J. Phys. Chem. 1985, 89, 2779-2786
2779
anion, Li+ > Na+ > K+. However, the cation seems predominant in the overall hydration effect of the electrolyte (see Table IV). The so-called hydration numbers of individual ions or electrolytes resulting from different measurements and different interpretations of various properties of solutions are considerably different. For example, Na+ has the following hydration numbers obtained by a number of investigator^:^ 71, 66, 44.5, 16.9, 16.2, 6-7, 4.5, 4, 2.5, 2, 1. Hinton and Amislo have given detailed surveys of the results obtained by investigators by use of various methods. Their work shows that the area of solvation is in a state of confusion. Some results are contradictory to others. The hydration parameter values presented in Table IV are somewhat close to the values reported by Baborovsky” using transference number measurements. For the corresponding ions, Baborovsky‘s values are as follows: Li+, 13-14; Na+ 8-9; K+, 5; C1-, 4; Br-, 3-4; I-, 2. However, the halogen ions are hydrated in the reverse order, Le., C1- > Br- > I-. The significant hydration parameters should be additive. It is true in this work that we may assign a value for each ion, Le., Li+, 16-17; Na+, 10; K+, 3-4; C1-, 3; Br-, 3-4; I-, 4. The only exception is LiI, which has a much bigger value of h. This discrepancy may not be due to the theory itself, but rather to the inaccuracy of the data, A and y+. The “effective” closest distance A cannot be reliably obtained independently. In general, its value is chosen to be greater than the sum of the crystallographic radii but less than the sum of the
hydrated radii of the cation and anion in the electrolyte. However, in calculating f& from eq 3, the value of A is assumed as the sum of the crystallographic radii of the ions since the hydration effect is not taken into consideration in the Debye-Hiickel theory. Elliottlz discussed the value of A in his report. In light of what he viewed as problems with adding ion sizes determined from other sources, he plotted data as deviations from a uniform assumed distance of interionic approach, thereby establishing both systematic deviations and correlations among the individual ions as they appeared in different salts. Indeed, Stokes and R ~ b i n s o nassumed ’~ a value of A = 4 A for typical strongly hydrated and highly soluble 1:l electrolytes. They found that the choice of A is more critical in dilute regions, but it has little effect at very high concentrations. Rasaiah et al.” used Monte Carlo techniques to study the restricted primitive model of 1:l electrolyte solutions. They chose A = 4.25 A in the calculations. The upturn of lnf+ data at concentrations up to 2 m is accounted for well by finite-ion-size effects alone. However, the primitive model is an idealized model. It is valid only for very dilute solutions where the short-range interactions are not important. It seems inappropriate to be used in dealing with real solutions, particularly in concentrated regions.
(9) Samoilov, 0.Ya. “Structureof Aqueous ElectrolyteSolutions and the Hydration of Ions”, Consultants Bureau: New York, 1981; p 75. (10) Hinton, J. F.; Amis, E. S. Chem. Rev. 1971, 71, 627. (11) Baborovsky, J. Z . Phys. Chem. 1934, AZ68, 135.
(12) Elliott, G. R. B. Report LA-3891, Los Alamos Scientific Laboratory, Los Alamos, NM, 1968. (13) Rasaiah, J. C.; Card, D. N.; Valleau, J. P. J . Chem. Phys. 1972, 56, 248.
Acknowledgment. The author thanks Dr. Guy R. B. Elliott and one of the reviewers for making suggestions to improve the manuscript. ~~~
~
~
Reversing-Pulse Electric Birefringence of Poly( p-styrenesulfonate) In Aqueous Solutions: Effects of Molecular Weight and Concentration on Anomalous Signal Patterns Arising from Fast- and Slow-Induced Ionic Dlpole Moments Kiwamu Yamaoka* and Koichiro Matsuda Faculty of Science, Hiroshima University, Higashisenda-machi, Naka- ku, Hiroshima 730, Japan (Received: November 19, 1984; In Final Form: February 1 1 , 1985)
By use of reversing-pulse electric birefringence. (RPEB) techniques, the dynamic behavior of the counterions (Na+ and Mg2+) of three poly@-styrenesulfonate) (PSS) samples with molecular weights of 1.6 X lo6, 1.1 X lo6, and 2.8 X los was studied in aqueous solutions without added salt at 20 O C and 535 nm. The MgPSS samples gave rise to the normal RPEB patterns of the negative birefringence. An in the buildup and reverse portions regardless of the molecular weights and concentrations, as reported previously. Thus the electric field orientation of MgPSS is due mostly to the fast induced electric dipole moment. The NaPSS samples showed the normal RPEB patterns with the negative sign for An at dilute concentrations, but they gave rise to anomalous RPEB signals with a hump or a dip in the reverse portion at higher concentrations. The dip appeared in the signal with the positive sign at extremely low pulse fields, while the hump appeared with the negative sign at higher fields. By use of a simple theory of W E B which takes into account the contribution of the timedependent ionic polarizabilities, the anomalous RPEB signals of NaPSS were simulated satisfactorily. The slow induced ionic dipole moments, caused by Na+ ions redistributing along the longitudinal and the transverse molecular axes, are responsible for the anomalies.
Introduction Reversing-pulse electric birefringence (RPEB) is a sophisticated electrmptical technique, which should be utilized in the dynamic study of the ionic properties of polyelectrolytes and biopolymers in aqueous solutions, not to mention the static study of conformations of un-ionic biopolymers in organic solvents.1-21 A (1) OKonski, C. T.; Pytkowicz, R. M. J . Am. Chem. SOC.1957, 79, 4815-4816. (2) OKonski, C. T.; Haltner, A. J. J . Am. Chem. SOC. 1957, 79, 5634-5649.
0022-3654/85/2089-2779$01.50/0
particularly interesting application of the RPEB method is the investigation of the counterion response to externally applied (3) Asai, H. J. Biochem. (Tokyo) 1961, 50, 182-189. (4) Colson, P.; Houssier, C.; Frederiq, C.; Bertolotto, J. A. Polymer 1974, 15, 396-397. (5) Colson, P.; Houssier, C.; Fredericq, E. Biochim. Biophys. Acra 1974, 340, 244-26 1 . (6) Grew, J.; De Heij, M. E. Biopolymers 1975, 14, 2441-2443. (7) DeLaney, E. E.; Krause, S. Macromolecules 1976, 9, 455-463. (8) Tricot, M.; Houssier, C.; Desreux, V.; van der Touw, F. Biophys. Chem. 1978, 8, 221-234.
0 1985 American Chemical Society
2780 The Journal of Physical Chemistry, Vol. 89, No. 13, 1985
electric field.22923 The orientation of an ionized polyelectrolyte results from the interaction of the electric field with the permanent dipole moment, the covalent (electronic and atomic) polarizability, and the ionic polarizability, all of which the polyion may possess. The ionic dipole moment arises from the translational diffusion of counterions over the polyion surface by the electric field. The RPEB theory was first formulated by T m m and Yamaoka about a quarter of a century ago,24on the basis of the roughly cylindrical model with two electric principal axes and two rotary diffusion coefficients (longitudinal and t r a n s v e r ~ e ) . According ~~ to this theory,24 the three types of electric moments can be differentiated. The counterion-induced moment may be classified to be either “fast” or “slow”, depending on the time scale of the relaxation time of the counterion redistribution relative to the rotational relaxation time of the whole molecule. The origin of this induced moment is a major subject in relation with the molecular conformation of high molecular weight D N A and other polynucleotides.26 Poly@-styrenesulfonate) (PSS) has been studied under various conditions by the ordinary square-wave electric birefringence meth~d.~’-~’ As a result, the orientational behavior of PSS is found to be closely related to that of DNA. The RPEB study of PSS was initiated by Yamaoka and Ueda,9*’7who found that the RPEB signal of a sample with a molecular weight of 2.8 X lo5 shows no extremum upon rapid reversal of electric pulse fields regardless of the counterions (Na’, Mg2+, and Ca2+)and ionic additives. It was then concluded that the field orientation of NaPSS and MgPSS is due to the fast induced ionic dipole moment. The objectives of this paper are (1) to further investigate the RPEB response of NaPSS and MgPSS samples with different molecular weights under various experimental conditions, since Ookubo et al. recently reported that a dip appears in an RPEB signal with the positive sign, instead of the usually observed negative sign, for an NaPSS with a higher molecular weight of 8.0 X lo5at an extremely tow field (2) to establish any correlation between anomalous RPEB signals and molecular weights, as already observed for high and low molecular weight D N A samples,”-’8 and finally (3) to confirm that such a dip, if indeed occurs, is due to the slow induced ionic dipole moment, by simulating the observed RPEB signals with theoretically calculated ones with due consideration of time-dependent polarizabilities. The anomalous RPEB patterns and the sign reversal of birefringence are unexpectedly complex, being
(9) Yamaoka, K.; Ueda, K. J. Phys. Chem. 1980,84, 1422. (10) Yamaoka, K.; Ichibakase, T.; Ueda, K.; Matsuda, K. J . Am. Chem. SOC.1980, 102, 5109-5110. (1 1 ) Yamaoka, K.; Matsuda, K. Macromolecules 1980,13, 1558-1560. (12) Takezoe, H.; Yu, H. Biochemistry 1981, 20, 5275-5281. (13) Elias, J. G.; Eden, D. Macromolecules 1981, 14, 410-419. (14) Yamaoka, K.; Ueda,K. J. Phys. Chem. 1982,86,406-413. (15) Yoshioka, K.; Fujimori, M.; Yamaoka, K.; Ueda, K. Int. J . Eiol. Macromol. 1982, 4, 55-61. (16) Roux, B.; Cassoly, B. Biophys. Chem. 1982,16, 193-198. (17) Yamaoka, K.; Ueda, K. Chem. Lett. 1983, 545-548. (18) Matsuda, K. J. Sei. Hiroshima Univ., Ser. A , 1983, 47, 41-65. (19) Ueda, K.; Nomura, M.; Yamaoka, K. Biopolymers 1983, 22, 2077-2090. (20) Ueda, K.; Mimura, M.; Yamaoka, K. Biopolymers 1984, 23, 1667-1681. (21) Ueda, K. Bull. Chem. Soc. Jpn. 1984, 57, 2703-2711. (22) Oosawa, F. “Polyelectrolytes”;Marcel Dekker: New York, 1971. (23) Selegny, E., Ed.‘Polyelectrolytes”; Reidel: Dordrecht, The Netherlands, 1974. (24) Tinoco, I., Jr.; Yamaoka, K. J . Phys. Chem. 1959, 63, 423-427. (25) Tinoco, I., Jr. J. Am. Chem. Soc. 1955, 77,4486-4489. (26) Balasubramanian, D.; Charney, E. J. Phys. Chem. 1981, 85, 1943-1947. Numerous references on this subject are cited therein. (27) Nakayama, H.; Yoshioka, K. Nippon Kagaku Zasshi 1964, 85, 177-182. (28) Nakayama, H.; Yoshioka, K. J . Polym. Sci., Part A 1965, 3, 813-825. (29) Kikuchi, K.; Yoshioka, K. J. Phys. Chem. 1973, 77, 2101-2107. (30) Tricot, M.; Houssier, C. Macromolecules 1982, 15, 854-865. (31) Yamaoka, K.; Ueda, K. Bull. Chem. Soc. Jpn. 1983,56,2390-2395. (32) Ookubo, N., presented in part at the 32nd Meeting of the Society of Polymer Science, Japan, Kanazawa, Japan, Oct 1983. Ookubo, N.; Hirai, Y.; Hayakawa, R. Polym. Prepr. Jpn. 1983, 32, 2381-2384.
Yamaoka and Matsuda
E
n (b)
2
--*t Figure 1. Schematic presentation of (a) molecular coordinates and (b) reversing-pulse field, RPEB signal with a dip, and time-dependent polarizability, the l iis permanent dipole moment; the aii,covalent polarizability; the ui, ionic polarizability;the gi, optical anisotropy factor; T,D, the relaxation time of the bound-but-mobile counterion; Oii, the rotary diffusion coefficient; ( i = 1 or 3). The 3-axis is the molecular symmetry axis. B, R, and D denote the buildup, reverse, and decay portions, respectively.
associated critically with such factors as polymer concentration, field strength, and the valence of counterion.
Experimental Section Materials. Two sodium poly@-styrenesulfonate) (NaPSS) samples with weight-average molecular weights, M,, of 1.6 X lo6 and 1.1 X 106, determined by light scattering, were kindly supplied from Toyo Soda Industries Co. These samples were used without further fractionation; hence, they are polydisperse regarding the molecular weights. Each sample was dissolved in doubly distilled water (ca. 50 mg/50 mL of water) and exhaustively dialyzed against salt-free water for 96 h by changing totally 8 L of fresh water to remove ionic additives. In order to replace the sodium counterion by the magnesium ion, the dialyzed NaPSS sample was dissolved in 0.05 M MgC12 solution (ca. 14 mg NaPSS/30 mL) and dialyzed, first, against two batches of 0.05 M MgClz solution for 48 h, and then for 72 h against distilled water (totally 6 L). Another NaPSS sample of a lower molecular weight of 2.8 X lo5 was the same as used in previous s t ~ d i e s . ~The ~ ~ molar ~~” concentrations of NaPSS and MgPSS were expressed as the residue concentration and determined photometrically at 262 nm by using a molar absorption coefficient e of 468 M-’ cm-1.34 Measurements. The RPEB signals were measured at 20 O C and 535 nm with two Kerr cells (one with a path length of 1.00 cm and an electrode gap of 0.207 cm, and the other with 2.00 cm and 0.330 cm)on an electric birefringence apparatus previously d e ~ c r i b e d . ~ +A’ ~quarter-wave *~~ plate was inserted before the analyzer to detect the sign of signals. The square-wave pulse electric dichroism was also measured at 20 OC with the above Kerr cells with an electric dichroism apparatus described e l ~ e w h e r e . ~ ~ ~ ~ ~ Since the signals were very weak in most cases, a large instrumental time constant was used to smooth out noises. As a result, only the steady-state dichroism signals were utilized. Data Analysis. The RPEB signals displayed on an oscilloscope screen were converted to the phase retardation, 6, in degrees or the birefringence, Anh4 The steady-state electric dichroism signal was expressed in terms of the parallel specific dichroism MI,/ A.33.34 Theoretical Calculations. On the basis of a general theoretical treatment for RPEB, the following expressions were derived for like molecules of roughly cylindrical symmetry (the symmetry axis is the 3-axis in Figure l a ) in the low-field (Kerr-law) region.24*25They are expressed with notations currently in use, together with some typographic corrections. For the moderately flexible PSS with ionized side chains, an ellipsoid of revolution (33) Yamaoka, K.; Matsuda, K. J. Sei. Hiroshima Univ.,Ser. A 1980, 43, 185-203. (34) Matsuda, K.; Yamaoka, K. Bull. Chem. SOC.Jpn. 1982,55,69-76.
The Journal of Physical Chemistry, Vol. 89, No. 13, 1985 2781
Reversing-Pulse Electric Birefringence or a cylinder has been shown to be a good mdel.ze31*34 The permanent electric dipole moment, p , of the polyanion is negligibly small, as compared with the counterion-induced electric dipole Therefore, the formalism is confined to the case of the polyion of axial symmetry which possesses no permanent dipole moment.24 By defining the molecular coordinates as in Figure l a , the buildup (or rise), reverse, and decay portions of a birefringence signal are given as follows: (i) Buildup: (a) 6 8 1 1 ~ 3#u 1 and 6 8 1 1 ~ # 1 u1;
+
AB =- An(t)/An(-) = 1 - A3S3exp(-t/r3") A I S l exp(-t/Tl") - (1 - A3S3+ AISI)exp(-6011t) (1) (b) 6 8 1 1 ~ 3=11 and 6 8 1 1 ~ = 1 u1;
AB = 1 - [ l
+ 619,~t(S~ - S,)] exp(-6811t)
An(-) = ( 2 G / 1 5 n ) ( g 3- g1)(2y3 - 2% + 2P3 - 2Pl) (ii) Reverse: (a) 6 8 1 1 ~ 3#u 1 and
AR
6811~1' #
(2) (3)
1;
+
An(t)/An(-) = 1 - A3S3exp(-t/~,') A l S l exP(-t/.r,") + (A3S3 - AlSI) exp(-6811t) (4)
(b) 6811~3' = 1 and 6811~1' = 1; AR = 1 - 6Bllt(S3- S,) exp(-6011t) An(-)
(2nCv/15n)k3 - gl)(273 - 271
+ 2P3 - 2Pl)
(5)
(6)
(iii) Decay:
AD = An(t)/An(-) = exp(-6811t)
(7)
An(-) in the buildup is equal to An(0) in the reverse. A1 = 6811~1"/(68l1~1" - 1) A3 = 6811~3'/(6811~3" - 1)
SI =
ul/[(a33
s3
a3/[(a33
=
- a l l ) + (u3 - u1)1 - all) + (u3 - .dl
(8) where An(t) = [nl,(t)- n,(t)] is the birefringence at time t, n is the refractive index of solution, C, is the volume fraction of solute, a33and allare the covalent (electronic and atomic) polarizabilities (2y3 = a3,E2/kTand 2yl = a l l E Z / k T )u3 , and ul are the ionic ) the polarizabilities (2p3 = u3E2/kTand 2p, = u l E Z / k T along 3- and 1-axes, (g3- g l ) is the optical anisotropy factor, and and 7,"are the relaxation times for bound-but-mobile counterions to redistribute along the 3- and 1-axes under the influence of an externally applied electric field E, and ell is the rotary diffusion coefficient of the whole molecule around the transverse axis (1axis) and is equal to (67)-I, where T is the rotational relaxation time. In this treatment, the time dependence of the ionic polarizability is phenomenologicallyassumed to be q ( t ) = ui(-)[l - exp(-t/~p)] ( i = 1 or 3), as shown in Figure lb. It is induced to saturate to be a,(-), vanishes instantaneously upon reversal of the pulse field, and is again induced in the same way as in the buildup process." It is possible to modify this concept of the ionic polarizability, which is caused by mobile counterions, by introducing, e.g., an additional relaxation time with which the polarized ions redistribute toward the other end of the macromolecular surface upon the electric pulse reversal.3s Actually, a more drastic concept was also put forward that the counterion-induced electric moment transforms to a permanent-like electric dipole moment in the buildup process and subsequently behaves as the permanent dipole moment upon the field reversal.I2 This situation can possibly be encountered in a complex aggregated molecular assembly like vesicles,12but it is difficult to realize in simple polyelectrolytes.
Results and Discussion RPEB Signals of High Molecular Weight NaPSS at Various Field Strengths. Figure 2 shows some RPEB signals of two high (35) Yamaoka, K., to be published.
molecular weight NaPSS samples at high, medium, and low fields. The signal patterns and signs of birefringence exhibit a variety of field-strength dependence, in contrast with a previously reported case of NaPSS with a lower molecular weight of 2.8 X lo5 in the concentration range of 0.33-3.3 mM.9,17 Surprisingly, the sign of the RPEB signals of those high molecular weight NaPSS samples is positive at extremely low fields. In addition, there appears an extremum (a dip in common usage) in RPEB pattern upon the reversal of pulse field (c and f).32 However, the sign of An changes to negative with the increase in field strength, as normally observed (a, b, d, and e).9,17*27-31Here, the most unexpected feature is the appearance of a characteristic hump in the reverse portion of RPEB signals. The buildup portion is either normal (b and e) or associated with a slight hump (a and d). Figure 2 thus clearly demonstrates an undisputable advantage of the RPEB method over the conventional single-pulse meWith a further increase in concentration of NaPSS up to 6 mM, no new signal pattern appeared. Since the output voltage of the RPEB signal was recorded on oscillograms, a direct comparison between photographs in Figure 2 is sometimes misleading. This point is illustrated in Figure 3, where the specific birefringence An/C is plotted. The combined effects of the quarter-wave plate and the nonlinear relationship between the photocurrent and the phase retardation 6 (or An) are obvious. The hump is apparently more suppressed in photograph~.~~ RPEB Signals of High Molecular Weight NaPSS at Lower Concentrations. Figure 4 shows oscillograms of high molecular weight NaPSS samples measured at high (a and c) and low (b and d) electric fields. Here, the effect of the NaPSS concentration on RPEB signals is striking; neither the sign of An nor the signal pattern in the buildup and reverse portions exhibits the anomalies that were observed in Figure 2. The disappearance of the extrema (hump and dip) may be related to the expansion of each NaPSS chain, considering that the dilution is the only factor which affects these drastic changes. The charge repulsion of ionized sulfonate groups is most probably responsible for the molecular expansion, which leads, in turn, to the faster mobility of counterions because of the penetration of solvent water molecules into the interior of NaPSS. The sign of An is also the same as that of a sample with an M, of 2.8 X 105.9J7It should be noted that both samples with M,'s of 1.1 X lo6 and 1.6 X lo6 show the anomalous signals similar to those observed in Figure 2 even in the concentration range between 0.6 and 0.7 mM. Concentration Effect on Lower Molecular Weight NaPSS. Now an important question arises on whether a mere increase of concentration induces such anomalous changes in birefringence signals as observed in Figure 2. A direct answer is given in Figure 5, where the signal pattern of the previous NaPSS sample with an M, of 2.8 X lo5exhibits a hump in the reverse portion at an increased concentration of 4.56 mM (Figure Sa) or more. Neither hump nor dip is, however, displayed at a concentration of 3.35 mM (Figure Sb) or less, as in previous studies (1.66-3.3 mM).9317 Thus, the high molecular weight is not the intrinsic factor for the anomalous signals. The appearance of the hump a t higher concentrations (>4.56 mM) is probably due to the inter- or intrachain entanglement of NaPSS polyanion(s), which produces an interior region for Na+ ions to redistribute only slowly by applied pulse field. The various RPEB patterns for NaPSS are summarized in Table I. RPEB Signals of MgPSS. It has been shown that the low molecular weight Mg- or CaPSS sample ( M , = 2.8 X lo5) gives rise to the field-free rotational relaxation time, ( T ) E B , much shorter than that of the corresponding NaPSS?J7 A similar correlation is also present between high molecular weight Na- and MgPSS samples (M, = 1.1 X lo6). Figure 6 shows RPEB signals of this MgPSS sample at two concentrations. Neither hump nor dip appears in the reverse portion over a wide range of concentrations and field strengths (5.29-0.079 mM and 5.2-0.20 kV/cm), the (36) Frederiq, E.; Houssier, C. "Electric Dichroism and Electric Birefringence";Clarendon Press: Oxford, 1973.
1181 The Journal of Physical Chemistry, Vol. 89, No. 13, 1985
Yamaoka and Matsuda
-
t%me 2 Anomalous RPEB signals of two NaPSS samples at different field strengths. ( a s ) : M. = 1.1 x IO6 and C = 1.98 mM. (d-f): M. 1.6 X 106 and C = 2.64 mM. The field strength. sweep time per division. and birefringene sign are as follows: (a) 2.62 kV/cm. 100 ps, and An < 0 (b) 1.33 kV/cm. 200 PS. and An < 0 (c) 0.1 I kV/cm. I ms. and An > 0; (d) 2.10 kV/cm. 200 ps. and An < 0 (e) 1.56 kV/cm. 200 ps, and An < 0 (00.052 kV/cm. I mi. and An > 0.
sign of An being always negative. Since MgPSS gives rise to a shorter ( T ) and ~ ~a weaker An than the corresponding NaPSS. it is probably in a compact conformation at a concentration as low as 0.079 mM. Thus, the field orientation of MgPSS is due mostly to the fast induced electric moment, which results from the redistribution of Mg’* ions over the polyanion surface. The Mg’+ ions in the interior region probably remain immobile even by externally applied high electric fields. Dependence of Steadystate Birefrinxence and Dichrdsm on Field Strength. Figure 7a shows the steady-state birefringence normalized by concentration, An/C, which is plotted against field strength for high molecular weight Na- and MgPSS samples. For NaPSS samples, the magnitude of An/C is larger when the molecular weight is higher and the concentration is lower. The
sign of An is positive at extremely low fields but changes to be negative at higher fields, the crossover point being in the range of 0 . 7 4 8 kV/cm. Electric linear dichroism was utilized in the study of NaPSS (Mw= 2.8 X IO1)and its dye complex.u The sign of M J A was found to be negative at the 262-nm band (B2J and the 224.5-nm band (B,J of the phenyl ring (cf. the formula in Figure 7) over a field strength range 0.12-1 1.6 kV/cm?‘ Figure 7 h c shows the field strength dependence of the parallel specific dichroism ( M , / A ) of an NaPSS sample with a high molecular weight of 1.1 X IO6 at 262 nm and 243 nm rather than 224.5 nm, where the 2 mM NaPSS solution absorbs light too strongly. Values of M I / A at the two absorption bands also change the sign from positive to negative between 0.7 and 0.8 kV/cm with the increase
Reversing-Pulse Electric Birefringence
The Journal of Physical Chemistry. Vol. 89, No. 13. 1985 2783
t /ms
,-----.€a-
0
az
a6
0.4
,.-.-
TABLE I: S " r y M
1.0
of Obaewed RPEB Signs a d Pancna of
-.Ih
M./IO" 1.6
1.1
0.28
[ N a M ] in monomer unit/mM 3-2 0.66 0.1 or lws low pos,dip pos. dip ncg. none high neg. hump neg, hump neg. hump neg. none low pos. dip poa,dip neg, none high neg. hump neg, hump mg. hump neg, none low neg. ? ncg, none neg, none high neg. hump neg. none neg, none E
5.54.5
* E stands for the field strength, which i t about 0.2 kV/cm for 'low" and above ca. I kV/cm for "high". P m and neg are the poaitive and negative steadyatate birefringence signs, respectively. Dip and hvmp denote the anomalous patterns. while none denote the normal pattern in the reverse portion of the RPEB signal; ? indicates that the anomaly could not bc determined due io the low signal/noirc.
+MZ
i
I
I
I
0
2
4
6
i
I /mr
Figme 3. Two typical anomalous RPEB signals of NaPSS with a molsular weight of 1.1 X I06 c x p d in terms of ths specific birefringene An/C. The upper and lower signals correspond to oscillograms a and c in Figure 2. rapcctively. Note that the time scale is different between two signals. Arrows indicate the positions where a pulse field is either reversed (R) or rcmwed (D).
in field strength Thus, the tilt and twist angles of the phmyl ring attached to the polymer chain an. as an average. less than fS4.7O
relative io the mi"tion mi3 of the NaF'SS molecule at extrunely low fields, but more than f54.7O at field strengths higher than 0.7-0.8 kV/cm. This higher-field behavior of the present sample resembles the result previously reported for the low molecular weight sample." Sign R e r s a l of Electric Birefringence and Dichroism. Figure 7 clearly shows that the sign of An reverses from positive to negative at abour 0.7-0.8 kV/cm. Some possible causes for this sign reversal will be discussed. Both specific birefringence An/C and reduced dichroism AA/A are expressed as the prcducl of the optical term (the optical anisotropy factor. Ag = (g, - g,). for birefringence and the saturated reduced dichroism, ( M I A ) , for dichroism) and the electric term (the orientation faclor, W3,ya)). In order tocaplain the positive sign of h / C o r M I A at extremely low fields. two combination%are pos%ible.as shown in Table II
Fl&"e 4. Oscillogramsof RPEB signals of high molecular weight NaFSS rampla ai dilute concentrations.
(a and b): M. = I .I X 106 and C = 0.091 mM. (c and d): M. = 1.6 X IO6 and C = 0.12 mM. The field strength and sweep time per division are (a) 9.84 kV/cm and 100 w:(b) 0.25 kV/cm and 500 11s: (c) 8.29 kV/cm and 100 ps: (d) 0.28 kV/cm and 200 ps. An < 0 in all CBSFS.
2784
The Journal of Physical Chemisrry, Vol. 89. No. 13. 1985
Yamaoka and Matsuda
Figure 5. Oscillograms of RPEB signals of a low molecular weight sample (M,, = 2.8 X IO') at different concentrations. The field strength, sweep time pr division, and concentration are (a) 1.87 kV/cm. IW ps. and 4.56 mM; (b) 2.06 kV/cm. 20 ws. and 3.35 mM. The sign of An is negative in t h e eases.
Figure 6. Oscillograms of RPEB signals of a high molecular weight MgF'SS sample (M. = 1.1 X IO). The field strength. sweep time p r division, and concentration are (a) 5.17 kV/cm. 20 ps, and 3.72 mM: (b) 0.65 kV/cm. 500 ,e. and 3.72 mM: ( c ) 5.09 kV/cm. 100 us. and 0.079 m M (d) 0.73 kV/cm, 200 ,as. and 0.079 mM. An < 0 in all cars. where the lowest field was about 0.2 kV/cm.
The uquivocal determination baween them is not easy. but some inferences can be made on the basis of the previous dak1?."*~~3' The overall shape of any lower molecular weight NaF'SS molecules can be approximated by a semiflexible rod in a salt-free solution;ze3' the chain flexibility should progressively increase with the increase in the molecular weight. Therefore, the overall shape of high molecular weight NaPSS molecules should be best ap
RTABLE 11: s m o( swnC BI~Aobsd AnlC or M I A ' Ag or ( M I A ) , positive at low fields negative at high fields
'AnIC.
K.:Cbmy. e. M a c d m l r r 1973.6.66-76. (38) Yamaoka. K.: Chamy, e. J. Am. C h m . Sa.Im 94.8963-8974. (37) Y8naOka.
-
positive negative negative positive
mCfim 9
positive negative positive negative
( 2 * / n ) ( n g ) and M I A = ( M I A ) , * .
proximated by a prolate ellipsoid of revolution; an oblate ellipsoid is an unlikely model. Hence. the molecular symmetry axis is most
The Journal of Physical Chemistry, Vol. 89, No. 13, 1985 2785
Reversing-Pulse Electric Birefringence E/(kV/cm) 0
1
I
0
2'
'3 I
4 I
5
E -6
U
- -4
-
c
1
U I
0
5-2 \ C
Q
-
Case Z 6 6 1 1 ~ 3=a and u3/ul = 0, i.e., the relaxation time for the counterion redistribution along the 3-axis is infinitely long as compared with the rotational relaxation time 1 /60, and hence the induced moment along the 3-axis is negligibly small. The ionic relaxation time along the 1-axis, T ~ "is, , however, comparable with , ~-). In this case, the steady-state bire1/60,, (0 < 6 8 , , ~ < fringence An(m) > 0, if Ag < 0, since 9 < 0. Case IZ 6811~1'= and ~ 1 / ~ 30, Le., the ionic relaxation time along the 1-axis is infinitely long, but the one along the 3-axis is comparable with l/6811. In this case, An(-) > 0, if Ag > 0, since @ > 0. Case IZZ 66ll~3'= 0 and ul/(u3 - al) > or < 0, Le., the ionic relaxation time along the 3-axis is very short as compared with 1/60,,, but the one along the 1-axis is comparable with 1/60,,. Case I K 6811~1'= 0 and u3/(u3 - u , ) > or < 0, Le., the ionic relaxation time along the 1-axis is very short as compared with 1/60,,, but the one along the 3-axis is comparable with l/6811. Since the overall shape of an NaPSS molecule can be assumed to be a prolate ellipsoid of revolution, the relation that (u3 - ul) > 0 probably holds for the counterion-induced dipole moments. This condition is obviously satisfied by cases I1 and 111, for which the following expressions can be derived from eq 1, 3, 4, 6, and 8. (For the sake of simplicity, the expressions, for which 6611~3' # 1 and 6 ~ 9 , # ~ ~1,~are ' considered here.) Case IZ
o +1
-
AR
-
=
1-
6811T3u
6811~3'- 1
exp(-t/r,") - ( 1 -
6011T3'
681 173' -1
)
exp(-601,t) (9)
7
0 t0.2
0
2 3 4 5 E/(kV/cm) Figure 7. Field-strength dependence of the specific birefringence An/ C and parallel specific dichroism U , , / Aof various PSS samples. Concentrations are at 1.98 mM (0)and 0.66 mM ( 0 )for NaPSS (M,= 1.1 X lo6) and at 3.35 mM (0)and 0.66 mM (dashed line) for NaPSS (M,= 2.8 X lo5),the last being taken from ref 31. MgPSS (M,= 1.1 X lo6) at 1.98 mM (A). Wavelengths are at (a) 535 nm, (b) 262 nm, and (c) 243 nm. The sign reversal from positive to negative occurs at 0.7-0.8 kV/cm in (a-c). probably the major semiaxis of the ellipsoid, along which the electric dipole moment (a u3) is induced by applied pulse field. In this case, the sign of the electric termis probably positive (Le., > 0) at any field strength. Then it should be the optical part that is responsible for the sign reversal of An/C or AA/A from positive at lower fields (Le., Ag or ( A A / A ) , > 0) to negative at medium-to-high fields (Le., Ag or ( A A / A ) , < 0), as can be seen in Table 11. There are two possible causes for the sign to be reversed: one is that the molecular shape is deformed with the increase in field strength, so that the angle between the orientation axis, which still coincides with the molecular symmetry axis, and the optical transition moments becomes, as an average, more than f54.7', and the other is that the orientation axis and the molecular symmetry axis deviate from each other, as the field strength is increased. This latter case, Le., the deviation of the electric principal axis from the hydrodynamic principal axis is the most probable cause for the sign reversal. Such a situation may be realized by the gradual increase of the transverse electric moment (a u , ) induced along the minor semiaxis of the ellipsoid. Thus, the resultant counterion-induced dipole moment, proportional to (u, - u,), directs away from the longitudinal molecular axis. Simulation of Observed RPEB Signals. The above conjecture may be justified by simulating the observed RPEB signal patterns correctly with the aid of eq 1-8. Here, four extreme but simple cases will be considered. (Both y3and y, are assumed to be zero.)
Case IIZ
=
2a = -cv(g3 15n
- g1)(2P3 -
2Pl)
(14)
In case 111, 0 1 / ( ~ 3 - ul) > 0, unless ul 0, because (u3 - ul) > 0 for the prolate ellipsoid. Figure 8 shows theoretical RPEB curves for the buildup and reverse portions. These curves were calculated for case I1 with 66'll~30 as the parameter (top) and for case I11 with 66'11~1u and ~ 1 / ( ~- 3u , ) as the parameters (middle and bottom). If the electric moment is induced by applied electric field only along the 3-axis of the prolate ellipsoid, Le., u3 >> u , , with a relatively slow time for the counterion redistribution, the RPEB curve rises smoothly and reaches the steady state, displaying a dip or minimum upon the pulse reversal (Figure 8a). The depth and time of this dip vary with the relative magnitude of T ~ ' and 1/60,,. These signal features reproduce well the observed RPEB patterns of the high molecular weight NaPSS samples at higher concentrations in the
2786 The Journal of Physical Chemistry, Vol. 89, No. 13, 1985 1.0
0.75 0.5
0.25
0
1
4
ta5 ;f
a 0.25
d,-d, 0.5
I
1.0
0.5
0
1
20
'
1
2
811 Figure 8. Theoretical RPEB curves calculated for counterion-induced fast and slow dipole moments. Both buildup and reverse portions are normalized by the steady-state birefringence An,(-) or An,(O). (a) 68,,rl" = and u3 >> ul with values of 66,173" as the parameter, (b) 681173u = 0 and u 1 / ( u 3- ul) = 0.5, and (c) 68,,1,"= 0 and uI/(u3 - 6,) = 1 .O with values of 681171u as the parameter. In (a), if Ag > 0, the sign of An is positive, and vice versa. In (b) and (c). if Ag < 0, the sign of An is negative, and vice versa.
-
extremely low field strength region, as are shown in Figure 2. It is now conceivable that the migration of the originally slow-moving counterions along the 3-axis will be accelerated, as the applied field strength increases, so that the time 7 3 ubecomes much shorter than the rotational relaxation time 1/68,,, Le., 6 8 1 1 ~ 3 u N 0. Under such circumstances, the electric moment may be saturated or approach saturation almost instantaneously; thus, the transverse dipole moment can possibly be induced (Le., uI > 0) on a time scale comparable with the overall molecular rotation of NaPSS, Le., 6 8 , , ~ >, ~0. For example, a value of 0.5 is given u varied (Figure 8b). The to u , / ( u 3- ul) and values of 6 B l l ~ lare calculated signal patterns show humps in the reverse portion, resembling those actually observed for NaPSS at field strengths above 0.8 kV/cm, at which the sign of An changes to negative. The theoretical profile also shows a hump in the buildup portion
Yamaoka and Matsuda which is shallower than the second hump in the reverse portion, if the ionic mobility is slower than the molecular rotation, e.g., 6 8 1 1 ~ 1=~2. These features are indeed confirmed in observed RPEB signals (cf. Figure 2). In summary, the RPEB curves simulated with the aid of simplified theoretical expressions (eq 9-14) reproduce satisfactorily the anomalous signal patterns with a dip or a hump in the reverse portion. The agreement between observed and calculated RPEB signals can undoubtedly be improved within the theoretical framework originally set forth by Tinoco and Y a m a ~ k a , if* ~the restrictions imposed on eq 9-14 are further removed. In order to facilitate the quantitative analysis of experimental signals, theoretical expressions should be developed, which take into account explicitly the dependence of both counterion-induced dipole moment and orientation function on applied electric fields. Signal-to-noise ratios should also be improved in the extremely low field region. Well-fractionated, nearly monodisperse PSS samples should be used for the accurate determination of field-on and field-off relaxation times.3g A quantitative study of the concentration effect on these relaxation times would make it possible to estimate the overlap threshold concentration in modern scaling theory.40 We believe that we have shown, in this work, the usefulness of RPEB techniques to elucidate the dynamic behavior of counterions bound to polyelectrolytes in aqueous solutions.
Conclusion The response of the counterions of PSS to externally applied electric field was clarified by the RPEB method. Three critical factors for a variety of RPEB signal patterns are (i) the valence of counterion, (ii) the concentration of PSS,and (iii) the strength of applied electric field. The MgPSS samples give rise to normal RPEB signals regardless of the molecular weight, concentration, and field strength, since only the fast-redistributing Mg2+ions on the exterior of the shrunk PSS chain contribute to the field orientation. On the other hand, NaPSS samples show anomalies in the reverse portion either with a dip at extremely low field strengths or with a hump at higher field strengths, unless the concentration is dilute. It is this dilution effect that is dependent on the molecular weight of NaPSS, and, hence, responsible for the complex signal patterns. The chain conformation is either contracted at higher concentrations or expanded to be penetrable by solvents at lower concentrations. When the redistribution time of the Na+ ions in the interior of the entangled PSS chain(s) becomes comparable with the overall rotational relaxation time, the slow induced dipole moment contributes to the field orientation. In addition to those anomalous RPEB patterns, the electric field strength affects the sign of the steady-state birefringence of higher molecular weight NaPSS samples, changing it from positive at low fields to negative at higher fields. Registry No. NaPSS, 28038-50-8; MgPSS, 86002-50-8. (39) Wijmenga, S. S. Ph.D. Dissertation, Leiden University, The Netherlands, 1984. A reviewer kindly informed us of this thesis work, in which some reversingpulsemeasurementswere carried out on close-to-monodisperse NaPSS in 0.01 M NaCl solution. (40) This possibility was pointed out by the same reviewer, to whom we are thankful.
