Langmuir 2001, 17, 5045-5058
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Smectic Phase of Fluid Membranes Decorated by Amphiphilic Copolymers Francisco Castro-Roman, Gre´goire Porte, and Christian Ligoure* Groupe de Dynamique des Phases Condense´ es, UMR CNRS/UM2 n° 5581, CC 26, Universite´ Montpellier II, F-34095 Montpellier Cedex 05, France Received January 29, 2001. In Final Form: May 3, 2001
We study the effect of amphiphilic block copolymers (poly(oxyethylene)-poly(oxypropylene)-poly(oxyethylene)) on the phase behavior and elastic properties of a lyotropic lamellar phase (made from a mixture of cetylpiridinium chloride/octanol/water) into which they are incorporated. The hydrophobic blocks adsorb onto the bilayers, whereas the hydrophilic tails remain in the aqueous solvent and decorate them. Small-angle (neutron and X-ray) experiments were carried out to characterize the structural and physical properties of the mixed system. We observe that the host phase, that is, the lamellar phase, remains preserved over large ranges of polymer and surfactant concentrations. The effects of the polymer layers are 2-fold. On one hand, they modify the elastic properties of the individual membranes: (i) they soften the bilayer’s mean curvature bending modulus and (ii) they stretch and therefore thin down the membranes; this thinning is due to a balance between the stretching of the polymer brush in the direction normal to the layers and the in-plane stretching of the bilayers. On the other hand, the presence of polymer layers grafted on the membranes strongly enhances the repulsive Helfrich interaction between the fluid membranes by increasing the effective bilayer thickness. This effect is strong, even at very low polymer concentration. The competition between the electrostatic and this “Helfrich’s modified” interaction is also studied. A simple model is built up which accounts almost quantitatively for both the strong stiffening of the Bragg peaks and the gradual insensitiveness to added salt of charged bilayers decorated by increasing amounts of amphiphilic copolymers.
I. Introduction Steric stabilizers of colloidal dispersions are usually amphiphilic copolymers, with a small lyophobic block (the anchor group) which attaches strongly to the particle surface and a lyophilic chain which trails freely in the dispersion medium; if the dispersion medium is a good solvent for the polymer lyophilic moieties, interdigitation of the grafted layers is not favored, and interparticle repulsion results. In recent years, this idea has been applied to biomaterials such as liposomes and vesicles, where the presence of terminally grafted polymers at the interface with biofluids dramatically enhances their lifetime in the bloodstream,1 allowing drug delivery to specific locations in the body.2 Due to their practical importance, several theoretical and experimental analyses have been performed to determine the range and magnitude of the steric barrier produced by attached polymers onto bilayers.3-9 For instance, Kuhl et al.3 have used a * Corresponding author. E-mail:
[email protected]. (1) Klibanov, A.; Maruyama, K.; Torchilin, V.; Huang, L. FEBS Lett. 1990, 268, 235. (2) Needham, D. et al. J. Liposome Res. 1992, 2, 411. (3) Kuhl, Τ. L.; Leckband, D. E.; Lasic, D. D.; Israelachvili, J. Biophys. J. 1994, 66, 1479. (4) Kenworthy, A. K.; Simon, S. A.; McIntosh, T. J. Biophys. J. 1995, 68, 1903. (5) Hristova, H. K.; Kenworthy, A.; McIntosh, T. J. Macromolecules 1995, 28, 7693. (6) Warriner, E. H.; Keller, S. L.; Idziak, S. H. J.; Slack, N. L.; Davidson, P.; Zasadzinski, J. A.; Safinya, C. R. Biophys. J. 1998, 75, 272. (7) Warriner, H. E.; Davidson, P. D.; Schmidt, H.-W.; Safinya, C. R. Macromolecules 1998, 31, 8503. (8) Warriner, H. E.; Davidson, P. D.; Slack, N. L.; Schellhorn, M.; Eiselt, P.; Idziak, S. H. J.; Schmidt, H.-W.; Safinya, C. R. J. Chem. Phys. 1997, 107, 3707. (9) Castro-Roman, F.; Porte, G.; Ligoure, C. Phys. Rev. Lett. 1999, 82, 109.
surface force apparatus to measure the interactions between lipid bilayers containing one particular poly(ethylene glycol) (PEG)-lipid (PEG 200) and deposed onto mica surfaces. They found that the force is repulsive but short range, because it results from the compression of the attached polymer layers coming into contact; the experimental results essentially confirm the theoretical prediction for the interaction forces between two polymer brushes. However, interaction between free membranes decorated by copolymers should certainly differ from those measured by Kuhl et al., since thermal fluctuations play a significant role. On the other hand, the presence of grafted polymers modifies not only the interactions between membranes but also the mechanical properties of the individual membranes themselves. A fluid membrane is characterized by three elastic constants: the mean and Gaussian bending moduli (denoted κ and κj, respectively) and the isothermal modulus of area compressibility which specifies the elastic response of a membrane surface area A to an isotropic tension τ. The bending moduli of membranes have been extensively studied both theoretically and experimentally, since their values are typically of a few kBT or less, and therefore their effects can be probed by analysis of thermal fluctuations. The influence of grafted polymers on the bending properties of fluid membranes has been also investigated; spontaneous formation of vesicles is expected10 and has been indeed observed.11 On the contrary, very little information has been collected on the elastic modulus . To our knowledge, two methods have been previously reported to measure (10) Porte, G.; Ligoure, C. J. Chem. Phys. 1995, 102, 4290. (11) Joannic, R.; Auvray, L.; Lasic, D. Phys. Rev. Lett. 1997, 78, 3402.
10.1021/la0101447 CCC: $20.00 © 2001 American Chemical Society Published on Web 07/07/2001
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this stretching modulus: the first one is to inspect visually the deformation of an individual giant vesicle of phospholipids under suction, using a micropipet technique.12 Typical values obtained for on phospholipid bilayers by this technique range from 100 to 230 mJ/m2.13 However, common surfactants usually do not form such macroscopic vesicles. The second method combines NMR and X-ray diffraction measurements14 using the osmotic stress technique; however, this method requires noncharged bilayers, since it has been shown recently that very long chains of water soluble polymers can be very easily incorporated between the bilayers of a lamellar phase made from ionic surfactants.15 Furthermore, it requires the use of perdeuterated lipids, and the analysis is based on the hypothesis of the change in area of the lipid bilayers with water concentration, which is not relevant for many common surfactants. In this context, it is very challenging to have a simple tool to estimate this elastic modulus for any type of synthetic or natural membranes. To estimate the maximal concentration of polymer-lipid that can be incorporated in a bilayer, Hristova and Needham16-19 have considered that the elastic energy stored in a polymer brush decorating a fluid membrane can be expressed in terms of a lateral tension between the chains and may generate area expansion or even membrane rupture. In this paper, we describe a pseudoternary lamellar system of cationic surfactant, cosurfactant, brine (or pure water), and amphiphilic triblock copolymer.9 The central hydrophobic group absorbs onto the fluid bilayers, while the two hydrophilic tails remain in the intermembrane aqueous region, decorating the bilayer. The existence of such lamellar phases of bilayers decorated by end-grafted polymers was first reported by Warriner and co-workers.20 Our system is different but exhibits a similar phase behavior. The lamellar LR phase is the simplest liquid crystalline phase of fluid membranes: it has the symmetry of a smectic-A lyotropic liquid crystal and consists of a onedimensional stack of bilayers of one or more amphiphilic components separated by solvent layers. It provides a powerful tool to investigate quantitatively both the interactions between fluid membranes and the elastic properties of the individual bilayers, using quantitative treatments of the small-angle neutron scattering (SANS) or small-angle X-ray scattering (SAXS) patterns of lamellar phases. The aim of the present paper is mainly to describe and interpret how end-grafted polymers modify the interactions between fluid membranes. We use quantitative treatments of neutron scattering patterns of the smectic phase, which allows us to measure the Caille´ parameter η characterizing the power-law singularities at the Bragg peaks. η is defined in terms of the smectic elastic constants by21 (12) Kwok, R.; Evans, E. Biophys. J. 1981, 35, 637. (13) Israelachvili, J. Intermolecular and Surfaces forces; Academic Press: London, 1992. (14) Koenig, W. B.; Strey, H. H.; Gawrisch, K. Biophys. J. 1997, 73, 1954 and references therein. (15) .Bouglet, G.; Ligoure, C. Eur. Phys. J. B 1999, 9, 137. (16) .Hristova, K.; Needham, D. J. Colloid Interface Sci. 1994, 168, 302. (17) .Hristova, K.; Needham, D. J. Colloid Interface Sci. 1994, 168, 302. (18) Hristova, K.; Needham, D. Macromolecules 1995, 28, 991. (19) Hristova, K.; Kenworthy, A.; McIntosh, T. J. Macromolecules 1995, 28, 7693. (20) Warriner, E. H.; Idziak, S. H. J.; Slack, N. L.; Davidson, P.; Safinya, C. R. Science 1996, 271, 969. (21) Caille´, A. C. R. Acad. Sci. Paris 1972, 274, 891.
Castro-Roman et al.
η)
qB2kT 8π xKB h
(1)
where K is the smectic curvature modulus (related to the bilayer bending modulus κ and to the smectic periodicity d according to K ) κ/d), B h is the smectic compression modulus related to the bilayer-bilayer interactions, and q0 is the position of the first Bragg singularity, qB ) 2π/d. The paper is organized as follows. In section II, we present the experimental system we have investigated and how the phase diagram of the host lamellar phase is modified by the addition of amphiphilic copolymers. In section III, we perform a careful analysis of the structural parameters of the decorated lamellar phase. Evidence is so obtained that grafted polymers induce a lateral stretching of the bilayers;22 this phenomenon provides a new simple method to estimate the stretching modulus of fluid membranes. In section IV, we report the measurements of the Caille´ parameter obtained from the analysis of the SANS pattern of the mixed lamellar phases.23-24 We have proceeded in two steps. First, we have incorporated increasing amounts of polymer in the undulationstabilized lamellar phase (solvent: brine with 0.2 mol/ NACl concentration) at a fixed membrane volume fraction. The polymer chains will form grafted layers on the membranes, whose structure depends mainly on the grafting density σ25 (dimensionless number of grafted chains per unit area of membrane). At low σ, the chains form separate “mushrooms”, each of size RG, the radius of gyration of the amphiphilic tail, and a brushlike structure forms in the opposite case. The smooth crossover σ* between these two regimes corresponds to the overlap limit of polymer mushrooms of area ∑* ) a2/σ* ) πRG2, where a is the Khun statistical length of the chain. In the brush regime, a stretched polymer layer is formed with a thickness larger than RG. Second, we have fixed both the bilayer and the polymer concentration at suitable values and vary only the salt concentration in the range 0 mol/L (pure water) to 0.2 mol/L. Doing so, we monitor the strength of the electrostatic interactions. Finally, in section V we built up a simple model which allows us to estimate the polymer-induced repulsion between adjacent membranes. The experimental results of the preceding section are then compared to our simple model. II. Experimental System Materials. The lamellar phases were obtained from a ternary mixture of cationic surfactant cetylpyridinium chloride (CPCl), octanol, and water (or brine). Water was doubly distilled and deionized. CPCl was purchased from Fluka and was recrystallized twice in ethanol/acetone mixtures. Octanol was purchased from Carlo Erba (“analyticals” grade) and used as received. The copolymer we used is a triblock amphiphilic copolymer with trade name Pluronics F68, a commercial product purchased from Fluka and used as received. It consists of two identical poly(oxyethylene) (PEO) blocks of 76 monomers each, symmetrically bound to a central shorter hydrophobic poly(oxypropylene) (PPO) block of 29 monomers. The PPO block is hydrophobic for temperatures higher than 15 °C, whereas PEO blocks are water(22) Castro-Roman, F.; Ligoure, C. Europhys. Lett. 2001, 53, 483. (23) Nallet, F.; Laversanne, R.; Roux, D. J. Phys. II France 1993, 3, 487. (24) Castro-Roman, F.; Porte, G.; Ligoure, C. In preparation. (25) De-Gennes, G. P. Macromolecules 1980, 13, 1069.
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the representation of Figure 1, the horizontal axis corresponds to the total weight fraction (CPCl +octanol) ws, whereas the vertical axis corresponds to the cosurfactant to surfactant weight ratio (moctanol/mCPCl). The LR phase, which is easily identified by simple observation through crossed polarizers, is stable in the range
{
5% < ws < 40% 0.844 < moctanol/mCPCl < 0.955
Figure 1. Phase diagram of the system octanol/CPCl/brine (0.2 mol/L NaCl) at 20 °C. X-axis: (surfactant + alcohol weight) fraction; Y-axis: octanol/CPCl weight fraction. L3: sponge phase; LR: lamellar phase; L4: unilamellar vesicles; LRΙΙ: coexistence of lamellar phase and multilamellar vesicles.
soluble in the temperature range 0-100 °C. We performed also some experiments with an hydrophobically modified PEO synthesized and purified in the laboratory (PEOm). The chemical formulas of the two polymers are the following:
For all experiments, we have fixed the cosurfactant to surfactant weight ratio: moctanol/mCPCl ) 0.95 in light water and moctanol/mCPCl ) 0.91 in heavy water. These samples which belong to the upper limit of the stability domain of the lamellar phase are transparent, slightly viscous, and show the typical textures of a cosurfactant-rich lamellar phase with numerous oily streaks in polarized light microscopy. The dry thickness δ0 of the membrane is obtained from the neutron (or X-ray) scattering patterns of the samples along a dilution line using the classical volume conservation condition qB ) (2πφs/δ0), where φs is the (cosurfactant + surfactant) volume fraction and qB is the position of the first Bragg singularity of the lamellar phase (qB ) 2π/d, where d is the smectic periodicity). One obtains δ ) 28.2 ( 0.4 Å. Preparation of the Samples. The samples are prepared by weight. CPCl is first dissolved in the solvent (water or brine); the polymer is then dissolved, and finally the alcohol is put in the sample. The samples are then stirred several times and left undisturbed at 20 °C for several weeks. The compositions of the samples are characterized by the membrane weight fraction, ws, the polymer to membrane weight ratio R, and the cosurfactant to surfactant weight ratio x, which is constant for all samples. The relevant definitions given below are used: ws ) φs )
moct + mCPCl moct + mCPCl + mpol + mbrine Voct + VCPCl Voct + VCPCl + Vpol + Vbrine
where mX and VX are the masses and the volumes of materials X. VX ) mX/FX, where FX is the density of material X. The membrane volume fraction φs is given by φs )
Small-angle X-ray scattering data are collected on the line ID2-BL4 of the European Synchrotron radiation facility in Grenoble, whereas small-angle neutron scattering data have been performed on line PACE at the “laboratoire Le´on Brillouin” in Saclay or on line D11 of the Institut Laue Langevin. For contrast reasons, H2O was replaced by D2O for the neutron experiments. We checked, provided that the density corrections are made so as to compare heavy and light water samples at the same composition in molecular units, that the change of solvent has no noticeable effect on the phase diagram. For neutron experiments, several incident wavelength and sample to detector distances have been used to collect data for all samples on a large enough q-range. All data were put on an absolute scale by dividing the scattering of H2O, and all treatments are very standard. The dilute part of the phase diagram of the mixture (octanol/CPCl/H2O, [NaCl] ) 0.2mol/L) at 20 °C is shown in Figure 1. Its geometry is very typical of a ternary system consisting of a surfactant and a cosurfactant in water. In
) [
(
MWPHILERws x 1 + w/ + (1 + x)Foct (1 + x)FCPCl s MWTOTFPHILE
MWPHOBRws MWTOTFPHOB
+
xws (1 + x)Foct
+
ws (1 + x)FCPCl
+
]
(1 - ws(1 + R)) Fwater
MWPHILE and MWPHOB are the molar mass of the hydrophilic and hydrophobic blocks, respectively, of the amphiphilic copolymer. All parameters necessary to calculate φs from the sample composition are collected in Table 1. The radius of gyration of one PEO tail in water is obtained from the experimental relationship RG ) 0.1078 26 M0.635 It gives RG ) 19 Å for the PEO tail of F68 and PEO . RG ) 24 Å for the PEO tail of PEOm. Phase Diagrams. Figures 2a,b and 3a,b) show how the phase diagram of the initial CPCl/octanol/brine 0.2 mol/L NaCl at 20 °C LR phase is modified by addition of F68 and PEOm, respectively; Figures 2a and 3a correspond to the phase behavior a few weeks after the preparation (26) Cabanne, B.; Duplessix, R. J. Phys. (Paris) 1982, 43, 1529.
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Figure 2. Phase diagram of the system CPCl/octanol/F68/brine at 20 °C. Fixed weight ratio moct/mCPCl ) 0.95. Salinity: [NaCl] ) 0.2 mol/L; x-axis: mass fraction of octanol + CpCl; y-axis: weight ratio R ) polymer/(octanol + CPCl). (a) A few weeks after the preparation of the samples; (b) several months after the preparation of the samples. Note the thinning of the gel domain after aging.
Figure 3. Phase diagram of the system CPCl/octanol/PEOm/ brine at 20 °C. Fixed weight ratio moct/mCPCl ) 0.95. Salinity: [NaCl] ) 0.2 mol/L; x-axis: mass fraction of octanol + CpCl; y-axis: weight ratio R ) polymer/(octanol + CPCl). (a) A few weeks after the preparation of the samples; (b) several months after the preparation of the samples. Note again the thinning of the gel domain after aging. Table 1. Densities and Molar Masses of the Compounds of the Mixed Lamellar Phase at 20 °C compound
density (g/cm3)
molar mass (g/mol)
H2O D2O octanol CPCl PEO (F68) PPO (F68) PEO (PEOm)
1.0 1.11 0.827 0.982 1.13 0.998 1.13
18 20 130 340 3344 1682 4928
of the samples, whereas Figures 2b and 3b correspond to the phase diagram 1 year after the preparation of the samples. The salient features are the following. First of all, for both systems there is a large domain where the lamellar phase is preserved upon the addition of copolymer. However, we note the existence (at intermediate values of R and ws) of a region in which the mechanical properties of the lamellar phase are those of a gel. The
criterion we use to qualify a given sample as a “gel” is to lay the test tube horizontal; if the sample does not flow after 15 s, we call it a gel. This gel phase (LR,gel) is identical, we think, to the “hydrogel” discovered by Warriner et al.20 in the lamellar phase of the DMPC/pentanol/water system, decorated by an end-modified PEO. Interestingly, we note that as in ref 20, this gelation is not correlated to any clear changes in the lamellar structure and smectic elastic moduli. Obviously, this “phase” is not a true equilibrium thermodynamic state; one year after the preparation of the samples, the gel domain of existence has shrunk, as evidenced by Figures 2b and 3b. To explain the existence of this gel, Warriner et al. have8,27 suggested that copolymer chains favor “edgelike” defects in the bilayers; they should spontaneously concentrate in high-curvature regions of the bilayers, that is, at the edge defect. The entanglement of the corresponding dislocations of the smectic order should induce in the system the elasticity of a gel. Preliminary results seem to indicate that in our system a consistent picture of the gel phase is rather that of close-packed onions (glass of soft spheres). A more detailed study of the physical properties of this intriguing soft glass will be the subject of a future publication.28 At low membrane concentration for the F68 system (we did not study in detail the PEOm system), we observe a classical sequence of lyotropic lamellar phases by increasing the polymer concentration, that is, LR, LRII (coexisting lamellar crystallites and multilayer vesicles), L4 (unilamellar vesicles), L1 (micelles). Note, however, that we cannot isolate the L4 phase. This sequence is reminiscent of the spontaneous vesiculation in lyotropic lamellar phases induced by the addition of amphiphilic copolymers.10-11 At high membrane concentration, SANS experiments show that the native lamellar phase in the F68 system separates into two distinct lamellar phases. This phase (27) Keller, L. S.; Warriner, H. E.; Safinya, C. R.; Zasadzinski, J. A. Phys. Rev. Lett. 1997, 78, 4781. (28) Ramos, L. et al. In preparation.
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Figure 5. Cartoon of the two possible scenarios to explain the decrease of the lamellar spacing due to the polymer. Left: membranes with holes without change of the membrane’s thickness; right: stretched membrane.
Figure 4. SAXS patterns of a series of lamellar samples of constant (surfactant + alcohol) weight ratio 16% for four different values of polymer to (surfactant + alcohol) weight ratio R (salinity ) 0.2 mol/L); moct/mCPCl ) 0.95.
separation occurs at decreasing polymer concentration as the membrane concentration progressively increases. Roughly, demixion occurs as soon as two opposite polymer layers, each of them having a thickness on the order of RG, come into contact; this indicates that the polymer layers do not want to be compressed or interdigitate. Figure 4 shows a series of SAXS patterns of lamellar samples at constant weight fraction of surfactants ws ) 16%, along a line of increasing polymer (F68) to amphiphile weight ratio R. Two interesting features are observed. First, the addition of amphiphilic copolymer correlates with the sharpening of the first Bragg peak and the emergence of higher harmonics in the scattering profile; this phenomenon deals with the polymer-induced interaction between bilayers and will be discussed in section IV. Second, the Bragg peaks of the lamellar phase progressively shift to higher q values as R increases; this phenomenon is discussed in the next section. Similar features are observed at all membrane concentrations where the lamellar phase is preserved and also when replacing F68 by PEOm. III. Lateral Stretching of the Membranes Induced by the Polymer This effect was reported in a previous paper.22 Three possible simple scenarios can be considered for the shift to higher q of the Bragg peaks at fixed membrane concentration upon increasing the copolymer to surfactant ratio R. Two of them are schematized in Figure 5. Either the copolymer induces the proliferation of holes in the bilayers, because of its tendency to bend the membrane (Figure 5a), or it thins the bilayers (Figure 5b). Indeed, in both cases, for the same volume of membrane, addition of polymer increases the effective area of the membrane, leading to a decrease of the smectic periodicity. However,
Figure 6. q4 times the neutron intensity profiles with fits to eq 2; system at ws ) 10%; diamonds: no polymer (R ) 0); circles: added polymer (R ) 0.6). The maximum of the first oscillation gives the thickness of the hydrophobic part of the bilayers.
in the first scenario, the thickness of the bilayer should not change, contrary to the second one. There is a third possible mechanism, since the ratio of true area/projected area could change as one adds copolymer, if the bending modulus significantly decreases as R increases. Again, in this case the bilayer thickness should not change. To discriminate, we have measured the bilayer’s thickness from SANS data taken at higher q values. In Figure 6, we have plotted q4 times the neutron intensity profiles as a function of the wave vector q at large q for a dilute sample (ws ) 10%), without (R ) 0) and with F68 polymer (R )0.6); this allows us to extract the neutron form factor of a bilayer P(q) in both cases using the following simple expression:23
q4I(q) ) Cq2P(q) ) C(1 - cos(δneutrq) exp(-q2(δneutr/4)2/2)) (2) where δneutr is the thickness of the hydrophobic part of the bilayer and C is a scaling factor. The position of the first maximum of the form factor oscillations is significantly shifted to a higher q value for the polymer-bearing membrane. From the fitting curves (full lines in Figure 5), one obtains accurate measurements of the thickness of the hydrophobic part of the bilayers, that is, δneut ) 2.37 nm for the bare membrane (the thickness of the hydrophilic part of the bilayer is then δh ) δ - δneut ) 2.82 - 2.37 ) 0.50 nm) and δneut ) 2.03 nm for the membrane decorated
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Figure 7. SANS spectra of the lamellar phase (membrane weight fraction 16%) without (mPPO/ms ) 0) and with PPO (mPPO/ ms ) 36%). The molecular weight of PPO is 2000 g/mol. There is no noticeable change in the position of the Bragg with PPO.
by polymeric brushes. This result unambiguously shows that the second scenario occurs; that is, the decrease of the smectic periodicity is directly related to thinning of the bilayer. We have checked (see Figure 7) by adding small PPO molecules (molecular weight 2000 g/mol) in the native lamellar phase, with the same proportion as in the triblock copolymer, that there is no shift of the Bragg peak and no significant variation of the form factor of the bilayer; this indicates that the stretching effect is not due to some interactions between the bilayers and the PPO block of the polymer. A similar thinning is also observed when incorporating PEOm rather than F68. For higher copolymer or membrane concentrations, we cannot measure the form factor of the membrane, because the structure factor interferes with it in the q range of our experiments. However, for these samples, the first Bragg peak position qB is well-defined; we interpret it in terms of the dry thickness of the bilayers: qB ) (φCPCl + φoct)2π/δ. We choose to report these results in terms of the more meaningful variations of σ/σ* as a function of the polymer to surfactant weight ratio; it can be expressed in two different ways: 2
σ/σ* )
πRG PQδ ) Vmemb
(3)
Vtot
P πR 2N Rδ σ 2 G A 1 x ) + σ* exp MWpol [1 + x)FCPCl (1 + x)Foct
[
first Bragg peak qB using the following expression:
() σ σ*
[
P πR 2N Rw d MWPHOBRws 2 G A s MWPHILERws ) + + MWpol MWpolFPHILE MWpolFPHOBE exp
]
-1
(4)
On the contrary, for hard samples the second equality of eq 3 is used, since the smectic periodicity d ) 2π/qB is defined with a very good accuracy from the position of the
]
(1 - ws(1 + R)) + + Fbrine (1 + x)FCPCl (1 + x)Foct ws
πRG2PQd
In eq 3, Q is the number of polymer chains in the sample, P ) 2 for F68 chains and P ) 1 for PEOm chains, Vtot is the volume of the sample, and Vmemb is the volume of the membrane (CPCl + octanol); these quantities can be easily expressed as a function of the weight fractions, molecular weights, and densities of the components for each sample. The first equality of eq 3 is used for the measurement of σ/σ* in the soft lamellar sample; in this case, the thickness of the bilayer δ is obtained from the fit of the form factor to expression 2 (δ ) δh + δneut with δh ) cste ) 0.50 nm) using the following expression:
()
Figure 8. Symbols: plot of the normalized grafting density versus polymer to (surfactant + alcohol) weight ratio for various dilutions; full line: corresponding expected variation of the normalized grafting density without membrane stretching (eq 6); (a) polymer F68 and (b) polymer PEOm
xws
-1
(5)
In the same manner, we can express the unperturbed ratio (σ/σ*)unp in terms of measurable quantities:
(σ/σ*)unp )
πRG2PQδ0 ) Vmemb
P πR 2N Rδ 2 G A 0 1 x + MWpol (1 + x)FCPCl (1 + x)Foct
[
]
-1
(6)
This gives for each polymer
(σ/σ*)unp,F68 ≈ 1.96R (σ/σ*)unp,PEOm ≈ 2.66R In Figure 8a,b, the experimental values of (σ/σ*) (symbols) for all investigated series are plotted as a function of R together with the hypothetical unperturbed values (σ/σ*)unp (full line). First, we observe that all experimental points belong to a same “master curve”
Effect of Copolymers on Fluid Membranes
Figure 9. Symbols: Evolution of the SANS spectra of the lamellar phases as a function of polymer/membrane weight ratio R for two different membrane dilutions. Full lines are fits (model of ref 23 for ws ) 25%; model of ref 24 for ws ) 10%).
whatever the surfactant concentration; this means that interactions between membranes do not play any significant role in the stretching properties of the bilayers. Second, the experimental master curve (σ/σ*)exp begins to deviate significantly from the unperturbed theoretical values (σ/σ*exp < σ/σ*unp) as soon as the brush regime is reached, that is, (σ/σ*) > 1. This result clearly confirms that grafted polymer layers are able to stretch fluid membranes16-19 and consequently to thin them, so as to relax polymer stretching normal to the bilayers in the brush regime. We can go beyond this qualitative analysis and measure the relative thinning, ∆ ) -((δ - δ0)/δ0) ) [(σ/σ*)unp/(σ/ σ*)exp] - 1, of the bilayers as a function of the normalized grafting density. This has been done in ref 22. Using a simple model, which predicts the thinning of a fluid membrane by an end-grafted polymer, we have been able to estimate the stretching modulus of the CPCl/octanol fluid membrane from the experimental values of ∆: ≈ 20 mJ/m2. For more details, the reader is invited to refer to ref 22. IV. Polymer-Induced Interactions between Membranes The sharpening of the first Bragg peak and the emergence of higher harmonics in the scattering profiles of the lamellar phase, which correlates with the addition of amphiphilic copolymers, is observed whatever the dilution of the lamellar phase. Figure 9 shows two series of SANS patterns corresponding to the peaks of a lamellar structure along a line of increasing polymer (F68) to amphiphile ratio at two extreme weight fractions of membrane, ws ) 10% and ws ) 25%. Note that, for even more dilute lamellar samples (ws ) 5 and 8%), this sharpening is partially masked by f finite size effects of the lamellar crystallites, which did not allow us to exploit quantitatively these profiles. For the PEOm, similar behaviors are observed. All SANS powder intensity spectra were fitted either with the simplified expression derived by Nallet, Laversanne, and Roux23 or by a more sophisticated numerical model we have developed.24 The simplified analytic expression and the more realistic numerical procedure
Langmuir, Vol. 17, No. 16, 2001 5051
actually give equally good fits for scattering patterns exhibiting sharp peaks. On the other hand, the model of Nallet et al. fails to fit the profiles of “soft” lamellar samples in which the first Bragg singularity is weak and the scattered intensity at low q is strong. For such dilute samples, we had to use our time-consuming numerical procedure which still provides very good fits even for very soft smectic order. Examples of such fits are depicted in Figure 9 (full lines). For the more concentrated samples (ws ) 16, 20, and 25%), the fits are obtained from the Nallet et al. formula; for the more dilute series (ws ) 10%), the fit is derived from our numerical procedure. Doing so, we have obtained the variations of the Caille´ parameter η (eq 1) as a function of the polymer to membrane weight ratio R for various series of lamellar phases at constant membrane volume fraction. They are shown in Figure 10. In each series, the maximal value of R corresponds to the upper limit of stability for the onephase smectic domain. For all membrane concentrations, the Caille´ parameter decreases sharply as R increases, until it reaches a plateau for R ≈ R* ≈ 0.5 (crossover to the brush regime). This saturation effect seems surprising since one expects that above R* the thickness of the polymer layer increases with the area density, but it may well be related to the in-plane stretching of the membrane which damps to the thickening of the polymer layer at high area density (see section III). In a second experiment, we have investigated eight series of samples at fixed amphiphile concentration ws ) 0.2. In each series, the polymer to membrane weight ratio R is kept constant (in the range 0-1 from one series to the other) as a function of the mean salinity (NaCl) cs in the range 0-0.2 mol/L. Figure 11 shows the evolution of the shape of the smectic order as a function of salinity for two series: R ) 0 and R ) 0.8. We have also plotted the corresponding Nallet et al. fits (full lines). The evolution of the scattering patterns is very different in the series without polymer and with polymer. Indeed, for the polymer-free samples, we observe a progressive softening of the smectic order: broadening of the harmonics of the smectic structure and increase of the scattering at small angles. This behavior is characteristic of the continuous transition from a smectic stabilized by long-range electrostatic forces to a smectic stabilized by much weaker undulation forces of Helfrich’s type.29 For the lamellar phase decorated by polymers, we observe no noticeable changes in the shape of the scattering patterns; the interaction between membranes is insensitive to the addition of salt. These qualitative features are quantitatively confirmed in Figure 12, which shows the variations of the Caille´ parameter η as a function of the mean salinity for each series of samples. For R ) 0 (no polymer), it appears that η increases monotonically with the relative strength of electrostatic versus undulation interactions. As some polymer is added, the situation is different: η still increases with ionic strength of the solvent, until it reaches a plateau. Both the plateau and the threshold salinity at which this plateau appears are smaller at higher R. V. Modelization Theoretical Background. Fluid membranes dispersed at equilibrium in a solvent are fluctuating objects. The undulations of a cylindrical portion of a bilayer of area S ) πL2 can be described by the Helfrich bending (29) Roux, D.; Safinya, C. R. J. Phys. France 1988, 49, 307.
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Figure 10. Plots of the Caille´ parameter η (obtained from the fitting of SANS spectra by the models of refs 23 or 24) as a function of the polymer/membrane weight ratio R for four different membrane dilutions.
free energy, evaluated for small deformations ∇u , 1,
H)
κ
κ
∫ d2F(∇2u(F))2 ) 2 ∑ q4|u(q)|2 2 s
(7)
q
where F is a 2-D vector on the base (planar) surface, κ is the bending modulus, and u(F) ) (xS/(2π)2) ∫s dqeiq.Fu(q) is the local displacement field, u(q) being the corresponding 2-D Fourier component. Using the equipartition theorem, one obtains from eq 5 the equilibrium spectrum of undulation: 〈|u(q)|2〉 ) (kBT)/(κq4). By integration over all q values, the average thermal fluctuations of the membrane are easily calculated:
〈|u(F)|2〉 )
kBT
L2
4π3κ
(8)
In a lamellar phase, each bilayer is confined between its two neighbors of the smectic stack. This phenomenon restricts the fluctuations of the membrane. Fluctuations h, of amplitude larger than d h are forbidden: 〈|u(F)|2〉 e d where d h ) d - δ is the average distance between the opposite surfaces of two adjacent bilayers. This inequality allows us to define a characteristic length in the plane of the membrane:
λc )
x
(2π)3κ d h ) Rcd h 2kBT
branes. Consider a smectic stack (periodicity d) of incompressible fluid bilayers of thickness δ without direct molecular interaction. Each bilayer of area S can be divided into portions of area λc2. By definition of the collision length, each of these portions does not feel the presence of the neighboring membranes (there is no restriction on the configurations at scales smaller than λc). The total free energy of the membrane is simply the sum of the curvature energies of each free portion of the membrane: F ≈ S/λc2 ∫∫λc2dxdy((∂2u)/(∂x2) + (∂2u)/(∂y2))2. By dimensional analysis, (∂2u)/(∂x2) ≈ (∂2u)/(∂y2) ≈ d h /λc2 ) 1/(Rc2d h ) and one 4 2 h ) ) CH[(kBT)2/κ](1/d h 2) which is the gets F/S = kBT/(Rc d expression of the long-range repulsive undulation interaction potential. The value of the numerical coefficient CH has been calculated by Helfrich:31 CH ) 3π2/128 ≈ 0.231. More recently, Monte Carlo simulations32-34 and field theoretical calculations35,36 give a value smaller by a factor of 2 for this constant: CH ≈ 0.106. We choose however the value originally estimated by Helfrich, because as shown below it fits perfectly our experimental data, contrary to the more recent one. The corresponding smectic compression modulus reads
9π2(kBT)2 1 64κ d h4
B h und ) d
(10)
Interestingly, when the interlamellar forces are dominated
(9)
λc is called the collision length and represents the average distance between two adjacent collisions of a membrane with its neighbors as shown in Figure 13. The notion of collision length allows us to reconstruct very rapidly30 the Helfrich interaction between mem-
(30) Chaikin, P. M.; Lubensky, T. C. Principles of condensed matter physics; Cambridge University Press: New York, 1995. (31) Helfrich, W. Z. Naturforsch. 1978, 33a, 305. (32) Gomper, G.; Kroll, D. M. Europhys. Lett. 1989, 9, 59. (33) Janke, W.; Kleinert, H.; Meinhard, M.; Phys. Lett. B, 1989, 217, 525. (34) Netz, R. R.; Lipowsky, R. Europhys. Lett. 1995, 29, 345. (35) Kleinert, H. Phys. Lett. A 1999, 257, 269. (36) Bachman, M.; Leinert, H.; Pelster, A. Phys. Lett. A 1999, 261, 127.
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Figure 11. Evolution of the SANS spectra of a lamellar phase (from bottom to top). Left: lamellar phase without polymer; softening of the smectic order. Right: lamellar phase decorated by F68 (R ) 0.8); the salt has no effect. In all cases, ws ) 0.2. Full lines are fits obtained from the model of ref 23.
by these entropic undulations, we expect from eqs 1 and 10 the Caille´ parameter η to follow the simple purely geometrical expression29
η)
4 δ 13 d
(
2
)
(11)
This means that the Caille´ parameter is a simple function of two structural parameters of the lamellar phase only: the smectic periodicity and the thickness of the bilayers. Renormalization of the Bilayer Thickness. In the first series of experiments, the measurements of the Caille´ parameter η are in agreement with the picture where the two grafted polymer layers on both sides of each bilayer
modify the Helfrich interaction by simply renormalizing the membrane thickness. This means that in eqs 10 and 11 the bare membrane thickness δ0 should be simply replaced by an effective membrane thickness δeff:
δeff ≡ δ0 + 2h
(12)
where h is the effective thickness of the polymer layer attached on each side of a bilayer membrane as depicted schematically in Figure 14. In Table 2, we report the measurements of η together with the expected theoretical values for three different membrane concentrations with and without polymer. For the samples without polymer, ηtheo is calculated from eq 11 with δ ) δ0 ) 28 Å being the
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Figure 12. Plot of the Caille´ parameter η as a function of the mean salinity of the lamellar solvent at different values of the polymer/membrane weight fraction R. In all experiments, the weight fraction of membrane is fixed, ws ) 20%. Table 2. Comparison between the Theoretical and Experimental Values of the Caille´ Parameter η in the Case of Naked Membranes (r ) 0) and in the Case r > r* ) 0.5 (Except for the Membrane Concentration ws ) 25%, for Which Phase Separation Occurs at r ) 0.4)a R
Figure 13. Cartoon in two dimensions of a stack of fluctuating membranes. Each lamella is confined between its two neighbors. The instantaneous average distance between two adjacent collisions of the membrane and its neighbors along the horizontal axes is called the collision length λC.
Figure 14. Schematic representation of a bilayer decorated by amphiphilic copolymers. The effective thickness of the decorated membrane is the sum of the bare membrane’s thickness and twice the thickness of a polymer layer decorating one side of the membrane.
bare membrane thickness. For the samples containing polymer, we measure η at a polymer to surfactant weight ratio R around 0.5, which corresponds to the mushroom/ brush transition. In this case, one expects the polymer layers to be uniform with a thickness of the order of the unperturbed size of a polymer coil in solution; that is, h ≈ RG ≈ 18Å. The expected theoretical values ηtheo are given again by eqs 11 and 12, with δ ≈ δeff ≈ δ0 + 2RG ≈ 65.2 Å. The results show perfect agreement between the experimental and theoretical values for the bare membranes (R ) 0) without polymer in brine. The lamellar phase is indeed stabilized by Helfrich’s interactions. Note that if we would choose the modified value for the Helfrich coefficient, CH ≈ 0.106,32-36 the theoretical values of the Caille´ parameter would be still given by eq 11, but with
ηexp
ηtheo
R
ηexp
ηtheo
0 0.6
ws ) 10% 1.03 0.64
1.1 0.63
0 0.6-2
ws ) 16% 0.83 ∼0.15
0.87 0.29
0 0.6-1.0
ws ) 20% 0.78 ∼0.15
0.78 0.22
0 0.4
ws ) 25% 0.64 0.15
0.60 0.10
a For R ) 0, the theoretical values of η are obtained from eq 9. For R * 0, the thickness of the bare membrane δ0 is replaced by the effective thickness δ0 + 2RG.
a different numerical prefactor, 1.97 rather than 1.33. However, our experimental results (Table 2) for the naked membrane unambiguously show that such values are clearly overestimated whereas the agreement with the data is nearly perfect with the original Helfrich coefficient,31 CH ) 0.231, at all membrane concentrations we have investigated. Previous experiments29,37,38 reported such a good agreement between experiments and the initial Helfrich value. Theoretical attempts39 to explain the agreement between the initial Helfrich value and these experiments invoked effects due to dissolved oil layers within the bilayers present in these systems. This explanation is not valid for our system, because there is no oil in the membrane. For the membranes decorated by polymers, the agreement between the predictions and the experiments is surprisingly good in comparison with the roughness of the model. Actually, the polymer layer is not as simple as a monolayer of surfactants: it is highly swollen by water and therefore highly compressible. So, modeling (37) Lei, N.; Safinya, C. R.; Bruinsma, R. F. J. Phys. II France 1995, 5, 1155. (38) Oda, R.; Lister, J. J. Phys. II France A 1997, 7, 815. (39) Netz, R. R.; Phys. Rev. E 1995, 52, 1897.
Effect of Copolymers on Fluid Membranes
Langmuir, Vol. 17, No. 16, 2001 5055 Table 3. Comparison between the Theoretical and Experimental Values of the Caille´ Parameter η in the Brush Regime (Calculations Obtained from Equations 11, 12) R
ηexp
0.6 0.8 1 1.3 1.6 2
ws ) 16% 0.17 0.14 0.13 0.13 0.125 0.125
ηtheo 0.33 0.28 0.24 0.18 0.13 0.09
R
ηexp
ηtheo
0.6 0.8 1
ws ) 20% 0.18 0.13 0.17
0.19 0.15 0.11
Figure 15. Plot of the effective thickness of a polymer layer h with respect to the polymer/membrane weight ratio R, obtained from eqs 11 and 12 and the measurements of η.
the effect of the polymer by an extra additive contribution to the effective thickness of the bilayer is then a crude approximation; however, a theory of collisions between compressible fluid membranes remains to be done. In Figure 15, we have plotted the variation of the effective polymer thickness heff obtained from the measurements of η: heff ) (d(1 - x(3/4)ηexp) - δ0)/2 for the series of weight fraction ws ) 20%. At low R, that is, in the very dilute mushroom regime, heff increases very rapidly as the polymer surface density increases, until it levels off at a plateau at the mushroom/brush transition. In the dilute mushroom regime, it is clear that the polymer layer decorating a membrane is not homogeneous and the simple picture we have proposed has to be refined. We will now discuss in more detail how the effective polymer layer thickness is related to the grafting density of copolymers in the two different regimes. Brush Regime. For the F68 copolymer, the transition between the mushroom and the brush regime occurs for R ≈ 0.51. In the brush regime, one expects the brush thickness to increase slowly with the polymer surface density:40
h = Nσ1/3a ) (σ/σ*)1/3RG ) (1.96R)1/3RG
(13)
Within the framework of our simple model, we should then expect a slow decrease of the Caille´ parameter as a function of the polymer to membrane weight ratio R. We therefore observe (see Figure 10) a plateau in the variation of η in the brush regime. As explained in the preceding section, the polymer brushes significantly thin the decorated bilayers. This effect induces a nonlinear variation of η with R. The experimental master curve (Figure 8a), (σ/σ*) versus R, can be fitted very well by the heuristic function (σ/σ*)exp ) 1.979R - 0.25R2. Putting this expression in eq 13 gives the expected theoretical thickness of the brush which takes into account the thinning effect. In Table 3, the corresponding theoretical variations of η obtained from eqs 11 and 12 are compared with the experimental values for the series ws ) 16 and 20%. Mushroom Regime. In this regime, the decorating layer of polymers on each membrane is highly inhomogeneous. However, as shown in Figure 15 there is a strong enhancement of the effective thickness at very low polymer grafting density. This effect seems to indicate that each (40) Alexander, S. J. Phys. (Paris) 1977, 38, 983.
Figure 16. Cartoon of a lamellar phase “bumped” by polymer mushrooms.
mushroom (which can be viewed as a cylindrical patch of height RG and area πRG2) induces a local thickening of the decorated membrane of the order of RG (the height of the patch) on an area much larger than the natural area of the patch πRG2. Indeed, because of the finite bending rigidity of the membrane, each mushroom of thickness RG adsorbed onto a confined membrane restricts the amplitude of its thermal fluctuations. This amplitude of restriction is on the order of x〈|u(F)|2〉 ≈ RG. From eq 9, this restriction concerns a portion of the membrane 2 ) π(RcRG2) centered around the mushroom of area πλmush
where Rc ) x((2π)3κ)/(2kBT) . 1. This means that, because of the finite bending rigidity of the membrane, a given mushroom not only affects the collision between two neighboring membranes at its own position but on the area of a “ghost” pancake centered around the position of the mushroom and of radius on the order of the collision length defined by the thickness of the mushroom. To take into account this effect, the decorated membrane is modeled as a uniformly bumped membrane (see Figure 16). Each bump, with axial symmetry, is centered around each mushroom, with a Gaussian shape of height RG, of width at half-height λ ) (3/4)λmush ) 3Rc2RG/4, and with a disk of area Σ ) πD2 being the base of a bump. D is the average distance between grafted chains, and Σ is the average area of membrane per mushroom. This truncation of the Gaussian shape allows us to avoid the nonphysical interferences between overlapping ghost pancakes in the mushroom regime. An approximate expression of the Caille´ parameter in the lamellar phase of such bumped membranes is given by eqs 11 and 12, where
h)h h)
1
∑
∫0D 2πrRGe-(r /λ ) dr ) 2
2
2 9 2 Rc RGφpol(1 - e-8/(9Rc φpol)) (14) 8
h h is the mean thickness of the bumped membrane, and φpol ) σ/σ*. The corresponding two asymptotic expressions for the Caille´ parameter are
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Figure 17. Plot of the variation of the Caille´ parameter in the mushroom regime (symbols) and model (full lines) given by eqs 11, 12, and 14. For all series, the same fitting parameter is used: Rc ) 3.
{
(
)
of Granek,41 give Rc ) x(2π)3/2 ≈ 11.4 in the same units. We observe that, despite its simplicity, our model of the bumped membrane quantitatively well describes the decrease of the Caille´ parameter in the regime of very dilute mushrooms, indicating a strong effect of the grafted polymers on the fluctuation spectrum of a stack of membranes. Interestingly, this strong effect has been recently confirmed in the case when the decorating polymer is a rigid peptide.42
Salt Effect. CPCl is cationic, so the membranes are positively charged. In the absence of added salt, the electrostatic interaction is repulsive, strong, and long range and dominates the Helfrich interaction. Addition of salt in the solvent progressively screens the electrostatic interaction. The corresponding screening length is the well-known Debye length λD; its numerical value for a monovalent electrolyte is given in a first approximation (neglecting the Donan effect) by λD (Å) ) 3.04/xcs where cs is the salt concentration in mol/L. This length corresponds approximately to the thickness of the counterion layer in the vicinity of the charged surfaces. Andelman43 has shown that, in the limit when λD , d, the electrostatic interaction is weak and strongly screened; its contribution to the total interaction between the membranes can be viewed as a correction to the Helfrich interaction: the effective thickness of the membrane is the sum of the bare membrane thickness and twice the thickness of the counterions (one for each side of the membrane), that is, 2 λD. The analogy with the decoration effect by copolymer layers is strong. In both cases, the guest component (salt or copolymer) modifies the Helfrich interaction by increasing the effective thickness of the bilayer, the extra contribution being the thickness of the counterion layers in the case of the salt and the thickness of the polymer layers in the other case. In the first case, this thickness decreases by increasing the amount of salt, whereas it increases by increasing the amount of adsorbed polymer in the latter. When both salt and copolymer are present, this analogy allows the simple following physical picture. At fixed polymer area density R and fixed concentration of membrane, the lamellar phase should be sensitive to the amount of salt at low ionic strength as long as the Debye
(41) Granek, R. J. Phys. II France 1997, 7, 1761. (42) Tsapis, N.; Urbach, W.; Ober, R.; Phys. Rev. E 2001, 63, 041903.
(43) Andelman, D. Handbook of Physics of biological systems; Elsevier Science B. V.: New York 1989; Vol. 1, Chapter 12.
2 φpol , 1 η ) 4 1 - δ0 + (9/4)Rc RGφpol 3 d δ0 + 2RG 2 4 φpol ≈ 1 η ) 1 3 2
(
)
2
(15)
In the regime of very dilute mushrooms, the effective thickness of the decorated membrane increases linearly with the surface fraction of polymers, whereas it reaches the plateau value δ0 + 2RG when the mushrooms begin to be very close together. In Figure 17, we have plotted (full line) the theoretical variation of η with respect to R, given by the set of eqs 11, 12, and 14, together with the experimental points (symbols) for four different membrane concentrations. Because we restrict the analysis to the dilute regime, we neglect the stretching effect of the membrane, that is, ψpol ) 1.96R. For all series, we used the same value Rc ) 3 as the single fitting parameter. This value has no real physical meaning, because the Gaussian shape for the bumps is arbitrary and was chosen for convenience; however, the value derived from the fit is quite reasonable. Chaikin and Lubensky propose30 Rc ) x2π ≈ 2.5 in xκ/kBT units, whereas our own calculations, identical to the calculation
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Figure 18. Plot of the variations of the Caille´ parameter (symbols: experiments; full lines: model from eq 16) as a function of the salinity of the lamellar solvent at constant weight fraction of membrane (ws ) 0.2).
length is larger than the thickness of the polymer layer (λD > hpol). An increase of the Caille´ parameter is seen upon increasing the salinity at low salt. At higher salt, however, as soon as λD < hpol the lamellar phase should be insensitive to further addition of salt and the Caille´ parameter should level off at a plateau value. The threshold salinity cs* between these two regimes is defined by λD(cs*) ≈ hpol(R). The measurements of η reported in Figure 11 actually confirm this behavior. Without polymer (R ) 0), the Caille´ parameter is an increasing function of the salt concentration, reflecting the gradual evolution from a lamellar phase stabilized by electrostatic interac-
tions to a lamellar phase stabilized by undulation interactions. When the membrane is decorated by a given amount of amphiphilic copolymers, the Caille´ parameter first increases with increasing the salt concentration, until a plateau is reached. The plateau value as well as the threshold salinity are decreasing functions of the polymer/ membrane ratio R. A more quantitative treatment of the competition between the polymer-modified Helfrich interaction and the electrostatic one is possible. We assume that both interactions are purely additive (this is a correct approximation in the case of a simple Helfrich interaction
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Table 4. Evolution of the Mean Bending Modulus K (Equation 16) with Respect to r (No Salt) R
κ(kBT)
R
κ(kBT)
