Langmuir 1996,11, 555-562
555
Adsorption Kinetics of Decanol at the Air-Water Interface Shi-Yow Lin,* Ting-Li Lu, and Woei-Bor Hwang Department of Chemical Engineering, National Taiwan Institute of Technology, Taipei, 106, Taiwan, R.O.C. Received April 18, 1994. In Final Form: August 23, 1994@ The adsorptionkinetics is studied for l-decanol by using pendant bubble tensiometry enhanced by video digitization. Cohesive forces between decanol molecules adsorbed onto an air-water interface play an important role in the sorption kinetics. Intermolecular cohesive forces among the adsorbed molecules raise the energy barrier for desorption of the adsorbed molecules into the bulk sublayer and correspondingly lower the desorption rate. For adsorption onto an initially clean interface, a reduction in the desorption rate affects the tail end of the process and diffusion is the rate-controlledstep. After the establishment of equilibrium,adsorbed monolayers at the air-water interface are compressed by shrinking the air bubble slightly. The subsequent re-equilibration,affected by the lower desorption rate during the entire period of the process, is controlled by both diffusion and sorption kinetics. A diffusion coefficient is computed by comparing the dynamic surface tension profiles of clean adsorption with numerical solutions of bulk surfactant diffusion equation and a generalized Frumkin adsorption isotherm. The adsorption and desorption rate constants are determined from the experimental relaxation of re-equilibration.
1. Introduction Adldesorption of bulk-soluble surfactants onto or out of a fluid interface occurs in two consecutive steps: (i) surfactant molecules diffuse between the bulk phase and the sublayer adjacent to the fluid interface; (ii) surfactant molecules exchange between the sublayer and fluid interface through the kinetic processes of adsorption and desorption. The description of these processes and the evaluation of the diffusion coefficient and rate constants are important in understanding the influence of surfactant adsorption on many processes. To date, diffusion coefficients and sorptive kinetic constants have been primarily determined from the measurement of surface tension relaxation due to surfactant exchange between the bulk and fluid interface. The experimental measurements are of three types: (i) adsorption onto a freshly created interface; (ii)dynamic surface phenomena of a fluid interface under a periodic expansion and contraction; (iii) re-equilibration from a sudden contraction (or expansion) of an interface originally at equilibrium. Some examples of the measuring technique include oscillatingjet,lI2expanding or compressing on a Langmuir Langmuir trough with a reversed funnel,8surface wave: and pendant drop method.1° Note that the relaxation is determined by the series processes of bulk diffusion and sorption kinetics. Fitting the data with two unknown constants is difficult. It is usually assumed that the process is either diffusion controlled or sorption kinetic controlled and therefore one is able to evaluate one of the parameters. When the adsorbing species consists of long slender hydrocarbon chains and small polar groups such as long-
* Author to whom correspondence
should be addressed. Abstract published inAdvance ACSAbstracts, January 1,1995. (1)Joos, P.; Serrien, G. J . Colloid Interface Sci. 1989,127,97. (2)Bleys, G.;Joos, P. J . Phys. Chem. 1985,89,1027. (3)Baret, J. F.; Bois, A. G.; Casalta, L.; Dupin, J. J.; Firpo, J. L.; Gonella, J.; Melinon, J. P. J . Colloid Interface Sci. 1975,53,50. (4)Panaiotov, I.; Dimitrov, D. S.; Ivanova, M. G. J . Colloid Interface Sci. 1979,69,318. (5) Panaiotov, I.; Sanfeld, A.; Bois, A,; Baret, J. F. J . Colloid Interface Sci. 1983,96,315. ( 6 )De Keyser, P.; Joos, P. J . Colloid Interface Sci. 1983,91,131. (7)Joos, P.; Bleys, G. Colloid Polym. Sci. 1983,261,1038. (8) Van hunsel, J.; Vollhardt, D.; Joos, P. Langmuir 1989,5,528. (9)Lucassen, J.;vandenTempe1, M. Chem.Eng. Sci. 1972,27,1283. (10)Lin, S.Y.; McKeigue, K.; Maldarelli, C.AIChE J . 1990,36,1785. @
0743-7463/95/2411-0555$09.00/0
chain n-alkanols, strong cohesive forces can be created at high surface concentrations. As the interface becomes crowded, the molecules become oriented perpendicular to the interface facilitating strong van der Waals bonding.'lJ2 In this case, the desorption becomes very slow because of the large energy barrier it is necessary to overcome to move the surfactant from the surface into the adjoining bulk sublayer. This effect bears significantly on developing an accurate methodology for measuring simultaneously the diffusion coefficient and the kinetic rate constants. When such surfactants adsorb onto a clean interface, this deceleration in the adsorption rate only affects the latter stages of the surface tension relaxation for which the accumulated surface coverage is reasonably high. On the other hand, when the interface is compressed or expanded from an equilibrium, relatively high surface coverage, the effective desorption rate governing the surface tension relaxation is much smaller than in the adsorption case. Therefore this re-equilibration relaxation is more kinetically limited than the adsorption relaxation. If the desorption rate becomes much smaller than the diffusion rate, the re-equilibration will be sorption kinetic controlled. When the desorption rate is the same order of magnitude as the diffusion rate, the re-equilibration relaxation is of mixed control, i.e., governed by both diffusion and sorption kinetics. Ideally one may choose conditions for which the adsorption relaxation is diffusion controlled and the reequilibration relaxation is of mixed control or of sorption kinetic control. Determination of the diffusion coefficient and the kinetic constants is then straightforward. The adsorption surface tension relaxation data are used to obtain the diffusion coefficient, and the mixed or kineticcontrolled re-equilibration relaxation is used to obtain the kinetic rate constants by use of the already determined value of the diffusion coefficient. Previous studies of the adsorption of surfactant onto an interface and the desorption from fluid interface onto adjoining bulk sublayer have neglected the influence of cohesive forces and have used the Langmuir model to formulate the kinetics. Ward and Tordai13developed, for a semi-infinite planar geometry, a linear convolution integral solution for the surface concentration as a function (11)Fainerman, V. B. Kolloidn. Zh. 1977,39,113. (12)Lin, S.Y.;McKeigue, K.; Maldarelli, C. Langmuir 1991,7,1055. (13)Ward, A. F. H.; Tordai, L. J. Chem. Phys. 1946,14,453.
0 1995 American Chemical Society
Lin et al.
556 Langmuir, Vol. 11, No. 2, 1995 of time in terms of the unknown subsurface concentration. Panaiotov et al.5analyzed the influence of a desorptioncontrolled interchange on dynamic surface tension excess at a uniformly compressed interface. For the case of diffusion control, the surface concentration is obtained by simultaneously solving the integral equation and the adsorption isotherm, while for mixed kinetics the sorption kinetic expression must be solved alongside the integral expression. Re-equilibration relaxation has been discussed by Van hunsel et a1.,8 but only in the context of diffusion control and small perturbations from equilibrium. In this case, the relaxation can be described by a modification of Sutherland's equations14 since the kinetics is linear for small deviations. The inclusion of cohesive forces was undertaken by Miller15for diffision-controlled adsorption by solving the convolution integral with the Frumkin and v a n der Waals isotherms. Borwankar and Wasan16obtained a solution for mixed adsorption using a modified Frumkin kinetic expression. While these last two studies clearly demonstrate the influence of the cohesive energy interaction parameter on the adsorption relaxation, they do not compare these curves with the corresponding re-equilibration relaxation. The aim of this paper is to verify the idea that, due to the effect of cohesive energies between adsorbed surfactant molecules, some surfactants may have a diffusioncontrolled adsorption and kinetic-diffusive-controlled reequilibration process. A video-enhanced pendant drop tensiometry was used to investigate the adsorption onto a clean interface and the re-equilibration relaxation of a suddenly compressed air-water interface which is originally at equilibrium. Profiles of the relaxation of l-decanol were measured. The influence of intermolecular cohesion on the sorption kinetics was incorporated by applying a generalized Frumkin adsorption isotherm in which the activation energy for desorption increases with t h e surface concentration. Comparison was made for the entire relaxation period. Therefore the diffusion coefficient can be determined from the adsorption experiments and re-equilibration experiments can be used to obtain the sorption rate constants. An outline of this paper is as follows. Section 2 describes briefly the pendant bubble experimental technique and details the relaxation profiles for the adsorption and reequilibrium experiments. The theoretical framework for the surfactant mass transfer process and the numerical solution procedure are given in section 3. In section 4, the experimental relaxation profiles are compared with theoretical solutions, which leads to computation of the diffusion coefficient and sorptive rate constants. The paper ends with a conclusion and discussion section.
2. Experimental Measurements 2.1. Materials. 1-Decanol(purity 99+%) was obtained from Aldrich Chemical Co. and used without modification. Acetone (HPLC grade) used to verify the measurement was obtained from Fisher ScientificCo. The water with which the aqueous solutions were made was purified via a Barnstead NANOpure water purification system, with the output water having a specific conductance less than 0.057 pS-l cm-l (uS/cm). The values of the surface tension of air-water and air-acetone, using the pendant bubble technique described below, were 72.3 and 23.2 dyn/cm, respectively at 22.70 & 0.02 "C. 2.2. Pendant Bubble Apparatus Design and Experimental Procedure. Pendant bubble tensiometry enhanced by video digitization was employed for the measurement of the 1-decanolsurface tension relaxation profiles and the equilibrium tensions. This particular application of the pendant bubble (14)Sutherland, K. Aust. J. Sei. Res. 1962, A5, 683. (15)Miller, R. Colloid Polym. Sei. 1981, 259, 375. (16) Borwankar,R. P.; Wasan, D. T. Chem. Eng. Sci. 1983,38,1637.
yH,
,-ysv
VTR
FCG
QP A
D
I
C
(IiD D
B
D
C
I
Vibration-Isolation table
Figure 1. Schematic ofpendant bubble tensiometry apparatus and the video digitization equipment: A, light source; B, pin hole; C, filter; CCD, video camera; D, plano-convex lens; DA, DIA Data Translation card; E, thermostatic air chamber, quartz cell, and suspending inverted needle; F, objective lens; FCG, frame code generator; FG, frame grabber; M, monitor; PC, personal computer; S, syringe; SP, syringe pump; SV, solenoid valve; VTR, video recorder.
++.
(dyn'cm)50
-i
**+.
*+*
5 H I
)O
Figure 2. Representative dynamic surface tensions (dydcm) for clean adsorption of 1-decanol aqueous solutions for CO= (1) 1.58 x (2) 1.97 x (3) 2.84 x lo+, (4)3.95 x (5) 6.32 x (6) 1.018 x and (7) 1.579 x mol/cm3. technique is described in detail in a previous study,'O and therefore only a brief description is given here. The equipment shown in Figure 1is used to create a silhouette of a pendant bubble, to video image the silhouette, and t o digitize the image. The image-forming and recording system consists of a light source (a halogen lamp with constant light intensity), a plano-convex lens system for producing a collimated beam, a quartz cell enclosed in a thermostatic air chamber, an objective lens, a video recorder (VO-9600,Sony)with frame code generator (FCG-700,Sony), and a solid state video camera (MS-4030CCD, Sierra Scientific Co.). The air thermostat is made of acrylic material of 1cm thickness (inside diameter is 18 x 53 x 36 cm) with a cooling copper pipe, a heater, and a fan inside. A cooling water of constant temperature passes through the copper pipe to take out energy. The heater is controlled by a computer with a PI control system, and the fan makes the air temperature uniform. The temperature stability of solution in the quartz cell placed in this air thermostat is better than ~k0.02K. The bubble-formingsystem consists of a stainless steel inverted needle (i.d. = 0.016 in.) which is connected to the normally closed port of a three-way miniature solenoid valve via V16 in. i.d. Teflon tubing. The common port of the valve is connected by the same Teflon tubing to a gas-tight Hamilton syringe placed in a syringe pump. The valve is controlled by the output signal of a D/A Data Translation card (DT2815). The video image digitizer (DT 2861 Arithmetic Frame Grabber, Data Translation), installed on a personal computer, digitizes the picture into 480 lines x 512 pixels and assigns to each one a level of gray with eight-bit resolution. An edge detection routine was devised to locate the interface contour from the digitized image, and a calibration procedure
Air- Water Interface Kinetics
'"
of
Decanol
Langmuir, Vol. 11, No. 2, 1995 557
1
I
1
40
oi
i
100
10
1000
t(s) Figure 3. Experimental values of the dynamic surface tensions (dydcm) for clean adsorption of 1-decanol solutions and the (b, bottom left) 6.32 x (c, top right) 1.018 x and theoretical predictions of the models for CO= (a, top left) 3.95 x (d, bottom right) 1.579 x lo-' moVcm3.
*
usingstainless steel balls (1.577 0.002 and 1.983 0.002 mm), calibrated by a Peacock digital linear gauge (PDN-12N,Ozakimfg Co.),was used to calculate the length between pixels along a row and along a column. The calibration procedure yielded values of 100.86 pixeYmm horizontally and 124.56 pixdmm vertically. The experimental protocol was as follows: The quartz cell was initially filled with the aqueous decanol solution, and the bubble-forming needle was positioned in the cell in the path of the collimated light beam. At this point, the solenoid valve was not energized. The syringe pump was turned on, and the air in the syringe was allowed to pass through the normally open port ofthe valve. Via input from the keyboard, the D/A card energized the valve and allowed the gas to pass through the needle, thereby forming a bubble of air. The valve was then closed (also by input from the keyboard) when the bubble achieved a diameter of -2 mm. The time required to create an air bubble of this size is -0.1 s. The bubble so created is one of constant mass. The change in volume, as the surface tension relaxes during the adsorption of surfactants onto the clean interface, is only a few percent over a time period of at least 1h. After the solenoid was closed and the bubble was formed, sequential digital images were then taken of the bubble, first at intervals of -0.1 s and then later in intervals of the order of seconds. After the relaxation of clean adsorption was complete, the valve was opened for 0.11 s (controlled by computer) while the syringe pump was off. A small part of the gas inside the bubble was allowed to pass through the solenoid valve and the surface area of gas bubble decreased 10-30%. The images were recorded during this process and also taken sequentially onto the computer. After the relaxation ofthe desorption was complete,the images on tape were processed to determine the bubble edge coordinates and the surface tension. It is noted that there is a nearly constant deviation of surface tension, -0.7 dydcm, between the images directly onto computer and those saved on tape.
The classical Laplace equation relates the pressure difference across a curved interfacel'J8
(1) where y is the surface tension, R1 and R2 are the two principal radii of curvature of the surface, and AP is the pressure difference across the interface. For the pendant bubble geometry, eq 1can be recast as a set of three first-order differential equations for the spatial positionsxl andxz and turning angle 4 of the interface as a function of the arc length s and then integrated by using a fourth-order Runge-Kutta scheme with boundary conditionsxl(0) =x2(0) = +(O) = 0. An objective function is defined as the sum of squares of the normal distance between the measured points and the calculated curve obtained from eq 1. The objective function depends on four unknown variables: the actual location of the apex (X10 andXzo),the radius of curvature at the apex (Ro), and the capillary constant ( B = ApgRo2/y). The surface tension is obtained from the best fit between the theoretical curve and the data points by minimizing the objective function. Minimization equations are solved by applying directly the NewtonRaphson method and from the optimal values of Ro and B the tension can be computed. As demonstrated by Lin et al.,1° the accuracy and reproducibility of the dynamic surface tension measurements obtained by this procedure are -0.1 dydcm. 2.3. Experimental Results. Relaxation in the surface tension due to decanol adsorption was measured at the airwater interface at 22.70 "C. Data were recorded up t o a few hours from the moment (referenced as t = 0) in which the valve was closed and the bubble was formed. Shown in Figure 2 are (17) Huh, C.; Reed, R. L. J. Colloid Interface Scz. 1983, 91, 472. (18) Rotenberg,Y.; Barouka,L.; Neumann, A. W. J.ColZoidInterfuce Scz. 1983, 93, 169.
Lin et al.
558 Langmuir, Vol. 11, No. 2, 1995
from the equilibrium value ( y e ; at this moment, surface area is A, and surface coverage is re)is referenced as the zero time in Figure 5, but in Figure 6, the moment with the lowest surface tension value (Yb;at this moment, surface area is Ab and surface coverage is rb)is set to be the zero time for the convenience of the theoretical calculation, as discussed in section 3.
3. Theoretical Framework Discussed in this section are the theoretical frameworks describing the unsteady adsorption and desorption of surfactant onto o r away from a pendant bubble interface and its effect on the surface tension. The pendant bubble is treated as a spherical bubble surrounded by an infinite medium containing a s u r f a ~ t a n t .The ~ ~ surfactant ~~~ is assumed not to dissolve into the gas phase of the bubble. Mass transfer between sublayer and bulk is described by Fick's law
generalized Frumkin
data from DCA
A A ~ A A
so 10
c
(mol/cms)
10 -'
Figure 4. Equilibrium surface tension (dydcm) for air-ldecanol aqueous solution and the theoretical predictions of Langmuir, Frumkin, and generalized Frumkin adsorption isotherms. 14
(in which the diffusion is assumed to be spherical symmetric and convection effects are neglected) with
initial and boundary conditions C(r,t)= C, ( r > b, t = 0)
18
\
'. ,
\
....** ..
# .
.c
12
L
C(r,t)= c, ( r
*
7
-
00,
t
> 0)
dT(t) -= D ( g ) ( r = b, t dt
v Bre
> 0)
11
u t ) = r b (t = 0) 10 0
Figure 5. Representative dynamic surface tension (dydcm; triangles) and surface area (mm2;squares) of pendant bubble for re-equilibration of 1-decanol aqueous solutions for CO= 2.16 x mol/cm3. representative dynamic surface tension profiles (for one selected bubble) of decanol aqueous solutions at seven different bulk 1.97 x 2.84 x 3.95 concentrations, C = 1.58 x x 10-8,6.32 x 10-8,10.18 x 10-8, and 15.79 x 10-8moUcm3.The reproducibility of these profiles is demonstrated in Figure 3a-d, where the results of several pendant bubbles at four concentrations are given. The equilibrium surface tensions for aqueous 1-decanol solutions at the aqueous-air interface are extracted from the longtime asymptotes in Figure 2 and are plotted as the circles in Figure 4. More equilibrium data were obtained by using the dynamic contact angle analyzer (DCA-322,Cahn) and are also shown (the triangles) in Figure 4. The re-equilibration process due to decanol desorption from a suddenly compressed air-water interface was measured and the images were recorded. A typical relaxation profile (the triangles) of surface tension is shown in Figure 5. The surface tension decreases from the equilibrium value (64.0 dyn/cm for C = 2.16 x lo-@mol/cm3) to a lower value (56.5 dyn/cm), corresponding to a surface coverage higher than the equilibrium one, in 5/60 s. The surface tension then increases and goes back to the equilibrium tension in several hundred seconds. The bubble surface area, also shown in Figure 5 (the squares), decreases 14% in 5/60 s and then keeps a nearly constant value for a few hundred seconds. Shown in Figure 6 were the other four desorption relaxation profiles at different bulk concentrations. They all show a similar behavior; surface tension increases smoothly up to its equilibrium value after an abrupt fall, although they were compressed with different percentage (12-27% as shown for AdA, in Table 2; Ab and A, denote the bubble surface area before and right after the compression, respectively). It is noted that the moment in which surface tension begins to deviate
(3)
where r and t are the spherical radial coordinate and time and D denotes the diffusion coefficient, C(r,t) the bulk concentration,r(t)the surface concentration,b the bubble radius, COthe concentration far from the bubble, and rb the initial surface coverage. By the Laplace transform, the solution of the above set equations can easily be formulated in terms of unknown subsurface concentration C,(t) = C(r=b,t);
To complete the solution for the surface concentration, the sorption kinetics must be specified. The model used here utilizes the Langmuir formalism.21 The adsorption rate is proportional to the subsurface concentration C , and the available surface vacancy (1 - T/T,), and the desorption rate is proportional to the surface coverage r
dr-- p exp(-E,/RnC,(r, - r) - a exp(-E,/Rnr dt
(5) where p, a,E,(T),andEd(T) are the pre-exponential factors and the energies of activation for adsorption and desorption, respectively. r, is the maximum surface concentration, T the temperature, and R the gas constant. To account for enhanced intermolecular attraction at increasing surface coverage, the activation energies are assumed to be r dependent and a power form is assumed; (19)Mysels, K.J. J.Phys. Chem. 1982,86, 4648. (20)Adamczyk,2.;Petlicki, J. J.Colloid Interface Sci. 1987,118,20. (21)Aveyard, R.;Haydon, D. A. A n Introduction to the Principles of Surface Chemistry; Cambridge University Press: Cambridge, U.K., 1973;Chapters 1 and 3.
Air- Water Interface Kinetics of Decanol
Langmuir, Vol. 21, No. 2, 1995 559
2
=lo'
I 1
Figure 6. Experimental values of the dynamic surfacetensions (dydcm) for re-equilibration of decanol solutions and the theoretical predictions of mixed controlled re-equilibration for different desorption rate constants of the generalized Frumkin model: Co = (a,top left) 1.71 x (b,bottom left) 1.97 x (c, top right) 2.16 x and (d,bottom right) 3.17 x mol/cm3. DC (dashed
line) denotes diffusion limited curves.
+ Ed = Ed" + E, =E,"
V,rn
(6)
Vdr"
where E,, Ed, va, and Ud are constants. Equation 5 in nondimensional form becomes
clx = K, exp(-v*pn)C,*(l dz
-x )
- Kd exp(-v*g"k
(7)
where x = IT,, z = tD/h2, h = TJCo, Ka = /? exp(-Eaol RT)Cd(D/h2),Kd = aexp(-Edo/RT)/(D/h2),ua* = v,T,"/RT, V*d = Vdr,"/RT, and c,* = c$co. At equilibrium, the time rate of change of r .vanishes and the adsorption isotherm that follows is given by
-r_ - x = r,
C a exp(kxn)
+C
(8)
where k = (v, - vd)T,"/RT and a = (a//?)exp[(E," - Edo)/ RTI. Equation 7 becomes the Frumkin i s ~ t h e r m ~ when n = 1and the Langmuir adsorption isotherm when ua= Ud = k = 0. The presence of cohesive intermolecular forces that increase with surface coverage and lower the desorption rate (relative to that of adsorption) is described by k < 0. Accounting for the dependence of the activation energies on through a power law has been undertaken (22) Frumkin, A. 2. Phys. Chem. (Lezpzig) 1925,116, 466. (23)Fainerman, V. B.Kolloidn. Zh. 1977,39, 113.
in the l i t e r a t ~ r e . ' ~It, ~has ~ been shown that the consideration of a power other than 1 is necessary in order to best account for the surface tension relaxation profiles for l-decanol.12 The theoretical framework that describes the unsteady bulk diffusion of surfactant toward an initially clean pendant bubble and its effect on the surface tension was formulated previously,1°J2 and therefore only a brief outline is given here. The pendant bubble is treated as a sphere surrounded by an infinite, quiescent medium which at time t = 0 contains a uniform concentration Co of surfactant. The concentration of surfactant on the bubble surface is assumed to be equal to be a constant initial surface concentration rb. r b = 0 for a clean adsorption process, in which the bubble was created suddenly (less than 0.1 s). There is a small amount of surfactant present on the air-water interface before bubble growth, and it was shown the effect on the dynamic surface tension relaxation is significant for the growing drop technique. This effect on our pendant drop method is ~ , ~ negligible ~ , ~ ~ because the bubble surface area increases -500 times and depletes the interface during the rapid growth of gas bubble. For the desorption process, the air-water interface was suddenly compressed for a small percentage of the surface area; rb is assumed to be equal to the surface coverage corresponding to the point with the lowest surface tension value, for example, the point (24)Hunter, J.R.;Kilpatrick, K.; Carbonell,R. G. J. Colloid Interface Sci. 1990,137, 462.
560 Langmuir, Vol. 11,No.2, 1995
Lin et al. Table 1. Constants of Optimal Fit for 1-Decanol Aqueous Solution
L at t = 0.083 s in Figure 5. So in Figure 6c, point L, plotted at the position o f t = 0.01 s, is referenced as the zero time corresponding to the beginningof the desorption process. When the surfactant solution can be considered ideal, the Gibbs adsorption equation dy = -TRT d In C and the equilibrium isotherm (equation 8)allow for the calculation of the surface tension explicitly in terms of r: y - yo = rJ3
4ln(1 -
x) -n + l
xn+ll
n
After integration, eq 4 is of the form
(11) where t, = nd and the constants bl and b2 are a function of the concentrations C,(O), C,(6), ..., C8(tn-1).Equation 7 is then solved numerically by first separating the variables and then integrating from x(t,-1) to z(t,) using Simpson's one-third rule.26 In this numerical integration, C,* is assumed to be a constant value during the small integration time step 6 and is set equal to the average value of C,*(t,-d and C,*(t,). By using Newton's method,26a value for x(t,) is found for which the surface concentration integral is equal to 6. The dynamic surface tension at time t, is obtained from eq 9 once the surface concentration x(t,) is known. (25)Miller, R.; Kretzschmar, G. Colloid Polym. Sci. 1980, 258, 85. (26) Canahan, B.; Luther, H. A.; Wilkes, J. 0.Applied Numerical Methods; John Wiley & Sons: New York, 1969.
(moYcm2) a (moVcm3) k n kCa 8.369 x 9.175 x -3.717 1 -4 9.703 x -5.031 0.259 -7.32
Frumkin a Critical values for the intermolecular interaction parameter k for which, below these values, discontinuities develop in the adsorption isotherm and the equation of state becomes nonmonotonic. k , = -(1
where z = I T , and yo is the clean surface tension. By fitting equilibrium data of the surface tension as a function of the bulk concentration using eqs 8 and 9, the equilibrium constants k,a , and n and the maximum coverage r, can be obtained. When the adsorption process is controlled solely by bulk diffusion, the surface concentration can be obtained by solving eq 4, describing the mass transfer between sublayer and bulk, and eq 8, the sorption kinetics between subsurface and interface. If the adsorption process is of mixed control, eq 7 instead of eq 8 is solved, coupled with eq 4,to find the surface concentration. The dynamic surface tension y ( t ) is calculated from eq 9. When solving these two equations (eqs 4 and 8 or eqs 4 and 7) numerically, the technique used is a modification of that used by Miller and K r e t ~ s c h m a r . First, ~ ~ the convolution integral of C, in eq 4 is integrated by first partitioning the interval (0, t ) into intervals of size 6 in time and integrating separately each interval by assuming a linear relation in time for C,
n-1
r,
Lagmuir 1.538 x Frumkin 6.334 x generalized 6.946 x
+ (lh~))l+~.
8-
e
n
"a
0
Y
8-
0
Figure 7. Diffusion coefficients (D, cm2/s)from the best fit between the clean adsorption relaxation and the model prediction for (L) Langmuir,(F)Frumkin, and (GF)generalized Frumkin models.
4. Comparisons of Experimental Relaxation Curves and Theoretical Profiles Relaxation in the surface tension due to decanol adsorption was measured at the air-water interface. Figure 4 presents the comparison between the decanol equilibrium surface tension and the best fit from the adsorption isotherms of Langmuir, Frumkin, and generalized Fnunkin. The model constants, as shown in Table 1,are obtained by adjustment so as to minimize the error between the model predictions and experimental values. The more exact agreement of the generalized Frumkin model indicates clearly that the relatively long, slender hydrocarbon chain of the 1-decanol molecule can give rise to strong intermolecular attraction as the surface coverage is high. Ifthe relaxation of decanol were assumed to be diffusion controlled, Figure 2 may be used to determine a diffusion coefficient for the surfactant molecules in the aqueous phase. The model constants (K,a , n, r,) used in obtaining the computed profiles represent the optimal fits of the y - In C equilibrium data. Numerical profiles were computed by adjusting the diffusion coefficient individually for each ofthe different bulk concentrations to achieve the best agreement with the data. As shown in Figure 3, the agreement between the theoretical relaxation profile obtained by using the Langmuir isotherm is poor. The Frumkin numerical profiles fit the experimental data better than the Langmuir, but these profiles are not very effective at predicting the data at the short, shoulder behavior times where the surface pressure is low. It is clear that the agreementbetween the numerical relaxation constructed from the generalized Frumkin model and the data is superior to that of the Langmuir and Frumkin models, especially in the short time range. The value of the diffusion coefficient,plotted in Figure 7, obtained using the generalized model is 6.6 x cm2/s.
Air- Water Znterface Kinetics of Decanol
Langmuir, Vol. 11, No. 2, 1995 561
I
2 3 4 diffusion control
I
54! 0.01
I
0.1
10
100
1000
0.01
1
, , , , *,,,,
, 1 1 1 1 1 1 ,
0.1
, , ,,,,,,,
1
t(s) Figure 8. Experimental values of the dynamic surface tensions (dydcm)for re-equilibration of decanol solution (CO= 2.16 x moVcm3)and the theoretical profiles of diffusion control of generalized Frumkin model forD = (2) 1x lo+, (3)4 x 10-6, cm2/s. Curve 1is the best fit of the mixed and (4) 6.6 x control of generalized Frumkin model forD = 6.6 x 10-6 c"Vs and Kd = 1000.
, , ,,,,,,,
, , ,,.
10
100
1000
10
100
1000
t(s)
Table 2. Values of Adsorption-Desorption Rate Constants and Experimental Conditions
I 1.713 1.974 2'163 3.177
3.712
66.5155.3 64'4/60'9 64.4/56.5 64.0/56.5 64.065.1 59.V41.9 56.9/42.9 56N45.6
0.75 o'88
0.82 0.86 0.82 0.75 0.73 0.76
1.09
i: ~~
1.07 1.04 0.89 0.83 0.85
1000
12
18
12
1700
20
33
21
looo
13
22
13
10
16
10
500 2oo
6
Consider next the re-equilibration process. An initially equilibrium-established air- water interface is suddenly compressed, and the surface tension relaxation profiles are shown in Figures 5 and 6. If this process were again assumed to be diffusioncontrolled,the diffusion-controlled relaxation profiles by using the generalized Frumkin model with the diffusioncoefficient obtained from the clean adsorption process are shown in Figure 6. It is clear that the desorption relaxation profiles depart significantly from the diffusion-limited curves in each of these bulk concentrations. Note that the agreement is still poor, as shown in Figure 8, even though a lower diffusioncoefficient value, 1 x or 4 x cm2/s,is assumed. According to the generalized Frumkin model, the intermolecular cohesive forces may lower the desorption rate when the surface coverage is high. It is believed that, right after a sudden compression for the adsorbed monolayer, decanol molecules become overcrowded at the air-water interface and have a lower desorption rate due to the intermolecular cohesive forces. This re-equilibration process becomes mixed (diffusive-kinetic)controlled. Relaxation profiles with a finite desorption rate constant, plotted in Figure 6, fit the desorption data quite well. The desorption rate constants obtained by using the generalized Frumkin model decrease with bulk concentration but the dependence is weak, as shown in Table 2. The average values of the kinetic rate constant are ,8 = 12 x lo6 cm3/(mol.s) c d s , which is the and a = 12 s-I. Here, PI', = 8.3 x same order of magnitude as Bleys and Joos obtained for the short-chain alcohols.z 6. Discussion and Conclusions The technique of pendant bubble tensiometry enhanced by video imaging is an effective tool for studying the
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Figure 9. Experimental values of the dynamic surface tensions (dyn/cm) for re-equilibration of decanol solution and the theoretical profiles of the Frumkin models for different &: Co = (a, top) 1.71 x lo-* and (b, bottom) 2.16 x
[email protected] denotes diffusion-limited curves. The dashed line denotes the best fit of generalized Frumkin model with Kd = lo3.
compression of adsorbed monolayer at the fluid interface. With the aid of a video recorder, it is able to monitor surface tension during the entire period, with a V30-s interval, of the re-equilibration process. The effect of temperature on the adsorption kinetics and the intermolecular cohesive energies can also be investigated using the pendent bubble technique. MacLeod and Radkez7have shown that, for a growing drop, a significant amount of surfactant is present on the remnant interface after breakout and it is the rapid stretching during early drop growth, not the breakout process, that actually depletes the interface. This also applies for the pendant bubble method. But the surface area of the bubble increases more than 500 times when the bubble grows and it usually takes only l/15 s or less for the bubble to grow from a hemisphere (with diameter that is equal to the inner diameter of needle) to a complete pendant bubble. So the effect of surfactants initially on the remnant interface after breakout for the pendant bubble method utilized in this study is negligible. The conceptz8that bulk-soluble surfactants with strong cohesion can have diffusion-controlled adsorption and mixed re-equilibration relaxation is verified in this study. 1-Decanol consists of long, slender hydrocarbon chains and small polar groups and are subject to strong, attractive (27) MacLeod, C. A.; Radke, C. J. J . Colloid Interface Sci. 1993,160, 435. (28) Lin, S. Y.; Mckeigue, K.; Maldarelli, C.Langmuir 1994,10,3442.
562 Langmuir, Vol. 11, No. 2, 1995 van der Waals forces when surface crowding causes interchain contact. Cohesive forces among the adsorbed molecules raise the energy barrier for desorption of the adsorbed molecules into the bulk sublayer. For adsorption onto an initially clean interface, the reduction in the desorption rate affects the tail end of the process because it is only at these extended times that the surface concentration becomes high enough for the cohesive energy effect to become important. However, when a concentrated equilibrium monolayer is perturbed, the subsequent reequilibration is affected by the lower desorption rate during the entire period of the process. The re-equilibration relaxation of 1-decanol is controlled by both diffusion and sorption kinetics. In the generalized Frumkin model, the activation energies are assumed to depend on the surface coverage with a power law (eq 6). The cases n = 0 (Langmuir model) and n = 1 (Frumkin model) are well-known, and the Frumkin adsorption isotherm fits the equilibrium surface tension data well also (see Figure 4). But the Frumkin model is not very effective at predicting the data of clean adsorption at the short, shoulder behavior times where the surface tension is larger than -67 dyn/cm (see Figure 3). As shown in Figure 9, the prediction of the Frumkin model is also worse than that of the generalized Frumkin model with n = 0.259 when compared with the relaxation profiles of the re-equilibrium experiment, especially at lower bulk concentration as shown in Figure 9a. It is not clear so far what the molecular mechanism is when n is a fraction, but a power law dependence on surface coverage for the activation energies does give a better prediction of the cooperative adsorption than that of n = 1. In section 4, the clean adsorption is assumed to be a diffusion-controlled process and the surface tension relaxation profiles are then used to determine a diffusion coefficient for the surfactant molecules in the aqueous phase. This assumption is reasonable under certain restrictions on the type of surfactants, the bulk concentration, and the forces of intermolecular attraction. For 1-decanol, Kd is -1000, i.e., a 1.2 x IO7 cm3/(mol-s). Figure 10 shows that the relaxation profile of clean adsorption with& = 500 is almost indistinguishable with that of diffusioncontrol. So,the assumption that the clean adsorption is a diffusion-controlled process is reasonable for 1-decanol in water a t the concentration range in this study. Video-enhanced pendant bubble tensiometry is capable of recording the entire dynamic behavior. Besides the surface tension relaxation, the relaxation of surface area of pendant bubble outside the inverted needle is also plotted in Figure 5. Bubble surface area decreases abruptly in 0.1 s for 14%when the solenoid valve is opened.
Lin et al.
1
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1 , 3 8 1 ,
1
,
, , , ,
,
IO
,
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, , ,,,, 100
, , , , ,,I 1 10
t(s) Figure 10. Experimental values of the dynamic surface tensions (dyn/cm) for clean adsorption of 1-decanol solution and the theoretical profiles for different Kd (10, 100,500) and the curve of best fit for diffusion control (the dashed line). CO = 6.32 x moUcm3;D = 6.6 x cm2/s.
M
Figure 11. Relaxation of surface concentration of 1-decanol molecules at the air-water interface of the pendant bubble. CO = 2.16 x moUcm3.
The surface concentration of 1-decanol on the air-water interface of the pendant bubble is also calculated, Tj(yj) from eq 9 and plotted in Figure 11.
Acknowledgment. This work was supported by the National Science Council of Taiwan, Republic of China (Grand NSC 81-0402-E-011-568). LA940324Z
