- Email: [email protected]
Desorption at constant surface pressure from slightly soluble monolayers
Desorption at Constant Surface Pressure from Slightly Soluble Monolayers P. DE K E Y S E R AND P. JOOS Universitaire Instelling Antwerpen, Department o f Cell Biology, B-2610 Wilrijk, Belgium Received January 19, 1982; accepted March 3, 1982 The desorption o f a slightly soluble monolayer, spread at the air/water interface, is considered by keeping the surface pressure constant while decreasing the area. As already observed by other investigators, the curve giving the logarithm o f the area with time can be derived in three regimes. In a first one, for small times, the logarithm o f the area decreases linearly with the square root o f time. In a second one, for large times, the logarithm o f the area decreases linearly with time. A n d finally a third one for the transition between both regimes (for intermediate times). For the first and second regimes a theory is presented, which allows the diffusion coefficient to be obtained from experimental data. The desorption process is diffusion controlled, and there is no need to postulate an adsorptiondesorption barrier, at least for the surfactants considered (lauric acid, decanol, and dodecanol).
decreases. This decrease is measured as a function of time. As long as the transfer o f surfactant to the bulk is diffusion controlled, the theory of Ward and Tordai (5) applies giving the adsorption P as a function o f time,
INTRODUCTION
The first desorption experiments at constant surface pressure o f slightly soluble monolayers were performed by Ter Minassian (1, 2). Later, similar experiments were done by Gershfeld and Patlak (3), while Baret et al. (4) presented a theoretical analysis. In our opinion the treatment of Baret is erroneous and leads to the false conclusion that desorption is not controlled by diffusion. It is our aim to reconsider the problem and to prove our theory by experiments from other investigators together with our own experimental data. Let us consider such a desorption experiment. A monolayer of a slightly soluble surfactant (e.g., lauric acid) is spread on a Langmuir trough on a clean water surface. After evaporation o f the solvent, the monolayer is suddenly compressed till an area Ao and a c o r r e s p o n d i n g surface pressure 11o are reached. Now from the soluble monolayer, surfactant molecules tend to dissolve in the underlying aqueous phase. Now two kind o f experiments can be done. In the first one, the area is kept constant and the surface pressure
P = F0 - 2
c s ( t - r ) d ( l " ) 1/2 , [11 '~0
with P0: adsorption at time t = 0, D: diffusion coefficient, c~: subsurface concentration, r: a d u m m y variable, and, 7r: 3.1415 . . . . In the second type o f experiment, considered here, the surface pressure is kept constant by further compression o f the m o n o layer, giving the area A as a function o f time. Usually the logarithm o f the relative area, is plotted as a function of t 1/2 or t (see Figs. 1 and 2). Plotting In (A/Ao) v s t 1/2 one sees that initially a linear relation is observed. Following Ter Minassian we define kv=
d In (A/Ao) d(t)v2
(small times).
[2]
On the other hand, plotting In (A/Ao) vs t one sees that after sufficiently long times a linear relation is again observed. This means that 131
Journalof Colloidand InterfaceScience.Vol. 91, No. 1, January 1983
0021-9797/83/010131-07503.00/0 Copyright© 1983by AcademicPress,Inc. All rightsof reproductionin any form reserved.
132
DE KEYSER AND JOOS
for in the diffusion equation. This is well discussed by Van Voorst Vader et al. (7). Neglect of this term leads to false conclusions. In the beginning of an experiment at constant surface pressure and monitoring the area as a function of time, this convective term is of minor importance. The appearance of convection was already recognized by Ter Minassian.
o•
1
2
3
~,
5
THEORY
~'f
FIG. 1. Dodecanol monolayer (H = 6 dyn cm-~). Logarithm of the relative area plotted vs t lie (t in minutes). Initially a linear relation is observed.
As already stated by Levich (8) and Van Voorst Vader (7), the boundary condition for conservation of mass at the surface reads dr
+ OF +
for long times the dilatation 0 defined as dln A 0- - -
dt
[31
is constant. In this paper we consider also a somewhat different experiment which is closely related to a desorption experiment at constant surface pressure. A monolayer of a slightly soluble monolayer is compressed with a constant relative compression rate 0. As seen from Fig. 3, in the beginning transient surface pressures are obtained, but after longer times the surface pressure remains constant, indicating a steady state. The relation between both experiments is evident. In the first one (measuring area as a function of time with constant surface pressure) a constant surface pressure is imposed to the system and after long times the dilatation 0 is measured. Whereas in the second experiment a constant dilatation is imposed and the surface pressure is measured in the steady state. Obviously, other experiments are possible. The simplest experiment seems to be that of M o to mu r a (6), where a slightly soluble monolayer (myristic acid) is compressed at constant dA/dt and the surface pressure is measured. For this type of experiment, however, the theoretical analysis is more difficult. For those experiments where the area changes with time, surface convection is generated and this convection term must be accounted Journal of Colloid and Interface Science, Vol. 91, No. 1, January 1983
D(OC) = O, \Oz] °
[4]
with c: concentration and z: coordinate normal to the surface. In Ter Minassian's experiments the surface pressure was kept constant; hence the adsorption also remains constant whence in Eq. [4], d F/dt = 0. Van Voorst Vader et al. have shown that
d l n A _(OVx~ 0 - --~
[5]
~,Ox]o,
with Vx: velocity along the surface and x: coordinate along the surface. Hence by changing the area convection is introduced and this convection must be accounted for in the diffusion equation. tn / ~
e*°, -02 °° °°° -..°°1 ,°
-1.5 t(min)
NO. 2. Dodecanol monolayer (H = 8 dyn cm-l). Logarithm of the relative area plotted (t in minutes). For sufficient long times a linear relation is found.
133
DESORPTION AT CONSTANT SURFACE PRESSURE
Van Voorst Vader also showed that by expanding an adsorbed monolayer with a constant relative expansion rate 0 a steady state is obtained. In the steady state the diffusion penetration depth 6v is independent upon time and is given by (film theory) ~F
=-
-(TrD m~ 0 r = 3 . 1 4 1 5 9
\ 20 ]
""
.).
[6]
Strictly, the theory of Van Voorst Vader, from which Eq. [6] results, is only valid for an expanding monolayer. For a compressed monolayer, the theory is far from being solved. On the other hand, Lucassen and Giles (9) showed that for small dilatations the jump in surface tension, Aa, brought about by expanding the monolayer at constant dilatation is given by
Aa _ RrrZ ( TrO]'/2 k2DI
[7]
(Aa = ad -- ~e; ~d: dynamic surface tension; and ag surface tension at equilibrium). He also showed that the modulus of elasticity It] for the case when diffusion is fully operating is approached by =
[81
(ox angular frequency) and it appears that the relationship between Aa and 0 is identical to that between I~1 and 2o~lrr. From this we conclude that, at least formally, 0 can be treated as a frequency. For a compressed monolayer, this frequency should be negative (since 0 < 0). Since a negative frequency has no meaning, we made the ad hoe assumption that the diffusion penetration depth is given by ~F = ~.21011
"
1(
[91
The diffusion flux in Eq. [4] can now be approximated for our situation (since for
S
•OZ]O
6F '
15
20
flminl
FIG. 3. Dodecanol monolayer compressed with Conslant relative compression rate 10[ = 1.34 × 10 -3 sec -1. After sufficiently long times a steady state is observed.
with cs: subsurface concentration. Substitution of Eqs. [10] and [9] in Eq. [4], remembering that d I'/dt = 0, yields
dln(A/Ao)_ dt
2D(~] 2 ~r \ P / "
[11]
Equation [11] predicts that after sufficiently long times, when the diffusion penetration depth is given by Eq. [9], the slope of d In (A/Ao)/dt becomes constant. This was observed by Ter Minassian, Gersfeld, and Baret and is also seen in Fig. 2. Assuming local equilibrium (no adsorption barrier) between the subsurface and the surface, at a given surface pressure the subsurface concentration c~ and the adsorption P are known from equilibrium data. Therefrom the diffusion coefficient can be calculated, which must be, for the surfactants used by us, of the order of 5 × 10-6 c m 2 sec- 1. Equation [ 11] was obtained by assuming that the diffusion penetration depth is given by Eq. [7]. This can only be true after sufficient long times. For shorter times, however, the diffusion penetration depth is not given by film theory, but by penetration theory (10)
6p = OrDt) m .
[ 12]
Substitution of Eq. [12] in Eq. [4], with dr~ dt = 0, accounting for Eq. [10] (rE replaced b y 6p) yields in A Ao
z - - - * o% c---* 0 ) ,
D(Oc] = D cs
10
[10]
2cs(Dt) 1/2 r
--
[13]
wherefrom Journal of Colloid and Interface Science, Vol. 91, No. 1, January 1983
134
DE K E Y S E R A N D JOOS
kv -
2c~ (Dr/2 r
d ln (A/Ao) _ d(t) 1/2
[14]
This equation applies for short times and is confirmed by the experiments of Ter Minassian, Gersfeld, and Baret. It is also in agreement with Fig. 1. Again assuming local equilibrium between surface and subsurface Eq. [14] can be used to calculate a diffusion coefficient. From Eqs. [11] and [14] results k 2 = 2[01.
[15]
Equation (12) remains valid if fiF > 6p and for Eq. [9], bF < 60. The transition between both regimes occurs when 6F -- bp; hence for t ~_ 1/21ol. EXPERIMENTAL
Products used in this investigation were of high purity and purchased from Aldrich (99.5% + Gold label). Surface tensions were
r
R E S U L T S A N D DISCUSSION
5O
/+0
--x-x
30
measured with a Wilhelmy plate connected to a Statham transducer (UC-3 Gold cell). The output was connected to a strip chart recorder or to a digital voltmeter. The depth of our Langmuir trough was 0.6 cm. For compressing dodecanol monolayer with a constant relative compression rate an apparatus similar to that described by Van Voorst Vader was used, but built for compression instead of expansion. A step motor was connected to a logarithmic spiral in order to obtain a constant compression rate. The reproducibility of the experiments was 1-1.5 dyn cm -l. For desorption experiments at a constant surface pressure, after spreading of the monolayer, evaporation of the spreading solvent, and compression to a desired surface pressure, this surface pressure was kept constant, within 0.1-0.2 dyn cm -1, by further manually decreasing the area. This area was measured as a function of time. The reproducibility of the resulting 0 values (for long times) was within 10%.
_xc___
c
xc
o
At a given surface pressure the corresponding adsorption and subsurface concentration must be known from the equilibrium data. For decanol and dodecanol the relations between surface pressure and concentration and between concentration and adsorption are given by the von Szyszkowski and Langmuir equations, respectively (11), II = RTF
2o!
r = r~c/(c
1(/ -4
• I -3
I ~2
I -1
Ioglel
FIG. 4. Dodecanol monolayer. Surface pressure 1I
plotted as a function of log 101. x: Compression of the monolayer with 101 constant, O: desorption experiments with II = constant. ×c: collapse. Journal of Colloid andlnterface Science. Vol. 91, No. 1, January 1983
~
In (1 + c / a ) , + a),
with I'~: saturation adsorption and a: Langmuir-von Szyszkowski distribution constant. The parameters are given for dodecanol (F °~ = 7 × 10-l° mole c m -2, a = 4.3 × 10 - 9 mole cm -3) by Arcuri (12) and for decanol (ro~ = 6.1 × 10-1° mole c m -2, a = 1.39 × 10-8 mole cm -3) by Vochten (13). Lauric acid did not follow the Langmuir and von Szyszkowski equations. Fortunately Ter
DESORPTION AT CONSTANT
X
X
I 5.10-3
I 10-2
1 15.10 "2
(0
{seclJ21
FIG. 5. L a u r i c acid. D e s o r p t i o n e x p e r i m e n t s at 11 = constant. ×: E x p e r i m e n t s o f T e r M i n a s s i a n , ©: o u r data.
Minassian presented enough equilibrium data to obtain for a given surface pressure the corresponding subsurface concentration and adsorption. A dodecanol monolayer was compressed with constant relative compression rates ranging from 1.34 × 1 0 - 3 s e c -1 t o 1.34 × 1 0 -1 s e c - 2 . The corresponding steady-state surface pressure was measured (an example is shown in Fig. 3). At high compression rates [0J > 2 X 1 0 - 2 s e c - l the monolayer collapses and Eq. [ 11] is invalid. Results are given in Fig. 4. For dodecanol we also did desorption experiments at constant pressure. The corresponding relative compression rates, for sufficiently long times, obtained from curves similar to those in Fig. 2 are also given in Fig. 4. Our experimental points for both
135
SURFACE PRESSURE
types of experiments could be fitted according to Eq. [11 ], assuming a diffusion coefficient D = 6 × 1 0 - 6 c m 2 s e c -1. Agreement between theory and experiment is evident. For lauric acid we repeated the experiments of Ter Minassian. Her results together with ours are given in Fig. 5. Her results compared with our data are different by a factor 2 to 3. Again our experimental points are described by Eq. [11] assuming a slightly too high diffusion coefficient (D = 10 -5 c m 2 sec-~). As will be shown later the systematic difference between the results of Ter Minassian with ours, and the fact that the apparent diffusion coefficient resulting from our data is slightly too high is due to the occurrence of spontaneous convection currents not accounted for in our theory. Finally we reconsidered Baret's desorption experiments from decanol monolayers. From Baret's curves we get the relative compression rate, J0J, for long times. Therefrom and using the adsorption parameter of Vochten, we calculate the diffusion coefficient, which as seen from Table I, is of the correct order of magnitude. As a second point we considered desorption at the initial stage of the desorption process. The parameter, kv was obtained from graphs similar to that given in Fig. 1 (ln (.4/ Ao) vs tl/2). In this time scale Eq. [12] applies, yielding a diffusion coefficient, which as seen from Tables I, II, and III is of the correct order of magnitude. On the other hand, having calculated kv for small times and J0J for long ones, then
TABLE I D e s o r p t i o n f r o m D e c a n o l M o n o l a y e r s (Baret) 17 (dyn cm-I ) 5 10 15 20
c, (mole cm-s) 0.55 1.31 2.37 3.84
× × × ×
10 s 10 -s 10 -s 10 -s
r (mole em-2) 1.71 2.94 3.82 4.45
× × × ×
10 -9 10 -9 10 -9 10 -9
Io[ (see-') 2.6 4.6 8.4 11.0
× × × ×
10 -s 10 -s 10 -s 10 -s
D, (era2 see-') 4 3.7 3.5 2.3
X X × ×
10 -6 10 -6 10 -6 10 -6
/~ (sec-la) 6.12 11.1 11.4 18.8
× × × X
10 -2 10 -2 10 -2 10 -2
D2 (cm~ sec-~) 2.9 4.9 2.7 3.7
× X × X
10 -6 10 -6 10 -6 10 -6
k~/10l 1.5 2.7 1.6 3.2
Note. DI c a l c u l a t e d f r o m 0 u s i n g Eq. [ 1 1 ] . / ) 2 c a l c u l a t e d f r o m / c , u s i n g Eq. [14]. Journal of Colloid and Interface Science, Vol. 9 l, No. 1, January 1983
136
DE KEYSER AND JOOS T A B L E II Desorption from Dodecanol Monolayers
n (dya em-') 2 4 6 8 10
c, (mole em-3) 0.52 1.11 1.77 2.51 3.34
X X X X X
10 -9 10 -9 10 -9 10 -9 10 -9
r (mole em-2) 0.76 1.44 2.04 2.58 3.06
× × × × ×
10 -1° 10 -1° 10 -1° 10 - m 10 -1°
101 (see-') 2.6 2.5 2.9 4.1 5.5
× x x X ×
D, (em2 sec-1)
10 -4 10 -4 10 -4 10 -4 10 -4
8.5 6.5 6.0 6.7 7.2
X x X X X
10 -6 10 -6 10 -6 10 -6 10 -6
k, (see-ta) 1.33 1.75 2.2 2.8 3.9
X × × X ×
10 -2 10 -2 10 -2 10 -2 10 -2
D2 (em2 see-') 2.9 4.0 5.1 6.3 10.2
× x X X x
10 -6 10 -6 10 -6 10 -6 10 -6
kS/101 0.7 1.2 1.7 1.9 2.8
Note. D1 c a l c u l a t e d f r o m 0 u s i n g Eq. [11]. D2 c a l c u l a t e d f r o m k~ u s i n g Eq. [14].
according to Eq. [15], k2vlO1-1 - 2. It appears this ratio is of the correct order of magnitude, nevertheless this ratio increases with increasing surface pressure. This discrepancy is not unexpected since it comes from the definitions of the diffusion penetration depths ~p and ~v and their use for simplifying the diffusion equation. For decanol (Baret's experiments) and laur~'c acid k 2101-1 is fairly equal to 2, whereas for decanol the scatter is much larger. This is probably due to experimental errors. Dimitrov et al. (14, 15) have shown that for certain circumstances the convective term due to the deformation of the surface is of minor importance. This also applies for the present experiments at the beginning of the desorption process. But in any case, this convection term operates and makes the expression for ~p a rather rough approximation.
For calculating diffusion coefficients after small times (In (A/Ao) vs t 1/2) or after long ones (In (A/Ao) vs t) errors can be introduced by the fact that the II - F and II - c~relations are not accurately enough known (16). A second source of errors, as already mentioned, are due to the occurrence of spontaneous convection currents, not accounted for in our theory, due to the temperature gradients in the liquid layer (Benard instability) and/or external vibrations. We think this is the reason why Ter Minassian's results for lauric acid are systematically different from ours (for long times see Fig. 5). The depth of the trough ofTer Minassian was 1.2 cm, whereas our trough depth was 0.6 cm. Ter Minassian observed that by decreasing the trough depth the rate of desorption decreases. The intensity of spontaneous convection currents becomes more important the deeper the trough.
T A B L E III Desorption from Lauric Acid Monolayers H (dya era-) 2 4 6 8 10 11 13 15
C= (mole em-3) 0.75 1.00 1.25 1.50 1.83 2.00 2.37 2.79
X X X X X X X X
10 -9 10 -9 10 -9 10 -9 10 -9 10 -9 10 -9 10 -9
r (mole em-2) 3.87 4.06 4.36 4.58 4.90 5.00 5.21 5.37
X X X X × X × ×
10 -1° 10 -1° 10 -1° 10 -1° 10 - m 10 -10
10 -1° 10 -1°
101 (sec-') 2.9 3.85 5.3 6.4 9.0 10.1 12.3 12.8
X X × × X × × ×
D, (cm2 see-')
10 -5 10 -5 10 -5 10 -5 10 -5 10 -5 10 -s 10 -5
1.21 1.00 1.00 9.4 1.12 9.9 9.3 7.4
X X × X X X × X
10 -5 10 -5 10 -5 10 -6 10 -5 10 -6 10 -6 10 -6
k~ (see-in)
4.7 5.5 6.5 7.5 9.5 10.6 12.8 15.2
Note. D1 c a l c u l a t e d f r o m 0 u s i n g Eq. [ 1 1 ] . / ) 2 c a l c u l a t e d f r o m k~ u s i n g Eq. [14]. Journal of Colloid and Interface Science, Vol. 91, No. 1, January 1983
× X X X × X X ×
10 -3 10 -3 10 -3 10 -3 10 -3 10 -3 10 -3 10 -3
D~ (era2 see-')
4.6 3.9 4.0 4.1 5.1 5.5 6.2 6.7
X X X X × X X X
10 -6 10 -6 10 -6 10 -6 10 -6 10 -6 10 -6 10 -6
~/10J 0.8 0.8 0.8 0.9 1.0 1.1 1.3 1.8
DESORPTION AT CONSTANT SURFACE PRESSURE
It is likely that the effect of this convection currents becomes more and more important, the slower the desorption process. Therefore, even in our less deep trough, for lauric acid desorption after long times, convection cannot be ruled out yielding a diffusion coefficient which is slightly too high (D = 10 -5 cm 2 sec -1 instead o f D - 5 × 10 -6 sec-l). From this investigation we can safely conclude that desorption from lauric acid, decanol, and dodecanol monolayers is controlled by diffusion and there is no need to postulate an adsorption barrier. ACKNOWLEDGMENT One of us (P.D.K.) is indebted to the Belgian IWONL for a grant. REFERENCES 1. Ter Minassian, L., J. Colloid Sci. 11, 398 (1956). 2. Ter Minassian, L., J. Chim. Phys. 52, 80 (1955). 3. Patlak, C. S., and Gershfeld, N. L., J. Colloid Interface Sci. 25, 503 (1967). 4. Baret, J. F., Bois, A. G., Casalta, L., Dupin, J. J., Firpo, J. L., Gonella, J., Melinon, J. P., and Rodeau, J. L., J. Colloid lnterface Sei. 53, 50 (1975).
137
5. Ward, A. F. G., and Tordai, L., J. Chem. Phys. 14, 453 (1946). 6. Motomura, R., Shibata, A., Nakamura, M., and Matuura, R., J. Colloid Interface Sci. 29, 623 (1969). 7. Van Voorst Vader, F., Erkelens, Th. F., and van den Tempel, M., Trans. Faraday Soc. 60, 1170 (1964). 8. Levich, B., "Physicochemical Hydrodynamics." Prentice-Hall, Englewood Cliffs, N. J., 1962. 9. Lucassen, J., and Giles, D., J. Chem. Soc. Faraday Trans. 1 71, 217 (1975). 10. Bird, R. B., Stewart, W. E., and Lightfoot, E. M., "Transport Phenomena." Wiley, New York, 1950. 11. Lucassen Reynders, E. H., and van den Tempe1, M., "Proceedings IVth International Congress Surface Activity Brussels (1964)," Vol II, p. 779, 1967. 12. Arcuri, C., "Colloquium grenslaagverschijnselen. Verhand. Koninkl. Vlaamse Acad. v, BelgiE, Brussels 1965." p. 149. 13. Vochten, R., Thesis, Ghent, 1966. 14. Dimitrov, D. S., Panaiotov, I., Richmond, P., and Ter Minassian-Saragu, L., J. Colloid Interface Sci. 65, 483 (1978). 15. Panaiotov, I., Dimitrov, D. S., and Ivanova, M. G., J. Colloid Interface Sci. 69, 318 (1979). 16. Lucassen Reynders, E. H., Lueassen, J., Garrett, P. R., Giles, D., and Hollway F., Advan. Chem. Sci. 144, 272 (1975).
Journalof Colloidand InterfaceScience.Vol.91, No. 1, January 1983
decreases. This decrease is measured as a function of time. As long as the transfer o f surfactant to the bulk is diffusion controlled, the theory of Ward and Tordai (5) applies giving the adsorption P as a function o f time,
INTRODUCTION
The first desorption experiments at constant surface pressure o f slightly soluble monolayers were performed by Ter Minassian (1, 2). Later, similar experiments were done by Gershfeld and Patlak (3), while Baret et al. (4) presented a theoretical analysis. In our opinion the treatment of Baret is erroneous and leads to the false conclusion that desorption is not controlled by diffusion. It is our aim to reconsider the problem and to prove our theory by experiments from other investigators together with our own experimental data. Let us consider such a desorption experiment. A monolayer of a slightly soluble surfactant (e.g., lauric acid) is spread on a Langmuir trough on a clean water surface. After evaporation o f the solvent, the monolayer is suddenly compressed till an area Ao and a c o r r e s p o n d i n g surface pressure 11o are reached. Now from the soluble monolayer, surfactant molecules tend to dissolve in the underlying aqueous phase. Now two kind o f experiments can be done. In the first one, the area is kept constant and the surface pressure
P = F0 - 2
c s ( t - r ) d ( l " ) 1/2 , [11 '~0
with P0: adsorption at time t = 0, D: diffusion coefficient, c~: subsurface concentration, r: a d u m m y variable, and, 7r: 3.1415 . . . . In the second type o f experiment, considered here, the surface pressure is kept constant by further compression o f the m o n o layer, giving the area A as a function o f time. Usually the logarithm o f the relative area, is plotted as a function of t 1/2 or t (see Figs. 1 and 2). Plotting In (A/Ao) v s t 1/2 one sees that initially a linear relation is observed. Following Ter Minassian we define kv=
d In (A/Ao) d(t)v2
(small times).
[2]
On the other hand, plotting In (A/Ao) vs t one sees that after sufficiently long times a linear relation is again observed. This means that 131
Journalof Colloidand InterfaceScience.Vol. 91, No. 1, January 1983
0021-9797/83/010131-07503.00/0 Copyright© 1983by AcademicPress,Inc. All rightsof reproductionin any form reserved.
132
DE KEYSER AND JOOS
for in the diffusion equation. This is well discussed by Van Voorst Vader et al. (7). Neglect of this term leads to false conclusions. In the beginning of an experiment at constant surface pressure and monitoring the area as a function of time, this convective term is of minor importance. The appearance of convection was already recognized by Ter Minassian.
o•
1
2
3
~,
5
THEORY
~'f
FIG. 1. Dodecanol monolayer (H = 6 dyn cm-~). Logarithm of the relative area plotted vs t lie (t in minutes). Initially a linear relation is observed.
As already stated by Levich (8) and Van Voorst Vader (7), the boundary condition for conservation of mass at the surface reads dr
+ OF +
for long times the dilatation 0 defined as dln A 0- - -
dt
[31
is constant. In this paper we consider also a somewhat different experiment which is closely related to a desorption experiment at constant surface pressure. A monolayer of a slightly soluble monolayer is compressed with a constant relative compression rate 0. As seen from Fig. 3, in the beginning transient surface pressures are obtained, but after longer times the surface pressure remains constant, indicating a steady state. The relation between both experiments is evident. In the first one (measuring area as a function of time with constant surface pressure) a constant surface pressure is imposed to the system and after long times the dilatation 0 is measured. Whereas in the second experiment a constant dilatation is imposed and the surface pressure is measured in the steady state. Obviously, other experiments are possible. The simplest experiment seems to be that of M o to mu r a (6), where a slightly soluble monolayer (myristic acid) is compressed at constant dA/dt and the surface pressure is measured. For this type of experiment, however, the theoretical analysis is more difficult. For those experiments where the area changes with time, surface convection is generated and this convection term must be accounted Journal of Colloid and Interface Science, Vol. 91, No. 1, January 1983
D(OC) = O, \Oz] °
[4]
with c: concentration and z: coordinate normal to the surface. In Ter Minassian's experiments the surface pressure was kept constant; hence the adsorption also remains constant whence in Eq. [4], d F/dt = 0. Van Voorst Vader et al. have shown that
d l n A _(OVx~ 0 - --~
[5]
~,Ox]o,
with Vx: velocity along the surface and x: coordinate along the surface. Hence by changing the area convection is introduced and this convection must be accounted for in the diffusion equation. tn / ~
e*°, -02 °° °°° -..°°1 ,°
-1.5 t(min)
NO. 2. Dodecanol monolayer (H = 8 dyn cm-l). Logarithm of the relative area plotted (t in minutes). For sufficient long times a linear relation is found.
133
DESORPTION AT CONSTANT SURFACE PRESSURE
Van Voorst Vader also showed that by expanding an adsorbed monolayer with a constant relative expansion rate 0 a steady state is obtained. In the steady state the diffusion penetration depth 6v is independent upon time and is given by (film theory) ~F
=-
-(TrD m~ 0 r = 3 . 1 4 1 5 9
\ 20 ]
""
.).
[6]
Strictly, the theory of Van Voorst Vader, from which Eq. [6] results, is only valid for an expanding monolayer. For a compressed monolayer, the theory is far from being solved. On the other hand, Lucassen and Giles (9) showed that for small dilatations the jump in surface tension, Aa, brought about by expanding the monolayer at constant dilatation is given by
Aa _ RrrZ ( TrO]'/2 k2DI
[7]
(Aa = ad -- ~e; ~d: dynamic surface tension; and ag surface tension at equilibrium). He also showed that the modulus of elasticity It] for the case when diffusion is fully operating is approached by =
[81
(ox angular frequency) and it appears that the relationship between Aa and 0 is identical to that between I~1 and 2o~lrr. From this we conclude that, at least formally, 0 can be treated as a frequency. For a compressed monolayer, this frequency should be negative (since 0 < 0). Since a negative frequency has no meaning, we made the ad hoe assumption that the diffusion penetration depth is given by ~F = ~.21011
"
1(
[91
The diffusion flux in Eq. [4] can now be approximated for our situation (since for
S
•OZ]O
6F '
15
20
flminl
FIG. 3. Dodecanol monolayer compressed with Conslant relative compression rate 10[ = 1.34 × 10 -3 sec -1. After sufficiently long times a steady state is observed.
with cs: subsurface concentration. Substitution of Eqs. [10] and [9] in Eq. [4], remembering that d I'/dt = 0, yields
dln(A/Ao)_ dt
2D(~] 2 ~r \ P / "
[11]
Equation [11] predicts that after sufficiently long times, when the diffusion penetration depth is given by Eq. [9], the slope of d In (A/Ao)/dt becomes constant. This was observed by Ter Minassian, Gersfeld, and Baret and is also seen in Fig. 2. Assuming local equilibrium (no adsorption barrier) between the subsurface and the surface, at a given surface pressure the subsurface concentration c~ and the adsorption P are known from equilibrium data. Therefrom the diffusion coefficient can be calculated, which must be, for the surfactants used by us, of the order of 5 × 10-6 c m 2 sec- 1. Equation [ 11] was obtained by assuming that the diffusion penetration depth is given by Eq. [7]. This can only be true after sufficient long times. For shorter times, however, the diffusion penetration depth is not given by film theory, but by penetration theory (10)
6p = OrDt) m .
[ 12]
Substitution of Eq. [12] in Eq. [4], with dr~ dt = 0, accounting for Eq. [10] (rE replaced b y 6p) yields in A Ao
z - - - * o% c---* 0 ) ,
D(Oc] = D cs
10
[10]
2cs(Dt) 1/2 r
--
[13]
wherefrom Journal of Colloid and Interface Science, Vol. 91, No. 1, January 1983
134
DE K E Y S E R A N D JOOS
kv -
2c~ (Dr/2 r
d ln (A/Ao) _ d(t) 1/2
[14]
This equation applies for short times and is confirmed by the experiments of Ter Minassian, Gersfeld, and Baret. It is also in agreement with Fig. 1. Again assuming local equilibrium between surface and subsurface Eq. [14] can be used to calculate a diffusion coefficient. From Eqs. [11] and [14] results k 2 = 2[01.
[15]
Equation (12) remains valid if fiF > 6p and for Eq. [9], bF < 60. The transition between both regimes occurs when 6F -- bp; hence for t ~_ 1/21ol. EXPERIMENTAL
Products used in this investigation were of high purity and purchased from Aldrich (99.5% + Gold label). Surface tensions were
r
R E S U L T S A N D DISCUSSION
5O
/+0
--x-x
30
measured with a Wilhelmy plate connected to a Statham transducer (UC-3 Gold cell). The output was connected to a strip chart recorder or to a digital voltmeter. The depth of our Langmuir trough was 0.6 cm. For compressing dodecanol monolayer with a constant relative compression rate an apparatus similar to that described by Van Voorst Vader was used, but built for compression instead of expansion. A step motor was connected to a logarithmic spiral in order to obtain a constant compression rate. The reproducibility of the experiments was 1-1.5 dyn cm -l. For desorption experiments at a constant surface pressure, after spreading of the monolayer, evaporation of the spreading solvent, and compression to a desired surface pressure, this surface pressure was kept constant, within 0.1-0.2 dyn cm -1, by further manually decreasing the area. This area was measured as a function of time. The reproducibility of the resulting 0 values (for long times) was within 10%.
_xc___
c
xc
o
At a given surface pressure the corresponding adsorption and subsurface concentration must be known from the equilibrium data. For decanol and dodecanol the relations between surface pressure and concentration and between concentration and adsorption are given by the von Szyszkowski and Langmuir equations, respectively (11), II = RTF
2o!
r = r~c/(c
1(/ -4
• I -3
I ~2
I -1
Ioglel
FIG. 4. Dodecanol monolayer. Surface pressure 1I
plotted as a function of log 101. x: Compression of the monolayer with 101 constant, O: desorption experiments with II = constant. ×c: collapse. Journal of Colloid andlnterface Science. Vol. 91, No. 1, January 1983
~
In (1 + c / a ) , + a),
with I'~: saturation adsorption and a: Langmuir-von Szyszkowski distribution constant. The parameters are given for dodecanol (F °~ = 7 × 10-l° mole c m -2, a = 4.3 × 10 - 9 mole cm -3) by Arcuri (12) and for decanol (ro~ = 6.1 × 10-1° mole c m -2, a = 1.39 × 10-8 mole cm -3) by Vochten (13). Lauric acid did not follow the Langmuir and von Szyszkowski equations. Fortunately Ter
DESORPTION AT CONSTANT
X
X
I 5.10-3
I 10-2
1 15.10 "2
(0
{seclJ21
FIG. 5. L a u r i c acid. D e s o r p t i o n e x p e r i m e n t s at 11 = constant. ×: E x p e r i m e n t s o f T e r M i n a s s i a n , ©: o u r data.
Minassian presented enough equilibrium data to obtain for a given surface pressure the corresponding subsurface concentration and adsorption. A dodecanol monolayer was compressed with constant relative compression rates ranging from 1.34 × 1 0 - 3 s e c -1 t o 1.34 × 1 0 -1 s e c - 2 . The corresponding steady-state surface pressure was measured (an example is shown in Fig. 3). At high compression rates [0J > 2 X 1 0 - 2 s e c - l the monolayer collapses and Eq. [ 11] is invalid. Results are given in Fig. 4. For dodecanol we also did desorption experiments at constant pressure. The corresponding relative compression rates, for sufficiently long times, obtained from curves similar to those in Fig. 2 are also given in Fig. 4. Our experimental points for both
135
SURFACE PRESSURE
types of experiments could be fitted according to Eq. [11 ], assuming a diffusion coefficient D = 6 × 1 0 - 6 c m 2 s e c -1. Agreement between theory and experiment is evident. For lauric acid we repeated the experiments of Ter Minassian. Her results together with ours are given in Fig. 5. Her results compared with our data are different by a factor 2 to 3. Again our experimental points are described by Eq. [11] assuming a slightly too high diffusion coefficient (D = 10 -5 c m 2 sec-~). As will be shown later the systematic difference between the results of Ter Minassian with ours, and the fact that the apparent diffusion coefficient resulting from our data is slightly too high is due to the occurrence of spontaneous convection currents not accounted for in our theory. Finally we reconsidered Baret's desorption experiments from decanol monolayers. From Baret's curves we get the relative compression rate, J0J, for long times. Therefrom and using the adsorption parameter of Vochten, we calculate the diffusion coefficient, which as seen from Table I, is of the correct order of magnitude. As a second point we considered desorption at the initial stage of the desorption process. The parameter, kv was obtained from graphs similar to that given in Fig. 1 (ln (.4/ Ao) vs tl/2). In this time scale Eq. [12] applies, yielding a diffusion coefficient, which as seen from Tables I, II, and III is of the correct order of magnitude. On the other hand, having calculated kv for small times and J0J for long ones, then
TABLE I D e s o r p t i o n f r o m D e c a n o l M o n o l a y e r s (Baret) 17 (dyn cm-I ) 5 10 15 20
c, (mole cm-s) 0.55 1.31 2.37 3.84
× × × ×
10 s 10 -s 10 -s 10 -s
r (mole em-2) 1.71 2.94 3.82 4.45
× × × ×
10 -9 10 -9 10 -9 10 -9
Io[ (see-') 2.6 4.6 8.4 11.0
× × × ×
10 -s 10 -s 10 -s 10 -s
D, (era2 see-') 4 3.7 3.5 2.3
X X × ×
10 -6 10 -6 10 -6 10 -6
/~ (sec-la) 6.12 11.1 11.4 18.8
× × × X
10 -2 10 -2 10 -2 10 -2
D2 (cm~ sec-~) 2.9 4.9 2.7 3.7
× X × X
10 -6 10 -6 10 -6 10 -6
k~/10l 1.5 2.7 1.6 3.2
Note. DI c a l c u l a t e d f r o m 0 u s i n g Eq. [ 1 1 ] . / ) 2 c a l c u l a t e d f r o m / c , u s i n g Eq. [14]. Journal of Colloid and Interface Science, Vol. 9 l, No. 1, January 1983
136
DE KEYSER AND JOOS T A B L E II Desorption from Dodecanol Monolayers
n (dya em-') 2 4 6 8 10
c, (mole em-3) 0.52 1.11 1.77 2.51 3.34
X X X X X
10 -9 10 -9 10 -9 10 -9 10 -9
r (mole em-2) 0.76 1.44 2.04 2.58 3.06
× × × × ×
10 -1° 10 -1° 10 -1° 10 - m 10 -1°
101 (see-') 2.6 2.5 2.9 4.1 5.5
× x x X ×
D, (em2 sec-1)
10 -4 10 -4 10 -4 10 -4 10 -4
8.5 6.5 6.0 6.7 7.2
X x X X X
10 -6 10 -6 10 -6 10 -6 10 -6
k, (see-ta) 1.33 1.75 2.2 2.8 3.9
X × × X ×
10 -2 10 -2 10 -2 10 -2 10 -2
D2 (em2 see-') 2.9 4.0 5.1 6.3 10.2
× x X X x
10 -6 10 -6 10 -6 10 -6 10 -6
kS/101 0.7 1.2 1.7 1.9 2.8
Note. D1 c a l c u l a t e d f r o m 0 u s i n g Eq. [11]. D2 c a l c u l a t e d f r o m k~ u s i n g Eq. [14].
according to Eq. [15], k2vlO1-1 - 2. It appears this ratio is of the correct order of magnitude, nevertheless this ratio increases with increasing surface pressure. This discrepancy is not unexpected since it comes from the definitions of the diffusion penetration depths ~p and ~v and their use for simplifying the diffusion equation. For decanol (Baret's experiments) and laur~'c acid k 2101-1 is fairly equal to 2, whereas for decanol the scatter is much larger. This is probably due to experimental errors. Dimitrov et al. (14, 15) have shown that for certain circumstances the convective term due to the deformation of the surface is of minor importance. This also applies for the present experiments at the beginning of the desorption process. But in any case, this convection term operates and makes the expression for ~p a rather rough approximation.
For calculating diffusion coefficients after small times (In (A/Ao) vs t 1/2) or after long ones (In (A/Ao) vs t) errors can be introduced by the fact that the II - F and II - c~relations are not accurately enough known (16). A second source of errors, as already mentioned, are due to the occurrence of spontaneous convection currents, not accounted for in our theory, due to the temperature gradients in the liquid layer (Benard instability) and/or external vibrations. We think this is the reason why Ter Minassian's results for lauric acid are systematically different from ours (for long times see Fig. 5). The depth of the trough ofTer Minassian was 1.2 cm, whereas our trough depth was 0.6 cm. Ter Minassian observed that by decreasing the trough depth the rate of desorption decreases. The intensity of spontaneous convection currents becomes more important the deeper the trough.
T A B L E III Desorption from Lauric Acid Monolayers H (dya era-) 2 4 6 8 10 11 13 15
C= (mole em-3) 0.75 1.00 1.25 1.50 1.83 2.00 2.37 2.79
X X X X X X X X
10 -9 10 -9 10 -9 10 -9 10 -9 10 -9 10 -9 10 -9
r (mole em-2) 3.87 4.06 4.36 4.58 4.90 5.00 5.21 5.37
X X X X × X × ×
10 -1° 10 -1° 10 -1° 10 -1° 10 - m 10 -10
10 -1° 10 -1°
101 (sec-') 2.9 3.85 5.3 6.4 9.0 10.1 12.3 12.8
X X × × X × × ×
D, (cm2 see-')
10 -5 10 -5 10 -5 10 -5 10 -5 10 -5 10 -s 10 -5
1.21 1.00 1.00 9.4 1.12 9.9 9.3 7.4
X X × X X X × X
10 -5 10 -5 10 -5 10 -6 10 -5 10 -6 10 -6 10 -6
k~ (see-in)
4.7 5.5 6.5 7.5 9.5 10.6 12.8 15.2
Note. D1 c a l c u l a t e d f r o m 0 u s i n g Eq. [ 1 1 ] . / ) 2 c a l c u l a t e d f r o m k~ u s i n g Eq. [14]. Journal of Colloid and Interface Science, Vol. 91, No. 1, January 1983
× X X X × X X ×
10 -3 10 -3 10 -3 10 -3 10 -3 10 -3 10 -3 10 -3
D~ (era2 see-')
4.6 3.9 4.0 4.1 5.1 5.5 6.2 6.7
X X X X × X X X
10 -6 10 -6 10 -6 10 -6 10 -6 10 -6 10 -6 10 -6
~/10J 0.8 0.8 0.8 0.9 1.0 1.1 1.3 1.8
DESORPTION AT CONSTANT SURFACE PRESSURE
It is likely that the effect of this convection currents becomes more and more important, the slower the desorption process. Therefore, even in our less deep trough, for lauric acid desorption after long times, convection cannot be ruled out yielding a diffusion coefficient which is slightly too high (D = 10 -5 cm 2 sec -1 instead o f D - 5 × 10 -6 sec-l). From this investigation we can safely conclude that desorption from lauric acid, decanol, and dodecanol monolayers is controlled by diffusion and there is no need to postulate an adsorption barrier. ACKNOWLEDGMENT One of us (P.D.K.) is indebted to the Belgian IWONL for a grant. REFERENCES 1. Ter Minassian, L., J. Colloid Sci. 11, 398 (1956). 2. Ter Minassian, L., J. Chim. Phys. 52, 80 (1955). 3. Patlak, C. S., and Gershfeld, N. L., J. Colloid Interface Sci. 25, 503 (1967). 4. Baret, J. F., Bois, A. G., Casalta, L., Dupin, J. J., Firpo, J. L., Gonella, J., Melinon, J. P., and Rodeau, J. L., J. Colloid lnterface Sei. 53, 50 (1975).
137
5. Ward, A. F. G., and Tordai, L., J. Chem. Phys. 14, 453 (1946). 6. Motomura, R., Shibata, A., Nakamura, M., and Matuura, R., J. Colloid Interface Sci. 29, 623 (1969). 7. Van Voorst Vader, F., Erkelens, Th. F., and van den Tempel, M., Trans. Faraday Soc. 60, 1170 (1964). 8. Levich, B., "Physicochemical Hydrodynamics." Prentice-Hall, Englewood Cliffs, N. J., 1962. 9. Lucassen, J., and Giles, D., J. Chem. Soc. Faraday Trans. 1 71, 217 (1975). 10. Bird, R. B., Stewart, W. E., and Lightfoot, E. M., "Transport Phenomena." Wiley, New York, 1950. 11. Lucassen Reynders, E. H., and van den Tempe1, M., "Proceedings IVth International Congress Surface Activity Brussels (1964)," Vol II, p. 779, 1967. 12. Arcuri, C., "Colloquium grenslaagverschijnselen. Verhand. Koninkl. Vlaamse Acad. v, BelgiE, Brussels 1965." p. 149. 13. Vochten, R., Thesis, Ghent, 1966. 14. Dimitrov, D. S., Panaiotov, I., Richmond, P., and Ter Minassian-Saragu, L., J. Colloid Interface Sci. 65, 483 (1978). 15. Panaiotov, I., Dimitrov, D. S., and Ivanova, M. G., J. Colloid Interface Sci. 69, 318 (1979). 16. Lucassen Reynders, E. H., Lueassen, J., Garrett, P. R., Giles, D., and Hollway F., Advan. Chem. Sci. 144, 272 (1975).
Journalof Colloidand InterfaceScience.Vol.91, No. 1, January 1983