Instabilities of the Stripe Phase in Lipid Monolayers

Abstract

In lipid monolayers consisting of two phases, the competition between line tension and electrostatic repulsion can give rise to a stripe phase. The stability of this phase with respect to small harmonic distortions is analyzed. It is shown that the stripe phase is just marginally stable when the stripes are of equilibrium width. As soon as the stripes exceed this width, coupled long wavelength distortions are energetically favored.Intermediate results concerning the stability of a single edge and a single stripe are also presented. IntroductionPhospholipid monolayers at the air-water interface can be readily prepared and observed in a two-phase regime.' The coexisting phases, which appear as dark and light phases under the fluorescence microscope, form a variety of spatial structures, such as circular or near-circular domains, parallel stripes, or more irregular, wormlike structures. Such structures are thought to arise from a competition between line tension and electrostatic r e p~l s i o n .~-~ In the simplest possible macroscopic model, the total free energy of the monolayer contains a line tension term, due to the interfacial energy between the two phases, and an electrostatic term, due to the long range repulsion forces between surface dipoles:6 d-AT ' -where A is the line tension, ,u is the difference in the dipole densities of the two phases, and A is a cutoff parameter that keeps the double line integral from diverging. Note that P is the total length of the interface between the two phases. The double line integral, which has the form of the Biot-Savart law between current loops, is taken over the interface. Equilibrium structures are those which minimize the free energy, usually under the constraint that the area of each phase is fixed. Mathematically, this translates into a challenging problem in variational calculus. Minimization of the total free energy leads to an integrodifferential equation, which in the general case is intractable.A more promising approach has been to postulate certain simple geometries for the two phases. The geometric parameters that characterize such geometries are then varied until the free energy is minimized. For example, if one assumes that one phase exists as circular islands of equal radius in a sea of the majority phase, it is straightforward to determine the radius which minimizes the total free energy:7(2)Similarly, for a phase with alternating black and white stripes of equal width w , the equilibrium stripe width is found to be8(3)The next logical step in this approach is to subject these @ Abstract published in Advance ACS Abstracts, April 1, 1995.geometries to various distortions and see whether the total free energy is thereby increased or decreased. Accordingly, in previous work, we have studied both the stability of circular domains with respect to small harmonic distortions9 and large nonharmonic distortions1° and the stability of a straight edge with respect to small deformations." We now apply the same method of (lowest order) stability analysis to a stripe...

Energy-driven pattern formation in planar dipole-dipole systems in the presence of weak noise

“…Our numeric simulations will corroborate this, as rectanglelike domains do indeed dominate in this regime. Finally, it is important to note previous calculations of the stability of isolated stripes [2,15,23], In our terms, they found that the critical width of a stripe, that is, the largest width for which an infinite stripe becomes unstable, is given by dy = eA+y+2, where y is Euler's constant. For an energy-minimized rectangle in the high-aspect-ratio limit, its width is given by tUrec = dy = e A + l, which is strictly less than the critical value for all values of A.…”

“…The constant itself can be roughly determined by sampling along the relatively constant region between -2. L ~ Lrec + (c + mn)wTec (15) for A sufficiently negative. This is a remarkably simple charac terization of complicated domain structure, but, as we will see, it indeed holds for domains that resemble the intricate structure of those seen in experiment.…”

“…Substantial work has been done in characterizing the physics, dynamics, and morphology of these systems . Langmuir monolayers [6][7][8][9][14][15][16]18,21,22,25] and ferrofluid confined to a HeleShaw cell [1][2][3][4][5][10][11][12][13]16,19,20,23,24,26,27] The inherent complexity of dipole-mediated systems has also inspired numerical simulations using dynamic evolution of some particular system's equations of mo tion [5,11,13,16,19,20,23,25,28,29]. For example, Cebers and co-workers [2,20] and Goldstein and co-workers [10,11] con sidered Hele-Shaw systems, providing analytic and asymptotic expressions for the energy and stability of a variety of domain structures, including circles and rectangles.…”

“…For example, Cebers and co-workers [2,20] and Goldstein and co-workers [10,11] con sidered Hele-Shaw systems, providing analytic and asymptotic expressions for the energy and stability of a variety of domain structures, including circles and rectangles. McConnell and co workers built an effective theoretical formalism for describing the energy of Langmuir domains and determined analytically the stability of circular and stripe domains to harmonic pertur bations [7][8][9]15]. de Koker and McConnell were able to show that the detailed physical parameters that describe Langmuir systems can be reduced to a single parameter [30].…”

Stability Analysis of Three-Dimensional Colloidal Domains: Quadratic Fluctuations

“…At high temperatures, thermal fluctuations can be described by density fluctuations of colloidal particles, 16,17,30 whereas at low temperatures, thermal fluctuations can be effectively described by shape fluctuations on sharp boundaries separating domains and dispersing medium. 15,[25][26][27][28][29][30][31][32][33] These two descriptions are both widely applied in the literature. In this paper, we focus on the low temperature and density regime where each domain can be treated as an isolated system.…”