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Activity coefficients of small ions in aqueous mixtures of polyelectrolytes and simple electrolytes
v&nne 41, number 3
ACTIWTY
1 August 1976
-CHEMICALPHYSICS LETTERS
COEFFICIENTS OF SMALL IONS IN AQUEOUS MKIXJRES
OF POLYELECTROLYTES
AND SIMPLE ELEC-I’FtOLYI-ES
M. RINAUDO and M. MILAS Centre de Reciwrches sur les ,Mccromolt?cz~lesVigirales (C.N.R.S.J. 38041 Grenoble Cede& Lance Received 23 February 1976 Revised manuscript received 12 Aprii 1976 The activity coefficients of coions Cl-, of counterions NaTand Ca*+ have been determined on Na-carboxymethylcelIuloseN&i and Ca-carboxymethylcellulose-CzClz systems for two charge densities of the polyelectrolytes (&,=.rched = 1.38 and 3.45); the mean activity coefficients of the external salt are then deduced. Their variations with the equivalent ratio X are investigated and the experimental values are compared to the theoretical ones @en by the Manning treatment.
1. Introduction
Many experimental and theoretical studies are available now which concern the activity coefficients of counterions and coions in aqueous mixtures of polyelectrolyte and simple electrolyte. Mock and Marshall [l ] show that the activity of coions is not modified by the polyelectrolyte in the system vinyltoluenestyrene copolymer sulfonic acid-HCI; the conclusions of Nagasawa et al. for sodium polyvinyl alcohol sulfateNaCl mixtures are identical [2] . For carrageenan [3], polystyrene sulfonic acid [4-61, and carboxymethylcellulose [7], it is found that ‘Jle mean activity coeffcients of the electrolyte or those of the coions are reduced by the polyelectrolyte. Noguchi and co-workers find a decrease of the activity of counterions in ionic dextran derivatives when simple electrolyte is added; it corresponds to a decrease of specific conductivity of the polyelectrolyte [8]. From a theoretical point of view, Manning’s limiting law is the only one to predict a decrease of coion activities [5,9]. The work of Wells allows the extension of the theory for fmite dilutions [ 70 j . In this work, the activity coefficients of the Cl-, NaT and Ca2+ Ions ’ have been determined on Na-carboxymethylcellulose-NaCl and Ca-carboxymethylcelluloseC&l, systems for two degrees of substitutio&of the car50xymethylcelluloses. Each polyelectrolyte is characterized by its charge 456
parameter X introduced by Lifson and Katchalsky [l 1] ; the variation of the conductance and the activity of ions are experimentally determined by potentiometry as a function of the quantity of simple electrolyte added and correlate to the X value. 2. Experimenta! The carboxymethylcelluloses (CMC) with substitution degrees (m) 1 and 2.5 have been prepared in our laboratory [12] . The simple electrolytes NaCi and CaCi2 are analytic grade products from Prolabo (France). The activities of ions are determined on a Sargent S 30000 potentiometer equipped with specific electrodes Orion: a crystalline membrane electrode (model 94.17) for Clions, a divalent cations selective liquid membrane electrode (model 92.32) for Ca2+ ions and an Ienagl_ass electrode for Na+ ions. The reference is a double junction electrode Orion (model 90.02). The conductivities of aqueous solutions are obtained with a Philips bridge (1000 cycles/s) and a conductivity measuring cell Philips (PW 9510, constant k = 9-7 cm-‘). All experiments
are done at 25 2 O.l°C.
3. Results and discussion In figs. 1 and 2, the r,/$ tion of X in ‘&e mixtures
values are given as a func-
respectively
with monovalent
Volume 41, number 3
0‘4 t t 0
1 August 1976
CHEMICAL PHYSICS LETTERS
1
f
s
IO
X
>
Fig. 1. Mean activity coefficients of simple electrolyte for the system NaCMC + NaCl. Solid line: Manning’s limiting laws. X
and e: experimental results for CMC DS = 1, h = 1.38 and Z% = 2.5, X = 3.45 respectively.
and divalent counterions (X = mp/ms -with mp the equivalent concentration of the polyelectrolyte and m, the equivalent concentration of the simple electrolyte). The ratio -y/r0 is obtained directly by comparison of-the electrochemical potential measured on the system polyY*
oa4.
.
0
Fig. 2. Mean activity coeffbients of simple electrolyte for the system CaCMC + CaCl2. Solid line: Manning’s Uniting laws. x and 0: experimental results for CMC a = 1, A = 1.38 and iE = 2.5, X = 3.45 respectively.
electrolyte-simple ekctrolyte and of the electrochemical potential of the simple ele&oIyte solution at the same ionic concentration; the +y” ratio shows the influence of the polyelectrolyte without account of the ionic strength and can be directly compared to the values obtained in Manning’s theory for infinite dilution [lo] . By conductometry, the variations of conductivity are compared for the same addition of simple electrolyte to a polyelectrolyte solution (A@ and to a simple electrolyte solution (A$) whose initial conductivities are equal. No modification of ultrasonic absorption by increase of site binding had been obtained previously on these Na-CMC-NaCl solutions [ 13 3 ; as a consequence the modification of conductivities can be attributed to a decrease of&e conductivity of external salt in presence of polyelectrolyte, but not to an increase of the counterions binding on the CMC. The experimental results are given with monovalent and divalent counterions respectively; the dependence with X shows a decrease of Ax,/Axv as those of the mean activity coefficients of the external salt (tables l-4). The experimental values of the activity coefficients can be compared to the theoretica values given by Manning’s limiting law. From this treatment, when a simple electrolyte MsYzl is added to the polyelectrolyte solution M-P (with u the number of ion in the salt molecule and z their valence), the Manning theory gives for X > l/z+ the following relations deduced in ref. [5] : In-/,
x/22+x
=x/z+x
lny_
=-
In yk = -
f
+ ln (X/z-%
f zfr)
Xi-z+r
z’r(l+r)
’
3X/2A x/z%
+ z*r(l+f)
rX12zfh x/2*x
i- z*r(l+r)
’
+
r l *r
In (X/z% +z+r) x+z+r
-
In these relations, y,, y_, and r* are respectively the ionic activity coefficient of the counterions, the coions and the mean activity coefficient of the simple electrolyte; r is equal to the ratio v+Iv_. The experimental and calculated values are given in tables 1-4. With monovalent-monovalent electrolyte, the theoretical values, y_= y, or y+_are always lower than the experimental ones; nevertheless, the differences become 457
Volume 4 X, number 3
CHEhllCAL PHYSICS LETl-ERS
Table 1 Experimental and theoretical activity coefficients (*/,, T_, 7+) in NaJXlC-NaCi
10 5
2.5 1.66 1.25 1 0.5 X-+-
0.685
0.680
0.515
0.760
0.595
0.888 0.890 0.900 0.908 0.922 0.920
0.900
0.725
0.705
0.890
0.785
01710
0.920
0.830
0,730
0.900
0.860
0.755
0.900
0.875
0.777
0.910
0.915
0.560 0.615 0.665 0.705 0.752 0.820 0.440
0.797 0.795 0.819 0.824 0.836 0.859
0.636 0.694 0.743 0.778 0.811 0.866
0.810 0.640
0.905 0.915 0.925 0.934 0.954 0.957 0.960
(L’bl
(3 +a+
0.880 0.900 0.905 0.910 0.920 0.930 0.940
0.750 0.810 0.870 0.90s 0.925 0.532 0.96s
3.33
2.5 1.66 1.25 1 0.666 X-+-J
458
0.880 0.895 0.925 0.930 0.935 0.950 0.955 0.958 0.970
0.800 0.865 0.900 0.900 0.890 0.915 0.930 0.940 0.940
0.909 0.91s 0.934 0.946 0.954 0.965
0.972 0.876 0.983
nkxtures (h = 3.45; mp = 2 X IF3
N)
yNa+
exp
0.375 0.410 0.475 0.525 0.580 0.615 0.790 0.350
Table 3 Experimental and theoretical activity coefficients (r+, T_, y+) in CaCMC-CaCl2
13.3 10 5
x l(rs_54)
0.850
A% AX:
13 5 2-S 1.66 1.25 1 0.S X+-
mixtures (h = 1.38; mp = 2
0.880
Table 2 Experimental and theoretical activity coefficients (v+, r_, y,) in NaCMC-NaCl
X
1 August1976
0.330 0.380 0.415 0.460 0.495 OS25 0.590 0.615 0.700 0.305
0.270 0.320 0.430 0.500 0.560 0.605 0.745 0.176
0.574 0.607 0.656 0.691 0.730 0.756 0.862
0.442 0.510 0.611 0.671 0.717 0.754 0.845
mixtures (A = 1.38; mp = 2 X IO-’ NI
0.277 0.295 0.356 0.408 0.451 0.521 0.575 0.618 0.695 0.220
0.595
0.657 0.695 0.719 0.732 0.760 0.799 0.816 0.852
0.612 0.628 0.678 0.714 0.743 0.786 0.816 0.838 0.875
Volume 41, number 3
CHEMICAL
1 August 1976
PHYSICS LETI’ERS
Table 4 Experimental and theoretical activity coefficients (-y+,7_, 7*) in CaCMC-CaClz mixtures (A = 3.45; mp = 2 x ZOW3 N)
13.3
0.880
0.820
0.932
1G 5 3.33 2.5 1.66 1.25 1 0.666 X-t=
0.880 3.885 0.915 0.9 0.950 0.960 0.968 0.978
0.880 0.900 0.930 0.935 0.935 0.940 0.940 0.950
0.940
0.175 0.210 0.250 0.310
0.960 0.970
0.360 0.435 0.500 0555 0.625 0.145
0.975 0.983 0.987 0.989 0.993
small when X is lower than 2 and 1 respectively X = 3.45 (a= 2.45) and 1.38 (a= 1).
for
With monovalent-divalent electrolyte, the r_ calculated are always larger than the experimental ones; the 7, calculated are lower than the experimental values but the agreement is relatively good as soon as X is lower than 5 and 2 respectively for X = 3.45 and 1.38. In figs. 1 and 2, the experimental-and theoretical values of the mean activity coefficients of the external salt are given as a function of X for monovalent and divalent counterions: - As for activity coefficients of coions and counterions separately, the agreement between experimental and theoretical values is better with divalent counterions and with the higher charge density polyelectrolyte. - Whatever the X value for divalent cougterions, the r+ calculated are practically equal to the y, experimental by a combination of both deviations on -y_ and 7, speciaily for large values of X (fig. 2). - In each case, the agreement with the theory is better when the charge density increases; one can see aiso that the decreases of the activity coefficient
is larger
when the charge density is higher.
4. Conclusion In this work, tlie carboxymethylcelluloses have been chosen to test the activity coefficients of coions and counterions as a function of the ionic strength and of the charge density. This polyelectrolyte used for a long time seems a good model to test the electrostatic proper-
0.154
0.490
0.174 0.244 0.303 0.353 0.435 0.497 0.548 0.639 0.088
0.546 0.587 0.645 0.680 0.724 0.762 0.788 0.826
0.512 0.541 0.608 O.$SS 0.696 0.749 0.785 0.812 0.857
ties of rigid polyelectrolytes as previously discussed [ 141. When X increases, the low decrease of the activity coefficient of the mcnovalent coions and of the mean activity coefficient is established; the agreement with the theoretical treatment proposed by Manning is relatively good. The agreement is better for divalent counterions. These conclusions are identical to those of Kwak on polystyrene sulfonate [5] _ Nevertheless, this work must be extended to other polyelectrolytas to estabhsh if this behaviour is general for polyelectrolytes whatever is the ionic site, the rigidity and chemical structure of the macromolecular backbone.
References [l] R.A. Mock and CA. Marshall,J. Polymer Sci. 13 (1954) 263. [2] M. Magasawa,M. Izumi and I. Kagawa, J. Polymer Sci. 37 (1959) 375. [3] T.J. Podlas and P. Ander, Macromolecules 2 (1969) 432. [4] J.C.T. Kwak, J. Phys. Chem. 77 (1973) 2790. [5] J-C-T. Kwak, hX.C.O’Brienand D.A. MacLean, J. Phys. Chem. 79 (1975) 2381. (61 T. Ueda and Y. Kobatake, J. Phys Chem. 77 (1973) 2995. [7] I. Kagawa and K. Katsuura, I. Polymer Sci. 9 (1952) 405. [8] H. Noguchi, K. Gekko and S. Makino, Macromolecules 6 (1973) 438. [9] G.S. Mann&, J. Chem. Phys. 51 (1969) 924. [lOI D. Wells, Biopolymers 12 (1973) 223. [ll] S. Lifson and A. Katchalsky, J. Polymer Sci. 13 (1954) 43. [12] M. Rmaudo and G. HudryClergeon, J. Chim. Phys. 64 (1967) 1746. [13] R. Zana, C. Tondre, M. Rinaudo and M. Mi!.as,J. Chim. Phys. 68 (1971) 1258. [ 141 M. M&s, Thesis, Universitkde Grenoble (1974).
459
ACTIWTY
1 August 1976
-CHEMICALPHYSICS LETTERS
COEFFICIENTS OF SMALL IONS IN AQUEOUS MKIXJRES
OF POLYELECTROLYTES
AND SIMPLE ELEC-I’FtOLYI-ES
M. RINAUDO and M. MILAS Centre de Reciwrches sur les ,Mccromolt?cz~lesVigirales (C.N.R.S.J. 38041 Grenoble Cede& Lance Received 23 February 1976 Revised manuscript received 12 Aprii 1976 The activity coefficients of coions Cl-, of counterions NaTand Ca*+ have been determined on Na-carboxymethylcelIuloseN&i and Ca-carboxymethylcellulose-CzClz systems for two charge densities of the polyelectrolytes (&,=.rched = 1.38 and 3.45); the mean activity coefficients of the external salt are then deduced. Their variations with the equivalent ratio X are investigated and the experimental values are compared to the theoretical ones @en by the Manning treatment.
1. Introduction
Many experimental and theoretical studies are available now which concern the activity coefficients of counterions and coions in aqueous mixtures of polyelectrolyte and simple electrolyte. Mock and Marshall [l ] show that the activity of coions is not modified by the polyelectrolyte in the system vinyltoluenestyrene copolymer sulfonic acid-HCI; the conclusions of Nagasawa et al. for sodium polyvinyl alcohol sulfateNaCl mixtures are identical [2] . For carrageenan [3], polystyrene sulfonic acid [4-61, and carboxymethylcellulose [7], it is found that ‘Jle mean activity coeffcients of the electrolyte or those of the coions are reduced by the polyelectrolyte. Noguchi and co-workers find a decrease of the activity of counterions in ionic dextran derivatives when simple electrolyte is added; it corresponds to a decrease of specific conductivity of the polyelectrolyte [8]. From a theoretical point of view, Manning’s limiting law is the only one to predict a decrease of coion activities [5,9]. The work of Wells allows the extension of the theory for fmite dilutions [ 70 j . In this work, the activity coefficients of the Cl-, NaT and Ca2+ Ions ’ have been determined on Na-carboxymethylcellulose-NaCl and Ca-carboxymethylcelluloseC&l, systems for two degrees of substitutio&of the car50xymethylcelluloses. Each polyelectrolyte is characterized by its charge 456
parameter X introduced by Lifson and Katchalsky [l 1] ; the variation of the conductance and the activity of ions are experimentally determined by potentiometry as a function of the quantity of simple electrolyte added and correlate to the X value. 2. Experimenta! The carboxymethylcelluloses (CMC) with substitution degrees (m) 1 and 2.5 have been prepared in our laboratory [12] . The simple electrolytes NaCi and CaCi2 are analytic grade products from Prolabo (France). The activities of ions are determined on a Sargent S 30000 potentiometer equipped with specific electrodes Orion: a crystalline membrane electrode (model 94.17) for Clions, a divalent cations selective liquid membrane electrode (model 92.32) for Ca2+ ions and an Ienagl_ass electrode for Na+ ions. The reference is a double junction electrode Orion (model 90.02). The conductivities of aqueous solutions are obtained with a Philips bridge (1000 cycles/s) and a conductivity measuring cell Philips (PW 9510, constant k = 9-7 cm-‘). All experiments
are done at 25 2 O.l°C.
3. Results and discussion In figs. 1 and 2, the r,/$ tion of X in ‘&e mixtures
values are given as a func-
respectively
with monovalent
Volume 41, number 3
0‘4 t t 0
1 August 1976
CHEMICAL PHYSICS LETTERS
1
f
s
IO
X
>
Fig. 1. Mean activity coefficients of simple electrolyte for the system NaCMC + NaCl. Solid line: Manning’s limiting laws. X
and e: experimental results for CMC DS = 1, h = 1.38 and Z% = 2.5, X = 3.45 respectively.
and divalent counterions (X = mp/ms -with mp the equivalent concentration of the polyelectrolyte and m, the equivalent concentration of the simple electrolyte). The ratio -y/r0 is obtained directly by comparison of-the electrochemical potential measured on the system polyY*
oa4.
.
0
Fig. 2. Mean activity coeffbients of simple electrolyte for the system CaCMC + CaCl2. Solid line: Manning’s Uniting laws. x and 0: experimental results for CMC a = 1, A = 1.38 and iE = 2.5, X = 3.45 respectively.
electrolyte-simple ekctrolyte and of the electrochemical potential of the simple ele&oIyte solution at the same ionic concentration; the +y” ratio shows the influence of the polyelectrolyte without account of the ionic strength and can be directly compared to the values obtained in Manning’s theory for infinite dilution [lo] . By conductometry, the variations of conductivity are compared for the same addition of simple electrolyte to a polyelectrolyte solution (A@ and to a simple electrolyte solution (A$) whose initial conductivities are equal. No modification of ultrasonic absorption by increase of site binding had been obtained previously on these Na-CMC-NaCl solutions [ 13 3 ; as a consequence the modification of conductivities can be attributed to a decrease of&e conductivity of external salt in presence of polyelectrolyte, but not to an increase of the counterions binding on the CMC. The experimental results are given with monovalent and divalent counterions respectively; the dependence with X shows a decrease of Ax,/Axv as those of the mean activity coefficients of the external salt (tables l-4). The experimental values of the activity coefficients can be compared to the theoretica values given by Manning’s limiting law. From this treatment, when a simple electrolyte MsYzl is added to the polyelectrolyte solution M-P (with u the number of ion in the salt molecule and z their valence), the Manning theory gives for X > l/z+ the following relations deduced in ref. [5] : In-/,
x/22+x
=x/z+x
lny_
=-
In yk = -
f
+ ln (X/z-%
f zfr)
Xi-z+r
z’r(l+r)
’
3X/2A x/z%
+ z*r(l+f)
rX12zfh x/2*x
i- z*r(l+r)
’
+
r l *r
In (X/z% +z+r) x+z+r
-
In these relations, y,, y_, and r* are respectively the ionic activity coefficient of the counterions, the coions and the mean activity coefficient of the simple electrolyte; r is equal to the ratio v+Iv_. The experimental and calculated values are given in tables 1-4. With monovalent-monovalent electrolyte, the theoretical values, y_= y, or y+_are always lower than the experimental ones; nevertheless, the differences become 457
Volume 4 X, number 3
CHEhllCAL PHYSICS LETl-ERS
Table 1 Experimental and theoretical activity coefficients (*/,, T_, 7+) in NaJXlC-NaCi
10 5
2.5 1.66 1.25 1 0.5 X-+-
0.685
0.680
0.515
0.760
0.595
0.888 0.890 0.900 0.908 0.922 0.920
0.900
0.725
0.705
0.890
0.785
01710
0.920
0.830
0,730
0.900
0.860
0.755
0.900
0.875
0.777
0.910
0.915
0.560 0.615 0.665 0.705 0.752 0.820 0.440
0.797 0.795 0.819 0.824 0.836 0.859
0.636 0.694 0.743 0.778 0.811 0.866
0.810 0.640
0.905 0.915 0.925 0.934 0.954 0.957 0.960
(L’bl
(3 +a+
0.880 0.900 0.905 0.910 0.920 0.930 0.940
0.750 0.810 0.870 0.90s 0.925 0.532 0.96s
3.33
2.5 1.66 1.25 1 0.666 X-+-J
458
0.880 0.895 0.925 0.930 0.935 0.950 0.955 0.958 0.970
0.800 0.865 0.900 0.900 0.890 0.915 0.930 0.940 0.940
0.909 0.91s 0.934 0.946 0.954 0.965
0.972 0.876 0.983
nkxtures (h = 3.45; mp = 2 X IF3
N)
yNa+
exp
0.375 0.410 0.475 0.525 0.580 0.615 0.790 0.350
Table 3 Experimental and theoretical activity coefficients (r+, T_, y+) in CaCMC-CaCl2
13.3 10 5
x l(rs_54)
0.850
A% AX:
13 5 2-S 1.66 1.25 1 0.S X+-
mixtures (h = 1.38; mp = 2
0.880
Table 2 Experimental and theoretical activity coefficients (v+, r_, y,) in NaCMC-NaCl
X
1 August1976
0.330 0.380 0.415 0.460 0.495 OS25 0.590 0.615 0.700 0.305
0.270 0.320 0.430 0.500 0.560 0.605 0.745 0.176
0.574 0.607 0.656 0.691 0.730 0.756 0.862
0.442 0.510 0.611 0.671 0.717 0.754 0.845
mixtures (A = 1.38; mp = 2 X IO-’ NI
0.277 0.295 0.356 0.408 0.451 0.521 0.575 0.618 0.695 0.220
0.595
0.657 0.695 0.719 0.732 0.760 0.799 0.816 0.852
0.612 0.628 0.678 0.714 0.743 0.786 0.816 0.838 0.875
Volume 41, number 3
CHEMICAL
1 August 1976
PHYSICS LETI’ERS
Table 4 Experimental and theoretical activity coefficients (-y+,7_, 7*) in CaCMC-CaClz mixtures (A = 3.45; mp = 2 x ZOW3 N)
13.3
0.880
0.820
0.932
1G 5 3.33 2.5 1.66 1.25 1 0.666 X-t=
0.880 3.885 0.915 0.9 0.950 0.960 0.968 0.978
0.880 0.900 0.930 0.935 0.935 0.940 0.940 0.950
0.940
0.175 0.210 0.250 0.310
0.960 0.970
0.360 0.435 0.500 0555 0.625 0.145
0.975 0.983 0.987 0.989 0.993
small when X is lower than 2 and 1 respectively X = 3.45 (a= 2.45) and 1.38 (a= 1).
for
With monovalent-divalent electrolyte, the r_ calculated are always larger than the experimental ones; the 7, calculated are lower than the experimental values but the agreement is relatively good as soon as X is lower than 5 and 2 respectively for X = 3.45 and 1.38. In figs. 1 and 2, the experimental-and theoretical values of the mean activity coefficients of the external salt are given as a function of X for monovalent and divalent counterions: - As for activity coefficients of coions and counterions separately, the agreement between experimental and theoretical values is better with divalent counterions and with the higher charge density polyelectrolyte. - Whatever the X value for divalent cougterions, the r+ calculated are practically equal to the y, experimental by a combination of both deviations on -y_ and 7, speciaily for large values of X (fig. 2). - In each case, the agreement with the theory is better when the charge density increases; one can see aiso that the decreases of the activity coefficient
is larger
when the charge density is higher.
4. Conclusion In this work, tlie carboxymethylcelluloses have been chosen to test the activity coefficients of coions and counterions as a function of the ionic strength and of the charge density. This polyelectrolyte used for a long time seems a good model to test the electrostatic proper-
0.154
0.490
0.174 0.244 0.303 0.353 0.435 0.497 0.548 0.639 0.088
0.546 0.587 0.645 0.680 0.724 0.762 0.788 0.826
0.512 0.541 0.608 O.$SS 0.696 0.749 0.785 0.812 0.857
ties of rigid polyelectrolytes as previously discussed [ 141. When X increases, the low decrease of the activity coefficient of the mcnovalent coions and of the mean activity coefficient is established; the agreement with the theoretical treatment proposed by Manning is relatively good. The agreement is better for divalent counterions. These conclusions are identical to those of Kwak on polystyrene sulfonate [5] _ Nevertheless, this work must be extended to other polyelectrolytas to estabhsh if this behaviour is general for polyelectrolytes whatever is the ionic site, the rigidity and chemical structure of the macromolecular backbone.
References [l] R.A. Mock and CA. Marshall,J. Polymer Sci. 13 (1954) 263. [2] M. Magasawa,M. Izumi and I. Kagawa, J. Polymer Sci. 37 (1959) 375. [3] T.J. Podlas and P. Ander, Macromolecules 2 (1969) 432. [4] J.C.T. Kwak, J. Phys. Chem. 77 (1973) 2790. [5] J-C-T. Kwak, hX.C.O’Brienand D.A. MacLean, J. Phys. Chem. 79 (1975) 2381. (61 T. Ueda and Y. Kobatake, J. Phys Chem. 77 (1973) 2995. [7] I. Kagawa and K. Katsuura, I. Polymer Sci. 9 (1952) 405. [8] H. Noguchi, K. Gekko and S. Makino, Macromolecules 6 (1973) 438. [9] G.S. Mann&, J. Chem. Phys. 51 (1969) 924. [lOI D. Wells, Biopolymers 12 (1973) 223. [ll] S. Lifson and A. Katchalsky, J. Polymer Sci. 13 (1954) 43. [12] M. Rmaudo and G. HudryClergeon, J. Chim. Phys. 64 (1967) 1746. [13] R. Zana, C. Tondre, M. Rinaudo and M. Mi!.as,J. Chim. Phys. 68 (1971) 1258. [ 141 M. M&s, Thesis, Universitkde Grenoble (1974).
459