J. Phys. Chem. 1993,97, 9831-9843
9837
Characterization of Field-Ordered Aqueous Liquid Crystals by NMR Diffusion Measurements J. Chug and J. H. Prestegard’ Department of Chemistry, Yale University, New Haven, Connecticut 0651I Received: February 19, 1993; In Final Form: July 1, 1993’
Anisotropic water diffusion in three lyotropic liquid-crystal systems, cesium perfluorooctanoate (CsPFO)/D*O,
dimyristoylphosphatidylcholine(DMPC)/CHAPSO/DzO, and DMPC/Triton X-100/D20, has been measured using the N M R pulsed field gradient spin echo (PFGSE) technique. Field gradients were applied in directions parallel to the magnetic field and perpendicular to the field, in order to obtain very precise water diffusion coefficients along orthogonal directions. The results have been analyzed using a diffudon model in which water molecules move stochastically over fixed sites in a lattice. Micelles which make up the liquid crystal are included as forbidden diskoidal regions of appropriate size and orientation in this lattice. The mean square displacement of a water molecule moving through the interstitial space of the obstacles is calculated as a function of time, and diffusion coefficients are extracted. From comparison of the simulated and the experimental results, the presence of diskoidal micelles of specific diameter and orientation is deduced, for two of the systems. In the CsPFO/D20 system where disk normals are parallel to the field, this observation agrees closely with published results obtained by different methods. In the case of DMPC/CHAPSO, where disk normals are perpendicular to the field, a useful characterization of a new liquid-crystalline system results. For the DMPC/Triton system, no adequate simulations of results could be obtained, suggesting that a more complex description of water interaction with micellar surfaces may be required.
Introduction In recent years, many amphiphilic molecules have been identifiedwhich, upon mixing with water in concentratedsolutions, form lyotropic liquid crystals with net macroscopic alignment in the presence of a magnetic field.1.2 These are systems of some fundamental interest as well as useful spectroscopic media. Characterization of their organizational state is important. Among the most throughly studied of these liquid crystals is the cesium perfluorooctanoate (CsPFO)/D20 systems5 and similar systems containing fluorinated fatty acid salts with large counterionssuch as cesium or ammonium.68 These perfluorinated fatty acids have been shown to form finite-sized diskotic bilayers in nematic phases, with directors parallel to the magnetic field direction. Although there is still some controversy regarding the precise morphology of the lamellar phase?JO small-angle X-ray scattering, deuterium NMR, and electrical conductivity have been used to show that the fundamental mesogenic particle in the nematic phase is the discrete diskotic micelle.11 In addition, a pulsed field gradient spin echo (PFGSE) diffusion study has been carried out in the ammonium perfluorononanoate (APFN)/D20 mixture that shows results which are consistent with the conductivity measurements in such a system.12 Less well-characterizedsystemsbased on field-orientedmicelles in aqueous media include the dimyristoylphosphatidylcholine (DMPC)/detergent/water systems where detergents such as 3-[(3-cholamidopropyl)dimethylammonio]-2-hydroxy1-propanesulfonate (CHAPSO)13or octylphenylpoly(ethy1ene glycol ether) (Triton X-100)1”16 have been used. Micelles in these systems are believed to maintain a bilayer-like morphology, but with the bilayer normal tilted 90° relative to the magnetic field direction. In addition to their very interesting morphology and phase behavior, the DMPC/detergent/water systems have found utility as models for biological membrane systems in NMR studies of conformational properties of membrane-bound glycolipids. Of primary importance in the latter study is the fact that the oriented micelles seem to maintain many of the characteristic qualities of pure phospholipid bilayer membranes.17-lg To whom correspondenceshould be addressed. Abstract published in Advance ACS Abstracts, September 1, 1993.
In an effort to gain further insight into the actual physical state of the fragments in the bilayer-likeoriented phases, we have carried out PFGSE diffusion measurements of water in the DMPC/detergent/water systems. To validate our analyses, we have alsoreproduceddata on themore thoroughly studied CsPFO/ D2O system. The PFGSE technique has been used extensively for the measurement of diffusion for many years.20 In theseexperiments, the movement of a species of interest (water) is monitored by observing the effect of magnetic field gradients on the extent to which an NMR signal, dephased by the field gradient, can be refocussed at a later point in time. In essence, the magnetic field in which the spin precesses is made to depend on position. If a molecule remains fixed in space, the field it sees is constant in time, and its signal can be nearly completely refocussed by appropriate rf pulses. If the molecule diffuses, the field varies and refocusing is incomplete. Analysis of the decay of the refocused signal as a function of time and gradient strength allows extraction of diffusion coefficients. Despite numerous applications to diffusion in isotropic media, there have been relatively few studies of anisotropic or restricted diffusion by this method, partly due to technical limitation^.^^-^^ With the recent development of commercial high-resolution solution NMR probes which are capable of delivering moderately strong pulsed field gradients along the direction of the external magnetic field as well as in the two orthogonal directions (the so-called “triple-axis”gradient probesz5),it is now possible to do much more precise measurements of diffusion, especially in anisotropic systems. The new probes allow for better measurements because the sample does not need to be rotated 90° in space in order to obtain diffusion coefficientsalong two orthogonal directions. To aid in the analysis of data in terms of the geometry of fragments in the micellar phase, we introduce a stochastic jump model of a single water particle freely diffusing in the space between a periodic lattice of disk-shaped obstacles representing themicelles. The model reproducesmuch of the theoreticalresults of numerical integration of Laplace’s equations used in the past to model anisotropic electrical conductivity measured in similar systems.6.26 The stochastic model, however, offers considerable
0022-365419312097-9837%04.00/0 0 1993 American Chemical Society
9838 The Journal of Physical Chemistry, Vol. 97, No. 38, 19‘93 flexibility in allowing the introduction of discrete layers of bound or diffusionally restricted molecules of water around the micelles. From a simple model with a single restricted layer of bound water, we obtain an idea of the size and orientation of the diskotic micelles, both those of the well-studied CsPFO system and the newer DMPC/detergent systems.
Experimental Methods SamplePreparation. Cesium perfluorooctanoate was prepared from pentadecafluorooctanoic acid (Aldrich Chemical Co., Milwaukee, WI) by titration with cesium hydroxide to a pH of 7.0 as described in ref 5. The sample used for the diffusion measurements was prepared by mixing equal weights of CsPFO and deuterium oxide and subjecting the mixtures to a series of freeze-thaw, heating, voltexing, and centrifuging cycles until a homogeneous, viscous solution was obtained. The DMPC:CHAPSO (3:l w/w) system, 30 wt % total lipid in D20, was prepared as published elsewhere.” The DMPC: Triton X-100 (3:l w/w) system, 30 wt % total lipid in D20, was prepared by mixing 135 mg of dimyristoylphosphatidylcholine (Sigma Chemical Co., St. Louis, MO), 45 mg of Triton X-100 (Boehringer Mannheim, Indianapolis, IN), and 420 FL of deuterium oxide in a 5-mm NMR tube and heating, freezethawing, and centrifuging repeatedly as described above. Volume Fraction Measurements. To a thin long test tube (7mm o.d.), with precalibrated volume, a known weight of lipid and D 2 0 was added. From the volume measurement and assuming a density of 1.11 g/mL for D20, volume fractions of the micelles were calculated. The volume fractions for the CsPFO (50 wt %)/D20 system and the DMPC:Triton (30 wt %)/D20 system were determined to be 0.37 f 0.05 and 0.28 f 0.05, respectively. Pulsed Field Gradient Spin Echo Diffusion Measurements. Measurements employed a gradient enhancement accessory for a General Electric Omega PSG N M R spectrometer operating at 500 MHz for pr0tons.2~The probe supplied with this accessory is equipped with the triple-axis pulsed field gradients described elsewhere.25 The strength of the pulsed field gradients were calibrated28 based on diffusion of protons in a standard deuterium oxide sample with less than 1% hydrogen oxide (H2O) and traces of gadolinium chloride to shorten the T1 relaxation time. Assuming a water diffusion coefficient of 1.90 X cm2/s29at 25 ‘C, the three gradient channels produce 28.6,27.0, and 33.0 G/cmalong thex,y, andzdirections, respectively (thez direction is chosen to coincide with the direction of the external magnetic field, Ho).The volume of the sample had to be kept to about 350 FL or less (in a 5-mm N M R tube), in order to minimize unwanted signal intensity arising from diffusion in nonlinear regions of the H1 rf field and the gradients field. This phenomenon was a problem especially at long values of the spin echo delay between the 90’ and 180’ pulses and at higher temperatures where diffusion over longer distances can occur. The data were acquired using a pulsed field gradient sequence described by Stejskal and Tanner.30 Here, after a recovery time of greater than 10T1, transverse magnetization is created with a 90’ pulse. A gradient of strength G, direction x, y , or z, and duration of 6 is applied near the beginning of a magnetization dephasing period T. A 180’ refocusing pulse is then applied, followed by a second gradient pulse which begins a t a time A from the initiation of the first gradient pulse. Optimal refocusing occurs as a spin echo at time T from the 180’ pulse. At this time signal is acquired as a free induction decay and Fourier transformed for analysis. The diffusion coefficients along the three orthogonal directions were obtained from the relation30
where Z(6) and Z(0) are the NMR spin echo intensities of the water proton signal following a 9Oo-r-18O0 sequence with and
Chung and Prestegard without gradients, respectively. y is the gyromagnetic ratio of the proton; DX,,,, are the diffusion coefficients along the directions, x, y , or z. For all of the measurements reported here, the spin echo delay T was kept at 10.0 ms, A was set equal to r , and the length of the gradient pulses, 6, was varied from 0 to 8.0 ms in 1-ms intervals. Typically only a single transient was collected, and the signal was processed with 5-Hz line broadening. The anisometric micelles posing as obstacles in the diffusional pathway of the water molecules to be studied present an example of restricted or bounded diffusion, where the echo attenuation would, in general, need to be described by an expression more complicated than eq 1.31 However, we assume here that the water diffusion obeys Fick’s law over the distances involved during the time scale of the spin echo experiment, and thus the echo attenuation data should fit eq 1 quite nicely for fixed values of the spin echo delay, T,as will be shown below. To study restricted diffusion in the systems to be discussed here, T would have to approach the time scale required for water molecules to acquire mean square displacements on the order of the micellar diameter. This time would be on the order of 100 ps, considerably shorter than T values accessible with our apparatus. Deuterium NMR Spectroscopy. The deuterium quadrupole splitting reported for the cesium perfluorooctanoate/DzO sample was obtained from the continuous wave (cw) lock sweep signal of the deuterium nucleus of water in the sample a t 76.7 MHz on the same spectrometer. ComputerSimulations. The computer calculations were carried out on Silicon Graphics (Mountain View, CA) Iris Indigo R4000 computer, with all programs coded in the C language. Simulations on a three-dimensional model, with as many as 5 12 points in the x and y dimension and 256 points in the z dimension, taken to 100000 steps 500 times, required approximately 10 min of computer time.
Results CsPFO provides a reasonably well characterized, fieldorientable micellar system with which to test our measurement and analsis procedures. The system has been shown to exist in a diskotic nematic (N) phase in the intermediate temperature region between the isotropic (I) phase and the lamellar (L) phase.5 The difference between the N and the L phases was convincingly demonstrated by deuterium N M R line shapes. In the N phase there is spontaneous reorganization of the mesophase director in response tochanges in the orientation of the sample in the magnetic field. In the L phase, the lamellar director is “locked” into the mesophase so that no significant reorientation occurs when the sample is rotated. The exact range of temperatures where the I to N and N to L transitions occur ( T ~ and N TNL, respectively) is concentration dependent. To ascertain which phases we are working with, deuterium quadrupole splitting of water in the phase is monitored as a function of temperature. A pronounced change in the slope of the splitting vs temperature curve is known to mark the conversion from the N to the L phase.5 In Figure 1, we show the deuterium quadrupole splitting (VQ) of D2O in our CsPFO/D20 sample, measured as a function of temperature. The results are consistent with similar plots previously shown for CsPFO (55 wt %)/D20.5~8.~1 Above 55 OC, TIN,there is an abrupt loss of quadrupole splitting, characteristic of a transition to an isotropic phase. Below 47 OC, TNL,the splitting vs temperature slope is relatively small and the system is believed to be in a lamellar phase. Between 55 and 47 ‘C, the slope is larger and the system is believed to be a nematic phase composed of diskoids. Diffusion measurements were made at several temperatures throughout the lamellar and nematic diskotic regions. At each point a t least 20 min of equilibration were allowed before each set of measurements. A typical signal decay curve is shown in Figure 2 for 38 “C. Gradients were first applied in the zdirection
Field-Ordered Aqueous Liquid Crystals
700
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, 35.0
The Journal of Physical Chemistry, Vol. 97, No. 38, 1993 9839
35.0 -
I
40.0
I
.
.
45.0
50.0
Temperature
(C)
40.0
45.0
50.0
55.0
I
Temperature
55.0
(C)
Figure 1. Plot of the quadrupole splitting of DzO in CsPFO(50 wt %)/
DzO as a function of temperature ("C). The filled diamonds and the open squares represent the lamellar and the nematic phases, respectively. The units for the quadruple splitting are arbitrary since the signal is observed through the lock channel of the spectrometer. The quadrupole splitting was measured while decreasingthe temperature about 1 OC/5 min.
n
n
B \
0.0 35.0
n
40.0
45.0
Temperature
0
100000
200000
-
300000
400000
y 2 6 2 62 (A 6 /3) (sedcm2) Figure 2. Typical plot of the decay of the water proton signal intensity as a function of gradient amplitude in CsPFO(5Owt %)/D20,for a fixed value of spin echo delay 7 . The upper and the lower lines are the decay curves for when the gradients perpendicular and parallel to the field direction are on, respectively. The diffusion coefficients(in units of cmz/ s) areread directly from theslopesofthe straight lines, whichareobtained by fitting to eq 1.
(diamonds) and then the y direction (squares). The diffusion coefficients are directly proportional to the slope and can be extracted by fitting to eq 1. As we can see from the result at 38 OC, there is a very large difference in the slopes when diffusion is along the direction of the field (parallel to the director), Dll, and when the diffusion is along the direction perpendicular to it, DL. In Figure 3a, we summarize all of the diffusion measurements as a function of temperature. Diffusion is highly anisotropic at all temperatures; it is the greatest at low temperatures where the ratio of diffusion constants for movement parallel and perpendicular to the field is almost a factor of 2.5. This is easiest to see in Figure 3b, where D I and Dll, normalized to the water diffusion constant, D H ~ oat , each temperature, is shown along with the experimental anisotropy, a, defined as 2(D11- Dl)/(Dll + D l ) . Previous NMR work had shown little variation in anisotropy with temperatures.l2 However, conductivity work had shown a pronounced variation with temperature and at low temperatures agrees in magnitude with our result." The better agreement
50.0
55.0
(C)
Figure 3. (A)Summary of D, and D,, the water diffusion coefficient in the two different directions, as a function of temperature for the CePFO (50 wt %)/DzO system. The open squares and the filled diamonds represent D, and D,, respectively. (B) Normalized D, and D, values, obtained by dividingthe values from A by h 2 oat each temperature. The filled squares represent the negative of the anisotropy, 2(Dz - D,)/(D, D,), defined in the text.
+
with conductivity may well result from the fact that our NMR PFGSE technique, like the conductivity measurement, does not require sample rotation to measure D , and 01. Previous NMR work did require rotation. The increased differences between conductivityin the parallel and perpendicular directions as temperature is lowered has been interpreted" as arising from the diameter of the disks increasing at lower temperatures. Qualitatively, diffusion or conduction normal to the disk surface is restricted compared to diffusion or conduction parallel to the disk surface, because the probability of encountering an obstacle is greater. As the disk increases in diameter, the probability of restrictive encounter in directions normal to the disk surface increases further, while the probability of encounter in directions parallel to the surface remains fixed. We aim to show below that data such as that in Figure 3 can be used to construct a more detailed model of the shape and size of the micelles in the nematic and lamellar phases when these probabilistic arguments are formalized. Before we describethe calculations, however, we show in Figure 4a the results of PFGSE diffusion measurement on DMPC:Triton X-lOO(3:l w/w)inDzO(30wt%totallipid)at35OC. Themost obvious difference from the CsPFO/water system is that the Dll is much larger than D L . The anisotropy a has consequently changed signs and is positive. In Figure 4b similar data are shown for DMPC:CHAPSO (3:l w/w) in DzO (30 wt % total lipid) at 40 OC. We see that a is much smaller than that for the Triton system but is also
9840 The Journal of Physical Chemistry, Vol. 97, No. 38, 1993
0
100000
y 2 G2 62 (A
0
200000
- 6 /3)
100000
300000
(sec/cm2)
200000
300000
-
y 2 62 62 (A 6 /3) (sec/cm2) Figure 4. Plot of the decay of the water proton signal intensity as a function of gradient amplitudefor (A) DMPC/Triton X-l00/D20 at 35 OC and (B) DMPC/CHAPSO/D20 at 40 OC. In both A and B,the open squares represent Dz(diffusion parallel to the field axis), and the filled squares and diamonds represent D, and D,. positive. The diffusion constants along the x, y , and z directions (in the lab frame), D,, D,, or D,, or the Dl and Dll (relative to the presumed director axis) are summarized in Table I, along with the anisotropy a for all three systems. For the CsPFO/D20 system, Dx, D,,, and Dl are used interchangeably, as are D,and Dll. For the other two systems, Dz= D l and D,, Dy = Dll.
Discussion It is clear from the results presented above that substantial anisotropies exist for diffusion of water in all three of the liquid crystals studied. Within the set, however, the most noticeable variation is that anisotropies are negative for the CsPFO/D20 system but positive for the DMPC/CHAPSO and DMPC/Triton systems. If we assume all three are diskoidal, this is indicative of a change in direction of preferred orientation for the diskoids. The change in preferential orientation is consistent with the expected diamagnetic anisotropy, Ax = XI] - xI, for these systems.18J9 Systems will tend to minimize the energy of interaction with a magnetic field. For diamagnetic substances, x is negative, implying that a field opposing dipole moment is induced. The energeticallyfavorableorientation is therefore that with thesmallest absolutexvalue parallel to the field. For DMPC, the smallest x is for a direction approximately perpendicular to the acyl chains of the bilayer. This would produce bilayer based diskoids oriented with the field in the plane of the bilayer. For perfluorooctanoatesthe smallest magnitude of x is in a direction parallel to the field, and bilayer-based diskoids would orient with
Chung and Prestegard normals parallel to the field. Beyond this qualitativeobservation, little can be deduced without a more quantitative analysis of diffusion and diffusion anisotropies. There have been a number of approaches to modeling diffusion in liquid-crystalenvironments, some of these quite sophisticated.32 In our case, however, we are primarily interested in geometry, as opposed to exact modeling of the rates of diffusion. A stochastic model in which water moves as an independent particle among discrete lattice sites would therefore seem adequate. Moreover, due to the large dimensions of the micelles we are studying (for example, DMPC/bile salt micelles have been reported to have diameters as large as 500 A”), an actual representation of the micellar lattice coupled with a sophisticated calculation such as a full molecular dynamics type of simulationwould be impractical. We represent the diffusing water as a single point randomly moving in a three-dimensional grid. The points of this grid are either occupied or empty depending on the shape, size, position, and the volume fraction of the micelles. For our most simplified model, the interaction of the water particle with the micelle obstacles is represented by a hard-sphere potential so that the water particlecan approach within one unit of, but cannot occupy, the spaces which are occupied by the micelle. We decide on the basis of the size of a water molecule that a single water jump step, represented by one unit on the grid, is 3 A; thus a cubic box of 5 12 points on each side represents over 1500 A in each direction, which is more than large enough to contain a suitable unit cell of close-packed micelles of diameter greater than 500 A. In the three-dimensional grid, we generate a unit cell of hexagonally close packed (ABCABC type) disk-shaped obstacles of desired radius, thickness, and volume density by representing occupied space by a l’s, and unoccupied space where the particle is free to diffuse, by the 0’s. In our calculationswe used the experimental volume fractions to fix the number of free points versus occupied points. To model CsPFO, we fix the thickness of a diskoid at approximately 22 A. This estimate is obtained from ref 11, but it is also consistent with the known bond distances for a fully extended length of opposed octanoyl chains. We also keep the distance of closest approach between the micelles within a plane at approximately 20 A. This leaves the radius of a disk as a primary variable. A system with 7 5 4 disks is represented in Figure 5 . Each A, B, and C layer is repeated seven times to reach the 2 2 4 thickness, and in this case each set of seven is separated by three layers of unoccupied space to reach the 38% occupancy indicated by the volume fraction measurements. Periodic boundary conditions were used to allow simulation of longer distances of diffusion in the matrix. The direction of the random move for the particle is determined by a random number generator,34assigningone-sixth of the range to each of the fx, f y , or fz directions. The period of the random number generator was determined to be greater than 108, and deviations from equal distribution among the six segments after a million steps was determined to be less than 0.005% To avoid correlations between any two directions the plus and minus directionalities were alternated with every other step. The diffusion coefficient along any one of the three directions is determined from the equation
where the mean square displacement is calculated from the trajectory as ( [ X ( N )- X(O)I2).32 The brackets represent the averaging over many starting points in one or more trajectories, and X ( N ) is the displacement at N number of steps. The slope of the mean-square displacement as a function of the number of steps at large numbers of steps is constant and gives the diffusion coefficient in units of (lattice distance*/step). In our case it was more than sufficient to go to an N of 100 000 steps in each trajectory and average over 500-1OOO starting points in one or more trajectories. The relative diffusion constants calculated in
The Journal of Physical Chemistry, Vol. 97, No. 38, 1993 9841
Field-Ordered Aqueous Liquid Crystals
TABLE I: Experimental Anisotropic Water Diffusion Coefficients
0
temp
Dx,(XIOs)b
DXYloHaO
38 45 48 54 55
2.07 2.27 2.30 2.40 2.38
0.855 0.816 0.797 0.762 0.741
35
1.09
0.470
40
1.90
0.748
Dz(X DzIoH20 CoPFO/D20 0.86 0.356 1.11 0.401 1.28 0.444 0.527 1.66 1.70 0.530 DMPC/Triton X-l00/D20 1.64 0.708 DMF’C/CHAPSO/D2O 2:10 0.825
a
S
a/s
-0.826 -0.686 -0,570 -0.364 -0.333
0.8
-0.712
0.403
1
0.403
0.10
0.5
0.20
In OC. b In cm2/s.
Figure 5. Three cross sections of a typical unit cell of hexagonally closepacked micelles (representedby the 1’s). The 0’s represent the free space which the water particle may occupy. Shown are the A, E, and Cplanes from micelles of radius 13 units, length 7 units, volume fraction of 38% in a unit cell of size 64 X 55 X 30 units.
this way are obviously dependent on the obstacles presented to diffusion in the various directions by the array of micelles in the model. For diskoidal obstacleswhen radius = thickness, isotropic
diffusion is approached. As the radius of a disk increases,diffusion in the direction perpendicular to the disk surface becomes increasingly impeded. Larger disks, thus, should produce qualitative agreement with experiment. The sign of the anisotropy for CsPFO is consistent with diskoids with normal along the field direction (z). However, we found that we were unable to obtain quantitatively the degree of anisotropyseen in the CsPFO/D20 system at 38 or 48 OC without changing the thickness or volume fractions as described above, or using unreasonable disk radii. For example, using a disk thickness of 7 units and a volume fraction of 37% (as determined experimentally), the radius of the disk had to be larger than 70 units in order to get a theoretical anisotropy of 4.46,whereas the experimental value of a is 4.57 at 48 OC. The diameter of the micelles at the lamellar to nematic transition (-47 “C) is reported to be about 64 A, or 21 units.’’ In fact, even at much larger values of radii the calculated anisotropy goes to an asymptotic value of only 4.48. Thus we modified our model from the mere hard-sphere type to one which allows some water particle and micelle interaction in the first water layer. First, we simply increased the viscosity near the micelle surface by doubling the residence time whenever the water molecule was within one unit of the micelle surface. No significant improvement could be obtained. Second, we allowed the particle to diffuse along the surface of the particle once it is “bound” to the surface but restricted movementaway from the surface. This is analogousto recognizing rapid movement of lipids and attached water within the micelle itself, or rapid exchange of water among adjacent lipid binding sites. The model is implemented by keeping the position of the particle fixed for one step count when the random number generated calls for movement along the normal to the surface (in either positive or negative directions) but allowingfree movement tangential to the surface when the random number calls for the other four directions. By implementing this “fast surface diffusion” model, we are able to obtain a theoretical anisotropy which can reproduce the experimental values. Weshow in Figure 6a typical plot of mean-squaredisplacement vs the number of steps for the three directions. The anisotropy is nearly 4.7,a value clearly large enough to span observed anisotropies. We note that the fraction of water bound to the surface (the number of times it is “stuckn for a given number of steps) varies depending on the volume density of the obstacles; the fraction bound can be fairly substantial, ranging from 5 to 15%. This value is quite consistent with water diffusion measurement for the determination of hydration and shape of protein molecules in solution.3s Thus the model that successfully reproduces the experimental anisotropiesdepends on a nontrivial interaction between the water and the micellar surface. Werealii that we have an additionalexperimentallyobtainable quantity that one can use to help fit to the theory. That is the ratio of the absolute values of the diffusion coefficients D L and Di or, to make modeling easier, the ratio of these coefficients to the pure water isotropicdiffusioncoefficient,&,o. We can obtain
Chung and Prestegard
9842 The Journal of Physical Chemistry, Vol. 97, No. 38, 1993 15
v -
0
20
40
60
00
100
Number of Steps (xl o-3) Figure 6. Typical mean-square displacement (MSD)along x, y , and z directions, plotted as a function of the number of random steps taken. This particular plot is obtained with a micelle radius of 15 units, length of 7 units in a 74 X 64 X 30 unit cell at 38% volume fraction by averaging 500 trajectories each taken to 100 OOO steps, for a total of 5 X lo7random move attempts, rquiring just 10 min of CPU time. The upper two lines represent Dxy (diffusion perpendicularto the director) and the lower line represents 0,.
-
0.8
0.6 0
-
0.4
0.2
3
0.0
of ref 26 in that the rate of decrease of DI as a function of diameter is not much larger than the corresponding increase in D I . The experimental data in Figure 3b, however, agree well with our simulations. Also shown in Figure 7 is the previously defined anisotropy as a function of the micellar diameter; this portion of the plot is identical to Figures 5 and 9 of ref 26. Our Figure 7 can be used to extract the diameter of the CsPFO micelles by comparing to the experimental plots of Figure 3b. A straightforwardapplication to our data at 48 OC would predict a value of 26 units, or a diameter of 78 A. This is in quite good agreement with the estimate from conductivity of 64 A.11 Measured anisotropies, however, should not be applied directly to the plot in Figure 7 because in a real sample, disks are not static but fluctuate in orientation about the director. These fluctations can be measured in terms of an order parameter, S, taken from measurements such as quadrupolar splitting of deuterium sites in the lipid molecules themselves. If we use a value of S 0.8 at 46 OC ( TNL),from Figure 6 of ref 11 and divide CY by 0.8, we obtain a value of the static anisotropy CY of -0.712 at 48 OC. Comparing the static CY to the themetical anisotropyof Figure7, weseethat thediameter at this temperature is approximately 35 units (correspondingto a diameter of 105 A). Thus, our results give diameters which are about 1.6 times the value obtained by scattering and conductivitymeasurements inref 11;ourvaluesfor thediameter/thickness ratioare, however, somewhat smaller than that obtained from calculation based on numerical integration of Laplace’s equation.6 Having refined our model and placed some limits on its accuracy, we now would like to apply the same model to the two DMPC/detergent/DzO systems. The DMPC/CHAPSO/D,O system is believed to be in the nematic phase at the temperature studied, 40 OC, with diskoids oriented perpendicular to the field. The structure of the DMPC/Triton/DzO is less well-defined, but given the sign of the anisotropy, the disks, if they exist, must have a similar orientation. For disks oriented with a director perpendicular to the field, there should be additional averaging that was not in our original model. Microdomains should exist with random orientation of directors in the x-y plane since the magnetic field would be parallel to the disk surface for any of these orientations. These domains are likely to be ordered at distances much larger than micellar dimensions in both x and y directions, but the equality of experimental diffusion constants in x and y directions implies that domains are small compared to our sampledimensions and randomly oriented in thex-y plane. In such a case, it is sufficient to model the anisotropic diffusion by taking the results obtained in Figure 7 and simply interchanging Dxyand Dz, except that the experimentalDxyshould be compared to the average of Dxy and D, in plots such as those of Figure 7. The measured anisotropy of the DMPC/CHAPSO system at 40 OC is 0.10. Using a bilayer thicknessof 5 1 A and volume fraction of 2895, new plots analogous to Figure 7 were constructed and a diameter less than 100 A estimated. Correcting for a micellar order parameter of 0.5’’ (and also correcting for the effects of perpendicular orientation), we predict a diameter of 207 A. In this case the diameter predicted is quite consistent with the 150500 A estimated previously for lecithin/bile salt micelles.” The results are, however, expected to be quite sensitive to the micellar concentration as well as the mole fraction of detergent. The DMPC/Triton/D20 system poses bigger problems, since the thickness and even the structure of the micelle is less certain. Moreover, from Table I, the value of a! of 0.403 and DZ/&,o of 0.7 1 could not be reproduced using a volume fraction of 28% and thicknesses of up to 88 A. A value of theoretical anisotropy of 0.403 can be obtained, but in analogy to Figure 7, it is coupled to a value of D,(Dll)/qi,o which is much larger than 0.71. Alternative shapes and packing geometry of the micelles can be imagined. Especially for the case of D, > Dxy, cylindrically shaped Ne+phases (using the nomenclature of refs 1 and 2) might
-
4 20
30
40
50
10
Diameter
(A)
Figure 7. Plot of the two normalized theoretical diffusion coefficients, Dx,/& (open squares),D,/Do (diamonds)and negative of the theoretical anisotropy,2(D, - Dxy)/(D,+ Ow) (filled squares) as a function of the micellar diameter,for a fixed thicknessof 7 units and fixed volumefraction of 38%.
a theoretical value of &,o, which we call DO, for a given number of diffusive jump steps by calculating the mean-square displacement without any obstacles. We show in Figure 7 the plot of theoretical values of Dx (=Dy = D I ) and Dz (=Dll), normalized by DO,as a function of the diameter of the micelle for a fixed thicknessof 7 units and volume fraction of 38% for a hexagonally close packed array. In this case the distance of closest approach between these layers was also fixed at 3 units, and the dimension of the unit cell along the x and y directions was varied along with the increase in the radius to keep the volume fraction constant. The result has some fluctuations due to volume fraction variation associated with digitizing finite radii circles on Cartesian planes, but the plot is similar to the previous theoretical results obtained by numerical integrations of Laplace’s equations, in Figures 4 and 8 of ref 26. It is also noteworthy that the variation of the anisotropy as a function of the distance of closest approach within a layer and between the layers was very small, as long as the ratio of these twodistances was kept near one. The important trend is that D I increases with increasing diameter, and the Dll decreases with increasing diameter. We notice that our result differs from that
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Field-Ordered Aqueous Liquid Crystals be expected to yield values of a and Dxys which can reproduce the experiments for reasonable radius to length ratios. However our preliminary investigationsfailed to reproduce DMPC/Triton/ DzOdata with cylinders and with infinite lamellar sheets with holes in them.6 There are some possible reasons for the discrepancy here. One is that the lower volume fractions of these disks (if they are disks) may be in error. The Triton hydrophilic headgroup is much more diffuse than that of CHAPS0 and may interact with a much larger layer ofwater than thevolume fraction would suggest. It may be that a more elaborate surface interaction representation is needed to model these systems.
Conclusions We have shown that precise measurementsof anisotropicwater diffusioncoefficientscan be obtained for the CsPFO/D20 system, the DMPC/CHAPSO/D20 system, and the DMPC/Triton/ DzO systemusingtheNMRPFGSE technique. Thereis potential utility in the use of water diffusion measurements in anisotropic systems such as these, since it can be performed on any aqueous lyotropic systems regardless of whether they contain electrolytes or not. Analysis of our data on the basis of a rather simple stochasticjump model for diffusion of water through a lattice of obstacles allows an estimate of particle geometry for at least some types of micelles. The version of our model which is the most successful in representing known micellar diameters is one where significant association of the diffusing water particle and the surface of the micelles is represented. Acknowledgment. We thank J. R. Tolman for his assistance discussed in in coding the computer program for the this work. This work was supported by a grant from the National Institutes of Health, GM33225, and benefitted from instrumentation provided by the National Science Foundation, DIR 9015967.
References and Notes (1) Boidart, M.; Hochapfel, A.; Laurent, 1988,154, 61.
M.Mol. Cryst. t i q . Crysf.
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