CHARLES LINDER AND IRVING F. MILLER
3434
Persistent Electrical Polarization in Polyelectrolyte Membranes172 by Charles Linder and Irving F. Miller* Polflnchnk Institute of Brooklyn, Brooklyn, New York llBOl
(Received February $3, 197.2)
Measurements of persistent electrical polarization as a function of composition, temperature, applied field, molecular weight of the polarizing component, and time were made on a series of membranes containing sodium polystyrenesulfonate in matrices of poly (vinyl alcohol), polyacrylamide, and polyvinylpyrrolidone. The resulis indicate that the process depends mainly on interactions between components, rather than the intrinsic nature of the components themselves. A model for the process has been developed in which the electret formation and stabilization mechanism involves an ion displacement in the direction of the applied field via a positive feedback between the local field and its reactive field component. The model is compatible with experimental data and explains observed saturation effects and the effects of such variables as membrane composition and polarizing temperature, as well as the fact that persistent polarization is compatible with electrical conductivity.
Introduction The phenomenon of persistent electrical polarization in materials has been of interest for many years. The phenomenon was firat conceived as an electrical analog of a magnet by Neaviside3 in 1885 and was first experimentally demonstrated by Eguchi4 in 1922 in studies involving carnauba wax.x. It has been postulated that such polarization phenomena underlie many biological processes such as nerve conduction,sr6 sight,’ and auditory proc(m.es.8 I n addition, the phenomenon has been used in many commercial devices such as microphones, electrophotography, electrometers, et^.^-" I n spite of the fact that persistent electrical polarization phenomena seem Lo appear in such a wide variety of places, these phenomena are not too well understood. Since a o s t of the early experimental studies were made with carnauba wax, most of the theoretical work has gone into it phenomenological description of the nature of the process in carnauba wax and the dynamics of its decay 9 , 1 2 , 1 3 As a result of this, most vorkers in the field still consider the phenomenon to take place essentially in dielectrics of a parallel plate condenser syYtern l 4 Based on our present understanding, two different types of phenomena can be distinguished. When polarization is -.st&lished by charging electrodes, and when the sign ~f the charge on the surface of the material is the aarne as the charge on the adjacent electrode, the material is referrcad to a8 a homocharged electret. When the charge on the surface is opposite to the charge on the adjacent electrode, the material is referred to as a heterocharged e l e ~ t r e t . ~Since , ~ ~ the rates of formation and decay of homocharged electrets and heterocharged elt~ctretsare quite different, it is possible to distinguish between them in the same materi a1 Studies of homocharged e l e c t r e t ~ ~ ~indicate -l~ that they are formed essmtially by absorption of charges by The Jozirnal of Phtsical Chrmistrg, Vol. 76,N o . $3, 1972
dielectric breakdown of the gas in the space between electrode and dielectric. This process, then, is essentially a contamination process and has not been considered in the present report. On the other hand, heterocharged electrets have been shown to involve volume polarizationla and have been shown to decay by ~ ~ cases mechanisms of dielectric r e l a ~ a t i o n ~In~ ,most it is not known whether such heterocharge formation is due to dipole orientation, ion displacement, electron trapping, or some combination of these. Most studies of heterocharged electrets have involved electrical insulators, and workers have postulated their formation (1) Based on a thesis submitted by Charles binder in partial fulfillment of the requirements for the Ph.D. degree, Polytechnic Institute of Brooklyn. (2) The authors gratefully acknowledge partial support for this work from the National Institutes of Health, under Grant No. GM12013. (3) 0. Heaviside, “Electrical Papers,” Vol. 1, Maemillan, New York, N. Y., 1892, p 488. (4) M. Eguchi, J a p . J. Phys., 1, 10 (1922). (5) L. Y. Wei, BdZ. Math. Bwphys., 31, 39 (1969). (6) D. Wobschali, J. Theor. Bwl., 21, 439 (1968). (7) H. C. Berg, Rwphys. J., 8 , 1051 (1968). (8) E. C. Hughes, Department of Surgery, University of Southern California, private communication. (9) V. M. Fridkin and I. 8 . Zheludev, “Photoelectrets and the Electrophotographic Process,” Consultants Bureau, New York, N. Y., 1960. (10) V. A. J. Carbaub, Electrochem. Technol., 6 , 3 (1968). (11) H. R. Anderson, E. A. Bartkus, and J. A. Reynolds, I B M J. Res. Develop., 15, 140 (1971). (12) M. M. Perlman and J. L. Mennier, J . AppZ. Phys., 36, 420 (1965). (13) F. Gutmann, Rem. Mod. Phys., 20, 457 (1948). (14) B. Gross, “Charge Storage in Solid Dielectrics,” Elsevier, Amsterdam, 1964. (15) J. R. Beeler, J. D . Stisnathan, and G. G. Wiseman, J. Chern. Phys., 32, 442 (1960). (16) R. A. Draughn and A. Catlin, “Electrets,” M. Perlman and 1,. M. Baxt, Ed., Electrochemical Society, New York, N. Y., 1967, p 93. (17) M. M . Perlman and C. W. Reedyk, ref 16, p 86. (18) B. Gross and R. J. DeMoraes, J . Chem. Phys,, 37, 710 (1962).
PERSIsTENT
'ELECTRICAL PQLARIZATION
I N POLYELECTROLYTE
to involve electron trapping1e in some cases, or ion disother ~ cases r ~ involving t ~ e s ~ ~ ~ a ~ofe ionic ~ ~ ~e n~ t ~ ~in ~ ~ ~ ~ ~ plast,ics,20 ~ a t i ~2~ i X gn one case, large amounts of perdistenb h e ~ ~ ~ ~ polarization ~ ~ i a r has ~ ebeen ~ observed ~n n p ~ ) ~ ~ e ~ I ?-polyethylene ~ ~ r o ~ y t e membrane22 ivhiela.was a good e ~ ~ ~semiconductor. ~ ~ r ~ c a ~ 111 H K R ~ cmes, ~ I W ~ Q U work S has involved phenomand qualitative e ~ o ~ d~~ ~~ ~ ~ns c r of ai pthe ~ ~ process ~ o s ~ ~ ~on ~uzicroneopic ~ n ~ models a t to~ characterize ~ ~ ~ such ~ h ~ t ~ ~Icel~elr, ~ rformation. ~ c It~ is the ~ purpose ~ ~ of ~ this report to dcscriil:e work undertaken to develop a 1 ~ ~for ~ 1heterocharged ~:~r elecguan litative ~ ~ 1 ~ 1 ~model t rete.
3435
MEMBRANES
~
~
I
PAC
~
~
~
e
~~~~~~~~
~
~
~
e
~
t
~
l
Polyelectrolyte nwrnbranes were prepared with sodium poly8tyrenr,Julfonatc (I'SSNa) and a polar polymer matrix. The polar polymers used were poly(viny1 alcohol) (PVA), polyacrylamide (PAC), and poly%.iiriylpyrrolndon(.(PVP). The structure of these materials i s sbow:i ia, Figure 1. The PSSNa, a linear polymcr of varying molecular weight, was obtained from Dow Chemical nith a 30% impurity of sodium hromide, 'lrh~material \vas purified by washing sevcrai Limes in a 90% methanol-water solution, dialyzing it f i v ~timen against distilled water, and then titrating mith carbonatc-free 0.01 N NaOH to pH 7. PVP was obtained from GeneraP Aniline and Film Corp. with the taesigna t,ion K90. PAC was obtained from American cy.-' d,yariamid Co. tvith the designation PAM 100, and PVA was obtained from Du Pont, with the designation 72-64) (99-100Yo hydrolyzed and of intermediate viscosity). All polymers were purified by five dialysis washings against cMAled water to remove impurities #and~ b e dried n iw t)acmfor 24 hr at 50". Siaxe all ingredients were water soluble, membranes were prepareci by first preparing aqueous solutions of the t srious ingredrents with deionized distilled water and mixing them in the proportion necessary to obtain the desired xr~cmbranccomposition. The membranes w r e then eilher m s t on a plate of polystyrene or poured into R ~ o ~ , y ~ ~ ~petri ~ y r dish. e n e The water was then allowed to evaporate in a dust-free environment to form the mcriibram The membranes were then preconditioned for 24 lar under vacuum a t 60" to drive off most of the water. Although infrared spectra of these membranes ~ridica(cdno characteristic water peaks, it is likely that solno residual water of hydration reinained in thcw membranes. However, since the prcstncc of small amounts; of water of hydration probably enhances the Pffects investigated, no attempt was made to completely dry thc mcmbrancs. lets \rere formed and characterized in a 1emperatur.e-controElecf vacuum oven, as shon n in Figure 2 . Thrl oven, a cylindrical chamber of naval bronze, was able to maintain a controlled temperature to within +=Q.P in thr range of 35-95". The chamber
I
HzC-CH~
PVP
Figure 1. Molecular structure of polymers used MEASURING CHAMBER
TEMPERATURE CONTROLPROBE
. \
i
3 ELECTRODE ASSEMBLIES IN MEASURING CHAMBER
MEMBRANE
Figure 2. Diagram of polarizing cell.
was designed to hold three membranes a t a time between electrodes, all of which were also rrmachined of naval bronze. The weight of the electrodes was used to ensure intimate contact with uniform and constant prcssurr between the electrodes and mcmbranes. All eIectrodc surfaces were coated nith Aquadag (an aqueous carbon suspension from the Anderson Colloids Co.) to reduce corrosion of the electrodes and subscquent membrane contamination. In an cxpcrimcnt, a membranr was placed between (19) (20) (21) (22)
11. Gerson arid J. H. Rohrbaugh, 9.Chem. Phya.* 23, 238 (1955). M. L. Miller, J . Polyrn. Sci., Part A-2, 4%685 (1966). D. K. Donald, ref 16, p 90. 2. Urban and R. Wallace, J. Electrochem. Soc., XIS, 518 (1968). The Journal of Ph,ysical Chernistrg, Vol. 76, No.
197d
CHARLES I;INDER AND IRVING F. MILLER
3436 the electrodes, the chamber was closed, and a vacuum was established by use of a Welsh Duoseal vacuum pump. The chamber was brought to the desired charging temperature (above ambient conditions), and the material was polarized by placing a 22.5-V battery in series with the membrane-electrode system for an arbitrary time (usually 90 min). During the charging cycle and, subseqrnently, during the stabilization and discharging cycle tl-e current flowing across the memto A) was monibrane (ranging from about tored with a Model 61OC Meithley electrometer. After a fixed time of charging, the system was cooled to room temperature atJabout 5"/min, using a fan and Dry Ice under the chamber while maintaining the charging ~i in the formation cycle, the membrane current and temperature were recorded throughout. At room temperature the voltage was removed, and a discharge current was measured with the electrometer. The discharge current, consisting of free capacitor charge and dielectric absorption not frozen in, decayed exponentially by zlboub three orders of magnitude over the 90 min in which the decay current was monitored. To measure the amount of charge frozen in, the membrane was reheated to 10" above the temperature of polarization aT, the sate of X"/min. The depolarization current was measured during the heating process and for an additional time required for the current to decay :it least 1.5 orders of magnitude from its peak value a t
r----
Temperature of Polarization
c3 7 6 8
A 865
* l
l
l
l
l
l
93.5
l
l
l
02 04 06 08 Mole Fraction of PSSNa
Figure 4. Polarization us. mole fraction of PSSNa in PAC.
the elevated temperature. The measured current integrated over the time of measurement was taken as the amount of frozen polarization or electret formed per unit volume of the membrane. I n all depolarization measurements the currents flowed in the opposite direction from the charging currents, thus indicating heterocharge formation with no sign of homocharge.
Results
A 86.5
*
I
DO
Figure 3.
93.5
69
02 04 06 08 Mole Fraction of PSSNa
IO
Polarization zs. mole fraction of PSSNa in PVA.
The Journal of Phfisical Chemistru, V a l . 76,N o . 23,1972
Polarization characteristics of the membranes were studied as a function of such variables afi membrane composition, molecular weight of the PSSNa, water content, membrane thickness, applied field, and time. The characteristics observed are reported below. Polarization as a Function of Composition. Figures 3,4, and 5 present data taken a t four diBerent temperatures of persistent polarization in coulombs per square centimeter as a function of the mole fraction of PSSNa in matrices of PVA, PAC, and PVP, respectively. The data presented were obtained OIL membranes approximately 0.005 cm thick under a potential of 22.5 V maintained for a period of 1.5 hr and represent an average of three separate measurements, with a reproducibility of f10%. The effect of a variation of these other variables will be discussed below. A number of observations can be made from a comparison of these threc figures. I n all three cases a dramatic change in the nature of charge storage seems to occur a t a mole fraction of PSSNa in the neighborhood of 0.3-0.4. The reason for
I\
d
I6'O
00
02 04 06 08 Mole Fraction of PSSNo
0 PVA-PSSNa h PVP-PSSNo
IO
Fig~1rc5 . Poiarization v s . mole fraction of PSSNa in PVP
16"
0.0
this apparent change is probably due to the fact that as the mole fyactron of PSSNa increases, the primary molecular process changes from one involving PSSNamatrix interaction 1 o one involving PSSNa-PSSNa interaction. This ifill be discussed in more detail below. A comparison of the three figures indicates that the PVA matrix membrane stores the most charge, the PVP matrix membrane stores the least charge, and the PAC matrix membrane is intermcdate It lis clezr " r ~ r nthe figures that the nature of the matrix has a profound rffect on the mcmbrane's charge storage capacity. The addition of PSSNa to PVA increases the charge storage capacity of the membrane by reverd o r d m of magnitude. On the other hand, thr additton of 1'8SNa to PYP in small amounts decreases the charge storage capacity of the membrane below that obtaiued for pure PVP. This difference in behavior 19 mor(' clearly shown on Figure 6, in 7%hich the charge storage. capsl city for Qv(lO5-cmthick membrarics under 22.3 V and 1.5 hr was measured for polarization O. The figurr prcsents data for a PVA-PSSIL'a membrane, a PVP-PSSrt'a membrane, and a third membrane in, u hich t h r matrix contained both PVP and PVA in a ratio of onc part of PVP to three parts of PVA (by m o m ) . From this figure it is evident that the PVP-I'SSF3Sa interaction IS substantially strongcr than the PI'A-PSSNa interaction. In spitr of the fact that the thrw camponcnt membrane contain5 substantially more IVJA than PVP, tho membrane bc-
0.2
0.4
0.6
0.8
0
Mole Fraction PSSNa
Figure 6. Polarization vs. mole fraction of PSSN a in matrices of PVA and PVP. Charging field is 4.5 kV/cm a t 70" for 1.5 hr.
haves much more like a PVP-PSSNa membrane than a PVA-PSSNa membrane. It might also be noted that as the mole fraction of PSSKa increases, all membranes approach a common behavior. The reason for this is that PSSNa-PSSNa interactions become predominant. Polarization as a Function of Applzecl Field. Figure 7 presents polarization results as a function of field intrnsity (volts per centimeter) for t hrre membranes containing PVA (0, 20, and 75y0PSSNa) at 70" and one membrane consisting of 33% 1'SSNa in PAC, with charging for a period of 90 min. The data prcsentrd for each membrane were obtained using three different membrane thicknesses, ranging from about 2 X lou3 ern thick t o about 1.5 X 1 0 - 2 cm thick. The results for the I T A membranes indicate that a saturation phcnomenon occur'r3 with the additional interesting result that those systems which saturate at a highcr level of polarization also reach their iat uratioii level at loner levels of applied field. At a given le-ircl of polaiizing firld the amount of polarization wcntially proportional to the numbrr of polarizable gi-oup~available (as determiried by the membrane volume). hi fhc casc of the PAC membrane, no haturation was obsca vcd, probably because thc firlds used were not strong enough The Journal of Physical Chemislry, Voi. 76, No. 19,1971
3438
CHARLES LINDERAND IRVING F. MILLER
Table I: Variation of Polarization with Molecular Weight of PSSNa in PVAa Mole fraction of PSSNa
61
0 20% PSSNo in PVA at 7OOC. A 75% PSSNa in PVA at 70%
*
:Ifl
Thiokness, om X 1 0 - 8
Density, g/cmn
Moleoular weight of PSSNa
(C/cm9
0.064
4.5 4.5 4.2 4.3 4.8
1.20 0.98 1.00 1.08 0.90
1.94 X 102 4 x 104 4 x 106 3 . 8 X 106 1 . 0 x 107
4.21 9.55 16.2 13.0 12.0
0.20
4.2 4.6 4.3 4.2
1.20 1.21 1.10 1.00
a
4 4 3.8
1.0
x x x x
104 106 106 107
Pe
x
107
60.5 370 295 210
Polariza,tion a t 70" and 22.5 V.
33% PSSNa in PVA at 86'C.
Table 11: Persistent Polarimtion us. Time of Application of 22 V across PSSNa-PVA Membranes 0.056 mm Thick at 69" 7
,
O 0
j L L - U I I 4 8 12 16 20 24
1
I
I
I
I
I
1
28 32 36 40 44 48 52
ELECTRIC FIELD INTENSITY (kV/crn)
Figure 7. Polarization density us. applied field for three membranes containing PVA a t 70" and one membrane containing PAC a t 86'.
Polarization as a Function of Molecular Weight of PSXNa. Table I presents data obtained a t 70" and a polarizing field of 5 kV/cm on two formulations of PSSNa in PV A. One formulation contains 6.4 mol yo PSSNa and the other contains 20 mol % PSSNa. In these forrriulstions the number-average molecular weight of the PSSn'a used varied from 194 (sodium p toluenesulfonate) to 3 X 10'. The amount of polarization occurring can be seen to increase up to a molecular neight of' aboul 4 X lo6 and then gradually to decrease. Thew results can be explained on the basis of the local density of PSSNa units in the membrane as the molecular weight changes. (The greater the molecular weight, t h ~greater the local density.) It might be noted that at molecular weights grcater than 4 X lo" the rncasned membrane density decreases slow.ly. This density decrease probably results in a reduction of iarter:tction between adjacent PSSNa units and as 2 result, the degree of polarization decreases s1ightl.t Polnrixatzon CIS a Functzon of Time. Table I1 presents data on the degree of polarization for three different PSSiVa--P't'A inembranes at 69" and a polarizing field of 4 kV/crn dii a function of time. Table III presents i;imilar results obtained with a membrane containing 25y0 1'SSXa in different matrices where polarization mas carried out at 86" in a field of approximately 8 ltV/crn. Ac. a gcncral observation it might be notrd The Journal of Physical Chemistry, Vol. 70, No. 23, 1972
-
Time, min
75% PSSNa
15 60 90 180
7.56 9.20 11.o 11.5
p
P o , C/cma X 10s 50 %
20%
PSSNa
PSSNa
5.78 6.0 6.18 6.35
2.70 3.0 3.16 3.36
7
Table I11 : Persistent Polarization vs. Time of Application of 44 V across Membranes Containing 0.25 Mole Fraction PSSNa in Different Matrices at 86"
- ~ - -
P o , C/cm2 X
Time, min
15 30 60 90 180 a
PVA
(0,049mm)"
823 946 986 1000
-
lo9
PAC (0.057mm)"
PVP (0.069mm)&
0.26 0.55
0.20 0.26
0.79 1.45
0.28 0.48
The thickness of the membrane is indicated in parentheses.
that those systems which polarize to a much greater degree also seem to reach saturation much more rapidly. Polarization vs. Conduction. As was mentioned earlier, it has been assumed for many years that persistent polarization and electrical conduction are incompatible in the same system. In all our experiments this was found not to be the case. I n fact, in these experiments it was found that systems which polarize to a greater degree also conducted to a greater degree, and the stability of these electrets did not decrease with increasing conductivity. Typical membrane conductivity data obtained for PSSNa in PVA are shown in Figure 8. A comparison of Figure 8 with Figure 3 indicates that, indeed, both processes seem to follow similar trends.
P E E S ~ - ELECTRICXI. S T ~ ~ POLARIZATION IN POLYELECTROLYTE MEMBRANES
3439
the absence of the out-of-phase persistent) polarization component, P,, eq 1 can be written a8
The dielectric constant, E , thus c h ~ r a c ~ e r t~ ~h esize s of the applied field necessary to give a certain pobrization. Since the external field is also rtqmnsible lor producing the persistent polarization component, p,, one can, by analogy, define an apparent l o c d dielectric constant, ee, by
2
Q
02
0.4
06
0.0
Mole Fraction of PSSNa
Figure 8. Membrane conductance us. mole fraction PSSNa in PVA.
The implicatior, of these results lies in the fact that the polarization process takes place by a mechanism which i s not antagonistic toward the dissipative processes involved in electronic conduction.
I n the case of a PSSNa membrane, is a measure of the effect that the local fields, as a function of the applied field, have on the Rodium sulfonate dipole with respect to any displacement of the sodium ion, and its consequent reaction with the local environment. In determining the local fidd intensity responsible for persistent polarization (electret formation) let us assume that the polarizing group (PSSXa) is in a spherical shell which represents the local environment and whose dielectric constant is e,. From classical eleetrostaticsjZ3the electric field arising in the tiphere of a local environment and polarizing di reference to
where E, is the local field in the absence of persistent polarization. In the presence of persistent polarization, P,, the actual local field, Ei, is given by
Bi Taken as a \vholc, the results presented above can be dcvelvp a molecular model for persistent polarization which is compatible with all the observations, along the follo.iring lines. The static diclectric polarizrttion of polar and ionic materials in a condensed phase may be characterized as having a component that follows the applied field, called the instantaneous polarization, P , , md another component, out-of-phase w rith the applicd fk’d, which reprrscnts persistent polarization, p,. ‘The instan tancous polarization, PI, is proportional to the applied electric field and disappears when the field i~ removed. A relationship between the total polarization pt, and the applied electric field, I?, is given byq3 U S C to ~
where E ips the static dielectric constant. In highly polarizing systems, E is usually quite large so that, in
=
B, Jr
4n --Pe
3
Combining eq 3, 4,arid 5 one obtains
E i=
e@ -
+ + 2Ee
3 ( 2 -4~
EeJ -= _____ b
(8)
which represents the local field a t a I’SSNs polarizable group when it interacts with o t h u specks (matrix molecules) in the environment. In the case where the PSSNa group is likely to be surrounded by the same configuration everywhere in the sample, ce =- eez and eq 6 reduces to
Let us now consider what happens when an external electric field is applied to a system which can support persistent electrical polarization. SUPPOWthat, ini(23) C. J. F.Bottoher, “Theory of Electrical Polarization,” Elsevier, Amsterdam, 1952.
The Journal
of
Physical Chemishy, Vol. 76, X o . 25, 19?’$
3440 tially, the mobile sodium ions are in an equilibrium position associated (vith a local sulfonate group. When an external iield is applied, the mobile sodium ion mill be displmxd hy the field and approach a new site in which the forces acting upon it are in balance. As the ion apywoacbes its new equilibrium position, persistent polarization, Be, begins to appear, and the local field intensity increases according t o eq 5. The process, therefore, is ticcelersted by a mechariism which is esrentiaily positive-feEdback in nature. That is, a small locai field est:tblishes a small degree of persistent polarization which, in turn, increases the local field intensity, which then increases local polarization and so on. The stability or” the system of displaced ions, matrix, and e l d r o d c r is increased by lowering the temperature and, P, hen the electrodes are removed, the new state remains in metastable equilibrium with the original configuration arid will eventually decay to the original corifiguratiorr in the presence of external disturbances. Thus, both the initial formation and the stability of the electrets formed depend on the local environments of the2 polarizing PSSNa units. Clearly, only t,hose PSSSa units that have a proper environment to support the positive-fwdback mechanism described above can participate i n forming electrets. The saturation phenomena observed during the experimental studies are now readily explainable. The fact that those systems which si”ppoi‘t the greateqt degree of persistent polarization ar” precisely those systems 1% hich come to the saturation lwei most rapidly (both as a function of time and as a funotion of external applied field) is a direct consequence of the positive-feedback mechanism i n which the loczl,lfield intensity increases as the degree of persistent polarization increases. The effect13 of locd mcmbrane morphology are clearly prerent in the l~roposcdmodel in terms of the participation of adjaccnr specks t o the polarizing groups through their local dir:lectric constants, ee,. The apparent change in rnechann;-ni for persistent clcctrical polarization ah the molt^ fraction of PSSNa increases is explainable in terms c d the vhanging probability of finding a polarizable I’SSNa group in the presence of a local matrix molecul~environment, as opposed to the probability of finding wch a PSSSa molecule in the presence of an environwtcrt of other I’SSNa molecules. Thr observation tnat persistent polarization and electronic condiretan em t o go hand in hand is also consistcnt with this cd. An increase in electronic conductance is an indication of an increased mobility of clectrons in the membrane, Since ion mobility is t ep in ion displacement and stabilization, it should riot be incompatible with an increase in electronic conduetanor. drlolecular Considerations. In the unperturbed state of the mcmbraiir a sodium ion may be considered as vibrating about an equifibrium poRition in the potential The Joiirnal of Physscal Chemzstry, Vol. 76, X o . 93. 19T2
CHARLES LINDERAND IRVING F. MILLER well of the sulfonate dipole. If the sodium ion is displaced to an adjacent site in the locai environment by the application of an external electric field, the criterion for whether or not the combination of the new site and the sodium ion will remain after the applied field is removed is that the combination of the new site and the sodium ion must produce a potential well of sufficient depth at a temperature below that of polarization. Before the external field is applied, the potential well depth containing the sodium ion i s characterized by the energy of a sodium sulfonate interaction, ps (in units of IcT). The depth of the unoccupied well to which the sodium ion will be displaced is zero, as the well is only produced when the ion is preaect. Upon displacement of the sodium ion, the well in the field direction has a depth of at least ps (assuming an eliectret structure is formed), and the original well nom has a depth of zero. The situation is that of a distortable double n ell system where the application. of a field produces occupancy of the wells lying in the direction of the field and where the combination of environmental distortion and the lowering of the temperature before removal of the field produces the electret state. The work required to separate the sodium ion from the sulfonate group is a function of the environment in which the dipole finds itself, as shown by EL8
e2 = ___-
4 markT
where e is 1.6 X 6,r is the distance of charge separation in meters, and e, Is the self-polarization dielectric constant of the environment. T o determine p s for the various species studied herein, namely PSSNa, PVA, PAC, and PVP, measurements of E , for these species were made by extrapolating to time = 0 the E, values measured for different polarization times over 5-min intervals for a range of 6 2 0 min at 10 V and 66”. I n the case of PSSNa, the measurements were made on a 94% PSSSa-PVA membrane because pure PSSNa is too brittle to work with. If the literature value of T o 1.94 X m for the PSSNa dipole in vacuo24 is assumcd, then ps values for thc different environments possible can be calculated. The results are presented in Table IV. The results presented in this table clearly indicate that electret formation is easiest in a PSSNa environment and becomes progressively more difficult as the environment shift8 from PSSNa to PVA, PAC, or PVP in keeping with the observations. The magnitude of persistent polarization observed can be taken as the volume density of induced dipoles. Since the only way to measure this polarization is by electrostatic induction of charges on the electrodes of a parallel plate condenser system, the only component of (24) “Tables of Interatomic Distances and Configurations in Molecules and ions,” Chem. SOC., s p e c . Pz‘bl., MQ. 11, M i 9 6 (1958).
Fmsr STFNT J~LECTRJCAL
POLARIZATIOS IN POLYELECTROLYTE MEMBRANES
3441
fraction of PT'A (X,= 1 - Xa)trations of PSSNa (X, > 0.33), N with X,according t o PS?
EnvironmenL
€a
PSSNa PYA PAC PVP
I48
1.73
102 40
2.46 6.40
13
N
units of kT a t T = 68'
15.0
9, that is observable is that component parallel to E . Thus, p , can bo written as (9) aherc N is the number of sodium sulfonate groups forming eleclret structures, e is the charge of a sodium or sulfonate ion, 1' i5 the volume, and r is the average displacemeril of the sodium ion parallel to the applied field, after tbe field is1 removed. The value of N is Some fraction of tlhe total number of PSSNa monomer units present. Thie fracj ion i s made up of simply those units 1% hich have f,hc proper environment for electret formation and, if the membrane is assumed homogeneous, this fraction is 3 function of the mole fraction of PSSNa in the membrane. Ii' the relationship between N and N , (the total number of PSSNa monomer units present) i s knonn, tlieri 1' can be calculated from eq 9. The nature of the 'unction depends, of course, on the detailed naturc cud' the typical electret forming unit, which i h imkno.i?n. However, an attempt was made to estimate the valw of r based on some reasonable models for the typ cai ele t forming unit for the system studied. T h e v mocids were developed from t h e shape of the experimcnt.al S U T V ~ Sof polarization us. mole fraction of PSS:\'a for thc three matrices as shown in Figures 3 , 4, arid 5 The results of these calculations are given below. PSXNa-PVA NeInbranes. I n the case of all three matriceh (PVA, PAC, and PTIP) the mechanism of clectret lormation seems to change when one goes from inole fractions of PSSSa below about 0.33 t o mole fractions of PSSKa above 0.33. The reason for this changeover i s that, ad Jow concentrations of PSSNa,the typical electret forming unit is in an environment of matrix groups; at) high concentrations of PSSNa the typical c&&-rt forming unit is in an environment of PSSNa groups. In the caw of the PVA matrix, at molc fractions of PSSXa bciow about 0.33, N , the number of PSSSa unit s participating in electret formation, was found to vary with ixiolc lraction according t o
N
=
N,Xb3X,
(10)
whcrc N , is tbr t o t a l number of PSSNa monomer units, X, i~ the molt1 Craction of PSSKa, and XI, is the molc
=
zv,x,s
(11)
Equations 10 and 11 imply that the probability of finding a particular molecular ~ o n ~ g u r a t is i oa~ furnction only of the mole fraction of the particular components present. The use of only X , in the ~ r o b a b i ~ ~ t y function assumes that thr PSSNa m i t s arc randomly distributed, which is only an a ~ p r o x in ~ poly~ ~ ~ ~ ~ ~ ~ meric materials. A polymer chain i 4 of finite dimensions even when completely entangled by a matrix polymer, so that at low eoncentrations there are arms in the membrane that contain o d y t b r matrix. The approximation gets better a8 the number of PSSNa polymer chains increases to the point wlaere the chains themselves are entangling and as the molecular decreases. Thus, the ~ r o b ofa havong ~ ~ ~ ~ ~ ~ units adjacent to each other should 132 higher than that calculated using X , , so that the aclual number of PSSNa units undergoing electret formiition via PSSN aPSSNa interactions will be higlicr t h m the calcwlattd value. Another assumption khat oc~:urs ~n using a probability function solely of X, i s t h ~ ifi a sodium sullonate group is surrounded by B surtable environment, it is automatically in the proper epatinl arrangement to form measurable electretd. Two ~ ~ [ ~ r ~arex ~ ~ ~ t xnvolved in this assumption. One is that if two PSSNa uriits are adjacent to each other, they are csrientecl SO that eketret formation rnay occur. ?'he stcoxkd 1s that while the PSSNa units may b~ propcrly oriwted wit,h respect t o each other, the enscmble must have a component parallel t o the applied heid. The first approximation is not a bad one as the clrctroc,tatic inrLeractisiis between the sodium sulfonate dipoles a1low for several associatmi structims whcre the ZXIOF.t ~ t , h l conr (the negative sulfonate ion associating v. itli tht positive sodium ion of the other dipole in a !zcad-to-tail nrrangenient along the dipole momenl, of both dipole-) is Lhr optimum arrangement for electrpt formation The error due i o the sccond approximatron decreasts the number of posRibie PSSNa units in\olveti in ckectrd formation more than that calculated using only X , ~ This decrease 1s opposite t o the ~ffec1h a t tlsi. approximation of complete spatial ranciomn of PSSXa monomer units has, such Ibat they mould, t o .mine ~ x t e n t , cancel each other, Tbt implications of cq 10 and I k nith regard to a molecular model for 1he typical electret structure are as follows. At low concentrations of P39'k;x t h typical clectret-forming structurc apparently consisti of one PSSNa molecule and three PYA moleoiaIrs. En this configuration, the charge kcparation of thc I'SSNa group is probably stabilized by hydrogen bonding t o the three surrounding PVA groups, At h conccntrat ionh, where the typical rlcclrct The Journal of Phvsical Chemiqtrg, Vd.76, 8Yo. 29, 19'73
CHARLES LINDERAND IRVING F. MILLER
3442
_--. Table V : The Theoretical Values of the Number of PSSNa Units and the Displacement Distance of Their Na" Ions
Involved in Electret Formation in PVA-PSSNa Membranes (66' and 22.5 V ) XS mole fraction of PSRNa
P,, C/crnz X 101
2.55 9.47 8.87 11.8 48.0
0.059
0.130 0.218 0.328 0.480
0.630
280
.V./A,a total moles of PSSNa x 10'
vx
105, cma
8.59 17.0 13.8 17.7 28.0 34.2
7.75 7.75 4.71 4.86 6.23 6.54
N ~ / A , ~ N/NS&td
0.43 0.55 0.63 0.68 0.50 0.50
NZ/A,b
4.25 x 1.46 X 1.44 x 1.75 x 1.89 x 1.09 x
10-6 10-6 10-5
10-6
r4?
T1,c
mol
a
6.0 X 5.95 x LO-@ 6.65 x 6.35 x 10-7 7.10 x 10-8 3.40 X 10+
0.98 0.87 0.54 0.57
mol
A
6.15 6.65
a A = Avogadro'~number. * VI and Nz are the number of PSSNa units involved in electret formation calculated from eq 10 and 11, respectively TI and Ta are the distances of N a + ion displacement calculated with eq 10 and 11, respectively.
volves interactions between adjacent PSSNa groups, eq 1I implies that the typical electret-forming structure involves five such YSSNa groups, in a stable configuration that also tenda t o insulate the electret group from the surrounding PYA, a slightly poorer electret forming structure than the I'SSNa, as is evidenced from Table
1 6- ~
IV.
1 6- ~
The values of r from eq 9,10, and 11were determined for the experiment illustrated as the lowest curve in Figure 3 , representing electret formation in a PVAPSSNa membrane a t 66" and an externally applied field of 22.5 V. The results of this calculation are presented in Table V. Since these conditions would not lead to saturation for all the membranes tested, eq 10 and 11 had to be modified slightly to account for the fact that not all configurations capable of forming electrets svoulcl actually do so. The appropriate coeffithe approach to saturation for the s w e also presented in the table. In the table, r l rcprcwnts the distance calculated from eq 9, where N is given by eq 10; r2 represents the distance calculated when the value of M in eq 9 is given by eq I l . For va,Iues of ;La w 0.33, the average value of X , = 0.33 the average value se values of ion displacement are in the ranqe of what one should expect if the proposed ion displaceincnt model for electret formation is correct. It might) k9e noted that r is considerably larger for the caw whore the chief mechanism involves I'SSNa-PS8Na intcractions than it does when t,he mechanism involves chiefly PSSNa-PVA interactions. This is Teasoxiable since the PVA monomer unit is smaller in sizc than a PSSSa unit and might be expected to be closer to the polarizing unit. A linear superposition of eq 9, 10, and 11, along with the average values of r for the range in which it is valid should give a s n g l e equation valid over the entire range of X,. Such B superposition leads to
Fe =
m "i4.04
v
x
l0--')Xa(l-
X,)3
+
4.25 X 10-5X,5] (12) The Journal
5 . f
Phvsical Chemistry, V d . 76,No. $3, 1.972
log
t 0
0.2
0.4
0.6
0.8
1.0
Mole Froction PSSNa
Figure 9. Polarization ws. mole fraction of PSSNa in PVA at 66.5" and 22.5 V. Measured values are compared to values predicted by eq 12.
where P, is polarization in coulombs per square centimeter, V is membrane volume under the electrodes in cubic centimeters, and m, is the number of moles of PSSNa present. In Figure 9, this equa,tion, representing variation of total persistent polarization as a function of mole fraction of PSSNa at 66.5' and an external field of 22.5 V, is plotted along with the measured values of polarization and the equation is found to fit the experimental data extremely well. As the temperature of electret formation increases above 66", the total amount of polarization was also found to increase. The question arises as to whether the polarization increase is accounted for by an increase
PEEBIBTENT ~~~~~~~~~~~,~~ POLARIZATION IN POLYELECTROLYTE MEMBRANES in the d ~ s p ~ ~of~ the c ~sodium ~ c ~ion t from its initial site, or xather by an increase in the number of PSSNa units involved From cq 8, it may he seen that an in4:reahe in ~ ~ ~should ~ lower c the~ potential a en~ c ~ g yberrier to inn displacement and thus should increaw the riunnbfr of dcctrets formed. At the same time, ho1ve3mx, a tcinperature increase also increases ~~~~~~i~~~~~~~ and h4pe rate in which the sodium ion would return t o i t s initial site. The measured amounts ob persistent ~ o l ~ r ias~ a ~function , ~ ~ of ~ temperature n and anole fraction of PSSNa in RVA are shown in Table V i . P t must tse uo6ad Irom this table that, as the temperature increasw from 66 to 93", the measured ~ ~ ~ oofu~ n t ~ ~for all~ membrane ~ ~formulations ~ t aaacrense by ahout a factor of 25. From Table V it must also be noted that in all cases a substantially larger t'ra,ctioa~thnn L/2s of the available Z'SSNa groups participate in electret formation (for all cases, the fracapproximately one in ten). Thus, while tinn ~nvolvr~s an increase izi tc mpcrnt ui.e perhaps does incrcasc the nurnbcr of PSSNa units contributing to the electret ~ ~ ~ c ~ ~ this o minicrcase ~ r ~ (in~number n ~ can only be a partial cxpilaneatioln for the temperature variation of
Table VI: Persistent I'olarization as a Function of Temperature and Mole Fraction of PSSNa in PVA (at 22.5 v for I 5 hr) -l.l__ll----.l_
oc
66 76.5 86.5 93
0.06
0.18
p e , C/cmz
_._-_.
2.55 12.9
34.0 61.0
9.47 42.4 I25 232
8.87 46.7 116 249
Table VII: The Variation of Ion Displacemelib with the Temperature of Polarization for PSSNa-PYA R.lembranes5
i
~ _ll_ll---.__rl ~ T, OC
65 76.5 86.5 93 a
n
_l--x _l-l_.l-.0.13
0.08
x
0.33 108-
11.8 56.3 151 295
s0.33
0.22
,j /
0.98 2.42 6.43 11.5
~
0.48
I
0.87 2.33
0.54
0.57
1.75
7.03 12.8
4.35 9.33
1.61 4.32 8.44
0 63
______
_____I__"
_ I _
6.15 14.5 53.5
7
6.65 14.1
207.0
Membranes of different PSSNa moie fractions.
N
-X* 0.22
is in the range of 16-86'. As the temperature increases from 65 to 93", the amount of eliain nrovemerlt in response to the thermal energy present increases ,significantly. ~ ~ Membrane ~ ~ conduetari r e n a~ also found to increase significantly a s the t e ~ ~ e r a t was ~ r e increased. From these consideratiorw then signifinant increases in ion displacement with ~ ~ ~ ~ eare~ toa t ~ r e be expected, rind the results proscrnled jn Table VJI are very reasonable
PSfNa-PVP Membyanes. In the same manner as with the PVA system, equations equivalent t o eq 10 and 11 can be derived for the PVP matrix system from an analysis of the shape of the lowest PSSXa-PVP curve in Figure 5. For mole fractions of PSSKa greater than 0.33 the appropriate equation is now
polar.milion.
?',
3443
0.48
0.63
48.0 162 597 120
230 700 ... ...
_____
Another possible explanation for the large increase in persistent polarization with temperature is that increasing temperature increases the number of possible environmental configurations in which a PSSNa unit may find itself. If such were the case, then the general shape of the ,pe 2)s. X,isotherm would change with temperature. However, it does not. It thus appears likely that the observed polarization increase with temperature is due to the further displacement of the Radium ion from its first jump site to anothcr one further down the field. If one assumes that a t the kigher Icmperatures all the membranes are at saturation, theii the variation of 1' with temperature can be calculated from the data in Table VI by use of eq 0, 10, axid 11. 'The rcwlts of these calculations are &own in Ta'nle 'L'Bli. Thcsci results indeed bear out the contontion that the average displacement of the sodium n ith increasing temperature. In the PSSSa-PVk syatem the glass transition temperature
=
NaXa10
(13)
For mole fractions of PSSKa belou 0.33 it was not possible to obtain a single function. However, in the range Xa < 0.14 the appropriate equation is
N
= NbXb3
(14)
where Nb is the total number of PVP monomer units and Xb its mole fraction. In the range 0.14 < X, < 0.33 the appropriate equation is
N
=
N,Xa
(15)
The significance of these equation? must be determined from an analysis of what they m a n on a molecular level. The inhibiting effect of PVP-PSSNa interactions on electret formation manifests itself by a minimum in the P, us. X, curves at low PSSNa concentrations and a rapid increase in polarization above a mole fraction of PSSNa of 0.33. At moic fractiorls of PSSNa greater than 0.33, the basic electret forming units of PSSNa monomers must be well insulated from any contact n i t h the strongly inhibiting PVI'. Equation 13 implics that any two interacting PSSNa groups must be insulated from their surroundings by at least eight other PSSNa groups if a local PVP lvioleculc j s not to prevent electret formation. I n other words, a, pair of interacting PSSNa groups must be insulated in a PVPfree region several shells deep. The Journal of Physical Chemistry, Vol. 7'6, N o . 23, 1872
CHARLES LINDERAND IRVING F. MILLER
3444
--Table ~ 1 1 1 : The Theoretical Values of the Number of PSSNa Units in Moles and the Displacement Distances of Their Na+ Ions Involved ir. Electret Formation in PVP-PSSNa Membranes (66’ and 22.5 V) Pa
7
C/omr X 10’0
1 .a3 65.5 2360 a
X , , mole fraction o f
N*/A
x
104,~
PSSNa
of PSSNa
0.33
2.35 3.71 2.79
0.52 0.71
A = Avogadro’s number.
_______----
vx
total motes
102, om8
7.6 9.1 6.4
W is calculated from eq 13.
=
N,X,7
( 16)
The exponent on X E Sindicates that the typical electretforming unit consists of seven PSSNa units. Thus, interacting PSSNa rnommer units must be insulated from their surroundings to a slightly greater degree than in the I’V4 systpm but not nearly as much as in the PVP system. 13eicnv a mole fraction of PSSNa of 0.33 the appropriate equation is identical with eq 10, indicating that PAC does not bind PSSNa to the point where it cannot) participate in electret formation a t all a t these low mcllr fractions, as in the case of the PVP system. In fact, at these l o x mole fractions, PAC behaves very much t h e > 0.33, th$ average value of r turnrd out to be 1’ = 2.90 I 0.44 A, a reasonable value. Again, it w:is not pci blc1 to determine r for values of T h e JournaE of Physical C’li-nistry, Vol. 76, 14‘0.
Z3,197X
N/A, mol
0.50
3.55 x 10-9 5.30 x 10-7 9.04 X 10-6
4.56 4.52 2.46
r is calculated from eq 9.
At mole fractions of PSSNa below 0.14 the measured amounts of paxistent polarization can be attributed exclusively t o I he polarization of PVP reacting in a configuration involving three PVP units. At these concentrations tho PSSNa is so tightly bound to the PVP that it cannot participate in electret formation at all. I n the range of mole fractions of PSSNa between 0.14 and 0.33 the presence of the PSSNa begins to make itzelf felt. Equations 8 and 13 were used to estimate r , the average amount of sodium ion displacement for PSSNaI’VP eiectret membranes above a mole fraction of PSSNa of O X < . The. results of these calculations are shown in Table VJBI. The average value for sodium ion displaccmentounder these conditions is seen to be r = 3.83 f 0.92 A , a reasonable number. An estimate for the value of r for mole fractions of PSSNa below 0.33 was not pospibio because it was not possible to determine the saturation level of polarization in these systems. PXSNa-PAC Membranes. The behavior of the PSSSn-PAC zystem turned out to be intermediate betneen the I’SSIVzt--kVA system and the PSSXaPVP system. For a mole fraction of PSSNa greater than 0.33, the appropriate equation for N to be used in eq 9 is LV
!v/Nsatdb
X , below 0.33 because polarization saturation for such membranes could not be achieved.
Conclusions A model for persistent electrical polarization in polymer membranes has been developed in which the eIectret formation and stabilization mechanism involves an ion displacement in the direction of the applied field uia a positive feedback between the local field and its reactive field component. The model has been used to explain experimental data involving persistent polarization in membranes consisting of PSSNa in matrices df PVA, PAC, and PVP at various mole fractions. Observed saturation effects have also been explained by the model. The experimental observations that! the chief variables in electret formation involve those variables affecting the interaction of components on a microscopic level rather than the intrinsic nature of the components themselves, art: completely in keeping with the proposed ion displacement model. This work has potentially highly significant implications in biological processes. One such area of application is in nerve excitation. In the currently most widely held view of how nerve cells conduct25the nerve cell membrane in the resting state is impermeable to sodium ion and slightly permeable to potassiuni ion. When a nerve impulse approaches, thc permeability of the membrane to sodium changes drastically and sodium is allowed to pass across the nerve membrane. In this view, the mcmbrane permeability to ions is a function only of the membrane potential and can change over several orders of magnitude when thc membrane potential is changed. I n the view of Tasaki and coworkersz6 in the resting state, the nerve membrane, containing fixed negatively charged sites, also contains divalent cationic countcrions (mostly e a 2 +and Mg2+). In the process of excitation, in this view, the membrane undergoes rapid reversible transition between two stable configurations which have different physiological properties. In the excited state, in fact, the counter(25) A. L. Hodgkin and A. F. Huxley, J . Phvswl., 116, 449, 473, 497 (1952); 117,500 (1952).
(26) I. Singer and I. Tasaki in “Biological Membranes,” D. Chapman, Ed., Academic Press, New York, N.Y . , 1968, Chapter 8.
~ ON
~
'POLYELECTROLYTE ' E CONCENTRATION ~ ~
~
~
~
-_____-
-.__
-__l_t___L___I(_^__
Table IX: The 'i'heorelical Values of the Number of PSSNa Units In Moles and the Displacement of the Na+ Ion Involved in Electret ~ ~ in PAC-PSSNa ~ Membranes ~ ~(66" and a22.5 V) ~ ~ ~ ~ Pep
cjornz
x
ID*
1.90 8.56
16.0 5
A
=
3445
~
xa mole lrnetioo of PIISNB
N * / A x 104,~ tot,al rnolRR of PSSNa
0 30 0" 39 0.57
2.12
~
Avogadro's number.
vx
2.80 3.90 6
N is calculated from eq 16.
102, cma
6.82 7.75 8.66 c
N/Nsatdb
0.94
NiA, mol
4.83 3.84 7.60
x x x
3.56
10" 10-7 10-8
r is calculated from eq 9.
--------,--
ions are chiefly univialent (Na+ and K+). More recent,lyz7-2@a, number of workers have suggested that the two stable states involve dipole orientation. The 10x1 displacement anoldci for persistent polarization is an extremely attractive one for use in the nerve excitation problem. I n xhis model the amount of persistent polarization P O S S ~ le is clearly a very strong function ol the nature of thc dispfaceable ion. The observation that the nerve membrane permeability change,.g occurring a,t excitation are a function only of membrane
i
a.44
2 70
~
-
potential is also compatible with the ion ~ ~ ~ s p l a c e ~ e n t model in terms of the effect that such external potentials would have on the microscopic e n ~ i r o n of ~e~~ mobile ions. Work on the application of this ion displacement model for persistent electrical poiarization in polymer membranes to the problem of nerve membrane conduction is continuing. (27) L. Y. Wei, Bull. Math. Bwphys., 31, 39 (1969). (28) D. Wobschall, J. Theor. Bbl., 21, 439 (1968). (29) B.B. Raniel and I. Zimmerman, Biophys. J., IO, 1329 (1970).
r the Dependence of pH on Polyelectrolyte Concentration
y lH[iro&i Maeda* Department of Chemistry, Faculty of Science, Nagoya University, Nagoya, Japan
and Fumio Oomwa Instit& of Molecular Biology and Department of Physks, Faculty of Science, Nagoya University, Nagoyaaz, Japan (Received April 4, 107%)
A thermodynamic theory is presented on the dependence of pH of polyelectrolyte solutions on the polymer concentmtion. This theory is based on both the additivity of the osmotic pressure or the counter ion activity in polyelectrolytes containing lowmolecular weight salts and the invariability of the osmotic coefficient or the counter ion activity coefficient with respect to the polymer concentration in salt-free solutions. I n salt-free solutions, pH is shown to change in proportion to the logarithm of the polymer concentration. This proportionsiity holds even when the salts are contained, if the change of the polymer concentration is caused by the removal or the addition of the solvent. I n the presence of excess salts, pH becomes independent of the polymer concentration, These results agree well with available experimental data. The theory also gives a procedure for obtaining the osmotic coefficient from the observed relation between the pH and the polymer concentration. lnterpretation of the theoretical result is made ni terms of the Donnan equilibrium between small ian~ free in the solvent and those bound on the polymer.
ntrod~ctio~i I n a solution of binary monovalent electrolytes, p~ expected to change in proportion to the square root of the total (b]e(*trolyl,ueoncentration, so long as the Dcbye-HBckel throry is valid. All interactions bet m e n small ions in such a solution are essentially
equivalent and may be said t o be symmetric. On the contrary, ionic interactions in a polyelectrolyte solution contribute to its thermodynamic propcrtks in different manners. The interaction of polyions with small ions plays a central role, whereas that between poiyions is of secondary importance and that betmcn srnail ions i s of The Journal of Ph.ysica1 Cizemistry, "01. 76, N o . 23, 1972
~
