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Langmuir 1993,9, 273-277
A Simple Test for Diffusion-Controlled Adsorption at an Air/Water Interface Qiang Jiang and Yee C. Chiew' Department of Chemical and Biochemical Engineering, Rutgers University, Piscataway, New Jersey 08855-0909 Received March 10,1992. In Final Form: August 28,1992 We proposed an experiment which can be used to test if bulk diffusion is the controlling mechanism for surfactant adsorption at an air/liquid interface. Thismethod involvesthe measurementaof the dynamic surfactant tension n(t)and the high-frequency surface elastic modulus e&), as a function of time, for a newly created fresh interface. A key feature of the proposed approach is that it does not require information on the surfactant bulk diffusion coefficient and the use of a surface equation of state. It is shown that the quantity tQ-1 dWdt must be equal to 0.6 for diffusion-controlled adsorption. This test was used to investigate the adsorption of two nonionic surfactants, C12M (hexaethyleneglycol mono-n-dodecylether) and Triton-X-100). may be neglected, and eq 1 simplifies to
1. Introduction
The characterization of the time-dependent behavior of adsorption of soluble surfactant molecules to the air/ water interface is a challenging and interesting problem that has attracted a great deal of attention.'-' It is a subject of considerablepractical relevance to a number of interfacial processes such as coating, wetting, foaming, detergency, and emulsification. The kinetics of surfactant adsorption is typically investigated through the observation of the dynamic surface tension, and its approach to equilibrium. It has been postulated6 that the kinetics of adsorption of surfactant molecules onto the air/water interface may be caused by the following three relaxation processee: (a)the transport of soluble surfactant molecules by diffusionlconvectinfrom the bulk to the "subsurface" immediately below the &/water interface, (b) the adsorption of surfactants from this subsurface layer to the interfaceover an "energy barrier", and (c)the reorientation of adsorbed molecules in the surface. In diffusioncontrolled adsorption, the surfactant concentration r(t ) at the interface, at time t, is considered to be determined solely by diffusive transport of surfactants from the bulk (with no adsorption barrier or slow equilibration process present in the interface). Theoretical modele for diffusion-controlledadsorption provide a quantitative descriptionof the time dependence of the surface concentration U t )of surfactanta, after the formationof a fresh interface. Ward and Tordais showed that, in diffusion-controlled adsorption, the surface concentration Ut) is given by r(t)
~ ( D / T ) ' ~ ~ -[ C c ' ~* C~ s'(/t ~)d(t - 2)'/'1
(1)
Here, C,(t) represents the surfactant concentration immediately below the interface, c b is the bulk surfactant concentration (constant) far from the interface, and D denotes the surfactant diffusion coefficient in the bulk solution. At small t, the integral in the above equation
* To whom correspondence should be addreseed.
(1) Lucaasaen-Reynders, E. H.; Lucasaen, J.; Garrett, P.; Giles, D.; Hollway, F. In Monolayers; Advaucea in Chemistry Series; Goddard, E. D., Ed.;American Chemical Society Washington, DC, 1972;Vol. 144,p 272. (2)Kretzachmar, G.; Mier, R. Adu. Colloid Interface Sci. 1991,36,
65. (3) Hua, X. Y.; Roeen, M. J. J. Colloid Interface Sci. 1988,124,652. (4)Semen, G.; Jooll, P. J. Colloid Interface Sci. ISSO, 139, 652. (5)van den Tempel, M.; Lucasaen-Reyndem, E. H. Adu. Colloid Interfuce Sci. 190, 18, 281. (6) Ward, A.; Tor&, L. J. Chem. Phys. 1946,14,453.
r(t)= 2(D/T)'/2Cbt'/2
(2)
Ruation 2 provides a quantitative description for, and may be used as a test for, diffusion-controlledadsorption. Because U t )cannot be easily measured, the conventional approach to the problem involves the direct measurement of the surface pressure II(t),as a function of time t, for a newly created fresh surface; this measured dynamic surface pressure II(t) is then used to obtain the surface concentration r(t)w t data through the use of a surface equation of state; i.e., II = II(r).The surface equation of state is obtained indirectly through equilibriummeasurements of surface pressure versus bulk concentration c b , and the thermodynamic Gibbs adsorption botherm. Because surfaceequationsof state are generally not known exactly,two different equations of state that fit the same set of equilibrium n-f& data may yield quite different results for the surfaceconcentrationr(t)-t.Furthermore, the surfactant diffusion coefficient in the bulk aqueous phase is not known accurately, and, in general, can only be determined within an order of magnitude. Consequently, uncertainties are inevitably introduced in such an approach. With the aim of obtaining a better understanding of adsorption at interfaces,we proposed an experiment that allows us to determine if diffusion is the dominant mechanism in adsorption without requiring knowledge of the surfactant surfaceequation of stateand bulk diffueion coefficient. This experiment involves the simultaneous measurements of the dynamic surface tension II(t) and the high-frequency surface dilational elastic modulus e,( t ) ,as a function of time t, for a newly created &/water interface. In this work, we examine the adsorption of two nonionic surfactants, Triton-X-100and C12E6, using the propoaed method. The theoretical basis for our experiment will be presentedin section 2. Experimentalmethods and apparatus are described in section 3. Major results obtained in this work are given in section 4. 2. Theoretical Background The theoretical basis of the proposed test will be considered in this section. The test involves the simultaneous measurements of the surface pressure n(t) and the high-frequency limit of the elastic modulus e&), at time t, during the course of adsorption. Because the surface elastic modulus plays an important role in this
0743-7463/93/2409-0273$04.00/0 (Q 1993 American Chemical Society
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274 Langmuir, Vol. 9, No. 1, 1993
test, we will briefly present some basic definitions and ideas relating to e&), followed by the derivation of the proposed test given by eq 7 below. The surface dilational viscoelasticity e is a quantitative measure of the response of the surfaceto local compression or expansion. It is defined as the ratio of the surface tension y variation to the fractional area (A) change by dy/d ln A = -dII/d ln A (3) Here, II represents the surface pressure that is defined by II = yo- y,where yois the surface tension of the air/water interface with no adsorption. Note that, in general, the quantity e is a function of the surfactant concentration at the interface, and depends on the time scale or frequency w of area compression/dilation. This frequency dependence is influenced by relaxation processes that affect the surface tension gradients at the interface as a result of area dilation/compression. In the case of diffusioncontrolled adsorption of soluble surfactants, the surface tension gradients are reduced by the transport of surfactants from the bulk. This effect may be represented by writing eq 3 as7
SURFACE WAVE PROFILER
I
e
e
= (dn/d ln I'),(-d ln r / d ln A ) ,
-
= eo = (dII/d ln r),
(5)
Note that the quantity eo, that is, the high-frequencylimit of the modulus e, may be identified as the thermodynamic surface elastic modulus. From the small time behavior of diffusion-controlled adsorption, given by eq 2, it is straightforward to show that d ln r(t)= (1/2) d ln t for small t If eq 5 is combined with eq 6a, we obtain
(6a)
A criterionfor diffusion-controlledadsorption is obtained (7) Lucaeeen, J.; Haneen, R. S. J. CoZZoid Interface Sei. 1967,23,319.
POWER
Pt plate
MPLlFlER TROUGH
(4)
Here, the first term on the right-hand side of eq 4 accounts for the effect of surface concentration r on 11,while the second term accounts for the additional effects of other relaxation processes on the surface concentration (and, consequently,the surface tension gradients) at the interface when the surface is subject to area dilation/compression at different frequencies. Clearly, in the case of insoluble surfactants, the product FA = constant, and -d In r / d In A is identically unity. In the case of diffusioncontrolled adsorption of soluble surfactants, the term -d ln r / d In A takes into account the variation of surfactant surfaceconcentrationsdue to diffusionas a result of surface area changes. The influence of diffusive transport on the modulus e is expected to be important if the frequency of surface deformation o is comparable to the characteristic frequency of diffusion WD = D(dC/dr)2/2.7 The influence of diffusion on e will decrease as w increases since the amount of surfactants transported from and to the surface will become smaller and less able to introduce significant changes in the surface tension. Hence, at sufficientlyhigh frequencies (i.e., w >> WD), the effect of surfactant bulk/ interfaceinterchangeon the surfacepressure and elasticity becomes negligible, the adsorbed interface behaves like an "insoluble"monolayer,and FA = constant. This implies that the quantity -d In r / d ln A approaches a value of unity, and that the surface dilational viscoelastic modulus e, in the limit of o -, may be written as e(--)
BE AM SPLITTER
LOCK-IN
FUNCTION
AMPLIFIER
ENERATOR
Figure 1. Schematicrepresentationof the electrocapillarywave generation and detection system.
by rearranging eq 6b to yield dIIW 1 forsmallt (7) dt 2 The equality in the above equation must hold if the adsorption process is diffusion-controlled. Since the quantities that occur on the left-hand side of eq 7 can be obtained experimentally (note that both the elastic modulus e&) and the rate of change of surface pressure dII/dt are experimentally measurable quantities), eq 7 can be used as a test for diffusion-controlled adsorption. A key feature of the proposed test is that it does not require information on the surface equation of state and the bulk diffusion coefficient. t
--=e()@)
3. Experimental Methods and Materials The dynamic surface pressure n(t)was measured using the Wilhelmyplate-Cahnbalancetechnique. The surface elasticity EO can be measured through electrocapillary ~lane8-'~ and cylindrica113J5propagating waves. In this investigation, plane capillary waves were excited, using the apparatus described earlier,14by applyinga sinusoidal and a de offset voltage between a metal blade and the water surface. The surface can be scanned over a distance of 80 mm by a computer-controlled linear translation motor. The waveswere detected by specularreflection of a focused laser beam from the surface to a photodiode. By using a lock-in technique, the wavelength X and damping coefficient@ can be measured accuratelyto within 0.1 96 and 1 7%, respectively, in the frequency range from 50 to lo00 Hz. A schematic of the apparatus is shown in Figure 1. The two nonionic surfactants C12E6 (hexaethylene glycol mono-n-dodecylether) (Nikkol, Japan, and supplied by Mitsui (8) Sohl,C. H.; Miyano, K.; Katterson, J. B. Rev. Sci. Instrum. 1978, 49,1464. (9) Nagarajan, N.; Webb, W. W.; Widom, B. J. Chem. Phys. 1982,77, 5771. (10) Miyano, K.; Abraham, B. M.; Tmg, L.; Wasan, D. T. J. Colloid. Interface Sei. 1983,92,297. (11) Stenvot, C.; Langevin, D. Langmuir 1988,4, 1179. (12) Vogel, V.; Mobius, D. Langmuir 1989,5, 129. (13)Ito, K.; Sauer, B. B.; Skarlupka, R. J.; Maaahito, S.; Yu, H. Langmuir 1990,6, 1379. (14) Jiang, Q.; Chiew, Y. C.; Valentini, J. E. J. Colloid. Sci., in prem. (15) Jiang, Q.; Chiew, Y. C.; Valentini, J. E. Langmuir 1992,8,2747.
Langmuir, Vol. 9, No.1,1993 276
Adsorption at an AirlWater Interface Table I. C12E6 and Triton-X-100Nonionic Aqueous Surfactant Solutions Uwd in the Exmriments ~
solution 1 2
3 4
surfactant c12E6 c12E6 Triton-X-100 Triton-X-100
~
0.23
~~
concn (mol/cm3) 4.5 x 10-10 4.5 x 10-10
temp ("C)
3.5 X 10-l0 4.5 X 10-l0
20 20
18 22
n
5
Y
5
p
P
2.00
0.22
5
1.50
0
0.21
0
\
C
0
0
F.
20
40
0
-a
T
Y
1.00
100
120
Figure 3. Measured values of the wavelength A(t) plotted as a function of time: open triangles, 4.5 x 10-10 mol/cms of C12E6 at 18 OC; open circles, 4.5 X 10-lomoUcm3of C12B at 22 O C ; open squares, 3.5 x 10-l0mol/cm3 of Triton-X-100 at 20 OC; crosses, 4.5 x 10-l0mol/cm* of Triton-X-100at 20 OC.
4
U
-8
80
time (min)
3
;
60
U
1
0.50 0,001 0.01
11
0.1
1
10
100
1000
frequency (Hz)
Figure 2. Measured values of the surface elastic modulus 4 plotted as a function of freqwcy: open triangles, 4.5 X 10-10 moUcm3 of Triton-X-100at 20 OC; open circles, 1.0 X 10-8 mol/ cm9of C12EX at 25 OC. The low-frequency data (w C 0.3 Hz)for C12EX are taken from ref 16. All other data were measured in our laboratory.
and Co.) and Triton-X-100 (purchased from Fisher Scientific) were used in the experiments. They were used as received without further purifiition. All solutions were prepared with doubledietilled water. Four surfactant solutions were used in this work (please refer to Table I for details). In our experiments,plane electrocapillarywavea were generated at 200 Hz. For the C12EX and Triton-X-100 surfactants considered here, the &/water interface behaves like an elastic monolayer with negligible surface dilational viscosity qd14J6 at this frequency. Furthemre, the surface elastic modulus t(w=200Hz) lies in the frequency-independentregime. This can be wan from Figure 2 in which measured valuea of e(w) for TritonX-100 (4.5 X WOmoVcma, 20 "C) andC12EX (1.0 X 10-8moVcm3, 25 O C ) are plotted as a function of frequency w. For w < 0.2 Hz, the modulus was obtained using the modiied Langmuir trough, a technique that is similar to the one designed and used by Lucacsen and Gilee.16 In the intermediate frequency range, i.e., 1Hz < w < 10 HE, the data were obtained through the damping of longitudinal surface waves.20 For w > a0 Hz,the modulus e was determined through the damping of plane electrocapillary wavea. A carefulexaminationof Figure 2 revealsthat, for TritonX-100 (open triangles),the modulus ettainsa constant value and becomes frequency independent at w = 0.1 Hz. In the case of C12E8,the modulusia obesrvedtobecome frequency independent at about 50 Hz. Since the concentretiom of the C12E6 systems (Le., solutions 1and 2 in Table I) used in our experiments are substantially (2 ort$ers of magnitude)rower than those shown in Figure2, one erpedte that the modulus for solutions 1and 2 will appro& the hi$h-frequency limit at a frequency below 50 Hz. These rauilts clemly damonstrate that, for the four solutions considered here, the h t i c modulus obtained by capillary wave (16) Lwa"n, J.; Gilea, D.d. C k m . Soc., Faraday *am. 1 1972,68,
2129.
m
:.
0.60
5
D
0.20
0
20
40
60
80
100
120
time (min)
Figure4. M~~valueaofthedampingcoafficientB(t) plotted as a function of time. The symbols are the same as in Figure 3.
measurementsat 200 Hz lies within the frequency-independent regime and corresponds to the high-frequency elastic limit EO. ThewavelengthA(t)anddampinecoaffcientB(t),attimet,&r the creation of a fresh interface (this is achieved by sweeping the surface with a Teflon barrier at t = 01, were measured at 200 Hz. The surface elasticity g(t) was then calculated from theae measured values of A(t) and B(t) by solving the dispersion of plane capillary waves in the high-frequency limit. The computational scheme for g(t) is clearly outlined in ref 7, and will not be reproduced here. In addition to d t ) , the computation ale0 yields calculatedvalues of II(t). It is noticed that the calculated surfacepmsure nisaccuratetoabout0.25dyn/om;moreaccurate surface pressure n(t)data obtained using the Wilhelmy plate technique (accurate to 0.06 dyn/cm) were used in the teat for diffusion-controlled adsorption.
4. Results and Dircussion Displayed in Figure 3 are measured valuea of the wavelength X(t) for C12Eg and Triton-X-100,at different concentrations and temperatures,plotted as a functionof timet. Itisobeervedthatthewavele~in~elishtly, and thendecrease rapidlywithtime. Thetimedependence of the damping coefficient B(t) is &own in Figure 4. For all four solutions, the damping coefficiemta B(t) are men
Jiang and Chiew
276 hngmuir, Vol. 9, No.1, 1993 51
4t
if' *++
I/
1
15
I
10
1
t
$
Y
3
f
+Tf
5
+ A
cn
k 2
3
u-
a
0 0
1
20
40
+I
25
80
100
120
time (mln)
0 0
60
50
75
100
125
time (min)
Figure 6. Dynamic surface pressure n(t)plotted as a function of time for 4.5 X 10-10mol/cm3of C12E6 at 18"C. Open triangles represent values of n(t)calculated from the corresponding wave parameters. Croeees represent values of n(t)measured using the Wilhelmy plate technique.
Figure 7. Surface dilational elasticity plotted as a function of time. The symbols are the same as in Figure 3. To avoid crowding in the figure, only the error bars for solution 2 (C12E6, 4.5 X 1 0 - l O mol/cm3at 22 OC) are included. The error bars for the other three systems are similar to those of solution 2. 0.80 j
0.60
I
0,20:
0.00 0
20
40
60
80
100
120
time (min) time (min)
Figure 6. Measured values of dynamic surface pressure n(t) plotted as a function of time: I, 4.6 X Womol/cmgof C 1 2 B at 22 "C; II,4.5 X 10-lomol/cm3 of Triton-X-100 at 20 "C; 111, 3.6 X 10-10 mol/cms of Triton-X-100 at 20 "C.
to increase initially, go through a maximum, and then decrease with increasing t. As d i s c 4 earlier, the timedependentsurface elasticity e&) and the dynamic surface pressure n(t),at time t , can be calculated from pairs of values of A ( t ) and @(t)on the basis of the computational scheme given in ref 7. Shown in Figure 6 is the calculated for C12E6 at a surface pressure II(t) (from A(t) and concentration of 4.6 X 10-lo moucm3 and 18 OC,along with the measured values of II(t)from the Wilhelmy plate technique. The two sets of data are in agreement within experimental errors. The measured dynamic surface pressures n(t)of the other three solutions are displayed in Figure 6. The calculatedvalues of the surfaceelasticity 4)areplotted in F w e 7 for all four solutionsconsidered. It is seen that the values of e&), which correspond to maximum damping (see F w e 31, lie in the range of 6-8 dyn/cm, as predictad by the theory.17JS The accuracy of (17) Lucerren, J.; Hanren, R 5. J. Colloid Interface Sci. 1967,22,32.
Figure 8. t(co(t))-l dWdt plotted as a function of time. The symbols are the same as in Figure 3. co is estimated to be 3% at maximum damping and about 10% when to = 2 dyn/cm.
For diffusion-controlled adsorption, the quantity E(t) t ( ~ ( t ) ) -dn/dt l is expected to be equal to l/2 at small t according to eq 7. Displayed in Figure 8are the values of E(t) plotted as a function of time t. An examination of the figure reveals that, in the case of C12E6,the quantity t ( ~ o ( t ) )dII/dt -~ remains at 0.5 in the fmt 50 min of adsorption before it deviates from the value of l/2. For Triton-X-100, this deviation occurs earlier when t > 20 min. This deviation is probably due to the fact that eq 7 no longer holds sinceit is only valid at emall times. These results clearly suggest that the adsorption of C12E6 and Triton-X-100is diffusion-controlled. As indicated earlier, the proposed test for diffusioncontrolled adsorption is only valid for small t. It ie useful to estimate the regime of validity of eq 7. In the caee of diffusion-controlledadsorption, it is reasonable toaseume that C,is in equilibriumwith the adsorbed surfactants (at concentration r). This suggests that the subsurface concentration C, may be obtained using the Szyszykoweki (18) Thimmn, D.;Scheludko, A. Kolloid 2.2. Polym.
M I , 218,139.
Langmuir, Vol. 9, No. 1, 1999 277
Adsorption at an AirlWater Interface equati0n:lQ II(t) = RZT- In (C,(t)/a+ 1) (8) In the situation encountered here, Le., c b
