J. Phys. Chem. 1996, 100, 9881-9891
9881
Multicomponent Diffusion of Distearyldimethylammonium Polyelectrolyte Solutions in the Presence of Salt: Coupled Transport of Sodium Chloride Michael Thies,† Shaun F. Clancy,*,‡ and Henrich H. Paradies*,† Biotechnology and Physical Chemistry, Ma¨ rkische Fachhochschule, P.O. Box 2061, D-58690 Iserlohn, Germany, and Safety, Health, and EnVironmental Affairs, Witco Corporation, One American Lane, Greenwich, Connecticut 06831-2559 ReceiVed: January 10, 1996; In Final Form: March 21, 1996X
A sensitive conductance method has been used to determine binary and ternary diffusion coefficients of distearyldimethylammonium chloride, bromide, and hydroxide (DSDMACl, DSDMABr, and DSDMAOH) in aqueous solutions in the absence and presence of salt. Experiments were conducted above and below the critical micelle concentration (CMC) at 25 °C, and the concentration of salt, when present, was 0.015 M NaX (X ) Cl, OH, Br). The experiments reveal that diffusion of DSDMA chloride, bromide, and especially hydroxide induces concurrent flow of NaCl, NaBr, and NaOH, respectively. Also, diffusion of DSDMAX in the presence of NaCl drives countertransport of DSDMACl or DSDMAOH. At pH 7.5 and an ionic strength of 0.015 M, each mole of diffusing DSDMACl (Nagg ) 72, R ) 0.15) cotransports 9.0 mol of NaCl. However, coupled flow of NaCl decreases as the solution pH approaches 5.2 and is almost zero at pH 4.5. The corresponding value for diffusing DSDMAOH at pH 7.5 is about 120 mol of NaCl. At pH 6.5 only 10 mol of NaCl are cotransported, and the value is almost zero at pH 5.5. However, at pH 4.5, diffusing DSDMAOH generates a counterflow of NaCl due to bound protons and a release of H2O (DSDMAOH + H+ T DSDMA+ + H2O), yielding a net positive charge; hence, the D21 value is negative. The cross-diffusion coefficient D21 of diffusing DSDMACl in the presence of 0.015 M (0.095-0.009) NaOH and 0.015 M NaCl yields a flux density of NaOH and NaCl, which is generated by the DSDMAX (X ) Cl + OH) gradient. A similar cross-diffusion coefficient, D21, is revealed when DSDMAOH, is used, instead of DSDMACl, in the presence of 0.015 M NaCl through induced coupled transport of salt. The coupled transport of NaCl or of NaCl and NaOH is primarily driven by the diffusion-induced electric field along the DSDMAOH and DSDMACl concentration gradients. For the DSDMAOH system, the direction and magnitude of the coupled flow of either NaCl or NaOH can be explained through the ionic mobility of DSDMA+ and OH-, in addition to the charge and degree of dissociation (Nagg ) 6400, R ) 0.33-0.35) at 25 °C. The tracer and mutual diffusion coefficients for the DSDMAX system were compared with those from light-scattering measurements of polyelectrolyte solutions of DSDMAX.
Introduction It is well-known that surfactant molecules self-assemble into discrete aggregates in aqueous solutions above a certain concentration, the critical micelle concentration (CMC). At dilute and semidilute concentrations, the aggregates can form spherical, ellipsoidal, or even wormlike structures in the presence of salt or large counterions.1,2 Aqueous solutions of di-nalkyldimethylammonium salts (DSDMAX; DSDMA ) A and X ) F, Cl, Br, I, OH) display a variety of structures, the form of which depends on the method of preparation, absence or presence of salt, temperature, and chemical nature of the anions.3-6 It has been found that the anions of AX dissociate only moderately. Furthermore, the apparent CMCs for A halides are rather low and are found to be 3.10 × 10-7 M for ACl and 4.28 × 10-5 M for ABr, whereas the CMCs for most cationic surfactants are in the range of ∼10-3 M. However, the degree of counterion dissociation, R, of ACl (0.13-0.15) is comparable with that of the bromide (0.11-0.12), but is still different from that for single-chain cationic surfactants. The salt dependence of R is less pronounced for both ACl and ABr micelles than for the corresponding vesicles, for which R is strongly dependent †
Ma¨rkische Fachhochschule. Witco Corporation. X Abstract published in AdVance ACS Abstracts, May 15, 1996. ‡
S0022-3654(96)00119-0 CCC: $12.00
on ionic strength.3,6 In several respects, AOH, which is believed to be a key intermediate in microbial biodegradation processes of ACl, displays a number of differences from the A halides studied so far. (1) AOH is highly water soluble and does not flocculate upon the addition of salt. (2) The degree of dissociation is on the order of 28-30%.6 (3) AOH displays a reversible change in aggregation structures in solution upon altering the concentration of AOH and/or salt, resulting in the formation of threadlike micelles within the overlapping concentration ranges, where the temperature is 18-25 °C.6 The results for AOH parallel studies of AY compounds in solution [Y ) organic counterion such as acetate, butyrate, or propionate, enantiomers of (S)-(+)- or (R)-(-)-ibuprofen, or (R)-(+)lactate], which show similar behavior.7 Due to its importance in the microbial biodegradation of AX through environmentally relevant microorganisms, the translocation of AX (and metabolites) across biological membranes is a basic mechanism for AX import into various cellular organelles, bacteria, and fungi. Although much progress has been made in understanding the shape, size, and vesicle formation of these surfactants, the physical aspects of the transport processes (self-diffusion or translocation mechanism at or through microbial membranes above and below the CMC) still remain unknown. Also unknown are the intracellular interactions of AX with nucleotides, e.g., ATP, ADP, etc., and binding © 1996 American Chemical Society
9882 J. Phys. Chem., Vol. 100, No. 23, 1996 of AX to outer bacterial cell membrane components including exopolymers.8 To make this study environmentally relevant, multicomponent effects have to be considered since mutual diffusion involves the interchange of solvent and solute by random thermal motions. This causes concentration gradients or concentration fluctuations to decay in time.9 For a polyelectrolyte such as AX with attendant counterions, the mutual diffusion coefficient is a complicated function of the diffusivities of the surfactant and counterions as well as of the conformation of the surfactant assemblies. The mobilities of AX micelles induce a relatively large electric field of a magnitude that generates a substantial coupled flow of added salt within the AX micellar gradient. Moreover, the mutual diffusion coefficient of the AX aggregates is proportional to the activity coefficient factor, γ ) 1 + c(d ln γ()/dc, which introduces variations in the driving force for nonideality. Such behavior has been noticed for diffusion in polyelectrolyte and salt systems,11 and for proteins12 and sodium dodecyl sulfate micelles in 0.1 M NaCl at 25 °C.13 The experiments reported here were conducted to learn more about the diffusivity of ACl, ABr, and AOH in salt solution on both sides of the CMC. To our knowledge, these experiments are the first dealing with this class of double-chain surfactants, despite their wide commercial and colloidal applications. In addition, the experiments will also yield information about the coupled transport of NaX during the diffusion of AX (X ) Cl, Br, OH). We will show that the diffusing double-chain surfactants, particularly AOH, can generate a coupled flow of salt that is strongly pH dependent. This solution behavior can be explained qualitatively by treating the double-chain surfactant systems (I ) AX in H2O; II ) AX and NaX in H2O) as mixtures of polyelectrolytes in the presence of added salt. Experimental Section Materials and Methods. Distearyldimethylammonium (A) chloride and bromide were prepared by Witco Corp. using standard manufacturing methods and were purified by recrystallization from chloroform. The hydroxide of A was prepared as described in ref 6. All measurements were made at 25 °C unless otherwise noted. AOH/ACl solutions having a different ratios of OH/Cl, e.g., 30% AOH and 20% (w/w) ACl, were prepared by batch exchange of a 30% by weight aqueous solution of ACl with IRA-400(OH) exchange resin (Ro¨hm & Haas). The effective exchange of the chloride for the hydroxide ion was 96-98%, resulting in a known composition of AOH/ ACl containing almost 30% (w/w) of AOH used in addition to the studies reported for pure AOH and ACl, respectively. The open-ended capillary tube method was used to determine binary and ternary diffusion coefficients for both systems I and II. Briefly, the apparatus consisted of two stirred chambers separated by a porous membrane (mean pore diameter ) 6.0 µm). The lower reservoir had a net capacity of 2.0 mL and was filled completely during operation. The upper reservoir had a capacity of 20 mL, but only 2 or 4 mL of solution were used during the experiments for systems I and II, respectively. The whole system was thermostated at 25 ( 0.1 °C. In addition, small electrodes were installed in the column walls to measure the electrical conductivities of the solutions. Micro ion selective electrodes, e.g., Cl-, and Br-, and a micro glass electrode for monitoring pH (List, Darmstadt, FRG) were also present. Two windows in the upper and lower reservoirs were installed for use in laser interferometry to measure concentrations by using two optical fibers: a sensing fiber and a reference fiber.14 A beam splitter was installed to divide the laser beam into two parts, which were launched into sensing and reference fibers.
Thies et al. The output from these fibers meets another beam splitter, and the two resulting beams are permitted to fall onto a photodetector. The fringe changes, which are proportional to the concentration changes within the measured sample, can be recorded by using real time holographic interferometry. The numbers of interference fringes are viewed in real time passing through the boundary region and are proportional to the refractive index difference, ∆n, which is related to the cell thickness d and the wavelength λ (632.8 nm) as
∆γ ) d(∆n)
(1)
d∆C2(t)/dt|t)0 ) -βD21∆C1(0), hence ∆n2(t)/dt|t)0 ) -βD21∆n1(0) (2) Here ∆Ci(t) or ∆ni(t) denotes the molar concentration of solute i (AX) in the lower chamber less its concentration in the upper chamber at time t, and β is the geometrical cell constant. The cross-diffusion coefficient D21 yields the flux density of NaCl brought about by the gradient in the concentration of AX. Apparent diffusion coefficients Dapp(t) obtained from conductance measurements vs time data are computed from
Dapp(t) ) (F/π2)(d ln(∆Λ(t))/dt)
(3)
F is the length of the diffusion channel (15.052 ( 0.0002 cm), ∆Λ(t) ) rΛB(t) - ΛT(t), where ΛB(t) and ΛT(t) are the electrical conductances determined at the bottom (B) and top (T) electrode pairs at time t, respectively, and r is the ratio of the electrode cell constants ()kB/kT). Diffusion was followed by using Harned’s restricted diffusion experiment13 at levels of one-sixth and five-sixths of the distance between the bottom and top of the solution column. Cell performance was checked by measuring diffusion coefficients of isoionic, salt-free human serum albumin (HSA) at 25 °C. The determined HSA binary diffusion coefficients were in close agreement (2-5%) with published values and those determined by interferometry.15 Compositions of AX (X ) Cl, Br) solutions were determined by HPLC using a lightscattering detector.6 For measuring electrical conductivities at AX concentrations well below the CMCs [0.001 mg mL-1 to 0.005 mg mL-1 in water and in the presence of NaCl or NaOH (NaBr)], the data were recorded by using a Model 2500A from Adeen-Hagerling, Inc. (Ohio), with a special option for conductance measurements as low as 3 × 10-7 S. Correction for solvent conductance (∼0.5 × 10-6 S) was found to be negligible. CMCs were determined by using the Ring method according to du Nuoy, as well as by the Wilhelmy plate technique. Static and Inelastic Light-Scattering Measurements. Static and inelastic light-scattering (quasielastic light scattering, QELS) experiments were performed on samples of AX for the determination of apparent aggregation numbers and the mutual diffusion coefficients under the specified conditions. The values obtained supplement previously determined data6 and are useful for comparison with ternary mutual diffusion coefficients that were determined conductometrically for AOH in this study. The static and inelastic light-scattering (QELS) experiments were conducted with an ALV-LSE-3000 goniometer (Langen, Germany) consisting of LSE/80 light-scattering electronics, a PM20 multiplier, and a monitor unit.16 Unpolarized laser light (NEC-Laser, 50 mW Ne) of 632.8 nm was used. An ALV3000 multibit, multi-τ correlator was operated with 25 simultaneous sampling times covering 8 decades in delay time by applying 220 exponentially spaced channels. All measurements
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J. Phys. Chem., Vol. 100, No. 23, 1996 9883
Λeq ) (Λ - Λ0)/cM
TABLE 1: Apparent Diffusion Coefficients of Aqueous DSDMAX Solutions below the CMC (25 °C) DSDMAX X ) OH
X ) Cl
X ) Br
data setsa (×10-9 m2 s-1) 3b c 1d 2d
103C1 (mM)
1b
2b
0.8 10.0 15.0 18.0 1.70 2.00 2.50 3.00 1.60 1.80 2.30
7.08 7.10 7.10 7.12 6.42 6.45 6.51 6.50 6.55 6.60 6.65
7.09 7.10 7.12 7.15 6.45 6.47 6.51 6.55 6.60 6.65 6.75
7.10 7.09 7.12 7.14 6.48 6.51 6.55 6.56 6.70 6.68 6.75
7.19 7.15 7.09 7.07 6.57 6.50 6.43 6.40 6.65 6.58 6.50
7.05 7.09 7.10 7.10 6.42 6.43 6.45 6.45 6.50 6.55 6.55
7.08 7.08 7.11 7.11 6.42 6.50 6.51 6.49 6.51 6.54 6.57
3d 7.06 7.09 7.11 7.15 6.41 6.50 6.49 6.48 6.50 6.55 6.56
a The data sets (1-3) were obtained by three different operators, and each data set is in triplicate. b Determined through the equivalent electrical conductances, by applying the Nernst limiting equation (eq 8). c Computed according to Onsager and Fuoss by applying eq 11.26 d Values were obtained from interferometry and electrical conductivity measurements.
were made in the homodyne mode, and the data were analyzed by the cumulants method. QELS measurements were performed at 90° and 120° on 1.5 mL samples contained in 6 mm diameter glass tubes, whose temperature was maintained at 25 ( 0.05 °C by a Peltier thermoelectric device. The apparent diffusion coefficient obtained by QELS is different from the mutual diffusion coefficient and is defined as
Dapp20 ) (kBT/f0)(H(K)/S(K)) ) D0(1 + (Kt + KH)φ)
(4)
where kBT is the thermal energy, f0 is the friction factor at infinite dilution, H(K) is the hydrodynamic interaction parameter, and S(K) is the solution structure factor. Kt and KH are the perturbation coefficients for thermodynamic and hydrodynamic interactions, respectively, which are proportional to the second virial coefficient17,18 with kH = 6.45 and kd = 1.52.18 Φ is the volume fraction of AX aggregates above the CMC. Within the limit of non-interacting particles, Dapp coincides with D0AX, so that Dm equals the AX assemblies. At especially low concentrations of AX, but above the CMC (particularly at these low CMCs), Dm = Dapp = D0AX.19 Above the CMCs for AX, micelle-micelle interactions are predominant over counterion diffusion, DX-, resulting in a mutual diffusion coefficient, Dm:
Dm ) DAX(H11/S11)((D0AX(1 - kHφ) = D0AX(1 + kdφ) (5) Results Apparent Diffusion Coefficients of DSDMA X below the CMC. Determinations of the apparent diffusion coefficients of salt-free AX solutions by using the conductometric technique below the CMCs of AX are listed in Table 1. Over the concentration ranges studied, e.g., for AOH between 0.001 and 0.0008 mg mL-1 (0.0018-0.00088 M), for ACl between 0.001 and 0.002 mg mL-1 (0.0017-0.0034 M), and for ABr between 0.001 and 0.0015 mg mL-1 (0.0016-0.0023 M), the apparent diffusion coefficients varied only slightly. A precise measurement of the solution conductivity at AX (X ) OH, Cl, Br) concentrations well below the CMCs (CMCOH ) 1.5 × 10-4, CMCCl ) 3.12 × 10-7, and CMCBr ) 4.28 × 10-5 M) in the concentration ranges noted earlier, permits us to compute the ideal diffusion coefficients of the two physical species: monomer (A+) and counterion (OH-, Cl-, Br-). Extrapolation of the equivalent conductivity according to
(6)
to zero monomer concentration (cM) yielded values for AOH of (3.60 ( 0.1) × 10-19 mS‚cm2, for ACl of (1.51 ( 0.08) × 10-19 mS‚cm2, and for ABr of (1.95 ( 0.05) × 10-19 mS‚cm2. Λ is the electrical conductivity of solutions at 25 °C, Λ0 is the conductivity of water, and cM is the number monomer concentration. We can apply the classical formulation for electrolyte solutions according to
D01 + D02 ) (kBT/e2)Λeq
(7)
where D01 is the ideal diffusion coefficient of the monomer of A+, D02 is the ideal diffusion coefficient for OH-, Cl-, or Br-, e is the charge unit, and kBT is the Boltzmann factor. The values of D01 + D02 obtained are as follows: AOH, 56.42 × 10-6 cm2 s-1; ACl, 24.23 × 10-6 cm2 s-1; and ABr, 24.62 × 10-6 cm2 s-1. Thus, for D01,A, a value of 3.8 × 10-6 cm2 s-1 is obtained by using D02 values for OH-, Cl-, and Br- of 52.7 × 10-6, 20.5 × 10-6, and 20.9 × 10-6 cm2 s-1 respectively, determined from another set of experiments. The reference values published are as follows: OH-, (52.45 ( 0.05) × 10-6 cm2 s-1; Cl-, (20.0 ( 0.08) × 10-6 cm2 s-1; Br-, (21 ( 0.05) × 10-6 cm2 s-1; and Na+, (13.0 ( 0.07) × 10-6 cm2 s-1. At zero AX concentration, the ideal Nernst diffusion coefficients according to
Dm = D0m ) [(2D1D2)/(D1 + D2)][1 + (d ln γ(/d ln c)] (8) are 7.09 × 10-6, 6.43 × 10-6, and 6.55 × 10-6 cm2 s-1 for AOH, ACl, and ABr, respectively. D0m is the mutual diffusion coefficient in the infinite dilution limit for the monomers of AX, which cannot be obtained reliably by quasielastic lightscattering measurements due to the low concentrations of AX below the CMCs. Similar values have been obtained by using capillary zone electrophoresis, applying the Taylor-Aris dispersion theory.20 Determinations of the apparent dissociation constants for AX below the CMC yielded values of Ka = 3.47 × 10-5 M for AOH, Ka = 1.79 × 10-9 M for ACl, and Ka = 3.8 × 10-7 M for ABr. The experimentally determined degrees of dissociation were Rapp ) 0.46 (AOH), 0.20 (ACl), and 0.11 (ABr). Since the experimentally determined values of the Nernst limiting diffusion coefficient are close to the theoretical values, progressive premicellization of AX monomers can be ruled out, as can the growth to dimers, trimers, or tetramers. This would affect the diffusion coefficient variation considerably, which is not observed in this study. Furthermore, the mutual diffusion coefficient is proportional to the activity coefficient factor by (1 + C1(d ln γ( /d C1), which reflects nonideality. We did not notice a significant contribution in C1 (∼C1/2) at these low concentrations of AX, as reported here (Figure 1). However, by adding 10-4 N NaOH to AOH or 10-4 M NaCl to solutions of ACl, we see a pronounced C11/2 dependence, so that the Debye-Hu¨ckel contribution cannot be neglected. In calculating the Debye-Hu¨ckel contribution according to known theory for AX in the presence of 10-4 NaX, particularly for X ) Cl, we notice that the shape of the theoretical curve follows the experimental curve with a slight deviation from the experimental data points of 5-9%. We found that the deviations in the experimental data curve from the complete theoretical curve were always greater than the deviations of the experimental data points. Therefore, our results confirm not only the DebyeHu¨ckel contribution but also the electrophoretic effect. However, the observed negative slope of the effective diffusion coefficient in these cases is probably due to the hydrodynamic
9884 J. Phys. Chem., Vol. 100, No. 23, 1996
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Figure 1. (A) Diffusion coefficients of DSDMAOH at 25 °C in water and in the presence of (a) 10-4 M NaOH and (b) 10-4 M NaCl. (B) Diffusion coefficients of DSDMACl at 25 °C in H2O (a) and in the presence of 10-4 M NaCl (b).
contribution of the AX (X ) OH, Cl) monomer perturbed by an attractive potential, as addressed in eq 5. Furthermore, the observed variation in the apparent diffusion coefficient at 10-4 M salt can also be affected by premicellization phenomena, since it is known that the CMC for this class of surfactants is shifted toward lower CMC values than those in salt-free solutions. Analysis of the apparent diffusion coefficients in this system also has to be considered as due to chemical exchange between monomers and micelles of AX, so that the contribution of the premicellar aggregates may have to be taken into account. This requires further investigation through measurements of the dependence of the diffusion coefficients close to the CMC because it should be possible to discriminate between the diffusion coefficient of aggregates close to the CMC and that for the monomer (DCMC ∼ 4.7 × 10-6 cm2 s-1 vs D1monomer ∼ 7.1 × 10-6 cm2 s-1). This also includes the hydrodynamic contributions, as well as the correction term in d ln γ(/d ln c1 close to the CMC. Table 1 lists the theoretical diffusion coefficients with concentrations of AX in this salt-free system (system I) treated
as a 1:1 electrolyte according to Onsager and Fuoss,21 expressed as
D ) [1 + C1(d ln γ(/d c1)][D0 + Σ∆n]
(9)
where D is the mutual diffusion coefficient of the electrolyte, γ( is the mean activity coefficient, C is the monomer surfactant concentration in molar, D0 is the Nernst-Hartley limiting value of the diffusion coefficient, and ∆n is the electrophoretic term. This term is defined as
∆n ) kBTAn ((Z1ntt20 + Z2ntt10)2/(an|Z1Z2|))
(10)
where An is a function of the dielectric constant, the viscosity of the solvent, the temperature, and the concentration dependent term (ka). Here k is the reciprocal of the average radius of the ionic atmosphere. The parameters t10 and t20 are the limiting transport numbers of the cations and anions, respectively; Z1 is the algebraic valency of the cation, and Z2 is the algebraic valency of the anion.
Diffusion of DSDMA Polyelectrolyte Solutions
J. Phys. Chem., Vol. 100, No. 23, 1996 9885
Figure 2. Determination of the apparent exchange rate of DSDMAX + OH- h DSDMAOH + X- for X ) Cl (a) in the presence of 10-4 M NaOH and a loading of DSDMAX 0.002 M and (b) a loading of DSDMABr of 0.002 M in the presence of 10-4 M NaOH. (c, d) Exchange rate reaction of DSDMACl/DSDMAOH both at concentrations of 0.002 M. is the fraction converted to DSDMAOH or exchanged by OH- from DSDMAX with X ) Cl or Br, respectively.
By applying these equations and assuming dilute solutions of AX, the density term, φ(d) is negligible,21 and by inserting the determined values for ΛA and ΛX, η0 ) 9.001 × 10-3 g‚cm2 s-1, and e ) 79.0, we obtained the theoretical D0 values listed in Table 1. By considering only the electrophoretic term of the first order (n ) 1) for nonsymmetrical electrolytes, Dapp can be compared with Dth on the basis of
D ) (D0 + ∆1)(1 + c(d ln γ(/dc))
(11)
Although not perfect, the theoretical and experimental values do not deviate significantly, and the difference between Dth and Dapp ranges between 0.10 and 0.12 for AOH, is ∼0.15 for ACl, and is ∼0.1 for ABr. The mean values obtained for the diffusion coefficients well below the CMC are 7.10 × 10-9 m2 s-1 for AOH, 6.47 × 10-9 m2 s-1 for ACl, and 6.63 × 10-9 m2 s1 for ABr. The contribution of the electrophoretic term, found by comparing Dth and Dapp, is on the order of ((0.5-0.7) × 10-9 m2 s-1. The reference frame for calculating diffusion coefficients used in our experiments has not been taken into account. However, model calculations on diffusion mixtures that might well be applicable to these experiments have been published by Friedman, Raineri, and Wood.34 Kinetics of the Exchange Reactions of DSDMAX (X ) Cl, Br) with OH-. By determining the concentrations of free chloride and bromide anions through the installed anion selective microelectrodes, we were able to determine the apparent exchange rate of AX + OH- T AOH + X- in the presence of 10-5-10-6 M NaOH, in addition to the electrical conductivity
of this system. The fraction () of AX (X ) Br, Cl) converted to AOH T A+ + OH- with time is shown in Figure 2. Within 15 min, half of the Cl- from ACl is released or exchanged in the presence of 10-4 M NaOH, and after 60 min approximately 85% of the chloride is released from the initial concentration of ACl. The same values were obtained from electrical conductance measurements. The corresponding values for ABr under the same conditions are t1/2 ) 8 min and t90% ) 35 min. AX experiments for X ) Cl, Br vs time were commenced with 0.001 and 0.015 mg/mL Cl- and Br-, respectively. By conducting the same experiments with an equimolar mixture of ACl/AOH or ABr/AOH, values of t1/2 ) 45 min for Cl-, t1/2 ) 25 min for Br-, t90% )120 min for Cl-, and t90% ∼ 100 min for Br- were determined. The apparent equilibrium constants for the exchange reactions for chloride and bromide are Kex ) (5.15 ( 0.07) × 10-5 and (1.09 ( 0.08) × 10-2, respectively, below the CMC, neglecting activity coefficients at this stage as well as chemical relaxation and physical effects due to ionic mobility. Consequently, the pKa values were calculated to be 8.54 for AOH, 7.58 for ABr, and 4.25 for ACl in this low concentration range, well below the CMC. These concentration ranges approximate conditions with respect to surfactant concentrations in vivo,22 but not necessarily the pH and counterion conditions. Apparent Diffusivities of DSDMA X Solutions above the CMC. Binary diffusion experiments have been conducted on AX solutions above the CMC in the absence of NaCl, and the results are listed in Table 2. We confined our measurements for AOH concentrations to between 40 and 150 mM and for
9886 J. Phys. Chem., Vol. 100, No. 23, 1996
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TABLE 2: Binary Diffusion Coefficients of Aqueous DSDMAX (X ) OH, Cl, Br) Solutions in the Absence of Salt above the CMC (25 °C) at Different DSDMAX Concentrations
TABLE 3: Comparison of Measured Binary Mutual Diffusion Coefficients (D11) and Apparent Diffusivities for DSDMAX (X ) OH, Cl, Br) in Water (25 °C) by Inelastic Light Scattering (QELS)
DSDMAX
103 C1 (mM)a
109 D11 (m2 s-1)
sample
103C1 (mM)a
109D11 (m2 s-1)
X ) OH
90 100 120 130 250 50 70 90 150 50 70 90 100
0.342 ( 0.004 0.375 ( 0.005 0.330 ( 0.004 0.319 ( 0.005 0.310 ( 0.004 0.242 ( 0.020 0.241 ( 0.020 0.239 ( 0.020 0.237 ( 0.025 0.229 ( 0.004 0.227 ( 0.004 0.225 ( 0.002 0.225 ( 0.002
DSDMAOH
90 150 50 90 50 90
0.342 ( 0.004 0.310 ( 0.005 0.242 ( 0.020 0.239 ( 0.020 0.229 ( 0.004 0.225 ( 0.002
X ) Cl
X ) Br
DSDMACl DSDMABr
a The concentrations [103C (mM)] are given as monomer concentra1 tions of DSDMAX. The actual micellar concentrations of DSDMAX are in the micromolar range when the apparent aggregation numbers are considered.
a 103C (mM) is given as monomer concentrations of DSDMAX. 1 The actual micellar concentrations of DSDMAX are in the micromolar range.
ABr from 50 to 100 mM. Over these concentration ranges, the binary diffusion coefficients obtained were almost constant and did not vary from day to day measurements. Each set of measurements was repeated three times for each AX concentration as was performed for ACl in the concentration range of 5-25 mM. The major solute species in these experiments are OH- , Cl-, or Br-, neglecting the concentrations of free H+ and OH- ions. Under these conditions, the Aaggn+ aggregates equilibrate rapidly by fast exchange with OH-, as well as with Cl- or Br-. By omitting monomer-micelle exchange, but assuming almost electroneutrality within the local equilibria of these micelles, only one solute flows: AX aggregates, which are independent of the monomer. The mutual diffusion coefficient can be assigned as
J1 ) -D11(dc1/dx)T
(12)
according to Fick’s law, where J1 denotes the molar flux density of AX components and D11 is its binary diffusion coefficient. Under these restrictions, the diffusion process (D11), with respect to the flow of the neutral AX rather than charged aggregates of AX, has the advantage of neglecting electric potential gradients from the transport equations. The measured D11 values for ACl were found to be (0.242 ( 0.002) × 10-9 and (0.237 ( 0.0025) × 10-9 m2 s-1 at concentrations of 50 and 150 mM, respectively, and D11 values for ABr are (0.229 ( 0.004) × 10-9 and (0.225 ( 0.002) × 10-9 m2 s-1 for concentrations of 50 and 100 mM, respectively. The narrow concentration ranges for ACl and ABr are due to solubility problems, which were experienced well above concentrations of approximately 0.15 mg mL-1 in the absence of salt and/or sonication. Furthermore, the history of the sample preparation is particularly crucial for ACl and ABr, but less so for AOH, which is more sensitive to surfactant concentration, ionic strength, and temperature. For example, sonication of aqueous solutions of ACl or ABr in the absence of salt yielded various structures in solution having different hydrodynamic sizes dependent on sonication time and temperature.5,6 Furthermore, fusion of unilamellar or multilamellar vesicles with time does occur after sonication to larger aggregates or to a heterogeneous size population of unilamellar vesicles. The effect of salt, as observed here by using 15 mM NaCl or NaBr for ACl or ABr surfactants, respectively, yielded unilamellar vesicles of different sizes (up to 1000 Å) as determined by QELS, molecular sieve chromatography (Sepharose 2B), and
laser light scattering.23 However, upon separation of the different molecular sizes of ACl or ABr aggregates under welldefined experimental conditions (T, n, c1), it is also possible to conduct similar diffusivity experiments with reasonable confidence. The corresponding D11 values for AOH, covering a much broader concentration range (90-250 mM) than for ACl or ABr, are (0.342 ( 0.004) × 10-9 m2 s-1 to (0.310 ( 0.004) × 10-9 m2 s-1. Due to the presence of more mobile OH- ions and to the dissociation of AOH, the value at the highest AOH concentration is lower, so that diffusion is more rapid than at the lower concentrations of AOH. For comparison, the apparent diffusion coefficients obtained from QELS measurements and interferometric measurements are (0.165 ( 0.005) × 10-9 and (0.170 ( 0.004) × 10-9 m2 s-1, respectively. The limiting diffusion coefficients at these low volume fractions, which were determined by extrapolation to zero AOH concentration, were found to be (0.175 ( 0.003) × 10-9 m2 s-1 by QELS and (0.178 ( 0.003) × 10-9 m2 s-1 by interferometric measurements. The corresponding values for ACl and ABr are (0.192 ( 0.003) × 10-9 and (0.180 ( 0.005) × 10-9 m2 s-1, respectively (Table 3). Clancy et al.6 used QELS and laser light scattering to measure apparent mutual diffusion coefficients for AOH in aqueous solutions at salt and surfactant compositions similar to those used in the present study. Table 3 compares the previous values obtained with QELS with additional ones generated by this study, also by QELS. The fact that very good agreement is obtained between these two sets of experiments is convincing evidence of the high quality of the AOH samples. The deviation from the values obtained by QELS measurements (15-19%) is basically due to different defined diffusion coefficients (see eq 5), which differ from the Dik phenomenological coefficients as defined by Fick’s law and measured through classical diffusion experiments.30 Ternary diffusion coefficients were determined for system II solutions, where X ) Cl, at various pH’s above the CMC and in the same surfactant concentration ranges as were used for salt-free solutions. The ternary diffusion coefficients are described according to
J1 ) -D11∇C1 + D12∇C2
(13)
J2 ) -D21∇C1 + D22∇C2
(14)
where Ji are the values of molar electrolyte fluxes, and Ci are the polyelectrolyte concentration values in moles per unit volume. Subscript 1 indicates AX and subscript 2 indicates NaX (X ) Cl, Br, or OH). The mutual ternary diffusion coefficient Dik yields the flux density of component i produced by the gradient (∇) in the concentration of component k, where the electrolytes have a common anion. We also conducted experiments with AOH (1) in the presence of 0.015 M NaCl at three different pH’s.
Diffusion of DSDMA Polyelectrolyte Solutions
Figure 3. Concentration differences ∆C2 for NaCl (0.001 M L-1) at mean DSDMAOH concentrations of 20 (O) and 40 mM (b), respectively, at 25 °C and pH 7.5 and at pH 5.5 for 20 mM DSDMAOH (0) and 40 mM DSDMAOH (9). 4 and 2 are measurements of concentration differences ∆C2 for NaCl (0.015 M L-1) at mean DSDMACl concentrations of 10 and 20 mM, respectively, at pH 7.5 (25 °C). 3 and 1 are measurements of concentration differences ∆C2 for the NaBr (0.015 M L-1) at mean DSDMABr concentrations of 10 and 20 mM, respectively, at pH 7.5.
Measurements were performed at AOH concentrations of 1020 mg mL-1 and at pH 7.5, 5.5, and 4.5. For all experiments, the mean cell concentration of NaCl was 0.015 M. Figure 3 shows the differences in concentration of NaCl across the cell diaphragm plotted against βt (eq 2). The data obtained for AOH + NaCl + H2O were fit to the quadratic equation ∆C2 ) A(βt) + B(βt)2 and a least-squares fit of the initial slope of ∆C2/d(βt) ≡ Αc2 (r ) 0.9994), from which the D21 values, the crossdiffusion coefficient, which yields the flux density of NaCl generated by the gradient in the concentration of AOH, are listed in Table 4, including calculated values according to eq 14, for the system AOH + NaCl + H2O. A. pH 7.5 Diffusivity of AOH from the lower to the upper cell compartments at pH 7.5 produced a concurrent flow of NaCl into the upper compartment. The D21 values for these AOH solutions are on the order of (8-10) × 10-9 m2 s-1 and are negative at pH 7.5 and an ionic strength of 0.015 M NaCl, which can be taken as a typical environmental value with respect to salt. The determined values of the ternary diffusion coefficients of AOH in the presence of 0.015 M NaCl are smaller than the D11 values for salt-free solutions, i.e., sometimes about 30%, and pH dependent (Tables 3 and 4). The binary diffusion coefficients of NaCl in this concentration range are from 1.52 × 10-9 to 1.54 × 10-9 m2 s-1. The ternary diffusion coefficient of NaCl (0.015 M) is almost 15% smaller than the D22 reported by Harned and Hildreth.24 The ratio of D21/D11 is related to the number of moles of NaCl cotransported per mole of AOH. It has been determined that, at pH 7.5, 50-60 mol NaCl is cotransported. At pH 5.5, 60-100 mol of NaCl is cotransported, and at pH 4.5 almost 100-120 mol of NaCl per mole of DSDAMOH moves. Since AOH is much less dissociated at pH 5.5 than at pH 7.5, the average positive charge on AOH is considerably reduced, and AOH can diffuse independently of the Na+ and Cl- ions. Therefore, it is not surprising that D11 is approximately the same as the DAOH of 0.415 × 10-9 m2 s-1, which is similar to the value determined for the threadlike micelles of AOH.6 This is also consistent with the value of (0.405 ( 0.008) × 10-9 m2 s-1 determined by QELS according to this study. Furthermore, by decreasing the ratio of salt to AOH, the values of D21 also decrease. This is not observed at pH 5.5 since the coupled flow of NaCl is decreasing, which is similar to that seen for the special surfactant composition of ACl/AOH (70/30 w/w) (see the following).
J. Phys. Chem., Vol. 100, No. 23, 1996 9887 Other colloidal polyelectrolytes generate a similar coupled flow of salt as shown for sodium dodecyl sulfate micelles in 0.1 M aqueous NaCl, which cotransports 120-170 mol of NaCl per mole of NaSDS-micelle.12 The dissociation of AOH into A+ and OH- species (R ) 0.25-0.30, Nagg = 6500) yields an A+ species having an average positive charge of 216.0 electron units at pH 7.5 and an ionic strength of 0.015 M. Considering the diffusitivity of the charged A+ aggregate, an electric field will be generated having a magnitude of (RT/FC1)|Z∇C1|, which will slow down the A+ cations and speed up the more mobile OH- counterions, thereby preventing considerable charge separation. Since the diffusion-induced electric field also drives a concurrent flow of Na+ co-ions, D21 takes large positive values. The major solute species at pH 7.5 are A+, OH-, Cl-, and Na+. Therefore, we can estimate D21 values by using the limiting expression for dilute ternary mixtures of two electrolytes having a common ion:
D21 ) tCl|Z|(DNa - DAOHZ)
(15)
where tCl is the transference number of Cl- ions, and DNa and DAOHZ denote the diffusion coefficients of Na+ and AOH ions. For solutions that contain an excess of NaCl, according to the present conditions, tCl can be approximated as DCl/(DCl + DCl). By applying eq 15 and the determined diffusivities for DCl ) 2.15 × 10-9 m2 s-1, DNa ) 1.32 × 10-9 m2 s-1, and DA ) 0.41 × 10-9 m2 s-1, we calculate D21 = 7.2 × 10-9 m2 s-1. Considering no corrections for electrophoretic terms and nonideal solution thermodynamics, the agreement between calculated and measured values is reasonable, having approximately the same magnitudes. B. pH 5.5. Lowering of the pH of the aqueous solution of AOH to pH 5.5 in the presence of 0.015 M NaCl makes the average charge of AOH close to zero, although 5-7 mol of NaCl is cotransported. Apparently this is close to the isoionic value of the AOH micelle, which was determined to be 5.54 by electrophoretic and conductivity methods.25 Furthermore, the degree of dissociation of AOH micelles is only on the order of 0.02-0.05. Since the electric field along the AOH concentration gradient vanishes, the diffusion of AOH at pH 5.0 does not produce a significant coupled flow of NaCl. Therefore, D21 values around pH 5.5 are relatively small. In addition, almost the same values were obtained for ACl at pH 5.5-6.5. Although the magnitudes for AOH and ACl are the same, the absolute values for AOH are 25 ( 5.5% higher than those determined for ACl, within the concentration ranges of these two AX concentrations studied. The situation for ABr at pH 5.0-5.5 is similar to that of ACl, although the D21 value for ABr is close to that for AOH. This is probably due to a higher degree of dissociation. C. pH 4.5. The results for AOH between pH 4.5 and 5.0 are particularly interesting since they reflect the physicochemical behavior of AOH within the aqueous environment, i.e., it reveals the different behavior of the AOH surfactant within the extracellular regime (pH ∼5.0) and the intracellular regime (pH ∼7.5). The AOH solution was acidified with 0.01 N HCl in the presence of 15 mM NaCl. The results are shown in Figure 4, revealing that diffusing AOH generated a substantial flow of NaCl into the lower cell reservoir; thus, the D21 values are significant and negative. This assumes that H+ ions interact with AOH according to
AOH + H+ T A+ + H2O
(16)
where the surfactant AOH carries a net positive charge since the pH is below the isoelectric value.33 It is safe to argue that,
9888 J. Phys. Chem., Vol. 100, No. 23, 1996
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TABLE 4: Determined and Calculated Ternary Diffusion Coefficientsa for Aqueous DSDMAOH Solutions (C1) in the Presence of NaCl (C2) at 20 °C (pH 7.5) 103C1 (mM) 20 40 60 20 (DSDMAOH/ DMAClb)
c
103C2 (mM)
109D11
109D12
109D21
109D22
D21 theorc
5.0 10.0 15.0 5.0 10.0 15.0 5.0 10.0 15.0 5.0 10.0 15.0
0.351 ( 0.003 0.352 ( 0.004 0.356 ( 0.004 0.349 ( 0.005 0.348 ( 0.007 0.349 ( 0.006 0.342 ( 0.005 0.378 ( 0.006 0.380 ( 0.004 0.410 ( 0.005 0.409 ( 0.005 0.405 ( 0.005
0.005 ( 0.003 0.009 ( 0.004 0.012 ( 0.006 0.006 ( 0.003 0.012 ( 0.004 0.014 ( 0.006 0.007 ( 0.004 0.009 ( 0.006 0.016 ( 0.007 0.010 ( 0.002 0.013 ( 0.003 0.017 ( 0.004
-7.9 ( 0.5 -6.5 ( 0.5 -2.4 ( 0.5 -7.2 ( 0.4 -6.3 ( 0.4 -2.5 ( 0.4 -6.9 ( 0.5 -4.3 ( 0.5 -2.7 ( 0.5 -6.9 ( 0.2 -5.0 ( 0.4 -2.9 ( 0.4
1.52 ( 0.04 1.52 ( 0.05 1.54 ( 0.04 1.50 ( 0.04 1.52 ( 0.04 1.54 ( 0.04 1.52 ( 0.05 1.52 ( 0.04 1.54 ( 0.04 1.52 ( 0.04 1.53 ( 0.04 1.54 ( 0.04
-8.52 -7.09 -3.12 -8.25 -7.05 -3.09 -7.85 -5.00 -3.05 -7.15 -6.05 -3.09
a Dik in m2 s-1. b Equimolar mixture of DSDMAOH and DSDMACl. Concentrations are expressed as the concentration of surfactant monomers. Dtheor was calculated using eq 15 or 17.
Figure 4. Concentration differences, ∆C2, for a NaCl concentration of 0.015 M L-1 at mean DSDMAOH concentrations of 20 (O) and 40 mM (b) at pH 4.5 (25 °C).
in our pH 4.5 solutions, diffusion of A+ cations and Clcounterions established an electric field that can act to speed up the cation tranport and slow down the chloride ions. Since the electric field transported Cl- ions counter to the flow of A, D21 values are negative. Similar experiments carried out with ACl or ABr at pH 4.5 are difficult to perform due to the low solubilities of ACl and ABr, i.e., precipitation occurred over the measuring time; hence, no meaningful results were obtained. The D21 values for AOH at pH 4.5 can be compared with values computed from an equation similar to eq 15: D21 ) |Z|tCl(DAOH - DCl), with the diffusivity DCl ) 2.15 × 10-9 m2 s-1 and DAOH ) 0.41 × 10-9 m2 s-1. By comparing the experimentally obtained results with those calculated by applying the diffusion coefficient for AOH at pH 4.5, as determined by QELS, close agreement is only obtained in a surfactant concentration range of 0.05-0.10 mg mL-1 in the presence of 15 mM NaCl, but not at higher surfactant concentrations between 0.20 and 0.30 mg mL-1. Diffusity of DSDMACl and DSDMAOH. Equimolar amounts of ACl + AOH above the CMC at pH 5.5 were studied with respect to diffusivity. These experiments circumvent the solubility problems encountered with ACl at pH 4.5-5.5. The concentration range studied for this equimolar ratio of ACl/ AOH was between 0.05 and 0.5 mg mL-1. The D21 values for this composition and in the presence of 0.015 M NaCl are relatively small, so that approximately 4-6 mol of NaCl is transported, which is almost the same value as that determined for ACl at pH 5.5. It is surprising that this ACl/AOH system behaves as a compact unit, revealing micellar aggregates containing AOH and ACl as a unit entity. If the contribution were due only to AOH at pH 5.5 in the presence of 0.015 M NaCl, we would expect a higher or lower D21 contribution. On
the other hand, we attribute the cotransport of NaCl to ACl because the electric field along the AOH concentration gradient has vanished. Therefore, the contribution for coupled flow of NaCl is insignificant, leaving only the cotransport of NaCl due to ACl (Table 4). QELS measurements of this particular system reveal an apparent mutual diffusion coefficient of (1.80 ( 0.55) × 10-7 cm2 s-1, which is the z-average diffusion coefficient. This value is close to that of AOH at pH 6.5 in the presence of 0.01 M NaCl, but different from that of ACl under the same conditions of 〈D〉z ) (2.91 ( 0.6) × 10-7 cm2 s-1. More reliable results may be obtained by QELS techniques.27 Diffusivity of the 30% (w/w) DSDMAOH/70% (w/w) DSDMACl System. This is a particularly interesting system since it has been used for the synthesis of mesoporous materials, including those of siliceous and molybdenum molecular sieves of pore diameters between ∼40 and 400 Å (Paradies, unpublished results, 1993). An unusual aspect of this micellar system [which has an aggregation number of 60, an apparent molecular weight of 60 000, and an apparent mutual diffusion coefficient of (5.95 ( 0.05) × 10-7cm2 s-1 as determined by QELS (a ) 0.18)] is the insensitivity against NaCl within a concentration range of 0.01-0.1 M NaCl. It should be noted that AOH is reported to be very sensitive against 0.01-0.1 M salt or surfactant concentration due to transitions from vesicular structures T threadlike micelles T spherical micelles.6 The determined mutual diffusion coefficient is naturally the z-average diffusion coefficient at a surfactant concentration of c1 ) 10 mM. Furthermore, considering the transition of AX (X ) Cl, Br) + OH- f AOH, given the equilibrium H2O ) H+ + OH-, the existence of the ACl/AOH (70/30 w/w) as a preferred and stable composition is quite surprising. The values determined for ternary mutual diffusion coefficients, e.g., D11 and D21 are listed in Table 5, together with the determined apparent mutual diffusion coefficients (QELS). At pH 7.5, diffusion of these particular aggregates produced counterflow of NaCl into the upper compartment. D21 for these solutions is on the order of 7.0 × 10-9 m2 s-1 due to the positive charge of the micellar species. The measured values for D11 at pH 5.4, which is the apparent pH when dissolving ACl/AOH (70/30) in water at 25°C, are (0.266 ( 0.0004) × 10-9 and (0.243 ( 0.006) × 10-9 m2 s-1 at surfactant concentrations of 0.25 and 0.35 mM, respectively. The corresponding value at pH 7.5 (20 °C) is quite different and is determined to be (0.369 ( 0.005) × 10-9 m2 s-1 at 0.25 mM surfactant. The measured ternary diffusion coefficients at 0.25 or 0.35 mM surfactant in the presence of 5.0, 10, and 15 mM NaCl are listed in Table 5, together with the binary values of D11 that were determined for salt-free solutions. Due to the relatively large molar conductance of these surfactant solutions compared to that of ACl or
Diffusion of DSDMA Polyelectrolyte Solutions TABLE 5: Measured Ternary Diffusion Coefficients for an Aqueous Solution of DSDMACl/DSDMAOH (C1) (20/30 w/w) in NaCl (C2) Solutions (25 °C)a 103C1 (mM)
pH
103C2 (mM)
109D11 (m2 s-1)
20 40 40 40 40 20 40 40 40 40
5.4 5.4 5.4 5.4 5.4 7.5 7.5 7.5 7.5 7.5
0.0 0.0 5.0 10.0 15.0 0.0 0.0 5.0 10.0 15.0
0.266 ( 0.004 0.243 ( 0.006 0.209 ( 0.015 0.15 ( 0.03 0.13 ( 0.03 0.369 ( 0.005 0.340 ( 0.005 0.245 ( 0.005 0.170 ( 0.004 0.120 ( 0.005
109D12 (m2 s-1)
-0.008 ( 0.005 -0.015 ( 0.003 -0.019 ( 0.003 -0.008 ( 0.003 -0.021 ( 0.003 -0.041 ( 0.003
a 103C (mM) is the concentration of the the monomer surfactant. 1 The actual micellar concentrations are in the micromolar range when the apparent aggregational numbers and weight-average molecular weights are considered. The mutual diffusion coefficient of DSDMACl/ DSDMAOH (70/30) at pH 5.4 (25 °C) in the presence of 0.015 M NaCl was determined by QELS to be (0.0595 ( 0.005) × 10-9 m2 s-1.
ABr, the values of D11 and D12 can be determined with more precision than the values of D21 and D22 for the NaCl component. The binary diffusivity of NaCl between 5 and 15 mM NaCl ranges from 1.54 × 10-9 to 1.55 × 10-9 m2 s-1 according to Harned and Hildreth,24 but here the values are about 15% smaller. The ratio of D21/D11 is related to the number of moles of NaCl cotransported per mole of surfactant and gives values between 20 and 30 at pH 7.5 and between 6 and 10 at pH 5.5. Since this particular surfactant has more counterions at pH 7.5 than at pH 5.5, the gradient at pH 7.5 generates a stronger electric field and it produces a larger coupled flow of NaCl compared to pH 5.5. If the ratio NaCl-ACl/AOH (70/ 30) decreases, so does the value of D21 because there cannot be a coupled flow of salt in a NaCl-free solution. The apparent mutual diffusion coefficients using QELS (Tables 3 and 4) are not necessarily the same as those determined by free diffusion; therefore, it is not surprising that precise agreement is not obtained. Since light scattering measures fluctuations in concentration, and micellar systems are particularly sensitive to salt, NaCl can contribute, in our case, to coupled scattering fluctuation and concentration fluctuation of the micelles at these concentrations. This can contribute directly to the autocorrelation function, particularly within the small K limit of QELS. Moreover, the first cumulant, Γ1 ) K2kBT/f0S(0), is related not only to the friction coefficient of the AOH micelles but also to the solution structure factor S(0). Since I(k) ∼ S(K) and I1 ∝1/I(K), the interaction strength between AOH micelles is also governed by the interaction potential and the hydrodynamic interaction tensor (eqs 4 and 5). Possible expansions of AOH micelles or AX (X ) OH, Cl, Br) aggregates are not taken into account by using lightscattering measurements, but they can have significant influence on D11 [see Tyrrell (ref 28)]. Discussion In a previous investigation by Quirion and Desnoyer,35 the authors reported a value of 2 for the ratio of the binding constants of Br and Cl to cetyltrimethylammonium (CTAB) micelles. This value is in accordance with a larger degree of counterion dissociation for the Cl than for the Br salt. The degree of counterion dissociation of CTAB decreases from 0.19 to 0.10, which is similar to numbers for the AX system reported here.36 These authors attributed the decrease to a postmicellar transition. A postmicellar transition in this double-chain micellar system seems to be possible, as indicated by recent
J. Phys. Chem., Vol. 100, No. 23, 1996 9889 cryotransmission electron microscopy experiments (Paradies, unpublished results). However, the double-chain cationic surfactants, particularly those of the C18 class, behave very differently from single-chain materials, as evidenced by differences in solubility, sensitivity to temperature and ionic strength, counterion dependence, and CMCs. The diffusion components have been treated as a binary or ternary mixture of 1:n and 1:1 electrolytes. It should be noted that, with this simplification (i.e., neglecting electrophoretic interactions between migrating ions, effects of changes in solution viscosity, and nonideality by considering activity coefficients), quite reasonable results are obtained despite the fact that quantitative estimates of these contributions to the Dik coefficients are not available. Notwithstanding these drawbacks, the theoretical values within the limiting regime for D11, D12, D22 and D21 can be calculated according to the following equations:
D11 ) DAOH + tDSDSMAOH(DNa - DAOH)
(17)
D12 ) tDSDSMAOH(DNa - DCl)
(18)
D22 ) DCl + tCl(DNa - DCl)
(19)
D21 ) ntCl(DNa - DAOH)
(20)
for the system AOH (1) + NaCl (2) in water. DAOH, DCl, and DNa are the limiting diffusion coefficients of micellar AOHn+, Cl-, and Na+ ions, respectively. We can express the ionic transference number as
ti ) CiZi2D1/∑CmZm2Dm
(21)
yielding the fraction of the total electric current due to ion i in an internally applied electric field. Zi and ci are the valence and molar concentrations of the ions present, respectively. These equations are only valid as the solute concentrations approach zero. However, D12 reflects the interactions between ions and micelles, which are supposed to be the major source of interactions even in dilute solutions, as shown very recently for another cationic micellar system.29 By using these approximations and the unresolved restrictions mentioned earlier, the self-diffusion coefficients of the Na+ ions, AOHn+ micelles, and Cl- ions are represented as DNa, DAOH, and DCl, respectively. In the calculations it was assumed that the mutual diffusion coefficient of An+ ions is identical with the diffusion coefficient of AOH micelles, as measured by QELS under the experimental conditions reported here. The average value is 0.410 × 10-9 m2 s-1 (pH 4.5, 〈D〉z = 0.391 × 10-9; pH 5.5, 〈D〉z = 0.401 × 10-9; pH 7.5, 〈D〉z = 0.415 × 10-9 m2 s-1). Under the assumption that the AOH micelles do not undergo considerable changes in shape or apparent aggregation numbers, and hence there is no molecular weight change, this approximation seems justified because no drastic changes in the concentration ranges of AOH micelles studied have been detected. The limiting diffusivity of Na+ and Cl- ions (1.33 × 10-9 and 2.03 × 10-9 m2 s-1), which are close to those determined here, was taken from Robinson and Stokes30 and compared to our measured values of 2.05 × 10-9 m2 s-1 for Cl- and 1.35 × 10-9 m2 s-1 for Na+. By increasing the NaCl concentration, the diffusivity of the AOH micellar component will drop from the binary Nernst value of
D11 ) (n + 1)(DAOHDCl)/(DCl + nDAOH) for C2/C1 ) 0 (22)
9890 J. Phys. Chem., Vol. 100, No. 23, 1996
Thies et al.
For salt-free AOH micellar solutions, the self-diffusion coefficient of AOHn+ ion becomes
D11 ) DAOH
(23)
when C1/C2 ) 0, for solutions containing excess NaCl with tAOH ) 0 (eqs 18 and 21). This actually results in a coupled flow of Na+ instead of speeding up AOHn+ ions. Furthermore, there is an increase in D21, which measures the coupled flow of NaCl produced by the AOH gradient with D21 ) 0, when C2/C1 ) 0:
D21 ) (nDNa + (DCl - DAOH))/(DNa + DCl)
(24)
D22, the diffusivity of NaCl, decreases from DCl (2.03 × 10-9 m2 s-1) to the binary value 2DNaDCl/(DNa + DCl) (1.61 × 10-9 m2 s-1) when the fraction of NaCl increases from 0 to 1, as observed in this study. D12 is a measure of the flux of AOH micelles generated by NaCl, ∇C1. At large concentrations of NaCl over AOH, D12 vanishes (C1/C2 f 0) despite the fact that AOH micelles at large salt concentrations collapse to small micelles having an aggregation number of Nagg = 61;6 hence, -J1 ) D11∇C1. This does not necessarily imply that D21 vanishes since the diffusion of AOH micelles will still cotransport NaCl. This situation raises another question: How do the cotransport properties of salt affect the diffusivity of AOH micelles due to different conformational states (e.g., vesicles, threadlike micelles, and spherical micelles) within certain pH ranges? Although more studies are needed, particularly at different temperatures (∼10-15 °C), with various salts [MCl (M ) Na, K) and CaCl2], and with AOH concentration, it is safe to claim that under these restricted surfactant concentrations and salt conditions the only important hydrodynamic species is the AOH aggregate with Nagg = 6000. By using measurements of the diffusion of AOH conducted by using refractive index measurements and comparing them with those determined from conductivity measurements, a difference was determined in the apparent diffusion coefficients of (0.1-0.15) × 10-9 m2 s-1 from the beginning of the experiment until the end of the experiment. It is assumed that the apparent diffusion coefficient of AOH is constant over the multicomponent diffusion, although this might be the reason for the difference since a significant portion of NaCl is ahead of the initial boundary, leaving behind a region depleted of NaCl. Some measured and predicted values of Dik are listed in Tables 2 and 4. The agreement is not overwhelming, although some are quite reasonable, but the theoretical values have the right orders of magnitude. pH and Ionic Strength Dependence. Figure 3 demonstrates the pH dependence of AOH micelles in the presence of 0.015 NaCl as determined experimentally. Also included are the calculated values of the ternary diffusion coefficient (D11, for AOH micelles as a function of pH and salt) determined by applying the appropriate equations and by using the proper assignments for Na+ and Cl-. Near pH 5.5, the average charge of AOH micelles is very low due to the low fraction of dissociated OH- (R ) 0.02), although the pKa of the monomer is 8.54. At pH 5.5, AOH micelles diffuse almost freely and are independent of salt. D11 equals DAOH = 0.405 × 109 m2 s-1. Figure 5 shows the calculated values of D11 for A+ in the presence of various concentrations of NaCl (C2) and different pH’s. By adding HCl or NaCl (0.015-0.001 M) to a solution of micellar AOH to obtain a pH of 4.5 or 5.5, respectively, we notice electrostatic coupling of the micelles with the mobile Cl- (pH 4.5) counterion and a considerable increase of the mutual diffusivity of AOHn+ micelles. A similar phenomenon
Figure 5. Calculated values of ternary diffusion coefficient D11 for DSDMAOH (C1) + NaCl (C2) solutions at 25 °C for different values of pH and concentrations of NaCl.
is seen at pH 7.5 for A+ micelles; however, the magnitude of D11 is lower when compared to the value at pH 4.5. The apparent changes in the AOH micellar diffusivities are a reflection of the hydrodynamic behavior (eq 5) and changes in S(K) are due to micellar interactions as seen by QELS or the observed transformation of AOH from vesicles to threadlike micelles (particularly at pH 7.5) or micelles with Nagg ) 61 at high salt concentrations.6 The work reported here supports the former view of a salt-induced change in AOH micelles, but not in ACl or ABr. DSDMAOH Diffusivities: Mutual, Self, and Apparent. The difference between measured and calculated mutual diffusion coefficients of proteins or other biological macromolecules is found in the assumption that the mutual diffusion coefficient equals the diffusion coefficient of the polyion. This is true for proteins if a large excess of supporting electrolyte is present (especially for light, neutron, and X-ray scattering experiments) to avoid Donnan equilibrium.31 This assumption does not hold for the system studied here, except for fairly small salt concentrations due to the transformation of AOH micelles from the vesicular state over a threadlike state to small micelles having an aggregation number of 60. These various states are both salt and surfactant concentration dependent. Therefore, the assumptions that excess supporting electrolyte allows electrophoretic effects to be neglected and that the activity coefficient of the diffusing salt that is constant along the diffusion path is unity may be not valid for AOH micelles. By treating AOH (1) and NaCl (2) as a multicomponent system rather than a simple binary system, it should be possible to estimate the apparent diffusivity of AOH when a salt is added. By assuming a constant pH (pH 4.5, 5.5, and 7.5) and an AOH gradient in a NaCl solution (0.015 M NaCl) with a uniform distribution of NaCl, diffusion of AOH will generate a coupled flow of NaCl, which is balanced by the NaCl flow back against the NaCl gradient.32 This yields J2 = 0 ) -D21∇C1 - D11∇C2. The induced gradient in NaCl is equal to (D12/D22)∇C1 (eqs 13 and 14). The ∇C2 gradient will in turn govern the flux of AOH, ∇C1, which is D11∇C2 ) (D12D11/D22)∇C1. This observation shows that the apparent diffusivity of the AOH with the abovementioned restriction differs from the true value by a factor of D12D21/D22. Furthermore, and more importantly for environmental concerns, it shows that a fraction of the AOH component is practically driven by a gradient brought about by NaCl, although there was no initial gradient of NaCl! This has been shown previously for SDS micelles in the presence of NaCl.12 Another interesting aspect of this investigation is a comparison of the mutual diffusion coefficients (as determined here) with the intradiffusion coefficients of AX, particularly for AOH. Whereas the mutual diffusion coefficient is important over a wide range of mass transport processes, the intradiffusion
Diffusion of DSDMA Polyelectrolyte Solutions coefficients (determined in the presence of D2O) will supply us with basic information about solvation as about well as the structure and dynamics of AOH solutions.33 Conclusions A detailed analysis of the diffusivity of DSDMAX (X ) OH, Cl, Br) has been conducted in the absence and presence of sodium chloride (0.015 M) at three pH values [4.5, 5.5, and 7.5 (25 °C)] over a broad surfactant concentration range covering concentrations below and above the CMC. By applying a multicomponent analysis for the measured diffusivities, the large coupled flow of NaCl in a pH dependent manner can be explained on the basis of known diffusion coefficients and concentrations of surfactant, counterions, and salt. The present work also shows that the apparent diffusion coefficients for the cationic micelles are higher than their true diffusivity due to the presence of NaCl. The salt and pH dependent diffusivities of DSDMAX, particularly for X ) OH, are of particular interest because DSDMAOH undergoes a reversible transformation from a vesicular state, through a threadlike state, to a micellar state. This process is also concentration and salt dependent. Acknowledgment. M.T. and H.H.P. acknowledge the support of grants from Euram-Brite (No. 4088) and Project 28211 (No. Par 3). S.F.C. acknowledges F. Friedli, J. Fuller, and the Witco Technical Center in Dublin, Ohio. References and Notes (1) Tanford, C. The Hydrophobic Effect; Wiley: New York, 1973. (2) Candau, S. J.; Hirsch, E.; Zana, R.; Adam, M. J. Colloid Interface Sci. 1988, 122, 430. Imae, T.; Ikeda, S. J. Phys. Chem. 1986, 90, 5216. Porte, C.; Appel, J.; Poggi, Y.; J. Phys. Chem. 1980, 84, 3105. Porte, C.; Appell, J. J. Phys. Chem. 1981, 85, 2511. Brown, W.; Johansson, K.; Algrem, M. J. Phys. Chem. 1989, 93, 5888. (3) Brady, J. F.; Evans, D. F.; Warr, G. G.; Grieser, F.; Ninham, B. W. J. Phys. Chem. 1986, 90, 1853. (4) Paradies, H. H. Angew. Chem., Int. Ed. Engl. 1982, 94, 793. Paradies, H. H. Angew. Chem. 1982, 1670-1681. (5) Fendler, J. H. Acc. Chem. Res. 1980, 13, 7. (6) Clancy, S. F.; Steiger, P. H.; Tanner, D. A.; Thies, M.; Paradies, H. H. J. Phys. Chem. 1994, 98, 11143. (7) Paradies, H. H.; Thies, M.; Hinze, U. Int. Symp. Mol. Chirality, Kyoto, Japan, 1994, 151-156. Paradies, H. H.; Clancy, S. F.; Steiger, P. H.; Theis, M. Materials Research Society Surface Symposium, December 1995, Paper N16; Functionalized Materials Symposium, National Academy of Science (Ukraine), 1995, 2, 91-101. (8) DSDMACl(OH) and ATP, ADP, GTP, and GMP form highly organized columnar rods in solution at DSDMACl(OH) concentrations of 1-2 mM and 10-20 µM nucleotides at pH 7.0-7.5 (25 °C) in HEPES or Na2HPO4 buffer. The DSDMA(Cl)OH‚ATP complexes, in the presence or absence of NaCl, form columnar mesophases; the columns are composed of a stacked array of ATP or ADP dimers held together by Coulomb interactions between the cationic surfactants and stacking interactions (Paradies, unpublished results, 1993). Moreover, there are strong interactions between DSDMAX and exopolymers produced by environmental microorganisms, which we feel have gone unnoticed by many environmental chemist (Paradies, H. H. In Physico-chemical Aspects of Metal-Biofilm Interactions; Gaylarde, C., Videla, H., Eds.; Cambridge University Press: Cambridge, UK, 1995; 196-269). (9) Vitagliano, V.; Laurentino, R.; Constantino, L. J. Phys. Chem. 1969, 73, 2456. (10) Leaist, D. G.; Lyons, P. A. J. Phys. Chem. 1982, 86, 1542.
J. Phys. Chem., Vol. 100, No. 23, 1996 9891 (11) Leaist, D. G. J. Phys. Chem. 1989, 93, 474. Keller, K. H.; Canales, E. R.; Yum, S. I. J. Phys. Chem. 1971, 75, 379. Gosting, L. J. AdV. Protein Chem. 1956, 11, 429. Tanford, C. Physical Chemistry of Macromolecules; Wiley: New York, 1961; 553. Tinoco, J., Jr.; Lyous, P. A. J. Phys. Chem. 1956, 60, 1342. Leaist, D. G. J. Solution Chem. 1987, 16, 805. Doherty, P.; Benedek, G. B. J. Chem. Phys. 1974, 61, 5426. Phillies, G. D. J.; Benedek, G. B.; Mazer, N. A. J. Chem. Phys. 1976, 65, 1883. Neal, D. G.; Parich, D.; Cannell; D. S. J. Chem. Phys. 1984, 80, 3469. Wagner, M. L.; Scheraga, H. A. J. Phys. Chem. 1956, 60, 1066. (12) Leaist, D. G. J. Colloid Interface Sci. 1986, 111, 240. (13) Harned, H. S.; Nuttall, R. L. J. Am. Chem. Soc. 1947, 69, 736. (14) Gower, I. Optical Communication Systems; Prentice Hall Intern.: Hewel Hempstead, 1984; Section 6.6. Wilson, I.; Hawkes, J. F. B. Optoelectronics; Prentice Hall: New York, 1989; p 414. (15) Creeth, J. M. Biochem. J. 1952, 51, 10. (16) Paradies, H. H.; Thies, M.; Hinze, U. Ber. Bunsen. Ges. Phys. Chem. 1994, 98, 938. (17) Batchelor, G. K. J. Fluid Mech. 1983, 131, 155. (18) Felderhof, J. Phys. A 1978, 11, 928. (19) Schmitz, K. S. Introduction Dynamic Light Scattering by Macromolecules; Academic Press: Boston, 1990. (20) Aris, R. Proc. R. Soc. London 1956, A235, 67. Taylor, G. Proc. R. Soc. London 1954, A225, 473. Taylor, G. Proc. R. Soc. London 1953, A219, 186. (21) Onsager, L.; Fuoss, R. M. J. Phys. Chem. 1932, 37, 2689. Fuoss, R. M.; Accascina, F. Electrolytic Conductance; Interscience: New York, 1959. (22) Clancy, S. F.; Tanner, D. A.; Thies, M.; Paradies, H. H. Extended Abstract, Division of Environmental Chemistry, San Francisco, CA, 203rd National Meeting of the American Chemical Society, 1992, 32 (1), 907909. (23) Paradies, H. H.; Tanner, D. A.; Clancy, S. F. Unpublished results, 1993. (24) Harned, H. S.; Hildreth, C. L. J. Am. Chem. Soc. 1951, 73, 650. (25) The pKa values were determined according to the equation:
ca - cs - [H+] 0.509xcA pKa ) pH + log + cs + [H+] 1 + xcA where cA) na/(VA + x), cs ) xcb/(VA+ x), and [H+] ) 10-pH. nA is the number of moles of DSDMAX, VA is the initial volume of the DSDMAX solution, x is the volume of added base solution, cb is the initial normality of base, cA is the concentration of acid, and cs is the concentration of base. (26) Barlin, G. B.; Perrin, D. O. In Techniques of Chemistry; Weissberger, A., Ed.; Wiley-Interscience: New York, 1972; Vol. IV, Chapter IX. (27) Phillies, G. D. J.; Stott, J.; Ren, S. Z. J. Phys. Chem. 1993, 97, 11563. (28) Tyrell, H. J. V.; Harris, K. R. Diffusion in Liquids; Butterworths: London, 1984; p 208. (29) (a) Thies, M.; Hinze, U.; Paradies, H. H. Colloids Surfaces 1995, 101, 261. (b) Paradies, H. H.; Thies, M.; Hinze, U.; Poschmann, M. Colloids Surfaces 1995, 101, 159. (30) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Butterworths: London, 1965; Chapter 11, Appendix 6.1. (31) Paradies, H. H. Eur. J. Biochem. 1981, 118, 187. Paradies, H. H.; Kagawa, Y. FEBS Lett. 1982, 137, 25. (32) Noulty, R. A.; Leaist, D. G. J. Solution Chem. 1987, 16, 813. (33) Leaist, D. G.; Ho, L. J. Phys. Chem. 1994, 98, 4702. (34) Friedman, H. L.; Raineri, F. O.; Wood, M. D. Chem. Scr. 1989, 29A, 49. Friedman, H. L.; Zhong, E. C.; Raineri, F. O. In Lectures on Thermodynamics and Statistical Mechanics; Gonzales, A. E., Vera, C., Medina-Noyola, M., Eds.; World Scientific Publishing Company: Singapore, 1989; p 185. Raineri, F. O.; Wood, M. D.; Friedman, H. L. J. Chem. Phys. 1990, 92, 649. (35) Quirion, F.; Desnoyers, J. E. J. Colloid Interface Sci. 1986, 112, 565. (36) Quirion, F.; Magid, L. J. J. Phys. Chem. 1986, 90, 5435.
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