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Colloid Phase Behavior
C H A P T E R
7 Colloid Phase Behavior Ulf Olsson Physical Chemistry, Lund University, Lund, Sweden
1 INTRODUCTION In this chapter, we discuss colloid phase behavior. Phase diagrams describe equilibrium, free energy minimum states and thus report on the thermodynamics of mixtures and the effective interactions. Attractive interactions often result in a “gas-liquid”-type phase separation where the colloidal particles “condense,” forming a concentrated phase, expelling excess water. Repulsive interactions, on the other hand, typically lead to the formation of crystals or liquid crystals. The colloidal systems discussed in this chapter include spherical, rodlike, and disklike colloids, macromolecules such as flexible polymers, and the association colloids, block copolymers, and surfactants (lipids), focusing in the latter case mainly on systems that can be described by flexible surfaces.
2 POLYMERS 2.1 Polymer Solutions Polymers are long-chain molecules typically made up of a repeating sequence. The smallest repeating sequence is called a monomer. There are both hydrophilic water-soluble polymers and hydrophobic polymers that are soluble in organic solvents. Because of the large molecular size, polymers typically have low solubility. In fact, polymers may be insoluble in a solvent where its monomers are highly soluble. This is due to the loss of translational entropy when the monomers are covalently connected into long chains, while the total monomersolvent interaction energy remains almost unchanged. A simple model describing the thermodynamics of polymer solutions or polymer blends is the so-called Flory-Huggins (FH) model [1,2] (described in Chapter 5), which is an extension of the Bragg-Williams (BW) model of regular solutions [3]. As in the BW model, the
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2014 Elsevier B.V. All rights reserved.
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monomer-solvent interactions are assumed to be short-ranged and characterized by an interaction parameter, w. The FH model free energy of mixing of a binary system is given by [4,5] DGFH f f ¼ wfA fB þ A lnfA þ B lnfB kB T NA NB
ð7:1Þ
where Ni and fi are the degree of polymerization (number of monomer units) and volume fraction of component i, respectively. In the BW model, NA ¼ NB ¼ 1. If we consider a polymer solution, typically, the solvent degree of polymerization NA ¼ 1, while for the polymer, NB can be a large number. The interaction parameter z 1 EAB ðEAA þ EBB Þ ð7:2Þ w¼ kB T 2 is discussed in some detail in Chapter 5. Briefly, z is the number of nearest neighbors of a segment of a solvent molecule (assumed to be the same) and Eij is the contact energy between particles i and j, where a particle is a solvent molecule or a polymer segment. Eij typically have a weak temperature dependence. Hence, to a reasonable approximation, we have w T1, implying that the polymer is soluble at higher temperatures, while there is phase separation at lower temperatures. From the model free energy, Equation (7.1), we can calculate the temperature-composition phase diagram, by, for example, considering the concentration dependence of the chemical potentials, mi ¼ @G/@ni. In the presence of a miscibility gap, the chemical potentials vary nonmonotonically with the concentration, and in the two-phase region, the chemical potential, mi, of each components is the same in the two coexisting phases. Equivalently, we can consider Equation (7.1) and obtain the phase boundaries from the so-called common tangent construction. In Figure 7.1, we show a schematic phase diagram for a polymer solvent system. Because of the large size difference between polymer and solvent, the critical point is located at low polymer concentrations. In fact, the critical concentration fp,c ¼ N1=2
ð7:3Þ
and the critical temperature corresponds to a critical interaction parameter, wc, that for large N is given by
Tc
L
T L′ + L″
fp,c
fp
FIGURE 7.1 Schematic Flory-Huggins phase diagram of a polymer solution. L is a homogeneous isotropic liquid phase that at lower temperatures separates into two coexisting phases, L0 and L00 , respectively. Tc is the critical temperature and fp,c is the critical composition.
2 POLYMERS
161
1 wc ¼ þ N1=2 2
ð7:4Þ
The latter should be compared to the regular solution value wc ¼ 2, which demonstrates that polymer solutions become more readily incompatible and phase separated than a corresponding mixture of small molecules. Phase separation conditions are often referred to as bad solvent conditions and full miscibility as good solvent conditions. Strictly, one defines a theta (y) temperature, which within the FH model corresponds to the critical temperature in the case an infinitely large polymer. Good solvent conditions corresponds to temperatures above the y temperature, and at that temperature, the solvent is referred to a theta solvent. The significance of the theta concept is that at theta conditions, the attractive segment-segment interactions compensate the polymer coil expansion due to excluded volume interactions so that the polymer conformation is described as a random walk.
2.2 Polymer Blends As mentioned earlier, polymers may, due to their high molecular weight, be only weakly soluble in a solvent with which the constituting monomers are completely miscible. From this, it is then clear that it is even more difficult to dissolve a polymer in another polymer. To describe high-molecular-weight polymer blends, we now also let NA 1 in Equation (7.1). Figure 7.2 shows the schematic phase diagram for the symmetric case NA ¼ NA ¼ N 1. By symmetry, the critical composition corresponds to fA ¼ fB ¼ 1/2. The critical temperature corresponds to wc ¼ 2/N [4], indicating that high-molecular-weight polymers require very high temperatures in order to mix. This inherent incompatibility of high-molecular-weight polymers represents a challenge in the recycling of plastic waste.
2.3 Ternary Systems As mentioned earlier, two polymers, A and B, in their melted state typically do not mix. Mixing may however be obtained, by adding a third component, C, typically a solvent that is a good solvent for both polymers. The effect can be understood as the replacement of
FIGURE 7.2 Schematic Flory-Huggins phase diagram of a symmetric polymer blend where the two polymers, A and B, have the same effective degree of polymerization or molecular weight. L is a homogeneous isotropic liquid phase that at lower temperatures separates into two coexisting phases, L0 and L00 , respectively. Tc is the critical temperature and the critical composition corresponds by symmetry to equal amounts of the two polymers, fB ¼ 0.5.
Tc L
T
L′ + L″
A
fB
B
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FIGURE 7.3 Schematic Flory-Huggins phase diagram of a ternary system composed of two immiscible polymers, A and B, and a solvent that is a good solvent for both A and B. At high solvent content, an isotropic homogeneous liquid phase, L, is formed. At lower solvent content, the system phase separates into two liquid phases, L0 and L00 , rich in polymer A and B, respectively.
unfavorable A-B contacts with more favorable A-C and B-C contacts. A schematic phase diagram is presented in Figure 7.3. An important application of such ternary systems is for the separation of biomolecules [6]. Here, a two-phase sample is prepared where the molecules one wants to separate have different affinities for the different polymers, e.g., dextran and poly(ethylene oxide).
2.4 Block Copolymers: Microphase Separation High-molecular-weight polymers, A and B, do not mix unless the interaction parameter, wAB, is very close to zero (or negative). Thus, obtaining mixing typically requires very large temperatures, >100 C. Consider a temperature, T, that is above the melting points of A and B but below the critical temperature, Tc, of the miscibility gap. Here, we expect only partial miscibility of A and B and a two-phase coexistence, unless the concentration of one of the polymers is very low. If we now connect, with a covalent bond, the two polymers, end to end, we have a so-called block copolymer, with blocks A and B. We then no longer have a mixture, but a single component. Consequently, there cannot be a two-phase coexistence. Instead of phase separating macroscopically, the block copolymer now solves the frustration by a “microphase separation” into A-rich and B-rich domains, respectively. This results in a structure of the melt on the colloidal length scale, the morphology of which depends on the polymer composition, for example, the volume fraction of block B, fB [7,8]. For symmetric block copolymers, fB 0.5, a lamellar phase is formed where the two blocks arrange themselves in alternating planar bilayers, giving the structure a periodicity of length d. The sharpness of the interface, that is, how much intermixing of the two blocks there is at the A-B interface, depends on wAB. The higher wAB, the sharper the interface becomes and, furthermore, the lower is the total interfacial area, or the area ap, per block copolymer molecule. There is a simple geometric relation between ap and d: d¼
2vp ap
ð7:5Þ
where vp is the volume of one block copolymer molecule. From this equation, we see that if ap decreases, d must increase. Hence, a decrease in ap due to an increase in wAB results in a
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stretching of the polymer chains and decrease of the conformational entropy. In fact, the structural behavior of block copolymer melts can be understood in terms of a compromise between an interfacial energy term that favors a smaller ap and a conformational entropy term that favors a larger ap. When fB deviates significantly from 0.5, there are other morphologies that are more stable than the lamellar phase. The unperturbed radius of gyration or end-to-end distance of a polymer chain typically varies as N1/2. When the two blocks no longer are equally long, their preferred lateral extension differs. The longer block prefers a larger lateral extension. This can be satisfied by having the interface curving towards the domain of the shorter block. For a given total degree of polymerization, the preferred curvature of the interface increases with increasing asymmetry. Thus, with increasing asymmetry, the lamellar phase is replaced by a phase of cylinders and a phase of spheres. The effective repulsion force between these objects is large, and they crystallize into a two-dimensional hexagonal phase and a cubic, BCC phase, respectively. In between the lamellar and hexagonal phases, one often observed a fourth morphology, a bicontinuous phase of cubic symmetry. This structure can be described as the shorter block forming a three-dimensionally continuous multiply connected surface that separates two interwoven and separate domains of the longer block. The minor block thus forms a bilayer, and the midplane of this bilayer represents a minimal surface and generally found to be the gyroid minimal surface. While the bilayer midplane has zero mean curvature, the mean curvature at the two A-B interfaces, which are almost parallel to the midplane minimal surface, is toward the domains of the larger blocks as they do not separate equal volumes on the two sides. In Figure 7.4, we compare a theoretical phase diagram [9] (left) with an experimental phase diagram obtained for the polystyrene-polyisoprene system [10].
3 COLLOIDAL CRYSTALS AND LIQUID CRYSTALS 3.1 Hard Spheres Hard spheres represent a highly important model system [11]. The model describes the behavior of many uncharged colloidal systems, and there exists an accurate analytic expression, due to Carnahan and Starling [11], for the equation of state. The phase behavior has been studied extensively [12–15], including phase diagram studies in zero gravity performed on the space shuttles Colombia and Discovery, respectively [16]. The solution phase is stable up to a volume fraction f ¼ 0.494, followed by a solution-crystal coexistence region up to f ¼ 0.545, above which a homogeneous crystal phase is formed. High-concentration crystals however have difficulties to form because, in addition, there is a glass transition occurring at f ¼ 0.58. The formation of the crystal phase is entropically driven. While the crystal phase has long-range order, this ordering actually increases the number of accessible particle configurations. By adding, e.g, polymer one can tune the effective colloid pair potential via a depletion interaction and induce liquid-liquid phase separation or gelation [17–19]. Liquid-liquid phase separation can also be studied with adhesive hard spheres and can also be obtained using polymer-coated particles and tune the solvent quality [20].
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FIGURE 7.4 Left: Theoretical mean field phase diagram of a diblock copolymer melt. DIS is a disordered melt. CPS corresponds to close-packed spheres (fcc or hcp); QIm3m is bcc phase (space group Im3m). H corresponds to a hexagonal phase and L to a lamellar phase. QIa3d is a bicontinuous cubic phase (space group Ia3d) that can be described in terms of a gyroid minimal surface. f denotes the volume fraction of one of the blocks. Right: Experimental phase diagram obtained for the poly(styrene)-poly(isoprene) block copolymer. Here, Im3m and Ia3d are bcc and bicontinuous cubic phases, respectively. HEX are hexagonal phases and LAM is a lamellar phase. HPL is a lamellar phase with hexagonally perforated layers. The dashed dotted line corresponds to the theoretical order-disorder line. Left: Reprinted with permission from Ref. [9]. Copyright 1996 American Chemical Society. Right: Reprinted with permission from Ref. [10]. Copyright 1995 American Chemical Society.
3.2 Charged Spheres At low ionic strength, the interactions between charged spheres become very long range, and they crystallize already at very low concentrations. Both BCC and FCC packings are observed, depending on the conditions [21–23]. Colloidal crystals have several interesting potential applications [24]. Mixing different sizes leads to very complex phase diagrams, depending on the size and charge ratio [25]. Crystals formed by oppositely charged colloids have also been observed, resembling NaCl or CsCl crystal, but on the colloidal length scale [26].
3.3 Rods Rodlike particles with large length-to-diameter or aspect ratios, L/D, undergo an isotropic-tonematic phase transition upon increasing the concentration. Examples range from tobacco mosaic virus [27], mineral particles [28], and peptide nanotubes [29]. The nematic phase is an example of a colloidal liquid crystal [28]. Onsager analyzed this transition for hard rods and calculated the location of the two-phase region to be 3.34 D/L f 4.49 D/L [30]. The Onsager phase
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FIGURE 7.5 Onsager phase diagram for hard rods,
300
showing the predicted relative stability of the isotropic and nematic phase, respectively.
Axial ratio L/D
250 200 Nematic 150 100 50 Isotropic fluid
0 0
0,05
0,1
0,15
0,2
f
diagram is shown in Figure 7.5. The orientational order in the nematic phase reduces orientational entropy, but it also significantly reduces the excluded volume interactions, and, similar to the hard sphere crystallization, it is the driving force for the phase transition. Charged rods have also been analyzed [31,32]. The long-range electrostatic interactions generally shift the transition to lower concentration although charged rods prefer to be perpendicular, not parallel.
3.4 Plates Also, platelike, e.g., clay particles undergo an isotropic-to-nematic phase transition [33]. The reason is essentially the same as for the rods. Alignment increases the free volume. Apart from the nematic phase, also other phases can be observed, such as columnar and cubatic [28,34].
4 SURFACTANTS Surfactants are molecules that typically are composed of a hydrophobic, water-insoluble alkyl chain and a hydrophilic “headgroup,” being a salt that can dissociate in water (ionic surfactant), or sugar group or a short block of a water-soluble polymer-like polyethylene oxide (nonionic surfactant) [5]. In the latter case, the surfactant can in fact be seen as a short block copolymer. Being composed of a strongly hydrophobic part and strongly hydrophilic part, these molecules are often referred to as being amphiphilic and are also called amphiphiles.
4.1 Binary Surfactant-Water Systems Because of the hydrophobic alkyl chain, surfactant molecules have a low solubility in water. However, instead of simply precipitating the surfactant at higher concentrations, the
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surfactant molecules self-assemble into thermodynamically stable clusters, called micelles, that are characterized by an equilibrium size distribution [35]. In addition, they are surface active and readily adsorb onto and cover hydrophobic surfaces in water, including the water-air interface. Their surface activity and their particular ability to self-assemble (which are related) are responsible for their usefulness in many practical applications, such as detergency and as stabilizers of hydrophobic colloids in water, e.g., hydrophobic pigments in water-based paints. Micelles form in dilute solutions when the surfactant concentration exceeds a critical value, the critical micellization concentration (cmc). For typical surfactants, micelle formation can be highly cooperative [35], and the transition, although not being a phase transition, is related to the “microphase separation” discussed earlier for block copolymer melts. Increasing the concentration above the cmc results in an increased concentration of micelles, while the monomer concentration remains constant or decreases with increasing concentration. The micelles are association (or self-assembly) colloids that interact and form ordered structures, colloidal liquid crystals at higher concentrations. However, there is an important difference. Because the micelles are fluid objects, they can change their shape and size to minimize the free energy. This additional degree of freedom gives rise to richer phase diagrams. Spherical or near spherical micelles crystallize at higher concentrations into a cubic lattice. For nonionic micelles, this occurs roughly at the hard sphere crystallization concentration. For ionic surfactants, the concentration is lower. As a response to the increased interactions with increasing concentration, the micelles first transform into cylinders and then into planar bilayers. This results in a sequence of lyotropic liquid crystalline phases. Cylindrical micelles form a hexagonal phase, ordered in two dimensions, while planar bilayers form a stack with onedimensional order. In between the hexagonal and the lamellar phases, one often also observes a bicontinuous cubic phase, the structure of which that can be described either as a consisting of two branched cylinder networks or, alternatively, in terms of a multiply connected bilayer. As an example, we present in Figure 7.6 the phase diagram of the binary system waterdodecyltrimethylammonium chloride [36]. With ionic surfactants, a quantitative analysis requires an accurate description of the electrostatic interactions. Jo¨nsson and Wennerstro¨m undertook a detailed analysis of ionic surfactant-water phase diagrams [37], where the relative stability of the different aggregate geometries, spheres, cylinders, and planes was assessed. For the electrostatic contribution to the free energy, they handled the different geometries in a mean field way by solving the Poisson-Boltzmann equation within a cell model. In addition, they considered contributions from hydrophobic interactions and entropy of mixing and were able to obtain accurate descriptions of ionic surfactant-water systems.
4.2 Ternary Surfactant-Water-Alcohol Systems The sequence of phases is the same as observed for block copolymer melts. However, the whole sequence from “normal” globular micelles to reverse micelles with a hydrophilic, water-containing core is not accessible in a binary surfactant-water systems. Upon the addition of an organic solvent, immiscible with water, additional phases can form. Phase diagrams of ternary systems were studied extensively by Ekwall and his coworkers Krister Fontell and Leo Mandell, during the 1960s and 1970s. As an example, the ternary phase
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FIGURE 7.6 Phase diagram of the binary water-dodecyltrimethylammonium chloride system. Here, F denotes an isotropic liquid phase. C0 is a cubic phase with discrete micelles distributed on a cubic lattice. M denotes the hexagonal phase. C is a bicontinuous cubic phase (space group Ia3d), and N is a lamellar phase. Reprinted with permission from Ref. [36]. Copyright 1969.
diagram of the ionic surfactant potassium decanoate (potassium caprate), water, and octanol [38] is shown in Figure 7.7a. Here, L1 is a water-rich liquid micellar phase and L2 is an alcohol-rich liquid phase containing reverse micelles. E and D are hexagonal and lamellar liquid crystalline phases. K is a liquid crystalline phase that later was shown to be a reverse hexagonal phase, F in Ekwall’s notation [40]. By adding the alcohol to the binary water-surfactant system, the aggregate structure becomes gradually inverted from normal micelles in water, via the planar bilayers in the lamellar phase, to reverse micelles in the alcohol. An additional complexity in the present system is that the alcohol, which is also slightly amphiphilic, acts simultaneously as a solvent and as a cosurfactant, participating in the surfactant aggregates.
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FIGURE 7.7
(a) Experimental phase diagram of the ternary system water-potassium octanoate (caprate)-octanol. L1 is a water-rich isotropic liquid phase, E is a hexagonal phase, D is a lamellar phase, K is a reverse hexagonal phase, and L2 is an alcohol-rich isotropic liquid phase. The surfactant forms normal micelles in the L1 phase and reverse micelles, with solubilized water in the interior, in the L2 phase. (b) Theoretical phase diagram of the same waterpotassium octanoate (caprate)-octanol system. Same notation as in (a) except that the reverse hexagonal phase is here denoted F. Panel a: Reprinted from Ref. [38]. Copyright 1969, with permission from Elsevier. Panel b: Reprinted with permission from Ref. [39]. Copyright 1987 American Chemical Society.
Jo¨nsson and Wennerstro¨m extended their model to allow for a third component and applied it to the potassium decanoate-water-octanol system [39]. Their calculated phase diagram is shown in Figure 7.7b. As can be seen, there is a good agreement between the calculated and the experimental phase diagrams.
4.3 Ternary Systems with Amphiphilic Block Copolymers The ternary Pluronic-water-oil systems essentially bridge the gap between the surfactant systems and the block copolymer melts. Here, Pluronics are widely used triblock copolymers composed of a hydrophobic poly(propylene oxide) (PPO) middle block connected to two hydrophilic poly(ethylene oxide) end blocks. Together with selective solvents for the two blocks, e.g., p-xylene and water, a wide range of liquid crystalline phases, including liquid micellar phases, are observed in the ternary phase diagrams. In the system water-EO19PO44EO19-pxylene (Figure 7.8), the full sequence with seven liquid crystalline phases and two isotropic liquid phases was observed [41]. Svensson analyzed ternary phase diagrams with different Pluronic copolymers [42]. When plotting the phase diagrams as a function of the total volume fraction of the hydrophobic components, oil and the PPO block, the different phase transitions occur at approximately the same volume fractions as in the case of block copolymer melts.
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FIGURE 7.8 Experimental phase diagram of the ternary water- EO19PO44EO19-p-xylene system at 25 C. A total of nine different phases are observed. L1 and L2 are isotropic liquid phases with normal and reverse micelles, respectively. I1 and I2 are cubic phases containing normal and reverse micelles, respectively. H1 and H2 are hexagonal phases containing normal and reverse cylindrical micelles, respectively. V1 and V2 are bicontinuous cubic phases. V1 has a reverse water-swollen bilayer separating two oil domains. V2 has a normal oil-swollen bilayer separating two water domains. La is a lamellar phase. The different structures are illustrated. Reprinted with permission from Ref. [41]. Copyright 1998 American Chemical Society.
4.4 Microemulsions: Phases of Fluid Surfaces Water and oil do not mix but can be mixed by adding a third component that is miscible with both water and oil. Figure 7.9 shows the phase diagram of the water-benzene-ethanol system [43], a system studied already by Bancroft and his student Taylor at the end of the nineteenth century [44]. With increasing concentration of ethanol, which is miscible with both water and benzene, there are less and less unfavorable water-benzene contacts and eventually a homogeneous phase is formed containing equal amounts of water and benzene. However, the amounts of alcohol required to mix water and oil are high, and if more hydrophobic alkanes are used instead of benzene, it is even higher.
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FIGURE 7.9 Phase diagram of the ternary water-ethanol-benzene system at 25 C. L is a homogeneous liquid phase that at lower ethanol concentrations separates into two coexisting phases, L0 and L00 , respectively. Drawn using data from Ref. [43].
Ethanol
L 0.5
0.5
L′ + L″
Water
0.5
Benzene
In the homogeneous phase water, benzene and ethanol molecules are almost randomly distributed, and the phase diagram can be understood within regular solution theory. With a proper surfactant instead of alcohol, the concentration needed to mix equal volumes of water and oil can be reduced by more than an order of magnitude. In this case, the distribution of molecules is far from random. While the mixture is homogeneous on the macroscopic scale, it is structured into water and oil domains on the colloidal length scale. Water and/or oil is dispersed into domains, with a dense self-assembled surfactant monolayer, with an area density of A fs ¼ V ls
ð7:6Þ
stabilizing the oil-water interface. Here, fs is the surfactant volume fraction and ls vs/as is the surfactant length that is defined as the surfactant molecular volume, vs, divided by the average area, as, that the surfactant molecule occupies at the water-oil interface. The homogeneous mixture is called microemulsion and its microstructure and phase behavior depends strongly on the spontaneous curvature, H0, of the surfactant film [45]. Microemulsions and related phases made up of flexible surfactant films can be understood in terms Helfrich curvature free energy. To leading order, the curvature free energy density is often written as [46] K gc ¼ 2kðH H0 Þ2 þ k
ð7:7Þ
Here, H ¼ (c1 þ c2)/2 and K ¼ c1c2 are the mean and the Gaussian curvature, with c1 and c2 be is the saddle ing the two principal curvatures. k > 0 is the bending rigidity of the film and k splay modulus that reports on the preferred topology of the film that can take either positive < 0, a spherically bent film is preferred (c1 and c2 having equal sign) or negative values. If k > 0, a locally favoring the formation of closed surfaces and droplet microemulsions. If k saddle-shaped surface is preferred favoring the formation of bicontinuous structures. k is typ is typically ically a few times kBT, low enough to be flexible but high enough so that H H0. k of the same magnitude as k, but negative.
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The total curvature free energy, Gc, for a given surface configuration is then formally obtained by integrating gc over the total surface, S: ðð dA gc ð7:8Þ Gc ¼ S
As mentioned earlier, the microemulsion structure and phase behavior depends strongly on H0. Counting curvature towards oil as positive, one finds oil-in-water microemulsions when H0 >> 0, and water-in-oil microemulsions when H0 0. For spherical droplets of radius, Gc has a minimum for R ¼ R0 where k R0 ¼ H01 1 þ ð7:9Þ 2k In both cases, the microemulsion phase is found at compositions where droplet radii R R0 (with a small correction due to the entropy of mixing, that is neglected here). For oil droplets in water, we have R¼
3f0 ls fs
ð7:10Þ
where f0 is the oil volume fraction. When R ¼ R0, the droplets are saturated with oil. Adding more oil results in a phase separation with an excess oil phase. Same things hold for negative spontaneous curvatures and water droplets in oil, as expected from the symmetry of Gc. For H0 0, the ternary water-surfactant-oil phase diagram is dominated by a lamellar phase, because with planar layers, H 0 irrespective of the composition. However when fw f0, H 0 can also be satisfied by a bicontinuous structure, which typically is found at lower surfactant concentrations. Particularly useful experimental model systems are those with nonionic surfactants oligo (ethylene glycol) alkyl ether, CmEn, where m is the number of carbons in the alkyl chain and n is the number of EO units in the polar part. The reason is the strong temperature dependence of the interactions between the water and the polar En chain. Water is a good solvent at lower temperatures and swells the En chains that are highly solvated. This results in H0 0. At higher temperatures, on the other hand, water-En interactions are less favorable and the En chains become less solvated, leading to H0 0. Thus, with these surfactants, the spontaneous curvature can be conveniently tuned by varying the temperature. A Taylor expansion around T ¼ T0, where T0 is the temperature, is the so-called balance temperature (equivalent to the phase inversion temperature, PIT, identified for emulsions stabilized by CmEn surfactants) where H0 ¼ 0 yields H0 ðT Þ ¼ aðT0 T Þ þ
3
1
1
ð7:11Þ
˚ K [45,47]. with a 10 A Adding temperature as the third dimension, the phase behavior a ternary surfactantwater-oil system can be represented by a phase prism. Different cuts through the phase prism can be chosen to illustrate the microemulsion phase behavior in a two-dimensional diagram. One particularly interesting cut is the Kahlweit’s “fish cut,” defined by fw ¼ f0 (Figure 7.10, left). The phase behavior typically observed in the fs T plane is schematically illustrated in Figure 7.10 (right). The microemulsion phase, L, coexists with an excess oil phase, O, at lower
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T
S
T0
W
O
W+L
L
W+L+O
T
Lα
L+O
f S*
fS
FIGURE 7.10
Left: Illustrating Kahlweit’s “fish cut” through the ternary phase prism, defined by equal amounts of water and oil. Right: Schematic phase behavior around the balance temperature, T0. L is the microemulsion phase. W and O are essentially pure water and oil phases, respectively. La is a lamellar liquid crystalline phase. fs is the surfactant volume fraction in the balanced microemulsion that coexists with excess water and oil at T ¼ T0. The lower the fs , the more efficient is the surfactant in solubilizing water and oil.
temperatures and with an excess water phase, W, at higher temperatures. At higher surfactant concentrations, the low-temperature and high-temperature microemulsion domains are separated by a lamellar liquid crystalline phase, La. At T ¼ T0, the lamellar phase is stable at higher fs. Upon decreasing fs, there is a phase transition to a microemulsion phase that here has a bicontinuous structure with H0 0. This phase can swell with water and oil down to a surfactant volume fraction, fs , corresponding to a separation limit for solubilizing equal volumes of water and oil. For fs < fs , not all water and oil can be solubilized and there is a separation into three phases, W þ L þ O. Another informative cut is the Shinoda cut, defined by a constant fs (Figure 7.11, left). Choosing fs to be between fs and the concentration where a lamellar phase is formed results in a phase diagram as the one shown in Figure 7.11 (right), corresponding to the plane f0/(fw þ f0) T. This is an experimentally determined phase diagram from the waterC12E5-tetradecane system at a constant surfactant weight fraction of 0.166 [48]. The oil fraction is also given as weight fraction. Here, one observes a microemulsion channel stretching from the water side at lower temperatures (H0 > 0) to the oil side at higher temperatures (H0 < 0). Interestingly, there is a second, more narrow channel having the opposite orientation and that crosses the main microemulsion channel around T ¼ T0. These more narrow channels correspond to the so-called L3 or “sponge phase” [49]. In contrast to the microemulsion, where a surfactant monolayer separates an oil domain from a water domain, the “sponge phase” has a bilayer structure where the bilayer separates two solvent domains of the same kind. At higher temperatures and higher water content, a normal, oil-swollen bilayer separates two water domains. At lower temperatures and higher oil content, a reverse water-swollen bilayer separates two oil domains. The bilayer midplane separates equal volumes on the two sides and here H 0. The surfactant film, on the other hand, has unequal volumes on its two sides and H 6¼ 0. In the water-rich “sponge phase” at higher temperatures, the surfactant film has H < 0, and the opposite holds for the oil-rich “sponge phase.” The third phase observed in the
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70
60
T / °C
S
W
O
T
W+L
50 Lα
L
Lα
40 L+O 30 0
Wt. fraction oil/(water + oil)
1
FIGURE 7.11 Left: Illustrating the “Shinoda cut” through the ternary phase prism, defined by a constant surfactant concentration. Right: Experimental phase diagram of the C12E5-water-tetradecane system in the plane defined by a constant surfactant concentration, 16.6 wt%. T0 for this system is 48 C. L denotes an isotropic liquid phase (microemulsion) and La is a lamellar liquid crystalline phase. Drawn using data from Ref. [48].
Shinoda cut is the lamellar phase. The phase diagram in Figure 7.11 (right) has a particular symmetry. Rotating 180 around the point f0/(fw þ f0) ¼ 0.5, T ¼ T0 returns the same phase diagram. This reflects the fact that the free energy is invariant upon exchanging water and oil domains (changing f0 to (1 f0)) and simultaneously reversing the sign of H0. Another particularly interesting cut is the one corresponding to H0 ¼ 0 (T ¼ T0). As pointed out earlier, the lamellar phase is then stable over a large range of compositions and, in addition, there is a microemulsion phase at fw f0 and lower surfactant concentrations and a three-phase triangle. In Figure 7.12a, we present an experimental phase diagram determined at T ¼ T0 of the water-C10E5-octane system [50]. Clearly, it is difficult to derive a free energy for a bicontinuous microemulsion with a flexible surfactant film. However, when H0 ¼ 0, the curvature free energy becomes length-scaleinvariant and it is possible to use a scaling approach. For a balanced microemulsion (fw ¼ f0), neglecting the surfactant film thickness, there is only a single length scale in the problem, the inverse interfacial area per unit volume (A/V)1 ¼ ls/fs. From the length-scale invariance of Gc, it then follows that G ¼ a3 f3s V
ð7:12Þ
where G here is the total free energy and a3 is a materials constant. However, the same scaling argument holds for the lamellar phase, Equation (7.12), and hence, the harmonic approximation of the curvature free energy, Equation (7.7), is not sufficient to explain the observed phase behavior, including the three-phase equilibrium at low fs. For the lamellar phase, Helfrich has derived a value of a3 that is positive and represents the repulsive undulation force
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S
C10E5 90
90
T = 44.6 °C
50
1
Lα
50 lamellar
2 2
1
10
3
H2O 10
(a)
2
2 3
50
3
10 90 n-octane
W
O
(b)
FIGURE 7.12 (a) Experimental phase diagram of the ternary C10E5-water-octane system at the balance temperature T0 ¼ 44.3 C. Concentrations are given as weight fractions. A lamellar phase, La, is stable at higher surfactant concentrations, over a wide range of water-oil ratios. 1, 2, and 3 denote single-phase, two-phase, three-phase regions, respectively. The microemulsion phase is stable for approximately equal amounts of water and oil. This phase coexists with excess water and oil at lower surfactant concentrations and with the lamellar phase at higher surfactant concentrations. (b) Theoretical phase diagram of a surfactant (S)-water (W)-oil (O) system for H0 ¼ 0, corresponding to T ¼ T0. The phase diagram was calculated based on parameters obtained from the C12E5-water-decane system and assuming a surfactant monolayer bending rigidity of 1 kBT. Panel a: Used with permission from Ref. [50]. Copyright 1993 The Deutsche Bunsen-Gesellschaft. Panel b: Reprinted with permission from Ref. [51]. Copyright 1995 American Chemical Society.
[52]. For the microemulsion phase, on the other hand, the finite swelling with phase separation at lower fs is consistent with a3 < 0. Expanding Equation (7.7) beyond the harmonic approximation, the next terms are quadratic in the curvatures, H4, H2K, and K2 (odd powers cancel by symmetry). Taking these into account, a first-order correction to Equation (7.12) becomes [53] G ¼ a3 f3s þ a5 f5s V
ð7:13Þ
The observed phase behavior implies a3 < 0 and a5 > 0 for the microemulsion and a3 > 0 and for the lamellar phase. a5 > 0 can be understood as a penalty for high curvatures. Modeling also a penalty for deviating from fw ¼ f0 [51] was able to calculate a phase diagram, presented in Figure 7.12b, that is in good agreement with experiments. Using the same framework, we can also understand the sponge phase, including its finite swelling and its coexistence with the lamellar phase [54]. Moreover, microemulsions can be used as model systems, analyzed in terms of a curvature free energy, to study a range of fundamental liquid-state phenomena [55]. Finally, we note that going from the flexible one-dimensional objects (polymers) to the flexible two-dimensional objects (surfactant film) corresponds to a significant increase in complexity.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49]
Huggins ML. Ann N Y Acad Sci 1942;43:1. Flory PJ. J Chem Phys 1942;10:51. Atkins P, de Paula J. Physical chemistry. 9th ed. Oxford: Oxford University Press; 2009. Flory PJ. Principles of polymer chemistry. Ithaca, New York: Cornell University Press; 1953. Holmberg K, Jo¨nsson B, Kronberg B, Lindman B. Surfactants and polymers in aqueous solution. 2nd ed. Chichester: John Wiley & Sons; 2002. Albertsson PA. Partition of cell particles and macromolecules. 3rd ed. New York: John Wiley & Sons; 1986. Bates FS. Science 1991;251:898. Bates FS, Fredrickson GH. Phys Today 1999;52:32. Matsen MW, Bates FS. Macromolecules 1996;29:1091. Khandpur AK, Fo¨rster S, Bates FS, Hamley IW, Ryan AJ, Bras W, et al. Macromolecules 1995;28:8796. Carnahan NF, Starling KE. J Chem Phys 1969;51:635. Alder BJ, Wainright TE. J Chem Phys 1957;27:1208. Pusey PN, van Megen W. Nature 1986;320:340. Royall CP, Poon WCK, Weeks ER. Soft Matter 2013;9:17. Parisi G, Zamponi G. Rev Mod Phys 2010;82:789. Cheng Z, Chaikin PM, Russel WB, Meyer WV, Zhu J, Rogers RB, et al. Mater Des 2001;22:529. Poon W. Science 2004;304:830. Poon WCK. J Phys Condens Matter 2002;14:R859. Anderson VJ, Lekkerkerker HNW. Nature 2002;416:811. Vrij A, Penders MHGM, Rouw PW, de Kruif CG, Dhont JKG, Smits C, et al. Faraday Discuss Chem Soc 1990;90:31. Robins MO, Kremer K, Grest GS. J Chem Phys 1988;88:3286. Lo¨wen H. Phys Rep 1994;237:249. Royall CP, Leunissen ME, van Blaaderen A. J Phys Condens Matter 2003;15:S3581. Sun Z, Yang B. Nanoscale Res Lett 2006;1:46. Lorenz NJ, Scho¨pe HJ, Reiber H, Palberg T, Wette P, Klassen I, et al. J Phys Condens Matter 2009;21:464166. Leunissen ME, Christova CG, Hynninen A-P, Royall CP, Campbell AW, Imhof A, et al. Nature 2005;437:235. Bawden FC, Pirie NW, Bernal JD, Fankuchen I. Nature 1936;138:1051. Lekkerkerker HNW, Vroege G. J Phil Trans R Soc A 2013;371:20120263. Bucak S, Cenker CC, Nasir I, Olsson U, Zackrisson M. Langmuir 2009;25:4262. Onsager L. Ann N Y Acad Sci 1949;51:627. Stroobants A, Lekkerkerker HNW, Odijk Th. Macromolecules 1986;19:2232. Vroege GJ, Lekkerkerker HNW. Rep Prog Phys 1992;55:1241. Langmuir I. J Chem Phys 1936;6:873. Veerman JAC, Frenkel D. Phys Rev A 1992;45:5632. Wennerstro¨m H, Lindman B. Phys Rep 1979;51:1. Balmbra RR, Clunie JS, Goodman JF. Nature 1969;222:1159. Jo¨nsson B, Wennerstro¨m H. J Colloid Interface Sci 1981;80:482. Ekwall P, Mandell L, Fontell K. Ternary systems of potassium soap, alcohol, and water: I. Phase equilibria and phase structures. J Colloid Interface Sci 1969;31:508. Jo¨nsson B, Wennerstro¨m H. J Phys Chem 1987;91:338. Neeson PG, Tiddy GJT. J Chem Soc Faraday Trans 1982;1(78):147. Alexandridis P, Olsson U, Lindman B. Langmuir 1998;14:2627. Svensson, B. PhD Thesis, Lund University; 2001. Bozeya A, Al-Bawab A, Friberg SE, Ge L, Rong G. J Colloid Interface Sci 2013;395:161. Taylor SF. J Phys Chem 1897;1:301. Olsson U, Wennerstro¨m H. Adv Colloid Interface Sci 1994;49:113. Helfrich W. Z Naturforsch 1973;28c:693. Strey R. Colloid Polym Sci 1994;272:1005. Olsson U, Shinoda K, Lindman B. J Phys Chem 1986;90:4083. Anderson DM, Wennerstro¨m H, Olsson U. J Phys Chem 1989;93:4243.
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Strey R. Ber Bunsenges Phys Chem 1993;97:742. Daicic J, Olsson U, Wennerstro¨m H. Langmuir 1995;11:2451. Helfrich W. Z Naturforsch 1978;33a:305. Wennerstro¨m H, Olsson U. Langmuir 1993;9:365. Le TD, Olsson U, Wennerstro¨m H, Schurtenberger P. Phys Rev E 1999;60:4300. Wennerstro¨m H, Olsson UCR. Chimie 2009;12:4.
7 Colloid Phase Behavior Ulf Olsson Physical Chemistry, Lund University, Lund, Sweden
1 INTRODUCTION In this chapter, we discuss colloid phase behavior. Phase diagrams describe equilibrium, free energy minimum states and thus report on the thermodynamics of mixtures and the effective interactions. Attractive interactions often result in a “gas-liquid”-type phase separation where the colloidal particles “condense,” forming a concentrated phase, expelling excess water. Repulsive interactions, on the other hand, typically lead to the formation of crystals or liquid crystals. The colloidal systems discussed in this chapter include spherical, rodlike, and disklike colloids, macromolecules such as flexible polymers, and the association colloids, block copolymers, and surfactants (lipids), focusing in the latter case mainly on systems that can be described by flexible surfaces.
2 POLYMERS 2.1 Polymer Solutions Polymers are long-chain molecules typically made up of a repeating sequence. The smallest repeating sequence is called a monomer. There are both hydrophilic water-soluble polymers and hydrophobic polymers that are soluble in organic solvents. Because of the large molecular size, polymers typically have low solubility. In fact, polymers may be insoluble in a solvent where its monomers are highly soluble. This is due to the loss of translational entropy when the monomers are covalently connected into long chains, while the total monomersolvent interaction energy remains almost unchanged. A simple model describing the thermodynamics of polymer solutions or polymer blends is the so-called Flory-Huggins (FH) model [1,2] (described in Chapter 5), which is an extension of the Bragg-Williams (BW) model of regular solutions [3]. As in the BW model, the
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#
2014 Elsevier B.V. All rights reserved.
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monomer-solvent interactions are assumed to be short-ranged and characterized by an interaction parameter, w. The FH model free energy of mixing of a binary system is given by [4,5] DGFH f f ¼ wfA fB þ A lnfA þ B lnfB kB T NA NB
ð7:1Þ
where Ni and fi are the degree of polymerization (number of monomer units) and volume fraction of component i, respectively. In the BW model, NA ¼ NB ¼ 1. If we consider a polymer solution, typically, the solvent degree of polymerization NA ¼ 1, while for the polymer, NB can be a large number. The interaction parameter z 1 EAB ðEAA þ EBB Þ ð7:2Þ w¼ kB T 2 is discussed in some detail in Chapter 5. Briefly, z is the number of nearest neighbors of a segment of a solvent molecule (assumed to be the same) and Eij is the contact energy between particles i and j, where a particle is a solvent molecule or a polymer segment. Eij typically have a weak temperature dependence. Hence, to a reasonable approximation, we have w T1, implying that the polymer is soluble at higher temperatures, while there is phase separation at lower temperatures. From the model free energy, Equation (7.1), we can calculate the temperature-composition phase diagram, by, for example, considering the concentration dependence of the chemical potentials, mi ¼ @G/@ni. In the presence of a miscibility gap, the chemical potentials vary nonmonotonically with the concentration, and in the two-phase region, the chemical potential, mi, of each components is the same in the two coexisting phases. Equivalently, we can consider Equation (7.1) and obtain the phase boundaries from the so-called common tangent construction. In Figure 7.1, we show a schematic phase diagram for a polymer solvent system. Because of the large size difference between polymer and solvent, the critical point is located at low polymer concentrations. In fact, the critical concentration fp,c ¼ N1=2
ð7:3Þ
and the critical temperature corresponds to a critical interaction parameter, wc, that for large N is given by
Tc
L
T L′ + L″
fp,c
fp
FIGURE 7.1 Schematic Flory-Huggins phase diagram of a polymer solution. L is a homogeneous isotropic liquid phase that at lower temperatures separates into two coexisting phases, L0 and L00 , respectively. Tc is the critical temperature and fp,c is the critical composition.
2 POLYMERS
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1 wc ¼ þ N1=2 2
ð7:4Þ
The latter should be compared to the regular solution value wc ¼ 2, which demonstrates that polymer solutions become more readily incompatible and phase separated than a corresponding mixture of small molecules. Phase separation conditions are often referred to as bad solvent conditions and full miscibility as good solvent conditions. Strictly, one defines a theta (y) temperature, which within the FH model corresponds to the critical temperature in the case an infinitely large polymer. Good solvent conditions corresponds to temperatures above the y temperature, and at that temperature, the solvent is referred to a theta solvent. The significance of the theta concept is that at theta conditions, the attractive segment-segment interactions compensate the polymer coil expansion due to excluded volume interactions so that the polymer conformation is described as a random walk.
2.2 Polymer Blends As mentioned earlier, polymers may, due to their high molecular weight, be only weakly soluble in a solvent with which the constituting monomers are completely miscible. From this, it is then clear that it is even more difficult to dissolve a polymer in another polymer. To describe high-molecular-weight polymer blends, we now also let NA 1 in Equation (7.1). Figure 7.2 shows the schematic phase diagram for the symmetric case NA ¼ NA ¼ N 1. By symmetry, the critical composition corresponds to fA ¼ fB ¼ 1/2. The critical temperature corresponds to wc ¼ 2/N [4], indicating that high-molecular-weight polymers require very high temperatures in order to mix. This inherent incompatibility of high-molecular-weight polymers represents a challenge in the recycling of plastic waste.
2.3 Ternary Systems As mentioned earlier, two polymers, A and B, in their melted state typically do not mix. Mixing may however be obtained, by adding a third component, C, typically a solvent that is a good solvent for both polymers. The effect can be understood as the replacement of
FIGURE 7.2 Schematic Flory-Huggins phase diagram of a symmetric polymer blend where the two polymers, A and B, have the same effective degree of polymerization or molecular weight. L is a homogeneous isotropic liquid phase that at lower temperatures separates into two coexisting phases, L0 and L00 , respectively. Tc is the critical temperature and the critical composition corresponds by symmetry to equal amounts of the two polymers, fB ¼ 0.5.
Tc L
T
L′ + L″
A
fB
B
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FIGURE 7.3 Schematic Flory-Huggins phase diagram of a ternary system composed of two immiscible polymers, A and B, and a solvent that is a good solvent for both A and B. At high solvent content, an isotropic homogeneous liquid phase, L, is formed. At lower solvent content, the system phase separates into two liquid phases, L0 and L00 , rich in polymer A and B, respectively.
unfavorable A-B contacts with more favorable A-C and B-C contacts. A schematic phase diagram is presented in Figure 7.3. An important application of such ternary systems is for the separation of biomolecules [6]. Here, a two-phase sample is prepared where the molecules one wants to separate have different affinities for the different polymers, e.g., dextran and poly(ethylene oxide).
2.4 Block Copolymers: Microphase Separation High-molecular-weight polymers, A and B, do not mix unless the interaction parameter, wAB, is very close to zero (or negative). Thus, obtaining mixing typically requires very large temperatures, >100 C. Consider a temperature, T, that is above the melting points of A and B but below the critical temperature, Tc, of the miscibility gap. Here, we expect only partial miscibility of A and B and a two-phase coexistence, unless the concentration of one of the polymers is very low. If we now connect, with a covalent bond, the two polymers, end to end, we have a so-called block copolymer, with blocks A and B. We then no longer have a mixture, but a single component. Consequently, there cannot be a two-phase coexistence. Instead of phase separating macroscopically, the block copolymer now solves the frustration by a “microphase separation” into A-rich and B-rich domains, respectively. This results in a structure of the melt on the colloidal length scale, the morphology of which depends on the polymer composition, for example, the volume fraction of block B, fB [7,8]. For symmetric block copolymers, fB 0.5, a lamellar phase is formed where the two blocks arrange themselves in alternating planar bilayers, giving the structure a periodicity of length d. The sharpness of the interface, that is, how much intermixing of the two blocks there is at the A-B interface, depends on wAB. The higher wAB, the sharper the interface becomes and, furthermore, the lower is the total interfacial area, or the area ap, per block copolymer molecule. There is a simple geometric relation between ap and d: d¼
2vp ap
ð7:5Þ
where vp is the volume of one block copolymer molecule. From this equation, we see that if ap decreases, d must increase. Hence, a decrease in ap due to an increase in wAB results in a
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stretching of the polymer chains and decrease of the conformational entropy. In fact, the structural behavior of block copolymer melts can be understood in terms of a compromise between an interfacial energy term that favors a smaller ap and a conformational entropy term that favors a larger ap. When fB deviates significantly from 0.5, there are other morphologies that are more stable than the lamellar phase. The unperturbed radius of gyration or end-to-end distance of a polymer chain typically varies as N1/2. When the two blocks no longer are equally long, their preferred lateral extension differs. The longer block prefers a larger lateral extension. This can be satisfied by having the interface curving towards the domain of the shorter block. For a given total degree of polymerization, the preferred curvature of the interface increases with increasing asymmetry. Thus, with increasing asymmetry, the lamellar phase is replaced by a phase of cylinders and a phase of spheres. The effective repulsion force between these objects is large, and they crystallize into a two-dimensional hexagonal phase and a cubic, BCC phase, respectively. In between the lamellar and hexagonal phases, one often observed a fourth morphology, a bicontinuous phase of cubic symmetry. This structure can be described as the shorter block forming a three-dimensionally continuous multiply connected surface that separates two interwoven and separate domains of the longer block. The minor block thus forms a bilayer, and the midplane of this bilayer represents a minimal surface and generally found to be the gyroid minimal surface. While the bilayer midplane has zero mean curvature, the mean curvature at the two A-B interfaces, which are almost parallel to the midplane minimal surface, is toward the domains of the larger blocks as they do not separate equal volumes on the two sides. In Figure 7.4, we compare a theoretical phase diagram [9] (left) with an experimental phase diagram obtained for the polystyrene-polyisoprene system [10].
3 COLLOIDAL CRYSTALS AND LIQUID CRYSTALS 3.1 Hard Spheres Hard spheres represent a highly important model system [11]. The model describes the behavior of many uncharged colloidal systems, and there exists an accurate analytic expression, due to Carnahan and Starling [11], for the equation of state. The phase behavior has been studied extensively [12–15], including phase diagram studies in zero gravity performed on the space shuttles Colombia and Discovery, respectively [16]. The solution phase is stable up to a volume fraction f ¼ 0.494, followed by a solution-crystal coexistence region up to f ¼ 0.545, above which a homogeneous crystal phase is formed. High-concentration crystals however have difficulties to form because, in addition, there is a glass transition occurring at f ¼ 0.58. The formation of the crystal phase is entropically driven. While the crystal phase has long-range order, this ordering actually increases the number of accessible particle configurations. By adding, e.g, polymer one can tune the effective colloid pair potential via a depletion interaction and induce liquid-liquid phase separation or gelation [17–19]. Liquid-liquid phase separation can also be studied with adhesive hard spheres and can also be obtained using polymer-coated particles and tune the solvent quality [20].
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FIGURE 7.4 Left: Theoretical mean field phase diagram of a diblock copolymer melt. DIS is a disordered melt. CPS corresponds to close-packed spheres (fcc or hcp); QIm3m is bcc phase (space group Im3m). H corresponds to a hexagonal phase and L to a lamellar phase. QIa3d is a bicontinuous cubic phase (space group Ia3d) that can be described in terms of a gyroid minimal surface. f denotes the volume fraction of one of the blocks. Right: Experimental phase diagram obtained for the poly(styrene)-poly(isoprene) block copolymer. Here, Im3m and Ia3d are bcc and bicontinuous cubic phases, respectively. HEX are hexagonal phases and LAM is a lamellar phase. HPL is a lamellar phase with hexagonally perforated layers. The dashed dotted line corresponds to the theoretical order-disorder line. Left: Reprinted with permission from Ref. [9]. Copyright 1996 American Chemical Society. Right: Reprinted with permission from Ref. [10]. Copyright 1995 American Chemical Society.
3.2 Charged Spheres At low ionic strength, the interactions between charged spheres become very long range, and they crystallize already at very low concentrations. Both BCC and FCC packings are observed, depending on the conditions [21–23]. Colloidal crystals have several interesting potential applications [24]. Mixing different sizes leads to very complex phase diagrams, depending on the size and charge ratio [25]. Crystals formed by oppositely charged colloids have also been observed, resembling NaCl or CsCl crystal, but on the colloidal length scale [26].
3.3 Rods Rodlike particles with large length-to-diameter or aspect ratios, L/D, undergo an isotropic-tonematic phase transition upon increasing the concentration. Examples range from tobacco mosaic virus [27], mineral particles [28], and peptide nanotubes [29]. The nematic phase is an example of a colloidal liquid crystal [28]. Onsager analyzed this transition for hard rods and calculated the location of the two-phase region to be 3.34 D/L f 4.49 D/L [30]. The Onsager phase
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FIGURE 7.5 Onsager phase diagram for hard rods,
300
showing the predicted relative stability of the isotropic and nematic phase, respectively.
Axial ratio L/D
250 200 Nematic 150 100 50 Isotropic fluid
0 0
0,05
0,1
0,15
0,2
f
diagram is shown in Figure 7.5. The orientational order in the nematic phase reduces orientational entropy, but it also significantly reduces the excluded volume interactions, and, similar to the hard sphere crystallization, it is the driving force for the phase transition. Charged rods have also been analyzed [31,32]. The long-range electrostatic interactions generally shift the transition to lower concentration although charged rods prefer to be perpendicular, not parallel.
3.4 Plates Also, platelike, e.g., clay particles undergo an isotropic-to-nematic phase transition [33]. The reason is essentially the same as for the rods. Alignment increases the free volume. Apart from the nematic phase, also other phases can be observed, such as columnar and cubatic [28,34].
4 SURFACTANTS Surfactants are molecules that typically are composed of a hydrophobic, water-insoluble alkyl chain and a hydrophilic “headgroup,” being a salt that can dissociate in water (ionic surfactant), or sugar group or a short block of a water-soluble polymer-like polyethylene oxide (nonionic surfactant) [5]. In the latter case, the surfactant can in fact be seen as a short block copolymer. Being composed of a strongly hydrophobic part and strongly hydrophilic part, these molecules are often referred to as being amphiphilic and are also called amphiphiles.
4.1 Binary Surfactant-Water Systems Because of the hydrophobic alkyl chain, surfactant molecules have a low solubility in water. However, instead of simply precipitating the surfactant at higher concentrations, the
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surfactant molecules self-assemble into thermodynamically stable clusters, called micelles, that are characterized by an equilibrium size distribution [35]. In addition, they are surface active and readily adsorb onto and cover hydrophobic surfaces in water, including the water-air interface. Their surface activity and their particular ability to self-assemble (which are related) are responsible for their usefulness in many practical applications, such as detergency and as stabilizers of hydrophobic colloids in water, e.g., hydrophobic pigments in water-based paints. Micelles form in dilute solutions when the surfactant concentration exceeds a critical value, the critical micellization concentration (cmc). For typical surfactants, micelle formation can be highly cooperative [35], and the transition, although not being a phase transition, is related to the “microphase separation” discussed earlier for block copolymer melts. Increasing the concentration above the cmc results in an increased concentration of micelles, while the monomer concentration remains constant or decreases with increasing concentration. The micelles are association (or self-assembly) colloids that interact and form ordered structures, colloidal liquid crystals at higher concentrations. However, there is an important difference. Because the micelles are fluid objects, they can change their shape and size to minimize the free energy. This additional degree of freedom gives rise to richer phase diagrams. Spherical or near spherical micelles crystallize at higher concentrations into a cubic lattice. For nonionic micelles, this occurs roughly at the hard sphere crystallization concentration. For ionic surfactants, the concentration is lower. As a response to the increased interactions with increasing concentration, the micelles first transform into cylinders and then into planar bilayers. This results in a sequence of lyotropic liquid crystalline phases. Cylindrical micelles form a hexagonal phase, ordered in two dimensions, while planar bilayers form a stack with onedimensional order. In between the hexagonal and the lamellar phases, one often also observes a bicontinuous cubic phase, the structure of which that can be described either as a consisting of two branched cylinder networks or, alternatively, in terms of a multiply connected bilayer. As an example, we present in Figure 7.6 the phase diagram of the binary system waterdodecyltrimethylammonium chloride [36]. With ionic surfactants, a quantitative analysis requires an accurate description of the electrostatic interactions. Jo¨nsson and Wennerstro¨m undertook a detailed analysis of ionic surfactant-water phase diagrams [37], where the relative stability of the different aggregate geometries, spheres, cylinders, and planes was assessed. For the electrostatic contribution to the free energy, they handled the different geometries in a mean field way by solving the Poisson-Boltzmann equation within a cell model. In addition, they considered contributions from hydrophobic interactions and entropy of mixing and were able to obtain accurate descriptions of ionic surfactant-water systems.
4.2 Ternary Surfactant-Water-Alcohol Systems The sequence of phases is the same as observed for block copolymer melts. However, the whole sequence from “normal” globular micelles to reverse micelles with a hydrophilic, water-containing core is not accessible in a binary surfactant-water systems. Upon the addition of an organic solvent, immiscible with water, additional phases can form. Phase diagrams of ternary systems were studied extensively by Ekwall and his coworkers Krister Fontell and Leo Mandell, during the 1960s and 1970s. As an example, the ternary phase
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FIGURE 7.6 Phase diagram of the binary water-dodecyltrimethylammonium chloride system. Here, F denotes an isotropic liquid phase. C0 is a cubic phase with discrete micelles distributed on a cubic lattice. M denotes the hexagonal phase. C is a bicontinuous cubic phase (space group Ia3d), and N is a lamellar phase. Reprinted with permission from Ref. [36]. Copyright 1969.
diagram of the ionic surfactant potassium decanoate (potassium caprate), water, and octanol [38] is shown in Figure 7.7a. Here, L1 is a water-rich liquid micellar phase and L2 is an alcohol-rich liquid phase containing reverse micelles. E and D are hexagonal and lamellar liquid crystalline phases. K is a liquid crystalline phase that later was shown to be a reverse hexagonal phase, F in Ekwall’s notation [40]. By adding the alcohol to the binary water-surfactant system, the aggregate structure becomes gradually inverted from normal micelles in water, via the planar bilayers in the lamellar phase, to reverse micelles in the alcohol. An additional complexity in the present system is that the alcohol, which is also slightly amphiphilic, acts simultaneously as a solvent and as a cosurfactant, participating in the surfactant aggregates.
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FIGURE 7.7
(a) Experimental phase diagram of the ternary system water-potassium octanoate (caprate)-octanol. L1 is a water-rich isotropic liquid phase, E is a hexagonal phase, D is a lamellar phase, K is a reverse hexagonal phase, and L2 is an alcohol-rich isotropic liquid phase. The surfactant forms normal micelles in the L1 phase and reverse micelles, with solubilized water in the interior, in the L2 phase. (b) Theoretical phase diagram of the same waterpotassium octanoate (caprate)-octanol system. Same notation as in (a) except that the reverse hexagonal phase is here denoted F. Panel a: Reprinted from Ref. [38]. Copyright 1969, with permission from Elsevier. Panel b: Reprinted with permission from Ref. [39]. Copyright 1987 American Chemical Society.
Jo¨nsson and Wennerstro¨m extended their model to allow for a third component and applied it to the potassium decanoate-water-octanol system [39]. Their calculated phase diagram is shown in Figure 7.7b. As can be seen, there is a good agreement between the calculated and the experimental phase diagrams.
4.3 Ternary Systems with Amphiphilic Block Copolymers The ternary Pluronic-water-oil systems essentially bridge the gap between the surfactant systems and the block copolymer melts. Here, Pluronics are widely used triblock copolymers composed of a hydrophobic poly(propylene oxide) (PPO) middle block connected to two hydrophilic poly(ethylene oxide) end blocks. Together with selective solvents for the two blocks, e.g., p-xylene and water, a wide range of liquid crystalline phases, including liquid micellar phases, are observed in the ternary phase diagrams. In the system water-EO19PO44EO19-pxylene (Figure 7.8), the full sequence with seven liquid crystalline phases and two isotropic liquid phases was observed [41]. Svensson analyzed ternary phase diagrams with different Pluronic copolymers [42]. When plotting the phase diagrams as a function of the total volume fraction of the hydrophobic components, oil and the PPO block, the different phase transitions occur at approximately the same volume fractions as in the case of block copolymer melts.
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FIGURE 7.8 Experimental phase diagram of the ternary water- EO19PO44EO19-p-xylene system at 25 C. A total of nine different phases are observed. L1 and L2 are isotropic liquid phases with normal and reverse micelles, respectively. I1 and I2 are cubic phases containing normal and reverse micelles, respectively. H1 and H2 are hexagonal phases containing normal and reverse cylindrical micelles, respectively. V1 and V2 are bicontinuous cubic phases. V1 has a reverse water-swollen bilayer separating two oil domains. V2 has a normal oil-swollen bilayer separating two water domains. La is a lamellar phase. The different structures are illustrated. Reprinted with permission from Ref. [41]. Copyright 1998 American Chemical Society.
4.4 Microemulsions: Phases of Fluid Surfaces Water and oil do not mix but can be mixed by adding a third component that is miscible with both water and oil. Figure 7.9 shows the phase diagram of the water-benzene-ethanol system [43], a system studied already by Bancroft and his student Taylor at the end of the nineteenth century [44]. With increasing concentration of ethanol, which is miscible with both water and benzene, there are less and less unfavorable water-benzene contacts and eventually a homogeneous phase is formed containing equal amounts of water and benzene. However, the amounts of alcohol required to mix water and oil are high, and if more hydrophobic alkanes are used instead of benzene, it is even higher.
170
7. COLLOID PHASE BEHAVIOR
FIGURE 7.9 Phase diagram of the ternary water-ethanol-benzene system at 25 C. L is a homogeneous liquid phase that at lower ethanol concentrations separates into two coexisting phases, L0 and L00 , respectively. Drawn using data from Ref. [43].
Ethanol
L 0.5
0.5
L′ + L″
Water
0.5
Benzene
In the homogeneous phase water, benzene and ethanol molecules are almost randomly distributed, and the phase diagram can be understood within regular solution theory. With a proper surfactant instead of alcohol, the concentration needed to mix equal volumes of water and oil can be reduced by more than an order of magnitude. In this case, the distribution of molecules is far from random. While the mixture is homogeneous on the macroscopic scale, it is structured into water and oil domains on the colloidal length scale. Water and/or oil is dispersed into domains, with a dense self-assembled surfactant monolayer, with an area density of A fs ¼ V ls
ð7:6Þ
stabilizing the oil-water interface. Here, fs is the surfactant volume fraction and ls vs/as is the surfactant length that is defined as the surfactant molecular volume, vs, divided by the average area, as, that the surfactant molecule occupies at the water-oil interface. The homogeneous mixture is called microemulsion and its microstructure and phase behavior depends strongly on the spontaneous curvature, H0, of the surfactant film [45]. Microemulsions and related phases made up of flexible surfactant films can be understood in terms Helfrich curvature free energy. To leading order, the curvature free energy density is often written as [46] K gc ¼ 2kðH H0 Þ2 þ k
ð7:7Þ
Here, H ¼ (c1 þ c2)/2 and K ¼ c1c2 are the mean and the Gaussian curvature, with c1 and c2 be is the saddle ing the two principal curvatures. k > 0 is the bending rigidity of the film and k splay modulus that reports on the preferred topology of the film that can take either positive < 0, a spherically bent film is preferred (c1 and c2 having equal sign) or negative values. If k > 0, a locally favoring the formation of closed surfaces and droplet microemulsions. If k saddle-shaped surface is preferred favoring the formation of bicontinuous structures. k is typ is typically ically a few times kBT, low enough to be flexible but high enough so that H H0. k of the same magnitude as k, but negative.
4 SURFACTANTS
171
The total curvature free energy, Gc, for a given surface configuration is then formally obtained by integrating gc over the total surface, S: ðð dA gc ð7:8Þ Gc ¼ S
As mentioned earlier, the microemulsion structure and phase behavior depends strongly on H0. Counting curvature towards oil as positive, one finds oil-in-water microemulsions when H0 >> 0, and water-in-oil microemulsions when H0 0. For spherical droplets of radius, Gc has a minimum for R ¼ R0 where k R0 ¼ H01 1 þ ð7:9Þ 2k In both cases, the microemulsion phase is found at compositions where droplet radii R R0 (with a small correction due to the entropy of mixing, that is neglected here). For oil droplets in water, we have R¼
3f0 ls fs
ð7:10Þ
where f0 is the oil volume fraction. When R ¼ R0, the droplets are saturated with oil. Adding more oil results in a phase separation with an excess oil phase. Same things hold for negative spontaneous curvatures and water droplets in oil, as expected from the symmetry of Gc. For H0 0, the ternary water-surfactant-oil phase diagram is dominated by a lamellar phase, because with planar layers, H 0 irrespective of the composition. However when fw f0, H 0 can also be satisfied by a bicontinuous structure, which typically is found at lower surfactant concentrations. Particularly useful experimental model systems are those with nonionic surfactants oligo (ethylene glycol) alkyl ether, CmEn, where m is the number of carbons in the alkyl chain and n is the number of EO units in the polar part. The reason is the strong temperature dependence of the interactions between the water and the polar En chain. Water is a good solvent at lower temperatures and swells the En chains that are highly solvated. This results in H0 0. At higher temperatures, on the other hand, water-En interactions are less favorable and the En chains become less solvated, leading to H0 0. Thus, with these surfactants, the spontaneous curvature can be conveniently tuned by varying the temperature. A Taylor expansion around T ¼ T0, where T0 is the temperature, is the so-called balance temperature (equivalent to the phase inversion temperature, PIT, identified for emulsions stabilized by CmEn surfactants) where H0 ¼ 0 yields H0 ðT Þ ¼ aðT0 T Þ þ
3
1
1
ð7:11Þ
˚ K [45,47]. with a 10 A Adding temperature as the third dimension, the phase behavior a ternary surfactantwater-oil system can be represented by a phase prism. Different cuts through the phase prism can be chosen to illustrate the microemulsion phase behavior in a two-dimensional diagram. One particularly interesting cut is the Kahlweit’s “fish cut,” defined by fw ¼ f0 (Figure 7.10, left). The phase behavior typically observed in the fs T plane is schematically illustrated in Figure 7.10 (right). The microemulsion phase, L, coexists with an excess oil phase, O, at lower
172
7. COLLOID PHASE BEHAVIOR
T
S
T0
W
O
W+L
L
W+L+O
T
Lα
L+O
f S*
fS
FIGURE 7.10
Left: Illustrating Kahlweit’s “fish cut” through the ternary phase prism, defined by equal amounts of water and oil. Right: Schematic phase behavior around the balance temperature, T0. L is the microemulsion phase. W and O are essentially pure water and oil phases, respectively. La is a lamellar liquid crystalline phase. fs is the surfactant volume fraction in the balanced microemulsion that coexists with excess water and oil at T ¼ T0. The lower the fs , the more efficient is the surfactant in solubilizing water and oil.
temperatures and with an excess water phase, W, at higher temperatures. At higher surfactant concentrations, the low-temperature and high-temperature microemulsion domains are separated by a lamellar liquid crystalline phase, La. At T ¼ T0, the lamellar phase is stable at higher fs. Upon decreasing fs, there is a phase transition to a microemulsion phase that here has a bicontinuous structure with H0 0. This phase can swell with water and oil down to a surfactant volume fraction, fs , corresponding to a separation limit for solubilizing equal volumes of water and oil. For fs < fs , not all water and oil can be solubilized and there is a separation into three phases, W þ L þ O. Another informative cut is the Shinoda cut, defined by a constant fs (Figure 7.11, left). Choosing fs to be between fs and the concentration where a lamellar phase is formed results in a phase diagram as the one shown in Figure 7.11 (right), corresponding to the plane f0/(fw þ f0) T. This is an experimentally determined phase diagram from the waterC12E5-tetradecane system at a constant surfactant weight fraction of 0.166 [48]. The oil fraction is also given as weight fraction. Here, one observes a microemulsion channel stretching from the water side at lower temperatures (H0 > 0) to the oil side at higher temperatures (H0 < 0). Interestingly, there is a second, more narrow channel having the opposite orientation and that crosses the main microemulsion channel around T ¼ T0. These more narrow channels correspond to the so-called L3 or “sponge phase” [49]. In contrast to the microemulsion, where a surfactant monolayer separates an oil domain from a water domain, the “sponge phase” has a bilayer structure where the bilayer separates two solvent domains of the same kind. At higher temperatures and higher water content, a normal, oil-swollen bilayer separates two water domains. At lower temperatures and higher oil content, a reverse water-swollen bilayer separates two oil domains. The bilayer midplane separates equal volumes on the two sides and here H 0. The surfactant film, on the other hand, has unequal volumes on its two sides and H 6¼ 0. In the water-rich “sponge phase” at higher temperatures, the surfactant film has H < 0, and the opposite holds for the oil-rich “sponge phase.” The third phase observed in the
173
4 SURFACTANTS
70
60
T / °C
S
W
O
T
W+L
50 Lα
L
Lα
40 L+O 30 0
Wt. fraction oil/(water + oil)
1
FIGURE 7.11 Left: Illustrating the “Shinoda cut” through the ternary phase prism, defined by a constant surfactant concentration. Right: Experimental phase diagram of the C12E5-water-tetradecane system in the plane defined by a constant surfactant concentration, 16.6 wt%. T0 for this system is 48 C. L denotes an isotropic liquid phase (microemulsion) and La is a lamellar liquid crystalline phase. Drawn using data from Ref. [48].
Shinoda cut is the lamellar phase. The phase diagram in Figure 7.11 (right) has a particular symmetry. Rotating 180 around the point f0/(fw þ f0) ¼ 0.5, T ¼ T0 returns the same phase diagram. This reflects the fact that the free energy is invariant upon exchanging water and oil domains (changing f0 to (1 f0)) and simultaneously reversing the sign of H0. Another particularly interesting cut is the one corresponding to H0 ¼ 0 (T ¼ T0). As pointed out earlier, the lamellar phase is then stable over a large range of compositions and, in addition, there is a microemulsion phase at fw f0 and lower surfactant concentrations and a three-phase triangle. In Figure 7.12a, we present an experimental phase diagram determined at T ¼ T0 of the water-C10E5-octane system [50]. Clearly, it is difficult to derive a free energy for a bicontinuous microemulsion with a flexible surfactant film. However, when H0 ¼ 0, the curvature free energy becomes length-scaleinvariant and it is possible to use a scaling approach. For a balanced microemulsion (fw ¼ f0), neglecting the surfactant film thickness, there is only a single length scale in the problem, the inverse interfacial area per unit volume (A/V)1 ¼ ls/fs. From the length-scale invariance of Gc, it then follows that G ¼ a3 f3s V
ð7:12Þ
where G here is the total free energy and a3 is a materials constant. However, the same scaling argument holds for the lamellar phase, Equation (7.12), and hence, the harmonic approximation of the curvature free energy, Equation (7.7), is not sufficient to explain the observed phase behavior, including the three-phase equilibrium at low fs. For the lamellar phase, Helfrich has derived a value of a3 that is positive and represents the repulsive undulation force
174
7. COLLOID PHASE BEHAVIOR
S
C10E5 90
90
T = 44.6 °C
50
1
Lα
50 lamellar
2 2
1
10
3
H2O 10
(a)
2
2 3
50
3
10 90 n-octane
W
O
(b)
FIGURE 7.12 (a) Experimental phase diagram of the ternary C10E5-water-octane system at the balance temperature T0 ¼ 44.3 C. Concentrations are given as weight fractions. A lamellar phase, La, is stable at higher surfactant concentrations, over a wide range of water-oil ratios. 1, 2, and 3 denote single-phase, two-phase, three-phase regions, respectively. The microemulsion phase is stable for approximately equal amounts of water and oil. This phase coexists with excess water and oil at lower surfactant concentrations and with the lamellar phase at higher surfactant concentrations. (b) Theoretical phase diagram of a surfactant (S)-water (W)-oil (O) system for H0 ¼ 0, corresponding to T ¼ T0. The phase diagram was calculated based on parameters obtained from the C12E5-water-decane system and assuming a surfactant monolayer bending rigidity of 1 kBT. Panel a: Used with permission from Ref. [50]. Copyright 1993 The Deutsche Bunsen-Gesellschaft. Panel b: Reprinted with permission from Ref. [51]. Copyright 1995 American Chemical Society.
[52]. For the microemulsion phase, on the other hand, the finite swelling with phase separation at lower fs is consistent with a3 < 0. Expanding Equation (7.7) beyond the harmonic approximation, the next terms are quadratic in the curvatures, H4, H2K, and K2 (odd powers cancel by symmetry). Taking these into account, a first-order correction to Equation (7.12) becomes [53] G ¼ a3 f3s þ a5 f5s V
ð7:13Þ
The observed phase behavior implies a3 < 0 and a5 > 0 for the microemulsion and a3 > 0 and for the lamellar phase. a5 > 0 can be understood as a penalty for high curvatures. Modeling also a penalty for deviating from fw ¼ f0 [51] was able to calculate a phase diagram, presented in Figure 7.12b, that is in good agreement with experiments. Using the same framework, we can also understand the sponge phase, including its finite swelling and its coexistence with the lamellar phase [54]. Moreover, microemulsions can be used as model systems, analyzed in terms of a curvature free energy, to study a range of fundamental liquid-state phenomena [55]. Finally, we note that going from the flexible one-dimensional objects (polymers) to the flexible two-dimensional objects (surfactant film) corresponds to a significant increase in complexity.
4 SURFACTANTS
175
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