Langmuir 1994,10, 4645-4650
4645
Equivalence between Two-Dimensional and Three-DimensionalPhases of Aliphatic Chain Derivatives I. R. Peterson*,+ Institute for Physical Chemistry, Johannes Gutenberg University, Jakob Welder Weg 11, 55099 Mainz, Germany
R. M. Kenn Institut Curie PC-PSI, 11, rue Pierre et Marie Curie, F-75231 Paris Cedex 05,France Received May 5, 1994. In Final Form: August 25, 1994@ We demonstrate a detailed structural equivalence between 7 of the 12 k n o w n condensed water-surface monolayer phases of long-chain fatty acids and 7 of the 11known bulk solid phases observed in long-chain alkanes. Two second-order phase transitions are common to both. Thermodynamic parameters are established in both systems for one of these and they are consistent with the proposed equivalence.
I. Introduction The aliphatic chain is the basic structural unit of many molecules which form two-dimensional phases a t interfaces. The most important of these lamellar phases are only partly ordered, for example, those of surfactants in detergents and food emulsions, and those of phospholipids in biomembranes. The bulk crystalline phases of aliphatic chain compounds have long been studied,lV2and the vast majority of them take the form of stacked lamellar phases. While the primary structure of aliphatic chains is simple, their packing behavior is quite varied. However it is possible to understand the latter in terms of the local packing of the chain^,^.^ as the same local packings recur in a large number of different crystals in spite of differing chain substituents. A number of authors have suggested that this scheme might be extended to include specific monolayer phases of certain long-chain compound^.^?^ Unfortunately, these proposed extensions are a t most suggestive and are difficult to extend to include all phases. As shown in Figure 1, the monolayers of simple amphiphiles like the fatty acids, alcohols, and their esters show at least 12, possibly as many as 17, distinct states of partial order (miscibility ~ategories).~ References to each phase are as follows (phase, reference numbers):
I /
I
(not to
scale)
*
T
Figure 1. A generic phase diagram showing all monolayer
phases for which either experimental or theoretical evidence exists.
CV4 7EZ, U.K.
AGS, 8; CS, 9-12; CS', 10;G, 13-15; L1,14,15;L'1,16-18; L2d, L2h, 9-12,15,19,22; L;, 9,lO;L2*/S*, 9-12,20,16; LS, 6,12,10,21;LS', 10,20; OV,23,24; 9-12; S', 10,20; X, 25,26. Because of this richness, it is possible to demonstrate many approximate correspondences between individual pairs of states which are in fact distinct. A more objective criterion for equivalence has recently been proposed,27involving both the packing of the chains and their contribution to the free energy. There were two parts in the experimental demonstration that such a detailed equivalence actually exists. The correspondence
(1) Muller, A. Proc. R . SOC.London, A 1928,102, 437. (2) Vand, V.; Bell, I. P. Acta Crystallogr. 1951, 465, 465. (3) Kitaigorodskii, A. I. Organic Chemical Crystallography;Consultants Bureau: New York, 1961. (4) Segerman, E. Acta Crystallogr. 1965,19, 789. (5)Lundquist, M. Chem. Scr. 1961, 1, 5. (6) Shih, M. C.; Bohanon, T. M.; Mikrut, J. M.; Zschack, P.; Dutta, P.Phys. Rev. A 1992,45, 5734. (7) Peterson, I. R. In The Molecular Electronics Handbook; May, V., Mahler, G., Schreiber, M., Eds.; Marcel Dekker: New York, 1994. (8) Albrecht, 0.; Gruler, H.; Sackmann, E. J.Phys. (Paris)1978,39, 301. (9) Lin, B.; Shih, M. C.; Bohanon, T. M.; Ice, G. E.; Dutta, P. Phys. Rev. Lett. 1990, 65, 191. (10) Lundquist, M. Chem. Scr. 1971, 1, 5, 197. (11) Stallberg-Stenhagen, S.;Stenhagen, E. Nature 1945,156,239. (12) Kenn, R. M.; Bohm, C.; Bibo, A. M.; Peterson, I. R.; Mohwald, H.; Kjaer, IC;Als-Nielsen, J. J. Phys. Chem. 1991, 95, 2092. (13) Adam, N. K. Proc. R . SOC.London, A 1926,110, 423. (14) Kim, M. W.; Cannell, D. S. Phys. Rev. A 1967,13,411. (15) Harkins, W. D.; Young, T. F.; Boyd, E. J. Chem. Phys. 1940,8, 954.
(16) Bibo, A. M.; Knobler, C. M.; Peterson, I. R. J.Phys. Chem. 1991, 95, 5591. (17) Bibo, A. M.; Peterson, I. R. Thin Solid Films 1992,210/211,515. (18) Peterson, I. R.; Kenn, R. M.; Goudot, A.; Fontaine, P.; Rondelez, F.; Bouwman, W. G.; Kjaer, K. Nature, in press. (19) Kaganer, V. M.; Indenbom, V. L. J. Phys. ZI 1993,3,813. (20) Lawrie, G. A.; Barnes, G. T. J. Colloid Interface Sci. 1994,162, 36. (21) Stenhagen, E. In Determination of Organic Structures by Physical Methods; Braude, E. A., Nachod, F. C., Eds.; Academic Press: New York, 1955; pp 325-371. (22) Kaganer, V. M.; Loginov, E. B. Phys. Rev. E, in press. (23) Overbeck, G. A.; Mobius, D. J. Phys. Chem., 1993,97, 7999. (24) Durban, M.; Malik, A.; Ghaskadvi, R.; Shih, M. C.; Zschack, P.; Dutta, P. J. Phys. Chem., in press. (25) Steitz, R.; Mitchell, E. E.; Peterson, I. R. Thin Solid Films 1991, 205, 124. (26) Shih, M. C.; Bohanon, T. M.; Mikrut, J. M.; Zschack, P.; Dutta, P. J. Chem. Phys. 1992,96, 1556. (27) Peterson, I. R.; Brzezinski, V.; Kenn, R. M.; Steitz, R. Langmuir 1992,8, 2995.
Abstract published in Advance ACS Abstracts, November 1, 1994. Present address: Nima Technology,The Science Park,Coventry @
0743-7463/94/2410-4645$04.50/00 1994 American Chemical Society
Peterson and Kenn
4646 Langmuir, Vol. 10, No. 12, 1994
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i 410
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d02/pm
Figure 2. Steitz plot, showing the degenerate d-spacing d11 (in-plane, units of pm) plotted against the nondegenerate d-spacing do (in-plane,units of pm)for a wide range oflamellar phases on water and solid substrates. The dashed line has equationdo2 =dll. Upright NN-distortedcrystallinemonolayer phase (circles): (open) docosanoic acid on water; (horizontal bar) tetracosanoic acid on water, (vertical bar) heneicosanoic acid on water. Upright NN-distorted monolayer mesophase (squares): (open) docosanoic acid on water; (vertical bar) heneicosanoic acid on water. Upright NNN-distorted crystalline monolayer phase (circles): (cross)heneicosanoicacid on water containing calcium ions;(leftfilled)docosanoic acid on Formvar; (top filled) docosanoic acid on PMMA. NNN-tilted monolayer rotator mesophase (diamonds): (open)docosanoicacid on water; (top filled) docosanoic acid on PMMA, (horizontal bar) tetracosanoic acid on water; (vertical bar) heneicosanoic acid on water. “-tilted monolayer rotator mesophase (right-pointing triangles): (open)docosanoic acid on water; (topfilled) docosanoic acid on PMMA; (horizontalbar) tetracosanoic acid on water. Langmuir-Blodgett films (upward-pointingtriangles): (vertical bar) lead(I1)octadecanoatemultilayer;(filled)free-standing cadmium eicosanoate bilayer; (rightfilled)cadmium eicosanoate bilayer on amorphous carbon; (open)cadmium hexadecanoate bilayer on amorphous carbon; (left filled) cadmium tetradecanoatebilayer on amorphous carbon. Bulk phases (downwardpointing triangles): (left filled)n-tertriacontane C33H68; (right filled)cyclodoheptacontane(CH2),2; (open)poly(docosy1acrylate) resin.
of packings was demonstrated by the Steitz plot of Figure 2, in which each lamellar packing is represented by a point whose coordinates are the two in-plane d-spacings of lowest order. The points plotted, corresponding to a range of lamellae with different chain lengths, headgroups, and lamellar environments whose details including literature sources are given in ref 27, all lie within experimental error on one of four straight lines. The correspondence of free energy contributions was shown by considering the surface pressure at the tilting transition of a number of aliphatic-chain monolayers at the airwater interface. Within experimental error, the variation with chain length was linear, with a slope Pchain independent of the particular headgroup. It should be mentioned that this “principle of corresponding states” has very little in common with the principle of the same name due to Prigogine, which is applicable to expanded states of small, nearly spherical molecules. The historical precedents of the present
principle are the work of Phillips and Chapman,28 of Kimelberg and Papahadjopo~los,~~ of Albrecht, Gruler, and Sackmann,8 and of C a d e d ~ e a d . ~ ~ Among the lamellar structures fitting the pattern of the Steitz plot were some for bulk alkanes. Compared to other aliphatic-chain derivatives, the alkanes have the advantages of structural simplicity, head-tail symmetry, and much greater ease of calculation of the interaction with their environment due to their lack of hydrogenbonding groups. While it has been thought in the past that such groups are necessary to allow the formation of surface monolayers, it is of particular interest that n-alkanes have recently been shown to do so under appropriate condition^.^^-^^ GID data show that the molecules in alkane surface monolayers are nearly densely packed and vertical; reflectivity data imply a layer thickness almost equal to the length of one molecule. Hence apart from the obvious molecular differences, a t the level considered here the alkane behavior is indistinguishable from that of an amphiphile. If an equivalence can be found between monolayer and alkane phases, it will permit faster progress in understanding biologically and technologically relevant monolayer structures. The phase behavior of the alkanes has been studied at least since 1930, and many groups have contributed to the subject, including Miiller,35,36 Ewen, Richter, et a1.,37-41 Luth, Nyburg, et a1.,42-44Doucet and c o - ~ o r k e r s , 4 ~ - ~ ~ Maroncelli, Snyder, and co-w0rkers,5~-~~ Ungar and MaBiC,53!54 D o r ~ e t and , ~ ~Sirota , ~ ~ et a1.57-59The alkane phase diagram of Figure 3 is a composite which includes the phase transitions reported by all workers. We use the Doucet and Sirota R- nomenclature for the phases at temperatures immediately below that ofthe isotropic melt, (28)Phillips, M. C.; Chapman, D. Biochim.Biophys. Acta 1968,163, 301. (29)Kimelberg, K; Papahadjopoulos,D. Biochem.Biophys.Acta 1971, 233,805. (30)Cadenhead,D.A. InStructuresandPropertiesofCellMembranes; Benga, G., Ed.; CRC Press: Cleveland, OH, 1985. (31)Earnshaw, J. C.; Hughes, C. J. Phys. Rev. A 1992,46,R4494. (32)Hughes, C. J.; Earnshaw, J. C. Phys. Rev. E 1993,47,3485. (33)Wu, X.Z.;Ocko, B. M.; Sirota, E. B.; Sinha, S. K.; Deutsch, M.; Cao, B. H.; Kim, M. W. Science 1993,261,1018. (34)Wu, X.Z.;Sirota, E. B.; Sinha, S. K.; Ocko, B. M.; Deutsch, M. Phys. Rev. Lett. 1993,70,958. (35)Muller, A. Proc. R.SOC. London, A 1930,127,417. (36)Muller, A. Proc. R. SOC. London, A 1932,132,514. (37)Piesczek, W.; Strobl, G. R.; Malzahn, KActa Crystal1ogr.B 1974, 30,1278. (38)Strobl, G.; Ewen, B.; Fischer, E. W.; Piesczek, W. J . Chem. Phys. 1974,61,5257. (39)Ewen, B.; Fischer, E. W.; Piesczek, W.; Strobl, G. J . Chem. Phys. 1974,61,5265. (40) Ewen, B.; Richter, D. J . Chem. Phys. 1978,69,2954. (41)Ewen, B.; Strobl, G. R.; Richter, D. Faraday Discuss. Chem. SOC. 1980,69,19. (42)Nyburg, S. C.;Luth, H. Acta Crystallogr. E 1972,28,2992. Nvbure. 9.C.: Robinson. P. M.: Scott. H. G. Mol. Crvst. (43)Luth. H.: Liq. Cryst. i974,i7,339. . (44)Gerson, A. R.; Nyburg, S. C. J . Appl. Cryst., in press. (45)Doucet, J.;Denicolo, I.; Craievich, A. J . Chem. Phys. 1981,75, 1523. (46)Doucet, J.;Denicolo, I.; Craievich, A,; Collet, A. J . Chem. Phys. 1981,75,5125. (47)Denicolo, I.; Doucet, J.; Craievich, A. J . Chem. Phys. 1983,78, 1465. (48)Denicolo, I.; Doucet, J.; Craievich, A.; Germain, C. J . Chem. Phys. 1984,80,1647. (49)Craievich, A. F.;Denicolo, I.; Doucet, J. Phys. Rev. B 1984,30, 4782. (50)Snyder, R. G.; Maroncelli, M.; Qi, S. P.; Strauss, H. L. Science 1981,214,188. (51)Maroncelli, M.; Qi, S. P.; Strauss, H. L.; Snyder, R. G.J . Am. Chem. SOC. 1982,104,6237. (52)Maroncelli, M.; Strauss, H. L.; Snyder, R. G. J . Chem. Phys. 1985,82,2811. (53)Ungar, G. J . Phys. Chem. 1983,87,689. (54)Ungar, G.; MasiC, N. J . Phys. Chem. 1985,89,1036. (55)Dorset, D. L. Macromolecules 1990,23,623.
Langmuir, Vol.10,No. 12, 1994 4647
Phases of Aliphatic Chain Derivatives
-* lllt
NNN
NN
n n n
$2
U
nn n n nn
==++=-
Figure 3. Acomposite phase diagram showing the temperature Tin "C of all alkane transitionsreported at atmospheric pressure as a function of chain length N . The phase nomenclature is explained in the text.
and the Maroncelli, Snyder, et al. Roman numeral nomenclature for the phases at intermediate temperature. For the low-temperature region we follow the Nyburg and Luth nomenclature 0, T, and M. The phase boundaries in this region are not shown because they display strong even-odd alternation, dependence on impurities, and slow kinetics. The symbols A, B, C , and D on the figure are those of Ewen and Strobl for tertriacontane. 11. Equivalent States
The phases of particular interest in the present work are those which give rise to diffraction spots, allowing the determination of the dimensions of the unit cell. For the states oftwo such aliphatic chain lamellae to be considered equivalent, their in-plane unit cell parameters, together with the chain tilt magnitude and direction, must be the same. It has been found that, as a phase oftilted molecules is compressed, the two-dimensional packing normal to the chains varies little while the tilt magnitude may change over a considerable range. Normal to the chains, the packing can be specified by the cross-sectional area per molecule A,, together with the magnitude 4 and direction d 2 of the distortion. The notation is chosen to agree with refs 19 and 22, in which a is an angle in the space of the distortion transform, which is a traceless symmetric 2 x 2 matrix. The latter two parameters are defined in terms of the ellipse passing through all six neighbors of a given molecule. The distortion magnitude 6 is (a2- b2)/(a2 b2),where a and b are the major and minor axes of the ellipse, respectively,while its direction d 2 is the azimuth of the minor axis. The three parameters A,, 6, and d 2 are conveniently expressed in terms of averages () of expressions involving the three distinct nearest-neighbor distances r
+
(56)Dorset, D. L.Mucromol. Chem. 1990,1,311. (57)Sirota, E.B., King, H. E., Jr.; Hughes, G. J.; Wan, W. K. Phys. Rev. Lett. 1992,68,492. (58) Sirota. E. B.: Kine, -. H. E., Jr.: Singer, - -D. M.; Shao, H. J . Chem. Phys. 1993,98,5809. (59)Sirota, E.B.;Singer, D. M.;King, H. E., Jr. J.Chem.Phys. 1994, 100, 1542. (60)Kenn, R. M. Doctoral thesis, Johannes Gutenberg University, Mninz. 1994. - ------, -- - (61)Brezesinski, G.; Mogel, H.J.Grenzfliichen und Kolloide;Spektrum Akad. Verlag: Heidelberg, 1993. (62)Broadhurst, M. G. J.Res. Natl. Bur. Stand. A 1962,66, 241. (63)Teare, P.W. Acta Crystallogr. 1959,12,294.
NNN
I I
!?8
Figure 4. Drawings representing the top view of the lamellar packing and categorizing the observed phases in terms of the tilt and distortion azimuths: NN, tilt or distortion toward the nearest neighbor (B = 0 or d 2 = 0, respectively); NNN, tilt or distortion toward the nearest neighbor (B = d 6 or d 2 = zl6, respectively); U, undefined (upright or undistorted); I, intermediate.
There are two cases of symmetrical distortion, with distortion azimuth NN ( d 2 = 01,and NNN ( d 2 = d 6 ) , respectively. In both cases the unit cell is centered rectangular, but they differ in that the ratio of the unit cell sides (its aspect ratio) is less or greater than 4 3 , respectively. Figure 4 shows drawings of the dominant molecular packing with different combinations of the directions of tilt and distortion, together with the monolayer and alkane phases known to be in each category. The numerical values of the packing parameters (except for the tilt angle, which is not characteristic) are shown in Table 1for most of the known bulk alkane and fattyacid monolayer phases. They were calculated from the data presented in refs 12,18,24,36,41,44,58,60,25,26, and 63,with the source in each case being shown in the table. The following seven structural correspondences can be seen: 0 to CS; RI to S; RIIto LS; RIIIto LI'; R I to ~ Ov; I11 to bh;and R, to S*/L2*. The difference in crosssectional areas is in some cases as large as 2%, slightly larger than a reasonable estimate of experimental error. This may perhaps be due to the fact that the packing normal to the chains does not remain exactly constant within a phase and that the data refer to states which are
Peterson and Kenn
4648 Langmuir, Vol. 10, No. 12, 1994 Table 1. Parameters Defining the Packing Normal to the Chains in a Number of Bulk Phases of the Alkanes and Water-Surf'ace Monolayer Phases of the Fatty Acids bulk alkane monolayer tilt area, dist phase[refl phase[refl a p nm2 distortionb az S*/L2* [12,16r "N 0.192 0.15 NN
NNN 0.183 NNN 0.195
0.16
NNN 0.198
0.02
"N 0.198 U 0.192 U 0.186 0 [361 U 0.185 U 0.195 Rr [581 LS [6,12Id U 0.198 U 0.198 RII [44,581 U 0.192 X [25,261 Lz" [601 NN 0.189 L2d [18,19,221 NN 0.198 L2h [12,18,19,223 NN 0.192 I11 [41] NN 0.192 Iv 1411 NN 0.188 RIII [581 I 0.198 LI' [I81 I 0.198 T [631 I 0.182
0.00
M [621 Rv [581 Rrv [581
Ov [241 s [I21 cs [121
0.08 0.11 0.14 0.15 0.13
0.00 0.00 0.14 0.16 0.05 0.09 0.11 0.13 0.04
0.03 0.26
NN NN U U NN NN NN"
NN U U
Partial differentiation with respect to the surface area per molecule u and subtraction from the surface tension of a clean water surface gives the surface pressure ~d developed by the monolayer 7c = 4
+ chain
The extension of eqs 4 and 5 to allow comparison with bulk states is relatively straightforward. The monolayer system considered in ref 27 consisted of a pure insoluble and involatile amphiphile at the interface between liquid water and its vapor. Both bulk phases consist of pure water (w), and it is easily shown that the total Helmholtz free energy F of the system on these assumptions is the sum of a bulk term and a surface excess term:
"N
NN N" N" N" "N
I I I
"U,undefined; NN, nearest neighbor; NNN, next nearest neighbor;I,intermediate. defined in eq 1. These two phases are indistinguishable in diffraction. Two regions LSI and LSIIof this phase with distinguishable peak profiles have been recognized.
F = n#c, - PV+ n f
(6)
where pw is the chemical potential of water and n, is the total number of water molecules in the system, P is the pressure and Vthe system volume, while n, is the number of molecules of surface-activeamphiphile and$ the surface excess free energy F" divided by n,. The formal procedure involving equivalent states of a homologous series can also be invoked for bulk lamellar phases of a pure substance, except that the intensive quantity f considered is the number of molecules divided into the total system free energy rather than its surface excess:
not precisely equivalent. The discrepancy might also be due to the influence of the interfaces, whose magnitude could be as large as D/9L,where L is the molecular length, f=fo+D (7) D the diameter, and 9 is an average Lennard-Jones exponent. As a result of the additivity expressed by eq 6, the The identification of most of these pairs has only recently asymptotic slope B must clearly be the same in any equivalent lamellar state of aliphatic chain compounds, become possible. Much of the alkane structural data whether monolayer or bulk. In addition, eq 7 may be referred to here was reported only last year.58 While the to the Gibbs-Duhem equation61 for a oneL1' phase was observed in isotherm studies 3 years a g ~ , ~ ~ compared J ~ component system with interfaces its molecular packing has only just been determined.18 The Overbeck-Mobius phase Ov was discovered in fattyF = yA - PV+ p n (8) acid monolayers only last year using Brewster-angle microscopyz3and grazing-incidence diffraction has now On dividing by the number of molecules n, it can be confirmed that its structure is distinct from that of seen that the term -PVgives an expression proportional neighboring phases.24 to the molecular length A, while the remaining two terms In the last correspondence, a pair of monolayer phases give p y u which has exactly the same form as a surface has been grouped together with a single alkane phase. excess energy per molecule.z7 This is justified because the most complete study of the To get the bulk pressure, f is differentiated with respect alkane phases employed only X-ray diffraction. The to the molecular volume u , which is more satisfactorily herringbone NNN-tilted monolayer phases S* and Lz* are expressed in terms of the lamellar thickness d than the not distinguished by X-ray measurements.12,60There is molecular length A: another related case. On the basis of X-ray diffraction results, a distinction was made between two regions of u = ad (9) the monolayer LS phase.6 In Figure 1these are labeled LSI and LSII. However the difference involves subtle In general, d depends on u, and this dependency changes in the profile of the single powder-averaged complicates the lamellar-bulk correspondence. However diffraction peak. Similar effects have been seen in the in states where the molecules are perpendicular to the alkanes but not interpreted as a phase t r a n ~ i t i o n . ~ ~ lamellae, d is essentially equal to the molecular length A and is independent of Q. Under these conditions the bulk 111. Thermodynamic Correspondence to Bulk pressure P is given by: States 4 If the above correspondence is sufficientlyclose for the P = Pchain(10) il principle of equivalent states to apply, then there is a thermodynamic parameter which should also correspond. Here, the intercept term has been expressed as an It has already been shown that if surface monolayers of interfacial pressure in analogy with eq 5 and 8. a homologous series of compounds are prepared in equivalent states, then asymptotically for large chain IV. Second-Order Tilting Transitions length A, the surface excess Helmholtz free energy per In the theory of equivalent states, second-order transimolecule f must vary linearly27 tions and particularly tilting transitions are of special importance, because they allow identification of corf=fo"+LB (4) responding states more accurately than is usually possible
+
+
Langmuir, Vol. 20,No. 12, 1994 4649
Phases of Aliphatic Chain Derivatives from u, which is often only known to a precision of a few percent. In the condensed phases, the variation of uacross a whole phase region may be less than this. A secondorder transition is signaled by a discontinuity in the derivatives of the free energy in a continuous sequence of states. From eq 4,the discontinuity occurs in one or both of foB and B , so that a discontinuity and hence a phase transition will occur at the same value of u for compounds of whatever chain length. This is independent of 1,so that transition states of different chain lengths must be equivalent. This is not necessarily the case for first-order transitions. Numerical data to check the theory exists for two transitions from upright to tilted chains. The RII-RN transition occurs only for chain length 26 and corresponds to the monolayer tilting transition LS-Ov. In previous studies of the latter transition, the Overbeck-Mobius phase was not distinguished from the LI’ phase, and both were considered t o be part of the L2 phase. However the thermodynamic parameters for the two phases appear to be essentiallyidentical, so that it is reasonable to take the internal chain pressure a t this transition as being P c h d = 0.24T MPa at T 0C,27 while the chain length , I= 127N Pm. Since there is only one chain length at which the transition occurs at ambient pressure, eq 10 can be made to fit the phase diagram of Figure 3 perfectly by choosing the appropriate value for 4, equal to -42 mN/m. The slope of the RII-RN transition line shown in Figure 3, which is clearly not fixed by the experimental data, was chosen to agree with the equation. However the slope shown accounts correctly for the fact that the RN and RIII phases are not observed for chain lengths of C25 or shorter, while the RII phase is not observed for C27 or longer. In spite of having one parameter to describe at present just one data point, eq 10 is not devoid of experimental content. It predicts the temperature of the phase transition to be much more sensitive to pressure than are the rotator-isotropic transitions. In particular, the RII-RN transition is predicted to occur in longer-chain alkanes at pressures slightly above atmospheric. In the C27 alkane, it should occur over the pressure range 0.5-1 MPa (5 to 10 bar) at the appropriate temperature. Unfortunately the pressure dependence of the rotator phases has as yet only been investigated in alkanes of chain length smaller than 26.59 The second tilting transition between phases in the correspondence is RI-Rv, whose pressure dependence has been investigated in the C23 alkane.59 Here the values for the monolayer transition are not known accurately: Peterson et al. report that the internal chain pressure Pchain at the transition is approximately -4 MPa27while Lawrie and Barnes find -7.4 MPa.20 It is difficult to reconcile the available data sets, which may indicate either that the monolayer data has large systematic error or that the transition is first order in the pure alkanes. V. Topology of the Phase Diagram The interfacial pressure 4 is presumably not exactly constant, depending on the state of compression of the monolayer, although no experimental data are yet available about its exact variatian. The internal chain pressure results from ambient pressure plus the action of 4 distributed over the chain length. If 4 remains negative, the internal pressure decreases with increasing chain length, and Figure 3 can be compared to Figure 1 by rotating it through 90”.It can be seen that all the relative positions in pressure and temperature are the same for the two systems. The only slight difference is that S*/L2* is adjacent t o L2h in the monolayers,but Rvis not adjacent
to 111. Nevertheless the relationship between them is as close as that between the monolayer phase diagrams of the long-chain fatty acids and their ethyl esters.16 The idea of topological similarity (the “generic phase diagram”) indicates an eighth possible match: the NNtilted monolayer phase L< occurs in a similar position relative to the other phases as V does in the alkane phase diagram.
VI. Difficulties with Translational Order In spite of these satisfactory aspects of the correspondence between the above pairs of monolayer and alkane phases, it appears to break down completely in respect of translational order, which is asserted by Sirota et al.58to be long-range in all five rotator phases they investigated. Of the monolayer phases listed in Table 1, CS and L2” are crystalline, while the rest have been shown using optical techniques to be m e s o m o r p h ~ u s . This ~~ means that they display long-range orientational order, while their translational correlations decay exponentially with separation. An increase of the translational order in the bulk phases of a given compound is quite plausible, because threedimensionality enhances the interactions between layer defects and thus favors a fully-ordered structure. This is only a tendency, not a universal law, because stacked hexatic phases are known.6s Since the stacked hexatics possess continuous inplane translational symmetry, the two sorts of phases are in different Landau categories. It is inconceivable that phases of different symmetry could be considered equivalent. On this basis, the true monolayer equivalent of the RIphase would have to be CS. However there is a considerable discrepancy between the unit cell dimensions of the latter pair. Moreover, the structure of the 0 phase corresponds very closely t o that of the CS phase, and there does not appear to be any more condensed alkane phase of related structure. Similarly, if the critical order parameter of the alkane transition I11 to IV is not translational, it is not at all clear why I11 corresponds so well to the known mesophase L2h. If the alkane rotator phases are true crystals as claimed, then these facts rule out the existence of a systematic correspondence, or one with any predictive power. Sirota et al. identified the alkane phases as crystalline on the basis of the radial fwhm of their diffraction peaks of approximately 0.002 A-l. However hexatic-B phases of calamitic moleculesnot uncommonly show radial fwhm’s of less than 0.04 A-1.66 This corresponds to a dislocation density of one per hundred molecules. Hexatic-B corresponds to the LS phase, which is the least ordered of the condensed monolayer phases, with a radial fwhm in the fatty acids ofbetween 0.04 and 0.10A-l.The peak widths observed in the more ordered monolayer mesophases can be as small as 0.01A-1.6J2,67 The terms short- and long-range to describe the translational order of mesomorphous and crystalline phases, respectively, do not refer to any particular length scale. They refer rather to qualitative differences in the way the translational correlation decays with increasing separation at thermodynamic equilibrium.68 When the peak width is as close to instrumental resolution as in the (64)Fisher, T.M.; Bruinsma, R.; Knobler, C. M. Phys. Rev E 1994, 50, 413.
(65)Pindak, R.;Moncton, D. E.; Davey, S. C.; Goodby, J. W. Phys. Reu. Lett. 1981,46, 1135. (66)Brock, J.D.; Noh, D. Y.; McClain, B. R.; Litster, J. D.; Birgeneau, R. J.; Aharony, A.; Horn, P. M.; Liang, J. C. 2.Phys. B 1989,74, 197. (67)Shih, M. C.;Bohanon, T. M.; Mikrut, J. M.; Zschack, P.; Dutta. P.J. Chem. Phys. 1992,97, 4485. (68)Nelson, D. R.;Halperin, B. I. Phys. Reu. B 1979, 19, 2457.
4650 Langmuir, Vol. 10,No.12, 1994
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monolayers. However inspection of their curve of surface present case, the decay law cannot be determined from monolayer thickness versus chain length shows a very X-ray measurements. In any case real crystalline maclear breakpoint a t a chain length of C28, and more terials never show the thermodynamically ideal decay law ~ s ~have l detected the associated tilt, which but instead one which is dominated by sample m o ~ a i c i t y . ~ ~ r e ~ e n t l y ~they is in the NN direction. While it is clear that correlation lengths in the alkanes Since the internal pressures at both the NN and “N are for some reason over an order of magnitude greater rotator phase tilt transitions appear from the monolayer than in calamitic mesophases, the results of Sirota et al. data to be equal, the parameter 4 at the R I I - R ~tilting do not prove that there is a qualitative difference and are transition for the surface monolayer phase of an alkane consistent with highly-ordered mesophases. is equal to -53 mN/m at 62 “C. If it is further assumed X-ray evidence can contribute to the question. For that 4 for the bulk phase lamellae is not strongly dependent example in a crystalline phase the mosaicity should on temperature, then the contribution to 4 for the methyldecrease on annealing, whereas a mesophase will most gas interface is -32 mN/m. Methyl-gas interfaces are likely show reproducible and reversible variation of peak a structural feature of many important water-surface widths. The optical observation of textures is perhaps monolayers, including the fatty acids, for which a total more conclusive as evidence for organizati01-1~~ and can value at this transition of probe much longer length scales than X-rays, although the mosaic texture of highly-ordered mesophases is 4 = 23 - 0.56TmNlm (11) essentially indistinguishable from a polycrystalline texhas already been determined.27 Hence the contribution ture, and this texture is the one which appears to occur from the headgroups can be separately deduced. Clearly most commonly in the alkanes. Most conclusive of all is the quality of the data is insufficient to allow any the comparison of X-ray and optical results. In nearconclusions to be drawn, but it can be seen that this equilibrium states with mosaic spread, a crystalline phase formalism can in principle reduce the complexity of the must show similar correlation lengths of translational and problem of understanding amphiphilic monolayers. orientational order, while a mesophase is characterized by a discrepancy between them of many orders of VIII. Conclusion magnitude. Of the alkane “rotator” phases, two (RI and Rv) show a It has been shown that both bulk straight-chain alkanes distortion significantly different from zero. Some authors and water-surface monolayers of amphiphiles have phase describe them as showing “hindered rotation”, in an diagrams of considerable complexity. A correspondence attempt to categorize the diffraction evidence for greater has been demonstrated between seven phases observed disorder than in the crystal. However this concept has no in both systems. This correspondence has considerable theoretical basis. In the case of monolayers, the S phase implications. For example, it leads to specific predictions shows broader peaks than the structurally quite similar about the pressure dependence of the tilting transition crystalline CS phase because it is mesomorphous. between the RIIand Rw phases of certain alkanes and the occurrence of this transition at chain lengths where it has VII. Implications for Surface Monolayers not yet been observed. It suggests a further possible correspondence between the V and L? phases. The value of -42 mN/m for the interfacial pressure 4, The narrow line widths observed from all the alkane which combines effects from the two sides of the lamella, phases represent a difficulty of the correspondence. is almost double the value of 21 mNlm for the surface However this is only a quantitative difficulty related to tension y of a liquid alkane. The correspondence is the line widths expected for the mesophases of substances probably fortuitous. y is the total surface excess free of different molecular structure and possibly levels of energy per unit surface with molecules free to enter or purity. “he successes are nevertheless impressive enough leave the interfacial region, while 4 is the derivative of to suggest that the alkane phases capable of existing in free energy with respect to the area per molecule, keeping equilibrium with the isotropic liquid may be highly-ordered the number of molecules at the interface constant. It is mesophases rather than true crystals. plausible that the two quantities should have the same A value for the lamellar interfacial pressure has been order of magnitude. It is in fact surprising that the deduced for the R I I - R ~tilting transition of alkanes. Once magnitude of 4 should be so large, because in the bulk the corresponding value for surface monolayers has been alkane the lamellar surfaces adjoin other identical lamelmeasured, it will be possible to calculate the contribution lae, and the interfacial tension of a liquid alkane-liquid from the methyl-air interface, which is a structural alkane interface is of course zero. feature of many important amphiphilic monolayers. If a neighboring lamella is removed from one side of the lamella under consideration, the stabilization due to van Acknowledgment. This work was supported finander Waals interactions is lost. However the above cially by the Fonds Henri de Rothschild and the French argument indicates that this stabilization is small, so that Ministere des Affaires fitranghres. The authors thank the interfacial pressure 4 for the surface monolayer on a Professor J. C. Earnshaw, Dr. V. M. Kaganer, Dr. L. Ter liquid alkane is expected to be more negative than -42 Minassian, Professor C. M. Knobler, and Dr. F. Rondelez mN/m, but not by much. Hence the molecules in the for useful discussions, Dr. E. B. Sirota for critically reading surface monolayer should be tilted at a chain length only the manuscript, and Professor H. Mohwald for support slightly higher than C26. In initial reports based on grazing-incidence diffraction measurements, Wu et a1,33,34 and encouragement. found no evidence for molecular tilt in the surface (70) Wu, X. 2.;Sirota, E. B.; Sinha, S. K.; Ocko, B.; Deutsch, M. Bull. ~~
(69) Dutta, P.; Sinha, S. K. Phys. Rev. Lett. 1981, 47, 50.
Am. Phys. SOC.1994, 39, 367. (71) Sirota, E. B., personal communication, 1994.
