J. Chem. SOC., Faraday Trans. I, 1982, 78, 255-271 Water Oxygen- 17 Magnetic Relaxation in Polyelectrolyte Solutions BY BERTIL HALLE* AND LENNART PICULELL Division of Physical Chemistry 1, Chemical Center, S-220 07 Lund 7, Sweden Received 16th February, 198 1 Water oxygen- 17 longitudinal and transverse relaxation rates have been measured at two frequencies for aqueous solutions of poly(acry1ic acid) and poly(methacry1ic acid) under conditions of variable temperature. concentration and degree of dissociation. The proton-exchange broadening of the ''0 resonance from these polyacid solutions has also been investigated. The data show that the residence time for water molecules interacting with the polyacid is of the order of s, during which time they reorient anisotropically, an order of magnitude slower than bulk water, and engage in rapid proton transfer with acidic groups.The motional perturbation of associated water is to a large extent electrostatically induced. A new method for determining an upper limit for the lifetime of 'bound' water from the proton-exchange broadening is employed and the necessary equations are derived. The perturbing effect of macromolecules in aqueous solution on the structure and dynamics of water, commonly referred to by the imprecise term ' hydration', has been intensely studied in recent years.l Before the ultimate goal of a full understanding of the interaction of water with complex biological structures can be accomplished, it will be necessary to study a wide variety of simpler model systems.One class of systems, which shares important features with many biological macromolecules, is aqueous solutions of synthetic linear polyelectrolytes. Of the various manifestations of the water-macromolecule interaction, effects on the rate and anisotropy of water reorientation, as well as the average residence time of a water molecule in the perturbed region, are probably most directly studied through the magnetic relaxation of solvent nuclei. In the present investigation of aqueous solutions of poly(acry1ic acid) (PAA) and poly(methacry1ic acid) (PMA) we have studied the magnetic relaxation of water oxygen-I7 nuclei. This nucleus is preferable to the other two accessible solvent nuclei, the proton and the deuteron, for several reasons. The 1 7 0 nucleus relaxes through the interaction of its electric quadrupole moment with the field gradient generated by the charge distribution of the water molecule.The virtual absence of intermolecular contributions2* l6 greatly simplifies the molecular interpretation of experimental relaxation rates. The large magnitude of the quadrupolar interaction leads to observable relaxation enhancements at comparatively low polyelectrolyte concentra- tions and makes the 170 relaxation rates much less sensitive to paramagnetic impurities, particularly in comparison with proton relaxation. Furthermore, it has been shown that, in aqueous solutions of PAA and PMA, the deuteron relaxation is determined by the rapidly exchanging acidic deuterons on the polyacids and thus cannot yield information on the state of water in these ~ y s t e r n s .~ ? ~ 255256 1 7 0 N.M.R. I N POLYELECTROLYTE SOLUTIONS EXPERIMENTAL CHEMICALS Aqueous solutions of poly(acry1ic acid) (25 %) and poly(methacry1ic acid) (20 %) were obtained from B.D.H. Chemicals. The degree of polymerization for the PAA preparation is ca. 3200 according to the manufacturer. Since no such information was available for the PMA preparation, we determined the intrinsic viscosity at 3O.O0C and pH 2.7. The value thus obtained, [q] = 5.8 cm3 g-l, corresponds5 to a degree of polymerization of ca. 90. Polyacid solutions were made by dilution with doubly distilled water (quartz apparatus) and titrated to complete dissociation (a = 1) with NaOH prior to relaxation measurements. The sodium-ion concentration was thus, at all degrees of dissociation, equal to or in slight excess of the monomer concentration.Polyacid concentrations are given as mol monomer per kg H,O (m), throughout this paper. Samples of ca, 4 g polyacid solution, enriched to ca. 1 % in 1 7 0 , were prepared in n.m.r. tubes with 12 mm outer diameter. H,O enriched to 10 atom% in 1 7 0 was obtained from Biogenzia Lemania, Lausanne. Propionic acid and isobutyric acid were from B.D.H. Chemicals and Th. Schuchardt Gmbh, respectively. TITRATIONS A N D pH MEASUREMENTS pH was varied by adding small volumes of concentrated HCI solutions to the n.m.r. samples. Relaxation rates from the titration experiments (fig. 3) were corrected for dilution assuming proportionality between excess rates and monomer molality.This correction, which is largest at a = 0, never exceeds 8%. pH was measured in the n.m.r. tubes at room temperature with Radiometer PHM 52 or PHM 64 pH meters, equipped with Radiometer GK2322C combination electrodes. Corrections for the sodium error were made according to a nomogram supplied by the manufacturer. The degree of dissociation, a, of the polyacids as a function of pH was determined through a separate potentiometric titration at each polyacid concentration. From the pH of the solution [H,O+] or [OH-] was calculated using operational activity coefficients (determined in the absence of polyacid). The fraction of charged monomers then follows from stoichiometric considerations. RELAXATION MEASUREMENTS Oxygen-17 magnetic relaxation rates were measured at 13.56 MHz on a modified Varian XL-100-15 Fourier-transform spectrometer and at 34.56 MHz on a home-built Fourier- transform spectrometer equipped with a 6 T wide-bore magnet from the Oxford Instrument co.Longitudinal relaxation rates ( R , ) were measured with the inversion recovery method (n-r-n/2 pulse sequences). Each R , value is the result of a least-squares fit of the magnetization plotted against delay time for ten different z values. Transverse relaxation rates ( R , ) were obtained from the linewidth (Avob,) at half amplitude of the absorption spectra according to R,. nhs = nAvohs. Reported R, values are averages of 2-4 spectra. The precision was better than & 2 s-l. The excess relaxation rates, defined by eqn (8), were calculated as Ri, ex = ' i , obs- R i , ref (i = 7 2).Ri, obs is the relaxation rate (or half-width times n) observed for a polyacid solution and Ri, rc,f is the same quantity for a sealed sample of H,O at pH 2.7, measured at the same temperature. Although R , = R , for pure water, Rz,ref > R l , r e f due to a contribution (ca. 3 s-l) to the linewidth from magnetic-field inhomogeneity. In the calcuation of Ri, ex this contribution cancels. From a large number of measurements we have found the true 1 7 0 relaxation rate in pure water [RF in eqn (8)] at 28.0 OC to be 127.3 & 2 s-l. The experiments were carried out at probe temperatures ranging from 27.4 to 28.5 OC and the data were subsequently normalized, using the temperature-dependence data in fig. 1, to 28.0 O C , unless otherwise stated.These corrections never exceeded 2 %. During each experiment the probe temperature was kept constant to within & 0.2 OC by the passage of dry thermostated air or nitrogen. The overall accuracy of the corrected and normalized excess data is estimated to be better than &3 s-l.B. H A L L E A N D L. P I C U L E L L 257 THEORETICAL BACKGROUND For a nucleus like oxygen-17, with spin I = $, the macroscopic magnetization decays, in general, as a weighted sum of three exponentiak6 No general analytical expressions exist for the three amplitudes or for the corresponding relaxation rates in terms of the spectral densities characterizing the molecular motion; rather the eigenvalues and eigenvectors of the relaxation matrix must be obtained numerically for each set of values for the spectral densities.However, in the systems under study the comparatively rapid motion of the ‘bound’ water and the large fraction of rapidly reorienting bulk water result in an effective spectral density that is only weakly frequency dependent in the experimental frequency range. The relaxation then becomes nearly exponential and approximate analytical expressions for the longitudinal and transverse rates can be deri~ed.~9 * For the present data, these expressions are accurate to better than 1 %. Since no 1 7 0 quadrupolar splittings were observed for the polyacid solutions, the molecular motion must average the quadrupolar interaction to zero on a time-scale of the order of 1 /x, where x is the quadrupole coupling constant. The relaxation rates can then be expressed as6 3n2 Rl = -x2 625 ( I +;) [~J((w,) + 8J(20,)] where coo is the resonance frequency and the reduced spectral density is Here it has been assumed that the molecular motion is isotropic, so that the correlation function for the field gradient decays exponentially with a correlation time 7,.(This assumption will be partly removed in the subsequent treatment.) For the H2170 quadrupole coupling constant we use the recent2f l6 estimate x = 6.67 MHz, while the asymmetry parameter is taken to be the same as in ice,9 q = 0.925. If the 1 7 0 nuclei exchange rapidly between two environments or states with different intrinsic relaxation rates but negligible chemical shift difference, then the total rates may be decomposed according tolo.l1 Ri = PFR,+PBRiB ( i = 1,2) (3) where PB = nm/55.5 is the fraction of nuclei in the B state and PF = 1 -PB. m is the monomer molality and n is the number of motionally perturbed water molecules per monomer. Eqn (3) corresponds to a two-state model with ‘free’ (F) and ‘bound’ (B) water. For ‘free’ water RIF = RZF = R,. ‘Rapid exchange’ here means that the average ‘lifetime’ z~~ of a water molecule in the B state is short compared with the intrinsic relaxation times in that state, i.e. that rlB 4 l/RiB. Due to the scalar spin-spin coupling between the 1 7 0 nucleus and the two protons, the water 1 7 0 resonance consists, in the absence of proton exchange, of a triplet12 with spin-spin coupling constant13 JOH = 90 Hz. At normal temperatures, however, the proton exchange rate Y > 2nJoH, making the 170 resonance nearly Lorentzian with a linewidth determined by Y, JOH and the transverse relaxation rate, R,, in the absence of exchange.In practice, R, is obtained from the linewidth at pH values where258 1 7 0 N.M.R. I N POLYELECTROLYTE SOLUTIONS Y 9 271JOH, from the proton decoupled linewidth, or from the longitudinal relaxation rate (provided that extreme narrowing conditions obtain). Starting from the extended Bloch equations,12 it is straightforward to derive an expression for the 1 7 0 lineshape as a function of Y, broadening of the linewidth due to proton exchange, Avexch AvObs- R2/71, can be obtained numerically. However, for the data simple expression, which is derived in Appendix approximation JOH and R,, from which the presented here, the following A, is valid to an excellent The rate of proton exchange in pure water may be YF = $k,[H3O+] + k,[OH- written as1, 1 ( 5 ) where k , and k , are the second-order rate constants for the acid- and base-catalysed transfer processes, respectively.With pH = - log(y'[H30+]), y' being an operational activity coefficient, and K, the ionic product of water eqn (5) becomes rF = ik;lO-pH +k;lK,IOPH (6) where k; = k,/y' and k;l = y'k,. In an aqueous polyacid solution additional mechanisms for water proton exchange may exist. Let rR be the rate of proton exchange for the fraction Pg of the water molecules which are engaged in proton transfer with prototropic groups on the polymer. [Note that Pg is not necessarily equal to the fraction P, appearing in eqn (3)].The proton-exchange kinetics is thus described by a six-state model with three spin states for each of the F and B states. However, provided that Pg < 1, the system can be treated as if there were proton exchange between only three (spin) states, with an effective rate where z g is the average lifetime in the B state for the fraction Pg of the water molecules. This result is derived in Appendix B. RESULTS In order to determine whether the fast exchange limit is applicable to the water 1 7 0 relaxation in our polyacid solutions, i.e. if eqn (3) is valid, we measured the linewidth as a function of temperature in the range 5-93 "C for solutions of PMA at two degrees of dissociation (fig. 1). Under very general conditions, an Eyring plot of ln(R,, ex/ T ) against 1 /Tyields a line of positive slope in the fast exchange limit.14 This is evidently the case for water interacting with either the coiled (a = 0.08) or the extended (a = 0.54) conformations (see below) of PMA.According to eqn (3) the excess relaxation rates may be written as Ri, ex G Ri - R , = P,(Ri, - RF). (8) The evaluation of Ri,ex as described in the Experimental section is equivalent to setting R, equal to the relaxation rate of pure water (127.3 s-l at 28.0 "C). The excess rates thus include a small contribution ( 5 3 s-l) from sodium and chloride ions (cf Discussion). Fig. 2 shows that the transverse excess relaxation rate is, within theB. HALLE AND L. PICULELL 259 0 0 2 7 2 9 3 1 3.3 3 5 37 lo3 KIT FIG. 1 .-Temperature dependence of the I7O transverse excess relaxation rate at 13.56 MHz for 0.55 rn PMA at a = 0.08 (0) and for 0.57 m PMA at o! = 0.54 (m).Error bars correspond to a 3 s-' uncertainty in 4, ex. experimental accuracy, proportional to the monomer molality, except for PMA at a = 0. Measurements on PMA at 0.19 and 0.59 m as a function of a (table 1) show that this non-linear behaviour becomes less pronounced as the charge density increases, but it persists up to at least a N 0.2. In the absence of polymer-polymer interaction, the intrinsic relaxation rates Ri, should be concentration independent, whereas P, and thus, according to eqn (8), Ri, ex should be proportional to the monomer molality. These findings therefore demonstrate the presence of polymer-polymer interaction in PMA solutions at low a in the investigated concentration range.Note that a linear concentration dependence, as was obtained for PAA and for PMA at a = 1 , is a necessary, although not sufficient, criterion for absence of polymer- polymer interactions. Indeed, our results show that the I7O relaxation rates in the concentration range < 0.6 m are unaffected by inter-polymer electrostatic repulsion. Since the non-linearity for PMA becomes more pronounced with decreasing charge density, it cannot have an electrostatic origin. As an explanation we suggest polymer-polymer association. That this occurs for PMA, but not for PAA (cf. also table 2), may be due to stronger van der Waals forces between the PMA chains with their additional methyl groups. A change in the degree of dissociation of the polyacid may influence the water 1 7 0 relaxation directly through the altered charge density or indirectly via conformational transitions or changes in counter-ion association.Indeed, fig. 3 reve$s a not-trivial a-dependence. For PAA we found that extreme narrowing conditions [J(w) = J(0) and thus, according to eqn ( I ) , R, = R,] obtain over the entire a range, whereas for PMA at low a values we observed considerable differences between longitudinal and260 1 7 0 N.M.R. I N POLYELECTROLYTE SOLUTIONS 40 - 30 - I 0 02 94 06 [monomer] /m FIG. 2.-Concentration dependence of the 1 7 0 transverse excess relaxation rate at 34.56 MHz and 27.7 OC for aqueous solutions of PAA and PMA at a = 0 (A) and a = 1 (0). Solid lines from least-squares fits.TABLE 1 .-RATIO OF NORMALIZED (TO r?.i = 1 ) TRANSVERSE EXCESS RELAXATION RATES AT 13.56 MHZ FOR TWO PMA CONCENTRATIONS AT DIFFERENT DEGREES OF DISSOClATlON a 0.19 R2, ,,(0.59 rn) 0,59 R2, ex(O. 19 rn) 1 .ooo 1.05 0.222 1.23 0.086 1.39 0.005 1.45 0.000 I .96 Data from the pH range of significant proton-exchange broadening, corresponding to 0.3 < a < 1 .O for 0.19 rn PMA, are not included.B. HALLE AND L. PICULELL 26 1 100 80 - 60 vl \ ?2 e 40 20 0 . .O O.0 Q o 0 m . 0 0 A & A A k 1 1 1 , , , , , , 1 , 0 0.5 1 .o a FIG. 3.--"0 excess relaxation rates at 28.0 O C plotted against degree of dissociation for 0.59 m PAA (A), 0.59 m PMA at 13.56 MHz (0, O), and 0.57 m PMA at 34.56 MHz (a, 0). Open symbols refer to longitudinal, filled symbols to transverse rates.At a values where only filled symbols are shown, measured R,. ex and R2, ex overlap. No data from the region of significant proton-exchange broadening are included. TABLE 2.-EXCESS RELAXATION RATES AT 0.59 rn AND 28.0 OC FOR MONOMERS AND POLYMERS monomer Rexa/s-l polymer R,,"s b / s - l CH,CH,COOH 10.5 PAA a = 0 11.3' (CH,),CHCOOH 14.4 PMAa=O (CH,),CHCOO- 30.4' PMA a = 1 56.7d CH,CH,COO- 23.6' PAA CI = 1 41.2d a The uncertainty in Re, is estimated to be 2 ssl. In all cases R, = R,. ' From the slopes ' The contribution of 3 s-l from 0.59 rn NaCl or Na+ (C1- makes a negligible A contribution of 1 s-l, corresponding to 'free' Na+ in fig. 2. contribution) has been subtracted. (see Discussion), has been subtracted. Frequency-dependent relaxation rates.transverse relaxation rates as well as a frequency dependence. This behaviour indicates that there are rapidly exchanging water molecules with motional components on a timescale of nanoseconds. Common to both polyacids is a nearly linear increase in relaxation rates with increasing charge density above a 5 0.5. In order to facilitate the interpretation of the 1 7 0 relaxation data for PAA and PMA, we studied the corresponding monomers, propionic acid and isobutyric acid, respectively. The results are shown in table 2 together with some polyacid data from fig. 2. The effect of the monomers is more than doubled upon dissociation of the carboxylic group. In the fully protonated state (a = 0) PAA produces the same effect as an equivalent concentration of monomers.This is not so for PMA, for which, as noted above, the relaxation rates are frequency-dependent and much larger than for the corresponding monomer solution. In the fully dissociated state (a = 1) we observe significantly larger effects from both of the polymers, as compared with the dissociated262 1 7 0 N.M.R. I N POLYELECTROLYTE SOLUTIONS monomers. Finally, we note that the difference between PAA and PMA at a = 1 is larger than the difference between the anionic monomers. The broadening of the 1 7 0 absorption curve around neutral pH, due to slow water-proton exchange, is drastically reduced on addition of even small amounts of PAA or PMA (fig. 4). This effect is evidence of additional catalytic mechanisms that are much more effective than those operating in bulk water around neutral pH.The shift of the linewidth maximum towards higher pH indicates that dissociation reduces thecatalyticefficiency of the polyacid. Furthermore, a reduction of the proton-exchange broadening to the level of the two intermediate, almost coincident, curves requires a 7-fold higher concentration of PMA as compared with PAA. PAA is thus a considerably more powerful proton-exchange catalyst than PMA. 80 60 ; d \ f 40 20 0 I I I I I I \ I f J 0 f’ / I I i 0 \. 6 7 8 9 PH FIG. 4.-Proton-exchange broadening of the 1 7 0 resonance plotted against pH at 28.4 *C for H,O (a), 1 . 5 0 ~ mPAA(.,dashedcurve), 10.2 x mPMA(A)and0.585 mPMA(A).Thecurvesare based on least-squares fits to theoretical expressions as explained in the text. DISCUSSION GENERAL CONSIDERATIONS The simplest picture consistent with the temperature and concentration dependences of the water 1 7 0 relaxation rate is a fast exchange of water molecules between two states: ‘free’ (F) and ‘bound’ (B) water. In attempting a more detailed interpretation one must, even within this relatively simple model, consider several dynamic processes : (1) reorientation of free water molecules, (2) chemical exchange between free and bound states, (3) reorientation (possibly anisotropic) and translation of bound water molecules, and (4) reorientation (possibly anisotropic) of entire polymer molecules (or aggregates) and segments thereof.Furthermore, all these processes may be influenced by the presence of counterions. To begin with, assume that water molecules in the B state are rigidly bound to theB.H A L L E AND L. P I C I J L E L L 263 polyelectrolyte, i.e. that processes of type 3 can be disregarded. Furthermore, assume that processes of types 2 and 4 can be described by a single effective correlation time zCH [cf. eqn (lo)]. With relaxation rates from fig. 3 for PMA at cc = 0 and with T , , ~ = 2.35 ps [from eqn (1) and (2) with R, = 127.3 s-l], eqn (I), (2) and (8) yield T ( , ~ = 3.3 10 monomer units. We regard this as an unreasonably low value for the number of interacting water molecules, even for a relatively compact polymer conformation. (A monolayer coverage of a compact sphere consisting of 90 monomers corresponds to ca. 3 water molecules per monomer.) These considerations led us to refine the model by allowing the bound water molecules some reorientational freedom, while still taking into account the inherent anisotropy of the water-polymer interaction. The local anisotropic reorientation will only partially average the quadrupolar interaction,lj the remaining part being averaged out by a slower motion, presumably chemical exchange and/or polymer reorientation.Since this extended two-state model involves more parameters than justified by our data, we will introduce several simplifying assumptions: (1) the local and the overall motions occur independently and on different timescales, (2) there is local axial symmetry and (3) the anisotropy is small. The spectral density for the B state may then be decomposed into components associated with the fast (0 and the slow (s) motionsl5? I 6 1.3 ns, corresponding to one bound water molecule for every 41 Zince the fast motion is in the extreme narrowing limit (see below), eqn (2) yields JRf = 2&.The residual anisotropy A , the magnitude of which is confined to the range [0, 11, is determined by the asymmetry parameter q and by the orientational probability distribution for the ‘bound’ water molecules with respect to the polymer chain.16 If the ‘ boznd ’ _water molecules were to reorient completely isotropically, then A = 0 giving J R = JRf, i.e. only the fast motion would contribute to the relaxation. From eqn (9), which is valid only for A2 6 1, it is seen that for the slow motion to contribute significantly to the relaxation rate, it must be of the order of a factor 1 / A 2 slower than the fast motion.On the basis of water 170 data [ref. (2) and unpublished data] from similar systems, we expect IAl to lie in the range 0.01-0.2. In the following, we will interpret the direct effect of carboxylic-group dissociation (indirect effects on polymer conformation and counter-ion association will be considered explicitly) on the ‘hydration’, as measured by the 170 relaxation rate, in terms of variations in reorientational rate and/or anisotropy for a fixed number of water molecules, rather than in terms of a changing ‘hydration number’. In view of the short-range character of the water-polyelectrolyte interaction (its most long-ranged component, the average charge-dipole potential, falls off as r4 to first order), we feel that this interpretation is physically reasonable and conceptually preferable.The other view may be more natural in connection with hydration studies through measurement of the water self-diffusion coefficient17 or of absorption and desorption isotherms.1s The results from such studies are, however, not simply related to the properties determining water magnetic relaxation rates. LOW a RANGE It has been documented with a variety of experimental t e ~ h n i q u e P ~ ~ that PMA, and to a much lesser extent also PAA, in aqueous solution undergoes a conformational transition from an extended form at high a to a more or less compact random coil264 1 7 0 N.M.R. I N POLYELECTROLYTE SOLUTIONS at lower a. This transition is undoubtedly the origin of the large 170 relaxation rates for PMA at low a values. The non-linear concentration dependence for PMA (fig.2 and table 1) implies that, in the investigated concentration range, association occurs also between chain segments situated in different polymer molecules. At a = 0, PAA has the same effect on the 1 7 0 relaxation as an equivalent concentration of monomers (table 2). This indicates that the local water reorientation is similar in these two systems and that more extensive motions do not contribute significantly at a = 0. For PMA, on the other hand, slow motions must be invoked to explain the much larger effect compared with the monomer solution (table 2), the inequality of R, and R,, and the frequency dependence (fig. 3). Since the rate and anisotropy of the local water reorientation should be similar for PMA and PAA at a = 0, the difference between PMA and its monomer must lie mainly in the slow correlation time z:~.This correlation time can be estimated if we assume that the local water reorientation is the same for PMA as for the monomer, as was found for PAA at a = 0. From the data in fig. 3 and table 2 we then find, using eqn (l), (2), (8) and (9), for PMA at a = 0: z S , ~ = 6 If: 2 ns and nA2 = 0.023. ‘Coordination numbers’ n in the range 2-10 thus correspond to anisotropies IAl in the range 0.1 1-0.05. The maximum in relaxation rates for PMA in the low a range can be understood as the result of two opposing factors. First, deprotonation of carboxylic groups should, through the charge-dipole interaction, increase the anisotropy of the local water reorientation.The factor A2 in eqn (9) is thereby increased and the slow motion is more heavily weighted, resulting in enhanced relaxation rates with increasing a. Secondly, as the polymer chain gradually unfolds with increasing charge density, the correlation time z S , ~ , which is at least partially (see below) determined by extensive polymer motions, decreases. This is seen from the decreasing ratio R2, ex/R1, ex with increasing a in the range 0.1-0.5 (fig. 3). The conformational transition thus has the effect of decreasing the slow motional contribution to the relaxation rates with increasing a. HIGH a RANGE Small-angle neutron and X-ray scattering studies of aqueous solutions of PMAZ5 and PAA,26 respectively, in the concentration range 0.1-0.4 rn and in the a range 0.4- 1 .O have revealed a peak in the plot of scattered intensity against scattering vector.This was taken as evidence for an electrostatically induced lattice-like structure with parallel extended polyelectrolyte chains. Since this ordering was found to persist down to the lowest investigated concentration (ca. 0.1 rn) and degree of dissociation (0.4 and 0.6), the linear dependence of the 1 7 0 relaxation rate on concentration at a = 1 (fig. 2) and on a above a N 0.5 (fig. 3) is not surprising. Thus, although the ordered structure is a direct result of electrostatic polymer- polymer interaction, this interaction is not expected to affect the water motion as long as the polymer segments remain extended and sufficiently separated.The monomer data in table 2 clearly demonstrate that the perturbation of the water reorientation, to a large extent, is electrostatically induced. The increase in relaxation rate with increasing a above a N 0.5 (fig. 3), where the polyacids exist in an extended conformation,19 26, 28 is thus readily accepted. The same situation, with charged residues (carboxylates in particular) contributing more than uncharged ones to the 1 7 0 relaxation rate, is found in aqueous protein solutions.2 In sharp contrast to our results, deuteron relaxation in PMA has been taken as evidence for a decreasing ‘hydration’ with increasing charge density. Later work3’ has, however, cast serious doubts on this interpretation. Our results clearly reinforce these doubts. The larger effect on the water 1 7 0 relaxation rate of charged groups is not a consequence of an increased electric field gradient at the oxygen nucleus.The waterB. HALLE A N D L. PICULELL 265 1 7 0 field gradient is, to an excellent approximation, of intramolecular origin. Indeed, recent quantum chemical calculations2* l6 have shown that the perturbation induced by a nearby charge is negligible. It remains for us to explain the substantially larger effect on the 170 relaxation rate of the fully dissociated polyacids as compared with the monomers (table 2). One difference between a polyelectrolyte and its separated monomers is that a fraction of the counter-ions associate with the polyelectrolyte, thereby reducing its effective charge density. This phenomenon can often be adequately described in terms of an ‘ion condensation’ m ~ d e l : ~ ~ t ~ ~ there exists a critical charge density or degree of dissociation, a,, below which all counter-ions are ‘free’ and above which all additional counterions become ‘bound’ to the polyion.For PAA and PMA a, = 0.35, according to this model. 0.5, the polymer (segmental) motions were sufficiently rapid for their contribution to the relaxation rate to be negligible [cf. eqn (9)], then one would expect ‘condensed’ counter-ions to produce roughly the same effect on the 1 7 0 relaxation rate as ‘ free’ counter-ions. We therefore conclude that the much larger effect of the polyacids in the high a range, relative to the monomers, is due to a contribution from slow motions. Consequently, the effect of carboxylic group dissociation on the 170 relaxation rate in the high a range can be viewed as the resultant of three electrostatic contributions.First, the charge-dipole interaction lowers the rate of local water reorientation, i.e. increases zfcB. This is the sole effect for the monomers. Secondly, the charge-dipole interaction enhances the anisotropy in the local water reorientation. As noted above, for PAA at a = 0 the anisotropy IAl is so small that the slow motion does not influence the relaxation significantly. For charged groups, on the other hand, the anisotropy is sufficiently large for the slow motion to contribute substantially according to eqn (9). Thirdly, association of counter-ions increases the number of water molecules associated with the polyelectro- lyte and thus ‘feeling’ the slow motion.(It has been that sodium ions retain their primary hydration sheath upon binding to carboxylate groups.) Finally, it is possible that the aforementioned electrostatic interactions are enhanced in the polyacid solutions because the adjacent hydrocarbon core of the polymer chain reduces the effective local dielectric permittivity through a ‘ dielectric shielding’ effect.34 Table 2 shows that there is a larger difference (15.5 s-l) between PMA and PAA at a = 1 than between the anionic monomers (6.8 s-l). In view of the previously discussed lattice-like structure in this a range, it is unlikely that this difference should be caused by the disparity in the degree of polymerization of our polyacid preparations (see Experimental section).As a check, we investigated the molar mass dependence of the 170 linewidth for 0.6 m solutions of PAA at a = 0 and at a = 1. No difference was found between degrees of polymerization 30, 90 and 3200. This leaves us with the conclusion that the additional methyl groups in PMA have a greater effect on the 1 7 0 relaxation rate than they have in the monomers. This would be the case if these methyl groups interfere with segmental chain motions, thereby increasing zEB, or if they, by adding to the bulk of the hydrocarbon core, promote the ‘dielectric shielding’ effect. Concluding the discussion of relaxation rates, it may be of interest to give an order-of-magnitude estimate of the rate of local water reorientation, i.e. of the correlation time zfc+ From the monomer data in table 2 we estimate the contribution from a carboxylate group to be ca.17 s-l. Using eqn (l), (2) and (8) we can then calculate the quantity n(zfcB-z,,), where zCF = 2.35 ps (see above). In this way we find that ‘coordination numbers’ n in the range 2-10 correspond to zfcB/zCF ratios in If, in the range a266 1 7 0 N.M.R. I N POLYELECTROLYTE SOLUTIONS the range 7-2. The rate of 'bound' water reorientation is thus less than an order of magnitude slower than in pure water. The same order of magnitude should apply for water associated with the polyacids. In fact, even if we disregard the contribution from slow motions at a = 1, we arrive at similar rates. Thus, for PAA at a = 1 we find that n values of 2-10 correspond to ztH/zCF ratios of 16-4.PROTON EXCHANGE In fig 4 is included the result of a redetermination of the rate constants k; and k; appearing in eqn (6). The upper curve, which resulted from a least-squares fit to the complete theoretical expression derived from the extended Bloch equations [the approximate eqn (4) yields identical results], corresponds to k ; = 13.1 k0.3 dm3 mol-1 ns-l and k; = 4.06 f 0.1 dm3 mol-l ns-l at 28.4 "C. To obtain these values, we have used JOH = 90 Hz,13 R, = 126 s-l and pK, = 13.88.35 Using Arrhenius activation energies36 of 10.9 kJ mol-1 ( k ; ) and 20.1 kJ mol-1 (k;), our results can be converted to 25.0 OC: k ; = 12.5 f 0.3 dm3 mol-1 ns-l and ki = 3.70f0.1 dm3 mo1-l ns-l, in close agreement with Meiboom's original results12 of 10.6f4 and 3.8 Studying the effect of acetic acid and sodium acetate on the kinetics of proton exchange in aqueous solution by means of the broadening of the proton resonance, Luz and Meiboom3' concluded that the dominant mechanism involves a hydrogen- bonded complex consisting of one undissociated acetic acid molecule and two water molecules.If this mechanism prevails also for the polyacids, an increased polyacid concentration should not only lower the exchange-broadening curve but also shift its maximum towards higher pH. This behaviour is confirmed by our experiments (fig. On the basis of these observations and the finding (table 2) that the COOH-water interaction is similar in monomer and polymer solutions, we assume that I'E in eqn (7) can be calculated as 2(1 -a)rn/55.5, corresponding to two water molecules per undissociated carboxylic group.[In contrast to PB in eqn (3) and (8), we have included a in the definition of PE.1 The three lower curves in fig. 4 were calculated from least-squares fits to the model defined by eqn (4), (6) and (7). We used cx values from separate potentiometric titration experiments and the values given above for J O H , K,, k ; and k ; . Thus we neglect the influence of counter-ions and added chloride ions on the rate of proton exchange in the F state. According to our studies of the salt effect on the rate of water proton exchange (unpublished data), this influence is completely negligible, at least for the two lower polyacid concentrations. From these data we obtain (1 /rH + zyB) = 5.6 f 0.6 ns for PAA and 3 1 5 ns for PMA.(The less accurate result from the higher PMA concentration is 40 12 ns.) Previous proton n.m.r. studies at 25 "C have resulted in values of 21 ns for acetic and 14-90 ns for PMA.4 The higher catalytic efficiency of PAA, as compared with acetic acid, suggests that another mechanism, presumably involving a chain of one or more water molecules linking a protonated with a deprotonated carboxylate group, contributes to the proton exchange in PAA solutions. This mechanism is expected to be less important for PMA, which was found to be less potent than acetic acid, since the methyl group has a stiffening effect on the polymer chain, as deduced from molecular models. Although the effect of solutes on the water proton exchange rate has been studied in a variety of systems [e.g.ref. (4) and (37)], no attempt has been made to extract information about the rate of water exchange from the experimental data. However, as shown by the analysis in Appendix B, the experiments yield the composite quantity (I/rH+~yH), which constitutes an upper limit for the average residence time zyB for those water molecules that exchange protons with the solute. 1.5 dm3 mol-1 ns-l, respectively. 4).B. HALLE A N D L. PICULELL 267 If the liO relaxation rate contains a contribution from slow motions, it may be possible to determine the slow correlation time zzB. If both polymer reorientation (correlation time zrB) and chemical exchange (average lifetime z,,) contribute to the slow motion, then this may be characterized by an effective correlation timell 1 1 1 - --+-.- - z:B zrB zlB This result is strictly valid only in the absence of orientational correlation between successive water-polymer encounters. For PMA at a = 0 we found (see above) zER = 6 f 2 ns. According to eqn (lo), this is a lower limit for the water lifetime in state B. Since the effect of PMA on the proton exchange broadening is due mainly, or entirely, to undissociated carboxylic groups, this value may be compared with the upper limit for z g obtained from the proton-exchange broadening. We can therefore conclude that water molecules associated with COOH groups in PMA at a = 0 have an average lifetime in the range of 4-36 ns. SUMMARY The present study shows that water 170 magnetic relaxation is a valuable tool for the understanding of dynamic processes in aqueous polyelectrolyte solutions.Water molecules associated with the polyelectrolyte are not rigidly bound; rather they reorient anisotropically with a rate that is merely an order of magnitude slower than in bulk water. This perturbation of the water reorientation is, to a large extent, electrostatically induced. Water molecules directly hydrogen-bonded to acidic groups are engaged in rapid proton transfer and have average residence times of the order of s. In addition, water li0 relaxation rates contain information about polymer dynamics which can be related to results obtained by other techniques. We are grateful to Maj-Lis Fontell for helping us with the potentiometric titrations and the viscosity measurements and to Hans Gustavsson and Torbjorn Drakenberg for experimental advice and helpful discussions. Grants from Stiftelsen Bengt Lundqvists Minne and from Kungl Fysiografiska Sallskapet i Lund are gratefully acknowledged .APPENDIX A: EFFECT O F RAPID PROTON TRANSFER ON THE 1 7 0 LINEWIDTH The observed 170 linewidth for pure water at 28 O C is strongly pH dependent with a maximum of ca. 115 Hz at pH 7.2 (fig. 4)’ whereas the ‘natural’ linewidth (due to quadrupolar relaxation) is merely ca. 40 Hz. Despite the considerable magnitude of this proton-exchange broadening, the lineshape remains Lorentzian, or very nearly so. This fact indicates that the proton-exchange broadening can be described, to high accuracy, by an equation which is simpler than the complete lineshape expression.Various limiting forms can be derived by introducing suitable approximations in the exact solution of the extended Bloch equations. Since this involves rather heavy algebra, we will choose a more elegant approach due to Wennerstrom.ll The essence of Wennerstrom’s treatment of chemical exchange is to write the spin Hamiltonian as (A 1) where the sum extends over all states (‘sites’). The random functionsf;(t) have the property of being unity if the nucleus considered is in state i at time t and zero otherwise. The ensemble average ( f i ( t ) ) , which we will denote by l$(t), is simply the fraction of nuclei in state i at time t . At equilibrium ( f i ( t ) ) , , = pi. H = C f i ( t ) H i 2268 1 7 0 N.M.R. I N POLYELECTROLYTE SOLUTIONS If the ‘natural’ linewidth is the same in all states, which, in the present case, must be true since the 1 7 0 field gradient is unaffected by the proton spin state, then it can be shown1’ that the exchange broadening is Av,,,.~ = - d7ZZ Cf,(O&(f+T)),,j Awi AWj (A 2 ) 2n - x i j where A o i == oOi - Ci pi woi, mOi being the resonance frequency in state i.Eqn (A 2) is valid only if the exchange rate is large compared with the relaxation rate and if the lineshape is Lorentzian. Furthermore, it is assumed that the exchange is slow compared with those motions (water reorientation in the present case) that are responsible for the ‘natural’ linewidth. The time correlation function in eqn (A 2) may be written6 as the product of the probability pi and the conditional probability pji(7) that the nucleus is in statej at time t + z , given that t .Thus (A 3) l X 2n --J. AvexCh = - dr Z p i p j i ( 7 ) A o i Awj. the exchange of 1 7 0 nuclei between different proton spin states is it was in state i at time The kinetic scheme for where Y is the proton exchange rate, i.e. the inverse of the average lifetime of a water molecule with a given pair ofprotons. The numerical factors follow from simple probability considerations and the principle of detailed balancing.I2 Furthermore, we have p 1 - 4 - 1 Aw, = -2nJ0, p 2 = $ and Aw, = O P 3 = a A o 3 = 2nJOH. The rate laws for the system (A 4) may be written in matrix notation as d -F(r) = -rF(t) dt where F(r) is the column vector with elements &(t) and 1 -1 0 -1 - 1 2 The formal solution to eqn (A 6) is F(t) = S exp( - A[) S-’ F(0) where S is the transformation that diagonalizes r according to A = S-I r S.After a little algebra we find 1 0 0 0 0 0 and The conditional probabilitypji(7) can now be obtained from the solution (A 8) as 4(7), subject to the initial condition &(O) = 6 k i . ThusB. HALLE AND L. PICULELL 269 Upon substitution of the elements of r7. and S (and its inverse) we find p , ,(7) = p3,(7) = f (e+ + 2e-r/r1/2 + 1 ) (A 13) pl,(7) = p3,(7) = (e-rlrl - 2e-rl*l12 + 1) where we have introduced absolute values in the exponentials to ensure that the correlation function becomes invariant with respect to time reversal. Conditional probabilities involving state 2 are not needed since Aw, = 0. It now remains only to substitute eqn (A 5) and (A 13) into eqn (A 3) and to carry out the integrations.The result is In order to assess quantitatively the accuracy of the approximate eqn (A 14), we have compared it with the exact numerical solution of the extended Bloch equations." For given values of r , R, and JOH we calculated the exact Av,,,,,, which was then inserted into eqn (A 14) to yield an approximate value of r . We used R, = 127.3 s-l and JOH = 90 Hz; however, the result is insensitive to the value of R,. Fig. 5 shows that eqn (A 14) is accurate to better than 1 % for the polyacid data in fig. 4. Avexch /Hz 90 80 70 60 50 40 30 20 1.10 , r . . . . . 1 r* 7 1.05 1.00 30 3.5 4.0 log ( r / s - l ) FIG. 5.-Ratio of proton exchange rate, r*, as calculated by eqn (A 14), to the exact rate r , plotted against log r and exchange broadening Av,,,~.The ranges of interest for pure water at 28 O C and for the polyacid data in fig. 4 are indicated. APPENDIX B : EFFECT OF TWO-STATE WATER EXCHANGE ON THE PROTON SPIN-STATE KINETICS Consider a system in which water molecules can reside in either of two states, defined by their respective proton exchange rates rF and r,. At equilibrium, the fraction of water molecules in each state is P, and P,, respectively. If the water molecules can exchange between the two states, with rates equal to the inverse of the respective lifetimes tF and t,, then kinetic scheme for the exchange of 1 7 0 nuclei between different proton spin F, Z$ F, % F, is r F I 2 r F l 2 rF/4 rF/4 states (indicated by a subscript) 1 _ _ 7F270 1 7 0 N.M.R.I N POLYELECTROLYTE SOLUTIONS Assume now that the equilibrium fraction of water molecules in the B state is very small, i.e. that P, < 1. (For the experimental data to be considered, i.e. the two intermediate curves in figure 4, P, < 2.5 x lop4.) In the absence of water exchange, the effect of the B state on the observed proton-exchange broadening would then be completely negligible. For non-zero water exchange rates, however, the B state can, despite its low population, affect the observed broadening. This is so because a transition between, for example, spin states F, and F, can occur not only by a proton exchange in the F state, but also via two water-exchange events and an intervening proton exchange in the B state. For this latter, indirect, route to contribute significantly to the observed proton spin-state kinetics, the water exchange as well as the proton exchange in the B state must be fast compared with the proton exchange in the F state, i.e.1 /tn. % rF and rB $- r?. It is thus clear that, in the limit P, Q 1 , the only effect of the B states on the proton spin-state kinetics is to increase the effective rate of transitions among the F states. Consequently, the scheme (B 1) reduces to rlz r / z F, $ F, + F,. r/J r/4 It remains to derive an expression for the effective proton-exchange rate r in terms of PB, z,, rB and rF. The principle of detailed balancing requires that, at equilibrium, where the 4 and Bi denote fractional populations. Furthermore, from the rate laws pertaining to scheme (B 1) we find that, at equilibrium Combination of eqn (B 3)-(B 5 ) leads to Consider now the return from an infinitesimal displacement of the equilibrium population of the species F,.According to the scheme (B 1) Since the actual populations are assumed to differ only infinitesimally from their equilibrium values, we may substitute B, from eqn (B 6) into eqn (B 7 ) to obtain According to the equivalent scheme (B 2) which, on comparison with eqn (B 8), yields the desired result Eqn ( 7 ) in the main text differs slightly in notation.B. HALLE AND L. P I C U L E L L 27 1 For an introduction to the vast literature, see Water-A Comprehensive Treatise, ed. F. Franks (Plenum Press, New York, 1975), vol. 4 and 5. B. Halle, T. Anderson, S. Forsen and B. Lindman, J. Am.Chem. Soc., 1981, 103, 500. J. J. van der Klink, J. Schriever and J. Leyte, Ber. Bunsenges. Phys. Chem., 1974. 78, 369. J. Schriever and J. C. Leyte, Chem. Phys., 1977, 21, 265. A. Katchalsky and H. Eisenberg, J. Polym. Sci., 1951, 6, 145. A. Abragam, The Principles of Nuclear Magnetism (Clarendon Press, Oxford, 1961). B. Halle and H. Wennerstrom, J. Magn. Reson., 1981, in press. A. D. McLachlan, Proc. R. SOC. London, Ser. A, 1964, 280, 271. D. T. Edmonds and A. Zussman, Phys. Lett. A, 1972, 41, 167. lo J. R. Zimmerman and W. E. Brittin, J. Phys. Chem., 1957, 61, 1328. H. Wennerstrom, Mol. Phys., 1972, 24, 69. l 2 S. Meiboom, J . Chem. Phys., 1961, 34, 375. l 3 L. J. Burnett and A. H. Zeltmann, J, Chem. Phys., 1974, 60, 4636. l4 B. Lindman and S. Forsen, Chlorine, Bromine and Iodine NMR; Physico-chemical and Biological Applications (Springer-Verlag, Heidelberg, 1976). l5 H. Wennerstrom, G . Lindblom and B. Lindman, Chem. Scr., 1974, 6, 97. B. Halle and H. Wennerstrom, J . Chem. Phys., 1981, 75, in press. l 7 Gy. Inzelt and P. Grof, Acta Chim. Acad. Sci. Hung., 1977, 93, 117. K . Hiraoka and T. Yokoyama, Polym. Bull., 1980, 2, 183. R. Arnold and J. Th. G. Overbeek, Rec. Trail. Chim. Pays-Bas, 1950, 69, 192. J. C. Leyte and M. Mandel, J. Polym. Sci., Part A2, 1964, 1879. 2o A. Silberberg, J. Eliassaf and A. Katchalsky, J. Polym. Sci., 1957, 23, 259. 22 M. Mandel, J. C. Leyte and M. G. Stadhouder, J. Phys. Chem., 1967, 71, 603. 23 0. Schafer and H. Schonert, Ber. Bunsenges. Phys. Chem., 1969, 73, 94. z4 A. Ikegami, Biopolymers, 1968, 6, 431. 25 M. Moan, J. Appl. Crystallogr., 1978, 11, 519. 2fi N. Ise, T. Okubo, Y . Hiragi, H. Kawai, T. Hashimoto, M. Fujimura, A. Nakajima and H. Hayashi, *’ J. J. v.d. Klink, L. H. Zuiderweg and J. C. Leyte, J. Chem. Phys., 1974, 60, 2391. 2H H. Gustavsson, B. Lindman, and B. Tornell, Chem. Scr., 1976, 10, 136. 29 H. Gustavsson, B. Lindman and T. Bull, J . Am. Chem. Soc., 1978, 100, 4655. 30 J. A. Glasel, J . Am. Chem. Soc., 1970, 92, 375. 31 F. Oosawa, Polyelectrolytes (Marcel Dekker. New York, 1971). 32 G . S. Manning, Ace. Chem. Res., 1979, 12, 443. 33 H. Gustavsson and B. Lindman, J. Am. Chem. Soc., 1978, 100, 4647. 3 3 J. D. Jackson, Classical Electrodynamics (Wiley, New York, 2nd edn 1975), sect. 4.4. 35 H. S. Harned and B. B. Owen, The Physical Chemistry of Electrolytic Solutions (Reinhold, New York, 36 A. Loewenstein and A. Szoke, J. Am. Chem. Soc., 1962, 84, 1151. N Z . Luz and S. Meiboom, J. Am. Chem. Soc., 1963, 85, 3923. J . Am. Chem. Soc., 1979, 101, 5836. 3rd edn, 1958), chap. 15. (PAPER 1 /255)