1352
Langmuir 1988, 4, 1352-1358
here, from NMR relaxation studies of interfacial water in other heterogeneous solution systems, e.g., normal ionic micelle^,^' p o l y m e r ~ , 2proteins,3O ~*~~ and colloidal silica.31 Finally, our data lend some support (although not conclusively) to the suggestionz3that small reversed micelles (27) Halle, B.; Carlstrom, G. J . Phys. Chem. 1981, 85, 2142. (28) Halle, B.; Piculell, L. J. Chem. SOC.,Faraday Trans. 1 1982, 78, 255. (29) Breen, J.; Huis, D.; de Bleijser, J.; Leyte, J. C. J. Chem. SOC., Faraday Trans. 1 1988,84, 293. (30) Halle, B.; Andersson, T.; ForsBn, S.; Lindman, B. J. Am. Chem. SOC.1981,103, 500. (31) Piculell, L. J. Chem. Soc., Faraday Trans. 1 1986,82,387.
coexist with the larger water droplets in the microemulsion. Acknowledgment. We are indebted to Ulf Henriksson for I7O relaxation measurements at frequencies below 13 MHz. Grants from the Swedish Natural Science Research Council are gratefully acknowledged. This work was not supported by any military agency. Registry No. AOT, 577-11-7;H20, 7732-18-5;D2, 7782-39-0; 1 7 0 , 13968-48-4;isooctane, 540-84-1. (32) Lankhorst, D.; Schriever, J.; Leyte, J. C. Ber. Bunsenges. Phys. Chem. 1982,86, 215.
Cholesterol Concentration Dependence of Quasi-Crystalline Domains in Mixed Monolayers of the Cholesterol-Dimyristoylphosphatidic Acid System W. M. Heckl,? D. A. Cadenhead,*tt and H. Mohwalds Physics Department (Biophysics) E22, Technische Universitat Munchen, 0-8046 Garching, Federal Republic of Germany, Johannes-Gutenberg- Universitat, Institute fur Physikalische Chemie, 0-6500 Mainz, Federal Republic of Germany, and Department of Chemistry, State University of New York at Buffalo, Buffalo, New York 14214 Received February 18, 1988. I n Final Form: July 5, 1988 The mixed monomolecular film system dimyristoylphosphatidic acid/cholesterol has been studied at 10 “C and pH 11 over the concentration range 0-25 mol % cholesterol by using a combination of classical surface pressure/area per molecule isotherm determinations and fluorescence microscopy. The latter technique permitted the direct observation of the quasi-crystalline condensed phases of this system and allowed both their number and shape to be documented. From the isotherms, the fluid/condensed-phase transition is observed to shift to higher surface pressures over the 0-20 mol % cholesterol concentration range. Broadening of the transition is detectable between 5 and 10 mol % and increases with increasing cholesterol concentration until at 25 mol % the transition is no longer detectable. The fluorescence micrographs parallel the isotherm behavior. Between 1 and 5 mol % cholesterolspiral condensed domains are observed. Below 1mol %,and between 7.5 and 20 mol %, other structures are found, but at 25 mol % condensed domains are no longer detectable. Interpretation of these results suggests that, at or below 5 mol % cholesterol, the sterol acts to reduce the line tension between the fluid mixed phase and a nearly pure phosphatidic acid condensed phase. Changing line width as a function of compression or degree of crystallization and quasi-crystallite nucleation is consistent with this picture. A t higher cholesterol concentrations the situation is complicated by probable cholesterol incorporation into the condensed phase, ultimately resulting in an elimination of the fluid/gel transition. The situation is further complicated by possible artifactural effects at high compression due to high fluorescentprobe concentrationsin the residual “fluid” phase. Introduction That membranologists have been fascinated by studies of cholesterol/phospholipid mixed systems is no surprise when it is realized that the erythrocyte membrane is among the most frequently studied and that the lipid composition of this membrane is approximately 42 mol % cho1esterol.l Many of these studies have utilized a monomolecular film membrane model system; indeed, one of the earliest studies which first described the condensing effect of cholesterol on fluid phospholipids was published in 192f1.~ More recently, a wide variety of techniques has been employed on both monolayer and bilayer systems, with the results generally paralleling one *Author to whom all correspondence should be addressed. + Technische Universitat Munchen. f State University of New York at Buffalo. 5 Johannes-Gutenberg-Universitat.
0743-746318812404-1352$01.50/0
A great deal of attention, particularly in monolayer studies, has focussed on possible molecular complexes, with 2:1,1:1, and 1:2 phospholipid/cholesterol complexes being the most commonly cited! However, many of these studies examined only the precise molar ratios of interest, and only two groups5v6 carried out such studies over the entire compositional range. Muller-Landau and Cadenhead5 in addition to examining a myristic acid/cholesterol mixed system also examined the dipalmitoylphosphatidylcholine (DPPC)/cholesterol mixed system over its entire compo-
s.
(1) O’Brian, J. J . Theor. Biol. 1967, 15, 307. (2) Leathes, J. B. Lancet 1925, 1, 853. (3) Demel, R. A.; de Kruyff, B. Biochim. Biophys. Acta 1976,457,109. (4) Cadenhead, D. A. In Structure and Properties of Cell Membranes; Benga, G., Ed.; CRC: Boca Raton, FL, 1985; Chapter 2, Vol. 111, p 21. (5) Miiller-Landau,F.; Cadenhead, D. A. Chem. Phys. Lipids 1979,25, 315. (6) Albrecht, 0.; Gruler, H.; Sackmann, E. J. Colloid Interface Sci.
1980, 79, 319.
0 1988 American Chemical Society
Langmuir, Vol. 4, No. 6, 1988 1353
CholesterollPhospholipid Mixed Systems sitional range a t pH 5.5. These authors observed an expanded liquid-crystalline/condensed gel-like phase transition for DPPC which shifted to higher pressures between 0 and 9 mol % cholesterol and then remained essentially constant, broadening and disappearing a t 22-23 mol % cholesterol. These results were interpreted in terms of molecular models with acyl chain/cholesterol ratios of 21:l and 7:l in which the acyl chains were arranged concentrically around the cholesterol and were presumed to form in an expanded or fluid phase. Maximum condensation was observed around 42 mol % cholesterol in the fluid phase, decreasing to lower cholesterol concentrations at higher surface pressures, and this too was interpreted in terms of each cholesterol being surrounded by seven shared acyl chains (an acyl chain cholesterol ratio of 3:1), an arrangement which could also produce maximum chain/ cholesterol contact. Above 42 mol % a pure cholesterol phase was assumed to form. Subsequently, Cadenhead4 suggested that the DPPC/cholesterol system could in part be described as an eutectic system. Albrecht et aL6 also studied the DPPC/cholesterol system a t pH 5.5 and obtained results which were similar to but not identical with, those of Miiller-Landau and CadenheadP The onset of the expanded/condensed transition (at) showed a similar shift to higher surface pressures with the first 5 mol % of cholesterol added but then broadened and shifted back to slightly lower pressures a t concentrations above 10 mol % cholesterol. This behavior was cited as being “not inconsistent” with a eutectic system, but it was also consistent with azeotropic behavior. For Albrecht et al. the expanded/condensed transition remained detectable up to 25 mol % cholesterol. Indeed, first isotherm differential plots suggested that the transition might persist as far as 30 mol % cholesterol. Once again maximum condensation was observed at cholesterol concentrations ranging from 48 mol % at low pressures to 20 mol % at high pressures. In both instances5g6significant cholesterol miscibility was postulated to occur in fluid or expanded phases, but limited miscibility was predicted in condensed, gel, or solid phases. In addition the Munich group6 studied the dimyristoylphosphatidic acid (DMPA)/cholesterol system at pH 5, where DMPA is singly ionized. The results were qualitatively similar to those obtained for the neutral DPPC/cholesterol system, but the expanded/condensed transition showed neither a plateau nor a maximum with increasing cholesterol concentration. Furthermore, the transition shifts by more than 5 dyn/cm, approximately a factor of 10 greater than that found for the DPPC/ cholesterol system. In both systems the transition broadens with increasing cholesterol content with the isotherm change in slope a t the onset of the transition (TJ decreasing until the transition is no longer detectable. More recently, fluorescence optical microscopy (EOM) has been applied to films of pure DPPC7v8 and of DMPA/cholesterol mixturesgJOby locating the film balance in the objective stage of an optical microscope and adding a trace amount of a fluorescent phospholipid probe. The approach has been used to visualize multimicron domains of a crystalline condensed phase in a fluid expanded phase by selecting a probe that has differing solubilities in the two phases, typically one which is essentially (7) Losche, M.; Sackmann, E.; Mohwald, H. Ber. Bunsen-Ges Phys. Chem. 1983.87, 848. (8) Weis, R. M.; McConnell, H. M. Nature (London) 1984, 310, 47. (9) Heckl, W. M.; Ldsche, M.; Cadenhead, D. A.; Mohwald, H. Eur. Biophys. J . 1986, 14, 11. (10) Heckl, W. M.; Mohwald, H. Ber. Bunsen-Ges. Phys. Chem. 1986, 90,1159.
insoluble in the crystalline phase. One interesting feature is that the solid domains formed from chiral molecules clearly reflect that chirality.a10 Specifically with DMPA, a t both pH 5 and 11 (singly and doubly ionized DMPA) a mixture of rounded and dendritic forms was observed? The addition of cholesterol to singly ionized DMPA resulted in an increase in the proportion of dendritic particles. More interesting was the observation that, in the 1-5 mol % cholesterol range, the presence of cholesterol induced spiral formation for both DPPC8 and doubly ionized DMPA.+” This extraordinary effect was explained in terms of the formation and stabilization of a tilted chain phospholipid condensed state. In the DPPC case the tilting arose because the bulky polar group was the limiting packing factor, whereas for doubly ionized DMPA the electrostatic repulsion between the polar head groups induced chain tilting.12J3 In both cases a ferroelectric state, produced by the in-plane dipole, favors spiral formation, which was stabilized by cholesterol reducing the line tension between the fluid and condensed phases. In this paper we examine the dependency of the main fluid/condensed-phase transition for the double ionized DMPA/cholesterol system as a function of cholesterol concentration in the 0-25 mol % range and demonstrate a correlation of the fluorescence microscopic observations with the more conventional surface pressure (,)/area per molecule ( A )observations. This correlation suggests that a t low cholesterol concentrations (1-5 mol % ) the condensed spiral formations consist of pure or nearly pure DMPA, in that the isotherm expanded/condensed transition, though shifted, remains sharp. The gradually changing condensed-phase structures above 5 mol % parallel transition broadening in the isotherm and are found to be consistent with the partial incorporation of cholesterol in these phases. Finally, the elimination of all contrast between condensed and fluid phases is noted at approximatelythe same cholesterol concentration at which the phase transition is eliminated (-25 mol %). At this concentration we postulate that cholesterol incorporation in the condensed phase has sufficiently fluidized it to make the ease of incorporation of the fluorescent probe into that phase equivalent to that of its incorporation into the fluid phase. This postulate is consistent with the concept that cholesterol expands the condensed phase and condenses the fluid phase until film compressibility no longer shows a discontinuity. Additionally, we describe the dependency of the condensed-phase nucleation on the cholesterol concentration at low cholesterol concentrations. At these concentrations (1-5 mol %) cholesterol is located, primarily or completely, in the fluid phase but additionally plays a significant role in stabilizing a condensed DMPA phase by locating at the fluid/condensed-phase interface. By treating the condensed-phase formation as a supercooled nucleation and by balancing bulk and interfacial free energies of formation, we have been able to predict an initial exponential dependency of nucleation on cholesterol concentrations, as is experimentally observed.
Experimental Section The fluorescence microscope and film balance have been described previ0us1y.l~ Fluorescence excitation is achieved via a (11) Mabrey, S.; Mateo, P. L.; Sturtevant, J. M. Biochemistry 1978,
-17. , -3AGA ---.
(12) Estep, T. N.; Montcastle, D. B.; Biltonen, R. L.; Thomson, T. E. Biochemistry 1978,17, 1984. (13) Knoll, W.; Schmidt, G.;b e l , K.; Sackmann, E. Biochemistry 1985, 24, 5240.
Heck1 et al.
1354 Langmuir, Vol. 4, No. 6, 1988
~
OL
05
0.6
0.7
08
09
molecular area pi I n d h d e c u ~ eOMPA]
Figure 1. Surface pressure (mN m-') versus area per molecule (nm') isotherms at 10 "C and pH 11of dimyristoylphosphatidic acid with 1, 5, 10, 15, and 20 mol % cholesterol, as indicated. water immersion objective in the bottom of a Langmuir trough, emission is imaged on a proximity focus image-intensified TV camera, and textures are photographed from a TV screen. The lateral resolution is better than 2 pm, and the sensitivity of the optical setup is sufficient to measure surface textures with dye concentrationsas low as 0.1 mol %. The contrast in the textures is observed because the dye used, DP-NBD-PE ((dipalmitoylnitrohenzoxadiazol)phosphatidyl)ethanolamine)(Avanti Polar Lipids Inc.), is soluble in the fluid phase hut not in the solid domains. T h e phospholipid bwdimyristoylphmphatidicacid (DMPA) (Sigma, Tautlirchen, FRG) was checked for punty chromatographically. Cholesterolwas purchased from Fluka AG, Buchs (Switzerland). The water used was distilled, deionized, and fikred hy using a Milli-Q system. pH was established by using NaOH (Fluka, Buchs, CH, pA grade). Contamination by divalent ions was prevented by using lod M EDTA (ethylenediaminetetraacetic acid, sodium salt, Sigma) in the subphase. The phospholipid/cholesterol/dye mixtures were spread from a chloroform/methanol (31) mixed solvent. Varying the dye content between 0.3 and 1.8 mol % showed no influence on the observed textures. Data analysis was performed hy using an Apple PC,and images recorded were analyzed via a custom-built analysis system, BAMBL" To compare images with different dye concentrations at a nominally identical fraction of one phase the reader should use the dark area ratio as a guide.
Results The surface premure/area per molecule (n/A)isotherms a t 10 OC for doubly ionized DMPA/cholesterol compositions of 1,5,10, 15, and 20 mol % cholesterol are shown in Figure 1. All the isotherms exhibit a readily detectable fluid/condensed-phase transition. The sharpness of the onset of the compressionaltransition (TJ diminishes with increasing cholesterol concentration. Essentially no broadening takes place between 0 and 5 mol %. Slight broadening takes place between 5 and 15 mol % while beyond 15 mol % broadening increases rapidly until at 25 mol % the transition is no longer detectable. Over a 20 mol % cholesterol addition rtalso shifts continuously to higher pressures by ahout 5 mN/m or by 0.25 mN/m per mol %. The shift is significantly larger than that found for the DPPCJcholesterol system!" The various solid domains formed are depicted in Figures 2-7. In the absence of cholesterol and at pH values in excess of 12, or less than 10, only a mixture of circular and dendritic domains is obtained. In Figure 2A-C domains in the transition are shown for a 0.25 mol % cho(14) Lbche, M.; Mbhwald, H.Rev. Sei. Instrum. 1984,55, 1968. (15) Duwe,H.-P., Diploma Thesis,TU Munich, 1985.
2 G Figure 2. Fluorescence micrographs of a 0.25 mol % cholesterol/DMPA f h at pH 11and 10 O C as it is compressed through the flnid/condensed transition depicted in Figure 1: (A) the onset of the transition; (B) an intermediate point; (C) the transition completion in the condensed region of the isotherm. The lighter regions represent a fluid mixed cholesterol-DMPA phase and the darker regions a nearly pure DMPA doubly ionized condensed phase. The darker stripe diagonally crossing the image is due to the camera opening time not being synchronized with the electron beam scanning of the TV monitor.
lesterol mixture. The initially formed domains tend to patch together, but the resultant patches are more circular than elongated. At 0.5 mol % the domains formed patches which were noncircular, more elongated but still could not be described as crescent moon shaped aggregates. Obviously there is also long-range order between domains. This develops gradually on increasing the surface pressure. Figure 3A-F illustrates the condensed-phase behavior at 1mol % cholesterol. The initially formed nuclei (Figure 3A) rapidly aggregate to form elongated domains (3B) on increasing surface pressure and then develop (3C,D) t o form crescent moon shaped aggregates. Upon a further decrease in the area/molecule and an increase in the surface pressure, the domain width decreases while the length increases, finally yielding spiral-shaped aggregates (Figure 3E,F). Very similar patterns are observed at 5 mol % (Figure 4) and at 7.5 mol % cholesterol (Figure 5) hut above this concentration the pattern changes. Thus, many of the spirals at 7.5 mol % exhibit branching, but a t 20 mol % the condensed phase assumes the form of rounded solid disks (Figure 6). The pattern change with concentration is depicted in Fimre 7. with 7A through 7E summarizing the-previously Zted data. Parts F andG of figure 7 show the behavior at 10 and 15 mol %, respectively, with circular-type structures, some of which appear to be closed,
CholesterollPhospholipid Mixed Systems
Figure 3. Fluorescence micrographs of a 1 mol W cholesterol/DMPA f h at pH 11 and 10 “C 88 it is “pressed through the fluid/condensed transition depicted in Figure 1: (A) the onset of the transition; (B-E) intermediate pints as the arm/molecule is decreased; (F) the transition completion in the c o n d e d region of the isotherm. The lighter regions represent a fluid mixed cholesterol-DMPA phase and the darker regions a near pure DMPA doubly ionized condensed phase. Table I. Condensed-PhaseDoubly Ionized DMPAfCbolesteml Structures at Differing Cholesterol Concentrations mol W cholesterol condensed-phase structure 0 circular plus dendritic 0.25 near-circularaggregates 0.5 elongated aggregates 1.0 spirals 7.5 spirals (with frequent branching) 10 spirals plus open circular 15 circular open aggregates 20 circular aggregates 25 undetectable hut it is difficult t o decide from the micrograph. At and above 25 mol % no condensed-phasepatches were visible. The various fluorescence observations are summarized in Table I.
Discussion Spiral Width. One interesting feature found during compression of 4 mol % cholesterol films is revealed in Figure 8, which shows the spiral width as a fundion of the degree of crystallization, 4, defined as the fraction of surface-adive molecules in the condensed phase. In a straightforward compression a linear $ versus A relation was obtained for values of 4 < 40%! In that case, where a lever rule holds, $ = (AL- A ) / ( A , - As), where A = mean molecular area, Ag = 42 A2,and AL = 64 A*,with S and
Langmuir, Vol. 4, No. 6, 1988 1355
%oi Figure 4. Fluorescence micrographs of a 5 mol 7 ‘ 0 cholesterol/ DMPA film at pH 11 and 10 “Cas it is compressed through the fluid/condensed transition depicted in Figure 1: (A) the onset of the transition; (EE)intermediate points as the area/molecule is de&, (F)the transition completion in the condensed region of the isotherm. The lighter regions represent a fluid mixed cholesterol-DMPA phase and the darker regions a near pure DMPA doubly ionized condensed phase.
L corresponding to the solid and fluid phases of this system, respectively. Since spiral width decreases with increasing pressure, we would expect a monotonic dependence of width on 4, and, clearly, this is not obtained. One explanation for the nonmonotonic behavior ohserved is tbat the condensed phase is itself compressible, and indeed there is evidence that this is the case. In order to explain the behavior of Figure 8, however, it would be necessary for the compressibility of the condensed phase to exceed that of the fluid phase, and this seems unlikely, at least for DMPA/cholesterol phases. An alternate explanation is that compensatory changes take place in fluid and condensed phases that are smaller than the observational limit (-2 pm), hut this too seems unlikely in that the bulk of the unobserved film would need to he condensed. A problem with this particular experimental approach is that the fluorescent dye used as a probe is of necessity concentrated in the residual “fluid” phase as compression approaches completion. Here it constitutes an impurity which essentially prevents complete condensation. For pure DMPA it would seem likely that this, at least in part, caused the problems experienced above 4 = 40%. With the DMPA/cholesterol system and its extensive spiral condensed/fluid interfacial regions, artifadural problems with fluorescent probes could he aggrevated. The most likely explanation is probably a combination of a somewhat
1356 Langmuir, Vol. 4, No.6, 1988
Heck1 et al.
E
/ .
-i ." .,
I'
I Bum Figure 5. Fluorescence micrographs of a 7.5 mol % choleat pH 11 and 10 OC as it is compressed through sterol/DMF'A fh the fluid/condensedtransition depiaed in Figure 1: (A) the onset of the transition; (B) an intermediate point; (C) the transition completion in the condensed region of the isotherm. The lighter regions represent a fluid mixed cholestero-DMPA phase and the darker regions a primarily DMPA doubly ionized phase. compressible condensed DMF'Afcholesterol phase coupled with a less condensable "fluid" phase arising through residual probe molecules. Spiral Nucleation. It is also of interest to quantitatively examine the question of how the number of crystallites formed varies as a function of the cholesterol concentration. The data obtained are illustrated in Figure 9 and show the behavior from (t20 mol % cholesterol. Between 0 and 1mol % the number of crystallites formed shows a distinct drop. This is due to the aggregation of the small near-spherical particles into crescent moon shaped aggregates, as was previously ~ ? ~ Between ' o 1 and 20 mol % cholesterol, the number of crystallites increased in an exponential fashion with cholesterol concentration. Above 20 mol %, crystallite formation, if it occurs at all, could not be detected due to the gradual loss of contrast between the condensed and fluid phases as this concentration is approached. The free energy change necessary for the formation of a nucleus results from the exothermic bulk formation and an endothermic crystal edge formation: AG = A G , d + y(2rrr) (1) In eq 1,r represents the radius of a spherical nucleus, AG, the decrease in bulk free energy per unit area of solid or condensed phase on crystallization, and y the line tension or edge energy per unit length of condensed phase. The first term on the right-hand side of eq 1 is negative and
2 G Figure 6. Fluorescence micrograph of a 20 mol % cholesterol/DMF'A f h at pH 11 and 10 OC as it is mmpressed through the fluid/condensedtransition depicted in Figure 1: (A) the onset of the transition; (B) an intermediate point; (C)the transition region in the condensed region of the isothenn The lighter regions represent a fluid mixed cholesterol-DMPA phase and the darker regions a mixed cholesterol-DMPA doubly ionized phase. must exceed the positive second term for nucleation to occur. But
-a=m- _ am AS-T- a s (2) aT aT aT We may m u m e aASfaT is small compared to other terms. If UHfBT is also small, then we may write9 A(AG) = -AS A T Since the edge energy will not be larger, and its temperature dependency will be small, we may say as a first approximation that
AGv = -AS A T (3) which relates the variation of the free energy of the formation of a nucleus to the necessary ordering (hsnegative) and the tendency to produce supercooling ( A T negative) before nucleation occurs. The role that cholesterol will play in affecting all these factors has not been quantified. No differential thermal analysis studies have been done on doubly ionized DMPAfcholesterol mixtures. There have been such studies on the DPPCfcholesterol system, which does behave in a similar way.".'* The results from both studies show that initially the addition of cholesterol (up to 5 mol (16) Chalmers, B. Principles of Solidification;Wiley: New York, London, Sydney, 1964.
Langmuir, Vol. 4, No. 6, 1988 1357
CholesterollPhospholipid Mixed Systems
%) depresses the gel/liquid crystalline transition temperature ( T a . From 5 to 10 and up to 23 mol % the sharp DPPC transition is broadened and eliminated, and around 4C-50 mol % the broad transition is also eliminated. It would seem that cholesterol initially shifts but does not broaden the transition. There is thus a small (less than 1"C) "supercooling" effect. At somewhere between 5 and 10 mol % cholesterol, cholesterol is incorporated into the condensed phase, resulting in a gradual elimination of the enthalpy and entropy of the transition. The elimination of the sharp transition could be equated with the elimination of "free" DPPC while the elimination of the broad transition is associated with the elimination of all but f d y condensed lecithin. As we have already pointed out, DPPC/cholesterol monomolecular films parallel the behavior of bilayer dispersions,' and since doubly-ionized DMPA/cholesterol f h s follow a similar pattern, it would seem likely that here too the AH and A S of the transition are also gradually eliminated. The loss of phase contrast and the broadening and elimination of the phase transition all support this postulate. However, at low cholesterol concentrations the main effect of cholesterol addition would seem to he the reduction of the fluid/condensedphase edge energy. It may be that the asymmetry of the cholesterol molecule plays a role here. It seems possible that a dense packing could be achieved on the flat a-face, with a less dense packing on the rounded &face. We may write (4) y = yo- ac where yois the edge energy at a zero cholesterol concentration c with the constant a > 0. The critical nucleus size, r*, is defined by d(AG)/dr = 0
-
20wm
Figure 7. Fluorescence micrographs at indicated cholesterol contents for comparable degrees of crystallization, $. A, 0.25%; B, 0.5%;C, 1%; D, 5%; E,7.5%;F, 10%; G, 15%; H,20%.
and may be derived from eq 1 as r* = r / A G , with the corresponding free energy AG(r*) = -RY /AGv
We now assume that the number of nuclei formed (N) during the growth period w i l l follow a Boltzmann law expression:" N = K exp(-AG(r*)/kT) (6)
+
I I
where K is a constant and k the Boltzmann constant. From eq 6,5, and 4 we then obtain
+ T -
N=Kexp
0 01
Ofl
02
03
(u
crystallization $ Figure 8. Condensed phase spiral width Oun) for a 4 mol % cholesterol-DMPA mixture as a function of the degree of crystallization, $.
b
z
D
(5)
, ? 51 10% 15% 201 ctK1e5teio1 consentmtion c
I
Figure 9. Number of crystallites (N)in a fixed monolayer mea as a function of cholesterol concentration (C).
The underlying basis of eq 4 was the assumption that lac1
