J. Phys. Chem. 1996, 100, 5035-5038
5035
Flexibility of the Membranes in a Doped Swollen Lamellar Phase Virginie Ponsinet* and Pascale Fabre Laboratoire de Physique de la matie` re condense´ e, URA-CNRS 792, Colle` ge de France, 11 place M. Berthelot, 75005 Paris, France ReceiVed: September 22, 1995; In Final Form: December 18, 1995X
We have measured the flexibility κ of the membranes of a quaternary lamellar phase of periodicity d, without and with particles included between the layers. We show that an increase in cosurfactant amount in the membranes leads to a decrease of the flexibility constant κ, for the two studied systems. On the other hand, the membrane flexibility κ is shown to be hardly different in nondoped and doped (ferrosmectic) systems. This implies that the enhanced value of the elastic bending modulus K, observed when the lamellar phase is doped with particles, cannot be attributed to a simple hardening of the membranes. This leads us to question the well-known relationship between the elasticity of a lamellar phase and the elasticity of its constituting layers: K ) κ/d, in the case of ferrosmectic phases.
Introduction Amphiphilic molecules in solution self-organize into assemblies of various shapes, forming what are called lyotropic systems. In systems of bilayers, or membranes, the macroscopic properties of the phases are related to the flexibility of the membranes, considered as continuous objects, and to their mutual interactions. The case of lamellar phases is of particular interest because the membranes are globally flat and stacked with a smectic order. The study of the properties of the smectic phase allows access to intrinsic features of the fluid membranes and, through appropriate modeling, makes possible the determination of the intermembrane interactions.1 A previous study2 had established that both the intermembrane repulsive interactions responsible for the stability of the lamellar phase and the bending elastic constant are considerably enhanced by the presence of intercalated solid nanoparticles. It was, at this point, fruitful to evaluate to what extent the strong action of the host particles is linked to a modification of the structural and elastic features of the membranes themselves. For this purpose and as is presented here, we studied the microscopic features of the system, e.g., the flexibility of the membranes, and relate them to the macroscopic properties of the phase, in the cases of the lamellar phase of an oil-swollen quaternary system, and of the corresponding lamellar phase doped with solid nanoparticles, known as the ferrosmectic phase.3 Theoretical Background Lamellar phases are lyotropic smectics obtained in numerous surfactant systems. They can be considered as two-component smectics if, forgetting the molecular details, the membranes are described as continuous quasi-two-dimensional objects. The free energy of such an isolated membrane of surface S is a free energy of bending elasticity, defined by
Fm ) ∫∫s2κc ds
When κ is of the order of the thermal energy kT, the membranes are flexible enough to exhibit large thermal undulations. Note that we neglect here the elastic energy of Gaussian curvature, as it is constant for membranes of nonvarying topology. On the other hand, the stacking of membranes of periodicity d is described by its density of elastic free energy including two terms:
(
X
Abstract published in AdVance ACS Abstracts, February 15, 1996.
0022-3654/96/20100-5035$12.00/0
( )
2
The first term is the energy of curvature, related to the local radii R1 and R2 of curvature of the stacking and characterized by the bending elastic constant K. The second term is the energy of compression, related to the departure of the layer positions from their equilibrium periodicity and to the elastic constant of compression, at constant chemical potential, B h . The measurements of the structural features of the lamellar phases and of their elastic constants K and B h are accessible by scattering techniques and by smectic defect observation. In the study presented here, we measured the elastic modulus κ and examined how the macroscopic elastic constants K and B h are related to the microscopic parameters of the smectic stacking. Both require us first to consider the repulsive stabilizing potential between membranes, which is different in the two types of lamellar phases we have studied. Entropically Stabilized Swollen Lamellar Phases. In the case of flexible membranes, the presence of neighboring sheets at a distance d restricts the thermal undulations of each particular membrane. This restriction induces a repulsive interaction potential which balances the van der Waals long-range attraction between membranes, thus ensuring the stability of the lamellar phase. The resulting intermembrane potential was calculated by Helfrich,4 with δ being the membrane thickness:
V(d) )
2
where c is the local mean curvature and κ is the mean bending elastic modulus of the membrane and is homogeneous to energy.
)
1 1 1 2 1 ∆d F) K + + B h 2 R1 R2 2 d
2 3π2 (kT) 128 κ(d - δ)2
(E1)
It is possible to deduce from (E1) the expression of the elastic constant of compression:5 2
B h)
d 9π (kT) 64 κ (d - δ)4
© 1996 American Chemical Society
5036 J. Phys. Chem., Vol. 100, No. 12, 1996
Ponsinet and Fabre
The elastic constant of curvature is also a function of the smectic periodicity d of the stacking and the elastic modulus of a single membrane κ:4
K)
κ d
The determination of the elastic modulus κ can be made independently. Keeping the membranes identical and changing the amount of solvent between them, one follows what is called a dilution line. On this line, one defines the swelling law as the relation between the periodicity d and the swelling ratio Φs, with
Φs )
solvent volume membrane volume
For flat membranes, the swelling law is
d(Φs) ) δ[1 + Φs]
(E2)
For flexible membranes, Helfrich calculated the magnitude of the thermal undulations,6 from which the swelling law of entropically stabilized lamellar phases has been determined.7,8 The relation now contains a logarithmic correction taking into account the area that the undulations of the membranes consume. To first order, one writes
[
d(Φs) ) δ[1 + Φs] 1 +
(x
kT ln γ 4πκ
κ δ (1 + Φs) kT a
)]
(E3)
where kT is the thermal energy, a is a typical molecular size (a ) 0.5 nm), and γ is a constant.9 Exploiting the small-angle scattering data, we determine on one hand the smectic periodicity d(Φs) from the Bragg peak position, along a dilution line, and on the other hand the thickness of the membrane δ from the line-shape analysis at large wave vector q. Fitting this result to (E3), one obtains the value of the membrane flexibility κ. Ferrosmectic or Doped Lamellar Phases. In ferrosmectic phases, our previous studies2 had shown that the intermembrane potential responsible for the stability of the lamellar phase is modified by the presence of solid particles. In order to interpret the experimental results, a phenomenological model was built,10 which assumed that there is a region forbidden to the membrane undulations. This volume was characterized by a length L(φ), related to the volume fraction φ in particles in the phase. With this assumption, we could calculate the new expression for the interaction potential
V(d,φ) )
(kT)2 3π2 128 κ(d - [δ + L(φ)])2
(E4)
along with the expression of the elastic constant of compression. This description accounts for our whole experimental results and leads to the quantitative determination of L(φ) as a function of the particle concentration. However, expression E4 reveals that the potential V itself depends on κ. L(φ) has thus been calculated with the assumption that κ is not a function of the particle volume fraction in the phase. The self-consistency of this procedure is checked in the following. As described in ref 10, it is possible to show that the length L(φ) increases between zero and the order of magnitude of the smectic periodicity, on the experimental range of particle volume fraction. The modification of the membrane undulations also implies that the swelling law differs from expression E2. The correction
Figure 1. Limits of the lamellar phase area in the oil-rich corner of the phase diagram of the quaternary system water-SDS-pentanolcyclohexane, with the weight ratio water/SDS at the value 2.5. Positions of the studied samples on three dilution lines. The first line corresponds to the minimum amount of alcohol in the phase. Starting from this first dilution line, the two other lines were obtained by keeping constant the ratio between the alcohol concentrations in the membrane and in the swelling solvent.
of the swelling law according to our assumptions leads to
d(Φs,φ) )
(x
[
kT ln γ 4πκ
δ[1 + Φs] 1 +
)]
κ δ(1 + Φs) - L(φ) kT a
(E5)
The determination of the membrane flexibility κ in the ferrosmectic phase relies as before on the measurement of d and δ, the swelling laws along dilution lines being now fitted to eq E5. Experimental Results Nondoped Lamellar Phase. Studies on the ternary system water + sodium dodecyl sulfate (SDS) + alcohol11 have shown that the more alcohol the membrane contains, the more flexible it is (low value of κ). It seems that the same tendency exists in quaternary systems, where the membrane includes a layer of second solvent.12,13 The lyotropic system studied here is a quaternary system, water + SDS + pentanol + cyclohexane, whose phase diagram has been explored before.10,12,14 In this work, we keep the weight ratio water/SDS at the value 2.5. The lamellar phase is composed of membranes separated by cyclohexane, in which a fraction of the pentanol is diluted. The membranes themselves are constituted of water surrounded by mixed bilayers of SDS and pentanol. The lamellar phase extends widely in the phase diagram, in terms of cyclohexane and pentanol concentration. We have studied three dilution lines, corresponding to different compositions of the membrane, i.e., different SDS/pentanol ratios. They are represented in Figure 1. The experiments were performed with powder samples, on a small-angle X-ray scattering pinhole apparatus, described in ref 15. Membrane thicknesses were measured for all samples by analyzing the form factor at large angles (wave vectors q such that 0.5 nm-1 < q < 1.5 nm-1). Due to the small extension of the surfactant heads and the close values of electronic density for the surfactant tails and cyclohexane, the scattering density profile was approximated by a single square function. We consider a membrane with thickness fluctuations, of thermal origin, described as a Gaussian distribution centered on δo and of width σ. The Porod’s law for the intensity scattered is then16,17
q4I(q) ∝ ∫0
( ) [(
)]
δ - δo 1 δ sin2 q exp 2 σ σxπ
∞
2
dδ (E6)
Membrane Flexibility in Lamellar Phases
J. Phys. Chem., Vol. 100, No. 12, 1996 5037 TABLE 2: Membrane Features for the Dilution Lines of the Studied Ferrosmectic Samples According to the Volume Fraction of Particles in the Ferrofluid Layers of the Phase φ, % 0.8 1.1 2 2.6 3.2 3.6 Np/Ns 1.9 1.9 2.8 2.9 2.7 2.7 δ, nm 4.7 ( 0.2 4.5 ( 0.2 4.3 ( 0.2 4.2 ( 0.2 4.2 ( 0.2 4.2 ( 0.2
Figure 2. Porod plot: q4I(q) as a function of q, for the nondoped lamellar phase Np/Ns ) 1.3, Φs ) 2.8. The X-ray scattering experiments were performed with powder samples. The line is the best agreement with (E6), giving the values σ ) 0.9 nm, δo ) 4.2 nm. The length of the surfactant tails must be added to this value in order to get the membrane thickness as given in Table 1.
TABLE 1: Membrane Features for the Three Studied Dilution Lines in the Nondoped Systema line
Np/Ns
δ, nm
κ/kT
1 2 3
1.3 2.0 2.7
4.7 ( 0.2 4.4 ( 0.2 3.9 ( 0.2
3.2 ( 1 2.3 ( 1 1.0 ( 1
a Np/Ns is the relative amount of molecules of pentanol and SDS in the membrane. δ is the thickness of the membrane and κ its flexibility, expressed in units of kT, the thermal energy.
Figure 3. Swelling law for swollen lamellar phases along the dilution line corresponding to Np/Ns ) 1.3, δ ) 4.7 nm. (a) In case of flat membranes, simple geometrical swelling (E2). (b) Experimental results obtained from the position of the quasi-Bragg peaks in small-angle scattering spectra. The fit is done with the swelling law for entropically stabilized swollen lamellar phases (E3), and gives the value κ ) 3.2 kT.
Measurements, made by least-squares fitting of expression E6 to the experimental spectra (cf. Figure 2), lead to the conclusion that the membrane thickness is constant on each line, as expected for dilution lines. Obtained values of σ are of the order of 1 nm. Thicknesses are summarized in Table 1: note that the evolution of the membrane thickness is easily linked to the respective areas per polar head of the two amphiphilic molecules.15 The smectic periodicities, deduced from the position of the Bragg singularities, are plotted as a function of the swelling ratios (cf. Figure 3). The data are analyzed by fitting expression E3 with a least-squares method. The good agreement between the experimental results and the theoretical expression confirms that the phases are stabilized by entropic repulsion (as described by (E1)). Extracted membrane elastic moduli are given in Table 1. Our results show that the elastic modulus of a membrane in the lamellar phase of the quaternary system water + SDS + pentanol + cyclohexane decreases from κ ) 4 to 1 kT as the
lamellar sector of the phase diagram is spanned by increasing the alcohol-to-surfactant molecular ratio Np/Ns in the membrane. This confirms the part played by the alcohol contained in the membrane in tuning its flexibility. Elastic modulus values are in good agreement with recent measurements in similar systems.18 Note however that our results differ from earlier results,13 giving lower values and an evolution of κ with the alcohol amount exhibiting a saturation effect when Np/Ns becomes higher than 1.6. Ferrosmectic Phase. As described previously,3,19 we obtain ferrosmectic phases when replacing, in the quaternary system, the oil solvent by a colloidal suspension of iron oxide nanometric particles in cyclohexane, known as a ferrofluid.20,21 The system can now be described as a smectic stacking of membranes in a ferrofluid solvent. Because, when using X-rays, the high contrast particles gives a signal drowning the lamellar signal, we used neutron scattering methods in order to probe the ferrosmectic phases. On the contrary, by using a contrast variation technique in neutron scattering, it is possible to completely match the particles and obtain the signal scattered by the membranes only. This is achieved by using partially deuterated cyclohexane, as described in ref 2. The small-angle scattering experiments have been performed on a neutron beam (PAXY) in the Laboratoire Le´on Brillouin, Saclay, France. The ferrosmectic samples studied were dilution line series, for different particle volume fractions. For each volume fraction, we chose the dilution line corresponding to the minimal alcohol ratio experimentally available in the phase. Because the phase diagram is slightly shifted as the amount of particles is increased, this means that the membrane composition Np/Ns is also different on each dilution line. When going from one dilution line to another, two parameters are thus varied simultaneously: the particle volume fraction and the membrane composition. From scattering experiments, as previously, we determined the membrane thickness for each dilution line. These values are reported in Table 2. The scattering spectra also give a determination of the smectic periodicities d. Values of d are plotted as a function of the swelling ratio of the phases. We then use the value of L(φ), determined independently,10 to adjust expression E5 to the experimental swelling laws of ferrosmectics. These fits, such as the one displayed in Figure 4, allow the determination of the flexibility κ. The values of κ as a function of the membrane composition for nondoped and doped phases are reported on Figure 5. Note that the rather large uncertainty on κ is usual and can be attributed to the weakness of the logarithmic deviation of the swelling law. One can make two observations based on these data: (i) the values of κ for ferrosmectics phases and nondoped phases are similar, and (ii) the variation of κ with the membrane composition is also analogous in both phases. This validates the selfconsistency of our analysis. We can thus conclude that the membrane flexibility κ is hardly not a function of the volume fraction of particles in the phase but varies mostly according to the membrane composition. At this stage, it is important to recall that, on the contrary, the bending elastic constant K is largely enhanced in doped phases and significantly varies with the particle concentration: for instance, K is higher by a factor 10 in a phase doped with
5038 J. Phys. Chem., Vol. 100, No. 12, 1996
Figure 4. Swelling law for ferrosmectic phases of particles volume fraction φ ) 2.6% along the dilution line corresponding to Np/Ns ) 2.9, δ ) 4.2 nm. (a) In case of flat membranes, simple geometrical swelling (E2). (b) Experimental results obtained from the position of the quasi-Bragg peaks in small-angle scattering spectra. The fit with the swelling law for entropically stabilized doped swollen lamellar phases (E5) with L ) 2.7 nm gives the value κ ) 0.4 kT. (c) Swelling law for entropically stabilized nondoped swollen lamellar phases (E3), with the same value κ ) 0.4 kT.
Ponsinet and Fabre alcohol in the membrane leads to a strong diminution of the value of κ, i.e., to a more flexible membrane. This effect of the cosurfactant was already pointed out in ternary systems. It is now evidenced in the studied quaternary system, where the membrane is indeed an inverse bilayer, and also in the ferrosmectic phases, an analogous system in which solid particles are incorporated between the membranes. Another important result is that the value of the membrane flexibility is not significantly modified by the presence of host particles. A striking consequence of this result is that the increase of the bulk bending elastic constant in ferrosmectics cannot be interpreted by an increase of the membrane rigidity itself. This implies a distinctive contribution of the particles to the macroscopic curvature elasticity of the doped phases. This additional energy of curvaturesnot fully understood yet in terms of an adequate microscopic mechanismshas to be emphasized, as well as the specific interaction potential proposed to account for the stability of the ferrosmectics. Both open new perspectives for the description of the stability of other hybrid systems, biologic or synthetic, in which fluid membranes and solid or polymeric objects are put together. Acknowledgment. We express warm thanks to Loı¨c Auvray and Raymond Ober for their cooperation in conducting the small-angle scattering experiments and to Madeleine Veyssie´ for numerous and fruitful discussions. References and Notes
Figure 5. Evolution of the flexibility κ with the membrane composition Np/Ns for the nondoped ([) and doped (9) lamellar systems.
2% volume fraction in particles than in the corresponding undoped phase.10 This strong increase of the bulk bending elasticity, whereas the membrane individual bending constant is unchanged, leads us to question, in the case of doped lamellar phases, the validity of the classical lamellar phase relationship4 between the macroscopic constant K and the flexibility κ: K ) κ/d. The departure from this classical law seems to suggest that, in such doped lamellar phases, the curvature energy of n smectic layers is not a simple addition of the energies of the n membranes but includes a supplementary rigidity due to the presence of the particles. Conclusion An experimental study of the membrane flexibility κ by X-ray and neutron scattering was performed in classical oil-swollen lamellar phases and in lamellar phases doped with solid particles. In both systems, it was shown that an increasing amount of
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