# «I. INTRODUCTION Gemini surfactants are composed of two monomeric surfactant molecules linked by a spacer chain. They constitute a new class of ...»

Models of Gemini Surfactants∗

Haim Diamant1 and David Andelman2

1

School of Chemistry and 2 School of Physics & Astronomy,

Beverly and Raymond Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel

arXiv:cond-mat/0302184v1 [cond-mat.soft] 10 Feb 2003

## I. INTRODUCTION

Gemini surfactants are composed of two monomeric surfactant molecules linked by a spacer chain. They constitute a new class of amphiphilic molecules having its own distinct behavior. Since their ﬁrst systematic studies over a decade ago, gemini surfactants have been the subject of intensive research (see Ref. [1] and references therein). Research has been motivated by the advantages of gemini surfactants over regular ones with respect to various applications, e.g., their increased surface activity, lower critical micelle concentration (cmc), and useful viscoelastic properties such as eﬀective thickening.

Besides their importance for applications, the behavior of gemini surfactants is qualitatively diﬀerent in several respects from that of regular surfactants, posing challenges to current theories of surfactant self-assembly. The main puzzles raised by gemini surfactants can be summarized as follows [1].

• Surface behavior. The area per molecule in a saturated monolayer at the water–air interface, made of gemini surfactants with polymethylene spacers (m-s-m surfactants, where s is the spacer length and m the tail length in hydrocarbon groups), has a non-monotonous dependence on s [2, 3]. For example, for tail length of m = 12 the molecular area at the water–air interface is found to increase with s for short spacers, reach a maximum

(2003).

2 In view of the amount of experimental work and its unusual ﬁndings, the number of theoretical studies devoted to gemini surfactants has been surprisingly small. In this chapter we have, therefore, two aims. The ﬁrst is to review the current state of theoretical models of gemini surfactants. The second, perhaps more important aim, is to indicate the considerable gaps in our knowledge and the key open questions awaiting theoretical work. In Sec. II we set the stage by reviewing several theoretical models of surfactant self-assembly. This will facilitate the discussion in Sec. III of the gemini surfactant models, which can be viewed as extensions to previous models of regular surfactants. Finally, in Sec. IV we conclude and summarize the open questions.

** II. MODELS OF SURFACTANT SELF-ASSEMBLY**

In this section we review several theoretical models pertaining to the self-assembly of soluble surfactants. This is not meant to be an exhaustive review of self-assembly theory but merely to provide the appropriate background for the models of gemini surfactants discussed in Sec. III.

** A. Surface behavior**

Let us start by considering an aqueous surfactant solution below the cmc. The soluble surfactant molecules selfassemble into a condensed layer at the water–air interface, referred to as a Gibbs monolayer (to be distinguished from Langmuir monolayers that form when insoluble surfactants are spread on the water–air interface) [22]. Since the surfactant is water-soluble, this layer exchanges molecules with the bulk solution and a nonuniform concentration proﬁle forms. Typical surfactants have strong surface activity, i.e., the energy gained by a molecule when it migrates to the surface is much larger than the thermal energy kB T. As a result, the concentration proﬁle drops sharply to its bulk value within a molecular distance from the surface (hence the term monolayer).

The number of molecules participating in a Gibbs monolayer per unit area, the surface excess Γ, is obtained from integrating the excess concentration (with respect to the bulk one) over the entire solution. Such a monolayer can be regarded as a separate sub-system at thermodynamic equilibrium and in contact with a large reservoir of molecules having temperature T and chemical potential µ. From the excess free energy per unit area of this system, γ(T, µ),

**which is by deﬁnition the surface tension of the solution, we get the number of molecules per unit area:**

∂γ Γ=−. (1) ∂µ T This is referred to as the Gibbs equation [22]. For dilute solutions µ ∝ kB T ln C, where C is the bulk surfactant concentration. (The constant of proportionality is one for nonionic surfactants and ionic ones at high salt concentration;

it has a higher value for salt-free ionic surfactant solutions, where strong correlations between the diﬀerent ions lead to non-ideal activity coeﬃcients [22].) Hence,

Because of the high surface activity of surfactant molecules, leading to a sharp concentration proﬁle at the water–air interface, Γ−1 is commonly interpreted as the average surface area per molecule, a. The second consequence of the high surface activity is that, already for C much smaller than the cmc, the monolayer becomes saturated, i.e., Γ stops increasing with C. Experimentally, the curve describing the change in γ as a function of ln C becomes a straight line with a negative slope proportional to −Γ.

We now wish to ﬁnd a simple estimate for the energy of lateral interaction between molecules in such a saturated monolayer (repeating a well-known result of Ref. [11]). Saturation implies that the molecules are packed in an energetically optimal density, such that there is no gain in adding or removing molecules. This optimum arises from a competition between attractive and repulsive interactions. The attractive interaction tries to decrease the area per molecule, and we can phenomenologically write its energy per molecule as proportional to a, γ1 a, where the proportionality constant γ1 has units of energy per unit area. Since the attraction comes mainly from the desire of the hydrocarbon tails to minimize their contact with water, γ1 should be roughly equal to the hydrocarbon–water interfacial tension (γ1 ≃ 50 mN/m). The repulsive interaction, on the other hand, tries to increase a and, at the same phenomenological level, we can write its energy per molecule as inversely proportional to a, α/a, where α is a positive constant. Minimizing the sum of these two contributions we get for the interaction energy per molecule 3

As the solution concentration is increased beyond the cmc the surfactant molecules start to form aggregates. Unlike simple solute molecules (e.g., alkanes), which undergo macroscopic phase separation upon increasing concentration or changing temperature, surfactants form micelles at the mesoscopic scale. The challenges posed to theories of surfactant self-assembly are to predict the micellization point as a function of concentration (i.e., the cmc, hereafter referred to by the corresponding volume fraction ϕcmc ) and temperature (Tm ), as well as the micelle shape and size.

The main complications come from the fact that micellization is not a macroscopic phase transition — the aggregate sizes are ﬁnite and polydisperse — and thus the well-developed theoretical framework of phase transitions does not strictly apply.

From a thermodynamic point of view, the diﬀerence between surfactant micellization and phase separation lies in the following observation [12]. For alkanes solubilized in water, for example, the (Gibbs) free energy per molecule in an aggregate of size N, gN, is a monotonously decreasing function of N — for N → ∞ gN tends to the free energy per molecule in the bulk alkane phase, g∞, while for smaller N gN g∞ due to unfavorable surface terms of the ﬁnite cluster. As a result, there is a critical concentration (or critical temperature) at which the favorable size changes discontinuously from monomers solubilized in water (N = 1) to a macroscopic phase of bulk alkane (N → ∞). The ﬁrst-order phase transition point is reached when the chemical potential of monomers exceeds min{gN } = g∞. In a dilute solution this implies that ϕc = eg∞ /(kB T ), kB Tc = g∞ / ln ϕ. (We have set the free energy of the N = 1 state as the reference, g1 = 0.) In the case of surfactants, by contrast, gN has a minimum at a ﬁnite aggregate size N ∗. As a result, when the chemical potential exceeds gN ∗, a large population of aggregates appears, whose sizes are distributed around N ∗. Hence, the micellization point can be estimated as

FIG. 1: Packing constraints on a surfactant in an aggregate. Each head group occupies an optimal area a0 on the aggregate surface; the tail chain occupies a volume v and cannot stretch beyond length l. These constraints deﬁne the packing parameter, P = v/(a0 l), which suggests the possible aggregate shape.

Once the shape is determined we can ﬁnd the maximum allowed aggregation number. For example, for spherical micelles N 4πl2 /a0 = (4π/3)l3 /v, and we get

We see how competing interactions between the molecules (giving rise to a0 ) together with the incompressibility of the micellar core lead to ﬁnite micelles. Since the tail chains usually should not stretch to their full extent, the actual aggregation number will be smaller than this upper bound.

Yet, these geometrical arguments cannot provide us with theoretical predictions as to the optimal molecular area (S) a0 itself or the aggregation free energy gN ∗, as well as their dependence on parameters such as temperature or salt concentration. In order to get such information and subsequently predict the micellization point, micelle shape and size, one needs a more detailed theory.

where a is the area per molecule on the aggregate surface and R the aggregate size (radius or width). Note that R is not an independent variable but is related to N and a via the aggregate geometry S, e.g., for spherical micelles, N a = 4πR2.

(i) The driving force for aggregation is the hydrophobic eﬀect, i.e., the free energy per surfactant molecule ghc (S) gained by shielding the hydrocarbon groups from water [24]. This contribution to gN is negative and, to a good approximation, independent of N and the aggregate geometry S. Namely, its contribution to the entire aggregate free energy is linear in N and tends to increase the aggregate size. The hydrophobic term ghc depends linearly on the number of hydrocarbon groups in the surfactant, with a reduction of roughly kB T per hydrocarbon group [12].

That is why, for regular surfactants, the cmc decreases exponentially with the number of hydrocarbon groups in the molecule and is reduced by a factor of roughly 2–3 per each additional hydrocarbon group.

(ii) The hydrophobic gain is corrected by an interfacial contribution gint due to the unfavorable contact between the hydrocarbon core and water,

** gint (a) = γ1 (a − amin ), (9)**

where γ1 is the interfacial tension of the core–water interface (roughly equal to the hydrocarbon–water interfacial tension), and amin is the minimum area per molecule, i.e., the interfacial area occupied by a head group. This contribution evidently acts to reduce the area per molecule.

(iii) If the surfactant head groups are charged, there is electrostatic repulsion between them, acting to increase a.

**Within the Poisson–Boltzmann theory this electrostatic contribution is given by [25]:**

where β = 4πlB λD /a is a dimensionless charging parameter depending on two other lengths, the Debye screening length λD and the Bjerrum length lB. The Debye screening length in the solution is λD = (8πlB csalt )−1/2, where csalt is the added salt concentration, taken here to be monovalent, and lB = e2 /(εkB T ) is about 7 ˚ for aqueous solution A with dielectric constant ε = 80 at room temperature. (For simplicity, a monovalent head group has been assumed.) Finally, c is the mean curvature of the aggregate (e.g., 1/R for spherical micelles).

(iv) There is also steric repulsion between head groups. From the (non-ideal) entropy of mixing per molecule we get for this contribution

Following Widom’s statistical-mechanical model of microemulsions [26], a host of lattice models were presented for treating surfactant self-assembly (see, e.g., Refs. [27, 28, 29, 30, 31]). These molecular “toy models” represent the water molecules and various groups in the surfactant as Ising spins on a discrete lattice. The various interactions between the groups are represented by ferromagnetic or antiferromagnetic couplings between nearest-neighbor spins (see Fig. 2). Evidently, this is a very crude description of surfactant solutions and is not expected to yield quantitative predictions. Another diﬃculty is attaining thermodynamic equilibrium in simulations of these self-assembling systems, which contain slowly relaxing, large aggregates. Such models, however, have been shown to correctly reproduce various qualitative features of amphiphilic systems, e.g., aggregate formation, aggregate shape, and the overall structure of phase diagrams. The main advantage of this statistical-mechanical approach is that, by tuning a small number of parameters, one can get from the MC simulations insight into molecular mechanisms that determine the overall system behavior. Here we brieﬂy present a model similar to that of Ref. [31]. It will serve us when we discuss gemini surfactants in Sec. III B.