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Andrei Dukhin
Research Article
Dispersion Technology Inc., Bedford Hills, NY, USA
Critical concentration of ion-pairs formation in nonpolar media
Received December 5, 2013 Revised January 15, 2014 Accepted January 26, 2014
It is known that nonpolar liquids can be ionized by adding surfactants, either ionic or nonionic. Surfactant molecules serve as solvating agents, building inverse micelles around ions, and preventing their association back into neutral molecules. According to the Bjerrum-Onsager-Fuoss theory, these inverse micelle ions should form “ion pairs.” This, in turn, leads to nonlinear dependence of the conductivity on the concentration. Surprisingly, ionic surfactants exhibit linear conductivity dependence, which implies that these inverse micelle ions do not form ion pairs. Theory predicts the existence of two ionic strength ranges, which are separated by a certain critical ion concentration. Ionic strength above the critical one is proportional to the square root of the ion concentration, whereas it becomes linear below the critical concentration. Critical ion concentration lies within the range of 10−11 –10−7 mol/L when ion size ranges from 1 to 3 nm. Critical ion concentration is related, but not equal, to a certain surfactant concentration (critical concentration of ionpairs formation (CIPC)) because only a fraction of the surfactant molecules is incorporated into the micelles ions. The linear conductivity dependence for ionic surfactants indicates that the corresponding CIPC is above the range of studied concentrations, perhaps, due to rather large ion size. The same linearity is a sign that charged inverse micelles structure and fraction are concentration independent due to strong charge–dipole interaction in the charge micelle core. This also proves that CIPC is independent of critical concentration of micelle formation. Nonionic surfactants, on the other hand, exhibit nonlinear conductivity dependence apparently due to smaller ion sizes. Keywords: Inverse micelle / Ion pairs / Ionization / Nonaqueous conductivity / Surfactant DOI 10.1002/elps.201300611
1 Introduction The subject of this paper is ionization in “nonpolar media,” which is defined as a liquid with a very low dielectric permittivity, such as below 10, following the classification suggested in the review [1] and modern handbooks on nonaqueous electrochemistry [2,3]. Examples of such nonpolar liquids include hydrocarbons such as heptane, hexane, toluene, etc. As solvents, the properties of such media differ vastly from those of water. The driving forces for the dissolution and disassociation of electrolytes, and the subsequent generation of ions, are entirely different from those in water. This, in turn, leads to profound consequences for the properties of colloidal systems and electrokinetic phenomena in nonpolar media including electrophoresis. Studying these systems, these phenomena, and the wide spectrum of practical applicaCorrespondence: Dr. Andrei Dukhin, Dispersion Technology Inc., 364 Adams Street, Bedford Hills, NY 10507, USA E-mail: [email protected] Fax: +1-914-241-4842
Abbreviation: CIPC, critical concentration of ion-pairs formation
C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
tion requires reliable and reproducible means for controlling the electrochemical properties of nonpolar systems. There is a recently published review [4] that presents the history of nonpolar liquids electrochemistry and the current state of affairs in this field. One of the most successful methods for controlling electrochemical composition of nonpolar liquids is related to the usage of surfactants. Surfactants are amphiphilic molecules with both parts playing important role in the ionization of nonpolar liquids. Their hydrophobic tails facilitate their easy dissolution in nonpolar liquid. Their polar hydrophilic parts carry dipole moments that can interact with electric charges similarly to how water molecules interact with ions in aqueous solutions. This ion–dipole interaction is responsible for the buildup of solvating layers around electric charges in water. Surfactant molecules create similar solvation layers around electric charges in nonpolar liquids. Objects that are sterically stabilized by surfactant are called “inverse micelles.” Inverse micelles with electric charge in the core, either positive or negative, serve as ions in nonpolar liquids.
Colour Online: See the article online to view Figs. 1, 3 and 4 in colour.
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monomers, fcs , is assembled into micelles with m monomers per micelle. Concentration of the inverse micelles, cm , equals: cm = cs
f m
(3)
Some fraction of them is charged, which can be determined with a dissociation constant ␣, similarly to the traditional electrolyte solution. The equilibrium constant is related to the dissociation constant ␣ with the equation: Mdisp =
Figure 1. Conductivity of heptane with additions of ionic surfactant AOT, reproduced with permission from [4].
There are many published papers [5–14] proving that the addition of surfactants to nonpolar liquid offers a simple and reliable way for controlling ionic strength and conductivity. However, some unresolved questions remained. One of the most striking ones is about the conductivity dependence on concentration of ionic surfactants. It is reported as linear in many studies [4–6, 11, 13, 14]. Figure 1 (reproduced from [4]) presents one example of such dependence. It is surprising because conductivity-concentration linear dependence is an attribute of a strong electrolyte, which is strange at a conductivity range that is several orders of magnitude lower than for aqueous systems. In his landmark review published in 1992 [5], Morrison expressed his surprise with these results: “ . . . What is surprising that conductivity is linear over quite a range of concentration . . . For weak electrolytes, the degree of dissociation varies with concentration and conductivity should vary (at low concentrations) with the square root of concentration . . . ” In order to explain this peculiar dependence, he suggested so-called “disproportionation model” of nonpolar liquids ionizations: “ . . . One possible explanation that the charged lecithin micelles are not formed by dissociation, but by the bimolecular collision of two uncharged micelles, and thus the creation of two charged micelles of opposite sign . . . .” This “disproportionation model” is widely used in the field currently. It seems that it explains linear conductivity dependence indeed. Here, how it can be demonstrated. The equilibrium mass action law for the disproportionation model reflects the balance of the two second-order reactions, which is quite different compared to the dissociation model: A+ + A− ↔ A + A
(1)
with equilibrium constant defined as the ratio of concentration products for neutral and charged micelles: Mdisp =
[A][A] [A+ ][A− ]
(2)
Let us denote surfactant molecules (monomers) concentration as cs . Let us assume that a certain fraction of these C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
(1 − ␣)2 ␣2
(4)
which in turn lead to the following expression for ionic strength J: J = ␣c m =
1+
1
f cs Mdisp m
(5)
This means that conductivity must be linear function of the surfactant concentration, whereas molar conductivity is constant, that is, independent on the concentration. It looks like this model has achieved explanation of experiments, one of which is shown in Fig. 1. However, this model applies number of assumption that must be discussed. For instance, this model ignores completely electrostatic interaction between cations and anions. This assumption is very hard, if not impossible, to justify. Such electrostatic interaction serves as the basis for the fundamental theory of ion pairs created in early 20th century by Bjerrum, Onsager, and Fuoss [15–18]. The notion of ion pairs still plays a central role in modern handbooks on nonaqueous electrochemistry [2,3]. The most detail description of this theory can be found in the classical Handbook on Electrochemistry by Bockris and Reddy [19]. Bjerrum was the first one who pointed out that cation and anion would stay together for some time after they occasionally collide while undergoing Brownian motion. He called such neutral entity “ion pair.” Ion pair differs from a neutral molecule because the cation and anion retain their solvating layers, completely or at least partially. The necessary condition for ion-pairs formation is usually formulated in terms of the restriction on the distance of the closest charges approach in ion pair l. This parameter must be smaller than certain critical distance q, which is usually referred to as Bjerrum distance: q =
z+ z− e 2 2εkT
(6)
where k is Boltzmann constant, T is absolute temperature, z± are ions valencies, e is electron charge, and ε is dielectric permittivity. Bockris and Reddy wrote, “ . . . Bjerrum concluded, therefore, that ion-pair formation occurs when an ion of one type of charge, for example, a negative ion, enters a sphere of radius q drawn around a reference ion of the opposite charge, for example, a positive ion . . . .” (p. 255 in [19]). We would like to stress that the possibility of ion-pair existence had been formulated in terms of the closest distance www.electrophoresis-journal.com
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of ions approach, and not in terms of ion sizes. Nevertheless, approximate ion size d can be used as an estimate for the l. According to this well-known theory, “inverse micelles ions” must also form ion pairs if their sizes are sufficiently small. They must be smaller than the Bjerum distance in nonpolar liquid, which is about 28 nm for a liquid with a dielectric permittivity 2. Published data on ion sizes formed by ionic surfactants indicate that these sizes are on scale of 2–10 nm, which is much smaller than Bjerrum distance in nonpolar liquids (28 nm). Therefore, ions formed by ionic surfactants must build up into ion pairs. This conclusion dramatically changes situation with conductivity-concentration dependence. The ionic strength in the solutions of ionic surfactants would be controlled by the equilibrium between free ions and ion pairs, instead of equilibrium suggested by the disproportionation model. The mass action law of ion/ion-pairs equilibrium is similar to the mass action law of the classical dissociation model. An equilibrium ionic environment is the result of a balance between a first-order reaction (breakup of the ion pairs) and a second-order reaction (association of ions into the ion pairs). It is reflected with the following mass action law: A+ + B − ↔ AB
Mdis s =
[AB] [A+ ][B − ]
(7)
(8)
where Mdiss is an equilibrium constant, and parameters in brackets in Eq. (8) are concentrations of the ion pairs and ions. This mass action law leads to a completely different conductivity dependence, compared to the disproportionation model predictions. The conductivity theory that is based on this mass action law was created by Onsager and Fuoss in 1930s [16, 17]. Their general theory can be substantially simplified when applied to the nonpolar liquids with rather low ionic strength. This possibility exists due to large distances between ions and very long Debye length. Therefore, the long-range interactions can be ignored, which leads to the simplified equation presented by Fuoss [17]: K = K ∞ c e − S(c e1/2 ) − K ∞ Mdiss c e2
(9)
where ce is electrolyte concentration and K is conductivity at extreme dilution. The term with the square root is the leading term of the relaxation and electrophoresis function affecting ions motion. It can be neglected as well due to the long Debye length compared to the ion sizes. This leaves only two terms being important in nonpolar liquids: 2 K = K ∞ c m − K ∞ Mdiss c m
(10)
We also replaced the concentration of electrolyte ce with concentration of inverse micelles cm , for applying this equation to surfactant solutions. C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Fuoss derived an equation for equilibrium constant Mdiss relating it to approximate ion sizes [17]: 2q 4 NAa 3 EXP (11) Mdiss = 3 a where NA is Avogadro number. Onsager–Fuoss theory predicts that conductivity must be a nonlinear function of ionic strength when ion pairs are formed, see Eq. (9). On the other hand, many scientists have observed linear conductivity dependence for ionic surfactants in nonpolar liquids. There is a clear contradiction between experiment and theory, which requires clarification. There is also another contradiction with regard to conductivity-concentration dependence. At the beginning of this introduction, we presented quote by Morrison where he refers to the “ . . . square root concentration dependence . . . ” [5]. This quote agrees with Bockris and Reddy [19], who wrote in page 450: “ . . . in the absence of significant association, conductivity was linearly dependent on c1/2 , as empirically shown by Kohlrausch . . . .” On the other hand, Onsager–Fuoss theory predicts dependence Eq. (9) that differs from the square root. What is the origin of that different prediction by well-known theories? Resolution of these two contradictions was suggested and discussed to some degree in the recently published review [4]. However, many important details remained beyond that paper scope due to it being structured as a review instead of dedicated new presentation. Here, we reproduce the main derivations suggested in Section 7 of [4] with following new important details. First of all, we discuss relationship between concepts of “strong–weak electrolyte” and “ion pairs.” We add this analysis to the discussion of the Ostwald dilution law as given in [4]. Then, we add formulation and discussion of two important necessary conditions for conductivity-concentration dependence being linear. These conditions are related to the structure of the inverse micelles ions at different concentration of the surfactants. We analyze applicability of the Onsager–Fuoss method for calculating ion sizes from the conductivity-concentration dependence. We conclude that this method is not applicable for estimating sizes of ions created by surfactants in nonpolar liquids. In addition, we present and discuss much wider experimental data collected and presented by different authors. Finally, we introduce here notion of “critical concentration of ion-pairs formation” (CIPC). This is critical concentration of surfactant, instead of critical ionic strength discussed in [4]. The last one is not measurable, whereas CIPC could be determined from the experimental data. Discussed here difference between such notions as “electrolyte concentration,” “concentration of inverse micelles ions,” “surfactant concentration” is critical for understanding nature of nonpolar liquids ionization with surfactants. We provide estimate of the CIPC values as a function of ion sizes (Table 1). Then, we compare these values with experiment data for ion-pair. www.electrophoresis-journal.com
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Table 1. Values of the Fuoss equilibrium constant and critical concentration of ion-pairs formation as a function of ion size, assuming that it is identical to the distance of the ions closest approach
Distance of the closest approach (l) [nm] Approximately ion size (d) [nm]
Fuoss constant Mdiss Critical concentration for Bjerrum distance of ion-pairs formation (CIPC), [mol/L] 28 nm Approximately critical conductivity [S/m]
1 2 3 4 5
3 646 318 737 233 24 256 149 769 818 176 85
2.7 × 10−13 4.1 × 10−8 1.3 × 10−6 5.6 × 10−6 1.2 × 10−5
2 Theory Mass action law for the dissociation model Eqs. (7) and (8) can be applied for describing the equilibrium of inverse micelle ions dissociation–association in ion pairs. We stress here that surfactant is not an electrolyte, as is usually assumed for substances described with the dissociation model. First of all, only a fraction of the surfactant molecules participate in ions formation. Second, they mostly perform function of a solvating agent. There is undergoing discussion into how exactly charge separation occurs in nonpolar liquids, that is, what is the chemical nature of the substance that dissociates. We do not address this complex issue here. Instead, following Onsager and Fuoss, we strive to derive some understanding of nonpolar liquid ionization from studying ions association. We postulate that there are inverse micelles with concentration cm in particular liquid, and our goal is to describe their aggregation into ion pairs, driven by pure electrostatic interaction. This approach was suggested by Onsager and Fuoss, who emphasized many times in their papers that investigation of ions association, not dissociation, would yield important insight in ionization process in general [16–18]. Equation (8) for the equilibrium constant of inverse micelle ions’ association into ion pairs can be re-written as the following, using notion of the dissociation constant ␣: Mdiss =
(1 − ␣) ␣2 c m
(12)
This equation is known as Ostwald dilution law. It is usually applied for deriving expression for the dissociation constant, which is less than 1 for a weak electrolyte. This derivation is achieved by assuming that dissociation constant ␣ is a small parameter. This eventually leads to the square root concentration dependence mentioned above in the introduction. There are several important comments that must be made before applying Ostwald dilution law to nonpolar liquids with surfactant. First of all, appearance of inverse micelles ions is not direct result of a certain electrolyte dissociation. Ionization in nonpolar liquids is limited by solvation. Therefore, buildup of steric solvation layers around ions C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
by surfactant molecules is important necessary step of this process. These layers could be structurally concentration dependent. This means that even structure of ions could vary with concentration, which is dramatically different from the aqueous solutions where solvation function is abundant. Saying this, we would apply Ostwald dilution law to inverse micelles ions only. We do not try applying this law for describing dissociation of some electrolyte that gives birth initially to the primary electric charges that are then become solvated by surfactants into inverse micelles. Second, we would like to stress difference between “weak electrolyte” and “solution with very low ionic strength, very small ions concentration.” Term “weak electrolyte” does not imply yet that concentration of ions is low. For instance, it is known that any electrolyte becomes completely dissociated at infinite dilution. This is a well-known Kohlrausch law. Therefore, dissociation constant becomes 1 at infinite dilution and electrolyte becomes a strong one, instead of weak one. This simple analysis shows that concentration of ions must be scaled with some independent parameter for judging about its relative value. Dissociation constant cannot serve this purpose because it is dimensionless. Equation (12) suggests another parameter for such role: reciprocal equilibrium constant (1/Mdiss ) which has the same dimension as concentration. Therefore, in order to derive asymptotic solution for ionic strength at low limit, we should use expansion of Eq. (12) with small dimensionless parameter (cm Mdiss ), instead of the dissociation constant ␣. We reproduce this derivation here following review [4]. As the first step, we reformulate Eq. (12) for introducing this new small parameter for the range of low ionic strength. We can replace the dissociation constant in this equation with ionic strength J, which is defined as following: J = ␣c m
(13)
There are several simple algebraic substitutions shown below: Mdiss =
(1 − ␣)c m c m − c m + (1 − ␣)c m cm − J = = 2 ␣2 c m c m ␣2 c m J2
(14)
which eventually lead to the following quadratic equation that relates ionic strength (fraction of free ions) with total concentration of inverse micelles ions: J 2 Mdiss + J − c m = 0
(15)
Equation (14) has a simple solution for ionic strength as function of inverse micelle ion concentration: √ 1 + 4Mdiss c m − 1 (16) J = 2Mdiss For some unknown reasons, this important equation has evaded text books on electrochemistry for a long time. It presents expression for ionic strength at any concentration of ions. Basically, it includes both weak and strong electrolytes ranges, low and high ionic strength. Calculation of the ionic strength for a given electrolyte concentration requires theory for the equilibrium constant www.electrophoresis-journal.com
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Mdiss . We will use Fuoss’ version Eq. (11), which seems the most accepted in modern electrochemistry. The combination of Eqs. (11) and (16) allows for calculation of the ionic strength at any concentration of inverse micelles ions for a given ion size and liquid dielectric permittivity. This statement is based on one major assumption that the ions retain their solvating layers when they collide. Modern electrochemistry ([2] p. 57) considers several situations that might occur during ion-pair formation. According to this book, there are three different types of ion pairs, depending on the structure of the solvating layers: solvent-separated ion pairs, solvent-shared ion pairs, and contact ion-pairs. It is not clear at this point what type of ion pairs would be formed by the collision of the inverse micelles ions. There are two asymptotic solutions for Eq. (16): one for the high concentration and one for the low concentrations of ions. They are separated by a critical concentration ccr : c cr =
1 Mdiss
This concentration range corresponds to the short distances between ions. Their interaction becomes important. This is the range where ionic interactions control the electrochemistry of the liquid. This is the range where the Debye– Huckel theory [20] of conductivity must be adopted. The asymptotic expression for ionic strength equals: √ √ cm 1 + 4Mdiss c m − 1 4Mdiss c m ≈ = (19) J = 2Mdiss 2Mdiss Mdiss This asymptote is identical to the solution for the weak electrolytes, mentioned by Morrison [5] and Bockris and Reddy [19]. However, it is valid only for sufficiently high concentration of ions. If ion concentration is low, then another asymptotic solution of Eq. (16) emerges. This second asymptotic solution for the low ionic strength is completely different. If we assume that: 1 (20) c m c cr = Mdiss then the two first terms of Eq. (16) when expanded would yield: (21)
This expression for the ionic strength allows us to estimate the concentration dependence of the conductivity. We can neglect relaxation and electrophoretic retardation effects on ion motion due to the very large Debye length compared to the ion sizes in low-conducting liquids where Eq. (20) is valid. Then, introducing conductivity at infinite dilution, we would arrive at the following equation: 2 K = K ∞ c m − 2K ∞ Mdiss c m
C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
This equation for molar conductivity is practically identical to Eq. (10) that was derived from the Onsager–Fuoss conductivity equation in the introduction. Similar derivation was conducted recently by D. Prieve, see [12]. There is coefficient 2 in the second term on the right-hand part of this equation, which makes it different somewhat from Eq. (10). We do not know reason of this difference. Original derivation by Onsager and Fuoss might contain simplification of including such coefficient into the equilibrium constant. Mathematics of their derivation is very complex because they took into account Debye–Huckel effect. This makes comparison quite cumbersome and not well defined. In any case, this coefficient does not affect results qualitatively. The second term in Eq. (22) reflects ion-pairs formation at higher concentrations of ions. If ion concentrations is much less than critical concentration ccr , then this second term becomes negligible and we arrive at a simple linear dependence of the conductivity on ion concentration:
(17)
Let us consider initially the range of the high ionic strength, which corresponds to the ions concentrations where: 1 (18) c m c cr = Mdiss
2 J = c m − 2Mdiss c m
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(22)
K c m c cr = K ∞ c m
(23)
This expression agrees with the well-known Kohlrausch law [19], which states that dissociation of electrolyte is complete at infinite dilution. The novelty of this paper is that we establish necessary condition for Kohlrausch law—concentration of ions must be much lower than the critical concentration, which is simply the reciprocal of the constant for ion/ion-pairs equilibrium. There is a very simple physical explanation for this dependence. The collisions of ions become increasingly rare with decreasing concentration. Eventually, the collision rate cannot compensate for the breakup of the ion pairs. Actually this balance between Brownian-driven collisions and ion pairs breakup is built into the Fuoss equilibrium constant. It was shown in [4] that this constant can be expressed as a ratio of two characteristic times: sm —Smoluchwski time of fast coagulation [21], and d ion-pair breakup time: Mdiss =
NA 2d N0 sm
(24)
Basically, this presentation of the Fuoss constant stresses the fact that electrochemistry at the low ionic strength range is controlled by the diffusion of ions and breakup of neutral ionpairs. The breakup becomes dominant at low ionic strength and all ions become free. Conductivity becomes linear function of ions concentration only if this concentration is much smaller than the critical concentration, expressed with Eq. (20). However, this does not mean it would be linear function of surfactant concentration. Concentration of ions cm is proportional to the surfactant concentration cs only if the number of monomers in the inverse micelle ion m and fraction of the monomers f that are incorporated into these micelles are independent on the surfactant concentration. Therefore, we can formulate three necessary conditions for conductivity being linear function of the surfactant: www.electrophoresis-journal.com
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(i) The concentration of ions is much smaller than the aforementioned critical concentration, which equals to reciprocal equilibrium constant ions–ion pairs; (ii) The number of monomers in the inverse micelle ion is independent of surfactant concentration; (iii) The fraction of the surfactant monomers that is incorporated in the inverse micelles ions is independent on the surfactant concentration. Only if all three conditions are valid, then conductivity would be linear function of the surfactant. In order to verify this theoretical prediction, we would first need an estimate of the critical ionic strength ccr . We can use Eqs. (11) and (17), calculating ccr for various values of ion sizes. We assume that ion size is approximately equal to the distance of closest approach of the electric charges in the ion pair. These values for ccr are shown in Table 1 for various ion sizes. Unfortunately, the concentration of inverse micelle ions and consequently critical ion concentration ccr are not measurable parameters. We should somehow relate this parameter to the experimentally measurable parameters, such as conductivity K [S/m] and concentration of the surfactant cs [%wt] in order to determine range of ion-pairs formation. In principle, one can use analogy with “critical concentration of micelles formation” and introduce critical “concentration of ion-pairs formation” (CIPC) in surfactant solutions in nonpolar liquids. Conductivity measured at different concentrations of surfactants would yield information for estimating CIPC. We will show this in the next section. Unfortunately, it is not sufficient for calculating ccr because two unknown parameters are involved in the relationship between CIPC and ccr : c cr = C I PC
f m
(25)
We do not know the fraction of surfactant that is incorporated into charged inverse micelles (f) and number of surfactant monomers in the single micelle (m). Instead of surfactant concentration, we can try using conductivity directly for estimating ccr . It is known that for aqueous systems conductivity expressed in Siemens (S)/m is about ten times higher than strong electrolyte concentration expressed in mol/L. Conductivity is reciprocally proportional to the ion sizes, which substantially increase in nonpolar liquids. Therefore, difference between these two numbers expressed in different units would be much smaller in nonpolar liquids. There is one more factor that would reduce this difference even more—viscosity. Conductivity is reciprocally proportional to viscosity according to the Walden rule [19]. Viscosity of nonpolar liquids is usually larger than viscosity of water. These two factors ion size and viscosity should make conductivity expressed in S/m and ionic strength expressed in mol/L much closer to each other, comparing to aqueous solutions. Even if they would be still different by two to three times, this is much smaller than the range of the conductivity variation due to the surfactants addition, which is several orders of magnitude. We suggest here to ignore this difference C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
and use conductivity numbers expressed in S/m as direct estimate of the ionic strength in mol/L. This assumption would allow us to test predictions of the conductivity linearity range at least qualitatively. Calculated values of ccr and corresponding Kcr are shown in Table 1. It is seen that the range of conductivity where it is supposed to be linear function of the surfactant concentration expands rapidly with increasing ion size. However, even for ions with sizes around 3 nm, there is a substantial range where conductivity is easily measurable and can be linear function. These conclusions can be verified with analysis of the existing published experiments.
3 Experiment: Review of published data As we mentioned in the introduction, conductivity measurements of various nonpolar solutions have been conducted since late 19th century. There is a recently published review of the early works on this subject [4]. These studies were usually made with simple substances that serve as electrolytes. Application of surfactants for controlling nonpolar liquids electrochemistry was introduced at the end of 20th century, initially with ionic surfactants, and much later with nonionic surfactants. Practically, all studies with ionic surfactant report a linear dependence of conductivity on the surfactant concentration. Here are these papers with reported values of the micelles sizes in some of them: (i) (ii) (iii) (iv) (v)
isopar with 7 nm lecithin micelles [13], benzene with AOT [14], dodecane with 5–10 nm OLOA micelles [6], isopar with 10 nm OLOA micelles [8], heptane with AOT [4].
We reproduce here figures from these published papers. Figure 1 corresponds to the AOT in heptane from the paper [4], Fig. 2A shows conductivity of OLOA in dodecane from the paper [6], and finally Fig. 2B depicts conductivity of OLOA in isopar from the paper [8]. Conductivity for all these measurements is ⬍10−6 S/m. At the same time, the size of micelles exceeds 3 nm for all these surfactants. Modern methods of measuring micelle sizes in these solutions confirmed the very early data by Mathews and Hirschhorn [22] for the size of AOT micelles in dodecane to be in the range from 3.4 up to 9.9 nm. This rather large size of micelles leads to a high value of the CIPC and critical conductivity, according to the data shown in Table 1. An ion size of 3 nm corresponds to a critical conductivity of 1.3 × 10−6 S/m. This critical value is higher than the conductivities measured in the systems (i)–(v) mentioned above. Thus, we can conclude that ionic surfactants exhibit linear conductivity dependence in nonpolar liquids because ionpairs formation is suppressed with sufficiently large ion sizes. Weak electrostatic attraction could not prevent the breakup of www.electrophoresis-journal.com
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Figure 2. Conductivity as a function of OLOA ionic surfactant. (A) Nonpolar liquid is dodecane, reproduced with permission from [6]. Error bars are based on the standard deviation of the measured conductivities. (B) Nonpolar liquid is isopar, reproduced with permission from [8]. Viscosity adjusted conductivity and dimensionless a for OLOA.
formed ion pairs, and collision by diffusion rate cannot compensate for the breakup rate due to the low concentration of ions. There are two more conditions formulated in the previous section for linear conductivity dependence: concentration independent micelle structure and fraction of surfactant incorporated in the charged micelles. Apparently, these conditions are valid for the systems (i)–(v) with ionic surfactants. This interesting conclusion sheds new light on micelle formation in nonpolar liquids. Nonionic surfactants, in contrary, exhibit nonlinear dependence on conductivity that agrees with the theoretical predictions of ion-pairs formation. Figures 3A and B and Fig. 4 copied from the paper [23] illustrate this statement. Similar nonlinearity was observed by Behren’s group [9, 10]. This nonlinearity indicates that positive and negative charges could approach closer when ions are bounded in the ion pair. This can happen, for instance, if ion size is smaller in the case of nonionic surfactants. There is some evidence that this might be the case. For instance, Behren’s group reported on ion size of 2.8 nm for SPAN 85 in hexane. The probability of building ion pairs is a very nonlinear function C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Figure 3. Conductivity of the various nonpolar liquids with nonionic surfactant, reproduced with permission from [23]. (A) Conductivity of various liquids with SPAN 80. d.p. stands for dielectric permittivity of the liquid. (B) Conductivity of the kerosene solutions with nonionic surfactants having various HLB numbers.
Figure 4. Conductivity of toluene with additions of nonionic surfactant SPAN 80, reproduced with permission from [4].
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of the ion size. According to Table 1, when reducing ion size from 3 nm down to 2 nm, CIPC drops 30 times. Therefore, even small reduction in the ion size causes a very large effect on ion-pairs formation. There is one alternative explanation that should be taken into account. Addition of surfactant could affect viscosity of the liquid. Conductivity decreases with increasing viscosity, according to the Walden rule [19]. Therefore, in principle, decreasing conductivity growth rate as shown in Fig. 4 could be associated with increasing viscosity of the liquid. There are two ways for resolving this ambiguity. First of all, direct viscosity measurement would yield the answer. According to the data shown in Fig. 4, viscosity should increase roughly two times at the 10% surfactant content for explaining observed decay in the conductivity growth rate. Unfortunately, viscosity data for these solutions are not available. There is also possibility of detecting ion pairs directly with other experimental methods. There is review of such methods given in the paper [4]. It seems that dielectric spectroscopy might be the best method for this purpose. Ion pairs should carry substantial dipole moment. There is a method of dielectric spectroscopy developed by Fuoss group in 1930s for monitoring this dipole moment. It is important to keep in mind possible alternative interpretations of conductivity data for future experiments. Onsager–Fuoss conductivity theory with ion pairs allows for fitting nonlinear conductivity dependence, as shown in Fig. 4. There have been many papers published in the mid20th century that used this fitting procedure for estimating ion size in nonaqueous solutions. Review of these papers can be found in the paper [4]. Unfortunately, this procedure does not work in case of surfactants and inverse micelle ions. The Onsager–Fuoss law analog for such systems would look like: ∞ K = K eq
f ∞ c s − K eq Mdiss m
f m
2 c s2
(26)
where we used Eq. (3) for relating ionic strength with surfactant concentration. This equation contains three unknown parameters: Keq , f/m, and Mdiss . On the other hand, the fitting procedure yields only two parameters, coefficients for the linear, and quadratic terms. This peculiarity of surfactant solutions reflects the fact that they perform function of solvating agent first of all. They cannot be considered as electrolytes in traditional aqueous electrochemistry sense.
4 Concluding remarks The addition of either ionic or nonionic surfactants to nonpolar liquids enhances conductivity by orders of magnitude. This effect can be used for controlling the electrochemical environment in these liquids, which is especially important for electrokinetic phenomena, such as electrophoresis. C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Surfactant performs function of a solvating agent. The resulting inverse micelle ions form ion pairs if concentration of the surfactant exceeds a certain critical value: the CIPC. Only a fraction of the surfactant molecules (f) is incorporated into inverse micelle ions, each ion require several surfactant molecules (m) for creating a solvating layer around the central electric charge. Consequently, concentration of ions is proportional, but not equal, to the concentration of surfactant. There is a critical concentration of ions that corresponds to the CIPC, above which ions form ion pairs. It is proportional to CIPC and unknown parameter f/m. On the other hand, this critical ion concentration is simply the reciprocal of the Fuoss constant for the ions/ionpairs equilibrium. This constant depends on the distance of the closest approach between cation and anion in the ion pair. This distance can be approximately estimated as ion size. Calculations indicate that critical ions concentration lies within a range of 10−11 –10−7 mol/L when ion size ranges from 1 to 3 nm. Therefore, even small variations in inverse micelle ion size cause large variation in the range of ion-pairs formation. This effect can be verified with conductivity measurements. The formation of ion pairs causes nonlinear dependence of the conductivity on the ion concentration, according to Onsager and Fuoss. This means that below the CIPC, conductivity should be a linear function of surfactant concentration, and becomes nonlinear above the CIPC. Experiment confirms that ionic surfactants exhibit linear conductivity dependence in a variety of nonpolar liquids. Estimated ion size in these systems varies from 3 to 10 nm and maximum conductivity does not exceed 10−6 S/m. This range of linearity agrees with the calculated linearity range for given values of ion sizes. This linear conductivity dependence for ionic surfactants indicates that parameters f and m are concentration independent. This means that the structure of inverse micelles and their fraction in the surfactant content remain constant and concentration independent in nonpolar liquids. This can be the result of strong charge–dipole interaction inside of the charged inverse micelle core. This interaction is much stronger than traditional dipole–dipole interaction of surfactant molecules polar parts that drive formation of neutral micelles. Therefore, CIPC and critical concentration of micelles formation are different and independent parameters. The first one reflects charge–dipole interaction within the micelles core, the second one depends on dipole–dipole interaction. Finally, experiments with nonionic surfactants show that conductivity becomes nonlinear function of the surfactant concentration above certain level, which depends on the nature of the surfactant and of the liquid. This might be explained assuming that ion sizes are smaller for nonionic surfactants. This would lead to the lower CIPC value for such systems. The author has declared no conflict of interest. www.electrophoresis-journal.com
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5 References [1] Lyklema, J., Adv. Colloid Interface Sci. 1968, 2, 65–114. [2] Izutsu, K., Electrochemistry in Nonaqueous Solutions, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany 2002. [3] Aurbach, D., Nonaqueous Electrochemistry, Marcel Dekker, New York 1999.
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[12] Fu, R., Prieve, D., Langmuir 2007, 23, 8048–8052. [13] Davis, T. G., Gibson, G. A., Luebbe, R. H., Yu, K., Proceedings of the 5th International Congress on Advances in Non-Impact Printing Techniques, SPSE, Sprigfield, VA 1989, pp. 404–416. [14] Denat, A., Gosse, B., Gosse, J. P. Rev. Phys. Appl. 1981, 16, 673–678.
[4] Dukhin, A. S., Parlia, S., COCIS 2013, 18, 93–115.
[15] Bjerrum, N., Proceedings of the 7th International Congress of Applied Chemistry, Section X, London 1929, pp. 55–60.
[5] Morrison, I. D., Colloids Surfaces A 1993, 71, 1–37.
[16] Onsager, L., Trans. Faraday Soc. 1927, 23, 341–349.
[6] Kim, J., Anderson, J. L., Garoff, S., Schlangen, L. J. M., Langmuir 2005, 21, 8620–8629.
[17] Fuoss, R. M., J. Am. Chem. Soc. 1957, 79, 3301–3303.
[7] Prieve, D. C., Haggard, J. D., Fu, R., Sides, P. J., Bethea, R., Langmuir 2008, 24, 1120–1132.
[19] Bockris, J., Reddy, A. K., Modern Electrochemistry. Vol. 1, A Plenum/Rosetta edition, New York 1977.
[8] Poovarodom, S., Berg, J. C., J. Coll. Interface Sci. 2010, 346, 370–377.
¨ [20] Debye, P., Huckel, E. J., Physikalische Zeitschrift 1923, 24, 185–206.
[9] Guo, Q., Singh, V., Behrens, S. H., Langmuir 2010, 26, 3203–3207.
[21] Dukhin, A. S., Dukhin S. S., Goetz, P. J., Adv. Colloid Interface Sci. 2007, 134–135, 35–71.
[10] Espinosa, C. E., Guo, Q., Singh, V., Behrens, S. H., Langmuir 2010, 26, 16941–16948.
[22] Mathews, M. B., Hirschhorn, E., J. Colloid Sci. 1953, 8, 86–96.
[11] Gacek, M., Bergsman, D., Michor, E., Berg, J. C., Langmuir 2012, 28, 11633–11638.
[23] Dukhin, A. S., Goetz, P. J., J. Electroanal. Chem. 2006, 588, 44–50.
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[18] Fuoss, R. M., J. Am. Chem. Soc. 1958, 80, 5059–5061.
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Andrei Dukhin
Research Article
Dispersion Technology Inc., Bedford Hills, NY, USA
Critical concentration of ion-pairs formation in nonpolar media
Received December 5, 2013 Revised January 15, 2014 Accepted January 26, 2014
It is known that nonpolar liquids can be ionized by adding surfactants, either ionic or nonionic. Surfactant molecules serve as solvating agents, building inverse micelles around ions, and preventing their association back into neutral molecules. According to the Bjerrum-Onsager-Fuoss theory, these inverse micelle ions should form “ion pairs.” This, in turn, leads to nonlinear dependence of the conductivity on the concentration. Surprisingly, ionic surfactants exhibit linear conductivity dependence, which implies that these inverse micelle ions do not form ion pairs. Theory predicts the existence of two ionic strength ranges, which are separated by a certain critical ion concentration. Ionic strength above the critical one is proportional to the square root of the ion concentration, whereas it becomes linear below the critical concentration. Critical ion concentration lies within the range of 10−11 –10−7 mol/L when ion size ranges from 1 to 3 nm. Critical ion concentration is related, but not equal, to a certain surfactant concentration (critical concentration of ionpairs formation (CIPC)) because only a fraction of the surfactant molecules is incorporated into the micelles ions. The linear conductivity dependence for ionic surfactants indicates that the corresponding CIPC is above the range of studied concentrations, perhaps, due to rather large ion size. The same linearity is a sign that charged inverse micelles structure and fraction are concentration independent due to strong charge–dipole interaction in the charge micelle core. This also proves that CIPC is independent of critical concentration of micelle formation. Nonionic surfactants, on the other hand, exhibit nonlinear conductivity dependence apparently due to smaller ion sizes. Keywords: Inverse micelle / Ion pairs / Ionization / Nonaqueous conductivity / Surfactant DOI 10.1002/elps.201300611
1 Introduction The subject of this paper is ionization in “nonpolar media,” which is defined as a liquid with a very low dielectric permittivity, such as below 10, following the classification suggested in the review [1] and modern handbooks on nonaqueous electrochemistry [2,3]. Examples of such nonpolar liquids include hydrocarbons such as heptane, hexane, toluene, etc. As solvents, the properties of such media differ vastly from those of water. The driving forces for the dissolution and disassociation of electrolytes, and the subsequent generation of ions, are entirely different from those in water. This, in turn, leads to profound consequences for the properties of colloidal systems and electrokinetic phenomena in nonpolar media including electrophoresis. Studying these systems, these phenomena, and the wide spectrum of practical applicaCorrespondence: Dr. Andrei Dukhin, Dispersion Technology Inc., 364 Adams Street, Bedford Hills, NY 10507, USA E-mail: [email protected] Fax: +1-914-241-4842
Abbreviation: CIPC, critical concentration of ion-pairs formation
C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
tion requires reliable and reproducible means for controlling the electrochemical properties of nonpolar systems. There is a recently published review [4] that presents the history of nonpolar liquids electrochemistry and the current state of affairs in this field. One of the most successful methods for controlling electrochemical composition of nonpolar liquids is related to the usage of surfactants. Surfactants are amphiphilic molecules with both parts playing important role in the ionization of nonpolar liquids. Their hydrophobic tails facilitate their easy dissolution in nonpolar liquid. Their polar hydrophilic parts carry dipole moments that can interact with electric charges similarly to how water molecules interact with ions in aqueous solutions. This ion–dipole interaction is responsible for the buildup of solvating layers around electric charges in water. Surfactant molecules create similar solvation layers around electric charges in nonpolar liquids. Objects that are sterically stabilized by surfactant are called “inverse micelles.” Inverse micelles with electric charge in the core, either positive or negative, serve as ions in nonpolar liquids.
Colour Online: See the article online to view Figs. 1, 3 and 4 in colour.
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monomers, fcs , is assembled into micelles with m monomers per micelle. Concentration of the inverse micelles, cm , equals: cm = cs
f m
(3)
Some fraction of them is charged, which can be determined with a dissociation constant ␣, similarly to the traditional electrolyte solution. The equilibrium constant is related to the dissociation constant ␣ with the equation: Mdisp =
Figure 1. Conductivity of heptane with additions of ionic surfactant AOT, reproduced with permission from [4].
There are many published papers [5–14] proving that the addition of surfactants to nonpolar liquid offers a simple and reliable way for controlling ionic strength and conductivity. However, some unresolved questions remained. One of the most striking ones is about the conductivity dependence on concentration of ionic surfactants. It is reported as linear in many studies [4–6, 11, 13, 14]. Figure 1 (reproduced from [4]) presents one example of such dependence. It is surprising because conductivity-concentration linear dependence is an attribute of a strong electrolyte, which is strange at a conductivity range that is several orders of magnitude lower than for aqueous systems. In his landmark review published in 1992 [5], Morrison expressed his surprise with these results: “ . . . What is surprising that conductivity is linear over quite a range of concentration . . . For weak electrolytes, the degree of dissociation varies with concentration and conductivity should vary (at low concentrations) with the square root of concentration . . . ” In order to explain this peculiar dependence, he suggested so-called “disproportionation model” of nonpolar liquids ionizations: “ . . . One possible explanation that the charged lecithin micelles are not formed by dissociation, but by the bimolecular collision of two uncharged micelles, and thus the creation of two charged micelles of opposite sign . . . .” This “disproportionation model” is widely used in the field currently. It seems that it explains linear conductivity dependence indeed. Here, how it can be demonstrated. The equilibrium mass action law for the disproportionation model reflects the balance of the two second-order reactions, which is quite different compared to the dissociation model: A+ + A− ↔ A + A
(1)
with equilibrium constant defined as the ratio of concentration products for neutral and charged micelles: Mdisp =
[A][A] [A+ ][A− ]
(2)
Let us denote surfactant molecules (monomers) concentration as cs . Let us assume that a certain fraction of these C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
(1 − ␣)2 ␣2
(4)
which in turn lead to the following expression for ionic strength J: J = ␣c m =
1+
1
f cs Mdisp m
(5)
This means that conductivity must be linear function of the surfactant concentration, whereas molar conductivity is constant, that is, independent on the concentration. It looks like this model has achieved explanation of experiments, one of which is shown in Fig. 1. However, this model applies number of assumption that must be discussed. For instance, this model ignores completely electrostatic interaction between cations and anions. This assumption is very hard, if not impossible, to justify. Such electrostatic interaction serves as the basis for the fundamental theory of ion pairs created in early 20th century by Bjerrum, Onsager, and Fuoss [15–18]. The notion of ion pairs still plays a central role in modern handbooks on nonaqueous electrochemistry [2,3]. The most detail description of this theory can be found in the classical Handbook on Electrochemistry by Bockris and Reddy [19]. Bjerrum was the first one who pointed out that cation and anion would stay together for some time after they occasionally collide while undergoing Brownian motion. He called such neutral entity “ion pair.” Ion pair differs from a neutral molecule because the cation and anion retain their solvating layers, completely or at least partially. The necessary condition for ion-pairs formation is usually formulated in terms of the restriction on the distance of the closest charges approach in ion pair l. This parameter must be smaller than certain critical distance q, which is usually referred to as Bjerrum distance: q =
z+ z− e 2 2εkT
(6)
where k is Boltzmann constant, T is absolute temperature, z± are ions valencies, e is electron charge, and ε is dielectric permittivity. Bockris and Reddy wrote, “ . . . Bjerrum concluded, therefore, that ion-pair formation occurs when an ion of one type of charge, for example, a negative ion, enters a sphere of radius q drawn around a reference ion of the opposite charge, for example, a positive ion . . . .” (p. 255 in [19]). We would like to stress that the possibility of ion-pair existence had been formulated in terms of the closest distance www.electrophoresis-journal.com
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of ions approach, and not in terms of ion sizes. Nevertheless, approximate ion size d can be used as an estimate for the l. According to this well-known theory, “inverse micelles ions” must also form ion pairs if their sizes are sufficiently small. They must be smaller than the Bjerum distance in nonpolar liquid, which is about 28 nm for a liquid with a dielectric permittivity 2. Published data on ion sizes formed by ionic surfactants indicate that these sizes are on scale of 2–10 nm, which is much smaller than Bjerrum distance in nonpolar liquids (28 nm). Therefore, ions formed by ionic surfactants must build up into ion pairs. This conclusion dramatically changes situation with conductivity-concentration dependence. The ionic strength in the solutions of ionic surfactants would be controlled by the equilibrium between free ions and ion pairs, instead of equilibrium suggested by the disproportionation model. The mass action law of ion/ion-pairs equilibrium is similar to the mass action law of the classical dissociation model. An equilibrium ionic environment is the result of a balance between a first-order reaction (breakup of the ion pairs) and a second-order reaction (association of ions into the ion pairs). It is reflected with the following mass action law: A+ + B − ↔ AB
Mdis s =
[AB] [A+ ][B − ]
(7)
(8)
where Mdiss is an equilibrium constant, and parameters in brackets in Eq. (8) are concentrations of the ion pairs and ions. This mass action law leads to a completely different conductivity dependence, compared to the disproportionation model predictions. The conductivity theory that is based on this mass action law was created by Onsager and Fuoss in 1930s [16, 17]. Their general theory can be substantially simplified when applied to the nonpolar liquids with rather low ionic strength. This possibility exists due to large distances between ions and very long Debye length. Therefore, the long-range interactions can be ignored, which leads to the simplified equation presented by Fuoss [17]: K = K ∞ c e − S(c e1/2 ) − K ∞ Mdiss c e2
(9)
where ce is electrolyte concentration and K is conductivity at extreme dilution. The term with the square root is the leading term of the relaxation and electrophoresis function affecting ions motion. It can be neglected as well due to the long Debye length compared to the ion sizes. This leaves only two terms being important in nonpolar liquids: 2 K = K ∞ c m − K ∞ Mdiss c m
(10)
We also replaced the concentration of electrolyte ce with concentration of inverse micelles cm , for applying this equation to surfactant solutions. C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Fuoss derived an equation for equilibrium constant Mdiss relating it to approximate ion sizes [17]: 2q 4 NAa 3 EXP (11) Mdiss = 3 a where NA is Avogadro number. Onsager–Fuoss theory predicts that conductivity must be a nonlinear function of ionic strength when ion pairs are formed, see Eq. (9). On the other hand, many scientists have observed linear conductivity dependence for ionic surfactants in nonpolar liquids. There is a clear contradiction between experiment and theory, which requires clarification. There is also another contradiction with regard to conductivity-concentration dependence. At the beginning of this introduction, we presented quote by Morrison where he refers to the “ . . . square root concentration dependence . . . ” [5]. This quote agrees with Bockris and Reddy [19], who wrote in page 450: “ . . . in the absence of significant association, conductivity was linearly dependent on c1/2 , as empirically shown by Kohlrausch . . . .” On the other hand, Onsager–Fuoss theory predicts dependence Eq. (9) that differs from the square root. What is the origin of that different prediction by well-known theories? Resolution of these two contradictions was suggested and discussed to some degree in the recently published review [4]. However, many important details remained beyond that paper scope due to it being structured as a review instead of dedicated new presentation. Here, we reproduce the main derivations suggested in Section 7 of [4] with following new important details. First of all, we discuss relationship between concepts of “strong–weak electrolyte” and “ion pairs.” We add this analysis to the discussion of the Ostwald dilution law as given in [4]. Then, we add formulation and discussion of two important necessary conditions for conductivity-concentration dependence being linear. These conditions are related to the structure of the inverse micelles ions at different concentration of the surfactants. We analyze applicability of the Onsager–Fuoss method for calculating ion sizes from the conductivity-concentration dependence. We conclude that this method is not applicable for estimating sizes of ions created by surfactants in nonpolar liquids. In addition, we present and discuss much wider experimental data collected and presented by different authors. Finally, we introduce here notion of “critical concentration of ion-pairs formation” (CIPC). This is critical concentration of surfactant, instead of critical ionic strength discussed in [4]. The last one is not measurable, whereas CIPC could be determined from the experimental data. Discussed here difference between such notions as “electrolyte concentration,” “concentration of inverse micelles ions,” “surfactant concentration” is critical for understanding nature of nonpolar liquids ionization with surfactants. We provide estimate of the CIPC values as a function of ion sizes (Table 1). Then, we compare these values with experiment data for ion-pair. www.electrophoresis-journal.com
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Table 1. Values of the Fuoss equilibrium constant and critical concentration of ion-pairs formation as a function of ion size, assuming that it is identical to the distance of the ions closest approach
Distance of the closest approach (l) [nm] Approximately ion size (d) [nm]
Fuoss constant Mdiss Critical concentration for Bjerrum distance of ion-pairs formation (CIPC), [mol/L] 28 nm Approximately critical conductivity [S/m]
1 2 3 4 5
3 646 318 737 233 24 256 149 769 818 176 85
2.7 × 10−13 4.1 × 10−8 1.3 × 10−6 5.6 × 10−6 1.2 × 10−5
2 Theory Mass action law for the dissociation model Eqs. (7) and (8) can be applied for describing the equilibrium of inverse micelle ions dissociation–association in ion pairs. We stress here that surfactant is not an electrolyte, as is usually assumed for substances described with the dissociation model. First of all, only a fraction of the surfactant molecules participate in ions formation. Second, they mostly perform function of a solvating agent. There is undergoing discussion into how exactly charge separation occurs in nonpolar liquids, that is, what is the chemical nature of the substance that dissociates. We do not address this complex issue here. Instead, following Onsager and Fuoss, we strive to derive some understanding of nonpolar liquid ionization from studying ions association. We postulate that there are inverse micelles with concentration cm in particular liquid, and our goal is to describe their aggregation into ion pairs, driven by pure electrostatic interaction. This approach was suggested by Onsager and Fuoss, who emphasized many times in their papers that investigation of ions association, not dissociation, would yield important insight in ionization process in general [16–18]. Equation (8) for the equilibrium constant of inverse micelle ions’ association into ion pairs can be re-written as the following, using notion of the dissociation constant ␣: Mdiss =
(1 − ␣) ␣2 c m
(12)
This equation is known as Ostwald dilution law. It is usually applied for deriving expression for the dissociation constant, which is less than 1 for a weak electrolyte. This derivation is achieved by assuming that dissociation constant ␣ is a small parameter. This eventually leads to the square root concentration dependence mentioned above in the introduction. There are several important comments that must be made before applying Ostwald dilution law to nonpolar liquids with surfactant. First of all, appearance of inverse micelles ions is not direct result of a certain electrolyte dissociation. Ionization in nonpolar liquids is limited by solvation. Therefore, buildup of steric solvation layers around ions C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
by surfactant molecules is important necessary step of this process. These layers could be structurally concentration dependent. This means that even structure of ions could vary with concentration, which is dramatically different from the aqueous solutions where solvation function is abundant. Saying this, we would apply Ostwald dilution law to inverse micelles ions only. We do not try applying this law for describing dissociation of some electrolyte that gives birth initially to the primary electric charges that are then become solvated by surfactants into inverse micelles. Second, we would like to stress difference between “weak electrolyte” and “solution with very low ionic strength, very small ions concentration.” Term “weak electrolyte” does not imply yet that concentration of ions is low. For instance, it is known that any electrolyte becomes completely dissociated at infinite dilution. This is a well-known Kohlrausch law. Therefore, dissociation constant becomes 1 at infinite dilution and electrolyte becomes a strong one, instead of weak one. This simple analysis shows that concentration of ions must be scaled with some independent parameter for judging about its relative value. Dissociation constant cannot serve this purpose because it is dimensionless. Equation (12) suggests another parameter for such role: reciprocal equilibrium constant (1/Mdiss ) which has the same dimension as concentration. Therefore, in order to derive asymptotic solution for ionic strength at low limit, we should use expansion of Eq. (12) with small dimensionless parameter (cm Mdiss ), instead of the dissociation constant ␣. We reproduce this derivation here following review [4]. As the first step, we reformulate Eq. (12) for introducing this new small parameter for the range of low ionic strength. We can replace the dissociation constant in this equation with ionic strength J, which is defined as following: J = ␣c m
(13)
There are several simple algebraic substitutions shown below: Mdiss =
(1 − ␣)c m c m − c m + (1 − ␣)c m cm − J = = 2 ␣2 c m c m ␣2 c m J2
(14)
which eventually lead to the following quadratic equation that relates ionic strength (fraction of free ions) with total concentration of inverse micelles ions: J 2 Mdiss + J − c m = 0
(15)
Equation (14) has a simple solution for ionic strength as function of inverse micelle ion concentration: √ 1 + 4Mdiss c m − 1 (16) J = 2Mdiss For some unknown reasons, this important equation has evaded text books on electrochemistry for a long time. It presents expression for ionic strength at any concentration of ions. Basically, it includes both weak and strong electrolytes ranges, low and high ionic strength. Calculation of the ionic strength for a given electrolyte concentration requires theory for the equilibrium constant www.electrophoresis-journal.com
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Mdiss . We will use Fuoss’ version Eq. (11), which seems the most accepted in modern electrochemistry. The combination of Eqs. (11) and (16) allows for calculation of the ionic strength at any concentration of inverse micelles ions for a given ion size and liquid dielectric permittivity. This statement is based on one major assumption that the ions retain their solvating layers when they collide. Modern electrochemistry ([2] p. 57) considers several situations that might occur during ion-pair formation. According to this book, there are three different types of ion pairs, depending on the structure of the solvating layers: solvent-separated ion pairs, solvent-shared ion pairs, and contact ion-pairs. It is not clear at this point what type of ion pairs would be formed by the collision of the inverse micelles ions. There are two asymptotic solutions for Eq. (16): one for the high concentration and one for the low concentrations of ions. They are separated by a critical concentration ccr : c cr =
1 Mdiss
This concentration range corresponds to the short distances between ions. Their interaction becomes important. This is the range where ionic interactions control the electrochemistry of the liquid. This is the range where the Debye– Huckel theory [20] of conductivity must be adopted. The asymptotic expression for ionic strength equals: √ √ cm 1 + 4Mdiss c m − 1 4Mdiss c m ≈ = (19) J = 2Mdiss 2Mdiss Mdiss This asymptote is identical to the solution for the weak electrolytes, mentioned by Morrison [5] and Bockris and Reddy [19]. However, it is valid only for sufficiently high concentration of ions. If ion concentration is low, then another asymptotic solution of Eq. (16) emerges. This second asymptotic solution for the low ionic strength is completely different. If we assume that: 1 (20) c m c cr = Mdiss then the two first terms of Eq. (16) when expanded would yield: (21)
This expression for the ionic strength allows us to estimate the concentration dependence of the conductivity. We can neglect relaxation and electrophoretic retardation effects on ion motion due to the very large Debye length compared to the ion sizes in low-conducting liquids where Eq. (20) is valid. Then, introducing conductivity at infinite dilution, we would arrive at the following equation: 2 K = K ∞ c m − 2K ∞ Mdiss c m
C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
This equation for molar conductivity is practically identical to Eq. (10) that was derived from the Onsager–Fuoss conductivity equation in the introduction. Similar derivation was conducted recently by D. Prieve, see [12]. There is coefficient 2 in the second term on the right-hand part of this equation, which makes it different somewhat from Eq. (10). We do not know reason of this difference. Original derivation by Onsager and Fuoss might contain simplification of including such coefficient into the equilibrium constant. Mathematics of their derivation is very complex because they took into account Debye–Huckel effect. This makes comparison quite cumbersome and not well defined. In any case, this coefficient does not affect results qualitatively. The second term in Eq. (22) reflects ion-pairs formation at higher concentrations of ions. If ion concentrations is much less than critical concentration ccr , then this second term becomes negligible and we arrive at a simple linear dependence of the conductivity on ion concentration:
(17)
Let us consider initially the range of the high ionic strength, which corresponds to the ions concentrations where: 1 (18) c m c cr = Mdiss
2 J = c m − 2Mdiss c m
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(22)
K c m c cr = K ∞ c m
(23)
This expression agrees with the well-known Kohlrausch law [19], which states that dissociation of electrolyte is complete at infinite dilution. The novelty of this paper is that we establish necessary condition for Kohlrausch law—concentration of ions must be much lower than the critical concentration, which is simply the reciprocal of the constant for ion/ion-pairs equilibrium. There is a very simple physical explanation for this dependence. The collisions of ions become increasingly rare with decreasing concentration. Eventually, the collision rate cannot compensate for the breakup of the ion pairs. Actually this balance between Brownian-driven collisions and ion pairs breakup is built into the Fuoss equilibrium constant. It was shown in [4] that this constant can be expressed as a ratio of two characteristic times: sm —Smoluchwski time of fast coagulation [21], and d ion-pair breakup time: Mdiss =
NA 2d N0 sm
(24)
Basically, this presentation of the Fuoss constant stresses the fact that electrochemistry at the low ionic strength range is controlled by the diffusion of ions and breakup of neutral ionpairs. The breakup becomes dominant at low ionic strength and all ions become free. Conductivity becomes linear function of ions concentration only if this concentration is much smaller than the critical concentration, expressed with Eq. (20). However, this does not mean it would be linear function of surfactant concentration. Concentration of ions cm is proportional to the surfactant concentration cs only if the number of monomers in the inverse micelle ion m and fraction of the monomers f that are incorporated into these micelles are independent on the surfactant concentration. Therefore, we can formulate three necessary conditions for conductivity being linear function of the surfactant: www.electrophoresis-journal.com
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(i) The concentration of ions is much smaller than the aforementioned critical concentration, which equals to reciprocal equilibrium constant ions–ion pairs; (ii) The number of monomers in the inverse micelle ion is independent of surfactant concentration; (iii) The fraction of the surfactant monomers that is incorporated in the inverse micelles ions is independent on the surfactant concentration. Only if all three conditions are valid, then conductivity would be linear function of the surfactant. In order to verify this theoretical prediction, we would first need an estimate of the critical ionic strength ccr . We can use Eqs. (11) and (17), calculating ccr for various values of ion sizes. We assume that ion size is approximately equal to the distance of closest approach of the electric charges in the ion pair. These values for ccr are shown in Table 1 for various ion sizes. Unfortunately, the concentration of inverse micelle ions and consequently critical ion concentration ccr are not measurable parameters. We should somehow relate this parameter to the experimentally measurable parameters, such as conductivity K [S/m] and concentration of the surfactant cs [%wt] in order to determine range of ion-pairs formation. In principle, one can use analogy with “critical concentration of micelles formation” and introduce critical “concentration of ion-pairs formation” (CIPC) in surfactant solutions in nonpolar liquids. Conductivity measured at different concentrations of surfactants would yield information for estimating CIPC. We will show this in the next section. Unfortunately, it is not sufficient for calculating ccr because two unknown parameters are involved in the relationship between CIPC and ccr : c cr = C I PC
f m
(25)
We do not know the fraction of surfactant that is incorporated into charged inverse micelles (f) and number of surfactant monomers in the single micelle (m). Instead of surfactant concentration, we can try using conductivity directly for estimating ccr . It is known that for aqueous systems conductivity expressed in Siemens (S)/m is about ten times higher than strong electrolyte concentration expressed in mol/L. Conductivity is reciprocally proportional to the ion sizes, which substantially increase in nonpolar liquids. Therefore, difference between these two numbers expressed in different units would be much smaller in nonpolar liquids. There is one more factor that would reduce this difference even more—viscosity. Conductivity is reciprocally proportional to viscosity according to the Walden rule [19]. Viscosity of nonpolar liquids is usually larger than viscosity of water. These two factors ion size and viscosity should make conductivity expressed in S/m and ionic strength expressed in mol/L much closer to each other, comparing to aqueous solutions. Even if they would be still different by two to three times, this is much smaller than the range of the conductivity variation due to the surfactants addition, which is several orders of magnitude. We suggest here to ignore this difference C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
and use conductivity numbers expressed in S/m as direct estimate of the ionic strength in mol/L. This assumption would allow us to test predictions of the conductivity linearity range at least qualitatively. Calculated values of ccr and corresponding Kcr are shown in Table 1. It is seen that the range of conductivity where it is supposed to be linear function of the surfactant concentration expands rapidly with increasing ion size. However, even for ions with sizes around 3 nm, there is a substantial range where conductivity is easily measurable and can be linear function. These conclusions can be verified with analysis of the existing published experiments.
3 Experiment: Review of published data As we mentioned in the introduction, conductivity measurements of various nonpolar solutions have been conducted since late 19th century. There is a recently published review of the early works on this subject [4]. These studies were usually made with simple substances that serve as electrolytes. Application of surfactants for controlling nonpolar liquids electrochemistry was introduced at the end of 20th century, initially with ionic surfactants, and much later with nonionic surfactants. Practically, all studies with ionic surfactant report a linear dependence of conductivity on the surfactant concentration. Here are these papers with reported values of the micelles sizes in some of them: (i) (ii) (iii) (iv) (v)
isopar with 7 nm lecithin micelles [13], benzene with AOT [14], dodecane with 5–10 nm OLOA micelles [6], isopar with 10 nm OLOA micelles [8], heptane with AOT [4].
We reproduce here figures from these published papers. Figure 1 corresponds to the AOT in heptane from the paper [4], Fig. 2A shows conductivity of OLOA in dodecane from the paper [6], and finally Fig. 2B depicts conductivity of OLOA in isopar from the paper [8]. Conductivity for all these measurements is ⬍10−6 S/m. At the same time, the size of micelles exceeds 3 nm for all these surfactants. Modern methods of measuring micelle sizes in these solutions confirmed the very early data by Mathews and Hirschhorn [22] for the size of AOT micelles in dodecane to be in the range from 3.4 up to 9.9 nm. This rather large size of micelles leads to a high value of the CIPC and critical conductivity, according to the data shown in Table 1. An ion size of 3 nm corresponds to a critical conductivity of 1.3 × 10−6 S/m. This critical value is higher than the conductivities measured in the systems (i)–(v) mentioned above. Thus, we can conclude that ionic surfactants exhibit linear conductivity dependence in nonpolar liquids because ionpairs formation is suppressed with sufficiently large ion sizes. Weak electrostatic attraction could not prevent the breakup of www.electrophoresis-journal.com
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Figure 2. Conductivity as a function of OLOA ionic surfactant. (A) Nonpolar liquid is dodecane, reproduced with permission from [6]. Error bars are based on the standard deviation of the measured conductivities. (B) Nonpolar liquid is isopar, reproduced with permission from [8]. Viscosity adjusted conductivity and dimensionless a for OLOA.
formed ion pairs, and collision by diffusion rate cannot compensate for the breakup rate due to the low concentration of ions. There are two more conditions formulated in the previous section for linear conductivity dependence: concentration independent micelle structure and fraction of surfactant incorporated in the charged micelles. Apparently, these conditions are valid for the systems (i)–(v) with ionic surfactants. This interesting conclusion sheds new light on micelle formation in nonpolar liquids. Nonionic surfactants, in contrary, exhibit nonlinear dependence on conductivity that agrees with the theoretical predictions of ion-pairs formation. Figures 3A and B and Fig. 4 copied from the paper [23] illustrate this statement. Similar nonlinearity was observed by Behren’s group [9, 10]. This nonlinearity indicates that positive and negative charges could approach closer when ions are bounded in the ion pair. This can happen, for instance, if ion size is smaller in the case of nonionic surfactants. There is some evidence that this might be the case. For instance, Behren’s group reported on ion size of 2.8 nm for SPAN 85 in hexane. The probability of building ion pairs is a very nonlinear function C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Figure 3. Conductivity of the various nonpolar liquids with nonionic surfactant, reproduced with permission from [23]. (A) Conductivity of various liquids with SPAN 80. d.p. stands for dielectric permittivity of the liquid. (B) Conductivity of the kerosene solutions with nonionic surfactants having various HLB numbers.
Figure 4. Conductivity of toluene with additions of nonionic surfactant SPAN 80, reproduced with permission from [4].
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of the ion size. According to Table 1, when reducing ion size from 3 nm down to 2 nm, CIPC drops 30 times. Therefore, even small reduction in the ion size causes a very large effect on ion-pairs formation. There is one alternative explanation that should be taken into account. Addition of surfactant could affect viscosity of the liquid. Conductivity decreases with increasing viscosity, according to the Walden rule [19]. Therefore, in principle, decreasing conductivity growth rate as shown in Fig. 4 could be associated with increasing viscosity of the liquid. There are two ways for resolving this ambiguity. First of all, direct viscosity measurement would yield the answer. According to the data shown in Fig. 4, viscosity should increase roughly two times at the 10% surfactant content for explaining observed decay in the conductivity growth rate. Unfortunately, viscosity data for these solutions are not available. There is also possibility of detecting ion pairs directly with other experimental methods. There is review of such methods given in the paper [4]. It seems that dielectric spectroscopy might be the best method for this purpose. Ion pairs should carry substantial dipole moment. There is a method of dielectric spectroscopy developed by Fuoss group in 1930s for monitoring this dipole moment. It is important to keep in mind possible alternative interpretations of conductivity data for future experiments. Onsager–Fuoss conductivity theory with ion pairs allows for fitting nonlinear conductivity dependence, as shown in Fig. 4. There have been many papers published in the mid20th century that used this fitting procedure for estimating ion size in nonaqueous solutions. Review of these papers can be found in the paper [4]. Unfortunately, this procedure does not work in case of surfactants and inverse micelle ions. The Onsager–Fuoss law analog for such systems would look like: ∞ K = K eq
f ∞ c s − K eq Mdiss m
f m
2 c s2
(26)
where we used Eq. (3) for relating ionic strength with surfactant concentration. This equation contains three unknown parameters: Keq , f/m, and Mdiss . On the other hand, the fitting procedure yields only two parameters, coefficients for the linear, and quadratic terms. This peculiarity of surfactant solutions reflects the fact that they perform function of solvating agent first of all. They cannot be considered as electrolytes in traditional aqueous electrochemistry sense.
4 Concluding remarks The addition of either ionic or nonionic surfactants to nonpolar liquids enhances conductivity by orders of magnitude. This effect can be used for controlling the electrochemical environment in these liquids, which is especially important for electrokinetic phenomena, such as electrophoresis. C 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Surfactant performs function of a solvating agent. The resulting inverse micelle ions form ion pairs if concentration of the surfactant exceeds a certain critical value: the CIPC. Only a fraction of the surfactant molecules (f) is incorporated into inverse micelle ions, each ion require several surfactant molecules (m) for creating a solvating layer around the central electric charge. Consequently, concentration of ions is proportional, but not equal, to the concentration of surfactant. There is a critical concentration of ions that corresponds to the CIPC, above which ions form ion pairs. It is proportional to CIPC and unknown parameter f/m. On the other hand, this critical ion concentration is simply the reciprocal of the Fuoss constant for the ions/ionpairs equilibrium. This constant depends on the distance of the closest approach between cation and anion in the ion pair. This distance can be approximately estimated as ion size. Calculations indicate that critical ions concentration lies within a range of 10−11 –10−7 mol/L when ion size ranges from 1 to 3 nm. Therefore, even small variations in inverse micelle ion size cause large variation in the range of ion-pairs formation. This effect can be verified with conductivity measurements. The formation of ion pairs causes nonlinear dependence of the conductivity on the ion concentration, according to Onsager and Fuoss. This means that below the CIPC, conductivity should be a linear function of surfactant concentration, and becomes nonlinear above the CIPC. Experiment confirms that ionic surfactants exhibit linear conductivity dependence in a variety of nonpolar liquids. Estimated ion size in these systems varies from 3 to 10 nm and maximum conductivity does not exceed 10−6 S/m. This range of linearity agrees with the calculated linearity range for given values of ion sizes. This linear conductivity dependence for ionic surfactants indicates that parameters f and m are concentration independent. This means that the structure of inverse micelles and their fraction in the surfactant content remain constant and concentration independent in nonpolar liquids. This can be the result of strong charge–dipole interaction inside of the charged inverse micelle core. This interaction is much stronger than traditional dipole–dipole interaction of surfactant molecules polar parts that drive formation of neutral micelles. Therefore, CIPC and critical concentration of micelles formation are different and independent parameters. The first one reflects charge–dipole interaction within the micelles core, the second one depends on dipole–dipole interaction. Finally, experiments with nonionic surfactants show that conductivity becomes nonlinear function of the surfactant concentration above certain level, which depends on the nature of the surfactant and of the liquid. This might be explained assuming that ion sizes are smaller for nonionic surfactants. This would lead to the lower CIPC value for such systems. The author has declared no conflict of interest. www.electrophoresis-journal.com
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