© Copyright 2000 American Chemical Society
JUNE 13, 2000 VOLUME 16, NUMBER 12
Letters Conformational Disorder Binds n-Alkanes into Surface Monolayers above the Normal Freezing Point A. J. Colussi,*,† M. R. Hoffmann,† and Y. Tang‡ W. M. Keck Laboratories, California Institute of Technology, Pasadena, California 91125, and Chevron Petroleum Technology Company, La Habra, California 90631 Received September 14, 1999. In Final Form: March 27, 2000 The monolayer arrays formed on liquid n-alkanes within ∆T ≈ 3 K of the normal freezing point TF are but the two-dimensional termini of a series of disordered solid phases. An enthalpy/entropy of fusion correlation shows that the arrays approach the melting behavior of molecules lacking internal rotors; i.e., liquid n-alkanes bind into monolayers with slight conformational entropy losses. Therefore, organized surface phases persist above TF because their end-chain torsional barriers are lowered from solid-phase values at such modest energy cost that net stabilization accrues from extra conformational disorder. The reported ∆T vs n diagram is consistent with about n/2 end-chain monolayer internal rotors gaining the ∆S ) 5.3 J K-1 mol-1 entropy increase associated with a reduction of the gauche/trans energy barrier from Vo ) 17 kJ mol-1 (as in packed, infinite polyethylene chains) to Vo < 6 kJ mol-1 (cf. with Vo ) 3.4 kJ mol-1 in liquid n-alkanes) at a cost of about 1.6 kJ mol-1.
Introduction The numerous phase transitions undergone by solids made of long, flexible molecules such as n-alkanes and lipids are generally accompanied by significant changes in rotamer populations.1-13 Considering that the ther† ‡
California Institute of Technology. Chevron Petroleum Technology Company.
(1) Ubbelohde, U. R. The Molten State of Matter. Melting and Crystal Structure; Wiley: Chichester, 1978; p 154. (2) Mandelkern, L.; Alamo, R. G.; Dorset, D. L. Acta Chim. Hung. 1993, 130, 415. (3) (a) Flory, P. J.; Vrij, A. J. Am. Chem. Soc. 1963, 85, 3548. (b) Dill, K. A.; Flory, P. J. Proc. Natl. Acad. Sci. U.S.A. 1981, 78, 676. (4) Small, D. M. The Physical Chemistry of Lipids; Plenum: New York, 1986; Chapter 7. (5) Snyder, R. G.; Maroncelli, M.; Qi, S. P.; Strauss, H. L. Science 1981, 214, 190. (6) Maroncelli, M.; Qi, S. P.; Strauss, H. L.; Snyder, R. G. J. Am. Chem. Soc. 1982, 104, 6237. (7) Snyder, R. G. J. Chem. Soc., Faraday Trans. 1992, 88, 1823. (8) Maroncelli, M.; Strauss, H. L.; Snyder, R. G. J. Chem. Phys. 1985, 82, 2811. (9) Clavell-Grunbaum, D.; Strauss, H. L.; Snyder, R. G. J. Phys. Chem. B 1997, 101, 335.
modynamic stability of the high-temperature phases issues not only from increased librational and rotational entropy but also from enhanced conformational disorder, low-dimensional arrays may well exist above the normal freezing point TF.14-22 This possibility has been recently (10) Forman-Kay, J. D. Nat. Struct. Biol. 1999, 6, 1086. (11) Williams, D. H.; Bardsley, B. Perspect. Drug Discovery Des. 1999, 17, 43. (12) Cavanagh, J.; Akke, M. Nat. Struct. Biol. 2000, 7, 11. (13) Zidek, L.; Novotny, M. V.; Stone, M. J. Nat. Struct. Biol. 1999, 6, 1118. (14) Sirota, E. B.; Herhold, A. B. Science 1999, 283, 529. (15) Ocko, B. M.; Wu, X, Z.; Sirota, E. B.; Sinha, S. K.; Gang, O.; Deutsch, M. Phys. Rev. E 1997, 55, 3164. (16) Sirota, E. B. Langmuir 1998, 14, 3133. (17) Smith, P.; Lynden-Bell, R. M.; Earnshaw, J. C.; Smith, W. Mol. Phys. 1999, 96, 249. (18) Yamamoto, Y.; Ohara, H.; Kajikawa, K.; et al. Chem. Phys. Lett. 1999, 304, 231. (19) Wu, X. Z.; Ocko, B. M.; Sirota, E. B.; Sinha, S. K.; Deutsch, M.; Cao, B. H.; Kim, M. W. Science 1993, 261, 1018. (20) Gang, O.; Wu, X. Z.; Ocko, B. M.; Sirota, E. B.; Deutsch, M. Phys. Rev. E 1998, 58, 6086. (21) Pfohl, T.; Beaglehole, D.; Riegler, H. Chem. Phys. Lett. 1996, 260, 82.
10.1021/la9912141 CCC: $19.00 © 2000 American Chemical Society Published on Web 05/17/2000
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confirmed by the finding that liquid n-alkanes (14 e n e 50) are indeed capped by partially ordered monolayers up to a few degrees above TF.14,15,19 Since the smaller interfacial energies of these marginally stable aggregates would facilitate their nucleation, conformational disorder may play an important role in the mechanism of crystallization of these materials.5,7 Odd alkanes with n g 9 and even alkanes with n g 22 exhibit stable rotator phases below TF.4 These rotator phases consist of packed layers of erect chains lacking long-range order with respect to rotation about the chain axis. The alkyl chains are packed in quasi-hexagonal arrays evolving from the more compact crystal structures through successive solid-phase transitions. Transient metastable rotator phases are also observed in the crystallization of n ) 16, 18, and 20 alkanes,14 confirming their universality and revealing the inverse dependence of sustainable disorder on chain length. The degree of conformational disorder, i.e., the departure from the most stable all-trans structures, also increases with temperature.2,5-9 The discontinuous jumps in the concentration of nonplanar conformers occurring in the solid-phase transitions in bulk n-alkanes were extensively documented by Snyder and others.5 The presence of gauche bonds impairs lateral packing and also affects the stacking of the rougher lamella.9,23,24 Surface tension and X-ray scattering measurements recently revealed the formation of ordered monolayers on the surface of liquid n-alkanes.15,25,26 Those observations were ascribed to fluctuations of the longitudinal mismatch among stretched chains, an effect that provides an entropy gain not entirely offset by the loss of stabilizing interactions.27,28 An alternative explanation suggests that known or estimated interfacial tensions among the various phases suffice to account for the existence of frozen monolayers in terms of macroscopic wetting interactions.15,29 Since infrared-visible sum-frequency generation vibrational spectroscopy unequivocally confirms the presence of a significant number of gauche bonds at the air/liquid n-eicosane interface,30 it is intriguing that neither argument ascribes a causative role to conformational disorder in the mechanism of surface freezing. Thus, the relevant issue at stake is not whether conformational disorder is involved in this phenomenon but, rather, what is the maximum number of gauche defects tolerated by longrange, two-dimensional ensembles above the melting point. In this paper we quantify the supplementary conformational disorder required by experimental observations and show that is entirely consistent with previous thermodynamic information on solid and liquid n-alkanes. We show that the enthalpy-entropy of fusion correlation for monolayers closely resembles that of organic molecules lacking internal rotors, actually merging with it above n ) 50.31a The implication is that n-alkane molecules in the (22) (a) Hughes, C. J.; Earnshaw, J. C. Phys. Rev. E 1993, 47, 3485. (b) Earnshaw, J. C.; Hughes, C. J. Phys. Rev. A 1992, 46, 4494. (23) Zerbi, G.; Del Zoppo, M. J. Chem. Soc., Faraday Trans. 1992, 88, 1835. (24) Kim, Y.; Strauss, H. L.; Snyder, R. G. J. Phys. Chem. 1989, 93, 7520. (25) Wu, X. Z.; Sirota, E. B.; Sinha, S. K.; Ocko, B. M.; Deutsch, M. Phys. Rev. Lett. 1993, 70, 958. (26) Sirota, E. B.; King, H. W., Jr.; Singer, D. M.; Shao, H. H. J. Chem. Phys. 1993, 98, 5809. (27) Tkachenko, A. V.; Rabin, Y. Phys. Rev. Lett. 1996, 76, 2527. (28) Tkachenko, A. V.; Rabin, Y. Phys. Rev. Lett. 1997, 79, 532. (29) Sirota, E. B.; Wu, X. Z.; Ocko, B. M.; Deutsch, M. Phys. Rev. Lett. 1997, 79, 531. (30) Sefler, G. A.; Du, Q.; Miranda, P. B.; Shen, Y. R. Chem. Phys. Lett. 1995, 235, 347. (31) (a) Searle, M. S.; Williams, D. H. J. Am. Chem. Soc. 1992, 114, 10690. (b) Gilbert, A. S. Thermochim. Acta 1999, 339, 131.
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Figure 1. Enthalpies of fusion of n-alkanes (from ref 4): 4, even n e 20; O, even n g 22; 1, odd n. Regression lines for even alkanes were calculated with eqs 1 and 3 (see text).
monolayers are almost as conformationally disordered as those in the liquid phase. In other words, looser chain packing at the monolayer boundaries preserves enthalpic stabilization at the core while providing for entropic stabilization at the edges. This arrangement is made possible by the sensitive dependence of internal rotational barriers, and hence of conformational entropy, to packing. These compensation effects may generally underlie molecular recognition phenomena.10-13,31 Results and Discussion In Figures 1 and 2 we plot the experimental enthalpy and entropy changes reported in the literature for the melting of normal even alkanes as functions of n, respectively.4 Discontinuities arise from the fact that even alkanes with n e 20 melt directly from triclinic phases, but the melting of the higher homologues is preceded by solid-phase transitions into hexagonal rotator phases. The regression lines in Figures 1 and 2 correspond to the following parameters:
∆HM,ne20(kJ mol-1) ) -10.23 + 3.93n
(1)
∆SM,ne20(J mol-1 K-1) ) 8.74 + 10.6n
(2)
∆HM,ng22(kJ mol-1) ) -13.51 + 2.79n
(3)
∆SM,ng22(J mol-1 K-1) ) -1.86 + 6.96n
(4)
They reflect the well-established linear additivity of CH2group contributions to the physical properties of nalkanes.4,31,32 In a series of studies Snyder et al. analyzed calorimetric and infrared/Raman spectroscopic data on conformational disorder in solid alkanes up to the melting point. In summary, Snyder concluded that end-gauche (gt) defects in the penultimate bond are present at all temperatures, while the more disruptive kink (gtg′) defects only appear a few degrees below the melting point.8 Discontinuous changes in spectral features associated with the occurrence of nonplanar conformations were observed at every phase transition. The concentration of conformational defects was found to increase exponentially with temperature (32) Benson, S. W. Thermochemical Kinetics, 2nd ed.; Wiley: New York, 1976; Section 2.17 and Table A.20. See also ref 34, Chapter 27.
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Figure 2. Entropies of fusion of n-alkanes (from ref 4): 4, even n e 20; O, even n g 22; 1, odd n. Regression lines for even alkanes were calculated with eqs 2 and 4 (see text).
between solid transitions and linearly with chain length at constant temperature. On the basis of this information, it is conceivable that additional conformational disorder could be compatible with long-range order in phases of lower dimensionality above TF. The free energies of the solid S and liquid L phases become identical at the melting point by compensation of enthalpic and entropic factors. At temperatures lower than TF the solid is more stable than the liquid for enthalpic reasons, whereas its larger entropy favors the liquid at higher temperatures. This argument implies that the entropy of detached monolayers must be larger than those stacked within the solid phase. From a thermodynamic point of view the extension of the solid phase is immaterial as long as its properties are preserved. We will denote as SL the unrelaxed lamella formally complying with the above condition. Hence, regardless of the actual mechanism of surface freezing, the thermodynamic conditions for the existence of surface monolayer phases ML (i.e., phases possessing smaller molar entropies than the liquid L) above TF is that their molar free energies be smaller and their molar entropies larger than those of the SL lamella: 5
6
S S [SL] S ML S L
(5,6)
That is ∆G5 ) ∆H5 - TF∆S5 < 0 and ∆S5 > 0. These conditions ensure that equilibrium between ML and L will be established at some higher temperature TS ) TF + ∆, ∆ g 0. Therefore, the temperature interval ∆ defining the domain of existence of the surface monolayers is generally given by
∆ ) TS - TF )
[
]
∆H6 ∆HM ∆S6 ∆SM
(7)
Notice that ∆H6 ) ∆HM - ∆H5 and ∆S6 ) ∆SM - ∆S5. We will assume that ∆H5 and ∆S5 increase linearly with chain length without any loss of generality (see above);4,32 i.e., ∆H5 ) A + Bn and ∆S5 ) C + Dn. In analyzing the phenomenon of surface melting in n-alkanes, it is valid to assume that a common set of melting parameters, ∆HM ) ∆HM,ng22 and ∆SM ) ∆SM,ng22 (eqs 3 and 4), applies to all homologues, in accord with the observation that the n ) 16, 18, and 20 alkanes actually freeze into transient metastable rotator phases.14 On the basis of these assumptions, a nonlinear least-squares regression of eq 7
Figure 3. Temperature differences, ∆, between the onset of surface freezing TS and the normal melting point TF: ∆ ) TS - TF, for n-alkanes: O, experimental data (from refs 15 and 19). Solid line is a fit of eq 7 to experimental data (see text).
to the ∆ data of Wu et al.15,19 for n-alkanes yields A ) -1.528 kJ mol-1, B ) 0.8046 kJ mol-1, C ) 8.876 J K-1 mol-1, and D ) 1.963 J K-1 mol-1, between 16 e n e 50 (Figure 3). These parameters encode the thermodynamics of the monolayers involved in the experimental phase diagram of Figure 3. We will now show that these parameters are consistent with conformational disorder as the dominant mechanism for surface freezing. A standing problem in the mechanism of ligand binding is to explain how the receptor cooperates to compensate the entropic cost associated with the loss of translational and external rotational degrees of freedom.10-13,31a A simple analysis of the thermodynamics of n-alkane melting and sublimation actually sheds useful insights into this important issue. Thus, from the verifiable linear dependences of ∆SM on n it is possible to infer that each internal rotor gains ∆SM ) 5.3 and 11.3 J K-1 mol-1 upon melting odd and even (n < 20) n-alkanes, respectively. 31a The reason more entropy per internal rotor is gained in the melting of short-chain even alkanes is that their triclinic solid phases are more compact than the rotator phases typical of odd members. In contrast, longer chain n g 22 even alkanes tend to behave as their odd counterparts. Thus, if molecular associations into organized assemblies bear any analogy with freezing phenomena, one would expect additional entropy losses from enhanced restriction to internal rotations. In this context, an enthalpy-entropy compensation plot for the fusion of organic molecules that lack internal rotors illustrates the reference case in which minimal entropy is exchanged during melting and freezing processes (Figure 4, RM).31a Molecules possessing numerous internal rotations that become more restricted in the solid than in the liquid phase will display larger freezing entropy losses per enthalpy unit. Only if conformational disorder were maintained during freezing would the corresponding enthalpy-entropy correlation approach the trend illustrated in Figure 4, RM. Let us compare the enthalpy-entropy compensation plots for short-chain even alkanes, eq 1 vs eq 2, for longchain even alkanes, eq 3 vs eq 4, and for the monolayers, i.e., ∆H6 vs ∆S6 that can be calculated from the parameters derived above. Confirming the preceding considerations, it is apparent that short-chain alkanes gain more entropy than their longer congeners. As expected, the melting correlation for the monolayers lies to the left of those corresponding to the solid phases. Notice that the slopes
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Figure 4. Enthalpy of fusion/entropy of fusion compensation plots: RM, for rigid molecules lacking internal rotors (∆HM ) -2.18 + 0.376∆SM, from Figure 4, ref 31a); ML, for n-alkane surface monolayers (∆H6 ) -7.72 + 0.397∆S6, see text); n > 22, for even n-alkanes with n g 22 (∆H ) -12.76 + 0.401∆S, from eqs 3 and 4); n < 22, for even n-alkanes with n e 20 (∆H ) -13.47 + 0.371∆S, from eqs 1 and 2).
are identical within experimental error. What is relevant is that monolayers closely approach the melting behavior of rigid molecules, which merge at the entropy change corresponding to the melting of pentacosane n ) 50. Let us consider the physical plausibility of the derived monolayer thermodynamic parameters. ∆H5 and ∆S5 are the net molar changes corresponding to the promotion of a certain number of end-chain internal rotors from trans to gauche conformations in the solid lamella SL at TF. Let and σ be the individual bond contributions to ∆H5 and ∆S5, respectively, and φ the fraction of the total number (n - 3) of bonds that could be promoted, i.e., ∆H5 ) φ(n - 3), and ∆S5 ) φ(n - 3)σ. A lower limit to σ is provided by the melting entropy increase per internal rotor in odd linear alkanes ∆S ) 5.3 J K-1 mol-1.31a Hence, we estimate upper limits of φ ) ∆S5/[5.3 (n - 3)] ≈ 0.5 and 〈〉 ) ∆H5/ [0.5 (n - 3)] ≈ 1.6 kJ mol-1. Snyder reports Vo ) 3.4 kJ mol-1 for the gauche-trans energy difference in liquid n-alkanes.7 The fact that Vo in liquid alkanes is five times smaller than the torsional barrier in packed, infinite polyethylene chains Vo ≈ 17 kJ mol-1 is of crucial importance to the present analysis.33 The latter value is representative of the torsional restrictions imposed by lamellar stacking to the rotation of C-C bonds in solid n-alkanes. By assuming that internal rotations behave as separable degrees of freedom, it is possible to evaluate the entropy increase of a single heavy rotor upon lowering the torsional barrier from Vo ) 20 kJ mol-1 to Vo ) 3 kJ mol-1 by standard statistical thermodynamic methods (Figure 5).32 It is apparent that about ∆S ≈ 5 J K-1 mol-1 is gained by lowering Vo from 7 kT to 2 kT, or from 2 kT to 0, i.e., to barrier relaxations from 17 down to 5 kJ mol-1 or from 5 down to 0 kJ mol-1 at T ≈ 300 K, respectively. In other words, the existence of monolayers is consistent with the loosening of chain ends upon lamella disjunction, the concomitant reduction of internal rotation barriers of the order of 5-12 kJ mol-1, and entropy gains of about 5.3 J K-1 mol-1 at the modicum of about 1.6 kJ mol-1 per bond. Therefore, the proposal that surface freezing of nalkanes is driven by the lower free energy of detached (33) (a) Fukui, K.; Sumpter, B. G.; Noid, D. W.; Yang, C.; Tuzun, R. E. J. Phys. Chem. B 2000, 104, 526 and references therein. (b) Colmenero, J.; Arbe, A. Phys. Rev. B 1998, 57, 13508.
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Figure 5. Entropy losses ∆S of a heavy internal rotor as function of the barrier to internal rotation Vo/kT (From Table 27-13, ref 34).
Figure 6. Excess entropy density of the bulk liquid relative to the frozen monolayers for n-alkanes: O, the experimentally measured derivatives of the surface tension of n-alkanes ∂γ/∂T between TF and TS (refs 15 and 19). Solid line shows values calculated with eq 8.
monolayers relative to those stacked within the solid due to conformational disorder is entirely consistent with first principles and available information. Notice that our interpretation of this phenomenon is formulated in purely microscopic terms, at variance with macroscopic explanations based on the balance of surface energies between semi-infinite phases separated by monolayers of molecular thickness. The distinct properties of surface monolayers are reflected in the lowered end-chain gauche/trans barriers. Monolayers could not stack without losing their entropic advantage, which issues from the looser motions at the liquid/monolayers interface. Since, by definition, the free energies of the solid and liquid phases are identical at TF, i.e., GS ) GL, the fact that ∆G5 < 0 implies that surface monolayers are the stable phase within TF and TF + ∆T. Notice that the coexistence of four phases (solid, liquid, vapor, and monolayer) in a one-component system is only possible for interfacial monolayers characterized by a surface tension γ, in addition to temperature and pressure variables.34 The regression to the data of Figure 3 yields excess surface entropies. The latter are related to the surface (34) Pitzer, K. S.; Brewer, L. Thermodynamics, 2nd. ed.; McGrawHill: Englewood Cliffs, NJ, 1961; p 152.
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tension γ by the basic thermodynamic relationship:15,19
SML - SL ∂γ )∂T A
(8)
where A ≈ 20 Å2 is the area of the alkane chain projection on the lamellar plane.15 In Figure 6 we show the excellent agreement between the ∂γ/∂T values calculated from eq 8 with (SML - SL) ) -∆S6 ) -(∆SM - ∆S5) and the experimental slopes ∂γ/∂T measured in the interval {TF,TS}.15,19 In other words, the derived thermodynamic properties of the surface monolayers account not only for their phase diagram but also for their surface tensions. The favored location of monolayers is at the liquid-air interface, rather than within the liquid. This condition essentially follows from the fact that the sum of the surface energies of two interfaces [liquid/monolayer (L, ML); monolayer/vapor (ML, V)] is generally smaller than that associated with three interfaces [liquid/monolayer (L, ML); monolayer/liquid (ML, L); liquid/vapor (L, V)]. An approximate assessment of the free energy change associated (35) Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1992; chapter 15. (36) (a) Gupta, S.; Koopman, D., C.; Westermann-Clark, G. B.; Bitsanis, I. A. J. Chem. Phys. 1994, 100, 8444. (b) Hammonds, K. D.; McDonald, I. R.; Ryckaert, J. P. Chem. Phys. Lett. 1993, 213, 27. (c) Liang, G. L.; Noid, D. W.; Sumpter, B. G.; Wunderlich, B. J. Phys. Chem. 1994, 98, 11739. (37) (a) Caira, M. R. Top. Curr. Chem. 1998, 198, 163. (b) Dunitz, J. D.; Bernstein, J. Acc. Chem. Res. 1995, 28, 193.
with the transfer of a monolayer from the bulk to the surface of the liquid is given by
∆Gt ) γML,V + γML,L - γL,V - 2γML,L ) γML,V - γML,L - γL,V (9) From the Dupre´ equation for alkane phases that only interact dispersively
γML,L ) γML,V + γL,V - 2(γML,VγL,V)1/2
(10)
we obtain35
∆Gt ) -2[γL,V - (γML,VγL,V)1/2]
(11)
The experimental values γL,V ≈ 28 mJ m-2 and 25 e γML,V/mJ m-2 e 28,15 lead to γML,L ≈ 0.02 mJ m-2 and to -3 e ∆Gt/mJ m-2 e 0. The latter condition confirms that the frozen monolayers are naturally confined to the interfacial region between the liquid and the vapor. The identity γML,L ≈ 0 is obviously consistent with the assumption that the conformationally disordered monolayer periphery merges smoothly into the liquid.36 Summing up, the conformational multiplicity of long alkyl chains allows the existence of ordered two-dimensional phases above the normal melting point. These phases, by providing crystal nuclei at negative supersaturations, may represent the key to crystallization control and crystal polymorphism.37 Further work is underway. LA9912141
