J . Phys. Chem. 1986, 90, 1721-1725 crystal. The effect of the left-right inequivalence can be seen by considering the torus-shaped crystal in Figure 5 . Define = X, and A,, = Xi and take A, < Xi corresponding to the outer and inner line tensions. W e use the following expression for the free energy of the torus-shaped crystal. F = 2aX,R,
+ 2~XiRi+ xK(R,,Ri)
(30)
Here R, and Ri are the outer and inner radii and aK(R,,Ri) is the deformation energy of the crystal. In terms of the variables R+ = R, + Ri, R- = R, - Ri, X+ = A, + Xi, and X_ = A, - Xi, this free energy is
+
F = rX+R+ aLR_
+ aK(R+,R-)
(31)
Under the constraint that the area of the crystal ( A , = aR+R-) is a constant, the free energy F is extremum when
R + / R _= (A-
+ dK/aR-)/(X++ aK/aR+)
(32)
The torus-shaped crystal cannot be in a state of thermodynamic equilibrium in the absence of a deformation energy ( K = constant) since the absolute values of R+/R-and XJX+ are greater than one and less than one, respectively. Note that if K varies as (RJR,)’, then aKIaR- is positive and aK/dR+is negative, so that in principle eq 32 can have real solutions. Torus-shaped crystals of DPPC can be formed experimentally in the presence of cholesterol and may represent equilibrium structures.13 The purpose of the above calculation is to show how one distorted crystal shape might be an equilibrium state, or at least a metastable equilibrium state. Similarly, it is possible that the spiral shapes result from the interplay of chiral line tensions and crystal distortion energies. In the case of the torous, assuming that X, and Xi are functions of cholesterol concentration, then a change in this concentration should lead to a change in R+/R.. and thus in both the width of the torus and its size (outer radius). Since cholesterol is optically active and has markedly different effects on the thickness of spiral monolayer crystals of R- and S-DPPC,” it is clear that the thickness of the spiral crystals is affected by line tension (assuming that cholesterol does not
1721
partition into the crytal phase). No detailed experimental studies have yet been carried out on the effects of cholesterol concentration on the shapes of the torus- and spiral-shaped crystals.
Conclusion In the present paper we have given a number of simple thermodynamic calculations that bear on the observed shapes of single two-dimensional crystals of the lipid DPPC at the air-water interface, in the presence of low concentrations of cholesterol. One important point is that the remarkable uniformity of the widths of the DPPC crystals in the presence of cholesterol is a consequence of the thermodynamics of the system. A second thermodynamic feature is the anomalous growth behavior of these crystals; on compression the crystals are calculated to grow in area but simultaneously become thinner, an effect which has been verified e~perimentally.’~ We believe that these simple calculations provide a useful background for more experimental studies that include the long-range effective forces between different crystals and presumably between different regions of an individual crystal. A discussion of the molecular basis of the effect of cholesterol in reducing the line tension between the crystal and fluid phases will be given elsewhere. Subsequent work will also describe in detail the reversible growth and decay of the monolayer c r ~ t a 1 s . l ~ Note Added in Proof. We have now extended our model15 to include a width-dependent crystal free energy that is dominated by long-range Coulomb (dipole-dipole) interactions in the monolayer. In this extended model the equilibrium crystal width w depends exponentially on the ratio Al/Aw2, where Aw is the difference in dipole density in the crystal and fluid phases. Acknowledgment. This work was supported by N S F Grant PCM80-21993 and the DOD Equipment Grant NO00 14-84-G0210. We have benefited from many helpful discussions with V. Moy and R. M. Weis. W e also acknowledge the Deutscheforschungsgemeinshaft for support of H. Gaub, and an NIH fellowship to D. J. Keller. (15) D. J. Keller, H. M. McConnell, and V . Moy, submitted to J . Phys. Chem.
Reversible Formation of Plastic Two-Dimensional Lipid Crystals H. E. Gaub, V. T. Moy, and H. M. McConnell* Stauffer Laboratory f o r Physical Chemistry, Stanford University, Stanford, California 94305 (Received: October 8, 1985)
Individual dipalmitoylphosphatidylcholine(DPPC) crystals formed in a monomolecular film of DPPC at the air-water interface have been studied by epifluorescence microscopy. For this purpose a new trough design was developed to enable the observation and tracking of a chosen crystal over periods of hours. We found that various observed crystal shapes are multiples of a basic domain structure. Each domain has a distinct in-plane director but no macroscopic symmetry. The complex crystal shapes were found to be reversible with respect to growth and decay. Trace amounts of cholesterol drastically alter the shapes of the crystals but change none of the above-mentioned properties. Upon monolayer compression, the crystals grow uniformly at the fluid-crystal interface while simultaneously elongating and thinning. This remarkable effect requires a high lateral plasticity of these two-dimensional crystals.
Introduction Monolayers of amphiphatic molecules at the air-water interface have been studied for a century,l-3 part]y due to the relative accessibility of a two-dimensional model system for thermodynamic
experiments. Furthermore, these systems have been4 and still ares useful tooh in studying biological membranes. Technological applications O f mOnOmOkCUlar films such as Coatings On Various substrates have become i m ~ o r t a n t . ~ . ’Recently, microscopic
(1) Pockels, A. Nature (London) 1891, 43, 437-439. (2) Langmuir, 1. J . Am. Chem. Soc. 1917, 39, 1848-1861. (3) Adamson, “Physical Chemistry of Surfaces”, 3rd ed.; Wiley: New York, 1973.
(4) McConnell, H . M.; Watts, T. H.; Weis, R. M.; Brain, A. A. BBA Biomembr. Rev.,submitted for publication. (5) Heck], W.; Losche, H.; Scher, H.; Mohwald, H. BBA Bioenergetics, in press.
0022-3654/86/2090-1721$01.50/0
0 1986 American Chemical Society
1722 The Journal of Physical Chemistry, Vol. 90, No. 8, 19
Gaub et al.
MICROSCOPE OBJECTIVE
- - -DPPC- DPPC/w
MOTORIZED MOTOR1 ACTUA
I 2% Cholesterol
rC
/
,
1
MOTORIZED X-Y
MICROSCOPE OBJECTIVE
STAGE
~
1
I
1
0.8 1.o 1.2 AREA (nrn2/molecule) Figure 2. Pressure-area diagram of R-DPPC with and without cholesterol. Surface pressure was measured by a pressure transducer and a Wilhelmy plate a t a relative film compression rate of 3.6 X 10-4/s. As our small trough area did not allow the direct determination of the molecular area we calibrated the horizontal axis using commonly accepted literature values.” The inner scale gives the area fraction of solid lipid in the coexistence region. 0.6
Figure 1. Schematic of flow-free trough
studies have revealed the heterogeneity of phospholipid monolayers in the coexistence regions.*-1° Crystalline domains having sizes up to e l 0 0 fim have been reported to have shapes related to the chirality of the phospholipids.’’ Low concentrations of cholesterol in the monolayer increases the length of the interface of the crystalline and the fluid phase.’? We have continued these studies by investigating the growth and decay of individual crystals in a R-DPPC monolayer containing 2% cholesterol in the fluid-crystal coexistence region. Our goals were to clarify the extent to which the shapes of these structures are determined by the dynamics of the cry~tallization,’~ and to trace the evolution of the observed crystal shapes. Throughout much of our following discussion we use the term “crystal” for brevity. It should be understood that these crystals may or may not be crystals in the usual sense. As discussed later, the crystals are solidlike domains. They may be single crystals in the usual sense, may be deformed single crystals, or may be a solidlike phase with only short-range molecular positional order, and long-range orientational order.
Materials and Methods Experimental Setup. Figure 1 shows the convection-free Langmuir trough that was developed to permit fluorescence microscopic observation of a single domain at the air-water interface. In order to scan all of the monolayer, a small trough was mounted on a motorized x-JJ stage. Both the trough and the movable barrier were milled from single blocks of Teflon. The depth of the trough was kept shallow (=4 mm) to minimize surface flow by convection. At full expansion, the surface dimension was approximately 25 mm by 80 mm. A large microscope coverglass (moved by the barrier) slides 1 mm above the air-water interface and closes the monolayer compartment. As a result the motion of the monolayer was reduced to
