Crossover Behavior of the Viscosity of Dilute and Semidilute

5736

J. Phys. Chem. B 2009, 113, 5736–5745

Crossover Behavior of the Viscosity of Dilute and Semidilute Polyelectrolyte Solutions B. Ashok† and M. Muthukumar* Department of Physics and Polymer Science & Engineering Department, Materials Research Science & Engineering Center, UniVersity of Massachusetts, Amherst, Massachusetts 01003 ReceiVed: September 10, 2008; ReVised Manuscript ReceiVed: February 12, 2009

We present theoretical results to describe the crossover behavior of the viscosity of dilute and semidilute polyelectrolyte solutions. We have resolved the puzzle of the experimentally observed nonmonotonic dependence of the reduced viscosity of polyelectrolyte solutions on the polymer concentration below the entanglement limit. We use a combination of the theory of triple screening of excluded volume, electrostatic, and hydrodynamic interactions in polyelectrolyte solutions and a thermodynamic theory of interchain interactions in obtaining the crossover formulas. The calculated results are in qualitative agreement with data in the experimental literature. I. Introduction The dynamical properties of a rigid sphere suspended in a solvent are very well-known. Many contributions have been made, from as early as Einstein’s work in 19061 to the theory of suspensions at finite concentration (for example, refs 2-11 and references therein). A natural extension of this would be a theory for the dynamics of macromolecules. At sufficiently large time scales and lengths, the dynamical behavior observed of polymer solutions is characteristic of the entire chain, and one does not observe any difference in the physics when compared to the simplest case of spherical Brownian particles. At dimensions corresponding to those within the polymer coil, these observations change. There are three regimes associated with the chain dynamics of a polymer chain in the dilute solution limit. The first corresponds to a short time regime and is characteristic of the properties of the monomer. Next we have the long time regime as described above, corresponding to the whole chain. Then we have the interesting intermediate case where the connectivity of the polymer chain influences the dynamical behavior. This last regime is the one where we seek to explore the global behavior of the chain, as, unlike in the short time regime, the chemistry and the particular chemical behavior of the monomers do not have any role in determining the dynamics, which are instead dominated by the connectivity of the chain. The theory for the viscosity and frictional properties of dilute polymer solutions has also been a very active field for a long time (see, for example, ref 12 and references therein). Over the past couple of decades, many workers have contributed to our understanding of the finite concentration regime as well.13-20 There is a vast and rich literature, theoretical as well as simulational and experimental, on the dynamics and rheological properties of polymer solutions (see, for instance, ref 17 and the reviews by Berry and Fox, Stockmayer, and Ferry21 and references therein). The situation for polyelectrolyte solutions has been much less satisfactory until recently (see, for example, refs 22-33 and * To whom correspondence should be addressed. E-mail: muthu@ polysci.umass.edu. † Current address: Advanced Centre for Research in High Energy Materials (ACRHEM), University of Hyderabad, Central University P.O., Gachibowli, Hyderabad-500046, India.

references therein). Polyelectrolyte viscosity seems to display apparently anomalous behavior when considered in comparison to that of uncharged polymers, and the literature reflects the attempts in explaining this, with several approaches being used for understanding comprehensively the dynamics of a polyelectrolyte solution. Concise reviews of the topic can be found by Fo¨rster and Schmidt25 and Barrat and Joanny,26 among others. More recent reviews include those by Holm et al.27 and Dobrynin and Rubinstein.28 Polyelectrolytes play an important role in industrial applications and even more importantly are present everywhere in the natural world, for example, as DNA and in proteins. The importance of correctly understanding and predicting polyelectrolyte behavior cannot therefore be overstated. There are many factors that influence the dynamics of a polymer solution. In the dilute case, hydrodynamic interaction predominates. The excluded volume interaction, as also the electrostatic interaction in the case of polyelectrolytes, is an additional feature. Intrachain entanglements also may be present, but we disregard these in considering our system. In the references mentioned above and in the literature cited therein, varied approaches have been taken to tackling the problem of a theory for polyelectrolyte solutions. One method has been to think of the polyelectrolyte chain as a necklace of blobs or globules strung together, with those of smaller size being located toward the middle of the chain.28,32 Another approach, which we follow, is to use a field-theoretical approach33,34 to calculate the chain configuration and dynamic properties of the polyelectrolyte system. Methods used by other researchers have been briefly mentioned in section III. Further details can be found in the aforementioned review articles. In this paper, we present the calculations we have performed using the multiple-scattering effective medium theory outlined in ref 33. We have explicitly calculated the specific viscosity of polymer solutions and polyelectrolyte solutions. We have additionally included a correction to the viscosity of polyelectrolyte solutions to account for the modification in the effective interaction potential being felt by the polyelectrolyte chains due to chain interactions. In the case of Θ solutions, remarkable similarity is observed between intrinsic viscosity values calculated theoretically and values obtained from the experimental literature. The transition from Zimm to Rouse regimes is clearly observed in our calculation. The first extensive use has been

10.1021/jp8080589 CCC: $40.75  2009 American Chemical Society Published on Web 04/03/2009

Viscosity of Polyelectrolyte Solutions

J. Phys. Chem. B, Vol. 113, No. 17, 2009 5737

made of Muthukumar’s double-screening theory31,34 in explicit calculation of the effective persistence length for polyelectrolyte chains in solution for both dilute and semidilute polymer concentration regimes and used as one of the inputs to numerically calculate the specific viscosity. It should be noted at this point that the original theory of Muthukumar on the dynamics of polyelectrolyte solutions,33 while capable of a full description, addressed only the asymptotic regimes of dilute (c , c*) and semidilute (c > c*) concentrations, with only scaling arguments in the entangled regime. The crossover between the dilute and the semidilute regimes was only sketched, with the actual crossover not being addressed. This crossover depends crucially on the interchain interactions for c < c*. This is what we address here. We explicitly include interchain interactions by means of an approximate radial distribution function and show in section IV that this gives qualitatively realistic behavior in the presence of added salt. The main new results in this paper are that these are the first explicit calculations using the double-screening theory34 in conjunction with hydrodynamic interactions to investigate the dilute to semidilute crossover regime in polyelectrolyte solutions. We provide numerical prefactors in describing the crossover regime in terms of polymer concentration, salt concentration, and degree of ionization and degree of polymerization of flexible polyelectrolytes. We show generally that the reduced viscosity exhibits a peak with respect to polymer concentration for the salt-free polyelectrolyte solution as well as the solution with added salt. This nonmonotonicity is derived to be a consequence of dynamical features of hydrodynamic screening and thermodynamic features of interchain interactions and screening of excluded volume and electrostatic interactions. We argue from our work that the explanation attributing the reduced viscosity to the polyelectrolyte coil expansion in the presence of added salt and requiring the salt-free case to be without a peak is not satisfactory. Our theory is different from an argument32 in the literature that the reduced viscosity has a peak at the polymer concentration cpP ) 2cs/R (cs being the salt concentration and R being the degree of ionization). We obtain a dependence of the polyelectrolyte peak concentration on the degree of ionization to be proportional to R-2.41.

(corresponding to the ith segment of chain R) is given at time t by

3kBT N-1 l2

∑ (2δi,j - δi,j+1 - δi,j-1)RRj + j)0

n

Ri

This includes a contribution from chain connectivity (the first (Rouse) term) and also the force due to the interaction potential V between the ith segment of the R chain and other segments of all chains. The interaction potential V between any two segments is given by

V(Rij) Zp2lB ) wδ(Rij) + exp(-κ|Rij |) kBT |Rij |

-η0∇ v(r) + ∇p(r) ) F(r)

(1)

n

-η0∇2v(r) + ∇p(r) ) F(r) +

N-1

∑ ∑ δ(r - RRi)σRi

(4)

R)1 i)0

Here, RRi is the position vector of the ith segment of chain R and σRi is the force exerted on the fluid by this chain segment, i.e., corresponding to the perturbation of the velocity flow on introduction of a polymer chain due to scattering off the chain segments. Therefore

|

r

|

(5) r

where G is the usual Oseen tensor. Assuming a no-slip boundary condition

∂RRi ) v(RRi) ∂t

(6)

With this constraint, the first of the coupled equations of motion can be written as

3kBT n-1 l2

∑ (2δi,j - δi,j+1 - δi,j-1)RRj + j)0

n

N-1

∑ ∑ ∇R V(RRi - RRj) ) FRi - σRi Ri

where η0 is the shear viscosity of the solvent, p is the pressure, and F(r) is the force arising from any potential field in the solution. When considering a system of n polyelectrolyte chains of N segments, the equation of motion for the position vector RRi

(3)

w is a short-ranged excluded volume interaction, lB ) e2/kBTε is the Bjerrum length, κ is the inverse Debye screening length, and eZp is the charge on each segment. The introduction of an object into the fluid (here, a polymer chain) causes the velocity field to be perturbed, which can be thought of as being scattered off the object. The equation of motion therefore becomes

v(r) ) G * F + G * σRi

For any description of a hydrodynamic system, a knowledge of the velocity field v(r) of the fluid is required, r corresponding to the position vector of any point in the fluid. This can be achieved by using the multiple-scattering technique. In this section we briefly review the effective medium theory used. Further details can be found in the original references.13,31,36 The assumptions made in considering the system are that the dimensions of the polymer chains are much larger than those of the solvent particles and that we can think of the solvent as a homogeneous, isothermal, and viscous continuum. In the absence of the polymer, the velocity field in the fluid can therefore be adequately described by the linearized Navier-Stokes equation

(2)

β)1 j)0

II. Model of the System and Methodology Used

2

N-1

∑ ∑ ∇R V(RRi - RRj) ) FRi

(7)

β)1 j)0

σRi can be eliminated and averaging done over all possible distributions of polymer chains to obtain the effective equation of motion for the polymer solution:

5738 J. Phys. Chem. B, Vol. 113, No. 17, 2009

-η0∇2〈v(r)〉 + ∇〈p(r)〉 +

Ashok and Muthukumar

∫ dr′ Σ(r - r′)〈v(r′)〉 ) F(r) (8)

where

Σ(r, r′)〈v(r′)〉 ) 〈σ(r)〉 Angular brackets denote averaging over all chain configurations. The self-energy of the fluid, the operator Σ(r,r′), contains information about the modified viscosity of the solution. Denoting the Fourier transform of Σ(r - r′) by Σ(k)

Σ(k) )

∫ dr Σ(r - r′) exp[ik(r - r′)]

nonzero concentrations of polyelectrolyte solutions and significantly affect the value of the effective persistence length for the polyelectrolyte chain which is itself an essential component in our work. This double screening must therefore be first considered before we can incorporate any triple screening, i.e., the effect of hydrodynamic interactions, in the dynamics of polymers. It can be shown that the free energy F of our system, consisting of n polyelectrolyte chains with N segments per chain, nc counterions, nγ ions of species γ of added salt, and ns solvent molecules, in volume Ω, with R degree of ionization, is34

( )

exp -

F ) kBT

(9)

n!nc!ns! [η0k 1 - Σ(k)]u(k) - ikp(k) ) F(k) 2

∫ ∏ D[RR] ∫

∏γ nγ !

{

and

(10)

R)1

n

exp -

∑ ∫0

3 2l R)1 n

L

dsR

(

n

∑∑∫

(11) n

∑ ∫0

1 l R)1

The change in shear viscosity of the solution due to the polymer chains is clearly given by -2

η - η0 ) lim k (1 - kˆkˆ)Σ(k) kf0

Σ(k) )

1cpL πN

J(q) )

∫

S(k, q) )

∫

∫

∞

2π/L

(

∞

-∞

2

-

dsR

d(s - s′) exp(iq(s - s′))〈exp(ik[R(s) - R(s′)])〉

(13)

from which one can calculate the change in viscosity and other related quantities. S(k,q) is the Fourier transform of the static structure factor. The effective medium theory for the equilibrium situation gives

〈exp(ik[R(s) - R(s′)])〉 ≈ exp(-k2l1 |s - s′|/6) (14) with l1 being the effective Kuhn length and given by 〈[R(L) R(0)]2〉 ≡ Ll1. The effective Kuhn length l1 plays a very important part in all our calculations. It is a nontrivial function of the excluded volume parameter w, strength of the electrostatic interaction wc, polymer concentration cp, chain length L, and inverse Debye length κ for the system. The effects of the bare excluded volume interaction as well as the screened Debye-Hu¨ckel potential are screened in

(15)

∑ Ups[RR(sR) - ri] -

i)1 ns

∑ ∑ Uss(ri - rj) -

R l R)1

∑ ∫0

L

dsR

j)1 nc+Σγnγ

∑

Vpi(RR(sR) - ri) -

i)1

1 2

nc+Σγnγ nc+Σγnγ

∑ i)1

∑

}

Vij(ri - rj)

j)1

Here

Zp2e2R2 1 V0(r) ) wpp(1 - R) δ(r) + εkBT r

)

dri ×

i

∫

2

jj dj j2 (2π)3 [1η0j2 + Σ(j)] 1-

L

1 2 i)1 n

dq S(k, q) J-1(q)

)

nγ

ns

ns

(12)

Using the effective medium theory, σ(k) is self-consistently evaluated with the assumption that Σ(k) from n chains is n times that from a single chain. We denote the polymer concentration by cp, the chain contour length by L, and the number of chain segments in a chain by N. Summing over Rouse modes q ) 2πP/L, (P ) 1, 2, 3,...), one eventually gets15,33

∂RR(sR) ∂sR

γ

L L 1 dsR 0 dsβ × 2 0 2l R)1 β)1 V0[RR(sR) - Rβ(sβ)] -

where

u(r′) ≡ 〈v(r′)〉

∑ ∏

nc+ns+

n

1

(16)

is the interaction energy between two chain segments separated by a distance r, with wpp being the intersegment excluded volume interaction strength, Uab ≡ wabδ(r) being short-range excluded volume interactions between solvent molecules and between the solvent and uncharged segments, and Vkl(r) being the electrostatic interaction between charged segments and ions. The free energy can be written explicitly as a sum of contributions from the background, Fb, and from the polymer chains, Fp, so that F ) Fb + Fp, with

Fb nγ ln Fγ - ns - nc ) ns ln Fs + nc ln Fc + kbT γ Ωκ3 1 nγ + ΩwssFs2 + ΩwpsFFs (17) 12π 2 γ

∑

∑

with Fc and Fγ being the number densities of the counterions and γth salt ion, F ) nN/Ω, Fs ) ns/Ω, and ln m! rewritten using Stirling’s approximation. The polymer chain contribution is given by Fp:


( )

exp -

Fp 1 ) kBT n!

∫ ∏ D[RR] R n

∑ ∫0

3 exp(2l R)1 n

n

∑ ∫0

1 2l2 R)1

L

L

dsR

∑ ∫0

L


〈[RR(L) - RR(0)2]〉 ) Ll1

×

dsR

(

∂RR(sR) ∂sR

)

2

In eq 15 ∆k is the effective interaction:

-

dsβ V[RR(sR) - Rβ(sβ)])

(

n

∑ ∫0

1 2l2 R)1

L

n

∑ ∫0

dsR

L

(18)

)

dsβ V[RR(sR) - Rβ(sβ)] )

β)1

[ ∫ D[φ] exp{-(i/l) ∑

n R)1

∫0L dsR φ[RR(sR)]

-

∫ dr dr′ φ(r) V-1(r - r′) φ(r′)} ]/ [ ∫ D[φ] exp{- 21 ∫ dr dr′ φ(r) V-1(r - r′) φ(r′)}] 1 2

(19)

so that

( )

exp -

[∫

Fp 1 ) D[φ] [G(φ)]n × kbT n! 1 exp dr dr′ φ(r) V-1(r - r′) φ(r′)} / 2 1 D[φ] exp{dr dr′ φ(r) V-1(r - r′) φ(r′)} (20) 2

[∫

{

∫

]

∫

]

with

G(φ) ≡

∫ D[R] exp{- 2l3 ∫0L ds ( ∂R(s) ∂s ) i L ∫ ds φ[R(s)]} l 0 2

(21)

The distribution function for n chains with each chain in the potential field created by other chains, then follows to be16

P(φ, R) ≡

{

∏ exp R

i l

-

3 2l

∫0L ds

(

∂RR(s) ∂s

)

2

-

}

∫0L ds ∫k φkeikR (s) - 21 ∫k φk2Vk-1 R

(22)

In the variational scheme of Muthukumar and Edwards,16 this is modeled by an effective Gaussian distribution, P0(φ,R)

P0(φ, R) ≡ exp(-H0) )

{

∏ exp R

-

∆k )

β)1

where V(R), the segment-segment interaction, includes both the short-range excluded volume interaction and the screened Coulombic interaction. To decouple chains from this equation, a field variable, φ, is introduced and using the Hubbard-Stratanovich transformation yields

exp -

3 2

∫0L dsl1

(

∂RR(s) ∂s

using an effective step length, l1

)

2

-

1 2

(24)

}

∫ki φk2∆k-1

(23)

2 1 D[R] D[φ] φk P(φ, R) Ω D[R] D[φ] P(φ, R)

(25)

Using a variational procedure as in ref 29 and maximizing the chain entropy eventually leads to the following expression for l1:

l15/2

(

)

1 2 1 ) 2 l l1 πl

6q B (l , q, w, w , κ, c ) λ 1

c

p

(26)

where Bλ is a complicated function of w, wc, κ, cp, and l1.34 We compute l1 and the radius of gyration Rg as a function of the polymer concentration, obtained using the double-screening theory. Figure 1a is a plot of l1 as a function of cp. As expected, the effective Kuhn length is maximum at low cp. With increasing polymer concentrations, the renormalized Kuhn length decreases in value until it equals that of the bare Kuhn length l. A comparison of the value of Rg calculated using the double-screening theory with that from simulation data points obtained by Liu et al.37 showed the two to be in good agreement. This is shown explicitly in Figure 1b. As has already been mentioned, the effects of electrostatic and excluded volume interaction are very significant and greatly affect the value of l1. It is therefore not surprising that l1 can be even a few orders of magnitude larger than the bare Kuhn length, depending upon the extent to which the interactions are screened in the polyelectrolyte solution. Also, the hydrodynamic screening length ξ is related to l1 through the expression33 ξ )(2/π)(cpll1)-1. Its behavior is intimately related to the Debye screening length κ-1. When κRg > 1, ξ ) (2/π)(1/33/4)(w + (4πlb/κ2))-1/4cp-3/4l-1/2. When κRg < 1, we find that ξ ) (8/(3)π)(π/62)2/3(l/4πlb)1/6cp-1/2l-1/2. Since κ2 is proportional to the counterion and salt concentrations, and the presence of the excluded volume interaction yields another screening length, ξe, that is proportional to cp-3/4 in the semidilute regime, apart from its dependence on κ, it becomes nontrivial to distinguish the contributions from each of the three screening effects separately. Further details may be found in the earlier work by Muthukumar.33,34 In calculating the renormalized Kuhn length l1(q), it has been implicitly assumed above that the first mode q ) 2π/L is predominant and that the other modes can be neglected.31 Performing an explicit calculation for flexible polyelectrolyte solutions of l1 as a function of q reconfirmed the assumption made in this paper of the predominance of the first mode over the others as a valid one. The value of the dielectric constant of water used in our calculations is ε ) 78.4, with a Bjerrum length lb ) 7.044 Å, at a temperature of 25 °C. Our results corresponding to the zero-salt case assume that no dissolved salt ions are present and there is no intrinsic solvent ionic strength. To summarize, in this section we have outlined the theory behind our calculations of polymer viscosity. Our formalism includes the effects of the excluded volume interaction, the electrostatic Debye-Hu¨ckel potential, and the hydrodynamic interactions. Uniform swelling of the polyelectrolyte chain throughout its contour length is assumed, and the new effective Kuhn length l1 for the chain in the presence of excluded volume and electrostatic interactions is calculated. This is used in



Figure 2. Specific viscosity ηsp as a function of the ratio of cp to the overlap concentration [η]cp for a Θ solution, showing transition from the Zimm regime (slope 1) to the Rouse regime (slope 2). Comparison with the experimental data (circles) for the Θ solution (ref 44) indicates a good fit for the theoretical points. Theory data have been explicitly calculated for various molecular weights, without any free, adjustable parameters.

Figure 1. (a) Renormalized Kuhn length l1 vs polymer concentration cp (N ) 5000, Mw ) 6.5208 × 106, R ) 0.21). (b) Comparative plots of Rg vs cp, calculated with double-screening interaction (solid points with a line) and simulation data (ref 37).

obtaining the viscosity of the solution within the effective medium theory. III. Viscosity of Polyelectrolyte Solutions The viscosity properties of polyelectrolytes are somewhat different from those of uncharged polymers. A review of polyelectrolyte solutions can be found, for example, in refs 25 and 26. In polyelectrolyte solutions, one also has to consider the concentration of the added salt, cs, apart from the polymer solution concentration cp, as another variable affecting the specific viscosity, ηsp ≡ (η - η0)/η0 (where η0 is the solvent viscosity). While for both salt-free and salty polyelectrolyte solutions this viscosity change is monotonic with concentration, the difference in behavior between the two is often dramatized by plotting, instead, the reduced viscosity ηred ≡ (η - η0)/η0cp ) ηsp/cp. In the case of polyelectrolyte solutions with zero or low added salt, a plot of ηred versus cp will show a peak.25,26,38-40 Fuoss’s interpretation for the increase in reduced viscosity with a decrease in polyelectrolyte concentration in the salt-free case was the premise that this was caused by an expansion of the polyelectrolyte coil.41 This explanation is unsatisfactory since

this maximal behavior of the reduced viscosity plot is also observed in suspensions of latex spheres.42 The Witten-Pincus approach considered the polyelectrolyte chains to be interpenetrating coils in seeking to explain observed polyelectrolyte viscosity behavior.30 Cohen, Priel, and Rabin29 have based their model explaining the experimental results on the work of Hess and Klein on the mode-mode coupling between strongly charged particles.43 A logical explanation for the occurrence of the reduced viscosity maximum is that because while ηsp is proportional to cp1/2 in the Rouse regime where the hydrodynamic interaction is screened, causing a decrease in the reduced viscosity vs cp plot, there is an effect of increasing the specific viscosity at low cp in the dilute Zimm regime, where the effect of interchain electrostatic and pairwise interactions between chains becomes highly important. The corresponding graph for high salt concentrations will not show this peak as ηsp is now proportional to c5/4, so that a log-log plot of ηred vs cp will have a positive slope (of 0.25).33 In the case of uncharged polymer solutions, the ηred plot is not so dramatic, since (η - η0)/η0 is directly proportional to cp at low polymer concentrations (Zimm regime) and to cp2 on crossover to the Rouse regime, for Θ solutions. This can be seen in Figure 2, where specific viscosity values are plotted as a function of cp[η]. The square data points represent values calculated from our theory for Θ solutions of polystyrene, while the round data points correspond to data from the experimental literature44 for Θ solutions of polystyrene in cyclohexane. As can be seen, there is good correspondence between theory and experiment. A representative plot of our computed values of the specific viscosity of polyelectrolyte solutions for the highsalt case is shown in Figure 3. At high salt concentrations, electrostatic interactions get screened out, and the solution begins to behave like an uncharged polymer solution. There is a significant point to be noted in calculating the viscosity of polyelectrolytes when using the procedure outlined in the previous section. The electrostatic repulsion between polyelectrolyte chains needs to be accounted for explicitly in some way or in the form of the radial distribution function for the polymers, in the multiple scattering formalism, by including the interaction potential in eq 14. The averages being evaluated would then have to be done with respect to this distribution



Figure 3. Reduced viscosity ηsp/cp as a function of cp for high added salt concentration (cs ) 0.8 M, T ) 298 K, N ) 1302.5, Mw ) 1.7 × 106, bare Kuhn length b ) 13.8 × 10- 8 cm, R ) 0.3).

function. Hence, an appropriate and correct choice of the structure factor and a proper accounting of interchain interactions should enable one to theoretically reproduce the experimental observations. In Figure 4a we plot reduced viscosities for different salt concentrations, without any explicit interchain interaction effect having been included in our calculations. The figure therefore corresponds to reduced viscosity taking only the intrachain contribution into account. Our theoretically calculated result is qualitatively consistent with the results obtained by the semiempirical approach of Nishida et al.,35 who deduce the plots for the intrachain contribution to the reduced viscosity of polyelectrolyte solutions by subtracting a calculated interchain contribution from the experimental values. The characteristic peak observed in polyelectrolyte specific viscosity plots is thus missing in Figure 4a, proving our central point that interchain interactions are the source for the peak. The overlap concentration c* for the different data sets is shown in the plots, c* being defined as the value of cp such that (4/3)πRg3F ) N, F being the corresponding monomer number density and Rg2 ) Nll1. One explanation that has been given32 for the reduced viscosity behavior is that, with increased dilution, the polymer size increases accompanied by an increase in the Rouse relaxation time and thus an increase in the reduced viscosity. Reaching the overlap concentration causes the relaxation time and reduced viscosity ηred to plateau. If salt is added to the solution, this causes screening of electrostatic interactions when cp ) 2cs/R (R being the degree of ionization), leading to a steep increase in the reduced viscosity at lower cp, with the plot peaking at 2cs/R. This reasoning, however, does not explain why, at zero salt, ηred vs cp plots should show a peak, nor does it explain why a suspension of latex spheres would show a peak which is observed experimentally.42 While the data of Dou and Colby45 on the reduced viscosity of a polyelectrolyte with zero salt do not show a peak when plotted against the polymer concentration, the experiments of Vink46 and Ganter, Milas, and Rinaudo40 of zero-salt polyelectrolyte solutions do show the oftreported ηred peak. Solutions of NaPSS and poly(styrenesulfonic acid) (HPSS) comprise the subject of Vink’s investigation of the rheology of dilute polyelectrolyte solutions (in particular, salt-free polyelectrolyte solutions). Vink states that while electrostatic expansion of polyions is given as a possible reason for the rapid increase in reduced viscosity, intermolecular electrostatic interactions between similarly charged polyions would also be present. As HPSS, a

Figure 4. (a) Uncorrected reduced viscosity for a polyelectrolyte solution in the absence of any explicit interchain potential corrections (N ) 1532, Mw ) 2 × 106, bare Kuhn length b ) 13.8 × 10- 8 cm, R ) 0.3) for different salt concentrations. Data points on the x axis shown as triangles correspond to the respective overlap concentrations c* for each of the three sets (from left to right, starting with cs ) 0, with increasing cs to the right). (b) Reduced viscosity as a function of cp for different salt concentrations, with inclusion of a correction due to interchain interactions (N ) 1532, Mw ) 2 × 106, bare Kuhn length b ) 13.8 × 10- 8 cm, R ) 0.3). Data points on the x axis shown as triangles correspond to the respective overlap concentrations c* for each of the three sets (from left to right, starting with cs ) 0, with increasing cs to the right).

strong acid, has the same charge density as NaPSS, it is expected to show similar polyion electrostatic expansion. In the dilute limit, electrostatic interactions between chains become important, especially in salt-free solutions. Vink has observed peaking behavior both in HPSS solutions and in NaPSS solutions, more so in the former (HPSS).46 He notes that polyacid solutions can be considered as being free of co-ions as self-ionization of water is suppressed in these, and they can be thought of as perfectly saltfree. The larger HPSS reduced viscosity peak (as compared to that of NaPSS) is a reflection of the fact that hydrolysis of NaPSS solutions can take place, causing NaOH formation to occur, so that trace amounts of salt can form in the NaPSS solutions, affecting the reduced viscosity detrimentally. It should be kept in mind that polyelectrolyte solutions made using freshly distilled water exposed to air cannot strictly be considered to be entirely salt-free as carbonic acid is known to form in water due to exposure to air. In ref 33, the Zimm regime is obtained from an approximated form, to leading order of cp, of the fluid self-energy Σ. At low



cp, this indicates a near plateau-like reduced viscosity. Our result, in the absence of interactions, is consistent with this. At the risk of repetition, lest the point be missed in a cursory reading, we reiterate the fact that interchain interaction was not discussed explicitly in earlier theory33 for cp < cp*, but was limited to a sketch of the expected behavior to the leading virial correction, not a numerically calculated plot. The position of the peak for reduced viscosity is critically dependent on the interchain structure factor. Keeping these in mind, it seems clear that our procedure of requiring a correction factor to the viscosity to be incorporated to account for interchain interactions (which leads to an increase of the form of eq 30 in the next section) is the correct one, as will now be demonstrated.

The distribution function g(r) in eq 27 can be approximated by the pair potential U(r)

g(r) ) exp[-U(r)/kBT]

(28)

Detailed expressions for the radial distribution function have been derived by many authors, including Fixman48 and Koyama,49 among others. In this paper, the friction coefficient ζ in eq 27 is approximated by using the expression for the translational friction coefficient for dilute polymer solutions. This is a concentration-dependent variable, and we calculate this using an expression analogous to the expression derived in ref 14:

IV. Incorporation of Interchain Interactions The interchain interactions of the polyelectrolytes have a profound impact on the viscous behavior of the solution. They cause the observed peaking behavior of the reduced viscosity when plotted as a function of the polymer concentration. Other workers such as Cohen et al.29 and Borsali et al.24 have attacked the problem of the calculation of the viscosity of polyelectrolyte solutions, by using a mode-mode coupling approach, with positive results. Cohen, Priel, and Rabin neglected hydrodynamic effects entirely and also approximated the static structure factor to unity in their evaluation. The effect of electrostatic screening at higher salt concentrations is also not reflected in their approach.29 Borsali, Vilgis, and Benmouna included hydrodynamic effects in their mode-mode coupling approach and treated the Rouse and Zimm models separately. In their work they assumed a regime with weak electrostatic interactions, employed the usual Stokes-Einstein relation for the monomer friction coefficient, and made a decision of using a constant radius of gyration in their calculation.24 There are several ways of incorporating the effect of interchain or interparticle interactions on the transport coefficients. This is usually done by inclusion of a radial distribution function for the pair correlation function for the interacting particles in the system. For a polyelectrolyte system, the relation between the change in viscosity and interchain structure factor can be obtained by several methods. We have assumed throughout that the polymer coil undergoes uniform expansion. We therefore chose charged spheres as a model for interchain interactions. One of the procedures first suggested was one by Rice and Kirkwood47 for the contribution to the shear viscosity of particles from the presence of an interparticle interaction potential U(r). They obtained

ηinter ) (MFP2/30ζ)

∫V r2[(∂2U/∂r2) + (4/r)(∂U/∂r)]g(r) d3r

(27)

Here, M, FP, ζ, and g(r) are the mass of a particle, number of particles per unit volume, friction coefficient, and radial distribution function of the polyelectrolyte chains, respectively. This expression is the correction to the viscosity due to interparticle interaction, not the viscosity itself. More recently, Nishida and co-workers have applied this method in the case of polyelectrolyte solutions.35 The Rice and Kirkwood expression, eq 27, is one of the simplest results open to easy adaptation. While other approaches and formulas may no doubt be readily used without any loss of generality, we chose this approach to simplify our calculations.

3η0

ζ)

8√2

(ll1N)1/2

(29)

As already explained earlier, it must be noted that the renormalized Kuhn length l1 (and hence Rg) is not a constant and depends upon various parameters including cp. We now consider the polymer chains as interacting spheres of radius Rg, the radius of gyration, which interact with each other through a Debye-Hu¨ckel potential. We therefore use a pair potential function of the form U(r) ) (wc/4πr)e-κr, κ being the inverse Debye screening length and wc ) 4πZp2e2R2/εkBT the strength of the interaction, where eZp is the charge on each of the RN segments of the polyelectrolyte chain. Substituting this into eqs 27 and 28, we obtain the correction to the shear viscosity of a polyelectrolyte solution due to the electrostatic interaction between polyelectrolyte chains to be

ηinter ) (MFP2/30ζ)

wc 4π

∫Rr

max

g

( (

(

))

-wc -κr e 4πr κ2 2 2κ - 2 - 3 d3r r r r

r2e-κr exp exp

)

×

(30)

The full viscosity of the polyelectrolyte solution is obtained by adding the above contribution to the shear viscosity from interchain interactions, which we calculated, to the value we calculated using the effective medium approach described earlier. We find that even this approximate method of incorporating chain interactions suffices to give meaningful results. It is worth repeating yet again at this point that the use of the Debye-Hu¨ckel interaction potential for U(r) is but a simplification to enable us to calculate the viscosity correction due to polyelectrolyte chain interactions. The true effective interaction potential being felt by the polyelectrolyte chains is ∆(r), whose Fourier transform ∆(k) is given by eq 25. ∆(r) includes the effect of screening of excluded volume and Coulomb interactions by counterions and salt ions as well as by the polyelectrolyte chains and was used to calculate the renormalized Kuhn length l1. Use of ∆(r) in obtaining the exact structure factor for use in eqs 13 and 14 being nontrivial, we have adopted the procedure outlined above. ∆(k) is also given by the expression

w+ ∆(k) ) 1+

F 2 2 k λk

(

wc k + κ2 wc w+ 2 k + κ2 2

)

(31)



Figure 5. Radius of gyration as a function of cp for different salt concentrations (N ) 1532, Mw ) 2 × 106, bare Kuhn length b ) 13.8 × 10- 8 cm, R ) 0.3). Data points on the x axis shown as triangles correspond to the respective overlap concentrations c* for each of the three sets.

λk is yet another complicated function itself of l1, κ, F, w, wc, and ∆(k). The reader is referred to ref 32 for details. We display our results in Figures 4b, 5, and 6, where the reduced viscosity is plotted against the polymer concentration. In agreement with experiments, the reduced viscosity shows a peak at a characteristic value of the polymer concentration for varying salt concentrations when the interchain effects are included (Figure 4b). An increased salt concentration reduces the peak and shifts it toward a higher polymer concentration. The contrast with Figure 4a, showing the reduced viscosity we calculated from the effective medium theory without including interchain interactions, is striking. As can be clearly observed, the incorporation of the interchain contribution to the viscosity is substantial and significantly affects the expected behavior of the solution. The radius of gyration Rg corresponding to the reduced viscosity plots of Figure 4 is shown in Figure 5. As expected, Rg for the zero-salt case is much larger than that in the presence of salt, at dilute polymer concentrations. In the absence of salt, the Debye sceening length κ-1 increases in proportion to cp-1/2, i.e., as r3/2, while the distance between the chains increases linearly (r). The interaction between the chains is therefore enhanced. In the presence of (low) salt, κ-1 ∝ cp-1/2 on dilution only up to, approximately, the overlap concentration, after which it begins to plateau off. The effect of interchain interactions is therefore diminished for the polyelectrolyte solution in the presence of salt. For the different cases (zero added salt and added salt), as the cp dependence of κ-1 with increasing polymer concentration after the overlap concentration approaches identical behavior and value, and so does the behavior of Rg, we would expect the reduced viscosity for the salt-free as well as addedsalt solutions to converge to the same value (while still in the semidilute regime). This is what we observe as well in Figure 4a,b. This can also be observed in the experimental data of, e.g., Nishida, Kaji, and Kanaya.35 Their theoretical model, however, was unsatisfactory as their calculated ηred for the salt and salt-free cases did not tend to converge toward the same value after coming down from the peak but instead had separate, distinct values, unlike the experimental data. The degree of ionization R of the polymer is also an important factor, with the reduced viscosity peak increasing in magnitude and shifting to lower polymer concentration with increasing R.

Figure 6. (a) Reduced viscosity as a function of cp for different degrees of ionization R (N ) 1532, Mw ) 2 × 106, cs ) 0, bare Kuhn length b ) 13.8 × 10- 8 cm). Data points on the x axis shown as triangles correspond to the respective overlap concentrations c* for each of the four sets (from right to left, for R ) 0.198, 0.3, 0.4, and 0.5, in that order). (b) cpP (cp corresponding to the ηred peak) as a function of R, plotted on a log-log scale.

This is illustrated by a plot for the zero-salt case in Figure 6. The dependence between cpP, the value of cp at which the ηred peak occurs, and the degree of ionization R for the zero-salt case can be quickly understood by the following reasoning. The principal cause of the peak is the correction factor introduced through eq 30 to the viscosity of the solution due to interacting polyelectrolyte chains. As has been observed previously,34 for polyelectrolyte solutions at low salt concentrations the dependence of the renormalized Kuhn length l1 on wc and cp is of the form l1 ∼ wc1/6cp-1/2. Since wc ∝ R2, and ζ depends on l1 through eq 29, the terms outside the integral in eq 30 contribute a dependence of cp5/4R11/6 for the reduced viscosity. Keeping in mind the increasing value of the Debye screening length κ-1 with increasing dilution, we find that the leading terms in the integral come from contributions proportional to l1 (i.e., proportional to R1/3cp-1/2) and l13/2κ (i.e., proportional to Rcp-1/4). The general, leading behavior at low and zero salt concentrations for the reduced viscosity correction with respect to the polymer concentration and the degree of ionization can be seen to be of the form ηred ∝ O(cp3/4R13/6) + O(cpR17/6) + O(cp5/4R9/2). cpP, corresponding to the maximal point of ηred, can easily be found, after some approximations, to vary roughly as cpP ∼ R-8/3, or cpP ∼ R-2.66. A more accurate number for this

5744 J. Phys. Chem. B, Vol. 113, No. 17, 2009 dependence can only be found on numerically evaluating the exact expression after including all the terms, and we find the actual dependence to be cpP ∼ R-2.41, as shown in Figure 6b, where cpP has been plotted as a function of R for the salt-free plots shown in Figure 6a. It is important to note that the dominance of the correction term due to interchain interactions over the uncorrected reduced viscosity does not occur uniformly. Indeed, at very low or high cp values, the correction term is negligible. It would be a mistake to assume that the correction term itself suffices to give the typical polyelectrolyte peak observed and discard the actual uncorrected values as insignificant, or assume our calculations of l1 in section II to be superfluous. Far from it! Over some decades of cp values, especially in the presence of added salt, the correction to the reduced viscosity is comparable to the uncorrected value itself. The importance of a systematic calculation is underlined by the actual form of the correction, which contains ζ, the translational friction coefficient. ζ has to be calculated first, and we have done so using eq 29. It will be noted that ζ is a function of the renormalized Kuhn length. Thus, an accurate calculation of l1 itself is an essential component of the final result and cannot be waved off as being irrelevant.

Ashok and Muthukumar concentration. From our calculations, the peak position clearly does not occur at the overlap concentration, but is a result due to the crossover between the interchain-interaction-mediated rapid rise and the weaker Rouse-like rise in specific viscosity. However, while our calculations assume a fixed degree of ionization regardless of the salt and polymer concentrations, increasing added salt would also have an effect on the ionization.50,51 Thus, effects such as the shifting of the polyelectrolyte peak to different polymer concentrations on changing the degree of ionization that are clearly evident from our calculations would manifest themselves in the experimental data as a shifting of the polyelectrolyte concentration corresponding to the reduced viscosity peak on changing the salt concentration instead. This does not in anyway detract from our explanation for the cause leading to the peak in the first place. Acknowledgment. We dedicate this paper to Professor Karl Freed for his significant contributions to polymer dynamics over the past four decades. Acknowledgment is made to the NIH (Grant No. R01HG002776) and NSF (Grant No. 0605833) and the MRSEC at the University of Massachusetts, Amherst. References and Notes

V. Discussion and Conclusions Using the above technique and the formalism outlined by Muthukumar,14,33 we have been able to calculate the specific viscosity of polymer and polyelectrolyte solutions. The effective persistence length of a polyelectrolyte chain was calculated selfconsistently in both dilute and semidilute regimes using the double-screening theory mentioned earlier.34 For flexible polymer solutions, remarkable coincidence of the theoretical results with the experimental data44 is obtained. Figure 2 illustrates this in ηsp vs [η]cp master plots. The Zimm to Rouse transition is evident in specific viscosity plots, with ηsp vs [η]cp log-log plots showing slopes changing from 1 to 2, as expected. In the case of polyelectrolyte solutions, the characteristic peak observed in the case of the experimental data of polyelectrolytes in aqueous solutions, for reduced viscosity plots, is observed theoretically as well, on inclusion of a correction term for the interchain viscosity contributed by an interchain interaction potential which accounts for the electrostatic interaction between chains, given by eq 30. The peak decreases in magnitude and shifts to higher polymer concentration values with increasing salt concentration. An increased degree of ionization causes an increase in the reduced viscosity peak and a shift to lower polymer concentrations. The polymer concentration at which the reduced viscosity peak occurs was found to have a dependence on the degree of ionization of the form cpP ∼ R-2.41. The interchain interaction was not discussed in earlier theory33 for low cp. The basis for the plot of the reduced viscosity peak sketched there was based on an expectation of the physical behavior of polyelectrolyte solutions and was not a numerically calculated plot. At low cp, in the absence of interchain interactions, the theory in ref 33 yields a plateau behavior for ηred as we have seen in this paper. Reference 33 addressed only the asymptotic regimes of dilute and semidilute concentrations, with the crossover between the two being sketched and not specifically addressed. The crossover depends on interchain interactions, which we have dealt with in this paper. It should be noted in conclusion that, in some experimental data, as for instance in those shown by Cohen, Priel, and Rabin,29 the polymer concentration corresponding to the reduced viscosity peak occurs at much lower concentrations than the overlap

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JP8080589

Crossover Behavior of the Viscosity of Dilute and Semidilute

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