spacing of the polyion chain.

For water at 30 °C, Q is 7.02 Å ( ) 78.5). For NaPSS

with a degree of sulfonation R ) 1, b is 2.46 Å. Thus, from eq 3, we obtain c* ) 0.147 g/L. By referring to Figure 3, the equivalent conductivity of the NaPSS solution was increased with decreasing concentration abruptly for c < cG*, the regime of which the polyions

have negligible interaction with each other.1_{For c* > c}

> cG* where the polyions remain rigid and interact

strongly, the equivalent conductivity was decreased with the concentration at a moderate slope. For c > c*, the equivalent conductivity was almost constant. Accord-ingly, the concentration of 0.1 g/L at which the dialysis was conducted in this study is in the dilute/semidilute region.

Uniformly Charged Domains. To interpret the

conductivity data, it is convenient for us to describe the swelling polyions and their aggregates as a uniformly charged domain and an undrained particle. Some of the dissociated counterions were trapped inside the domains and screened by the polyions. It is rationale to assume that those counterions bound inside the domains have no contribution to the conductivity. Only those outside the domains have a chance to move out and contribute to the conductance.

In coagulation, the free energy of mixing between polyions and water,∆Gm, is larger than zero. To simplify

the case, the effect of counterions on ∆Gm was not

included first. Owing to the Donnan-type equilibrium, ∆Gmwill be counterbalanced by the free energy

differ-ence,∆Gi, due to the difference in counterion

concentra-tion in and out the aggregated domains. That is,∆Gm

+ ∆Gi) 0. Therefore,

or

where (∆µ1)mis the chemical potential of water due to

the mixing and (µ1out)i- (µ1in)iis the chemical potential

difference of water resulting from the difference in counterion concentration in and out the aggregated domains. The free energy change of mixing between polyions and water in local-scale coagulation may be expressed by∆Gm) ∆Hm- T∆Sm, with

and

where n1and n2are the number of moles of water and

polyions in the solution, φ1 and φ2 are the volume

fractions of the water and polyions, m is the averaged number of polyions in an aggregated domain, andχ1is

the interaction parameter expressed as

where z is the number of contacts per water molecule,

V1and Vsare the molar volumes of water and segments

of polyions, and ∆ω12 is the change in energy of

Figure 3. Equivalent conductivity of purified NaPSS aqueous solution as a function of concentration at 30 °C.

(6) ∆Hm) RTχ1n1φ2 (7) χ1) z∆ω12V1 RTV_s (8)

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formation of the unlike contact pair expressed as

whereω12,ω11, andω22are the energy of bond formation

for those respective pair contacts. Because the solution forms aggregated domains, the interaction energyω22

between polymer segments must be substantially high. Some researchers have suggested that the hydrophobic backbone of polyelectrolytes might be responsible for the aggregation.25-28_{However, it has been questioned that,}

in the case of strongly charged polyelectrolytes, the effect of the poor solvation of a chain backbone should be fully overcompensated by the strong solvation of numerous charged groups along the chain.29 _{A “pure}

electrostatic interaction” mechanism of domain forma-tion was also suggested as often indicated that the intermediary of oppositely charged counterions is re-sponsible for the attractive interaction between polyions in aggregated domains.2,5,6,9 _{We believed that both}

mechanisms contribute toω22so that it may be divided

into two components,ω22,Wan der Waalandω22,electrostatic.

Thus, the chemical potential of water due to mixing of polyions and water in small-scale coagulations can be derived as follows,

and

where x is the number of segments in a polymer chain. On the other hand, to estimate the (µ1out)i - (µ1in)i,

the following conditions may be assumed: (1) the molar volume of the water is equal to that of the counterions (Na+), and (2) the osmotic coefficient, ζ, may be

ex-pressed by30

and

whereξ is the charge density parameter and Q is the Bjerrum length. If f is the ratio of counterions that diffused out of the aggregated domains compared to the total dissociated counterions, then

whereφcinis the volume fraction of counterions inside

the domains, expressed as

if n1 . n2xV1/Vs and φcout is the volume fraction of

counterions outside the domains, expressed as

where R is the degree of sulfonation in NaPSS. By inserting eqs 10, 14, 15, and 16 into eq 5, f may be expressed as

Thus, less than half of the dissociated counterions diffused out of the polyion-aggregated domains. A very recent paper by Schmitz31_{has also predicted that the}

counterion concentration within a cluster is greater than the external concentration according to the juxtaposition of potential fields for multibody interaction systems. From eq 17, f is increased with the osmotic coefficientζ

and the degree of sulfonation R. Manning30_{pointed out}

thatζ is the fraction of couterions that is free and does not interact with the polyion including the Debye-Huckel interactions. According to his ref 31,ζ ) 0.175

(ξ ) 2.85) may be obtained for the salt-free NaPSS (R ∼ 1) aqueous solution at 30 °C. Besides, f decreased with increasingχ1(orω22),φ2, m, and x.

The equivalent conductivity of polyelectrolyte solu-tion,Λ (S cm2_{/monomol), may be expessed as}

where J is a faraday of electricity, uois the mobility of

counterions which is the same as in solutions of simple electrolytes, and uaggregate is the average mobility of

segments in polyion aggregates. If the coagulated domains of polyions were treated as a spherical particle carrying q ) mxRfe negative charges and only electro-phoretic mobility was considered here, uaggregatemay be

derived by balancing the electric force (qE) with the viscous force (6πηRv):32

whereη is the viscosity of water and R is the radius of

domains. Equation 20 has many different forms de-pending on drainage of the particle33_{and the thickness}

of electrical double layers in the particle surface.32_Since

we have assumed that the particles are an undrained sphere andκR , 1 (κ is the reciprocal Debye screening

length) can be applied to the salt-free polyelectrolyte solution (low electric charge density in solution) in the dilute/sedilute region, eq 20 may be applied to our system. Thus, by inserting eqs 17 and 20 into eq 18,Λ may be expressed by

Equation 21 indicates that the equivalent conductivity is decreased with the concentration and molecular weight of polyelectrolytes, the trend of which is consis-∆ω12) ω12 -1 /₂(ω₁₁+ ω₂₂) (9) (∆µ1)m) ∂∆Gm ∂n1 ) RT

[

(χ1-1/2)φ2 2_- V1 mxV_sφ2

]

(10) χ1> 1 2+ V₁ mxV_sφ2 (11) ζ ) (2ξ)-1_, _{ξ ) Q/b ) e}2 /kTb > 1 (12) ζ ) 1 - ξ 2, ξ < 1 (13) (µ₁out)_i- (µ₁in)_i= RTζ(φ_cin- φ_cout) (14) φcin₎ mxR(1 - f) (n₁m/n₂) + (mxV₁/V_s) + mxR(1 - f) =n2xR(1 - f) n1 (15) φc out₌n2xRf n1 (16) f )1 2- 12Rζ

[

(

χ1- 12

)

]

}

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tent with the experimental data reported by Wandrey34

for the NaPSS aqueous solution in the dilute/semidilute region. By referring to Figure 3, the slope that the equivalent conductivity decreased with concentration may be expressed by

where F is the mass density of NaPSS. The equivalent conductivity of sodium ion at 30 °C is 55.6 (S cm2_/mol).35

If F∼ 1 (the mass density of NaPSS, g/cm3_{) and R}_{∼ 1}

are assumed, we obtained f∼ 0.435, uaggrerage∼ 0.117uo,

andχ1 ∼ 227 by inserting φ ) 0.0001 and Λ ) 30.7 (S

cm2_{/monomol) into eqs 21 and 22. Apparently,}_ω 22,elec-trostatic should be responsible for the high value ofχ1.

ω22,Wan der Waal contributed little toχ1. The early work

on the phase separation of polyelectrolyte solutions by Michaeli et al.36 _{also suggested that the electrical}

interactions only may suffice to bring about phase separation. On the other hand, eq 21 also suggests that the equivalent conductivity be decreased with the domain size and the number of polyions in an ag-gregated domain.

The foregoing derivation has eliminated the effects of counterions on∆Gm, although their contributions to

∆Gmmay be included in the high value ofχ1. To consider

the dissociated counterions as the third component that is assumed to be well dispersed in the aqueous solution, eq 10 may be modified as

whereχijis pair interaction parameters corresponding toχ1in eq 8. Thus,

As a result, the conductivity may be modified as

The trends that f andΛ decreased with m but increased with R andζ are similar to those indicated by eqs 17 and 21. Besides, f is also less than one-half.

Effect of Dialysis on Conductivity and Aggrega-tion. When the polyelectrolyte solution is dialyzed

against water, a chemical potential difference of water, (µ1w)d- (µ1s)d, across the membrane will be produced,

resulting from the difference of number of polyion domains and concentration of unbound counterions between solution and water side of the membrane. Since the molecular weight of polyions was much higher than the cutoff molecular weight, few polyions with their

counterions diffused across the membrane to the water side. Thus, we assumed that their concentration in the water side can be neglected so that (µ1w)d ) µ1o.

However, water can diffuse across the membrane to dilute the solution. As a result, n1′, the moles of water

molecules in the solution after dialysis, will be larger than n1 andφ2′(the volume fraction of polyions after

dialysis) will be slightly less thanφ2. Thus, (µ1w)d- (µ1s)d

may be expressed by

Therefore, the osmotic pressureπ1that applied to the

solution will be

Since the total pressure on the solution has been increased by an osmotic pressure, the chemical potential of water in the system will be changed to

i.e.,

By inserting eqs 14, 15, 16, 23, and 26 into eq 29, we obtain

By rearrangement of eq 30, f′may be expressed as

1 - V1 mxV_s

)

φ2 φ3 + (χ12φ2+ χ13φ3)φ2+ φ3 φ3 -χ23φ2V1 xV_s

Concluding Remarks

Uniformly charged aggregation model of polyions is a crude model, because it neglects many important factors affecting the coagulation such as volume shrink-age of polyelectrolyte solution during mixing and drain-age of domains. However, it has successfully interpreted the unsolved conductivity problems of salt-free poly-electroyles in the dilute/semidilute region, such as the increase of equivalent conductivity by decreasing the concentration and molecular weight. It also clarifies the dispute about the nature of attractive interactions between the polymer segments in aggregates. In addi-tion, it provides another interpretation for the long-disputed coagulation phenomenon of polyions observed by dynamic light scattering. Above all, it interprets the observed unexpected phenomenon that the conductivity of salt-free polyelectrolyte solution is increased by dialysis in the dilute/semidilute region.

Acknowledgment. The authors acknowledge the

financial support of the National Science Council in Taiwan, Republic of China, through Grant NSC 87-2216-E-002-001.

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