Langmuir 1995,11, 334-340
334
On the Area Neutral Surface of Inverse Bicontinuous Cubic Phases of Lyotropic Liquid Crystals R. H. Templer Department of Chemistry, Imperial College, London SW7 2AY, England Received August 10, 1994. I n Final Form: November 4, 1994@ The location of the area neutral surface in lyotropic liquid crystalline membranes plays a critical role in the phase behavior of all the mesophases which possess curved interfaces. In this paper a mathematical formalism is developed which allows the experimental determination of the area neutral surface from X-ray structural measurements of the inverse bicontinuous cubic phases. The theory is used to determine the location of the area neutral surface in monoolein (Chung, H.; Caffrey, M. Biophys. J. 1994,66,377381) and didodecyl-j3-D-glucopyranosyl-ruc-glycerol (Turner,D. C.; Wang, Z.-G.; Gruner, S. M.; Mannock, D. A.; McElhaney, R. N. J.Phys. 11 Fr. 1992,2,2039-2063). The results indicate that the area neutral surface is located between the C1 and C3 positions on the hydrocarbon chains. However, the X-ray and gravimetric data currently available on these systems appears to be of too low a precision t o allow one to check for the presence of any net stretching. An error analysis of the theory is used to provide an outline prescriptionfor a new generation of experiments which will allow us to determine the presence of stretching in bicontinuous cubic phases.
Introduction Under certain circumstances there may exist a surface within an amphiphilic monolayer whose area does not change when the monolayer is bent. In the definition that will be used in this paper this surface is defined by the locus of points associated with the average location of some part of the amphiphilic molecule, whose crosssectional area does not change upon hydration. This means that in this very restrictive definition the average amount of matter either side of the area neutral surface must remain constant. The neutral surface has been defined in this restrictive way here for a number of reasons which we now go on to discuss. The area neutral surface and its location are of crucial importance in the modeling of the energy associated with the curvature of amphiphilic monolayer^.^-^ In particular, this paper will be concerned with its use in measurements of curvature elastic parameters in the inverse bicontinuous cubic phases of amphiphilic lipids in ~ a t e r . ' , ~ ?The ~~' curvature elastic models for these mesophases assume that the curvature elastic parameters remain constant while the interface is being bent. This should be true as long as the system has pure bending about the neutral surface, i.e., there has been no net stretching. In theory, it should be possible, using a relaxed definition of the neutral surface, to deal with the case in which there is net stretching, but this appears to be quite a complex problem and has yet to be done.8 In the preceding paragraph the curvature elastic model was rather generically defined for pure bending about the area neutral surface. However, such pure bending is @
Abstract published in Advance ACS Abstracts, December 15,
1994. (1)Chung, H.; Caffrey, M. Biophys. J. 1994,66, 377-381. (2) Turner, D. C.; Wang, Z.-G.; Gruner, S. M.; Mannock, D. A.; McElhaney, R. N. J . Phys. II Fr. 1992,2,2039-2063. (3) Gruner, S. M. J. Phys. Chem. 1989,93,7562-7570. (4) Rand, P. R.; Fuller, N.; Gruner, S. M.; Parsegian, A. V. Biochemistry 1990,29,76-87. (5) Szleifer, I.; Kramer, D.; Ben-Shaul, A.; Gelbart, W. M.; Safran, S . A.J . Chem. Phys. 1990,92,6800-6817. (6) Templer, R. H.; Seddon, J. M.; Warrender, N. A. Biophys. Chem. 1994,49,1-12. (7) Templer, R. H.; Turner, D. C.; Harper, P.; Seddon, J. M. J . Phys. II Fr., submitted for publication. (8) Kozlov, M. M.; Winterhalter, M. J . Phys. II Fr. 1991,I , 10771084.
0743-746319512411-0334$09.00/0
probably restricted in practice to experiments in which only the water composition is varied. This can be inferred from the observation that the phase boundaries for the inverse bicontinuous cubics at the excess water point are in general not vertical. This indicates that at least one of the curvature elastic parameters is temperature dependent, which means that the curvature elastic models cannot be used. This of course renders measurements of area neutral surface by variation in the temperature a somewhat pointless task since it is expected that the moduli and area neutral surface are in general ~ o u p l e d . ~ More significantly for the purpose of making neutral surface measurements, if the excesswater phase boundary gradient is not excessively steep it implies that the energetic cost of removing water from the system is rather small. This is taken to mean that the water expelled in raising the temperature is not strongly bound to the head groups, but simply fills the aqueous volume set by the system's equilibrium interfacial curvature. This implies that the local intermolecular interactions in the headgroup region are likely to be unaltered, at least to first order, through a limited range of dehydration from the excess water point. Of course, it is as yet unclear when this approximation breaks down; only by making measurements on a range of systems will we be able to devise more quantitative rules to enable us to identify which systems may be modeled by the methods described in this paper. As has already been stated, it is the purpose of this paper to develop a mathematical description of the position of the area neutral surface in an inverse bicontinuous cubic phase, which is consistent with the models of curvature elastic energy which have been reported previously. Specifically, this means that we will assume that both the area neutral surface and the lipiawater interface lie parallel to the underlying periodic minimal surface. As has been pointed out previously, one would expect this to be a close approximation to reality when the bilayer thickness is much smaller than the lattice ~ a r a m e t e r . ~ For bilayer thicknesses comparable to the lattice parameter the increasing inhomogeneity in the curvature per unit area of a parallel film would be expected to drive the interface toward one which has constant mean curvature. (9) Anderson, D. M.; Gruner, S. M.; Leibler, S. Proc. Natl. Acad. Sci. U S A . 1988,85,5364-5368.
0 1995 American Chemical Society
Langmuir, Vol. 11, No. 1, 1995 335
Area Neutral Surface of Lyotropic Liquid Crystals However, it will never be able to achieve constant mean curvature, because the lipid chain extension becomes inhomogeneous in this case. In other words, competition between inhomogeneity in the molecular shape and the molecular length means a compromise between a parallel interface and a constant mean curvature interface must be achieved. So far all curvature elastic modeling of the inverse bicontinuous cubic phases has used the parallel interface, both because it affords us relatively straightforward analytical solutions to the differential geometry and because one anticipates that in many cases the departures from parallelism are relatively small. The parallel interface model will be used to determine the variation in the monolayer thickness and the distance to the area neutral surface as a function of the unit cell dimensions of the cubic mesophase. The variation of monolayer thickness with lattice parameter can be measured experimentally using X-ray diffraction in conjunction with lipid volume fraction measurements and then fitted with this theory. This is done for published data on monooleinl and
[email protected] An examination ofthe experimental data and the theory leads one to the inevitable conclusion that the precision of structural data obtained by X-ray diffraction and volume fraction measurements is insuf%cient to determine whether any net stretching has actually occurred, i.e., the absence of the area neutral surface. A simple error analysis is made, and the precision of the next generation of measurements is estimated. Where the area neutral surface does not exist current models of the curvature elastic energy of the bicontinuous cubic phases do not hold. If this were the case the less restrictive definition of the neutral surface given by Kozlov and WinterhalteP would have to be used. This states that the neutral surface is that location where the bending and stretching terms in the total elastic energy of the system are uncoupled. Their model has been applied to inverse hexagonal phases, but still remains to be developed for the case of bicontinuous cubic phases.
apply the rules of differential geometry to determine the position of the neutral surface. We begin with expressions for the area of a parallel patch at a distance z from the minimal surface and the volume swept out in this projectionll
Theory
It is now possible to substitute for E in eq 2. After some rather tedious algebra one obtains
Current knowledge indicates that the inverse bicontinuous cubic phases consist of an amphiphilic bilayer draped onto an infinite periodic minimal surface.1° The minimal surface lies on the average location of the bilayer mid-plane. It is possible to model the geometrical characteristics of the bilayer by assuming that all molecularly defined surfaces, e.g., the polarhonpolar interface, lie parallel to the minimal surface. This assumption is used in all the current models of the curvature elastic energy, and it is therefore a t least consistent to use the same assumption when we attempt to determine the location of the neutral surface for use with these same models. We imagine the area neutral surface to be at a perpendicular distance E from the minimal surface, with cross-sectional area A(E) and enclosing a volume per molecule u ( 6 ) between the minimal and neutral surfaces. During bending will vary, but the quantity of matter between the minimal surface and the area neutral surface will not. If we assume that the molecular density does not vary, then u ( c ) should be constant. Of course, A(() is constant, and furthermore, the total volume per molecule, equivalent to u(l),where 1is the monolayer thickness, will also remain constant. Using these constraints we can ~~~
(10)Seddon. J. M.; Templer, R. H. Phil. Trans. Roy. SOC.London A 1993,344,377-401 and references therein.
+ u(z) = A(O)z(l + sKz2) 1 A(z) = A(O)(1 K z 2 )
(la)
(lb)
whereA(0)is the area of the patch on the minimal surface and K is the Gaussian curvature of the patch on the minimal surface (K is the product of the principal curvatures). The mean curvature (given by half the sum of the principal curvatures) does not appear in these expressions because it is identically zero at all points on a minimal surface. We can use eq l b to express the molecular volume at the neutral surface, z = 6, and at the polarhonpolar interface, z = 1, and then determine their ratio. However, we should note that because the Gaussian curvature varies in magnitude on the minimal surface we must calculate the surface averaged values ofA and u at these distances. We will denote the surface average of any quantity by enclosing it in carets (( )).
We now wish to substitute for in order to leave us with an equation which only contains the variables ( K ) and 1. This is done using eqs l a and l b withz = 6 and combining to eliminate A(O), which gives
(3)
(4)
+
where SZ = l(1 (“3). When the Gauss-Bonnet theorem is applied, the surface averaged Gaussian curvature can be determined experimentally from
(@=a ua2 where is the Euler characteristic of the minimal surface in the unit cell, uis the dimensionless area ofthe minimal surface in the unit cell, and a is the unit cell lattice parameter. Values of x and u have been calculated for a number of the minimal surfaces underlying the bicontinuous cubic mesophases.12 The monolayer thickness is found experimentally by measuring the lattice parameter of the cubic cell as a function of the water content and then solving (11)Hyde, S.T.J. Phys. Chem. 1989,93,1458-1464. (12)Anderson, D.M.;Wennerstrom, H.; Olsson, U.J. Phys. Chem. 1989,93,4243-4253.
Templer
336 Langmuir, Vol. 11, No. 1, 1995 Area neutral surface
17.5 Increasing lattlce parameter
17 16.5 Y
16
15
1
-@
100 150 200 250 300 350 400 450 500 alA
Increasing lattlce parameter
kea neutral surface
Figure 2. A simple, generic model for the average shape of an
CV(C)>
= 200 A3
-I J
" " " " " " ' ~ ' " " " " " " " " " ~ " ' ~ -
100 150 200 250 300. 350 400 450 500 alA
Figure 1. (Top) dependence of monolayer thickness as a function of lattice parameter for the P surface cubic (spacegroup Im3m). The calculations have been made for a neutral surface area of 30 Az and a total molecular volume of 500 A3 for the range ofneutral surface volumes shown. (Bottom)dependence of neutral surface distance as a function of lattice parameter for the same parameters.
where 41is the lipid volume fraction.2 In other words, by measuring 1 as a function of a we can use eq 4 to fit our data and thereby determine (A(5))and ( ~ ( 5 ) )This . will be done in the section which follows, but before passing on to this we will examine the behavior we might expect to observe in measuring the monolayer thickness as a function of lattice parameter. Using eq 4,we have calculated 1 = l(a)for a molecular volume of 500 A3,a neutral surface area of 30 A,2 and a volume to the neutral surface which runs from 200 to 400 A3in 50 A3steps. By using eq 3 5 has then been calculated as well. The resultinggraphs are shown in Figure 1.What is immediately evident is that 1 may either increase or decrease as the cubic lattice swells. A n increase occurs when the neutral surface lies toward the polar end of the molecule and a decrease when the neutral surface lies toward the nonpolar end of the molecule. This result can be arrived a t in a qualitative fashion if one considers the generic model of molecular shape in Figure 2. As the cone like molecular shape tends to the cylindrical, i.e., lattice swelling, the total molecular length must increase if the neutral surface is at the head group and decrease if it is at the position of the chain termini. Clearly a t positions in between the same behavior will be observed, with one neutral surface position somewhere along the molecule leading to no variation in thickness a t all. Since the model for molecular shape we have used does not constrain the molecular geometry within the bounds of a
amphiphilicmolecule in a monolayerwith interfacial curvature. (Top)with the neutral surface at the head group end a decrease in molecular splay leads to an increase in molecular length. (Bottom) with the neutral surface at the chain ends a decrease in molecular splay causes a decrease in molecular length.
Table 1. Structural and Gravimetric Data for Monoolein at 25 "C in the Gyroid Bicontinuous Cubic Phase' 91 UJA 1IA 0.7694 0.7400 0.7244 0.7242 0.7240 0.7235 0.7161 0.7122 0.7113 0.7110 0.7100 0.7040 0.6831 0.6829 0.6740 0.6654
125.35 131.56 133.93 132.79 135.16 135.43 135.39 136.06 137.36 135.47 137.08 139.73 142.77 143.36 146.61 147.59
17.43 17.40 17.24 17.09 17.39 17.41 17.18 17.15 17.29 17.04 17.215 17.37 17.11 17.18 17.29 17.14
mesophase topology, these statements are equally true for other phases, but the precise behavior will vary from case to case. By contrast to the monolayer thickness, the distance to the neutral surface always increases as the phase is hydrated. This is important, since the curvature elastic energy does not vary unless 5 does.7 However, the magnitude of the variation in 5 is coupled to the position of the neutral surface, with the largest variations being seen with the neutral surface lying toward the polar end of the molecule. Experimental Analyses
Let us first reanalyze the data on monoolein in water published by Chung and Caffrey.l Their data of the lattice parameter of the gyroid cubic phase (space group Za3d) in monoolein over a range of water weight fraction extending from 0.2306 to 0.3346 at 25 "C is reproduced in Table 1. Assumingthat the lipid density ~ using an atomic weight for monoolein of 356 is 1 g - ~ m -and gmol-l, 9~has been calculated and 1 determined using eq 6 and included in Table 1. In Figure 3 the surface averaged Gaussian curvature has been determined using eq 5 and plotted against S2 = 1(1 (K)Zz13). The plotted data has then been least squares
+
Langmuir, Vol. 11, No. 1, 1995 337
Area Neutral Surface of Lyotropic Liquid Crystals
Table 2. Structural and Gravimetric Data for Didodecyl-/3-D-glucopyranosyl-ruc-glycero12@ cwlT 0.115 0.16 0.20 0.224 0.253 0.275 0.29 0.312
45 "C 93.0115.7 97.7115.3
50 "C 96.0116.25 97.6115.3 125.6116.8 127.1116.3
127.3116.3 127.7116.0 136.8116.5
0.29 0.312 0.335 0.356
55 "C 97.0116.4 96.9115.2 107.3115.7 110.8115.5 125.9/16.8
82.7116.8 82.4116.1 83.7115.7 84.3115.2
83.5116.3 86.W16.1 86.9116.7
86.6116.2 89.U16.1
91.6116.5
60 "C 95.4116.2 96.7h5.1 107115.6 109.6115.4 122116.3 123.9115.9
65 "C
105.5115.4 108.0115.2 122.2116.3
70 "C 94.6114.8 106.1115.5 107.6115.1 120.6115.5
79.7116.2 81.1115.8 81.5115.3 81.9114.8
79.U15.5 80.1115.0
The values for the lattice parameter and monolayer thickness are separated by a space and tabulated as a function of temperature and water weight fraction, cw. The lipid volume fraction is determined via & = 1 - [ l ew/el(l/c, - l)]-l, where the lipid density as a function of temperature has been determined to be el = 1.0371 - 0.00079T, where T i s in degrees centigrak2 The data above the single line boundary are in the gyroid phase (space group Za3d) and below it are in the D cubic (space group Pn3m). a
+
Table 3. Neutral Surface Volume and Area for -7.5i o 4
-9.510.4 -1
10.3
0
45 50 55 60 65 70
1
1
15.5
Didodecyl-/3-D-glucopyranosyl-ruc-glycerol
j 0:
15.6
15.7
15.8
15.9
16
a/A Figure 3. Neutral surface theory fit to the data of Chung and Caffrey' for monoolein. fitted t o eq 4 using the Levenberg-Marquardt algorithm as implemented on Kaleidagraph v. 3.0.1, giving us (A(())l(u(l))= 0.055 f 0.002 A-l and (u(())/(u(l))= 0.76 f 0.09. The use of Chung and Cafiey's calculation for ( ~ ( 1 ) = ) 593 A3 results in (A(()) = 32.6 f 1.2 bzand (~((1)= 451 f 53 A3. In the fluid phase the volume of a CH3 group is approximately 54 A3 and of a CH2 group is approximately 27 A3.13 This places the neutral surface at or about the C3 position on the hydrocarbon chain. According to the calculationsof Szleifer and his colleagues5 this is the chain position which is predicted to exert the greatest outward pressure in the chain region and, hence, be the most incompressible part of the chain. It therefore appears quite reasonable to have the area neutral surface located a t or close to this position. The precise locationwill of course also be affected by the lateral pressure in the head group and the interfacial tension. For example, an extremely incompressible head group region would shift the neutral surface toward the head group, while a very compressible moiety here should move the area neutral surface back toward the most incompressible part ofthe chains. In their analysis of the neutral surface position Chung and Cafieyl did not use a requirement for constant molecular mass between the neutral and minimal surfaces. Instead they determined the molecular cross-sectional area as a function of distance from the minimal surface and ascertained the average distance a t which these curves crossed each other. In doing this it was assumed that the monolayer thickness was not varying with hydration, which is a reasonable assumption on the basis of the data, but by assuming that the distance to the minimal surface was also constant they have imposed a constraint that is inconsistent with the differential geometry of these phases, see Figure 1. Indeed, since this distance is held constant it implies that the amount of matter between this location and the minimal surface must be varying, which is contrary to the definition of the neutral surface which we require. (13)Reiss-Husson, F.; Luzzati, V. J.Phys. Chem. 1964,68, 35043511.
900 f 1 903 1 906 f 1 909 f 1 913 f 1 917 f 1
*
737 f 30 737 f 66 773 f 80 789 f 82 1050 f 280 748 f 70
50.9 f 1.2 50.8 f 2.5 50.1 f 3.4 50.4 f 3.5 38.8 f 19.9 54.6 f 2.8
Notwithstanding these difficulties let us compare the location ofthe neutral surface calculated by Chung and Caffrey with that found here. With a monolayer thickness 1 = 17.3A and ( = 8.8 A it is clear that from their calculations the neutral surface is much closer to the minimal surface. For example, according to the analysis used here, a t full hydration the neutral surface is located a t ( 12.6 A. A value of ( e 8.8 A locates the neutral surface somewhere near the middle of the hydrocarbon chain, Le., around CS. This is a rather surprising location for the neutral surface, since the lateral pressure in this location should be much less than that at C S . ~This means that in pivoting around C g the most incompressible regions of the chain, around C3, would in fact have to be compressed, a process which appears to be energetically highly unfavorable. We now turn to an analysis of the data accumulated by Turner and his colleagues2on didodecyl-/3-D-glucopyranosyl-ruc-glycerol, reproduced in Table 2. The data have been fitted, using the same method as before, for the isotherms 45,50,55,60,65, and 70 "C, Figure 4, and the resultant values for (A(()) and (~(5))are listed in Table 3. The lipid volume, (u(Z)), has been calculated from the measured lipid density and molecular weight. If we exclude the result for T = 65 "C, where the data is poor, it appears that within experimental error the locationofthe neutral surface is constant with respect to the temperature, with a weighted mean value of (u(())/(v(Z)) = 0.825 f 0.001. Using the same assumptions as before, this places the neutral surface approximately at the C1 position. Again this appears to be a reasonable location as far as the chain compressibility is concerned, since this location is close to the least compressible part of the chain group. The fact that the neutral surface is closer to the head group than monoolein may indicate that the glucopyranosylhead group is less compressible than that of the monoglyceride. The data appear to indicate that the neutral surface area increases with temperature. A weighted linear least-squares fit t o the data, excluding that at 65 "C, gives a coefficient of area expansion of 0.11 f 0.06 A ~ s K - ~ .
Comments on the Precision of Neutral Surface Measurements In the preceding section we h a v e fitted experimental data t o the neutral surface theory derived in this report and thus obtained values for the location of the neutral surface. However, even a cursory glance at t h e fitted data in Figures 3 a n d 4 indicates that the scatter on the measurements is too large t o enable us t o confirm or deny
338 Langmuir, Vol. 11, No. 1, 1995
Templer
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Figure 4. Neutral surface theory fit t o the data of Turner and his colleagues2for didodecyl-b-D-glucopyranosyl-rac-glycerol at the temperatures indicated on the graphs. the existence of the neutral surface. At present, our term, so that for the purposes of an error analysis it suffices determinations ofneutral surface locationand area depend to write 1 x aq$/2a. Noting that 41 = (1 + mweJmlel)-', to a large degree on an untested belief in the existence of where ml is the lipid mass, m , is the water mass, el is the the surface itself. lipid density, and e, is the water density, and after some In this section we address this issue and ask ourselves algebra, we obtain the following expression for AQ/Q what levels of experimental precision would be needed to observe stretching of the putative neutral surface. Let us first determine expressions for the fractional error in ( K ) and SZ and then determine the best precision we might reasonably require, in the light of the available experi. . . - . mental data. The expression for the fractional error in ( K ) , A(K)/(K), is
--
(€0
- 2/24 a
(7)
where Aula is the fractional error in the measurement of lattice parameter (henceforth A d x will be taken to be the fractional error in the quantityx). The determination of the fractional error in SZ is somewhat more involved since it includes a measurement ofl, given by the cubic equation in (6). We note, however, that the term which is cubic in 1is at least 1 order of magnitude smaller than the linear
This may be further simplified, since for reasonable values
Area Neutral Surface of Lyotropic Liquid Crystals
Langmuir, Vol. 11, No. 1, 1995 339 realistic assessment of the presence or otherwise of a neutral surface using current instrumentation. As has already been stated, it is believed that the existence of an area neutral surface is most likely to occur when the cubic phases are swollen, i.e., large lattice parameter and relatively small monolayer thickness. Under these conditions the term (mleJm41 U2 in (9) will tend to unity, and therefore, all error terms are equally significant. It therefore seems reasonable to suppose that we should require a precision of 1 part in lo4 in all our measurements. This precision is certainly feasible in measurements of density and mass; one simply requires an increase in the sample volumes that are used. As far as the author is aware most workers make X-ray measurements on gravimetrically prepared samples whose total volume is of the order of 10 pL. To achieve the required precision in making mass measurements would need approximately 100 mg of sample, which can be weighed t o a precision of 10 pg (this specification complies with a number of commercial balances). Densities may also be measured to the required precision of 1part in lo4with an oscillating U-tube density meter, but this requires sample volumes of 7 mL. Having such large sample volumes will of course also reduce systematic errors such as sample drying during handling, and the only drawback is one of sample cost. Making measurements of lattice parameter to 1 part in lo4 seems a more problematical undertaking. In most cases, and especially when lattice parameters are large, there are rather few Bragg peaks to be detected. This means that the error associated with measuring the lattice parameter is essentially the error in measuring an individual Bragg peak, which is given by
+
-1.610.~ 13.6 13.7 13.8 13.9
14
n/A
14.1 14.2 14.3
Figure 5. Neutral surface curves both having a molecular volume of 900 A3 and a neutral surface volume of 742.5 A3,but with neutral surface areas of 50.85 and 50.96A2. of a and 1 the numerator in the final term is of the order of 1000 times greater than the numerator on the penultimate term. Hence, we get
(9) We are now ready to proceed toward an estimate of the precision we will require in measuring a,ml, m,, el, and e,. According to Figure 1it is clear that we can expect t o have the most difficulty in discerning systematic variations in cross-sectional area at the supposed neutral surface when the lattice parameter is quite large. However, before we consider this situation we will use the data from didodecyl-P-D-glucopyranosyl-rac-glycerol to make estimates of the experimental precision required when the lattice parameter is small. In Figure 5 we have plotted a air of neutral surface curves using (4)with (VU))= 900 (u(E))l(u(l))= 0.825, and (A(@)= 50.85Azand 50.96Az.These values represent the expected changes in the neutral surface area in didodecyl-P-D-glucopyranosyl-rac-glycerol for a 1"Cjump in temperature from 45to 46 "C. To discriminate between these curves we would require A(K)I(K) x f 0.017 and AQIQ x f 7.1 x So if we wish to be able to detect systematic variations away from a calculated neutral surface which are equivalent to the change we might expect to observe for a 1"C step, we must measure Q and ( K )to at least the precisions quoted above. It is clear that the minimum precision with which the lattice parameter must be measured is set by the maximum allowable error in Q and not (K). For the case of didodecyl-/3-D-glucopyranosyl-rac-glycerol the denominator in the final term of (91,(mlewlmwel 112,varies from 80 to 8. This means that the fractional error in measuring the lattice parameter dominates the total error and must be less than f 7 x The error in each of the other measurements could only be of the order of 10times greater than this. A reasonable estimate of the fractional errors we would need to aim for are Aala = f 5 x Amdm1 = AmJm, = Aellel = AQJQ, xz f 7 x As far as the author is aware, most laboratories make measurements of lattice parameter to a precision which f3 x and the lipid and is no better than hala water masses and densities are usually measured to no better than 1part in lo3. By the reasoning set out here we should therefore not expect to be able to make any
H3,
+
&?= a
(10)
where 1 is the wavelength of the X-rays being used, x is the distance between the sample and plane in which the diffraction is recorded, and y is the real space diameter of a diffraction ring. In principle, we can make x as large as we like and thereby increase y such that the errors from these terms become negligible. However, if we are to do this we must use an X-ray beam which is either highly collimated or has a very large depth of field. Furthermore, the camera must be capable of providing an X-ray beam with a monochromaticity of A M These considerations immediately preclude the use of simple mirror systems, because of their poor monochromaticity, or curved crystal monochromators, because of their limited depth of field and inadequate monochromaticity. The choice then narrows down to the use of a double bounce monochromator, where one can obtain monochromaticities of the order MA = 2 x and good collimation (beam divergences of the order of 0.005O).
Concluding Remarks A mathematical model of the area neutral surface in bicontinuous cubic phases has been presented, which is consistent with the presence of a location on the amphiphiles, whose cross-sectional area is invariant during bending. In an analysis of data from two lipidwater systemsthe position of the neutral surface has been located close to the most incompressible section of the hydrocarbon chain, i.e., close to the top ofthe chain. However, the data from which these calculations have been made exhibit such a large degree of scatter that it is not possible to demonstrate unequivocally that there has been no net stretching during bending. In order to be able to demonstrate the absence of bilayer extension during bending,
Templer
340 Langmuir, Vol. 11, No. 1, 1995 far greater precision than is normally used is required in measurements of mass, density and lattice parameter. These conclusions suggest the need for a new generation of higher precision measurements. At this stage a number of points should be made. The analysis used here is model dependent, with the important stricture being that the neutral surface is parallel to the minimal surface. This is less likely to be true at low hydrations than a t high hydrations. Secondly, we have to use bulk lipid density in our calculations of the neutral surface location, which will always lead to a systematic error in its location. Finally, it should be noted that although high precision is a necessary prerequisite to detect stretching in the hydration of bicontinuous cubics it is not a sufficient condition. A number of systematic errors may well add noise to measurements of Q. Sample drying, inhomogeneity, and phase metastability are frequently encountered problems in these phases. A more intractable difficulty may also arise due to the presence of appreciable numbers of defect sites and grain boundaries. We know little about these
structures in inverse bicontinuous cubic phases, but we may anticipate that they might act as sinks for water and render our measurements extremely imprecise. We often observe higher than normal levels of incoherent scattering a t very low angles with these mesophases, which may well be indicative of a high level of such defects, but at this point we await direct experimental evidence before coming to any premature conclusions.
Acknowledgment. This work was supported by the SERC/EPSRC and the Royal Society, the latter in the form of a Research Fellowship for the author. The author wishes to thank Hesson Chung and David Turner for allowing him access to their experimental data and John Seddon for many useful discussions. In addition, the author would like to thank the reviewers for their questions and challenges, which resulted in a more rigorous treatment of the model in the introduction. LA9406261
