DETERMINATION OF SURFACTANT CONCENTRATION AT AIR-WATERIXTERFACE
Nov., 1963
isoenergetic redistribution of the available free volume as the origin of voids. Table I1 lists the values of the mean displacement a* calculated from eq. 4 with average values of D and with y = 1. The free volume was obtained from eq. 3, deg.-l, and Om = 16.48 X with TO= 0, Q = 1.34 X lO-Z4 cm.a; the latter values for the mean coefficient of thermal expansion and the mean atomic volume of liquid zinc in the temperature range of the measurements were calculated from density data in smith ell^.^ TABLE I1 VALUESFOR COHEN-TURNBULL EQUATION (LIQUIDZINC) D
x
105 om.’
u, 10-4 am.
T , OK.
see. -1
see. -1
720 775 826 873
2.31 2.91 3.72 4.G6
5.24 5.44
5.61 5.77
of
x
loa4
1.590 1.711 1.824 1.928
cm.8 y*
x
IO* om.
0.90 .89 .8G .82
With y arbitrarily set equal to unity in these calculations, the particle radius (r* = ( 3 / 4 m * ) 1 / a and onehalf the mean displacement a* are equal. Their value (3) C. J. Smithells, “Metals Reference Book,” Val. 2. Butterworths Scientific Publications, London, 1555, p. 640.
2355
ranges from 0.82 X to 0.90 X cm. and is reasonably close to the ionic radius of Zn+* (0.74 X ~ m . ) : Its ~ apparent dependence upon temperature may not be correct, since the coefficient of expansion (and consequently the free volume) probably increases with temperature. The interesting conclusion is that the diffusing particle appears to be the ion core in zinc metal, in agreement with the observations of Cohen and Turnbull and of other workers6-’ who have shown that the Stokes-Einstein relation is folloxred by liquid metals when the ion radius is used. Equally important is the fact that the results are satisfactorily interpreted in terms of a simple free volume theory which avoids the drawbacks of the activated state approach. Acknowledgments.-The authors wish to express their thanks to the Air Force Office of Scientific Research for partial support of this work under Grant AFOSR 62-231. (4) L. Pauling, “The Xature of the Chemical Bond,” Cornell University Press, Xew York. N. Y., 1945, p. 346. (5) R. E. Hoffman, J . Chem. Phgs., ZO, 1567 (1952). (6) X. H. Nachtrieb and 3 . Petit, %bid.,24, 746 (1956) (7) R. E. JIeyer and N. H Nachtrieb, zbad., 23, 1851 (1555).
DIFFUSION CONTROLLED BUILDUP OF SURFACE ACTIVE MATERIAL AT T E E AIR-WATER INTERFACE DURING EVAPORATION BY F. L. JACKSON AND F. P. KRAUSE M i a m i Valley Laboratories, The Procter & Gamble Company, Cincinnati, Ohio Received M a y 6, 1963 The buildup of sodium dodecyl sulfate (SDS) a t the air-water interface during evaporation of the water was followed using SS6-tagged surfactant and a counter for counting the surface activity of solutions in the inverted position, that is, with the air-exposed surface a t the bottom. Buildup of SDS was found to be essentially linear with time of evaporation, the slope depending upon evaporation rate. The amount of SDS ac~ being by a factor of six greater than the initial concumulating in the air-water interfacial region w a substantial, centration after 25 min. of evaporation a t the rate of 1.68 X 10-4 g./sec./cm.2 at GO”. A mathematical formula was derived which allowed calculation of diffusion coefficients from the rate of surfactant buildup at the airwater interface. Comparison of calculated diffusion coefficients with bulk solution diffusion coefficients, obtained by the self-diffusion open-end capillary method, led to the conclusion that the rate of buildup of SDS a t the air-water interface during evaporation is essentially diffusion controlled.
Introduction Many investigators have used radiotracer techniques to determine the excess surface concentration of surfactants under equilibrium conditions. Reference to these works is given in Kilsson’s’ article. The present work is concerned with determination of excess concentration of surfactants at the air-water interface under nonequilibrium conditions when evaporation of the solute is allowed to proceed continuously. Mathematical analysis of the system and of the experimental data enabled conclusions to be drawn regarding the basic factors responsible for surfactant buildup a t the air-water interface during evaporation. Within the authors’ knowledge, there have been no studies published previously which describe the effect of evaporation OS water on the concentration of surfactants a t the air-water interface of a detergent solution. However, it may be of interest to cite briefly some related studies concerned with the mechanism responsible for changes in interfacial tension, of solutions containing surface active material, immediately after the for(1) a. Nilsson, J . Phys. Chem.. 61, 1135 (1957).
mation of the air-water interface. Over a short time period evaporation is negligible. Flengas and RideaP concluded, as a result of their study of the surface aging of C14-taggedsodium stearate solutions, that the process is very slow and is diffusion-controlled. Defay and Hommelen,3.4 using the vibrating jet technique, found that normal alcohols (C6--Clo)and decanoic acid exhibit a surface tension lowering with time that is essentially diffusion-controlled. Azelaic acid, however, produced a slower change in surface tension with time than was explicable on the basis of diffusion, leading to the conclusion that an energy barrier exists in this system. Defay and Hommelen4 concluded that their data (with the exception of azelaic acid) and those of Bervichian5 for aqueous solutions of undecylic acid, those of Kalousek and Blahnike for eosine solution a t water-mercury interface, and those of Saraga7 for lauric acid aqueous ( 2 ) R. N. Flengas and E. Rideal, Trans. Faraday Soe.. 55, 339 (1959) (3) R. Defay and J . R. Hommelen, J. CoZIozd Sci., 13, 553 (1958). (4) R. Defay and J. R. Hommelen, ibid., 14, 411 (1559). (5) D. G. Dervichian, Koliozd-Z., 146, 98 (1956). (6) M. Kalousek and R. Blahnik, Trau. ehzm. Tehccoslov., 20, 782 (1055). (7) L. Ter Minassian-Saraga, Thesis, Paris, ‘1956.
F.L. JACKSON AND F.P. KRAUSE
2356
solutions are consistent in indicating that the diffusion mechanism controls changes in surface tension with time. On the other hand, Ward and Tordai,B using data of Addisong for normal alcohols in aqueous solutions, found calculated diffusion coefficients to be smaller than accepted diffusion coefficients. On this basis, Ward and Tordai concluded that an energy barrier existed a t the surface. The effect of evaporation on surface tension a t the air-water interface was the subject of papers by Schwarz and Woodloand by Hommelen. l 1 Schwarz and Wood described an experimental setup designed to eliminate evaporation during the measurement of radiotagged-surfactant concentration a t the air-water interface. Hommelen discussed the avoidance of errors, due to the evaporation of the solute, in the determination of the surface tension of solutions of long chain alcohols in water. Keither of the works by Schwarz and Wood nor Hommelen considered the buildup of surfactant a t the air-water interface or the factors which control such a buildup. Theoretical The derivation of a formula which allows calculation of diffusion coefficients from the rate of increase of surfactant concentration a t the air-water interface as a function of time is based on the model indicated. .Surface j 1j ,Solution
TJ 0
%+AX
X
,1 x-axis
In this model the solution moves toward the surface with the veIocity v. Consider a solution elemesit 1 whose sides are taken to be of unit area. The amount of solute moving across the plane a t z during the short time At is dependent upon Fick’s first law of diffusion and the movement of the solution
Vol. 67
which, on taking the limit with At becomes the differential equation
+.0 and
4%4 0,
z c bc - = D -3x2 bt + v ~
bC
It will be noted that eq. I is Fick’s second law of diffusion plus a variable factor which takes into account the movement of internal elements of solution toward the surface reference point as a Consequence of evaporation of the solution. Initial and Boundary Conditions.-The initial conditions of the system described by (I) are given as c(z,O) = co,
c(z,O)
=
2
0,2
> o\ = 7 hP P2@ cl) and inversion of LA(??), which can be carried out with the aid of tables,'* gives the surface activity A .
+
A(t) =
+
+
2Kcob2k _ _ I
m
+
~
[Dt -k JoD' erf (bdi)dz)]'5
dA
-=
at
+
X (3)
where
=
lim-dA = Kcov2 t-mdt v+kD ~
which, on substituting K from eq. 1 and rearrangement, becomes
rn
=
L2
+ 2kb
(13) See for instance K. V. Churchill, "Operational Calculus," McGrawHill Book Co., Inc., New Yorlc, N. Y., 1958. (14) A . Erdelyi, Editor, "Tables of Integral Transforms," McGraw-Hill Rook Co., Ino.,New York, M.Y.. 1954 (notably item 22, Vol. I, p. 235). (16)
mDA - KkDc(0,x)
which must be satisfied by eq. 3 as it indeed is. Calculation of Difision Coefficients.-Direct calculation of diffusion coefficients from eq. 3 is obviously impractical. Fortunately, for sufficiently long evaporation times, the diffusion coefficients can be calculated conveniently from the slope of the asymptote of eq. 3
KCO 2Kcok(b 1 ~ ) ~ erf (bds) IC m2
+
Equation 3 can be verified as follows. Equation 11, which has the form of a Laplace transformation, is used to transform eq. I with respect to x in the same way as eq. I is transformed with respect to t (or x ) in the usual Laplace transformation. The concentration a t the surface, c(O,t), needed for the lower integration limit, is given by eq. 5 later. This transformation leads to the ordinary differential equation
erf ( b 4 i ) d z =:
(Dt - 2 L ) erf ( b d / D t ) + --
(4) an expression that relates D in simple terms with experimentally measurable quantities. The slope of the surface activity curve approximates rapidly the slope of the asymptote. Theoretically, the approximation becomes good a t the critical time t, = bW2D-', that is, when the exponents in the dominant exponential terms of A become -1. Typical critical
E'. L. JACKSON AND F. P.ICRAUSE
2358
Vol. 67
The concentration is highest a t the surface. Away from the surface it falls off to co approximately proportionately with e-br for sufficiently long evaporation times. Experimental
Fig. 2.-Surface activity of 0.1% HzS3504solution during evaporation a t 60"; cell 1 (diameter 3.7 mm.); high rate of evaporation. I
I
I
I
I
1
x,
40
I
b -373 c p m 400-
Ab
9
v 200-
0.
* 0.29 q m . / w c .
.
D
0
1.21
x IOPCrn/.&.
LB X 10~'an2/ee
eo
lo
I
0
I
0
20
10
40
Y1""l.l
Fig. 4 . S u r f a c e activity of 0.1% HnS3504solution during evaporation a t 60'; cell 2 (diameter 6.1 mm.).
times are 10-20 min., which are easily exceeded in practice. The surface activity curve in Fig. 6, which was calculated from a set of typical constants, illustrates how good the approximation is not only a t t, but even earlier. Solute Concentrations.-The formula for the solute concentration a t the surface is obtained from eq. 2 by setting 2 = 0 and interpreting the transform in t. c ( ~ , t )= co
+ 2c erf(bdD2) + 2cb [Dt + SoDt erf(bdi)dz]''
(5)
The apparatus for counting solution surfaces during evaporation is shown in Fig. 1. The solutions were held in the inverted position in round glass cells 3.7 and 6.1 mm. in diameter and about 5 cm. long (cells 1 and 2). A horizontal capillary connected the upper end of the cell with a plastic reservoir from which, by means of an adjustable clamp, solvent could be forced manually into the cell t o replace evaporated solvent. A mercury plug in the capillary served to measure the solvent flow, from which the evaporation rate was computed. The shielding assembly consisted of three stainless steel rings with aligned central openings 4 mm. in diameter and rested on the window frame of a Geiger-Muller tube. The assembly collimated the radiation emanating from the solution and thereby blocked out any stray radiation from the cell rim. A tightly fitting movable glass sleeve on the outside of the assembly permitted closure of the evaporation space and discontinuation of evaporation. The end of the large cell was tapered from the outside to a sharp edge in order to facilitate shielding, which was adequate for the large cell. However, this provision proved to be inadequate for the small cell. The end of the small cell was, therefore, ground flat and covered with a matching piece of Teflon sheet 0.5 mm. thick, which, since Teflon is not wetted by detergent solution, prevented the deposition of radioactive solute a t the cell rim and thus made shielding unnecessary. The surface counter was mounted inside an air temperature box so that only the water reservoir remained outside. Inside the box a small fan provided as vigorous an air circulation as was possible to attain without causing noticeable vibration. The humidity inside the box was kept from rising by admitting some outside air t o the box. During evaporation tests the air temperature in the box was 60 0.5". The cell, the capillary, and the reservoir were filled outside the box while in an upright position, inverted and put in place with the cell rim 1.5-2 mm. above the upper shielding ring and approximately 16 mm. above the tube window. After the sleeve was closed and the meniscus of the solution adjusted so as to be level with the cell rim, the base count, i.e., the surface activity of the unevaporated solution, was taken a t room temperature. This was necessary because evaporation could not be completely prevented a t 60" and the base counts taken a t that temperature were too high, notably with the small cell. The base count a t 60°,A b , was obtained from the count taken a t room temperature by multiplication with 1.06. The correction factor was computed from the air-density change, the length of the air path, and the mass absorption coefficient of the S3%adiation. A similar correction factor was obtained experimentally with an 8 mm. diameter cell. Because the diffusion coefficient is not computed from the surface activity but from its rate of increase, the slow evaporation in the closed system is, except for its effect on the base count, of little consequence to the evaporation experiment. The evaporation was usually begun 10 min. after the air temperature in the box had reached 60". Counts were taken continuously for 1-min. periods and meniscus adjustments made every 0.5 or 1 min., depending on the rate of evaporation. The delays of resetting the scaler, about 2 . 5 sec./min. counting time, were allowed for in the computations. The displacement of the mercury plug was read every 10 min. Total evaporation time was usually 40 min. Low evaporation rates were obtained by raising the sleeve only to the level of the second shielding ring. The evaporation rates were constant during the tests except for the first few minutes of evaporation when they were about 20% higher. Assuming, as a first approximation, that the rate of evaporation is proportional to the vapor pressure of the solvent, the surface temperature of the rapidly evaporating solutions is estimated a t 5 5 " . The sulfuric acid-S35 was used as received from Oak Ridge after removal of the bulk of HC1 by evaporation and appropriate dilution with inactive sulfuric acid. The sodium dodecyl sulfate5 3 5 (SDS) was a t least 95Ycpure, the remainder being 1-2% each of NazSOa and NaCl with up to 1% water.
Nov. , 1963
DETERMINATION OF SURFACTANT CONCENTRATION AT AIR-WATERINTERFACE
2359
Results Typical surface activity curves for evaporating solutions are shown in Fig. 2-5. The first three are those of 0.1% sulfuric acid solutions tested a t different evaporation rates and in different cells. The fourth belongs to a 0.5% sodium dodecyl sulfate solution. The base counts, Ab, are indicated by short horizontal lines preceding the beginning of the evaporation. All curves rise with increasing evaporation times. The diffusion coefficients are computed as follows. A straight line is drawn through the points of the curve disregarding those for the first 15 min. of evaporation. I n the anomalous case of the SDS curve a tangent is drawn instead a t the 15-min. point. The slope of the straight lines or the tangent, S, is expressed in terms of activity increase per second, that is, in counts/min./sec. From 8,the base count A b , the evaporation rate v in terms of the solution advancement in the cell, in cm./ sec., and the absorption coefficient of the radiation k in cm. -1, the diffusion coefficient is calculated by means of the formula
D
=
A ib
--
S
v2
- k2,
TABLE I DATAFOR THE COMPUTATION OF DIFFUSION COEFFICIENTS BY THE EVAPORATION METHODAND STANDARD DIFFUSION COEFFICIENTS
Solution
O.l%H&04 .l%HzS01 .l%HzS04 .5'%SDS
2 3 4
5
439 373 2663 109
0.95 0.29 3.19 0.30
I
i Ab .IO9 cpm.
s V
0.MCpm.h.C. rn
1.68 X 10~'cw~h0.
D * 0.95 X 4 . '
Ylnures
20
x)
on'/aez 40
Fig. 5 . 4 u r f a c e activity of 0.5% sodium dodecyl sulfate (Sa9 solution during evaporation a t 60"; cell 2 (diameter 6.1 mm.).
[~m.~/sec.]
whose derivation was given in the Theoretical section. The diffusion coefficients for the curves are listed in Table I along with the figures from which they were computed. For comparison, diffusion coefficients determined-by , the open-ended capillary method are also listed.
S, Ab, c.p.m./ Figure c.p.m. sec.
B
V D X-105, in"cm.z/sec. X 104, kiSa5 Evaporacm./ in HzO), tion Standard see. om.-' method method
2.33 1.21 1.58 1.68
230 230 230 230
2.4 1.8 2.0 0.95
2.2 2.2 2.2 0.82
Discussion It is seen that the diffusion coefficients estimated from the surface activity increase during evaporation do not differ by more than 10-20% from their standard values. This is a satisfactory agreement, for the very nature of the evaporation experiment is such that disturbing factors like vibration, impact of air currents, and surface cooling cannot be adequately controlled or eliminated. For high evaporation rates the cell diameter has no effect on the diffusion coefficient as long as it is not larger than 6 mm. When it was increased to 8 mm., the diffusion coefficients became too large, presumably because the solution in a wide cell is more easily disturbed than in a narrow cell. To test the diffusion coefficient formula experimentally, the diffusion coefficients of sulfuric acid were determined a t different evaporation rates (Fig. 2 and 3 ) . The formula requires that the slope of the surface activity curve should increase almost with the square of the evaporation rate. The experimental agreement with this requirement is fair.
01
I
0
I
10
NlnuM.
20
I
30
1
4a
I
Fig. 6.-Theoretical surface activity curve eq. 3: co = 10-3 g . / ~ m . ~K; = 2.3 X lo7c.p.m. cm.2/g.; 1.1 = 2.30 cm.-l; A b = 100 c.p.m.; v = 2 X cm./sec.; D = 10-6 cm.Z/sec.
*i5
10
0
1. 2 s
H' 2
Mlnuloa,
Fig. 7.-Theoretical surface concentration curve eq. 5 : co = 10-3 g./cm.S; v = 2 X cm./sec.; D = cm.Z/sec.
The theory also requires a virtually1 constant skrface activity increase for solutes with constant diffusibn coefficients. This is actually observed with sulfuric acid solutions. It is interesting that the detergent' SDS, whose diffusion coefficient decreases with incrkasing concentration above 0.5%, should give a n upward curving surface activity curve, thus reflecting the diffusion coefficient decrease. From the agreement between experiment and theory with respect to the surface activity, it follows that the surface concentration of the solute, too, should be as given by the theory (eq. 5 ) . Figures 6 and 7 show that there is a close parallelism between the calculated surface activity and the calculated surface concentration, the latter merely increasing somewhat more steeply than the former.
2360
B. J. FONTANA
Conclusions From the agreement between the conventional diffusion coefficients and those estimated by the evaporation procedure follows that the solute buildup a t the surface of evaporating solutions is diffusion controlled. This holds equally for electrolytes and surface active compounds. When the radioactive source is a weak p
Vol. 67
emitter, the surface activity increase approximates the surface concentration increase. After a short initial transition period, both increase continuously and virtually linearly with the time of evaporation. Away from the surface the concentration falls off approximately exponentially from its highest value a t the surface to the bulk concentration.
THE CONFIGURATIOX OF AX ADSORBED POLYMERIC DISPERSANT BY INFRARED STL‘DIES BY B. J. FONTAXA California Research Corporation, Richmond, California Received May 6, 1963 The fraction of attached ester segments of an alkyl methacrylate-polyglycol methacrylate copolymer molecule adsorbed onto silica was determined directly by infrared spectrometry. The exclusion of ester segments from attachment to the surface because of preferential adsorption of the polyglycol ether segments is demonstrated. It is suggested that the resultant polymer configuration is thus more extended away from the surface and that this accounts for the enhanced colloid stabilization properties reported for such polar-substituted polymers.
Introduction The steric mechanism of stabilization of colloids requires the adsorption on the particles of a surfactant film of a thickness sufficient to increase the distance of closest approach and thus decrease the van der Waals energy of attraction to below the thermal energy. Such an effect has been demonstrated experimentally by van der Waarden’ and others and treated theoretically by Mackor aiid van der Waals and by T‘oldS2 Adsorbed macromolecules would be expected to possibly function well in this respect. Heller and Pugh3 have postulated such a mechanism for aqueous gold sols stabilized by high molecular weight polyethylene glycols. With regard to hydrocarbon media, it has been found that appropriate modification of alkyl methacrylates can result in outstanding dispersants for use in automotive lubricating oils. The introduction of a relatively small fraction of polar substituents into the polymer molecule results in a great enhancement of the dispersance. In a previous study5 it has been shown that polylauryl methacrylate, PLMA, adsorbed on silica, is attached to the surface through about 40% of the segments. In t’his case then, a relatively flat adsorbed film results, estimated to be -30 A.thick. Incorporating about 17 mole % N-vinyl-2-pyrrolidoiie into PLMA as a copolymer, PAM-VP, increased the adsorption a t t’otalsurface coverage and gave a thicker film (-200 A. as determined from sedimentation measurements) .6 The PAM-VP was shown to adsorb uia both t’he ester and pyrrolidone carbonyl groups. The enhanced dispersant property of the PARII-VP4 then is apparently due to the thicker adsorbed film, aiid it appears reasonable to assume that the thicker film results from the preferential adsorption of the smaller number of more strongly polar pyrrolidone groups. The latter behavior is in accord with the predictions of the recent’ statistica,l (1) ill. van der Waarden, J. Colloid Sci., 6 , 317 (1950); 6, 443 (1951). (2) E. L. Maokor, ibid., 6 , 492 (1951); E. L. RIackor and J. H . van der Waals, ibid., 7, 536 (1952): hl. J. Vold, ihid., 16, 1 (1961). (3) W. Heller and T. L. Pugh, J. Polymer Sci., 47, 203 (1960). (4) A. L. Lyman and F. W. Kavanaph, Proc. A m . Petrol. Inst., Sect. III, 39, 296 (1959). (6) B. J. Fontana and J. R. Thomas, J . Phys. Ch6m., 6 6 , 480 (1981).
analysis of polymer adsorptioii by Silberberg.6 It was not possible to demonstrate this effect in the previous studies because of the overlapping of the pertinent infrared bands. In the present study the effect has been quantitatively determined for an alkyl methacrylatepolyglycol methacrylate copolymer, PA3I-PG. The latter belongs to a class of outstanding dispersants used as ashless detergents in engine oils and specifically designed to be efficient hydrogen bonding detergents.’ Experimental P-4RI-PG was prepared by direct copolymerization of the alkyl methacrylate and polyethylene glycol tridecyl ether methacrylate with 2,2’-azobis(2-methylpropionitrile) initiator in refluxing benzene. The alkyl groups were of mixed length averaging 14 carbon atoms. The polyethylene glycol residue [CH&H20] was 1600 average molecular weight and was capped Kith a 13carbon alkyl. The present sample was a narrow fraction obtained by solvent precipitation. The mole ratio of alkyl methacrylate to polyethylene glycol in this fraction was 20:l as determined by both infrared and elemental analysis. The copolymer molecular weight Yas 4 1 0 , 0 0 0 as determined from intrinsic viscosity (29 in isooctane, 85 in benzene at 2 5 ’ ) . The measurement of the adsorption isotherms and of the infrared spectra of the adsorbed species on silica is described in a previous publication.6 The details of the determination of the pertinent extinction coefficients and calculations of the fraction of adsorbed ester segments, p, are also given therein. The carbonyl extinction coefficient, e, for free polymer PAM-PG in n-dodecane solution was 1120 as compared to 1465 for PLMA (and 1180 and 1510, respectively, in decalin). This ratio (0.773 0.009) is accounted for by the difference in equivalent weights. Hence, the e,% value used for adsorbed carbonyl was corrected accordingly ew
(PARI-PG)
=
1650 X 0.773
2
1275
The infrared frequencies for the free and adsorbed carbonyl had exactly the same values as previously observed for PLlIA.
Results and Discussion In comparing the behavior of the polyglycol substituted polymer with the simple polyalkyl methacrylate, it should be noted that both have identical methacrylate “backbones.” The basic structural formula of PARI-PG is given on the following page. (6) A. Silberbarg, zbzd., 66, 1872 (1982). (7) W. T. Stewart, F. 4 Stuart, and J. A . Miller, Division of Petroleum Chemistry. American Chemical Society, Preprints, Vol 7, No. 4, 1962, P. B-19.
