J. Phys. Chem. 1995, 99, 13505-13508
13505
Lipid Domain Instabilities in Monolayers Overlying Sublayers of Finite Depth H. A. Stone*?+and H. M. McConnelF Division of Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, and Department of Chemistry, Stanford University, Stanford, Califomia 94305 Received: February 28, 1995; In Final Form: June 27, 1995@
Phase-separated, circular monolayer domains at the air-water interface are unstable due to dipolar repulsions that tend to elongate the domain shapes. The corresponding monolayer motions are viscously coupled to the underlying liquid that acts to resist these shape changes. The flow field in a sublayer of finite depth is calculated and the growth rate of small domain shape disturbances is determined using linear stability analysis. The special case of a very shallow aqueous sublayer is considered, where the monolayer flow field is irrotational except close to the domain boundary. The calculations are applicable to fluid lipid monolayers.
I. Introduction Lipid monolayers at the air-water interface show thermodynamic phase separations at various applied two-dimensional pressures. These phases are often observed using fluorescence microscopy by incorporating a fluorescent lipid probe into the monolayers at a low probe concentration. Frequently, one phase is broken up into a large number of separate domains each of which is surrounded by a second continuous phase. The domains themselves may be ordered in the monolayer plane, forming superstructure phases, such as the hexagonal and stripe phases.',* In principle, there can be thermodynamic transitions between these superstructure phases, such as the stripe to hexagonal phase transition. Also, individual, relatively isolated, domains undergo shape transitions. The early stages of these transitions are often referred to as shape instabilities. There have been a number of theoretical and experimental studies of the transitions of circular lipid domains to domains with shapes of lower ~ y m m e t r y . ~In - ~lipid monolayers these transitions are driven by long-range dipole-dipole electrostatic repulsions and are opposed by line tension that stabilizes circular shape^.^ Examples of shape transitions are those in which a circular domain transforms at early times into a domain with an n-fold (sinusoidal) rotation symmetry, where n = 2, 3, .... The dynamics of these shape transitions for fluid monolayer phases is determined by competition between a driving force within the monolayer and the viscous drag of the aqueous subphase. In recent work we have treated the hydrodynamics of these systems theoretically to obtain the rates of shape change at short times for domains overlying an infinitely deep ~ublayer.~ In earlier work we discussed this same problem, but assumed the flow in the monolayer was irrotational.6 As indicated below, this irrotational flow approximation is only valid when the aqueous subphase is very shallow. The present paper treats the problem of a shallow aqueous subphase in more detail to obtain a quantitative expression for the subphase drag and the growth rate of the domain shape instability. The lipid domains of interest which undergo shape changes are typically small (radii R 10-100 pm), shape distortions with magnitude a fraction of the radius occur over a period of minutes, and the sublayer liquid is water, so we model the hydrodynamics of the sublayer flow as a viscously dominated, low Reynolds number fluid motion. The principal analytical
' Harvard University:
[email protected].
* Stanford University:
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Abstract published in Advance ACS Abstracts, August 1, 1995
0022-3654/95/2099-13505$09.00/0
difficulty with this class of flow problems is that motion in the monolayer drives motion throughout the sublayer. Indeed, although monolayers typically have effective Newtonian viscosities about a factor of IO2 larger than the underlying aqueous phase, the fact that monolayers are thin relative to the domain radius implies that the dominant resistance to motion of the domain is contributed by the s ~ b l a y e r . ~ .Hence, '~ a detailed hydrodynamic analysis must necessarily couple the monolayer and sublayer flow fields. We are able to do this exactly for circular domains assuming that the monolayer is infinite and remains planar. In the calculation reported here, we first obtain a velocity field representation for the sublayer Stokes flow consistent with the form of the assumed harmonic domain distortions responsible for motion. Viscous stresses exerted on the lipid layer by the flowing subphase couple the dynamics of these two distinct fluid layers. Following the original analysis of Saffman" (see also Hughes et al.;'* for a related application see Brochard et al.I3), the equations of motion for the monolayer are Stokes equations modified to include (i) an electrical body force accounting for dipolar repulsions of the monolayer molecules and (ii) a force which effectively resists vertical gradients in the monolayer flow and so balances viscous stresses exerted by the sublayer (see eq 6). Assuming a small perturbation to a circular shape and incorporating the effect of the perturbation as a ring forcing in the monolayer momentum equations lead to an exact solution for the sublayer and monolayer velocity fields. Balancing the rate of energy viscously dissipated by the fluid motion with the rate-of-change of the electrostatic-line tension energy responsible for the shape change yields the growth rate of small disturbances, which is the usual goal of linear stability theory. The effects of a lower boundary on the deforming domain, while straightforward in principle to include in the analysis, are nevertheless in practice rather tedious to calculate. The final equation for the disturbance growth rate requires numerical integration, but does incorporate the degree of domain distortion, the sublayer depth, and the relative viscosity of the monolayer and subphase. Not surprisingly,the effect of the lower boundary is to slow the typical evolution of the domain shape change and will be most important for sublayers with thicknesses comparable to or smaller than the domain radius. Also, the functional dependence of the disturbance growth rate on spatial frequency and domain radius is dependent on the sublayer depth. Overall, the value of the analysis lies in the fact that an explicit formula for the growth rate, derived on the basis of a complete 0 1995 American Chemical Society
13506 J. Phys. Chem., Vol. 99, No. 36, I995
Stone and McConnell The unknown function A(k) is determined by considering the equation of motion for the monolayer. Since the velocity field is assumed to be continuous across the interface separating the lipid layer from the sublayer, the monolayer velocity field Um(r,t9,Z)satisfies
NEWTONIAN FLUID (SUBPHASE)
p, Vz
Figure 1. Nearly circular lipid domain coexisting with a second monolayer phase at the planar air-water interface. The aqueous sublayer has depth H. The distortions from the circular shape are sketched.
analysis of the viscous flow equations and accounting for a lower boundary, can be derived, and asymptotic formulas for the deep and very shallow sublayer limits can be derived which agree with previously published results. 11. Problem Description and Solution A circular lipid domain of radius R exists in equilibrium with a second lipid phase at the air-water interface. The lipid monolayer of thickness h rests on an aqueous layer with height H and viscosity p, as shown in Figure 1. It will be convenient to utilize a cylindrical (r,e,z) coordinate system with z = 0 located at the monolayer-sublayer interface and z =- 0 directed into the sublayer. A. Velocity Fields. We consider harmonic distortions of a nearly circular monolayer domain whose radius evolves according to rint(e,t) = R [1
+ en@)
COS
n e ] , n = 2 , 3, ...
(1)
where the azimuthal wavenumber n is an integer and E&) is a dimensionless measure of distortion. The n = 0, 1 modes are not included since they correspond, respectively, to a uniform expansion and translation. We assume for this linearized analysis that Icn(t)l > 1, we may consider the thin sublayer limit, H/R R, the circular shape is energetically unstable to an n-fold shape distortion. @d represents the effective dipole density of the monolayer. Equating eqs 14 and 16 provides an estimate for the growth rate of domain shape changes that are assumed to evolve according to ~ ~ ( t ) In particular, the growth rate t, may be represented in the convenient dimensionless form 0~
= n2(n2- l)Af,(H/R,A)
(Note that there are typographical errors in our earlier publication (Stone and McConnell, 1995). Equation 4.1 should not have a factor of 2 in the denominator, below eq B3 V should be defined as V = 21n(8R/A), and eq B5 should have a factor of 4 (not 2) in the numerator.) Equation 17 is the principal analytical result of this study. The logarithm term has been scaled with R2 as indicated for the ease of experimentalists who measure R2 (e.g., Lee and McConnel16). The numerical evaluation of the infinite integralf,, which involves products of Bessel functions, is performed using the recently developed method and routines of Lucas.I4
This result clearly identifies NUH as the appropriate viscosity ratio parameter for the thin sublayer limit. We further note by comparing (18) and (19) that the n dependence of f,, which enters into the growth rate formula 17, is different for a deep sublayer than for a shallow sublayer.
111. Growth Rate as a Function of Sublayer Depth From eq 17 we observe that the perturbation growth rate has a functional dependence on the dimensionless domain size (W R2) that is independent of the layer depth. The primary effect of the finite sublayer is thus to simply decrease the growth rate In Figure 2 we plot the function Af,(H/R,A), relative to the infinite sublayer value (18), versus the sublayer depth H/R and consider n = 2,4, 6. Two representative values of the viscosity ratio parameter, A = 1, 100, are considered. In Figure 2, small deviations of the ordinate from unity for the A = 100 calculations are due to the neglect of higher order error terms in eq 18. From these numerical calculations we see that as the sublayer thins, the growth rates are substantially reduced, with t, 0 as H/R 0, as is to be expected on physical grounds and is seen in eq 19. The growth rates are substantially diminished for H/R < 1 and attain the infinite sublayer values for H/R > 3. Furthermore, we see that the asymptotic approximation for H/R 0.1, and (iii) for very shallow layers the growth rates are substantially diminished.
IV. Discussion It is instructive to consider separately the effect of a very shallow aqueous subphase. In this case, the sublayer exerts on the monolayer a viscous stress, denoted f in eq 6 , which may be approximated as that shear stress arising from a local shear flow.I6 Hence, for shallow sublayers m
f,lK h H
Substituting into (6) we see that for shallow sublayers the monolayer flow conforms to
expand into a region of finite vorticity of width (hHpmlp)ll2 around both sides of the interface. An approximation equivalent to eq 23 has been used previously for monolayers and bilayers by Brochard et al.,I3 Evans and Sackmann,16 and Andelman et a1.I8 For the special geometry treated here and the model used to represent the monolayer-domain geometry, the calculation we describe is exact as it solves the requisite boundary value problem for the fluid velocity. Thus, the global nature of the flow problem, i.e. that motion on one part of the interface affects other parts of the interface, is taken into account. It will be interesting to determine whether the “exact” calculation developed here and in Stone and McConnel19can be generalized so as to incorporate the detailed pattern formation theory developed recently by Goldstein and Jacksod for the dipole-line tension competition. There is increasing practical interest in performing laboratory experiments using monolayers with shallow aqueous subphases. This interest is motivated, in part, by the need to dampen surface motions (e.g. fluctuations) as much as possible while still retaining a fluid aqueous subphase. The present calculations then provide a quantitative basis for predicting the effect of depth on monolayer flow. Very shallow aqueous phases below lipid monolayers (or bilayers) are also of importance for biophysical studies (McConnell et al.I9). In such systems the aqueous subphase may be as thin as 30 8, (Johnson et aL20), in which case subphase drag can potentially have a large effect on membrane dynamics.
Acknowledgment. We thank Dr. S. K. Lucas for making available numerical routines for integrating the Bessel function integrals and Dr. R. E. Goldstein for providing us with preprints of recent research. H.A.S. acknowledges support from NSFPresidential Young Investigator Award CTS-8957043, and H.M.M. acknowledges the sponsorship of NSF Grant DMB 900555. References and Notes
F h
e, - cos ne 6(r - R ) = 0 (21) Velocity gradients in the monolayer occur on the scale of the domain R so that comparing the two viscous resistance terms (u, is a characteristic velocity that typifies the fluid motion), monolayer viscous stress - O@mUcJR2) - pmh H - 1 H sublayer viscous stress O@u,lhH) p R R A R (22) As A = @lo2),then provided H/R < 0(1), we expect that the sublayer stress term is dominant throughout the monolayer, in which case the monolayer flow reduces to
which is simply Darcy’s equation as arises, for example, in Hele-Shaw flows. Hence, everywhere away from the domain boundary the flow is necessarily irrotational. The shallow trough, irrotational flow approximation was used previously by McConnellI7 and Lee and McConnelL6 The calculations in the present paper allow the determination of the drag coefficient y used in the previous work as y = pIH. It should be noted that the irrotational flow approximation leads to a physically unacceptable vortex sheet at the domain boundary between the two liquid phases. When the subphase viscosity is taken into account, we estimate that this vortex sheet should
(1) McConnell, H. M. Annu. Rev. Phys. Chem. 1991, 42, 171-95. (2) Seul, M.; Andelman, D. Science 1995, 267, 476-83. (3) McConnell, H. M.; Moy, V. T. J . Phys. Chem. 1988,92,4520-5. (4) McConnell, H. M. J . Phys. Chem. 1990, 94, 4728-31. ( 5 ) Langer, S. A.; Goldstein, R. E.; Jackson, D. P. Phys. Rev. A 1992, 46, 4894-904. (6) Lee, K. Y. C.; McConnell, H. M. J . Phys. Chem. 1993,97,95329. (7) de Koker, R.; McConnell, H. M. J . Phys. Chem. 1993, 97, 1341924. (8) Goldstein, R. E.; Jackson, D. P. J . Phys. Chem. 1994, 98, 962636. (9) Stone, H. A.; McConnell, H. M. Proc. R. SOC. London A 1995, 448, 97-111. (10) Klingler, J. F.; McConnell, H. M. J . Phys. Chem. 1993,97,6096100. (1 1) Saffman, P. G. J . Fluid Mech. 1976, 73, 593-602. (12) Hughes, B. D.; Pailthorpe, B. A,; White, L. R. J . FluidMech. 1981, 110, 349-72. (13) Brochard, F.; Joanny, J. F.; Andelman, D. In Physics ofAmphiphilic Layers; Meunier, J., Langevin, D., Boccara, N., Eds.; Springer-Verlag; Berlin, 1987; pp 13-9. (14) Lucas, S. K.J . Comput. Appl. Math., in press. (15) Watson, G. N. A Treatise on the Theory of Bessel Functions; Cambridge University Press: Cambridge, 1945. (16) Evans, E.; Sackmann, E. J . Fluid Mech. 1988, 194, 553-61. (17) McConnell, H. M. J . Phys. Chem. 1992, 96, 3167-9. (18) Andelman, D.; Brochard, F.; Knobler, C.; Rondelez, F. In Micelles, Membranes, Microemulsions and Monolayers; Gelbard, W. M., Ben-Shaul, A., Roux, D. A., Eds.; Springer-Verlag; New York, 1994; pp 559-62. (19) McConnell, H. M.; Watts, T. H.; Weis, R. M.; Brian, A. A. Biochim. Biophys. Acta 1986, 864, 95- 106. (20) Johnson, S. J.; Bayerl, T. B.; McDermott, D. C.; Adam, G. W.; Rennie, A. R.; Thomas, R. K.; Sackmann, E. Biophys. J . 1991,59, 289-94. JP950572J
