2996
Langmuir 1992,8,2995-3002
Equivalent States of Amphiphilic Lamellae I. R. Peterson,*V. Brzezinski, R. M. Kenn, and R. Steitz Institut fiir physikalische Chemie, Johannes- Gutenberg- Universitiit, Jakob- Welder- Weg, 11, W-6500 Mainz, Germany Received March 5, 1992. In Final Form: July 10, 1992
There are many different condensed phases of amphiphilic monolayers, which occur both at the air water and at the airsolid interface. Thermodynamicreasoning is applied to the results of recent X-ray structural investigations and is used to justify the subdivision of the total surface tension in these phases into a purely interfacialterm and a term proportionalto the bulk pressure internal to the monolayer. This analysis has implications for the variation of the region of stability of each phase, and it is shown that these are consistent with literature reports of phase diagramsfor amphiphileswith different chain length and head groups. The underlyingassumptionshave also been crm-checkedusing diffractionmeasurements and shown to be valid on different subphases or substrates. The results provide support for the principle of corresponding states. The states of two different lamellae can be considered equivalent if their hydrophobic chains have the same periodic structure and if the details of the chain packing are the same. 1. Introduction
The technique of Langmuir-Blodgett (LB) deposition
has attracted considerable attention as a way of fabricating low-defect ultrathin organic films with promise of applications in high te~hnology.l-~ Recent investigations of the optical and electrical characteristics of films of some commonly-used materials have concluded that these parameters are significantly influenced by statistical fluctuations of film structure and by film textures, both of which are characteristic for smectic me so phase^.^* These reports also demonstrated that the films display specificdeleterious defects, which are formed in the water surface monolayer shortly after spreading and conserved on deposition.’+ Hence improvements in film performance are expected to result from advances in the understanding of the condensed phases of the watersurface monolayer from which they are deposited. These concepts and expectations motivated recent investigations which have led to a much deeper understanding of the condensed phases. Using a series of novel measurement techniques, it has been shown that conventional surfactants such as the fatty acids and the phospholipids show at least five such phases on the water surface, each with distinct molecular packing symmetry.le12 Figure 1showsa composite?r-T phase diagram for a series of even-chainfatty acids,showing the Harkins-Stenhagen nomenclature for these phases and the variation of their regions of stability with chain length. At least one of them, the CS phase, is crystalline in the strict sense of longrange translational order, but the phases most suitable (1) Roberts, G. G., Ed.Langmuir-Blodgett Film; Plenum: New York, 1990. (2) M6bius, D., Ed.Thin Solid F i l m 1988,154-160. (3) Fukuda, K., Sugi,M., Eds. Thin Solid F i l m 1989,178-180. (4) Zanoni, R.; Naselli, C.; Bell,J.; Stegeman, G. I.; Seeton, C. T. Phys. Rev. Lett. 1986,57, 2838. ( 5 ) Peterson, I. R. J. Chim. Phys. 1988, 85,997. (6) Peterson, I. R.: Steitz,R.:K w ,H.: VoiatMartin,I. J.Phys. (Paris) 1990,51,1003. (7) Peterson, 1. R. Br. Polym. J. 1987, 19, 391. (8) Peterson. I. R.: Earls..J. D.:. Girlinn, -. I. R.: Russell, G. J. Mol. Cryst. Liq. Cryst. 1987, 147, 141. (9) B i b , A. M.; Peterson, I. R. Thin Solid F i l m 1989,178,81. (10) B i b , A. M.; Peterson, I. R. Adu. Mater. 1990,2,309. (11) Lin, B.; Shih,M. C.; Bohanon, T. M.; Ice, G. E.; Dutta, P. Phys. Rev. Lett. 1990,65,191. (12) Kenn, R.M.;B 6 h , C.; B i b , A. M.; Peterson, I. R.; M6hwald,H.; Kjaer, K.; Ale-Nielsen, J. J. Phys. Chem. 1991,95, 2092.
I 0
c22’&24
20
40
60
80 X
Figure 1. A composite phase diagram of the even-chain fatty acids from tetradecanoic((214)to tetracosanoic (C24),compiled from data given in refs 10 and 27. The abscissa z representa temperature, with one division representing 1 K, but with a different offset for each fatty acid. For chain length N,x = T [K]+ (16- 166N + 3 f l ) / 8 . The phase regions are labeled with their Harkins-Stenhagen symbol, and only those phase tramitions recognized by Stenhagen are shown.
for LB deposition are all mesophases and can be assigned to known smectic categories.12J3 Now that the monolayer structure of a number of substances has been established on the water surface, it has becomefeasibleto investigatethe relationship between the molecular packing before and after deposition. It has turned out that the structures are usually not the same, but that they are not completely unrelated either.14JsThe structure of the first monolayer deposited on a substrate always correspondsto one which has been observed on the water surface, but possibly under different conditions of surface pressure, temperature, and subphase composition. It has recently been shownthat the well-knownprinciple of corresponding states can be refined to provide a frameworkfor understanding the observed changes of film structure on varying chain length, head group, and monolayer environment.16 Clearly, the phase which is (13) Bibo, A. M.; Knobler, C. M.; Peterson, I. R. J. Phys. Chem. 1991, 95, 5591. (14) Steitz, R.; Mitchell, E. M.; Peterson, I. R. Thin Solid Film 1991, 205, 124. (15) Engel, M.; Merle, H. J.; Peterson, I. R.; Riegler, H.; Steitz,R. Ber. Bunsen-Ces. Phys. Chem. 1991,95, 1514.
0743-7463/92/2408-2995$03.00/0 0 1992 American Chemical Society
Peterson et al.
2996 Langmuir, Vol. 8, No. 12, 1992
stable under any given set of conditions is the one with lowest free energy. A range of experimental data on monolayers of a number of common amphiphiles indicated that it should be possible to approximate to the variation with chain length of the Helmholtzfree energy by ita longchain asymptote. The aim of the present work is systematically to check this approximation against the available literature data. 2. Theory
Suppose a thermodynamic system has an interface of total area A separating a bulk phase 1from a bulk phase 2. Then the Gibbs' surface excess value PS of a given extensive thermodynamic parameter P is given by17
P = P-aP,/v-pP2/v
Suppose that the chain length of the amphiphileis now increased without limit at constant u and T. Because of the effective thermodynamic independence at large separations mentioned above, it follows that, asymptotically for large molecular length A, p must be linear in X
P(u,T,X)= f;(u,T)
CW= ydA - F d T + pdn
(2) where y is the interfacial tension and p the chemical potential of the amphiphile. The differential of eq 2 can be transformed to express each of the three coefficients y, Ss, and p as derivatives of P, in particular (3)
Equation 2 can also be integrated at constant y, p, and
T to express P in terms of ita own derivative@ = yA + pn = n(yu + A (4) where u Aln is the surface area per molecule. Note that eq 2 ignores line tension effects. Both it and eq 4 express the fact that macroscopically separated parts of a system are essentially independent, so that each extensive thermodynamic parameter for the surface as a whole is proportional to a well-defined intensive density independent of n for constant u and T. The intensive density corresponding to P is the surface excess Helmholtz free energy per molecule f
F
t" 1 F l n (5)
Replacing the extensive parameters P and A in eq 3 by the intensive parameters f and u leads to an alternative (16) Peterson, I. R. Polym. Aduan. Technol., in press. (17) Chattorej, D. K.; Birdi, K. S. Adsorption and the Gibbs' Surface Excess; Plenum: New York, 1984. (18) Aveyard, R.; Haydon, D. A. An Introduction to the Principles of Surface Chemistry;Cambridge University Press: Cambridge,1973;p 11.
+ X B(u,T)
(7)
In experimentwork on monolayers, it is usual to report, not the surface tension y, but the surface pressure T , defied by
(1)
where PIIV and PdV are the (well-defined) volume densities of the parameter P in the two individual phases. a and /3 are chosen to bring to zero both the surface excess volume VB and the surface excess number of molecules nms of one of the major system component substancesm chosen as reference. Supposethat the interface between water and ita vapor is covered by an adsorbed monolayer of an amphiphile, which is involatile and insoluble in the water. Since the bulk phase concentrationsare both zero, the surfaceexcess number of molecules ns is equal to the total number n of molecules of the amphiphile in the system and is independent of the particular reference substance. The condition for conservation of energy at equilibrium is given in terms of the surface excess Helmholtz free enrgy P by17J8
=yU+Cc
definition for y, more useful for present purposes
= ro(T)- y
(8)
where yo(T) is the surface tension of a clean water surface. For this reason it is convenient to introduce a function I(u,T) corrected for the free energy associated with an equivalent clean water surface I(U,T)
= f;(Q,T) - Q Yo(l?
(9)
The free energy I and gradient B will be called the interfacial and bulk Helmholtz coefficients, respectively. It should be noted that the correct statistical thermodynamic treatment of interfaces is not straightforward, even in the absence of adsorbed amphiphiles.19 However in spite of these difficulties, the concept of interfacial tension as a thermodynamic variable related to the free energy per unit interface is well established. Irrespective of future theoretical advances in methods of calculating p for model molecular systems, it is expected that eq 7 will correctly describe the asymptotic variation as a function of chain length in the limit of long chains. Note that a similar linear dependence of free energy on chain length has been used successfully to describe bulk isotropic condensed phases of long-chain compounds.20 Harkins pictured the head groups of lipids in the liquid condensed phase as being close-packed, with the chains showing liquid-like order.21 If this were true, the packing in this phase of many important lipide would be dominated by the head groups. Clearly, in the long-chain limit, the head groups can at most be a small perturbation to an arrangementdominated by the chains. Hence, on Harkins' picture, the asymptote of eq 7 would only be valid for chain lengths far in excess of those encountered in amphiphiles displaying a liquid condensed phase. However Harkins provided very little experimental evidence to support his view, which has since ita proposal been disputed by other investigator^.^^^^^ From the X-ray structure data and recent fluorescencemicroscopy results% it is now known that the region called liquid-condensed actually comprises a number of distinct phases, in all of which the chains are close-packed with long-range order. The chains are also close-packed and ordered in the other condensed phases. For a system conforming to eq 7 the surface pressure of the monolayer-covered interface is given by (19) Rowlinnon, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon: Oxford, 1982. (20) Williamson, A. G.; Scott,R. L.Trans. Faraday SOC.1970,66,336. (21) Harkins, W. D.; Boyd, E. J. Chem. Phys. 1940,8,129. (22) Dervichian, D. G. J. Chem. Phys. 1940,4347. (23) Joly, M. J. Colloid Sci. 1960,5, 49. (24) Qiu, X . ; Rub-Garcia, J.; Stine, K. J.; Knobler, C. M.; Seliiger, J. V. Phys. Reu. Lett. 1991, 67, 703.
Langmuir, Vol. 8, No. 12, 1992 2997
Equivalent States of Amphiphilic Lamellae
from the concept presented here that the pressure P12 or
P for a given first- or second-order transition, respectively, should depend on temperature only and should be
where
and aB
P = -(-)ab T P has the dimensions of a bulk pressure and is closely related to the bulk pressure in the limit of large chain length. It may be considered to be a bulk pressure internal to the monolayer. Just as the pressure inside a cylinder need not be the same as the pressure outside as a result of the confining action of the cylinder walls, the monolayer internal bulk pressure need not be equal to the bulk pressure in the monolayer environment because the molecules are confined to the air-water interface by hydrophilic-hydrophobic interactions. Another analogous system is the Earth’s atmosphere, which exerts a pressure of lo5Pa at the Earth’s surface, in spite of being in contact with the vacuum of outer space. The constant offset cp may be considered to be a pressure of the lamella interfaces and will be called the interfacial pressure. As its name would suggest, 9 tends to zero in the monolayer gas phase in the limit of zero surface density. In a second-ordertransition, the phase remains uniform, and only its symmetry changes as the threshold is crossed. There is no discontinuity of the free energy$ or of its first derivatives, although the occurrence of the transition is marked by a discontinuity of the second or higher derivatives. The surface pressure at which the phase transition occurs varies with chain length according to eq 10. If the transition between two phases 1 and 2 is first order, the surface pressure a12 at which the transition occurs is determined by the equality of the chemical potential p in the two phases. Note that the derivative of p with respect to a is equal to the molecular area u. On the assumption that these areas in the two phases at the monolayer transition differ from their respective values at the asymptotic transition pressure by significantlyless than the difference between the two phases, the transition surface pressure is given approximately by r 1 2 zz -M/Au
+ hp12
(11) where P12 is the asymptotic transition pressure for infinitely long chains, which is determined completely by the bulk Helmholtz coefficients BI and B2, Au = ul(P12) - uz(P12) and is the difference in the molecular areas u of the two phases at the asymptotic transition pressure, and Al = Il(u1) - I2(uz) and is the difference in the interfacial Helmholtz coefficients of the two phases, each evaluated at the appropriate area above. In a second-order transition, the internal monolayer pressure is the value P in eq 10, which is the value at which the transition would occur in the long chain limit. In a first-order transition, however, the internal monolayer pressure is in general not equal to the asymptotictransition pressure P12 and is not the same in the two phases. To the above approximation, these are given by
In all the materials to be investigated in this work, the hydrophobicchain of the molecule is aliphatic. It follows
independent of the particular head group. The dependence on head group and environmentwill be reflected by the interfacial pressure cp. In general, cp will also depend on the chemical nature of the chain. Lundquist’s work revealed significant alternation between the phase diagramsof even and odd members of the one homologous series, an effect which is well-known in bulk studies of long-chain compounds. It is due to axial coherence along the molecules and indicates that there is only a low density of chain kinks which would otherwise allow the molecular ends on both sides of the lamella to take up their energetically most favorable configuration independent of the length of chain. Because of this effect, the interfacial pressure cp is expected to differ between compounds with the same head group but different chain parity. In this work the chain length of an acid or ester will be classified as even or odd depending on the total number of carbon atoms. However, note that in a closepacked monolayer the sp3 oxygen forms an extension of the all-trans chain, so that the conformationof the terminal groups resembles that in an alkane of opposite parity.
3. Experimental Determination 3.1. General Information. The parameters of
this
theory are most directly accessible experimentally for second-ordertransitions. Accordingto eq 10,the transition surface pressure at a particular temperature should vary linearly with chain length. The gradient of this variation is the internal pressure at the transition. Clearly,to define a straight line, the phase transition must exist at that temperature for at least two different chain lengths. Confidence in the predictive power of the theory will of course be much stronger if the linear dependence is confirmed for more than two. In view of the long-term dispute over the nature of the liquid-expanded to liquid-condensed transition, it is necessary to be cautious about assigning the order of transitions between the condensed phases. In particular, isotherm data can never be used in isolation. If available, X-raydiffraction data are much more conclusive. In Kenn et al.’s investigation of docosanoic acid,12it was possible to classify two transitions as second order, namely, L z L S and L2’-S. Both of these transitions are between phases with tilted and upright molecules, respectively, and have been called26“tilting transitions”. It may be considered that even this very precise study could not rule out the possibility of an extremely small discontinuity in parameters across the above two transitions, so that they might still be very weakly fiit order. This problem is intrinsic, because it is theoretically poesible for a true second-order transition to be converted to very weakly first order and vice versa by the action of small amounts of impurities, which can never be completely eliminated. However the above theory is unaffected by this uncertainty, because eqs 11and 12tend continuously to eq 10 in the limit of a weak first-order transition. Not only second-order transitions but also those of first order should show a transition pressure varying linearly with chain length. However, complete analysis of the isotherm data in the latter case requires additional information. If there is a point in common with a secondorder transition definingthe internal pressure in one phase, eq 12 can be used to determine AIIAu. In view of the (25) Shih, M. C.; Bohanon, T.M.; Mikrut, J. M.;Zechack, P.;Dutta, 1992,45, 5734.
P.Phys. Reu. A
Peterson et al.
2998 Langmuir, Vol. 8, No. 12, 1992
Table I. Regression Line Parameters and Literature Sources for the LhLS Phase Transition
mN/m +
+ c22
+c24
I
1
c20
I,,,f ,,;, #i', , / ,
,,
, (
10
,,,,,,,
20
30 T,OC
Figure 2. r-T phase diagrams for the even chain fatty acids from C14 to C24 plotted on the same surface pressure and temperaturescales,showingtwo phase transitionsof secondorder. The L2-LS transition is the continuous heavy line, and the Lz'-S transition is the alternate heavy and light line.
parity even
odd
substance
+ aT [refl
"C. Within experimental accuracy,the transition surface pressure r a t T "C and chain length N (i.e. A = 127 N pm) is given by
+
23 - 0.56T 0.03NT (13) This is compatible with eq 10. Comparison of the two allows deduction of the following parameters: A
experimental difficulties in determining molecular areas and compressibilities from isotherms, further analysis requires the use of more reliable techniques, for example X-ray diffraction. In the first proposal above for determining the parameters of the theory, a second-orderphase transition is used as amarker for the equivalence of states between materials of differing chain length or head group. This is not applicable for a general state. However it can be seen from section 2 that u is also a marker, which is generally valid for the determination of equivalent states. In principle, u can be determined from isotherms, although there are many experimental difficulties. The area of the unit cell determined by diffraction methods is believed to be very good approximation to the thermodynamic parameter a, and it can be determined much more accurately. For the tilted phases, which can exist over a range of u values of more than 20% at a given temperature, an accuracyof 2 % would be adequate. For the upright phases, an accuracy of at least 0.1 % is required. Diffraction techniques also provide an independent cross-checkfor the originalhypothesis that the free energy of the monolayer can be well-approximated by the longchain limit. In this limit, the molecular packing is determined by the chains, with no influencefrom the head groups. Diffraction provides not just the cell area u but also the individual unit cell vectors a and b and their subtended angle y. States of identical u in materials of different chain length or head group can be considered truly equivalent if the ratio b/aand the unit cell angle y are also the same. 3.2. Tilting Transitions. Figure 2 shows the phase diagrams for the even-chain acids from C14 to C24 in the temperature range 10-40 O C , with the L2-LS transition shown by a continuous heavy line. It can be seen that at all temperatures in this range there are at least three and in some cases five substances in the homologous series displaying this transition. Note that to within experimentalaccuracy,the transition surface pressure varies linearly with temperature for each of the substances, with negative gradient for chain length C18 and shorter, and positive gradient for chain length C20 and longer. The thin dotted lines are the extrapolations of this linear dependence outside the range of the phase transition for each substance. It can be seen that all of them intersect at the point A = 22.8 mN/m, T = 0
regression line r = TO
C ~ ~ H Z ~ C O O H22.2 - 0.112'[lo];19.7- 0.112'[28] 22.8- 0.012'[lo];23.0- 0.082'[261 ClsH31COOH Ci,H&OOH 23.2 + 0.052' [lo] 22.8 + 0.122'[lo];22.8 + 0.042'1271 CigH3sCOOH 22.8 + 0.182'1101;22.8 + 0.102'1271 CzlH43COOH CzsH4,COOH 22.8 + 0.162'[271 C H ~ C O O C ~ 21.0 ~ I + 0.002' [291 CHsCOOCzzHls. 21.0 + 0.062'1291 C~SH~~COOCZHS 11.4- 0.132'1301 C~,H&OOC~HS 11.4- 0.072'[30] ClgHs&OOCzHs 11.4- 0.012'[301 ClrH&OOH 21.8- 0.092' 1281 CleH&OOCzHs 9.4 - 0.122'[301 Cl&,COOCzHs 9.4- 0.062'[30] C&ilCOOCzHs 9.4+ 0.012'[301
cp
= 23 - 0.56TmN/m
P = 0.24T MPa
(14)
Data for this transition of the fatty acids are available from a number of sources, and in Figure 2 the particular diagram for each substance has been chosen to make the agreement with the theory clearest. Table I shows the equations for the regression lines obtained from other sources. It can be seen that although there are significant differences between the data of Bib0 and Peterson, Akamatsu and Rondelez, Stenhagen, and Nutting and Harkins, often amounting to more than the difference between two adjacent members of a homologous series, these are considerablysmaller than the variation between the shortest- and longest-chain materials. Moreover all the data sets show the same general trend within themselves, so that these differences can perhaps be ascribed to systematic differences in measurement technique or material purity. The L2-LS transition has been observed by Lundquist in two even-chain acetate esters, of eicosanoland docosanol, respectively. Her phase diagrams are shown in Figure 3, and the regression line parameters are shown in Table I. It can be seen that the variation of transition surface pressure with temperature is also linear in this case. The extensions of the lines for the two esters intersect at T = 0 OC, and the difference in their ordinates agrees very well with the common difference 0.06T observed for the fatty acids. Thisindicatesthat the packing in the corresponding phases of the two homologous series is similar, i.e., that the packing is dominated by the aliphatic chains, as also indicated by the X-ray data. The parameters of eq 8 are given by (26) Nutting, G. C.; Harkins, W. D. J. Am. Chem. SOC.1939,61,2040. (27) Stenhagen,E. In Determination of Organic Structures by Physical
Methods; Braude, E. A., Nachod, F. C., Eds.;Academic Press: New York, 1955; Chapter 8. 128) Akamatau, S.; Rondelez, F. J. Phys. 11 1991, 1, 1309. (29) Lundquist, M.Chem. Scr. 1971, 1 , 6. (30)Lundquist, M. Chem. Scr. 1971, 1 , 197.
Langmuir, Vol. 8, No. 12, 1992 2999
Equivalent States of Arnphiphilic Lamellae
?Ti mN/m c22 I
1s;:
15
L, 1' 1 I 1
10
1 ,
'
'
'
20
' 1
1
'
I 1
'
1
1 '
30
+
+
+ ' 1
'
z
L
'1''
"
40
' 1
' ' $ 1 "''1"'
T,OC
= 21 - 0.66T mN/m
P = 0.24T MPa
(15)
The interfacial pressure cp at all accessibletemperatures is less than that of the fatty acids. Considering that an acetate ester has a terminal methyl group in direct contact with water where a fatty acid has an OH group, one factor in this reduction is likely to be the smaller polar interactions between esters and water. However, if this were the only factor, the difference with respect to the fatty acids would appear surprisingly small. There may be a second, steric contribution, which should increase cp for bulkier head groups. A reduction of cp due to the lack of steric hindrance between head groups could explain the absence of a tilted condensed phase in monolayers of perfluorinated carboxylic acids.31 (It cannot be due to the perfluorinated chains by themselves, because fluoroalkyl amphiphiles with bulkier head groups do display tilted phases.32) The data for the even-chain ethyl esters are given in Figure 4. This class of substance has been the subject of a monolayer miscibility comparison with the fatty acids. In that study, the phase diagram of the C22 ester was measured onlyup to 30 "C, at which temperature the tiltedto-upright transition was L$-LS. However the Lz-Lz' boundary is upwardly curved. Examination of Lundquist's phase diagramsshows that for each of the even-chain ethyl esters, the nearly horizontal part of the lower LS boundary is composed of two straight-linesegments. Hence the hightemperature segment can be tentatively identified as LZLS. Consistent with this identification, the extensions to these segments intersect at T =e 0 "C, and the differences between their ordinates are closely equal to the value for the L2-LS transition of the fatty acids. The eq 10 parameters are cp
= 11.4 - 0.67T mN/m
P = 0.24T MPa
FJ
'c21,
(16)
Here, the interfacial pressure is significantly less than that for the acids and acetates, reflecting the reduced interaction between head groups and water. (31) Barton,S. W.;Goudot,A.;Boulouasa,O.;Rondelez,F.;Lin,B. H.; Novak, F.; Acero, A.; Rice, S. A. J . Chem. Phys. 1992,96, 1343. (32)Wolf,S. G.; Deutach, M.; Landau, E. M.; Lahav, M.; Leiserowitz, L.; Kjaer, K.; Ala-Nielaen, J. Science 1988, 242, 1286.
+
+
i
+
+
+
/
l5 0
'
Figure 3. r T phase diagrams for the acetate esters of total carbon number C22 and C24, showingthe two second-orderphase transitions also shown in Figure 2. cp
1
Y+ 4+
I
l
L
Figure 4. Phase diagrams for even and odd chain ethyl esters from total carbon number Cl8 to C23, showing the two secondorder phase transitions also shown in Figure 2. Comparison of the interfacial pressures for even and odd chain lengths is possible for Lundquist's study of the ethyl esters, from which the lower graph of Figure 4 has been drawn. Instead of the two linear segmentsbounding the LS phase, three can be clearly seen in Lundquist's data for the C21 and C23 esters. It should be noted that Qiu et al.33 have observed rather complicated phase behavior of the odd-chain esters. Agreement with the values already tabulated for the L2-LS transition is obtained if this phase transition is taken to be the central segment. With this identification, the eq 10 parameters are cp
= 9.4 - 0.69T mN/m
P = 0.24T MPa
(17)
Once again, the internal monolayer pressure is consistent with that of the even-chain fatty acids. The interfacial pressure is slightly lower than that for the even-chain ethyl esters, as expected from the relative orientations of the two terminal groups of an all-trans molecular chain. A phase diagram is also available for pentadecanoic acid.28 Although clearly these data must be interpreted with caution, the following interfacial pressure may be deduced = 21.8 - 0.54T mN/m (18) The temperature variation is essentially the same as for the even-chainacids, while the intercept is slightly smaller. Within experimental error, the odd-eventrend is the same as for the ethyl esters. The parameter sets given in eqs 13 to 18 are summarized in Table 111. There is literature data for the Li-S transition of some fatty acids, acetates, and ethyl esters in the work of Stenhagen and Lundquist. These data are not nearly as satisfactory as that for the L z L S transition, because in any one substance the temperature range of the transition is always less than 5", and there are no cases of overlap between these regions in different compounds with the same head group and chain parity. The data of Tables I11 and IV resume the best fit interfacial tensions and internal pressures for the transition on the assumption that the cp
(33) Qiu, X.; Ruiz-Garcia, J.; and Knobler, C. M. MateriaLP &!Search Society Symposium Series, in press.
Peterson et al.
3000 Langmuir, Vol. 8, No. 12, 1992 Table 11. Regression Line Parameters and Literature Sources for the 4’-5 Phase Transition DaritY substance regression line r = ro + a T [refl 20 + 0.44T[27] even ClsHmCOOH 19 + 0.44T[271 Czi&COOH 18 + 0.44T [271 C&&OOH 17 + 0.42T [29] CHsCOOCdii 16 + 0.42T [29] CH&OOCz& 12 + 0.31T [30] C17H3sCOOCzHs 11 + 0.31T [30] CisH,qgCOOCzHs 18.7 + 0.44T[ll] odd CdiiCOOH 7 + 0.32T 1301 Ci8H37COOCzHs 6 + 0.32T [30] CmHilCOOCzHs Table 111. Rerume of Bulk and Interfacial Parameters for the LrLS Transition of Long-chainAmphiphiles AliDhatic Chain, P = 0.24T MPa cp.
mN/m odd21.8- 0.54T
even acid
acetate ethyl ester
23 - 0.56T 21 - 0.66T 11.4- 0.67T
9.4- 0.69T
Table IV. Beet-Fit Bulk and Interfacial Parameters for the La’-S Transition of Long-chain Amphiphiles Aliphatic Chain, P = -4 MPa cp,
even acid
acetate ethyl ester
30 + 0.44T 28 + 0.44T 22 + 0.44T
mN/m oda 29 + 0.44T 17.5 + 0.32T
transition surface pressures vary linearly with temperature and that the theory described in section 2 is correct. Although the values of Tables I11 and IV are quite uncertain, it is still possible to make some definite statements. Whereas the internal pressure is positive at the Lz-LS transition at all accessible temperatures, it is negative at the Lz’-S transition. The latter transition nevertheless occurs at somewhat higher surface pressures than the former in all known substances because the interfacial pressures are substantially higher. This might reflect the fact that the packing at this transition, although more closely approaching the crystalline packing of aliphatic chains, is less favorablefor the slightlymore bulky head groups, or it might be due to less favorable geometry for interaction of the head groups with water. The internal pressures at these two transitions are large in comparison with atomospheric pressure. However condensed matter has a density roughly lo3times that of the atmosphere and considerablyless free volume. Hence the characteristic pressure scale for phase transitions in condensed matter is in excess of 100 MPa. From this viewpoint, both tilted-to-upright transitions occur at essentially zero pressure. This is in agreement with the predictions of a recent mean-field model for the hexatic rotator phases of m0nolayers.3~ Both Stenhagenand Lundquist report a phase transition from the LS phase at high surface pressures, to a phase which Lundquist symbolizes LS’. The isotherms are similar to those of a second-order phase transition. While the monolayers in this region are so unstable that no X-ray structural investigation on the water surface has been carried out, it could conceivablybe identical to the X phase (34)Kaganer, V. M.; Osipov, M. A.; Peterson, I. R. J. Chem. Phys., in press. (35) Bibo, A. M.; Peterson, I. R. Thin Solid Films 1992,210, 515. (36) Tippmann-Krayer, P.; MBhwald, H., Langmuir 1991, 7, 2303.
recently observed in monolayers on solid s u b ~ t r a t e s ~ ~ J ~ and on calcium-containingsubphases.3’ However the data of Lundquist and Stenhagen do not agree on the sign of the internal pressure. 3.3. First-Order Transitions. Although the change with chain length of the surface pressure at the tiltedto-upright transitions is small, this is not the case for the first-order transitions visible in the phase diagrams of Figures 1-3. As derived in eq 13,the bulk pressure deduced from this change is not the actual internal monolayer pressure but the asymptotic value at infinite chain length. Since the line of the S-LS transition is nearly vertical, and its temperature changes significantly with chain length, the change of surface pressure at constant temperature must be large and negative, although there is no overlap between ita temperature ranges in different materials. Extrapolating linearly, the surface pressure change can be estimated to be -80 mN/m for a change of chain length of 254 pm at approximately 20 OC,but the error bars are considerable. This correspondsto a pressure of -300 MPa. At the Lz’-S-LS triple point, the internal pressure in the monolayer is -4 MPa, so that A U A u is roughly -80 mN/m. Since these values are so large, it is not obvious that the approximation of eq 11 is justified. The pressure of the CS-S transition is also negative and even greater in magnitude, so that the temperature range of stability of the S phase decreases with increasing chain length. Note that the CS phase is crystalline, and that the L2” phase probably is too. Hence in the long-chain limit, mesophasesare only thermodynamically stable with respect to crystalline phases at extremely large negative pre8sures, where they are in any case unstable with respect to the gas phase. Their occurrence in monolayers can be explained by the favorable negative sign of the AI/ Au term in eq 12,which shifts the transition to accessiblepressures for sufficiently short chain length. The asymptotic pressure of the LrL2’ transition is also negative, although an order of magnitude smaller at approximately -30 MPa. The fact that this transition shows two linear segments in phase diagrams for the and for the acetates%suggests that the LZregion contains more than one distinct phase. Independent evidence for such a subdivision is provided by the kinetics of defect rec~mbination~~ and by the large difference in unit cell parameters of apparently nearest-neighbor-tilted (smectic I) phases observed in eicosanoic acidw and docosanoic acid12monolayers. Not all asymptotic transition pressures deduced in this way are negative, for example in the transition between LS and Lz’. It has already been proposed that the very clear kink in this transition line is due to the subdivision of the next-nearest-neighbor-tilted Ld region into smectic F and smectic H sub phase^.'^ In general, as a result of the uncertainties in phase identification, the approximations involved in the derivation of eq 11,and the lack of temperature overlap in the (37) Kitaigorodskii, A. I. Organic Chemical Crystallography; Consultants Bureau: New York, 1961. (38)Shih, M. C.;Bohanon, T. M.; Mikrut, J. M.; Zechack, P.; Dutta, P. J. Chem. Phys. 1992,96, 1656. (39) Schlossman, M. L.; Schwartz, D. K.; Perehan, P. S.; Kawamoto, E. H.; Kellogg, G. J.; Lee, S. Phys. Reu. Lett. 1991,66,1599. (40) Stephens,J. F.;Tuck-Lee,C. J.Appl. Cryst. 1969,2,1. Prakaah, M.; Dutta, P.; Ketterson,J. B.; Abraham, B. M. Chem. Phys. Lett. 1984, 111,395.
(41) Brzezinski, V. Unpublished transmission electron diffraction results. (42)Ewen, B.; Strobl, G.R.; Richter, D. Discuss. Faraday SOC.1980, 69, 19. (43)Lieser, G.;Lee, K.-S.;Wegner, G. Colloid Polym. Sci. 1988,266, 419.
Equivalent States of Amphiphilic Lamellae
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400 410 420 430dll/pm Figure 5. Plot of nondegenerate d-spacing do2 (in-plane, units of pm) versus degenerate d-spacing dll (in-plane,units of pm) for monolayer phases on water, poly(methy1methacrylate), and other monolayerswith unit cell aspect ratio normal to chains bla = 1.5. The dashed line has equation dm = dll. Orthogonal conformationallydisordered crystalline phase: 0, CS phase of docosanoicacid on pure water;12a,U phase of tetracosanoicacid on water;= e, A phase of heneicosanoic acid on water." Orthogonal biaxial mesophase, smectic E 0 , S phase of docosanoic acid on pure water;128 , A' phase of heneicosanoic acid on pure water." NNN-tilted rotator mesophase, smectic F 0 ,Lz'phase of docosanoic acid on pure water;lZe, Lz' phase of docosanoic acid on a PMMA s ~ b s t r a t e ; ~9, ~ JF~ phase of tetracosanoic acid on pure water;= e ,C phase of heneicosanoic acid on pure water.ll Multilayer structures: V, lead(I1) octadecanoate multilayers;'OA,freestanding cadmium eicosanoate bilayer;" A,cadmium eicosanoate bilayer on amorphous carbon 4, cadmium hexadecanoate bilayer on amorphous carbon substrate;" I>, cadmium tetradecanoate bilayer on amorphous carbon substrate;" V, magnetic-field annealed cast f i b of poly(docoey1acrylate);414,tertriacontane(n-C&@),bulk crystalline phases with and without conformational disorder.42 available data, it is not worthwhile to analyze any of the first-order transitions further. 3.4. Diffraction Cross-Check of State Correspondence. Figures 5 and 6 are compilations of diffraction data from a number of lamellarsystemsof aliphatic chains, includingfatty acid and soap monolayers on a water surface and on solid substrates. In Figure 5, the solid substrates include cases which are commensurate, i.e. other monolayers in a bilayer or multilayer stack. In all cases, two of the lowesborder reflections are degenerate; i.e. they have the same d-spacing, indicating a rectangular unit cell. Hence it is only necessary to plot two parameters to c o n f i i that the molecular area u fully defines the structure a t a given temperature. In both figures, the degenerate spacing d11is plotted along one axis, and the nondegenerate spacing do2 along the other. Note that for convenience the choice of abscissa and ordinate is different in the two figures. The dashed line in both cases is the line do2 = dll corresponding to hexagonal symmetry. The solid lines have been drawn as a guide to the eye. A given point was plo,tted in Figure 5 if the packing was characterized by a unit cell aspect ratio bla normal to the aliphatic chains of significantly less than 1.73 and approximately equal to 1.5. The packing in these cases is believed to be related to the Kitaigorodskii orthorhombic
6 + 370 + + 420 430 440 450 d02/pm Figure 6. Plot of degenerate d-spacing d11 (in-plane,units of pm) versus nondegenerate d-spacing dm (in-plane, units of pm) for monolayerphases on water,poly(vinylformal)and poly(methy1 methacrylate) with unit cell aspect ratio normal to chains b/a = 2. Note that the axes are exchanged with respect to Figure 4. The dashed l i e has equation d o ~= dll. Orthogonal conformationallydisorderedcrystalline phase: 0,X phase of henecoeanoic acid on water containing Ca2+ions;%0, X phase of docosanoic acid on a Formvar substrate;" 8,X phase of docosanoic acid on a PMMA sub~trate.~~ "-tilted rotator mesophase, smectic I 0 ,LZphase of docosanoic acid on pure water?* e, LZphase of docosanoic acid on a PMMA suhtrate;16 9,I phase of tetracosanoic acid on water.% Multilayer structures: 4 , phase C of cyclodoheptacontane (CHZhZ (the other phases have oblique 2D unit cells).'8 subcell packing3' (R)of aliphatic chains. If a packing showed a unit cell aspect ratio normal to the chains of significantly greater than 1.73 and approximately equal to 2, it was plotted in Figure 6. It has been propoeed14 that these packings are related to the Kitaigorodskii monoclinic subcel13' (MI.There are also phases characterized by bla approximately equal to 1.73. On the water surface, these are LS26728*29 and L1'.13135936The former can be definitely assigned to smectic category BH and the latter tentatively to smectic L. These are believed to be related to Kitaigorodskii hexagonal subcell packing H. Since the hexagonal symmetry in these cases completely determines the unit cell parameters once u is given, they do not provide any confirmation of the principle discussed in this paper and have not been plotted. Not all the data points of Figures 5 and 6 correspond to the same temperature, although all lie nominally in the range 1 to 25 "C. Strictly speaking, the principle of equivalent states as developed here allows the unit cell parameters to depend on temperature T as well as molecular area u. However in the X-ray measurements no consistent variation with temperature can be seen, so it is probable that such variation is small. In the case of the electron diffraction measurements, it is impoesible to rule out local heating of the sample by the electron beam, which may explain the somewhat large statistical scatter
Peterson et al.
3002 Langmuir, Vol. 8, No. 12, 1992
of these data points compared to the X-ray measurements on the water surface. It can be seen from the two figures that the expectations based on the principle are essentially fulfilled. To within a reasonable estimate of experimental error, all points (&I, d02) lie on a one-dimensionallocus. The statistical scatter is much smaller when only the X-ray data are considered, which may be a result of the better temperature control possible in this case. Hence the unit cell parameters are completely determined by the molecular area u and temperature T, even though the interactions between head groups and substrate or subphase differ considerably. The allocation of the points to Figure 5, Figure 6, or not at all was made on the basis of the aspect ratio of the unit cell perpendicular to the aliphatic chains, whether near 1.5, 2, or 1.73, respectively. The fact that the locus of Figure 6 requires two regression lines probably indicates that there are two distinctly different packing modes in this category, meaning a total of four packing modes in all. It is interesting that Kitaigorodskiilists a total of four possible local packings. In addition to R, M, and H, there is also T (triclinic subcells), which is very commonly encountered in aliphatic chain crystals. The fact that the symmetry of the diffraction patterns is rectangular does not rule this out, as higher macroscopic symmetry may result from the presence of conformational disorder. For example, this sort of disorder is known to explain the differing microscopic and macroscopic symmetries of the smectic B phases." It is tempting to identify the two regression lines of Figure 6 with local M and T packing, although a firm identification will require an increased understanding of conformational disorder in systems of aliphatic chains. the polymorphism of aliphatic As previously chainsshown in Figures 5 and 6 differs from the commonlyencountered case of crystalline polymorphism in that the unit cell of each form R, M, H, and possibly T has a range of variation, depending on the internal bulk pressure. The difference may merely lie in the much greater range of internal pressure possible in a monolayer. It is clear that the principle of equivalent states cannot be true under all circumstances. Although the molecular packing may be independent of chain length for infinitely long chains, there must be a finite perturbation from the terminal groupsin practical cases. An interesting problem for further study will be to determine the limits of validity of the principle, with respect both to structure and to thermodynamics,and to quantify the perturbations to unit cell parameters and to eq 10 caused by changes in head group and environment. (44) Luckhurst, G. R.; Simpson, P.; Zannoni, C. Liq. Cryst. 1987, 2,
313. (45) Peterson, I. R.;Russell,G. J.;Earls,J. D.; Girling, I. R. Thin Solid Films 1988, 161, 325.
4. Conclusions
A new statement of the principle of correspondingstates has been developed for monolayers of long-chain amphiphiles in condensed phases. The deductions from this principle convincingly fit the literature phase diagrams for the Lz-LS transition of single-chainamphiphiles with three different head groups. They are consistent with the phase diagrams reported for the Lz'-S transition and with diffraction studies of fatty acid monolayers both on the water surface and on solid supports. In the condensed phases of long-chain amphiphiles, the molecules pack in an arrangement characteristic for the chains, so that even if two substances have different head groups and their monolayers are located in different environments, it is possible for their structures to be equivalent. The head group and environmentdo, however, significantly influence the arrangement adopted by the chains. The free energy associated with the head group packing and the head group-environment interaction can swing the balance in favor of one or other possible chain conformations. In a condensed phase of a monolayer, it is possible to talk unambiguously about the internal bulk pressure. The characteristic pressure scale for phase transitions is 100 MPa, and second-order phase transitions occur at characteristic values of this pressure. On this scale, two transitions between upright and tilted moleculeshave been experimentally observed to occur near zero internal pressure. The principle of corresponding states was originally enunciated in connection with the expanded phases of monolayers,which are extremely important in detergency and biology. However, the methods of the present study do not apply to them, since expanded phases do not give rise to diffraction peaks, and their major phase transitions are first order. The comparisonof members of homologous series is also inapplicableto that very important biological amphiphile, cholesterol. This does not mean that it is impossible to talk about equivalent states, but that the criteria for equivalence will be different. The picture of monolayer structure revealed here differs significantly from previously held ideas and could conceivablystimulate further progress in these areas. Acknowledgment. This work was partly financed by the Deutsche Forschungsgemeinschaft(SFB262)and the Bundesministerium far Forschung und Technologie (Ultrathin Polymer Film Project). We thank Professors C. M. Knobler, P. S. Pershan, and P. Dutta for preprints of their work, Drs. D. MBbius, V. Kaganer, and M. Seul for useful discussion,and Professor H. M6hwald for support and encouragement.
