J. Phys. Chem. 1992, 96, 1401-1409
1401
Tilt Order in Langmuir Monolayers A. M. Somoza and Rashmi C.Desai* Department of Physics, University of Toronto, Toronto, Ontario M5S 1A7, Canada (Received: July 18, 1991;
In Final Form: September 5, 1991)
Tilt (smectic-C) order has been observed both in experiments and simulations of Langmuir monolayers. We have constructed a mean-field model, based on a system of rigid molecules with both repulsive and attractive interactions, to study this kind of order. Different phase diagrams as well as the limit of infinitely long hard rods at vanishing density (Onsager limit) are studied. We also comment on the possible influence of other degrees of freedom not included in our model.
1. Introduction
Monolayers of amphiphilic molecules on a water surface (Langmuir monolayers) display a rich thermodynamic phase and have evoked scientific interest since the turn of the century.3 Recent advances in experimental techniques have qualitatively improved the reproducibilityand accuracy of various measurementseI0 relevant to the phase diagram and structure of these monolayers. And the phenomenal increase in the available computing power has led recently to some serious simulation studiesl1-l3of monolayer models at the molecular level. In this paper, we describe a framework which we believe can be used to understand qualitatively some unexplained aspects of the complex phase diagram of lipid monolayers. The object of our attention is the orientational ordering of molecules in such monolayers. In general the molecules are tilted with respect to the direction perpendicular to the surface; we are interested in knowing not only by how much the molecules are tilted but more importantly in the possible symmetry breaking, due to long-range order in the direction of tilt. There is evidence of such order from both laboratory experiments and computer simulation. This orientational order has been referred to in the literature variously as tilt6,* or ~ m e c t i c - C (in ~ . ~analogy to liquid crystals). The transition relevant to tilt order is referred to as the LI-Lz trans i t i ~ n , liquid-expanded ~J~ (LE)-liquid condensed (LC) transi(1) Knobler, C. M. In Advances in Chemicul Physics; Prigogine, I., Rice, S. A., Us.; Wiley: New York, 1990; p 397. (2) Bibo, A. M.; Knobler, C. M.; Peterson, I. R. J . Phys. Chem. 1991, 95, 5591. (3) Pockels, A. Nature 1891, 43, 437. (4) Albrecht, 0.;Griiler, H.; Sackmann, E. J . Phys. (France) 1978, 39,
--.. qni
(5) (a) Helm, C. A.; MBhwald, H.; Kjaer, K.; Als-Nielsen, J. Europhys. Lett. 1987, 4, 697; Biophys. J. 1987, 52, 381. (b) Vaknin, D.; Kjaer, K.; Als-Nielsen, J.; Losche, M. Biophys. J . 1991, 59, 1324. (6) (a) MBhwald, H.; Kenn, R. M.; Degenhardt, D.; Kjaer, K.; AlsNielsen, J. Physica A 1990,168, 127. (b) Kenn, R. M.; Bohm, C.; Bibo, A. M.; Peterson, I. A.; Mbhwald, H.; Als-Nielsen, J.; Kjaer, K. J . Phys. Chem. 1991, 95, 2092. (c) Lin, 9.; Peng, J. B.; Ketterson, J. B.; Dutta, P.; Thomas, B.; Buotempo, J.; Rice, S. A. J. Chem. Phys. 1989.90,2393. (d) Jacquemain, D.; GrayerWolf, S.; Leveiller, F.; Lahav, M.; Leiserowitz, L.; Deutsch, M.; Kjaer, K.; Als-Nielsen, J. J . Am. Chem. Soc. 1990, 112, 7724. (7) (a) Rasing, T.; Shen, Y. R.; Kim, M. W.; Grubb, S. Phys. Rev. Lett. 1985,55,2903. (b) Rasing, Th.;Hsiung, H.; Shen, Y. R.; Kim, M. W. Phys. Rev. A 1988, 37, 2732. (8) Moy, V. T.; Keller, D. J.; Gaub, H. E.; McConnell, H. M. J . Phys. Chem. 1986, 90, 3198. (9) Bercegol, H.; Gallet, F.; Langevin, D.; Meunier, J. J . Phys. (Paris) 1989, 50, 2277. (10) Moore, 9. G.; Knobler, C. M.; Akamatsu, S.; Rondelez, F. J . Phys. Chem. 1990, 94, 4588. (11) Harris, J.; Rice, S. A. J . Chem. Phys. 1988,89, 5898. (12) (a) Bareman, J. P.; Cardini, G.; Klein, M. L. Phys. Reu. Lett. 1988, 60, 2152. (b) Hautman, J.; Klein, M. L. J . Chem. Phys. 1990, 93, 7483. (13) Kreer, M.; Scheringer, M.; Binder, K.; Kremer, K.; Hilfer, R. Modeling of orientational Ordering in Lipid Monolayers. Preprint, May 1991. (14) Harkins, W. D.; Boyd, E. J . Phys. Chem. 1941, 45, 20. For the Harkins-Stenhagen-Lundquist nomenclature, see: (a) Stenhagen, E. In Determination of organic structures by physical methods; Brande, E. A., Nachod, F.C., Eds.;Academic Press: New York,1955; p 325. (b) Lundquist, M. Chem. Scr. 1971, I , 197.
0022-3654/92/2096- 1401$03.00/0
tion,1,6Jsor as the main transition: in analogy to the phenomena in bilayers. The amphiphiles which constitutevarious experimentally studied monolayers are long-chain molecules with head group just under the water surface (denoted by the XY plane at z = 0) and the attached chain(s) extended in the air above ( z > 0 half space). The molecular heads are mobile within the z = 0 plane. At high values of the mean density po (Po = No/Aowhere No is the total number of molecules within a macrampic area Ao),the molecular tail is expected to be rigid (chains in the all-trans state) with its orientation defined by the polar angles (e,(p,x) with respect to the layer normal (the 2 axis). For high pressures and low temperatures, the monolayer phase is crystalline in which the molecular heads form a 2-Dtriangular lattice and the tails are extended straight up parallel to the 2 axis (e = 0). At the other extreme, in the gas phase, the molecules in the monolayer are independent of one another, and their orientation is determined only by the interaction with water. The role that chain flexibility (gauche defects along the chains) and chain orientation play in the description of molecular order within various phases at intermediate pressures and temperatures is one of the central issues in understanding the thermodynamics and structure of Langmuir monolayers. The nature of positional correlations among the molecular heads in the XY plane is also of great importance in this regard. The framework that we describe and use in this paper can in principle be used to include both the orientational and positional correlations in monolayers. However in the specific calculations that we present, we integrate over positions in order to focus attention on the consequences of orientational correlations. At high pressures, the packing of molecules make the positional correlations of the heads quite important, and at low pressures, the chain flexibility acquires a greater relevance. We believe that at intermediate pressures, for the monolayers in which the molecular chains are long enough for chain-chain excluded volume to be important, the orientational correlations may play a dominant role in the thermodynamicphase behavior and structure. The results obtained in this paper are developed for such physical situations. In Figure 1, we sketch a schematic phase diagram in the surface pressure (II)-temperature ( T ) plane taken from ref 2. The dashed and solid lines display their proposed phase boundaries for the monolayers of fatty acids and their esters (we have also included the low-pressure gaseous phase). In each phase, we also indicate the liquid-qstalline label that represents the anticipated molecular structure and correlations. If one envisages integrating over positional correlations, the distinction between many of the phases (which are different only in respect of such correlations) is removed; one then obtains the solid line in Figure 1. It separates phases in which there is some tilt order from the phases in which there is none. The specific calculations presented in this paper attempts to understand the qualitative nature of such a phase boundary. ( 15) Adam, N. K. The Physics and Chemistry of Surfmes; The Clarendon Press: Oxford, 1930.
Q 1992 American Chemical Society
Somoza and Desai
1402 The Journal of Physical Chemistry, Vol. 96, No. 3, 1992 I
I s (S, 1;
t
LS (S,")
/ I / /
I
I
I
I
T
Figure 1. Schematic phase diagram for long-chain acids, acetates, and ethyl esters taken from ref 2. The lines (solid and dashed) display the proposed phase boundaries. Each phase is indicated by the HarkinsStenhagen-Lundquist nomenclatureI4and (in brackets) by the equivalent smectic category. The solid line separates the regions with and without tilt order. We have also included the gas-liquid expanded transition (dotted line) and its critical point (triangle).
Evidence for tilt order is presented in many experiments such as fluorescence imaging using polarized ill~mination:,~ specular reflection of X-rays and neutron^,^ and most convincingly in-plane grazing incidence diffraction of X-raysO6Similar evidence is also found in computer s i m u l a t i ~ n of ~ ~model J ~ Langmuir-Blodgett monolayers. There is also a recent reportk6of canted spin configuration in ultrathin Fe layers on &( 100) which could be viewed as a magnetic analogue of the tilt order in monolayers. Theoretical understanding of structure and thermodynamic behavior of Langmuir monolayers has been evolving. Some authors17JShave assumed that the L2 L1 (LC LE) transition is associated with the chain flexibility and is a sort of chain melting transition which creates disorder by addition of gauche defects. At the other extreme, some authors1e25have used a monolayer model of grafted rigid rods at a mean field level to explore the possibility of phase transitions in such a model. Also for such a model, using a spin-1 Ising variable Hamiltonian, a renormalization group analysis has been pursued recently.26 Finally, Shin et al.27have constructed a fairly detailed lattice model of liquid-supported monolayer of amphiphile molecules in which they include (i) free energy of mixing of water molecules and chain molecules, (ii) free energy of interaction in the surface layer, and (iii) free energy of interaction above the surface layer. They find two first-order phase transitions which they identify with gas-LE (L,) and LE-LC (Ll-L2) transitions. The LI-L2 transition is fairly robust with respect to the kind of modeling of the interaction energy for the chains above the surface layer: any one of the three possibilities considered, viz., flexible chains, stiff chains, or mean-field interaction model are qualitatively satisfactory. Despite the large number of models discussed above, the orientationalorder is usually studied imposing complete azimuthal symmetry and looking at the nematic order parameter. The possibility of anbut isotropic order had been proposed some time ago,19,21,23*24*25 explicit estimations for tilt order are lacking. Instead due to an artifact of the model (discretization of the orientational degrees of freedom), another kind of order, called biaxial, has been con~ i d e r e d The . ~ ~biaxial ~ ~ ~ phase is characterized by two elements
- -
(16) Pappas, D. P.; Gmper, K.-P.; Hopster, H. Phys. Rm. Lett. 1990.64, 3179. (17)CaillC, A,; Pink, D.; de Vertenil, F.; Zuckermann, M. J. Can. J . Phys. 1980,58, 58 1. (18) Doniach, S.J . Chem. Phys. 1978,68,4912. (19) Boehm, R. S.;Martire, D. E. J. Chem. Phys. 1977,67,1061. (20)Halperin, A.; Alexander, S.; Schechter, I. J. Chem. Phys. 1987,86, 6550. But see refs 21 and 22. (21) Moore, B. G. J . Chem. Phys. 1989, 91, 1381. (22) Halperin, A.;Alexander, S.;Schechter, I. J . Chem. Phys. 1989,91, 1383. (23) Chen, 2.-Y.;Talbot, J.; Gelbart, W. M.; Ben-Shaul, A. Phys. Rev. Lett. 1988,61, 1376. (24)Wang, 2.-G. J . Phys. (France) 1990, 51, 1431. (25)Kramer, D.; Ben-Shaul, A.; Chen, 2.-Y.;Gelbart, W. M. Preprint, 1991. (26)Costas, M. E.;Wang, 2. G.; Gelbart, W. M. Preprint, 1991. (27)Shin, S.;Wang, 2.-G.; Rice, S.A. J . Chem. Phys. 1990, 92, 1427.
of symmetry: an axis, like tilt order, and an inversion point (lacking in a uniformly tilted phase). Whereas a biaxial order as considered in refs 23-25 has an inversion point and is like a two-dimensional nematic, what we call tilt order in this paper, does not have such an inversion point and is analogous to a single monolayer of a smectic C. In previous models with discrete o r i e n t a t i o n ~ ~the ~ q distinction ~~ between biaxial and tilt order disappears,but it exists and is important in models with continuous orientations in order to estimate the possible existence of anisotropic phases. A biaxial phase (two-dimensional nematic) is observed in a system of twdmensional ellipses,28but it is expected in a system of long grafted rods only if, for every molecule, 0 E;: r/2. Thus, a large external field (or adsorption to the water substrate) is necessary to make a biaxial phase table.^^,^^ In this paper we study the role of tilt order in the monolayer phase diagram (we neglect biaxial order). We conceptualize the monolayer by a system of grafted rods which are (i) mobile on the surface and (ii) have both hard (excluded volume repulsion) and attractive interactions. The framework takes into account the half-space symmetry of the monolayer system and uses the ideas from the density functional theory successfully used in molecular fluids and liquid crystal^.^^^^^ In section 2 (and the Appendix), we describe the theory, which uses the approach of Landau theory of phase transitions. In section 3, we consider the limiting behavior for low density of long-chain molecules (Onsager limit) in order to choose the coefficients appearing in the theory. In section 4, we present the results and discuss them in the context of other the~ries.'~-~' What emerges is a useful qualitative understanding of the role that various interactions and symmetry play in creating orientational (tilt) order in monolayers. The nature (order) of the phase transitions is also explored. 2. Qualitative Study of the Phase Diagram Let us consider a system of N rodlike molecules with one edge grafted on a planar surface (XUplane at z = 0) of area A. We
assume that each of the molecules has a cylindrical symmetry about its axis, but the two ends are different, giving the molecule a directed orientation Ci. The assumed symmetry implies that the unit vector ri can be uniquely specified by two angles, e and cp: 9 is the angle between ri and the normal 2 to the monolayer surface, and cp is the azimuthal angle between 3 and the projection of ri in the XYplane. The position of the grafted end of the molecule on the XYplane is denoted by the twedimensional vector i, The orientationally averaged local density distribution p(7) (such that N p d = Jp(7) d2r) can be used to study the positional order of various monolayer phases. This is described in the Appendix. Here we consider the orientational distribution P(6) which is normalized: I d 6 P(6)
l "0h c p l ' d0 ( c o s
e) P(ri) = 1
(1)
and which can be defined as P(ri) = l d 2 r p ( F , 6 ) / ~ d r i ~ d p(i,ri) 2r
(14
where p(i,ri)d2r dii is the number of molecules with heads within the area d2r of i and with orientation vector within dri of 6. Clearly, p(7) = Sd6 $(F,ri). We shall denote the average of the physical variables over the orientational distribution function P(h) by (...). As we outlined in the Introduction, monolayers are made up of flexible long-chain molecules, and three major elements are important in understanding their phase behavior. These are chain melting (gauche defects), orientational (in particular nematic and tilt) order, and positional order. It is the combination of these aspects which makes the problem complex. In the body of the paper, we concentrate on the orientational degrees of freedom, especially the tilt order in order to complement existing stud(28) Cuesta, J. A.; Frenkel, D. Phys. Rev. A 1990,42, 3442. (29)Parsons, J. D.J . Chem. Phys. 1979,87,4972. (30)Somoza, A. M.;Tarazona, P. Phys. Reu. Lett. 1988,61,2566.
The Journal of Physical Chemistry, Vol. 96, No. 3, 1992 1403
Tilt Order in Langmuir Monolayers ies17J8927 which emphasize the role of chain melting in monolayer phase transitions. (We indicate briefly in the Appendix how certain aspects of positional correlations can be incorporated in the description below.) With this in mind, we define a free energy functional for the system being considered: FbS(b)l = Fid + Mr + M a t t + Fext (2) which respectively consist of ideal gas, repulsive, attractive, and external field terms (see also eqs 3 and 6). The repulsive term hF, is the excess free energy due only to the repulsive interactions, e.g., of the excluded volume type. The explicit expression that we use for M r is valid for the homogeneous phases and uses a straightforward modification of the decoupling approximation for the hard-body (rigid molecules) systems.29 It is related to the excluded area A(ii,ii’) between a pair of molecules with orientations ii and ii’, and to the excess free energy A$ of a system of hard disks as u r [ ~ ~ , P ( d=) lNkTA$(p$o) ( ( A ( 4 i ? ) ) /4a0 (3) where a,is the average area occupied by a molecule in the closest packing configuration and ( (...)) indicates an average over P(d) and P ( i ? . Explicit expressions for A(ii,ii? exist in the literaturem for the case of long thin grafted rods. For the excess free energy of hard disks, we use the scaled particle theory result;31 (4)
where co = poa0 is the dimensionless mean density. We believe that any reasonable expression for A$(co) would produce the same qualitative results. A more systematic expression for AFr which includes the position dependence of p(?) is given in the Appendix. Equation 3 can be obtained by approximating the pair correlation function g(F,ii,ii?as
we suppress F,,,for simplicity in obtaining qualitative results. Thus the free energy per particle that we use is the following functional: F -=f= N
(7) Assuming that g,(F,ii,ii? additionally does not depend on temperature, T,and the distributions p(?), p(@,and P(ii? explicitly, eq 7 is essentially the generalized van der Waals model for the monolayers which uses a local approximation for the short-range repulsive interactions and a mean-field level description for the longer range attractive interactions. Given the currently available knowledge of real systems, we believe this to be a good starting point for a theoretical analysis, which we describe below, in order to get some feeling for the validity of various approximations. Computer simulations for rigid molecules would be useful in order to test and improve this free energy functional. Following a standard procedure in molecular liquids,36we and A(ii,ii? in spherical harmonics. Note expand both &&,ii? that since ii and ii’span only the upper half space ( z > 0) above the water surface, an expansion in the full set of spherical harmonics F,,,(B,cp)leads to a problem of overcompletness. One has to restrict the set with a surface condition: two simple possibilities are to choose either a set for which I + m is even or a set for which 1 + m is odd, although many other linear combinations (of these two surface conditions) are also possible. We have chosen the condition t3~db~)/t3%=,/2 =0
(8)
+
where ghd(X) is the pair correlation function for hard-disks and d(?/lq,ii,ii? is the in-plane contact distance between a pair of molecules with orientations ii and P a s they approach each other in the direction F/lfl. This approximation reproduces correctly the second virial coefficient and provides reasonable values for the higher ones. It has been widely used in the liquid-crystals literature (see ref 32 for a review) and for two-dimensional ell i p s e ~with ~ ~quite good results. And it can also be generalized to study positional order as it has been done for liquid crystals.30J23 Finally, the term AFaain eq 2 is due to the attractive interactions between the molecules. We conceptualize the intermolecular interactions as a sum of pairwise terms and treat the repulsive part in AFr and the attractive part in a manner analogous to that used in standard thermodynamic perturbation theories developed for simple liquids.35 The result is, at the first order of perturbations
Matt= %Nco( (4,fd&@/a0))
which implies that only the spherical harmonics with 1 m = “even” are permitted. We remark that eq 8 is only a mathematical necessity to define a complete set of orthonormal functions in terms of which t$,&ii,ii? and A(ii,P) can be expanded. If no further approximations were made, the final results would not depend on the particular choice as in eq 8. In practice we need to truncate the series at some point and in this case, for a given number of terms, one particular set (determined by eq 8) could be better than the others but, for enough number of terms, the qualitative results should not change. It is straightforward to write the expansions in the forms A(ii,ii? -400
- E,
1
- -Ell sin 8 sin 8’ cos y 2
4,dG ? -- -U, - U,,sin 8 sin 8’ cos y an
(6)
where
&d@? = I d 2 r gr(F,@?&tt(F,,ci,ii? One may use eq 5 in principle to evaluate this term from +.Ji,ii,ii?.In practice however, poor knowledge about various microscopic details of monolayer systems may make it intractable. We approach the problem with a view to obtain qualitative results. The effect of interactions between amphiphilic molecules and water can be usefully introduced in eq 2 though the external field term Fat. A way to do this is illustrated by Shin et al.27 But here (31) Re&, H.; Frisch, H. L.; Lebowitz, J. L. J. Chem. Phys. 1959,31,369. (32) Somoza,A. M.; Tarazona, P. J . Chem. Phys. 1989, 91, 517. (33) Cuesta,J. A.; Tejero, C. F.;Baus, M. Phys. Rev.A 1989,39,6498. (34) Somoza, A. M.; Tarazona, T. Phys. Rev.A 1989,40,4161; 1990,41, 965. (35) Hamn, J. P.; McDonald, I. R. Theory of Simple Liquids; Academic: London, 1986.
where y = cp - d . Various signs in eq 9 are chosen to make the coefficients E,, and vi. numerically positive: due to the excluded area nature of A&, E,,,,is positive; as the orientational order is increased, one expects the excluded area to decrease which leads to the negative signs in front of Ell, Em etc. Since 4dt((i,Cj?is related to attractive interactions, the leading sign in front of U, is negative; we have also put negative signs in front of VI, and Uz0,but in principle U,,and/or UZ0could turn out to be negative numbers depending on the competition between short- and long-range interactions. Many in the literature assume U, to be zero,but this is not correct: since if it is nonvanishing, however small, it arises by symmetry before U2,and U31 in the spherical harmonics expansion. In the (36) Gray, G.;Gubbins, K. Theory of Molecular Fluids; Clarendon Press: New York. 1986.
1404 The Journal of Physical Chemistry, Vol. 96, No. 3, 1992
analysis below, we truncate the expansions in eq 9 to the terms shown. This may be quantitatively not good but it gives a good qualitative perspective for the phase diagram. For a specific model of long thin rods, from the work of Halperin et al.,zOor for other particular geometric model of hard-core molecules, one may obtain the coefficients E.. numerically; the coefficients U may be estimated by stdying separately the contribution of the multipolar interactions, dispersion forces, etc., as is done for molecular liquids.36 To obtain the qualitative features of the phase diagram, the six coefficients in eq 9 are minimum necessary: E , and U, are responsible for the gas-liquid expanded (LI) transition, E l l and VI, induce the tilt (in-plane orientational) order, and Ezo and Uzo induce the perpendicular (to the monolayer) alignment of the molecules. Note that for three-dimensional systems, E,, is zero but for monolayers it is not. The usual term associated with UZ2,which is the Maier-Saupe term for nematic liquid crystals in three dimensions, is not essential for monolayers since there is no phase transition related to the nematic order parameter (see next section). The inclusion of further terms in the expansion merely complicates the calculations without adding new qualitative features. Quantitatively, however, the convergence of the expansion in eq 9 could be poor and could lead to poor results if for specific models, A(6,d') and I#Je,,@,d') depend very strongly on 0 and &'. Note that due to truncation (in order to have reasonable expressions for A(d,d') and I#Je;Ld,d')) we need to impose new constraints on E, and Vi,.For example, for 6 = 8' = 0, using truncated version of eq 9, the excluded area is E , - E20,whereas for 0 = 8' = 7r/2, cp = d , it is E , - 1 / 2 E l l 1/2E203 One expects the former to be a minimum and thus the truncation leads to a restriction E l l < 3E20, which we impose on the range of Eij used in our study. Similar considerations for &@,a') in eq 9b lead to an additional restriction, UII< 3/2U20. We remark that we are not studying biaxial order. For this purpose, we should keep additional terms like E2z and Uzzsuch that, for certain values of the parameters, a phase with zero tilt order parameter, q (see eq 1 l), but broken azimuthal symmetry is possible. In our treatment (eqs 8-18) the natural order parameter related to such a biaxial phase would be v = ( sinZ6 cos 2cp), which is qualitatively similar to the biaxial order parameter used in previous s t ~ d i e s In . ~this ~ ~sense, ~ ~ we can say that our study is complimentary to that of Wang24since the tilt order is neglected and biaxial order is considered in ref 24. Thus eq 7 with the truncated series in eq 9 has to be viewed as a semiphenomenological starting point, although the order parameters, q and S, defined below, as well as the density and temperature dependence are based on molecular considerations. The free energy functional, fi in eq 7 is a functional of the orientational distribution function P(6). We now minimize f with respect to P(d) subject to the normalization constraint, eq 1, on P(d) (A is a lagrange multiplier):
+
--sf
6P(ii)
ASP(;) dii = 0
with the result that
where the tilt order parameter q is q = (sin 0 cos cp)
(11)
and the angular brackets stand for the average Over P(d) as given in eq 10. The partition function Z in eq 10 is the normalization constant in order to satisfy the constraint eq 1:
=
1
d(cos e) ~ o [ ~ lsin l qe] exp
3 cosz e - 1
where the integration over cp can be explicitly performed. Zo(x)
Somoza and Desai (and Zl(x) below) are modified Bessel functions. In eqs 10 and 12 (13)
From the definition eq 1 1 of q, one obtains by performing integration over cp q = ZISId(cos 0
e) sin 0 Zl[aIlq sin e] exp
3 cosz e - 1
1
(15)
Equation 15 is an integral equation for q, the solution of which would minimize the free energy, eq 7; note that the partition function 2 also depends on q . For the nematic order parameter, S, one obtains
= (3 1
= Z - I S d(cos e) 0
- 1)
3 cosz e - 1 Z0[allq sin e] X 2
Neither eq 12 nor eq 15 involves S, the nematic order parameter. Equation 16 can be used to obtain S,once the minimization procedure through eq 15 yields the equilibrium solution for q. One may then also obtain the minimized free energy and the equation of state n, which are explicitly given as
f = kT[log CO - 1 - log Z
+ AJ/(cO)(EW+ %ElIq2)] + XC0(&0+ ullllz) (17)
n = P O W 1 + ~ o ~ v ( c o ) ( E-o E20S o - Y2E11t1z)1(co2/2a0)(U,
+ u20S + ullllz) (18)
Our definition of the tilt order parameter q = (sin 0 cos cp) (and not (cos c p ) ) makes it evident that the tilt order cannot exist at complete perpendicular alignment. In the next section, we consider the limit of low density of amphiphiles in the context of Onsager model before exploring (in section 4) the results, eqs 15-1 8 for some sample parameter values. Section 4 also includes the discussion of the results. 3. Possibility of Tilt Order in the Onsager Model For a system of N cylindrical, rigid rods of length L and diameter d (L >> d), in which one end of each of the rods is grafted on a planar surface of area A and in which the only interaction between rods is the steric repulsion, the possibility of a sharp alignment-ordering transition has been investigated in literat ~ r e . ~ The " - ~general ~ conclusion is that to have a discontinuous change in alignment, one needs either attractive forces between rods or/and large adsorption energy with the substrate (external field in the sense of section 2). In these references,%24the authors have considered the order parameter analogous to S,the nematic order parameter. In some instances,u*" the molecular orientation is modeled to have discrete directions such that the directions x and -x are indistinguishable and so are y and -y. Such an introduction of a point of inversion implies a zero value for the tilt order parameter q. In this section, we reexamine this question for the case of continuous orientation. Halperin et a1.2*22 have investigated this question using continuous orientation for the nematic order parameter S and after some initial minor false leadsZohave come to the same conclusion21-2z that one needs a sufficiently strong external bias field to induce a sharp, alignment-ordering transition. In this section, we report the rtsult for both the order parameters S and q. And we also come to the same conclusions for nematic and tilt orders in the Onsager limit.37
Tilt Order in Langmuir Monolayers
The Journal of Physical Chemistry, Vol. 96, No. 3, 1992 1405
The virtue of the Onsager model is a free energy expression that is simpler than eq 7. One follows the standard well known steps of the Onsager model for bulk nematics: assume P(4) is almost isotropic, take the limits L / d and po 0 (keeping p a of order unity), and retain only the leading terms; as a result, the virial expansion is truncated after the second virial coefficient. In terms of the dimensionless density co = (2/r)Ldp0 and the dimensionless area A* = (r/Z)(A(ii,ii’)/Ld), the free energy per molecule is
- -
f F/N
bo
+ kT[log CO - 1 + u + ‘/ZC~((A*(ii,ii’)))] (19)
where u = (log (2rP(ci)) is the orientational entropy and bois essentially the ideal gas chemical potential. The (dimensionless) surface pressure, fi, is
We use the explicit expressions for A(@’) obtained by Halperin et al.” in the specific calculations. The minimization procedure outlined in the last section can again be applied to the free energy in eq 19 with the result that the orientational distribution function satisfies the integral equation:
P(d) = Z0-I exp[-coJdii’P(ii’) A*(ii,ii’)]
(21)
Zo = JdB” exp[-coJdii’ P(ci’) A*(ii”,ii’)]
(21a)
where
Even though the distribution is not isotropic and the higher order terms in virial expansion (or in A(ci,ii’)) may become important, the model has provided insights for the three-dirrmional nematics. Halperin et a1.20looked at the azimuthally symmetric phase for this model, for which P(4) = Po(@,independent of cp. Equation 21 simplifies to Po(e) = exp[-cOL1d(mse’)Po(U A*,(B,V] 2rJ1d(cos 6”) exp[-coJ1d(cos 0 V ) Po(V) A*,(B”,e’)]
~ ( c i )= Po(e)
+ wl(e) COS cp + wz(e) COS 2cp+ ...
We expect the bifurcation to occur into a tilted phase (6Pl(8) # 0); in this case it is easy to show that, for a bifurcation analysis, all other terms (with SP,(O), i 1 2) are negligible and can be dropped. Substituting in eq 21 and linearizing with respect to 6P1(8),we get
wl(e) = c,po(e) S,ld(coS e’)wl(e’)
A*,(e,e’)
(24)
where A*l(e,e’) = -Xzrdy A*(e,e’,y) cos
(24a)
with y = (cp- d). Without loss of generality, we define the new functions
me) = ~ P l ( e ) / q x z ~ ~ ( ~ =~ &Z&/‘XEA*l(e,e~) ;e,u
(254 (25b)
in order to transform eq 24 to a simpler form: 1
sP(e) = C O S , d(cos e’) sP(e’)Al(co;e,e’)
(26)
This is the basic bifurcation equation, which is an eigenvalue problem. There will be a nontrivial solution only if 1/co is equal to an eigenvalue of the integral operator:
8 S,ld(cW e’) Al(co;e,e’)
(27)
Note that AI depends on co through Po(e),eq 25b, and there is no a priori guarantee that a bifurcation solution will occur. We proceed by expanding SP(8) in a complete orthogonal set of functions for 8 E [0,7r/2], with weight function sin 8:
sP(e) = C(sP),Pli(cos e) i
(28)
where P1,(cos e), i = 1, 3, 5 , ..., are the associated Legendre functions, and the sum over i is only over odd integral values. Substituting eq 28 in eq 26, we get a set of infinite algebraic equations: (SP>i= coF(6B),,aij(co)
where
(23)
(29)
where i and j take only odd integral values: A*o(e,V) = x z r d y A*(ii,ci’)
(22d
To carry out the numerical integrations, we have to discretize in the variable 8 and then apply an iterative method. This algorithm is described in more detail in ref 38, where it is applied to Onsager’s model for nematics and where the authors also study its accuracy depending on the number of discrete values used. We have used a grid of 301 points in 0 which ensured a relative numerical error lower than for all the physical quantities studied. Our numerical study of eqs 19-22 does not show any sharp transition or any inflection point related to nematic order in agreement with earlier studies. We have also investigated the possibility of tilt order: numerically,# for a given value of density c,,, eq 22 provides only one solution, as we find, then we are assured that it is the azimuthally symmetric distribution Po@)with lowest free energy. In the wider space of functions of (e,+), the solution Po(@)so found could be a saddle point with a minimum in the restricted space of functions of 8, but not in the wider space. There could Wtist anisotropic solutions of eq 21 with P(6) I P(8,cp) which depend on 0 and which have a lower free energy. To investigate this possibility, we have made a bifurcation analysis: we have looked for the point in the phase space where eq 21 admits an infinitesimally anisotropic solution. We assume (37) Onsager, L.Ann. N.Y. Acad. Sei. 1949, 51, 627. (38) Lee, S.-D.;Meyer, R.B. J . Chem. Phys. 1986,84, 3443.
since b is a real symmetric operator, all its eigenvalues are real. Also, for low enough densities co, the azimuthally symmetric solution is expected to be the minimum energy solution and then one expects all the eigenvalues to be lower than l/co. The bifurcation point, if it exits, will correspond to the solution of th,e equation (coXl(co)= 1) where XI is the largest eigenvalue of 0. We obtain approximate value of XI by inverting a truncated version of the infiiteorder square-matrix ail(co). As the size of the matrix is increased, the approximate XI approaches the exact one monotonically, as seen in Figure 2, where we show a plot of coXl(co) as a function of co for the truncated matrix 5, of sizes 1 X 1 to 6 X 6. Although the full convergence has not yet occurred, it is clear that the maximum of the function coXl(co) is far short of unity and there will not be a bifurcation point. As the density co is increased, the tilt order is favored but the aligment of the molecules perpendicular to the monolayer surface diminishes this possibility and the entropic term dominates along
1406 The Journal of Physical Chemistry, Vo1.96, No. 3, 1992
Somoza and Desai
= x* = 2.924 and has a value t(x*) = t,,,= 0.790 with 2/5;, = 2.531. This information is particularly useful in the two limits of very high and low temperature. In both cases the ratio aI1/am tends to a constant, while am may be varied arbitrarily (see eqs 13 and 14). The most favorable situation corresponds to a2,= x*; thus in these two limits a bifurcation point may exist only if a l l / u mL 2/f,,,. At high temperatures we expect a region of intermediate densities with q # 0 if El1
1 2.531E20
(33d
while for low temperatures (and densities) this region will appear if 1 1.266U20 0
1
3
2
4
5
CO
Figure 2. Variation of cOhl(c0)with co for the truncated matrix Z,,(co) of sizes 1 X 1 to 6 X 6 . As the size of the matrix is increased, coXl(co) curve monotonically moves up. A bifurcation point would correspond to c0X,(co) = 1.
the whole range of densities. And so the conclusion is that, in the Onsager model, tilt order is not possible. Nevertheless, repulsive interactions, even though not enough by themselves, contribute quite significantly to get the system closer to a bifurcation point which could be achieved with a little help from attractive intermolecular interactions. 4. Results and Discussion
In this section, we numerically investigate the role of orientational correlations in the monolayer phase diagram using the results obtained in section 2 (eqs 12-18). Without loss of generality, by choosing proper units of length and energy, we can set a. = U, = 1. In principle, one should determine the remaining five U,j and Eij from the molecular interactions. In this paper, our purpose is not to compare results to experiments on real systems but to obtain qualitative information about the phase diagram. We therefore fix these coefficients to some reasonable values. To see for which values of the parameters the system may have a tilted phase, we perform a bifurcation analysis of eq 15: we inquire when eq 15 has a solution for nonzero q which is infinitesimally small. If a second-order phase transition exists, the bifurcation point becomes a critical point, and if a first-order transition occurs, then one sees an end of the metastable region. If the bifurcation point exists, the temperatureand density at that point are related by the following integral equation which is obtained by setting the second derivative of the free energy to zero: 1=
22(9=0+)
)]
Jld(cos 8) sin2 0 exp[ a2,( 3 ~ 0 ~ 2 e - i
(31) Since we are looking for the bifwcation point (v=O+), the partition function Z(q=O+)is seen from eq 12b to depend only on a2,,so that eq 31 is of the form 1 = 1/2allt(a20)r where the function 3. is independent of all;it is convenient to rewrite this as 1 = (~11/2a2o)t(~20) with t(a20) = azOt(a20) =
-
It is easy to see that XO) = 0 and as x XX) approaches 2/3 from above. Thus X x ) increases from zero to a maximum before beginning its decrease toward the asymptotic value of 2/3 for large values of x. The maximum of { ( x ) occurs approximately at x +=,
(33b)
These considerations give us a qualitative sense for the phase diagram, but the line of bifurcation points ( p * , , P ) (if it exists) must be obtained numerically from eq 3 1. To determine whether the phase boundary corresponds to a first- or second-order phase transition, one has to either look at the fourth derivative of the free energy at the boundary points ( p * , P ) or by explicitly computing the free energy in the neighborhood of the phase boundary on both sides. We have followed both procedures. Among a variety of parameter choices that we have explored, we present the results for two sets since they appear to qualitatively resemble some of the experimental phase diagrams in literature. When we use the values Em = 4 E20 = 1 Ell = 2.8 (34) u, = 1 lJ2, = 0.2 VI, = 0.22 we find the region of anisotropy (nonzero tilt order parameter) of the phase diagram as shown in Figure 3 (solid lines correspond to second-order phase transition and dashed lines to first order; also see discussion below). This phase diagram looks similar to one proposed by Albretch et a1.4 If VII is made slightly larger, the region of nonzero q extends all the way to zero temperature. Note that E11/E20= 2.8 > 2.531 and U l l / U m= 1.1 C 1.266, so that the tilt order is achieved through E,. A qualitatively different phase diagram emerges if we assume that a system of hard molecules does not undergo a phase transition but misses it by a small amount, and a small amount of attractive interaction is needed to achieve the transition (as is the case for Onsager model: see section 3 and Figure 1). Such a case occurs in Figure 4, where we have used the values Em = 4 E20 = 1 Ell = 0.8 (35) u, = 1 u,, = 0 VI, = 0.2 which qualitatively changes the ratio U l l / U mto infinity and the ratio E l l / E 2 ,= 0.8 < 2.531. In both Figures 3 and 4, there is a new intermediate phase with a nonzero tilt order q. The results of section 3 show that the steric interactions between rigid particles are not enough to induce tilt order (at low densities). Thus one expccts this phase to disappear at high temperatures. This would make the phase diagram in Figure 4 to be more appropriate for systems in which the Onsager model can be validated. In some systems, the special nature of polar heads and its bond to the chain@)could destroy the validity of the Onsager model. In such instances, with our approach it is difficult to give an unambiguous qualitative result for the phase diagram. The Onsager model gives us a way to fix the parameters E,, to reasonable values. In obtaining Figure 4, the choice of parameters Em,E2,, and Ell was made such that the calculation using E,, in eq 35 but with all U, set to zero approximately reproduced the maximum of c o X 1 ( c ~(see Figure 2). The values of parameters VI, and U, then are chosen so as to obtain n o m m tilt order regions. The pair of values for U20and VII used in eq 35 are clearly not unique. In Figure 4, several qualitative features should be noted. Solid lines indicate a second-order and dashed lines a first-order transition. The critical temperature T,for the gas-LE transition is about 0.034. (The dimensionless variables used in Figures 3 and
The Journal of Physical Chemistry, Vol. 96, No. 3, I992 1407
Tilt Order in Langmuir Monolayers 6
4
4.5
3.5
N
0 4
x
8 3
3
4
\
5-
3.5
2.5
3
2
2.5
1.5
, , , , , , , , , ,,,, 1
2
3
4
5
6
1
2
A/%
3
4
5
6
A/%
5
4
I
X
8
3
-2 E
lr]
2
0
0 4
X
8 3
B E
[ , , , L f Y 3
3.5
4
4.5
5
kT/Uoox 1 0-2
1.5
2
2.5
3
3.5
kT/Umx 1O-*
Figure 3. Tempcrature-area (a) and pressure-temperature (b) phase diagrams obtained from eqs 15-18 with values of the parameters given by eq 34. The dashed and solid lines indicate first- and second-order
Figure 4. Temperature-area (a) and pressurttempcrature (b) phase diagrams obtained from eqs 15-18 with values of the parameters given by eq 35. The dashed and solid lines indicate first- and second-order
phase transitions respectively. The triangle in (b) shows the liquid expanded-gas critical point, and the solid point shows a tricritical point. The regions of liquid expanded, nonzero tilt order and liquid expandedgas coexistence are indicated.
phase transitions respectively. The triangle in (b) shows the liquid expanded-gas critical point, and the solid point shows a tricritical point. Comparing with Figure 1, we have indicated the regions of liquid expanded, liquid condensed, and their coexistence with gas. We have also conjectured where various phases with additional positional order would appear in a more general calculation.
4 are Tz = kT/U,, A* = A/ao,and ll* = na0/U,.) Note an extremum in the phase boundary at PoN 0.029, ASoN 1.9 and IIs0N 0.07. If one considers a system at'7 < PoN 0.029 and compresses it from a lowdensity gas-phase state, one goes through G-L, cotxistence first, then into the L1(LE) phase if T+ > 0.026; further compression leads to the phase transition to a nonzero tilt phase which we tentatively identify with L(LC) phase. This phase transition is second order in our calculations, but other features of real systems, not included here, are expected to turn it into a first-order transition. Various possibilities are discussed below. Further compression of the system leads it to undergo a second phase transition which could be a second (first)-order transition if P > 0.02 (T+ C 0.02). Such a transition is analogous to a second 'kink" observed in some experimental isotherms, e.&, the experimentdoon pentadecanoic acid. As is clear from Figure 4a, our calculations cannot distinguish between the L, phase and the other ordered phases LS,S,and CS sketched in Figure 1. This is due to our having integrated over the positional order. Whether the molecules are lying nearly parallel to the water surface in an azimuthally isotropic manner (as in the Ll phase) or they are packed nearly perpendicular to the water surface with possible positional order (as in the LS,S,and CS phases), we would get 7 = 0. Using Figure 1 as a guide, we have also conjectured in
Figure 4b where various phases with additional positional order would occur if positional correlations were built into our model. In particular we believe the high-density phase to the left of L2 phase in Figure 4a to be another ordered phase which our theory cannot describe in full detail (see also Appendix for possible future calculations in this regard). Some aspects of real monolayer systems are not included in our phenomenology. We now list and discuss these aspects: (i) Adsorption at the Water-Air Interface. If the molecular chains are attracted by the interface, their perpendicular alignment is partially destroyed and tilt order is indirectly induced. Such additional adsorption energy term can be introductd as an external field term (see eq 2) which will be linear in the external field and (after an expansion in spherical harmonics) linear in the nematic order parameter S. Within our model free energy (eqs 7 and 9), this effect, for a fixed density, will essentially decrease the numerical value of the parameter U,, and the resultant phase diagram would be qualitatively similar to that in Figure 4. (ii) NonuniaxiPUity of the Molecules. If a molecule is not symmetric axially either due to rigid shape or due to flexibility, then it is easier to induce tilt order (as for example in a collection
Somoza and Desai
1408 The Journal of Physical Chemistry, Vol. 96, No. 3, 1992
of long flat objects versus long cylindrical objects). The coupling between the twist angle x and cp is expected to favor tilt order. (i) Positional Correlations of Molecules in the Surface. This neglected aspect is certainly important, and we indicate in the Appendix how it can be included in our analysis. Since we have integrated over the position variable P in the beginning of our analysis, phases with different positional order but the same tilt order would become degenerate in the phase diagram (compare Figures 1 and 4). Long-range tilt order induces also nonzero hexatic order parameter,39so that some coupling between these two order parameters is also expected in real systems. (iv) Flexibility of the Molecules. The LE-LC transition has been explained as an effect of the molecular In these models the degrees of freedom of the chains are usually described by the nematic (or equivalent) order parameter and another one related to the number of conformationaldefects. The feature of the system is determined by these parameters and the coupling between them. Intuitively, the connection between orientation of the chains and number of defects is clear: a compression of a system of randomly oriented chains induces, in the first instance, new defects, while if the molecules are mainly parallel the lateral interaction between them favors an “all-trans” configuration. This simple picture suggests that, instead of the nematic order parameter, the relative angle between molecules is a more appropriate parameter to couple with the number of conformational defects. In this case, tilt order could acquire great importance. (v) Fluctuations. We have presented a mean-field model for tilt order in monolayers. In a two-dimensional system, fluctuations could be very important, but we believe that our analysis is still correct. Fluctuations could destroy long-range order in two dimensions, but as we comment below we expect that they do not destroy (could even favor) the phase transitions that we obtain. Also it is known that the order of a phase transition can change from a second to first order due to fluctuations. But in a real system, to obtain a large discontinuity in density (as seen experimentally), surely it is more important to take into account the coupling of the tilt order parameter t) with other ordering degrees of freedom such as those commented above. Finally it is also useful to consider the magnetic analogy of the monolayer system: since the mean orientation of an amphiphilic molecule in Langmuir monolayers is described in terms of the two angles (O,q),the order parameter for monolayers is analogous to the magnetic system represented by a two-dimensional Heisenberg model. Just as in the Langmuir monolayers, this phase transition which creates a nonzero tilt order has been experimentally observed16 in ultrathin iron films on Cu( 100). According to the well-known Mermin-Wagner theorem, fluctuations would destroy long-range order in a two-dimensional system. However, the monolayer systems are somewhat different as can be seen from a recent studym by Pescia and Pokrovsky, who studied a system of ferromagnetic monolayer and showed that “as a consequence of the strongfluctuations of the 2D Heisenberg model, a temperature TRmay exist at (above) which the magnetization of the perpendicularly oriented ground state turns into the plane of the monolayer”. The spins in the 2D Heisenberg model are free to rotate in full 3D-space. In contrast, molecular orientations in the Langmuir monolayer are restricted to the half space away from water. For the Heisenberg model, an external field normal to the ferromagnetic monolayer creates a broken symmetry whereas it is inherent in the Langmuir monolayer. It is the broken symmetry in both systems that should make then qualitatively equivalent, and we expect the conclusions of Pescia and Pokrovski to be valid for the Langmuir monolayer. In conclusion, we have initiated a new approach based on half space symmetry of the monolayers and the tilt order parameter t) to investigate the thermodynamic behavior of Langmuir monolayers. As the density is increased, we find two phase transitions which are associated with tilt order. We tentatively identify the (39) Acppli, G.; Bruisma, R. Phys. Rev. k r r . 1984, 53, 2133. (40) Pescia, D.; Pokrovsky, V. L. Phys. Rev. Leu. 1990, 65, 2599.
first one as Ll-L2 (although surely tilt order is not the only mechanism which forces the phase transition). The second transition, at higher densities, may already have been ~bserved.~~~ Our model predicts decreasing pressure at coexistence when temperature is increased for this transition (although the coupling to other degrees of freedom might screen the effect). Acknowledgment. This wprk was supported by the Natural Sciences and Engineering Research Council of Canada and by a postdoctoral fellowship to A.M.S. by the Ministerio de Education y Ciencia (Spain). We would like to thank C. M. Knobler, W. M. Gelbart, Z.-G. Wang, J. Ruiz-Garcia, and J. Als-Nielsen for useful discussions.
Appendix In this Appendix we show how eqs 3-5 can be extended in order to study positional order. The procedure exposed here is a generalization of that used for liquid ~ r y s t a l s . ~ ~ ~ ~ ~ ~ ~ ~ Equation 3 relates the excess free energy of hard disk (HD) with that of grafted hard rods. Thus, the first step is to construct a nonlocal density functional model for two-dimensional hard disks. A quite successful way to do this is by defining the following free energy:
Fhdb(F)]= Fidb(F)] +
Sd2r dflA*hd@(F))
(AI)
where is the excess free energy per particle of HD in the liquid phase and
p(F) = S d 2 r ’p ( f ’ ) W(IP - 71,p(F))
(A2)
is an average density which takes into account spatial correlations. The weighting function W(r,p)is constructed in order to guarantee an accurate direct correlation f u n ~ t i o n . 4 For ~ ~ ~a qualitative study, the simple prescription used in ref 41 should be useful. With the proper selection of the diameter of the disk, the correlations of this system should be a good first approximation to that of the real one, at least in the isotropic phase. If there is tilt order, the anisotropy induced both by the new order and by the molecular geometry should be also included. A first approach to include this effect may be to simply scale in one direction the correlations between particles in eqs AI and A2. That means transforming the system of hard disks to one of hard parallel ellipses (HPE). The thermodynamics cannot be affected by the scaling; thus a new density functional for HPE is obtained from eqs A1 and A2 by the corresponding deformation of the weight function.” Within this considerations, a generalization of eq 3 is straightforward:
hF[P(f,ii)l = ]d2rSdb P(f,ii)A*hpc(P(fl) X S d 2 r ‘ S d 6 ‘ p(P’,d’) M(f
- f’,ii,ii’)
S d 2 r ’p ( t ’ ) Mhp(P- 7’)
(A31
where M(f,i,ii’)and Mhp(fl are the Mayer functions of the grafted rod and the HPE, respectively. An extremely useful and physically reasonable approximation (in the absence of interfaces) consists in assuming that the angular distribution function, P(6), does not depend on position, Le., p(f,ri) p ( i ) P(6). The only thing left in order to fully define the functional is a prescription to choose the area and shape of the HPE. A reasonable criterion may be constructed by defining a symmetric tensor:
-
A , = Jd2rJd6]dh‘ (41) (42) (43) (44)
r,r,P(C) P(3’) M(j,6,6’) (A4)
Tarazona, T. Mol. Phys. 1984, 52, 81. Tarazona, T. Phys. Rev. A 1985, 31, 2672. Curtin, W. A.; Ashcroft, N. W. Phys. Reu. t r r r . 1986, 56. 2775. Mederos, L.; Sullivan, D. E. Phys. Rev. A 1989, 39, 854.
J. Phys. Chem. 1992, 96, 1409-1417 with i, j = 1, 2 and rl = x, r2 = y . Now, the principal axes of the ellipse, ux, by, may be fixed in order to verify that Jd2r' r'/,Mhpe(F';uz,uy)= A,,
645)
where with Mhpc(~ux,u,,) we explicitly show the dependence of the Mayer function on the principal axes. This criterion takes into account both the macroscopic symmetry (through the angular
1409
distribution function) and the molecular geometry and it fixes both the orientation of the principal axes and their magnitude. In summary, in this Appendix we have provided a nonlocal density functional model for grafted hard rods which needs as input the excess free energy and direct correlation function for HD, as well as the Mayer function of the grafted rods. We believe that such a model can be used to introduce positional correlations in the model for Langmuir monolayers described in the main text.
Adsorption and Thermal Decomposition of Benzene on Ni( 110) Studied by Chemical, Spectroscopic, and Computational Methods D.R. Huntley,* S . L.Jordan,' Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 -6201
and F. A. Grimm Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996- 1600 (Received: July 25, 1991; In Final Form: September 20, 1991)
The chemisorption and reactions of benzene on Ni( 110) have been studied by temperature-programmed desorption (TPD) including isotopic labeling, X-ray photoelectron spectroscopy(XPS), high-resolution electron energy loss spectroscopy(HREELS), and low-energy electron diffraction (LEED) as a function of coverage and adsorption temperature between 100 and 300 K. At saturation, 70430% of the benzene is irreversibly chemisorbed, and C-H bond scission commences at 320 K. For high exposures, molecular desorption competeswith decomposition. A ~ ( 4 x 2LEED ) pattern is observed at saturation coverage of chemisorbed benzene (0.2 monolayer by XPS). HREEL spectroscopy indicates that the benzene ring lies parallel to the surface. Semiempirical molecular orbital calculations have been made and predict the most likely adsorption site for benzene chemisorption to be the atop site at a height of about 1.75 A or the short bridge site at 1.90 A. Upon annealing above 300 K, the benzene decomposes, evolving H2and forming a surface carbide. Additionally, a species forms which ultimately desorbs as benzene at 460 K but also undergoes H-D exchange with benzene-& An unambiguous identification of this fragment has not been made, but the vibrational spectroscopy and isotopic exchange data are consistent with the assignment of a phenyl or benzyne group. The major effects of coadsorbed sulfur and oxygen are to inhibit dissociation and to weaken the interaction between the benzene and the surface.
Adsorption and reactions of benzene on clean, partially oxidized and partially sulfided Ni( 110) were examined with two major goals. The first was to help understand the interactions of the aromatic molecule with the Ni( 110) surface, as an extension to a study of the interactions of benzenethiol with nickel.' The second was to compare the reactivity of benzene on the corrugated Ni( 110) face with previous studies of benzene on the closepacked Ni( 111) face2s3and with the Ni( 100) Benzene chemisorption on metal surfaces has been extensively studied on Ni( 111),2*3*611 Ni( 100),"6*8-11Ni( 110),'Jo and other metal ~ u r f a c e s , ' ~usually - ~ ~ on overlayers adsorbed near 300 K. Fewer studies have been made following low-temperature ads ~ r p t i o n . ~ ~In* general, ' ~ - ~ ~ chemisorbed benzene is thought to be x bonded, with the molecular plane parallel to the surface. The only proposed exception to that model was for Pd(1 lo), where the benzene molecule apparently lies tilted about 10-20° into the troughs.I2 This possibility also exists for Ni(ll0) since it is isostructural with Pd(ll0). Previous studies of the decomposition of benzene on Ni( and Ni( 111)2have indicated some interesting differences. In both casts, the only desorption products observed by heating a chemisorbed overlayer were hydrogen and benzene. On Ni(100), H2 (actually D2since these studies concentrated on perdeuterobenzene) was evolved prior to molecular benzene. The hydrogen peak temperature was 472 K with a low-temperature tail, while the benzene peak desorption temperature was 483 K. In contrast, To whom correspondence should be addressed. Great Lakes Colleges Association Science Semester participant.
Steinriick et al. have made similar studies of benzene on Ni( 111) and find that benzene desorption precedes hydrogen desorption.2 (1) Huntley, D. R. Submitted for publication in J. Phys. Chem. (2) Steinriick, H. P.; Huber, W.; Pache, T.; Menzel, D. Surf. Sei. 1989, 218, 293. (3) Huber, W.; Steinrrlck, H. P.; Pache, T.; Menzel, D. Surf. Sci. 1989, 217, 103.
(4) Blass, P. M.; Akhter, S.; White, J. M. Surf. Sci. 1987, 191, 406. (5) Myers, A. K.; Benziger, J. B. Lungmuir 1987, 3, 414. (6) Jobic, H.; Tardy, B.; Bertolini, J. C.; J. Electron Spectrosc. Relat. Phenom. 1986, 38, 55. (7) Bertolini, J. C.; Massardier, J.; Tardy, B.; J . Chim. Phys. 1981,78,939. (8) Bertolini, J. C.; Dalmai-Imelik, G.; Rousseau, J. Surf. Sci. 1977,67, 478. (9) Bertolini, J. C.; Rousseau, J. Surf. Sci. 1979,89, 467. (10) Friend, C. M.; Muetterties, E. L. J. Am. Chem. Soc. 1981, 103,773. (11) Meyers, A. K.; Shoofs, G. R.; Benziger, J. B. J . Phys. Chem. 1987, 91. 2230. (12) Netzer, F. P.; Rangelov, G.; Rosina, G.; Saalfeld, H. B.; Neumann, M.; Lloyd, D. R. Phys. Rev. E 1988, 37, 10399. (13) Polta, J. A.; Thiel, P. A. J. Am. Chem. Soc. 1986, 108, 7560. (14) Jakob, P.; Menzel, D. Surf. Sci. 1988, 201, 503. (15) Jakob, P.; Menzel, D. Surf. Sci. 1989, 220, 70. (16) Liu, A. C.; Friend, C. M. J . Chem. Phys. 1988,89,4396. (17) Grassian, V. H.; Muetterties, E. L. J . Phys. Chem. 1987, 91, 389. (18) Waddill, G. D.; Kesmodel, L. L. Phys. Reu. E 1985, 31, 4940. (19) Ohtani, H.; Bent, B. E.; Mate, C. M.; van Hove, M. A,; Somorjai, G. A. Appl. Surf. Sci. 1988,33134, 254. (20) Koel, B. E.; Crowell, J. E.; Mate, C. M.; Somorjai, G. A. J . Phys. Chem. 1984,88, 1988. (21) Koel, B. E.; Crowell, J. E.; Bent, B. E.; Mate, C. M.; Somorjai, G. A. J . Phys. Chem. 1986, 90, 2949. (22) Mate, C. M.; Somorjai, G. A. Surf. Sci. 1985, 160, 542. (23) Surman, M.; Bare, S . R.; Hofmann, P.; King, D. A. Surf. Sci. 1987, 179, 243.
0022-3654/92/2096-1409$03.00/00 1992 American Chemical Society
