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Measurement of the Surface Elasticity of an Adsorbed Monolayer by Continuous Surface Deformation
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
183, 559–567 (1996)
0580
Measurement of the Surface Elasticity of an Adsorbed Monolayer by Continuous Surface Deformation PAUL JOOS, 1, * PETER PETROV,*
AND
JIPING FANG **
*Department of Chemistry, University of Antwerp, U.I.A., Universiteitsplein 1, B-2610 Wilrijk, Belgium; and **Max-Planck-Institut, Kolloid-und Grenzflachenforschung, Rudower Chaussee 5, D-12489 Berlin, Germany Received March 11, 1996; accepted June 19, 1996
The elasticity of a monolayer adsorbed from a surfactant solution is considered by continuous surface deformation. For all kinds surface deformations a single relation between the elasticity and the effective time should be obtained (master curve). This effective time depends on the rate of deformation and on the kind of deformation. The relation between elasticity and effective time was proven experimentally for a decanoic acid solution at a concentration 10 07 mol cm03 . The following surface deformations are considered: (i) expansion at a constant dilatation rate, (ii) compression at a constant dilatation rate, (iii) linear expansion, (iv) linear compression, and (v, vi) compression and expansion in a some what more complicated way. From these data the Gibbs elasticity and the diffusion relaxation time are obtained. These parameters are also obtained by longitudinal wave method of Lucassen–Van den Tempel. Both set of parameters are comparable. Finally the situation at a growing drop with constant flow rate is considered (Triton X-100) to show that also here the approach is followed. q 1996 Academic Press, Inc.
Key Words: surface elasticity by continuous surface deformation.
INTRODUCTION
Let us consider a monolayer of a surfactant solution in equilibrium with the bulk. Initially the area of the monolayer is V0 , the equilibrium surface tension se (equilibrium surface pressure Pe ), adsorption Ge , and the concentration c0 is uniform over the whole solution. Now this area is deformed continuously so that the area is a function of time: V(t). The exact function of V(t) depends of course on the experimental conditions (on the way and the kind of deformation), for instance, linear expansion, linear compression, expansion at a constant dilatation rate, or compression at a constant dilatation rate, etc. By the area deformation the equilibrium between surface and the bulk is broken and the system will tend to re-establish its equilibrium by diffusion of surfactant from the surface to the bulk (compression) or vice versa 1
(expansion). During area deformation, the surface tension is different from its equilibrium value and depends on time: s(t). We are merely interested in the jump of the surface tension s(t) defined as Ds(t) Å s(t) 0 se . If the relaxation process of the equilibrium re-establishing is diffusion, it is plain the diffusion equation must be considered with corresponding boundary conditions. Since the surface is deformed continuously, a convection term must be accounted for in the diffusion Eq. [1]. In this way the jump in surface tension with time is predicted and the theoretical curve has to be compared with experimental data. To describe this curve two parameters, the Gibbs elasticity, e0 , and the diffusion relaxation time, tD , are involved. From these experiments also the elasticity of the monolayer is obtained, which depends on time. If the surface deformation is fast, the diffusion process has no time to operate and the surface is purely elastic; for large times, the diffusion operates and the monolayer becomes viscoelastic. One should expect that for different ways of area changes, the same elasticity should be found. It is the scope of this paper to show this, and to do this we considered a surfactant solution at the same experimental conditions (decanoic acid in water at a concentration c Å 10 07 mol cm03 , pH 2, at the air/ water interface) which is subjected to different kinds surface deformations. The area changes considered are linear expansion and compression, expansion and compression with a constant dilatation rate, and even more complicated surface deformations. MacLeod and Radke (2) reported also experiments of the variation of the interfacial tension, at a growing drop with constant flow rate. These experiments are considered to prove that our approach is also applicable to this situation. THEORY
The dilatation rate u, being defined as
uÅ
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accounts for the convection. For an expanding surface the convective diffusion equation reads (1) as Ìc Ìc Ì2c 0 uz ÅD , Ìt Ìz Ìz 2
uú0
[2]
Ds Å e0
uõ0
[3]
1/
1 4teff ptD
F G
teff Å
S D*S D 2
V0 V
t
S D
dG Ìc / uG Å D dt Ìz
0
zr`
and
c r c0
dc ( G 0 Ge ). dG
* S VV D dt. 2
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[10]
0
In Eq. [8] DV is the change in area, defined as DV Å V(t) 0 V0 .
[11]
For expansion DV ú 0 and for compression DV õ 0. The parameters involved are the Gibbs elasticity, e0 , and the diffusion relaxation time, defined as
[7]
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e0 Å 0
ds Ge dG
1 D
S D
and tD Å
dG dc
[12]
2
.
[13]
If the surface elasticity, e, is defined as eÅ
ds V, dV
[14]
then from Eqs. [8] and [14] we obtain eÅ
[6]
can be obtained, but involves the numerical evaluation of the convolution integral in the generalized Ward and Tordai equation. In order to avoid this numerical calculations we approximated the convolution integral. The final solution for expansion is (4, 6)
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teff Å
[5]
In this way the jump in adsorption as a function of time DG(t) Å G(t) 0 Ge
[9]
For compression the same Eq. [8] is obtained (5, 7) but with effective time now defined as
[4]
In these equations z is a coordinate normal to the surface, c is the concentration, G is the adsorption, D is the diffusion coefficient, and t is the time. The integration of these equations results in a generalized Ward and Tordai equation ( 4 – 6 ) . Further it is assumed that there is a local equilibrium between the surface and the subsurface. This means that the relation between the adsorption and the subsurface concentration, cs , is given by the adsorption isotherm. If we restrict ourselves to small jumps in surface tension, this adsorption isotherm is linearized:
cs 0 ce Å
dt.
0
and taking into account the boundary conditions far away from the surface at
[8]
2
V V0
t
These differential equations are integrated (4, 5) taking into account the boundary conditions for conservation of the mass at the surface
1/2
with teff , the effective time, defined as
It was shown (3) that for compressed surfaces this equation should be written as Ìc Ìc Ì2c / uz ÅD , Ìt Ìz Ìz 2
DV V
e0 . 1 / [4teff / ptD ] 1 / 2
[15]
If a given surfactant solution is subjected to various kinds of area deformations, as indicated by Eqs. [9] and [10], different expressions for the effective time are obtained, but if the elasticity obtained from all these different experiments is plotted as a function of teff (we calculated it as a function q of teff ), then the same master curve should be obtained. We considered linear expansion and compression, expansion and compression at constant u for decanoic acid monolayer at the same concentration. For these kinds of deformations, the expressions for the area with time and the effective
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TABLE 1 Area Variation and Effective Time for Different Kinds of Surface Deformations Deformation at
Kind of deformation
Area variation
Relative deformation
Effective time
Expansion u constant
ut
V Å V0e
d ln V dt
uÅ
DV Å 1 0 e0ut V
teff Å
1 (1 0 e02ut) 2u
(u ú 0) dV constant dt
aÅ
1 dV V0 dt
V Å V0(1 / at)
DV at Å V 1 / at
teff Å
x 3 / 3x 2 / 3x 3a(1 / x)2
(x Å at, a ú 0) m constant
mt
mt
V Å V0( be 0 g)
–
DV b(e 0 1) Å V bemt 0 g
teff Å
b2(e2y 0 1) 0 4bg(e y 0 1) / 2g 2y 2m( bey 0 g)2
(y Å at, m ú 0) F constant
aÅ
V Å V0(1 / at)2/3
F V0
DV (1 / at)2/3 0 1 Å V (1 / at)2/3
teff Å
3(1 / x)7/3 0 1 7a(1 / x)4/3
(x Å at, a ú 0) Compression u constant
0ÉuÉt
V Å V0e
d ln V dt
uÅ
DV Å (1 0 eÉuÉt ) V
teff Å
1 [1 0 e02ÉuÉt] 2ÉuÉ
(u õ 0) dV constant dt
aÅ
1 dV V0 dt
V Å V0 (1 / at)
–
V Å V0 (2emt 0 1)
DV at Å V 1 / at
teff Å
t 1 / at
(a õ 0) m constant
mt
DV 2(e 0 1) Å V 2emt 0 1
teff Å
FS
D
G
1 e mt 2(e mt 0 1) ln / m 2e mt 0 1 2e mt 0 1
( m õ 0)
times are given in Table 1. Effective times are obtained from Eqs. [9] and [10] using the appropriate functions of V. J. Fang and K. Lunkenheimer considered other types of area deformations in a Langmuir trough with two moving barriers (10): one for compression V Å V0 (2e mt 0 1)
mõ0
[16]
m ú 0,
[17]
and one for expansion V Å V0[ ge mt 0 b]
where b and g are constants depending on the initial area ( V0 ) and the final area ( V` ).
(In our experimental conditions, for expansion V0 Å 22.5 cm2 and V` Å 225 cm2 , whence b Å 11 and g Å 10), and m is a constant which determinate the speed of moving of the barriers. For compression m õ 0 and for expansion m ú 0. The area deformations and effective times are also given in Table 1. The experiments of MacLeod and Radke (2) are also considered for growing drop at constant flow rate, as analyzed previously (11). The expressions for the area change with time and the effective times are also given in Table 1. If small periodic amplitude area deformations are applied [this is the longitudinal wave method of Lucassen and Van den Tempel (8, 9)], then the modulus of elasticity ÉeÉ can be measured as ÉeÉ Å
V0 / V` bÅ V0
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ÉDsÉ V0 ÉDVÉ
[18]
with ÉDsÉ as the amplitude of the surface tension variation
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FIG. 1. Compression according to Eq. [16]. DP as a function of time, t for decanoic acid monolayers (c0 Å 10 07 mol cm03 ) at different compression speeds: ( s ) m Å 01.33 1 10 02 s 01 ; ( L ) m Å 01.00 1 10 02 s 01 ; ( n ) m Å 06.67 1 10 03 s 01 ; ( h ) m Å 03.33 1 10 03 s 01 ; ( / ) m Å 01.67 1 10 03 s 01 . Solid lines are calculated according to Eq. [8] with e0 Å 57.2 mN m01 , tD Å 0.45 s, and the corresponding expressions for teff and DV / V (Table 1).
and ÉDVÉ as the amplitude of the area deformation. It was shown that for a diffusion-controlled adsorption process, the modulus of elasticity is given by (8, 9) ÉeÉ Å
with zÅ
e0 (1 / 2z / 2z 2 ) 1 / 2
S D S 1 2vtD
1/2
Å
[19]
1 4pntD
D
1/2
,
[20]
A comparison between longitudinal wave method and the other ones presented here should be made. Both theories apply to a diffusion relaxation process using the same boundary conditions at the surface, Eqs. [4] and [6], and also in both the elasticity of the surface is defined in the same way (Eq. [14]). In the longitudinal wave method, the surface is subjected to small amplitude deformation, resulting in a complex elasticity depending on the frequency: e(iv ) Å
1/
where v is the radial frequency (rad s 01 ) and n the common frequency (Hz).
e0 1 ivtD
S D
1/2
.
[21]
FIG. 2. Expansion according to Eq. [17]. Ds as a function of time for decanoic acid monolayers ( c0 Å 10 07 mol cm03 ) at different compression speeds: ( s ) m Å 1.67 1 10 02 s 01 ; ( L ) m Å 1.33 1 10 02 s 01 ; ( l ) m Å 1.00 1 10 02 s 01 ; ( n ) m Å 6.67 1 10 03 s 01 ; ( h ) m Å 3.33 1 10 03 s 01 ; ( l ) m Å 1.67 1 10 03 s 01 . Solid lines are calculated according to Eq. [8] with e0 Å 57.2 mN m01 and tD Å 0.45 s.
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q
FIG. 3. Elasticities, e, obtained from compression with constant u of decanoic acid monolayers (c0 Å 10 07 mol cm03 ) as a function of teff : ( l ) u Å 02.62 1 10 01 s 01 ; ( / ) u Å 01.31 1 10 01 s 01 ; ( m ) u Å 06.55 1 10 02 s 01 ; ( s ) u Å 02.62 1 10 02 s 01 ; ( L ) u Å 01.31 1 10 02 s 01 ; ( h ) u Å 06.55 1 10 03 s 01 ; ( n ) u Å 02.62 1 10 03 s 01 . The solid line is calculated according to Eq. [15] with e0 Å 57.2 mN m01 and tD Å 0.45 s.
Since small amplitude surface deformations are considered, the convection term need not be accounted for in the diffusion equation. In the other methods the continuously deformed surface (consequently the convection term in the diffusion equation must be accounted for) resulting in the elasticity as a function of the effective time, given by Eq. [15]. This equation is in the same form as Eq. [21]. However, Eq. [21] assume no approximations for small amplitudes surface deformations, whereas our expression [16] involves an approximation of the convolution integral in the generalized Ward and Tordai equation. The longitudinal wave method of Lucassen–Van den Tempel and the discussed ones for continuous surface deformations are complementary each other. It is expected that
for a fixed surfactant concentration, all kinds of continuous surface deformations give the same e 0 teff curve (master curve) which is described by the same values of the Gibbs elasticity and the diffusion relaxation time. These parameters are also obtained by longitudinal wave technique. It is expected that the parameters obtained in both ways at least should be comparable. It is not expected that they should be the same because our Eq. [16] involves an approximation and Eq. [21] is exact. MATERIALS AND METHODS
Compression and expansion experiments with constant dilatation rate and constant speed were reported previously
q
FIG. 4. Elasticities obtained from expansion with constant u experiments as a function of teff (decanoic acid, c0 Å 10 07 mol cm03 ): ( l ) u Å 1.31 1 10 01 s 01 ; ( / ) u Å 6.55 1 10 02 s 01 ; ( l ) u Å 2.62 1 10 02 s 01 ; ( n ) u Å 1.31 1 10 02 s 01 ; ( s ) u Å 6.55 1 10 03 s 01 . The solid line is calculated according to Eq. [15] with e0 Å 57.2 mN m01 and tD Å 0.45 s.
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FIG. 5. Elasticities obtained from linear compression experiments (decanoic acid, c0 Å 10 07 mol cm03 ): ( l ) a Å 02.49 1 10 02 s 01 ; ( n ) a Å 01.45 1 10 02 s 01 ; ( h ) a Å 06.30 1 10 03 s 01 ; ( L ) a Å 03.23 1 10 03 s 01 ; ( / ) a Å 01.65 1 10 03 s 01 ; ( m ) a Å 06.67 1 10 04 s 01 ; ( l ) a Å 03.30 1 10 04 s 01 . The solid line is calculated according to Eq. [15] with e0 Å 57.2 mN m01 and tD Å 0.45 s.
(4–6, 12), and the experimental conditions are not repeated here. We will remark only that all these experiments are carried out in a Langmuir trough with one or two movable barriers. For the experiments with constant dilatation rate deformations the barrier was moved with exponentially varying speed, and for the experiments with constant speed the barrier was moved with constant linear speed. Compression and expansion according to Eqs. [16] and [17] were performed in Berlin (MPI) also using a Langmuir trough with programable moving barriers (10) (KVE instrument). The barrier movements in this case are more complicated functions of the time. The modulus of elasticity was also measured using the longitudinal wave method of Lucassen (8, 9).
In all of these experiments decanoic acid (Aldrich, 99/% purity) at a concentration c0 Å 10 07 mol cm03 in water was used. In experiments performed in Berlin the decanoic acid was additionally purified by the method of Lunkenheimer (13). The water solutions of this organic acid contained 0.01 M HCl (pH 2) and 0.09 M NaCl (ionic strength É 0.1 M) to suppress the ionization of the acid. RESULTS AND DISCUSSION
We considered here experimental data for expansion and compression with a constant dilatation rate and expansion with a constant speed–peaktensiometry [these data are taken from previously published work (4–6)], compression with
FIG. 6. Elasticities obtained from linear expansion experiments (decanoic acid, c0 Å 10 07 mol cm03 ): ( s ) a Å 4.00 1 10 02 s 01 ; ( j ) a Å 3.34 1 10 02 s 01 ; ( L ) a Å 2.16 1 10 02 s 01 ; ( / ) a Å 1.66 1 10 02 s 01 ; ( l ) a Å 1.11 1 10 02 s 01 ; ( h ) a Å 7.05 1 10 03 s 01 ; ( m ) a Å 4.54 1 10 03 s 01 . The solid line is calculated according to Eq. [15] with e0 Å 57.2 mN m01 and tD Å 0.45 s.
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565
FIG. 7. Elasticities obtained by compression according to Eq. [16] (decanoic acid, c0 Å 10 07 mol cm03 ): ( m ) m Å 1.33 1 10 02 s 01 ; ( n ) m Å 1.00 1 10 02 s 01 ; ( L ) m Å 6.67 1 10 03 s 01 ; ( h ) m Å 3.33 1 10 03 s 01 ; ( s ) m Å 1.67 1 10 03 s 01 . The solid line is calculated according to Eq. [15] with e0 Å 57.2 mN m01 and tD Å 0.45 s.
constant speed, compression and expansion according to Eqs. [16] and [17], periodical deformation by longitudinal waves, and some data for expansion by growing drop published by MacLeod and Radke (2). These experiments are carried out at different speeds of surface variation, namely, at different dilatation rate u, speed of deformation a, constant m, radial frequency v, and flow rate F. For compression and expansion at constant dilatation rate, u, a steady state is attained after a sufficiently long time. For expansion with constant u, the effective time, teff r 1/2u for t r ` . For expansion with constant speed, a maximum in the Ds 0 t curve is observed (decreasing u ), therefore we called this method peaktensiometry. For compression with constant
speed (7) the surface pressure increases with time (increasing u ). The compression and expansion curves at a constant dilatation rate and a constant speed are not given, most of them are earlier presented or are similar to already published data. Only the compression and expansion curves according to surface deformation given by Eqs. [16] and [17] are given in Figs. 1 and 2. There is a significant difference between the calculated curves and the experimental data at some speed of deformation. Nevertheless the elasticities obtained from all of these data follow the same master curve (see Figs. 7 and 8). For all above mentioned experimental data the elasticities of the adsorption monolayers were calculated. The results,
FIG. 8. Elasticities obtained by expansion according to Eq. [17] (decanoic acid, c0 Å 10 07 mol cm03 ): ( j ) m Å 1.67 1 10 02 s 01 ; ( s ) m Å 1.33 1 10 02 s 01 ; ( n ) m Å 1.00 1 10 02 s 01 ; ( L ) m Å 6.67 1 10 03 s 01 ; ( h ) m Å 3.33 1 10 03 s 01 ; ( l ) m Å 1.67 1 10 03 s 01 . The solid line is calculated according to Eq. [15] with e0 Å 57.2 mN m01 and tD Å 0.45 s.
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FIG. 9. Modulus of elasticity, ÉeÉ, as a function of frequency, n, for decanoic acid monolayers (c0 Å 10 07 mol cm03 ). The solid line is the best fit according to Eq. [19], the broken line is calculated elastisities with e0 Å 57.2 mN m01 and tD Å 0.45 s according to Eq. [19].
q
e vs teff , are shown in Figs. 3–8. All of these data are on the same master curve described by Eq. [15] with the same parameters of e0 Å 57.2 mN m01 and tD Å 0.45 s. There is no theoretical argument to prefer one surface deformation to another. We can choose the method that is experimentally the most convenient. It is only necessary to keep in mind that the accuracy of the methods can be different in the time domain in which we are interested. It was shown previously (14) that q
e 0 tD Å
RT G 2 . q c0 D
[22]
Using for the diffusion coefficient D Å 5 1 10 06 cm2 s 02 ,
we obtain for adsorption G Å 5.87 1 10 010 mol cm02 . This value compares well with this obtained from equilibrium surface tension concentration curve: G Å 5.18 1 10 010 mol cm02 (the agreement between both is within 13%). Also the modulus of elasticity as a function of the frequency is measured by small periodical amplitude variations of the surface area. The experimental data are fitted using the Lucassen equation [19] with e0 Å 60 mN m01 and tD Å 0.61 s. The result is shown in Fig. 9. The parameters for describing the elasticities during a continuous surface deformations and those for small amplitude periodic surface deformation are rather close, but different. In Fig. 9 we also have plotted the calculated values for the modulus of elasticity using the parameters for continuous
FIG. 10. Elasticities obtained by growing drop technique from Triton X-100 solution, c0 Å 3.09 1 10 08 mol cm03 [results of MacLeod and Radke (2)]: ( s ) a Å 8.46 1 10 01 s 01 ; ( h ) a Å 6.04 1 10 01 s 01 ; ( L ) a Å 2.49 1 10 01 s 01 ; ( n ) a Å 2.02 1 10 01 s 01 . The solid line is the best fit according to Eq. [23].
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surface deformations (broken line). The reason for this difference in the values of the elasticities can be that the theory for the method of Lucassen is exact, whereas our Eq. [8] comes from an approximation of the convolution integral in the equation of Ward and Tordai. A second reason may be that in the method of Lucassen the linearization is certainly allowed, whereas in our experiments it is not so evident. Finally the results of MacLeod and Radke for Triton X100 with concentration c0 Å 3.09 1 10 08 mol cm03 at a dodecane/water interface were analyzed. From Fig. 10 it is seen that the data for different expansion rates F/V0 , F being the flow rate and V0 the initial volume of the drop, the same q e 0 teff curve is obtained. In this case the data are fitted by Eq. [15], which is approximated as q
e Å e 0 tD
because
S D
S D 4teff ptD
p 4teff
1/2
[23]
@ 1.
q
The value of e0 tD is 21 dyn cm01 s 1 / 2 for this system. Using again a diffusion coefficient D Å 5 1 10 06 cm2 s 02 , with Eq. [22] we obtain an adsorption G Å 2.41 1 10 010 mol cm02 , which is of the expected order of magnitude.
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CONCLUSION
As a conclusion we can say that the elasticity of the surface for different kinds of surface deformations is well described by Eq. [15], and qall of these experiments give approximately the same e 0 teff curves and are described with the same parameters e0 and tD . Of course the effective time depends on the speed of deformation and kind of deformation where the effect of convection is concluded. REFERENCES 1. Van Voorst Vader, F., Erkens, F., and Van den Tempel, M., Trans. Faraday Soc. 60(6), 1170 (1964). 2. MacLeod, C. A., and Radke, C. J., J. Colloid Interface Sci. 160, 435 (1993). 3. Joos, P., and Van Uffelen, M., J. Colloid Interface Sci. 155, 271 (1993). 4. Van Uffelen, M., and Joos, P., Colloids Surf. A 85, 119 (1994). 5. Van Uffelen, M., and Joos, P., Colloids Surf. A 100, 245 (1995). 6. Van Uffelen, M., and Joos, P., Colloids Surf. A 85, 107 (1994). 7. Petrov, P., and Joos, P., J. Colloid Interface Sci. (accepted for publication). 8. Lucassen, J., and Van den Tempel, M., Chem. Eng. Sci. 17, 1283 (1972). 9. Lucassen, J., and Van den Tempel, M., J. Colloid Interface Sci. 41, 491 (1972). 10. Jiping, F., and Lunkenheimer, K., private communications. 11. Van Uffelen, M., and Joos, P., J. Colloid Interface Sci. 171, 297 (1995). 12. Horozov, T., and Joos, P., J. Colloid Interface Sci. 173, 334 (1995). 13. Lunkenheimer, K., and Miller, R., J. Colloid Interface Sci. 120, 176 (1987). 14. Li, B., Joos, P., and Horozov, T., Colloids Surf. A 94, 85 (1995).
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0580
Measurement of the Surface Elasticity of an Adsorbed Monolayer by Continuous Surface Deformation PAUL JOOS, 1, * PETER PETROV,*
AND
JIPING FANG **
*Department of Chemistry, University of Antwerp, U.I.A., Universiteitsplein 1, B-2610 Wilrijk, Belgium; and **Max-Planck-Institut, Kolloid-und Grenzflachenforschung, Rudower Chaussee 5, D-12489 Berlin, Germany Received March 11, 1996; accepted June 19, 1996
The elasticity of a monolayer adsorbed from a surfactant solution is considered by continuous surface deformation. For all kinds surface deformations a single relation between the elasticity and the effective time should be obtained (master curve). This effective time depends on the rate of deformation and on the kind of deformation. The relation between elasticity and effective time was proven experimentally for a decanoic acid solution at a concentration 10 07 mol cm03 . The following surface deformations are considered: (i) expansion at a constant dilatation rate, (ii) compression at a constant dilatation rate, (iii) linear expansion, (iv) linear compression, and (v, vi) compression and expansion in a some what more complicated way. From these data the Gibbs elasticity and the diffusion relaxation time are obtained. These parameters are also obtained by longitudinal wave method of Lucassen–Van den Tempel. Both set of parameters are comparable. Finally the situation at a growing drop with constant flow rate is considered (Triton X-100) to show that also here the approach is followed. q 1996 Academic Press, Inc.
Key Words: surface elasticity by continuous surface deformation.
INTRODUCTION
Let us consider a monolayer of a surfactant solution in equilibrium with the bulk. Initially the area of the monolayer is V0 , the equilibrium surface tension se (equilibrium surface pressure Pe ), adsorption Ge , and the concentration c0 is uniform over the whole solution. Now this area is deformed continuously so that the area is a function of time: V(t). The exact function of V(t) depends of course on the experimental conditions (on the way and the kind of deformation), for instance, linear expansion, linear compression, expansion at a constant dilatation rate, or compression at a constant dilatation rate, etc. By the area deformation the equilibrium between surface and the bulk is broken and the system will tend to re-establish its equilibrium by diffusion of surfactant from the surface to the bulk (compression) or vice versa 1
(expansion). During area deformation, the surface tension is different from its equilibrium value and depends on time: s(t). We are merely interested in the jump of the surface tension s(t) defined as Ds(t) Å s(t) 0 se . If the relaxation process of the equilibrium re-establishing is diffusion, it is plain the diffusion equation must be considered with corresponding boundary conditions. Since the surface is deformed continuously, a convection term must be accounted for in the diffusion Eq. [1]. In this way the jump in surface tension with time is predicted and the theoretical curve has to be compared with experimental data. To describe this curve two parameters, the Gibbs elasticity, e0 , and the diffusion relaxation time, tD , are involved. From these experiments also the elasticity of the monolayer is obtained, which depends on time. If the surface deformation is fast, the diffusion process has no time to operate and the surface is purely elastic; for large times, the diffusion operates and the monolayer becomes viscoelastic. One should expect that for different ways of area changes, the same elasticity should be found. It is the scope of this paper to show this, and to do this we considered a surfactant solution at the same experimental conditions (decanoic acid in water at a concentration c Å 10 07 mol cm03 , pH 2, at the air/ water interface) which is subjected to different kinds surface deformations. The area changes considered are linear expansion and compression, expansion and compression with a constant dilatation rate, and even more complicated surface deformations. MacLeod and Radke (2) reported also experiments of the variation of the interfacial tension, at a growing drop with constant flow rate. These experiments are considered to prove that our approach is also applicable to this situation. THEORY
The dilatation rate u, being defined as
uÅ
To whom correspondence should be addressed. 559
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accounts for the convection. For an expanding surface the convective diffusion equation reads (1) as Ìc Ìc Ì2c 0 uz ÅD , Ìt Ìz Ìz 2
uú0
[2]
Ds Å e0
uõ0
[3]
1/
1 4teff ptD
F G
teff Å
S D*S D 2
V0 V
t
S D
dG Ìc / uG Å D dt Ìz
0
zr`
and
c r c0
dc ( G 0 Ge ). dG
* S VV D dt. 2
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[10]
0
In Eq. [8] DV is the change in area, defined as DV Å V(t) 0 V0 .
[11]
For expansion DV ú 0 and for compression DV õ 0. The parameters involved are the Gibbs elasticity, e0 , and the diffusion relaxation time, defined as
[7]
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e0 Å 0
ds Ge dG
1 D
S D
and tD Å
dG dc
[12]
2
.
[13]
If the surface elasticity, e, is defined as eÅ
ds V, dV
[14]
then from Eqs. [8] and [14] we obtain eÅ
[6]
can be obtained, but involves the numerical evaluation of the convolution integral in the generalized Ward and Tordai equation. In order to avoid this numerical calculations we approximated the convolution integral. The final solution for expansion is (4, 6)
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teff Å
[5]
In this way the jump in adsorption as a function of time DG(t) Å G(t) 0 Ge
[9]
For compression the same Eq. [8] is obtained (5, 7) but with effective time now defined as
[4]
In these equations z is a coordinate normal to the surface, c is the concentration, G is the adsorption, D is the diffusion coefficient, and t is the time. The integration of these equations results in a generalized Ward and Tordai equation ( 4 – 6 ) . Further it is assumed that there is a local equilibrium between the surface and the subsurface. This means that the relation between the adsorption and the subsurface concentration, cs , is given by the adsorption isotherm. If we restrict ourselves to small jumps in surface tension, this adsorption isotherm is linearized:
cs 0 ce Å
dt.
0
and taking into account the boundary conditions far away from the surface at
[8]
2
V V0
t
These differential equations are integrated (4, 5) taking into account the boundary conditions for conservation of the mass at the surface
1/2
with teff , the effective time, defined as
It was shown (3) that for compressed surfaces this equation should be written as Ìc Ìc Ì2c / uz ÅD , Ìt Ìz Ìz 2
DV V
e0 . 1 / [4teff / ptD ] 1 / 2
[15]
If a given surfactant solution is subjected to various kinds of area deformations, as indicated by Eqs. [9] and [10], different expressions for the effective time are obtained, but if the elasticity obtained from all these different experiments is plotted as a function of teff (we calculated it as a function q of teff ), then the same master curve should be obtained. We considered linear expansion and compression, expansion and compression at constant u for decanoic acid monolayer at the same concentration. For these kinds of deformations, the expressions for the area with time and the effective
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TABLE 1 Area Variation and Effective Time for Different Kinds of Surface Deformations Deformation at
Kind of deformation
Area variation
Relative deformation
Effective time
Expansion u constant
ut
V Å V0e
d ln V dt
uÅ
DV Å 1 0 e0ut V
teff Å
1 (1 0 e02ut) 2u
(u ú 0) dV constant dt
aÅ
1 dV V0 dt
V Å V0(1 / at)
DV at Å V 1 / at
teff Å
x 3 / 3x 2 / 3x 3a(1 / x)2
(x Å at, a ú 0) m constant
mt
mt
V Å V0( be 0 g)
–
DV b(e 0 1) Å V bemt 0 g
teff Å
b2(e2y 0 1) 0 4bg(e y 0 1) / 2g 2y 2m( bey 0 g)2
(y Å at, m ú 0) F constant
aÅ
V Å V0(1 / at)2/3
F V0
DV (1 / at)2/3 0 1 Å V (1 / at)2/3
teff Å
3(1 / x)7/3 0 1 7a(1 / x)4/3
(x Å at, a ú 0) Compression u constant
0ÉuÉt
V Å V0e
d ln V dt
uÅ
DV Å (1 0 eÉuÉt ) V
teff Å
1 [1 0 e02ÉuÉt] 2ÉuÉ
(u õ 0) dV constant dt
aÅ
1 dV V0 dt
V Å V0 (1 / at)
–
V Å V0 (2emt 0 1)
DV at Å V 1 / at
teff Å
t 1 / at
(a õ 0) m constant
mt
DV 2(e 0 1) Å V 2emt 0 1
teff Å
FS
D
G
1 e mt 2(e mt 0 1) ln / m 2e mt 0 1 2e mt 0 1
( m õ 0)
times are given in Table 1. Effective times are obtained from Eqs. [9] and [10] using the appropriate functions of V. J. Fang and K. Lunkenheimer considered other types of area deformations in a Langmuir trough with two moving barriers (10): one for compression V Å V0 (2e mt 0 1)
mõ0
[16]
m ú 0,
[17]
and one for expansion V Å V0[ ge mt 0 b]
where b and g are constants depending on the initial area ( V0 ) and the final area ( V` ).
(In our experimental conditions, for expansion V0 Å 22.5 cm2 and V` Å 225 cm2 , whence b Å 11 and g Å 10), and m is a constant which determinate the speed of moving of the barriers. For compression m õ 0 and for expansion m ú 0. The area deformations and effective times are also given in Table 1. The experiments of MacLeod and Radke (2) are also considered for growing drop at constant flow rate, as analyzed previously (11). The expressions for the area change with time and the effective times are also given in Table 1. If small periodic amplitude area deformations are applied [this is the longitudinal wave method of Lucassen and Van den Tempel (8, 9)], then the modulus of elasticity ÉeÉ can be measured as ÉeÉ Å
V0 / V` bÅ V0
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ÉDsÉ V0 ÉDVÉ
[18]
with ÉDsÉ as the amplitude of the surface tension variation
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FIG. 1. Compression according to Eq. [16]. DP as a function of time, t for decanoic acid monolayers (c0 Å 10 07 mol cm03 ) at different compression speeds: ( s ) m Å 01.33 1 10 02 s 01 ; ( L ) m Å 01.00 1 10 02 s 01 ; ( n ) m Å 06.67 1 10 03 s 01 ; ( h ) m Å 03.33 1 10 03 s 01 ; ( / ) m Å 01.67 1 10 03 s 01 . Solid lines are calculated according to Eq. [8] with e0 Å 57.2 mN m01 , tD Å 0.45 s, and the corresponding expressions for teff and DV / V (Table 1).
and ÉDVÉ as the amplitude of the area deformation. It was shown that for a diffusion-controlled adsorption process, the modulus of elasticity is given by (8, 9) ÉeÉ Å
with zÅ
e0 (1 / 2z / 2z 2 ) 1 / 2
S D S 1 2vtD
1/2
Å
[19]
1 4pntD
D
1/2
,
[20]
A comparison between longitudinal wave method and the other ones presented here should be made. Both theories apply to a diffusion relaxation process using the same boundary conditions at the surface, Eqs. [4] and [6], and also in both the elasticity of the surface is defined in the same way (Eq. [14]). In the longitudinal wave method, the surface is subjected to small amplitude deformation, resulting in a complex elasticity depending on the frequency: e(iv ) Å
1/
where v is the radial frequency (rad s 01 ) and n the common frequency (Hz).
e0 1 ivtD
S D
1/2
.
[21]
FIG. 2. Expansion according to Eq. [17]. Ds as a function of time for decanoic acid monolayers ( c0 Å 10 07 mol cm03 ) at different compression speeds: ( s ) m Å 1.67 1 10 02 s 01 ; ( L ) m Å 1.33 1 10 02 s 01 ; ( l ) m Å 1.00 1 10 02 s 01 ; ( n ) m Å 6.67 1 10 03 s 01 ; ( h ) m Å 3.33 1 10 03 s 01 ; ( l ) m Å 1.67 1 10 03 s 01 . Solid lines are calculated according to Eq. [8] with e0 Å 57.2 mN m01 and tD Å 0.45 s.
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q
FIG. 3. Elasticities, e, obtained from compression with constant u of decanoic acid monolayers (c0 Å 10 07 mol cm03 ) as a function of teff : ( l ) u Å 02.62 1 10 01 s 01 ; ( / ) u Å 01.31 1 10 01 s 01 ; ( m ) u Å 06.55 1 10 02 s 01 ; ( s ) u Å 02.62 1 10 02 s 01 ; ( L ) u Å 01.31 1 10 02 s 01 ; ( h ) u Å 06.55 1 10 03 s 01 ; ( n ) u Å 02.62 1 10 03 s 01 . The solid line is calculated according to Eq. [15] with e0 Å 57.2 mN m01 and tD Å 0.45 s.
Since small amplitude surface deformations are considered, the convection term need not be accounted for in the diffusion equation. In the other methods the continuously deformed surface (consequently the convection term in the diffusion equation must be accounted for) resulting in the elasticity as a function of the effective time, given by Eq. [15]. This equation is in the same form as Eq. [21]. However, Eq. [21] assume no approximations for small amplitudes surface deformations, whereas our expression [16] involves an approximation of the convolution integral in the generalized Ward and Tordai equation. The longitudinal wave method of Lucassen–Van den Tempel and the discussed ones for continuous surface deformations are complementary each other. It is expected that
for a fixed surfactant concentration, all kinds of continuous surface deformations give the same e 0 teff curve (master curve) which is described by the same values of the Gibbs elasticity and the diffusion relaxation time. These parameters are also obtained by longitudinal wave technique. It is expected that the parameters obtained in both ways at least should be comparable. It is not expected that they should be the same because our Eq. [16] involves an approximation and Eq. [21] is exact. MATERIALS AND METHODS
Compression and expansion experiments with constant dilatation rate and constant speed were reported previously
q
FIG. 4. Elasticities obtained from expansion with constant u experiments as a function of teff (decanoic acid, c0 Å 10 07 mol cm03 ): ( l ) u Å 1.31 1 10 01 s 01 ; ( / ) u Å 6.55 1 10 02 s 01 ; ( l ) u Å 2.62 1 10 02 s 01 ; ( n ) u Å 1.31 1 10 02 s 01 ; ( s ) u Å 6.55 1 10 03 s 01 . The solid line is calculated according to Eq. [15] with e0 Å 57.2 mN m01 and tD Å 0.45 s.
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FIG. 5. Elasticities obtained from linear compression experiments (decanoic acid, c0 Å 10 07 mol cm03 ): ( l ) a Å 02.49 1 10 02 s 01 ; ( n ) a Å 01.45 1 10 02 s 01 ; ( h ) a Å 06.30 1 10 03 s 01 ; ( L ) a Å 03.23 1 10 03 s 01 ; ( / ) a Å 01.65 1 10 03 s 01 ; ( m ) a Å 06.67 1 10 04 s 01 ; ( l ) a Å 03.30 1 10 04 s 01 . The solid line is calculated according to Eq. [15] with e0 Å 57.2 mN m01 and tD Å 0.45 s.
(4–6, 12), and the experimental conditions are not repeated here. We will remark only that all these experiments are carried out in a Langmuir trough with one or two movable barriers. For the experiments with constant dilatation rate deformations the barrier was moved with exponentially varying speed, and for the experiments with constant speed the barrier was moved with constant linear speed. Compression and expansion according to Eqs. [16] and [17] were performed in Berlin (MPI) also using a Langmuir trough with programable moving barriers (10) (KVE instrument). The barrier movements in this case are more complicated functions of the time. The modulus of elasticity was also measured using the longitudinal wave method of Lucassen (8, 9).
In all of these experiments decanoic acid (Aldrich, 99/% purity) at a concentration c0 Å 10 07 mol cm03 in water was used. In experiments performed in Berlin the decanoic acid was additionally purified by the method of Lunkenheimer (13). The water solutions of this organic acid contained 0.01 M HCl (pH 2) and 0.09 M NaCl (ionic strength É 0.1 M) to suppress the ionization of the acid. RESULTS AND DISCUSSION
We considered here experimental data for expansion and compression with a constant dilatation rate and expansion with a constant speed–peaktensiometry [these data are taken from previously published work (4–6)], compression with
FIG. 6. Elasticities obtained from linear expansion experiments (decanoic acid, c0 Å 10 07 mol cm03 ): ( s ) a Å 4.00 1 10 02 s 01 ; ( j ) a Å 3.34 1 10 02 s 01 ; ( L ) a Å 2.16 1 10 02 s 01 ; ( / ) a Å 1.66 1 10 02 s 01 ; ( l ) a Å 1.11 1 10 02 s 01 ; ( h ) a Å 7.05 1 10 03 s 01 ; ( m ) a Å 4.54 1 10 03 s 01 . The solid line is calculated according to Eq. [15] with e0 Å 57.2 mN m01 and tD Å 0.45 s.
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565
FIG. 7. Elasticities obtained by compression according to Eq. [16] (decanoic acid, c0 Å 10 07 mol cm03 ): ( m ) m Å 1.33 1 10 02 s 01 ; ( n ) m Å 1.00 1 10 02 s 01 ; ( L ) m Å 6.67 1 10 03 s 01 ; ( h ) m Å 3.33 1 10 03 s 01 ; ( s ) m Å 1.67 1 10 03 s 01 . The solid line is calculated according to Eq. [15] with e0 Å 57.2 mN m01 and tD Å 0.45 s.
constant speed, compression and expansion according to Eqs. [16] and [17], periodical deformation by longitudinal waves, and some data for expansion by growing drop published by MacLeod and Radke (2). These experiments are carried out at different speeds of surface variation, namely, at different dilatation rate u, speed of deformation a, constant m, radial frequency v, and flow rate F. For compression and expansion at constant dilatation rate, u, a steady state is attained after a sufficiently long time. For expansion with constant u, the effective time, teff r 1/2u for t r ` . For expansion with constant speed, a maximum in the Ds 0 t curve is observed (decreasing u ), therefore we called this method peaktensiometry. For compression with constant
speed (7) the surface pressure increases with time (increasing u ). The compression and expansion curves at a constant dilatation rate and a constant speed are not given, most of them are earlier presented or are similar to already published data. Only the compression and expansion curves according to surface deformation given by Eqs. [16] and [17] are given in Figs. 1 and 2. There is a significant difference between the calculated curves and the experimental data at some speed of deformation. Nevertheless the elasticities obtained from all of these data follow the same master curve (see Figs. 7 and 8). For all above mentioned experimental data the elasticities of the adsorption monolayers were calculated. The results,
FIG. 8. Elasticities obtained by expansion according to Eq. [17] (decanoic acid, c0 Å 10 07 mol cm03 ): ( j ) m Å 1.67 1 10 02 s 01 ; ( s ) m Å 1.33 1 10 02 s 01 ; ( n ) m Å 1.00 1 10 02 s 01 ; ( L ) m Å 6.67 1 10 03 s 01 ; ( h ) m Å 3.33 1 10 03 s 01 ; ( l ) m Å 1.67 1 10 03 s 01 . The solid line is calculated according to Eq. [15] with e0 Å 57.2 mN m01 and tD Å 0.45 s.
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FIG. 9. Modulus of elasticity, ÉeÉ, as a function of frequency, n, for decanoic acid monolayers (c0 Å 10 07 mol cm03 ). The solid line is the best fit according to Eq. [19], the broken line is calculated elastisities with e0 Å 57.2 mN m01 and tD Å 0.45 s according to Eq. [19].
q
e vs teff , are shown in Figs. 3–8. All of these data are on the same master curve described by Eq. [15] with the same parameters of e0 Å 57.2 mN m01 and tD Å 0.45 s. There is no theoretical argument to prefer one surface deformation to another. We can choose the method that is experimentally the most convenient. It is only necessary to keep in mind that the accuracy of the methods can be different in the time domain in which we are interested. It was shown previously (14) that q
e 0 tD Å
RT G 2 . q c0 D
[22]
Using for the diffusion coefficient D Å 5 1 10 06 cm2 s 02 ,
we obtain for adsorption G Å 5.87 1 10 010 mol cm02 . This value compares well with this obtained from equilibrium surface tension concentration curve: G Å 5.18 1 10 010 mol cm02 (the agreement between both is within 13%). Also the modulus of elasticity as a function of the frequency is measured by small periodical amplitude variations of the surface area. The experimental data are fitted using the Lucassen equation [19] with e0 Å 60 mN m01 and tD Å 0.61 s. The result is shown in Fig. 9. The parameters for describing the elasticities during a continuous surface deformations and those for small amplitude periodic surface deformation are rather close, but different. In Fig. 9 we also have plotted the calculated values for the modulus of elasticity using the parameters for continuous
FIG. 10. Elasticities obtained by growing drop technique from Triton X-100 solution, c0 Å 3.09 1 10 08 mol cm03 [results of MacLeod and Radke (2)]: ( s ) a Å 8.46 1 10 01 s 01 ; ( h ) a Å 6.04 1 10 01 s 01 ; ( L ) a Å 2.49 1 10 01 s 01 ; ( n ) a Å 2.02 1 10 01 s 01 . The solid line is the best fit according to Eq. [23].
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surface deformations (broken line). The reason for this difference in the values of the elasticities can be that the theory for the method of Lucassen is exact, whereas our Eq. [8] comes from an approximation of the convolution integral in the equation of Ward and Tordai. A second reason may be that in the method of Lucassen the linearization is certainly allowed, whereas in our experiments it is not so evident. Finally the results of MacLeod and Radke for Triton X100 with concentration c0 Å 3.09 1 10 08 mol cm03 at a dodecane/water interface were analyzed. From Fig. 10 it is seen that the data for different expansion rates F/V0 , F being the flow rate and V0 the initial volume of the drop, the same q e 0 teff curve is obtained. In this case the data are fitted by Eq. [15], which is approximated as q
e Å e 0 tD
because
S D
S D 4teff ptD
p 4teff
1/2
[23]
@ 1.
q
The value of e0 tD is 21 dyn cm01 s 1 / 2 for this system. Using again a diffusion coefficient D Å 5 1 10 06 cm2 s 02 , with Eq. [22] we obtain an adsorption G Å 2.41 1 10 010 mol cm02 , which is of the expected order of magnitude.
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CONCLUSION
As a conclusion we can say that the elasticity of the surface for different kinds of surface deformations is well described by Eq. [15], and qall of these experiments give approximately the same e 0 teff curves and are described with the same parameters e0 and tD . Of course the effective time depends on the speed of deformation and kind of deformation where the effect of convection is concluded. REFERENCES 1. Van Voorst Vader, F., Erkens, F., and Van den Tempel, M., Trans. Faraday Soc. 60(6), 1170 (1964). 2. MacLeod, C. A., and Radke, C. J., J. Colloid Interface Sci. 160, 435 (1993). 3. Joos, P., and Van Uffelen, M., J. Colloid Interface Sci. 155, 271 (1993). 4. Van Uffelen, M., and Joos, P., Colloids Surf. A 85, 119 (1994). 5. Van Uffelen, M., and Joos, P., Colloids Surf. A 100, 245 (1995). 6. Van Uffelen, M., and Joos, P., Colloids Surf. A 85, 107 (1994). 7. Petrov, P., and Joos, P., J. Colloid Interface Sci. (accepted for publication). 8. Lucassen, J., and Van den Tempel, M., Chem. Eng. Sci. 17, 1283 (1972). 9. Lucassen, J., and Van den Tempel, M., J. Colloid Interface Sci. 41, 491 (1972). 10. Jiping, F., and Lunkenheimer, K., private communications. 11. Van Uffelen, M., and Joos, P., J. Colloid Interface Sci. 171, 297 (1995). 12. Horozov, T., and Joos, P., J. Colloid Interface Sci. 173, 334 (1995). 13. Lunkenheimer, K., and Miller, R., J. Colloid Interface Sci. 120, 176 (1987). 14. Li, B., Joos, P., and Horozov, T., Colloids Surf. A 94, 85 (1995).
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