Effect of Microheterogeneous Environments on Reaction Rate: Theoretical Explanation through Different Models

Abstract

Microheterogeneous systems (typically micelles and reverse micelle systems) refer to aggregate systems in which different structural shapes are formed by amphiphiles in water or other similar media. Amphiphilic molecules undergo a self-assembly process known as micellization through hydrophobic or H-bonding interactions. The Microenvironment consisting surfactants changes the physicochemical behavior of the system and can catalyze or inhibit reaction rates. In addition, thermodynamic parameters such as standard enthalpy of activation, standard entropy of activation, energy of activation etc. may vary in this environment. Thus, for last few decades, the enormous interest among the researchers in the study of the effect of microheterogeneous environments on reaction rate. It is also challenging how to fit experimental results with different models. Every model has its own significance and limitations.

1. Introduction

For last few decades, there has been enormous interest among the researchers in the study of the effect of micro-heterogeneous environments on reaction rate. Amphiphile molecules generally form such environments due to their hydrophilic head and hydrophobic tail groups, they can able to interact with both polar and nonpolar solvents (/compounds) and easily form micelle or reverse micelle structures [1]. Reactants accommodated in surfactant-based organized assemblies, such as micelles, microemulsions, and vesicles, often achieve a greater degree of organization compared to their geometries in homogeneous continuous solution, can mimic reactions in biosystems, and have potential for energy storage [2]. Micelles can cause an acceleration (/catalyze) or inhibition of a given chemical reaction relative to the equivalent reaction in reaction medium due to the concentration effect in the micellar pseudophase and can change the reaction pathway [3, 4, 5]. Reverse micelles also play an important role to alter the reaction pathway as well as rate of reactions. Surfactants in nonpolar solvents can solubilize considerable amounts of water with the formation of inverted or reversed micelles. The size of the spherical reverse micellar cavity known as the water pool (w = [H₂O]/[Surfactant]) [6, 7] carries the most important structural feature of the reverse micellar system and controls the reaction rate. Coulombic, hydrophobic and charge transfer can form a variety of interactions between the above-mentioned organized media and incoming species. All of these interactions can play an important role in altering rates and reaction pathways by altering the microenvironment around the reacting species. Therefore, micro heterogeneous environments are very important topic to many researchers. Thermodynamic parameters for the micelle formation (i.e. the standard free energy of micelle formation,${\mathrm{G}}_{\mathrm{m}}^0$; the standard enthalpy of micelle formation, ${\mathrm{H}}_{\mathrm{m}}^0$ etc.) obtained from the temperature dependence of the CMC and the degree of counter ion binding have been reported in several works [8, 9]. Most kinetic data in kinetic studies are interpreted with one of the proposed micellar models such as pseudo-phase (PP) model [3, 4], ion-exchange PP model [10], Columbic model [11], enzyme kinetic type model [12], mass-action and multiple separation micellar models [13, 14]. The chapter deals with the effect of micro-heterogeneous environments on reaction rates and briefly discusses various aspects of different kinetic models for treating micellar catalysis or inhibition on the rate of reactions.

2. Surfactants

Surfactants are surface-active agents, active on surfaces. They are compounds that lower the surface tension (/interfacial tension) due to the adsorption at fluid surfaces and interfaces and change the property of the interface considerably. Some example of surface and interface are solid-vapor (surface), liquid-vapor (surface), solid–liquid (interface), solid–solid (interface), liquid–liquid (interface) etc. They are usually organic compounds that are amphiphilic in nature (thus sometimes called amphipathic structure) that means they contain both tails (lyophobic groups) and heads (lyophilic groups). Their heads are polar, or hydrophilic, and their tails hydrophobic (Figure 1). They are soluble in both organic solvent and water. When the solvent is water, the lyophobic and lyophilic parts are called hydrophobic and hydrophilic, respectively. Surfactant is one of the most frequently used chemical in the everyday life with a wide range of industrial application such as in detergents [15], pharmaceuticals [16], cosmetics [17], textiles [18], medicine [19], chemistry [20] etc.

Figure 1.

Different types of surfactants: Their hydrophobic and hydrophilic parts.

2.1 Chemical structure of surfactants

The aliphatic, aromatic hydrophobic part of the surfactant contains 8 to 22 carbons in linear or branched chain form. The hydrophobic part is usually obtained from fats, oils, synthetic polymers or synthetic alcohols. The surfactant molecules also have a functional group called the hydrophilic group or polar group. The choice of polar group and comparative size of the hydrophobic and polar groups are important factors in determining the properties and surfactant physiochemical behavior in water. Depending on the type of charge on the head groups attached to their chains, surfactants are mainly divided into two categories, ionic and non-ionic, also zwitterionic (or amphoteric) and Gemini surfactants. A very brief description of the different types of surfactants is given below:

2.1.1 Cationic surfactants

Cationic surfactants are substances whose head groups are positively charged. Thus, these are useful for adsorption on negatively charged surfaces. Some well-known cationic surfactants are long chain amines ($\mathrm{R}\stackrel{+}{\mathrm{N}}{\mathrm{H}}_3{\mathrm{X}}^{-}$), quaternary ammonium salts ($\mathrm{R}\stackrel{+}{\mathrm{N}}{\left({\mathrm{CH}}_3\right)}_3{\mathrm{X}}^{-}$) and quaternary salts of polyethylene oxide amine derivatives ($\mathrm{R}\stackrel{+}{\mathrm{N}}\left({\mathrm{CH}}_3\right){\left\{{\left({\mathrm{C}}_2{\mathrm{H}}_4\mathrm{O}\right)}_x\mathrm{H}\right\}}_2{\mathrm{Cl}}^{-}$). The most common cationic surfactants are shown schematically in Figure 2.

Figure 2.

Chemical structures of some common cationic surfactants.

2.1.2 Anionic surfactants

The head groups (or functional groups) of anionic surfactants are negatively charged and neutralized by alkali metal cations. The head groups of anionic surfactants are based on carboxylates (e.g. soaps, RCOO⁻Na⁺), sulphates (RSO₄⁻Na⁺), phosphates (${\mathrm{RPO}}_4^{2-}2{\mathrm{Na}}^{+}$) and benzene sulphonates (RC₆H₄SO₃⁻Na⁺). Soaps are the largest single type of anionic surfactants obtained by saponification of natural oils and fats. The use of anionic surfactants are very sensitive to hardness of water because some of the anionic surfactants (e.g. salts of fatty acids) are precipitated from the aqueous solution in presence of salts containing Ca²⁺ and Al³⁺ ions. Figure 3 shows the most useful anionic surfactants.

Figure 3.

Chemical structures of commonly used anionic surfactants.

2.1.3 Non-ionic surfactants

In these types of surfactants, they do not have any significant charge on their surface-active part. Therefore, there is no electrical interaction between the head groups. These surfactants are stable in presence of electrolysis. They usually generate less foam than the ionic surfactants. They are insensitive to water hardness. They are mainly used as dispersing agents for pigments in paints, foam control agents, also used in textile detergents, metal cleaning and shampoos. These surfactants have covalently bonded oxygen containing hydrophilic groups, which are bonded to hydrophobic parent structures (Figure 4). The common examples of non-ionic surfactants are polyoxypropylene glycols, polyoxyethylene mercaptants [RS (C₂H₄O)_xH], alkyl phenol ethoxylene [RC₆H₄ (OC₂H₄)_xOH]; alcohol ethoxylates [R(OC₂H₄)_xOH] etc.

Figure 4.

Chemical structures of common non-ionic surfactants.

2.1.4 Zwitterionic surfactants

Zwitterionic surfactant consists of two oppositely charged groups, both charges, positive and negative present on the surface-active part of the same molecule. The cationic part is based on primary, secondary, tertiary amines or quaternary ammonium cations (Figure 5). The anionic part can be more variable and include sulfonates, phosphates etc. Among them, the long chain amino acids (RN⁺H₂CH₂COO⁻) are the well-known examples of the zwitterionic surfactants (Figure 5). Their main advantage is that they are compatible with both anionic and cationic surfactants due to the presence of both positive and negative charges. Most of these surfactants are sensitive to pH. They are considered zwitterionic at pH 7. However, they show the properties of anionic surfactants at high pH whereas they behave as cationic surfactants at low pH.

Figure 5.

Chemical structures of some zwitterionic surfactants.

2.1.5 Gemini surfactants

A surfactant contains one polar group. In recent times, there have been considerable research interest in certain dimeric surfactants, which contain two or three hydrophobic tails and two head groups linked together with a short spacer (Figure 6). They are referred to as Gemini surfactants. The properties of these surfactants are highly dependent on the structure of these three parts of the molecule. The interfacial effects of these surfactants can be significantly greater than that of surfactants with single hydrophilic and hydrophobic groups [21]. Gemini surfactants can be ionic with negative, positive or both types of charge or can be non-ionic. These surfactants are less water soluble due to their long hydrophobic part and have an affinity for adsorption at the interface.

Figure 6.

Chemical structures of some Gemini surfactants.

3. Surfactant aggregation (/self-assemble): Formation of micelle and reverse micelle

Surfactants have polar hydrophilic parts, called head groups, and non-polar hydrophobic parts, called tail groups. This amphipathic character of surfactants may produce ensembles viz., micelles, vesicles, fibers, discs, lamellae, bicontinuous phases, liquid crystals, and 3D networks [22, 23, 24, 25, 26]. In micelle, the outer core of it is made up of hydrophobic part while the polar head groups are positioned at the inner micelle-water interface (Figure 7c). The polarity of the medium is an important factor for surfactant association. Usually, micellization occur in aqueous solution. But in nonpolar medium (benzene, heptane, octane, cyclohexane etc.), formation of micelle is totally absent; rather a reverse orientation of surfactants occurs, where the tail groups are outside and head groups are inside the micelles, called ‘reverse micelles’ (Figure 7d). A trace amount of water helps easier and stable formation of reverse micelle. The other variations in shape in micelle clusters are lamellar, cylindrical, vesicle, liposome (Figure 7) etc.

Figure 7.

(a) monomer, (b) monomer layer and different surfactant aggregations: (c) micelle (spherical shape) (d) reverse micelle (e) lamellar (f) micelle (cylindrical shape) (g) vesicle (h) liposome.

Micelles are formed only above a minimum concentration that are characteristics of the system.

The lower limit of concentration of the surfactant at which micellization starts spontaneously is called as critical micelle concentration (CMC). The common names and CMCs of different surfactants are shown in Table 1. It is significant that micelle formation becomes easier for surfactants with greater hydrophobicity (or increased chain length) in a homogeneous series. Other factors that can affect the formation of micelle are solvent polarity and type, temperature, pressure, presence of additives (e.g. salts) etc.

Surfactant type	Name and structure of surfactant	CMC (mol L^−1)
Ionic surfactants	CTAB,	(0.98–1.04) × 10^−3 [27, 28]
	TTAB,	(3.42–3.50) × 10^−3 [4, 29, 30]
	DTAB,	(13.81–14.59) × 10^−3 [3, 30, 31]
	CPC,	(0.9–1.03) × 10^−3 [32, 33]
	SDS,	(6.81–8.6) × 10^−3 [27, 34]
Non-ionic surfactants	AOT, sodium bis (ethylhexyl) sulfosuccinate	(2.66–2.68) × 10^−3 [32, 35]
	Brij-35	(1.0–2.0) × 10^−4 [36]
	Triton X 100	(0.25–0.27) × 10^−3 [37, 38]
	Tween 20	(4.88–4.99) × 10^−5 [39, 40]
	Tween 40	(2.26–3.33) × 10^−5 [39, 40]
	Tween 60	(1.67–2.1) × 10^−5 [39, 40]
	Tween 80	(1.5–1.89) × 10^−5 [39, 41]
Zwitterionic surfactants	CAPB,	(2.8–3.1) × 10^−3 [42, 43]
	CHAPS,; 3-[(3-Cholamidopropyl) dimethylammonio]-1-propanesulfonate hydrate	(4.0–5.65) × 10^−3 [44]
Gemini surfactants	Anionic gemini; Sodium2,3-didodecyl-1,2,3,4-butane-tetracarboxylate; Cationic gemini; 1,10-(alkane-1, s-diyl)bis(1-dodecyl; pyrrolidinium)bromide (C12-C3-C12)PB	8.9 × 10^−5 [45]; (0.41–0.66) × 10^−3 [46]

Table 1.

Essential surfactants and their CMC values.

4. Structure of micelle

The hydrocarbon chain of the surfactant molecule can form the inner part of the micelle and its radius is about the range of a fully extended hydrophobic chain as shown in Figure 8. A stern layer covers the core and exists between the core surface and the hydrodynamic shear surface of ionic micelles. The stern layer contains the n fully ionized head groups and (1−α)n counter ions, where α is the degree of ionization and for most ionic micelles, it is 0.2 to 0.3; n is the aggregation number (the number of monomers required to form an aggregate is known as aggregation number) [47]. The remaining ‘αn’ counter ions neutralizing the micelle charge are distributed outside the shear surface in a Gouy-Chapman diffuse ionic atmosphere. The Gouy-Chapman layer is diffuse and surrounds the Stern layer i.e. the outer surface of the Stern layer.

Figure 8.

A two dimensional schematic representation of a spherical ionic micelle.

The structure of micelles formed by non-ionic surfactants is almost identical to micelles formed by ionic surfactants except the counter ions. In ionic micelles, the counter ions are found in the outer regions, while in non-ionic micelles there are loops of hydrated polyethylene oxide chains. They are larger than ionic micelles and sometimes extend into ellipsoidal or rod-like structures. They contain a hydrophobic core, formed by the hydrocarbon chains of the surfactant, surrounded by a shell (palisade layer) made by the oxyethylene chains of the surfactant.

5. Structure of reverse micelle

In non-aqueous solution, the micelles are formed, known as reverse or inverted micelles. Reverse micelles are spherical molecular aggregates dispersed in an oil phase. The polar solvent within the droplet is covered by the polar head groups of surfactant (Figure 7d). For most of the systems, a co-surfactant (usually long chain alcohol) is added to increase the stability. The size of the spherical reverse micellar cavity corresponds to hydrodynamic radius (r_h) is reflected by the molar ratio of water to surfactant, which is known as water pool (w) [6].

$$w=\frac{\left[{\mathrm{H}}_2\mathrm{O}\right]}{\left[\text{Surfactant}\right]}$$

The temperature, nature of oil (non-polar solvent) and the dispersed phase volume fraction (ϕ) have slight effects on the size of reverse micellar cavity. The size of the water pool can be precisely controlled by w on the nanometer scale. The parameter, w is the key factor controlling the most important structural features of the reverse micellar system [48].

6. Factors affecting CMC

6.1 Nature of chain and counter ion

It is imperative that micelle formation becomes easier for surfactants having greater hydrophobicity (or increased chain length) in a homologous series. Increase in the number of carbon atoms in unbranched hydrocarbon chain leads to decrease in CMC, since

$$log\left(\mathrm{CMC}\right)=\mathrm{A}–\mathrm{B}\times \mathrm{m}$$

Where, m is the number of carbon atoms in the hydrophobic chain [49]. Increased hydrophobicity provided by increasing chain length leads to an increase in micellar size. The aggregation number increases with increase in hydrocarbon chain length of the surfactant molecules (Table 2). The CMC decreases with the increase in chain length.

Surfactant	Aggregation number
Cationic surfactants:
C10H21N^+(CH3)3Br^−	36
C12H25N^+(CH3)3Br^−	50
C14H29N^+(CH3)3Br^−	75
Anionic surfactants:
C10H21SO4^−Na^+	50
C12H25SO4^−Na^+	80

Table 2.

Effect of the hydrophobic chain length on aggregation number in water at 298 K.

The CMC also depends on the size of the hydrophilic group. As the size of the hydrophilic group gets larger, the repulsion between them increases. The change of counter ion to one of greater valence leads to a decrease in CMC and increase in aggregation number. An increase of CMC is noted with increase in hydrated radius.

6.2 Solvent polarity and type

The nature of the solvent can affect the CMC. The polarity of the medium favors surfactant association. Nonpolar medium offers environment similar to the surfactant tail so that their tendency of self-association is reduced. In a good nonpolar medium like cyclohexane, carbon tetrachloride, heptane, octane, decane etc., favors for reverse orientation of the surfactants instead of normal micelle.

6.3 Temperature

Temperature is an important factor that can affect the micellization phenomenon, although the effect of temperature on micellization may not be straightforward. The effect of temperature on the formation of micelles depends largely on how temperature affects the solubility and other behaviors of surfactants in solution. In general, the solubility of surfactants in water does not increase dramatically with temperature. Desolvation and change in solvent structure play an essential role in this regard. Typically, micelle formation is favored by an increase in temperature in the lower temperature range compared to the higher temperature range.

The formation of micelle may be prevented by a strong electrostatic repulsion of the desolvated head groups. The situation is complicated by the change in polarity of the medium at high temperature. At high temperature, the solution becomes cloudy due to phase separation resulting from desolvation of non-ionic surfactant polar head groups. The temperature at which the phenomenon begins is known as the cloud point. Their cloud points often refer the temperature stability of non-ionic surfactants. Aqueous solutions of non-ionic surfactant micelles exhibit thermoreversible phase separation phenomena on heating/cooling through a cloud point (Tc) [50]. Over the past few years, the cloud point phenomenon has been significantly applied in extraction and separation science. It has also been successfully used for the recovery of various species such as organic compounds, inorganic metal ions, biological analytes, etc. Watanabe et al. [51] illustrated the extraction of soluble metal [Zn(II)] chelates from aqueous solutions by cloud point extraction (CPE). When modulating the cloud point, it seems easier to reduce than to increase. Many common salts, such as NaCl, are highly effective cloud point suppressants. In contrast some salts, such as those containing I⁻¹, [Fe(CN)₅NO]⁻² and SCN⁻¹ anions, can raise the cloud point.

The surfactants have characteristic temperature dependent phase behaviors, shown in Figure 9. The Kraft temperature is a point of phase change below which the surfactant (monomer) remains in crystalline form, even in aqueous solution (Figure 9, zone C). In zone A, only surfactant monomers occur in solution and in zone B, monomers remain in equilibrium with micelles. At the point P, all the three phases (monomer, crystal and micelle) co-exist, known as invariant point and the corresponding temperature is called Kraft temperature (T_K). At this temperature, the micelles are formed and the solubility is significantly increased and equal to the surfactant’s CMC at the same temperature. Actually, micelles form only above the CMC and above the Kraft temperature. Below the Kraft temperature, the maximum solubility of the surfactant will be lower than the CMC, meaning micelles will not form. Kraft temperature increases with a long hydrophobic carbon chain as its solubility in water decreases. Meanwhile, the Kraft points have no relationship with a series of methylene linking groups. The Kraft point of the Gemini surfactant containing benzyl is 1°C higher than those containing methylene series, although the hydrophobic carbon chain is the same in both cases [52]. Usually, electrolytes raise the Kraft temperature, whereas there is no general trend for the dependence on counter ions. However, the Kraft point is typically much higher in presence of divalent counter ions than monovalent counter ions. For cationic surfactants, the Kraft point is typically higher for bromides than chlorides, and still higher for the iodides.

Figure 9.

The schematic representation of the different phases of surfactant at different temperatures and T_K represents a ‘Kraft temperature’.

6.4 Pressure

Pressure also has a significant effect on the self-organization of surfactants. Pressure initially retards the association and after a certain limiting value, known as threshold value (100–200 MPa), the process is favored. The release of surfactant monomers from the micelles in the lower range of pressure and their association at higher pressure together with the changed dielectric constant of the solution play the specific roles in surfactant organization. This has been supported by the measurement of aggregation number, which shows a minimum for ionic surfactants and a rapid initial decrease for non-ionic surfactants with respect to pressure. The effect of pressure on CMC for ionic surfactant solutions is easily determined from the conductivity measurements of the solution at elevated pressures [53]. On the other hand, no direct experimental determination of the effect of pressure on CMC for non-ionic surfactant solutions has been reported, except for high-pressure NMR techniques. For example, ¹³C-NMR experiments have been reported in the literature [54] to obtain CMC and micelle aggregation numbers for non-ionic surfactant solutions at ambient pressure. The CMC’s for non-ionic surfactant solutions at elevated pressures have been calculated, however, based on measured aggregation numbers at ambient pressure and estimated hydrocarbon compressibilities in the micelle core. These calculations indicate that the cmc increases with increasing pressure [55].

6.5 Presence of additives

Additives may have significant effects on surfactant self-organization [56, 57]. A salting out effect of salts influencing the surfactant activity to assist easier aggregation may arise. The study of salt effects is important as amphiphiles are mostly handled in electrolyte environments in chemical research. Non-electrolytes may both increase and decrease micellization tendency of surfactants. The matter is complicated because additives can affect solvent structure and polarity and interact directly with surfactants. Urea and guanidine hydrochloride are obvious in this case, they greatly inhibit micellization and can break the water structure [57].

7. Determination of CMC

There are various techniques/methods have been frequently used for determination of CMC values such as Conductometric, tensiometry, cyclic voltammetry, densiometry, dye solubilization, UV–visible and fluorescence spectroscopy etc. Different mathematical approaches have also been applied to determine the CMC values from the raw experimental data. In this chapter, some techniques are discussed briefly.

7.1 Conductometric

Conductivity is a common technique to determine CMC for ionic surfactants, which behave as electrolytes in water. This technique cannot be used for non-ionic surfactants since these surfactants have a negligible effect on the conductivity of the solution. Conductivity measurements are usually carried out using a digital conductivity meter (typically cell constant 1.0 cm⁻¹) of surfactant solution in water. The conductivity meter is calibrated using aqueous potassium chloride (KCl) solutions in the proper concentration range before measurement. The measured conductivity values have been multiplied with cell constant to obtain the specific conductivity (κ, kappa) values.

Two linear segments with different slopes can be determined and identified to the monomeric and micellar states of surfactants in solution. In particular, the increase in conductivity with concentration is greater when the surfactants in solution are present as unimers than in the presence of micelles [34, 58]. CMC can be calculated from the breakpoint of specific conductivity versus concentration curve shown in Figure 10.

Figure 10.

Variation of conductance with different concentrations of CTAB.

7.2 Tensiometry

The surface tension of surfactants at various concentrations is measured by using tensiometer. The tensiometer should be properly cleaned before the measurement and the reliability of the instrument is verified by measuring the surface tension of pure water. The value of pure water should be 71.97 mN/m at 298.15 K. Typically, a surfactant stock solution is added to pure water, the solution is stirred for 1 min before the surface tension is measured, and then the reading is taken. The surface tension is linearly dependent on the logarithm’s surfactant concentrations [30, 32]. However, above the CMC, the surface tension is independent of the surfactant’s concentration. Mostly, CMC is the point of intersection between two slopes. The intersection points are deteriorations of the straight line of the linear dependent region and a straight line passing through the flat terrain where no change in surface tension as shown in Figure 11.

Figure 11.

The observed surface tension vs. concentration profile for CTAB.

7.3 Fluorometry

Fluorescence measurements are taken using a fluorimeter of a 10 mm path length quartz cuvette. A fluorescence probe (pyrene) is used in very low concentration around 2 μM. In this method 1st excited the probe in its excitation wavelength and then measure the emission spectra. The slit widths are kept in fixed for excitation and emission.

The fluorescence spectra of pyrene is shown in Figure 12a in presence of AOT. The spectra showed five emission peaks at 375 (I₁), 380 (I₂), 385 (I₃), 390 (I₄) and 395 (I₅) nm for pyrene [58, 59]. In polar solvents i.e. water, MeOH etc., pyrene shows high fluorescence intensity for emission peaks I₁ and I₃. Below the CMC (absence of micelles), pyrene senses the polar environment of methanol (MeOH) molecules. Consequently, the ratio of fluorescence emission intensities corresponding to the first and third vibrational peaks (I₁/I₃) is high. But above the CMC (presence of micelles), pyrene molecules are solubilized in the interior micellar phase due to high hydrophobicity of pyrene. This is a non-polar like solvent, so the environment sensed by pyrene is less polar. Therefore, the ratio I₁/I₃ decreases (Figure 12b). Such a decrease indicates that the microenvironment around fluorescent probe changes with surfactant concentrations becoming more hydrophobic, because of probe (pyrene) interactions with the surfactant micelles. Similar sigmoid shaped curve have been obtained in presence of SDS, SDDS (N-lauroyl sarcosine sodium salt) [60].

Figure 12.

(a) Fluorescence spectra of pyrene in presence of AOT in MetOH (b) I₁/I₃ vs. AOT concentration profile for pyrene. The sigmoid line corresponds to the Sigmoid-Boltzmann equation (SBE) fitting.

7.4 Refractive index

In refractive index experiment, the CMC of surfactant is determined by the following method. Firstly, a concentrated surfactant solution is added over time to a fixed volume of distilled water in a beaker with an automatic burette. Secondly, the solution in the beaker is stirred using a magnetic stirrer. The control program automatically measures the refractive index of the sample at regular time intervals (typically 3–5 s) with the addition of surfactant solution. A plot of refractive index versus concentration is also displayed in the program window. In fact, the automatic burette is operated at a constant speed and the time axis of the chart can be converted to concentration units. The CMC of the surfactant is measured by a breakpoint from the plot under investigation at constant temperature.

The working principle of CMC detection using the refractive index depends on two factors: (i) higher refractive index of micelles than monomers and (ii) surfactant molecules adsorption on the sensing tip. The hydrophobic fiber core is made of silica and the bare core in the sensing region is immersed in the sample solution. The surfactant molecules begin to aggregate at the air–solution interface with increasing concentration. The hydrophilic parts of the surfactant molecules begin to interact with the hydrophilic surface of the sensing tips above the CMC [61]. A clear indication of CMC at this point can be obtained due to the combined effects of micelle formation and surfactant adsorption that increases the refractive index more rapidly with concentration. Figure 13 is the plot of concentration vs. refractive index to depict the CMC of SDS. The CMC value is determined from the intersection of the lines above and below the breakpoint.

Figure 13.

Plot of refractive index versus concentration of SDS at 298 K. The intersection of the breakpoint indicates the CMC of SDS.

7.5 Viscometry

The critical micelle concentration of the surfactant is determined by measuring viscosity as a function of surfactant concentration using the viscometer by the Poiseuille equation,

$$\eta =\frac{\pi \mathrm{\Delta }{Pr}^4}{8l\mathrm{V}}\mathrm{\Delta }t$$

Whereas ΔP denotes the pressure in tube, r is tube radial, l is tube length and V is tube volume of a viscometer. All these parameters are kept constant during the measurements. Δt is the flow time. The viscosity of the pure solvent (η₀) and the viscosity of the surfactant solution (η_s) can be estimated by the flow time ratio,

$$\frac{{\eta }_0}{{\eta }_{\mathrm{s}}}=\frac{\mathrm{\Delta }{t}_{\mathrm{s}}}{\mathrm{\Delta }{t}_0}$$

The flow time of the solution containing surfactant increases relative to that of a pure solvent and hence the viscosity increases. The concentration of the surfactants is plotted against the estimated viscosity shown in Figure 14.

Figure 14.

Plot of viscosity vs. concentration of the surfactant.

From Figure 14, it is clear that there is a sharp change in the linear viscosity increase in both surfactant systems, AOT and SDS [59]. The slope of the plot line changes after a critical point. A critical point is initially reached, with a gradual increase in the surfactant concentration of the dispersion. Afterwards, the surfactant molecules redistribute into the solution to form micelles. The viscosity of the solution increases sharply in micelles as the micellization process leads to a more compact solution structure. It is then possible to estimate the CMC from the intersection of the two straight lines in Figure 14.

7.6 Density measurements

Density is an important physical parameter of solutions that varies with the state of aggregation of surfactants. In particular, the increase in the density of a monomer solution of the surfactant per unit mass is greater than its aggregation state where micelles are present [62]. It is related to the different volume fractions of monomers and micelles in solution. Monomers have a higher volume fraction than micelles due to higher hydration. Therefore, water is more bounded in the presence of monomers than micelles. Micellization, indeed, is a dehydration process. It releases more free water than bound water. Consequently, the increase in volume per unit mass of the surfactant as monomers is less than that of micelles.

The CMC of the surfactant can be determined from the deflection point [62] of the density versus concentration plot shown in Figure 15. The extent of the increase in density in the monomer state (slope) depends on the different grade of hydration and the molecular weight of the surfactant.

Figure 15.

Schematic representation of density versus surfactant concentration, the defection point shows the CMC of the surfactant.

However, the density measurement technique is not suitable for CMC determination of non-ionic surfactants because the density variation is not appreciable with low surfactant concentrations and becomes comparable to pure water.

8. Different models for micellar catalyst reactions

Several models have been developed to explain the surfactant effects on reaction rates, both pre-micellar and post-micellar. A few models are briefly discussed in this section.

8.1 Merger and Portnoy model (/simple distribution model)

Merger and Portnoy [63, 64] proposed the first kinetic model, which related micelles as enzyme like particles and successfully fitted inhibited bimolecular reactions like basic hydrolysis of p-nitrophenyl acetate, mono-p-nitrophenyl dodecanedioate and p-nitrophenyl octanoate in presence of laurate micelle.

8.1.1 The fundamental concepts and assumptions of this model

In a homogeneous surfactant solution (above the critical micelle concentration), the reactive site of a substrate may exist in one or more of the following environments: the micelle interior, the micelle-water interface, and the bulk solvent. The nature of the micelle interior, formed by the lyophobic portion of the surfactant, is very complicated. From high-resolution NMR experiments, it appears that (i) the centre of the micelle resembles a liquid hydrocarbon and (ii) water can penetrate the micelle so that part of the alkyl chain (possibly the first five carbons from the ionic group) is exposed to the solvent. Based on different site of micellar phase, Menger and Portnoy in their model have been assumed that the variation of the rate with surfactant concentration depends on the distribution of substrate between aqueous and micellar pseudophase and they proposed the following Figure 16.

Figure 16.

Different steps in Merger and Portnoy model.

Where, D_n, S and D_nS represent micellar surfactant (/detergent), free substrate (e.g. ester) and adsorbed substrate respectively. k₁ and k₂ are the first order rate constants for the reaction in the aqueous and micellar pseudophase, respectively. In order to evaluate the micelle concentration, they make use of the ‘phase separation’ concept, which assumes that the unassociated surfactant concentration remains constant above the CMC.

If the average number of molecules per surfactant micelle is n (i.e., aggregation number of the micelle), then the micelle concentration is approximated by Eq. (1)

$$\left[{\mathrm{D}}_{\mathrm{n}}\right]=\frac{{\left[\mathrm{D}\right]}_{\mathrm{T}}-\mathrm{CMC}}{\mathrm{n}}$$E1

Where, [D]_T signifies the total concentration of surfactant and [D_n] is the micellized concentration of surfactant. If the average number of molecules per laurate micelle is 33 [64] then the micelle concentration is approximated by Eq. (1) as

$$\left[{\mathrm{D}}_{\mathrm{n}}\right]=\frac{{\left[\text{laurate}\right]}_{\mathrm{T}}-\mathrm{CMC}}{33}$$

The rate constant, k₂ can be now determined in the following way:

The binding constant (K) according to Figure 16 is given by

$$\mathrm{K}=\frac{\left[{\mathrm{D}}_{\mathrm{n}}\mathrm{S}\right]}{\left[{\mathrm{D}}_{\mathrm{n}}\right]{\left[\mathrm{S}\right]}_{\text{free}}}$$E2

(Or,) $\left[{\mathrm{D}}_{\mathrm{n}}\mathrm{S}\right]=\mathrm{K}\left[{\mathrm{D}}_{\mathrm{n}}\right]{\left[\mathrm{S}\right]}_{\text{free}}$and K expressed in terms of the concentration of micelle rather than micellized surfactant.

Now, the initial concentration of substrate written as

$$\begin{array}{l}{\left[\mathrm{S}\right]}_0={\left[\mathrm{S}\right]}_{\text{free}}+\left[{\mathrm{D}}_{\mathrm{n}}\mathrm{S}\right]\\ ={\left[\mathrm{S}\right]}_{\text{free}}+\mathrm{K}\left[{\mathrm{D}}_{\mathrm{n}}\right]{\left[\mathrm{S}\right]}_{\text{free}}\left(\text{substitute}\left[{\mathrm{D}}_{\mathrm{n}}\mathrm{S}\right]\text{from}\mathrm{eq}.2\right)\\ =\left(1+\mathrm{K}\left[{\mathrm{D}}_{\mathrm{n}}\right]\right){\left[\mathrm{S}\right]}_{\text{free}}\end{array}$$

$$\therefore {\left[\mathrm{S}\right]}_{\text{free}}=\frac{{\left[\mathrm{S}\right]}_0}{1+\mathrm{K}\left[{\mathrm{D}}_{\mathrm{n}}\right]}$$E3

Again, rate of reaction,

$$\begin{array}{l}\mathrm{r}=k_1{\left[\mathrm{S}\right]}_{\text{free}}+k_2\left[{\mathrm{D}}_{\mathrm{n}}\mathrm{S}\right]\\ =k_1{\left[\mathrm{S}\right]}_{\text{free}}+k_2\mathrm{K}\left[{\mathrm{D}}_{\mathrm{n}}\right]{\left[\mathrm{S}\right]}_{\text{free}}\\ =\left(k_1+k_2\mathrm{K}\left[{\mathrm{D}}_{\mathrm{n}}\right]\right){\left[\mathrm{S}\right]}_{\text{free}}\end{array}$$

Substituting [S]free from Eq. (3), we get,

$$\mathrm{r}=\left(k_1+k_2\mathrm{K}\left[{\mathrm{D}}_{\mathrm{n}}\right]\right)\frac{{\left[\mathrm{S}\right]}_0}{1+\mathrm{K}\left[{\mathrm{D}}_{\mathrm{n}}\right]}=k_{\mathit{obs}}{\left[\mathrm{S}\right]}_0$$E4

Where,

$$k_{\mathit{obs}}=\frac{\left(k_1+k_2\mathrm{K}\left[{\mathrm{D}}_{\mathrm{n}}\right]\right)}{\left(1+\mathrm{K}\left[{\mathrm{D}}_{\mathrm{n}}\right]\right)}$$E5

Now,

$$k_{\mathrm{obs}}-k_1=\frac{\left(k_1+k_2\mathrm{K}\left[{\mathrm{D}}_{\mathrm{n}}\right]\right)}{\left(1+\mathrm{K}\left[{\mathrm{D}}_{\mathrm{n}}\right]\right)}-k_1=\frac{\left(k_1+k_2\right)\mathrm{K}\left[{\mathrm{D}}_{\mathrm{n}}\right]}{1+\mathrm{K}\left[{\mathrm{D}}_{\mathrm{n}}\right]}$$

$$\frac{1}{\left(k_{\mathrm{obs}}-k_1\right)}=\frac{1}{\left(k_2-k_1\right)}+\frac{1}{\mathrm{K}\left(k_2-k_1\right)}\times \frac{1}{\left[{\mathrm{D}}_{\mathrm{n}}\right]}$$E6

Eq. (6) is very useful to determination of both k₂ and K from the linear plot of 1/(k_obs – k₁) vs. 1/[D_n]. Its linearity is remarkable in view of the assumptions made in this analysis: (i) substrate and surfactant monomer do not form complex; (ii) the micellation process does not perturbed by substrate; (iii) substrate and micelle associate with a 1: 1 stoichiometry; (iv) micellation happens exactly at CMC rather than in a small concentration range; (v) Eq. (1) is valid.

Eq. (5) has a form similar to the Michaelis–Menten equation of enzyme kinetics; although enzyme reactions are generally followed with substrate in excess over the enzyme whereas in the normal reaction generally the surfactant is in large excess over the substrate. Eq. (6), which is similar to the Lineweaver-Burk equation of enzyme kinetics.

Eq. (6) was first applied to the inhibition of the saponification of p-nitrophenyl alkanoates by micelles of sodium laurate [64] and was subsequently applied to unimolecular hydrolyses of dinitrophenyl phosphate dianions and 2,4-dinitrophenyl sulphate monoanion catalyzed by cationic micelles. These observations suggest that the model is useful in analyzing micellar catalysis and inhibition. There are few reactions (Figure 17), whose kinetic data satisfied the above rate expression as well as the model [65, 66].

Figure 17.

(a) Micellar inhibition of bimolecular reaction; (b) cationic micellar catalysis of unimolecular reactions.

8.1.2 Merits and demerits of this model

The simple distribution model (Eqs. (5) and (6)) adequately treats micellar effects upon unimolecular reactions and inhibited bimolecular reactions because these micellar effects can be treated in terms of the distribution of only one reactant between the aqueous and micellar pseudophases. This model also fitted spontaneous reactions [67, 68]. However, this simple model could not simulate the rate maxima characteristic of micellar-enhanced bimolecular reactions, where rate constants generally go through maxima with increasing surfactant concentrations, because the distribution of two reactants must be considered, and this problem is considered in the next model. Berezin and co-workers [68] developed the first general treatment based on the pseudophase model and successfully simulated spontaneous and bimolecular reactions between neutral organic reactants. Later the pseudophase model was modified to simulate bimolecular, ion-molecule, reactions.

8.2 Pseudophase (PP) Model (or Berezin Model)

Berezin and co-workers [68] developed the first general treatment based on the pseudophase model and successfully simulated spontaneous and bimolecular reactions between non-ionic reactants and with partial success to reactions between neutral organic substrates and hydrophobic anionic nucleophiles. All pseudophase kinetic models are supported by two common assumptions. These are: (i) the micelles and water are different phases and (ii) the rate constants depending on the distribution of substrate (S) and nucleophile (N) in the micelles and aqueous phases in addition to surfactant and salt. In a few studies, it is also assumed that changes in micelle size and shape are not very important so that only those factors, which control the distribution of reactants, will significantly affect the observed reaction rate. The model correctly predicts the rate maxima, which are typically observed with micellar catalyzed bimolecular reactions. Micelle formation is a highly cooperative phenomenon. When the surfactant concentration exceeds the critical micelle concentration (CMC), all additional surfactant forms micelles. Thus, the micellized surfactant can be defined as follows:

$$\left[{\mathrm{D}}_{\mathrm{n}}\right]={\left[\mathrm{D}\right]}_{\mathrm{T}}–\mathrm{CMC}$$E7

Where, [D]_T is the stoichiometric surfactant (/detergent) concentration and the CMC is that obtained under experimental reaction conditions. Often [D]_T > > CMC and the CMC can be neglected. According to this model, the micellized surfactant and the solute are in thermal equilibrium throughout the reaction. The second-order rate constants (k₂^w and k₂^m) can be considered as the sum of the concurrent reaction rates in each pseudophase shown in Figure 18. In Figure 18, subscripts m and w indicate the micellar and aqueous pseudophases, respectively.

Figure 18.

Different steps in Pseudophase model.

Micellar binding of substrate is governed by both columbic and hydrophobic interactions and is generally described by a binding constant, K_s:

$${\mathrm{K}}_{\mathrm{S}}=\frac{\left[{\mathrm{S}}_{\mathrm{m}}\right]}{\left[{\mathrm{S}}_{\mathrm{w}}\right][\left[{\mathrm{D}}_{\mathrm{n}}\right]}$$E8

Values for K_s can usually be estimated, in the absence of N, by spectroscopy, solubility, liquid chromatography, or ultrafiltration, and they increase with increasing substrate hydrophobicity (R is the length of alkyl chain of the substrate). Spectroscopically it is shown that the bound polar substrates are on the micellar surface rather than in the hydrophobic core. The rate expression can be obtained from Figure 18 using Eq. (9) as

$$k_{\mathrm{obs}}=\frac{k_2^w\left[{\mathrm{N}}_w\right]+k_2^m{\mathrm{K}}_s{\mathrm{N}}_m\left[{\mathrm{D}}_n\right]}{1+{\mathrm{K}}_s\left[{\mathrm{D}}_n\right]}$$E9

Where, the term N_m is the local molar concentration of the ionic reactant in the micellar pseudophase and can be written as Eq. (10).

$${\mathrm{N}}_m=\frac{\left[{\mathrm{N}}_m\right]}{\left[{\mathrm{D}}_n\right]{\mathrm{V}}_m}$$E10

Similarly, V_m is the molar volume of the reactive region and the micellar fractional volume where the reaction occurs is denoted as [D_n] × V_m. In the aqueous phase, the concentration of ions and molecules is usually expressed in molarity of the total solution volume since the micellar pseudophase volume exceeds about 3% (for about 0.1 M surfactant) of the total solution volume.

The relative proportions of the overall reaction occurring in the aqueous and micellar pseudophases are dependent on both [D_n] and K_s [3, 4, 69, 70] as indicated in Eq. (8). Also, Eq. (9) illustrates a fundamental property of all pseudophase models, that reaction rate within the micellar pseudophase depends upon the local concentration of N within the micellar pseudophase and not its stoichiometric concentration.

8.2.1 Demerits of the oseudophase model

The pseudophase model for the micellar-catalyzed bimolecular reactions is based on various assumptions about the interactions of solutes with micelles and the appropriate units of concentration. However, there are some questions about the validity of assumptions about whether the reaction occurs in the aqueous pseudophase or in micelles. It is also not clear whether a reactant in water attacks another reactant in the micelle. Furthermore, how are the overall rate constants in the two pseudo-phases so generalized in Eqs. (5) and (9)? It also appears that this generally accepted hypothesis fails in certain special cases.

The model accurately predicts the observed maximum rate of micellar catalyzed bimolecular reactions. However, it has not been thoroughly tested for micellar catalyzed bimolecular reactions involving hydrophilic ions, with the exception of reactions involving hydrogen ions and nucleophilic addition to carbocations.

The pseudophase model explicitly assumes that changes in micelle size and shape are not very important, so that only those factors, which control the distribution of reactants, will significantly affect the observed reaction rate. Buffer effects are particularly difficult to understand, so the model applicable to the selected reactions where would not require use of buffers. Finally, the presence of both reactive and unreactive counter ions results in uncontrolled variations in distribution of them between the micellar and aqueous pseudo phases. Thus, it is further complicated to interpret the CMC.

8.3 Pseudophase ion exchange (PIE) model

The treatment of micellar catalysis and inhibition is based on the assumption that reaction occurs in the micellar and aqueous pseudo-phases and that equilibrium is maintained between reactants in the two pseudo-phases. The binding constant is given by Eq. (8) and the corresponding rate constant is given by Eq. (9). Now the problem is nonreactive counter ions. A widely used approach is to assume that counter ions bind to a micelle according to an ion-exchange equation

$${\mathrm{N}}_{\mathrm{m}}+{\mathrm{X}}_{\mathrm{w}}\rightleftharpoons {\mathrm{N}}_{\mathrm{w}}+{\mathrm{X}}_{\mathrm{m}}$$

A useful physical picture for interpreting counter ion effects is based on Stigter’s model, which interprets the effect of added salt on the surface potential of micelles [71]. The counter ions are supposed to be distributed in two discrete sections of an ionic bilayer. These are (i) the Stern layer strongly bound with the counter ions that move with the micellar aggregate (the kinetic micelle), and (ii) the Gouy-Chapman layer that is loosely bound with the remaining counter ions according to the Boltzmann distribution law in the aqueous phase (Figure 19). Similar double-layer models describe counter ion distributions around planar interfaces, charged electrodes, and lyophobic colloids. Since the potential drop across the Stern layer is relatively insensitive to increasing the concentration of counter ions, the counter ions concentration in the Stern layer and the fraction of counter ions bound to the micelle surface will be approximately constant (β) [72, 73]. The theoretical concept of Stigter enables the calculation of the value of β [72, 73].

Figure 19.

Model of a hypothetical cationic micelle showing the locations of head groups, surfactant chains, and counter ions.

According to Figure 19, N is a counter ion to ionic micelles; binding substrate, S brings it into a microenvironment. Consequently, the local concentration of N increases and hence increases the observed rate constant (k_obs). Again, when N is a co-ion, binding S brings it into a microenvironment with a much lower concentration of N, which reduces k_obs. Two assumptions are considered to estimate the concentration of N in micelles in the PIE model. The assumptions consist of (i) the micellar surface, known as a selective ion exchanger, and a relationship between inert counter ions (X), and reactive counter ions (N), given by Eq. (11).

$${\mathrm{K}}_{\mathrm{N}}^{\mathrm{X}}=\frac{\left[{\mathrm{N}}_{\mathrm{w}}\right]\left[{\mathrm{X}}_{\mathrm{m}}\right]}{\left[{\mathrm{N}}_m\right]\left[{\mathrm{X}}_{\mathrm{w}}\right]}$$E11

(ii) the fraction of the surface occupied by the two counter ions, X and N is constant and given by the degree of counter ion binding, β, expressed in Eq. (12).

$$\beta =\frac{\left[{\mathrm{N}}_m\right]+\left[{\mathrm{X}}_{\mathrm{m}}\right]}{\left[{\mathrm{D}}_n\right]}$$E12

The value of β varies with the variety of counter ions and head groups, and the experimental values are around 0.6–0.9 [72, 73]. This can be estimated from the fractional micellar charge ɑ (ɑ = 1 − β). β is usually insensitive to surfactant and salt concentration, indicating significant specific interactions at the micelle surface. Thus if [N_m] and [N_w] are calculated from Eqs. (11) and (12), the variation of k_obs with [D_n] (Eq. 9) can be used to estimate$k_2^m$. In practice, non-linear simulation in computer base is usually used to fit the variation of k_obs with [D_n] in terms of the various parameters in Eq. (9). However, the above ion-exchange pseudophase model based on Eq. (11) involves the assumption that β does not change as reactive counter ion, e.g., N is added or the surfactant concentration changed. This assumption appears to fail when the only anions in the solution are very hydrophilic, e.g., OH⁻ and F⁻. In these systems the kinetic data can be fitted by a mass-action model in which assumes that β increases with increasing [OH⁻] or [F⁻] in Eq. (13). β appears to increase significantly, with their total concentrations indicating that they interact mostly coulombically and that appreciable fractions of these ions are in the diffuse double layer.

$${\mathrm{D}}_{\mathrm{n}}+{\mathrm{OH}}_{\mathrm{w}}^{-}\stackrel{{\mathrm{K}}_{\mathrm{OH}}}{\rightleftarrows }{\mathrm{OH}}_{\mathrm{m}}^{-}$$E13

The binding of inert counter ion, e.g., Br⁻, could similarly follow Eq. (14)

$${\mathrm{D}}_{\mathrm{n}}+{\mathrm{Br}}_{\mathrm{w}}^{-}\stackrel{{\mathrm{K}}_{\mathrm{Br}}}{\rightleftarrows }{\mathrm{Br}}_{\mathrm{m}}^{-}$$E14

Thus,

$${\mathrm{K}}_{\mathrm{OH}}=\frac{\left[{\mathrm{OH}}_{\mathrm{m}}^{-}\right]}{\left(\left[{\mathrm{D}}_{\mathrm{n}}\right]-\left[{\mathrm{OH}}_{\mathrm{m}}^{-}\right]-\left[{\mathrm{Br}}_{\mathrm{m}}^{-}\right]\right)\left(\left[{\mathrm{OH}}_{\mathrm{T}}^{-}\right]-\left[{\mathrm{OH}}_{\mathrm{m}}^{-}\right]\right)}$$E15

$$\text{and},{\mathrm{K}}_{\mathrm{Br}}=\frac{\left[{\mathrm{Br}}_{\mathrm{m}}^{-}\right]}{\left(\left[{\mathrm{D}}_{\mathrm{n}}\right]-\left[{\mathrm{OH}}_{\mathrm{m}}^{-}\right]-\left[{\mathrm{Br}}_{\mathrm{m}}^{-}\right]\right)\left(\left[{\mathrm{Br}}_{\mathrm{T}}^{-}\right]-\left[{\mathrm{Br}}_{\mathrm{m}}^{-}\right]\right)}$$E16

OH⁻ ion appears to bind very weakly to cationic micelles so that in presence of CTAB,$\left[{\mathrm{OH}}_{\mathrm{T}}^{-}\right]>>\left[{\mathrm{OH}}_{\mathrm{m}}^{-}\right]$. With this assumption and Eqs. (12) and (13), and mass balance, the value of $\left[{\mathrm{OH}}_{\mathrm{m}}^{-}\right]$, is given by

$$\begin{array}{l}\left({\mathrm{K}}_{\mathrm{Br}}+{\mathrm{K}}_{\mathrm{Br}}{\mathrm{K}}_{\mathrm{OH}}\left[{\mathrm{OH}}_{\mathrm{T}}^{-}\right]\right){\left[{\mathrm{OH}}_{\mathrm{m}}^{-}\right]}^2+\left(1+{\mathrm{K}}_{\mathrm{OH}}\left[{\mathrm{OH}}_{\mathrm{T}}^{-}\right]+{\mathrm{K}}_{\mathrm{Br}}\left[{\mathrm{Br}}_{\mathrm{T}}^{-}\right]-{\mathrm{K}}_{\mathrm{Br}}\left[{\mathrm{D}}_{\mathrm{n}}\right]\right)\left({\mathrm{K}}_{\mathrm{OH}}\left[{\mathrm{OH}}_{\mathrm{T}}^{-}\right]\right)\left[{\mathrm{OH}}_{\mathrm{m}}^{-}\right]\\ -{\mathrm{K}}_{\mathrm{OH}}\left[{\mathrm{OH}}_{\mathrm{T}}^{-}\right]\left[{\mathrm{D}}_{\mathrm{n}}\right]\\ =0\end{array}$$E17

The quadratic Eq. (17) can be solved for given concentrations of OH⁻ and CTAB, assuming that K_OH and K_Br are independent parameters and also assuming their values. Using the calculated value of $\left[{\mathrm{OH}}_{\mathrm{m}}^{-}\right]$ from Eq. (17), the observed rate expression becomes

$$k_{\mathrm{obs}}=\frac{k_2^{\mathrm{w}}\left[{\mathrm{OH}}_{\mathrm{w}}^{-}\right]+k_2^{\mathrm{m}}{\mathrm{K}}_{\mathrm{S}}\left[{\mathrm{OH}}_{\mathrm{m}}^{-}\right]\left[{\mathrm{D}}_{\mathrm{n}}\right]}{1+{\mathrm{K}}_{\mathrm{S}}\left[{\mathrm{D}}_{\mathrm{n}}\right]}$$

This treatment makes no assumptions regarding the value of β, which in principle can range from 0 to 1. Both the ion-exchange model in Eq. (11) and the mass action model in Eqs. (13) and (14) are limiting models, which fit the kinetic data, but only approximate to reality.

8.4 Piszkiewicz co-operativity model

There are important similarities between micelle-catalyzed reactions and enzyme-catalyzed reactions. The structures of both micelles and enzymes are similar in that they have hydrophobic cores with polar groups on their surfaces. The structures of micelles are disrupted by common protein denaturing agents such as urea and guanidinium salts. Both catalytic micelles and enzymes bind substrates in a noncovalent manner. The kinetics of micellar catalysis resemble that of enzymatic catalysis in that the micelle may be saturated by the substrate; and, conversely, the substrate may be saturated by the micelle [74].

Piszkiewicz showed further similarities in micellar and enzyme reactions [75, 76, 77]. The sigmoid curves are obtained by plotting the rate constants of micelle-catalyzed reactions with respect to the detergent concentration. This behavior is referred to as positive cooperativity (or positive homotropic interactions) of enzyme catalyzed reactions. The Hill model, a kinetic model [78], also describes the sigmoid dependence of rate on substrate concentration. Piszkiewicz in his model demonstrated that micelle-catalyzed reactions also are models of enzyme-catalyzed reactions that show positive cooperativity [4, 77]. In most cases, the rate of micellar reaction is much faster than that of other common observations. The micelle-catalyzed reactions occur in two steps. The first step is the complex formation of detergent-substrate (DnS) which is the fast step and the other is the catalytic step, which is the rate limiting step. Therefore, two rate constants (k_w and k_m) are required to describe these reactions. The terms, k_w and k_m, denotes the rate constants in absence and presence of detergent respectively and k_m/k_w is equal to the rate acceleration effected by micellar catalysis.

However, this model assumes that a substrate, S, and a number of detergent molecules (nD), aggregate to form catalytic micelles, D_nS, which may then react to yield product shown in Figure 20.

Figure 20.

Reaction steps involved in the Piszkiewicz’s model.

K_D is the dissociation constant of the detergent-substrate (DnS) complex towards its free components. k_w and k_m are the rate constants in absence and presence of detergent. Indeed, there are several steps and sequential equilibrium steps involved in the aggregation of detergent and substrate molecules to form the catalytic micelle, D_nS. For convenience, the steps are presented as a single association step (Figure 20) in this article. For the reaction indicated in Figure 20, the observed rate constant is expressed as a function of the concentration of detergent by Eq. (18).

$$k_{\mathit{obs}}=\frac{k_m{\left[\mathrm{D}\right]}^n+k_{\mathrm{w}}{\mathrm{K}}_{\mathrm{D}}}{{\mathrm{K}}_{\mathrm{D}}+{\left[\mathrm{D}\right]}^n}$$E18

This equation may be rearranged and its logarithmic form can be written as

$$log\left(\frac{k_{\mathit{obs}}-k_{\mathrm{w}}}{k_{\mathrm{m}}-k_{\mathit{obs}}}\right)=\mathrm{n}log\left[\mathrm{D}\right]-{\text{logK}}_{\mathrm{D}}$$E19

From Eq. (19) above, a linear curve having a slope of n is obtained by plotting of log [(k_obs − k_w)/(k_m − k_obs)] vs. log [D] for a micelle-catalyzed reaction. When log [(k_obs − k_w)/(k_m − k_obs)] = 0, n log[D] = log K_D. Also, at log [(k_obs − k_w)/(k_m − k_obs)] = 0, catalysis by detergent shows one-half of its maximum effect on rate constant i.e. the log of the detergent concentration at which half-maximal velocity is obtained. For convenience, the value of log [D] at this point is designated as log[D]₅₀, and it is equal to (log K_D)/n.

8.4.1 Effect of temperature variation and detergent structure

Several experiments show that the value of n depends on temperature. The average value of n appears to increase as temperature decreases. E. H. Cordes and his co-worker [79] were determined an average value of n of 1.87 at 40°C for the hydrolysis of methyl orthobenzoate in presence of sixteen different detergents. In contrast, the hydrolysis of the substrate catalyzed by six detergents at 25°C had an average ‘n’ value of 3.48. However, three detergents with an average ‘n’ value of 1.41 inhibited the hydrolysis of methyl orthobenzoate [77]. Only a few are listed in Table 3.

Detergents	Temp. (°C)	No. of kobs values used	Log [D]50	n
Sodium octyl sulfate	25	4	−0.91	3.30
Sodium decyl sulfate	25	3	−1.42	5.46
Sodium dodecyl sulfate	25	8	−1.96	3.67
Sodium dodecyl sulfate	40	4	−1.90	5.86
3-Hexadecyl sodium sulfate	40	5	−2.45	1.55
5-Hexadecyl sodium sulfate	40	5	−2.35	1.82
Sodium 2-hexadecyloxy-1-methylethyl sulfate	40	7	−2.69	1.63
Disodium 2-sulfo-2-methyloctadecanoate	40	4	−2.52	2.14
Sodium methyl-ɑ-sulfostearate	40	4	−2.56	1.43
Sodium 2-dodecylbenzenesulfonate	40	4	−2.49	1.75
Dimethyldodecylammonium propanesulfonate^a	25	6	−1.68	1.03
Dimetliyldodecylphosphine oxide^a	25	4	−1.90	1.38
Dimetliyldodecylammonium acetate^a	25	3	−1.85	1.82

Table 3.

Observed rate constants and values of n for the hydrolysis of methyl Orthobenzoate in the presence of various detergents.

^a

Hydrolysis of methyl orthobenzoate was inhibited by these detergents [80].

8.4.2 Effect of changes of substrate structure

Several series of reactions have been studied in which a single detergent was employed within the series, but slight variations in the structure of the substrate were made. One such series of reactions were reported by Dunlap et al. [80] in which hydrolyses of para-substituted benzaldehyde diethyl acetals occur in presence of sodium dodecyl sulfate. Table 4 summarizes the constant values, log [D]₅₀ and n, for these reactions. It is clear from the data shown in Table 4 that the value of log [D]₅₀ remains virtually constant as the inductive substituent constant (σ) of the para substituent is varied from −0.069 for p-methyl to +0.710 for p-nitro. Also, show that there is no consistent variation of n with σ.

Substituents	σ	No. of kobs values used	Log[D]50	n
p-NO2	0.710	2	−2.06	4.47
p-Cl	0.373	4	−2.03	5.02
p-F	0.337	5	−1.96	3.74
p-OCH3	0.115	4	−2.01	4.52
p-H	0.00	4	−1.98	5.41
p-CH3	−0.069	5	−1.99	4.33

Table 4.

Derived values of log [D]₅₀ and n for hydrolyses of Para-substituted benzaldehyde diethyl acetals in presence of sodium dodecyl sulfate.

A somewhat different situation is seen for the solvolyses of several esters by the nucleophilic detergent p-trimethylammonio benzyl decylamine chloride as reported by Bruice et al. [81]. Table 5 summarizes the parameters, log [D]₅₀ and n, for these reactions are clearly a function of the structure of the substrate. Generalized data shown in Table 5 can be explained in different ways. Firstly, log [D]₅₀ increases as the chain length of the carboxylic acid portion of the negatively charged ester increases. It is clear from the result that the better binding of the longer chain substrates within the catalytic micelle complex (D_nS) due to hydrophobic interactions. Secondly, ‘n’ appears to decrease as the chain length of the carboxylic acid moiety increases, though there are few exceptions. In this reaction, the substrate has a much longer aliphatic chain than does the detergent and thus this is different from other series of reactions. This structural difference probably radically changes the mechanism of the formation of the catalytic micelle (D_nS).The catalytically active complex may form firstly the aggregation of substrate molecules then include the nucleophilic detergent molecules.

Esters	No. of kobs values used	log[D]50	n
3-Nitro-4-acetoxy benzenesulfonate	7	−2.34	4.14
3-Nitro-4-hexanoyloxy benzenesulfonate	6	−2.69	3.01
3-Nitro-4-octanoyloxy benzenesulfonate	5	−3.05	1.63
3-Nitro-4-decanoyloxy benzenesulfonate	5	−3.03	1.25
3-Nitro-4- hexadecanoyloxy benzenesulfonate	4	−2.76	4.87
3-Nitro-4-acetoxyphenyltrimethylammonium iodide	10	−2.38	1.91
3-Nitro-4-octanoyloxyphenyltrimethylammonium iodide	9	−2.03	1.80
o-Nitrophenyl acetate	10	−1.92	1.58

Table 5.

Nucleophilic solvolyses of different esters derived from p-Trimethylammonio benzyl decylamine chloride and n values in micelle.

From observing the effects of changing substrate structure, it is clear that changing log [D]₅₀ or n requires a major structural change. The change in electronic inductive effects may not be sufficient. Moreover, the relation between the nature of changes in log [D]₅₀ and n is a complex function of the nature of structural changes.

8.4.3 Limitations

The use of log [(k_obs − k_w)/(k_m − k_obs)] causes significant scale expansion at both ends, but very little distortion in the mid-range. In the range where the rate constants to be subtracted are about an order of magnitude from each other, a reasonable value of n can be predicted. However, the precision of linear plot points outside this range degrades sharply. This degradation of precision at the extremes may contribute to the lack of precision in the value of n. The term, log K_D is equal to n log [D]₅₀. As n log [D]₅₀ is the product of two terms, it reflects the uncertainty in both, K_D and n. Moreover, the K_D units vary with n reaction to reaction. These factors severely limit the utility of log K_D in comparing different micelle-catalyzed reactions.

8.5 Raghavan and Srinivasan’s model

P.S. Raghavan and V.S. Srinivasan [82] observed that the cationic micelles of cetyltrimethylammonium bromide (CTAB), cetyldibenzylammonium chloride (CDBAC) and cetylpyridinium chloride (CPC) stabilize the tetrahedral intermediate formed on the hydrolysis of carboxylic esters. Based on this observation a model for bimolecular micellar catalyzed reactions was developed. The model predicts the constancy of k_obs values at high detergent concentrations of and can be used to estimate the binding constants of reactants. Figure 21 shows the distribution of both reactants and nucleophiles in the aqueous and micellar phases proposed as by Raghavan and Srinivasan.

Figure 21.

Reaction steps involved in the Raghavan-Srinivasan’s model.

The formations of products are assumed to result from decomposition of ternary complex in micellar phase as well as from the reaction between the substrate and the nucleophile in aqueous medium. After analyzing the data based on this model, they concluded that almost all the nucleophile is present in the bulk phase.

Where D, S and N refer to the detergent monomer, substrate and the nucleophile, respectively, while D_nS and D_nSN are the binary and ternary complexes, respectively. The rate law for the above Figure 21 is

$$\text{rate}=k_{\mathrm{w}}{\left[\mathrm{S}\right]}_0{\left[\mathrm{N}\right]}_0+k_{\mathrm{m}}\left[{\mathrm{D}}_{\mathrm{n}}\mathrm{SN}\right]$$E20

Where, subscript ‘0’refers to free species. Now,

$$\left[{\mathrm{D}}_{\mathrm{n}}\mathrm{SN}\right]={\mathrm{K}}_2\left[\mathrm{D}{}_{\mathrm{n}}\mathrm{S}\right]{\left[\mathrm{N}\right]}_0\text{and}\left[{\mathrm{D}}_{\mathrm{n}}\mathrm{S}\right]={\mathrm{K}}_1{\left[\mathrm{D}\right]}^{\mathrm{n}}{\left[\mathrm{S}\right]}_0$$

Again, free concentration of substrate,

$${\left[\mathrm{S}\right]}_0={\left[\mathrm{S}\right]}_{\mathrm{T}}-\left[{\mathrm{D}}_{\mathrm{n}}\mathrm{S}\right]$$

Where, [S]_T is the total concentration. Thus, the concentration of D_nS and D_nSN can be given as

$$\left[{\mathrm{D}}_{\mathrm{n}}\mathrm{S}\right]=\frac{{\mathrm{K}}_1{\left[\mathrm{D}\right]}^{\mathrm{n}}{\left[\mathrm{S}\right]}_{\mathrm{T}}}{1+{\mathrm{K}}_1{\left[\mathrm{D}\right]}^{\mathrm{n}}}$$

$$\left[{\mathrm{D}}_{\mathrm{n}}\mathrm{SN}\right]=\frac{{\mathrm{K}}_1{\mathrm{K}}_2{\left[\mathrm{D}\right]}^{\mathrm{n}}{\left[\mathrm{S}\right]}_{\mathrm{T}}{\left[\mathrm{N}\right]}_0}{1+{\mathrm{K}}_1{\left[\mathrm{D}\right]}^{\mathrm{n}}}$$

Assuming [N]₀ = [N]_T − [D_nSN]. On introducing [S]₀, [N]₀ and [D_nSN] in Eq. (20), it becomes

$$\text{rate}=\frac{k_{\mathrm{w}}{\left[\mathrm{S}\right]}_{\mathrm{T}}{\left[\mathrm{N}\right]}_{\mathrm{T}}+k_{\mathrm{m}}{\mathrm{K}}_1{\mathrm{K}}_2{\left[\mathrm{D}\right]}^{\mathrm{n}}{\left[\mathrm{S}\right]}_{\mathrm{T}}{\left[\mathrm{N}\right]}_{\mathrm{T}}}{1+{\mathrm{K}}_1{\left[\mathrm{D}\right]}^{\mathrm{n}}\left\{1+{\mathrm{K}}_2{\left[\mathrm{S}\right]}_{\mathrm{T}}\right\}}$$

$$\text{or},k_{\mathrm{obs}}=\frac{k_{\mathrm{w}}+k_{\mathrm{m}}{\mathrm{K}}_1{\mathrm{K}}_2{\left[\mathrm{D}\right]}^{\mathrm{n}}}{1+{\mathrm{K}}_1{\left[\mathrm{D}\right]}^{\mathrm{n}}\left\{1+{\mathrm{K}}_2{\left[\mathrm{S}\right]}_{\mathrm{T}}\right\}}$$E21

Eq. (21) may be rearranged in the form

$$\left\{\frac{k_{\mathrm{obs}}-k_{\mathrm{w}}}{k_{\mathrm{obs}}}\right\}\frac{1}{{\left[\mathrm{D}\right]}^{\mathrm{n}}}={\mathrm{K}}_1{\mathrm{K}}_2\left(\frac{k_{\mathrm{m}}}{k_{\mathrm{obs}}}\right)-{\mathrm{K}}_1\left\{1+{\mathrm{K}}_2{\left[\mathrm{S}\right]}_{\mathrm{T}}\right\}$$E22

Eq. (22) predicts a linear relationship between {(k_obs^_k_w)/k_obs)}(1/[D]ⁿ) and (k_m/k_obs). In this model, the value of the cooperativity index (n) used is taken from the Piszkiewicz’s cooperativity model. The values of the binding constants (K₁ and K₂) and K_D (reciprocal of K₁) were calculated from the intercept and slope of the linear plot. It is apparent that the binary complex (D_nS) dissociation constant, K_D in the Piszkiewicz model is actually the reciprocal of K₁ in the Raghavan and Srinivasan model. The K_D values assessed in this model are consistent with those obtained in the Piskiewicz model [4, 83]. However, this model does not provide an explanation of the effect of counter ions or the interaction of products with micelles. When the proposed model is applied to several systems, the value of K₁ and the intercept are approximately equal to K₂[S]_T ≪ 1 in all cases. The K₁ values evaluated are also consistent with those found in the Piskiewicz model. An abnormal value of K₁ (or, high value of K₁) from both the models would be due to the assumption that the substrate induces micellization in which case there should be very strong association of the substrates with the micelles and thus the equilibrium favors the formation of binary complex or the catalytic micelle, D_nS. Again, the low values of K₂ (K₂ ≪ K₁) suggest that [N]₀ = [N]_T, and therefore, the nucleophile is present almost in full in the bulk phase. Thus, one may suggest that the reaction takes place between the substrate solubilized into a catalytic micelle and the nucleophile residing at the interface.

9. Concluding remarks

This chapter can be divided into two parts: the first part of this chapter addresses the concept of microheterogeneous systems and then various model concepts, their importance and limitations. By microheterogeneous system, one refers to an aggregated system in which the structure of the constituent molecules along with the solvent or other surrounding medium determines the structure of the aggregate. Most of the aggregates are self-assembling and form micelle, reverse micelle, vesicle, liposome etc. Although, each aggregate contains a large number of molecules but the overall aggregate is small enough to form colloidal dimensions and therefore called a microheterogeneous system. These systems have various aspects but the most important feature for the researcher is the study of the effect on reaction kinetics. Surfactants are the most commonly used species that form microheterogeneous systems. The functionality, non-functionality and inertness of a surfactant generally depend on the head group of the surfactant. Most functional surfactants contain nucleophilic, or occasionally basic, residues and form micelles that have a high concentration of reactive groups on their surface. Another very common microheterogeneous system is the reverse micellar system. There are three types of water molecules in the reverse micellar cavities: bound, trapped and free. The water inside the water pool possesses some typically different properties like micro polarity and micro viscosity etc. The rate of the reaction in reverse micelle is generally enhanced as compared to the aqueous medium as well as micelle medium.

It is impossible to carry out micellar chemistry without models and pictures. Thus, this chapter analyses the microheterogeneous systems with the different kinetic models. Several models describe the distribution of solutes and the mutual diffusion of reactants in micellar and reverse micellar media. The distribution of solutes is essentially statistical in nature.

The ‘pseudophase model (PP)’ is well-treated to explain bimolecular reaction rates. According to this model, in most cases, the rate increase is due to an increase in the concentration of reactive species locally in the micelles (inner or interface region) rather than an increase at the surface. Whereas, in the ‘pseudophase ion exchange model (PIE)’, adsorption of ionic reactants on the micelle surface occurs and exchange of counter ions is possible which determines the nature of their distribution. Salt also plays an important role in changing the reaction rate. In this case, the micelles are treated as ion exchange resins and competing binding/exchange parameters are used to estimate the rate increases. Paszkiewicz described the positive cooperativity behavior of enzyme-catalyzed reactions in his model and explained the sigmoid shape of reaction rates. Raghavan and Srinivasan proposed the distribution of both reactants and nucleophiles in the aqueous and micellar phases for bimolecular micellar catalysis reactions and products from the micellar and aqueous phases. In most cases, binding constants and dissociation constants in both models, Paczkiewicz and Raghavan–Srinivasa corroborate each other. Although each model has limitations, the models should not have too much trouble interpreting kinetic raw data and to understand the role of models on reaction kinetics requires further studies.

Acknowledgments

I thank Sree Chaitanya College, Habra for giving permission to write the dissertation. In addition, I thank Professor Ambikesh Mahapatra for very useful discussions on all the major points of this chapter. The author acknowledges Sourita Mandal and Soumini Mandal for providing sufficient time for literature survey and chapter writing.