pubs.acs.org/Langmuir © 2009 American Chemical Society
Nonequilibrium Phase Transformations at the Air-Liquid Interface Christoffer A˚berg,*,† Emma Sparr,† Karen J. Edler,‡ and Ha˚kan Wennerstr€om† †
Physical Chemistry, Lund University, SE-221 00 Lund, Sweden, and ‡Chemistry Department, University of Bath, Bath BA2 7AY, U.K. Received March 11, 2009. Revised Manuscript Received July 12, 2009
A theoretical model is presented for the formation of an ordered phase close to the air-liquid interface of an open binary aqueous solution. The chemical potential of water in the liquid phase is, in general, not equal to the chemical potential of water in the ambient atmosphere. There are therefore nonequilibrium conditions close to the air-liquid interface. There is also a gradient in the chemical potential of water, which could lead to the formation of a new interfacial phase. The formation of an interfacial phase is analyzed in terms of the equilibrium phase behavior corresponding to the local water chemical potential. The possibility of forming an interfacial phase is strongly dependent on the ambient conditions, bulk composition, and diffusive transport properties of the phases in question. Explicit calculations are presented for the formation of a lamellar liquid-crystalline phase close to the air-liquid interface of an isotropic surfactant solution with parameters chosen from the sodium bis(2-ethylhexyl)sulfosuccinate (AOT)/water system. We consider the relevance of the model to neutron reflectivity studies of the interface between air and surfactant/ water systems, as well as to surfactant/polymer/water systems.
1. Introduction When a liquid sample is exposed to ambient air, there are typically nonequilibrium conditions at the interface. Volatile compounds in the liquid tend to evaporate into the air, while nitrogen, oxygen, and carbon dioxide in the air can dissolve in the liquid phase. The driving force for such events is a gradient in the chemical potential of the different components. Such transport processes can give rise to local thermodynamic properties in the interfacial part of the liquid that differ from bulk values. One example of this is the slight dip in temperature that occurs at the interface between (pure) water and air with a relative humidity (RH) below 100%.1 For aqueous solutions and water/solvent mixtures, the selective evaporation of water in general gives rise to local changes in concentration, which can have important practical consequences.2 For the case in which the bulk system is close to a phase separation, the interfacial gradients in concentration and thus in chemical potential can result in the local formation of new phases. In this work, we explore this possibility for binary systems of water and an amphiphile. Spontaneous film formation at the air-liquid interface of mixtures of a cationic surfactant with polyethylenimine (PEI) has previously been reported by one of us.3-6 The films can be made highly ordered and macroscopically (micrometer) thick. A similar film can also be seen in solutions with a mixture of cationic and *Corresponding author. Tel: þ353 1 716 2447. Fax: þ353 1 716 2415. E-mail:
[email protected]. (1) Cammenga, H. K.; Schreiber, D.; Rudolph, B. E. J. Colloid Interface Sci. 1983, 92, 181–188. (2) Kabalnov, A.; Wennerstr€om, H. Manuscript in preparation. (3) O’Driscoll, B. M. D.; Fernandez-Martin, C.; Wilson, R. D.; Knott, J.; Roser, S. J.; Edler, K. J. Langmuir 2007, 23, 4589–4598. (4) Comas-Rojas, H.; Aluicio-Sarduy, E.; Rodrı´ guez-Calvo, S.; Perez-Gramatges, A.; Roser, S. J.; Edler, K. J. Soft Matter 2007, 3, 747–753. (5) O’Driscoll, B. M. D.; Milsom, E.; Fernandez-Martin, C.; White, L.; Roser, S. J.; Edler, K. J. Macromolecules 2005, 38, 8785–8794. (6) Edler, K. J.; Goldar, A.; Brennan, T.; Roser, S. J. Chem. Commun. 2003, 2003, 1724–1725. (7) O’Driscoll, B. M. D.; Nickels, E. A.; Edler, K. J. Chem. Commun. 2007, 2007, 1068–1070. (8) Edler, K. J.; Wasbrough, M. J.; Holdaway, J. A.; O’Driscoll, B. M. D. Langmuir 2009, 25, 4047–4055.
Langmuir 2009, 25(20), 12177–12184
anionic surfactants and a water-soluble polymer.7,8 In this case, the resulting film is sufficiently robust to be removed from the airliquid interface, even without the help of a cross-linking agent. From these studies, we emphasize in particular the observation that film formation is prevented by increasing the relative humidity in the container. This supports the view, central to the current work, that the formation of the film is due to a difference in water chemical potential between the ambient atmosphere and the bulk solution. In other words, the system is not at equilibrium, and the relevant situation is instead the steady state that forms after an initial induction period. However, we assume that locally at each position there is equilibrium, so that the part of the solution closest to the air-liquid interface will be in local equilibrium with the atmosphere. Dry ambient conditions therefore correspond to a considerably lower chemical potential of the water close to the interface than in the bulk solution. From this point of view, it is not surprising that the more ordered phases, which form at low water chemical potentials in surfactant systems, can be found close to the air-liquid interface. In essence, a thin layer of the condensed phase can potentially span the full equilibrium phase diagram, from the bulk solution composition to the composition that represents equilibrium with the ambient air. We note that an analogous situation is found in the so-called penetration scan method of monitoring phase equilibria in binary water/amphiphile systems,9 as well as studies of surfactant dissolution in water.10-12 A related problem is the dynamic behavior of an oil phase in contact with an aqueous surfactant solution. Miller and co-workers have conducted extensive studies of this problem and have noted the common appearance of an intermediate phase close to the original oil-water interface.13 The paper is organized as follows. In the next section, we outline a general theoretical model for the description of the (9) Rendall, K.; Tiddy, G. J. T.; Trevethan, M. A. J. Chem. Soc., Faraday Trans. 1 1983, 79, 637–649. (10) Chen, B.-H.; Miller, C. A.; Walsh, J. M.; Warren, P. B.; Ruddock, J. N.; Garrett, P. R.; Argoul, F.; Leger, C. Langmuir 2000, 16, 5276–5283. (11) Chen, B.-H.; Miller, C. A.; Garrett, P. R. Colloids Surf., A 2001, 183, 191– 202. (12) Arunagirinathan, M. A.; Roy, M.; Dua, A. K.; Manohar, C.; Bellare, J. R. Langmuir 2004, 20, 4816–4822. (13) Lim, J. C.; Miller, C. A. Langmuir 1991, 7, 2021–2027.
Published on Web 09/15/2009
DOI: 10.1021/la900867k
12177
A˚berg et al.
Article
structure close to the air-liquid interface. Central to this model are an assumption of a local equilibrium and a description of the diffusion through the phases formed close to the interface. The essential idea is demonstrated by making some simplifying assumptions. We then continue with a more detailed model for the formation of a lamellar liquid-crystalline phase close to the interface. This example is particularly simple to analyze, and we show some numerical results with parameters relevant for the sodium bis(2-ethylhexyl)sulfosuccinate (AOT)/water system, which serves as a basis for making explicit quantitative predictions. Finally, other systems in which this mechanism potentially plays a role are presented, and applications are discussed.
2. General Theoretical Model 2.1. Diffusion with Solvent Evaporation. Consider a twocomponent solution with an interface exposed to the ambient atmosphere. For the sake of simplicity, we will refer to the solvent as water, though the formalism applies to any volatile solvent. We assume that the vapor pressure of the solute is negligibly small, so that the total amount of solute in the condensed phase is constant. In contrast, the water can be transported between the liquid and the gas phases driven by a difference in the chemical potential. In the typical case, there is an evaporation of water into the gas phase. We are focusing on the nonequilibrium effects found in the steady state that is established during evaporation. A more detailed account of both transient and steady-state concentration profiles is found in ref 2. The solvent evaporation causes a net material transport in the solution. This has two important consequences. First, it results in an increased concentration of the solute close to the interface. Second, the location of the air-liquid interface changes with time. The increased solute concentration clearly leads to a local increase in the solute chemical potential. We are in particular interested in the case in which such a change in chemical potential can trigger the local formation of another phase. To find a quantitative description of this phenomenon, we need a description of the combined diffusive and bulk liquid flow. A model for the bulk thermodynamic behavior describing potential phase changes is also required. A transport equation is most simply based on an equation of continuity for the mass of the two components. For the sake of simplicity, we assume equal densities of the components and use mass fractions, Xi, as variables DXi D þ ðvi Xi Þ ¼ 0 Dt Dz
ð1Þ
where vi is the local velocity of component i. viXi is the total mass fraction flux of i, which is composed of a bulk flow moving at the center of mass velocity, v, and a diffusive flux, Ji14 vi Xi ¼ vXi þJi
ð2Þ
where w and s denote water and solute, respectively. By virtue of the condition Xw þ Xs = 1, it follows from eq 1 that v is independent of z. We furthermore note that the definition of the center of mass velocity, eq 2, ensures that the diffusive fluxes are equal in magnitude and opposite in direction (Jw = -Js). (14) de Groot, S. R.; Mazur, P. Non-Equilibrium Thermodynamics; Dover Publications, Inc.: New York, 1984.
12178 DOI: 10.1021/la900867k
There are two basically equivalent ways to arrange the experimental setup. In one version, the system under study is coupled to a reservoir that keeps the hydrostatic pressure constant. Then the material lost by evaporation is supplied from the reservoir leading to a net flow of liquid through the system under study. The second alternative, illustrated in Figure 1a, is to have a closed bulk liquid system that loses material through evaporation. In this case, the location of the air-liquid interface will drift slowly downward and there is no bulk liquid flow (v = 0), while at each z coordinate below the interface there is a diffusive flux. In the following, we will only discuss the latter scenario. A complication is that in a coordinate system fixed on the container a steady state cannot be established. It is more convenient to use a coordinate system such that the moving interface is stationary z0 ¼ z - sðtÞ where s(t) denotes the position of the air-liquid interface (see Figure 1a). Under steady-state conditions, the mass fractions (Xi) are, in terms of the new coordinate z0 , independent of time. Furthermore, the evaporation rate is constant, and thus, the interface _ Using the condition that v = 0, the moves at a constant velocity s. continuity equation, eq 1, transforms into DXi : DXi DJi -s 0 þ 0 ¼ 0 Dt Dz Dz so that in the steady state D : ðJi - sXi Þ ¼ 0 Dz0
The center of mass velocity is given by v ¼ vw Xw þvs Xs
Figure 1. (a) Schematic illustration of an open aqueous solution with evaporation of the solvent. Because of the evaporation, the position of the air-liquid interface, s(t), changes with time. (b) Formation of an interfacial phase, β, close to the air-liquid interface. For the two-phase coexistence, an R phase of composition Xtr,R is in equilibrium with a β phase of composition Xtr,β s s . These are therefore the local compositions at the point z0 = ztr0 , which marks the R-β interface. L measures the size of the unstirred layer, so that at z0 = L the bulk concentration is reached.
ð3Þ
For the sake of brevity, we will in the following dispense of the primes on the new coordinate and write z0 simply as z. At the interface, the solute molecules are forced to move with the same velocity as the interface, while solvent can escape through evaporation. This observation provides a boundary condition, vs(z = 0) = s_ at all times. Integration of eq 3 then yields : Js ¼ sXs
ð4Þ
To arrive at a quantitative description, it is necessary to model the diffusive transport explicitly. When diffusion in Langmuir 2009, 25(20), 12177–12184
A˚berg et al.
Article
inhomogeneous systems is described, it is usually advantageous to use a generalized form of Fick’s law15 D dμ Xs s ð5Þ Js ¼ RT dz This form of Fick’s law emphasizes the fact that it is the gradient in chemical potential that is the driving force for diffusive transport. For example, at a phase boundary, there is an abrupt change in concentration, but a diffusive flow occurs only for a nonequilibrium situation with a gradient in chemical potential. To illustrate the basic consequences of solvent evaporation on the concentration profiles, we will assume that the condensed phase is a solution with ideal mixing and that the diffusion coefficient, D, is constant. These assumptions are in general not true. However, the simplifying assumptions enable us to express the general idea more clearly without complications of a technical nature. The explicit model presented in the following section is more detailed, and a discussion of how to deal with a composition-dependent diffusion coefficient is also given in that context. For now, we accept the assumptions of ideal mixing and a constant diffusion coefficient, under which eq 5 reduces to the original form of Fick’s law Js ¼ -D
dXs dz
ð6Þ
Inserted into eq 4, this gives the simple differential equation : s dXs ¼ - Xs D dz
ð7Þ
For a constant D, eq 7 has a simple solution in terms of an exponentially decaying mass fraction profile with characteristic length D/_s : Xs ðzÞ ¼ C exp½ -ðs=DÞz
ð8Þ
The integration constant, C, in eq 8 is to be determined by a boundary condition. The evaporation is a transport-limited nonequilibrium phenomenon, and to arrive at a quantitative prediction of its consequences, it is necessary to make a choice regarding the boundary condition. For a finite container with evaporation at the surface, there are two reasonable alternatives. The most obvious is to consider a diffusional flow throughout the liquid phase. In such a system, a true steady state will not appear since the concentration at the bottom will change slightly with time. Another possibility is to recognize the experimental complication that it is extremely difficult to prevent convective flow in a bulk liquid in a container of reasonable size. As a consequence, we introduce the concept of an unstirred layer of thickness L close to the surface. In the bulk, . This provides the convection occurs, so that for z > L, Xi = Xbulk i missing boundary condition that determines C in eq 8, and : : Xs ðzÞ ¼ Xsbulk exp½ -ðs=DÞðz -LÞ ¼ Xs ð0Þ exp½ -ðs=DÞz ð9Þ When the length D/_s is much shorter than L, the solute concentration decreases exponentially over a length scale D/_s close to the interface. In the other limit when D/_s is large relative to L, i.e., in the limit of slow evaporation compared to diffusion, the concentration profile is linear : Xs ðzÞ ¼ Xs ð0Þ½1 -ðs=DÞz ¼ Xs ð0Þ þ ½Xsbulk -Xs ð0Þðz=LÞ ð10Þ
In eqs 9 and 10, the evaporation rate, described through the velocity s_ of the interface, is still undetermined. To find this quantity, it is necessary to specify the composition at the airliquid interface. We assume that equilibration is fast in the air through convection, so that the chemical potential of water is uniform in the gas phase. Under the assumption of local equilibrium, it then follows that the water content in the liquid at the z = 0 boundary is determined by matching the chemical potential of water in the air and locally at the interface. Assuming ideal behavior in both the gas phase and the condensed phase16 pw 1 -Xs ð0Þ ð11Þ ¼ RH ¼ xw ð0Þ ¼ θ 1 - ð1 -Mw =Ms ÞXs ð0Þ pw where pw and pθw are the partial pressure of water in the gas phase and that in a gas phase in equilibrium with pure water, respectively. The first equality defines the relative humidity; the second equality relates the relative humidity to the mole fraction of water (Raoult’s law) locally at the interface, xw(0), and the third converts the mole fraction to mass fraction. Mi is the molar mass of component i. Equation 9 shows that the evaporation rate is given by : D ð12Þ s ¼ ln½Xs ð0Þ=Xsbulk L with Xs(0) from eq 11. To summarize, a difference in the chemical potential of water between the liquid and gas phases drives evaporation of water in an open system. Under steady-state conditions, the evaporation rate is constant so that the air-liquid interface moves with a constant velocity, and a nonuniform concentration profile develops close to the moving interface. Under the simplifying assumptions of ideal mixing and a constant diffusion coefficient, a quantitative description is given by eq 12 for the evaporation rate and by eqs 8-10 for the concentration profile. 2.2. Formation of a New Phase Close to the Air-Liquid Interface. The concentration profile close to the interface, exemplified by eqs 8-10, also implies that there is a profile in the chemical potential of the solute and the solvent. Thus, in addition to the effect of an enhanced solute concentration close to the surface, there is also the possibility of a local change in phase behavior. This would most readily occur when the bulk conditions are such that one is close to two-phase conditions. To evaluate this possibility, we studied the case in which the ambient conditions are such that the chemical potential of water in the bulk solution and the gas phase are on opposite sides of a phase boundary. In a bulk description (the validity of which is discussed below), there is under these circumstances the formation of an interfacial phase, β, close to the air-liquid interface. At z = ztr, the two-phase coexistence β-R, (where R is the bulk phase) is reached. Figure 1b illustrates the situation. The assumption of a local equilibrium implies that the compositions of both phases at z = ztr are known from the equilibrium phase diagram and they are and Xtr,β denoted Xtr,R s s for the R and β phases, respectively. For a single phase with ideal mixing and a constant diffusion coefficient, eq 12 holds. Under the same simplifying assumptions, the analogous equations for a two-phase system read : sztr ¼ Dβ ln½Xs ð0Þ=Xstr;β
: sðL - ztr Þ ¼ DR lnðXstr;R =Xsbulk Þ
ð13Þ
as for diffusion across a planar membrane. (15) Evans, D. F.; Wennerstr€om, H. In The Colloidal Domain: Where Physics, Chemistry, Biology and Technology Meet; Wiley-VCH: New York, 1999; Chapter 1.
Langmuir 2009, 25(20), 12177–12184
(16) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. In Transport Phenomena; John Wiley & Sons: New York, 2002; Chapter 18.
DOI: 10.1021/la900867k
12179
A˚berg et al.
Article
where Dφ is the diffusion coefficient in phase φ. Here all parameters on the right-hand side are assumed to be known: and Xtr,β from the equilibrium phase behavior, Xs(0) Xtr,R s s from the from eq 11 and the ambient conditions, and Xbulk s preparation of the sample. Equation 13 is an equation system _ with the solution in the two unknowns ztr and s, ztr Dβ ln½Xs ð0Þ=Xstr;β ¼ tr;R L DR lnðXs =Xsbulk Þ þ Dβ ln½Xs ð0Þ=Xstr;β DR lnðXstr;R =Xsbulk Þ þ Dβ ln½Xs ð0Þ=Xstr;β : s ¼ L
ð14Þ
Equation 14 gives the thickness, ztr, of the interfacial β phase. We based this simple estimate of the occurrence of the interfacial phase and its thickness on a bulk thermodynamic description of the phases, and the classical form of Fick’ law (eq 6), with a constant diffusion coefficient. It neglects several complications: For a bulk description of the β phase to be approximately valid, the estimated ztr should be significantly larger than a microscopic structural distance in the phase. If this is not the case, the β phase does not exist. Additionally, the usage of the classical form of Fick’s law implies, as discussed in connection with eq 6, an ideal mixing description of the two phases, R and β. In the following section, we will use a more realistic thermodynamic model for the specific case of an AOT/water system. There we also discuss how to deal with a composition-dependent diffusion coefficient. Ideal behavior also implied the usage of Raoult’s law for relating the chemical potential of water in the gas phase and locally at the interface. A more realistic assumption would demand a more complicated description, though we expect the same qualitative behavior as long as the general trend is that the water activity is lower for a lower ambient relative humidity. Finally, by introducing a new phase, one also introduces a second interface. There is an additional free energy contribution from the surface free energy. The interfacial effects result in a free energy contribution Gs ¼ areaðγβ=air þ γβ=R -γR=air Þ which should be balanced by a bulk contribution17 Gb ¼ area ztr GβR The contribution from the interfacial free energy, Gs, can inhibit the appearance of a new phase. However, if a new phase is formed, the Gs contribution is constant and does not affect the thickness of the layer. The consequence is that if the surface energy gives a positive free energy contribution, then the new phase appears in a steplike manner. In other words, when the new phase appears, it has a finite thickness at steady state. , is close to a bulk coexistence If the bulk mass fraction, Xbulk s value, Xtr,R s , the propensity of forming an interfacial phase is greater. This is also seen in eq 14. Indeed, as the bulk concentration approaches the phase line, the thickness of the β phase approaches L. A more sophisticated model of the transport processes in the bulk phase is required to arrive at a quantitatively accurate prediction in this limit. Assuming the interfacial phase β exists, then eq 14 shows that the faster transport is through the β phase the thicker it is at steady (17) Evans, D. F.; Wennerstr€om, H. In The Colloidal Domain: Where Physics, Chemistry, Biology and Technology Meet; Wiley-VCH: New York, 1999; Chapter 5.
12180 DOI: 10.1021/la900867k
state, all other parameters being equal. This is the expected result, since the larger the permeability of a phase, the thicker it has to be to sustain a certain gradient. There is then the theoretical possibility that more than one phase forms successively from bulk solution to the air-liquid interface, if the transport properties of the phases are similar. There is nothing that prevents the formalism described above to be extended by a generalization of eq 13, though we refrain from doing so here.
3. Formation of an Interfacial Lamellar LiquidCrystalline Phase Several reports of the formation of new phases at the air-liquid interface using neutron reflectivity exist.3-8,18-28 The appearance of a new phase is more easily observed for phases that have a distinctly different character than the bulk phase, as when a lyotropic liquid crystal separates from an isotropic solution. Above we have developed the formalism for a binary system. In practice, it is more common to study ternary or multicomponent systems. However, to avoid undue complications, we illustrate the general theoretical concept by making an explicit study for the binary water/AOT [sodium bis(2-ethylhexyl)sulfosuccinate] system. Thomas and co-workers have reported the formation of lamellar liquid-crystalline structures at the interface between air and solution for this system.19-21 Furthermore, we have previously presented a quantitative model of the bulk phase behavior29,30 based on a Poisson-Boltzmann description of the strong electrostatic interactions in the system.We also take advantage of previous studies in which we developed a model describing diffusion of water through a lamellar system with a gradient in the chemical potential of water.31 AOT is an ionic short double-chained surfactant. It readily forms a lamellar structure, and the two-phase coexistence iso1 = 1.3% (w/w, tropic solution-lamellar phase occurs at Xtr,L s 32 20 20 °C), not far from the CMC at ∼2.5 mM [0.11% (w/w)]. On the basis of these facts and the general considerations presented in the previous section, we anticipate the possibility of the formation of a lamellar liquid-crystalline phase at the air-liquid interface from a bulk isotropic surfactant solution above the CMC. AOT forms a monolayer at the air-liquid interface for all phases. If a lamellar phase forms at the interface, it will develop parallel to the interface following the preexisting monolayer. When discussing the transport of water through the interfacial region, one should (18) Lu, J. R.; Simister, E. A.; Thomas, R. K.; Penfold, J. J. Phys. Chem. 1993, 97, 13907–13913. (19) Li, Z. X.; Lu, J. R.; Thomas, R. K.; Penfold, J. Faraday Discuss. 1996, 104, 127–138. (20) Li, Z. X.; Weller, A.; Thomas, R. K.; Rennie, A. R.; Webster, J. R. P.; Penfold, J.; Heenan, R. K.; Cubitt, R. J. Phys. Chem. B 1999, 103, 10800–10806. (21) Li, Z. X.; Lu, J. R.; Thomas, R. K.; Weller, A.; Penfold, J.; Webster, J. R. P.; Sivia, D. S.; Rennie, A. R. Langmuir 2001, 17, 5858–5864. (22) McGillivray, D. J.; Thomas, R. K.; Rennie, A. R.; Penfold, J.; Sivia, D. S. Langmuir 2003, 19, 7719–7726. (23) Penfold, J.; Sivia, D. S.; Staples, E.; Tucker, I.; Thomas, R. K. Langmuir 2004, 20, 2265–2269. (24) Penfold, J.; Tucker, I.; Thomas, R. K.; Zhang, J. Langmuir 2005, 21, 10061– 10073. (25) Penfold, J.; Tucker, I.; Thomas, R. K.; Taylor, D. J. F.; Zhang, J.; Bell, C. Langmuir 2006, 22, 8840–8849. (26) Penfold, J.; Thomas, R. K.; Dong, C. C.; Tucker, I.; Metcalfe, K.; Golding, S.; Grillo, I. Langmuir 2007, 23, 10140–10149. (27) Penfold, J.; Tucker, I.; Thomas, R. K.; Taylor, D. J. F.; Zhang, J.; Zhang, X. L. Langmuir 2007, 23, 3690–3698. (28) Penfold, J.; Thomas, R. K.; Zhang, X. L.; Taylor, D. J. F. Langmuir 2009, 25, 3972–3980. (29) J€onsson, B.; Wennerstr€om, H. J. Colloid Interface Sci. 1981, 80, 482–496. (30) Khan, A.; J€onsson, B.; Wennerstr€om, H. J. Phys. Chem. 1985, 89, 5180– 5184. (31) Sparr, E.; Wennerstr€om, H. Colloids Surf., B 2000, 19, 103–116. (32) Fontell, K. J. Colloid Interface Sci. 1973, 44, 318–329.
Langmuir 2009, 25(20), 12177–12184
A˚berg et al.
Article
dence of the diffusion coefficient is rather moderate so that the diffusion coefficient can be regarded as constant. Below we find that the interesting concentration interval is actually very small, and the assumption of a constant diffusion coefficient is therefore a good approximation. As in eq 9, we then arrive at Xs ðztr Þ ¼
Figure 2. (a) Schematic of an interfacial lamellar phase consisting of n bilayers formed close to the air-liquid interface of an isotropic surfactant solution. The thickness of aqueous layer number i is denoted by hi, and the thickness of the bilayers is denoted by l. L denotes the thickness of the unstirred layer. (b) System in the twophase coexistence region, where a liquid-crystalline lamellar phase LR has sunk to the bottom of the container because of a higher density of the surfactant than water. On top is an isotropic solution L1 of thickness H.
include the effect of the monolayer. Transport of water through a monolayer or bilayer is at least a factor of 1000 slower than through bulk water. The reason is not that the molecular motion is slower but that the local volume concentration is much lower, making the net transport less effective. 3.1. Theoretical Description. As in section 2.2, we are interested in the situation in which the chemical potential in the bulk solution and the gas phase are on opposite sides of a phase boundary. Specifically, in this context, the bulk solution is an isotropic solution, L1, whereas the chemical potential of water in the gas phase corresponds to a liquid-crystalline lamellar phase, LR. Within a bulk description, there is then the formation of the lamellar phase in the interfacial region. The situation is illustrated in Figure 2a where, in addition to the monolayer, there are n bilayers of thickness l on top of the isotropic solution. At a distance L from the interface, we assume that equilibrium conditions are met due to convection. The bulk description of the lamellar phase breaks down if it is not sufficiently thick, which for this system means that the lamellar phase has to consist of at least a couple of bilayers. If this is not the case, then the model predicts the absence of an interfacial phase. See the discussion following eq 14. To estimate the thickness of the lamellar phase, it is necessary to model separately the diffusion in the two regions: z < ztr and z > ztr. In the isotropic part of the system (z > ztr), the surfactant is dilute, and diffusive transport is dominated by surfactant selfdiffusion in an aqueous medium. We base our description on the original Fick’s law (eq 6), which, despite the fact that it is a simplification, has proved to be capable of describing diffusion in surfactant systems.10 On this basis, we rewrite eq 7 as Z
Xsbulk
DL1 : dXs ¼ -s X s Xs ðztr Þ
Z
L
: dz ¼ -sðL - ztr Þ
ð15Þ
ztr
We note that an analogous integral can be written also for a more general transport equation, such as eq 5, based on a gradient in the chemical potential. If the concentration dependence of the diffusion coefficient is known, one can perform the integration in the left-hand side of eq 15. However, the opposite is more common, and we therefore assume that the concentration depenLangmuir 2009, 25(20), 12177–12184
Xstr;L1
¼
Xsbulk
: s exp L ðL - ztr Þ D 1
ð16Þ
where DL1 is approximately equal to the self-diffusion coefficient of the surfactant. For diffusion across the lamellar region (z < ztr), we note that within a bilayer the water can be considered as a dilute solute. One can therefore use the original form of Fick’s law, as in eq 6, with a constant diffusion coefficient, provided one interprets Xw as the mass fraction in the bilayer. This concentration is related to the concentration in the adjacent aqueous phase by a bilayer-water partition coefficient for water. We formulate the partitioning in terms of mole fractions (cf. Raoult’s law) ¼ Kxaq Xwbilayer ≈ðMw =Ms Þxbilayer w w
ð17Þ
where the first equality is a good approximation due to fact that the mass fraction of water in the bilayer is very small and the second equality defines a partition coefficient, K. At the interface (z = ztr) between the isotropic phase and the lamellar phase, eq 17 shows that Xwbilayer ðztr Þ ¼ Kxaq w ðztr Þ ¼ K
1 - Xstr;L1 1 -ð1 - Mw =Ms ÞXstr;L1
ð18Þ
As in section 2.2, we assume that there is a local equilibrium at the air-liquid interface (z = 0), so that eq 17 here reads Xwbilayer ð0Þ ¼ K RH
ð19Þ
since we approximate the activity of water in the gas phase with the relative humidity. Because the partition coefficient, K, is small, the gradient in water chemical potential over the aqueous layers of the lamellar phase is negligible, and we can simplify the description by accounting for only the bilayers. Equation 9 then shows that 1 -Xwbilayer ð0Þ
¼
½1 -Xwbilayer ðztr Þ
" : # s 1 exp L n þ l D R 2
ð20Þ
where DLR is to a good approximation equal to the self-diffusion (0) and Xbilayer (ztr) are given by eqs coefficient of water and Xbilayer w w 19 and 18, respectively. Since the diffusion of water across the bilayers is slow, a non-negligible fraction of the transport could occur through defects in the lamellae.33,34 For the sake of simplicity, we neglect this complication. It is our experience from other systems that this effect starts to be a major complication in gel-like lamellae, where water diffusion is even more hindered than for liquid crystals. To obtain the thickness of the lamellar phase, the thickness of the aqueous layers separating the bilayers also remains to be considered. The swelling of the lamellar phase depends on the chemical potential of the water so that each layer, thickness hi, (33) Costa-Balogh, F. O.; A˚berg, C.; Sousa, J. J. S.; Sparr, E. Langmuir 2005, 21, 10307–10310. (34) A˚berg, C.; Pairin, C.; Costa-Balogh, F. O.; Sparr, E. Biochim. Biophys. Acta 2008, 1778, 549–558.
DOI: 10.1021/la900867k
12181
A˚berg et al.
Article
becomes successively thinner the closer one is to the surface. In general, a lamellar phase with n bilayers (and one monolayer) extends over a distance ztr ¼
X i
1 hi þ n þ l 2
ð21Þ
One can calculate the swelling of the lamellar phase for AOT on the basis of the counterion-only case and the nonlinear regime of the Poisson-Boltzmann equation so that17 h ¼
pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi kT 2π εr ε0 =ðZeÞ pffiffiffiffiffiffiffiffiffiffiffi Πosm
ð22Þ
where the osmotic pressure Πosm is simply related to the chemical potential of the water,15 and Z = 1 for monovalent counterions. To avoid numerical complications, we replace the sum over hi in eq 21 with an average so that X
hi ≈n=2ðhmax þ hmin Þ nhav
ð23Þ
i
where the thicknesses of the aqueous layers at the top and bottom are obtained from eq 22 since the chemical potentials are known boundary values. Combining eqs 16, 18-21, and 23, one can solve for the _ and n. The result is that the number of bilayers, unknowns ztr, s, n, is given by nþ
L þ 12 hav 1 ¼ 2 hav þ lð1þ rÞ
ð24Þ
lamellar phase, and the evaporation rate stays essentially constant. This effect is analogous to the one found for the transport of water across human skin35,36 and was also predicted from a model for diffusive transport in responding lipid membranes.37,38 Some experimental studies have been performed for a system in the two-phase area with a lamellar phase resting at the bottom of the container. The AOT surfactant has a density greater than that of water (1.13 103 kg/m3 at 30 °C39), and in a two-phase system, the lamellae should therefore rest at the bottom of the container, as depicted in Figure 2b. Because of the gravitational contribution, there is a weak dependence of the chemical potential on the vertical position. At the location of the unstirred layer, the value of the chemical potential depends on the thickness H of the isotropic phase (see Figure 2b). The contribution to the chemical potential of water from the gravitational field is ¼ -V w ðFs - Fw ÞgH Δμgrav w This is a small contribution but implies that the concentrationdependent part of the chemical potential of water at equilibrium is slightly above the chemical potential of water at the phase boundary. For AOT, a corresponding negative change is obtained from the Gibbs-Duhem relation. In effect, the gravitational contribution lowers the surfactant concentration at the position of the unstirred layer, Xs(L). Assuming that the osmotic pressure of the micellar solution can be approximated by the contributions from the monomers and corresponding counterions at the CMC and the micelles Πosm ¼ RTð2CMC þ cmic Þ the resulting change is given by
where r is the ratio r ¼
L1
D
lnðXstr;L1 =Xsbulk Þ h i
DLR ln
1 - KRH 1 - Kxw ðztr Þ
Xs ðLÞ ¼ Xstr;L1 -ðFs =Fw -1Þ ð25Þ
Above, we have neglected the gradient in water chemical potential over the aqueous layers of the lamellar phase, due to the low bilayer-water partition coefficient for water, K (defined in eq 17). It is then consistent to retain only the first-order term in K of eq 24 nþ
1 L þ hav =2 KDLR xw ðztr Þ -RH þ ≈ ¼ 2 l DL1 lnðXstr;L1 =Xsbulk Þ 3 3 3 ≈
L KDLR xw ðztr Þ -RH l DL1 lnðXstr;L1 =Xsbulk Þ
ð26Þ
where xw(ztr) is given by eq 18, and we have finally also neglected (half) the average aqueous layer hav in comparison with L. It follows from eqs 20, 24, and 25 that the evaporation rate in the same approximation is given by ! Xstr;L1 : DL1 ð27Þ ln s≈ L Xsbulk From eq 27, we see that, in the limit of a low partition coefficient K, the evaporation rate is insensitive to variations in the ambient relative humidity once the protecting lamellar phase has formed. The net result is as follows. For very humid ambient conditions, the evaporation rate at steady state varies with RH. However, at a certain RH, there is the abrupt formation of the interfacial 12182 DOI: 10.1021/la900867k
Ms gH nagg RT
ð28Þ
where nagg is the micelle aggregation number. For a system in the in eqs 26 and 27 should be two-phase coexistence region, Xbulk s replaced by Xs(L) from eq 28. A consequence of eqs 26 and 28 is that the thickness of the interfacial lamellar phase depends on the thickness of the isotropic phase, H. The latter is determined both by the size of the container and by the total surfactant concentration via the lever rule. It is clear that the thickness of the isotropic phase decreases with an increase in total surfactant concentration. Equations 26,28 therefore imply that the interfacial lamellar phase increases with total surfactant concentration within the two-phase coexistence region. A concomitant moderate decrease in the evaporation rate follows from a combination of eqs 27 and 28. 3.2. Results Exemplified by the AOT/Water System. To apply the results quantitatively to the AOT/water system, it is necessary to specify the parameters. The CMC occurs at around 2.5 mM,20 and the phase boundary between the isotropic solution and the two-phase coexistence isotropic micellar solution and 1 = 1.3% at 20 °C. The two-phase lamellar phase occurs at Xtr,L s region is rather broad; the lamellar phase appears around 10 wt % and is stable up to around 70 wt %.32 Unless very dry ambient air is considered, it is sufficient to consider only the isotropic and (35) Blank, I. H.; Moloney, J., III; Emslie, A. G.; Simon, I.; Apt, C. J. Invest. Dermatol. 1984, 82, 188–194. (36) McCallion, R.; Po, A. L. W. Int. J. Pharm. 1994, 105, 103–112. (37) Sparr, E.; Wennerstr€om, H. Biophys. J. 2001, 81, 1014–1028. (38) Sparr, E.; A˚berg, C.; Nilsson, P.; Wennerstr€om, H. Soft Matter 2009, 5, 3225-3233. (39) Ghosh, O.; Miller, C. A. J. Phys. Chem. 1987, 91, 4528–4535.
Langmuir 2009, 25(20), 12177–12184
A˚berg et al.
Figure 3. Thickness of the interfacial lamellar phase as a function
of the bulk surfactant mass fraction, Xbulk . As the phase boundary s 1 Xtr,L = 1.3% is approached, the thickness of the lamellar phase s increases. The effect is more pronounced for lower ambient RH. The ztr scale on the right is for RH = 60%, though in practice it is essentially correct also for RH = 95 and 99%.
lamellar phases. The molecular weight (Ms) of AOT is 444.56 g/ mol, and the bilayer thickness l equals 20 A˚.32 Assuming spherical micelles with radii corresponding to half the bilayer thickness, we estimate that the micelle aggregation number (nagg) equals 6. The self-diffusion coefficient of AOT in the isotropic solution (DL1) is on the order of 10-10 m2/s,40 and the self-diffusion coefficient of water in the bilayer (DLR) is on the order of 10-9 m2/s.41 We use a bilayer-water partition coefficient (K) of 2 10-6, which is a reasonable value if one compares it with the solubility of water in alkanes.42 We furthermore estimate the thickness of the unstirred layer in the solution (L) to be on the order of 100 μm.43,44 The formalism is set up to calculate the steady-state conditions that will occur if the system is left for a sufficient time to evolve. Establishing a lamellar phase in the interfacial region requires that there be enough time to transport AOT from the reservoir at the bottom of the container. It is questionable if experiments have been performed for such lengths of time, and it is more likely that the systems are studied during the buildup of the steady state. We note that the assumption of convectional flow within the bulk of the system corresponds well to the experimental setup normally used for reflectivity measurements. If the trough holding the sample is thermostated, this is done by circulating water underneath. If the temperature is kept constant via Peltier control, this is also achieved on the bottom of the trough. Since the top and sides of the container are not temperature-controlled, there are certainly going to be convectional flows during the measurement. One can also note that due to the presence of a monolayer at the interface the formation of a lamellar phase results in a very small increase in surface free energy. Both before and after the formation of a lamellar structure. the isotropic solution is exposed to a charged amphiphile monolayer, and the same holds for the lamellar phase. Figure 3 shows the thickness, in terms of both the number of bilayers (n) and the total thickness (ztr), of the interfacial lamellar . There is phase as a function of bulk surfactant mass fraction Xbulk s an exponential growth of the lamellar phase as the two-phase 1 . coexistence is approached from the isotropic solution side, Xtr,L s The effect is, as expected, larger for lower relative humidities. We find that the formation of an interfacial lamellar phase is expected (40) (41) (42) (43) (44) 1409.
Stilbs, P.; Lindman, B. J. Colloid Interface Sci. 1984, 99, 290–293. Mills, R. J. Phys. Chem. 1973, 77, 685–688. Schatzberg, P. J. Phys. Chem. 1963, 67, 776–779. Brinck, J.; J€onsson, B.; Tiberg, F. Langmuir 1998, 14, 1058–1071. Pohl, P.; Saparov, S. M.; Antonenko, Y. N. Biophys. J. 1998, 75, 1403–
Langmuir 2009, 25(20), 12177–12184
Article
Figure 4. Thickness of the interfacial lamellar phase as a function of ambient relative humidity, RH, for a two-phase bulk solution. The different curves show the effect of the isotropic bulk solution thickness, H.
for only compositions very close to the phase boundary, even at low relative humidities. This provides an a posteriori justification that the interesting concentration interval in eq 15 is small and that regarding the diffusion coefficient as a constant is justified. Since the gravitational contribution in eq 28 is also small, the same is true within the two-phase coexistence region. In contrast to the situation in which the bulk concentration is within the isotropic one-phase region, Figure 4 shows that the formation of an interfacial lamellar phase is most probable for a two-phase sample. As expected from eq 26, the figure shows that the number of bilayers increases with a decrease in relative humidity. In other words, the larger the difference in water chemical potential between the ambient air and the bulk solution, the larger the lamellar phase. It is clear from the figure that the thickness of the lamellar phase can be rather substantial, and that an interfacial lamellar phase is formed for moderate relative humidities. In Figure 4, the dependence on the thickness of the isotropic phase, H, is made explicit. We see an increase in the thickness of the interfacial lamellar phase with H. The consequence is that the interfacial lamellar phase increases with total surfactant concentration within the two-phase coexistence region, as discussed above.
4. Comparison with Experimental Observations of Interfacial Film Formation In the previous sections, we have developed a theoretical model that predicts that under certain circumstances one can observe a separate phase in the interfacial region of a system undergoing water (solvent) evaporation. The propensity of forming such a phase depends on a number of variables. It is larger with a lower relative humidity in the air. The closer the bulk phase is to a phase separation, the easier it is for the new phase to appear. The higher the diffusion coefficient in the new phase, the thicker it is. There are several reports of the observation of liquid-crystalline films at the interface between air and aqueous amphiphile systems. However, a detailed comparison with experimental data is hampered by the fact the effect of relative humidity has not been studied in a systematic way. Cevc et al. reported the appearance of a stack of ∼10 bilayers at the air-liquid interface of dimyristoylphosphatidylcholine (DMPC) vesicle suspensions.45 The interface structure was investigated by X-ray reflection and revealed an induction time before the appearance of the interfacial bilayers. Furthermore, a spatially dependent dehydration from the bulk to the air-liquid (45) Cevc, G.; Fenzl, W.; Sigl, L. Science 1990, 249, 1161–1163.
DOI: 10.1021/la900867k
12183
A˚berg et al.
Article
interface was inferred from the temperature variation of the positions of the peaks in the reflectivity curves. These observations are all similar to what has been noted for the AOT case described above. Edler et al. have made extensive studies of film formation in ternary water/amphiphile/polymer systems. In this case, it has been demonstrated that the formation of the phase depends on the existence of air that is not saturated with water.3,5,6 It is a definitive conclusion from this study that the film is due to the qualitative effect discussed in this paper. However, the case of a ternary system is more complex to describe. The main difference is that there is one more thermodynamic degree of freedom so that two phases can coexist along a line in composition space. We leave this interesting case for future studies. There are several observations in the extensive studies by Thomas, Penfold, and co-workers that can be explained within our model. However, to make a definite conclusion requires more information about the relative humidity and how it is controlled during the experiment. A complicating factor related to the humidity is that evaporation of water across the interfacial film is very slow due to the low water permeability, and the time needed to reach full saturation in a closed gas volume is expected to be very long (cf. eq 27).
5. Conclusions For an open aqueous solution, the water chemical potential corresponding to the ambient relative humidity will, in general, not match the water chemical potential of the solution. There are thus nonequilibrium conditions at the air-liquid interface, and a description of the system can be based on steady-state rather than equilibrium conditions. If the bulk water chemical potential is close to a phase transition, there is the possibility of forming an interfacial phase close to the air-liquid interface, where the water chemical potential gradient is located. We have presented a quantitative model for the possibility of forming such a phase. The main qualitative features of the model are as follows. (i) The formation of an interfacial phase is more probable for a system where the bulk composition is close to a phase separation. (ii) A formed interfacial phase is thicker if diffusive transport through it is higher, or if the bulk composition is close to a phase separation. (iii) Because of the role of surface free energy, the transition from a single phase to the appearance of a surface phase at steady state is abrupt. In the unusual case that the new phase completely wets the interface, the situation is reversed and an interfacial phase can appear even at equilibrium with saturated air. We analyzed in more detail the case of a lamellar liquidcrystalline phase forming at the air-liquid interface of an isotropic surfactant solution. This case allows for a detailed modeling of the transport process. Because of the slow transport of water through the lamellae, such a phase forms only under favorable conditions when the bulk system is close to a phase boundary. Specifically for a system with a lamellar phase with a mass density higher than that of the isotropic solution, the small gravitational contribution makes it possible to fine-tune the chemical potentials, and the appearance of a lamellar phase at
12184 DOI: 10.1021/la900867k
the interface is predicted for realistic values of the relative humidity in the ambient air. We see several possible applications of the concept of transport-generated interfacial phase separation. Edler and co-workers have been able to produce polymer surfactant films that are robust enough to be separated from the liquid support.7 These films have predominantly a normal two-dimensional hexagonal structure. As an extension of this concept, one can use two-phase lamellar isotropic systems to form interfacial films of multiple oriented bilayers. Because of the slow rate of formation, these bilayers are expected to show few defects if the purity of the system is controlled. By using a polymerizable amphiphile, one could produce a film that is sufficiently robust to be removed. A number of other variations of the theme appear to be feasible. The concepts used in this study might also have implications for the formation of liquid-crystalline phases when dip or spin coating surfactant solutions. In a similar way as described above, the nucleation of the new phase might start at the evaporating surface, though alternative mechanisms have been suggested.46-48 We note that the assumption of steady-state evaporation made in the study presented here likely does not apply to the dip-coating problem, but that a similar analysis might be conducted. We leave this as a possible future application. Surface films of amphiphiles have been used to reduce the level of evaporation from water reservoirs in hot areas.49 By having a lamellar phase that settles, as in the AOT case discussed in the paper, one has a system that forms the protecting film on demand in a reversible way. Thus, as the relative humidity decreases below a certain value, the lamellar phase forms and prevents additional increases in the evaporation rate. If the humidity rises again, the film dissolves and sinks to the bottom only to reappear when it becomes dry again. For a surfactant-stabilized foam in ambient air, drying is an important destabilization mechanism. It is conceivable that a few bilayers might form close to the air-liquid interface due to the mechanism discussed in this paper. As shown above, this would lead to a drastic decrease in the evaporation rate with a concomitant prolongation of the lifetime of the foam. Acknowledgment. E.S. and H.W. gratefully acknowledge financial support from the Swedish Research Council (Vetenskapsra˚det) both through regular grants and through a Linneaus programme. E.S. acknowledges The Swedish Foundation for Strategic Research (Stiftelsen f€or Strategisk Forskning) for financial support. K.J.E. acknowledges funding from the EPSRC (Project EP/E029914/1). We thank Lennart Piculell for initiating this collaboration. The application of the mechanism described in this paper to the stability of foams is due to an anonymous reviewer. (46) Huang, M. H.; Dunn, B. S.; Soyez, H.; Zink, J. I. Langmuir 1998, 14, 7331– 7333. (47) Huang, M. H.; Dunn, B. S.; Zink, J. I. J. Am. Chem. Soc. 2000, 122, 3739– 3745. (48) Lu, Y.; Ganguli, R.; Drewien, C. A.; Anderson, M. T.; Brinker, C. J.; Gong, W.; Guo, Y.; Soyez, H.; Dunn, B.; Huang, M. H.; Zink, J. I. Nature 1997, 389, 364– 368. (49) Mer, V. K. L., Ed. Retardation of Evaporation by Monolayers; Academic Press: New York, 1962.
Langmuir 2009, 25(20), 12177–12184
