Nanoscale surface relaxation of a membrane stack
aa r X i v : . [ c ond - m a t . s o f t ] O c t Nanoscale surface relaxation of a membrane stack
Hamutal Bary-Soroker ∗ School of Physics & Astronomy, Raymond & Beverly Sackler Facultyof Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Haim Diamant † School of Chemistry, Raymond & Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel (Dated: May 29, 2007)Recent measurements of the short-wavelength ( ∼ ∼ . − ), which fits theknown baroclinic mode of bulk lamellar phases, and a slower one ( ∼ µ s − ) of unknown origin.We show that the latter is accounted for by an overdamped capillary mode, depending on the surfacetension of the stack and its anisotropic viscosity. We thereby demonstrate how the dynamic surfacetension of membrane stacks could be extracted from such measurements. PACS numbers: 82.70.Uv,61.30.St,68.03.Kn
Self-assembled stacks of membranes are encountered invarious industrial and biological systems. They consistof parallel bilayers of amphiphilic molecules separated bymicroscopic layers of solvent — a structure with the sym-metry of a smectic A liquid crystal [1]. Such stacks formlyotropic lamellar phases [2], on which many cleaning andcosmetic products are based. Lamellar bodies are foundalso in the lung [3] and as multilayer vesicles (“onions”)[4]. Membrane stacks made of phospholipids have beenwidely used to study properties of biological membranes,whereby the large number of identical, equally spacedmembranes helps enhance the signal and allows the studyof membrane–membrane interactions (e.g., [5]).The elasticity of membrane stacks is equivalent to thatof single-component (thermotropic) smectics [1] and hasbeen extensively studied. The elastic moduli of the stackcan be extracted from its equilibrium fluctuations using,e.g., x-ray line shape analysis [6]. By contrast, the hydro-dynamics of membrane stacks [7, 8], because of their twomicro-phase-separated components, differs from that ofthermotropic smectics [9]. An additional hydrodynamicmode appears — the baroclinic (slip) mode — along witha unique dissipation mechanism, in which the membranesand solvent layers develop different average velocities [7].Experimental studies of hydrodynamic modes in mem-brane stacks have been rather scarce, the prevalent tech-nique being dynamic light scattering [8], whose spatialresolution is limited by the wavelength of light.In a recent experiment using neutron spin-echo spec-trometry, Rheinst¨adter, H¨außler, and Salditt (RHS) haveprovided a first look at the relaxation of membrane stacksat short wavelength (1–100 nm) and short time (1–10 ns) [10]. Their system consisted of several thousandsof dimyristoylphosphatidylcholine (DMPC) phospholipid ∗ Present address: Department of Condensed Matter Physics, Weiz-mann Institute of Science, Rehovot 76100, Israel. † Electronic address: [email protected] bilayers, self-assembled into a stack of d ∼ ∼ . − ) could be well fitted inthe fluid-membrane case to that of the baroclinic mode ofa bulk lamellar phase [11], while the slower mode (decayrate of ∼ µ s − ) was left unexplained. We demon-strate below that this slower relaxation is well accountedfor by a surface mode, i.e., a perturbation which is local-ized within a finite penetration depth from the surface ofthe stack.In a recent publication [12] we have addressed the sur-face dynamics of membrane stacks, highlighting the qual-itative differences from the surface dynamics of both sim-ple liquids and thermotropic smectics [13]. These differ-ences arise from the slip dissipation mechanism, which isabsent in simple liquids and thermotropic smectics but isusually dominant in lyotropic lamellar phases. Althoughthe formulation in Ref. [12] is general, its analysis is fo-cused on a very different domain (larger wavelengths andslower rates) from that sampled by RHS. In that do-main the slip dissipation dominates and, consequently,the surface relaxation is governed by an overdamped dif-fusive mode, whose decay rate Γ increases quadraticallywith the wavevector q . In this Brief Report we presenta slight adaptation of that theory for a large- q , high-Γregime such as that of RHS.The general surface dynamics of membrane stacks isquite complex, depending on several restoring and dis-sipation mechanisms [12]. Three moduli are associatedwith the restoring forces: the compression modulus B ,bending modulus K , and surface tension γ . Viscous dis-sipation is characterized (in the limit of incompressibleflow) by three viscosity coefficients [7], denoted η M , η T ,and η V . The coefficient η M , associated with differencesin the lateral velocity across layers (sliding viscosity), ismuch smaller than the other two, which correspond tothe viscous response to deformations of the lipid mem-branes. We use the parameter Θ = 2( η T + η V ) /η M tocharacterize this viscosity anisotropy; it is typically oforder 10 –10 [7, 14]. The aforementioned slip motionrequires another transport coefficient [7], µ ≃ d / (12 η ),where η is the viscosity of the solvent (water) layer.In view of this richness it is helpful to begin by iden-tifying the dominant contributions to the slower modeof Ref. [10]. First, for the typical parameters of thatcase — q ∼ − nm − , Γ ∼ µ s − , η M ∼ − Pa s,and mass density ρ ∼ — one gets a negligibleReynolds number, Re ∼ ρ Γ / ( η M q ) ∼ − , implyingthat inertial modes [13] are irrelevant in the current case.Second, to determine the dominant dissipation mecha-nism one should compare the friction due to slip, µ − v ( v being a characteristic relative velocity), with that dueto viscous stresses, η T , V q v , i.e., the dimensionless pa-rameter S = ( η M µq ) − is to be compared with Θ [12].We find S ∼ ≪ Θ. Thus, unlike the mode focusedon in Ref. [12], in the current large- q case viscous dissi-pation is dominant. Finally, the relative importance ofthe three restoring mechanisms depends not only on thesurface perturbation wavevector q but also on its pen-etration depth α − . Since the value of α is unknown apriori, all three mechanisms should be considered in prin-ciple. However, to keep the analysis as simple as possiblewe shall assume that the surface tension is the dominantfactor. This ansatz is motivated by the experimental factthat the rate of RHS’s slower mode is linear in | q | at small q (see Fig. 1); the way to get such a linear overdampeddispersion relation is to balance a surface tension stressagainst a viscous one, γq u ∼ ηq Γ u ( u being the ampli-tude of the surface deformation). We will return to theconsistency of this assumption later on.The continuum theory formulated in Ref. [12] is validfor wavelengths much larger than the inter-membranespacing, qd ≪
1. RHS’s experiment, however, samplesthe range 0 . < qd <
4. To obtain an extrapolationof the analysis to large q we introduce one last modifi-cation to the theory — the distance z from the surfaceinto the stack is discretized, z → − dn ( n = 0 , , , . . . ),turning the differential equations of Ref. [12] into finite-difference ones (similar to the analysis of high- q acousticmodes in a crystal). The lateral position x parallel tothe membranes is kept continuous, and we consider, forsimplicity, a surface perturbation which is uniform in thesecond lateral direction y .Within these assumptions Eq. (11) of Ref. [12] yieldsthe following surface mode for the vertical displacementsof the membranes, u n ( x, t ): u n = ( C + e − α + dn + C − e − α − dn ) e iqx − Γ t α ± = 2 d sinh − (cid:18)
12 Θ ± / | q | d (cid:19) . (1)For sufficiently small q ( qd ≪ Θ − / ) the spatial decaycoefficients are α ± ≃ Θ ± / | q | , i.e., the mode containstwo terms of disparate penetration depths, α − − ≫ α − . (A qualitatively similar result was obtained for the sur-face mode analyzed in Ref. [12], yet in the current casethe origin of the two differing penetration depths is thelarge viscosity anisotropy rather than the strong slip dis-sipation.) In the other limit of qd ≫ Θ / , as expected,both contributions become localized within a distance oforder d from the surface, α ± ≃ (2 /d ) ln(Θ ± / | q | d ).The dispersion relation Γ( q ) is set by the boundaryconditions for the stress tensor at the stack surface, assummarized in Eq. (13) of Ref. [12]. Substituting in thatequation the expressions for α ± obtained above, we get,within the same approximations,Γ( q ) = 2 γ Θ η M d × (cid:20) sinh − (cid:18)
12 Θ / | q | d (cid:19) + sinh − (cid:18)
12 Θ − / | q | d (cid:19)(cid:21) . (2)Equation (2) is the main result of our current analysis.For large wavelengths this dispersion relation becomesΓ( qd ≪ Θ − / ) ≃ γ η M ( η T + η V ) / / | q | . (3)Equation (3) is equivalent to the dispersion relation ofan overdamped capillary mode at the surface of a simpleliquid having effective viscosity η eff = [ η M ( η T + η V ) / / .In the opposite, short-wavelength limit we getΓ( qd ≫ Θ / ) ≃ γ ( η T + η V ) d/ | q | d ) . (4)In this quasi-two-dimensional limit the dependence onthe smaller (sliding) viscosity, η M , disappears, and aneffective two-dimensional viscosity emerges, η = ( η T + η V ) d/ q > . − are considered less reli-able due to scattering by defects in the stack [10].) Thestack periodicity was measured as d = 5 . . T = 30 ◦ C (fluid membranes) and 19 ◦ C(gel-like membranes), respectively [10]. The value of thesliding viscosity at 30 ◦ C, η M = 0 .
016 Pa s, was indepen-dently found from a fit of the faster mode [10]. We arethus left with two fitting parameters in Eq. (2), Θ and γ .For the fluid-membrane case we find Θ = 110 and γ = 5 . η T , V ∼ Θ η M ∼ η ∼ η T , V d ∼ − –10 − Pa s m,which agrees well with measurements of the surface vis-cosity of fluid DMPC membranes [18].The applicability of the theory to stacks of solid, gel-like membranes should be questioned, as such stacks have q (nm -1 ) Γ ( µ s - ) FIG. 1: Dispersion relations for the slower relaxation mode ofstacks of DMPC lipid membranes at 30 ◦ C (circles) and 19 ◦ C(squares). (Data taken from Ref. [10].) The solid lines arefits to Eq. (2) with d = 5 . η M = 0 .
016 Pa s, Θ = 110,and γ = 5 . d = 5 . η M = 0 . γ = 28 mN/m (upper curve). The valuesof d and η M are taken from Ref. [10]; Θ and γ are fittingparameters. additional intra-membrane elasticity. The same concern,in fact, should be raised regarding the fluid-membranecase as well, since at the high frequencies consideredhere the individual membranes are expected to have aviscoelastic response. The fits obtained in Fig. 1 (in par-ticular, the linear behavior for small q ) suggest, however,that these additional restoring forces are negligible com-pared to the surface tension and do not affect the sur-face relaxation. The fit for T = 19 ◦ C yields significantlylarger values for both the viscosity anisotropy and thesurface tension, Θ = 350 and γ = 28 mN/m, which isthe expected trend for stiffer membranes [19]. (In the fitwe have assumed that the sliding viscosity η M does notchange much with temperature.)The elasticity of membrane stacks gives rise to an ef-fective static surface tension, γ el = ( KB ) / [1, 20, 21].The values of K and B in the fluid-membrane state wereextracted by RHS from the fit of the faster mode as K ≃ . × − N (corresponding to a membrane bend-ing modulus κ = Kd ≃ . k B T ) and B ≃ . × Pa. This yields γ el ≃ . α − − < Θ / /q < ∼ nm, which is at least 1–2orders of magnitude smaller than the thickness of RHS’sfilms ( ∼ µ m). Second, for surface tension to be thedominant restoring force, one should have γ > γ el . Thiscondition can be obtained rigorously [24] but is also real-ized upon demanding that the stress arising from surfacetension, γq αu , be larger than both the compression one, Bα u , and the bending one, Kq u . As described above,we actually have γ ∼ γ el and, thus, the assumption canbe only marginally fulfilled. Moreover, the omission ofthe bending terms requires also that Kq /η M be smallerthan Γ [12], which is satisfied only for the lowest end ofthe sampled q range, q < ∼ . − . The apparent suc-cess of the simplified theory over the extended q range(Fig. 1), therefore, is somewhat surprising. We note thatthe stacks of RHS are densely packed. The thickness ofa DMPC bilayer at 30 ◦ C is 4 . q surface perturbationsthe stack might not follow the usual description of linearsmectic elasticity but respond merely as an anisotropicviscous liquid with surface tension.In summary, the relaxation of nanoscale fluctuationsin finite membrane stacks seems to occur via two dis-tinct overdamped modes — a bulk baroclinic mode anda slower surface mode. The dispersion relation of the sur-face mode provides access to the dynamic surface tensionof the stack, which should be hard to measure otherwise.Supplementing such an experiment with measurementsat larger wavelengths (e.g., using dynamic light scatter-ing), yielding a value for Θ, may allow the accurate ex-traction of the dynamic surface tension. Acknowledgments
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