Orientational ordering in lipid monolayers: A two-dimensional model of rigid rods grafted onto a lattice

Transcription

1 Orientational orering in lipi monolayers: A two-imensional moel of rigi ros grafte onto a lattie M. Sheringer, ) F Hilfer, an K. Biner nstitut ftir Physik, Johannes Gutenberg-Universittit Maim, Postfah 39 8, 65 Mainz, Germany (Reeive 1 August 199 1; aepte 23 Otober 1991) A simple moel for lipi monolayers on water surfaes at high spreaing pressure is investigate in this work. n this moel, the hyrophili hea group of the lipi moleules form a rigi regular triangular lattie, an the hyrophobi alkane hains (assume to be in an alltrans state) are represente by rigi ros with two angular egrees of freeom (6, p,). The ros onsist of effetive monomers, an between the effetive monomers on neighboring ros a ennar-jones interation is assume. The moel is stuie by exat groun-state alulations, mean-fiel theory, an Monte Carlo simulations. Basi parameters are ro length a an lattie onstant b. The groun-state phase iagram shows the following phases: for small b, the ros are oriente perpeniularly to the monolayer plane (no-tilt phase, (8 ) = ); for somewhat larger b, a sixfol egenerate uniform-tilt state ours with all ros tilte towars one of their next-nearest neighbors. For still larger b, the ros are tilte nonuniformly an form a stripe struture. These unexpete phases o not our if we allow a retangular istortion of the lattie. For T>, the simplest mean-fiel theory preits a graual isorering of the uniformtilt state via a seon-orer phase transition. For the transition region, the Monte Carlo results isagree with this piture. nstea they show a strong asymmetri first-orer phase transition with pronoune hysteresis. The transition temperature inreases with inreasing ro length a, qualitatively similar to experiment. 1. NTRODUCTON AND DEFNTON OF THE MODE ipi monolayers at the air-water interfae are of great interest beause they are moel systems for more omplex biologial membranes an beause they exhibit various kins of orere phases. - The lipi moleules forming these layers have a hyrophili hea group to whih one or two alkane hains (CH,),CH, with 12<M<22 are attahe. While there has been enormous ativity on the main transition, i.e., the melting of the layer (see e.g., Ref. 8 for a review on moels of this transition), somewhat less attention has been pai towars the unerstaning of the transitions between various soli phases.9y These transitions our at high spreaing pressure3, - an exhibit ifferent kins of orientational orer. Beause of the omplexity of the moleules, it is iffiult to evelop mirosopially realisti analyti theories for suh systems, an even atomisti omputer simulations an yiel only qualitative insight into the phase behavior., -3 Following the arguments of Refs. 8-1, we onsier here only a simplifie oarse-graine moel, where the alkane hains are esribe as rigi ros. n orer that the moel remains tratable, eah ro has only two orientational egrees of freeom, q~ [Fig. 1 (a)]. The ros have the length A an are grafte on a rigi triangular lattie with lattie onstant B. Within the lattie, there are no further egrees of freeom. For the interation between the alkane hains, we as- ) Present aress: aboratorium fiir physikalishe Chemie, ETH-Zentrum, 892 Ziirih, Shweiz. sume a ennar-jones potential monomers U, () between effetive U, () = ljo{(a/) l2-2(a/)?, (1) where U, esribes the strength, u the range of this potential, an is the istane between effetive monomers. Assuming that there are v. effetive monomers per ro, an measuring all lengths in units of a, we have the reseale ro length a = A /aan the reseale lattie parameter b = B /B. The total potential energy between a pair of ros beomes, f. Fig. 1 (b) VZ (6,pl,e2,p2 1 = (a2/ 1 2 [ (lj,21) - 2 j,= 1 - (2j,,t > - * (2) Here the units of energy an temperature have been hosen suh that the onstant U, in Eq. ( 1) is absorbe in these units. U2, is the istane between monomerj at ro 1 an monomer at ro 2, where C,,, 2 = (iah j2 + (la/v, 2 + ( (ila2/ S, *S, + 2b,, [ (ia/vo S, + U~o)S2], (3) with Si = (os qi sin Bi, sin pi sin ei, os ei ) enoting an unit vetor along ro i, an b12 = [b os (k?r/3), b sin ( kn-/3 ), 1, with k =, 1, *** 5 a vetor onneting lattie sites 1 an 2 [Fig. 1 (b) 1. Finally, the moel is fully speifie by the restrition to interations between nearestneighbor ros. n previous work on the one-imensional version of this J. Chem. Phys. 96 (3), 1 February Amerian nstitute of Physis 2269

2 227 Sheringer, Hilfer, an Biner: Orientational orering in lipi monolayers FG. 1. (a) Shemati rnoelling of a lipi moleule (labele by inex i) : the hea group lies in the air-water interfae whih is oriente in the xy plane, while the alkane hain in its all-truns state is haraterize by an unit vetor S, or its assoiate polar angles ( O,,qi). (b) The interation between two ros 1 an 2 ours between effetive monomers j ( lg<v, ) an ( 1 <<v, ). f we hoose new polar oorinates suh that the polar axis oinies with the vetor b,, (onneting the two hea groups) an that thexaxis lies in the plane forme by the vetors b,, an S,, the istane,,,,, an be expresse in terms of the two polar angles (x,, a,, an one azimuthal angle fi, for ro 2. () Top view of the layer with uniform-tilt struture. Hea groups form a regular triangular lattie, an the projetions of the unit vetors S, into the xy plane (arrows) point towars their next-nearest neighbors. moel it was foun that in the grounstate there are two istint phases: a (nonegenerate) state where the ros are oriente perpeniularly to the lattie (no-tilt state), an a (twofol egenerate) state where the ros are uniformly tilte (i.e., 8, = + t9, for all i). At nonzero temperature T, flutuations estroy the long range orer of the latter state, an the remaining regions of uniform tilt exten only over a finite orrelation length. n the present two-imensional ase, we expet that in aition to the no-tilt state there shoul be a (sixfol egenerate) uniform-tilt phase [Fig. 1 () ] whih shoul be stable over a nonzero temperature range. There exists also a thir orere phase where the ros lie in the xy plane. This phase will remain outsie of onsieration here beause it oes not orrespon to any of the experimentally observe phases. (a) b (l We stuy the moel in Se. by exat groun-state alulations an in Se. by the simplest version of a mean fiel theory. Setion V presents our Monte Carlo metho an isusses the numerial results. Beause for v. = 15 the reorientation of a single ro insie its wreath of nearest neighbors requires to evaluate 6tio = 135 ennar-jones interations even the simulation of this very simple moel is not at all straightforwar. We have foun it onvenient an feasible to isretize angular spae an store the neessary pair energies between neighboring ros in a huge table. n Se. V we ompare our results briefly to orresponing experimental ata an isuss possible improvements of the present moel.. GROUND STATE ANAYSS f we assume a struture of uniform-tilt type, the total potential energy U epens only on two angular variables, an Q)~. t is straightforwar to alulate U(,,po ) numerially. From its partial first an seon erivatives we loate the minimum in orer to obtain the groun state struture.1 As in the one-imensional ase, the no tilt phase is the groun state for small enough b, i.e., b < 6, (a,v, 1. At the ritial line b = b, (a,vo) a seon-orer transition to the struture with uniformly tilte ros ours. This struture is haraterize by, - [b - b, (a,vo ) ] 2 an p. =?r/6, if, beause of the sixfol symmetry of the problem, p is restrite to the interval <~,<~/3, The funtion 6, (a,vo) is foun to be iential to the one-imensional ase. While for imensionality = 1 no other phases nee to be onsiere, this is no longer true for = 2. We have isovere this fat from Monte Carlo alulations at low temperatures. These alulations were starte from the uniformtilt onfiguration, an in a broa regime in the (a,b) plane, they generate onfigurations with a lower energy than the energy of the uniform-tilt onfiguration. Visual inspetion of these onfigurations reveale various patterns of strutures with nonuniformly tilte ros. This fat suggeste to inlue a two-sublattie stripe phase into the groun-state analysis. n this phase the triangular lattie is eompose into rows, where the vetors Si are haraterize by angles f3,,, poo alternating with rows where the vetors Si are haraterize by angles e,,, pal,. Using simplex-ownhill minimization routines, l5 we loate the minima of the resulting fourimensional energy hypersurfae for some points of the (a,b) plane. n Table the energy of this two-sublattie struture is ompare to the energy of the uniform-tilt state. t is seen that the two-sublattie struture in this ase (a = 5) is lowest in energy from about b= 1.8 to about bz For other values of b the twosublattie solution onverges also to the uniform-tilt state. However, Monte Carlo simulations for some values of b lea to even lower energies than those of the two-sublattie state (Table ). The nature of these phases with nonuniformly tilte ros is iffiult to ientify, sine the variation of the energy with the lattie size is nonuniform. For arbitrary values of, strutures with sublatties will not fit to the lattie, beause, ue to the perioi bounary onitions, a misfit ours if is not an integer multiple of the perio of the stru- J. Chem. Phys., Vol. 96, No. 3,l February 1992

3 Sheringer, Hilfer, an Biner: Orientational orering in lipi monolayers 2271 TABE. Potential energy per lattie site for unifortn an ouble-stripe onfigurations; a = 5.. b ture. The small energy ifferenes between o an even suggest that possibly long perio superstrutures might our. We have not investigate the groun-state phase iagram (Fig. 2) further, sine this woul be a formiable numerial problem. Any results of that kin are probably irrelevant beause only no-tilt phases an various uniformtilt phases have been etete so far in experiments on fatty ais. Experimentally, the triangular lattie has the full hexagonal symmetry only in the no-tilt phase: in the uniform-tilt phase, an orthorhombi istortion ours so that a entere retangular lattie results rather than a stanar triangular lattie. Allowing suh orthorhombi istortions of the lattie [whih are measure by a istortion fator f with (au/af = 1, we obtain Table. n this alulation the elasti energy of the eformation is neglete in omparison with the ennar-jones interations. t shows that the lattie ontrats in the iretion perpeniular to the tilt, an that the phase with uniformly tilte ros is always stable for b > 6, (a). n this ase, the phase iagram for T = is again simple (Fig. 3). The angle, in the uniform-tilt phase epens only very little on whether the lattie is eformable or not (see Table ). This iniates a onsierable robustness of our moel against hanges of etails.. MEAN-FED THEORY A prerequisite for a mean-fiel alulation is the ientifiation of the orering that one wishes to stuy. Here we restrit attention to the uniform-tilt state only. This yiels two self-onsisteny onitions for the two angular egrees of freeom of the entral ro. With the notation ai = ( ei,vi ) (i = 1,2) one obtains as a self-onsisteny onition S,=(6,)= %%,61 Z=, exp[ - U,(Q,&)/T] - AJO exp[ - u,(8,,8,)/t] n our notation we have alreay expresse the fat that for the numerial implementation the integrals over the angles of the entral ro (f9r,pr ) in the fiel of its six neighbors (whih all have orientation 6, ) are isretize. The prout : = o represents the eomposition of the total energy into six ontributions U, of single nearest-neighbor pairs [ Eqs. (2) an (3) 1. Note that we have hosen k, E 1. The numerial solution of Eq. () is laborious beause of the omplexity of the energy surfae 8: = o U, (6,,6, ), see Fig. for an illustrative example. One an see that the uniform-tilt minim urn isappears for, >.8, iniating the importane of states with nonuniform tilt also on the meanfiel level. This must be expete, sine suh states appeare in the T = alulation. At low temperatures it is learly permissible to exploit the approximate egeneray of Eq. () in the q variable. Restriting qr,q2 as q, = q2 = n/6 leas to an one-imensional self-onsisteny equation 3, = (8, ), whih an be solve straightforwarly (Fig. 5). This proeure obviously leas to a seon orer transition. We shall ompare the results of this treatment to the Monte Carlo ata in the next setion. V. MONTE CARO STUDES As has been mentione in Se., for Q = 15 alreay 6 = 135 ennar-jones interations must be onsiere for reorienting a single ro. Consequently, suh a Monte Carlo algorithm is very slow: for a small 2 1 X 2 1 lattie, lo Monte Carlo steps (MCS) per ro take alreay 3 min CPU time on a CRAY-XMP or Siemens-Fujitsu VP 1 omputer. Therefore, even a stuy of this very simple moel by Monte Carlo methos using ontinous angles (Bi,~i ) woul () TABE. Simulation results for various system sizes at T =.1 (potential energy per lattie site). b =5 =8 = 11 = 1 = 17 =

4 2272 Sheringer, Hilfer, an Biner: Orientational orering in lipi monolayers - :, rg : M al -u 7 :\ ? / /,,* * \ *.- :,a,,1,,,,,(, \ ~ r,,,,,,! )1lp;llll % lattie onstant b / 9 2b UNFORM TT. m ) DEFORHATON TO ORTHORHOHEC aj U.- ).95 l-.9 F,,,, ) ro length a FG. 2. Groun-state phase iagram for the moel with rigi regular triangular lattie. ines separate ifferent phases in the plane of variable ( (resale ro length) an b (reseale lattie spaing). 1 labels the no-tilt phase, 2 labels the uniform-tilt phase, 3 ontains (possibly several) phases with strutures of nonuniformly tilte ros. This phase iagram orrespons to the hoie v,, = 15 [in the limit v, - CO the bounary between the no-tilt an uniform-tilt phase woul be at b = b, (a) = 11. take exessive amounts of omputer time. n the one-imensional ase it has been foun that isretizing the angles {6,} into 5 values, whih must be situate in a suitably hosen interval, yiels a program speeup by a fator of 25. Diret extension of this tehnique to the two-imensional moel is not possible, beause eah ro has now two egrees of freeom ( Bi,pi ). n orer to isretize the four angles (Or, pr, O,, pz ) of a pair of neighboring ros an to store the orresponing interation energies, we first trie to use the largest possible table fitting into the omputer memory. But it was foun that there were still systemati isrepanies between this isretize program version an the orret alulation using ontinuous angles. However, as iniate in Fig. 1 (b), for eah pair of ros it is possible to TABE. Grounstate energy U, an tilt angle 6Je for the orthorhombi lattie; a = 5.. U; an B are the orresponing quantities for the uniform tilt state of the uneforme lattie. b uo f u: : FG. 3. Groun-state phase iagram for the moel with eformable lattie. Nature of the phases is iniate in the figure. arry out a oorinate transformation from the angles (19~,~~,19~,p2 ) to a new set (a,, a,, pz ) involving three angles only. A isretization where a,, /3* take 199 values an a2 takes 398 values has been foun to reproue with suffiient auray the results of the Monte Carlo program whih uses ontinuous angles. Although this isretization involves a lookup table of 6 MByte for the pair energy, it is still faster than the ontinuous program version by a fator of 2. n both versions of the Monte Carlo program the stanar Metropolis algorithm was use,16 an the triangular lattie was eompose into three sublatties so that straightforwar vetorization (in analogy with the hekerboar algorithm for the square lattie ) was possible. The following quantities were reore (N = x is the number of ros) : () The potential energy per ro ( U ). (ii) The tilt angle (,) of eah ro as well as the average tilt angle (e)=wiv) 5 (~9~). i= (iii) The polar angle (pi ) of eah ro as well as the average (v)=(l/n) i: (Pi). i= 1 (6) (iv) Thezomponent ofthe orientational vetor Sj, (ST), as well as its average (S ) = (l/n) 2 (ST). (7) i= 1 (v) The root mean square value of the transverse omponents S :, S j, whih an be interprete as an orer parameter for the uniform-tilt-no-tilt transition, (KY) = ((S Y + (syy)o.s, (S ) = (l/n) i (ST), i= 1 (SY) = (l/n) i: (ST). (8) i= 1 (5)

5 Sheringer, Hilfer, an Biner: Orientational orering in lipi monolayers x18 n12 lt 3x12 2T (a) 9 () 3rl8 lx 2 T 2n n: 2 n: lT (b) (P () p FG.. Contour plot of the energy surfae log,, {Xi= U, (6,,a, ) - U,,,,, + 1) for the hoie q, = 15, C = 5, b = 1, as a funtion of 8, (y axis) an q~, (X axis). Case (a) orrespons to ~9, =, q2 =.2, ase (b) to, =.35, q2 =.2, ase () to, =.7, pz =.2, an ase () to, = 1., pz =.2. Contours orrespon to values i/2, i = 1,...2. n ase (a) the minima our at q = kn/6, k =, * * 5, the global minimum at = an further minima at 2. (vi) The heat apaity V is alulate from energy flutuations as usual, C =N((U2) - (u)2)/t2. (vii) The orientational pair orrelation funtion (K,)=(1/6N) 5 i (3(os2{@(r,) i= 1 k=o - e(ri + Abt2 (k) )> - )/29 (1) where A = 1,2,... /2 an b,, (k) is a nearest-neighbor vetor as efine in $. ( 3 ), while ri is the lattie vetor labeling (9) the site i, f3( ri ) E 19~. n pratie, we have alulate the average of all orrelations with the same value of the istane A (measure in units of the lattie onstant b = lb,, ). For the parameter values a = 5, b = 1, all simulations were arrie out for three lattie sizes 15 X 15, 21 X 21, an 36X 36. Beause the finite size effets were foun to be smaller than the statistial errors, simulations for a = an a = 6 were arrie out for one lattie size ( = 15) only. We now briefly esribe our numerial results. Figure 6 (a) an 6 (b) show the average tilt angle ( ) an orer parameter (R, ) over the full temperature range stuie an J. Chem. Phys., Vol. 96, No. 3, 1 February 1992

6 227 Sheringer, Hilfer, an Biner: Orientational orering in lipi monolayers FG. 5. Plot of the simplifie mean fiel onstrution (8, ) -, vs, (zeros in this plot orrespon to selfonsistent solutions). Parameters hosen are v, = 15, a = 5, b = 1, an the various temperatures as iniate in the figure. -.P 7 f.j * 15 A Theory OJ 9 m ok i t \ \ (a) temperature T A x A - 21 hi Theory Ki a.l OJ u o- Q lllllllllll~~~~~l~~~j~~j~~~~~ b ( (b) temperature T FG. 6. Average tilt-angle () (a) an orer parameter (R,) (b) plotte vs temperature for the moel with V, = 15, b = 1 an z = 5. Points enote Monte Carlo ata for = 15,21, an 36, while the full urves are the result of mean-fiel theory, whih implies a seon-orer transition at Tz2.6. r $ t ompares these results to the mean fiel alulation of the previous setion. t is seen that for TS 2 Monte Carlo an mean fiel results are in very goo agreement, while for T> 2 there is marke isagreement: both orer parameter (R,) an the angle (6 ) stay roughly onstant, until at TZ (i.e., a temperature muh higher than the mean fiel transition temperature T y z 2.6) a pronoune first orer transition ours. This transition leas to a isorere high-temperature phase, an after all, in the temperature range stuie, the moel shows three ifferent regimes. n Fig. 9 we isplay equilibrate onfigurations at T = 1, T = 3, an T = 7, orresponing to these three regimes. We interpret the surprising isrepany between meanfiel theory an Monte Carlo simulations as an iniation that the simple orer with uniform tilt an orientation persists only for T< 2, while for T> 2 a nonuniform type of orering takes over. This interpretation is onfirme by Figs. 9 (a) an 9 (b). t is interesting to note that the angular orrelation (K ) stays nearly perfet in the intermeiate regime [Fig. 7(a) 1. n the isorere high-temperature phase (K) ereases steaily with temperature, however. Not unexpetely, the struture of this high-temperature phase epens ritially on the restrition of the moel to interations between nearest-neighbor ros: there are intersetions of not neighbouring ros [see Fig. 9 () 1. Both from (KA) an from the potential energy (U) [Fig. 7(b)] we notie pronoune hysteresis at the firstorer transition to the isorere phase. The first-orer harater of this transition is also evient from the lak of signifiant finite size effets in both (U ) an the heat apaity y [Fig. 7() 1. Only in the orer parameter (R,) in the isorere phase there is a systemati size effet, whih is trivially unerstanable sine (R, ) in Eq. ( 8 ) is efine as a nonnegative quantity an the flutuations measure there are the larger the smaller the system. This first orer harater beomes somewhat less pronoune for longer ro length a (Fig. 8). The transition temperature inreases with the ro length a, qualitatively onsistent with experiment. V. DSCUSSON n the present paper, we stuie a moel of orientational orering in ense lipi monolayers by investigating the groun state by exat analysis an the finite-temperature behavior by Monte Carlo simulations. n the moel the alkane hains are treate as perfetly rigi ros interating with eah other by a ennar-jones potential; the hea groups of the lipi moleules were assume to form a perfet an rigi triangular lattie. The ros are isretize into effetive monomers an the interations were restrite to nearest- neighboring ros. Of ourse, this rue moel is no longer relate to any partiular material, but it may still apture some of the essential physis of har-ro-like moleules grafte on a planar surfae at high surfae ensity. We have foun that alreay in the groun state several phases our-a phase where the ros are oriente perpeniularly to the surfae (no-tilt phase), a sixfol egenerate phase where the ros are uniformly tilte in the iretion towars a next-nearest neighbor, an various phases with

7 Sheringer, Hilfer, an Biner: Orientational orering in lipi monolayers eaeeam C J : 7 C m al u (a) oa-5 # AA-~ temperature $ T > t-l x : a m u 3 ii A (l temperature T -5 - _ m A? lx.2-15 A o k l m a, a (b) temperature T % Co a k - - : () temperature T FG. 7. (a) Angular orrelation funtion (KA) plotte vs temperature, for A = 5 an A = 6. (b) Potential energy (U) plotte vs temperature, for three ifferent lattie sizes. () Speifi heat,, plotte vs temperature, for three values of. () Orer parameter (R,) plotte vs temperature, for three values Of. - 2 oa-5 A-,:-~ A a- &.,.%!-..A 2-5 oa-6,a. ;J&" z al t- A.A.A i e-q---- (3-Q OJ -6 --* A- - e- a r~,~,,,r,,,,,,,1,r~~,,,,,,,,rl,,,,,,,,, A O- A.' a a, temperature FG. 8. Potential energy (U) plotte vs temperature for = 15 an three hoies of ro length a. T nonuniformly tilte ros. Whih phase is stable epens on the two parameters of the moel, the ro length a an the lattie spaing b (measure in units of the W-interation range). Thus at T =, the orering is onsierably more omplex than in the orresponing one-imensional version of the moel, where only the no-tilt phase an a twofol egenerate uniform-tilt phase our. At nonzero temperature, we have fouse attention on the isorering of the uniform-tilt phase only. t turns out that three ifferent regimes an be istinguishe: in the first regime, the harater of the sixfol egenerate orering is still the same as at T =, only the tilt angle ereases with inreasing temperature. This orering is well aounte for by a mean-fiel theory, whih is in quantitative agreement with our Monte Carlo results. n the seon regime, the symmetry breaking with respet to the azimuthal angle is estroye, as an be verifie by iret inspetion of Fig. 9 (b), an thus there is onsierable isorer with respet to this azimuthal variable, while the polar tilt angle stays nearly

8 2276 Sheringer, Hilfer, an Biner: Orientational orering in lipi monolayers is an unphysial feature of our moel an has probably no ounterpart in real systems, where there is repulsive interation between hea groups an the ros. Changing the ro length, we fin that the temperature of the transition to the fully isorere phase inreases signifiantly with ro length. Qualitatively, but not quantitatively, the same tren is observe in reent experiments on fatty ais. The present work shoul only be onsiere as a first step, an further refinement of the moel woul be esirable to allow for abetter omparison with experiments: the lattie of the hea groups shoul not be treate as rigi an stritly triangular, but one shoul allow for elasti eformations towars entere retangular strutures. The latter are observe experimentally. As mentione above, fores between the monomers of the ros an the hea groups shoul be aounte for. We have not yet trie to implement an stuy suh improve moels, sine alreay the moel stuie here took an extremely large amount of CPU time. This amount of CPU time oul not be further reue in spite of our working tehnique of using a isretize instea of a ontinous angular spae. With the latter, our simulations woul have taken CPU times larger by.a fator of 2. Thus the formulation of tratable oarse-graine moels, by whih phase transitions in lipi monolayers or other systems ontaining ros grafte on surfaes an be stuie, remains a hallenge for every omputational treatment. ACKNOWEDGMENTS We are grateful to H. MGhwal an. R. Peterson for stimulating isussions. This work was supporte in part by Sonerforshungsbereih 262 er Deutshen Forshungsgemeinshaft. One of us (R.H.) is supporte by the Bunesministerium fur Forshung un Tehnologie (BMFT), Grant No D. We thank the Regionales Hohshulrehenzentrum Kaiserslautern (R-RK) for a generous grant of omputer time at the Siemens-Fujitsu VP 1 vetor proessor. FG. 9. Equilibrate onfigurations of a 15 x 15 lattie for V, = 15, b = 1, ana = 5. (a) T= 1; (b) T= 3; () T= 7. inepenent of temperature. This phase is somewhat reminisent of the floating phase of the well-known six-state lok moel. The thir regime is haraterize by a omplete loss of long range orer. The transition between regime an is strongly of first orer, with pronoune hysteresis, while the transition between regime an is ontinuous. n regime there is a pronoune inrease of the average tilt angle with temperature, beause some of the ros are strongly tilte an almost reah the xy plane. The fat that suh an orientation may be energetially favorable ertainly G.. Gaines, nsoluble Monolayers at iqui Gas nterfaes (Wiley, New York, 1966). N.. Gershfel, Ann. Rev. Phys. Chem. 27,39 ( 1976). D. Albreht, H. Gruler, an E. Sakmann, J. Phys. (Paris) 39, 31 (1978). F. W. Wiegel an A. J. Cox, Av. Chem. Phys. 1,195 (198). A. Bibs an. R. Peterson, Av. Mater. 2,39 (199). B. m, M. C. Shih, T. M. Bohanon, G. E. e, an P. Dutta, Phys. Rev. &t. 65, 191 (199). R. Kenn, C. Biihm, A. Bibo,. R. Peterson, H. Mohwal, J. Als-Nielsen, an K. Kjaer, J. Phys. Chem. 95, 292 (1991), H. Miihwal (private ommuniation).. G. Mouritsen, Computer Simulation of Cwperative Phenomena in ipi Membranes, in Moleular Desription ofbioiogia1 Membrane Components by Computer-Aie ConformationaZAnalysis, eite by R. Brasseur (CRC, Boa Raton, 199). 9 S. A. Safran, M.. Robbms, an S. Garoff, Phys. Rev. A 33,2 186 ( 1986). M. Kreer, K. Kremer, an K. Biner, J. Chem. Phys. 92,6195 ( 199). Z-Y. Chm, J. Talbot, W. M. Gelbart, an A. Be&haul, Phys. Rev. ett. 61, 1376 (1988).

9 Sheringer, Hilfer, an Biner: Orientational orering in lipi monolayers J. P. Bareman, G. Carini, an M.. Klein, Phys. Rev. ett. 6, 2152 (1988). D. J. Tilesley (private ommuniation). More etails an be foun in M. Sheringer, Diplomarbeit, Universitiit Mainz, 199 (unpublishe). S W H. Press, B. P. Flannery, S. A. Teukolsky, an W. T. Vetterling, Nu-. meriul Reipes (Cambrige University, Cambrige, 1986). 6Ante Carlo Methos in Statistial Physis, eite by K. Biner (Springer, Berlin, 1979). W. Oe, Appl. nformatis 7, 358 ( 1982). J. Chem. Phys., Vol. 96, No. 3,i February 1992