56
A . P. BRADS
SURFACE TENSIONS OF AQUEOUS SOLUTIONS O F TWO FOAM-FR.4CTIOnTATED DETERGEKTS' 1.P. BRADY Srnrrfor d Research I n s t i t u t e , S t a n f o r d , CnliforiAi(b Receiued AliqJi(st 19, 1948 INTRODUCTION
Two disturbing aspects of the curves uf surface tension uerszcs concentration for aqueous solutions of soaps, synthetic detergents, and wetting agents have been the subject of a great deal of theoretical discnssion: ( I ) The surface tension drops slowly over long periods of time (2, 5, 8), especially in more dilute solutions, a'nd ( 2 ) the ciir\-e of surface tension uerslLs concentration passes through a minimum (32) atj a concentration more or less characteristic for each material. The first of these phenomena may be due to complex and slow orientation effects not taken into account in diffusion theory, and so possibly is easily dismissed; the second, on the other hand, leads to an apparent contradiction between the observed adsorption a t the surface and thermodynamic theory. The theory requires that for a solution containing only one solute the adsorption must he zero at t)heniinimum in the curve of surface tension versiis concentration, and negative in t'he rising portion. Foaming-off experiments (13) and the microtome ( I 4) have amply demonstrat.ec1,however, that the adsorption of the surfaceactive agents is al\mys positive. This aont'ratliction can be resolved by assuming that the supposedly single soluk consists in realit'y of two components, because of either an impurity or hydrolysis. Ordinary soup in bulk solution is correctly regarded as a single component, even thorigh it hydrolyzes, but it cannot be so regarded if the surface is considered, because it has been sho\vn that ut certain concentrations acid soap is strungly concentrated in t8hesurface, leaving free allcali in the bulk. Similar minima in the surface tension curves are exhibited, however, by non-hydrolyzable s sulfonic acid, sodium a,lkyl sulfates, or alkyl quaternary detergent's such i ~ lauryl ammonium salts. On the other hand, Miles and Shedlovslcy (16) have shown for certain sodium alkyl sulfates t'hat purification of the detergent by repeated ext twtion with ether t'ends t'o eliminate t,his minimum. The same workers, ant1 -7. \-. Kobinson in the Stanford University Laboratory, found that deliberate atidit ion oi lauryl alcohol to sume of these detergents greatly intensified the minimum. These facts strongly suggest that the minimum may arise here from the presence of a relatiTrely small amount of surface-active impurity. The presence of an impurity is not, p e r se, enough to nccoiintmf u r n minimimi in t,he sui-facetension ciii8ii.e;however, such a minimum could arise from the well1 Pi,esent,ed a t the Twent,y-secoiid Satioiial Colloid Syniposiuin, wliicli was held under the auspices of the Division of Colloid Cheinistry of the American Chemical Society a t Caiiibridgc, Jlassncltusetts, Julie 23-25, 1948. 2 This study was conducted under a contract between the Office of S a v u l Kvsearcli aud Stanford Research Iiistitute, under the supervision of Professor J . W. McBaiii.
kiio\vn "aolitbilizing" ac,t'ion of associative colloids. T h w the impurit,y could he free aiid ahle t'o Io\ver the surfme tension in dilute solution, I)ut after the soap or tlet,ergent h:is increwetl sufficiently iii concentration to contain colloid, islie impririt>r m:iy lie i,emoi.etl f m n the scene of :ict8ioiiby iiimiporntioii Ivithiti f he colloidal micelles.:' If this concept is correct, t.he impiirity shoiild lie most stronglj, adsorbed at the intwface at ;I concent'ration just, I)elo\\, the critical concenti-at'ion for micelles of t,he detergent solution, that' is, at 01' helo\\- t'lie concentration for which the surface tension is a t a ininimiun. It shoiild then be removable by frothing off it solution nt or near this concentration. The n.orli of >riles and Shedlovsky has already indicated this t o be t,he case, since these authors foiind a ma,rked increase in siirface tension \\.hen a solution at t'he minimum in surface tension \\'as frothed, whereas the same treatment, a t R, higher concent)ration, follo\\-etl by dilution t o a concent,rntion corresponding to the minimum, sho\\-etl little or no cliangein surface tension. The present paper describes a fract ionntion of t'\\-o detergent sollitions using this pi*inciple,and the results of surface tension nieasrirements ovei' a \vide concentration range of the purified materials. This principle does not, lio\\.ei.er, avvouiit tor t \ v o minima ivportec! (9)101, l.lie s i i r f ~ v etensior~-c.oncent.l'atioii ('iirves in wrtaiii synthetic (letergents, [inless possibly more than one kind 01 impurity is p n w n t , each iwno\.ctl lrom t'he sphere of act,ion b>*(1iffPrent kinds of inicdles ocmi-ring in the tliltt'ercnt' ronccritrntioiis of the same tleteigent. EXPERIRlRS'I'i\L
The t'\vo detergents used for these, experiments were lauiyl srilfonic acid (pi-epared by 31. Synerholm) and piiritietl sodium dotlecgl sulfat'e (supplied by Procter & Gamble). These m:tt,erials \\.ere chosen because of other data extant on their soliit,ions, aiid because their crit'ical concent'ration for micelles (and tlie minimum in t h e surface tension ctirve) occurs n t nil esperimentdly oonirenient concentration. 'l'he foam fract,ionation apparat,iis cunsist,ed of a Pyres tiihe, 1 meter long and about, one-t'liird filled with the solut,ion t o be fractionated. The foam \\.as formed by passing in humidified nitrogen through ,z sintered-glass disk. The appnrat'us was run either continuously, by bubbling at, a, rnte suitsable for the foam to have drained t o the desired extent at the top, or intermit tenily, in \\,liich case the t'ube \\-as filled \vith foam, chaiiiage allo\ved t'o proceed, and then the old foam rephcetl by ne\\- (the line of cleniarcat,ion remains qiiitc sharp). The progress of the fractioiiat'ion \vas follo\~edby measuring boirli the surface tension and the concentration-the latter by men,ns of a Zeiss intei,ferometei-. .\-otc ciddscl i j i pcoo/': Heceiitl>, Reicheiibwy (Trans. Paruda). &IC. 43, 467 (1947)) l ~ a s published a t Ilcoi,etical consideration of t lie ininitnn obsarveti i n surface tension curves of uolloitial electrolytes. The main thesis of t'he paper, t81iat inininin ilia!. arise from multivalent ions i l l the distilled mater, obviousl). does not upply here. He did, liowevei,, also suggest that, surface-active impurities might be :L factor, i n harinony witli ilic views exIressed i n the present paper.
5e
A . P. B R h D T
When the foaming \vas considered to have proceeded t80the desired extent, the residual solut'ion was fi-ozen in solid carbon dioxide and the ice sublimed off nntler vacuum, t'o give the dry solut8e. This procedure obviated Iioth foaming difficulties and the danger of part'ial decomposition nt highei, temperutmuises. Surface tensions were measured at room temperature (29'C. f 0.5") \vith a Cenco du Nouy tensiometer equipped \vith R I-cm. ring. The solutions were cont'ainetl in a silica dish. Both t,he dish and the ring were flamed before each measurement. I n order t o minimize evaporation and nccidental contamination, the silica dish was placed in a larger Petri dish rontaining \vet blotting paper, and covered by a n inverted crystallizing dish of int8ermediatesize wit'h a small hole bored in t,he l)ott,om to admit the \\.ire attached t o the ring. The tensiometer readings were corrected for meniscns shape from the table of Haykins and Jordan (6). The surface tension of t,he conductivity of lvat'er iisetl \\.as frequently checked and aln-ays gave 71.6 i. 0.1 dynes/cm. TABLE 1 8uij"arc l e n s i o n s o,f soltilions of joarti-j'raclionatcd sodium laui,y1 sulfate and lauryl s u ( f o n i c acid
.
~
-
-
~
LAURYL SULFONIC ACID
SODIUM LAURYL SCLFATE
_ _ -~ S u r f a c e tension
.___
Concentration
Surface tension
mulcs/liler
dyiicslcia,
0.00161 0,00315 0.00570 0.00687 0.00867 0.0105 0.0138 0.0283
50.7 50.0 43.4 40.5 38.7 38.7 38.6 38. 1
-1
Concentration
~.
niolcs/li/o-
dyncs/cm, ~
I
0.00144 0.00272 0.00560 0.00887 0.0119 0.0265
I
69.6 50.8 11.5 37.7 37.8 37.6
RESCLTS
Table 1 and figure 1 give the results obtained \\.it11 sodium lauryl su1fat.e purified by the intermittent, foaming method. The data of Miles and Shedlovsky on an ether-extracted sample of t'he same compound are also included in the figure. The solution placed in the foaming apparatus \vas 0.00707 N , t'he collapsed foamed-off portion 0.0118 N , and the residue 0.0064 X, corresponding t.o 12 per cent of t'he solut'e foamed off. It is at once e\+lent' from the curves that, fractionation has taken place; t'he minimum is eliminated in the surface tension of the residual solute, but it is more pronounced than before in the foamed-off portion. The predent data on sulfate purified hy foam'ng agree within experimental error \\-it11those of Miles and Shedlovsky on ether-estractetl material. I t is also \\.ort'hy of not'e that t,he original material had the strong odor characteristic of commercial lauryl alcohol, whereas the purified material was quite odorless. The data in figure 1 are for surfaces aged 15 min. It \vould have heen better
50
SLiRF.\('E TENSIOSS O F FO.\~I-FI~.\C'TIOPi.I'rED DETERGENTS
to inirestigate surfaces of giwter age, h i t ) in tlie present apparatus the chances of contamination \vere too great. Fortrinately, the effect of t,iine \vith the purified sample is much less tlinn ivit'li t,he original matjerial, indicating that the long aging periods foiiiicl necessary in otliei. \vorli 011 surface-active agents ( 2 ) inqr arise at' least, in part f i r m penetration of t'he impurity to t,he surface. Thus, in the original sample at, 0.0028 *Ir, the sriiface t'ension at' 3 min. age \\.as 28.6 dynes ,'cm.; after 15 min., 2 6 . i dynes; n paitially purified sample 0.0025 *V gave 3 . 1 dynes at 3 min. :~iitl 52.0 dynes at 13 min., while the final purified material tit, 0.0034 AT g a \ ~50.9 tllmes at 3 min. niicl 30.9 dynes a t 15 niin. This last result, is not to he tnkeii ns slio\ving nl)solutel~* 110 effect) of age, Ilut rather that the aging effect, hat1 h e n retliicetl t o less than 0.2 clyne,'cm. a t that concentration. The i-ie\\.that the aging effect is largely due to the impurity present
d
I\
0 ORIGINAL MATERIAL
+
COLLAPSED FOAM RESIDUE FROM FOAMING X MILES 8 SHEDLOVSKY, ETHER- EXTRACTED MATERIAL
0
W 0 1
"I v)
28r,
0.0I
0.02
a 13
CONCENTRATION, MOLESILITER CONCENTRATION, MOLES/LITER FIG.1. Surface tension r m u l t s on foani-fi,actiorint~d sodiuni lauryl sulfate
is supported hy the obser\.at'ion (13) that if surface tensions are talteii 011 relatiirely fresh surfaces of t'liese solutions (aft'er a few seconds) the curves so oht'ained show no minimum and gii-e reasonable results from direct applicat,ioii of t,he Gihhs equation in its t'n.o-coinponent form. The result's oht8ainecl \\.ith another surface-active agent, lauryl sulfonic acid, are summarized in table 1 niicl figwe 2. The oiiginal soliit'ion \vas 0.0072 'V, the collapsed foam 0.0091 .V, and the iaesidual solut'ion 0.0056 N , corresponding t o 46 per cent of the solute foamed off. Here again it, is clea,r that a fract,ionation has t'alieii place, i v i t l i the originnl minimum in tlie siiiface tension curve enhanced in the foam fract'ion and flattened in the residue. The separntjion in this case, lio\\w.er, does not appenr to he so n e d y complete a s for the sodium Intiryl sulfate, for tnx i'emoiis: The h s t is that the surface tension is independent, of c*oncentixtion \\.ithin espeiimentnl error nbo1.e 0.01 .IT, \vhicli is just nhout, :LS
60
A . P. B R h D l
bad t'lieoretically as a iiiiiiiiiiiiiii-irideecl, a minimum may well exist at greater surface ages. The second is that the time effect was not nearly so diminished, although again the purified material showed less aging effect,: a drop from 52.3 to 50.5 t1ynes~'cm.nt 0.0027 AV,\rherens the original at the same concentrat'ion moved from 40.8 t o 43.4 dynes,'cm. The fiwtionation in figrtrc 2 \ r x s done by the continrrous bubbling method, \rliich is more rapid than t'lie intermittent, method hut is difficult to mal-Le as efficient regarding drainage. ;hi utt1eoipt tJouse the intermittent method with 1au1yl sitltonic acid, so soucessfril \\.itli sodium 1nur.yl sulfate, resulted in little impro\.enient, in t'he corresponding time. Thus 0.007 .V sulfonic scicl purified by tJhe intermittent met'liotl gnye il surt'ace tension of 37.0 dynes 'cin., halfway lietween t'he \ d u e s for tlie original and the purified samples in figrire 2. This
leads one to suspect tli:it, :i certain amountj of impurity is formed by reaction ill the surface of the foam. The end product wonld then be a result of a steady st,ate, and would be the purer the more rapidly the extended surface is renioved. The data pi,esent8etlin figures 1 and 2 lend considerable suppoit to the premise that. t'he stirface behairior of all practica,l solutions of detergents, and of all solutions ul^ soaps and other hydrolyzing mnt>erials,is complicated Iiy the presence a t the surface of t\\.u 01' inore surface-active species. Furt'her, these species often st.i*onglyinteract, as is evidenced both by the maximum surface tension lowering in the system lauryl alcohol-sodilun lauryl sulfate-water being much greater t'han that obtainable with either substance alone in vat'ei', and by the strong influence of lauryl nkohol on the foam stability of the same system (If.). A large decrease in foam stability as t,he impurity is removed was qnalitntii-ely yuit'e evident in t,he foani frnctionu.tio11experiments reportNedhere.
SURFACE TEXSIONS O F F O S ; L I - F R , ~ C T I O N ~ ~ TDETERGESTS ~D
61
APPLICdTION O F T H E GIBBS EQUATION
Xt constant pressure and temperature the Gibbs adsorption theorem takes the form dr
where
5
=
- rodpo -
is the surface tension,
rl dpl - r2dp2 - . .
ro the
(1)
surface adsorption of the $olvent, and
T1,rz,etc. the adsorption of the solutes present. I t i5 the usual convention to
set8ro= 0 as a standard of reference. then rediices to
I n the c a v of n single solute, equation 1
where CLIdenotes tlie act8ivityof the solute. For an ideal solution of imi-univalent electrolyt,e, equation 2 simplifies t o equation 3. du - - r l d 1 1 1 ~ -. (3)
2RT
1
I n this equation cI is the molal concentration of the solute. The Gibbs-Duheni relation requires the activity to increase with concentration. Hence the previously mentioned requirement that the adsorption of a single solute be zero a t a minimum in the curve of surface tension versus concentration, and negative in a rising portion, may be deduced from equation 2. Below the critical concentration for micelles, it is known (9) that colloidal electrolytes behave to a sufficient approximation as ordinary mi-univalent electrolytes, so that one may use the Debye-Hiiclrel expression for the activity coefficient, In yl = -1.1~5 \/’cl \I
here y l is tlie niem activity coefficient. Equation 2 then becomes dg
= =
-2F1 d In ylcl -2r1(1 - 0.G
t/’Ll)
d In cl
Equation 4 inay be used to evaluate the adsorption of purified material below the critical concentration. +Abovethe critical concentration, the activity differs \\ idely from that predicted by ionic atmosphere effects alone. However, certain iegularities noted elsewhere may be used. It has been found (10) that the osmotic coefficient vs. concentration curves of colloidal electrolytes can be divided into classes according to the value of the derivative dg/d In c1 = f(g), where g is the osmotic coefficient and f(g) a function of Q alone. All the straight-chain +alti for \\ hich tlnta itre LLvnilnble huve the same wlue of f’(g); all the branchedchLLiii salts, Including the Aerosols and the kinked-chain potassium oleate, another. Firrther, it \\vis forind that to R good approximation (3, 4)
where /3 is a constlantjchaiwteristic of tfhe class (i.e., f(g) = @ - g ) .
62
.4. P. B R h D Y
The Gibbs-Duhem relation in dilute solution takes the form
Comparing equations 5 and 0 it is seen that
h
E
X
AEROSOL Ot
+
LAURYL SULFONIC ACID, /cO.OS
In m/m,.o.s FIG.3. Activity coefficient us. concentration for some colloidal electrolytes
According to equation 7, a, log-log plot of the activity us. concentration or, nlternatively of the activity coefficient LIS. concentratlion, should give a straight. line. Figure 3 is a test of this relat'ion, from which it can be seen to hold quite s a h factorily. The activities of the Aerosols \yere t8akenfrom 3IcBain and Boldiian (10) and those for lauryl sulfonic acid from ,Johnston ( 7 ) . The concentration units, 7 r ~ / r n , , = ~ .are ~ , actual molalities divided b y the molality, at \vhich the osmotic coefficient is 0.5. This is used here simply for convenience in plott,ing and does not influence t,he slope. The value of p is about, O.0G for lauiyl sulfonic acid, and presumahly ahout' the same for sodium lauryl sulfate. T h e Gihlis eqtiatioii above the critical concent,ration t,herefore may be expressed in the form :
Figiiw -1 i- :i wnilogai.ithmic* plot of surface tension aguiubt concentration. The ciir1.e for purified sodium l ~ i i y sulfate l shows an abrupt change in slope a t 0.0073 N , i. e., the region of the minimum for the unfractionatetl material shown in figure 2. This change in slope is to be expected a t the critical concentration
SODIUM LAURYL SULFATE 0 THIS RESEARCH 0 MILES a SHEDLOVSK
--
POTASSIUM LAURATE 0 - DERIVED DATA (See Text) 2-
2 v)
E 45 c W
;aa 3 v)
35
?&I
'V
'
.Ob2 .Ob3;04
iObS.doeb0
b2
b3
.A4
CO NC E NT R A T ION, MOLES/ LITER
ds dl
F I G .4. Se~nilogarith~nic plot of surface tension vs. concentration
for micelles from consideration of equations 3 and 8. Below the critical concentration the slope is given by
\\+iilc nbm-e thih concentration the slop: is given by
The critical concent ration arises from the distribution of long-chain ions changing finom :I i*antlom aimngement t hrougliout the ~ o l u t i o nt o more 01' less localized
64
A . P. BRADY
clusters. Since the ions are already fairly well concentrated on the surface below the critical concentration, no large change in rl would be expected t o accompany the agglomeration within the solution; the slope of the surface tension ciirve should therefore change about sixteenfold. The data for lauryl sulfonic acid are not included in figure 5 , because of the apparent incomplete purification mentioned in the last section, but the curve shows such an abrupt change of
FIG.5 . Reciprocal of thc surface tension lonei,ing us. the area per iiiolcculc for sotliuiu lauryl sulfate. TABLE 2 Calculated adsorption of s o d i w n lauryl siilfato
rx
F
CONCENTRATION .
--
iu-io
-_ _ .\.z
___
PER Jlnl.T.CULE
.
dy,lrsjclls.
0.0016 0.0025
0.0035 0.006 0.012
12 18 21 2T 34
1.8 2.5
2.9 3.21 3
90 66 .58 51.6 55
slope a t 0.008 N , in fair agreement with other determinations of the critical concentration (0.0075 N by dye spectra (18); 0.009 AT by condncti.ility (3)). Table 2 gives the results of the application of equations 4 and 8 to estimate the surface adsorption of sodium lauryl sulfate froni the surface tension data. It is evident that the film so calculated is of the expanded type, since the F A product is of the order of 1000. The functional relation between F and A is quite interesting. Plots of A es. F , F A 11s.F , or 1/,4 vs. F give curved lines. I n a
phi ui 1, F 1's. A, ho\vever (given in figure 5 ) , the points belo\\- the critical coilventration fit the equ :I t 1011 '
F ( A - 20)
=
21;T
~ v i ~ h iexperimental n error. Until more data are a ~ ~ a i l a b lif-e is probably best to regard this as merely a n empirical relation, butothe factor of 2 in front of the I:?" term, the estmpolated moleciilar area of 20 at "infinite" filin pressure, and the lack of a n Poterm are extremely suggestive for this soluble ionized film. Because of doubtful purity the surface tension curve for lauryl sulfonic acid is not included in figure 4. Just belo\\- the critical pncentration the curvc lias a dope corresponding t'o rl 2.8 X lo-'" or about GO h.'per molecule. Ail)ot.ethe critical cwncent'ration the previously nient ioned fiat region, ascril)etf to resitliial impurities, \\.odd, of coiirse, correspond to zero adsorption. It is of int'erest to compare these results i\.it,h the direct measurement of :idsorption on lauryl sulfonic acid (uiipiirified) 1)y t'he miciatonie :uitl iiiterferoinct er methods (M). -41,0.002 S these gal-e til per molecule. respecti-i.ely; at 0.006 AT and 53 the t\\'o methotls disagreed, the microtome gi1,ing 31 and the interferometer per 33 .I.. At, the critical concentration, 0.008 -V, the microtome og;t~'e4s molecule, anti a t still higher concentrat'ions, 0.0115 X, about 32 d.? Thus below t8hecritical concentrat'ion, the surface tension results agree \\.it11 the microtome m d interferometcr measurements as \vel1 as these latter methods agree with each other. -1hoi.e the critical concentration, ho\vever, the niirrotome gixres very much lo\\er areas per molecule; n o interferometer reaiilts are a i d a b l e in this concentration re-'0'1o11. A41soincluded in figure 4 is a plot of surface tensions derived from data of Poivney (17). H e found t'liat the addition of a srnall amounl of potassium carbonate to a solution of potassium laurate raised the surface tension; with further additions the surface tension fell slo\\.ly and linearly \vitli t.he :Imount of potassiuni carbonate added. If we assume u i t h Adam (1) that the back extrapolation of the linear portion to zero potassium carbonate gives the surface tension of a hypot'lietmicalunhydi~olyzed binary system potassium laumte-\\.atjer, the Gibbs adsorption equation can be applied to tlie curire so obtained. The points in figure 4 are such derived surface tensions. I t may be seen that the general shape of the curve is similar to that for sodium lauryl sulfate. The calculated adsorption near the critical concentration (0.024 N ) is 2.5; X moles, mi.?, correspoilding t'o G5 k2per molecule. h similar calculatioii'f~oinPo\\,ney's data has been made by Adani ( I ) , but lie apparently neglected the factor of 2 appearing in equation 4, and thus computed the area per molecule to be about 32 The above applications of tlie G i b h equation inc1icat)e that the surface of the purified detergent soiut ions is by no iiieaiis close packed, since the croz:-section as compared ivitli the 30-60 calcuof a hydrocarbon ehain is only about 20 lated foi, the area per molecule a t the detergent solution surface. It sliould be fairly easy, therefore, for an u t d ~ a r g e dsurface-active species to penetrate to the surface. Further, this should result in a large change in the properties of the surface layer, because of ion-dipole interaction. I t has tieen shorrn (18) that a
-
-I.',
GG
.i. P. E R I D Y
miscd film of cetyl alcohol-sodium cetyl sulfate is solid a t the air-wter interface, under coiidit'ions where t>hefilm of neither substance alone is solid; the strong influence of h r y l alcohol on the foam stability of sodium lauryl sulfate solutions, c i t ed before, is therefore qui t'e unders tandahle. SUMMARY
Tn.0 detergent's which were fairly piire, but still sho\ved strong minima in the surface tension 1's. concentration c'urve, were further purified by foam fractionation. The foamed-off port,ions sho\ved very strong minima indeed, \vhile the imidues showed either a greatly reduced minimum or none. The result supports the hypot'hesis that t'lie minima arise from the adsorption of n small amount of surface-nctiye material belo\\ the critical concentrat'ion of the main solute, and it's effective removal from the surface above the critical concentration through solubilization wit'hin the colloidal micelles of detergent. It was also nDted that thc effect of time on the surface t'ension of solutions of the purified detergent was either giently reduced or eliminated for surfaces aged for more than 1 or 2 min. The ciirve of surface tension LIS. t'he logarithm of t,he concentration of a purified detergent sho\vs nn abrupt change in slope a t t'he critical concentration, a result to be expected from a ronsideration of t,he thermodynamics of these solutions. The area occupied per molecule adsorbed at the surface of purified detergent solution \vas calculat~edto reach a minimum of about 50 A,', by applying the G i b h adsorption equation. This is a. fairly expanded Elm, since the crosssectional area of the hydrocarbon chains is only at)out 20 .I.?The surface tension measurement8s are not in serious disagreement with the results of direct measurement of adsorpt'ion b y the microtome and interferometer methods, esr:pt at, concentrations above t8he critical, \\.here the microtome gives about 32 A.2 per molecule. REFEREXC
>
( I ) .\D.
