The Impact of Micelle Formation on Surfactant Adsorption–Desorption

Abstract

The monomer–micelle equilibrium is shown to be responsible for an asymmetry between surfactant adsorption and desorption rates. When a solution containing micelles is brought into contact with a solid surface, the micelles dissociate to supply monomers that adsorb to the surface. When the same surface is subsequently exposed to a surfactant-free solution, desorption occurs slowly because of the higher affinity of the monomers to remain to the surface than to form micelles. As a result, the number of monomers that desorb is limited by the critical micelle concentration (CMC) of the surfactant. This effect is particularly pronounced for surfactants with low CMC values and in systems with high surface-to-volume ratios, such as porous media. A generic model is developed and applied to simulate the Ca²⁺-mediated adsorption and desorption of surfactants in limestone cores.

Introduction

The adsorption of surfactants at solid–liquid interfaces has been extensively studied due to its important role in applications such as detergency, mineral flotation, corrosion inhibition, dispersion of solids, and chemical enhanced oil recovery (EOR).¹ It is generally driven by two factors: the change in free energy when changing a surface–water contact into a surface–surfactant contact and the escaping tendency of the surfactant hydrocarbon, fluorocarbon, or siloxane moiety from the aqueous environment.² Adsorption typically involves single monomers rather than micelles and may occur through ion exchange, ion pairing, entropy gain, polarization of π electrons, or van der Waals forces.³⁻⁶ Many factors influence adsorption behavior such as the nature and charge of the surfactant headgroup and solid surface, the nature and length of the surfactant chain, and parameters such as temperature, salinity, and pH.

The adsorption behavior of surfactants can be captured by a simple model considering monomer/micellar diffusion and micellar dissociation.⁷⁻⁹ In this model, micelles are assumed to contribute to adsorption only by releasing monomers that in turn diffuse to the surface and not by direct adsorption of micelles. When local equilibrium is reached between the adsorbed surfactant and the concentration near the solid surface, the net adsorption rate is zero, or alternatively the adsorption rate is equal to the desorption rate. Here, we focus on the rate at which surfactants desorb from a surfactant-loaded surface that is exposed to a surfactant-free solution. This process is relevant in, for example, chemical EOR, where surfactants are injected into oil reservoirs to reduce the oil–water interfacial tension and to mobilize residual oil toward producing wells.¹⁰ Experiments have shown that continued injection of a surfactant-free solution leads to slow desorption of the adsorbed surfactants, and it was proposed that this could be utilized for surfactant reactivation.¹¹ Asymmetries between adsorption and desorption rates have also been observed in experiments with a quartz crystal microbalance with dissipation monitoring (QCM-D) sensors.^12,13 Here, adsorption rate refers to mass adsorbed during the continuous injection of a surfactant solution (typically at concentrations above the CMC), and desorption rate refers to mass desorbed during the injection of a surfactant-free solution. To date, however, there has been no mechanistic explanation of this phenomenon. We note that in some cases, surfactants may also be effectively irreversibly adsorbed.¹⁴ In porous media, one might expect diffusional limitation by pore blocking effects to play a role, which arise as a consequence of the micropore morphology and connectivity. This has been observed for the adsorption–desorption behavior of gases in three-dimensional disordered pore networks such as porous glasses or polymer networks.^15,16 However, this cannot account for the observations in QCM-D experiments, where the surfactant solution flows past the surface of a coated sensor. The same holds for other mechanisms relying on pore space geometries such as micellar exclusion. This points toward a more general mechanism responsible for limiting the rate of surfactant desorption. Here, we show that the local chemical equilibrium between the adsorbed monomers and the monomers and micelles in solution is responsible for the low surfactant desorption rate, which is particularly pronounced in systems with high surface-to-volume ratios such as porous media.

Results

Surfactant Adsorption in Core Flooding Experiments

Core flooding experiments were performed to investigate surfactant adsorption and desorption rates. Three solutions were sequentially injected into an Estaillades limestone core: (i) brine with surfactant, (ii) brine, and (iii) a 1:1 mixture of brine and isopropyl alcohol (IPA). The injection of brine/IPA leads to fast desorption of the surfactant as it increases the activity of the aqueous phase and therefore the solubility of the surfactant. The brine composition is shown in Table 1.

Table 1. Brine Composition.

ion	concentration (mg/L)	concentration (mmol/L)
[Na+]	1664	72.38
[K+]	28	0.72
[Ca2+]	26	0.65
[Mg2+]	11	0.45
[Cl–]	2670	75.31
TDS	4399

Effluent fractions were collected throughout the experiment, and the surfactant concentration was measured by titration. The surfactant used (ENORDET J771) is based on a (Shell proprietary) branched C12–C13 alcohol and contains a sulfate group.^17,18 The molecular weight is about 700 g/mol on average, and the CMC is 0.025 mmol/L. Other experimental conditions are included in the Methods. In the first step, 2 pore volumes (PV) of 0.56% J771 surfactant were injected into the core (2031 mg in total). The resulting concentration is shown as a function of the number of PVs injected in Figure 1. Note that it would take approximately 1 PV for the injected surfactant solution to arrive at the core outlet (reaching 50% of the injected concentration) in case no adsorption would occur.

Figure 1.

Surfactant concentration versus pore volumes injected. Three solutions were injected sequentially: (i) brine with surfactant, (ii) brine, and (iii) a 1:1 mixture of brine and IPA. The dashed line indicates the injected surfactant concentration.

The surfactant concentration reaches 50% of the injected value at approximately 1.2 PV and 100% (dashed line) at 2 PV, indicating that adsorption is complete. Brine injection starts at 2 PV, and about 1.5 PV later the produced surfactant concentration has decreased below the experimental resolution. The amount of surfactant produced in the first two injection steps can be determined from the area under the curve and amounts to 1741 mg. Subtracting this from the injected amount leads to a surfactant adsorption value of 24.1 mg/100 g rock. Injection of brine and IPA in a 1:1 ratio recovers the remaining surfactant (274 mg) within experimental error.

The experiment shows that the surfactant either does not desorb or desorbs very slowly from the calcite surface when brine with the same ionic composition is injected. This has been reproduced in other core flooding experiments (not shown here). Surfactant adsorption–desorption experiments in other systems, such as surfactant solutions flowing past the surface of silica- or calcite-coated sensors, show that adsorption is in fact reversible but that the desorption rate is very low.^12,13 Other experiments showing slow desorption of surfactants from Berea rock surfaces confirm that this behavior does not only apply to divalent ion bridging of surfactants to calcite surfaces.¹¹ In the following, we will show that the low rate of desorption can be understood by considering the equilibrium between adsorbed surfactants and the surfactant monomers/micelles in solution.

Static Equilibrium between Surfactants, Ca²⁺ Ions, and the Rock Surface

To understand the asymmetry in adsorption–desorption rates, a simplified chemical model is worked out in which the CMC plays a key role. We assume that the only divalent ion type in solution is calcium (Ca²⁺) and that the only adsorption mechanism is divalent ion bridging to negatively charged surface sites. Van der Waals adsorption of neutral surfactant complexes and adsorption to positively charged surface sites are not included. The relevant equilibrium reactions are shown schematically in Figure 2.

Figure 2.

Reaction pathways of surfactant monomers in an aqueous solution containing Ca²⁺ interfaced with a solid surface.

The complexation of Ca²⁺ ions by surfactant monomers (Surf^–) is given by

1

and the adsorption of surfactant–Ca²⁺ complexes to negative surface sites (>^–) can be expressed as

2

The formation of surfactant micelles is described using a closed association model, meaning that only the monomers and monodisperse micelles with aggregation number n are present in the solution. This can be expressed with the following equilibrium reaction

3

where Surf_n^n– is a micelle consisting of n monomers. The value of K_mic is such that, above the CMC, all additional monomers are aggregated into micelles. Ca²⁺ complexation by micelles (considering complete charge neutralization) can be expressed as

4

Note that other ions such as Na⁺ will also screen some of the charge of the surfactant headgroups and of the rock surface. Inclusion of such reactions will change the values of the equilibrium constants and is not considered here for simplicity.

Mechanistically, the adsorption process can be understood as follows. When injecting a surfactant solution at a concentration above the CMC, surfactants are present as monomers and aggregated into surfactant micelles. When the solution is in contact with the surface, the monomers can adsorb to the surface bridged by Ca²⁺. To maintain the monomer concentration levels in solution at the CMC, the micelles dissociate to supply additional monomers to the solution. At the flood front, the surfactant concentration drops below the CMC and equilibrium is established between the monomers, the available Ca²⁺, and the surface sites. When the surfactant solution has reached the core outlet and adsorption is complete, equilibrium is reached between the monomers – with their concentration at the CMC, since micelles are also present – the available Ca²⁺, and the surface sites. In the following, we will derive an equation with which the resulting equilibrium concentrations can be determined, requiring only a few input parameters. The mass action equations and equilibrium constants for reactions 1 and 2 are given by

5

6

The overall mass balance over the negative surface sites is given by

7

For Ca, the balance is

8

and for surfactants, it is

9

in which

10

For the relation between the free monomers and the total amount of monomers in the solution (free monomers and monomers in the micelles), we use

11

This equation captures that, in the absence of micelles, [Surf^–] is equal to [Surf*], and for concentrations above the CMC, [Surf^–] is limited by [Surf_CMC^–] (the CMC value) and the remaining monomers are located within the micelles. In the following, the above equations are used to solve for [SurfCa⁺], resulting in an intrinsic equation that can be solved numerically. Solving eq 6 for [>^–] and substituting in eq 7 give

12

Equation 9 can be rewritten as

13

which provides a functional connection to [SurfCa⁺] after substitution of eq 12

14

Extracting [SurfCa⁺] from eq 5 and substituting for [Surf^–] through the combination of eqs 11 and 14 give

15

which is an intrinsic equation for [SurfCa⁺]. For the solution of this equation, a Python script was used in combination with the “brentq” solver. In this solution scheme, [Ca²⁺] is an externally provided parameter, which is not attempted to solve for. The total required concentration of calcium to arrive at this value of [Ca²⁺], i.e., [Ca]⁰, can be calculated from eq 8. Once the numerical value of [SurfCa⁺] is obtained, the value of [Surf*] can be calculated from eq 13. By substituting the outcome in eq 11, [Surf^–] can be calculated. To calculate the value of [>SurfCa], eq 13 can be used. The required input parameters are listed in Table 2.

Table 2. Input Parameters.

parameter	description	determination	value (example case)
[Ca2+]	equilibrium Ca2+ concentration	calcite dissolution calculations	0.1 mmol/L (1 mmol/L is used as the high case)
[SurfCMC–]	CMC value of the surfactant	measurement or provided by supplier	0.025 mmol/L (17.5 mg/L)
[Surf]0	total surfactant concentration	experimental input	0.5%
[>–]0	number of negative surface sites	surface area measurements	0.75 mmol/100 g
KSurfCa+	complex formation constant	spectrophotometric measurements	log K = −2.2

Application to Surfactant Adsorption in Limestone Cores

The discussion thus far has been applicable to brines containing Ca²⁺ and anionic surfactants and the adsorption of SurfCa⁺ complexes to a negatively charged surface. The input parameters for the above model (listed in Table 2) are now discussed for the J771 surfactant, calcite, and the brine composition in Table 1. The parameters are controlled in the experiment, measured, or provided by the supplier of the chemicals. The CMC of the surfactant (0.025 mmol/L) is measured in deionized water at room temperature; note that this value is relatively low and can be higher for other surfactants. An initial surfactant concentration of 0.5% is used, corresponding to a concentration of 7.14 mmol/L given an average molecular weight of 700 Da.

The number of surface sites is derived from surface area measurements. The Brunauer–Emmett–Teller (BET) surface area of Estaillades limestone cores was determined by krypton adsorption to be 0.79 m²/g. Together with an adsorption value of 24.1 mg/100 g rock, this translates to an adsorption density (at maximum adsorption) of 1 surfactant monomer per 3.8 nm². The surface area can also be converted to a surface site density expressed as mol/g rock. The predominant cleavage plane of calcite is the (104) surface, which exposes equal numbers of >Ca⁺ and >CO₃^– sites (approximately 5 sites/nm² or 8.22 × 10^–6 mol/m²).¹⁹⁻²⁵ A surface area of 0.79 m²/g therefore yields a total surface site density of about 1.5 mmol/100 g rock (>Ca⁺ sites + >CO₃^– sites). Since we consider only negative surface sites, a surface site density of 0.75 mmol/100 g is used. For ease of interpretation, this surface density is transferred to an effective concentration in the liquid-filled pores. 100 g of rock corresponds to a bulk rock volume of 0.1/ρ_rock m³; 1 m³ of bulk rock corresponds to a pore volume of φ/(1 – φ) m³. The pore volume for 100 g is thus 0.1/ρ_rock × φ/(1 – φ) m³. Assuming that ρ_rock = 2500 kg/m³ and φ = 0.3, the pore volume that relates to 100 g of rock is 17 cm³ and the “volumetric” surface density is of the order of 0.044 mol/L.

The values of the two equilibrium constants (K_SurfCa⁺ and K_>SurfCa) are also required for evaluating eq 15. The surfactant–Ca²⁺ complex formation constant has been measured independently by spectrophotometric techniques.²⁶ Based on those results, we set this value to the approximate value of log K = −2.2. The equilibrium constant for the surface reaction with the surfactant complex has not been measured independently and is tuned to log K = −3.95 to achieve an adsorption of 25% of the supplied surfactants. The resulting equilibrium between the surfactant solution and the calcite surface is shown in Figure 3.

Figure 3.

(a) [Surf]_micelles, [Surf^–], [>SurfCa], and [Ca]_total versus total surfactant concentration for the case where [Ca²⁺] = 0.1 mM. (b) Surfactant free monomer concentration ([Surf^–]) versus total surfactant concentration. (c) Schematics of the equilibrium situation at concentrations indicated by I and II in panel (a).

Figure 3 shows that both [Surf^–] and [>SurfCa] increase linearly with the total surfactant concentration until the maximum adsorption value is reached. When this value is reached, [Surf^–] stabilizes at the CMC and micelles start to form. In our example, this occurs at a total surfactant concentration of about 2 mmol/L (1400 mg/L). From that concentration onward, the number of adsorbed surfactants is in equilibrium with [Surf^–] (which equals the CMC) and will not increase further; all added surfactant will therefore aggregate into micelles. This reflects that the affinity of surfactants to adsorb to the surface is higher than the affinity to form micelles.

The dashed line indicated by II reflects the equilibrium situation in the first stage of the core flooding experiment, in which brine with the surfactant is injected (Figure 1). When the injected surfactant solution is in contact with the rock surface ([Surf]_total = 7.14 mmol/L), adsorption has reached its maximum value and most of the surfactants in solution are aggregated into micelles. Microscopically, this means that the micelles dissociate to supply monomers that adsorb to the rock surface.

For desorption, we consider that the surfactant-loaded surface is initially in equilibrium with a high-concentration surfactant solution, which is then replaced by the same but surfactant-free solution. This is the situation in the second stage of the core flooding experiment, when brine is displacing the surfactant solution. As chemical equilibrium is assumed, part of the adsorbed surfactants will go into solution and a Surf^– concentration will establish, which is the situation indicated by I. Micelles do not form because adsorption is favored over micellization. Since [Surf^–] is much smaller than the adsorbed surfactant concentration (≤0.025 versus 1.8 mmol/L), only a very few surfactant monomers have to desorb to obtain equilibrium. To remove all the absorbed surfactants, the surfactant-free brine solution needs to be replaced many times. In fact, when the process progresses, the Surf^– concentration to obtain equilibrium decreases and so the ability to store released surfactants diminishes exponentially. For this case, it would take hundreds of pore volumes of brine to achieve full desorption of the adsorbed surfactants (see Figure 4). Note the value of the CMC is crucial: in case the CMC for a particular surfactant would be in the mmol/L range, fewer pore volumes of brine would be required for complete surfactant desorption. Note that [Surf]_micelles is nonzero because the formation of micelles is approximated by an exponential function (eq 11) and is therefore gradual.

Figure 4.

(a) [Surf]_micelles, [Surf^–], and [>SurfCa] and (b) fraction of adsorbed surfactants vs PV of brine injected.

The rate of desorption will also be higher in systems with a smaller surface-to-volume ratio, such as liquid flowing past the surface of a calcite-coated sensor. Although the equilibrium concentrations are the same, the larger volume means that the surfactant-free brine solution needs to be replaced fewer times to lower the adsorbed surfactant concentration. For modeling purposes, this can be reflected in a lower value of the volumetric surface density.

Implementation into Numerical Simulators

In the following, we discuss how the monomer–micelle equilibrium can be implemented into a numerical simulator. PHREEQC is used for the definition of chemical reactions and the calculation of the equilibrium concentrations. Coupled to a flow simulator such as MoReS, this can be extended to calculate concentrations in a dynamic setting, such as a core flooding experiment. As discussed previously, the aggregation of surfactant monomers into micelles can be described by an equilibrium reaction with an associated equilibrium constant

16

A value for the equilibrium constant can be derived that depends on the CMC value of the surfactant. The total number of monomers is given by

17

which can be combined with the expression for K_mic to yield

18

The CMC value can be linked to an approximate K_mic by considering that at the CMC

19

Combining this with the expression for K_mic yields

20

Using the CMC value of J771 (0.025 mmol/L) and n = 6, we obtain an equilibrium constant of K_mic = 5.9 × 10^–23. The resulting concentration of free surfactant monomers and monomers in micelles is shown in Figure 5a as a function of total surfactant concentration. The transition becomes more abrupt at higher n (e.g., n = 60, shown in Figure 5b).

Figure 5.

Concentrations of free surfactant monomers and monomers in micelles as a function of total surfactant concentration. (a) n = 6. (b) n = 60. The CMC is indicated by the gray dashed line.

We include the micellization of surfactants in PHREEQC by defining the equilibrium reaction (eq 16) and assigning an approximate equilibrium constant (log K_mic = −22 for n = 6). Note that when we consider that most of the surfactants are in micelles, an equilibrium reaction also needs to be included for their interaction with divalent ions (eq 4). Here, the value of K_mic–Ca was tuned to match the experimentally observed Ca²⁺ concentration in solution.

Modeling without Micellization

We first investigate surfactant adsorption and desorption when modeling surfactant and brine injection into a calcite core without considering micellization. Reactions 1 and 2 were included in MoReS–PHREEQC, and the equilibrium constant of the Ca²⁺ complexation reaction (K_SurfCa⁺) was set to log K = −2.2. The resulting surfactant concentration is shown for different values of K_>SurfCa in Figure 6.

Figure 6.

Surfactant adsorption and desorption without considering micellization.

A delay of 0.23 PV is obtained with log K_>SurfCa = – 2.0, which is in agreement with the core flooding experiment shown in Figure 1. However, using equilibrium reactions to describe adsorption results in complete desorption of the surfactant when brine is injected, which is at odds with the experimental data. At high equilibrium constants, desorption even gives rise to a hump-like feature extending to 4 PV. Considering the low rate of desorption, we could regard adsorption as irreversible in the modeling by describing it with a Langmuir adsorption isotherm. However, in this case it is no longer possible to use equilibrium reactions for its description and no surfactant desorption would be observed at all.

Modeling with Micellization

We now look at the results of the model when including the micellization reaction (eq 16) with an equilibrium constant of log K_mic = – 22.0. The resulting surfactant concentration profiles are shown in Figure 7a for different values of log K_>SurfCa. Here, [Surfactant] represents the total surfactant concentration: monomers, SurfCa⁺ complexes, and monomers in micelles. The delays are about 0.1 PV for log K_>SurfCa = – 4.0 and 0.22 PV for log K_>SurfCa = – 5.0. The surfactant concentration decreases sharply at 1 PV after the brine is injected but tails off at a much lower rate. This is in good agreement with the core flooding experiment shown in Figure 1. Figure 7b shows that the surfactant concentration remains constant at a low value because the monomers adsorbed on the surface are in equilibrium with the monomers in solution.

Figure 7.

Surfactant adsorption and desorption considering micellization. (a) Surfactant concentration using different equilibrium constants. (b) Closer view of the surfactant concentration for log K = −4.0.

Since the surfactant monomers are bridged to the surface by Ca²⁺, the Ca²⁺ concentration in solution also affects the desorption rate. Modeling the injection of deionized water into the calcite core shows that part of the surfactant indeed desorbs when DI water is injected (see the Supporting Information). Essentially, the low concentration of Ca²⁺ in solution causes the surfactant to desorb and dissociate following the reaction path >SurfCa → SurfCa⁺ → Surf^– + Ca²⁺. The adsorbed surfactant now functions as a source of Ca²⁺ ions, which desorbs to compensate for the low concentration of Ca²⁺ in solution. A new equilibrium is established where the remaining surfactant monomers desorb at a low but nonzero rate if more pore volumes of DI water are injected.

Conclusions

A mechanistic model was put forward that captures the asymmetry between surfactant adsorption and desorption rates. Due to the lower affinity of surfactant monomers to form micelles than to adsorb to the surface, no micelles form when a surfactant-free solution flows past a surface on which a layer of surfactants is adsorbed. Desorption is therefore limited by the CMC value of the surfactant and the desorption rate will be particularly low in systems with high surface-to-volume ratios. The model was applied to the specific case of Ca²⁺-bridging of anionic surfactants to calcite surfaces, but the conclusions apply to other systems and modes of adsorption.

Methods

The 30 cm long, 2″ diameter Estaillades limestone core consisted of 99.2% calcite and 2% quartz. The porosity and brine permeability were determined to be 30.4% and 76.9 mD, respectively. The pore volume was approximately 180 mL, and the dry weight was 1135.12 g. Core flooding experiments were performed at a temperature of 46 °C and with a flow rate of 3 ft/day (0.375 mL/min). The pH of brine was adjusted to 9.0 by adding NaOH or HCl prior to injection. 5 mL of effluent samples was collected in single-use vials. The surfactant concentration was measured using a surfactant-sensitive electrode (SSE) with a graphite rod and an ion carrier.

The MoReS–PHREEQC simulator package was used to model reactive transport. MoReS handled fluid transport, while the core of the aqueous chemical solver was performed by PHREEQC. Transport was modeled through a 1D system with 100 grid blocks. Injection occurred in grid block 1, and the fluid was produced from grid block 100. Fluids with different compositions were injected sequentially, and the produced surfactant concentration was recorded. Parameters such as porosity, permeability, and liquid injection rate were based on experimental values.

Rock–brine interaction was taken into account by considering (i) the dissolution and precipitation of calcite and (ii) ion exchange in the electric double layer. The negative surface charge density (expressed in mmol/100 g rock) was estimated based on surface area measurements. The composition of the injected fluids was defined in PHREEQC, and the pH was adjusted to 9.0 by varying the Na⁺ concentration. Reactions were included that describe (i) divalent ion complexation, (ii) micelle formation, and (iii) the adsorption of SurfCa⁺ complexes to negatively charged surface sites. The PHREEQC database contained equilibrium constants for the solution equilibrium and ion exchange reactions.

Glossary

ABBREVIATIONS

BET

Brunauer−Emmett−Teller

CMC

critical micelle concentration

DI

deionized

EOR

enhanced oil recovery

IPA

isopropyl alcohol

PV

pore volume

QCM-D

quartz crystal microbalance with dissipation monitoring

SSE

surfactant-sensitive electrode

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.0c05532.

• The Supporting Information contains results from modeling two example cases with different Ca²⁺ concentrations and from modeling the injection of DI water. The reaction for the formation of neutral Surf₂Ca complexes is also included (PDF).

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

The authors declare no competing financial interest.