Determination of
Micelle Aggregation Numbers
The Journal of Pbysical Chemistty, Vol. 83, No. 20, 1979 2621
(16) 0.L. Bertrand and T. E. Burchfield, Anal. Calorlmetry, Proc. Symp., 3, 283 (1974). (17) L. G. Hepler and D. V. Fenby, J . Chem. Thermodyn.,5 , 471 (1973). (18) W. Guschlbaver, “Nucleic Acid Structure”, Springer-Verlag, New Yak, 1976. (19) C. R. Noller, “Chemistry of Organic Compounds”, 3rd ed., W. B. Saunders, Phlladelphia, Pa., 1965, p 658. (20) H. A. Sober, Ed., “Handbook of Blochemlstry”, 2nd ed., CRC Press, Cleveland, Ohio, 1970. (21) J. S. Binford and D. M. Holloway, J. Mol. Biol., 31, 91 (1968). (22) E. Osawa and 2 . Yoshida, Spectrochim. Acta, Part A , 23, 2029 (1967). (23) L. J. Bellamy, K. J. Morgan, and R. J. Pace, Specfrocblm. Acta, 22, 535 (1965). (24) G. C. Pimental and A. L. McClellan, “The Hydrogen Bond”, W. H.
Freeman and Co., San Francisco, Calif., 1960. (25) G. D. Frederick and C. D. Poulter, J. Am. Chem. SOC.,97, 1797 (1975). (26) C. D. PouRer and 0.D. Frederick, TetrahedronLeft.,26,2171 (1975). (27) D. D. Perrin, ”Dissociation Constants of Organic Bases In Aqueous Solution”, Butterworths, London, 1965. (28) H. Iwahashi and Y. Kyogoku, J. Am. Chem. Soc., 99, 7761 (1977). (29) G. R. Wiley and S. I. Miller, J. Am. Chem. SOC.,94, 3287 (1972). (30) A. DAlbis, M. P. Wickens, and W. B. Gratzer, SiOpo&mrs, 14, 1423 (1975). (31) J. Alvarez and R. Biltonen, Biopolymers, 12, 1815 (1973). (32) R. A. Newmark and C. R. Cantor, J . Am. Chem. Soc., 90, 5010 (1968). (33) H. A. Nash and D. F. Bradley, J. Chem. Phys., 45, 1380 (1966). (34) J. Donohue and K. Trueblood, J. Mol. Biol., 2, 363 (1960).
Isopiestic Compositions of Aqueous Ionic Surfactant Systems as a Measure of Preferential Interactions. Application to the Determination of Micelle Aggregation Numbers by Equilibrium Ultracentrifugation Daryl A. Doughty Department of Energy, Bartlesville Energy Technology Center, Sart/esvllle, Oklahoma 74003 (Received January 29, 1979) Publication costs assisted by the U.S. Department of Energy
The determination of reliable micelle aggregation numbers for ionid surfactants by means of equilibrium ultracentrifugation requires a correction for the preferential interactions which occur in multicomponent charged systems. Isopiestic distillation experiments on solutions containing the volatile solvent and the nonvolatile supporting electrolyte and varying amounts of the ionic surfactant lead directly to the desired correction for preferential interactions. Results for the surfactant, sodium dodecyl sulfate (SDDS), in various NaCl(aq) background solutions show that a substantial correction, increasing with increasing NaCl concentration,is required. Results on the surfactant, sodium decylsulfonate (SDS), show a similar correction above the critical micelle concentration (cmc). Data for SDS below the cmc show a much different trend from that above, a condition not evident with SDDS because of the much lower cmc. If the interaction is interpreted as arising strictly from micelle charge, values for the micelle charge can be obtained. Results for SDDS indicate a fractional charge ranging from 0.356 for SDDS in 0.1 m NaCl(aq) to 0.418 in 0.3 m NaCl(aq). Sedimentation equilibrium experiments on SDDS in NaCl(aq) solutions gave aggregation numbers in good agreement with published values though generally higher. These higher results are consistent with the more reliable corrections for charge effects or preferential interactions available from the isopiestic distillation experiments. The aggregation numbers for SDDS increased with increasing NaCl concentration, consistent with published results. Precision density measurements, using a magnetic float densimeter having a precision of 1ppm, were used to obtain accurate values for the partial specific volumes (L?) of SDDS in NaCl(aq) and water for use in the ultracentrifuge calculations. The results show a slight increase in 0 with NaCl concentration. Values for the cmc obtained from the density measurements are in excellent agreement with published values. A model relating the increasing fractional charge on the micelle to decreasing micelle surface area to micelle volume ratio as a function of NaCl concentration is presented.
Introduction for the interpretation of sedimentation equilibrium data obtained from experiments on systems containing charged Ultracentrifugation has been used to determine micellar particles and/or several components. The system of inweights and aggregation numbers of surfactant micelles terest will be an aqueous surfactant system containing an and also can give information about micelle polydisperionic surfactant together with a low molecular weight, ~ i t y . l - ~However, the ionic nature of the surfactants uni-univalent electrolyte having an ion in common with currently of interest to enhanced oil recovery research the surfactant. In all discussions and equations which presents difficulties as far as the interpretation of results follow, I will use the standard convention of identifying is concerned. These difficulties arise from the charged solution components by subscripts with the subscript 2 nature of the surfactant micelles and the resulting inreferring to the ionic surfactant, the subscript 3 referring fluence of this charge on the distribution of the various solution components in the sample during ~entrifugation.~ to the uni-univalent supporting electrolyte, and the subscript 1referring to the primary solvent, water in this Correcting for these effects is a major obstacle to the case. Also, the equations as presented will apply strictly routine application of ultracentrifugation to the study of only under conditions of constant temperature and asionic surfactant systems. My discussion will be restricted sumed incompressibility of the solution. The first conto sedimentation equilibrium ultracentrifugation because dition may be controlled by the experimenter and the of the more rigorous theoretical foundation which exists This article not subject to
U.S.Copyright. Published 1979 by the American Chemical Society
2022
Daryl A. Doughty
The Journal of Physical Chemistry, Vol. 83, No. 20, 1979
second condition is reasonable for aqueous systems at the lower speeds usually employed for equilibrium ultracentrifugation. Under conditions of sedimentation equilibrium the following equation summarizes the determination of molecular weight in a ternary system in the limit of vanishing concentration of component 2:5
In eq 1 m2* represents the apparent molal concentration of component 2 as a function of radial position r in the sample cell at equilibrium, u2 is the partial specific volume of component 2 measured in the background solution of component 3 in component 1, po is the density of the background solution, w is the ultracentrifuge rotor speed in radian/s, R is the gas constant, T is the absolute temperature, and N*M2 is the apparent molecular weight of component 2. For convenience the micellar weight of component 2 is represented as N*M2 where N* is the apparent aggregation number of a surfactant micelle composed of monomers of molecular weight M2. A quantity proportional to m2* such as the number of fringes from an interference photograph may be substituted for m2* in eq 1. The concentration of component 2 is considered apparent as determined from a photograph because it includes the effects of redistribution of components 1 and 3 caused by various “preferential interactions” which can occur in a solution containing more than two components, particularly if one or more of the components are ionizeda5 The apparent micelle weight N*M2 is related to the actual weight by the following equation in the limit of vanishing concentration of component 2:5
In eq 2 M3 and d3 are the molecular weight and partial specific volume, respectively, of component 3 and To is the “binding coefficient”. The superscript zero indicates the terms are evaluated at vanishing m2. The use of the term “binding” does not mean that an actual physical bond occurs between the components but only that they interact in some manner which influences their distribution in addition to the effects of the centrifugal fielda5If To can be determined for the system then actual values of the aggregation numbers can be determined for ionic surfactant micelles in aqueous solution. Eisenberg has shown that r0 can strictly only be determined for solutions in dialysis equilibrium through a membrane impermeable to component 2 but through which components 1 and 3 are diffusiblea6 He defines a density increment ( a p / a ~ where ~ ) ~ c2 is the molar concentration of component 2 and the subscript p implies the constancy of chemical potentials p1and yLawhich exists at dialysis equilibrium on both sides of the membrane. Using the density increment, he has shown that in the limit of vanishing c2 d In c2* (3) NM2 dr2 Thus, actual values of the aggregation numbers of ionic surfactant micelles could be obtained from sedimentation equilibrium experiments if (ap/ac2),,0 can be evaluated. Dialysis equilibria are impractical for surfactant systems, however, because of the existence of the monomer-micelle equilibrium which occurs at concentrations above the
critical micelle c~ncentration.~ A membrane impermeable to the micelle most likely will be permeable to the monomeric surfactant present and no true dialysis equilibrium can be achieved.8 The density increment can be expressed in other ways, however, which are more useful for this situation. If the additional assumption is made that the density of the solution, at low surfactant concentration, is linear in c2 theng
(z); u 2 4 = (1-
1+
(:)I( -)]
(4)
The term (ag3/dg2): is the weight of component 3 in grams to be removed from the solution, per gram of component 2 added, in order to maintain constancy of pl and p3. If this expression for (ap/ac2): is inserted in eq 3, comparison of the result with eq 1 and 2 shows that these are equivalent expressions in the limit of vanishing c2 (or m2) with
A brief digression appears in order at this point to deal with a characteristic of aqueous surfactant systems, already referred to, which may have introduced some confusion to the reader. Frequent mention has been made of evaluating an expression in the limit of vanishing concentration of component 2. The centrifugeable species in these surfactant systems is the micelle. The monomeric surfactant present (because of the monomer-micelle equilibrium) at a concentration equal to the cmc exists in a completely ionized form for these ionic surfactants. Therefore, making the assumption that the monomeric surfactant undergoes negligible redistribution at sedimentation equilibrium, I am treating the monomeric surfactant as part of component 3.3 The limit of vanishing concentration of component 2 then occurs at the cmc. The cmc’s of ionic surfactants of interest to oil recovery are generally low, particularly in the presence of salt. Thus, any uncertainty arising from this assumption is probably very small. The determination of (ag3/agZ),O cannot be obtained without dialysis equilibrium, but the technique of isopiestic distillationlo permits the determination of an approximate form which can be shown to be essentially identical under the conditions encountered in aqueous surfactant syst e m ~ . ~ Isopiestic l~J~ distillation is applicable if only one of the solution components is volatile. By varying the amounts of surfactant placed in different vials containing the same background solution and allowing the solutions to jointly reach isopiestic equilibrium, the following expression may be eva1~ated:~J’ =
(2)( ,I):
The subscript yl represents the constancy of chemical potential of the primary solvent which exists at isopiestic equilibrium. Constant pressure and temperature are assumed. The following relationship permits the conversion of the above derivative to the desired result?
The term 1 / N in eq 7 replaces 1in the equation presented by Hade and Tanford. This arises because m2 in my equations represents the concentration as monomeric surfactant, whereas in the references quoted m2represents
Determination of Micelle Aggregation Numbers the concentration of the centrifugeable species, which is a factor of N smaller applied to surfactant micelles. In eq 7 y3 is the activity coefficient of component 3 on the molar scale. The term (a In y3/amJ0,evaluated in the limit of vanishing m2,depends only on the activity coefficient of component 3 in component 1and will normally be known for typical supporting electrolytes. Thus, as long as the surfactant systems studied fulfill the conditions specified, the binding coefficient may be evaluated and actual values of micelle aggregation numbers should be obtainable. A convenient equation for the calculation of N may be found by combining eq 1, 2, and 5 and rearranging to obtain
where S = (d In mz*)/dr2and A2 = M2(1- b 2 p o ) 0 2 / 2 R T . One final consideration is in order for ionic surfactants. If the preferential interactions are assumed to arise completely from the charged nature of the micelles, then an estimate of the micelle charge' can be obtained. Under this assumption Fujita has shown that I'O = - 2 / 2 where 2 is the effective number of charges on the micelles5 Introducing this term into eq 5 and simplifying, one obtains the following:
where i is the effective charge fraction on the micelle. I felt it was necessary to investigate a well-characterized surfactant in order to properly evaluate the isopiestic distillation technique as a means of determining preferential interactions. Thus, sodium dodecyl sulfate (SDDS) was selected as the surfactant with NaCl as the supporting electrolyte. Experimental Section The SDDS used was reagent grade material as supplied by Eastman Organic Chemicals. No additional procedures were used to further purify it with the exception that some of the material was dried under vacuum for 5 h. No significant differences in density of the solutions made from this dried SDDS compared to those made from the material as received were detected. The NaCl used was certified ACS quality as supplied by Fisher Scientific Co. The water used was laboratory deionized water distilled from permanganate solution and degassed under vacuum prior to use. Densities were measured with a magnetic float densimeter having a precision of fl ppm.12 The densimeter was immersed in a water bath maintained at a temperature of 25.00 "C. The bath temperature was controlled to better than f0.005 "C as monitored by a Beckman thermometer. Isopiestic distillation experiments were conducted by using procedures similar to those of Hade and Tanford." There were some modifications of apparatus and chamber evacuation procedures. Aluminum was used in place of silver-plated copper for the sample block. Nineteen flat-bottomed holes were machined in the block with the central hole being drilled through to accommodate the evacuation fitting on the isopiestic chamber base plate. Aluminum sample vials were machined, with snug fitting lids, to be a close slip fit in the block. An aluminum alloy, NO. 6061-T651, resistant to possible NaCl corrosion was selected for the vials. The chamber consisted of a polycarbonate vacuum jar resting on an aluminum base plate
The Journal of Physical Chemlstty, Vol. 83, No. 20, 1979 2623
with a neoprene gasket in between. The base plate had a central fitting connected to a valve for chamber evacuation. To speed up the equilibration process, which requires more time for the more dilute solutions used in these experiments, only 1.5 g of solution was placed in each vial. The solutions were paired, as in the Hade and Tanford studies, giving a reliable check on the degree to which equilibration had been achieved. All weighing5 of vials were made with the lids in place and weights were corrected to vacuum. The sample block with the vials in place was cooled to near freezing prior to evacuation to retard foaming and splattering of the solutions. The chamber evacuation was accomplished by using the procedure of Scatchard, Hamer, and Wood13 which gave much better control of this process. Any loss of sample from the vials due to splattering would require preparing new solutions. Most experiments with these dilute solutions required 2-3 weeks to achieve equilibration. The isopiestic chamber was submerged in a water bath maintained at 25.00 f 0.005 "C during equilibration. Ultracentrifuge experiments were conducted with a Spinco Model E analytical ultracentrifuge fitted with interference optics. The sample cells had centerpieces of aluminum-filled Epon and sapphire windows. A four-place An-F aluminum rotor was used providing for the simultaneous running of three samples. The concentrations of SDDS investigated varied from 0.1 to 0.005 m in the various salt backgrounds. The typical solution column length was about 3.5 mm. The reference solution in each case was the corresponding NaCl(aq) background solution (0.1,0.2,or 0.3 m, respectively). No oil was layered in the bottom of the cells because of the risk of contaminating the surfactant solutions. The technique of overspeeding was used to reduce the time required to achieve sedimentation eq~i1ibrium.l~ Angular speeds used were about 1900 radian/s with the overspeed being 1.44 times as high. The time of overspeeding was dependent on initial conditions and was typically 2 h in 0.1 m NaC1, 2.5 h in 0.2 m NaC1, and 3.0 h in 0.3 m NaC1. During a run slit photographs were taken at the "hinge point" at 8-min intervals to monitor the change in concentration during centrifugation. A full photograph was taken at about 7.5 h and a final one at about 23 h. Comparison of several of these pairs of photographs showed very little change the last 16 h of the runs. The final photographs were used for analysis with confidence that equilibrium had been essentially attained after 23 h. Rotor temperature control was somewhat erratic due to contact resistance between the thermistor needle and the mercury pool. Sample temperatures at completion of a run were typically 24-27 "C. A double-sector synthetic boundary cell with quartz windows and aluminum-filled Epon centerpiece was used in an An-D rotor to obtain initial concentrations of the various SDDS solutions in fringes for use in the ultracentrifuge calculations. This synthetic boundary cell data, together with the concentration shift obtained at the reference point from analysis of the slit photographs, was used to obtain the absolute concentration distribution in fringes for each sample. The spread of the concentration distribution was about 60 fringes for the highest SDDS concentrations studied down to 3 fringes for the lowest SDDS concentrations. Results The results of the density measurements on solutions of SDDS in water and various background NaCl(aq) concentrations are summarized in Figure 1, where the partial molal volume is plotted as a function of the square
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The Journal of Physical Chemistry, Vol. 83, No. 20, 1979
Daryl A. Doughty
TABLE I: Experimental Results for the Parameters of Interest for SDDS in Water and Aqueous NaCl cmc cmc (obsd) ( calcd)a v2 “solvent” m X l o 3 r n l~o 3 (abovecmc) V,O (ag3/ag2)U, (ag3/ag2)U9 .., .
H,O 0.01 m NaCl
0.1 m NaCl
0.2 m NaCl 0.3 m NaCl a
8.20 5.53 1.50 0.73
248.05 247.80 248.50 248.90 249.35
8.16 5.58 1.45 0.68 0.34
N
L
236.9 236.8 -0.0385 -0.0427 -0.0443
-0.0361 -0.0406 -0.0424
91.3 t 5.3 104.7 k 1.2 118.5 i: 2.3
These values were calculated by using the empirical equation of Yan.I5
250 2351
I
0
*
.. I
005
I
I
0 IO
0 I5
I
0 20
0 25
m Flgure 1. Varlatlon in P, the partial molal volume of SDDS,as a function of SDDS concentration, expressed as the square root of molality. The Curve labels represent the “solvent”, either water or the designated molality of NaCl(aq). For clarity, the lndlvldual data are not shown for the SDDS solutions In NaCl(aq).
root of molality of SDDS. The scatter apparent in the SDDS/water data is typical of that obtained for all solutions at comparable SDDS concentrations. Because the partial molal volume calculation is derived from the difference in density between the solution and the solvent background, the limiting precision of the density measurements represents a proportionately greater uncertainty in this difference at low concentrations of SDDS. Thus. the scatter increases greatly at lower concentrations. The dramatic increase in the partial molal volume at the cmc is evident in all solutions, particularly in water and the lower NaCl concentrations. The cmc is a function of the NaCl background concentration, decreasing with increasing salt concentration. This relationship can be represented by the following empirical equation16 cmc’ = u2(cmc’+ m3)-1/2 + b2 where cmc’ is the shifted cmc at salt concentration m3. The constants u2 and b2 were found to be 8.47 X and -1.21 X respectively, for SDDS in NaCl(aq) with concentrations in molalities. The calculated values of the cmc a t various salt concentrations agree closely with the experimentally derived values in those cases where they could be measured. These experimental values were obtained from relative density vs. molality plots at each salt concentration and represent the SDDS concentration at the intersection of the extrapolated trends from below the cmc and from above the cmc. The calculated values of the cmc were used for the higher salt concentrations where experimental values were not available. These cmc values are presented in Table I along with other data whose significance will be explained shortly. As Figure 1 indicates, the partial molal volumes of SDDS are essentially constant above the cmc at each salt concentration and, except for the result for 0.01 m NaC1, show a gradual increase in value with increasing salt concentration. The difference between the water and 0.01 m
0
4
8
12
16
20
24
g2vg/kg
Figure 2. Variation of Ag3, the change in weight molality of NaCI, as a function of SDDS concentration, g2, for several initial NaCl(aq) concentrations: (A) 5.930 g of NaCVkg of H,O; (B) 12.067 g of NaCVkg of H20; (C) 17.938 g of NaCl/kg of H20. For clarity, the data of curves B and C have been displaced downward by 0.1 and 0.2 g/kg, respectively. The error bars represent the difference in concentration existing between members of each pair of solutions at isopiestic equilibrium.
NaCl curves is on the borderline of being statistically significant, considering the scatter in the data, so the exceptional relationship of this data to the rest may not be real under further investigation. The numerical values of the partial molal volumes above the cmc at each salt Also listed are concentration are listed in Table I as 7,. the partial molal volumes at infinite dilution, V20, for the water and 0.01 m salt data, where sufficient measurements below the cmc were made to obtain a reasonable trend. The results of the isopiestic distillation experiments on SDDS are shown in Figure 2. Here, the change in weight molalities (defined as grams of solute per kilogram of water) of NaCl are plotted vs. weight molalities of SDDS. As is apparent from Figure 2, a linear relationship best describes the data at each salt concentration, particularly if the data point at zero SDDS concentration (the reference salt solution) is neglected. The straight line drawn through each data set in Figure 2 is the least-squares fit neglecting this point in each case. The reasons for not fitting to this point will be given later in the discussion. The slopes of these lines are given in Table I as (ag3/ag2),. The magnitude of (ag3/ag2),, increases somewhat with increasing NaCl concentration but some indication of leveling off at higher salt concentrations is apparent. This parameter, obtained directly from the isopiestic distillation results, must be evaluated in the limit of vanishing m2 and converted to (ag3/ag2),,0before it can be utilized in sedimentation equilibrium calculations. Because of the linear relationship between Ag3 and g2 in Figure 2, this limiting value is easily obtainable. As is apparent from eq 6 and 7, the relationship between these two parameters is dependent on the value of N , the aggregation number of the SDDS micelle. Therefore, a successive approxi-
Determination of Micelle Aggregation Numbers mation procedure is required in calculating the value of N from the ultracentrifuge data and at the same time utilizing the interaction parameter, (ag3/ag,),,O, to correct the calculation. In practice, this successive approximation quickly converges to a limiting value of N in six or seven steps. For comparison, the value of (ag3/ag2),O, calculated by using eq 6 and 7 and the limiting value o! N obtained with the ultracentrifuge results, is included in Table I. The results of the ultracentrifuge measurements are listed in Table I as N. The values shown are the infinite dilution values obtained by extrapolation from plots of In N vs. (mz- cmc) which were essentially linear for all NaCl backgrounds. The term (m2- cmc) represents the effective SDDS concentration. Values of N at each SDDS concentration were obtained by using eq 8. Values of S for use in the calculations were obtained from plots of In (m2* - cmc), in fringes, vs. r2. The relationship between concentration in molality and in fringes of light on the photographic plate was obtained from the synthetic boundary cell measurements, a plot of which showed that the number of fringes was proportional to the SDDS molality for all SDDS solutions, regardless of salt background. The slope of this plot represents the refractive index gradient of SDDS, (dn/dm2),and a value of 0.0321 was obtained over the SDDS concentration range of 0.0-0.1 m. The plots of In (m2*- cmc) vs. r2 were not linear in all cases, At higher ratios of m2/m3 the value of S decreased with increasing r while at low ratios the opposite trend was observed. The value of S used in the calculation of N was the concentrat_ion-weightedaverage value determined by the equation S = CniiSi/Cmiwhere Si is the slope over an increment of the In (m2*- cmc) vs. r2 plot (usually one or two fringes) and mcis the concentration, expressed in fringes, a t the midpoint of the increment. This method of determining an average value of S from a plot of In (m2* - cmc) vs. r2which shows some curvature is consistent with the definition of weight-average molecular weight for a polydisperse system.6
Discussion Table I1 provides some of the previously reported measurements of SDDS solution parameters corresponding to those reported by me. All values, where applicable, were converted to concentrations in molalities if they were not originally reported in those terms. The cmc value listed for SDDS in water is the average of 12 separate determinations. The crnc value listed for SDDS in 0.01 m NaC1, while reported for a temperature of 21 "C, is probably representative of the value at 25 "C. The temperature dependence of the cmc for SDDS solutions in water passes through a minimum near 25 "C and at 20 and 30 "C is only 1-1.5% higher than the minimum value.l6J7 The shape of this temperature dependence is very likely similar for the 0.01 m NaCl solution and the minimum probably occurs at a temperature also near 25 "C. Thus, the change in cmc that would result from changing the temperature is very likely less than the listed uncertainty. The cmc values in Table I are essentially identical with those listed in Table 11. The one apparent exception is the value for SDDS in 0.2 m NaCl where a literature value of 9.1 X lov4is reported. However, the probable uncertainty in both values makes the difference insignificant. Because the cmc is sensitive to many types of impurities, as is evident from the effect of NaCl on the cmc, some confidence in the purity of the SDDS used in this investigation is gained. However, some types of impurities apparently have much smaller effectsls so the absolute purity is still somewhat uncertain.
The Journal of Physical Chemistry, Vol. 83, No. 20, 1979 2625
TABLE 11: Literature Values for SDDS Solution Parameters of Interest cmc (18)" "solvent"
mx
lo3
H*O
8.23 t 0.11
0.01 m NaCl
5.47 t 0.25c
v2
246.4 ( 1 9 ) 247.8 (20) 248.8 ( 2 2 ) 250.2 ( 2 3 ) 237.9 (2) 246.6 ( 2 )
-
TI?o
N
234.4 ( 1 9 ) 236.8 (20) 238.0 (22) 237.2 (23) 234.7 (21)
35 ( 2)b 62 ( 2 6 )
50 ( 2)d
67 ( 26)e 77 (2)b 76 ( 3 ) 86 ( 26)e 0.2 m NaCl 0.91 98 (2)d 101 (26) 116 (26)e 0.3 m NaCl 0.4 m NaCl 249.7 ( 3 ) 119 ( 3 ) ' 128 ( 26)e a The numbers in parentheses identify the literature reference for the value. These values were obtained by plotting the data as In N vs. ( m- cmc) and fitting to a straight line. This value represents data obtained at 21 This value was estimated from a "C instead of 25 "C. single measurement but with a trend similar to that I obtained. e These values were estimated from a graph because data at corresponding NaCl concentrations were not available.
0.1 m NaCl
1.50
246.6 (2) 248.9 ( 3 )
*
The partial molal volumes in Table I are also comparable to most of the corresponding values listed in Table 11. Considering the precision of the density measurements used in deriving the values for 7, listed in Table I, the probable uncertainty in V2 above the cmc is f0.2. Below the cmc the uncertainty increases rapidly so the uncertainty in V? in Table I is approximately f l . Several of the values listed in Table I1 were derived from densities measured by pycnometers2i3having a precision no better than 10 ppm and probably somewhat worse. The resulting U2 values are likely to have uncertainties an order of magnitude greater than those I obtained. Some of the values for V2 and V: in water were obtained by a Cartesian diver methodlgand others by a dilatometer method.2O The value of 234.7 for V? listed in Table I1 was obtained with a magnetic float densimeter having a reported precision of 1 ppm.21 However, these data, reported as 4", the apparent molal volume, vs. c1I2, is not representative of other studies in the region near and above the cmc. Thus, no attempt was made to list a value for V2above the cmc from this report. Their data below the cmc are representative. The most recent values listed in Table I1 were obtained with a vibrating tube densimeter having a preI find it interesting that the cision of 322 or 5 respective values reported for V2 and V? are also the highest listed for SDDS in water. This might imply some type of bias with this mode of measurement. The reported densities of Musbally, Perron, and DesnoyersZ2for SDDS in water were 5-10% lower than those I obtained at corresponding molalities. No explanation for this discrepancy is available. They reported no density data for SDDS below the cmc so the source of their reported value for V20is unknown. The partial specific volume u2 appears in the ultracentrifuge calculations in the buoyancy term, (1 - bzpO). Because p", the background solvent density, is very near 1 and Uz for SDDS is near 0.86, the buoyancy of SDDS is approximately 0.14. Therefore, a high degree of precision in the value of bz is required to obtain reasonable precision in the buoyancy term. Considering the uncertainty in b2 obtained from my results, and substituting appropriate values for po, the uncertainty in the buoyancy is f0.5%,
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The Journal of Physical Chemistty, Val. 83, No. 20, 1979
Daryl A. Doughty
Thus, the buoyancy is unlikely to be the limiting factor in determining the precision in N . The basis of the isopiestic method is the comparison of the thermodynamic properties of nonvolatile solutes at constant solvent activity. The results of the isopiestic distillation experiments on SDDS in NaCl(aq) imply a simple relationship exists between SDDS and NaCl in mixed solutions at constant water activity. However, the micellar properties of SDDS and the existence of the monomer-micelle equilibrium suggest a profound alteration in isopiestic characteristics should occur for SDDS at the cmc. These data give only a hint of this in the displacement of the points at zero SDDS concentration from the regular trends observed at higher concentrations. This displacement is generally more than can be accounted for on the basis of experimental uncertainty. A comparison of my results with those obtained for other mixed elect trolyte systems should be helpful. 0 0 4 0 06 0 008 4 0 002 004 There have been many studies of mixed electrolyte m2 system^.^^^^^ However, if the systems considered are reFlgure 3. Variation of Am,, the change in molality of NaCI, as a function stricted to binary mixtures of 1:l electrolytes having an of SDS concentration, m2, for several initial NaCl(aq) concentratlons: ion in common and studied by isopiestic distillation (0)0.1186 m NaCI; ( 0 )0.3095 m NaCI. For clarity, the data for 0.3095 techniques in which solutions of different compositions are m NaCl have been displaced downward by 0.002 m . Curves A and compared at constant water activities, the number is B represent the variation of Am,,, as a function of m K C for l the limited. No investigations in which one of the components respective, initial NaCl(aq) concentrations. The error bars represent the difference in m, existing between members of each pair of solutions was a micellar material have been found. Robinson reat isopiestic equilibrium. ported data on the KCl/NaCl system for various reference NaCl concentrations, the lowest of which was 0.5 m cmc and, considering the low concentrations below the cmc NaCl(aq).25While his data are reported as R, the isopiestic and the strong electrolyte characteristics of SDS, this is for the isopiestic mixtures) vs. ratio (where R = rnref/mbtal a reasonable assumption below the cmc. The interaction x , mole fraction of KC1, in the isopiestic mixtures, they can of the trends from above the cmc and from below the cmc be converted to the form equivalent to Figure 2. At the X m SDS in 0.1 m NaCl and 1.36 X occur at 2.19 lower concentrations used in my experiments, Robinson’s m SDS in 0.3 m NaC1. The value in 0.1 m NaCl (actually data suggest an essentially linear relationship exists be0.0975 m NaCl at the intersection) compares very favorably tween and g K C l for mref5 0.3 m and x < 0.5. The with the published value for the cmc of SDS in 0.1 m NaCl study of Scatchard, Hamer, and Wood compared isopiestic (am3/am2),,above listed above. The values obtained for solutions of single electrolytes at various concentrati~ns.~~ the cmc are -0.189 and -0.221, respectively, for soiutions Using their data for NaCl and KC1 at concentrations of in 0.1 and 0.3 m NaC1. Below the crnc the respective values 0.1-0.3 m NaCl and assuming the trends obtained by are -0.957 and -0.962. Robinson for the mixed solutions can be extrapolated to Comparing these values with those for SDDS, one finds lower NaCl concentrations, one can obtain a reasonable that above the cmc they are essentially identical, neglecting estimate of (gNaCl/gKC1)M. The data are more meaningfully any temperature dependence. Below the cmc the values compared if converted from a weight basis to molalities, for SDS are comparable to those for KC1 in NaCl(aq). The is -0.190 i.e., (am3/amz)pl.For SDDS in NaCl (am3/am2)pl cmc data for SDS suggest a slightly lower cmc at 25 “C for 0.1 m NaCl solutions and -0.219 for 0.3 m NaCl socompared to that at 30 oC.18 This would tend to make the lutions. The respective values for KCl and NaCl are -0.986 results below the cmc even more comparable to the and -0.977. KCl/NaCl results. The resulta indicate that above the cmc For a mixture of two “ideal” 1:l electrolytes the value micelle formation causes a marked departure from ideality would be -1.000. NaCl and KC1 approach of (am3/amJp1 while below the cmc the surfactant behaves essentially as this ideal behavior, particularly at low concentrations. any other strong 1:l electrolyte. The isopiestic results for SDDS is far from ideal even to very low concentrations less SDS support the conclusion that little if any premicellar than 0.01 m because of the association of SDDS into association exists, at least for this surfactant. Extending micelles. The low cmc’s of SDDS in the various NaCl these conclusions to SDDS/NaCl solutions would appear backgrounds prevents any isopiestic studies below the cmc. to be reasonable considering the close agreement of the isopiestic results above the crnc for the two surfactants. As preparation for future ultracentrifuge studies of other As was mentioned in the Introduction, the parameter surfactants, isopiestic distillation experiments have been (ag3/agz)plmust be evaluated in the limit of vanishing m2 preformed with the surfactant sodium decyl sulfonate for use in the ultracentrifuge calculations. The radical (SDS) in NaCl(aq). These experiments were preformed change in the value of this parameter at the cmc would at 30 “C because of solubility problems at 25 “C. The appear to complicate this determination. However, for published cmc’s for SDS are 4.41 X m in water and m in 0.1 m NaCl(aq),ls considerably higher than 2.22 X purposes of sedimentation equilibrium calculations, this those for SDDS. Isopiestic data below the cmc was obdisappearance of the micelles at or below the cmc reptainable. The results are shown in Figure 3 for SDS in 0.1 resents infinite dilution of the centrifugeable species. Thus, an extrapolation of the parameter to the cmc efand 0.3 m NaC1, only this time the data are plotted as change in molality of NaCl vs. molality of SDS at isopiestic fectively accomplishes the desired result. The assumption that the monomeric surfactant, present at a concentration equilibrium. Assuming that a linear relationship exists both above and below the cmc, I fitted the curves shown equal to the cmc, behaves as part of the supporting in Figure 3 to the data. These data suggest this above the electrolyte would appear to be supported by the isopiestic
Determination of Micelle Aggregation Numbers
distillation results on SDS. No ultracentrifuge experiments on SDDS solutions a t concentrations below the cmc were attempted to determine the extent of any redistribution that may have occurred in the monomeric surfactant. In the worst case studied in my investigations, SDDS in 0.1 m NaC1, the monomeric surfactant concentration represents only 1.5% of the supporting electrolyte under this assumption. Appreciable redistribution of the monomeric surfactant would have to occur to have a noticeable impact on the results. The values of N listed in Table I represent the weight-average values for the SDDS micelles.6 If the micelles were monodisperse the plot of In (m2*- cmc) vs. r2 would be linear under ideal conditions. Nonideality effects arise at finite concentrations of micelles in the region where the ratio, m2/m3,is not much less than 1. The value of S decreases at higher r (and resulting higher micelle concentrations). These effects were noted in my studies, particularly at the lower NaCl concentrations. Polydispersity would appear as a reverse effect in the plots with S increasing at higher r values. Experiments at intermediate SDDS concentrations gave plots having essentially constant S. However, the lowest concentration experiments, particularly in 0.2 and 0.3 m NaC1, showed the effects of polydispersity. The linear plots in intermediate cases probably represent a masking of polydispersity by the effects of nonideality. Other studies have shown that polydispersity generally occurs in these micellar ~ystems.~ No attempt was made to quantitatively determine the degree of polydispersity in my results. Linearity of these plots can be influenced by the analytical treatment of the data as well as by the effects of nonideality and polydispersity. Uncertainties in the concentration shifts at the reference point, the relationship between fringes and molality from synthetic boundary cell data, and the cmc values would all influence the values obtained for the absolute concentration distribution as a function of radius. If the values of (m2* - cmc) were decreased by errors in these quantities, they would exhibit apparent nonideality effects unless masked. Conversely, errors increasing the values of (m2*- cmc) would imitate the effects of polydispersity. Of the three sources of uncertainty mentioned, the first two are most likely the dominant factors. Consideringthe precision of my density measurements and the close agreement of the resulting cmc values with literature values, an uncertainty in the cmc value, expressed as fringes, of as much of 0.1 fringe would be unlikely. Table I1 lists some of the published values for the aggregation numbers of SDDS in NaCl(aq) at corresponding NaCl concentrations. My results are generally close to the reported values of Mysels and Princen which were obtained by light scattering,26 though their results are consistently lower. Considering the uncertainties of our individual results this difference may not be significant. The values of N listed in Table I1 that were obtained by ultracentrifugation, however, are significantly An examination of eq 8 shows that neglecting the charge correction would give a lower value for N. This implies that attempts to correct results for charge in previous ultracentrifuge studies have been inadequate if attempted a t all. The values obtained for the interaction parameter, (~%!3/%!2),,, can be used to calculate a value for the electric charge on the micelle if the assumption is made that the interaction parameter arises solely from the charged nature of the micelle and is not affected by preferential solvation or salt binding. Using my values for this parameter listed
The Journal of Physical Chemistry, Vol. 83, No. 20, 1979 2627
in Table I and converting them to (am3/dm2),,, one obtains the following values for i using eq 9: 0.356 in 0.1 m NaC1, 0.400 in 0.2 m NaC1, and 0.418 in 0.3 m NaC1. These results are much higher than previous values obtained by light scatteringz6 or ultracentrifugation3 but compare favorably with results obtained with other techniques.’ Dielectric measurements on SDDS solutions in water have shown that the micelles are surrounded by a layer of “bound” water.z7 This preferential solvation cannot be distinguished from the effects of salt rejection ascribable to the Donnan equilibrium which arises from the charged nature of the m i ~ e l l e .If~a portion of the experimentally obtained interaction parameter does arise from preferential solvation, the actual values of i would be smaller than my results indicate. Assuming my calculated values for i are valid, the results indicate an increase in i with increasing aggregation number. My partial molal volume results show only a slight increase with increasing aggregation number which implies that the volume of the micelle is essentially proportional to the value of N.The ratio of micelle surface area-to-volume decreases as the micelle volume increases. Thus, the surface area per surfactant head group decreases with increasing N. The resulting increased crowding of the bound counterions which tend to neutralize the micelle charge causes an additional few to be repulsed, increasing the fractional charge on the micelle. If a spherical micelle is assumed in dilute solutions, the ratios of surface area-to-volume can be calculated for micelles having volumes proportional to my aggregation numbers. The ratio of the bound counterion (1- i) to the micelle surface area (A)/volume (V) ratio is found to be nearly constant, having an uncertainty of 1.4%. This may be compared to the variation in the A/ V ratio of 8.7% for the spherical micelles. Thus, the model can account for much of the apparent variation in i with aggregation number. Even if part of the interaction parameter is contributed by preferential solvation, a reasonable assumption is that this would be a constant fraction of the total in the dilute solutions involved here. Therefore, the portion arising from charge effects would still show the same percentage variability with N and the above model would still be valid. Acknowledgment. I acknowledge the many helpful suggestions of R. L. Berg and C. W. Dwiggins, Jr., of the Center in the investigations and preparation of the manuscript. I also acknowledge the contribution of E. A. Pavelka for machining parts of the magnetic float densimeter and the sample block and vials for the isopiestic distillation apparatus.
References and Notes
(1 1) (12) (13) (14) (15) (16)
C. W. Dwiggins, Jr., and R. J. M e n , J. phys. Chem., 66, 574 (1962). K. Kakiuchi, K. Hattori, and T. Isemura, Bull. Chem. SOC. Jpn., 36, 1250 (1963). E. W. Anacker, R. M. Rush, and J. S. Johnson, J . Phys. Chem., 68, 81 (1964). H. K. Schachman, “Ultracentrifugationin Biochemistry”, Academlc Press, New York, 1959. H. FuJita,“Foundations of UltracentrifugalAnalysis”, Wiley, New Yo&, 1975. H. Eisenberg, “Biological Macromolecules and Polyelectrolytes in Solution”, Clarendon Press, Oxford, 1976. L. R. Fisher and D. G. Oakenfull, Chem. Soc. Rev., 6 , 25 (1977). H. Coll, J . Phys. Chem., 74, 520 (1970). G. Cohen and H. Elsenberg, Biopolymers, 6, 1077 (1968). R. A. Robinsonand R. H. Stokes, “Electrolyte Solutions”. Butterworths, London, 1955. E. P. K. Hade and C. Tanford, J. Am. Chem. Soc.. 89. 5034 (1967). F. J. Millero, Jr., Rev. Sci. Instrum., 38, 1441 (1967). G. Scatchard, W. J. Hamer, and S. E. Wood, J . Am. Chem. Soc., 60, 3061 (1938). G. J. Howlett and L. W. Nichol, J . Phys. Chem., 7 6 , 2740 (1972). J. F. Yan, J . Colloid Interface Sci., 22, 303 (1966). E. D. Goddard and 0. C. Benson, Can. J . Chem., 35, 986 (1957). .
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2628
The Journal of Physical Chemisfry, Vol. 83, No. 20, 1979
J. R. Morton, K. F. Preston, and
(17) B. D. Flockhart, J . Colloid Sci., 16,484 (1961). 118) P. Murkerlee and K. J. Mvsels. “Critical Micelle Concentrations of .
S.J.
Strach
(22) G. M. Musbally, G. Perron, and J. E. Desnoyers, J. ColioM Interface Sci.. 40. 494 (19741. (23) T. S: Brun, H. Hdland, and E. Vikingstad, J. Colloid Interface Sci., 63, 89 (1978). (24) K. S. Pltzer and J. J. Kim. J. Am. Chem. Soc.. 96.5701 119741. . , (25j R. A. Robinson, J . fhys. Chem., 65, 662 (1961). (26) K. J. Mysels and L. H. Prlncen, J. Phys. Chem., 63,1696 (1959). (27) R. B. Beard, T. F. McMaster, and S. Takashlma, J. ColloM Interface Sci., 48,92 (1974).
I
Aqueous -Surfactant Syitems”, National Bureau of Standards, Washington, D.C., 1971. (19) J. M. Corkill, J. F. Goodman, and T. Walker, Trans. Faraday Soc.,
63,768 (1967). (20) K. Shlnoda and T. Soda, J . fhys. Chem., 67, 2072 (1963). (21) F. Franks, M. J. Quickenden, J. R. Ravenhill, and H. T. Smith, J. fhys, Chem., 72, 2668 (1968).
Temperature-Dependent Hyperfine Interactions of Pyramidal Radicals Trapped in KPF,+ J. R. Morton,” K. F. Preston, and S. J. Strach’ Division of Chemistty, National Research Council of Canada, Ottawa, Ontario, Canada K I A OR9 (Received May 30, 1979) Publication costs assisted by the National Research Council of Canada
A study has been made of the temperature dependence of the central atom hyperfine interaction of various radicals trapped in KPF6. It is concluded that certain tetratomic, pyramidal species are inverting at 300 K and above, and that the barrier height is approximately 2 kcal/mol. Introduction I t has been known for many years that paramagnetic defects can be formed and trapped in NH4PF6and KPF6 a.t room temperature by ionizing radiation.’ These species include2 PF, (first thought1 to be PF4),and various other species (FP02-, derived from low concentrations of hydrolysis products. An interesting property of NH4PF6 and KPF6 is that, at room temperature, they are “rotator solids”, and electron paramagnetic resonance spectra of radicals trapped therein are isotropic. Individual spectral lines are sharp (AH 2 G), and exact measurement and analysis are possible. Of course, the hexafluorophosphates are not unique in this regard: SF6 is the rotator solid pur excellence over the temperature range 93-140 KS4 The unique property of KPF6, however, is that it is a rotator solid at room temperature, and that isotropic EPR spectra can be detected in it over the temperature range 250-475 K. The ammonium salt has a similar, although lower, range over which isotropic EPR spectra are exhibited: 190-390 K. These matrices therefore offer the possibility of measuring the temperature dependence of the EPR spectral parameters (a) very accurately and (b) over a wide, and relatively high-temperature range. Because of recent interest in the temperature dependence of hyperfine interactions, we report herewith data for the following species which we have observed in y-irradiated KPF6: PFC, AsFc, FPO,, HP02-, P032-,and AsOs2-. The latter two species have previously been studied in a calcite m a t r i ~ the ,~ (anisotropic) hyperfine interactions being studied over the temperature range 4-300 K.
TABLE I : EPR Parameters of Radicals at 295 K
Trapped in KPF,
hfi, MHz ~
-
Experimental Section Potassium hexafluorophosphatewas obtained from PCR Inc., Gainesville, Fla. After irradiation at 300 K in a 6oCo y cell, the EPR spectra not only of PFs-but also (weakly) of AsF6-, FP02-,and PO:- were obtained. This indicates (a) that the KPF6 contained traces of KAsF6,and (b) that some hydrolysis had taken place. The spectra of HP02and As02- were obtained by recrystallizing the KPF6 from an aqueous solution containing 2% KHzPOz (Pfaltz and NRCC No. 17727. t NRCC Research Associate 1977-1979. 0022-3654/79/2083-2628$01 .OO/O
central atom
ligand
PF,-
2.00174(2)‘
3800.8(2)
554.1(5)b
hF,-
2.00060(5)
5063.(5)
FPO,-
2.00129(2) 2.00319(1) 2.00163( 2) 2.00328(2)
1890.2(2) 1178.1(2) 1684.3(2) 1558.5(1)
505.4(5)b 15.(1) 491.6(2) 262.1(2)
HP0,PO,’-
N
~
g factor
radical
941)
As0,’Numbers in parentheses are the estimated errors in the last digit. Four equivalent 19Fnuclei have this
’
hyperfine interaction.
Bauer). This process apparently brought about hydrolysis of the AsF6- impurity. The EPR spectra of the irradiated samples were detected and measured with a Varian E-12 spectrometer equipped with a variable temperature accessory. The microwave frequency was measured with a Systron-Donner Model 6054 frequency counter, and the magnetic field was measured with a Cyclotron Corp. Model 5300 NMR gaussmeter. The spectra were analyzed by standard methods, the spectral parameters being obtained by iterative computer diagonalization of the spin matrix. In many cases the precision was sufficient to determine the relative signs3 of the central atom and ligand hyperfine interactions. The samples were contained in 4-mm 0.d. thin-walled Suprasil tubes. These tubes were equipped with a reentrant capillary tube which enabled a fine copperconstantan thermocouple to be inserted and imbedded in the center of the sample. Overall temperature measurement and control were thought to be accurate to better than 1 K. Results and Discussion In Table I we collect the spectral parameters of the various radicals we have detected in y-irradiated KPF6 at 295 K. Most of these species have been reported elsewhere, although not necessarily in a KPF6 matrix. The identity of PF5-(originally misidentified as PF4)is now well es0 1979 American Chemical Society
