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Surface tension relaxation in a surface containing surfactant particles
Surface Tension Relaxation in a Surface Containing Surfactant Particles F. A. VEER AVID M. VAN D E N T E M P E L Unilever Research Vlaardingen, Olivier Van Noortlaan 120, Vlaardingen, Th~ Netherlands
Received May 9, 1972; accepted July 20, 1972 At the surface of aqueous systems containing slightly soluble surfactants, slow relaxation processes occur that are accompanied by high values of the surface dilational modulus. A detailed investigation into these phenomena was carried out in systems containing dodecanol, tetradecanol and hexadecanol, using longitudinal surface waves and stress-relaxation experiments. The state of the surfaces involved resembles a monolayer after collapse: particles are present which do not consist of the thermodynamically stable bulk phase of the surfactant. A quantitative interpretation of the surface tension relaxation is given in terms of two processes: surfactant exchange in the surface between the monolayer and the particles, and diffusional exchange of surfactant between monolayer and bulk solution. The parameters characterizing these processes were measured; the values obtained are in satisfactory agreement with the results of other experiments.
INTRODUCTION
Collapse of a monolayer leads to the formation of molecular aggregates consisting of water and surfactant in a layered structure (1,2). The presence of such aggregates is accompanied by the occurrence of time-dependent processes in a surface (3): the change in surface tension resulting from a change in surface area depends on the time scale of the area change. Longitudinal surface waves are a means to study such phenomena. This type of experiment (4-6) characterizes a surface by means of a dilational modulus ]e I and a phase angle 0; magnitude and time dependence of these parameters are controlled by the relaxation processes occurring in or near the surface. Experiments on a hexadecanol layer spread at the air-water interface, revealed the effect of collapse (Fig. 1). Before collapse, at a surface tension of 48 dyne. cm-1, the surface exhibits almost purely elastic behavior in the time scale investigated: the modulus is hardly frequency dependent and the phase angle
equals zero. At 48 dyne. cm-1 surface tension changes resulting from a small decrease in surface area (1%), relaxed within several minutes only, indicating the onset of collapse phenomena. After a large area decrease (25%) the surface tension relaxed towards 29 dyne.cm -1 within several minutes, and remained at this level sufficiently long to perform another longitudinal wave experiment. Freshly collapsed material will now be present at the surface; it is readily seen from Fig. 1 that in this situation the surface is characterized by a very high value of the modulus, and by phase angles exceeding 7r/4 at low frequencies. The latter phenomenon indicates the occurrence of a relaxation process that is not diffusion controlled, because in such a case phase angles between 0 and 7r/4 will be found only (6). Similar surface behavior has been found on aqueous systems containing fatty acids, fatty alcohols and monoglycerides with straight carbon chains of twelve atoms or more. These 418
Journal of Colloid and Interface Science. Vol. 42. No. 2. February 1973
Copyright ~ 1973 by Academic Press, Inc. All rights of reproduction in 0,ny"fqrrg reserved.
SURFACE TENSION RELAXATION surfactants were dissolved in water at a temperature above their melting point. On cooling, gel-like particles precipitated and the presence of these particles in the surface affected the surface rheology in the same way as particles resulting from collapse phenomena did. The aqueous systems prepared as described here, will further be called "suspensions." The surface behavior mentioned has only been found when non-crystalline material was present; the type of particles involved will be termed "mesomorphous," thus emphasizing that they probably possess a layered structure and that they do not consist of the thermodynamically stable bulk phase of the surfactant. A detailed investigation of the surface tension relaxation was carried out in suspensions containing fatty alcohols. A model is developed to describe the kinetics of this process as it affects longitudinal surface waves and stress-relaxation in the surface. THEORY Two possibilities arise in regard to the influence of mesomorphous particles on relaxation phenomena in the surface: 1. The particles exchange surfactant with the surrounding monolayer, thus affecting the composition and properties of the monolayer. 2. The particles are arranged in a twodimensional network that yields on compression and expansion. The model to be developed here is based upon the first possibility. Consider the surface of an aqueous suspension of a surfactant. In this surface mesomorphous particles are present, but the monolayer part of the surface determines the surface tension. The adsorption in the monolayer part m a y then be regarded as the solubility limit in the surface of the mesomorphous material. If, in this situation, the adsorption is changed by a change in surface area, two surface relaxation mechanisms m a y occur: 1. Surfactant exchange between monolayer and underlying solution by means of diffusion.
419
3.5
t,f t , '~-~/'~
z~~
f
~'
after I~l beforecollapse
°~6-I°~ ° ' "
°~
o
2.5
tan0 after collapse ~ D -t~ before
o~
¢
_~
J
4
• tog co(co in s "1)
Fro. 1. Results of longitudinal wave experiments on spread hexadecanol layer before (at cr = 48 dyne. cm-1) and after (at ¢ = 29 dyne. cm-1) collapse, o~= radial frequency. 2. Surfactant exchange between the monolayer and the particles in the surface. If the former process is assumed to be unaffected by the presence of mesomorphous particles in the bulk solution, its effect on wave behavior is known (5). The process in the surface will be described by a first-order rate process which leads to the following material balance for the monolayer part of the surface:
dr/dt
= -- r(g -- £o) --
D(Oc/Oy)~=o -- r ( d l n A / d t ) .
[11
Here, I' denotes the adsorption (moles.cm -2) in the monolayer part of the surface, in the undisturbed surface this quantity equals Po; r is a rate constant (sec-1), D the bulk diffusion coefficient (cm2.sec -1) and c the bulk concentration (moles. cm-3). The coordinate y is perpendicular to the surface, y = 0 at the surface and y is negative in the solution. The monolayer area is denoted by A (cm2), the time by t(sec). The relaxation process in the surface which is characterized by the parameter r, m a y be regarded as growth of particles during compression and dissolution during expansion of the surface. Assuming the rate of these processes to be proportional to the amount of
Journal of Colloid and Interface Science, VoI. 42, No. 2, February 1973
420
VEER AND VAN DEN TEMPEL
super- or subsaturation, is a simplification. This simplification, however, is justified for growth and evaporation of crystals from the vapor phase, if small values of super and subsaturation are involved (7). In a bulk phase growth and dissolution are processes proceeding in two steps: diffusional transport towards or from the particle controlled by a concentration gradient, and a rate process at the surface of the particle: At the air-water interface the latter process will be rate-detet-mining, as the transport within the interface proceeds relatively fast due to Marangoni-effects. Introducing parameter r as is done in [-1-], implies that r will also be affected by the number of particles present at the surface, because the surface process introduced may be regarded as a heterogeneous reaction. The behavior of the surface during the passage of a longitudinal wave, and during stress-relaxation experiments, will now be described in terms of the parameters introduced in [-1].
frequency of the barrier motion (sec-1) and n = (1 + i)(~/2D)½. At any moment the subsurface concentration cv=0 is assumed to be in equilibrium with the adsorption F; as a consequence the following relationship may be formulated: cl = (dc/dP)v=oG
[-5~
The quantity (dc/dr)~=o represents an equilibrium property of the system, viz. the inverse of the slope of the adsorption isotherm. In this quantity the subscript y = 0 will further be omitted. Substituting I-2-]-I-5] in [-1-] gives: ioaG = -- rG -- (1 + i) dc/dF
X [G(~D/2)~] - ( F / A ) i ~ A I .
[-6]
After the introduction of two dimensionless parameters p = (dc/dr)(D/2co)~ and p = r/oo,
Equation [-6] reads:
[7]
iG = -- pG -- (p + i p ) a - i ( r / A ) A 1 .
So
Longitudinal Surface Waves
In this type of experiment a surface is subjected to a periodic deformation imposed by a surface barrier. The deformations propagate along the surface, introducing spatial variations in the surface composition. The experimental techniques available (4-6) provide means to correct for these propagation effects, thus permitting measurement of the frequency dependence of the surface tension variation. This quantity will now be calculated starting from the frequency dependent variations in I', c and A, for which the following reIations will be valid, if small deviations from equilibrium are considered only: F = ro -Jr- AF = Yo -}- Ge ~
[-2]
c = Co + Ac = Co + cle"~e ¢°~t
[-3]
A = Ao + A A = Ao + A l e i %
[-4]
G=
-- i F A J A { p
+ p + i ( I + P)}.
[-8-]
Now, a relation should be formulated between the measurable quantity--the surface tension variation--and G the quantity which determines the variation in the adsorption. As small variations are considered only, a linear relationship may be used: ~
= (d~/dr)~r
= (d~/dr)aei%
E9]
where ~ is the surface tension in dyne. cm-1. The quantity d r / d r is an equilibrium property of the surface, and therefore this formulation implies that at any moment equilibrium is assumed between surface tension and adsorption in the monolayer. Using [-8] and F9] we find for the complex modulus e: A A d~ e = --/X~ G AA A1 d r
Here, subscript o denotes the values in the absence of surface deformations; G, c~ and A t are complex quantities; w represents the radial Journal of Colloid and Interface Science, Vol. 42, No. 2, F e b r u a r y 1973
~;eo
=
,
p +f
+ i ( 1 +~-)
Bo-1
SURFACE TENSION RELAXATION where eo = - (@~din r). The real part er and the imaginary part ei of this complex modulus are:
= ~
o[1 + KI/[(p +
+ (1 +
= eo[P 4- P-1/[(p 4- p)2 4- (1 4- p)2]].
dAV/dt = -- rAP -- (dc/dr)(D/rrt)~AP.
1-123
Integration gives:
tan 0 = (p 4- ~')/(1 + ~') =
i + [(p
-
equilibrium, may be written as - D ( d c / d F ) X (ar/(~rDt)~). So the equation to be solved is:
1-113
This results in the following expressions for the surface dilational modulus I el and the phase angle 0:
421
in [ ~ r / ( ~ r ) , = o ] = -- rt - 2(dc/dI')(Dt/rc)~.
El6]
[173
If a linear A~ -- Ar relationship is assumed, the right hand side of [17] will equal In
8/(I
+
If p and ~ are of the same order of magnitude, both relaxation processes will affect the surface properties. This will roughly be the case in a frequency range of one decade at both sides of the frequency for which p = ~'. At higher frequencies diffusion will be more dominant (/->> p) and [13] and [14~ will degenerate into the known expressions for diffusion controlled relaxation (6). At lower frequencies p >> ~', the relaxation mechanism in the surface will then be rate-determining. From E13~ it follows that for oJ--+ 0, [ e [ --+ co~p, so the slope in a log ] e[ vs log coplot will approach 1 at low frequencies, whereas for the diffusioncontrolled case (p = 0) the corresponding value equals ½. Equation F14] predicts values of the phase angle between 0 and 4- 7r/2; for f = 0 the tan 0 values should vary~inversely proportional to the frequency. The diffusion controlled case allows values between 0 and 4- 7r/4 only. Stress-Relaxation In stress-relaxation experiments the surface area is subjected to a sudden change after which the surface tension relaxation is measured. For this experiment Eq. [-17 simplifies to:
dr~dr = -- r ( r -- to) - D(Oc/Oy)~=o.
[15]
The diffusion term will be approximated by --DAc/(rrDt)i which, for small deviations from
as well. It is emphasized that, according to this model, particles in a surface affect the surface tension only as a result of exchange of surfactant between the particles and the rnonolayer. This action does not affect the equilibrium between surface tension, adsorption in the monolayer and subsurface concentration; at any moment and location the relationships between these quantities are given by the equilibrium surface equation of state. EXPERIMENTAL Longitudinal wave experiments and surface tension relaxation experiments were performed with aqueous suspensions containing dodecanol, tetradecanol or hexadecanol. The water used was deionized and twice distilled; the dodecanol was ex. Fluka puriss, grade; the tetradecanol was ex. Fluka pract, grade; the hexadecanol sample was prepared in our laboratory. The suspensions contained 20, 10 or 4 rag. 1-1 of the respective alcohols and were prepared by shaking the alcohols with water for 1 hr at a temperature (40, 60 and 70°C, respectively) well above its melting point. On cooling towards room temperature gel-like particles precipitated and the suspension thus obtained was poured into a Langmuir trough (Teflon, 60 X 8 cm). The experiments started about 1 hr after formation of the surface. Experiments with dodecanol were performed at 15°C; the other experiments at 23°C.
Journal of Colloid and Interface Science, Vol. 42, No. 2, February 1973
422
VEER AND VAN DEN TEMPEL I I I
Two techniques were used for longitudinal wave measurements. The technique at high frequency (c0 ~ 103 sec -1) w a s described in Ref. 4. At these frequencies the wavelength of the longitudinal wave is short in comparison with the trough length which permits a direct measurement of damping coefficient and wave number; from these parameters l eI and tan 0 m a y be calculated. The method at low frequencies (10-2 < co< 1 sec-~) was analogous to the one described in Ref. 6. In our apparatus the roughened glass Wilhelmy plate was suspended from a Cahn R G electrobalance. A light-sensitive resistance generated a signal representing the barrier motion. Barrier motion and surface tension were recorded on a Hewlett-Packard 135 AM x - y recorder. From the ellipse thus obtained, the surface dilational modulus and the phase anglemay be calculated
C~2
2
\~n C14
\
I~ log co(coin s M)
Fie. 3. Measured (
) and calculated (. . . .
) phase
angle for aqueous suspensionsof dodecanol, tetradecalml and hexadecanol.
3.5
.......
,o ......""
3 s s .....'*
,2
, / " o/
/4"/
2.~
2
t5
directly, because at these frequencies, the longitudinal wave length is much longer than the length of the trough. The barrier amplitude varied from 0.11 to 0.14 cm. The effective trough length was 32.5 cm. In the relaxation experiments the surface area was changed by moving a surface barrier driven by a motor. The area change varied from 0.5 to 1% in about one second. After the area change the surface tension relaxation was recorded with the combination Wilhelmy plate - - C a h n R G electrobalance--Servoscribe recorder.
t I
-
\
RESULTS
1
T 1' tog co(coin s4)
) and calculated ( - - - - ) surFIO. 2 . 3 d e a s u r e d ( face dilational modulus for aqueous suspensions of dodecanol, tetradecanol and hexadecanol. (~ = 25, 31 and 29 dyne.era -I, respectively).
Figures 2 and 3 give the results of the longitudinal wave experiments. At the lower frequencies the surface dilational moduli were calculated assuming the surface area, occupied by the mesomorphous particles, to be negligible. The data will be interpreted using [13-] and [14-]. From [14-] it follows that the parameter r equals the value of co for which
Journal of Colloid and Interface Science, VoI. 42, No. 2, F e b r u a r y 1973
SURFACE TENSION RELAXATION
10
Cl~o
/
8
5 a
C12
[]
O
3
0
I 2
1
I 3
FIO. 4. Determination of (dc/dP) (D/2)~ using Eq. [-18], for the surface of a dodecanol (O) and a tetradecanol (K]) suspension. tan 0 = 1. Rearranging [,,14] gives:
I+
/--/ ~\2~/
[18]
tanO-- 1
Plotting the right hand side of El8] against (l/w)} should result in a straight line with a slope that equals (dc/dr)(D/2)}; this line should go through the point [-(i/c0)½ = 0, (p-1)/(tan0--1) = 1] (see Fig. 4). The values of r and (dc/dP)(D/2)~ thus obtained m a y now be introduced into [,13] together with the measured [e I values in order to calculate co. The resulting average value of eo is recorded in Table I. The equilibrittm parameter eo equals the surface dilational modulus at frequencies too high for the occurrence of any relaxation process (p = f = 0). Obviously, these conditions were satisfied at o~ ~ 10 ~ sec-I, where tan 0 values were found to be near zero. The I el values measured at this ireTABLE
quency m a y therefore be compared with the calculated eo value. The procedure outlined here was applied to the data of the dodecanol and tetradecanoI suspensions. For the hexadecanol suspension the influence of the relaxation process in the surface on the measured tan 0 values happened to be rather small, which prevented an accurate determination of the parameter r. For hexadecanol the parameters r, (dc/dr)(D/2)} and eo were therefore calculated by extrapolation from the corresponding values for the lower alcohols, assuming linear relationships between chain length and the logarithm of these parameters. From Co, r and ( d c / d r ) ( D / 2 ) } theoretical curves m a y be calculated representing the frequency dependence of [e I and tan0. The results are shown in Figs. 2 and 3 to enable comparison with the experimental curves. Finally the parameters r and (dc/dr)(D/2)} m a y be introduced into [-17] to calculate a theoretical surface tension relaxation curve. I n Fig. 5 these curves are plotted together with the experimental ones.
C16 ~' % o05
--
C~
Cx4
--0.24
-- 1.48
0.58 in s-~
1.0
C12, oO.E 2
LONGITUDINAL WAVE EXPERIMENTS
( l o g ~o) t . n 0 = l
l ~
I
SIJR~ACE PARAMETERS OBTAINED FROlk{
r in s-1
423
0.033
C~o --
0.0019
I~time(s)
dP 0.41 0.092 0.021 eo in dyne. cm-I 1350 1800 2400 [e [~=1oain dyne. cm-1 1800 2300 2800
FIG. 5. Measured surface tension relaxation curves ( ) after compression and expansion; curves predicted from longitudinal wave results (. . . . . ).
Journal of Colloid and Interface Science, Vol. 42, No. 2, F e b r u a r y 1973
424
VEER AND VAN DEN TEMPEL DISCUSSION
By comparing measured and calculated longitudinal wave parameters in Figs. 2 and 3 it may be concluded that the model presented here is useful for a quantitative description of the data. Largest deviations occur for hexadecanol, which is not surprising as in this case the parameter values were obtained by extrapolation. Nevertheless, the predicted curves agree qualitatively with those actually measured. The relaxation curves calculated from the longitudinal wave results provide correct information regarding the time scale of the relaxation-process; it may therefore be concluded that the given interpretation of the longitudinal wave data leads to the determination of physically relevant quantities. Exact prediction of the relaxation behavior was not attained. This will partly be due to the simplifications introduced in the derivation of Eq. [-17]; for instance the use of linear A c - A l l - Aa relationships will only be justified if small deviations from equilibrium are involved. Moreover for greater deviations from equilibrium the rate of the process in the surface will not be proportional to Ap. Rate of spreading and rate of particle growth may then differ considerably, and the r-value measured by means of longitudinal waves will represent an average between these two rates. This may explain why the predicted relaxation curve for tetradecanol seems to be an average of the curves measured after compression and expansion. The same phenomenon was not found for the hexadecanol surface which may be caused by the inaccurate surface parameters available. In the theoretical part of this work the parameter r was related to a rate process at the periphery of the mesomorphous particles. At a fixed temperature such a process is expected to proceed faster as the chain length of the alcohol decreases, which may explain the measured shift in the values of r. However, r may also be affected by the number of particles present at the surface, and the structure and
TABLE
II
THE INVERSE OF THE SLOPE OF THE ADSORPTION ISOTttERE
in dyne . cm -1
d c / d r , experimental values, in crn -~ dc/dP, calculated from [24-], in cm -~
C~
C14
C~
25 260 850
31 58 31
29 13 4
composition of these particles, but a detailed investigation regarding these possible influences, was not performed. Introducing another process together with diffusion relaxation raises the question whether a satisfactory interpretation might also be given in terms of adsorption-desorption barriers. However, such a mechanism should lower the rate of diffusional exchange between surface and bulk solution and as a consequence it will cause the phase difference between surface tension and deformation to be smaller than in the diffusion controlled case. Here, the opposite effect should be accounted for; the measured phase angles exceed the limit set by diffusion relaxation, which indicates the occurrence of a relaxation process faster than diffusion under the given circumstances. The question remaining to be answered is, why diffusion proceeds relatively slowly in the studied systems, thus giving the opportunity for another relaxation process to occur as well. The rate of diffusion is mainly controlled by the value of dc/dI', the inverse of the slope of the adsorption isotherm. The analysis of the longitudinal wave results yields the value of (dc/dP)(D/2)~ and assuming a diffusion coefficient of 5 X 10-6 cm ~ sec-1, an experimental value of dc/dr may be calculated (Table II). These values are surprisingly low for highly condensed monolayers, where the slope of the adsorption isotherm is known to become practically zero. Analysis of the adsorption isotherms in the region of saturation adsorption, however, shows that the values reported in Table II are perfectly reasonable. This analysis must be based on a surface equation of state, which is the relationship be-
Journal of Colloid and Interface Science, VoL 42, No. 2, February 1973
SURFACE TENSION RELAXATION tween surface tension, adsorption and bulk concentration. For surfaces showing regular instead of perfect behavior, it has been shown (8) that the Frumkin and the Langmuir isotherms may be generalized to: ~r° - - a = - - R T F ~
E
×
I n (1 -
x~') + (x~') ~.
425
For poorly soluble surfactants the bulk mole fraction will be much smaller than one, and if the bulk activity coefficient 72 is assumed to equal unity, c = (1/18)'r2x2% If furthermore P* is taken to be 5.5 X 10-l° moles cm -2, F = 5.5 X 10-1°x2~, and [-23] m a y be transformed into:
[-193
dc
I~°-al
- 108aexp
2-----
dF
72X 2a
--
_
exp
_
2x2.
.
.
[-24]
R T F ~ .J
[-20]
x2~
Values for the parameter a are not avaiiable for the fatty alcohols; for the Cs and C12 fatty Here, the compositions of the solution (superacids a is known to be 7 X 10.7 and 7 X 10.9 script a) and of the surface (superscript s) are moles cm -a respectively (8). The C12 value will given in terms of mole fractions of surfacebe used here for dodecanol; for the C14 and C16 active solute; hence alcohol the a values will be calculated from the • ~ = rd(r, + r ~ ) = r ~ / r ~. given values, assuming a linear relationship between chain length and log a. The calculated Component 1 is water and 2 is solute. I '~ is values for d c / d r have been recorded in Table the saturation adsorption of the surfactant, ~° II; the surface tensions used are the values the surface tension of pure water, and 72 is the measured for the suspensions. bulk activity coefficient. The adsorption energy As regards the order of magnitude and the parameter a is a measure of the surface activity of the solute, and H accounts for non-ideal be- chain length dependence of the d c / d F values havior in the surface. I t is easily verified that in Table II, the agreement is satisfactory. Ob[-19] and [-201 reduce to the well-known Frum- viously, a high degree of surface activity which kin and Langmuir equations, respectively, for is expressed in a low value of a, induces low H=0. values of d c / d r even near saturation adsorpIn the present investigation, the surfaces tion. As a consequence diffusion between surare assumed to be in a state analogous to a face and bulk proceeds relatively slowly in spread monolayer after collapse. Under these these systems. conditions x2*--~ 1 and [-19] and [-20] are simThe applicability of the surface equation of plified to: state [-19~ and [-207 is further substantiated by comparing the measured values of eo (Table I) .o _ ~ = _ R r r ~ In (1 - - x2 *) + [-21-] with a value predicted by these equations. Differentiating [-19] gives: i
--
E
a
--2H
1 -- X2 s
1
dcr 6o
d(r
--
d in F
d
in x2~
~2s~
These equations give for the inverse of the slope of the adsorption isotherm: d'y2oc2 ~ dx2 ~
-
aexp
I
~° -- (r~
2-----
RTF ~ A
.
1-23~
To obtain an expression containing measurable
Journal of Colloid and Interface Science,
VoI. 42, No.
2, F e b r u a r y
1973
426
VEER AND VAN DEN TEMPEL
quantities only, (1 -- x2~) may be eliminated from [-25] using [-21]. For x28 ~ 1, [-25], reads:
~o = R T £ ~
I2 --
ACKNOWLEDGMENTS Thanks are due to Mr. J. Benjamins who skillfull performed the experiments.
H RT
REFERENCES H
+ exp ( + . \ RT R T £ oo/ A
[-26-]
As had already been recognized by LucassenReynders (8), the surface interaction parameter H strongly affects the value of ~o: a change in H of 0.1 R T induces a change in eo of about 10%. For dodecanoic acid H = 2.0 R T (8) and [26] gives, at a surface tension of 25 dyne. cm-1, ~o = 3350 dyne-cm-t. This value is of the same order of magnitude as the values measured for the alcohols, so the use of [-19] and [20] seems justified.
1. R1ES, H. E., AND KIMBALL, W. A., Proc. of the 2n Intern. Congr. Surface Activity I, 23 (1957). 2. GAINES, G. L., JR., "Insoluble monolayers a liquid-gas interfaces," p. 144, Interscience Pu[ lishers, New York, (1965). 3 RABINOVITCH,W., ROBERTSON,R. F., AND MASOi~ S. G., Can. J. Chem. 38, 1881 (1960). 4. LtTCASSEn,J., Trans. Faraday. Soc. 64, 2221 (1968; 5. LUCASSEN,J., ANDVANDIN TEI~PEL, M., Chem. En~ Sci. 27, 1283 (1972); LUCASSEN, J., AnD VA] DEN TEMPEL, M., to be published. 6. LUCASS,cN-REYNDERS, E. H., AnD LUCASSEn, J. Adv. Colloid Interf. Sci. 2, 347 (1969). 7. KITCHENER, S. A., AND STR1CKLAND CONSTABLt R. F., Proc. Roy. Soc. (London) A245, 93 (1958) 8. LUCASSEN, J., AND LUCASSEN-REYNDERS, E. I-I. J. Colloid Interf. Sci. 25, 496 (1967); LUCASSEN REYNDERS,E. H., unpublished.
dournat of Colloid and Interface Science, Vol. 42, No. 2, February 1973
Received May 9, 1972; accepted July 20, 1972 At the surface of aqueous systems containing slightly soluble surfactants, slow relaxation processes occur that are accompanied by high values of the surface dilational modulus. A detailed investigation into these phenomena was carried out in systems containing dodecanol, tetradecanol and hexadecanol, using longitudinal surface waves and stress-relaxation experiments. The state of the surfaces involved resembles a monolayer after collapse: particles are present which do not consist of the thermodynamically stable bulk phase of the surfactant. A quantitative interpretation of the surface tension relaxation is given in terms of two processes: surfactant exchange in the surface between the monolayer and the particles, and diffusional exchange of surfactant between monolayer and bulk solution. The parameters characterizing these processes were measured; the values obtained are in satisfactory agreement with the results of other experiments.
INTRODUCTION
Collapse of a monolayer leads to the formation of molecular aggregates consisting of water and surfactant in a layered structure (1,2). The presence of such aggregates is accompanied by the occurrence of time-dependent processes in a surface (3): the change in surface tension resulting from a change in surface area depends on the time scale of the area change. Longitudinal surface waves are a means to study such phenomena. This type of experiment (4-6) characterizes a surface by means of a dilational modulus ]e I and a phase angle 0; magnitude and time dependence of these parameters are controlled by the relaxation processes occurring in or near the surface. Experiments on a hexadecanol layer spread at the air-water interface, revealed the effect of collapse (Fig. 1). Before collapse, at a surface tension of 48 dyne. cm-1, the surface exhibits almost purely elastic behavior in the time scale investigated: the modulus is hardly frequency dependent and the phase angle
equals zero. At 48 dyne. cm-1 surface tension changes resulting from a small decrease in surface area (1%), relaxed within several minutes only, indicating the onset of collapse phenomena. After a large area decrease (25%) the surface tension relaxed towards 29 dyne.cm -1 within several minutes, and remained at this level sufficiently long to perform another longitudinal wave experiment. Freshly collapsed material will now be present at the surface; it is readily seen from Fig. 1 that in this situation the surface is characterized by a very high value of the modulus, and by phase angles exceeding 7r/4 at low frequencies. The latter phenomenon indicates the occurrence of a relaxation process that is not diffusion controlled, because in such a case phase angles between 0 and 7r/4 will be found only (6). Similar surface behavior has been found on aqueous systems containing fatty acids, fatty alcohols and monoglycerides with straight carbon chains of twelve atoms or more. These 418
Journal of Colloid and Interface Science. Vol. 42. No. 2. February 1973
Copyright ~ 1973 by Academic Press, Inc. All rights of reproduction in 0,ny"fqrrg reserved.
SURFACE TENSION RELAXATION surfactants were dissolved in water at a temperature above their melting point. On cooling, gel-like particles precipitated and the presence of these particles in the surface affected the surface rheology in the same way as particles resulting from collapse phenomena did. The aqueous systems prepared as described here, will further be called "suspensions." The surface behavior mentioned has only been found when non-crystalline material was present; the type of particles involved will be termed "mesomorphous," thus emphasizing that they probably possess a layered structure and that they do not consist of the thermodynamically stable bulk phase of the surfactant. A detailed investigation of the surface tension relaxation was carried out in suspensions containing fatty alcohols. A model is developed to describe the kinetics of this process as it affects longitudinal surface waves and stress-relaxation in the surface. THEORY Two possibilities arise in regard to the influence of mesomorphous particles on relaxation phenomena in the surface: 1. The particles exchange surfactant with the surrounding monolayer, thus affecting the composition and properties of the monolayer. 2. The particles are arranged in a twodimensional network that yields on compression and expansion. The model to be developed here is based upon the first possibility. Consider the surface of an aqueous suspension of a surfactant. In this surface mesomorphous particles are present, but the monolayer part of the surface determines the surface tension. The adsorption in the monolayer part m a y then be regarded as the solubility limit in the surface of the mesomorphous material. If, in this situation, the adsorption is changed by a change in surface area, two surface relaxation mechanisms m a y occur: 1. Surfactant exchange between monolayer and underlying solution by means of diffusion.
419
3.5
t,f t , '~-~/'~
z~~
f
~'
after I~l beforecollapse
°~6-I°~ ° ' "
°~
o
2.5
tan0 after collapse ~ D -t~ before
o~
¢
_~
J
4
• tog co(co in s "1)
Fro. 1. Results of longitudinal wave experiments on spread hexadecanol layer before (at cr = 48 dyne. cm-1) and after (at ¢ = 29 dyne. cm-1) collapse, o~= radial frequency. 2. Surfactant exchange between the monolayer and the particles in the surface. If the former process is assumed to be unaffected by the presence of mesomorphous particles in the bulk solution, its effect on wave behavior is known (5). The process in the surface will be described by a first-order rate process which leads to the following material balance for the monolayer part of the surface:
dr/dt
= -- r(g -- £o) --
D(Oc/Oy)~=o -- r ( d l n A / d t ) .
[11
Here, I' denotes the adsorption (moles.cm -2) in the monolayer part of the surface, in the undisturbed surface this quantity equals Po; r is a rate constant (sec-1), D the bulk diffusion coefficient (cm2.sec -1) and c the bulk concentration (moles. cm-3). The coordinate y is perpendicular to the surface, y = 0 at the surface and y is negative in the solution. The monolayer area is denoted by A (cm2), the time by t(sec). The relaxation process in the surface which is characterized by the parameter r, m a y be regarded as growth of particles during compression and dissolution during expansion of the surface. Assuming the rate of these processes to be proportional to the amount of
Journal of Colloid and Interface Science, VoI. 42, No. 2, February 1973
420
VEER AND VAN DEN TEMPEL
super- or subsaturation, is a simplification. This simplification, however, is justified for growth and evaporation of crystals from the vapor phase, if small values of super and subsaturation are involved (7). In a bulk phase growth and dissolution are processes proceeding in two steps: diffusional transport towards or from the particle controlled by a concentration gradient, and a rate process at the surface of the particle: At the air-water interface the latter process will be rate-detet-mining, as the transport within the interface proceeds relatively fast due to Marangoni-effects. Introducing parameter r as is done in [-1-], implies that r will also be affected by the number of particles present at the surface, because the surface process introduced may be regarded as a heterogeneous reaction. The behavior of the surface during the passage of a longitudinal wave, and during stress-relaxation experiments, will now be described in terms of the parameters introduced in [-1].
frequency of the barrier motion (sec-1) and n = (1 + i)(~/2D)½. At any moment the subsurface concentration cv=0 is assumed to be in equilibrium with the adsorption F; as a consequence the following relationship may be formulated: cl = (dc/dP)v=oG
[-5~
The quantity (dc/dr)~=o represents an equilibrium property of the system, viz. the inverse of the slope of the adsorption isotherm. In this quantity the subscript y = 0 will further be omitted. Substituting I-2-]-I-5] in [-1-] gives: ioaG = -- rG -- (1 + i) dc/dF
X [G(~D/2)~] - ( F / A ) i ~ A I .
[-6]
After the introduction of two dimensionless parameters p = (dc/dr)(D/2co)~ and p = r/oo,
Equation [-6] reads:
[7]
iG = -- pG -- (p + i p ) a - i ( r / A ) A 1 .
So
Longitudinal Surface Waves
In this type of experiment a surface is subjected to a periodic deformation imposed by a surface barrier. The deformations propagate along the surface, introducing spatial variations in the surface composition. The experimental techniques available (4-6) provide means to correct for these propagation effects, thus permitting measurement of the frequency dependence of the surface tension variation. This quantity will now be calculated starting from the frequency dependent variations in I', c and A, for which the following reIations will be valid, if small deviations from equilibrium are considered only: F = ro -Jr- AF = Yo -}- Ge ~
[-2]
c = Co + Ac = Co + cle"~e ¢°~t
[-3]
A = Ao + A A = Ao + A l e i %
[-4]
G=
-- i F A J A { p
+ p + i ( I + P)}.
[-8-]
Now, a relation should be formulated between the measurable quantity--the surface tension variation--and G the quantity which determines the variation in the adsorption. As small variations are considered only, a linear relationship may be used: ~
= (d~/dr)~r
= (d~/dr)aei%
E9]
where ~ is the surface tension in dyne. cm-1. The quantity d r / d r is an equilibrium property of the surface, and therefore this formulation implies that at any moment equilibrium is assumed between surface tension and adsorption in the monolayer. Using [-8] and F9] we find for the complex modulus e: A A d~ e = --/X~ G AA A1 d r
Here, subscript o denotes the values in the absence of surface deformations; G, c~ and A t are complex quantities; w represents the radial Journal of Colloid and Interface Science, Vol. 42, No. 2, F e b r u a r y 1973
~;eo
=
,
p +f
+ i ( 1 +~-)
Bo-1
SURFACE TENSION RELAXATION where eo = - (@~din r). The real part er and the imaginary part ei of this complex modulus are:
= ~
o[1 + KI/[(p +
+ (1 +
= eo[P 4- P-1/[(p 4- p)2 4- (1 4- p)2]].
dAV/dt = -- rAP -- (dc/dr)(D/rrt)~AP.
1-123
Integration gives:
tan 0 = (p 4- ~')/(1 + ~') =
i + [(p
-
equilibrium, may be written as - D ( d c / d F ) X (ar/(~rDt)~). So the equation to be solved is:
1-113
This results in the following expressions for the surface dilational modulus I el and the phase angle 0:
421
in [ ~ r / ( ~ r ) , = o ] = -- rt - 2(dc/dI')(Dt/rc)~.
El6]
[173
If a linear A~ -- Ar relationship is assumed, the right hand side of [17] will equal In
8/(I
+
If p and ~ are of the same order of magnitude, both relaxation processes will affect the surface properties. This will roughly be the case in a frequency range of one decade at both sides of the frequency for which p = ~'. At higher frequencies diffusion will be more dominant (/->> p) and [13] and [14~ will degenerate into the known expressions for diffusion controlled relaxation (6). At lower frequencies p >> ~', the relaxation mechanism in the surface will then be rate-determining. From E13~ it follows that for oJ--+ 0, [ e [ --+ co~p, so the slope in a log ] e[ vs log coplot will approach 1 at low frequencies, whereas for the diffusioncontrolled case (p = 0) the corresponding value equals ½. Equation F14] predicts values of the phase angle between 0 and 4- 7r/2; for f = 0 the tan 0 values should vary~inversely proportional to the frequency. The diffusion controlled case allows values between 0 and 4- 7r/4 only. Stress-Relaxation In stress-relaxation experiments the surface area is subjected to a sudden change after which the surface tension relaxation is measured. For this experiment Eq. [-17 simplifies to:
dr~dr = -- r ( r -- to) - D(Oc/Oy)~=o.
[15]
The diffusion term will be approximated by --DAc/(rrDt)i which, for small deviations from
as well. It is emphasized that, according to this model, particles in a surface affect the surface tension only as a result of exchange of surfactant between the particles and the rnonolayer. This action does not affect the equilibrium between surface tension, adsorption in the monolayer and subsurface concentration; at any moment and location the relationships between these quantities are given by the equilibrium surface equation of state. EXPERIMENTAL Longitudinal wave experiments and surface tension relaxation experiments were performed with aqueous suspensions containing dodecanol, tetradecanol or hexadecanol. The water used was deionized and twice distilled; the dodecanol was ex. Fluka puriss, grade; the tetradecanol was ex. Fluka pract, grade; the hexadecanol sample was prepared in our laboratory. The suspensions contained 20, 10 or 4 rag. 1-1 of the respective alcohols and were prepared by shaking the alcohols with water for 1 hr at a temperature (40, 60 and 70°C, respectively) well above its melting point. On cooling towards room temperature gel-like particles precipitated and the suspension thus obtained was poured into a Langmuir trough (Teflon, 60 X 8 cm). The experiments started about 1 hr after formation of the surface. Experiments with dodecanol were performed at 15°C; the other experiments at 23°C.
Journal of Colloid and Interface Science, Vol. 42, No. 2, February 1973
422
VEER AND VAN DEN TEMPEL I I I
Two techniques were used for longitudinal wave measurements. The technique at high frequency (c0 ~ 103 sec -1) w a s described in Ref. 4. At these frequencies the wavelength of the longitudinal wave is short in comparison with the trough length which permits a direct measurement of damping coefficient and wave number; from these parameters l eI and tan 0 m a y be calculated. The method at low frequencies (10-2 < co< 1 sec-~) was analogous to the one described in Ref. 6. In our apparatus the roughened glass Wilhelmy plate was suspended from a Cahn R G electrobalance. A light-sensitive resistance generated a signal representing the barrier motion. Barrier motion and surface tension were recorded on a Hewlett-Packard 135 AM x - y recorder. From the ellipse thus obtained, the surface dilational modulus and the phase anglemay be calculated
C~2
2
\~n C14
\
I~ log co(coin s M)
Fie. 3. Measured (
) and calculated (. . . .
) phase
angle for aqueous suspensionsof dodecanol, tetradecalml and hexadecanol.
3.5
.......
,o ......""
3 s s .....'*
,2
, / " o/
/4"/
2.~
2
t5
directly, because at these frequencies, the longitudinal wave length is much longer than the length of the trough. The barrier amplitude varied from 0.11 to 0.14 cm. The effective trough length was 32.5 cm. In the relaxation experiments the surface area was changed by moving a surface barrier driven by a motor. The area change varied from 0.5 to 1% in about one second. After the area change the surface tension relaxation was recorded with the combination Wilhelmy plate - - C a h n R G electrobalance--Servoscribe recorder.
t I
-
\
RESULTS
1
T 1' tog co(coin s4)
) and calculated ( - - - - ) surFIO. 2 . 3 d e a s u r e d ( face dilational modulus for aqueous suspensions of dodecanol, tetradecanol and hexadecanol. (~ = 25, 31 and 29 dyne.era -I, respectively).
Figures 2 and 3 give the results of the longitudinal wave experiments. At the lower frequencies the surface dilational moduli were calculated assuming the surface area, occupied by the mesomorphous particles, to be negligible. The data will be interpreted using [13-] and [14-]. From [14-] it follows that the parameter r equals the value of co for which
Journal of Colloid and Interface Science, VoI. 42, No. 2, F e b r u a r y 1973
SURFACE TENSION RELAXATION
10
Cl~o
/
8
5 a
C12
[]
O
3
0
I 2
1
I 3
FIO. 4. Determination of (dc/dP) (D/2)~ using Eq. [-18], for the surface of a dodecanol (O) and a tetradecanol (K]) suspension. tan 0 = 1. Rearranging [,,14] gives:
I+
/--/ ~\2~/
[18]
tanO-- 1
Plotting the right hand side of El8] against (l/w)} should result in a straight line with a slope that equals (dc/dr)(D/2)}; this line should go through the point [-(i/c0)½ = 0, (p-1)/(tan0--1) = 1] (see Fig. 4). The values of r and (dc/dP)(D/2)~ thus obtained m a y now be introduced into [,13] together with the measured [e I values in order to calculate co. The resulting average value of eo is recorded in Table I. The equilibrittm parameter eo equals the surface dilational modulus at frequencies too high for the occurrence of any relaxation process (p = f = 0). Obviously, these conditions were satisfied at o~ ~ 10 ~ sec-I, where tan 0 values were found to be near zero. The I el values measured at this ireTABLE
quency m a y therefore be compared with the calculated eo value. The procedure outlined here was applied to the data of the dodecanol and tetradecanoI suspensions. For the hexadecanol suspension the influence of the relaxation process in the surface on the measured tan 0 values happened to be rather small, which prevented an accurate determination of the parameter r. For hexadecanol the parameters r, (dc/dr)(D/2)} and eo were therefore calculated by extrapolation from the corresponding values for the lower alcohols, assuming linear relationships between chain length and the logarithm of these parameters. From Co, r and ( d c / d r ) ( D / 2 ) } theoretical curves m a y be calculated representing the frequency dependence of [e I and tan0. The results are shown in Figs. 2 and 3 to enable comparison with the experimental curves. Finally the parameters r and (dc/dr)(D/2)} m a y be introduced into [-17] to calculate a theoretical surface tension relaxation curve. I n Fig. 5 these curves are plotted together with the experimental ones.
C16 ~' % o05
--
C~
Cx4
--0.24
-- 1.48
0.58 in s-~
1.0
C12, oO.E 2
LONGITUDINAL WAVE EXPERIMENTS
( l o g ~o) t . n 0 = l
l ~
I
SIJR~ACE PARAMETERS OBTAINED FROlk{
r in s-1
423
0.033
C~o --
0.0019
I~time(s)
dP 0.41 0.092 0.021 eo in dyne. cm-I 1350 1800 2400 [e [~=1oain dyne. cm-1 1800 2300 2800
FIG. 5. Measured surface tension relaxation curves ( ) after compression and expansion; curves predicted from longitudinal wave results (. . . . . ).
Journal of Colloid and Interface Science, Vol. 42, No. 2, F e b r u a r y 1973
424
VEER AND VAN DEN TEMPEL DISCUSSION
By comparing measured and calculated longitudinal wave parameters in Figs. 2 and 3 it may be concluded that the model presented here is useful for a quantitative description of the data. Largest deviations occur for hexadecanol, which is not surprising as in this case the parameter values were obtained by extrapolation. Nevertheless, the predicted curves agree qualitatively with those actually measured. The relaxation curves calculated from the longitudinal wave results provide correct information regarding the time scale of the relaxation-process; it may therefore be concluded that the given interpretation of the longitudinal wave data leads to the determination of physically relevant quantities. Exact prediction of the relaxation behavior was not attained. This will partly be due to the simplifications introduced in the derivation of Eq. [-17]; for instance the use of linear A c - A l l - Aa relationships will only be justified if small deviations from equilibrium are involved. Moreover for greater deviations from equilibrium the rate of the process in the surface will not be proportional to Ap. Rate of spreading and rate of particle growth may then differ considerably, and the r-value measured by means of longitudinal waves will represent an average between these two rates. This may explain why the predicted relaxation curve for tetradecanol seems to be an average of the curves measured after compression and expansion. The same phenomenon was not found for the hexadecanol surface which may be caused by the inaccurate surface parameters available. In the theoretical part of this work the parameter r was related to a rate process at the periphery of the mesomorphous particles. At a fixed temperature such a process is expected to proceed faster as the chain length of the alcohol decreases, which may explain the measured shift in the values of r. However, r may also be affected by the number of particles present at the surface, and the structure and
TABLE
II
THE INVERSE OF THE SLOPE OF THE ADSORPTION ISOTttERE
in dyne . cm -1
d c / d r , experimental values, in crn -~ dc/dP, calculated from [24-], in cm -~
C~
C14
C~
25 260 850
31 58 31
29 13 4
composition of these particles, but a detailed investigation regarding these possible influences, was not performed. Introducing another process together with diffusion relaxation raises the question whether a satisfactory interpretation might also be given in terms of adsorption-desorption barriers. However, such a mechanism should lower the rate of diffusional exchange between surface and bulk solution and as a consequence it will cause the phase difference between surface tension and deformation to be smaller than in the diffusion controlled case. Here, the opposite effect should be accounted for; the measured phase angles exceed the limit set by diffusion relaxation, which indicates the occurrence of a relaxation process faster than diffusion under the given circumstances. The question remaining to be answered is, why diffusion proceeds relatively slowly in the studied systems, thus giving the opportunity for another relaxation process to occur as well. The rate of diffusion is mainly controlled by the value of dc/dI', the inverse of the slope of the adsorption isotherm. The analysis of the longitudinal wave results yields the value of (dc/dP)(D/2)~ and assuming a diffusion coefficient of 5 X 10-6 cm ~ sec-1, an experimental value of dc/dr may be calculated (Table II). These values are surprisingly low for highly condensed monolayers, where the slope of the adsorption isotherm is known to become practically zero. Analysis of the adsorption isotherms in the region of saturation adsorption, however, shows that the values reported in Table II are perfectly reasonable. This analysis must be based on a surface equation of state, which is the relationship be-
Journal of Colloid and Interface Science, VoL 42, No. 2, February 1973
SURFACE TENSION RELAXATION tween surface tension, adsorption and bulk concentration. For surfaces showing regular instead of perfect behavior, it has been shown (8) that the Frumkin and the Langmuir isotherms may be generalized to: ~r° - - a = - - R T F ~
E
×
I n (1 -
x~') + (x~') ~.
425
For poorly soluble surfactants the bulk mole fraction will be much smaller than one, and if the bulk activity coefficient 72 is assumed to equal unity, c = (1/18)'r2x2% If furthermore P* is taken to be 5.5 X 10-l° moles cm -2, F = 5.5 X 10-1°x2~, and [-23] m a y be transformed into:
[-193
dc
I~°-al
- 108aexp
2-----
dF
72X 2a
--
_
exp
_
2x2.
.
.
[-24]
R T F ~ .J
[-20]
x2~
Values for the parameter a are not avaiiable for the fatty alcohols; for the Cs and C12 fatty Here, the compositions of the solution (superacids a is known to be 7 X 10.7 and 7 X 10.9 script a) and of the surface (superscript s) are moles cm -a respectively (8). The C12 value will given in terms of mole fractions of surfacebe used here for dodecanol; for the C14 and C16 active solute; hence alcohol the a values will be calculated from the • ~ = rd(r, + r ~ ) = r ~ / r ~. given values, assuming a linear relationship between chain length and log a. The calculated Component 1 is water and 2 is solute. I '~ is values for d c / d r have been recorded in Table the saturation adsorption of the surfactant, ~° II; the surface tensions used are the values the surface tension of pure water, and 72 is the measured for the suspensions. bulk activity coefficient. The adsorption energy As regards the order of magnitude and the parameter a is a measure of the surface activity of the solute, and H accounts for non-ideal be- chain length dependence of the d c / d F values havior in the surface. I t is easily verified that in Table II, the agreement is satisfactory. Ob[-19] and [-201 reduce to the well-known Frum- viously, a high degree of surface activity which kin and Langmuir equations, respectively, for is expressed in a low value of a, induces low H=0. values of d c / d r even near saturation adsorpIn the present investigation, the surfaces tion. As a consequence diffusion between surare assumed to be in a state analogous to a face and bulk proceeds relatively slowly in spread monolayer after collapse. Under these these systems. conditions x2*--~ 1 and [-19] and [-20] are simThe applicability of the surface equation of plified to: state [-19~ and [-207 is further substantiated by comparing the measured values of eo (Table I) .o _ ~ = _ R r r ~ In (1 - - x2 *) + [-21-] with a value predicted by these equations. Differentiating [-19] gives: i
--
E
a
--2H
1 -- X2 s
1
dcr 6o
d(r
--
d in F
d
in x2~
~2s~
These equations give for the inverse of the slope of the adsorption isotherm: d'y2oc2 ~ dx2 ~
-
aexp
I
~° -- (r~
2-----
RTF ~ A
.
1-23~
To obtain an expression containing measurable
Journal of Colloid and Interface Science,
VoI. 42, No.
2, F e b r u a r y
1973
426
VEER AND VAN DEN TEMPEL
quantities only, (1 -- x2~) may be eliminated from [-25] using [-21]. For x28 ~ 1, [-25], reads:
~o = R T £ ~
I2 --
ACKNOWLEDGMENTS Thanks are due to Mr. J. Benjamins who skillfull performed the experiments.
H RT
REFERENCES H
+ exp ( + . \ RT R T £ oo/ A
[-26-]
As had already been recognized by LucassenReynders (8), the surface interaction parameter H strongly affects the value of ~o: a change in H of 0.1 R T induces a change in eo of about 10%. For dodecanoic acid H = 2.0 R T (8) and [26] gives, at a surface tension of 25 dyne. cm-1, ~o = 3350 dyne-cm-t. This value is of the same order of magnitude as the values measured for the alcohols, so the use of [-19] and [20] seems justified.
1. R1ES, H. E., AND KIMBALL, W. A., Proc. of the 2n Intern. Congr. Surface Activity I, 23 (1957). 2. GAINES, G. L., JR., "Insoluble monolayers a liquid-gas interfaces," p. 144, Interscience Pu[ lishers, New York, (1965). 3 RABINOVITCH,W., ROBERTSON,R. F., AND MASOi~ S. G., Can. J. Chem. 38, 1881 (1960). 4. LtTCASSEn,J., Trans. Faraday. Soc. 64, 2221 (1968; 5. LUCASSEN,J., ANDVANDIN TEI~PEL, M., Chem. En~ Sci. 27, 1283 (1972); LUCASSEN, J., AnD VA] DEN TEMPEL, M., to be published. 6. LUCASS,cN-REYNDERS, E. H., AnD LUCASSEn, J. Adv. Colloid Interf. Sci. 2, 347 (1969). 7. KITCHENER, S. A., AND STR1CKLAND CONSTABLt R. F., Proc. Roy. Soc. (London) A245, 93 (1958) 8. LUCASSEN, J., AND LUCASSEN-REYNDERS, E. I-I. J. Colloid Interf. Sci. 25, 496 (1967); LUCASSEN REYNDERS,E. H., unpublished.
dournat of Colloid and Interface Science, Vol. 42, No. 2, February 1973