J. Phys. Chem. B 1997, 101, 3653-3664
3653
ARTICLES Ionic Polarization of Carrageenans with Different Charge Densities As Studied by Electric Birefringence Measurements Kazuyoshi Ueda,*,† Shigenobu Sato,† Hiroshi Ochiai,‡ and Akira Imamura† Department of Chemistry, Faculty of Science, and Department of Science Education, Faculty of Education, Hiroshima UniVersity, Kagamiyama, Higashi-Hiroshima, Hiroshima 739, Japan ReceiVed: April 17, 1996; In Final Form: January 29, 1997X
Ionic polarizability of the κ-, ι-, and λ-carrageenans have been investigated by the electric birefringence method. These carrageenan samples were selected because they have different numbers of sulfuric acid residues on their linear polysaccharide chains; that is, their charge densities are different from each other. The analysis of both the polymer concentration dependence and the added salt concentration dependence showed that these three carrageenans have almost the same ionic polarizability irrespective of their different charge densities when they were compared in the same conditions. The results were analyzed by using two groups of theories. That is, one is based on the “loosely bound” counterion polarization, and the other is based on the “condensed” counterion polarization. The ionic strength dependence of the experimentally obtained polarizability could be fitted by Manning’s and Rau and Charney’s “loosely bound” counterion polarization theories.
Introduction Polyelectrolyte molecules in aqueous solution can be oriented by an externally applied electric field and shows large electrooptical phenomena, such as electric birefringence and electric dichroism. The origin of this orientation is thought to be a result of the torque caused by an interaction between an induced ionic polarization moment and the external electric field. Mandel1 early derived an equation to account for this polarization moment, which treated the bound counterions as distributed in the discrete charge potential well in accordance with the Boltzmann distribution under the electric field. Subsequently, Oosawa considered the thermal fluctuation of the counterions’ distribution and derived an equation for polarization by expanding the fluctuation in a Fourier series.2 Since then, many theoretical treatments have been presented to account for this polarization mechanism.3-10 However, the polarization of polyelectrolyte molecule is not still well established. As for the behavior of the bound counterions around a polyion, some portion of the counterions may be attracted and bound to the vicinity of the polyion due to the high electric potential of a polyion. Manning showed in his counterion condensation theory that those bound counterions are classified into two types of bound states.11 That is, some of them are tightly bound to the vicinity of the polyion to cancel the charge of the polyion at some extent. These counterions are called “condensed” counterions. The other type of bound counterions are loosely attracted to the moiety around the polyion by the Debye-Hu¨ckel type electrostatic force. These counterions are called “loosely” bound counterions. These types are classified by the charge density parameter of the polyion ξ ()e2/kTb), where e is the elementary electric charge, is the dielectric constant of the solvent, k is the Boltzmann constant, T is the †
Department of Chemistry, Faculty of Science. Department of Science Education, Faculty of Education. X Abstract published in AdVance ACS Abstracts, April 15, 1997. ‡
S1089-5647(96)01113-3 CCC: $14.00
Kelvin temperature, and b is the distance between the charged sites on the polyion. If a charge density parameter ξ of a polyelectrolyte molecule is smaller than unity, only the loosely bound type counterion exists. On the contrary, if ξ > 1, two types of bound counterions coexist, that is, some of the counterions are tightly bound to the polyion (condensed) and reduce the net charge of the polyion until the values of ξ tend to unity. The remaining counterions are “loosely bound” to the polyion. There have been developed many theories of the ionic polarization of a polyion where Manning’s counterion condensation theory was treated as a basic concept to determine the polarizable counterions. In some polarization theories, the “tightly bound” (condensed) counterions in the condensation theory were considered to be attributable to the polarizable counterions by the external electric field.4-6 Other theories were based on the “loosely bound” counterions around the vicinity of the polyion as polarizable counterions.7-9 However, whether the counterions responsible for the ionic polarization are “tightly bound” counterions at the vicinity of the polyion and/or “loosely bound” counterions has not been clarified yet, and is still controversial. From the experimental point of view, there are few discussions on the polarization mechanism and the polarizable ions that are attributed either to the condensed counterions or to loosely bound counterions in the ionic atmosphere. Most of the previous works have been carried out mainly by using the samples of synthetic polyelectrolytes12-15 and polynucleotides.5,16,17 Usually, these polymers were highly charged along the polyion chain and, therefore, the values of ξ of them were larger than unity. Hence, the ionic polarizability has been generally interpreted as caused only due to the condensed counterions. Charney first measured an electric dichroism to focus on the loosely bound counterion polarization by using the hyaluronic acid sample whose value of ξ is smaller than unity.18 He showed that the polarizability of this molecule was © 1997 American Chemical Society
3654 J. Phys. Chem. B, Vol. 101, No. 19, 1997 well explained by his ionic atmosphere polarization theory. We also performed a reversing pulse electric birefringence experiment of κ-carrageenan, which has no condensed counterion, i.e., ξ < 1, and showed that this molecule has a large fast-induced ionic polarizability.19 Carrageenan is a family with different numbers of sulfuric acids on the linear polysaccharide chains. κ-Carrageenan has only one sulfuric acid residue on its disaccharide repeating unit. The types of ι- and λ-carrageenans have roughly two and three sulfuric acid residues on their repeating units, respectively. Therefore, the carrageenans can be considered as suitable samples for the investigation of the effect of charge density on the polarization mechanism because we can select several carrageenan samples with differing charge density parameter ξ. In another communication,20 we compared these three different types of carrageenans and briefly reported that these samples had the same amount of polarization irrespective their different values of ξ. The result suggested the possibility that the contribution of the “loosely bound” counterions on the polarization cannot be ignored or rather should be considered as a main mechanism for the ionic polarization under the external electric field. In this paper, we proceeded with further investigation on the mechanism of the polarization by measuring the effect of added salts on the ionic polarization using κ-, ι-, and λ-types of carrageenans. Several theories have been applied to investigate the mechanism of ionic polarization. The previous works were also summarized and thoroughly analyzed with the present data. Hydrodynamic properties of the carrageenans were also considered to account for the polarizability more quantitatively. The hydrodynamic properties were obtained by analyzing the electric field free decay curve of the electric birefringence signals after the applied external electric pulse terminated. These signals were able to obtain simultaneously in the experiment with the steady-state signals which were used for the analysis to obtain the ionic polarizability. Experimental Section Materials. κ- and ι-carrageenans were extracted from Eucheuma cottonii and Eucheuma spinosa, and were kindly gifted from Mitsubishi Rayon, Co. Ltd. (Tokyo). λ-Carrageenan was purchased from Sigma Chemical Co. Ltd. (Lot No. 21H0322). These samples were sonicated in aqueous solution to reduce their molecular weight with a Tommy Seiko UR200P sonicator at 0 °C and a power level of 200 W for 5 min, then followed by a 5 min interval for bubbling helium gas. This radiation and bubbling cycle was repeated to a total radiation time of 2 h.19,21 The effect of irradiation time on molecular weights was examined by gel permeation chromatography (GPC) with small portions of samples at appropriate time intervals during irradiation as was shown in the previous paper.19 The sonicated samples were then adjusted to 0.2 N NaCl solution and repeatedly fractionated by the stepwise precipitation with 2-propanol until samples with desired molecular weights and narrow molecular weight distributions were obtained. A sample fraction with a similar degree of polymerization was selected from the fraction for κ-, ι-, and λ-carrageenan, respectively. These samples were exhaustively dialyzed against distilled water, and were first converted to the acidic form through a mixed-bed ion-exchange column (Amberlite IR-120, IRA-402). Then, the potassium form sample was obtained by rapid neutralization to pH 7.0 with aqueous potassium hydroxide by potentiometric titration followed by the procedure of Gekko et al.22 Finally, the samples were freeze-dried. The degree of polymerization (DP) and the ratio of weight- to number-average molecular weights (Mw/Mn) of the samples were determined by
Ueda et al. TABLE 1: Characteristic Molecular Parameters of Various Carrageenans DP Lc (nm) DS Mw/Mn b (Å) ξ
κ-type
ι-type
λ-type
156 160 0.9 1.2 11.4 0.64
177 182 1.8 1.2 5.7 1.26
179 184 2.4 1.1 4.3 1.68
gel permeation chromatography with small angle laser light scattering (GPC-LALLS) by using an apparatus Tosoh LS-8000 system (Tokyo, Japan) with pullulan as a standard. The sulfate contents per disaccharide unit of carrageenans (DS) were determined by the potentiometric titration. These characteristic data for three types of carrageenans are listed in Table 1. The contour lengths, Lc, were calculated by using 10.3 Å for a disaccharide unit length of the nonhelical and elongated conformation of carrageenans.23 By using the values of DS and disaccharide unit length, the distance b between the charges for three types of carrageenans was calculated and is shown in Table 1. The value of charge density parameter ξ for each sample was calculated from the data above and is also listed in Table 1. Measurements. Electric birefringence measurements were performed at 30 °C and at 633 nm on a laser electric birefringence apparatus and the details are given in elsewhere.24,25 The electric field was applied as a single squared pulse with 50-100 ms duration pulse width, and the strengths were varied from 0 to ca. 13 kV/cm. The Kerr cell was made of Kel-F and the optical path length of 1.0 cm was used. The signals were stored on a transient wave memory and accumulated in arbitrary numbers and then averaged for analysis. A typical electric birefringence signal can be seen for example in previous papers.15,26 Analysis. The steady-state value of the electric birefringence ∆n is expressed as follows as a function of external electric field strength E.27
∆n ) (2πCv/n)(g3 - g1) Φ (E)
(1)
Here, Cv is the volume fraction of the polymer, n is the refractive index of the solution, (g3 - g1) is the optical anisotropy factor, and Φ(E) is the orientation function. As for the orientation function of Φ(E), a new theoretical orientation function, the SUSID (saturable-unsaturable induced dipole) function Φ(Fs,F,γ′), which was derived by Yamaoka and Fukudome for a rodlike and ionized polymer without permanent dipole moment, was used in this study.28,29 The three parameters used in this function are expressed as follows.
Fs ) (∆σE0/kT)E F ) (∆σ/kT)E2
(E g E0) (E e E0)
γ′ ) (∆R′/kT)E2
(2) (3) (4)
where ∆σ is the saturable polarizability anisotropy, ∆R′ is the unsaturable polarizability anisotropy, and E0 is the critical electric field discussed below. The notations were all according to Yamaoka and Fukudome.29 The SUSID function assumes that the electric moment consists of two terms. One is a saturable induced ionic dipole moment ∆σE, which is saturated at the critical electric field E0, and thereafter behaves as the permanent-like dipole moment ∆σE0. The concept of the existence of the saturable ionic moment was first introduced by Yoshioka.30 The other is the unsaturable induced dipole
Ionic Polarization of Carrageenans
J. Phys. Chem. B, Vol. 101, No. 19, 1997 3655
Figure 1. (a). Electric field strength dependence of the steady-state electric birefringence ∆n of different types of carrageenans in aqueous solution without added salts. Solid curves were theoretically calculated ones by using the SUSID orientation function. The details are shown in the text. The polymer concentration of the solution is shown in the figure. (b) Low-field behavior of the steady-state electric birefringence ∆n of different types of carrageenans in aqueous solution without added salts. The slopes show Kerr’s law.
TABLE 2: Concentration Dependence of Kerr Constants and Calculated Electrooptical Parameters for Various Types of Carrageenans Obtained by Using the Theoretical Orientation Function SUSID Cp (mM)
B × 1011 (cm/V2)
(g3 - g1) × 103
∆R′ × 1032 (F‚m2)
∆σ × 1032 (F‚m2)
E0 (kV/cm)
∆σE0 × 1027 (C‚m)
∆R × 1032 (F‚m2)
κ-type
0.59 0.31 0.15 0.07
18.1 16.6 12.2 7.2
2.2 2.3 2.5 2.7
1.2 2.3 3.6 3.6
3.1 5.9 8.9 9.0
2.3 1.7 1.4 1.4
7.2 10.0 12.3 12.4
2.5 4.7 7.2 7.7
ι-type
0.46 0.23 0.12 0.06
20.4 22.7 20.6 14.1
3.2 3.2 3.8 4.2
1.0 2.5 3.8 4.4
2.5 6.1 9.5 11.1
2.6 1.6 1.3 1.2
6.5 10.1 12.6 13.6
2.0 4.9 7.6 8.9
λ-type
0.46 0.23 0.12 0.06
24.0 26.3 20.2 12.6
3.3 3.4 3.8 4.4
1.1 2.3 3.4 3.9
2.7 5.8 8.6 9.7
2.5 1.7 1.4 1.3
6.7 9.8 12.0 12.7
2.1 4.6 6.8 7.7
moment, ∆R′E, which is not saturated at any electric field strengths.28 This orientation function was applied to DNA molecules and was shown to reproduce satisfactorily the field strength dependence of DNA over an entire electric field strength region.28 The total value of ionic polarizability, ∆R, which is the sum of the saturable and unsaturable ionic polarization, is calculated by using the following equation.
∆R ) ∆R′ + (∆σE0)2/kT
(5)
The fitting procedure between the theoretical SUSID function and field strength dependence of the steady-state electric birefringence is shown in detail in previous papers.28,29 Results Figure 1a shows the electric field strength dependence of the
steady-state values of the electric birefringence (∆n). They were measured at various concentrations in aqueous solution for three types of carrageenans. All samples exhibit the similar dependence of ∆n on the square of the electric field strength E2, which tend to saturate at high electric field strength. At low electric field strength region, the value of ∆n is proportional to E2 as was shown in Figure 1b; that is, the Kerr law holds. The Kerr constant B, which was defined as (∆n/λE2)Ef0 was calculated from the slope of ∆n versus E2, and the obtained values are listed in Table 2. In the above equation, λ is the wavelength of the incident light. The Kerr constants tend to decrease as the concentration decreases. The specific Kerr constant, B/c, was plotted in Figure 2 against the mass concentration c in g/cm3. Yoshioka and O’Konski discussed the specific Kerr constants of polymer solution and divided them into classes.31
3656 J. Phys. Chem. B, Vol. 101, No. 19, 1997
Ueda et al.
Figure 2. Polymer concentration dependence of the specific Kerr constant obtained for different types of carrageenans.
That is, the specific Kerr constants are practically independent of the concentration when the macromolecules are rigid.32 By contrast, flexible polyions have usually large specific Kerr constants at low polymer concentration, and exhibit a large dependence on the concentration as shown in Figure 3 of Kikuchi and Yoshioka’s paper for polystyrenesulfonate.14 This is due to the increased repulsion between the charged groups and the resulting expansion of the chain at low polymer concentration. This can be considered as a measure of the chain flexibility. In our case, the specific Kerr constants in Figure 2 showed a concentration dependence, but not so large as that of polystyrenesulfonate.14 This indicates that the carrageenan molecule can be regarded neither as a rigid chain nor as a fully flexible chain, but should be considered as a somewhat flexible polymer. Once the Kerr law holds, the field strength dependence of the electric birefringence can possibly be analyzed by using eq 1. The previous experiment showed that κ-carrageenan does not have a permanent dipole moment and/or a slow-induced ionic dipole moment which was performed by using the reversing pulse electric field,33 and only possesses a fast-induced ionic polarizability.19 This satisfies the requirement of the SUSID orientation function because it was derived for the ionized polymer without the permanent dipole moment. The SUSID orientation function recently extended to a form which includes permanent dipole orientation.34 However, the original SUSID function is more suitable for this work because it can neglect the permanent dipole moment term and this can reduce the number of parameters for fitting procedure. In order to obtain the ionic polarizability for the carrageenan samples, the observed field strength dependence of ∆n was fitted to the theoretical curve of the SUSID orientation function. The best fitted theoretical curve for each sample is shown by the solid line in Figure 1a. The SUSID parameters which were used to fit the theoretical curves to the experimental curves are listed in Table 2. The obtained values of ∆R′ and ∆σE0 and also the total ionic polarizability ∆R for all types of carrageenans were quite similar to each other when the values were compared at the same concentrations. The threshold electric field strength of the saturable ionic moment saturated, E0, was around 2.0 kV/cm and slightly decreased as the concentration decreased. This is close to the result obtained for the previous case of DNA.29 The optical anisotropy factor (g3 - g1) was not changed with concentration in accordance with the result as found by Kikuchi and Yoshioka.14 The total ionic polarizabilities, ∆R, for three types of carrageenans are plotted against the polymer concentration in Figure 3. Both the magnitude and the concentration dependence of the polarizability were quite similar to each other in spite of their large difference in charge densities of the polymer chains. The values of the total ionic polariz-
Figure 3. Polymer concentration dependence of the total ionic polarizability ∆R of different types of carrageenans in aqueous solution without added salts. Carrageenan samples are κ-type (0), ι-type (O), and λ-type (4), respectively.
abilities at infinite dilution which were obtained by extrapolation to zero concentration, ∆R0, were 8.8, 10.2, and 8.9 for κ-, ι-, and λ-carrageenans, respectively. It should be emphasized again that these values are surprisingly similar to each other irrespective their large difference in charge densities. Effect of Added Salt Concentration. In order to investigate the added salt concentration dependence of the ionic polarizability, electric birefringence of carrageenan with added KCl salt solution was measured. Figure 4 shows the electric field strength dependence of the electric birefringence of three types of carrageenans measured at different added KCl concentrations. The values of the electric birefringence ∆n decrease with the increment of the concentration of added KCl. In the low electric field region at around E2 < 20 (kV/cm)2, Kerr’s law holds in all cases as shown in Figure 4b. The experimental data were fitted by using the SUSID orientation function. Solid curves in Figure 4a represent the theoretical curves calculated by using the SUSID orientation function. The theoretical curves well reproduced the electric field strength dependence of the experimental electric birefringence values. The parameters used for these fitting are listed in Table 3. The values of the total ionic polarizability ∆R were also obtained from the ∆R′ and ∆σE0, and are listed in Table 3. In the calculation of the ionic strength, the contributions of both simple salt ions and the counterions of the polyion have to be taken into account. In this study, the ionic strength of each solution was calculated by using the following equation.35
I ) Cs + (1/2)fCp′
(6)
where Cs is the equivalent concentration of simple salt, Cp′ is the polyion concentration expressed in equivalent/liter (DS × Cp), and f is the uncompensated fraction of the polyion charge; that is, f ) 1 at ξ < 1 and f ) ξ-1 at ξ > 1 for monovalent counterions. The dependence of the total ionic polarization on the ionic strength is plotted in Figure 5. The popular dependence of ∆R on ionic strength can be seen; that is, the values of ∆R decreased with the increment of ionic strength. Although the values for κ-carrageenan in Figure 5 are slightly smaller than those for other types of carrageenans, no significant difference can be seen in the magnitude of the values and the dependence of ∆R on ionic strength in these three carrageenans. Analysis of the Relaxation Process. The magnitude of the ionic polarization of polyion molecules depends on the hydrodynamic size and shape of the polyion in solution.36 The decay
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J. Phys. Chem. B, Vol. 101, No. 19, 1997 3657
Figure 4. (a). Electric field strength dependence of the steady-state electric birefringence ∆n of different types of carrageenans in aqueous solution with addition of KCl salts. Solid curves were theoretically calculated ones by using the SUSID orientation function. The details are shown in the text. Polymer concentrations of the solutions were kept at the constant values of 0.15, 0.12, and 0.12 for κ, ι, and λ, respectively. Salt concentrations of the solutions are shown in figures. (b) Low-field behavior of the steady-state electric birefringence ∆n of different types of carrageenans in aqueous solution with addition of KCl salts. The slopes show Kerr’s law. Salt concentrations of the solution are shown in the figure. Polymer concentrations of the solutions were kept at the constant values of 0.15, 0.12, and 0.12 for κ, ι, and λ, respectively.
TABLE 3: KCl Salt Concentration Dependence of Kerr Constants and Calculated Electrooptical Parameters for Various Types of Carrageenans Obtained by Using the Theoretical Orientation Function SUSIDa Cs (mM)
κ × 103 (Å)-1
B × 1011 (cm/V2)
(g3 - g1) × 103
∆R′ × 1032 (F‚m2)
∆σ × 1032 (F‚m2)
E0 (kV/cm)
∆σE0 × 1027 (C‚m)
∆R × 1032 (F‚m2)
κ-type
0.0 0.13 0.25 0.40 0.65 1.0 1.5
2.8 4.7 5.9 7.2 8.9 10.8 13.1
12.2 7.4 6.3 4.6 3.8 2.5
2.5 2.3 2.3 2.3 2.3 2.2 2.2
3.6 2.2 1.7 1.3 0.94 0.65 0.55
8.9 5.5 4.3 3.2 2.3 1.6 1.4
1.4 1.8 2.0 2.3 2.7 3.2 3.5
12.3 9.6 8.5 7.3 6.3 5.3 4.8
7.2 4.4 3.4 2.5 1.9 1.3 1.1
ι-type
0.0 0.20 0.40 0.64 0.99
3.0 5.5 7.2 8.9 10.9
20.6 10.1 7.4 5.4 4.4
3.8 2.9 2.8 2.7 2.6
3.8 2.5 1.7 1.3 1.0
9.5 6.2 4.4 3.3 2.6
1.3 1.6 2.0 2.3 2.5
12.6 10.2 8.5 7.4 6.6
7.6 4.9 3.5 2.6 2.1
λ-type
0.0 0.20 0.40 0.64 0.99
2.9 5.5 7.3 8.9 10.9
20.2 16.2 10.1 8.0 5.8
3.8 3.6 3.4 3.4 3.3
3.4 2.5 1.7 1.3 0.95
8.6 6.3 4.3 3.3 2.4
1.4 1.6 2.0 2.3 2.7
12.0 10.3 8.5 7.4 6.3
6.8 5.2 3.5 2.6 1.9
a
Polymer concentrations of the solutions were kept at the constant values of 0.15, 0.12, and 0.12 for κ, ι, and λ, respectively.
portion of an electric birefringence signal after the removal of the rectangular electric pulse is related to the rotational relaxation time of a solute molecule. Generally, relaxation time is a sensitive quantity to investigate the molecular shape because the relaxation time is roughly proportional to the cubic power of the length of the solute molecule. In this study, a rotational relaxation time was calculated from the decay signal of the electric birefringence by the area method.37 By using this method, the electric birefringence average relaxation time, 〈τ〉EB, can be obtained from the area surrounded by the base line and the relaxation curve of the electric birefringence signal after the electric pulse terminated. Figure 6 shows the plots of 〈τ〉EB
obtained for three types of carrageenans at four concentrations against the square of the applied electric field strength just before the electric pulse terminated. The values of 〈τ〉EB increased as the concentration decreased. If a solute molecule is a thoroughly rigid molecule, 〈τ〉EB does not show any concentration dependence or slightly increases as the concentration increment because of the solute-solute interaction.24,25 The reverse tendency shown in here may indicate that the carrageenan molecules are not rigid but flexible. The bulk conformation of their molecular chain may expanded toward a more extended conformation as the concentration decreased. Values of 〈τ〉EB also depend on the electric field strength. The relaxation times
3658 J. Phys. Chem. B, Vol. 101, No. 19, 1997
Ueda et al. radius of gyration,41 〈S2〉1/2, can be calculated by the following equations.
〈R2〉 ) 2q[Lc - q(1 - exp(-Lc/q))] 〈S2〉 )
Figure 5. Ionic strength dependence of the ionic polarizability ∆R of carrageenans in KCl solution. The samples are κ-type (0), ι-type (O), and λ-type (4), respectively.
gradually decreased and approached the steady-state values as the electric field strength increased. This dependence is attributable to the polydispersity and also to the flexibility of the samples. The rigorous treatment of the effect of polydispersity on the analysis of the decay curve has been given for rigid helical polypeptide in a previous paper.24 However, the carrageenan samples used in this study must have mixed effects of polydispersity and flexibility of the polyion chain. Therefore, the previous method cannot be applied in this study. Instead, the extrapolated value of the relaxation time, 〈τ〉EB,Ef∞, was obtained and used in the following analysis. These values were obtained by extrapolating the electric birefringence average relaxation time to the infinite electric field strength by plotting the values of 〈τ〉EB against E-2. The values of 〈τ〉EB,Ef∞ are known to give the weight-average value of the electric birefringence relaxation time 〈τ〉w.38 The obtained values of 〈τ〉w are listed in Table 4. The length of the molecule was estimated by using a wormlike chain model derived by Hearst, which treated polymers as connected spheres of diameter a with distance D between the spheres from the following equation.39
〈τ〉w )
[ ()
Lc η0qLc2 0.126 12kT q
1/2
()
+ 0.159 ln
2q D
0.387 + 0.160
[ ( ) ( ) ( )(
qLc q q 2 q 1-3 +6 -6 3 Lc Lc Lc
3
(8)
( ))]
1 - exp -
Lc q
(9)
The calculated values are also listed in Table 4. Those values are plotted against concentration in Figure 7. The effects of the addition of KCl salt on the relaxation time were also investigated. The obtained values of 〈τ〉w and the calculated values of q, 〈R2〉1/2, and 〈S2〉1/2 are all listed in Table 5 and shown in Figure 8. It can be seen that κ-carrageenan did not show any dependence on the ionic strength. For ι- and λ-carrageenans, the calculated values of q, 〈R2〉1/2, and 〈S2〉1/2 depend on the ionic strengths. However, those values converged to the same value for κ-carrageenan at ionic strength higher than 0.5 mM. Odijk42 and Skolnick and Fixman43 defined the persistence length of a polyion as a sum of the intrinsic rigidity of the molecule at a neutral state, qn, and the electrostatic portion, qe, that is, q ) qn + qe. From the analogy of this definition, it can be considered that the persistence length of κ-carrageenan gives the intrinsic rigidity, qn, because it did not depend either on the polymer concentrations or on the ionic strengths. Judging from the converged value of q at the higher ionic strength, carrageenans may have a common intrinsic rigidity irrespective of their types. The intrinsic rigidity, qn, may be interpreted by Manning’s.44 He predicts that the local structure of the polyelectrolyte chain in the range ξ > 1 is fully extended. On the other hand, the chain has locally folded structure in the range of ξ < 1. This folded structure may be corresponding to the structure with intrinsic rigidity, qn, and it resists the electrostatic expansion of the chain.44 Discussion
( )] D a
-1
(7)
where η0 is the viscosity of the solvent, q is the persistence length, which is a measure of the chain rigidity, and Lc is the contour length. The parameters a and D were put as equal. The calculated values of q are shown in Table 4. The concentration dependence of the persistence length obtained for three types of carrageenans is plotted in Figure 7. The values of q for κ-type carrageenan did not show so much dependence on the concentration. On the other hand, ι- and λ-carrageenans showed rather large dependence on concentration. The interesting point is that these two carrageenans showed similar magnitudes and dependency on the concentration irrespective their difference of the charge density of the polyion chains. In the case of κ-carrageenan, the charge distance is 11.4 Å, which means the electric repulsion between the charges is small. This may make the length of q small and reduce the concentration dependence. The similarity of ι- and λ-carrageenan may be explained by the ion condensation theory; that is, the values of the charge density parameter ξ for both polymers were reduced to ξ ) 1 by ion condensation, and as a result, the apparent charge of both samples becomes the same. This would result in the same magnitude of charge repulsion in both carrageenans. By using the values of q and contour length Lc, the values of the mean square end to end length,40 〈R2〉1/2, and the mean square
Comparison with Manning’s Condensation Theory. Manning’s limiting theory can theoretically be applied to calculate the fraction of the free counterion γ and condensed (tightly bound) counterion φ at zero concentration from the charge density parameter as follows.11
ln γ ) -ξ/2
(ξ < 1)
ln γ ) -1/2 - ln ξ and φ ) 1 - ξ-1
(10) (ξ > 1)
(11)
By using the calculated values of γ and φ, the number of free, loosely bound, and tightly bound counterions can be calculated and the values are shown in Table 6. κ-Carrageenan has no tightly bound counterions because the charge density parameter ξ is less than unity. If we suppose that only tightly bound counterions can contribute to the polarization, it is expected that κ-carrageenan cannot have a polarization from the point of this theory. Furthermore, if we suppose that the ionic polarization is simply proportional to the number of tightly bound counterions,1 the polarizability of λ-type carrageenan should be 2.5 times larger than that of ι-type. However, the experimental ∆R0 for κ-type is not zero and is much the same for all types of carrageenans irrespective of their number of tightly bound counterions. On the other hand, if only the loosely bound counterions can contribute to the polarizability, the experimental results of ι- and λ-types can be qualitatively interpreted. They have the same number of loosely bound counterions, and they show nearly the same ionic polarization. However, the relatively large ∆R0 value of κ-type carrageenan cannot be interpreted
Ionic Polarization of Carrageenans
J. Phys. Chem. B, Vol. 101, No. 19, 1997 3659
Figure 6. Field strength dependence of the electric birefringence relaxation time 〈τ〉EB of κ-, ι-, and λ-type carrageenans in aqueous solution without added salts. Polymer concentrations of the solution are shown in the figure.
TABLE 4: Weight-Average Electric Birefringence Relaxation Time, 〈τ〉w, Obtained from Field Free Decay Curves of Electric Birefringence Signals and Calculated Hydrodynamic Properties of Carrageenans in Aqueous Solution without Added Salts Cp (mM)
〈τ〉w (µs)
q (nm)
〈R2〉1/2 (nm)
〈S2〉1/2 (nm)
κ-type
0.59 0.31 0.15 0.07
1.5 1.6 1.8 1.9
3.2 3.4 3.7 3.9
32 32 34 35
13 13 14 14
ι-type
0.46 0.23 0.12 0.06
3.2 7.3 7.5 8.6
5.1 9.9 10.2 11.6
42 58 59 63
17 23 23 24
λ-type
0.46 0.23 0.12 0.06
3.5 6.4 8.0 8.2
5.4 8.8 10.6 10.8
44 56 61 61
17 22 23 24
satisfactorily, since the calculated number of loosely bound counterions for κ-type is about one-third those for ι- and λ-types. As can be seen from the above discussion, the magnitude of the ionic polarization can be fairly well interpreted by Manning’s counterion condensation theory by considering that the loosely bound counterions are the main factor for the ionic polarization. However, there is still a discrepancy between the experimentally obtained ∆R0 and the calculated number of loosely bound counterions. In order to further discuss the origin of the ionic polarizability, the effects of the ionic strengths and the molecular chain conformation on the polarization are discussed in the following sections. Ionic Strength Dependence of the Ionic Polarization. Rau and Charney derived a theory for an induced polarization of a Debye-Hu¨ckel type ion atmosphere cloud surrounding a charged polyion with the length of L.7,16 Their theory showed that the dependence of ∆R on the Debye screening parameter κ is given by the following equations.
∆R ) (constant)L1.85/κ1.15 ∆R ) (constant)L/κ2
(κL < 12) (κL > 12)
(12) (13)
where κ is calculated from the following equation.
κ2 )
(
)
8πNAe2d I 1000kT
(14)
Figure 7. Polymer concentration dependence of persistence length q, end-to-end distance 〈R2〉1/2, and radius of gyration 〈S2〉1/2 in aqueous solution, which were calculated by using the weight-average electric birefringence relaxation time 〈τ〉w. The samples are κ-type (0), ι-type (O), and λ-type (4), respectively.
In the above equation, NA is Avogadro’s number, e is the elementary electric charge, d is the density of the solvent, is the dielectric constant of the solvent, k is the Boltzmann constant, and T is the absolute temperature. I is the ionic strength which was defined in eq 6. The calculated values of κ are shown in Table 3. Equations 12 and 13 indicate that the dependence of log ∆R on log κ should have a different slope before and after the critical value of κL ) 12, if the origin of the ionic polarizability is caused by Debye-Hu¨ckel type loosely bound counterions. Figure 9 shows the double logarithmic plots
3660 J. Phys. Chem. B, Vol. 101, No. 19, 1997
Ueda et al.
TABLE 5: Weight-Average Electric Birefringence Relaxation Time, 〈τ〉w, Obtained from Field Free Decay Curves of Electric Birefringence Signals and Calculated Hydrodynamic Properties of Carrageenans in Added KCl Salt Solutionsa Cs (mM)
〈τ〉w (µs)
q (nm)
〈R2〉1/2 (nm)
〈S2〉1/2 (nm)
κ-type
0.0 0.13 0.25 0.40 0.65 1.0
1.8 1.6 1.5 1.5 1.4 1.3
3.7 3.4 3.3 3.2 3.0 3.0
34 33 32 32 31 31
14 13 13 13 12 12
ι-type
0.0 0.20 0.40 0.65 1.0
7.5 3.2 1.7 1.3 1.0
10.2 5.0 3.1 2.5 2.1
59 42 33 30 27
23 17 13 12 11
λ-type
0.0 0.20 0.40 0.65 1.0
8.0 4.3 2.6 1.6 1.6
10.6 6.4 4.2 2.9 2.9
61 48 39 32 32
24 19 15 13 13
Figure 9. Double logarithmic plots of the ionic polarizability ∆R and the Debye screening parameter κ for various samples of carrageenans in KCl solution. The samples are κ-type (0), ι-type (O), and λ-type (4), respectively. Dashed lines indicate the point at κL ) 12 for κ-, ι-, and λ-carrageenans as indicated in the figure.
TABLE 6: Ionic Polarizability at Infinite Dilution, Number of Charged Sites of Carrageenans, and Number of Free, Loosely, and Tightly Bound Counterions Calculated from Manning’s Theory
a
Polymer concentrations of the solutions were kept at the constant values of 0.15, 0.12, and 0.12 for κ, ι, and λ, respectively.
∆R0/ 10-32 (F‚m2)
charged sites
free
loosely bound
tightly bound
8.8 10.2 8.9
140 319 430
102 153 155
38 100 101
0 66 174
κ-type ι-type λ-type
TABLE 7: Slopes in Double Logarithmic Plots of ∆r and Debye Screening Parameter, K, for Different types of Carrageenans κL < 12 κL > 12
Figure 8. Ionic strength dependence of persistence length q, end-toend distance 〈R2〉1/2, and radius of gyration 〈S2〉1/2 of carrageenans in KCl solution, which were calculated by using the weight-average electric birefringence relaxation time 〈τ〉w. The samples are κ-type (0), ι-type (O), and λ-type (4), respectively. Polymer concentrations of the solutions were kept at the constant values of 0.15, 0.12, and 0.12 for κ, ι, and λ, respectively.
of the ionic polarizability ∆R and the Debye screening parameter κ for all types of carrageenans. The critical values of κ at κL ) 12 are shown for all three samples in Figure 9 as dashed lines. The values of log ∆R can be fitted by linear lines with different slopes in the region of log κ above and below the value
κ-type
ι-type
λ-type
-0.93 -1.62
-0.71 -1.25
-0.58 -1.51
of κL ) 12. The slopes were obtained by using the least-squares method for all carrageenans, and their values are listed in Table 7. Although the slopes are a little smaller than the theoretically predicted values of -1.15 and -2.0 for both regions of κL < 12 and κL > 12, respectively, the tendency of the slopes in the two regions was in good agreement with the theory. Recently, Charney reported experimental work to check his theory by using the electric dichroism method with hyaluronic acid.18 His data were in good accordance with the predictions showing a dependence of the polarizability on the -0.9 power of the ionic strength in the region of κL < 12. The charge density parameter ξ of his polyion is lower than unity. This is the same situation as our case of κ-carrageenan. Our result for κ-carrageenan shown in Table 7 gives a value quite similar to the one obtained by Charney. Moreover, ι- and λ-carrageenans, which have condensed counterions within the framework of Manning’s theory, showed ionic strength dependence similar to the case of κ-carrageenan. It can be said that the ionic polarization obeys the behavior of the Debye-Hu¨ckel type counterion atmosphere polarization, irrespective of whether the counterions are tightly bound. Comparison with Polarization Theories. There have been proposed various theories attempting to interpret the ionic polarizability of polyions. One of the most accepted concepts for ionic polarization is that it arises from the thermal perturbation of the bound counterions from the equilibrium position around the polyion. Mandel first developed an equation based on the perturbation of the counterions.1 Subsequently, Oosawa introduced an electrostatic repulsion between the bound counterions to Mandel’s equation and derived the following equation:2
∆R )
(
e2L3 φ 12bkT 1 - 2ξφ ln(r/R)
)
(15)
Ionic Polarization of Carrageenans
J. Phys. Chem. B, Vol. 101, No. 19, 1997 3661
TABLE 8: Polymer Concentration Dependence of Polarizability in Aqueous Solution without any Added Salts and Calculated Polarizability by Using Manning’s Equation for Condensed Counterion Polarization Obtained for Rod and Ellipsoid Models ∆R × 1032 (F‚m2) Cp (mM)
κ × 103 (Å)-1
experimental
rod model
ellipsoid sphere model
ι-type
0.46 0.23 0.12 0.06
5.9 4.2 3.0 2.1
2.0 4.9 7.6 8.9
40.5 38.1 35.9 33.9
0.5 1.3 1.2 1.4
λ-type
0.46 0.23 0.12 0.06
6.0 4.2 2.9 2.1
2.1 4.6 6.8 7.7
51.2 47.4 43.8 41.3
0.7 1.3 1.6 1.5
TABLE 9: KCl Salt Concentration Dependence of Polarizability in Aqueous Solution and Calculated Polarizability by Using Manning’s Equation for Condensed Counterion Polarization Obtained for Rod and Ellipsoid Models ∆R × 1032 (F‚m2) Cs (mM)
κ× (Å)-1
experimental
rod model
ellipsoid sphere model
ι-type
0 0.20 0.40 0.65 1.0
3.0 5.5 7.2 8.9 10.9
7.6 4.9 3.5 2.6 2.1
35.9 39.9 42.1 43.9 45.7
1.2 0.5 0.3 0.2 0.2
λ-type
0 0.20 0.40 0.65 1.0
2.9 5.5 7.3 8.9 10.9
6.8 5.2 3.5 2.6 1.9
43.0 49.3 52.6 55.2 58.1
1.5 0.8 0.5 0.3 0.3
103
where e is the elementary electric charge, L is the length of the polyion, b is the distance between charge sites on the polyion, R is the cylindrical radius of the volume of solution per single rodlike polyion, and r is the radius of the cylindrical volume where the fraction of the polarizable bound counterions φ is located. ξ is the charge density parameter. This equation gave a fairly good explanation for the experimental results of the polyion polarization.13 Within the same theoretical framework, Manning proposed an equation of the ionic polarization which arises from the condensed counterions.4 In order to discuss within the same framework of Manning’s ion condensation theory, we estimated the ionic polarizability by using Manning’s equation as shown below.
∆R )
(
)
e2L3 φ 12bkT 1 - 2ξφ ln(κb)
(16)
where κ can be calculated by using eq 14. Other notations are the same as used in eq 15. If we compare eqs 15 and 16, it can be seen that the only difference is the logarithmic part of these equations. The term of ln(r/R) in Oosawa’s equation means that the polarizable bound counterions are contained within the cylindrical layer of radius r, whereas R is the cylindrical radius of the volume of solution per single polyion where all counterions are included in this volume. With the analogy of Oosawa’s equation, ln(b/κ-1) in Manning’s (eq 16) could be regarded that the polarizable bound counterions are contained within the cylindrical layer, whose radius is the same order of the charge distance b. The length L should be the contour length Lc if a polyion is a rigid rodlike molecule. However, as was discussed in the analysis of the electric birefringence decay signal, carrageenan
Figure 10. Comparison between the polymer concentration dependence of the experimentally obtained ionic polarizability ∆R and the theoretically calculated ones, which takes into account Manning’s condensed counterion polarization. The symbols are experimental (9), rod model (O), and ellipsoid model (]), respectively.
Figure 11. Comparison between the ionic strength dependence of the experimentally obtained ionic polarizability ∆R and the theoretically calculated ones, which takes into account Manning’s condensed counterion polarization. The symbols are experimental (9), rod model (O), and ellipsoid model (]), respectively.
molecules are not rigid but flexible, and therefore, two models were used for L in the calculation of eq 16. The first is a rod model in which contour length Lc is used as L. In this model, the radius of the rod is assumed to be 15 Å by taking into account the hydration of the carrageenan molecule. The other model is an ellipsoid model whose longitudinal axis is coincided to the mean square end-to-end length 〈R2〉1/2 obtained from the decay signal of the electric birefringence. Axial ratio of the ellipsoid was determined to have the same volume with the rod model. The values of other parameters in eq 16 were taken as the same in both models. That is, the charge distances b used are listed in Table 1 and the fractions of the condensed counterion φ were calculated from eq 11 and the values of ξ in Table 1. The calculated values of the condensed counterion polarization for ι- and λ-carrageenans obtained for two hydro-
3662 J. Phys. Chem. B, Vol. 101, No. 19, 1997
Ueda et al.
TABLE 10: Polymer Concentration Dependence of Ionic Polarizability in Aqueous Solution without any Added Salts and Calculated Polarizability by Using Manning’s Equation for Loosely-Bound Counterion Polarization Obtained for Rod and Ellipsoid Models rod model
ellipsoid model
∆R × 10 (F‚m ) p ) 0.85 p ) 0.9 p ) 0.95 32
2
∆R × 1032 (F‚m2) p ) 0.85 p ) 0.9 p ) 0.95
Cp (mM)
κ × 103 (Å)-1
experimental ∆R × 1032 (F‚m2)
κ-type
0.59 0.31 0.15 0.07
5.7 4.1 2.8 2.0
2.5 4.7 7.2 7.7
0.57
4.0 7.6 16.0 31.9
2.6 4.9 10.2 20.5
1.2 2.4 4.9 9.9
1.8 1.8 1.7 1.7
2.5 4.7 10.2 20.7
1.6 3.0 6.5 13.3
0.8 1.5 3.1 6.4
ι-type
0.46 0.23 0.12 0.06
5.9 4.2 3.0 2.1
2.0 4.9 7.6 8.9
0.77
5.7 11.3 22.6 44.8
3.6 7.2 14.5 28.7
1.8 3.5 7.0 13.9
2.2 1.9 1.9 1.8
3.8 8.8 17.7 36.0
2.5 5.6 11.3 23.1
1.2 2.7 5.5 11.1
λ-type
0.46 0.23 0.12 0.06
6.0 4.2 2.9 2.1
2.1 4.6 6.8 7.7
0.78
5.7 11.4 24.8 45.2
3.7 7.3 15.9 28.9
1.8 3.5 7.7 14.0
2.2 2.0 1.9 1.9
3.9 8.7 19.6 35.8
2.5 5.6 12.5 22.9
1.2 2.7 6.1 11.1
up/us
up/us
TABLE 11: KCl Salt Concentration Dependence of Ionic Polarizability in Aqueous Solution and Calculated Polarizability by Using Manning’s Equation for Loosely-Bound Counterion Polarization Obtained for Rod and Ellipsoid Models rod model
ellipsoid model
∆R × 10 (F‚m ) p ) 0.85 p ) 0.9 p ) 0.95 32
2
∆R × 1032 (F‚m2) p ) 0.85 p ) 0.9 p ) 0.95
Cs (mM)
κ × 103 (Å)-1
experimental ∆R × 1032 (F‚m2)
κ-type
0 0.13 0.25 0.40 0.65 1.0
2.8 4.7 5.9 7.2 8.9 10.8
7.2 4.4 3.4 2.5 1.9 1.3
0.57
16.0 5.7 3.6 2.5 1.6 1.1
10.2 3.7 2.3 1.6 1.0 0.7
4.9 1.8 1.1 0.8 0.5 0.3
1.8 1.7 1.8 1.8 1.8 1.8
10.2 3.6 2.3 1.5 1.0 0.7
6.5 2.3 1.4 1.0 0.6 0.4
3.1 1.1 0.7 0.5 0.3 0.2
ι-type
0.0 0.20 0.40 0.65 1.0
3.0 5.5 7.2 8.9 10.9
7.6 4.9 3.5 2.6 2.1
0.77
22.6 6.6 3.9 2.5 1.7
14.5 4.2 2.4 1.6 1.1
7.0 2.1 1.2 0.8 0.5
1.9 2.3 2.5 2.7 2.8
17.7 4.5 2.3 1.4 0.9
11.3 2.9 1.5 0.9 0.6
5.5 1.4 0.7 0.4 0.3
λ-type
0.0 0.20 0.40 0.65 1.0
2.9 5.5 7.3 8.9 10.9
6.8 5.2 3.5 2.6 1.9
0.78
24.8 6.6 3.8 2.6 1.7
15.9 4.2 2.4 1.6 1.1
7.7 2.0 1.2 0.8 0.5
1.9 2.1 2.4 2.6 2.6
19.6 4.7 2.5 1.5 1.0
12.5 3.0 1.6 1.0 0.7
6.1 1.5 0.8 0.5 0.3
up/us
dynamic models are shown in Tables 8 and 9 for no salt and added salt systems, respectively. Debye screening parameters κ, which were used in this calculation, were obtained by using eq 14 and are listed in the same tables. The polymer concentration dependences of these calculated values are shown in Figure 10. The calculated values for the assumed rod model are 5-10 times larger than the experimental ones. Moreover, the dependence of the calculated values on the concentration is opposite the experimental results as was already pointed out by Manning4 and Charney et al.5 On the other hand, the calculated values for the ellipsoid model are about one-third the experimental values. However, in this model, the calculated polarizability decreased with the increment of the polymer concentration in accordance with the dependence of the experimental values on concentration. By contrast, if we consider the polyion dimension as an adjustable parameter, we can estimate the polyion dimension so as to fit the experimental values of ∆R by using eq 16. In this case, the polyion dimension was estimated to be roughly 3 times larger than 〈R2〉1/2, which was obtained by the electric birefringence decay signals. Salt concentration dependence of the ionic polarization is plotted in Figure 11. The results show behavior similar to that of the concentration dependence discussed above. In the rod model, the calculated ionic polarizability increased with the increment of the salt concentration. On the other hand, the
up/us
calculated ionic polarizability decreased with the increment of the salt concentration in the ellipsoid model. Manning also proposed a theoretical equation where the Debye-Hu¨ckel type ionic atmosphere contributes to the ionic polarization. This was expressed as follows.6,9
∆R )
(
e2L3 3(1 - p) up 1 bkT 2p + 1 us ξκ2L2
)
(17)
where up is the electrophoretic mobility of the polyion and us is an average electrophoretic mobility of the salt ions defined by us-1 ) (1/2)(u1-1 + u2-1), where u1 is the electrophoretic mobility of the counterion and u2 is the one for co-ion, respectively. The parameter p is a measure of the anisotropy of the mobility of the macroion. The value unity for p means that the Debye polarization is isotropic and cannot contribute to the polarization. Other notations are the same as before. For a rodlike polyion, the mobility up,para along the axis of the rod is larger than the mobility along the perpendicular direction of the axis up,per and is defined as up,para ) pup,per (p < 1). In order to calculate the above equation, the ratio of up/us and the value of p are needed. However, it is difficult to estimate the anisotropy measure p. The accurate value of p is not available, but is expected to be close to the isotropic value unity.6 Therefore, three values of 0.85, 0.9, and 0.95 were used here as p to estimate the variation of the resulting polarizability. The
Ionic Polarization of Carrageenans
Figure 12. Comparison between the polymer concentration dependence of the experimentally obtained ionic polarizability ∆R and the theoretically calculated ones, which takes into account Manning’s loosely bound counterion polarization. The symbols are experimental (9), rod model (O), and ellipsoid model (]), respectively.
estimation of the ratio of up/us is rather complicated. We first estimated the electrophoretic mobility of a polyion by using the equation up ) ze/f, where f is a frictional coefficient of a polyion and z is the total effective charge of a polyion which corresponds to the number of free counterions of a polyion because the bound counterions cancel the polyion charge and reduce the apparent charge of a polyion. The frictional coefficient f for the rod model was calculated by using the Kirkwood and Riseman equation,45 modifying the rod as a chain with an equivalent length of connected spheres of radius 15 Å. For the ellipsoid model, Perrin’s equation was used to estimate the frictional coefficient.46 The mobilities of the counterion and coion were calculated by using the equation ui ) |zi|e/6πηrs,i, where η is the viscosity of the solvent and rs,i is the Stokes radius for the ith ion species. The calculated values are 8.2 × 10-8 kg-1 s2 A for K+ and 15.2 × 10-8 kg-1 s2 A for Cl-, respectively. The average electrophoretic mobility of salt ions, us, is 10.6 × 10-8 kg-1 s2 A. The obtained values of up/us for the rod model are 0.57, 0.77, and 0.78 for κ-, ι-, and λ-carrageenans, respectively. For the ellipsoid model, the values of up/us are listed in Tables 10 and 11. Manning estimated up/us to be about 0.83, which is close to the calculated values for the case of our rod model. However, if we use the ellipsoid model, the values of up/us become large. The resulted polarizability calculated for some values of p are listed in Tables 10 and 11. We compared the calculated values to the experimentally obtained polarizability, and the calculated values for p ) 0.9 and 0.95 are in fairly good agreement with the experimental values. Figure 12 shows the results of the concentration dependence of the calculated polarizability for p ) 0.9. It can be seen that the experimental values at higher concentration were well reproduced by the
J. Phys. Chem. B, Vol. 101, No. 19, 1997 3663
Figure 13. Comparison between the ionic strength dependence of the experimentally obtained ionic polarizability ∆R and the theoretically calculated ones, which takes into account Manning’s loosely bound counterion polarization. The symbols are experimental (9), rod model (O), and ellipsoid model (]), respectively.
parameter of p ) 0.9. The difference of the used models between rod and ellipsoid was small. On the other hand, the ionic strength dependence can also be well reproduced by the parameter of p ) 0.9 as shown in Figure 13. In the above calculations, both models can be fitted by using the same value of p ) 0.9. The physical adequacy whether anisotropy parameter p takes similar values in both models cannot be elucidated in this work, and further experimental work will be needed. Although there are still many ambiguities in the choice of parameters, we can roughly say that Manning’s loosely bound counterion theory can be used to explain both the magnitude and the dependence on the concentration and ionic strength of the experimental values of ∆R. Conclusion The ionic polarizability has been obtained for carrageenans with different numbers of sulfuric acid on the polyion chain. They have been found to have similar values of ionic polarizability irrespective of their charge density parameters of the polyion chains. At a first glance of this result, we can conclude that the concept in which only the condensed counterions proposed by Manning contribute to the ionic polarization is not sufficient to account for the experimental results from the following two points. The first point is that κ-carrageenan experimentally has a large ionic polarizability and has no condensed counterions as long as Manning’s condensation theory is valid for this system. The second point is that the magnitude of the polarization does not depend on the number of the condensed counterions as was seen in the case of the comparison of ι- and λ-types of carrageenans. Although the number of condensed counterions of λ-carrageenan is 2.5 times
3664 J. Phys. Chem. B, Vol. 101, No. 19, 1997 larger than that of ι-type, the values of their experimentally obtained ionic polarizabilities were rather equal. When we compared the polymer concentration and added salt concentration dependence of the polarizability with two groups of theories, which were based on loosely bound counterion and condensed counterion polarization, the Debye-Hu¨ckel type loosely bound counterions were found to reproduce the experimental values as shown in Figures 9, 12, and 13. Manning also discussed the possibility of the onset of mechanical instability of local folded structures at the counterion condensation point. According to this theory, the polyelectrolyte chain with ξ < 1 takes a folded structure and, therefore, its actual value of ξ approaches unity. Even if this occurred in the κ-carrageenan, ξ cannot exceed unity as was discussed by Manning. This also supports that κ-carrageenan does not have condensed counterions. As for Manning’s theory, there are still some discussions in the theoretical foundation, especially about the distribution of the bound counterions around the polyion. Le Bret and Zimm discussed the location of the condensed counterions around the polyion by numerically analyzing the Poisson-Boltzmann equation and claimed that they are not condensed in the small volume at the narrow region around the surface of a polyion but actually form a diffuse cloud.47,48 This means that even “condensed” counterions defined by Manning’s condensation theory may possibly behave like loosely bound counterions. This also means these counterions can be treated by the theory of the Debye-Hu¨ckel type ionic polarization. It is not clear that there still remain some counterions that locate very close to the polyion, and they should be treated by the polarization theory of condensed counterions. Most of the theories developed for the condensed counterions were of little or rather of opposite dependence on the ionic strength. Although our experiment still cannot clarify the origin of the polarizability, it seems difficult to assign the counterions attributable to the polarizability to one of the two types of counterions, that is, condensed counterions and loosely bound counterions. Yoshida et al. performed a Monte Carlo simulation of the ionic polarization of a polyion.49 They showed that not only the condensed counterions but also the free counterions can produce a large ionic polarizability. Further investigation of the nature of the loosely bound counterions seems to be needed. Experimentally, we should pay more attention on the charge density parameter of polyions and more experiments should be needed to perform on the polyions with different charge densities. Theoretically, the method of computer simulation will be more important to investigate the polarization behavior of the counterion around the polyion. Acknowledgment. The authors thank Prof. Kiwamu Yamaoka and the colleagues of his laboratory for kind permission to use the electric birefringence apparatus and help in carrying out this work. The authors also thank Prof. Kazuo Kikuchi, University of Tokyo, for precious advice on this paper. The authors also thank Daicel Chemical Co., Ltd., for financial support.
Ueda et al. References and Notes (1) Mandel, M. Mol. Phys. 1961, 4, 489. (2) Oosawa, F. Biopolymers 1970, 9, 677. (3) Schwarz, G. Z. Phys. Chem. 1959, 19, 286. (4) Manning, G. S. Biophys. Chem. 1978, 9, 65. (5) Charney, E.; Yamaoka, K.; Manning, G. S. Biophys. Chem. 1980, 11, 167. (6) Manning, J. S. J. Chem. Phys. 1993, 99, 477. (7) Rau, D. C.; Charney, E. Biophys. Chem. 1981, 14, 1. (8) Hornick, C.; Weill, G. Biopolymers 1971, 10, 2345. (9) Manning, G. S. J. Chem. Phys. 1989, 90, 5704. (10) Plum, G. E.; Bloomfield, V. A. Biopolymers 1990, 29, 1137. (11) Manning, G. S. J. Chem. Phys. 1969, 51, 924. (12) Tricot, M.; Houssier, C.; Desreux, V. Eur. Polym. J. 1976, 12, 575. (13) Tricot, M.; Houssier, C.; Desreux, V. Biophys. Chem. 1978, 8, 221. (14) Kikuchi, K.; Yoshioka, K. J. Phys. Chem. 1973, 77, 2101. (15) Yamaoka, K.; Ueda, K. J. Phys. Chem. 1980, 84, 1422. (16) Rau, D. C.; Charney, E. Biophys. Chem. 1983, 17, 35. (17) Matsuda, K.; Yamaoka, K. Bull. Chem. Soc. Jpn. 1982, 55, 1727. (18) Charney, E. In Dynamic BehaVior of Macromolecules, Liquid Crystals and Biological Systems by Optical and Electro-Optical Methods; Watanabe, H. Ed.; Hirokawa Publishing Co.: Tokyo, 1989; p 163. (19) Ueda, K.; Ochiai, H.; Itaya, T.; Yamaoka, K. Polymer 1992, 33, 429. (20) Ueda, K.; Sato, S.; Ochiai, H.; Imamura, A.; Yamaoka, K. Chem. Lett. 1994, 763. (21) Fukudome, K.; Yamaoka, K.; Nishikori, K.; Takahashi, T.; Yamamoto, O. Polym. J. 1986, 18, 71. (22) Gekko, K.; Mugishima, H.; Koga, S. Int. J. Biol. Macromol. 1985, 7, 57. (23) Rochas, C.; Rinaudo, M. Biopolymers 1980, 19, 1675. (24) Yamaoka, K.; Ueda, K. J. Phys. Chem. 1982, 86, 406. (25) Ueda, K. Bull. Chem. Soc. Jpn. 1984, 57, 2703. (26) Yamaoka, K.; Ueda, K. Chem. Lett. 1983, 545. (27) O’Konski, C. T.; Yoshioka, K.; Orttung, W. H. J. Phys. Chem. 1959, 63, 1558. (28) Yamaoka, K.; Fukudome, K. J. Phys. Chem. 1988, 92, 4994. (29) Yamaoka, K.; Fukudome, K. J. Phys. Chem. 1990, 94, 6896. (30) Yoshioka, K. J. Chem. Phys. 1983, 79, 3482. (31) Yoshioka, K.; O’Konski, C. T. J. Polym. Sci., Part A-2 1968, 6, 421. (32) Ueda, K.; Mimura, M.; Yamaoka, K. Biopolymers 1984, 23, 1667. (33) Tinoco, Jr., I.; Yamaoka, K. J. Phys. Chem. 1959, 63, 423. (34) Sasai, R.; Yamaoka, K. J. Phys. Chem. 1995, 99, 17754. (35) Paoletti, S.; Smidsrod, O.; Grasdalen, H. Biopolymers 1984, 23, 1771. (36) Tricot, M.; Houssier C. In Polyelectrolytes; Frisch, K. C., Klempner, D.; Patsis, A. V., Eds.; Technomic Publishing Co.: Westport, CT, 1976; p 43. (37) Yoshioka, K.; Watanabe, H. Nippon Kagaku Zasshi 1963, 84, 626. (38) Yamaoka, K.; Fukudome, K. Bull. Chem. Soc. Jpn. 1983, 56, 60. (39) Hearst, J. J. Chem. Phys. 1963, 38, 1062. (40) Kratky, O.; Porod, G. Recl. TraV. Chim. 1949, 68, 1106. (41) Benoit, H.; Doty, P. J. Phys. Chem. 1953, 57, 958. (42) Odijk, T. J. Polym. Sci., Polym. Phys. Ed. 1977, 15, 477. (43) Skolnick, J.; Fixman, M. Macromolecules 1977, 10, 944. (44) Manning, G. S. J. Chem. Phys. 1988, 89, 3772. (45) Kirkwood, J. G.; Riseman, J. In Rheology; Eirich, F., Ed.; Academic Press: New York, 1956; Vol. 1. (46) Perrin, J. J. Phys. 1934, 5, 498. (47) Le Bret, M.; Zimm, B. H. Biopolymers 1984, 23, 287. (48) Le Bret, M. L.; Zimm, B. H. Biopolymers 1984, 23, 271. (49) Yoshida, M.; Kikuchi, K.; Maekawa, T.; Watanabe, H. J. Phys. Chem. 1992, 96, 2365.
