Langmuir 1994,10, 2965-2971
2965
Electrostatic Model for the Viscoelasticity of Ionic Surfactant Monolayers A. Bonfillon and D. Langevin* Laboratoire de Physique Statistique de l’ENS,f 24 Rue Lhomond, 75231 Paris Cedex 05,France Received February 4, 1994. In Final Form: June 3, 1994@ In a previous paper (Langmuir 1993,9,8), we have reported the measurement oflow-frequency dilational viscoelasticity for monolayers of an ionic surfactant (sodium dodecyl sulfate (SDS)) at the interfacebetween water and dodecane. We have compared the results with a model in which it is assumed that the exchanges between the surface and bulk are diffusion controlled, the surfactant undergoing simple Fickian diffusion. In the presence of added salt, very good agreement was found with the model, whereas large discrepancies arised in the absence of salt. In this paper, we present similar experimental data with another ionic surfactant (sodium bis(ethylhexy1) sulfosuccinate (AOT))and we propose a new model in which we take into account the role of the electric field created by the charged monolayer on the diffision process. This new model leads to very good agreement with the experiments on SDS and AOT monolayers, when assuming that the surfactant molecules at the oil-water interface are fully ionized. These results demonstrate that no charge condensation occurs in these monolayers in the absence of salt and that the surface charge plays a very important role in surface rheology.
I. Introduction The presence of a surfactant monolayer at the liquidgas or liquid-liquid interface allows it to resist different kinds of perturbations and imparts dilational properties to the interface. The relevance of these properties to phenomena such as foam stability has already been emphasized. It is well known for instance that the stability of sodium dodecyl sulfate foams increases when a small This phenomenon is amount of dodecanol is correlated with an increase of the dilational elasticity. The dilational properties may be characterized by a surface dilational modulus E
= dy/d(lnA)
(1)
which describes the change in the surface tension y with the fractional change in the area A of a given surface element. In real systems, relaxation processes such as adsorption or desorption of soluble surfactant may be involved and the response of a surface element to expansion or compression will depend upon the time scale of the event (the surface will therefore exhibit viscoelastic rather than purely elastic behavior). The measurement of dilational viscoelasticity is then a good way to study the nonequilibrium properties of surfactant solutions. The understanding of these dynamic properties is important because all the applications of surfactant in processes are invariably concerned with nonequilibrium situations. This happens in technological procedures as emulsification or foaming but also in physiological processes as breathing. Lucassen and Van den TempeP4pioneered the work on the dilational viscoelasticity of surfactant monolayers. They have elaborated a model allowing the calculation of the viscoelastic coefficient supposing that the exchanges between the bulk and the surface were governed by Fickian’s diffusion. This model, called aftward the LT model, is generally in good agreement with the measure~~
t Associated with
CNRS,Universities Paris 6 a n d 7.
* Abstract published in Advance ACSAbstracts, August 1,1994.
(1)Reynders, E.H.L., Ed. Anionic Surfactants; Surfactant Science Series; Marcel Dekker: New York, 1981;Vol. 11, p 173. (2)Djabbarah, N. F.;Wasan, D. T. Chem. Eng. Sei. 1982,37,175. ( 3 )Lucassen, J.;van den Tempel, M. Chem.Eng.Sci.1972,271,1283; J. Colloid Interface Sci. 1972,41, 491. (4)Lucassen, J.;Giles, D. J.Chem. Soc. Faraday Trans. 1,1976,71, 217.
ments done with nonionic surfactants, but large discrepancies have been found for ionic ~urfactants.~5 In order to investigate this problem, we have studied the role of the surfactant charge in a previous paper.’ Our findings indicate that the dilational properties of sodium dodecyl sulfate are highly dependent on electrostatic interactions. When salt is added to the solution, electrostatic interactions are screened and the LT model is obeyed, in contrast to the case of no added electrolyte. In this paper, we analyze the precise role of the surfactant charge and we show that the electric field generated by the charged monolayer has a large effect on the viscoelasticity. In the following we will first recall the LT model and then derive a modified model supposing that the exchanges between the bulk and surface are always governed by diffision but taking in account the electrical field created by the charged monolayer. We will then compare these models with our experiments with two ionic model surfactants: sodium dodecyl sulfate (SDS) and sodium bis(ethylhexy1) sulfosuccinate (AOT).
II. Theoretical Background 11.1. Lucassen and van den Tempel Model. The dilational properties may be characterized by a surface dilational modulus
4
1
dy-dym
‘=dlnA
d l n r dlnA
which describes the change in the surface tension y with the fractional change in the area A of a given surface element. l-’is the surfactant concentration at the interface. Due to the existence of relaxations a t the interface, the dilational modulus depends on the surface dilation rate o and can be conveniently represented by a complex number: E
= E,
+iq
(3)
where E*, the real part of the modulus, is the dilational elasticity and K = E i l o is the dilational viscosity. (5)Maru, H.C.;Wasan, D. T. Chem. Eng. Sci. 1979,34,1295. (6)Lemaire, C.;Langevin, D. Colloids Surf: 1992,65, 101. (7)Bonfillon, A,; Langevin, D. Langmuir 1993,9, 8.
0743-746319412410-2965$04.50/00 1994 American Chemical Society
Bonfillon and Langevin
2966 Langmuir, Vol. 10, No. 9, 1994 The relaxation processes that occur in the surface monolayer, such as molecular reorientation, affect the first term dy/d(ln r) of eq 2, whereas the second one, d(ln r)/ d(ln A) is related to surfactant exchanges between the bulk and surface. In the following we will suppose that the relaxation processes within the surface are highfrequency processes and that -dy/d(ln r)can be considered as frequency independent and taken equal to EO calculated from the I'variation of the equilibrium surface tension. In order to calculate the second derivative d(ln IYd(lnA), one may write for the area a t time t A = A,,
+ GAeiwt
(44
where A,, is the value of the area at equilibrium and dA the amplitude of the surface dilatation. Similar forms are taken for the bulk concentration c and for the surface concentration r:
c
r = re,+ 6reiwt
(4b)
= c e , + &e iwte ikz
(4c)
where z is the direction of the normal to the interface. The surfactant being ionic s only present in the water phase situated in the regionz 0 and a t the interface z = 0. The oil phase is lighter and occupies the region z > 0. The wavevector k is obtained by solving the diffusion equation (second Fick's law)
to note that at low frequency (typically 1 Hz) c;Z is large and then cr ~i = KO. 11.2. Modified Model. The LT model supposes that the exchanges between the bulk and surface are governed by diffusion, but it does not take into account the electric field that may be created by the monolayer when the surfactant is ionic. In order to take this into account, it is necessary to consider the role of the electric field in the surfactant flux. In the first paragraph we are going to recall briefly the Gouy-Chapman theory that will help us later to model the electrical potential, and we will then derive a diffusion model taking into account the electric field. 11.2.1. Gouy-Chapman Theory. The calculation of the electrical potential in a solution containing positive and negative ions in contact with a plane-charged surface can be found in many basic books.8 In the following the ions will be treated as point charges. As in the experiments, the surfactant will be assumed to be monovalent and anionic: the charge per molecule is then -e, e being the elementary charge. For a point at the distance z from the surface, where the electrical potential is Y, the potential energy of an ion of charge -e is -eY. The probability of finding an ion at this particular point is proportional to the Boltzmann factor e(eylkm. The z variation of the ionic concentration is given by (eYIkl7
c = cbe
where Cb is the ionic concentration far from the surface. Poisson's law allows where D is the surfactant diffusion coefficient . For z < 0, the solution to be used in eq 4c is the root with a positive The final step is the mass real part: k = -(i - 1)(0/20)~/~. conservation equation at the interface:
This leads to
to be written, where K =(21cb)l12,K - ~is the Debye length, and A is 4x times the Bjerrum length : 1 = e2/cdkr),where Ed is the dielectric constant (1= 90 A in water at 25 OC). Far from the surface, Y = 0, and at the surface Y(z = 0) = Yo < 0. This equation is integrated taking into account that the potential has to go to zero a t large z and that a t z = 0 the electric field is given by (dY/dz), = o = d / E d where u is the surface charge density: u is equal to -el- for a fully charged monolayer. This leads to
exp(%l E,
= Eo
Ei
= Eo
l+Q
1
+ 2Q + 2Q2
1
+ 2Q + 2Q2
Q
and
This model presents two interesting limits. It is clear that if D vanishes or w goes to infinity, the exchange of matter between the bulk and interface is too slow and the monolayer behaves as if it were insoluble: the interface is purely elastic. On the other hand, if UJ goes to zero, there is no resistance to compression, equilibrium can be reached quickly, and = ci = KO = 0. It is also important
+
1 tanh(eYd4kT) exp(Kz) = 1- tanh(eY,-j4kT) exp(Kz)
(10)
where YOis the potential a t the surface (z = 0) and is given by eYdkT = -lnWr2/2cb) in the approximation of large eYdkT. The variation of Y as a function of z is given in Figure 1. In the following we will use this very simple formula to model the electrical potential. It is important to notice that a t small distances and for large potentials, this treatment runs into difficulties mainly because size effects of the ions are not taken into account. Equation 10 always overestimates the potential, nevertheless giving a good approximation for it. 11.2.2. Modified Diffusion Model. The calculations that we present model the adsorption of an ionic surfactant in the presence of added salt. The flux of surfactant in the presence of an electric field is given by (8)Adamson. Physical Chemistry of Surfaces, 5th ed.; WileyInterscience: New York, Chapter V.
Viscoelasticity of Ionic Surfactant Monolayers
Langmuir, Vol. 10, No. 9, 1994 2967 and making an expansion to first order in the perturbation, we obtain
-e
2
6
54 d2dfldz2= k k l ( Z ) eXp(feq)
+ 21(cb f c,)dfch(feq)
(16~)
Combining eqs 16b and 16c and taking into account eq 9, we obtain 0
2
1
3
4
d
5
6
Figure 1. Electrical potential versus distance to the interface in debye length units. The vertical bars represent the simplified potential used in the numerical calculations. The smooth curve corresponds to the Chapman-Gouy theory for a solution of SDS, cb = 1 g/L. j = - - Dc kTgradp=--
Dc grad(kT In c - e Y ) = kT Dce -D grad c -grad Y (11) kT
+
where ,u is the chemical potential of the surfactant. With f = eY/kT, eqs 5 and 6 become (12)
We have now to solve eq 17 and to bring back the results in eq 16a. The solution can be conveniently obtained by replacing feq by a piecewise constant function. For the calculations a five-step function was found to very conveniently describe the potential (Figure 1). Within this approximation, the solution for each step is of the form
+ +
dcl(z) = dcb(Al exp(ik,z) + A 2 exp(ikp) A, exp(ikg) A, exp(ik,z)) The Poisson equation allows us to write another relation between f and c:
a2flaz2 = n(c - C+
+ c,-
- c,+)
(14)
where c+ is the bulk concentration of the positive counterions, cs+and c8- are the concentrations in, respectively, added positive and negative salt ions. Counterions and salt ions are currently much smaller than the surfactant ions and their diffusion coefficients are much larger. We will suppose that they are always at their equilibrium position in the electric field. Their density profile then obeys the Boltzmann distribution c+ = cb exp(-f)
c,+ = c, exp(-f)
c,- = c, e x p o
where Cb is as before the bulk surfactant concentration and c8the concentration in added salt, both taken far from the interface. If we now submit the interface to a sinusoidal dilatation, the area will vary as in eq 4a and r, c, and f will be given by
r = res+ dreiUt
+
c = ceq(z) 6cl(z) exp(f,,)eiUt
(154
+ cbdflz) expVeq)eiUt (15b)
+
f = feq(z) dRz)eiU'
(154
We have chosen a special form for c in order to simplify the equations; let us recall that, in the case ofthe presence of an electric field, the equilibrium values ofc and f depend on z . When reporting these expressions in eqs 12-14,
where 121, ka, k3, and k4 are the roots of the fourth-order equation associated with eq 17 (see Appendix), dCb is the amplitude of the modulation of concentration at z = -6K1, and the four constants Ai have to be determined by taking into account the boundary conditions between each couple of adjacent steps. Far from the interface (at IzI greater th'an 6 ~ - l K, - ~being the Debye length), f e s can be taken equal to zero and the modulation of concentration has to decrease to zero whenz goes to minus infinity. This allows us to take in this step the two constants Ai that have positive imaginary parts equal to 0. The two other constants are determined as a function of dcb by using eqs 16b and 16c (see Appendix). All the constants can be determined in each step by writing that dcl and its three first derivatives have to be continuous and the result can be brought back in eq 16a. This leads to
1 (1+ (acdar)exp(feq)(D/cu)(Alkl+ A&,
+ A3k, + A4k4N (18)
where f, and all the constants AI,Az, AB,andA4 are those of the step at z = 0. In order to check the influence of the number of steps, we have made a few trials with eight steps instead of five. The result of the calculations was the same within 5%. We have also verified that the model is consistant with the LT model at large salt concentration. In this case, fes goes to zero in each step and the only physical root of eq 17 goes to (1 - i)(o/2D)1/2.It is also easy to check that in each step the constant A associated with this mode is 1 (see Appendix). Equation 18 goes to the LT model (see eq 8a):
Bonfillon and Langevin
2968 Langmuir, Vol. 10, No. 9, 1994
Table 1. Calculated Elasticities and Viscosities for an SDS Monolayer at the Dodecane-Water Interface and = 0.4 g/Lfor Different Degrees of Surface Ionization a
E=-Y ‘d:n
A
1 1 (&,,/ar) (1 - i)(D/2w)l12
+
In the following, we will analyze the experiments performed at frequencies o % 1 Hz and with surfactant solutions with large enough bulk concentrations so that (War)exp(f,,)(D/o)(A&1 +A2kz+A3k3+ A d 4 )is always going to be greater than 1, and then
1 0.9 0.5
2
22 20 12 Interfacial pressure Adydcm) 5 1 45
In the case of no added salt, and assuming that the monolayer is fully ionized, the Gibbs equation takes the form 2kZT = -dy/d(ln cb) and we obtain
0.1 0
1.8 1.1
0.15
/
l0I 35
25
0.22
SDS+(O.1MA NaCI)/Dodecane
1
,
I
, ,
,
, ,
I
-3
-4
(20)
1.7 0.15
I
I
-2
,
I
I
I
I
0
-I
1
In (C)
In the above model, we have assumed that the surfactant is fully ionized: o = -eT. The model can be extended to partially ionized monolayers: u = -a&, where a is the ionization degree. For this purpose, the surface concentration r has to be increased according to Gibbs’ relation: (1 a)kZT = -dy/d(ln cb). The potential is still given by eq 10 with eYd(2kT) = - ~ i n h - ~ ( ~ a ~ T ~ / 8 cInb )eq ~ ”20, . the factor 2 is to be replaced by 1 a and new A, coefficients, corresponding to the new potential, have to be calculated. We have performed these calculations for a typical system studied in the present work: an SDS monolayer at a dodecane-water interface for Cb = 0.4 4. The results are reported in Table 1. The numbers for a = 0 are those given by the LT model. One sees that the influence of a on the viscoelasticity is considerable. Even for a = 0.9, the elasticity is lowered by 10%. The difference in elasticities for a fully ionized monolayer and a neutral one amounts to 2 orders of magnitude; the difference in viscosities is smaller: 1 order of magnitude. Similar differences were observed for other values of cb as will be discussed in the following.
+
+
111, Experimental Procedure and Data Analysis 111. 1. Materials and Experimental Procedure. Sodium dodecyl sulfate was purchased from BDH (specially pure), sodium chloride from Merck, and AOT from Fluka. All were used as received. The solutions were made with ultrapure water (MilliQ system). The oil is dodecane (Normapur Label, Prolabo) and was filtered through an aluminum column to remove surface-active impurities. In order to study the viscoelastic coefficient of the monolayers at the oil-water interface, we use a longitudinal wave apparatus. Details about this apparatus can be found el~ewhere.~” 111. 2. Measurement of the Equilibrium Surface Tension. The values for the equilibrium surface tension have been obtained either from the excited capillary waves device associated with the longitudinal waves apparatus or with a pendent drop tensiometer. They do not differ by more than 0.2 dydcm. The results of these measurements for different surfactant bulk concentrations are given in Figures 2-4. The slope of these curves allows us to calculate through Gibb’s law the value of the surface coverage r.
Figure 2. Surface pressure vs bulk concentration for SDS in solution with 0.1 mol/L NaCl at the interface between dodecane and water. The line is the fit with eq 21. Interfacial pressure (dydcm) 45
SDS/Dodecane
1
254
-2
7
’. -1.5
-1
-0.5
0
0.5
1
1.5
In (C)
Figure 3. Surface pressure vs bulk concentration for SDS at the interface between dodecane and water. The line is the fit with eq 21. No minimum close to the cmc is observed in the surface tension curves, which is indicative of the purity of the samples. Let us recall that the study of SDS is easier at the oil-water interface than at the air-water one: a t the air-water interface, due to the hydrolysis of SDS into water-insoluble dodecanol,the measured properties (surface tension, viscoelasticity) are those of mixed SDS/ dodecanol monolayers. At alkane-water interfaces, dodecanol no longer accumulates because it is very soluble in the oil phase, and the SDSmonolayer is thus unaffected by surfactant hydroly~is.~ In the case of the AOT the surface tension continues to decrease after the cmc. This may be due to the fact that the concentration range for the micellar phase is small. 111. 3. Measurements of the Viscoelastic Coefficients. Figures 5-7 show the results of the measurements of the viscoelastic properties. In order to compare the experiments with the model, we need to relate T and c . For this purpose, we have used the Langmuir equation of state: (9) Joos, P.;Vollhardt, 525.
D.;Vermeulen, M. Langmuir
1990, 6, 524-
Viscoelasticity of Ionic Surfactant Monolayers Interfacial pressure (dyn/cm) 50 45
Langmuir, Vol. 10, No. 9,1994 2969 E (dydcm)
SDS I Dodecane 0.6 Hz
AOT/Dodecane
1
i
/
20 - , , , , , , , ,
/ , , ,
, , ,
,,,,
Modified LT model taking into account the electrical field
, , , , , , , ,
0
0.5
I Concentration in gil
Kw(dyn/cm)
1.5
2
SDS / Dodecane 0.6Hz
Modified LT model taking into account the electrical field
0
o~,,,,l,,,,,,,,,l,,,,,,~,,l,,,,, 0
0.05
0.1
0.15
0.2
0.25
0.3
Concentration in (gil) K W (dydcm)
\
SDS 0.1 M NaCl I Dodecane 0.6 hz
'*I b
10
0.5
1
Concentration in gil
1.5
2
Figure 6. (a, top) Dilational elastic modulus vs bulk concentration for SDS at the interface between dodecane and water. The frequency of the compression waves is 0.6 Hz.(b, bottom) Dilational surface viscosity times w vs bulk concentration for SDS at the interface between dodecane and water. The frequency of the compression waves is 0.6 Hz.
if the surfactant is nonionic and 2 if the surfactant is fully ionized. The parameters a and r, are determined by fitting the surface tension data curves with eq 21. In the case of aqueous SDS solutions containing sodium g/L, and r, = 3.75 x 10-lo chloride, n = 1, a = 1.4 mol/cm2. The value for the diffusion coefficient D is 6 x cm2/s.l0 Equation 21 leads to C
dT
r2 a r,
q , = n k Z T w - and - = - a dc c2 Concentration in (911)
Figure 5. (a, top) Dilational elastic modulus vs bulk concentration for SDS in solution with 0.1 mom NaCl at the interface
between dodecane and water. The line is the prediction from the LT model. The frequency of the compression waves is 0.6 Hz. (b, bottom) Dilational surface viscosity times w vs bulk concentration for SDS in solution with 0.1 mom NaCl at the interface between dodecaneand water. The line is the prediction from the LT model. The frequency of the compression waves is 0.6 Hz.
(21) where I7 is the surface pressure (II= y - yowlyow being the oil-water interfacial tension in the absence of surfactant), r, is the adsorption limit a is the Szykowski concentration, and n is a parameter which is equal to 1
The viscoelastic coefficients calculated (Figure 5) are in good agreement with the LT model (eq 8). In the case of SDS and AOT without salt (Figures 6 and 71, the LT model is no longer able to fit the data. This can be seen even without going through the calculations: in the frequency range probed here, the LT model leads to ci = cr, and except for the solutions with salt, the experiments do not fullfil1 this relationship (without salt, c?is currently 10 times larger than ci). We can see on the other hand that when the electrostatic term is properly taken into account, the improved model is in very good agreement with the experiments for both SDS and AOT in the absence of salt. (The two diffusion coefficients D are taken equal to 6 x cm2/s.) We have to stress here that the model does not introduce any adjustable parameter. Small deviations are found for low surfactant (10) Lindman, B.; Puyal, M. C.; Kamenka, N.; RymdBn, R; Stilbs, P.
J.Phys. Chem. 1984,88,5048.
Bonfillon and Langevin
2970 Langmuir, Vol. 10, No. 9, 1994 AOT / Dodecane 0.6 Hz
Modified LT model taking into account 15
5 I , ,, , , , ,,,
0 KO
10
0.1
(dydcm)
,
,
(
,",",",
0.3 Concentration in fl 0.2
0.4
0.5
AOT I Dodecane 0.6 Hz
--I
Modified LT model taking into account the electrical field
0
0.1
0.2 0.3 Concentration in gll
0.4
0.5
Figure 7. (a, top) Dilational elastic modulus vs bulk concentration for AOT at the interface between dodecane and water. The frequency of the compression waves is 0.6 Hz. (b, bottom) Dilational surface viscosity times o vs bulk concentration for AOT at the interface between dodecane and water. The frequency of the compression waves is 0.6 Hz.
concentrations: these deviations may be explained by the fact that for the high potentials the Gouy-Chapman theory runs unto difficulties and overestimates the potential. This can result in a too large elasticity and a too small viscosity. This leads us to the conclusion that, in these systems, the exchanges of surfactant between surface and bulk are governed by diffusionand that there is no need to introduce intralayer relaxation mechanisms to describe the viscoelastic response to periodic surface dilatation. The role of the electric field can also be viewed as introducing an adsorptioddesorption barrier, which can be predicted quite accurately by using the simplified electrostatic model described here. The model does not introduce any adjustable parameter, and assumes that all the surfactant molecules are dissociated at the interface. In view of the very good agreement with the experiments, it can be said that this is an argument against counterion association at the interface, at least in the absence of salt: a = 1. Surface potential determinations on ionic surfactant solutions support this conclusion,"J2 as well as recent comparisons between the measured area per molecule by neutron reflectivity and the one deduced from surface tension data with n = 1 a = 2 in the Gibbs equation for tetradecyltrimethylammonium bromide.13 This is also suggested by a simple calculation made by
+
(11)Pethica, B. A.; Few, A. V. Discuss. Faraday SOC.1964,258,lS. (12) Fromherz, P.; Masters, B. Biochim. Biophys. Acta, 1974,356 270.
Alexander et al.14 They assume that surfactant counterions will associate when their chemical potential at the surface will be equal to their chemical potential in the bulk. In the bulk, the chemical potential of the counterions is mainly associated with configurational entropy: PI, = KT In(#) (4 being the counterion volume fraction). At the surface, their chemical potential is dominated by electrostatic attraction from the surfactant ions ,use1 = eYs. The minimum value ofp:l is then on the order ofpb,leading to Ysmin x (kTle)In(#). For bulk surfactant concentrations on the order of 1 mM and counterions of molecular weight on the order of 50,# x 5 x and Y, min x -10kTle x -250 mV. Taking again In(#) - 10, this leads to an are? per surface charge on the order of 20 Az (with d = 90 A in water and K - ~x 100 A for c x 1 d). In our experiment, the area per surfactant limit is on the order of 60 A2 for SDS. This explains why counterion association is unlikely here. Let us note that it is frequently mentioned in the literature that there is a certain degree of counterion association in systems of ionic surfactants. However, this can be partly due to the use of linearized PoissonBoltzmann equations such as in the Debye-Huckel theory. The calculated potentials from linearized and nonlinearized (the Gouy-Chapman theory used here) equations are similar far from the surface, but the Debye-Huckel potential is smaller for z = 0. As a consequence, the apparent surface charge is smaller. We have used here the Gouy-Chapman theory, and we can deduce the real degree of counterion association ( 1- a),which would arise from geometrical intercalation of surfactant ions and counterions. Our sensitivity to the degree of association is also due to the fact that we are measuring a surface property, the viscoelasticity.
IV. Conclusion We have measured the low-frequency dilational viscoelasticity for monolayers of surfactants at the interface between water and dodecane. We have compared our results with models in which it is assumed that the exchanges between the bulk and surface are diffusion controlled. These models are in good agreement with all the experiments. This demonstrates that the diffusion processes are responsible for the main part of the viscoelastic response of many soluble surfactant monolayers, and that the contribution of the intramonolayer relaxations are insignificant at low frequencies. In the case of charged monolayers, we have worked out a diffusion model for the viscoelasticity, which takes explicitly into account the role of the surface potential in the diffusion process near the surface. The model has no adjustable parameters, and the agreement with the experimental data supports the assumption that the monolayers are fully ionized. The role of the electric field created by the charged monolayer is specially important on the surface elasticity, which is typically 2 orders of magnitude larger than the value predicted in the absence of charge (LT model), the surface viscosity being typically 1 order of magnitude larger.
Acknowledgment. We are grateful to A. Asnacios who performed the experiments on AOT. This work was supported in part by RhBne-Poulenc. (13)Lu, J. R.; Thomas, R. K.; Aveyard, R.; Binks, B. P.; Cooper, P.; Fletcher, P. D. I.; Sokolowski, A.; Penfold, J. J.Phys. Chem. 1992,96, 10971. (14)Alexander, S.; Chaikin, P. M.; Grant, P.; Morales, G. J.; Pincus, P. J. Chem. Phys. 1984,80, 57766.
Langmuir, Vol. 10,No.9,1994 2971
Viscoelasticity of Ionic Surfactant Monolayers
V. Appendix
de is the debye length
In this appendix we will present the details of the calculation. The fourth order equation associated with eq 17 for each step is
&b
exp(-ik16de) - k;A,
= -$-k12A, D
exp(-ik16de)
+ (:k
CbZ”
exp(-ik16de)
The roots of this equation can be found by using a mathematical software, but approached values can be estimated analytically due to the values of the constants. This leads to
-i sh(fe42) f i(-8 sh(f,,)
+ 2 ch(f,,) + 2Y2
2
(2kb
+
exp(-feq) 2C, ch(f,,) Cs) ch(f,,) - 4(Cb + Cs) ShCf,,)
)”(1 - i)
In the limit of large quantities of added salt, k1,z is not a physical root and k3,4 goes to f ( ~ / 2 D ) ~-~i).( l In order to determine the constants Ai, we write eqs 16b and 16c in a region far from the interface where the equilibrium potential can be taken equal to zero. In this region, the modulation of concentrations has to decrease to zero at large z; the two constants associated with the roots with positive imaginary parts (z is negative) must be equal to zero. dcl is then equal to
dc,(z) = 6cb(A, exp(ik,z)
+ A, exp(ikp))
Reporting in 16b and 16c we obtain for z = -6de, where
+ k?)A,6cb x
+ A, exp(-ik36de))
(22)
The solution of this system gives us the value of the constants in the region far from the interface. One finds that A1 is close to zero, and A3 = 2 in this first step in the case ofno salt added andAs equals 1when a large quantity of salt is added. The values of the other constants in each step are then determined by writing the continuity of the function and of its three first derivatives. This leads us to solve in each step the following system
+
+
A, exp(ik,z) +A, exp(ikp) A, exp(ikp) A, exp(ik,z) = B , exp(ik’,z) B, exp(ik’p) B, exp(ik’p) B, exp(ik’,z)
+
Cb
x
+
+
+
+
+
A,k, exp(ik,z) A,k, exp(ikp) A,k, exp(ikp) A4k4exp(ik,z) = B,k, exp(ik’,z) B,k, exp(ik’.g) B3k3exp(ik’p) B4k4exp(ik’,z) A,kI2 exp(ik,z) B,k;
+
+
+
+ A2kz2exp(ikp) + A,k3’ exp(ikp) + +
A,k; exp(ik,z) = Blk12exp(ik’,z) exp(ik’,z) B,k; exp(ik’+) B,k; exp(ik’,z)
+
+
+
+
A,k13 exp(ik,z) A,k; exp(ikg) A,k,, exp(ikp) A&: exp(ik,z) = B,k13 exp(ik’,z)
B,k; exp(ik’9)
+ B3k,, exp(ik’p) + B,k:
+
+
exp(ik’,z)
where Bi are the constants of the precedent step, k‘i the wave vector associated with Bi,Ai the constants to be determined, and z the position of the frontier between the two steps. All these systems have been solved with the help of the mathematical software “Mathematica”.
