Micellization and Physicochemical Properties of CTAB in Aqueous Solution: Interfacial Properties, Energetics, and Aggregation Number at 290 to 323 K

Abstract

This work reports a detailed investigation of the micellization of the cationic surfactant cetyltrimethylammonium bromide (CTAB) over the temperature range of 290–313 K. Conductometry, tensiometry, fluorimetry, and isothermal titration calorimetry (ITC) were used to study the overall solution behavior of amphiphilic self-aggregation. The CMC values showed a trend of first declining and then rising with a minimum temperature of 298 K. The adsorption at the air–water interface and micellization processes of CTAB are both spontaneous. The enthalpy of CTAB micellization is negative at 290 K and increases negatively as the temperature rises. The interfacial parameters—maximum surface excess concentration (Γ_max), minimum area per molecule (A_min), standard free energy of adsorption (A_min), and surface pressure at CMC (Π_CMC)—were calculated using surface tension data. The aggregation numbers (N) of CTAB micelles and others (SDS and CHAPS) determined at different [surfactant]>CMC by the static fluorescence quenching method were used to find out the N values at CMC (or N^CMC). The results revealed that the N^CMC values were 48, 54, and 4 for CTAB, SDS, and CHAPS micelles, respectively. Temperature-dependent N^CMC by the ITC method was also examined for the studied surfactants. Additionally, the ITC-determined specific heat of micellization was used to find out the extent of water penetration into the micelle interior of up to 8, 7, and 3 hydrocarbons for CTAB, SDS, and CHAPS, respectively.

1. Introduction

Surfactants are amphiphilic molecules characterized by a distinct head–tail structure, consisting of a hydrophilic head group and predominantly hydrophobic alkyl chains [1]. These molecules have a wide range of applications, including their use as disinfectants, antiseptic agents, antistatic agents (e.g., in hair conditioners), textile softeners, stabilizers, solubilizers, corrosion inhibitors, foam depressants, antibacterial agents, and emulsifiers, and in processes such as oil recovery, nanomaterial synthesis, and drug delivery [2,3,4,5,6,7]. Typically, the hydrophilic head interacts with water, while the hydrophobic tail avoids it, leading to the alignment of surfactant molecules at the air–water interface, with their tails oriented toward the air. This behavior is consistent across all classes of surfactants. At higher concentrations, surfactant molecules aggregate in bulk solution to form micelles. The concentration at which this aggregation begins is known as the critical micelle concentration (CMC). According to IUPAC, the definition of the CMC of a surfactant is “There is a relatively small range of concentration separating the limit below which virtually no micelles are detected and the limit above which virtually all the additional surfactant forms micelles” [8]. The stability of these micelles in solution is governed by soft interactions, including hydrogen bonding, hydrophobic effects, electrostatic interactions, and van der Waals forces. These interactions are highly dynamic, and even minor changes in the system can significantly influence the physicochemical and surface properties of the surfactant solution. Factors such as additives, temperature, and solvent composition play crucial roles in altering these properties. The self-assembly process of amphiphiles has been extensively studied in colloid and interface science since its introduction by McBain in 1913 [9]. However, despite substantial progress, there remains significant scope for improving the quantitative understanding of micellar systems through the development of advanced techniques and theoretical models.

A physical or chemical process’s viability and sustainability are determined by the related thermodynamics. The process of micellization is not an exception. It is necessary to determine this information using both enthalpy change and free energy knowledge. The approach used to determine the enthalpy of the micelle-forming process has been found to have a significant influence on this issue. Most of the time, there is a discrepancy between the direct measurement of enthalpy in a calorimeter and the indirect measurement of the same by measuring the critical micelle concentration (CMC) at various temperatures and processing the findings by the van’t Hoff standard [3]. This is an important issue that needs to be clearly understood. Isothermal titration calorimetry (ITC) has become a valuable technique for studying the thermodynamics of micellization. It enables direct measurement of enthalpy changes and, when combined with complementary analyses, allows the determination of key parameters such as the critical micelle concentration (CMC), free energy, entropy, and aggregation number in a single experiment. Studies in this area are still limited.

The aggregation number (N) is an important physical parameter for self-aggregated surfactant systems or micelles. The static light scattering (SLS), static and time-resolved fluorescent quenching (SFQ and TRFQ), and small angle neutron scattering (SANS) methods are mostly used for the determination of N [10,11,12,13,14]. The physicochemical properties of micelles are reported and analyzed with reference to the CMC point, where surfactant monomers just transform into aggregates. But the above mentioned evaluation methods are not enough sensitive at [surfactant]=CMC, because of the formation of a very low concentration of micelles at that point (at CMC, only ~2% of surfactant monomers assemble). Hence [surfactant]>CMC is required to increase the sensitivity and accuracy of measurements. Based on the “mass action model,” Blume et al. [15] proposed a model for the evaluation of N at CMC using the isothermal titration calorimetry (ITC) method, which was also limitedly used for other surfactants [16,17,18]. The method yielded N values that were fairly lower than those obtained by fluorimetry or other methods. We suspected the difference arose because of the use of a higher concentration (>CMC) than with other methods. Wadekar et al. [19] also proposed a simple relationship to determine N from the ITC enthalpogram profile, where N is related to the slope of the ITC curve between the monomer and the micellar state.

The micellization of cetyltrimethylammonium bromide (CTAB), a cationic surfactant, is a widely studied model system due to its well-defined molecular structure and extensive applications in diverse fields. It is used as an emulsifier, corrosion inhibitor, disinfectant, antiseptic, and biocide, and in numerous industrial processes and medication formulations [20]. Furthermore, due to its natural anticancer characteristics, CTAB appears to target mitochondrial tumors [21]. It is widely used in the production of silver and gold nanoparticles [5]. Examining the micellization process of CTAB, particularly from a thermodynamic perspective, offers valuable insights into the driving forces behind micelle formation and the behavior of these aggregates under various conditions. Key parameters, including the aggregation number, the extent of water penetration, and the water-accessible surface area, play crucial roles in the design of surfactant-based applications, yet they remain inadequately documented in the literature. Additionally, many interfacial phenomena—such as maximum surfactant adsorption, micellar solubilization, and surface tension—are closely tied to the critical micelle concentration (CMC). A comprehensive understanding of CTAB’s micellization behavior is essential for optimizing above mentioned applications. Bielawska et al. [22] and Kumar et al. [23] measured the conductivity in order to examine the thermodynamic characteristics of CTAB micellization. All of them used the indirect van’t Hoff method to calculate energetic parameters. There are, however, insufficiently thorough studies of the calorimetric and tensiometric methods. According to my literature search, no investigation has been performed on the temperature-dependent aggregation number of CTAB micellization at CMC.

In this work, an elaborate attempt has been made to determine the different physical and thermodynamics properties of the surfactant cetyltrimethyalammonium bromide (CTAB) as a model system. In recent years, in addition to providing thermodynamic information on surfactant self-aggregation in solution, the ITC method has been shown to provide information on water penetration in to the micelle, which affects the micelle’s physicochemical properties as well as the evaluation of the aggregation number of micelles at CMC, which has been not explored to the best of our knowledge. The above two aspects of CTAB micelles will also be presented and discussed. For additional support, we have attempted to obtain the above information on the anionic surfactant sodium dodecylsulfate (SDS) studied earlier [24], but not treated, to estimate the water penetration into the micelle and the micelle aggregation number by ITC, as herein proposed. The results of the zwiterionic surfactant 3-[(3(cholamidopropyl)dimethylammonio]-1-propanesulfonate (CHAPS) have been also considered for a comparison with CTAB and SDS, with reference to aggregation number and water penetration into micelles. Overall, this study seeks to investigate, with a focus on the interfacial properties, the thermodynamics of micellization, aggregation number, and water molecule penetration into micelles of CTAB, SDS, and CHAPS by employing ITC.

2. Experimental Section

Materials: The surfactants cetyltrimethylammonium bromide (CTAB) and sodiumdodecyl sulfate (SDS) were products of Sigma, Mumbai, India. CHAPS was a product of Dojindo Laboratories, Tokyo, Japan. They did not show minima in their surface tension–concentration profiles and, hence, were free from contamination by other surface active impurities. Cetylpyridinium chloride (CPC) and pyrene were AR grade products of Sigma. They were used as received. Double-distilled water (κ = 2–4 μs cm⁻¹ at 303 K) was employed for the preparation of all solutions.

Methods: Conductometry: The conductivity measurements were obtained using a Labman conductometer, Chennai, India in a double-walled glass container in the temperature range of 290–323 K maintained by a Labman RCB 620 (Chennai, India) circulating water bath. Concentrated solution of surfactant was progressively added in steps using a Tarsons micropipette (Tarsons Products Ltd., Kolkata, India) in water or salt solution. At each step, the specific conductance was measured in a dip-type cell with a cell constant = 0.1 cm⁻¹. The accuracy of the measurements was within ±0.1 mS cm⁻¹. Every measurement was repeated, and the mean was noted and applied.

Tensiometry: A Krüss K6 tensiometer (kruss Scientific, Hamburg, Germany), based on the du Noüy platinum ring detachment method, was used to measure the surface tension (γ) at the air/solution interface. A concentrated solution of CTAB (30 mM) was added stepwise, as required, with the help of a Tarsons micropipette in water (taken in a double-walled container) maintained at the required temperature by a Labman RCB 620 circulating water bath. After adding the CTAB solution to the container, the surface tension was measured after about ten minutes of equilibrium. The accuracy of the γ values was within ±0.1 mN m⁻¹. To ensure reproducibility, the tests were repeated, and mean values were employed for data collection and processing.

Isothermal Titration Calorimetry: An OMEGA isothermal titration calorimeter (ITC) from Microcal (Northampton, MA, USA) was used for thermometric measurements. During the measurements, the temperature was kept constant by circulating water in a Neslab RTE100 (Thermo Fisher Scientific, Waltham, MA, USA) water bath at 5 K below the temperature in the calorimetric cell, as per the recommended procedural requirement. The heat released or absorbed at each step of dilution of the surfactant solution in water was recorded and the enthalpy change per mole of injectant was calculated using ITC software. The reproducibility was checked from repeat experimentations [10].

Fluorimetry: Fluorescence measurements were obtained using a Perkin-Elmer LS 55 (Shelton, CT, USA) luminescence spectrometer. The aggregation number of the surfactant was determined by the SFQ (static fluorescence quenching) method at 374 nm using pyrene (0.1 μM) as the probe and CPC (0.005–0.10 mM) as the quencher in a total surfactant concentration of 5–25 mM. The excitation wavelength of pyrene was 332 nm, and the emission spectra were recorded in the range of 350–500 nm. The fluorescence intensity was measured at each stage of addition of CPC in the surfactant solution after thorough mixing and equilibration.

N can be determined in the studied temperature range using the following relationship [10,25]:

$$\ln ({\displaystyle \frac{{\mathrm{F}}_0}{\mathrm{F}}})={\displaystyle \frac{\mathrm{N}\left[\mathrm{CPC}\right]}{{\left[\mathrm{Surfactant}\right]}_{\mathrm{t}}-\mathrm{CMC}}}$$

(1)

where F₀ and F represent non-quenched and quenched fluorescence intensities respectively, [Surfactant]_t is the total concentration of surfactant in solution, and [CPC] is the concentration of quencher in solution, which was varied. The linear plots between ln(F₀/F) vs. [CPC] yielded N from the slope. [Surfactant]_t herein used to obtain N was >CMC and variable. Extrapolation of N up to CMC was performed to determine N at CMC.

Since the CMC of CPC and CTAB were reported to be 1.0 mM at 30 °C, it was anticipated that they would not affect one another’s CMC. Less than 0.1 CMC was the maximum quencher (CPC) concentration that was utilized. Consequently, it had a very slight effect on [surfactant]_t and, accordingly, [micelle]_t. Hence, the micelle concentration of the CTAB system was essentially unaffected by the distribution of CPC between the micelle interior and the water outside. In this experiment, the pyrene concentration (10⁻⁷ M) was maintained low to prevent any complexicity due to excimer formation.

3. Results and Discussion

Critical micelle concentration (CMC): The CMC values of CTAB were determined by conductometry, tensiometry, and calorimetry methods at six different temperatures between 290 and 323 K. Typical experimental findings are illustrated in Figure 1A–C (CMC points are indicated in the figure).

For conductometric titration, concentrated surfactant solution was added in water. Initially, the amphiphilic molecules dissociated to furnish monomeric ions and counter ions. Upon successive addition, the concentration of ions increased and, therefore, the conductance (κ) also increased until the CMC was reached. At CMC, the monomeric ions aggregated to form micelles with the adhered counter ions. The number of free ions thereby decreased. The rate of increase in κ was also retarded. Two straight lines with different slopes before and after CMC were obtained. From the intersection point of these two lines, CMC was determined, as shown in Figure 1A, inset. There, the procedure for finding β (fraction of counter ions binding to the micelle) is indicated from the magnitudes of the slopes S₁ and S₂ (discussed latter) [26].

Surface tension measurement is a classical approach for studying the critical micelle concentration (CMC) of surfactants [1]. Figure 1B illustrates the relationship between surface tension (γ) and [CTAB] at different temperatures. As the surfactant concentration increased, the surface tension gradually decreased due to the adsorption of surfactant molecules at the air–water interface. Beyond a certain concentration, the surface tension stabilized, indicating that the air–liquid interface was saturated with surfactant molecules. The resulting plot of γ versus log[CTAB] exhibited a distinct breakpoint. This breakpoint corresponded to the CMC, with the derived values provided in

Table 1. Critical micelle concentration (CMC), fraction of counterion binding (β), standard free energy change (${\Delta \mathrm{G}}_{\mathrm{m}}^0$), standard entropy change (${\Delta \mathrm{S}}_{\mathrm{m}}^0$) and standard enthalpy change (${\Delta \mathrm{H}}_{\mathrm{m}}^0$) of micellization of CTAB at different temperatures in aqueous medium ^a.

T/K	CMC/mM			α	β	$\mathbf{\Delta }{\mathit{G}}_{\mathit{m}}^0$	$\mathbf{\Delta }{\mathit{S}}_{\mathit{m}}^0$	$\mathbf{\Delta }{\mathit{H}}_{\mathit{m}}^0$
	Cond.	Surface Tension	ITC			ITC	ITC (ITCVH)	ITC (ITCVH)
290	1.02	1.20	1.33	0.29	0.71	43.9	139 (161)	−3.50 (2.86)
293	0.97	1.10	1.25	0.30	0.70	44.3	131 (130)	−6.02 (−6.10)
298	0.94	0.98	1.18	0.32	0.68	44.8	122 (81)	−8.37 (−20.6)
303	0.97	1.08	1.19	0.33	0.67	45.2	112 (35)	−11.2 (−34.7)
308	1.03	1.15	1.35	0.34	0.64	44.6	106 (−12)	−12.0 (−48.3)
313	1.10	1.31	1.44	0.37	0.63	44.8	93 (−53)	−15.6 (−61.4)
323	1.25	-	1.60	0.41	0.59	44.6	80 (−130)	−18.7 (−86.6)

$\Delta G_m^0$/kJ mol⁻¹, $\Delta S_m^0$/J K⁻¹mol⁻¹, $\Delta H_m^0$/kJ mol⁻¹; ^a errors in CMC_ave, β, $\Delta G_m^0$, $\Delta H_m^0$, and $\Delta S_m^0$ were ±5, ±8, ±3, ±6, and ±10%, respectively.

.

The CMC of a surfactant can be determined precisely by microcalorimetry, which is based on thermal compensation method [27]. It has an advantage over other methods as the associated enthalpy change during the micellization process can also be evaluated. A representative ITC plot is shown in Figure 1C. The enthalpy of dilution courses consisted of three different regions, with two more or less horizontal regions intervened by a declining one; the height between the horizontal regions corresponded to the enthalpy of micellization and the inflection point indicated the CMC point (see inset, Figure 1C). The CMC point is typically considered to be the point of maximum slope on a sigmoidal curve using the Sigmoidal—Boltzman (SB) statistical relationship. Usually, the SB approach minimizes inconsistency by producing the CMC from data fitting. Better fit and more accurate findings would arise from a steep drop in enthalpy values between the two asymptotes. This inflection point is also corroborated with the minima from the first derivative plot, which is sensitive toward the transition point (see inset, Figure 1C). A comprehensive analysis of how to determine a non-ambiguous CMC is provided in our previous publication for different methods [28].

The CMC values obtained by different methods are shown in Table 1. The temperature dependence of CMC showed a minimum at 298 K for CTAB. The variation inCMC with temperature is presented in Figure 2A. The CMC values from ITC were higher than for conductance and surface tension; CMC is known to be a method-dependent property [29]. ITC is based on the enthalpy change of the involved process, whereas conductometry depends on the availability of ions in solution. On a comparative basis, all processes had a similar trend in CMC values: first CMC progressively decreased up to 298 K, and afterward it gradually increased up to 323 K (temperature limit of the study). Ionic surfactants like SDS [30] and bile salt (sodium cholate, CHAPS) [31] also showed similar temperature-dependent CMC minima. Non-ionic amphiphiles initially showed a CMC increase with temperature and, after a certain elevated temperature, the CMC decreased by way of dehydration of the polar head groups mostly leading to clouding [32]. In Figure 3A, the CMC values of Blume et al. [15] are also compared with our results, but they did not observe a distinct CMC minimum with temperature and their values were closer to conductometric CMC. Differences in the results reported from different laboratories on the same system are not uncommon in the literature [30].

Table 2. Method-dependent CMC values of CTAB at 303 K.

Method	CMC/mM	Reference Number
Tensiometry	1.14 (1.08) a	[33]
Conductometry	1.11 (0.97) a	[33]
Viscometry	1.01	[33]
Densiometry	0.86	[33]
Calorimetry	1.15 (1.19) a	[24]
Spectrophotometry	0.91	[34]
Fluorimetry	0.90	[35]
Voltametry	0.88	[36]
Static light scattering	1.0	[37]
Electron paramagnetic resonance	1.1	[38]

^a Parentheses are values from this study.

shows the method-dependent CMC of CTAB at 30 °C.

As observed in Table 2, the CMC values of CTAB obtained by various techniques were somewhat different. This confirmed that CMC is not one pinpoint value but is a range of concentrations at which aggregates are formed, as mentioned by IUPAC in its definition of CMC [8]. It reflects the onset of micelle formation rather than an abrupt phase transition as the system transitions gradually from monomeric to aggregated surfactant species. As mentioned earlier, CMC is indeed technique dependent, as different methods are sensitive to distinct changes in the system’s properties. Thus, these techniques can yield slightly different CMC values based on the physical or chemical property being measured. It might be mentioned that the CMC values of CTAB reported in this study by different methods were within the range reported in the literature.

For ionic surfactants in solution, the ionic head groups repel one another, hindering their association [30]. At lower temperatures, water molecules formed strong hydrogen bonds around the hydrophobic tails of the CTAB molecules. Breaking these structured hydration shells to allow micelle formation required significant energy, and the hydrophobic effect (which drives micellization) was weaker because the entropy gain from releasing the water molecules around the hydrophobic groups was not sufficient to overcome the enthalpic penalty, resulting in an increased CMC. As the temperature increased, the hydrophobic effect became stronger due to the entropy gain from water release, and the hydration shells around the hydrophobic tails were more easily disrupted. The balance between enthalpy and entropy has made micelle formation favorable (discussed later) at lower surfactant concentrations. At higher temperatures, the hydration of the hydrophilic head groups decreased, and thermal agitation weakened the cohesive forces driving micellization. For ionic surfactants, the electrostatic repulsion between charged head groups becomes more significant, raising the energy barrier for micelle formation. As a result, the CMC is increased. The temperature for a minimum CMC should be somewhere in between, which for ionic surfactants normally ranges between 293 and 313 K [18,30].

Counterion binding: The fraction of counterion binding (β) of micelles is obtained conductometrically by the slope ratio method. The counterions released from surfactant monomers are electrostatically attracted to the charged micelles and they are absorbed in the inner electrical double layer, partially neutralizing the micellar charge. The extent of charge neutralization, which is known as counterion binding, is evaluated from the ratio of the post- and pre-miceller slopes, ${\mathrm{S}}_2$ and ${\mathrm{S}}_1$, respectively, which are obtained from the conductometric plots, shown in Figure 1A, inset. The ratio ${\displaystyle \frac{{\mathrm{S}}_2}{{\mathrm{S}}_1}}$ is taken as the degree of ionization (α) of the micelle. Table 1 illustrates that α increased from 0.29 to 0.41 as the temperature rose from 290 to 323 K. This increase could be attributed to the interplay between columbic and thermal forces [22]. The columbic forces pulled the counterions toward the polar surfactant head groups, while the thermal forces promoted the dissociation of counterions from the surfactant head groups. Evidently, the thermal forces dominated over the columbic forces, resulting in the ionization of the surfactant CTAB and, consequently, the higher α values observed with increasing temperature.

The fraction of counterion binding is given as follows [26]:

$$\mathsf{\beta }=1-{\displaystyle \frac{{\mathrm{S}}_2}{{\mathrm{S}}_1}}$$

(2)

Generally, the β value depends on the nature (effective hydrodynamic radius, charge, polarizability of the counter ion), concentration of electrolytes, and micelle aggregation number [18]. The obtained values are shown in Table 1, column 4, and Figure 2B. The β values progressively decreased with increasing temperature. The increasing degree of ionization with temperature decreased β.

3.1. Energetics of Micellization

The standard enthalpy, Gibbs free energy, and entropy of micellization are among the thermodynamic characteristics of micellization in aqueous solutions that are of great importance. These thermodynamic variables are important for comprehending how structural and environmental factors affect the CMC values. In the evaluation of the energetics of micellization, the standard enthalpy of micellization (${\Delta \mathrm{H}}_{\mathrm{m}}^0$) values are directly obtained from ITC experiments [5,15]. The standard free energy change (${\Delta \mathrm{G}}_{\mathrm{m}}^0$) and standard entropy change ($\Delta S_m^0$) of micellization are calculated from equations 3 and 4, given below [10]:

$$\Delta G_m^0=(1+\beta )RT\ln X_{CMC}$$

(3)

where β and $X_{cmc}$ are the fraction of counterion binding and mole fraction concentration of CMC, respectively.

$$\Delta S_m^0=(\Delta H_m^0-\Delta G_m^0)/T$$

(4)

The standard enthalpy change of micellization can also be obtained from the van’t Hoff rationale [30].

$${\displaystyle \frac{d(\Delta G_m^0/T)}{d(1/T)}}=\Delta H_m^0$$

(5)

Plots between $\Delta G_m^0/T$ and 1/T were nonlinear (Figure 3A). Hence, a polynomial equation of the following form was used to calculate $\Delta H_m^0$.

$$\Delta G_m^0/T=A+B/T+C/T^2$$

(6)

Thus,

$$\Delta H_m^0={\displaystyle \frac{d(\Delta G_m^0/T)}{d(1/T)}}=B+2C/T$$

(7)

where A, B, and C denote the coefficients of the polynomial relation. The values of A, B, and C were −872.37, 12, and 6909, respectively, for the calorimetry-based CMC.

Also,

$$\Delta C_{p,m}^0={\displaystyle \frac{d(\Delta H_m^0)}{dT}}$$

(8)

where $\Delta C_{p,m}^0$ is the standard specific heat change of micellization.

It was evaluated from the slope of $\Delta H_m^0$ versus T plot (Figure 3B).

The energetic parameters of CTAB based on calorimetry are presented in Table 1. A negative value of $\Delta G_m^0$ is the measure of the spontaneity of micelle formation. The $\Delta H_m^0$ became more and more exothermic with increasing temperature by both methods (direct ITC and indirect van’t Hoff). With increasing temperature, the three dimensional structure of the medium and amphiphilic hydration were partially disrupted, and, consequently, the endothermic processes became weaker, making the overall aggregation largely exothermic. Normally, changeover of endothermic courses to exothermic with increasing temperature is common in ionic-amphiphilic self-aggregation. Blume et al. [15] studied DTAB, TTAB, and CTAB in the temperature range of 293–328 K but did not observe endothermicity; the observed course in the studied temperature range was exothermic. We considered that the surfactant head group was responsible for the nature of the thermogram because we observed exo-endo transition for SDS [30] and SDBS (sodium dodecylbenzenesulfonate) [16]. There may be such crossovers in the low temperature range. $\Delta H_m^0$ (van’t Hoff) has shown such a prospect (Figure 3B).

It was observed that $\Delta H_m^0$ directly determined by microcalorimetry (ITC) on the whole was higher than that evaluated by the van’t Hoff procedure using the ITC-based CMC. The dependence of $\Delta H_m^0$ on temperature presented in Figure 3B was linear. The $\Delta C_{p,m}^0$ (found from the slopes) values were −2.7 and −0.45 kJ mol⁻¹K⁻¹ by van’t Hoff and direct ITC, respectively. The negative values of $\Delta C_{p,m}^0$ indicated less exposure of the hydrophobic part of the surfactant to water molecules upon micellization [39]. Also, $\Delta C_{p,m}^0(calorimetry)<<\Delta C_{p,m}^0(va{n}^{,}tHoff)$.

The thermodynamics of CTAB micellization showed significant differences as assessed by van’t Hoff and calorimetric techniques. Calorimetric analysis is an integral process, while the van’t Hoff method is a differential process. Thus, it was not expected that the two methodologies would agree. There are a number of potential causes for the discrepancy, but the problem has received little attention. The latter includes the contributions of additional related or unrelated activities in addition to the amphiphilicself-aggregation process. In the biological domain, environmental changes (solvent structure alteration, solvation, desolvation, molecular orientation, component dissociation, molecular organization, etc.) are always linked to physicochemical processes (such as macromolecule-to-small-molecule interactions or macromolecular association, denaturation, etc.). All of these should contribute to the total heat or enthalpy change; as a result, the integral heat that results will be visible in a calorimeter. However, only one process is covered by the van’t Hoff relationship. Similarly, a sensitive calorimeter, such as ITC, can be used to measure the combined effect of changing the solvent structure, solvating the head groups, breaking icebergs around the amphiphilic tails, orienting the amphiphilic molecules to end up with a particular geometry, inter-micellar interaction, etc. during the micellization process. However, understanding the different kinds of impacts that are present in a system and separating out the distinct effects is a tough and demanding endeavor. Therefore, it was likewise not predicted that the van’t Hoff enthalpy and calorimetric enthalpy changes for the micellization process would agree.

The above differences in the two sets of $\Delta H_m^0$ were also reflected in $\Delta S_m^0$. The calorimetric $\Delta S_m^0$ values were more positive (except at 290 K). Although self-organization leads to orderedness, other processes (over and above releasing the hydrophobic hydration around the amphiphilic tail and the increased lability in the micellar interior) contribute to the increased entropy of the process. The entropies and heat capacities during micelle formation mainly arise from the change in the “hydrophobic” and “hydrophilic” hydration of the ionic surfactant monomers and their counter ions. During micellization, both the hydrophobic hydration of the amphiphilic monomers and the hydrophilic hydration of the ionic head groups and the counter ions (due to their mutual association) decrease. The processes are expected to be associated with the positive entropy change normally observed [15,17]. The significantly higher entropy values in the direct calorimetric measurements suggested the contributions of other associated processes operative in the system, as mentioned earlier. The large negative enthalpy change by van’t Hoff could even make a negative entropy change at higher temperature (see Table 1). It is important to note that the significant increase in the values of T$\Delta S_m^0$ compared to $\Delta H_m^0$ in this study indicated that micellization is driven by entropy, as the hydrophobic groups tend to orient themselves toward the micelle interior away from the bulk solvent.

The $\Delta H_m^0$ and $\Delta S_m^0$ values described above showed a good linear compensative correlation between them (Figure 3C). This phenomenon is considered to consist of two parts, a desolvation part for the dehydration of the hydrocarbon part of the surfactant molecules ($T_c$) and a chemical part for the aggregation of the hydrocarbon part of the surfactant molecules to form a micelle ($\Delta H_m^{*}$) [40,41]. Thus,

$$\Delta H_m^0=\Delta H_m^{*}+T_c\Delta S_m^0$$

(9)

T_c is proposed as a measure of the desolvation part, and the intercept $\Delta H_m^{*}$ is considered as an index of the chemical part of the process. The T_c values for CTAB micellization were found to be 304.13 and 253.61 K by the van’t Hoff and direct ITC methods, respectively. The average experimental temperature of the studied system was 304 K. T_c by ITC (direct) was much different from T_c by van’t Hoff. The $\Delta H_m^{*}$ values were found to be −45.56, and −39.13 kJ mol⁻¹ for CTAB by the van’t Hoff and direct ITC methods, respectively. More studies are required for the physicochemical justification of the parameters.

3.2. Interfacial Properties

In order to understand the behavior of CTAB solutions and their quantitative properties, it is essential to examine the solution’s interfacial parameters. These interfacial properties are the key to determining the surface activity. By plotting surface tension against [CTAB] solution, various interfacial parameters, such as the maximum surface excess concentration (Γ_max), minimum area per surfactant molecule (A_min), and surface pressure at the critical micelle concentration (Π_CMC), can be derived from the surface tension data. Specifically, the slope of the sigmoidal curve (dγ/dlogC), where C represents the surfactant concentration in mol L⁻¹, provides valuable insight into these properties.

The maximum surface excess concentration (Γ_max) at the air/water interface can be calculated by applying Gibb’s adsorption equation, as follows:

$${\Gamma }_{\max }=-{\displaystyle \frac{1}{2.303nRT}}\underset{C\to CMC}{\lim }{\left({\displaystyle \frac{d\gamma }{d\log C}}\right)}_T$$

(10)

where n is the number of particles per molecule of the surfactant. It was taken as 2 due to the fact that dissociation of the surfactant leads to the formation of two species. (dγ/dlogC)_T represents the slope of the plot of γ vs. log[CTAB] at constant temperature below CMC.

Assuming complete monolayer formation at CMC, the minimum area per surfactant molecule at the air–water interface (A_min) is calculated using the following equation:

$$A_{\min }={\displaystyle \frac{{10}^{18}}{N{\Gamma }_{\max }}}$$

(11)

where N = Avogadro number.

The value of the surface pressure at the CMC (Π_CMC) is obtained as follows:

$${\Pi }_{\mathrm{CMC}} = {\gamma }_0 - {\gamma }_{\mathrm{CMC}}$$

(12)

where γ₀ and γ_CMC are the values of the surface tension of water and the surfactant solution at the CMC, respectively.

Finally, the standard free energy of adsorption ($\Delta G_{ads}^0$) at the air–water interface can be evaluated from the following relationship:

$$\Delta G_{ads}^0=\Delta G_m^0-{\displaystyle \frac{{\Pi }_{CMC}}{{\Gamma }_{\max }}}$$

(13)

The parameters derived from the surface tension measurements at different temperatures are summarized in

Table 3. Maximum surface excess concentration (Γ_max), surface pressure at the CMC (Π_CMC), minimum area per surfactant molecule (A_min), and standard free energy of adsorption ($\Delta G_{ads}^0$) at the air–water interface of CTAB at different temperatures in aqueous medium.

T/K	10−6 Γmax	ΠCMC	Amin	$-\Delta {\mathit{G}}_{\mathit{a}\mathit{d}\mathit{s}}^0$
290	1.40	40	1.18	72.5
293	1.35	39.5	1.23	73.5
298	1.25	39	1.33	76.0
303	1.20	37	1.38	76.1
308	1.17	36.5	1.42	75.8
313	1.04	36	1.60	79.4

Γ_max/mol m⁻²; A_min/nm² molecule⁻¹; Π_CMC/mN m⁻¹, $\Delta G_{ads}^0$/kJ mol⁻¹.

.

The data showed that the Γ_max as well as Π_CMC values (Table 3) decreased with increasing temperature. This behavior underscored the interplay of molecular dynamics, thermal agitation, and solubility with rising temperature in CTAB solution, making it energetically favorable for more CTAB molecules to remain in the bulk phase rather than adsorbing at the interface. The A_min values increased from 1.18 to 1.60 nm² molecule⁻¹ with increasing temperatures, which denoted that the surfactant molecules occupied more surface areas at the air–water interface. With a rise in temperature, the surfactant molecules at the air–water interface experienced increased thermal agitation, which probably disrupted the tight packing of the molecules. As a result, the molecules occupied a larger area at the interface. Similar trends were observed in a previous study [42].

Negative values of $\Delta G_{ads}^0$ indicate that the adsorption of surfactant molecules at the air–water interface is spontaneous. The increasing $\Delta G_{ads}^0$ values with rising temperature suggested that the adsorption process involved the expansion of dehydrated hydrophilic groups. However, since the hydrophilic groups of the surfactant were insufficiently hydrated, the adsorption process occurred with minimal energy input as the temperature increased. This surface adsorption phenomenon was more spontaneous than micellization due to a larger negative value than $\Delta G_m^0$. Upon increasing the temperature, the $\Delta G_{ads}^0$ values became more negative, which indicated the higher spontaneity of adsorption of CTAB molecules on the surface. Such trends were also observed in the literature [42].

4. Water Penetration

Water penetration into the micelle has been an important aspect of studying micelle behavior in solution. Spectral measurements (specially NMR spectra) have been used to understand the phenomenon, and there are reports of water penetration of up to seven methylene groups of the buried alkyl chain inside the micelle [43]. Recently, Kresheck [44] proposed to evaluate a solvent accessible area ($AS{A}_{\exp }$) parameter from ITC measurements, which may be correlated with the extent of hydration of the micelle interior, n (or the number of water exposed carbon atoms in the micelle) according to the ‘fjord’ model of Menger [43]. We used Kresheck’s relationship to evaluate or assess the water penetration extent into the cationic micelle of CTAB along with anionic (SDS) and zwitterionic (CHAPS) micelles.

The proposed $AS{A}_{\exp }$ of Kresheck is related to $\Delta C_{p,m}^0$ by the following relationship:

$${\mathit{ASA}}_{\exp }=-{\displaystyle \frac{\Delta C_{p,m}^0 \left({\mathrm{J} \mathrm{mol}}^{-1}{ \mathrm{K}}^{-1}\right)}{1.463 \left({\mathrm{J} \mathrm{mol}}^{-1} {\mathrm{K}}^{-1}{ \AA }^2\right) }}$$

(14)

$\Delta C_{p,m}^0$ by ITC (−0.45 kJ mol⁻¹) of CTAB yielded ($AS{A}_{\exp }$) = 308 Ų from Equation (14). Such $AS{A}_{\exp }$ values of SDS and CHAPS ($\Delta C_{p,m}^0$ = −0.589 and −0.241 kJ mol⁻¹, respectively) were found to be 402.6 and 170 Ų, respectively. Since $AS{A}_{\exp }$ relates to n by the relationship [44], n = ($AS{A}_{\exp }$ − 89)/29, the calculated n values for CTAB, SDS, and CHAPS were 8, 7, and 3, respectively. Thus, water penetrated up to 8, 7, and 3 carbon atoms in their micelles. With reference to Menger’s report [30] of water penetration of up to 7 carbon atoms into the mixed micelles of CTAB and 8-keto CTAB in the molar ratio of 4.3:1, the present findings were reasonable. Kreshek’s n values from ITC for alkyldimethylphosphineoxide micelles were 5, 6, 9, 11, and 12 for 8, 9, 10, 11, and 12 alkyl chains, respectively [44].

In a recent study [45] following Kresheck’s rationale, n values for decyl-, dodecyl- and tetradecyltrimethyl ammonium chloride micelles in water and in NaCl solution to the extent of 4–8 were reported. Although this report and our findings looked reasonable, they are to be considered with some reservations since the relationships used for calculation were semi-empirical and they were formulated for non-ionic surfactants, while the surfactants used by here (except CHAPS) and by Rogac et al. [45] were all ionic. More studies in this direction are warranted.

5. Micelle Aggregation Number (N)

The extent of monomers present in a micelle is important information about surfactant behavior. Different methods are used to determine the aggregation number (N), and each method has its limitations. Unambiguous results are seldom obtained using different methods and concepts. In most of the past work, the static light scattering (SLS) method was used for determining N [11]. The estimated micellar molar mass was divided by the monomer molar mass to obtain N. Because of counterion binding of micelles and micellar solvation, the N values found from SLS in principle should be greater than actual values. Other than SLS, the fluorescence quenching method has been frequently used to obtain the N values of micelles wherein counter-ion association and solvation hardly affect the emission characteristics of the probe and, hence, the magnitude of N [13]. Both dynamic and static quenching methods are used, of which the former gives more accurate results than the latter, although for the sake of simplicity and easier equipment availability, the latter method is most frequently used. In the quenching procedure, surfactant solutions above CMC (~10 times or more) are employed. In SANS experiments, surfactant concentrations ~1 wt% are required on the lower side to obtain reasonable measurement sensitivity; concentrations much higher than CMC are used in practice [14]. In most of the methods, [Surfactant] > CMC is used to increase measurement accuracy. Determination at CMC (N^CMC) mostly remains unperformed. To obtain N^CMC, results of measurements at different [surfactant] > CMC have to either be graphically plotted to find N^CMC by way of extrapolation or evaluated using appropriate polynomial relations. This is seldom done, and hence the results remain non-normalized and non-consistent. In recent literature, newer approaches (principles) and methods are emerging to evaluate N^CMC. ITC is one of them. Thinning of the surfactant film in stages has been also used to determine N not N^CMC. In the present study, we considered the SFQ and ITC methods to estimate the N^CMC values of surfactants of different types, like CTAB, SDS, and CHAPS, at a fixed temperature. The temperature effect on N was also studied.

SFQ findings: The fluorescence spectra of pyrene at a constant [surfactant]_t, i.e., of CTAB in the presence of varied [CPC], are presented in Figure 4A (inset). The ln(F₀/F) vs. [CPC] plot in the main diagram helped to obtain N from the slope using Equation (1). The N values obtained at different concentrations of the surfactant were then plotted against ([surfactant]_t-CMC) to obtain the aggregation number at CMC, or N^CMC. Such plots for CTAB, SDS, and CHAPS at 303 K are presented in Figure 4B. The plot of SDS was linear whereas those of CTAB and CHAPS were curved (two degree polynomial relations were required to obtain N^CMC values from the data). The N^CMC values of SDS, CTAB, and CHAPS at 303 K were found to be 54, 48, and 4, respectively. The [Surfactant]_t-dependent N values herein obtained are presented in

Table 4. Concentration-dependent aggregation number (N) of CTAB, SDS, and CHAPS at 303 K ^a.

CTAB		SDS		CHAPS
Concentration	N b	Concentration	N	Concentration	N
0 (at CMC, by extrapolation)	48	0 (at CMC, by extrapolation)	54	0 (at CMC, by extrapolation)	4
4.9	53	40	64	25	6
9.8	63	80	78	50	15
14.7	66	120	83	75	20
19.6	67	160	97	100	22
25	69	200	104	125	24
30	71	240		150	27

^a Errors in N were ±8%. ^b From reference [10].

. In this context, we mention that an elaborate assessment of the N of CTAB by different methods (TRFQ, SFQ, SANS, and SLS) was performed earlier by our laboratory [10], with special reference to the evaluation of N^CMC and its comparison with different methods.

It might be added that pyrene can also be quenched at higher concentration by itself without the need for a quencher. Here, pyrene concentrations in the range of mM were used to track the strength of excimer formation, which in turn helped to calculate the aggregation number of micelles [46,47]. Kwiatkowski et al. [46] reported the micellization aggregation number of potassium oleate and in the presence of hydrophobic polymer poly(4-vinylpyridine) to monitor self-quenching of pyrene at varying concentrations.

ITC Findings: Blume et al. [15] proposed a simulation method using ITC results based on the “mass action model” of micelle formation to evaluate the N^CMC values of a number of ionic surfactants. We also used the method on several cationic, anionic, and non-ionic surfactants [16,17,18]. We experienced lower values of N^CMC than those obtained by the SFQ method (where [surfactant] = 10CMC was used in the experiments), which reasonably accounted for the difference. The results of Blume et al. on CTAB at different temperatures are presented in Table 3.

In a recent study, Wadekar et al. [19] used the ITC technique to evaluate the micelle aggregation number, N^CMC, considering an approximate model [48,49]. They used the slope between ΔH (enthalpy of demicellization) and the concentration C, i.e., (dΔH/dC) around C, i.e., CMC. So that

$$N={\displaystyle \frac{8C}{\Delta H_m^0}}{\displaystyle \frac{d\Delta H_m^0}{dC}}$$

(15)

We also used Equation (15) to obtain theN values of CTAB, SDS, and CHAPS at different temperatures. The results are presented in the

Table 5. Aggregation number (N) of CTAB, SDS, and CHAPS at different temperatures by ITC and fluorimetry.

T/K	CTAB		SDS		CHAPS
	ITC (at CMC)	Fluorimetry (10 mM)	ITC (at CMC)	Fluorimetry (80 mM)	ITC (at CMC)	Fluorimetry (50 mM)
290	32	69		100	5	16
293	33 (16) a	-	23		5	15
298	34	66		86	5	15
303	33 (24) a	63 (48) b	21	78 (54) b	5	18 (4) b
308	33	-	21		5	17
313	35 (36) a	58	20	60	6	18
318			20
323	35 (27) a	53		37	6	18

^a Parenthesis values of Blume et al. [15] by simulation procedure. ^b Values obtained by extrapolation from concentration dependent N.

.

The N values of all three surfactants (CTAB, SDS, and CHAPS) were found to be significantly dependent on [surfactant] (Table 3). The derived values at CMC were 48 and 54 for CTAB and SDS, respectively, whereas that of CHAPS was much smaller, i.e., only 4. CHAPS being structurally comparable with cholate showed a low N value like bile salts. The larger head group in CTAB produced a lower N value than SDS, which has a relatively lower head group size.

The temperature-dependent N of the studied surfactants was interesting. Fluorimetry evidenced a decreased N value at 10CMC for both CTAB and SDS with increased temperature, which meant a decreased micelle size, which was in line with the literature [18,50,51]. Whereas for CHAPS at 10CMC, within experimental error, N was practically independent of temperature. The decrease in β value with temperature also nicely correlated with decreasing N value; smaller micelles have lower surface charge and consequently lower β. The ITC results were at variance; the results found from Wadekar et al.’s procedure were independent of temperature for all three surfactants. The ITC findings of Blume et al. [15] on CTAB (shown in parenthesis), although non-consistent, evidenced a trend of increasing up to 313 K and declining at 323 K. So, on the whole, the ITC method was still not beyond reservation to accept. The methods need improvement, particularly that of Wadekar et al., which showed only very little variation in N^CMC value for all three surfactants.

In this connection, we may refer to the work of Anackkor et al. [52] to determine the N value of ionic surfactants from stepwise thinning of foam films. They showed via SFQ and other methods an increase in N value with increasing [surfactant], viz., SDS, CTAB, CPC, etc., which generally tallied with past reports. The method has prospects and requires refinements. In addition to N, it has been reported that the method can also assess the micelle ionization degree and micelle charge.

6. Conclusions

The interfacial properties and micellization of CTAB at different temperatures by tensiometry, conductometry, and calorimetry were studied, and systematic and detailed studies are presented at a large number of temperatures that were seldom performed in the past. Temperature-dependent CMC of CTAB progressively decreased from 290 to 298 K, and afterward it gradually increased up to 323 K, displaying a U-shaped trend. Basically, repulsion among positive trimethyalammonium head groups and hydrophobic hydration around non-polar tails hinders aggregation, resulting in a high CMC at low temperatures. As temperature rises beyond this point, reduced hydrophobic hydration and entropy effects facilitate aggregation, lowering the CMC. Beyond 298 K, thermal energy and electrostatic repulsion controlled micelle formation, causing the CMC to increase.

The $\Delta G_{ads}^0$ and $\Delta G_m^0$ values suggested that CTAB monomer adsorption at the air–water interface is more spontaneous than micellization. The spontaneity of CTAB adsorption at the air–water interface arises from its immediate impact on the interfacial tension with a lower energy barrier and its occurrence at lower concentrations than micellization. At higher concentrations, micellization is still thermodynamically favorable with a more complex balance of enthalpic and entropic contributions. ITC-determined CMC-based $\Delta H_m^0$ found from the van’t Hoff indirect rationale was found to be more exothermic with temperature increment compared to $\Delta H_m^0$ directly determined by calorimetry. The difference was considered to arise from the integral heat registered in the calorimeter (which takes care of varied possible specific and nonspecific processes in the reacting system), whereas the van’t Hoff method takes care of only the monomer↔micelle equilibrium. Hence, the energetics of the two methods of evaluation should not match unless the monomer↔micelle equilibrium process is the main contributor. The values of T$\Delta S_m^0$ were significantly greater than those of $\Delta H_m^0$. This indicated that micellization is mainly controlled by the entropy gain resulting from the disintegration of the structured water molecules surrounding the hydrophobic groups of the surfactants when these groups are moved from the bulk solvent to the micelle’s interior, where they feel freer due to the non-polar environment.

In micelle self-aggregation, N is an important parameter that is normally determined at [surfactant] >>CMC, whereas other parameters (essentially energetics) are determined at CMC. We therefore also determined N^CMC by fluorimetry. An ITC-based N^CMC determining method was also used. It was observed that ITC produced N^CMC values that were practically temperature independent. This unexpected observation requires more experimentations and analysis. Another ITC-evaluated parameter was the information on the degree penetration of water into the micelle interior. This was evaluated on all three studied surfactants, CTAB, SDS, and CHAPS, and the n values (number of methylene groups in the alkyl chains where water molecules penetrates) were found to be 8, 7 and 3, respectively, which was not unusual as per literature reports and the ‘fjord’ micelle model. The results presented herein warrant scope for further studies.

By employing a multi-technique approach, this work revealed a deeper understanding of how temperature modulates the surface activity, aggregation number, and energetics of the micellization (in comparison between van’t Hoff and calorimetry) process. These results advance the understanding of amphiphilic behavior and contribute to the broader field of surfactant science, particularly in the context of temperature-sensitive applications in industrial and pharmaceutical systems. The findings presented herein not only advance the fundamental knowledge of CTAB behavior but also provide a robust dataset that can serve as a reference for future studies and modeling efforts. This work underscores the importance of systematic exploration of temperature effects to fully elucidate the complex physicochemical phenomena underlying surfactant systems.