J. Phys. Chem. B 1997, 101, 381-388
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Cloverleaf Monolayer Domains Sara Perkovic´ and Harden M. McConnell* Department of Chemistry, Stanford UniVersity, Stanford, California 94305 ReceiVed: June 19, 1996; In Final Form: October 9, 1996X
Two-component monolayers of dipalmitoylphosphatidylcholine (DPPC) (or dimyristoylphosphatidylcholine (DMPC)) and dihydrocholesterol (DChol) form cloverleaf domains at a surface pressure just above 0 mN/m and for monolayer concentrations of less than 30% mol DChol. A cloverleaf domain consists of DChol-rich lobes attached to a small gas cavity. Upon further expansion of the monolayer, at constant surface pressure, the gas cavity in the center of the cloverleaf domain expands. The stability of a cloverleaf domain is studied using a line tension-dipole repulsions model. In this model, the equilibrium shape of a constant-area monolayer domain arises from a competition between the line tension of the domain boundary and the dipole-dipole repulsions between the molecular dipoles in the monolayer. The energies of the three-leaf, four-leaf, and five-leaf clover domains are compared with the energies of three, four, and five circular domains, respectively. The results show that a cloverleaf domain is stabilized by the small gas cavity to which the DChol-rich lobes are attached.
1. Introduction The phases and phase transitions in a lipid monolayer at the air-water interface have been traditionally studied by determining the pressure-area isotherms of the monolayer. In 1981, von Tscharner and McConnell1 incorporated a small amount of a fluorescent dye into a dipalmitoylphosphatidylcholine (DPPC) lipid film and used fluorescence microscopy to visualize the phases present in the monolayer. Since the fluorescent dye partitions preferentially in some phases and not in others, a phase containing a higher concentration of the fluorescent dye is brighter than a phase with a lower dye concentration. This difference in brightness can be video recorded using a lowlight-level camera attached to the fluorescence microscope. Fluorescence microscopy has since been successfully applied to the study of monolayer phase coexistence, monolayer domain shapes, and shape transitions.2 Furthermore, it also led to the development of a theoretical model that explains the sizes and shapes of monolayer domains, as well as domain shape transitions. Using fluorescence microscopy, Lo¨sche et al.3 visualized all the phases in DPPC and dimyristoylphosphatidylcholine (DMPC) lipid monolayers. The model phase diagrams for these systems have been studied by Andelman et al.4,5 by including in their model energy for a monolayer longrange dipole-dipole repulsions. Solid-phase lipid domains coexisting with a fluid lipid phase have been reported,6,7 as well as the formation of two-dimensional chiral lipid crystal domains in monolayers composed of chiral phospholipids.8,9 Keller et al.10 developed a line tension-dipole repulsions model to explain the various shapes of the lipid domains. This simple phenomenological model attributes the size and shape of a monolayer domain to a competition between the line tension of the domain boundary and the electrostatic dipole-dipole repulsions between the molecular dipoles associated with the lipid molecules in the monolayer. This model has, for example, been used to study the contribution of long-range dipolar interactions to the formation of chiral domain shapes.11 Monolayers consisting of mixtures of phospholipids and cholesterol (Chol) also show phase separation at low surface pressures. Subramaniam and McConnell12 studied DMPCX
Abstract published in AdVance ACS Abstracts, December 15, 1996.
S1089-5647(96)01818-4 CCC: $14.00
Chol monolayers and observed phase separation of the monolayer, at low surface pressures, into DMPC-rich and Chol-rich liquid phases. Studying the same system, Rice and McConnell13 observed domain shape transitions from circular Chol-rich domains to domains of less symmetrical shapes. The line tension-dipole repulsions model has been successfully used to study the transition from a circular shape to a shape with 2-fold symmetry14 and the transition between a square and a rectangular monolayer domain.11 A study of domain shapes by Lee and McConnell15 in a DMPC-dihydrocholesterol (DChol) system has led to the observation and theoretical description of domains of lower rotation symmetry and the phase transitions between these domains and circular ones. These calculations have been performed numerically16 as well as analytically.15,17,18 Very near 0 mN/m surface pressure, a DChol/lipid monolayer may have a third, gas phase, coexisting with two liquid phases. Hagen and McConnell19 studied the formation of gas domains at the interface between the phospholipid-rich and the DCholrich liquid phases. From the three phase arrangements they determined the relative line tensions of the three two-phase interfaces. Their work involved the study of monolayers of very large DChol concentrations (larger than 60 mol % DChol). At smaller DChol concentrations, in monolayers of DMPC-DChol and DPPC-DChol and at a surface pressure just above 0 mN/ m, we have observed the formation of cloverleaf domains. One such domain consists of DChol-rich lobes attached to a small gas cavity. After expanding the monolayer at constant surface pressure, the gas cavity expands to form a gas phase. In this paper, the methods of fluorescence microscopy are used to study cloverleaf domains in monolayers of DMPCDChol and DPPC-DChol, with a 1% Texas Red dye as the fluorescent probe (discussed below). The stability of a cloverleaf domain is studied by calculating its energy using the line tension-dipole repulsions model,20 with the assumption that no gas cavity is present. The energies of the three-leaf, four-leaf, and five-leaf clover domains are compared to those of the three, four, and five circles, respectively, that would be produced if the lobes in each of the cloverleaf domains would separate from the central gas cavity. Under the experimental conditions, a cloverleaf domain does not represent a stable shape in the absence of the gas cavity. In the next section, the theoretical background associated with © 1997 American Chemical Society
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the study of monolayer domain shapes and shape transitions is reviewed. In that section, we define the model energy that is used to determine the stabilities of cloverleaf monolayer domains. In section 3, we describe the experimental setup and the experiment, and we report our observations of cloverleaf domain shapes. In section 4, the model energy described in section 2 is used together with model curves to calculate the stabilities of the cloverleaf domains. We describe and discuss the results in section 5. We end, in section 6, with a summary of this work. 2. Theoretical Background
Etotal ) Eb + Eλ + Eel
(2.1)
where Eb is the bulk energy of the domain, Eλ is the interface energy of the domain, and Eel is the electrostatic energy due to the long-range repulsive interactions between the molecules, not already taken into account by the first two terms in (2.1). The energy terms in (2.1) are given by21
µ2 2
E ) λP -
µ2 2
∫C ∫C [(br
1
db r 1‚db r2 -b r 2)2 + ∆2]1/2
(2.7)
3. Experimental Section
A monolayer composed of lipids at the air-water interface is modeled as a two-dimensional array of molecular dipoles. These dipoles are considered to be on average perpendicular to the air-water interface, so they exert repulsive dipole-dipole interactions between themselves. When phase separation occurs, the boundary between the two newly formed phases will have a tendency to form the shortest length possible in order to minimize the line energy. However, due to the repulsive longrange interactions between the molecular dipoles in the monolayer, these dipoles will favor being as far away as possible from each other. This competition between the line tension of the boundary between two phases and the repulsive dipoledipole interactions between the molecular dipoles in the monolayer causes the formation of domains of one phase surrounded by the other phase. The size and shape of these domains are the direct result of the line tension-dipole repulsions competition. Therefore, to determine the equilibrium shape of a monolayer domain at constant area, a model energy that includes a line tension-dipole repulsions competition is needed. Minimization of that energy with respect to the domain shape will result in the equilibrium shape of the domain, for a given domain size. The model energy of an isolated monolayer domain can be written as21
Eel ) -
where 0 and λ0 are hypothetical bulk and line energies per unit area and length, respectively, in the absence of dipolar forces. For a domain of constant area, the first term in (2.1) is constant and is not affected by the shape of the domain. Therefore, we omit this term in the study of equilibrium domain shapes in section 4. In that case, the shape-dependent energy of an isolated monolayer domain is given by21
Eb ) A
(2.2)
Eλ ) λP
(2.3)
∫C ∫C [(br
db r 1‚db r2
1
-b r 2)2 + ∆2]1/2
(2.4)
where is the bulk energy per unit area; A is the area of the domain; λ is the line tension (line energy per unit length of the domain’s perimeter P ); µ is the difference in dipole moment densities between the two phases; and ∆ is a cutoff parameter that prevents the integral in (2.4) from diverging and is associated with the distance between two neighboring dipoles in the model. The line integrals in (2.4) are over the perimeter C of the domain. The energies and λ are given by22,23
µ2π ∆
(2.5)
λ ) λ0 - µ2
(2.6)
) 0 +
3.1. Materials. The phospholipids L-R-dimyristoylphosphatidylcholine (DMPC) and L-R-dipalmitoylphosphatidylcholine (DPPC) were purchased from Avanti Polar Lipids. Dihydrocholesterol (DChol) was obtained from Sigma. The dye N-(Texas Red sulfonyl)dipalmitoyl-L-R-phosphatidylethanolamine (TR-DPPE) was purchased from Molecular Probes. All substances were used without further purification. The monolayers were spread from a 1 mM solution in chloroform and contained 1 mol % of the dye TR-DPPE. All the experiments were carried out on a subphase of double-distilled water at 2325 °C (room temperature). 3.2. Setup. A monolayer is spread on the water subphase contained in a small Teflon Langmuir trough of dimensions 10 cm × 6 cm. The trough has two depths: a deeper side (2 cm), where a plunger is present, and a shallower side (1 cm), where the monolayer is observed. The plunger adjusts the water level with a precision of about 1 µm. To reduce convection, and therefore drift of the monolayer film, a rectangular piece of solid Teflon of height 0.5 cm is placed in the shallow side of the trough, thus decreasing the depth of the subphase. To obtain essentially drift-free monolayers, a stainless steel collar (diameter 7 mm, height 2.5 mm) with a sharp upper edge is placed on the rectangular Teflon piece.24 Using the plunger, the water level can be carefully lowered until the monolayer gets caught on the circular upper edge of the steel collar. The monolayer is still continuous if there is a lack of visible difference in the monolayer inside and outside the collar, as the surface pressure is changed.24 The surface pressure is adjusted with a manually movable barrier that sits on top of the Teflon trough and is measured with a Wilhelmy plate film-balance apparatus. The entire trough is mounted on an x,y-directions movable stage that allows one to look at different parts of the monolayer. The monolayers are observed using a Zeiss Photomicroscope III with a Nikon 40X long-distance microscope objective. An HBO mercury lamp is the light source. The white light is passed through a 546 nm interference filter, which generates green light used to illuminate the monolayer. The fluorescence of the TRDPPE dye in the monolayer is observed in the red region with a long-pass filter of wavelengths greater than 590 nm. The images of monolayer domains are recorded on a JVC S-VHS video recorder (Model BR601MU) with a low-light level camera (Cohu) attached to the microscope. 3.3. Results: Monolayer Domains. The monolayers were spread from a 1 mM chloroform solution containing 1 mol % of the dye TR-DPPE. The following compositions were studied: 10 mol % DChol-89 mol % DPPC; 15 mol % DChol-84 mol % DPPC; 20 mol % DChol-79 mol % DPPC; 25 mol % DChol-74 mol % DPPC; and 30 mol % DChol-69 mol % DPPC, as well as 20 mol % DChol-79 mol % DMPC; 25 mol % DChol-74 mol % DMPC; and 30 mol % DChol69 mol % DMPC. Once spread, the monolayers were left to equilibrate for 10-15 min. Before any study or measurements were done, the monolayers were compressed at a rate of 0.5
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Figure 1. Schematic phase diagram of a phospholipid-DChol monolayer. The surface pressure Π of the monolayer is plotted against the mol % DChol concentration. The boundary line between the onephase and two-phase regions determines the phase transformation pressures.
mN/m per minute until the phase transformation pressure was exceeded (see below). After a 10-15 min equilibration time, the monolayers were then expanded at the same rate as above to almost 0 mN/m, where gas-phase domains were formed. The phase transformation pressure is the pressure at which there is a transition between a region in the thermodynamic space where two phases are at coexistence and a region where only one homogeneous phase is present (Figure 1 ). The values of the phase transformation pressures for the DChol-DMPC and the DChol-DPPC systems were obtained by Benvegnu and McConnell25 and by Slotte,26 respectively. For pressures above 0 mN/m, in the absence of the gas phase, the 30 mol % DChol-69 mol % DPPC monolayer consists of DPPC-rich (bright) circular domains on a DChol-rich (dark) background. After decreasing the surface pressure toward 0 mN/m, gas-phase domains nucleate at the boundary between the liquid DChol-rich and DPPC-rich phases. The phase arrangements that are formed are reminiscent of the ones reported by Hagen and McConnell.19 They studied the threephase (two liquids and a gas phase) arrangements of a diphytanoyl-L-R-phosphatidylethanolamine (DPhPE)-DChol monolayer. They reported that the three phases will arrange in such a way that the line tension of each of the two-phase boundaries will be less than the sum of the other two line tensions. Such relations between the line tensions represent the mechanical conditions for equilibrium of three two-dimensional phases meeting at a point.27 From these inequalities, Hagen and McConnell determined the relative line tensions of the three two-phase interfaces: DChol/gas, DPhPE/gas, and DChol/ DPhPE interfaces. For all the other compositions of DChol-DPPC studied, at a constant surface pressure just above 0 mN/m (but in the absence of gas phase), DChol-rich (dark) circular domains are observed surrounded by the DPPC-rich (light) phase. The difference in size of circular domains has recently been studied by McConnell and de Koker by including in their model energy a term allowing transfer of lipids across the domain boundary.28 They determined that there is an infinite number of metastable equilibrium sizes for circular domains (determined by local minima in the energy), at a given surface pressure. Under fixed experimental conditions, the global minimum of the energy might not be reached due to a large activation energy between the metastable and stable states, so a distribution of sizes is present instead. As the above monolayer is expanded at constant surface pressure and small gas cavities are formed, new domain shapes are created. These are cloverleaf domains, each formed by the attachment of DChol-rich lobes to a gas cavity (Figure 2a). These lobes are not in contact with each other. As the
Figure 2. (a, top) Monolayer of 84% DPPC-15% DChol-1% Texas Red dye at a surface pressure just above 0 mN/m. A cloverleaf domain consists of dark DChol-rich domains attached to a small gas cavity. The bright phase is DPPC-rich. The size of a typical domain is given by R ≈10 µm, where R determines the area A of the domain, A ) πR2. (b, bottom) After expanding the monolayer in part a (top) at constant surface pressure, the gas cavity in the center of a cloverleaf domain expands. A “flower-like” domain consists of DChol-rich lobes attached to a central gas-phase domain.
monolayer is further expanded at constant surface pressure, the gas cavity in the center of a cloverleaf domain expands while the DChol-rich domains stay attached to it, forming “flowerlike” domains (Figure 2b). This process is reversible and shown schematically from top to bottom in Figure 3: as the monolayer of “flower-like” domains is compressed, cloverleaf domains are recovered in the limit when the gas phase in the center of each domain becomes a small cavity. With further compression, there is no more gas phase present and the cloverleaf domains separate into circular domains (formerly, the cloverleaf lobes). Moore et al.29 have reported “flower-like”-shaped domains, but in their case, the central domains were the liquid phase while the lobes were the gas phase. We cannot determine quantitatively the difference between a surface pressure of “just above 0 mN/m” and one that is equal to 0 mN/m. We distinguish these states qualitatively only: the former has coexistence of liquid and gas phases, while the latter has only a gas phase present. Similar cloverleaf domains have been observed in the DMPC-DChol monolayer, for DChol concentrations of 30 mol % or less, but are not shown here. The experimental expansion and compression steps are discrete. At any moment in the cycle, the monolayer consists of either two liquid phases (bottom picture in Figure 3) or two liquids and a gas phase (three top pictures in Figure 3). The formation of the cloverleaf domain from circular domains has not been observed. However, the reverse process has been observed as the detachment of lobes from two-leaf and threeleaf clover domains, producing two circular domains in the
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Perkovic´ and McConnell pressures, the dipole-dipole repulsions determine the equilibrium domain shape. At a surface pressure just above 0 mN/m, circular DCholrich domains are present. The shape-dependent energy of a circular domain of fixed area A ) πR2, where R is the radius of the circle, has been determined from (2.7) to be15
8R E ) 2πRλ - 2πRµ2 ln 2 e∆
(4.1)
The equilibrium radius Req of the circular domain is obtained by minimizing the energy in (4.1) with respect to the radius R, at constant area A, to obtain11,22
Req )
Figure 3. Schematic representation of the reversible process of forming a cloverleaf domain (third domain from top) from a “flower-like” domain (top domain) by compressing the monolayer. The central black domain represents the gas phase. The gray domains are DChol-rich, and the white background represents the DPPC-rich phase. The bottom circular domains are formed from the cloverleaf lobes once the gas phase is no longer present.
former case and a two-leaf clover domain and a circular domain in the latter case. We expect a cloverleaf domain to be stabilized by the presence of the small gas cavity. In its absence, the lobes of the cloverleaf domain would dissociate to form circular domains. The instability of a cloverleaf domain in the absence of a small gas cavity is verified in the next section. We use model curves, namely, the rose curves, to describe the shapes of the cloverleaf domains, with the assumption that no gas phase is present. We then determine, within the line tension-dipole repulsions model, the energies of the three-leaf, four-leaf, and five-leaf clover domains and compare them to the energies of the corresponding three, four, and five circular domains. The shapes with the lowest energy determine the equilibrium shapes of the domains, at a given surface pressure. 4. Model: Rose Curves Within the theoretical model, the equilibrium shape and size of a monolayer domain arises from a competition between the line tension of the domain boundary and the dipole-dipole electrostatic repulsions between the dipoles in the monolayer. A large line tension, compared to the dipole-dipole repulsions, favors a shape with a minimum domain boundary, namely, a circular shape, while large dipole-dipole repulsions, compared with the line tension, favor a distorted shape that maximizes the distance between the dipoles. In the two-phase coexistence region, the equilibrium domain shape is determined by the line tension at lower surface pressures, while at higher surface
e3∆ λ/µ2 e 8
(4.2)
After expanding the monolayer at constant surface pressure, cloverleaf domains are created, each composed of DChol-rich lobes attached to a small gas cavity. (Figure 2a). The cloverleaf domains are different from the noncircular domains studied in earlier works,15,21 as they are not formed by a transition of one circular domain to another noncircular shape, but rather, they represent systems of a gas cavity to which several lobelike domains are attached. The DChol-rich lobes in the cloverleaf domains are shaped in a way to minimize their energy. We imagine that a cloverleaf domain consists of the DCholrich lobes without the gas cavity. The shapes of these cloverleaf domains are approximated with rose curves.30,31 Therefore, we treat a cloverleaf domain as if it were one domain. This simplification is an approximation allowing us to neglect the term in the energy that describes the interaction between two domains. The cloverleaf domains consist of lobes of similar size. This aspect is approximated well with rose curves. However, while rose curves have lobes meeting at a point, the lobes of a cloverleaf domain meet at a very small gas cavity. The shapes of the three-leaf, four-leaf, and five-leaf clover domains are described with the rose curve, given in polar coordinates by30,31
b r ) a cos(nθ)eˆ r
(4.3)
where b r is the parametrization of the rose curve, a is the maximum value of ||r b||, n ) 2, 3, 5 for the four-leaf, threeleaf, and five-leaf curves, respectively, θ is the angle whose values are 0 e θ e 2π for n even, and 0 e θ e π for n odd, and eˆ r is the radial unit vector. The two-leaf clover domain is not studied here because its shape is described by the two-leaf rose curve b r ) eˆ r where r2 ) a2 cos 2θ, which is of a different form than the curve in (4.3). Cloverleaf-like structures with a greater number of lobes are not studied here either. So, in what follows, we limit ourselves to three-leaf, four-leaf, and fiveleaf clover domains (Figure 4). The rose curves for the three-leaf, four-leaf, and five-leaf clover domains at a constant area A are given, respectively, by
b r ) 2R cos(3θ)eˆ r
(4.4)
b r ) x2R cos(2θ)eˆ r
(4.5)
b r ) 2R cos(5θ)eˆ r
(4.6)
Substituting one of the rose curve expressions from (4.4)(4.6) into the energy equation (2.7) to determine the energy of that specific cloverleaf domain is not practical. The value of
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J. Phys. Chem. B, Vol. 101, No. 3, 1997 385 further than is possible for other, general shapes (as for example the ovals of Cassini).21 However, the energy values for the three different cloverleaf domains must still be calculated numerically (see Appendix). 5. Results The energy in (A.16) can be rewritten as a function of the ratio R/Req. Such an expression for the energy is convenient for comparing the theoretical and experimental results. The quantity R can be measured directly from the micrographs of the monolayers: it is the radius for a circular domain, or it is proportional to the longest axis of a lobe in a cloverleaf domain. The quantity Req, defined in (4.2), is the equilibrium radius of a circular domain and is obtained from the measurements of the line tension λ and the dipole moment density difference µ, at a constant surface pressure. For a circular domain, the energy is given by (4.1). Inserting into that expression the equation for Req, one obtains the following expression for the dimensionless energy of a circular domain of radius R:
R Req E ) ln 2πµ2Req Req eR
(5.1)
In this work, the energy of the cloverleaf domain is compared to the energy of the appropriate number of circular domains that would be created if the lobes of the cloverleaf domain would separate from the gas cavity. In this comparison, the areas of the cloverleaf domains and of the systems of three, four, and five circular domains are kept constant at A ) πR2 (i.e. the area of three circles equals the area of the three-leaf clover domain, and similarly for the four-leaf and five-leaf clover domains and four and five circles, respectively). So, at a given constant area A, the energies of the systems of three, four, and five circular domains are given, respectively, by the following expressions:
Figure 4. Rose curves, given by r ) a cos nθ, used to model the shapes of cloverleaf domains in Figure 2: (a, top) three-leaf rose curve (a ) 2R, n ) 3, and θ ) 0 f π), (b, middle) four-leaf rose curve (a ) x2R, n ) 2, and θ ) 0 f 2π), and (c, bottom) five-leaf rose curve (a ) 2R, n ) 5, and θ ) 0 f π ). R is the radius of a hypothetical circle whose area is equal to πR2.
the energy depends on the parameter ∆, which for different values will give different values for E. De Koker and McConnell21 noted that the value of ∆ should only affect the bulk and interface energies. This led them to manipulate the expression for the shape-dependent energy in (2.7), to obtain an expression that contains ∆ explicitly only in the interface energy term. Such an energy expression is useful for understanding the ∆-dependence of E as well as for numerical calculations of the energy of domains of general shape (only the energy of simple shapes, such as circular shapes, can be determined analytically). In the Appendix, we follow their presentation to determine an analytical expression for the energy of cloverleaf domains. Due to the simplicity of the rose-curve equations, the energy expression can be developed somewhat
E R x3Req ) x3 ln 2 Req eR 2πµ Req
(5.2)
E R 2Req ) 2 ln 2 R eR 2πµ Req eq
(5.3)
E R x5Req ) x5 ln 2 Req eR 2πµ Req
(5.4)
A similar analysis can be done for the cloverleaf domains, starting from (A.16). The dimensionless energy of the threeleaf, four-leaf, and five-leaf clover domains as a function of R/Req is given by
(
)
P ˜ (n) R 8Req (E˜ (n)/P˜ (n))-3 E e ) ln 2 2π Req R 2πµ Req
(5.5)
We first analyze the theoretical results. In Figure 5a we have plotted the dimensionless energy E/(2πµ2Req) versus the dimensionless size parameter R/Req for one circular domain of area A ) πR2 (eq 5.1), three circular domains of total area A ) πR2 (eq 5.2), and the three-leaf clover domain whose area is also equal to A ) πR2 (eq 5.5). As mentioned in the Appendix, P ˜ (n) and E ˜ (n) in (5.5) are determined numerically and are constants for a given domain shape. The equilibrium size of a circular domain is R ) Req (see section 4 ). At that value
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Figure 6. Plot of the dimensionless energy E/(2πµ2Req) versus the parameter R/Req for one circular domain, five circular domains, and the five-leaf clover domain (n ) 5) whose total area equals the sum of the areas of the five circular domains and is also equal to the area of the single circular domain (A ) πR2).
Figure 5. Plot of the dimensionless energy E/(2πµ2Req) versus the parameter R/Req: (a, top) for one circular domain, three circular domains, and the three-leaf clover domain (n ) 3) whose total area equals the sum of the areas of the three circular domains and is also equal to the area of the single circular domain (A ) πR2); (b, bottom) an enlargement of a section of the plot in part a.
of R, the circular domain shape is the stable one, when compared with the three circular domains and the three-leaf clover domain (Figure 5a). Figure 5b, which is an enlargement of a section from Figure 5a, shows that for 1.349 < R/Req < 1.519 three circular domains represent the most stable system, while for R/Req > 1.519, the three-leaf clover domain is the energetically favorable domain shape. This trend is in accordance with the line tension-dipole repulsions model: for larger domain sizes, the dipole-dipole repulsions are more important than the line tension of the domain boundary. Similar results are found for the five-leaf clover domain. In Figure 6, we plot the dimensionless energy E/(2πµ2Req) versus R/Req for the one and five circular domains, as well as for the five-leaf clover domain, at constant area A. For R ) Req, the circle is the equilibrium domain shape. It is the stable shape until R/Req ) 1.577 is reached. For 1.577 < R/Req < 3.527, the stable configuration consists of five circles, while for R/Req > 3.527, the stable shape is the five-leaf clover domain. The plot of the energy for the four circles and four-leaf clover domains is not shown here. It is only different from the earlier plots in that the value of R/Req at which the four-leaf clover domain becomes more stable than the four circular domains is very large and equals 351. This is understood by comparing the energies of the cloverleaf domains in Figure 7. The fourleaf clover domain (n ) 2) is always less stable than either the three-leaf (n ) 3) or the five-leaf (n ) 5) clover domains. The explanation for this unexpected result probably lies in the shapes of the rose curves used to model the cloverleaf domains. The lobes of the four-leaf clover domain (Figure 4b) are very close to each other, almost touching, while the lobes in the three-leaf and five-leaf clover domains are well separated from each other
Figure 7. Plot of the dimensionless energy E/(2πµ2Req) versus the parameter R/Req for the three-leaf (n ) 3), four-leaf (n ) 2), and fiveleaf (n ) 5) clover domains at constant area A ) πR2.
(Figure 4a,c, respectively). The close proximity of the lobes in the four-leaf rose curve creates more dipole-dipole repulsion between the lobes, which can lead to a higher calculated energy for the four-leaf clover domain. We now compare the theoretical with the experimental results, using measurements performed by Benvegnu and McConnell. They determined experimental values for the line tension λ of the domain boundary in a DMPC/DChol monolayer as a function of the surface pressure.25 They also measured the values of the dipole moment density difference µ between coexisting DMPC-rich and DChol-rich liquid phases, as a function of the surface pressure.32 The order of magnitude of the experimental quantity λ/µ2 at 0 mN/m surface pressure is λ/µ2 ≈ O(17). We expect this order of magnitude to be true for our monolayer as well. The value of ∆ in Req is taken to be equal to ∆ ) 1 nm. Then, Req at 0 mN/m surface pressure is equal to Req ≈ 60,000 µm, a very large value. The typical size of a domain in Figure 2 is given by R ≈ 10 µm. Therefore, under the conditions of our experiment, R/Req < 1 and the circular domains are more stable than the cloverleaf domains (Figure 5 and Figure 6). Each cloverleaf domain is stabilized by a small gas cavity located at its center. It should be noted again that the predicted theoretical stability of the cloverleaf domains, for larger values of R/Req, compared to that of the circular ones is consistent with the line tension-dipole repulsions competition model. For large R/Req values, the dipoledipole repulsions are more important than the line tension, thus favoring elongated domains with more boundary. However,
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these results do not predict the existence of cloverleaf domains. Other domain shapes might have lower energies for these larger values of R/Req.16 6. Conclusions Cloverleaf domains are present at a surface pressure just above 0 mN/m, in DPPC-DChol and DMPC-DChol monolayers. A cloverleaf domain consists of DChol-rich lobes attached to a small gas cavity. When the monolayer is expanded at constant surface pressure, the amount of gas phase increases. From these observations, we conclude that the initial formation of cloverleaf domains is driven by the subsequent favorable domain arrangement of the gas and liquid phases.19 The stability of a cloverleaf domain is studied by describing its shape with a model rose curve, r ) a cos nθ. Then, the energy of the domain is calculated and compared to the energy of circular domains that would form if the lobes of the cloverleaf domain would separate from the gas cavity. Our theoretical calculations show that the cloverleaf domain is not a stable shape when compared to circular domains, in the case when the gas cavity in its center is ignored. Therefore, a cloverleaf domain appears to be stabilized by the small gas cavity in its center. This argument is further supported by the presence of “flowerlike” domain arrangements of the gas and DChol-rich liquid, upon further expansion of the monolayer (Figure 2b). The modeling of cloverleaf domains with rose curves does not change the qualitative picture of the region of stability of these domains relative to the region of stability of circular domains. Only the specific energy values and transition points depend on the model curves used. The experiments and the calculations in this work raise questions about the interactions between the three phases (two liquids and a gas phase) in monolayers. The effects of the dipole moment densities in each of the phases and of the line tensions of the two-phase boundaries on the three-phase arrangement might add understanding to the phenomenon of spreading of one phase onto another. A. Appendix Here, we determine the expression for the energy E of a cloverleaf domain and describe how to calculate it numerically. We start with the shape-dependent energy of an isolated monolayer domain of constant area, given in (2.7) by
µ2 E ) λP - I(∆) 2
(A.1)
where I(∆), for n even, is given by
I(∆) )
∫0
2π
∫0
dφ1
2π
db r db r (φ )‚ (φ ) dφ 1 dφ 2 dφ2 [(b(φ r 1) - b(φ r 2))2 + ∆2]1/2
(A.2)
In (A.2), the two integrals are over the polar angles φ1 and φ2. When n is odd, the rose curves are obtained when φ1 and φ2 range from 0 f π only. What follows is a derivation of the energy of cloverleaf domains, for n even only. For n odd, the derivation is a straightforward extension of the one for n even. We seek to determine an analytical expression for the energy of cloverleaf domains in the limit of ∆ f 0 (∆ is introduced in the electrostatic energy expression to keep the integrals from diverging). In the limit ∆ f 0, the double integral in (A.2) diverges. This divergence can be separated out by defining θ ≡ φ2 - φ1 and rewriting I(∆) in terms of θ and φ1 to obtain21
I(∆) )
db r db r (φ )‚ (θ + φ1) dφ 1 dφ dφ1 -πdθ (A.3) [(b(φ r 1) - b(θ r + φ1))2 + ∆2]1/2
∫0
2π
∫
π
In the limit of θ f 0, the Taylor expansions of the numerator and denominator in the integrand of I(∆), taken to their first approximation, yield the integrand
[(r′)2 + r2]1/2
(A.4)
[θ2 + ∆2/((r′)2 + r2)]1/2
where r ≡ r(φ1) and r′ ≡ dr/dφ1. The double integral I(∆) then becomes
I(∆) )
[(r′)2 + r2]1/2
∫02πdφ1∫-ππdθ[θ2 + ∆2/((r′)2 + r2)]1/2 + I′(∆)
(A.5)
where I′(∆) is given by
I′(∆) )
[
∫02πdφ1∫-ππdθ ×
]
db r db r (φ )‚ (θ + φ1) [(r′)2 + r2]1/2 dφ 1 dφ [(b(φ r 1) - b(θ r + φ1))2 + ∆2]1/2 [θ2 + ∆2/((r′)2 + r2)]1/2 (A.6)
These results are equivalent to eq 19 and eq 20 of ref 21. The first term in (A.5) can be integrated with respect to θ to obtain, in the limit of ∆ f 0,
2
2π[(r′)2 + r2]1/2 ∆
∫02πdφ1[(r′)2 + r2]1/2 ln
(A.7)
Substituting for r the general expression for the rose curve, r ) a cos nφ1, for n even and n > 0, one can solve the above integral to obtain
2πa 4a 2 (n + 1)K(m) + 8an ln + 4an ln n E(m) n ∆e
(
( )
)
(A.8)
where m ≡ (n2 - 1)1/2/n. The calculation for n odd and n > 0 yields
πa 2a 2 (n + 1)K(m) + 4an ln + 2an ln n E(m) n ∆e
(
( )
)
(A.9)
In (A.8) and (A.9), K and E are the complete elliptic integrals of the first and second kind, respectively. The value of n for the four-leaf, three-leaf, and five-leaf curves is n ) 2, 3, 5, respectively, while a ) x2R for n even and a ) 2R for n odd. The substitution of n ) 1 in (A.9) describes a circular domain, r ) a cos φ1, whose origin is at (a/2,0) and whose radius is a/2. In that case, the first term in I(∆), given by (A.9), equals 2P ln(P/∆) (equivalent to eq 22 of ref 21), where P is the perimeter of the circular domain, P ) πa. The second term in I(∆), given by I′(∆) in (A.6), does not diverge for θ f 0 and ∆ f 0. In the limit of θ f 0, I′(∆) f 0 as well. In the limit of ∆ f 0, I′(∆) f finite limit.21 In the limit of ∆ f 0, we keep the first approximation, I′(∆) ≈ I′(0). This quantity is determined numerically (see below). The first term in (A.5), given by (A.8) for n even and by (A.9) for n odd, may be further simplified. The perimeter of a rose curve is equal to P(n) ) 4anE(m) for n even and is equal to P(n) ) 2anE(m) for n odd, where m was defined above. Defining G(n) as
388 J. Phys. Chem. B, Vol. 101, No. 3, 1997
G(n) )
(n2 + 1)K(m) n2E(m)
Perkovic´ and McConnell
π (A.10) + ln n + 2 ln 2enE(m)
one obtains for the first term in (A.5), for both n even and n odd,
R P P(n) G(n) + 2P(n) ln + 2P(n) ln ∆ R
(A.11)
where R is the radius of a hypothetical circle of area A ) πR2, which is also equal to the area of a cloverleaf domain. Then, the shape-dependent energy is given by
E) λP(n) -
µ2 I(∆) 2
[
]
I′(0) 1 R - P(n) G(n) ) P(n)λeff + µ2 P(n) ln 2 P(n) 2
(A.12)
where λeff ≡ λ + µ2 ln(∆/R). In terms of the dimensionless quantities21
P ˜ (n) ≡ P(n)/R
(A.13)
˜I′(0) ≡ I′(0)/2R
(A.14)
1 E ˜ (n) ≡ -P ˜ (n) ln P ˜ (n) - I′(0) - P ˜ (n) G(n) (A.15) 2 the shape-dependent energy is given by
[
E ) µ2R
λeff P ˜ (n) + E ˜ (n) µ2
]
(A.16)
The energy E of a cloverleaf domain is determined as a function of the dimensionless parameter λeff/µ2, representing the competition between the line tension of the domain boundary and the dipole-dipole repulsions between the dipoles in the monolayer. For a circle, the value of G is G(n)1) ) 0, and the main result of ref 21 is recovered. The quantities P ˜ (n) and E ˜ (n) in (A.16), which depend only on the shape of a domain and not on its size, are calculated numerically for each cloverleaf domain (see below). P˜ (n) was obtained analytically as well (see above), but for consistency, the numerical values are used in E. We now describe the numerical calculation of the energy E/2πµ2Req in (5.5) for the three-leaf, four-leaf, and five-leaf clover domains. The quantities to be determined are the numerical values of the integral I′(0), given in (A.6), and the perimeter of the leafed domain P(n). These are the quantities that are used to define P ˜ (n) and E ˜ (n) in (5.5). The integral I′(0) in (A.6), even though finite at ∆ f 0, is difficult to solve numerically because the angle difference θ in the denominator can be 0. Therefore, we use the following method to determine I′(0), which is described here for the fourleaf clover domain. The calculations are similar for the threeleaf and five-leaf clover domains. One substitutes (4.5) into (A.2) to obtain the integral I(∆/R), which is then determined for various, decreasing values of the parameter ∆/R. The first term in I(∆/R) can be determined analytically (eq A.7). However, in order for the calculations to be consistent, we
determine this first term numerically for various values of ∆/R f 0. Subtracting that first term from I(∆/R), one obtains I′(∆/ R), which is then extrapolated to ∆/R f 0. The values of I′(0) for n ) 2, 3, 5 are -111.55R, -94.60R, and -191.70R, respectively. The numerical integrations were done using the Romberg integration method. We use the minimum number of segments in the integration interval that is required for the results to be independent of segment number. The smallest value of ∆/R that is used in the integration method is ∆/R ) 0.01. Smaller values require a larger number of segments and a substantial increase in computational time. For consistency, the perimeter is also calculated numerically to give Pnum(n)3) ) 13.36R, Pnum(n)2) ) 13.70R, and Pnum(n)5) ) 21.01R. With the values of P(n) and I′(0), for n ) 2, 3, 5, the dimensionless energy E/2πµ2Req of the cloverleaf domains is then calculated. Acknowledgment. This work was supported by the National Science Foundation, Grant MCB9316256. We thank John P. Hagen for helpful suggestions and discussions concerning the experimental part of this work and Rudi de Koker for discussions on monolayer domain shapes and stability calculations. References and Notes (1) von Tscharner, V.; McConnell, H. M. Biophys. J. 1981, 36, 409. (2) Knobler, C. M. Science 1990, 249, 870. (3) Lo¨sche, M.; Sackmann, E.; Mo¨hwald, H. Ber. Bunsen-Ges. Phys. Chem. 1983, 87, 848. (4) Andelman, D.; Broc¸ hard, F.; de Gennes, P.-G.; Joanny, J.-F. C. R. Acad. Sci. Paris. 1985, 301, 675. (5) Andelman, D.; Broc¸ hard, F.; Joanny, J.-F. J. Chem. Phys. 1987, 86, 3673. (6) Peters, R.; Beck, K. Proc. Natl. Acad. Sci. U.S.A. 1983, 80, 7183. (7) McConnell, H. M.; Tamm, L. K.; Weis, R. M. Proc. Natl. Acad. Sci. U.S.A. 1984, 81, 3249. (8) Weis, R. M.; McConnell, H. M. Nature 1984, 310, 47. (9) Gaub, H. E.; Moy, V. T.; McConnell, H. M. J. Phys. Chem. 1986, 90, 1721. (10) Keller, D. J.; McConnell, H. M.; Moy, V. T. J. Phys. Chem. 1986, 90, 2311. (11) McConnell, H. M.; Moy, V. T. J. Phys. Chem. 1988, 92, 4520. (12) Subramaniam, S.; McConnell, H. M. J. Phys. Chem. 1987, 91, 1715. (13) Rice, P. A.; McConnell, H. M. Proc. Natl. Acad. Sci. U.S.A. 1989, 86, 6445. (14) Keller, D. J.; Korb, J. P.; McConnell, H. M. J. Phys. Chem. 1987, 91, 6417. (15) Lee, K. Y. C.; McConnell, H. M. J. Phys. Chem. 1993, 97, 9532. (16) Vanderlick, T. K.; Mo¨hwald, H. J. Phys. Chem. 1990, 94, 886. (17) McConnell, H. M. J. Phys. Chem. 1990, 94, 4728. (18) Deutch, J. M.; Low, F. E. J. Phys. Chem. 1992, 96, 7097. (19) Hagen, J. P.; McConnell, H. M. Colloids Surf. A 1995, 102, 167. (20) McConnell, H. M. Annu. ReV. Phys. Chem. 1991, 42, 171. (21) de Koker, R.; McConnell, H. M. J. Phys. Chem. 1993, 97, 13419. (22) McConnell, H. M.; de Koker, R. J. Phys. Chem. 1992, 96, 7101. (23) McConnell, H. M.; Bazaliy, Y. B. Proc. Natl. Acad. Sci. U.S.A. 1995, 92, 8823. (24) Klingler, J. F.; McConnell, H. M. J. Phys. Chem. 1993, 97, 2962. (25) Benvegnu, D. J.; McConnell, H. M. J. Phys. Chem. 1992, 96, 6820. (26) Slotte, J. P. Biochim. Biophys. Acta 1995, 1238, 118. (27) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon Press: Oxford, 1982; Chapter 8. (28) McConnell, H. M.; de Koker, R. Langmuir 1996, 12, 4897. (29) Moore, B. G.; Knobler, C. M.; Akamatsu, S.; Rondelez, F. J. Phys. Chem. 1990, 94, 4588. (30) Lawrence, J. D. A Catalog of Special Plane CurVes; Dover: New York, 1972. (31) Artobolevskii, I. I. Mechanisms for the Generation of Plane CurVes; Macmillan: New York, 1964. (32) Benvegnu, D. J.; McConnell, H. M. J. Phys. Chem. 1993, 97, 6686.
