Effect of Ionic Strength on the Dynamic Mobility of Polyelectrolytes

Langmuir 1999, 15, 5237-5243

5237

Effect of Ionic Strength on the Dynamic Mobility of Polyelectrolytes Charlotte Walldal* and Bjo¨rn Åkerman Department of Physical Chemistry, Go¨ teborg University and Chalmers University of Technology, S-412 96 Go¨ teborg, Sweden Received October 20, 1998. In Final Form: April 14, 1999 The electrokinetic sonic amplitude (ESA) technique and density measurements have been used to determine the dynamic mobility of a cationic polyacrylamide, CPAM (mol wt 5 × 106), and two cationic polyamines (mol wt 5 × 104 and 5 × 105). For all three polyelectrolytes, the ESA signal increased linearly with increasing polymer concentration up to 4 mg/mL. The dynamic mobility was higher for the larger polyamine than for the smaller, but after correction for its higher charge density, it was found that the dynamic mobility was essentially independent of molecular weight for the polyamines. At low ionic strength the dynamic mobility of the polyacrylamide was 5 and 6 times lower than those for the two polyamines, in agreement with its 5 and 6 times lower charge density, respectively.The dynamic mobility decreased with increasing ionic strength for all three polymers, as expected. However, the corresponding electrokinetic charge fraction of the polymers, calculated by modeling the polyelectrolytes as cylinders, decreased with increasing ionic strength. This in contrast to the constant charge fraction evaluated from the dynamic mobility of DNA (Rasmusson, M.; Åkerman, B. Langmuir 1998, 14, 3512), which shows that the atypical behavior of the polyacrylamide and the polyamines is not an inherent property of the dynamic mobility of polymers. The apparent persistence length of the polyacrylamide was evaluated from viscosity measurements. From comparison with the electrophoretic behavior of other polymers, it is concluded that CPAM is free-draining in ESA measurements, which shows that the cylinder model is applicable. The decreasing charge fraction thus most likely reflects a real change in the electrokinetic charge of the polymer with increasing salt concentration.

Introduction Polyelectrolytes play an important role in nature and have widespread use in many industrial processes. One application of polyelectrolytes is in phase separation processes in aqueous systems, for example as flocculants in water processing and the mining industry or as retention aids in papermaking. Electrostatic interactions generally play a dominant role in these processes, but hydrophobic forces and hydrogen bond formations may also be important. At increased ionic strength, electrostatic interactions are screened and the electrostatic repulsion between the polymer segments is reduced, which affects the conformation of the polymer and also the ability to flocculate. It is therefore very important to characterize the electrostatic properties of polyelectrolytes which are used in such processes. Electrophoresis is a suitable method for monitoring effective charges of polymers in solution. Electrophoretic mobilities of polyelectrolytes can be measured by several techniques. Nagasawa et al.1 performed electrophoretic measurements on sodium poly(vinyl sulfate) (NaPVS) using a moving-boundary method. More recently, Bernard et al.2 determined the electrophoretic mobility of a polyelectrolyte (heparin) by radioactive labeling of the polyions. Electrophoretic light scattering3 and fluorescence recovery after photobleaching4 are other techniques which have been used to measure the free solution mobility of DNA and other polyelectrolytes. * Corresponding author. Phone: 46317722815. Fax: 4631167194. E-mail: [email protected]. (1) Nagasawa, M.; Soda A.; Kagawa, I. J. Polym. Sci. 1958, 31, 439451. (2) Bernard, O.; Turq, P.; Blum, L. J. Phys. Chem. 1991, 95, 95089513. (3) Hartford, S. L.; Flygare, W. H. Macromolecules 1975, 8, 80-83.

These techniques are usually used to measure mobilities in constant fields. Here we measure the dynamic mobility in an alternating field, µd, of three different polyelectrolytes using the electrokinetic sonic amplitude (ESA) technique.5 The ESA technique is based on measurements of the amplitude of the pressure waves which are generated when the alternating electric field, in our case of frequency 1 MHz, is applied to a dispersion of charged particles. For dilute colloidal suspensions (up to approximately 5% volume fraction), the relation between the ESA effect and the dynamic mobility of the charged particles is given by6

ESA(ω) ) A(ω) φ(∆F/F0)µd

(1)

where ESA is measured in Pa/(V/cm-1), A(ω) is an instrument factor, φ is the volume fraction of the particles, and ∆F is the density difference between the particle density (Fp) and that of the solvent (F0). The main advantages of the ESA technique are that it is fast and not limited to dilute suspensions. Most ESA studies so far have been performed on rigid particles, but recently the ESA technique has been shown to be suitable for electrokinetic characterization also of polyelectrolytes. The ESA technique has been applied to DNA,7 an anionic semirigid polymer, and to cationized amylopectin8 a branched polymer. In the present work, we study two other important classes of polyelectrolytes, cationized polyacrylamide and cationic polyamine, which are used as flocculating agents. From the dynamic mobility deter(4) Tinland, B.; Pernodet, N.; Weil, G. Electrophoresis 1996, 17, 10461051. (5) Oja, T.; Petersen, G.; Cannon, D. U.S. Patent 4 497 207, 1985. (6) O’Brien, R. W. J. Fluid Mech. 1990, 212, 81-93. (7) Rasmusson, M.; Åkerman, B. Langmuir 1998, 14, 3512-3516. (8) Larsson, A.; Rasmusson, M. Carbohydr. Res. 1997, 304, 315323.

10.1021/la981475v CCC: $18.00 © 1999 American Chemical Society Published on Web 06/23/1999

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Walldal and Åkerman 1 and 4 × 10-8 m2/(V s) (see below), the amplitude of the motion of the polymers in this study was typically 5-20 Å. The wavelength was 1.5 mm. The cell was thoroughly rinsed with water, and the ESA signal was checked (ESA ) 0) before the cell was rinsed and filled with the appropriate sample. It was observed that the polymers adsorb to the cell wall. For each sample, the measurement was made by taking out and reinjecting the same sample in the cell. Every data point is the average of 50 measurements. The background signal from the electrolyte was measured after each sample was measured, and the ESA signal from the sample was corrected according to a procedure outlined by Desai et al.11 The measured ESA (ESAmeas) is a vector sum of the true ESA for the polymers (ESAtrue) and the ESA for the background electrolyte (ESAbkgd):

ESAmeas ) ESAtrue + ESAbkgd

(2)

ESAtrue can be calculated by resolving eq 2 into its two components:

Figure 1. Chemical structures of CPAM (a) and the polyamine (b). Table 1. Properties of the Polyelectrolytes

polymer CPAM PolA PolB a

mol wt

Lc (µm)

mol wt of repeating unit

5 000 000 50 000 500 000

17.5 a a

798 120 120

charge dist (Å)

mequiv/g

28 8.6 7.5

1.2 6.4 7.2

Contour length not given due to unknown degree of branching.

mined by ESA, we calculate the ζ potential and the electrokinetic charge fraction, R, using a cylinder model. For our purposes, the most important conclusion of the DNA study is that the dynamic mobility as measured by the ESA technique is very similar to the constant-field mobility. This allows us to compare the mobilities measured here with those of other synthetic polymers measured in constant fields.

|ESAmeas| cos θ ) |ESAtrue| cos β + |ESAbkgd| cos φ (3) |ESAmeas| sin θ ) |ESAtrue| sin β + |ESAbkgd| sin φ

(4)

where θ, β, and φ are the phase angles of the measured, true, and background ESA signals, respectively. At high ionic strength, the contribution to the ESA signal from the electrolyte is much larger than that from the polymers. Therefore, only samples with an electrolyte concentration up to 60 mM NaCl were investigated. Density Measurements. The density of the polyelectrolyte samples and electrolyte solutions was measured using an Anton Paar density meter. As noted by Wade et al.,12 φ∆F in eq 1 is equal to Fsuspension - F0, which means that the factor φ(∆F/F0) in eq 1 can be obtained by measuring the density of the polymer suspension and the density of the electrolyte solvent. Viscosity Measurements. Viscosity measurements were performed on CPAM with a Brookfield rotation viscometer where the experimental data were extrapolated to zero shear rate. The intrinsic viscosity was obtained from the intercept at C ) 0 by plotting ηspec/C against the polymer concentration C (g/cm3). Conductivity Measurements. The conductivities were measured using a CDM 83 conductivity meter from Radiometer, Copenhagen, operating at a frequency of 4.69 kHz.

Experimental Section Results and Discussion Polyelectrolytes. All three polyelectrolytes were gifts from Eka Chemicals, Bohus, Sweden. Two of them were branched polyamines (consisting of epichlorhydrin, dimethylamine, and ethylendiamine) with molecular weights of 50 × 103 (PolA) and 500 × 103 (PolB), respectively. The third polymer was a linear cationic polyacrylamide (CPAM) with 10% of the acrylamide monomer replaced by (trimethylammonio)ethyl acrylate chloride and with a molecular weight of 5 × 106. All molecular weights are number averages according to the manufacturers. Figure 1 shows the chemical structures of the studied polyelectrolytes, and Table 1 summarizes their properties. The total charge of the polyelectrolytes was determined by titration with potassium poly(vinyl sulfate).9 From these data an average contour distance between the charges was calculated. In some experiments, the polyelectrolyte samples were dialyzed against the solvent prior to the measurements. Cationic polyacrylamide is sensitive to hydrolysis which reduces the charge density,10 therefore the dialysis time for CPAM was limited to 72 h. All experiments were performed at room temperature at pH 5. Methods. ESA Measurements. The ESA measurements were performed on an ESA 8000 instrument (Matec Applied Sciences) using a high-sensitivity PPL-80 cell, which requires about 4 mL of sample. The field strength was about 500 V/cm, and the field frequency was 0.9-1.0 MHz. With a dynamic mobility between (9) Horn, D. Prog. Colloid Polym. Sci. 1978, 65, 251-264. (10) Aksberg, R.; Wågberg, L. J. Appl. Polym. Sci. 1989, 38, 297304.

Dynamic Mobility. Figure 2 presents the electrolytecorrected ESA values (ESAtrue; see Methods) for the three studied polyelectrolytes plotted vs total ionic strength ()RIpolyelectrolyte + INaCl) for different polymer concentrations. The ESA values at high ionic strengths are not shown for the lowest polymer concentrations because the signal resulting from the polymer was too low compared to the signal from the electrolyte. The results presented in Figure 2 show that the corrected ESA signal for the polyelectrolytes decreases with increasing ionic strength and increases with increasing polymer concentration. The calculated phase angle β varies by (2° at low ionic strength (e5 mM) and by (10° at higher electrolyte content. The relative constancy of the calculated β lends credibility to the phase angle measurements and the background correction procedure.11 The dynamic mobility of the polyelectrolytes can be calculated using eq 1 if φ(∆F/F0) ) (Fsuspension - F0)/F0 is known. Figure 3 shows the density of the polyelectrolyte solutions as a function of ionic strength, together with the density of the background NaCl solutions. (11) Desai, F. N.; Hammand, H. R.; Hayes, K. F. Langmuir 1993, 9, 2888-2894. (12) Wade, T.; Beattie, J. K.; Rowlands, W. N.; Augustin, M.-A. J. Dairy Res. 1990, 63, 387-392.

Dynamic Mobility of Polyelectrolytes

Figure 2. (a) Corrected ESA signal (ESAtrue) for CPAM as a function of ionic strength (mM), at the concentrations 0.5 mg/ mL ([), 1 mg/mL (9), 2 mg/mL (2), 3 mg/mL (×), and 4 mg/mL (b). (b) Corrected ESA signal (ESAtrue) for PolA as a function of ionic strength, at the concentrations 1 mg/mL ([), 2 mg/mL (9), 3 mg/mL (2), and 4 mg/mL (×). (c) Corrected ESA signal (ESAtrue) for PolB as a function of ionic strength, at the concentrations 0.8 mg/mL ([), 1.6 mg/mL (9), 2.4 mg/mL (2), and 3.2 mg/mL (×).

Figure 4 shows the dynamic mobility calculated from the data of Figures 2 and 3, plotted as a function of electrolyte concentration. As expected, the dynamic mobility decreases with increasing electrolyte concentration. The overall changes in dynamic mobility with increasing ionic strength were similar for the two polyamines. The dynamic mobility values were at most 20% higher for PolB than for PolA. A major part of this difference can be ascribed to the 12% higher charge density on PolB (Table 1). The mobilities of the two polyamines are therefore very similar at all ionic strengths even though there is a difference in molecular weight by a factor of 10. This indicates that the dynamic mobility of the polyamines is nearly independent of molecular weight. We conclude that the polyamines are free-draining in electrophoresis at the ionic strengths studied here. CPAM has 5 and 6 times lower charge density than the polyamines, and at low ionic strengths, the dynamic mobility of CPAM is indeed about 5 times lower than that

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Figure 3. (a) Density (g/cm3) of CPAM solutions as a function of ionic strength, for polymer concentrations of 0.5 mg/mL ([), 1 mg/mL (9), 2 mg/mL (2), 3 mg/mL (×), and 4 mg/mL (b), and the density of the NaCl solution (O). (b) Density (g/cm3) of PolA solutions as a function of ionic strength, for polymer concentrations of 1 mg/mL ([), 2 mg/mL (9), 3 mg/mL (2), and 4 mg/mL (b), and the density of the NaCl solution (O). (c) Density (g/cm3) of PolB solutions as a function of ionic strength, for polymer concentrations of 0.8 mg/mL ([), 1.6 mg/mL (9), 2.4 mg/mL (2), and 3.2 mg/mL (b), and the density of the NaCl solution (O).

for the polyamines. This indicates that, in the cases studied here, the dynamic mobility is mainly dictated by the density of charge and that structure and chemical composition are less important for the dynamic mobility of these polyelectrolytes. In Figure 5 the evaluated dynamic mobilities of the polyelectrolytes at fixed ionic strength are plotted versus the polyelectrolyte concentrations. As can be seen, the mobility does not exhibit any systematic concentration dependence for concentrations up to 4 g/dm3 (PolA), 3.2 g/dm3 (PolB), and 4 g/dm3 (CPAM). This indicates that, at these concentrations, the dynamic mobility is not affected by interactions between different chains. The Cylinder Model and the Charge Fraction. Following its successful application to the dynamic

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Walldal and Åkerman

Figure 4. Dynamic mobility (µd‚108) as a function of ionic strength (mM) for CPAM ([), PolA (9), and PolB (2). Curves are guides for the eye only.

Figure 6. Charge fraction, R, vs ionic strength for CPAM ([), PolA (9), and PolB (2), calculated from the data in Figure 4 as described in the text. Table 2. ζ Potentials at Different Ionic Strengths CPAM

PolA

PolB

Itot (mM)

ζ (mV)

Itot (mM)

ζ (mV)

Itot (mM)

ζ (mV)

1 5 10 25 50

23 15 11 8 6

1 5 10 25 50

77 64 47 29 19

1 5 10 25 50

97 78 52 39 27

Schellman and Stigter16 defined the charge fraction R as

R ) σkin/σ0 Figure 5. Dynamic mobility (µd‚108) plotted against polyelectrolyte concentration for CPAM ([) at 5 mM, PolA (9) at 7 mM, and PolB (2) at 5 mM NaCl.

(7)

where β is a correction factor which incorporates the nonlinear charge-potential relationship,  is the dielectric constant of the solvent, 0 is the permittivity in a vacuum, κ is the inverse Debye length, and K0 and K1 are Bessel functions. The ζ potential is the potential at the slip plane.14 The ζ potential was calculated with the computer program “Mobility” based on the O’Brien-White theory.15 The results are found in Table 2. The same ζ potentials were obtained with the Hu¨ckel equation, which indicates that the relaxation effect was small. This is in accordance with the ζ potential being low for CPAM compared to, for example, DNA, where correction for relaxation effects was essential.7

which is a measure of the net polymer charge in electrokinetic measurements, or in other words a measure of how large a fraction of the counterions is located outside the slip plane. In Figure 6, the charge fraction is plotted as a function of electrolyte concentration. It is seen that R increases with decreasing electrolyte concentration, which according to the cylinder model can be interpreted as a decrease of the fraction of counterions which remain inside the slip plane as the ionic strength decreases. Such a behavior is in disagreement with the cylinder model with constant charge density, which predicts a constant charge fraction.16 Relationship between Dynamic and Electrophoretic Mobilities. One possible source of the above discrepancy is that the cylinder model was developed for constant fields, whereas we measure the mobility in alternating fields. We deem this explanation unlikely for two reasons. First, according to Mangelsdorf and White,17 a sphere of radius smaller than 200 nm has a dynamic mobility at 1 MHz, which equals its zero frequency electrophoretic mobility within 2% if the ζ potential is less than 100 mV. Ohshima18 has shown that the dynamic mobility of a cylinder equals the mobility of a sphere with a radius 1.5 times the cylinder radius. The cylinder radius for polyacrylamide was chosen to be 1.4 Å, which is half the length of the monomer unit. For the polyamines, half the distance between the closest nitrogen atoms, 2.1 Å, was chosen to be the radius. Since the cylinder radii multiplied by 1.5 were less than 200 nm and since the ζ potentials were less than 100 mV (Table 2), the dynamic mobility can be expected to be equal to the electrophoretic (constant field) mobility. Second, when the cylinder model was applied to data on the dynamic mobility of DNA, the same type of analysis gave rise to a charge fraction which

(13) Stigter, D. J. Colloid Interface Sci. 1975, 197, 296-306. (14) Lyklema, J. Colloids Surf. 1994, 92, 41-50. (15) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 77, 1607-1626.

(16) Schellman, J. A.; Stigter, D. Biopolymers 1977, 16, 1415-1434. (17) Mangelsdorf, C. S.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1992, 88, 3567-3581. (18) Ohshima, H. J. Colloid Interface Sci. 1997, 185, 131-139.

mobility of DNA,7 we use the cylinder model to interpret the data of Figure 4. The polymer is modeled as a (rigid) cylinder with a uniform charge density, given by

σ0 ) ze/2πab

(5)

where e is the elementary charge, z is equal to 1 if b is the axial distance between the charges, and a is the cylinder radius. According to Stigter,13 the relation between the kinetic charge density, σkin, and the ζ potential is

σkin )

ζκ0β K1(κa) K0(κa)

(6)

Dynamic Mobility of Polyelectrolytes

Langmuir, Vol. 15, No. 16, 1999 5241

if anything decreases with decreasing salt at low ionic strength.7 This observation seems to rule out the idea that the increase in R at low ionic strength observed for CPAM, PolA, and PolB is an inherent property of the dynamic mobility of polymers. As supported by the results on DNA,7 we will therefore base the following discussion on the assumption that the dynamic and electrophoretic mobilities of the studied polyelectrolytes are equal. The Possibility of Non-Free-Draining. The cylinder model applies to a flexible polymer only if the different polymer segments do not interact electrostatically or hydrodynamically, because only then can the segments be treated as independent rigid cylinders. A second possible explanation for the increase in R going toward low ionic strengths therefore arises if the cationic polymers are not free-draining in electrophoresis at low ionic strength. Hydrodynamic interactions between segments at low salt concentration would decrease the friction coefficient, and as a consequence, the mobility would become higher than the free-draining mobility. If the cylinder model still is applied, as here, this increase in mobility would result in an apparent increase in R. Since a polymer is free-draining only if the double layer is thin compared to the typical distance between different polymer segments, this is indeed most likely to fail at low salt concentrations when the double layer is thickest. This possibility is also supported by the fact that DNA, which does not exhibit the increase in R, is known to be freedraining in electrophoresis.19 However, since it was earlier concluded that the polyamines are free-draining at all ionic strengths, the increase in R they also exhibit at low ionic strength is not likely due to hydrodynamic interactions. Still it was deemed essential to investigate the possibility that CPAM was not free-draining at low salt concentrations. The most straightforward method, to compare different molecular weights as done here with the polyamine, was not available to us for the CPAM case. We instead choose to apply two theoretical criteria for free-draining behavior. It has been suggested that the degree of overlap between the double layers of different segments can be judged by replacing the polyelectrolyte with a cylinder having a volume equivalent to that of the polymer coil20

πrfree2L ) (4/3)πRcoil3

(8)

where L is the contour length of the polymer and

〈Rcoil2〉 ) (5/3)〈RG2〉

(9)

where RG is the radius of gyration. 2rfree can be seen as the diameter of a cylindrical volume around the polymer which is in effect free of other polymer segments. If this diameter is much larger than the thickness of the ionic atmosphere 1/κ, where κ is the inverse Debye screening length, electric and hydrodynamic overlap between different parts of the chain are negligible, and the cylinder model can be used.20 More quantitatively, for κrfree ) 4, the potential at rfree is only 0.2% of the surface potential of the cylindrical polyelectrolyte.20 Second, according to de Gennes and co-workers,21 a polymer will be free-draining in electrophoresis if the persistence length is much larger than the Debye length. (19) Olivera, B. M.; Baine, P.; Davidson, N. Biopolymers 1964, 21, 245-257. (20) Van der Drift, W. P. J. T.; De Keizer, A.; Overbeek, J. Th. G. J. Colloid Interface Sci. 1979, 71, 67-78, 79-92. (21) Brochard-Wyart, F.; de Gennes, P.-G. C. R. Acad. Sci., Ser. II 1988, 307, 1497-1500.

Figure 7. Intrinsic viscosity as a function of ionic strength for CPAM. Table 3. rfree and rfreeK Values for CPAM Itot (mM)

rfree (nm)

rfreeκ

3 5 9 14

42 35 30 26

7.2 8.2 9 10

Itot (mM)

rfree (nm)

rfreeκ

17 30 45

24 19 16

10.3 10.9 11.1

The persistence length Lt is a measure of polymer stiffness, in terms of the characteristic length over which the directional correlation between the polymer segments disappears. The de Gennes criterion therefore reflects the fact that the overlap between different segments will be small if the polymer cannot bend back on the length scale of the double layer. Since the double-helical DNA is an unusually stiff polymer, there is clearly the possibility that the CPAM case may be different in terms of hydrodynamic overlap since this polymer can be expected to be more flexible. To apply these criteria, we measured the intrinsic viscosity of CPAM, from which RG and subsequently Lt were evaluated. The models used for polymer coil dimensions are strictly valid only for linear chains, so the corresponding analyses for the polyamines were not performed due to their branched structure. Viscosity Measurements. Figure 7 shows that the intrinsic viscosity for CPAM decreases with increasing salt concentration, which reflects how the screening of the polymer charges results in smaller coil dimensions.22 The relation between intrinsic viscosity [η] and the radius of gyration RG for a nondraining coil is

[η] ) 63/2Φc〈RG2〉3/2/M

(10)

where Φc = 2.87 × 1023 if [η] is in units of cm3 g-1 and M is the molecular weight.22 Table 3 presents the values of rfree and κrfree calculated from the viscosity data for CPAM. It can be seen that κrfree is considerably larger than 1, indicating a free-draining behavior for CPAM. To apply also the de Gennes criterion, persistence lengths were evaluated from the RG data. For a nonbranched wormlike chain, the radius of gyration is related to the total persistence length, Lt, and the contour length L by22

〈RG2〉 ) LLt/3 - Lt2 + 2Lt3/L - 2(Lt4/L2)(1 - e-L/Lt) (11) (22) Dautzenberger, H.; Jaeger, W.; Ko¨tz, J.; Philipp, B.; Seidel, Ch.; Stscherbina, D. Polyelectrolytes; Hanser Verlag: Munich, Vienna, New York, 1994.

5242 Langmuir, Vol. 15, No. 16, 1999

Figure 8. Persistence length as a function of ionic strength for CPAM, calculated from the data of Figure 7 as described in the text.

where L ) Mb/m, with M and m being the polymer and monomer masses, respectively, and b is the contour length per monomer. Since no account is taken of excludedvolume effects in eq 11, Lt values derived using it will be denoted the apparent persistence length, Lt′. Excludedvolume effects have been invoked23 in order to explain the deviation between theory and experiment regarding the effect of salt on the persistence length. OSF (Odjik, Skolnick, and Fixman) theory predicts Le ∝ I-1 whereas when experiments were interpreted using (11), they often gave Lt ∝ I-1/2.22 If electrostatic excluded-volume effects were taken into account, better agreement with experiment was found.23 However recent theories24 for flexible polymers predict the observed I-1/2 dependence without involving excluded-volume effects. In this context, a flexible polymer is one where the electrostatic persistence length (Le) is long compared to the bare persistence length (L0) of the uncharged equivalent chain. Since this is the case for the polymers used here (see below), we choose to neglect excluded-volume effects and use (11) to calculate an apparent persistence length. Figure 8 shows the Lt′ values calculated for CPAM using eq 11, eq 10, and the data on [η] from Figure 7. From the plot against ionic strength, as in Figure 8, the two contributions to the total apparent persistence length Lt′ can be distinguished. At low ionic strength, there is a large contribution from the electrostatic effects, whereas at high ionic strength, where the graph tends to level out, the main contribution is the bare persistence length. A log-log plot of Lt′ vs I (not shown) is linear and has a slope of -0.45 ( 0.02, in agreement with slopes around -0.5, which are commonly observed for polyelectrolytes22 and predicted by recent theory24 for flexible polyelectrolytes (In fact, simulations25 indicate that the value of the slope can be even lower than -0.5.) If Lt′ is plotted vs I-0.45 (not shown), the extrapolated persistence length at 1 M ionic strength is 8 Å. This is an estimate of the bare persistence length L0′ of the present polyacrylamide, since electrostatic interactions in polyelectrolytes are in effect screened out at this ionic strength.26 Thus CPAM is flexible in the sense that the electrostatic contribution (Le′) is considerably larger than the bare L0′ in the present range of salt concentrations, which explains the observed approximate I-0.5 dependence, Theoretical Support for CPAM Being Free-Draining. Comparison of the two parameters Lt and rfree with (23) Reed, W. F.; Ghosh, S.; Medjahdi, G.; Franc¸ois, J. Macromolecules 1991, 24, 6189-6198. (24) Ha, B.-Y.; Thirumalai, D. Macromolecules 1995, 28, 577-581. (25) Micka, U.; Kremer, K. Phys. Rev. E 1996, 54, 2653-2662. (26) Norden, B.; Lincoln, P.; Tuite, E.; Åkerman, B. Met. Ions Biol. Syst. 1996, 33, 177-252.

Walldal and Åkerman

Figure 9. Persistence length/Debye length (Lt′‚κ) vs ionic strength for CPAM ([), PVS (×), PMA (+), and DNA (b). The values for PVS, PMA, and DNA were calculated from RG data from refs 1, 20, and 27, respectively, as described in the text.

κ-1 provides a test for free-draining behavior, albeit a qualitative one. To make a more quantitative statement, the values of Ltκ and rfreeκ for CPAM were compared with those for polymers of known drainage behavior in electrophoresis. Poly(vinyl sulfate)1 (PVS) and DNA19 are known to be free-draining in (constant-field) electrophoresis and so is probably polymethacrylate20 (PMA). In Figure 9, the Ltκ values of four different linear polyelectrolytes, including CPAM, are compared at different ionic strengths. For CPAM, the ratio increases slightly with increasing salt concentration from a value of 1.4 at low salt concentration but then is almost constant; i.e., the persistence length and the Debye length decay essentially in parallel. By comparison, for DNA27 the ratio is overall considerably larger and also increases much more strongly with increasing ionic strength. The stiffness of the DNA molecules is less sensitive to added electrolyte, because the double-helical nature of DNA gives it an unusally long L0 of about 500 Å. The markedly different curve shape compared to that of DNA reinforces the picture of a comparatively low bare stiffness of CPAM. Using Lt′ values calculated from published data on 〈RG2〉, Figure 9 also shows that, for PVS, the ratio has a magnitude and ionic strength dependence which are very similar to those of CPAM. For PMA, the ratio drops more markedly at low ionic strength, but the ratio is overall higher than those for PVS and CPAM, presumably because of a stronger Le contribution due to the rather high charge density under the condition used20 (50% dissociation). Importantly, at all included ionic strengths, the value for CPAM falls between those of the two known free-draining polymers DNA and PVS. This strongly indicates that CPAM is freedraining too. A similar conclusion is reached on the basis of the rfree parameter, as can be seen from the comparison of the rfreeκ values in Figure 10. The CPAM studied here again falls between the free-draining cases of PVS, which has values only just above 1, and DNA, which has typical rfreeκ values of 50 or larger.24 Also, PMA has rfreeκ considerably larger than 1 at all salt concentrations used here, in agreement with the proposed free-draining behavior during electrophoresis. Thus, by both criteria, CPAM falls between two established cases of electrophoretically free-draining polymers, and we conclude CPAM is free-draining in electrophoresis. It is noteworthy that rfreeκ g 2.5 and Ltκ g 1 seem to suffice for a free-draining behavior. The two criteria are not independent: a stiff polymer will have a large RG, and by (27) Sobel, E. S.; Harpst, J. A. Biopolymers 1991, 31, 1559-1564.

Dynamic Mobility of Polyelectrolytes

Figure 10. rfreeκ is vs ionic strength for CPAM ([), PVS (×), PMA (+), and DNA (b). The values for PVS, PMA, and DNA were calculated from RG data from refs 1, 20, and 27, respectively, as described in the text.

both criteria, this favors a free-draining behavior. They are not completely dependent, however, as can be seen from the behavior of CPAM being similar to that of PMA in terms of rfree but closer to PVS in terms of Lt. Compared to the rfree parameter, the de Gennes criterion is formally a local one, since it does not involve the global parameter contour length. It should be remembered however that the apparent Lt′ values we use may include excludedvolume interactions, which do depend on the contour length. The somewhat different pictures that emerge with rfreeκ (Figure 10) and Lt′κ (Figure 9) thus likely reflect different weightings of short- and long-range overlaps of double layers. Interestingly, the free-draining behavior explains for dynamic mobility measurements the apparent absence of polymer-polymer interactions (Figure 5), which is surprising in the case of CPAM because the polymer

Langmuir, Vol. 15, No. 16, 1999 5243

concentrations we have used are above the overlap concentration. From the RG value, we calculate a c* ) 1/[η] of 0.17 mg/mL for the highest salt concentration (largest coils). However, for free-draining polymers, hydrodynamic interactions between different coils will be suppressed for the same reason that hydrodynamic interactions between different segments in a given coil are.20 The polyamines are free-draining too but have considerably smaller RG values so those measurements were performed below the overlap concentration anyway. A quantitative analysis20 can be made by assuming that all polymer in the sample forms one single molecule and by calculating an effective rfree value for the radius of a cylindrical free space available for this hypothetical polymer. For the highest CPAM concentration of 4 mg/ mL (Figure 5), this gives an rfree of 10.9 nm, which is about twice the value of κ-1 at the lowest salt concentration of 3 mM. By this rough estimate, the overlap between the double layers of different polymers is thus not negligible but still small, which explains why polymer-polymer interactions are not detected in the dynamic mobility also above the overlap concentration. Another possibility is that ESA is less sensitive to such interactions because the molecules move short distances compared to their dimensions. More importantly, since CPAM and the polyamines are free-draining during electrophoresis, hydrodynamic interactions cannot explain why the charge fraction increases at decreasing salt concentrations in all three cases. The cylinder model should thus be applicable, and one possible source of the nonconstant R is then that the assumption of a constant charge density is not fulfilled. Acknowledgment. Dr. Mikael Rasmusson is thanked for valuable discussions mainly regarding electrokinetics. LA981475V