Remarks Concerning the Relation between Rotator Phases of Bulk n

Figure 2 Phase diagram showing the rotator phases of binary mixtures of alkanes with Δn ≅ 0.5 (i.e., n:n + 1 mixtures). For this small Δn, the pha...
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Langmuir 1997, 13, 3849-3859

3849

Remarks Concerning the Relation between Rotator Phases of Bulk n-Alkanes and Those of Langmuir Monolayers of Alkyl-Chain Surfactants on Water E. B. Sirota* Corporate Research Science Laboratories, Exxon Research and Engineering Company, Route 22 East, Clinton Township, Annandale, New Jersey 08801 Received February 28, 1997. In Final Form: May 13, 1997X Substantial advances have recently been made in characterizing the structures and phase diagrams of Langmuir monolayers of alkyl-chain surfactants on water. These allow detailed comparisons to be made with the bulk phases formed by n-alkanes, in particular, the rotator phases. It is shown that the untilted Langmuir phases denoted CS, S, and LS, correspond to the herringbone-crystal, distorted rotator without long-range herringbone order (RI) and hexagonal rotator (RII), respectively. The quantitative behavior of the distortion and area/molecule in the Langmuir monolayers, their derivatives and jumps at the transitions are shown to correspond more closely to the behavior of n-alkane mixtures where the interlayer interaction is weakened than to that of the pure n-alkanes. It is shown that the tilted phases can be associated with the same three categories regarding distortion and herringbone order that characterize the untilted phases, if the distortion is measured perpendicular to the chain-axis. An “anomalous” reduction of positional order with decreasing temperature in the hexagonal phase of Langmuir monolayers is shown to correspond to the same phenomena observed in bulk n-alkane mixtures. While, due to experimental limitations, it has not been possible to observe the long-range herringbone order in the Langmuir monolayers, the quantitative comparisons to bulk systems where the weak herringbone peak has been observed allow us to associate the S-CS transition with the onset of long-range herringbone order.

Over the past few years a great deal of progress has been made in elucidating the phase diagram and structures formed by Langmuir monolayers of alkane-derivative surfactants,1-5 including fatty acids,6-15 alcohols,16-20 esters,21 phospholipids,22-25 and mixtures.26-28 Likewise,

characterization and understanding of the rotator phase structures formed by the n-alkanes,29-39 has recently been advanced, both for the bulk40-45 as well as ordered monolayers at their liquid-vapor interface.46-48 While many authors have noted relations of the Langmuir monolayers to the alkane phases, a detailed comparison

* Email: [email protected]. X Abstract published in Advance ACS Abstracts, July 1, 1997.

(22) Kenn, R. M.; Kjaer, K.; Mohwald, H. Colloids Surf. 1996, 117, 171. (23) Gehlert, U.; Vollhardt, D.; Brezesinski, G.; Mohwald, H. Langmuir 1996, 12, 4892. (24) Brezesinski, G.; Scalas, E.; Struth, B.; Mohwald, H.; Bringezu, F.; Gehlert, U.; Weidmann, G.; Vollhardt, D. J. Phys. Chem. 1995, 99, 8758. (25) Bringezu, F.; Brezesinski, G.; Nuhn, P.; Mohwald, H. Biophys. J. 1996, 70, 1789. (26) Fischer, B.; Teer, E.; Knobler, C. M. J. Chem. Phys. 1995, 103, 2365. (27) Bibo, A. M.; Knobler, C. M.; Peterson, I. R. J. Phys. Chem. 1991, 95, 5591. (28) Shih, M. C.; Durbin, M. K.; Malik, A.; Zschack, P.; Dutta, P. J. Chem. Phys. 1994, 101, 9132. (29) Ewen, B.; Strobl, G. R.; Richter, D. Faraday Discuss. Chem. Soc. 1980, 69, 19. (30) Doucet, J.; Denicolo, I.; Craievich, A. J. Chem. Phys. 1981, 75, 1523. (31) Denicolo, I.; Doucet, J.; Craievich, A. F. J. Chem. Phys. 1983, 78, 1465. (32) Doucet, J.; Denicolo, I.; Craievich, A. F.; Germain, C. J. Chem. Phys. 1984, 80, 1647. (33) Craievich, A.; Denicolo, I.; Doucet, J. Phys. Rev. B 1984, 30, 4782. (34) Denicolo, I.; Craievich, A. F.; Doucet, J. J. Chem. Phys. 1984, 80, 6200. (35) Ungar, G. J. Phys. Chem. 1983, 87, 689. (36) Ungar, G.; Masic, N. J. Phys. Chem. 1985, 89, 1036. (37) Snyder, R. G.; Srivatsavoy, V. J. P.; Cates, D. A.; Strauss, H. L.; White, J. W.; Dorset, D. L. J. Phys. Chem. 1994, 98, 674. (38) Stewart, M. J.; Jarrett, W. L.; Mathias, L. J.; Alamo, R. G.; Mandelkern, L. Macromolecules 1996, 29, 4963. (39) Dirand, M.; Achour, Z.; Jouti, B.; Sabour, A.; Gachon, J. C. Mol. Cryst. Liq. Cryst. 1996, 275, 293. (40) Sirota, E. B.; King, H. E., Jr.; Singer, D. M.; Shao, H. H. J. Chem. Phys. 1993, 98, 5809. (41) Sirota, E. B.; Singer, D. M.; King, H. E., Jr. J. Chem. Phys. 1994, 100, 1542. (42) Sirota, E. B.; Singer, D. M. J. Chem. Phys. 1994, 101, 10873. (43) Sirota, E. B.; King, H. E., Jr.; Shao, H. H.; Singer, D. M. J. Phys. Chem. 1995, 99, 798. (44) Jimenez, R.; Kruger, J. K.; Prechtl, M.; Grammes, C.; Alnot, P. J. Phys. Condens. Matter 1994, 6, 10977. (45) Yamamoto, T.; Nozaki, K. Polymer 1994, 35, 3340.

(1) Peterson, I. R.; Brzezinski, V.; Kenn, R. M.; Seitz, R. Langmuir 1992, 8, 2995. (2) Jacquemain, D.; Wolf, S. G.; Leveiller, F.; Deutsch, M.; Kjaer, K.; Als-Nielsen, J.; Lahav, M.; Leiserowitz, L. Angew. Chem., Int. Ed. Engl. 1992, 31, 130. (3) Jacquemain, D.; Leveiller, F.; Weinbach, S. P.; Lahav, M.; Leiserowitz, L.; Kjaer, K.; Als-Nielsen, J. J. Am. Chem. Soc. 1991, 113, 7684. (4) Bohanon, T. M.; Lin, B.; Shih, M. C.; Ice, G. E.; Dutta, P. Phys. Rev. B 1990, 41, 4846. (5) Li, M.; Rice, S. A. J. Chem. Phys. 1996, 104, 6860. (6) Shih, M. C.; Bohanon, T. M.; Mikrut, J. M.; Zschack, P.; Dutta, P. Phys. Rev. A 1992, 45, 5734. (7) Lin, B.; Shih, M. C.; Bohanon, T. M.; Ice, G. E.; Dutta, P. Phys. Rev. Lett. 1990, 65, 191. (8) Kaganer, V. M.; Peterson, I. R.; Kenn, R. M.; Shih, M. C.; Durbin, M.; Dutta, P. J. Chem. Phys. 1995, 102, 9412. (9) Bommarito, G. M.; Foster, W. J.; Pershan, P. S.; Schlossman, M. L. J. Chem. Phys. 1996, 105, 5265. (10) Durbin, M. K.; Malik, A.; Ghaskadvi, R.; Shih, M. C.; Zschack, P.; Dutta, P. J. Phys. Chem. 1994, 98, 1753. (11) Schwartz, D. K.; Schlossman, M. L.; Pershan, P. S. J. Chem. Phys. 1992, 96, 2356. (12) Riviere, S.; Henon, S.; Meunier, J.; Schwartz, D. K.; Tsao, M.W.; Knobler, C. M. J. Chem. Phys. 1994, 101, 10045. (13) Riviere-Cantin, S.; Henon, S.; Meunier, J. Phys. Rev. E 1996, 54, 1683. (14) Peterson, I. R.; Kenn, R. M.; Goudot, A.; Fontain, P.; Rondelez, F.; Bowman, W. G.; Kjaer, K. Phys. Rev. E 1996, 53, 667. (15) Kenn, R. M.; Bohm, C.; Bibo, A. M.; Peterson, I. R.; Mo¨hwald, H.; Kjaer, K.; Als-Nielsen, J. J. Phys. Chem. 1991, 95, 2092. (16) Barton, S. W.; Thomas, B. N.; Flom, E. B.; Rice, S. A.; Lin, B.; Peng, J. B.; Ketterson, J. B.; Dutta, P. J. Chem. Phys. 1988, 89, 2257. (17) Lin, B.; Peng, J. B.; Ketterson, J. B.; Dutta, P.; Thomas, B. N.; Buontempo, J.; Rice, S. A. J. Chem. Phys. 1989, 90, 2393. (18) Shih, M. C.; Bohanon, T. M.; Mikrut, J. M.; Zschack, P.; Dutta, P. J. Chem. Phys. 1992, 97, 4485. (19) Overbeck, G. A.; Honig, D.; Mobius, D. Langmuir 1993, 9, 555. (20) Overbeck, G. A.; Mobius, D. J. Phys. Chem. 1993, 97, 7999. (21) Foster, W. J.; Shih, M. C.; Pershan, P. S. J. Chem. Phys. 1996, 105, 3307.

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Figure 1. Schematic phase diagram for the Langmuir monolayers, showing both the common nomenclature for the phases and that used in this paper. Various phases may be squeezed out allowing, for example, direct transitions from the “U” to “I” phases missing the “F”. The schematics of the phases are projections along the chain axis, and the arrows represent the direction of tilt. The distortions are depicted as compressions with respect to the chain axis. The distortion in the tilted II phases is a compression in the direction of tilt, which is opposite to the trivial stretching in the direction of tilt apparent when looking in the surface plane.

has not yet been made. Here we show that the behavior of the high-pressure phase sequence of the Langmuir monolayers exhibits a remarkable similarity to the untilted alkane phases, as do the transitions to the tilted phases. We will show that the phase sequence of the untilted phases hexagonal (RII ) LS)-distorted hexagonal (RI ) S)-herringbone (X ) CS) is the same with regard to distortion and area/molecule, their derivatives, and their jumps at the transitions. Furthermore, a disordering phenomena with “anomalous” temperature dependence is shown to occur in both hexagonal phases. The correspondence becomes much closer when the behavior of the monolayer is compared with n-alkane mixtures, where the interlayer coupling is reduced. We show that this correspondence does not require identification of the phases in either or both systems as being hexatics. We then show how the tilted phases in both surface and bulk relate to the untilted phases. We argue that the distortion order-parameter must be defined perpendicular to the chain axis, since the ordering sequence involves freezing out of rotational disorder about the chain axis. Untilted Alkane Phases In most monolayer systems, the untilted phase sequence which occurs at high surface pressures (π) progresses from a herringbone crystal at low temperatures (CS) to a hexagonal rotator structure at high temperatures (LS), with an intermediate distorted phase (S) as shown in the schematic phase diagram of Figure 1. At this point we will summarize the pertinent characteristics of the n-alkane phase sequence to which the Langmuir monolayers will to be compared. Figure 2 is the phase diagram for n:n + 1 (∆n ) 0.5) binary mixtures of n-alkanes.43 The temperature axis has been shifted by the interpolated melting temperatures of the pure n-alkanes, which are (46) Wu, X. Z.; Sirota, E. B.; Sinha, S. K.; Ocko, B. M.; Deutsch, M. Phys. Rev. Lett. 1993, 70, 958. (47) Wu, X. Z.; Ocko, B. M.; Sirota, E. B.; Sinha, S. K.; Deutsch, M.; Cao, B. H.; Kim, M. W. Science 1993, 261, 1018. (48) Wu, X. Z.; Ocko, B. M.; Tang, H.; Sirota, E. B.; Sinha, S. K.; Deutsch, M. Phys. Rev. Lett. 1995, 75, 1332.

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Figure 2. Phase diagram showing the rotator phases of binary mixtures of alkanes with ∆n = 0.5 (i.e., n:n + 1 mixtures). For this small ∆n, the phase diagram is very close to that of pure n-alkanes; however, the even-odd effect at the transition to the crystal phase disappears. The temperatures are shifted to be relative to the melting temperature of the interpolated pure n-alkane (Tm ) -184.79 + 21.90n - 0.7820n2 + 0.01398n3 (9.884 × 10-5)n4 with T in °C).43 Shown above are the surface crystalline phases observed just above the melting temperature for the pure n-alkanes.80 Mixtures of n:n + 1 showed little deviation from this.48

Figure 3. (top) Schematic showing long-range herringbone order in the crystal (X) phase. The diagonal lines show the planes which give rise to the “herringbone” peaks. (bottom) Schematic of 2D reciprocal space showing the first-order inplane reflections as solid dots and the herringbone peaks as circles. The peaks represented by open circles are absent in the RI phase.

well-defined quantities. As ∆n increases from 0, this phase diagram varies, as is discussed below.43 The untilted rotator phases exist for the shorter chain lengths (n e 26) in pure n-alkanes.29-36 The lowtemperature non-rotator crystalline phase (X) has orthorhombic (distorted-hexagonal) packing within a layer, as well as long-range herringbone order of the rotational degree of freedom of the backbones (Figure 3). A “rotatorphase”, as the term is used here, refers to a phase where there is no long-range order with respect to the rotational degree of freedom of a molecule about its long axis (modulo 180°).49 Thus, a rotator phase need not be hexagonal. For a molecule whose preferred low-temperature packing is herringbone-ordered orthorhombic, a structure will be in a rotator phase if the herringbone order does not extend to long range. In this generic sequence, the highest temperature phase is the RII in which the molecules are

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Figure 4. Schematic showing 3-fold orientationally disordered orthorhombic domains and the average hexagonal structure it produces, representing the RII phase.

in an average hexagonal arrangement. The molecules may locally pack in a distorted structure. However, the size of correlated regions may only be a few molecules, and thus the phase has macroscopic hexagonal symmetry as illustrated in Figure 4. The term “free rotator” is often mistakenly used for this phase. For there to be truly free uncorrelated rotation of the molecules, the area/molecule (A) would have to be >25 Å2. The intermediate phase is the RI, in which the molecules are packed in a distorted hexagonal structure, where there is not long-range herringbone order. This is illustrated schematically in Figure 5 where both the left and right domains have the same distortion and both have local herringbone packing, but there lacks long-range herringbone order. The positional order of the centers of mass may, nevertheless, be maintained. The “domains” in Figure 5 cannot be annealed away since they are entropically stabilized. The use of the term “herringbone” by some authors, to describe such a phase has led to a great deal of confusion. Figure 5 is not meant to describe the nature of the defects in molecular packing. However, it is likely that some of these defect packings are the parallel configurations characterisitic of the subcell of the triclinic crystalline phase which occurs in pure even-n-alkanes.50,51 It is extremely useful to quantify the distortion in terms of an order-parameter which goes to 0 in the hexagonal phase and is finite in the distorted phases. Such a parameter was originally defined as D ) 1 - (b/a) where a and b are the major and minor axes of an ellipse drawn through the six nearest neighbors.40 Kaganer et al.8 have made a slight modification to this, defining the distortion ξ ) (a2 - b2)/(a2 + b2). These are nearly the same for small distortions where ξ ) D*(1 - D/2)/(1 - D + D2/2). Thus, D ) ξ ) 0 in the RII phase, and in the orthorhombic herringbone crystal phase the distortion is essentially fixed at D ) 0.133, ξ ) 0.142. In tilted phases, it must be defined whether the distortion is measured with respect to the layers surface (i.e., center of mass) or a projection with (49) As determined by Ewen et al. (ref 29) there is an additional transition between two crystalline herringbone states in pure bulk n-alkanes where the 180° rotational disorder freezes out giving longrange order with respect to that rotation as well. Since the in-plane packing is equivalent with respect to a 180° rotation and a 1-carbonlength translation, this ordering would be very sensitive to interlayer interactions and would unlikely occur in a monolayer or in crystals of mixed chain lengths. (50) Kitaigorodskii, A. I. Molecular Crystals and Molecules; Academic Press: New York, 1973. (51) Yamamoto, T. J. Chem. Soc., Faraday Trans. 1995, 91, 2559.

Figure 5. Schematic of the RI phase, showing two regions with the same distortion and each with local herringbone order. In the middle is a schematic showing the definitions of the distortion order parameter. Below is a schematic illustrating the average symmetry of the phases. The nature of the local packing at the defect is not intended to be addressed by this schematic. Such defects do not necessarily destroy the crystalline positional order.

respect to the long axis of the molecules. As we will elaborate on below, in this paper we always consider the molecular axis projection. The area/molecule in the RII phase is A = 19.7 Å2 with a thermal expansion of (dA/dT)/A = 1.3 × 10-3 °C-1. The isothermal compressibility in the plane of the layers is -(dA/dP)/A = 4 × 10-5 bar-1 ) 4 × 10-11 cm2/dyn and the heat capacity is about 1 kJ/mol.41,42 On cooling, the pure n-alkanes undergo a transition to the RI phase where ξ becomes finite. This transition is always first order, albeit weakly so; in Landau theory language, this is because a positive and negative ξ yield inequivalent situations42 giving rise to a finite ξ3 term in the free energy. The jump in ξ at the transition for pure n-alkanes is ∆ξ = 0.02 and the decrease in area is ∆A = 0.02 Å2. The RI phase itself is characterized by a dramatic increase in ξ with decreasing temperature and a very large thermal expansion (dA/dT)/A = 2.3 × 10-3 °C-1, compressibility -(dAdP)/A = 6.2 × 10-5 bar-1 ) 6.2 × 10-11 cm2/dyn, and heat capacity ∼1.5 kJ/ mol. The RI phase is interesting in that these derivatives themselves are strongly temperature dependent.52 At still lower temperatures, occurs a first-order transition to the herringbone-crystal phase which is characterized by the doubling of the unit cell and development of a long-range herringbone peak shown in Figure 6 and schematically in Figure 3. Since this peak is often very weak, the most measured characteristics of this transition are the jump in ξ to 0.142 where it no longer changes with temperature, the jump decrease in A of 0.6-1.0 Å2 in the pure n-alkanes, a drop in the thermal expansion to (dA/dT)/A = 5 × 10-4 °C-1, the compressibility to -(dAdP)/A = 2 × 10-5 bar-1 ) 2 × 10-11 cm2/dyn, and heat capacity to ∼0.5 kJ/mol. Mixtures. With the production of binary mixtures of different length n-alkanes, the phase diagram (Figure 2) (52) Shao, H.; King, H. E., Jr.; Sirota, E. B. To be submitted for publication.

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Figure 6. Scans from a 50:50 n-alkane C21:C23 binary mixture in the (top) herringbone crystal phase (23 °C) and (bottom) RI phase (28 °C). The “herringbone peak” {210} at q = 2.1 Å-1 is clearly present in the low-temperature phase and absent in the rotator phase. The y-axis is in log scale to make the {210} peak apparent.

is modified.34,36,43 The principal effect of the multiple chain lengths is to produce a more disordered interlayer interface and thus reduce the interlayer coupling. Similar effects were seen with intercalated gas between layers.41 It is this regime which we expect to behave most like the Langmuir monolayers. Chain-length distributions have a dramatic effect on the RII phase and on the RI-RII phase transition. The temperature of the RI-RII phase transition is decreased significantly, thereby increasing the temperature range of the RII phase. Thus, while in pure alkanes the RII phase has a temperature range of stability (∆T) of 2-5 °C, in mixtures ∆T can be >15 °C. This allows the RII phase to occur with an area/molecule 0.2 Å2 smaller than “usual”. As temperature is reduced through the RII phase, a striking decrease in the intensity of the hexagonal in-plane Bragg peak is observed, allowing the diffuse scattering under the peak to increasingly dominate.53 The large amount of diffuse scattering under a rather weak Bragg peak causes the decreasing intensity of the Bragg peak to appear as a broadening when a low or moderate resolution is used in performing the X-ray scattering. This is illustrated by the cooling sequences in Figure 7 taken at both high and low resolution. This intensity reduction is continuous, and at low enough temperature, no trace can be found of the Bragg component (within our resolution), leaving only the broader diffuse peak. This suggests short-range positional order and possible hexatic behavior. This phenomenon was found to be a reversible equilibrium effect, not associated with chain-length segregation.43,53 The transition to the RI phase involves the appearance of two peaks which are, again, sharp. When the RII phase is thus extended down in temperature, the transition to the RI phase takes on a different character. Instead of the jump in the distortion being ∆ξ ) 0.02, it can be as high as ∆ξ ) 0.088. The distortion is plotted in Figure 8a for a 50:50 mixture of C23 and C28 n-alkanes which illustrated this effect. Similarly ∆A and ∆H for this transition are larger as well. The mechanism for this disordering effect on decreasing temperature has been attributed to the frustration caused by increasingly larger orthorhombic domains packed in different orientations to give hexagonal symmetry.43,53 Evidence for this effect has also been seen in bulk mixtures of 1-alcohols,54 which have the same generic untilted phase sequence. (53) Sirota, E. B.; King, H. E., Jr.; Hughes, G. J.; Wan, W. K. Phys. Rev. Lett. 1992, 68, 492. (54) Sirota, E. B.; Wu, X. Z. J. Chem. Phys. 1996, 105, 7763.

Figure 7. (a) A series of low-resolution powder scans for a 50:50 weight fraction mixture of C24 and C30 n-alkanes at temperatures 50.0, 48.3, 46.5, 44.5, 43.1, 42.3, and 41.2 °C. The curves are offset to the right by 0.01 Å-1 for clarity. The resolution fwhm was ∆q ) 0.01 Å-1 for this measurement. (b) A series of high-resolution powder scans (∆q ) 0.0006 Å-1) for a 50:50 C23:C28 mixture at (R) T ) 48.5 °C, (β) 43.7 °C, (γ) 40.0 °C, and (δ) 37.5 °C in the RII phase. Here the curves are offset vertically for clarity. A finer scan shows two resolved peaks separated by ∆q ) 0.002 arising from the bilayer stacking in this phase.53

We now discuss the effect of chain-length mixtures on the RI phase and the RI-X phase transition.43 The transition temperature is substantially lowered, allowing a much larger rotator phase range, where the RI phase itself can be stable for ∆T > 15 °C. In Figure 8b we show the distortion for a 50:50 mixture of C21 and C23 n-alkanes. With an increased range of the RI phase, which has a higher heat capacity (cp) than the lower temperature crystal phase, the first-order jumps of ∆A, ∆ξ, and ∆H are substantially reduced compared to the pure n-alkanes. This is opposite to the RII-RI transition where cp is higher in the lower temperature phase, causing larger jumps. Specifically ∆A can become as low as 0.2 Å2, and ∆ξ can become 0) and ideally no distortion with respect to the chain axis (ξ ) 0). The FI has finite distortion (ξ * 0) and the FX has long-range herringbone order. (This discussion pertains equally to the “I” phases with NN tilt direction). We say that FII has ξ ) 0 “ideally” because in practice ξ is finite but weak in these phases, in both the Langmuir and bulk systems. A weak compression with respect to chains is observed in the direction of tilt in the RIV phase of the alkanes40 as it is in the Ov(τ) phase of the ester.21 In the L2d and Ov phases of the fatty acid, Durbin et al.10 do not claim a deviation from hexagonal larger than their error bars; however, their data are consistent with both the direction and magnitude of the distortion observed in the alkane and ester. A symmetry problem arises when rigorously considering the difference between the FII and FI phases: There is technically no difference in symmetry between θ > 0, ξ ) 0, and θ > 0, ξ * 0, so there cannot rigorously be a second-order transition between the two with ξ as the order parameter. If, however, we write ξ as the order parameter, we find that the finite coupling between the tilt magnitude (θ) and the distortion (ξ) (which occurs in the mean field theory as a θ2ξ term8), will induce a weak, yet finite distortion in the plane perpendicular to the chainaxis even in the high-temperature (FII) phase. This will technically destroy the possibility for a rigorous secondorder transition between the two phases, consistent with the symmetries of the two phases stated above. Thus, there is technically no difference in symmetry between the FI and FII phases as defined in the above paragraph: However, in practice, they can be distinct and transitions between them can occur. This is because the coupling is weak, which can be related to the fact that the only reason that the θ > 0, ξ ) 0 phase is of no higher symmetry than the θ > 0, ξ > 0 phase is due to the lack of overlap of chain ends during the appropriate symmetry operation. For long chains with chain-end disorder,75 this lack of overlap clearly becomes less relevant. In fact, when considering only projections along the chain axis, a finite ξ does break the symmetry. As an experimental observable, this distinction is very much dependent on experimental resolution; since there will be a minimum detectable ξ, the observation of a transition from ξ ) 0 to ξ * 0 will depend on the quantitative ability to measure it. This is very much analogous to the smectic-C and smectic-I phases, where the smectic-I is a tilted hexatic and the smectic-C is supposed to be a liquid, but rigorous symmetry arguments say that it must be a hexatic. This is because a finite coupling between tilt and bond-orientational orderparameters induces hexatic order in the smectic-C phase.76,77 If there is a transition between such phases it must be first order. However, there can be what appears as a second-order transition smeared out in a similar way that a magnetic field smears out the Curie transition. In other words, the virtual transition can be considered as (75) Maroncelli, M.; Strauss, H. L.; Snyder, R. G. J. Chem. Phys. 1985, 82, 2811. (76) Bruinsma, R.; Nelson, D. R. Phys. Rev. B 1981, 23, 402. (77) Nelson, D. R.; Halperin, B. I. Phys. Rev. B 1980, 21, 5312.

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the temperature where ξ would extrapolate to zero if the θ2ξ coupling were not there. Similarly, in the case of the smectic-I and smectic-C transition, the second-order transition can still be effectively studied.78 For any given tilt direction one can then expect up to three phases which should always occur in the order of II, I, and X on decreasing temperature. The phase diagram for methyl eicosanate21 is a clear example of this description of the phases, where the rotator phases all have NNN tilt and the crystal phase has NN tilt. This shows a transition from the FI (L2′) to FII (τ, Ov) on heating. Anisotropy of Peak Widths. In the tilted I and II type phases, an anisotropy of the peak widths is generally observed in Langmuir monolayers.8,15,21,63 Regardless of whether it is a NN or NNN tilt, the widths are considerably broader in the direction of the tilt. This phenomenon, not unique to monolayers, is general for weakly ordered tilted phases, occurring in bulk liquid crystalline materials as well. In the smectic-F and smectic-I phases which are stacked hexatics, the correlation lengths are also shorter in the direction of tilt.58 These smectics also show anisotropy in the interlayer correlations. A comparison of the stacked hexatic (smectic-F) and the rotator phase analogue (crystalline-G) shows that the anisotropy which appears in the peak widths of the hexatic manifests itself in the relative level of diffuse scattering under the Bragg peak in the corresponding rotator phase.58 The dependence of the tilt magnitude (θ) on temperature and pressure is one place where the Langmuir films differ from the bulk systems. The most direct way of decreasing the tilt magnitude for Langmuir films is by increasing π. This is because π couples directly to the projected surface area As ) A/cosθ (A is measured perpendicular to the molecule’s long axis). Since there is more entropy associated with decreasing A than decreasing θ, as A involves not only the rotational degrees of freedom, but also chain defects,75 it is θ which is most easily reduced. While pressure affects the untilted phases of bulk alkanes in a similar way to Langmuir monolayers, the effect on the tilted phases is much different. Pressure couples to the volume, which is most affected by the area as measured normal to the chains. Thus, reducing tilt does not reduce this area directly. In the bulk C26 alkane, tilt arises from the RII phase on both decreasing and increasing temperature. On increasing temperature the RIV (or FII) phase occurs. The increased tilt in this temperature range has been attributed42 to the increased effective size of the chain-ends due to chain-end gauche bonds becoming more prevalent at higher temperatures.75,79 At lower temperatures the RV (or FI) phase occurs. While NN tilt has not been found in bulk n-alkanes, it does occur as a surface rotator phase for longer chain lengths.80 While a direct RIV-RV transition does not occur in the pure n-alkanes, it does occur in binary mixtures43 (Figure 2) as well as in hydrated 1-alcohols.54 In longer chain lengths of pure n-alkanes (n > 29) a NNN tilted herringbone ordered crystal (FX) exists between the untilted crystal (UX) and the rotator phases.29 In Langmuir monolayers, however, the tilted phase is in the NN direction (IX ) L2′′). There are some suggestions that the phase denoted L2′ is, in fact, two phases27 (possibly FX and FI); however, there is no X-ray structural confirmation of this. (78) Brock, J. D.; Noh, D. Y.; McClain, B. R.; Litster, J. D.; Birgeneau, R. J.; Aharony, A.; Horn, P. M.; Liang, J. C. Z. Phys. B: Condens. Matter 1989, 74, 197. (79) Jarrett, W. L.; Mathias, L. J.; Alamo, R. G.; Mandelkern, L.; Dorset, D. L. Macromolecules 1992, 25, 3468. (80) Ocko, B. M.; Wu, X. Z.; Sirota, E. B.; Sinha, S. K.; Gang, O.; Deutsch, M. Phys. Rev. E 1997, 55, 3164.

Relation between Bulk Alkanes and Langmuir Monolayers

The transitions from untilted to tilted phases within the same category (II or I) in Langmuir films appear to be nearly second order. This is consistent with the secondorder character of the RII-RIV and RI-RV transitions in bulk alkanes.42 For example, the transition from UI to FI is observed to be continuous7 which is consistent with the second-order nature of the RI-RV transition in the bulk alkanes where tilt sets in an already distorted system. The transition which appears first order from the lower temperature region of the UII phase into a NN tilted phase (L2)19 is likely to be into the II phase, which is analogous to the first-order transition observed in bulk C26 between the RII and the RV.42 Tilt Orientation. It is apparent that the distortion normal to the chain axis and herringbone ordering are the predominant order parameters driven by temperature. The tilt magnitude (θ), coupling directly to the projected area/molecule, is strongly driven by surface pressure. The tilt direction is related to subtler effects and has associated with it extremely low entropy differences when the transition only involves the tilt direction and no other order parameters.81,82 In understanding the phase diagram and the transitions, it is important to first characterize the phases in terms of II, I, and X. If a transition involves a tilt reorientation, but also a transition between the II, I, and X categories, it is not primarily a “swiveling” transition but rather a “distortion” or “herringbone” transition, where the tilt reorientation is a subtlety. This view is supported by the extremely low latent heat of the crystal-J to crystal-G transition in liquid crystals.81 If, however, the tilt orientation is the only parameter effected at the transition, then it can rightly be called a “swiveling transition”. In the liquid-crystal literature, it is found that when such a transition occurs within one of the categories (i.e., hexatic I/F, rotator J/G, herringbone K/H), the NN tilt (i.e., I, J, K) is always the one at the higher temperature.58,81-83 This was explained in terms of an entropy arguement tied to the symmetry of the hexagonal lattice.82,84 Even though some reported swiveling transitions in Langmuir monolayers occur where the NNN tilt appears at higher temperature (and lower π) than the NN, a careful examination of these show that in such cases a more significant change in the distortion or herringbone ordering is likely occurring. For example the L2′′-L2′ transition on increasing temperature reported by Bommarito et al.9 is a IX-FI transition whose entropy is comparable to that of rotator-crystal transitions and is very similar to such a transition observed on surface-crystallized monolayers of hexatriacontane.48 In monoglycerides,23 the evolution from the II to the FI on increasing pressure was observed. The evolution from III to FII on increasing pressure was observed in fatty acids.10 For hydrated mutlilayers of dialkylphosphatidylcholines, Sun et al.85 have observed what appears to be an IX-FI transition. Thus, by distinguishing the phases in terms of II, I, and X it may be possible to see more consistent trends in the T and π dependence of the tilt orientation transitions. From the schematic phase diagram (Figure 1) it is apparent that in the IX and II phases there is a difference in the couplings between the bond/tilt direction and the (81) Budai, J.; Pindak, R.; Davey, S. C.; Goodby, J. W. J. Phys. Lett. 1984, 45, L1053. (82) Sirota, E. B. J. Phys. (Paris) 1988, 49, 1443. (83) Gane, P. A. C.; Leadbetter, A. J.; Wrighton, P. G.; Goodby, J. W.; Gray, G. W.; Tajbakhsh, A. R. Mol. Cryst. Liq. Cryst. 1983, 100, 67. (84) Sirota, E. B.; Safinya, C. R.; Smith, G. S.; Plano, R.; Roux, D.; Clark, N. A. In Geometry and Thermodynamics; Toledano, J. C., Eds.; Plenum: New York, 1990; p 255. (85) Sun, W. J.; Tristram-Nagle, S.; Suter, R. M.; Nagle, J. F. Biochim. Biophys. Acta 1996, 1279, 17.

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Figure 10. (a) Schematic showing 2-fold disorder of the distortion of the FI phase, which can give rise to the distortion of the II phase. (b) and (d) are two examples of “chains” of these regions. (d) shows that the alternation need not be regular. (c) is a superposition of (b)-like regions showing the intrinsic disorder characteristic of the rotator phases. In other words: the disorder is not harmonic vibrations about a single welldefined position but possibly multiple minima for each molecule.

direction of the distortion. The distortion in the IX phase is as one would expect, given the local packing associated with herringbone ordering; when viewed down the long axis of the molecules it is the same as in the UX phase. In the rotator phases, there is no long-range herringbone order, and therefore, the dominant coupling of the distortion is to the tilt direction. If locally the packing is herringbone in the II phase, how can the distortion involve compression in the NN direction? As shown for the hexagonal phase in Figure 4, the rotator phases are not well-packed crystal structures, but have excess area/ molecule which allows multiple local packings to give rise to a structure with a different symmetry. In the case of the II phase, a possible local packing situation is illustrated schematically in Figure 10. Here local distortions in the NNN direction, oriented 60° apart, exist in equal numbers, giving an average compression along the NN direction. In a rotator phase, each molecule does not have a single welldefined lowest energy position around which it experiences nearly harmonic vibrations, but rather, a series of local minima, both rotational and positional. Within this model, it is clear that the NN distortion is a less ordered structure and, thus, a higher temperature state than the NNN distortion. From a microscopic point of view, the ability to maintain the coupling between the tilt and distortion directions in the rotator phases is also explained. The switch in the distortion direction upon entering the IX phase is now clear: Unlike the rotator phases, the herringbone crystal phase does not allow such disorder as portrayed in Figure 10; packing must be essentially complete. Thus, the distortion must be in the same direction as the herringbone order which is tied to the bond-orientational order of the lattice. In mean-field language, the coupling between the herringbone order and the distortion direction is much stronger than that between the tilt and the distortion directions. In comparing the tilted herringbone phases in the bulk and the Langmuir systems, we see that while the distortion is always in the same direction with respect to the lattice, the Langmuir films exhibit a IX phase with varying tilt angle, while the bulk exhibits only a FX phase with a single

3858 Langmuir, Vol. 13, No. 14, 1997

Figure 11. (a) Schematic showing regular offsets along the chain axis giving rise to the 18.5° tilt in the bulk FX phase, where the interlayer interface is smooth. (b) Schematic showing situations where the tilt is ∼6° and the interface is less regular. Since the plane spacing is smaller for NN tilt, 〈u2〉 will be smaller, as discussed in the text.

tilt angle locked-in at 18.5°.29,86 Such a structure is derived from the untilted herringbone structure by regular displacements of a projected CH2 length (1.27 Å) and 180° rotations, along and around the molecular axis, with each positional step in the NNN direction (i.e., spacing of ∼31/2a/ 2). Such an ordered set of displacements leaves the molecular ends in a single plane (as shown schematically in Figure 11a) and makes possible the well-packed interlayer stacking associated with the bulk herringbone crystalline phases.50 Thus, as distinct from the bulk rotator phases, the bulk herringbone phases only allow specific quantized tilt angles. In a monolayer, the herringbone phase only requires well-defined order in twodimensions. Thus, the lower tilt angles are allowed, due to the allowable disorder along the molecular axis (nearly normal to the surface). Such disorder will indeed have some energetic cost due to the microscopic contribution of the surface tension, however, it is not prohibited by packing considerations as in the bulk. To maintain the required proper in-plane packing, displacements along the molecular axis are still quantized. To produce a continuous reduction in the tilt angle requires that the average lateral displacement before a longitudinal translation will be greater than one plane spacing and that it cannot be totally regular. This will result in a situation where the molecular ends will not be coplanar (Figure11 b). The deviation from a single plane might be approximated as producing an energetic cost ∝ 〈u2〉, i.e., the mean-squared displacement from the average surface. One can easily show that the smaller the lateral distance between lattices planes, the smaller 〈u2〉 will be for a given tilt. (Imagine N objects uniformly distributed from z ) -w/2 to z ) +w/2. For N ) 2, 〈u2〉 ) w2/4; for N ) 3, 〈u2〉 ) w2/6; and for N ) ∞, 〈u2〉 ) w2/12.) Since, for a NN tilt the spacing is ∼a/2, compared to ∼31/2a/2 for NNN tilt, the NN tilt would yield a smoother surface for the small tilt angles which occur in compressed monolayers. Thus, we would expect the appearance of the NN tilt IX phase, as is observed, for example, by Foster et al.21 with a tilt angle of ∼6°. Intermediate Tilt. In hydrated phospholipid multilayers, the progression from NNN (LβF) to NN (LβI) tilt direction occurs with increasing temperature and humidity.73,74,87 This, however, was found to proceed by two second-order phase transitions through a phase with (86) Heyding, R. D.; Russell, K. E.; Varty, T. L.; St-Cyr, D. Powder Diffr. 1990, 5, 93. (87) Smith, G. S.; Sirota, E. B.; Safinya, C. R.; Plano, R. J.; Clark, N. A. J. Chem. Phys. 1990, 92, 4519.

Sirota

intermediate tilt direction which was termed LβL, and the “liquid-crystal” (FGH, IJK) nomenclature was extended to include LMN for the possible phases of intermediate tilt.73,74,87 Selinger and Nelson then extended theoretical work on tilted hexatics to include such phases of intermediate tilt.88 Such phases have also recently been observed in thermotropic liquid crystals.89,90 Bulk n-alkanes show a rotator phase of intermediate tilt (RIII) in which the tilt direction varies away from the symmetry directions. In C27, the sequence on heating is RV-RIII-RIV and the X-ray data suggest that the RV-RIII transition proceeds continuously and a very slight structural discontinuity exists at the RIII-RIV transition.40 However, in longer alkanes it is the RIII-RIV transition which appears to be second order.40,42 It is, therefore, not clear whether the RIII phase is II-like or I-like or whether there may be, in fact, two different “RIII” phases.40 Furthermore, in alkanes there has not been a continuous evolution observed from the NN to NNN tilt direction. In Langmuir monolayers, intermediate tilts have been observed,14,23 and an interesting partly continous evolution between the two symmetry directions (II and FI) has recently been reported.91 Discussion and Conclusions The structures of the Langmuir films have been shown to correspond to those of the alkanes, where the primary ordering transitions and classification of structures are according to the distortion and long-range herringbone ordering as viewed down the chain axis. The Langmuir films behave similar to mixtures of alkanes where the interlayer coupling is reduced. In both systems the hexagonal phase is extended down in temperature to a regime where positional order decreases with decreasing temperature. A first-order transition occurs to a phase with long-range distortion and that distortion increases with decreasing temperature until a first-order transition occurs to a nonrotator crystalline phase with long-range herringbone order. The nature of the tilting transitions is also shown to be similar to those in the alkanes and the tilted phases undergo the same distortion and herringbone ordering sequences, with distortion measured perpendicular to the chain axis. The interpretation of the broad peaks in some phases as hexatic is questioned based on finite resolution. However, comparisons with bulk alkanes, which are 3D crystals, led us to argue that the distinction between hexatic and 2D crystal was not a critical distinction between these phases with regard to the distortion, tilt, and herringbone transitions. Because of the reduced dimensionality, it is possible that while the corresponding bulk systems have a weak long-range positional order, the monolayers may indeed be hexatics. It must be pointed out, however, that the hexagonal RII phase of surface crystalline alkane monolayers in equilibrium with their melt appear to be 2D crystals rather than hexatics.80 On the theoretical side, Kaganer and co-workers8,92-94 have developed a detailed mean-field theory to incorporate all of the observed phases. Competing effects of tilt, which tends to stretch the projected cell in the direction of tilt, and the intermolecular distortion, which usually com(88) Selinger, J. V.; Nelson, D. R. Phys. Rev. Lett. 1988, 61, 416. (89) Maclennan, J. E.; Sohling, U.; Clark, N. A.; Seul, M. Phys. Rev. E 1994, 49, 3207. (90) Chao, C. Y.; Hui, S. W.; Maclennan, J. E.; Chou, C. F.; Ho, J. T. Phys. Rev. Lett. 1997, 78, 2581. (91) Durbin, M. K.; Malik, A.; Richter, A. G.; Ghaskadvi, R.; Gog, T.; Dutta, P. J. Chem. Phys. 1997, (92) Kaganer, V. M.; Loginov, E. B. Phys. Rev. Lett. 1993, 71, 2599. (93) Kaganer, V. M.; Loginov, E. B. Phys. Rev. E 1995, 51, 2237. (94) Kaganer, V. M.; Indenbom, V. L. J. Phys. II 1993, 3, 813.

Relation between Bulk Alkanes and Langmuir Monolayers

presses in the direction of tilt, create uneventful zero crossings of the distortion in the plane of the surface. It was thus apparent that such a measured distortion was not suitable order-parameter. We, therefore, would encourage the development of a comparably complete mean-field theory in terms of the distortion perpendicular to the molecules, the tilt and the herringbone order. The coupling term, θ2ξ, the temperature dependence of the tilt direction, as well as coupling between the distortion, tilt and herringbone directions could allow the tilt azimuth to change at a first-order transition in ξ between, for example, FII and II phases. From a scattering perspective, it is clear that the observables at the two main transitions are the distortion and the amplitude of the herringbone peak. An alternative point of view which may, in fact, be more microscopically valid, is to consider the orientational distribution of the backbone planes. This is more accessible with other techniques including simulations.51,95 As described by Schofield and Rice,68 the order parameter for the herringbone state is defined modulo 180° rotations, with a doubling of the unit cell, giving every other molecule a 90° phase shift. In that work, unfortunately, the two transitions were combined into one, so the non-herringboneordered distorted phase (UI ) S) was not included, and only the UII-UX transition was considered. By recognizing the distinction between long-range herringbone order and short-range herringbone order directed predominantly in one of the three equivalent directions on the hexagonal lattice, one can consider a rotational order parameter for the intermediate phase. This would simply be defined modulo 90°, with all sites equivalent. From a microscopic perspective, the lattice compression occurs normal to lines of molecules which are oriented in the same direction. We would, therefore, encourage the extension of the densityfunctional theory to include the two distinct transitions.68 While this rotational order parameter is possibly more fundamental to the transition, it is the distortion which is currently most experimentally accessible and for which most data exists. (95) Ryckaert, J. P.; Klein, M. L.; McDonald, I. R. Mol. Phys. 1994, 83, 439.

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In addition to the possible order parameters described above, it is known that substantial chain defects populate chain molecules in these phases.75,96-98 Such defects occur in all the phases, increase in number with increasing temperature, and jump in number at the phase transitions. It is clear that they contribute substantially to the heat capacity and latent heat of transitions and are possibly responsible for driving transitions first-order (such as the herringbone ordering transition UI-UX). However, their finite number in all these phases makes them unsuitable as order parameters for the transitions. One question, which has not yet been addressed theoretically or through simulations, is the increased positional disorder which occurs on cooling in the hexagonal phases. While we have qualitatively attributed this to local regions of lower symmetry, more detailed calculations have been lacking, as well as predictions for the form of the diffuse scattering, and descriptions of the defect regions. Some suggested further experimental measurements are as follows: the effect of π on A, ξ, and the phase boundaries in the untilted phases of relaxed monolayers; the measurement of the scattering at the position of the herringbone peak, facilitated by the availablility of new generation synchrotron sources. Acknowledgment. I would like to acknowledge fruitful discussions over the years with, and comments on the manuscript by, many of the people working in this area: particularly P. S. Pershan, V. Kaganer, M. Dirand, and P. Dutta, as well as T. C. Lubensky and J. V. Selinger. I also acknowledge the very important interactions with my collaborators on studies of alkanes: X. Z. Wu, H. Shao, H. Gang, S. K. Sinha, M. W. Kim, B. M. Ocko, M. Deutsch, and especially H. E. King, Jr. LA9702291 (96) Maroncelli, M.; Qi, S. P.; Strauss, H. L.; Snyder, R. G. J. Am. Chem. Soc. 1982, 103, 6237. (97) Kim, Y.; Strauss, H. L.; Snyder, R. L. J. Phys. Chem 1989, 93, 485. (98) Clavell-Grunbaum, D.; Strauss, H. L.; Snyder, R. G. J. Phys. Chem. B 1997, 101, 335.