Effect of the counterions on the surface properties of surfactant solutions: kinetics of the surface tension and surface potential

Colloids and Surfaces A: Physicochemical and Engineering Aspects 163 (2000) 177 – 190 www.elsevier.nl/locate/colsurfa

Effect of the counterions on the surface properties of surfactant solutions: kinetics of the surface tension and surface potential I.U. Vakarelski, C.D. Dushkin * Laboratory of Thermodynamics and Physico-Chemical Hydrodynamics, Faculty of Chemistry, Uni6ersity of Sofia, 1126 Sofia, Bulgaria Received 6 January 1999; accepted 18 June 1999

Abstract The kinetics of adsorption from anionic surfactant solutions, containing counterions of different valence, is investigated by measuring the surface tension as a function of time and the surface potential. The surfactant used is sodium dodecyl polyoxiethylene-2 sulfate; the electrolytes are chlorides of sodium, calcium and aluminium at a constant total ionic strength of 0.024 M. A strong effect of the counterions on the adsorption kinetics is found. The increasing counterion valence leads to a faster relaxation of the surface tension toward its equilibrium value. A theoretical model for diffusion controlled adsorption of charged surface active ions in the presence of counterions is proposed. Asymptotic equations for the surface tension as a function of time are derived accounting for the contribution of micelles present in the solution. The theoretical predictions agree quantitatively with the experimental data. The calculated model parameters, characteristic diffusion and micellization time-constants are of reasonable magnitude. The results from surface tension experiments are in accordance with the surface potential measurements, which give an increase of the surfactant adsorption in the presence of multivalent ions. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Kinetic surface tension; Surface potential; Multivalent ions; Sodium dodecyl polyoxyethylene-2 sulphate; Micelles

1. Introduction

* Corresponding author. Present address: Non-Equilibrium Laboratory, Yokohama Research Center, Mitsubishi Chemical Corporation, 1000 Kamoshida-cho, Aoba-ku, Yokohama 227-8502, Japan. Tel.: +81-45-9634364; fax: + 81-459634366. E-mail address: [email protected] (C.D. Dushkin)

Here we study the kinetics of adsorption from a (water) solution of an anionic surfactant, which contains also an electrolyte providing counterions of different valence. It is well established that such ions strongly affect the equilibrium surface properties of the solution [1,2]. For example, the surface tension of solutions of sodium dodecyl

0927-7757/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 9 9 ) 0 0 3 0 6 - 4

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sulfate (SDS) has been appreciably lowered by very small amounts of AlCl3 [1] or CaCl2 [2] (about 10 − 4 M or less). Similarly, the critical micelle concentration (CMC) of the same solutions has been decreased. Various metal ions of larger ionic strength (up to 0.02 M) exist in the tap water, which can affect the applications of a surfactant in the detergency. Since important for the practical processes is the time variation of the surface properties, we study here the effect of the counterion type on the surface tension kinetics. The model surfactant used is sodium dodecyl polyoxyethylene-2 sulfate (SDE2S), which has better resistance against precipitation by metal ions than SDS. For this reason ethoxylated surfactants are widely included in detergent compositions which are commonly dosed at concentrations exceeding CMC. In such a case the micelles act both as the material sources supplying the interface with surfactant molecules (monomers) and as the negatively charged particles adsorbing part of the counterions in the solution. The last effect seems negligible at SDE2S concentration close to CMC. Experimentally we found that the addition of electrolyte makes the relaxation of the surface tension s(t) faster. The respective data plots suggest that the adsorption kinetics is diffusion controlled at large time t. Based on this observation we proposed asymptotic expressions for s(t) valid below and above CMC. For this purpose the original approach of Dukhin and Shilov [3] is used to solve the problem for diffusion in charged solutions. However, our equations for the adsorption kinetics differs from later results published in Refs. [4–8]. We assume that, at small deviations from equilibrium, the subsurface concentrations of species can be approximated by their values derived from the concentration profiles in the diffusion layer rather than by the values in the electric double layer. This assumption allows calculating also, the surface potential as a function of time. The subsurface concentrations are then related by an adsorption isotherm of a general type depending on both the surfactant and the electrolyte concentration. The contribution of micelles to the diffusion of charged species is accounted for by assuming a pseudo-first-order reaction mechanism for the micellization kinetics

[9,10]. Using the theoretical expressions we calculated the characteristic times of diffusion and micellization, which decrease with increasing surfactant concentration from hundreds of seconds to several seconds. At a constant surfactant concentration these time constants also alter depending on the type of counterion. To obtain the adsorption of SDE2S from solutions with added salts the surface potential (DVpotential) is measured. These measurements suggested a new empirical isotherm relating the adsorption with the electrolyte concentration. Although there was no appreciable effect of the counterion valence on the surface potential, the surfactant adsorption increases because the multivalent ions decrease that part of the potential coming from the electric double layer. This fact explains the surface tension decrease observed experimentally.

2. Materials and methods

2.1. Materials The surfactant, sodium dodecyl polyoxiethylene-2 sulphate (SDE2S) is a commercial product purchased from Albright and Wilson. Counterions of different valence are added as NaCl, CaCl2 and AlCl3, so that the total ionic strength of the solution stays constant, 0.024 M. Although the multivalent ions could change the CMC of SDE2S as well, it was not determined for all compositions used. For example, the CMC of purified SDE2S without added salt is 2.9×10 − 3 mol l − 1 [11]. The CMC in the presence of 0.024 M NaCl was found to be about 1.33× 10 − 4 mol l − 1. Two of our solutions (4× 10 − 5 and 6× 10 − 5 mol l − 1) are below this value, whereas the other two solutions (2× 10 − 4 and 4 × 10 − 4) are above it.

2.2. Surface tension The surface tension was measured as a function of time using the Wilhelmy plate method (Kru¨ss Tensiometer K10T). Soon after the surfactant solution had been poured in the vessel, a platinum

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plate was dipped into and pulled out of the solution to counterbalance the surface tension force. Such alignment did not allow recording surface tension variations at an initial time of less than 1–2 min. For this period the surface tension, even for a slow adsorbing surfactant like SDE2S, dropped appreciably from its initial value s0, especially at a high surfactant concentration. Although no sweeping of the water/air interface was applied prior the measurement, the way the surfactant solution has been loaded assures always a fresh interface at the beginning of adsorption. The temperature was kept 30°C throughout the experiments.

2.3. DV-potential The DV-potential was measured by the vibrating plate method [12] as the difference between the surface potential of the surfactant solution and the reference potential of pure water. It was not possible to record the potential earlier than 5 min after a fresh interface had been created. This time delay, together with fluctuations in the detected signal, did not allow measuring a reliable kinetics of the surface potential as in Refs. [13,14]. For this reason we present below only equilibrium values for the DV-potential.

179

Fig. 1. The surface tension s of SDE2S solutions as a function of time t. The surfactant concentration is 4 × 10 − 5 mol l − 1, without added salt or in the presence of various multivalent ions (total ionic strength of the solution 0.024 M). The solid lines are theoretical fits of the data by Eq. (2).

3. Results

tained are typically 0.5–1 dyn cm − 1 less than the last experimental point in the plot. The equilibrium surface tension of pure surfactant is always higher than the respective values with added electrolyte (Fig. 5). For this reason the relaxation curves for SDE2S lie above the respective curves for solutions with electrolyte thus implying on a faster surfactant adsorption in the latter case. As seen from the figures, there is a strong effect of the counterion type on the surface tension kinetics. The Ca2 + or Al3 + ions present in the

The results for s(t) are shown in Figs. 1–4. Each figure represents data at a fixed SDE2S concentration and different type of counterions (Na+, Ca2 + or Al3 + ) of a constant ionic strength. For comparison are given also data obtained without added salt. The solid curves in these figures are numerical fits drawn by the theoretical model for the kinetics of adsorption considered below. Fig. 5 summarises data for the equilibrium surface tension of solutions. They have been found from the plot of experimental values for s(t) versus 1/ t, which gives straight line especially at large times, i.e. at small 1/ t. The equilibrium value s is the intercept of a straight line with the ordinate axis. The data points thus ob-

Fig. 2. Kinetics of the surface tension of SDE2S solutions of concentration 6 ×10 − 5 mol l − 1. The other notations are the same as in Fig. 1.

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Fig. 3. Time dependence of the surface tension of SDE2S solutions of concentration 2 ×10 − 4 mol l − 1. The solid lines represent fits by Eq. (2) (without salt and with NaCl) or by Eq. (7) for the other two curves.

solution significantly lower both the equilibrium surface tension and the relaxation time compared to the solutions with only Na+ ions. The most pronounced effects are shown in Figs. 1 and 2, where the calcium and aluminium ions decrease the equilibrium surface tension with 9–5 dyn cm − 1 compared to the value with sodium ions. Above CMC the difference in s is about 2 dyn cm − 1 (Figs. 3 and 4) while the surface tension relaxation is much faster. The Al3 + ions caused precipitation at these concentrations. After 6–8 days, however, the precipitated solutions recov-

Fig. 4. Kinetics of the surface tension of SDE2S solutions of concentration 4 × 10 − 4 mol l − 1. The empty figures with AlCl3 represent data for a precipitated solution. The solid curves are drawn by Eq. (7), except the case without added salt where Eq. (2) is used.

Fig. 5. The equilibrium surface tension of the solutions represented in Figs. 1 – 4.

ered becoming clear again. Nevertheless, the surface tension relaxation of precipitated solution (empty figures in Fig. 4) differs from the one of clarified solution (solid figures). This is an evidence for certain irreversibility of the precipitation–recovery process. Two types of measurements of DV-potential have been carried out. First, at constant surfactant concentration of 4× 10 − 5 M and variable amount of NaCl providing solutions of different ionic strength (electrolyte concentration ce). In this case the DV-potential increases with increasing of ce (Fig. 6) which is in accordance with the surface electrostatics (see below). Second, at fixed SDE2S concentration and ionic strength in the presence of different counterions. For both sur-

Fig. 6. The DV-potential of SDE2S solutions as a function of the electrolyte concentration ce. The solid figures represent data collected at a concentration below CMC, while the empty figures — above CMC.

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factant concentrations, below CMC, 4× 10 − 5 M, and above CMC, 4 × 10 − 4 M, the highest value of potential is the one with calcium ions whereas the lowest one is the value with aluminium ions. This effect is more pronounced above CMC where the micelles adsorb on their surface part of the counterions.

4. Discussion To interpret the experimental data from Figs. 1–4 we derived equations describing the dependence of the surface tension on time and accounting for the contribution of ions to the adsorption kinetics. The observation that s(t) follows the line 1/ t at long time is a clear indication for diffusion control of the adsorption kinetics, both below and above CMC. First we consider the simpler case of solutions without micelles, which is then generalised for the effect of micellization kinetics. Our treatment is based on the model of Dukhin and Shilov [3] for the diffusion of two types of charged species. In contrast to their mathematics, however, a rigorous asymptotic approach is used here, which allowed deriving of new equations for the surfactant concentration in the double electric layer and for the electric potential. The asymptotic equations are then used to describe the adsorption kinetics. The idea of asymptotic approach is to split the liquid adjacent to the adsorbed layer into two regions of different thickness (Fig. 7). The inner region is the electric double layer where the electric potential 8 decays from its surface value, 80, to zero in the bulk of solution. This layer is of a characteristic thickness k − 1 where k is the reverse Debye length. The outer region is the diffusion layer created by concentration gradients provoked by the disturbance of the adsorption layer. This region is of a characteristic length dD as known from the usual adsorption kinetics (see e.g. Ref. [10]). The definitions of k − 1 and dD, given below, depend on the particular system. What is known, however, is that dD k − 1, which allows one defining the small parameter

Fig. 7. Model of the diffusion and the adsorption of anionic surfactant molecules on a charged interface.

a= (kdD) − 2  1

(1)

Expanding the unknown functions in series of a one can seek for asymptotic solutions of the respective diffusion equations. Based on them we derived equations for the bulk concentrations of the surfactant ions and counterions, for the adsorption and surface tension and for the electric potential (see Appendix A). The final result for the surface tension is s(t)=s¯ + (s0 − s¯ )E( t/tD)

(2)

Here s0 = s(0); tD = d 2D/D

(3)

is the usual characteristic diffusion time [10]; dD is the diffusion length, which turns out to be dD =



#G( #G( 1 z− + z+ z+ + z− #c¯ + #c¯ −



(4)

D is an apparent diffusivity given as [3] D=

D + D − (z + + z − ) z+D+ + z−D−

(5)

In the case of anionic surfactant c − is the concentration of the surfactant ion of charge z − , while c + and z + are the respective quantities of the counterion; G is the adsorption. Bar above the letter denotes equilibrium quantity. Eq. (2) is derived for a symmetrical electrolyte. It resembles the Sutherland’s equation [15] (see also [10,16]), however the parameters D and dD are of different representation. At a large value of

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Table 1 Characteristic diffusion time tD and characteristic micellization time tMa (in s) calculated for SDE2S solutions without salt or in presence of different multivalent ions of total ionic strength 0.024 M SDE2S (M)

No salt

NaCl (0.024 M)

CaCl2 (0.008 M)

AlCl3 (0.004 M)

4×10(5 6×10(5 2×10(4 4×10(4

213.0 60.6 54.2 13.8

167.3 23.1 9.3 9.3

56.2 16.2 8.7 8.0

27.3 10.5 1.1 3.7

a

(16.2)

(7.0) (2.0)

(5.1) (3.7)

Given in parenthesis.

the argument (y = t/tD 1), the function on the right hand side of Eq. (2) can be approximated as E(y) $1/ py [17], therefore leading to s− s¯ $ s0 −s¯

'

tD pt

(6)

Assuming a surface equation of state in the form s0 − s =RTG [18,19] and a diffusion length dD $ G/c¯ s [20], Eq. (6) transforms in the known approximation s −s $ (RTG2/c¯ s) p/Dt [21]. The last equation is valid for 1:1 ionic surfactant without electrolyte. In the case of indifferent electrolyte added at a concentration c¯ e, the right hand side of the same equation should be multiplied by h=1 + c¯ s/(c¯ s +c¯ e) [22], where c¯ s = c¯ − is the surfactant concentration. In our case, c¯ e  c¯ s, one has h$ 1, whereas for a non-ionic surfactant h= 2. For micellar solutions we derive in Appendix B a generalized counterpart of Eq. (2): s(t)= s¯ +(s¯ −s0)

 ' 1+ G 2



1 − t e tM (1 + G)E 2G

 ' n

t 1 −G −(1 −G)E tD 2

t tD

(7)

where G= 1 +4tD/tM, with tM being a characteristic time of demicellization. The form of Eq. (7) resembles that known for the micelles of a non-ionic surfactant [10], but the meaning of parameters is somewhat different. A discussion pointing out the difference between Eq. (7) and other equations available in the literature [23,24] is given in Ref. [25]. In the fit of experimental data by Eqs. (2) and (7) we used for s0 the surface tension of pure water. It should correspond to the one of a

newly created interface after loading the solution in the tensiometer (time t= 0), which is confirmed also by the fits. The applied computational procedure comprises a minimization of the difference between calculated and experimental data points using a modification of the least square method [20,26]. In the case of surfactant with added multivalent ions, shown in Figs. 1–4, such a difference is negligible, which assures the good quality of fitting. This finding supports the applicability of the theoretical model, derived assuming small deviations from equilibrium, to treat the adsorption kinetics at conditions, which might not obey this assumption. Although in most of our measurements is captured the longtime decay, where the values of s(t) are close to the equilibrium one: (s−s)/s  0.1–0.3 (cf. Figs. 1–5). The same fact has been established already for experiments with more drastic perturbation of the interface such as the maximum bubble pressure method [20,25,26]. Table 1 lists the results for the time constants tD and tM stemming from the fit of experimental data by Eqs. (2) and (7). The characteristic diffusion time tD is of the order of a few to hundred of seconds. Qualitatively, tD exhibits two trends seen in Figs. 1–4. First, with increasing of the surfactant concentration at a fixed electrolyte, tD decreases implying on a smaller length dD, i.e. faster adsorption (if D does not vary so much [20]). For the micellar solutions (the last two points in columns 3–5 of Table 1), tD does not change appreciably which is in accord with the constancy of free monomers concentration above CMC [26]. Second, tD significantly decreases with increasing of the electrolyte valence. The calculated micellization time tM in Table 1

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Table 2 Parameters for various counterions Counterion

Ion radius (nm)a

D+ (cm2 s−1)b

D (cm2 s−1)c

dD (cm)d

Na+ Ca2+ Al3+

0.095 0.099 0.050

2.34×10−5 2.24×10−5 4.44×10−5

9.2×10−6 6.7×10−6 6.4×10−6

0.039 0.019 0.013

a

Ref. [30]. Calculated by the Stokes–Einstein equation. c Calculated by Eq. (5) at D− = 5×10−6 cm2 s−1 [31]. d Values, obtained from the data in Table 1 at 4×10−5 M SDE2S. b

testifies lowering of CMC in the presence of multivalent ions (cf. Fig. 5). For example, the solutions of 2×10 − 4 M SDE2S contain micelles (either pure or with NaCl), although these with multivalent electrolytes do not. The typical values for tM of several seconds seem decreasing with the counterion valence. The diffusion time constant tD is an effective quantity [20,26], related to the diffusivity and diffusion length (equilibrium adsorption). Here we calculate first the diffusivity of counterions D + and the overall diffusivity D by Eq. (5) (Table 2) which then gives the diffusion length dD. The values of dD decrease with increasing of the ion valence (fixed surfactant concentration of 4× 10 − 5 mol l − 1). This means that the multivalent ions easily expel surfactant from the solution thus decreasing the thickness of diffusion layer. Mathematically it can be understood by considering the definition of dD. The derivative #G( /#c¯ + #G( /#c¯e, estimated from the adsorption isotherm (Eq. (8)), is 8.5× 10 − 7 cm — much smaller than the value of the first derivative in Eq. (4). Nevertheless, to calculate dD one needs the dependence of G on cs, unknown for our systems. Therefore, from the apparent decrease of dD in Table 2 one can only conclude that the slope of the adsorption isotherm #G( /#c¯s is decreasing at increasing counterion valence. Note that this conclusion is valid at a number of simplifications, which have not been verified in our study. Concerning the micellization time tM, it is of the right order of magnitude compared to the values for similar surfactants reported in literature

[26,27]. It corresponds to the so-called slow relaxation time associated with the disintegration of a micelle [10,28]. The results of surface tension measurements are in agreement with the data for the surface potential in Fig. 6. With increasing of the electrolyte concentration ce the DV-potential increases too, which leads to an increase of the adsorbed amount of surfactant according to the empirical adsorption isotherm



G B c = 1+ ln e c*e A G

(8)

where A and B are constants (see the derivation in Appendix C). Subsequently, the surface tension of the solutions containing salt decreases compared to the values without added electrolyte. Below CMC there is no appreciable effect of the counterion valence on the surface potential. This observation is consistent with other results for both anionic and cationic surfactants [29], which means that the electrostatic interactions do not measurably affect the ideality of the surface layer. Above CMC, however, the calcium ions increase the surface potential compared to the sodium ions, whereas the aluminium ions decrease it. This suggests a stronger ability of the SDE2S micelles to bind aluminium ions than calcium ions. In both cases, however, the surfactant adsorption increases, as the multivalent ions are known to decrease the surface potential [30,31]. The latter is in agreement with the vast decrease of surface tension caused by the calcium and aluminium ions compared to the sodium ions.

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5. Concluding remarks We experimentally studied the kinetics of adsorption of sodium dodecyl polyoxiethylene-2 sulfate in the presence of counterions of different valence and constant ionic strength (0.024 M). The surface tension of the solutions has been measured as a function of time and, independently, the surface potential. The results of measurements showed that the counterions significantly affect the kinetics of surface tension compared to the case of surfactant with no salt added. The aluminium chloride causes the surface tension to relax in the fastest way, followed by the calcium chloride and then by the sodium chloride. This qualitatively means that counterions of higher valence expel more effectively the surfactant molecules out from the solution in dynamic conditions. To quantitatively explain these results, a theoretical model for a diffusion-controlled adsorption of charged surface-active ions in the presence of counterions is proposed. The derived asymptotic expressions for the surface tension account also for the micellization kinetics. The model parameters are the characteristic diffusion and micellization time constants. Their values calculated from the experimental data are of a reasonable magnitude. Based on the surface potential measurements, a new empirical adsorption isotherm is formulated relating the adsorbed amount with the electrolyte concentration. There is no appreciable effect of the counterion valence on the surface potential, especially below the critical micelle concentration. However, the surfactant adsorption increases because the multivalent ions decrease the surface potential, in correspondence with the respective decrease of the surface tension.

Acknowledgements I.U. Vakarelski is indebted to Dr Stoyan Russev (University of Sofia) for the support in carrying out of the surface potential measurements. The valuable suggestions of late Professor Paul Joos (University of Antwerp) on improving the

manuscript are greatly acknowledged. This work was supported by a grant of Colgate-Palmolive Company.

Appendix A. Kinetics of the surface tension below CMC The equations describing the diffusion of two type of ions are:

   

#2c + F # #8 #c + = D+ + D+z+ c+ #t #x 2 RT #x #x

(A.1)

#2c # #8 F #c − = D− − − D−z− c− 2 #t #x #x #x RT

(A.2)

The first terms on the right hand sides of Eqs. (A.1) and (A.2) represent the Fickian diffusion due to concentration gradients in the solution. The second terms give the electrodiffusion due to the migration of charged species in an external electric field of a potential 8. The space distribution of electrical potential is given by the Poisson equation: # 28 4pF =− (z + c + − z − c − ) o #x 2

(A.3)

Eqs. (A.1), (A.2) and (A.3) form a non-linear set for three unknown functions: the concentration of surface-active ions c − (x, t), the concentration of counterions c + (x, t) and the potential 8(x, t). For analytical solution, one needs further simplification based on the physical dimensions of the system. The characteristic scales can be determined by considering two simple examples: A.1. Equilibrium distribution of ions in the 6icinity of interface Here we briefly reminded of the Gouy–Chapman theory for the electrical double layer (see e.g. Refs. [18,30,32]). The respective expressions will be referred later at considering the case of a non-equilibrium charge distribution. Using the Boltzmann distribution of ions



c 9 = c¯ 9 exp 

z 9 F8 RT



(A.4)

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with the ion concentrations obeying the electroneutrality condition (A.5)

z + c¯ + − z − c¯ − =0

far from the interface, one can resolve Eq. (A.3) with respect to 8. Let us assume a small electric potential, z 9 F8/RT  1, which means z 9 8 26.1 mV (at absolute temperature T = 303 K, Faraday constant F = 96484.56 C mol − 1 and universal gas constant R = 8.314 ×107 erg mol − 1 K − 1). For small 8, Eq. (A.3) simplifies to #28 = k 28 #x 2

(A.6)

where k is the reverse Debye length k2=

4pF 2 2 2 (z + c¯ + +z − c¯ − ) oRT

(A.7)

The solution of Eq. (A.6) is the well-known expression [32] 8(x)=80e − kx A.2. Non-equilibrium distribution of nonionic surfactant near an interface The diffusion of nonionic surfactant is described by the equation #c #2c =D 2 #x #t

(A.8)

where D is the diffusivity of surfactant molecules. The solution of Eq. (A.8) at small deviations from equilibrium ( c − c¯ c¯ ) gives for the surfactant distribution [10] c(x, t)=c¯ +(c0 −c¯ )e

x2 − 4Dt

E



x

2 Dt

+

' t tD

185

X= x/d, where d= dD states for the outer region and d= k − 1 for the inner region shown in Fig. 7. Time variable Y= t/tD. Concentrations c 9 = c¯ 9 (1+j 9 ), where j 9 are the relative deviations from the equilibrium values. Electric potential F =8/8*, where 8*= RT/F(z + + z − ) is a characteristic potential. Hence, the scaled equations become:

 

d 2 D #j + dD D + #Y



n

(A.10)



n

(A.11)

#2j + # z+ #F + (1+j + ) #X 2 z + + z − #X #X 2 d D #j − dD D − #Y

=

 

# #2j − z− #F − (1+j − ) #X 2 z + + z − #X #X 2 1 #F = − k 2(j + − j − ) d 2 #X 2 =

(A.12)

where D is given by Eq. (5). There are two asymptotic solutions of the equation set depending on the space scale d: inner, valid in the electric double layer and outer, for the diffusion layer. First we consider the inner solution assuming d= k − 1. Eqs. (A.10), (A.11) and (A.12) then read



n

D #j + #2j + # z+ #F = + (1+j + ) D + #Y #X 2 z + + z − #X #X (A.13)

a



n

D #j − #2j − z− # #F − (1+j − ) = #X D − #Y #X 2 z + + z − #X (A.14)

a

#2F = − (j + − j − ) #X 2

(A.15)

A.3. Inner asymptotic solution

where a is the small parameter defined by Eq. (1). Let us represent the unknown functions as the (0) (1) asymptotic series j 9 = j 9 + aj 9 + O(a 2) and (0) (1) 2 F= F + aF + O(a ). Introducing the latter in A.13–A.15 one derives the zero-order approximation

In view of points A.1 and A.2, the following dimensionless variables turned out to be appropriate for asymptotically solving the diffusion problem for an ionic surfactant. Space variable

(0) # 2j + z+ + 2 #X z+ + z− (0) # 2j − z− − 2 #X z+ + z−

(A.9)

2

where E(y)=ey erfc(y), tD is defined by Eq. (3) and dD by Eq. (4).

 

n n

# #F(0) (0) (1+j + ) =0 #X #X # #F(0) (0) (1+j − ) =0 #X #X

(A.16) (A.17)

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#2F(0) (0) (0) = − (j + −j − ) #X 2

(A.18)

Neglecting the thickness of the Stern layer, i.e. d:0 (Fig. 7), we obtain for the concentrations of ions

!

c 9 (x, t)=c 9 (0, t)exp 

and for the potential



tanh

n

"

z9F [8(x, t) − 8(0, t)] RT (A.19)



n

zF8(x, t) zF8(0, t) − kx =tanh e 4RT 4RT

(A.20)

Here c 9 (0, t) are the respective subsurface concentrations and 8(0, t) is the surface potential. Although Eq. (A.19) resemble Eqs (A.4), the former equations hold for non-equilibrium distributions of the concentrations and potential. The same is true for Eq. (A.20), which has also an equilibrium counterpart [32]. A.4. Outer asymptotic solution After setting d = dD, (A.10) – (A.12) give



n

z+ # #F D #j + #2j + = + (1 + j + ) 2 #X z + +z − #X #X D + #Y (A.21)



n

D #j − #2j − # z− #F = − (1 + j − ) D − #Y #X 2 z + +z − #X #X (A.22) #2F a 2 = − (j + −j − ) #X

(A.23)

The zeroth approximation of Eq. (A.23) is (0) (0) j+ = j− , which represents the local electroneutrality condition (0) (0) z+c+ =z − c −

(A.24)

(1) − The first approximation #2F(0)/#X 2 = −(j + (1) j − ), however, requires knowledge of the func(1) tions j 9 . This can be avoided if one multiplies Eq. (A.21) by z + D − and Eq. (A.22) by z − D + and then sums them up:

#F D − −D + #j = 2 #X z + D + +z − D − #Y 2

(0)

(0)

(A.25)

Eq. (A.25) relates the surface potential with an effective concentration j(0, t), defined as c= (0) c¯ (1 + j) where c (0) = c + (0)/z − = c − /z + in view of Eq. (A.24). To find the concentration j(0, t) let us multiply Eq. (A.21) by D − and Eq. (A.22) by D + and then sum them up: #j (0) #2j (0) = #Y #X 2

(A.26)

Eq. (A.26) has the same form as Eq. (A.8), with D being the apparent diffusivity (Eq. (5)). Since Eqs. (A.25) and (A.26) are valid in the outer region, the correct way of their solution excludes the usage of boundary conditions assigned to the plane x= 0. This means that the constants of integration should be determined by matching of the outer solutions for j 9 (x, t) and 8(x, t) with the inner ones. Since this procedure is rather complex, to avoid it we determined the constants of integration for the outer solution by using the boundary conditions for the adsorption at the fluid interface. The disregarding of the diffusion contribution in the double electric layer leads to an error in the concentration of the order of c(k − 1, t)− c(0, t) 1  = a  1 c¯ − c(0, t) kdD Hence, c(0, t):c(k − 1, t), i.e. one can assign as subsurface concentration the one defined at x =0. This error should be even smaller for multivalent ions, which strongly compress the electric double layer. The boundary conditions for the adsorption of ions read #c + dG + = D+ dt #x #c dG − = D− − dt #x

)

+ x=0

)

− x=0

)

F #8 D z c RT + + + #x x = 0

)

(A.27)

F #8 D−z−c− RT #x x = 0 (A.28)

Summing up these equations one obtains for the total adsorption, G= G + + G − ,

I.U. Vakarelski, C.D. Dushkin / Colloids and Surfaces A: Physicochem. Eng. Aspects 163 (2000) 177–190

)

dG #c = (z − D + +z + D − ) dt #x x = 0 +



)

F #8 z + z − (D + −D − )c¯ RT #x x = 0

(A.29)

An equation for the surface charge r =z + G + − z − G − can also be derived by multiplying Eq. (A.27) by z + and Eq. (A.28) by z − and summation:

)

#c dr = z + z − (D + −D − ) #x x = 0 dt +

)

F #8 z + z − (z + D + +z − D − )c¯ RT #x x = 0 (A.30)

One needs one more relationship between G and the ionic concentrations, which can be thought as an adsorption isotherm of the general type G= G(c + , c − )

(A.31)

At small deviations from equilibrium one can expand Eq. (A.31) in the series #G( [c + (0, t) − c¯ + ] G(t)= G( + #c¯ + +

#G( [c − (0, t) − c¯ − ] #c¯ −

which can then be written in the dimensionless form g(Y)=(c¯ /G( )dD(z + +z − )j(0, Y) at G= G(1 + g). Hence, Eq. (A.29) finally takes the form G( D dg #j =(z − D + +z + D − ) c¯dD dY #X x = 0

)

)

#F #X x = 0

+z + z − (D + −D − )

(A.32)

The Laplace image of the concentration is j(X, s)= j0e − X s/(s + s) where s is the transformation parameter. Therefore, the original for the bulk concentration is j(x, t)= j0E



x

' t tD

(A.33)

2Dt where j0 = j(0, 0). At x =0, Eq. (A.33) yields for the adsorption g(t) =g0E( t/tD), which is in fact Eq. (2) for the surface tension. Note, that g0 = g(0) and g/g0 =(s −s¯ )/(s0 −s¯ ). Eqs. (A.25) and (A.33) give for the bulk potential +

x

'

t t D

2Dt and for the surface potential (at x= 0) F(x, t)= F0E

+

F(0, t)= F0E( t/tD)

187

(A.34)

(A.35)

where F0 = F(0, 0). Eq. (A.35) represents the part of DV-potential depending on the electric double layer (see below). It should vary with time in the same manner as the surface tension, cf. Eq. (2), although it was not possible to experimentally detect such dependence. The versatility of our approach could be understood, for example, by comparison with model incorporating a non-linear adsorption isotherm [6] and numerical integration of the kinetic equations for surfactant molecules. This requires more computational efforts, while the interrelation between two species (surfactant ions and counterions) remains unclear. Appendix B. Kinetics of the surface tension above CMC In this case we postulate the following constitutive equations for the diffusion of species: #2c # #8 F #c + = D+ + + D+z+ c+ 2 #t #x RT #x #x

 

1 (c − c¯ + ) tM + #2c # #8 F #c − = D− − − D−z− c #t #x 2 RT #x − #x −

 

(B.1)

1 (c − c¯ − ) (B.2) tM − The effect of micelles is accounted for by the last terms in Eqs. (B.1) and (B.2), in similarity with the way it has been done for nonionic surfactants [9,10]. The rate of demicellization is assumed proportional to the respective concentrations of ions like in a pseudo-first-order-chemical reaction. Since tM is a characteristic time of demicellization, it is a complicated function of the micelle size distribution and of the equilibrium surfactant and electrolyte concentrations [33,34]. Without micelles, tM “ , Eqs. (B.1) and (B.2) transform into Eqs. (A.1) and (A.2), respectively. Multiplying −

I.U. Vakarelski, C.D. Dushkin / Colloids and Surfaces A: Physicochem. Eng. Aspects 163 (2000) 177–190

188

Eq. (B.1) by D − and Eq. (B.2) by D + and summing them up, one obtains the equation 1 #c #2c =D 2 − (c −c¯ ) #x tM #t which reads in a dimensionless form #j #2j = −bj (B.3) #Y #X 2 here b =tD/tM is a parameter accounting for the contribution of micelles. The latter prevails at b ] 1 and vanishes at b 1. The Laplace image for the adsorption, satisfying Eq. (B.3), is

s* g(s* −b)= g0 (B.4) s* s* +s* −b s* + Db where s* is a new transformation parameter defined as s* =s +b. The parameter D accounts for the difference in the diffusivities of two ions z + z − (D + −D − )2 D= (B.5) (z + + z − )2 D + D − The original of Eq. (B.4) is p 21 E(p1 Y) g(Y)= g0e − bY (p1 −p2)(p1 −p3) p 22 + E(p2 Y) (p2 −p1)(p2 −p3)



n

2 3

p (B.6) E(p3 Y) (p3 −p1)(p3 −p2) where the constants pi obey the following relationships: p1 + p2 +p3 =1, p1p2 +p1p3 +p2p3 = − b and p1p2p3 = Db. To simplify the calculations one can assume nearly equal diffusivities of the ions, D + : D − , i.e. D= 0 — cf. Eq. (B.5). Although this assumption is not true in the view of Table 2, it allows vast simplification of Eq. (B.6), because in this case p3 =0, p1 =(1 +G)/2 and p2 =(1− G)/2, where G = 1 +4b. The final form of approximate equation for the adsorption kinetics becomes +

g(Y) = g0





1 − bY 1 +G (1 + G)E e

Y 2G 2

−(1−G)E



1 −G

Y 2

n

 (B.7)

compared with Eq. (7). It resembles the respective equation for nonionic micelles derived in Ref. [10].

Appendix C. Calculation of the surface potential Generally, the DV-potential can be split into two terms [31]: DV = 4pmG+ 80

(C.1)

Here m is the so-called apparent dipole moment of an adsorbed surfactant monomer. 80 is the surface potential given by [18]





2RT FG sinh − 1 − (C.2) F 2 2o0oRTce where o0 = 8.854×10 − 21 C2 erg − 1 cm − 1 is the dielectric permittivity of vacuum, o= 81 is the relative permittivity of water. Introducing dimensionless adsorption, u= G/G , and a characterisF 2G2 /8o0oRT, tic electrolyte concentration, c *= e one can rewrite equation Eq. (C.2) as 80 =

80 =

2RT ln(− z+ z 2 + 1) F

(C.3)

where z= u c*/c e e. Since in our experiments G = 4.46× 10 − 10 mol cm − 2, then c *= 12.8 mol e l − 1. The latter value is rather high compared to the actual electrolyte concentrations in Fig. 6. Moreover, u is close to unity as calculated from the adsorption isotherm (8). For these reasons z 1, which allows replacing the logarithmic function in Eq. (C.3) with ln(1/2z). The final result for the potential is DV = 4pmG u−



2RT RT c ln(2u)+ ln e F F c*e

(C.4)

Eq. (C.4) predicts a linear dependence of the DV-potential on the logarithm of electrolyte concentration, if the first two terms on the right hand side does not change significantly with ce. However, by plotting DV, obtained experimentally, versus ln ce we found a slope of 50.8 mV. This value is twice as large as the slope RT/F=26.1 mV predicted by Eq. (C.4). It means that the relative adsorption u should also depend on ce. To account for this dependence one estimates the

I.U. Vakarelski, C.D. Dushkin / Colloids and Surfaces A: Physicochem. Eng. Aspects 163 (2000) 177–190

189

contribution of each term on the right hand side of Eq. (C.4). The last term varies between − 236 mV (at ce =0.0015 M) and − 153 mV (at ce = 0.038 M). The second term has a maximum, −36 mV, at u = 1. Bearing in mind that the experimental values for DV lie between 220 and 460 mV (Fig. 6) this term can be neglected. In view of Eq. (C.4) one can assume the following relationship: 4pmG u =A+ B ln





ce ce RT =DV − ln c*e c*e F

(C.5)



which gives in turn for the potential RT c DV = A+(B+ )ln e F c*e

(C.6)

The constant A can be expressed from Eq. (C.5) as A= 4pmG at ce =c *e and u = 1. The constants obtained from the numerical fit of the experimental data for DV in Fig. 8 are A = 710 mV and B=24.7 mV. This enables one estimating the dipole moment of adsorbed molecules: m =706 mD, which is comparable with the known values of 220 mD for normal alcohols [31] and 400 mD for polyoxyethylene n-dodecanol [35]. A rich set of data for m of normal alcohols is published in Ref. [13] ranging between 242 mD for butanol and 810 mD for hexanol. Any difference between our value for m and those quoted in literature can be due to the charge redistribution along the SDE2S molecule, which has a charged head. Also, m is an effective quantity including contributions from the reorientation of water molecules, which is sensitive to the surfactant type [31].

Fig. 9. Empirical adsorption isotherms, drawn by Eq. (8) at different ratio B/A.

Based on the above considerations we propose the empirical adsorption isotherm (Eq. (8)) valid for the adsorption of SDE2S in the presence of symmetrical monovalent electrolyte. The constants A and B in Eq. (8) will vary depending on the surfactant concentration. Plot of the adsorption as a function of electrolyte concentration is given in Fig. 9 at different values of the parameter B/A. The solid line corresponds to the value B/A= 0.035 obtained experimentally at SDE2S concentration 4×10 − 4 M. With increasing of the surfactant concentration the ratio B/A decreases, which leads to better saturation of the adsorbed layer if the electrolyte concentration increases. Similar adsorption isotherm can be derived also for multivalent ions from respective measurements of the DV-potential.

References

Fig. 8. Logarithmic plot of the potential of 4 ×10 − 5 M SDE2S solutions with different NaCl content. The experimental points are from Fig. 6; the solid line is a fit by Eq. (8).

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