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Colloids and Surfaces A: Physicochem. Eng. Aspects 221 (2003) 185 /195 www.elsevier.com/locate/colsurfa
Surface dilatational properties of mixed sodium dodecyl sulfate/dodecanol solutions ¨ rtegren K.-D. Wantke *, H. Fruhner, J. O Max-Planck-Institute of Colloids and Interface, Am Mu ¨ hlenberg, D-14476 Golm, Germany Received 22 May 2002; accepted 27 March 2003
Abstract The surface dilatational properties of aqueous solutions of sodium dodecyl sulfate (SDS) and n -dodecanol are investigated in the frequency range 1 5/f5/500 Hz using the oscillating bubble method. The results demonstrate that a pure dodecanol solution has an elastic surface without viscous effect whereas the surface of a SDS solution without added dodecanol exhibits a strong viscoelastic behavior. Mixtures show graduated properties. The time behavior of their surface dilatational moduli demonstrates that dodecanol molecules drive the SDS molecules slowly out of the surface. Therefore, the known one-component model describing the surface dilatational modulus can be used also for these mixtures. A simple theoretical consideration explains this effect. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Sodium dodecyl sulfate; n -Dodecanol; Oscillating bubble; Surface dilatational modulus; Surface rheology
1. Introduction The rheological properties of fluid surfaces and interfaces have a decisive influence on the behavior of many complex systems and technological processes. Examples are coating processes, multiphase flow, and stability of foams and emulsions. The prerequisite for the investigation of the specific mechanisms in such complex systems is a detailed knowledge about surface and interface properties.
* Corresponding author. Tel./fax: /33-1-567-9233/9202. E-mail address: [email protected] (K.-D. Wantke).
In particular, dilatational effects are of major importance because the creation of a new surface or interface is a highly energetic process. In addition, these properties depend in a strong manner on the chemical composition of the systems. Methods and problems of this field are described in many books and papers [1 /11]. In the last years great progress could be reached by investigation of dilatational properties of fluid surfaces applying oscillating bubbles and drops [12 /18]. The authors contributed to this work by the development of a new version of the oscillating bubble method which allows simple and reliable measurements of the dilatational modulus of fluid
0927-7757/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0927-7757(03)00135-3
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surfaces in the frequency range 15/f5/500 Hz [19,20]. The modulus is defined by the equation o(f ; cn )E(f; cn )exp(i8 (f; cn ))
ADg DA
(1) G(f)
where DA/A represents the relative change in surface area, Dg the change in surface tension, f the frequency, and cn the concentration of the components. The modulus describes the response of the surface tension to a sinusoidal surface dilatation, DA /jDAjexp(i2pft). Such measurements exhibit various characteristics in o(f, cn) due to the different dynamics of the molecules at the surface. The modulus covers elastic, viscous, and transport properties. In previous papers several surface effects of solutions containing only one soluble surfactant are documented and interpreted [21 /24]. Here we investigate the surface behavior of mixed sodium dodecyl sulfate (SDS)/n -dodecanol solutions.
2. Theoretical considerations The oscillating bubble method applies the following measuring principle: within a closed chamber, which is filled with a surface active solution, a small hemispherical bubble is produced at the tip of a capillary. A piezoelectric driver at the wall of the chamber causes a small sinusoidal oscillation of the bubble surface via the solution. A change in the size of the bubble produces a change of the pressure in the chamber. At the pressure transducer the sinusoidal contribution to the pressure, Dp, is given by Dp
2 r0
Dg
2g (r0 )2
DrG(f)
relevant information about the dynamic surface tension. For an ideal spherical bubble with a fixed center the viscous and inertial effects can be calculated easily [20,25]. It leads to
(2)
where r0 describes the mean radius of the bubble, Dr the change in the radius, and G(f) the influence of the fluid dynamics, that means the influence of the bulk viscosity and the inertia of the solution [20]. According to Eq. (1) the surface dilatational modulus results from the first term on the left side in Eq. (2). Therefore, the pressure measurements must be corrected by the last terms to obtain a
8pihf r Drr(2pf)2 r0 1 0 Dr: r0 r1
Here h represents the bulk viscosity, r the density of the solution, and r1 the distance of the pressure transducer to the center. However, the geometrical conditions in a real set up deviate from the ideal geometry and the correction term G(f) must be calculated via a path integral between the bubble surface and the pressure transducer using the solution of the Navier /Stokes equations. We apply calibration measurements for the correction to avoid this complex calculations. The necessary correction is the smallest in the case of the half sphere where the center is not fixed. Then, Dr becomes approximately zero [20]. Proper calibration systems are solutions with a purely elastic surface where the constant elasticity can be determined in the low frequency range. The difference between this value and the real pressure at the transducer measured with same solutions in the higher frequency range yields the experimental correction curve. It includes all correction terms of Eq. (2) for a real chamber. The curves presented in this paper show only pure surface effects that means corrected values. For the interpretation of these effects we consider a surface as a layer with a thickness of a few molecules. All physical properties of such layer should be projected onto a two-dimensional interface. Then the definition of the surface concentration depends on the thickness of the layer. In an equilibrium state, the influence of the thickness on the surface concentration can be eliminated using an invariant scale [26]. The related monolayer model allows only the consideration of lateral structures, whereas various internal structures are possible within a vertically extended layer. Then, a surface deformation can lead to nonequilibrium states and dissipative losses due to molecular exchange processes between structures. Such assumptions are necessary for the explanation of measured dilatational properties of fluid inter-
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faces. This should be demonstrated by a simple consideration. The surface dilatational modulus of a solution with one surfactant results from the following formula o(f;
cn )o m n (Gn (cn ))
A
DGn (f; cn )
Gn (cn )
DA
i2pf kn (cn ):
(3)
The first term characterizes the influence of change in surface concentration, DG(f, cn), on the surface tension. It depends only on the surface concentration Gn(f, cn) /Gn(cn)/DGn(f, cn), and therefore, some authors introduce the name compositional effects [27,28]. In our notation the expression om n (Gn (cn )) Gn Dg=DGn
(4)
represents the experimental Gibbs elasticity of SDS (n/1) or dodecanol (n/2). Recent results have documented the presence of trace components of n-dodecanol within aqueous SDS solutions [29,30]. We discuss this problem in Section 3. The second term characterizes an intrinsic surface viscous effect, which is caused by dissipative processes within the surface layer. A prerequisite of such an effect is a nonequilibrium state near or within the surface layer and the term becomes zero if there the thermodynamic equilibrium is instantaneously established. For the evaluation of dilatational moduli one must take into account that the first term of Eq. (3) also includes an imaginary part due to the influence of bulk diffusion on the change in surface concentration, DGn(f, cn). In these cases the dissipative processes are located within the adjacent bulk phases and have a weaker influence on complex systems like foams [23]. Therefore, the separation of the intrinsic surface dilatational viscosity from the effect of the bulk diffusion is an aim of our experiments. It requires the calculation of DGn. A harmonic oscillation of a bubble leads after a short time to an established state at its surface, and all properties can be considered as functions of the dilatational state characterized by DA(t) / jDAjexp(ivt). Then, for k/0, the surface dilatational modulus follows from
o(f ; cn )o m n
187
dln Gn dln A
(5)
dg : dln Gn
(6)
with om n (Gn (cn ))
The relation between the change in surface concentration, DGn, and the change in surface area, DA(t), results from the law of mass conservation which is given for a monolayer model by Gn dA dGn Gn dA dGn dcn @cn 1 A dt dt A dGn dcn dcsn @t x0
j
Dn
@cn @x
j
:
(7)
x0
Dn represents the diffusion constant, csn /cn(x /0, t) a real sublayer concentration, and cn a fictive sublayer concentration resulting from the isotherm equation Gn /Gn(cn ). For a bulk diffusion controlled process, which means that the surface is in an equilibrium state, the concentrations become equal (csn /cn ). Consequently, their difference is a scale for the molecular exchange flow jn $ kn (csn cn):
(8)
This equation defines the molecular exchange constant kn [20]. With the solution of the diffusion equation cn (x; t)c0n ½Dcsn ½exp[(1i)an xivtib?n ]
(9)
(an wave number of the diffusion wave, bn? phase angle resulting from the boundary condition, v / 2pf) for an oscillating process in the area xB/0 we obtain the relation hn (f ; cn )
dcsn
kn pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dcn kn (1 i) vDn =2
½hn (f ; cn )½exp(ibn (f ; cn ))
(10)
for the characterization of the molecular exchange balance between sublayer and layer. The Eqs. (3), (5), (7) and (10) lead to the expression o(f ; cn )o m n with
1 z?n iz?n ivkn 1 2z?n 2(z?n )2
(11)
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sffiffiffiffiffiffiffi vm Dcn 2 n m ; vn Dn z?n hn (f ; c)zn ; zn ; 2v DGn (12) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi vDn =2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi and bn arctan kn vDn =2 [20 /22]. In a final step the general Eq. (11) can be transformed into the following formula pffiffiffi om 2zn ½hn ½exp(i(p=4 bn ))) n (1 o(f ; c) 1 2zn ½hn ½(cos(bn ) sin(bn )) 2(zn ½hn ½)2 ivkn (13) which describes the dilatational modulus of surfactant solutions of one component. The verification of this equation by experiments is comprehensively discussed in [20 /22]. An equilibrium state is instantaneously established at the surface if the exchange constant kn becomes large ((kn)2 /Dnv), whereas molecular exchanges can be neglected for (kn)2 /Dnv. In the first case hn(f, cn) becomes one and kn should be zero. Then zn? /zn and Eq. (11) describes the known Lucassen/van den Tempel (vdT) modulus [6]: o(f ; c)o m n
1 zn izn : 1 2zn 2(zn )2
the experimental Gibbs elasticity, om n , and the experimental diffusion relaxation time, tn /1/vm n, but also the intrinsic surface dilatational viscosity, kn, and the molecular exchange parameter kn. Surfaces of SDS solutions without added dodecanol show such typical viscoelastic behavior and their moduli can be approximated only by the complete Eq. (13) for a nonionic surfactant although SDS is an ionic one. Generally, solutions of ionic surfactants have the same surface dilatational properties as nonionic surfactants. This can be explained also with the assumption of an expanded surface layer. For high ionic strength and frequencies B/1000 Hz the solution of the diffusion equation outside of the electric double layer behaves in the same way as for nonionic surfactant [24]. Besides, at an oscillating surface a relation between the monolayer concentration Gm(DA(t)) and the concentration G /G1 of the complete surface layer is established, and therefore, in Eq. (7) we can replace dGn/dA by dG dG d( Gm ) dA d( Gm ) dA
(15)
with G(DA(t)) G G 9 Gm 9 Gd
(14)
The surface dilatational properties of many solutions can be characterized by this modulus [21,22]. However, investigations demonstrate also that the experimental values of the parameters om and vm differ widely from the calculated parameter o0 and v0. Obviously, the assumptions of the calculation are not fully correct. The calculation is based on the surface tension isotherm equation and the monolayer model of the surface. The difference of the parameters can be explained using the model of an extended surface layer due to the different definition of the surface concentration [22]. On the contrary, the surfaces of other surfactants exhibit a typical viscoelastic behavior. In [22] the viscous effect is explained by dissipative losses due to molecular exchanges between monolayer and sublayer in a nonequilibrium state. The fitting procedure of these measurements yields not only
0 9
m
g
G (c9 (x; t)c1 )dx:
(16)
d
Because of dG/d( Gm) "/1 we obtain different values as well for om and o0 as for vm and v0. In the static case the integration over the concentration c9(x, t) leads to the known expression for the integral 0 9
g
d
G (c9 (x; t)c1 )dx d
$
2c1 eC 1 : exp 9 2kT k
(17)
Gd represents the contribution of the diffuse layer to the equilibrium surface excess concentration, k* the Debye reciprocal length parameter, and C the electrostatic potential at the border of the diffuse part of the double layer. The assump9
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tion of an extended surface layer requires for ionic surfactants that the Debye length is small in comparison to the wave length l /2p/a1 of the diffusion wave which is fulfilled for concentrations c1 ]/104 mol l 1 (1/k*5/30 nm, l /1 mm). The problem of the thickness of the layer is discussed in [22] and the effect of the oscillation on the concentration distribution within the double layer is calculated in [24]. There only low concentrated solutions of ionic surfactants are investigated in which the intrinsic viscosity can be neglected. Here we do not consider all details of this model since the integral formula Eq. (16) allows a simple evaluation of experimental results in the case of higher concentrated solutions. Then it is only necessary that relations between Dcn, DGm n , DGn are established, and therefore, the functions hn(f) and zn(f) are clearly defined. An intrinsic surface dilatational viscous effect of ionic surfactant solutions is also a hint to molecular exchange processes in a nonequilibrium state between monolayer and diffuse part of the extended surface layer. This consideration explains the similar curves of surface dilatational moduli of ionic and nonionic surfactant solutions. The surface of a pure dodecanol solution characterized by the index n/2, is elastic as demonstrated in Fig. 2. The phase angle of the modulus is approximately zero and its amount is independent of the frequency. That means vm 2 , and k2 are zero and the level value of o represents the experimental Gibbs elasticity om 2. In the case of a mixture of two surface active components with the volume concentration c1 and c2 the equilibrium surface tension g becomes a function of the related surface concentrations G1(c1, c2) and G2(c1, c2). At least in a small concentration range we can introduce an effective mean value Gg /aG1/bG2 of the surface concentration and use the function gg(Gg DGg ) with DGg aDG1 bDG2 :
(18)
For example, some authors describe the surface tension of SDS/dodecanol mixtures by the Frumkin equation
189
g(t)g0 RTGg ln(1G1 =G1 G2 =G2 ) (19) where Gn characterizes the maximum of the surface concentration of the component [31 /33]. If these relations are valid also for dynamic conditions we obtain by development of Eq. (18) the expression Dg
dg dGg
DGg :
(20)
The bubble oscillation leads after a short time to an established state at the surface and all dynamic properties become functions of the surface deformation, DA(t). Therefore, we can write gg(Gg DGg (DA)) g(Gg )
dg dGg dG1 dGg dG1 DA
DA
(21)
and the surface dilatational modulus is given by o(f )o m
dln G1 dln A
(22)
with o m (G1 ; G2 )
dg dln Gg dln Gg dln G1
(23)
The exchange of dodecanol molecules is very slow in comparison to SDS molecules, and therefore, the number of dodecanol molecules at the surface, N2, remains approximately constant during the oscillation. Then, the negative value of the relative change in the concentration DG2/G2 is equal to the relative change in the surface area. That means, in Eq. (7) we must only replace Gn by Gg /(N1/N2)/A and the derivations dGn/dcn, dcn/ dcsn, @cn/@t, and @cn/@x by @Gg/@c1, @c1/@cs1, @c1/@t, and @c1/@x, respectively. Here all partial derivations fulfill the conditions c2 /constant and N2 / constant. Then, the rearrangement leads again to Eq. (11) or Eq. (13) with different meaning, however, constant values of the parameters om n and vm n . The experimental results confirm this assumption and for our special mixture the equations calculated for the solution of one surfactant remain valid. Nevertheless, one must bear in mind that the parameters om, vm, k, and k, as well the equili-
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brium surface concentrations G1(c1, c2) and G2(c1, c2) are functions of the independent values of the bulk concentrations c1 and c2. This demonstrates the adjustment of the measured surface dilatational moduli of SDS/dodecanol solutions to Eq. (13).
3. Experimental results Experiments with the oscillating bubble method require careful preparations of the measuring cell. In a first test the purity of the cell must be verified. This is possible by filling the cell with clean, pure water and monitoring the pressure signal. The system responds at the pressure transducer with a doubling of the frequency in the half sphere case because a water surface has neither elasticity nor viscosity. The effect can be demonstrated easily using an oscilloscope. An unclean water does not show such behavior. The next step is a calibration measurement. We use normally a solution of decanoic acid, which reaches for frequencies f / 100 Hz a known, constant elasticity value. In the range f /300 Hz the measured pressure deviates from this value. The differences describe the correction terms mentioned above. The cell and the capillary must be cleaned by flushing and all small air bubbles within the filled cell must be eliminated. The real measurement with the closed cell can be started after 30 min waiting for the tempering which is necessary to reach a stable bubble. In some cases this time is not long enough for the establishing of an equilibrium state at the surface. However, after 1 h all presented measurements were reproducible. Mixed and unmixed dodecanol solutions needs approximately 50 min for this process, whereas, for unmixed SDS solutions only 30 min are required. The experiments were carried out with four differently concentrated SDS solutions, four differently concentrated dodecanol solutions and 11 mixtures of both. Surface dilatational moduli were measured in the frequency range 1 5/f 5/500 Hz [19] and a ring tensiometer from Lauda was applied to determine the surface tension isotherms. SDS and dodecanol were supplied by Merck as
received. Measurements of isotherms are a simple test of the degree of purity. In Fig. 1 the isotherm of SDS shows a small minimum near the CMC. Such minima hint at impurities of approximately 0.02 mol% dodecanol [34,35]. The evaluation of the purity of SDS samples is controversially discussed in the literature [36 /38]. For the estimation of its influence on the dynamic surface tension we have compared the surface dilatational modulus of a highly purified solution (surface chemically pure) with measurements using an unpurified Merck product. The results demonstrate that in the medium frequency range the effect of impurities on the surface dilatational modulus is much smaller than on the static surface tension. The reason is probably the small contribution of these molecules to the exchange dynamics at the surface, which depends on the concentration, and the influence of very low concentrated impurities on the modulus can, therefore, be neglected. In this paper the term ‘‘unmixed SDS solutions’’ is used for the description of a SDS solution without added dodecanol. The theoretical values of the parameters o0 and v0 are also not presented here because of the unrealistic results of such calculations. The reasons for that are discussed in [22]. As mentioned above the modulus o(f, cn)/E(f, cn)exp(i8(f, cn)) describes the response of the surface tension to a sinusoidal dilatation of the surface. It is a complex function of the frequency with the amount E(f, cn) and the phase angle 8(f, cn) which depend also on the concentrations cn of
Fig. 1. Equilibrium surface tension isotherm of SDS.
K.-D. Wantke et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 221 (2003) 185 /195
the components of the solution. The surface of pure water does not exhibit any rheological effect because the molecular exchange processes very quickly reach an equilibrium state, whereas solutions of surface active substances show various surface effects. For example, all dodecanol solutions have purely elastic adsorption layers as demonstrated in Fig. 2. The amount E is constant and the phase angle 8 approximately zero in the complete frequency range. These surfaces behave like insoluble, homogeneous monolayers. Such behavior is possible in the frequency range used if the exchange rate k2 of dodecanol molecules is small (k2 /1 cm s 1) and their diffusion relaxation time large (1/vm /1/v). The behavior of unmixed SDS solutions is much different. Figs. 3 and 4 demonstrate that these moduli do not reach a constant level in the complete frequency range and the phase angles increase after passing a minimum. The reason for both effects is a detectable value of the intrinsic dilatational viscosity of the surface characterized by the last term in Eq. (11). According to this equation the phase angle of the modulus approaches zero, and the amount remains constant (E /om) for increasing frequencies as long as the viscous term is vanishing. The reasons for decreasing phase angles in the low frequency range are
191
Fig. 3. Amount E(f) of the surface dilatational modulus of SDS solutions: (1/j) c1 /5/10 3 M, (2/') c1 /3/10 3 M, (3/ %) c1 /1/10 3 M, ( */) according to Eq. (11) using parameters of Table 1.
Fig. 4. Phase angle 8(f) of the surface dilatational modulus of SDS solutions: (1/j) c1 /5/10 3 M, (2/') c1 /3/10 3 M, (3/%) c1 /1/10 3 M, ( */) according to Eq. (11) using parameters of Table 1.
Fig. 2. Amount E(f) and a few phase angles 8(f) of the surface dilatational modulus of dodecanol solutions: ( /j /) c2 /1.2/ 10 5 M, ( /m /) c2 /8/10 6 M, ( /' /) c2 /6/10 6 M, ( / % /), c2 /5/10 6 M.
bulk diffusion (v /vm) and kinetic effects (h"/1) [22], whereas the rising curves in the frequency range f /100 Hz are caused by a surface viscosity k /0. All curves of unmixed, higher concentrated SDS-solutions have this characteristic form. In Table 1 the parameters of the moduli are documented. They have the same tendencies as the parameters of nonionic solutions [22].
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Table 1 Parameters of the surface dilatational modulus of selected solutions c1 (mol dm 3)
om (mN m 1)
vm (s 1)
k (mN s m 1)
/k1 (cm s 1)
Mixtures of c1 SDS/c2 dodecanol 0.005 0 0.005 0.05 0.005 0.06 0.005 0.1 0.005 0.4 0.003 0.0 0.003 0.05 0.003 0.1 0.003 0.4 0.001 0.0 0.001 0.02 0.001 0.05 0.001 0.1 0.001 0.4
25.5 42 85 127 125.5 32.5 72.5 130 143 25.5 25.5 106 129 135
250 150 240 140 45 75 200 35 40 5 12 9 11 5
0.0088 0.0071 0.0025 0.0017 0.0 0.0037 0.0069 0.0 0.0 0.0041 0.0024 0.0022 0.0006 0.0
0.5 0.15 0.1 0.07 0.04 1.9 0.1 0.15 0.08 / / / / /
Unmixed SDS-solutions: 0.006 0 0.005 0 0.003 0 0.001 0
20.5 25.5 32.5 25.5
330 250 75 5
0.0079 0.0088 0.0037 0.0041
0.3 0.5 1.9 /
Mol% dodec. (100c2/c1)
Fig. 5. Amount E(f) of the surface dilatational modulus of mixed 5/10 3 M SDS/x mol% dodecanol solutions: (1/') x/0.4 mol%, (2/j) x /0.1 mol%, (3/m) x/0.06 mol%, (4//) x/0.05 mol%, (5/%) x/0 mol%, ( */) according to Eq. (11) using parameters of Table 1; (6/ /" /) pure 1.2/10 5 M dodecanol.
The Figs. 5/10 show the surface dilatational moduli of mixed SDS/dodecanol solutions. These mixtures reach the equilibrium state at the surface approximately 50 min after the bubble at the tip of
Fig. 6. Phase angle 8(f) of the surface dilatational modulus of mixed 5/10 3 M SDS/x mol% dodecanol solutions: (1/') x/0.4 mol%, (2/j) x/0.1 mol%, (3/m) x/0.06 mol%, (4//) x/0.05 mol%, (5/%) x/0 mol%, ( */) according to Eq. (11) using parameters of Table 1.
the capillary is formed. Surfaces exhibit a greater viscous effect at the beginning than in the final state. However, these first measurements of the moduli are not reproducible. The effect gives a hint to the very slow adsorption rate of dodecanol
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Fig. 7. Amount E(f) of the surface dilatational modulus of mixed 3/10 3 M SDS/x mol% dodecanol solutions: (1/') x/0.4 mol%, (2/j) x /0.1 mol%, (3/ /) x/0.05 mol%, (4/%) x/0 mol%, ( */) according to Eq. (11) using parameters of Table 1.
Fig. 8. Phase angle 8(f) of the surface dilatational modulus of mixed 3/10 3 M SDS/x mol% dodecanol solutions: (1/') x/0.4 mol%, (2/j) x /0.1 mol%, (3/ /) x/0.05 mol%, (4/%) x/0 mol%, ( */) according to Eq. (11) using parameters of Table 1.
molecules in comparison to SDS molecules due to the very low concentration and the great surface activity of dodecanol. At the beginning of the measurements the surface concentration of SDS is higher than the surface concentration of dodecanol, whereas in the following time the replacement of these molecules by dodecanol molecules leads to a more elastic surface. Such interim states are verified for a few frequencies. This effect corre-
193
Fig. 9. Amount E(f) of the surface dilatational modulus of mixed 1/10 3 M SDS/x mol% dodecanol solutions: (1/') x/0.4 mol%, (2/j) x /0.1 mol%, (3/m) x/0.05 mol%, (4/%) x/0.02 mol%, (5/") x/0.0 mol%, ( */) according to Eq. (11) using parameters of Table 1.
Fig. 10. Phase angle 8(f) of the surface dilatational modulus of mixed 1/10 3 M SDS/x mol% dodecanol solutions: (1/') x/0.4 mol%, (2/j) x /0.1 mol%, (3/m) x/0.05 mol%, (4/%) x/0.02 mol%, (5/") x/0.0 mol%, ( */) according to Eq. (11) using parameters of Table 1.
sponds with the results of Vollhardt et al. [39 /42]. The authors have investigated the dynamic surface tension of SDS/dodecanol solutions in the temperature range 5/15 8C. They found a break point in the slope of the adsorption kinetics, which indicates a first-order phase transition. In addition, the formation of domains was verified by using Brewster angle microscopy (BAM) images. The time behavior of these formations also docu-
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ments the replacement effect. Our experiments were carried out at 23 8C, which is above the demonstrated phase transition. Nevertheless, the replacement of SDS molecules by dodecanol molecules should be widely independent of the temperature. In Table 1 the fitting parameters of the Eq. (13) to surface dilatational moduli measured are presented. A few values of k1 are missing since for small vm 1 the influence of k1 becomes negligible. For the same reason the values of k1 listed in Table 1, are not very exact. They represent only a rough approximation of the exchange rate of SDS molecules at the surface. The trends of the other parameters correspond to known results [21,22]. However, one must bear in mind that a certain separation of different effects is not possible by fitting in some cases. A low amount of dodecanol in the mixture leads to E values, which are higher than the values of comparable pure dodecanol solutions. In addition, the related phase angles are small, however, not zero. This means, both substances are adsorbed at the surface but the number and the exchange rate are noticeably reduced for SDS molecules due to the great surface activity of dodecanol. Nevertheless, the behavior of phase angle in the high frequencies range exhibits the influence of surface dilatational viscosity for mixtures with a low amount of dodecanol. These phase angles can be used as scale for the presence of SDS molecules at the surface because the viscous behavior depends on the molecular exchange rate. According to our interpretation, the effect is caused by dissipative losses due to molecular exchange processes between monolayer and sublayer in a nonequilibrium state [22]. Only the exchange rate of SDS molecules is obviously large enough for such effect.
4. Conclusions The surface dilatational modulus is an appropriate function for the characterization of rheological properties of fluid surfaces. In a medium frequency range it can be determined in a fast and reliable manner using the oscillating bubble method. The evaluation of such measurements
leads to information about the molecular exchange mechanism at the surface and its influence on the surface rheological properties. The results of mixed SDS/dodecanol solutions demonstrate the different influence of the components. The surface of a pure dodecanol solution is elastic without a viscous effect, whereas the surface of an unmixed SDS solution shows a marked viscoelastic behavior. Mixtures exhibit graduated properties. In addition, the time influence on the dilatational modulus demonstrates that dodecanol molecules drive the SDS-molecules slowly out of the surface. Such time behavior allows the theoretical description of the dynamics at the surface of these mixed solutions by the one-component model.
Acknowledgements The authors thank I. Bartsch for careful measurements and Professor Mo¨hwald for his support of this work. Financial assistance from the Deutsche Forschungsgemeinschaft (WA 1438/1) is gratefully acknowledged.
References [1] J.C. Slattery, Interfacial Transport Phenomena, Springer, New York, 1990. [2] D.A. Edwards, T. Brenner, D.T. Wasan, Interfacial Transport Processes and Rheology, Butterworth-Heinemann, Stoneham, MA, 1991. [3] R.L. Kao, D.A. Edwards, D.T. Wasan, E. Chen, J. Colloid Interf. Sci. 155 (1993) 118. [4] S.S. Dukhin, G. Kretzschmar, R. Miller, in: D. Mo¨bius, R. Miller (Eds.), Studies in Interface Science, vol. 1, Elsevier, Amsterdam, 1995. [5] B.A. Noskov, Adv. Colloid Interf. Sci. 80 (1996) 63. [6] J. Lucassen, M. van den Tempel, Chem. Eng. Sci. 27 (1972) 1283. [7] C.A. MacLeod, C.J. Radke, Langmuir 10 (1994) 3555. [8] C.A. MacLeod, C.J. Radke, J. Colloid Interf. Sci. 166 (1994) 73. [9] Q. Jiang, J.E. Valentini, Y.C. Chiew, J. Colloid Interf. Sci. 174 (1995) 286. [10] L. Arnaudov, N.D. Denkov, I. Surcheva, P. Durbut, G. Bronze, A. Mehreteab, Langmuir 17 (2001) 6999 /7010. [11] A. Battacharyya, F. Monroy, D. Langevin, et al., Langmuir 16 (2000) 8727. [12] S. Hall, M. Bermel, Y. Ko, H. Palmer, G. Enhorning, R. Notter, J. Appl. Physiol. 75 (1993) 467.
K.-D. Wantke et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 221 (2003) 185 /195 [13] D.O. Johnson, K.J. Stebe, J. Colloid Interf. Sci. 182 (1996) 21. [14] C.-H. Chang, E.I. Franses, J. Colloid Interf. Sci. 164 (1994) 107. [15] C.-H. Chang, K.A. Coltharp, S.Y. Park, E.I. Franses, Colloids Surf. A 114 (1996) 185. [16] Y. Tian, R.G. Holt, R. Apfel, Fhys Fluids 7 (1995) 2938. [17] Y. Tian, R.G. Holt, R. Apfel, J. Colloid Interf. Sci. 187 (1997) 1. [18] V.I. Kovalchuk, J. Kra¨gel, E.V. Aksenko, G. Loglio, L. Liggieri, in: D. Mo¨bius, R. Miller (Eds.), Studies in Interface Science, vol. 11, Elsevier, Amsterdam, 2001, p. 485. [19] H. Fruhner, K.-D. Wantke, Colloids Surf. A 114 (1996) 53. [20] K.-D. Wantke, H. Fruhner, in: D. Mo¨bius, R. Miller (Eds.), Studies in Interface Science, vol. 6, Elsevier, Amsterdam, 1998, p. 327. [21] K.-D. Wantke, H. Fruhner, J. Fang, K. Lunkenheimer, J. Colloid Interf. Sci. 208 (1998) 34. [22] K.-D. Wantke, H. Fruhner, J. Colloid Interf. Sci. 237 (2001) 185. [23] H. Fruhner, K.-D. Wantke, K. Lunkenheimer, Colloids Surf. A 162 (2000) 193. [24] P. Warszynski, K.-D. Wantke, H. Fruhner, Colloids Surf. A 139 (1998) 137. [25] H. Wong, D. Rumschitzki, C. Maldarelli, J. Fluid Mech. 356 (1998) 93.
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[26] A.W. Adamson, Physical Chemistry of Surfaces, Wiley, New York, 1993. [27] H.C. Maru, D.T. Mohan, D.T. Wasan, Chem. Eng. Sci. 34 (1979) 1283. [28] H.C. Maru, D.T. Wasan, Chem. Eng. Sci. 34 (1979) 1295. [29] J.R. Lu, I.P. Purcell, M. Lee, E.A. Simister, R.K. Thomas, A.R. Rennie, J. Penfold, J. Colloid Interf. Sci. 174 (1995) 441. [30] R.N. Ward, P.B. Davies, C.D. Bain, J. Phys. Chem. B 101 (1997) 1594. [31] J.P. Fang, P. Joos, Colloids Surf. 65 (1992) 113. [32] J.P. Fang, P. Joos, Colloids Surf. 65 (1992) 121. [33] V.B. Fainerman, R. Miller, Colloids Surf. A 97 (1995) 65. [34] G. Czichocki, D. Vollhardt, H. Seibt, Tenside Detergents 18 (1981) 6. [35] G. Czichocki, D. Vollhardt, H. Much, J. Colloid Interf. Sci. 95 (1983) 275. [36] K. Lunkenheimer, J. Colloid Interf. Sci. 160 (1993) 509. [37] K. Lunkenheimer, H. Fruhner, F. Theil, Colloids Surf. A 76 (1993) 289. [38] J.D. Hines, J. Colloid Interf. Sci. 180 (1996) 488. [39] D. Vollhardt, G. Emrich, Colloids Surf. A 161 (2000) 173. [40] V.B. Fainerman, D. Vollhardt, G. Emrich, J. Phys. Chem. B 104 (2000) 8536. [41] V.B. Fainerman, D. Vollhardt, J. Phys. Chem. B 105 (2001) 4324. [42] D. Vollhardt, G. Brezesinski, S. Siegel, G. Emrich, J. Phys. Chem. B 105 (2001) 12061.
Surface dilatational properties of mixed sodium dodecyl sulfate/dodecanol solutions ¨ rtegren K.-D. Wantke *, H. Fruhner, J. O Max-Planck-Institute of Colloids and Interface, Am Mu ¨ hlenberg, D-14476 Golm, Germany Received 22 May 2002; accepted 27 March 2003
Abstract The surface dilatational properties of aqueous solutions of sodium dodecyl sulfate (SDS) and n -dodecanol are investigated in the frequency range 1 5/f5/500 Hz using the oscillating bubble method. The results demonstrate that a pure dodecanol solution has an elastic surface without viscous effect whereas the surface of a SDS solution without added dodecanol exhibits a strong viscoelastic behavior. Mixtures show graduated properties. The time behavior of their surface dilatational moduli demonstrates that dodecanol molecules drive the SDS molecules slowly out of the surface. Therefore, the known one-component model describing the surface dilatational modulus can be used also for these mixtures. A simple theoretical consideration explains this effect. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Sodium dodecyl sulfate; n -Dodecanol; Oscillating bubble; Surface dilatational modulus; Surface rheology
1. Introduction The rheological properties of fluid surfaces and interfaces have a decisive influence on the behavior of many complex systems and technological processes. Examples are coating processes, multiphase flow, and stability of foams and emulsions. The prerequisite for the investigation of the specific mechanisms in such complex systems is a detailed knowledge about surface and interface properties.
* Corresponding author. Tel./fax: /33-1-567-9233/9202. E-mail address: [email protected] (K.-D. Wantke).
In particular, dilatational effects are of major importance because the creation of a new surface or interface is a highly energetic process. In addition, these properties depend in a strong manner on the chemical composition of the systems. Methods and problems of this field are described in many books and papers [1 /11]. In the last years great progress could be reached by investigation of dilatational properties of fluid surfaces applying oscillating bubbles and drops [12 /18]. The authors contributed to this work by the development of a new version of the oscillating bubble method which allows simple and reliable measurements of the dilatational modulus of fluid
0927-7757/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0927-7757(03)00135-3
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surfaces in the frequency range 15/f5/500 Hz [19,20]. The modulus is defined by the equation o(f ; cn )E(f; cn )exp(i8 (f; cn ))
ADg DA
(1) G(f)
where DA/A represents the relative change in surface area, Dg the change in surface tension, f the frequency, and cn the concentration of the components. The modulus describes the response of the surface tension to a sinusoidal surface dilatation, DA /jDAjexp(i2pft). Such measurements exhibit various characteristics in o(f, cn) due to the different dynamics of the molecules at the surface. The modulus covers elastic, viscous, and transport properties. In previous papers several surface effects of solutions containing only one soluble surfactant are documented and interpreted [21 /24]. Here we investigate the surface behavior of mixed sodium dodecyl sulfate (SDS)/n -dodecanol solutions.
2. Theoretical considerations The oscillating bubble method applies the following measuring principle: within a closed chamber, which is filled with a surface active solution, a small hemispherical bubble is produced at the tip of a capillary. A piezoelectric driver at the wall of the chamber causes a small sinusoidal oscillation of the bubble surface via the solution. A change in the size of the bubble produces a change of the pressure in the chamber. At the pressure transducer the sinusoidal contribution to the pressure, Dp, is given by Dp
2 r0
Dg
2g (r0 )2
DrG(f)
relevant information about the dynamic surface tension. For an ideal spherical bubble with a fixed center the viscous and inertial effects can be calculated easily [20,25]. It leads to
(2)
where r0 describes the mean radius of the bubble, Dr the change in the radius, and G(f) the influence of the fluid dynamics, that means the influence of the bulk viscosity and the inertia of the solution [20]. According to Eq. (1) the surface dilatational modulus results from the first term on the left side in Eq. (2). Therefore, the pressure measurements must be corrected by the last terms to obtain a
8pihf r Drr(2pf)2 r0 1 0 Dr: r0 r1
Here h represents the bulk viscosity, r the density of the solution, and r1 the distance of the pressure transducer to the center. However, the geometrical conditions in a real set up deviate from the ideal geometry and the correction term G(f) must be calculated via a path integral between the bubble surface and the pressure transducer using the solution of the Navier /Stokes equations. We apply calibration measurements for the correction to avoid this complex calculations. The necessary correction is the smallest in the case of the half sphere where the center is not fixed. Then, Dr becomes approximately zero [20]. Proper calibration systems are solutions with a purely elastic surface where the constant elasticity can be determined in the low frequency range. The difference between this value and the real pressure at the transducer measured with same solutions in the higher frequency range yields the experimental correction curve. It includes all correction terms of Eq. (2) for a real chamber. The curves presented in this paper show only pure surface effects that means corrected values. For the interpretation of these effects we consider a surface as a layer with a thickness of a few molecules. All physical properties of such layer should be projected onto a two-dimensional interface. Then the definition of the surface concentration depends on the thickness of the layer. In an equilibrium state, the influence of the thickness on the surface concentration can be eliminated using an invariant scale [26]. The related monolayer model allows only the consideration of lateral structures, whereas various internal structures are possible within a vertically extended layer. Then, a surface deformation can lead to nonequilibrium states and dissipative losses due to molecular exchange processes between structures. Such assumptions are necessary for the explanation of measured dilatational properties of fluid inter-
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faces. This should be demonstrated by a simple consideration. The surface dilatational modulus of a solution with one surfactant results from the following formula o(f;
cn )o m n (Gn (cn ))
A
DGn (f; cn )
Gn (cn )
DA
i2pf kn (cn ):
(3)
The first term characterizes the influence of change in surface concentration, DG(f, cn), on the surface tension. It depends only on the surface concentration Gn(f, cn) /Gn(cn)/DGn(f, cn), and therefore, some authors introduce the name compositional effects [27,28]. In our notation the expression om n (Gn (cn )) Gn Dg=DGn
(4)
represents the experimental Gibbs elasticity of SDS (n/1) or dodecanol (n/2). Recent results have documented the presence of trace components of n-dodecanol within aqueous SDS solutions [29,30]. We discuss this problem in Section 3. The second term characterizes an intrinsic surface viscous effect, which is caused by dissipative processes within the surface layer. A prerequisite of such an effect is a nonequilibrium state near or within the surface layer and the term becomes zero if there the thermodynamic equilibrium is instantaneously established. For the evaluation of dilatational moduli one must take into account that the first term of Eq. (3) also includes an imaginary part due to the influence of bulk diffusion on the change in surface concentration, DGn(f, cn). In these cases the dissipative processes are located within the adjacent bulk phases and have a weaker influence on complex systems like foams [23]. Therefore, the separation of the intrinsic surface dilatational viscosity from the effect of the bulk diffusion is an aim of our experiments. It requires the calculation of DGn. A harmonic oscillation of a bubble leads after a short time to an established state at its surface, and all properties can be considered as functions of the dilatational state characterized by DA(t) / jDAjexp(ivt). Then, for k/0, the surface dilatational modulus follows from
o(f ; cn )o m n
187
dln Gn dln A
(5)
dg : dln Gn
(6)
with om n (Gn (cn ))
The relation between the change in surface concentration, DGn, and the change in surface area, DA(t), results from the law of mass conservation which is given for a monolayer model by Gn dA dGn Gn dA dGn dcn @cn 1 A dt dt A dGn dcn dcsn @t x0
j
Dn
@cn @x
j
:
(7)
x0
Dn represents the diffusion constant, csn /cn(x /0, t) a real sublayer concentration, and cn a fictive sublayer concentration resulting from the isotherm equation Gn /Gn(cn ). For a bulk diffusion controlled process, which means that the surface is in an equilibrium state, the concentrations become equal (csn /cn ). Consequently, their difference is a scale for the molecular exchange flow jn $ kn (csn cn):
(8)
This equation defines the molecular exchange constant kn [20]. With the solution of the diffusion equation cn (x; t)c0n ½Dcsn ½exp[(1i)an xivtib?n ]
(9)
(an wave number of the diffusion wave, bn? phase angle resulting from the boundary condition, v / 2pf) for an oscillating process in the area xB/0 we obtain the relation hn (f ; cn )
dcsn
kn pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dcn kn (1 i) vDn =2
½hn (f ; cn )½exp(ibn (f ; cn ))
(10)
for the characterization of the molecular exchange balance between sublayer and layer. The Eqs. (3), (5), (7) and (10) lead to the expression o(f ; cn )o m n with
1 z?n iz?n ivkn 1 2z?n 2(z?n )2
(11)
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sffiffiffiffiffiffiffi vm Dcn 2 n m ; vn Dn z?n hn (f ; c)zn ; zn ; 2v DGn (12) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi vDn =2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi and bn arctan kn vDn =2 [20 /22]. In a final step the general Eq. (11) can be transformed into the following formula pffiffiffi om 2zn ½hn ½exp(i(p=4 bn ))) n (1 o(f ; c) 1 2zn ½hn ½(cos(bn ) sin(bn )) 2(zn ½hn ½)2 ivkn (13) which describes the dilatational modulus of surfactant solutions of one component. The verification of this equation by experiments is comprehensively discussed in [20 /22]. An equilibrium state is instantaneously established at the surface if the exchange constant kn becomes large ((kn)2 /Dnv), whereas molecular exchanges can be neglected for (kn)2 /Dnv. In the first case hn(f, cn) becomes one and kn should be zero. Then zn? /zn and Eq. (11) describes the known Lucassen/van den Tempel (vdT) modulus [6]: o(f ; c)o m n
1 zn izn : 1 2zn 2(zn )2
the experimental Gibbs elasticity, om n , and the experimental diffusion relaxation time, tn /1/vm n, but also the intrinsic surface dilatational viscosity, kn, and the molecular exchange parameter kn. Surfaces of SDS solutions without added dodecanol show such typical viscoelastic behavior and their moduli can be approximated only by the complete Eq. (13) for a nonionic surfactant although SDS is an ionic one. Generally, solutions of ionic surfactants have the same surface dilatational properties as nonionic surfactants. This can be explained also with the assumption of an expanded surface layer. For high ionic strength and frequencies B/1000 Hz the solution of the diffusion equation outside of the electric double layer behaves in the same way as for nonionic surfactant [24]. Besides, at an oscillating surface a relation between the monolayer concentration Gm(DA(t)) and the concentration G /G1 of the complete surface layer is established, and therefore, in Eq. (7) we can replace dGn/dA by dG dG d( Gm ) dA d( Gm ) dA
(15)
with G(DA(t)) G G 9 Gm 9 Gd
(14)
The surface dilatational properties of many solutions can be characterized by this modulus [21,22]. However, investigations demonstrate also that the experimental values of the parameters om and vm differ widely from the calculated parameter o0 and v0. Obviously, the assumptions of the calculation are not fully correct. The calculation is based on the surface tension isotherm equation and the monolayer model of the surface. The difference of the parameters can be explained using the model of an extended surface layer due to the different definition of the surface concentration [22]. On the contrary, the surfaces of other surfactants exhibit a typical viscoelastic behavior. In [22] the viscous effect is explained by dissipative losses due to molecular exchanges between monolayer and sublayer in a nonequilibrium state. The fitting procedure of these measurements yields not only
0 9
m
g
G (c9 (x; t)c1 )dx:
(16)
d
Because of dG/d( Gm) "/1 we obtain different values as well for om and o0 as for vm and v0. In the static case the integration over the concentration c9(x, t) leads to the known expression for the integral 0 9
g
d
G (c9 (x; t)c1 )dx d
$
2c1 eC 1 : exp 9 2kT k
(17)
Gd represents the contribution of the diffuse layer to the equilibrium surface excess concentration, k* the Debye reciprocal length parameter, and C the electrostatic potential at the border of the diffuse part of the double layer. The assump9
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tion of an extended surface layer requires for ionic surfactants that the Debye length is small in comparison to the wave length l /2p/a1 of the diffusion wave which is fulfilled for concentrations c1 ]/104 mol l 1 (1/k*5/30 nm, l /1 mm). The problem of the thickness of the layer is discussed in [22] and the effect of the oscillation on the concentration distribution within the double layer is calculated in [24]. There only low concentrated solutions of ionic surfactants are investigated in which the intrinsic viscosity can be neglected. Here we do not consider all details of this model since the integral formula Eq. (16) allows a simple evaluation of experimental results in the case of higher concentrated solutions. Then it is only necessary that relations between Dcn, DGm n , DGn are established, and therefore, the functions hn(f) and zn(f) are clearly defined. An intrinsic surface dilatational viscous effect of ionic surfactant solutions is also a hint to molecular exchange processes in a nonequilibrium state between monolayer and diffuse part of the extended surface layer. This consideration explains the similar curves of surface dilatational moduli of ionic and nonionic surfactant solutions. The surface of a pure dodecanol solution characterized by the index n/2, is elastic as demonstrated in Fig. 2. The phase angle of the modulus is approximately zero and its amount is independent of the frequency. That means vm 2 , and k2 are zero and the level value of o represents the experimental Gibbs elasticity om 2. In the case of a mixture of two surface active components with the volume concentration c1 and c2 the equilibrium surface tension g becomes a function of the related surface concentrations G1(c1, c2) and G2(c1, c2). At least in a small concentration range we can introduce an effective mean value Gg /aG1/bG2 of the surface concentration and use the function gg(Gg DGg ) with DGg aDG1 bDG2 :
(18)
For example, some authors describe the surface tension of SDS/dodecanol mixtures by the Frumkin equation
189
g(t)g0 RTGg ln(1G1 =G1 G2 =G2 ) (19) where Gn characterizes the maximum of the surface concentration of the component [31 /33]. If these relations are valid also for dynamic conditions we obtain by development of Eq. (18) the expression Dg
dg dGg
DGg :
(20)
The bubble oscillation leads after a short time to an established state at the surface and all dynamic properties become functions of the surface deformation, DA(t). Therefore, we can write gg(Gg DGg (DA)) g(Gg )
dg dGg dG1 dGg dG1 DA
DA
(21)
and the surface dilatational modulus is given by o(f )o m
dln G1 dln A
(22)
with o m (G1 ; G2 )
dg dln Gg dln Gg dln G1
(23)
The exchange of dodecanol molecules is very slow in comparison to SDS molecules, and therefore, the number of dodecanol molecules at the surface, N2, remains approximately constant during the oscillation. Then, the negative value of the relative change in the concentration DG2/G2 is equal to the relative change in the surface area. That means, in Eq. (7) we must only replace Gn by Gg /(N1/N2)/A and the derivations dGn/dcn, dcn/ dcsn, @cn/@t, and @cn/@x by @Gg/@c1, @c1/@cs1, @c1/@t, and @c1/@x, respectively. Here all partial derivations fulfill the conditions c2 /constant and N2 / constant. Then, the rearrangement leads again to Eq. (11) or Eq. (13) with different meaning, however, constant values of the parameters om n and vm n . The experimental results confirm this assumption and for our special mixture the equations calculated for the solution of one surfactant remain valid. Nevertheless, one must bear in mind that the parameters om, vm, k, and k, as well the equili-
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brium surface concentrations G1(c1, c2) and G2(c1, c2) are functions of the independent values of the bulk concentrations c1 and c2. This demonstrates the adjustment of the measured surface dilatational moduli of SDS/dodecanol solutions to Eq. (13).
3. Experimental results Experiments with the oscillating bubble method require careful preparations of the measuring cell. In a first test the purity of the cell must be verified. This is possible by filling the cell with clean, pure water and monitoring the pressure signal. The system responds at the pressure transducer with a doubling of the frequency in the half sphere case because a water surface has neither elasticity nor viscosity. The effect can be demonstrated easily using an oscilloscope. An unclean water does not show such behavior. The next step is a calibration measurement. We use normally a solution of decanoic acid, which reaches for frequencies f / 100 Hz a known, constant elasticity value. In the range f /300 Hz the measured pressure deviates from this value. The differences describe the correction terms mentioned above. The cell and the capillary must be cleaned by flushing and all small air bubbles within the filled cell must be eliminated. The real measurement with the closed cell can be started after 30 min waiting for the tempering which is necessary to reach a stable bubble. In some cases this time is not long enough for the establishing of an equilibrium state at the surface. However, after 1 h all presented measurements were reproducible. Mixed and unmixed dodecanol solutions needs approximately 50 min for this process, whereas, for unmixed SDS solutions only 30 min are required. The experiments were carried out with four differently concentrated SDS solutions, four differently concentrated dodecanol solutions and 11 mixtures of both. Surface dilatational moduli were measured in the frequency range 1 5/f 5/500 Hz [19] and a ring tensiometer from Lauda was applied to determine the surface tension isotherms. SDS and dodecanol were supplied by Merck as
received. Measurements of isotherms are a simple test of the degree of purity. In Fig. 1 the isotherm of SDS shows a small minimum near the CMC. Such minima hint at impurities of approximately 0.02 mol% dodecanol [34,35]. The evaluation of the purity of SDS samples is controversially discussed in the literature [36 /38]. For the estimation of its influence on the dynamic surface tension we have compared the surface dilatational modulus of a highly purified solution (surface chemically pure) with measurements using an unpurified Merck product. The results demonstrate that in the medium frequency range the effect of impurities on the surface dilatational modulus is much smaller than on the static surface tension. The reason is probably the small contribution of these molecules to the exchange dynamics at the surface, which depends on the concentration, and the influence of very low concentrated impurities on the modulus can, therefore, be neglected. In this paper the term ‘‘unmixed SDS solutions’’ is used for the description of a SDS solution without added dodecanol. The theoretical values of the parameters o0 and v0 are also not presented here because of the unrealistic results of such calculations. The reasons for that are discussed in [22]. As mentioned above the modulus o(f, cn)/E(f, cn)exp(i8(f, cn)) describes the response of the surface tension to a sinusoidal dilatation of the surface. It is a complex function of the frequency with the amount E(f, cn) and the phase angle 8(f, cn) which depend also on the concentrations cn of
Fig. 1. Equilibrium surface tension isotherm of SDS.
K.-D. Wantke et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 221 (2003) 185 /195
the components of the solution. The surface of pure water does not exhibit any rheological effect because the molecular exchange processes very quickly reach an equilibrium state, whereas solutions of surface active substances show various surface effects. For example, all dodecanol solutions have purely elastic adsorption layers as demonstrated in Fig. 2. The amount E is constant and the phase angle 8 approximately zero in the complete frequency range. These surfaces behave like insoluble, homogeneous monolayers. Such behavior is possible in the frequency range used if the exchange rate k2 of dodecanol molecules is small (k2 /1 cm s 1) and their diffusion relaxation time large (1/vm /1/v). The behavior of unmixed SDS solutions is much different. Figs. 3 and 4 demonstrate that these moduli do not reach a constant level in the complete frequency range and the phase angles increase after passing a minimum. The reason for both effects is a detectable value of the intrinsic dilatational viscosity of the surface characterized by the last term in Eq. (11). According to this equation the phase angle of the modulus approaches zero, and the amount remains constant (E /om) for increasing frequencies as long as the viscous term is vanishing. The reasons for decreasing phase angles in the low frequency range are
191
Fig. 3. Amount E(f) of the surface dilatational modulus of SDS solutions: (1/j) c1 /5/10 3 M, (2/') c1 /3/10 3 M, (3/ %) c1 /1/10 3 M, ( */) according to Eq. (11) using parameters of Table 1.
Fig. 4. Phase angle 8(f) of the surface dilatational modulus of SDS solutions: (1/j) c1 /5/10 3 M, (2/') c1 /3/10 3 M, (3/%) c1 /1/10 3 M, ( */) according to Eq. (11) using parameters of Table 1.
Fig. 2. Amount E(f) and a few phase angles 8(f) of the surface dilatational modulus of dodecanol solutions: ( /j /) c2 /1.2/ 10 5 M, ( /m /) c2 /8/10 6 M, ( /' /) c2 /6/10 6 M, ( / % /), c2 /5/10 6 M.
bulk diffusion (v /vm) and kinetic effects (h"/1) [22], whereas the rising curves in the frequency range f /100 Hz are caused by a surface viscosity k /0. All curves of unmixed, higher concentrated SDS-solutions have this characteristic form. In Table 1 the parameters of the moduli are documented. They have the same tendencies as the parameters of nonionic solutions [22].
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Table 1 Parameters of the surface dilatational modulus of selected solutions c1 (mol dm 3)
om (mN m 1)
vm (s 1)
k (mN s m 1)
/k1 (cm s 1)
Mixtures of c1 SDS/c2 dodecanol 0.005 0 0.005 0.05 0.005 0.06 0.005 0.1 0.005 0.4 0.003 0.0 0.003 0.05 0.003 0.1 0.003 0.4 0.001 0.0 0.001 0.02 0.001 0.05 0.001 0.1 0.001 0.4
25.5 42 85 127 125.5 32.5 72.5 130 143 25.5 25.5 106 129 135
250 150 240 140 45 75 200 35 40 5 12 9 11 5
0.0088 0.0071 0.0025 0.0017 0.0 0.0037 0.0069 0.0 0.0 0.0041 0.0024 0.0022 0.0006 0.0
0.5 0.15 0.1 0.07 0.04 1.9 0.1 0.15 0.08 / / / / /
Unmixed SDS-solutions: 0.006 0 0.005 0 0.003 0 0.001 0
20.5 25.5 32.5 25.5
330 250 75 5
0.0079 0.0088 0.0037 0.0041
0.3 0.5 1.9 /
Mol% dodec. (100c2/c1)
Fig. 5. Amount E(f) of the surface dilatational modulus of mixed 5/10 3 M SDS/x mol% dodecanol solutions: (1/') x/0.4 mol%, (2/j) x /0.1 mol%, (3/m) x/0.06 mol%, (4//) x/0.05 mol%, (5/%) x/0 mol%, ( */) according to Eq. (11) using parameters of Table 1; (6/ /" /) pure 1.2/10 5 M dodecanol.
The Figs. 5/10 show the surface dilatational moduli of mixed SDS/dodecanol solutions. These mixtures reach the equilibrium state at the surface approximately 50 min after the bubble at the tip of
Fig. 6. Phase angle 8(f) of the surface dilatational modulus of mixed 5/10 3 M SDS/x mol% dodecanol solutions: (1/') x/0.4 mol%, (2/j) x/0.1 mol%, (3/m) x/0.06 mol%, (4//) x/0.05 mol%, (5/%) x/0 mol%, ( */) according to Eq. (11) using parameters of Table 1.
the capillary is formed. Surfaces exhibit a greater viscous effect at the beginning than in the final state. However, these first measurements of the moduli are not reproducible. The effect gives a hint to the very slow adsorption rate of dodecanol
K.-D. Wantke et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 221 (2003) 185 /195
Fig. 7. Amount E(f) of the surface dilatational modulus of mixed 3/10 3 M SDS/x mol% dodecanol solutions: (1/') x/0.4 mol%, (2/j) x /0.1 mol%, (3/ /) x/0.05 mol%, (4/%) x/0 mol%, ( */) according to Eq. (11) using parameters of Table 1.
Fig. 8. Phase angle 8(f) of the surface dilatational modulus of mixed 3/10 3 M SDS/x mol% dodecanol solutions: (1/') x/0.4 mol%, (2/j) x /0.1 mol%, (3/ /) x/0.05 mol%, (4/%) x/0 mol%, ( */) according to Eq. (11) using parameters of Table 1.
molecules in comparison to SDS molecules due to the very low concentration and the great surface activity of dodecanol. At the beginning of the measurements the surface concentration of SDS is higher than the surface concentration of dodecanol, whereas in the following time the replacement of these molecules by dodecanol molecules leads to a more elastic surface. Such interim states are verified for a few frequencies. This effect corre-
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Fig. 9. Amount E(f) of the surface dilatational modulus of mixed 1/10 3 M SDS/x mol% dodecanol solutions: (1/') x/0.4 mol%, (2/j) x /0.1 mol%, (3/m) x/0.05 mol%, (4/%) x/0.02 mol%, (5/") x/0.0 mol%, ( */) according to Eq. (11) using parameters of Table 1.
Fig. 10. Phase angle 8(f) of the surface dilatational modulus of mixed 1/10 3 M SDS/x mol% dodecanol solutions: (1/') x/0.4 mol%, (2/j) x /0.1 mol%, (3/m) x/0.05 mol%, (4/%) x/0.02 mol%, (5/") x/0.0 mol%, ( */) according to Eq. (11) using parameters of Table 1.
sponds with the results of Vollhardt et al. [39 /42]. The authors have investigated the dynamic surface tension of SDS/dodecanol solutions in the temperature range 5/15 8C. They found a break point in the slope of the adsorption kinetics, which indicates a first-order phase transition. In addition, the formation of domains was verified by using Brewster angle microscopy (BAM) images. The time behavior of these formations also docu-
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ments the replacement effect. Our experiments were carried out at 23 8C, which is above the demonstrated phase transition. Nevertheless, the replacement of SDS molecules by dodecanol molecules should be widely independent of the temperature. In Table 1 the fitting parameters of the Eq. (13) to surface dilatational moduli measured are presented. A few values of k1 are missing since for small vm 1 the influence of k1 becomes negligible. For the same reason the values of k1 listed in Table 1, are not very exact. They represent only a rough approximation of the exchange rate of SDS molecules at the surface. The trends of the other parameters correspond to known results [21,22]. However, one must bear in mind that a certain separation of different effects is not possible by fitting in some cases. A low amount of dodecanol in the mixture leads to E values, which are higher than the values of comparable pure dodecanol solutions. In addition, the related phase angles are small, however, not zero. This means, both substances are adsorbed at the surface but the number and the exchange rate are noticeably reduced for SDS molecules due to the great surface activity of dodecanol. Nevertheless, the behavior of phase angle in the high frequencies range exhibits the influence of surface dilatational viscosity for mixtures with a low amount of dodecanol. These phase angles can be used as scale for the presence of SDS molecules at the surface because the viscous behavior depends on the molecular exchange rate. According to our interpretation, the effect is caused by dissipative losses due to molecular exchange processes between monolayer and sublayer in a nonequilibrium state [22]. Only the exchange rate of SDS molecules is obviously large enough for such effect.
4. Conclusions The surface dilatational modulus is an appropriate function for the characterization of rheological properties of fluid surfaces. In a medium frequency range it can be determined in a fast and reliable manner using the oscillating bubble method. The evaluation of such measurements
leads to information about the molecular exchange mechanism at the surface and its influence on the surface rheological properties. The results of mixed SDS/dodecanol solutions demonstrate the different influence of the components. The surface of a pure dodecanol solution is elastic without a viscous effect, whereas the surface of an unmixed SDS solution shows a marked viscoelastic behavior. Mixtures exhibit graduated properties. In addition, the time influence on the dilatational modulus demonstrates that dodecanol molecules drive the SDS-molecules slowly out of the surface. Such time behavior allows the theoretical description of the dynamics at the surface of these mixed solutions by the one-component model.
Acknowledgements The authors thank I. Bartsch for careful measurements and Professor Mo¨hwald for his support of this work. Financial assistance from the Deutsche Forschungsgemeinschaft (WA 1438/1) is gratefully acknowledged.
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