2244
The Journal of Physical Chemistry, Vol. 83, No. 17, 1979
R. Van
den Bogaert and
P. Joos
Dynamic Surface Tensions of Sodium Myristate Solutions R. Van den Bogaert and B. Joos" Universitaire Instellhg Antwerpen, Department of Cell Biology, Universiteltsplein 1, 26 10 Wl/rijk, Belgium (Received November 13, 1978; Revised Manuscript Received March 14, 1979) Publication costs assisted by Unlversitaire Instelling Antwerpen
A method is described for investigating the adsorption kinetics of a surfactant at the solution-air interface, in a time scale from 50 ms to 2 s. This method bears some analogy with the oscillating jet and consists of flowing a surfactant solution (sodium myristate) over an inclined plate while measuring the surface tension as a function of the distance from the inlet. Experimental data are analyzed by means of the theory of Ward and Tordai. It was found that the adsorption kinetics of sodium myristate is diffusion controlled.
Introduction As a result of a sudden extension of the surface of a solution containing surface active molecules, a new surface is created with a very different composition, hence a different surface tension, from that which is observed at equilibrium. Experimentally it is found that the equilibrium surface tension is reached after some time. Between the initial state of the freshly formed surface and the final equilibrium state the surface phase passes through a series of equilibrium states, each state corresponding to a dynamic surface tension. A suitable method to study these dynamic surface tensions is the oscillating jet method. Here the age of the surface ranges from a millisecond to several hundreths of a sec0nd.l In this paper a method is described to investigate the dynamic surface tensions in a time scale between a tenth of a second to a few seconds. In this way this technique supplements the oscillating jet. The present method bears some analogy with the oscillating jet. It consists of flowing a thin layer of a surfactant solution over an inclined plate. At the inlet a fresh surface is formed and gradually surfactant molecules adsorb with increasing distance. In this way the surface tension decreases at increasing distance from the inlet. This surface tension is measured as a function of the distance by means of a Wilhelmy plate connected to a transducer. A similar setup was already used by Posner and Alexander2 but they measured the surface potential. Experimental Section A surfactant solution, in the present study a sodium myristate solution, was allowed to flow over an inclined glass plate with the angle of inclination, a, varying between 15 and 4O (Figure 1). The length of the glass plate was about 2 m. At the side edges two other small glass plates, having the same length, were fixed. In this way the surfactant solution flowed in a canal, having a width of 2.0 cm. The surfactant solution was pumped over the glass plate and the flow rate was measured by a suitable rotameter (Brooks Instrument Type R 6-15-B; sapphire ball). At the inlet a fresh surface is formed and with increasing distance from the inlet the surface tension was measured. The flowing liquid layer was rather thin (less than 1 mm) but the surface tension could be easily measured by using a Wilhelmy plate, firmly connected to the sidearm of a Statham transducer (Gold Cell with microscale accessory). Since the Wilhelmy plate is inserted on the surface of a flowing solution, drag effects may complicate the mechanical response of the plate. However, the plate orients itself with its length (2.00 cm) parallel to the flow direction. The thickness of the plate was less than 0.5 mm. With a 0022-3654/79/2083-2244$0 1.OO/O
flowing film of pure water, the correct surface tension was obtained. Therefore, we are confident that the present device for surface tension measurements on a flowing liquid layer is satisfactory. A typical graph of the surface tension, a, as a function of the distance, x , is shown in Figure 2, for a flowing surfactant solution. All experiments were carried out a t room temperature (22 f 1 "C). As usual in the oscillating jet technique, from the surface tension the adsorption and the subsurface concentration were calculated by using suitable equations of state. For sodium myristate it was found that the equilibrium surface tension as a function of concentration c obeys the von Szyszkowski e q ~ a t i o n : ~ a. - a
= RTr" In [ l f c/a]
(1)
with a. the surface tension of the pure solvent; I?" the saturation adsorption; and a the Langmuir-von Szyszkowski constant. The other symbols have their usual meaning. If eq 1applies it can be shown that the relation between adsorption and surface tension is given by Frumkin's equation: 0
-
go
= RTr" In [1-
r/rm]
(2)
The parameters I'" and a were determined from the equilibrium surface tension measurements (see Figure 3). We found F" = 4.12 X mol/cm2 and a = 2.9 X mol ~ m - From ~ , the saturation adsorption a limiting area per molecule of 40 A2 is obtained. This seems reasonable since, on alkaline substrates, fatty acid monolayers are expanded, whereas they are condensed on neutral or acid substrates. Previously,14 under similar experimental conditions, we found for sodium laurate a saturation adsorption of 4.1 X mol cm-2,in excellent agreement with the present data for sodium myristate. The parameter a is employed for calculating the subsurface concentration by means of eq 1. As with the oscillating jet, the evaluation of the surface age, t , at some distance, x , from the inlet is a more serious problem. At increasing distance from the inlet more surface active material is adsorbed and a surface tension gradient, da/dx, is present. This surface tension gradient acts as a shearing stress on the underlying liquid and the velocity at the surface is reduced. This is the well-known Marangoni e f f e ~ t .For ~ our experimental conditions, the velocity a t the surface can be calculated (see later). However, because of the presence of this surface tension gradient, adsorbed material is conveyed by surface convection from regions with a lower surface tension (far from 0 1979 American Chemical Society
The Journal of Physical Chemistry, Vol. 83, NO. 77, 7979
Surface Tensions Dynamics
near to the inlet, the surface is expanding, that is, new surface elements are formed, and the surface is diluted. This remains true as long as d2a/dx2 > 0. However, Hansen5 has shown that both opposite effects cancel out exactly, and the surface age, t , at a distance, x , may be evaluated provided the surface velocity, ug, at this point is known:
Inset eter
t = x/v,
Flgure 1. Sketch of the apparatus. The insert shows the magnification of the inlet.
d2v + pg sin dz2
LO 10
15
20
i5
30
35
48
x[m)
Figure 2. Surface tension, u (mN m-’), as a function of increasing distance, x (cm), from the inlet. Sodium myristate. Bulk concentration co = 3 X lo-’ mol ~ m - angle ~ , of inclination a = 8 O , flow rate F = 9.45 cm3 s-l. 0 [dyne crn-ll
70
(3)
Moreover, PBtri9 explicitly states that the surface age, defined in this way, does not depend on mechanisms which tend to equalize the surface velocity. Davies7reexamined the surface age of jets and finds that, by comparison with published entry length corrections for gas absorption, x / v , gives the most reasonable age of the surface at any distance x along a jet of pure water. For surfactant solutions, however, still, according to Davies, the Marangoni effect should accelerate the adsorption rate. We think this last statement is incorrect because he overestimates the Marangoni effect, moreover it contradicts the elegant analysis of H a n ~ e n . ~Obviously, in our experimental situation, the surface age can be defined as according to eq 3. The surface velocity will now be calculated. For the present situation, the Navier-Stokes equation reads 7-
5
2245
(Y
=0
(4)
where q is the viscosity; u the velocity; z the coordinate perpendicular to the glass plate; p the density; and g the acceleration of gravity. In this equation the acceleration term, pv(dv/dx), which is only important in the entrance region, is neglected, as well as the term 7(d2v/dx2)which was considered small as compared with 7(d2u/dz2). This is because the velocity changes more rapidly along the z than along the x coordinate. This equation must be integrated by taking into account proper boundary conditions. First a t z = 0 (the glassliquid interface) the velocity is zero, u = 0. Secondly, at the air-liquid interface at z = h
In our coordinate system du/dx < 0. Further (y is the coordinate lying in the surface but perpendicular to x), we assume the surface shear viscosity coefficient y is zero. Integration yields
60.
The flow rate F is given by 501
l
6 -7 -‘6 log c [c in mol/mJl Flgure 3. Equilibrium surface tension of sodium myristate as a function of concentration (mol cm-7. -9
the inlet), to regions with a higher surface tension (nearer to the inlet). Hence a surface element at a distance x is contaminated by older elements on the surface. Secondly,
where b is the width of the canal. This equation holds in the nonaccelerating range, but not in the entrance region. This i s because the terms pv grad u and 7 d2v/dx2 are omitted in the Navier-Stokes equation (eq 4). The flow rate is measured by a rotameter and hence from eq 7 the layer thickness is obtained. Subsequent substitution of the layer thickness, h, in eq 6 yields, at z = h, the surface velocity. In this simple calculation effects on the edges are neglected. As was shown by Ahmad and Hansen,8 and confirmed by our own computer calculat i o n ~this , ~ is allowed as long as h / b
