J. Phys. Chem. 1994, 98, 11846- 11857
11846
Fluid Structure in Supercritical Xenon by Nuclear Magnetic Resonance Spectroscopy and Small Angle X-ray Scattering D. M. Pfund, T. S. Zemanian, J. C. Linehan, J. L. Fulton, and C. R. Yonker* Chemical Sciences Department, Pacific Northwest Laboratory,+Richland, Washington 99352 Received: April 6, 1994; In Final Form: August 14, 1994@
Chemical shifts for 129Xeat 25 “C are presented for densities between 5 x lop3and 2.2 x moYcm3, corresponding to pressures between 50 and 1000 bar. Also presented are small angle X-ray scattering results for xenon at 28 and 45 “C and pressures between 30 and 400 bar. The scattering results reveal the expected increase in aggregation and correlation length near the critical density. The chemical shifts exhibit deviations from the second-order virial equation over a broad range of densities. These deviations are also evident in calculations from integral equation theory. A departure from the cage model of solvent nearest neighbors due to repulsive solvation effects is discussed as a possible source of these deviations.
Introduction With the continuing interest in supercritical fluids as solvents in organic reactions, separations processes, and hazardous waste destruction, the fundamental understanding of fluid solution structure and chemistry is crucial for the continued advancement of this technology.’ The current level of understanding of fluids ranges from the characterization of their bulk properties to a rudimentary description of structures in solution on an atomic or molecular level. The structure of supercritical fluids has been studied using theory, molecular simulation, and experimental investigations. Each has contributed to the understanding of solution structure and solvation dynamics in supercritical fluids. Initial experiments studying supercritical fluids as solvents have focused on the understanding of solvation dynamics and cluster formation in these s y ~ t e m s . ~Infrared, -~ Raman, fluorescence, and UV spectroscopic studies helped quantify solvent strength, cluster formation, and solute solubility.6-’0 Other techniques, such as NMR, have been used to characterize molecular dynamics in supercritical fluids.’ Integral equation theories for the radial distribution functions were used in early studies of the microscopic structure of fluids. These theories have been coupled with small angle X-ray (SAXS) or neutron ( S A N S ) scattering to correlate the predicted distribution functions for the mixture with those obtained experimentally from the scattering data. Studies have focused on the investigation of inert gases and their solution structure at various densities and pressures.’* In this work we describe the application of SAXS and NMR spectroscopy to the study of the microscopic structure of supercritical xenon. The combination of these two techniques was used to examine two regions of solvation in supercritical Xe. NMR was used to probe the first solvation shell, which has the greatest influence on the chemical shift of the xenon nucleus. Solvent effects on the shielding constant of the nuclei in the fluid are mainly dependent on the bulk magnetic susceptibility of xenon and the van der Waals interactions between the xenon “solute” atom and its neighboring xenon “solvent” atoms. A new method of pressure containment was used which allowed, for the first time, measurement of the chemical shift over the full range of densities from gas-like to liquid-like. SAXS results for Xe were used to investigate the
* Corresponding author. @
Operated by the Battelle Memorial Institute. Abstract published in Advance ACS Abstracts, October 15, 1994.
0022-365419412098-11846$04.50/0
long-range density fluctuations in the fluid which make the major contribution to the xenon cluster ~ i z e . ~ ~Use . ’ ~of a modern synchrotron-based SAXS beamline allowed the first collection of scattering data for xenon at very small angles, very close to the critical point. Both SAXS and NMR data were obtained near the critical point for xenon to study the region of cluster formation. The NMR and SAXS results obtained were modeled using radial distribution functions estimated using the integral equation method. This is the first time that NMR and SAXS data have been modeled in this way; only within the past decade have theories existed which predict pressures, compressibilities, and radial distribution functions with sufficient accuracy. The scattering results revealed the expected increase in aggregation and correlation length near the critical density. The NMR chemical shifts exhibited deviations from the second-order virial equation over a broad range of densities. These effects were also evident in calculations from integral equation theory, implying that they were not an experimental artifact. Similar nonlinearities as a function of density have been observed in other properties of probe molecules in supercritical fluids. However, in this case the departure of the chemical shift from linearity (as measured by extrapolating the shift at low pressure) was maximized not near the critical density but at a much higher density. Therefore, the observed nonlinearity was not a critical clustering phenomenon. In addition, the data imply a reduction, rather than an enhancement, of short-range xenon-xenon correlations. A departure from the cage model of nearest neighbors in the solvent due to “repulsive” solvation effects is considered as a possible source of the nonlinearity in the chemical shift versus density.
Experimental Section NMR Experiments. We have developed a simple, safe, and inexpensive method of obtaining NMR spectra of fluids at high pressures and have employed this technique to investigate the state dependence of the chemical shift for xenon. Most of the existing NMR data on compressed gases have been taken in pressure vessels of beryllium, copper, or sapphire15because of their nonmagnetic nature and mechanical strength. A few studies have described the use of Pyrex capillary vessels holding liquids at pressures up to 4 kbar.16 Our cell consists of a fused silica capillary tube bent repeatedly to permit multiple passes (up to 40) in an existing NMR probe. The spectrometer used 0 1994 American Chemical Society
Fluid Structure in Supercritical Xenon
J. Phys. Chem., Vol. 98, No. 46, 1994 11847
Pressure Transducer
Pressure Readout
Hand Operated Syringe Pump View Cell with Stir Bar
J
1
i 11-1
Fused Silica Capillary Cell
Figure 2. Equipment used in high-pressure NMR experiments.
14-5
+-I."
was a Varian VXR 300 pulsed NMR spectrometer with a 7.04-T superconducting magnet operated at 82.6 MHz to probe for 129Xe. The cells used in this study were constructed from fused silica tubing (Polymicro Technologies) having an outer diameter of 360 pm and an inner diameter of 75-200 pm. As shown in Figure 1, the tubing was bent back and forth so the resultant cell would fit into a standard NMR tube. The capillary tubing, as received, was coated with polyimide to strengthen it. Where the coating was removed during bending, it was replaced with cyanoacrylate. The open ends of the capillary cell were glued into Vespel ferrules (Alltech Associates) using cyanoacrylate and were connected to high-pressure pumps via standard stainless steel fittings. The internal volume of a cell was approximately 7 pL. These cells had an approximate failure rate of 30% at 4 kbar, while the previously reported Pyrex cells had an 80% failure rate at the same pressure. Because of the small internal volume of the capillary, no hazard results from leaks or breaks, and replacement of the cell is simple and inexpensive. The cells are designed to fit easily into 5-mm Wilmad NMR tubes. This allows use of the capillary cell in the NMR probe without modification of the probe. The volume inside the NMR tube and exterior to the capillary cell may be filled with deuterated liquid (in this case, with D20) to lock the NMR signal. This technique has the added advantage of putting the "external-internal" standard in the NMR field without exposing the standard to high pressure or introducing solvent shifts caused by interactions of the standard with the analyte. Single lengths of fused silica capillary, sealed at one end, may also be inserted in the 5-mm tube along with the cell. A single length of tubing
containing ethylene glycol was inserted for temperature calibration and measurement. Reference compounds may likewise be introduced into the 5-mm tube, although in this work the chemical shift was measured relative to the frequency of the excitation pulse. The high-pressure apparatus is shown in a schematic diagram in Figure 2. Pressure generation was accomplished using a hand-operated syringe pump (HIP) rated for a maximum pressure of 15 000 psig. High-pressure valves (HIP) allowed the syringe pump to be filled and the cell to be isolated from the remainder of the equipment to allow cell replacement. The temperature was kept constant to within f O . l "C using the air bath controller of the spectrometer. The deuterated liquid in which the capillary cell was immersed provided an additional measure of temperature stability. The temperature of the air stream surrounding the probe was measured using the readout from the probe temperature controller. This readout was calibrated by acquisition of proton NMR spectra from a sample of ethylene glycol spiked with a trace amount of HCl. The glycol solution was held in a single length of capillary tubing that was sealed at one end and placed in the probe with the cell. The temperature was calculated from the separation of the peaks corresponding to the methylene and hydroxyl proton^.^^-^^ The HC1 was added to reduce the multiplicity of the peaks. This technique measured the temperature of the D20 bath surrounding the cell. A calibration curve relating the air stream temperature to the D20 temperature was generated, and the temperatures reported by the probe controller were corrected. The remaining uncertainty in the temperature measurement of f 0 . 2 "C can be expected to cause some scatter in the near critical region. The pressure transducer (Precise Sensors) was calibrated against a new unit, which was in turn calibrated in the factory. The transducer exhibited a maximum absolute percent error of three-tenths of a percent, which corresponds to an error of at most 3 bar at the maximum pressure investigated. In the critical region (P = 58.4 bar) the maximum error in the pressure measurement was approximately 0.18 bar, or 2.5 psi. The digital
11848 J. Phys. Chem., Vol. 98, No. 46, I994
-
Scattered beam
Window nut assembly containing diamond wiindow
View port
Figure 3. Schematic of the high-pressure SAXS scattering cell.
pressure readout reported pressures to within f 0 . 5 psi. The combined error in the pressure measurement was negligible, even in the highly compressible near-critical region. Chemical shifts were measured for xenon (Linde, >99.995%) at pressures between 54.6 and 1039.6 bar and at a temperature of 24.1 "C. Several spectra were taken for pressures between 55 and 75 bar, where xenon is very compressible. The critical temperature, pressure, and density of xenon are 16.6 "C, 58.4 bar, and 0.0085 mol/cm3, respectively. Densities of the fluid were calculated at the measured temperatures and pressures using corresponding states with an accurate equation of state for methane*O and published critical constants for methane and xenon. The corresponding states procedure, applied at the conditions of Michels et a1.,21 reproduced their measured densities of xenon to an average absolute percent error of 0.3% for temperatures between 25 and 50 "C and pressures between 25 and 700 bar. SAXS Experiments. The X-ray source, optics, and detector were capable of making rapid measurements on weakly scattering systems. The X-ray diffraction data were taken at the Time-Resolved Diffraction Facility (station X12B) at the National Synchrotron Light Source at Brookhaven National Laboratory. A custom-built two-dimensional gas delay line detector was used with a surface area of 10 x 10 cm containing 512 x 512 pixels. The detector was interfaced to a real-time histogramming data collection system. The optical system provided a doubly focused monochromatic X-ray beam with a spot size of 0.5 x 0.5 mm (fwhm) and a bandpass of 5 x M/il over wavelengths il between 0.9 and 1.5 A. For these experiments the wavelength was 0.9 A. The q range for these experiments was 0.04-0.74 A-l, where q = 4t(sin 8)/1 and 28 is the scattering angle. All detector images were circularly integrated about the beam center in bins of 1-pixel width. A new SAXS cell designed for high pressure was used in these experiments. The SAXS cell, illustrated in Figure 3, was a high-pressure stainless steel monoblock containing two diamond SAXS windows and a single sapphire view port. Each window nut assembly (supplied by Omley Industries, Inc.) contained a 3-mm-diameter by 0.5-mm-thick natural diamond window brazed to the window retaining nut. Diamond is an excellent window material for SAXS studies because, being an isotropic single crystal of low atomic number atoms, it has low absorbance and low scattering power, although imperfections in the diamond or in the brazing can cause significant small angle scattering. The view port contained a sapphire window that was sealed to the metal block by a 2.54-cm diameter goldplated metal V-ring seal (Parker, No. 8812-2001-0050). This window provided a means of viewing the sample to determine
Pfund et al. the number of phases. The SAXS path length at atmospheric pressure was 90 pm. The path length increased linearly with pressure to 96 p m at 400 bar, due to bowing of the diamond windows. The cell had a volume of 10 mL and had a maximum pressure rating of 500 bar. Temperature and pressure were monitored carefully to ensure accurate results for very compressible fluids. Because the X-ray scattering experiments with synchrotron radiation and a twodimensional detector could be performed rapidly, variations in temperature and pressure during a run were small. Fluid pressure was monitored to f0.07 bar with an electronic transducer (Precise Sensors, Inc., No. C45 l), which was calibrated against a deadweight tester (Ashcroft, No. 1305-D). Pressures generally drifted 'less than 0.17 bar during the experiment, variations being larger only at high xenon densities. The temperature of the SAXS cell was controlled to within f 0 . 5 "C using a three-mode controller with a platinum resistance probe (Omega, No. N2001). The temperature was also monitored to f O . 1 "C with a platinum resistive thermometer (Fluka, No. 2108A). The probes were calibrated against a mercury thermometer over the temperature range 25-50 "C. Temperature variations during the course of an experiment were generally less than 0.2 "C. Pressures and temperatures at the beginning and end of each 10-min run were recorded and averaged to determine the conditions for the experiment. Other variables affecting the outcome of the experiments were identified and controlled. Prior to comparison, the results from separate experiments made under different sample and instrument conditions were scaled to remove differences caused by detector nonuniformity, beam intensity, sample transmission, path length variations, etc. The information from each experiment consisted of a matrix of pixel responses from the twodimensional detector. The data were corrected for detector nonuniformity by multiplying the intensity array by a correction matrix determined from the isotropic scattering for a concentrated solution of polyethylene glycol. The data were circularly integrated (as discussed above) to obtain the intensity as a function of scattering angle. During the experiment the intensity of the incident beam was measured by a scintillation counter positioned upstream from the cell. Results from each experiment were scaled to a common beam intensity. Nonzero detector response time and limited histogrammer throughput limited the rate at which events could be detected. Therefore, these experimental results were also scaled to a common detection dead-time. Equivalent empty cell blanks were subtracted from the data at each condition to obtain the scattering from xenon alone. The transmission through the cell was known from measurements made by upstream and downstream ionization monitors. All intensities were scaled to those which would result for perfect transmission (zero absorbance). Finally, xenon scattering intensities acquired with different path lengths (due to the bowing of the windows with pressure) were scaled to a common path length. Experimental conditions spanned a broad range of states; emphasis was placed on characterizing the compressible region near the critical point of xenon. SAXS experiments were conducted at nominal temperatures of 28 and 45 "C and nominal pressures of 400, 350, 200, 120, 100, 80, 70, 50, and 30 bar. The densities examined ranged from liquid-like to gas-like: 0.018 mol/cm3 to 0.001 29 mol/cm3. The xenon was research grade (with quoted purity of 99.995%) from Linde. NMR Results A nucleus in an external magnetic field H will experience a slightly different local field HL due to interactions between the
Fluid Structure in Supercritical Xenon
J. Phys. Chem., Vol. 98, No. 46, 1994 11849
field and neighboring electrons. The nuclear shielding constant a is defined by the relation HL = H(l - a). An NMR active nucleus will resonate at a given frequency at a particular local magnetic field, which is directly proportional to the frequency of resonance. In practice, one typically holds the magnetic field constant and searches for the resonant frequency, but the analysis is more straightforward if one assumes that the frequency is held constant and the field swept in search of resonance. Thus, the external magnetic field required for resonance at a given frequency is dependent upon a;H = Hr/( 1 - a). The chemical shift 6, which is the quantity actually measured by the spectrometer, is defined as the difference between the necessary extemal field for a particular sample and that of a reference material, divided by the field required for a bare nucleus:
If we consider only pairwise interactions, then
+
where a,(r) a&) is the effect on the nuclear shielding of a xenon atom due to the presence of another xenon atom at position r (relative to the central atom) and the brackets denote an average over the volume of the system. Because of the spherical symmetry of this system, these averages are integrals over all separation distances, (7) where the subscript x denotes w or e, N is Avogadro’s number, is the molar density, and g(r) is the radial distribution function.24 Neg(r) is the local number density of xenon atoms at a distance r from an arbitrary central atom in the homogeneous fluid. NMR experiments provide a weighted probe of the radial distribution function for those separation distances where the a,(r) are nonzero. We proceed under the assumption that the exchange contribution a, is negligible. The recent ab initio calculations of JamesonZ3have demonstrated that this term is not sampled by the range of interatomic separations encountered in experiments of chemical shifts in inert gases. The problem reduces to calculating the effect of van der Waals interactions on the nuclear shielding constant. The general theory of R a m ~ e y ~ ~ demonstrates that the effect of van der Waals interactions on a is proportional to the mean of the square of the fluctuating electric field at the nucleus I?. Calculation of a,(r) is then reduced to calculation of a proportionality constant B:
e
where HRef is taken here to be the required field for the species of interest at very low density, HRef = H(T,e=O) = HJ(1 URef). Then, the relationship between 6 and a is
We find (since a and aRef are much less than one) that 6 w a - OR& Thus, the chemical shift and the nuclear shielding constant differ only by an additive constant. The chemical shift reflects the local magnetic field at the nucleus and thus is dependent upon the neighboring electronic environment. The nuclear shielding constant of a compressed gas is typically described by a truncated virial e q ~ a t i o n ? ~ , ~ ~
where e is the density of the fluid. The chemical shift may be split into several contributing terms:22
6 =0 - 00 =a b
+ + + + 0, 0,
0,
0,
(4)
where o b represents the term due to bulk magnetization of the sample, Oa represents the contribution from magnetic anisotropy, OE represents the effect of an associated permanent electric dipole, a, represents the effect of van der Waals dispersion interactions (which are due to fluctuations in the instantaneous dipoles), and a, represents the effect of short-range exchange interactions (which occur upon atomic overlap). The bulk susceptibility term ab is a small contribution to the total susceptibility of xenon. Xenon exhibits chemical shifts which vary with density by more than lo4 (ppm*cm3/mol), whereas the bulk susceptibility term (for spectrometers with the base magnetic field parallel to the sample cylinder, as in this case) is much smaller:
oble= -4nx,,,/3
= 190.6 (ppm*cm3/mol)
a,(r) = BE^
(8)
Bothner-By26treated the effect as a distortion of the surrounding electron cloud due to dispersion energy and concluded that the mean square of the fluctuating electric field is inversely proportional to the intermolecular separation to the sixth power,
E’
r-6
(9)
a result also used by Raynes et aLZ2 This approximation was verified by the ab initio calculations of J a m e ~ o n .This ~ ~ was then used with eq 7, approximating g(r) with the Boltzmann factor exp[-u(r)/ka,
an approximation which is accurate at low den~ities:~’
(5)
Here, xm is the molar susceptibility, which for xenon is -45.5 ( ~ p m * c m ~ / m o lDespite ) . ~ ~ the limited magnitude of a b , it shall be included in calculations of the shielding coefficient in this paper. a, may be neglected, as xenon is monatomic, and thus exhibits no magnetic anisotropy. UE may be neglected for the same reason; an inert gas atom does not have a permanent electrical dipole. Thus, the density dependence of the chemical shift is controlled by a,, Ue, and a b .
In eqs 10 and 11, k is Boltzmann’s constant and u(r) is the intermolecular potential, which Raynes et al. modeled with a Stockmayer potential. Since we are interested in the state dependence of the shielding, we shall not describe methods for estimating the constant B, a priori. Trappeniers and O l d e n ~ i e l ~ ~ also used the Boltzmann factor approximation for g(r) but used a Lennard-Jones potential for u(r) and obtained results for the proton resonance of methane and that of ethylene in approximate agreement with experimental data. Note (by eq 11) that the use of the Boltzmann factor exp(-u(r)/kn does not account
11850 J. Phys. Chem., Vol. 98, No. 46, 1994
Pfund et al.
Calculated, using Boltzmann factor approx.
to g(r) (low density approx).
Calculated, using g(r) from HMSA theory with modified Barker potential.
01' 0
1
0.005
0.01
0.07.
0.015
0.025
Xe density, mole/cm3 Figure 4. Calculated and experimental density dependence of NMR shielding in xenon at 25 "C; Comparison of extrapolated low-density behavior
with the result obtained using the density dependent radial distribution function. 0.002
0.0015
h
0-
Omstein-Zemike-Debye relation: l/I(q) = 0.06638q2t (4.358x
s
v
ln[I(q)] = -60.597q2t 7.4 0.001
0
0.005
0.01
0.015
0.02
0.0005 0.025
q2, A-2
Figure 5. S A X S data for xenon at 28.4 "C and 71.2 bar: Guinier and Omstein-Zemike-Debye plots.
for the effect of density on the structure of the fluid surrounding the central atom. At low densities this is a useful approximation. At higher densities the terms g,(r) can become significant, where each g,(r) gives the effect that n atoms surrounding the central pair (the position of each averaged over the volume of the system) have on the probability of finding the pair a distance r apart. The sum of these terms describes the formation of second and higher coordination shells and describes the development of long-range correlations near the critical point. The result of the approximate eq 10 is a predicted chemical shift that is linear in density. Our experimental results exhibited significant nonlinearity. A result of applying eq 10 for the shielding is the cage model of NMR shifts, in which the chemical shift is linear in d e n ~ i t y . ' ~ ,Because ~ ~ . ~ ~ the factor r-6 decays rapidly with distance, only the first coordination shell affects the shielding
to a significant extent. Under the Boltzmann factor approximation to g(r) the structure of the first shell, or cage, is not affected by the density of the fluid and retains a gas-like structure under all conditions. This solvent cage is envisioned to consist of a fixed number of available sites at fixed distances from the central atom, which are populated in proportion to the density. Nonlinearities in the observed shift with changes in density can be considered to result from distortions of the cage structure with density. Such distortions are described by the deviation of g ( r ) from the Boltzmann factor. The solvent-solvent "clusters" (as are now commonly defined30) that appear near the critical point result from changes in the long-range behavior of g(r),13 as will be discussed in conjunction with the SAXS results below. Thus, they cannot directly affect the NMR shielding. The long-range behavior of g ( r ) does influence the PVT properties of the fluid via the compressibility and can influence the shielding indirectly.
J. Phys. Chem., Vol. 98, No. 46, 1994 11851
Fluid Structure in Supercritical Xenon TABLE 1: Results of SAXS Experiments on Xenon at Several Supercritical States
W),
T,"C P, bar experimentalunits 28.5 28.5 28.5 28.4 45.0 45.3 44.7 45.0
118.7 100.7 80.0 71.2 200.4 120.9 100.4 80.2
343 (67) 489 ( f 8 ) 1243 (f22) 2295 (f51) 217 ( f 4 ) 522 ( f 7 ) 903 ( f l l ) 694 (f9)
correlation Reatto-Tau E3, length L, 8, Ab 11.88(f0.39) 3375 13.88(f0.27) 3371 21.92 (f0.20) 3406 30.23 (f0.19) 3335 (6191) 9.22 (&0.49) 3288 14.06 (f0.22) 3373 18.31 (f0.16) 3159 16.98(f0.19) 3320
The estimate of (a,) may be improved by use of g(r) values obtained from integral equation theory or simulation:
densities indicates (via eq 12) that the radial distribution function within the first coordination shell is (on average) less than the Boltzmann factor. Thus, at increased density, the structure of the solvent cage is altered relative to the structure at low densities. As a further test, the method was able to predict the data of Kanegsberg et al.,35which were taken at 20 and 80 "C and covered a density range 0.0013-0.0134 mol/cm3. The average absolute percent errors for the two isotherms were 3.4 and 3.5%, respectively. The absolute error increased with density. The absolute error at a density of 0.0137 mol/cm3 was 1.8 ppm at 20 OC and was 1.9 ppm at 80 "C.
SAXS Results The scattering from a pure fluid is typically expressed as the product of a function P ( q ) giving the scattering per particle and a function S(q) related to the structure of the fluid,
The hybrid mean spherical approximation (HMSA) for g(r)31 fU(q) = S(q) (13) used with the accurate Barker pair potential3*for xenon provided a good model of chemical shift as a function of density. The where leu is the intensity per unit of volume in electron units, Omstein-Zernike equation was solved simultaneouslywith the e is the number density of particles, and P is the square of the HMSA closure using methods discussed p r e v i o u ~ l y .The ~ ~ ~ ~ ~ scattering amplitude per particle. Because of the influence of HMSA theory is known to produce radial distribution functions, P(q), SAXS experiments can provide information about the size, pressures, and intemal energies that are in good agreement with shape, and intemal structure of particles in the fluid. S(q) is simulations of painvise additive models of inert gases. The the structure factor: scheme used in the HMSA of forcing thermodynamic consistency between the virial and compressibility routes to the equation of state may help to make this theory particularly appropriate for simultaneously modeling NMR and SAXS data. As discussed elsewhere in this paper, NMR results depend on In the Zernike-P~ins~~ equation (14), g(r) is the radial distributhe structure of the first coordination shell, as does the vinal tion function; g(r) - 1 = h(r) is referred to as the total pressure, whereas SAXS results depend on the long-range correlation function. The integral is its three-dimensional structure of the fluid, as does the compressibility. The Barker Fourier transform, after applying spherical symmetry. Because potential model was originally developed from a simultaneous of the influence of S(q), SAXS experiments indicate at what fit of gas viscosities and second virial coefficients, differential conditions local fluid densities at some distances from an scattering cross sections, energy level spacings for xenon dimers, arbitrary central atom differ from the bulk average (as indicated and lattice spacing and cohesive energy in solid xenon. In this by &(r) - 11 being nonzero). NMR provides a F4-weighted work the well depth d k and the distance to the minimum r, in sampling of the local density, eg(r). SAXS provides an the Barker model were adjusted slightly to bring the HMSAr-weighted sampling of the difference between the local and calculated pressures at 28 and 45 "C in closer agreement with bulk densities. Therefore, SAXS experiments can provide those of real xenon. The optimized well depth d k was 255.867 information about small density fluctuations separated by large K (adjusted from 281.0 K), and the optimized distance to the separation distances r. For randomly distributed particles S(q) potential minimum r, was 4.4136 A (adjusted from 4.3623 A). is 1. The P ( q ) and S(q) factors provide two possible sources The average absolute percent error in the calculated pressure of inhomogeneities which can result in the scattering of X-rays. was 3.1% at 28 and 45 "C over a pressure range 30-400 bar. One source is the development of statistical (or dynamic) We then estimated values of g(r) for xenon at 25 "C and density fluctuations near the critical point of the fluid, which pressures from 7.6 to 2187 bar and used these values in the affects S(q). These fluctuations are related to the compressibility integrand of eq 12. The calculated values of -(a,) are plotted of the fluid and the intercept of the scattering curve by versus density in Figure 4. Also shown are the experimental chemical shifts for xenon and the calculated linear shift that results from assuming the low-pressure limit of g(r) = exp(W2)(15) S(q=O) = k q g ) T = 1 Qh(0)= u(r)/k7'). The calculated value of Ob from eq 5 has been 0 subtracted from the experimental data. The experimental data at 24.1 "C and densities between 0.003 73 and 0.009 66 mol/ where the number density e = (h9/V.30,37The quantity S(0) cm3 yielded a second virial term (al), of -1.249 (f0.191 at 1 has the units of number of atoms and has been interpreted as the 95% confidence level) x lo4 ppm.cm3/mol, a result that the number of excess solvent atoms (above the bulk average) agreed with the determination of Jameson et al. of -1.238 x surrounding another solvent atom, or the "cluster size".30 104 p p m * ~ m ~ / m o Using l . ~ ~ this result, the constant B, was Changes in both short-range and long-range structure influence estimated at -1.748 (f0.267) x lo5 ppmA6/atom by matching this number. However, by eq 14, small density fluctuations the calculated and experimental slopes at low density. With separated by large distances r are considerably more effective the aid of the B , determined at low density and the g(r) from in influencing this number than are large density fluctuations the HMSA theory, the chemical shift was predicted over the separated by small r. This long-range contribution has a certain full range of densities examined. The calculated density universal character in that its contribution is scalable from one dependence of the chemical shift was in good agreement with species of central particle to another on the basis of the strength experiment. The curvature in the data at intermediate to high of the particle-solvent attraction and on the particle-solvent
em)
+
o2
Pfund et al.
11852 J. Phys. Chem., Vol. 98, No. 46, 1994
excluded v01ume.'~~'~ Thus, a study of pure solvent provides information about cluster formation in mixed solute-solvent systems. Because of the reciprocal nature of r and q space, large scattering at small q is the result of interparticle interference between xenon atoms separated by large r, which implies that the total Correlation function is nonzero at large r (by eq 14) in such cases. At the critical point, g(r) decays very slowly to 1 with increasing I , making the compressibility diverge. The average size of the fluctuations (or correlation length) L was defined by D e b ~ as e~~
Expanding the sine term in eq 14, restricting the result to small q values, and considering only states near the critical point (where S(0) >> 1)yields the Omstein-Zemike-Debye relation for scattering at small angle^:^^.^^ (small q, near critical point)
(17)
The SAXS studies c o n f i i the known increase in the correlation length of density fluctuations in compressible fluids near the critical point. Shown in Figure 5 is a plot of l/Z(q) versus q2 made using data taken at a temperature of 28.4 "C and at a pressure of 71.2 bar. Then, by eq 17 the slope provides the square of the correlation length. The data were correlated well by the Omstein-Zemike-Debye relation. Given in Table 1 are the intercepts of the scattering curves (in experimental intensity units) and the correlation lengths for a number of nearcritical states, including confidence intervals at the 95% level. The regressions were made using the 58-69 data points in the q range 0.045-0.15 A-l. As the intercept (and therefore the compressibility of the fluid) increased, so did the correlation length. The longest correlation length observed was 30.23 A (f0.19 A) at a temperature of 28.4 "C and a pressure of 71.2 bar (TR = 1.041, PR = 1.220, @ R = 0.874). This length was much larger than the radii of any of the well-developed coordination spheres in the fluid. From the intercepts Z(0) and the known compressibilities and P(0) for xenon40 we have estimated the conversion factor necessary to obtain the intensities per unit volume in electron units at small angles from the experimental intensities. Another possible source of inhomogeneities is the organization of xenon atoms to form isolated static clusters with their own particle factors P(q).37*41Scattering from clusters obeys the Guinier law at small angles, In[P(q)] = In[P(O)] - q2R,2/3
(18)
or, given the assumption that interactions between clusters are negligible, ln[Z(q)] = ln[Z(O)] - q2R,2/3
(19)
where R, is the radius of gyration of the cluster. Included in Figure 5 is a plot of ln[Z(q)] versus q2 made using data taken at a temperature of 28.4 "C and at a pressure of 7 1.2 bar (the condition of maximum observed scattering). The scattering at small angles shows significant deviation from the Guinier law, demonstrating that static clusters were not the principle source of scattering. Reatto and Tau42 have proposed a method for determining the departure from pairwise additivity due to Axilrod-Teller
triple dipole forces in simple fluids from small angle scattering data. London0 et al. recently attempted to estimate the magnitude of these forces in supercritical neon and krypton.12 They were unable to c o n f i i the predictions of Reatto and Tau. We have pursued these studies using xenon, as the triple dipole forces should be more noticeable in it. First, the Reatto-Tau function 1(q) was obtained:
This function is defined in terms of the Fourier transform of the direct correlation function by A(q) = [E(q) - E(0)]/q2. Equation 20 is obtained from this definition after applying eqs 13 and 14 and the Omstein-Zemike definition of c(r). The intensities in electron units were obtained from the experimental intensities by dividing by both the conversion factor discussed above and the polarization factor that appears in the Thompson equation.43 The particle factor of xenon (P(q))included the small real and imaginary parts of the dispersion correction.u Reatto and Tau have proposed that, over a certain range of q, 1(q) is linear in q,
+
A(q) R5 I , I,q (21) where E2 and E3 are coefficients in the polynomial expansion of Z(q) and the linear term E 1 is 0. The q range over which A is linear depends on the state of the fluid. This range seems to be widest near the critical density at supercritical temperatures.6 For a purely pairwise interaction, I3 is density i n d e ~ e n d e n t . ~ ~ Axilrod-Teller forces manifest themselves through a density dependence on E3 which is nearly linear. In addition, E3 is thought to be inversely proportional to the temperature. Plotted in Figure 6 is 1(q) from eq 20 for a temperature of 28.4 "C and at a pressure of 71.2 bar. The function was linear for q between 0.2 and 0.75 A-l. The value of E 3 for this state was 3335 A6 ( f 1 9 1 A6). The (95%) confidence interval includes the statistical uncertainties in the intensity conversion factor, in the intercept of the scattering curve, and in the slope of the plot of 1 versus q. The prediction of Reatto and Tau is 4851 A6. Listed in Table 1 are the E3 coefficients obtained for eight near-critical states. We were unable to confirm the predicted state dependence of E3, since the confidence interval contains the E3 values obtained for the other temperatures and densities. Thus, we were unable to determine the magnitude of Axilrod-Teller triple dipole forces using the Reatto-Tau method. The HMSA integral equation theory used with the modified Barker potential provided a good model of the small angle scattering data, except for the states closest to the critical point. Radial distribution functions in xenon were calculated as discussed above in the N M R Results section. Small angle X-ray scattering intensities in electron units per unit of volume were then calculated using eqs 13 and 14. These were then converted to experimental units, as discussed above. The calculated intensities are in qualitative agreement with experiment. In Figure 7 the calculated and experimental intensities at 28 "C and 30, 80, and 400 bar are plotted versus q. The predicted scattering at the lowest and highest pressures matches experiment to within the accuracy of the data. Results for Z(0) near the critical density are greatly affected by very small errors in the calculated (aP/a@),which approaches 0 near the critical point. Discussion The observed increase in scattering at small angles as the critical point is approached demonstrates the development of
Fluid Structure in Supercritical Xenon
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J. Phys. Chem., Vol. 98, No. 46, 1994 11853
0
q,
A-'
Figure 6. L(q) versus q for xenon at 28.4 "C and 71.2 bar. 2000
.-x
:.P ,
Y
v)
-E
Small angle x-ray scattering from xenon
T = 28.4 O C = 71.2 bar
1000
,? 500
% ' .*-
T = 28.4 O C P = 30.2 bar
0 0.025 0.075 0.125 0.175 0.225 0.275 0.325 0.375 0.425 0.475 0.525 0.575 0.625 0.675 0.725
q, A-' Figure 7. Comparison of calculated and experimental S A X S intensities at 28 OC in experimental intensity units. Calculations are from HMSA theory and modified Barker potential. Xenon particle factors are from ref 40.
long-range correlations in the fluid. Small angle scattering is a useful probe of long-range structure, but it conveys little information about short-range structure. Because NMR shielding is proportional to the integral,
o(T,e)= ~ L + ~ r ~ g ( r ) 4dnr r * and this integral depends largely on the value of g ( r ) at small separation distances, NMR experiments are useful for probing short-range structure. This observation is confirmed by plotting the cumulative effect of separations less than R on the NMR integral:
u*(R;T,Q)= LRr-6g(r)4nr2dr a*(R;T,g)is plotted in Figure 8 versus sphere radius R at three
different densities at 25 "C. The corresponding pressures were 30.6,69.6, and 444.9 bar. The radial distribution functions were estimated from the HMSA theory, as discussed above. Separation distances less than about 5.4 8, contribute 75% of the shielding; distances less than 7.25 8, contribute 90%. Thus, it is the structure of the first coordination shell that determines the chemical shift. This observation confirms one aspect of the cage model. However, the structure of the cage is not independent of density. If it were, g ( r ) would be independent of density and equal to the low-pressure (Boltzmann factor) value and Q* would be independent of density, which it is not. The cage structure was most effective at shielding the central atom at low densities, though that structure was not fully occupied. At higher densities the cage was slightly less effective, though it was more fully occupied, the shielding being an increasing function of the density.
11854 J. Phys. Chem., Vol. 98, No. 46, 1994
C A
Pfund et al.
I
1 Approx. distance to 2nd
e,
-8
j neighbor
.e c1
m
uE
0.02-
0m
In
In
N
In
L3
i
Figure 8. Effect of xenon atoms within a sphere of radius R on NMR specific shielding. Calculations are from HMSA theory and modified Barker potential at 25 "C.
0.1
I
0
I
0.0025
0.005
0.0075
t
0.01
M~~ comprcssibili~y
I
,
I
0.0125
0.015
0.0175
0.02
O.(
t
Min compressibility factor
Xe density, mole/cm3 Figure 9. Calculated NMR specific shielding and compressibility factor of xenon versus density at 25 "C.
o(T,e) is an increasing function of the bulk density e. It is helpful to consider the specific shielding,
cso e =J+"r6g(r)4nr2 d r which depends on how g(r), and therefore the structure of the cage of nearest neighbors, varies with temperature and density. The specific shielding and the compressibility factor PIekT at 25 "C are plotted versus density in Figure 9. Both the integral and the compressibility factor were minimized at a density of about 0.0125 moVcm3 (at 25 "C). This is the density at which the fluid becomes very incompressible, a density that is considerably higher than that at which the compressibility is maximum (which is close to the critical point). Therefore, the
observed minimum in the shielding integral was not a phenomenon resulting from being close to the critical point. At densities below 0.0125 moVcm3, ole decreased with increasing density. Therefore (by eq 24), the height of g(r) was reduced as density was increased for distances r within the first coordination shell. At higher densities u/@ and the height of g ( r ) within the first shell began to increase with increasing density, indicating that the sphere of first neighbors was being compressed. Calculated radial distribution functions g ( r ) at 25 "C are plotted versus separation distance in Figure 10. The corresponding pressures were 30.6,69.6, 101.8, and 444.9 bar. The pressure 69.6 bar was close to the condition of maximum compressibility; 101.8 bar was close to the condition of minimum compressibility factor and specific shielding. At short range (separations less than about 7.25 A) the magnitude of
Fluid Structure in Supercritical Xenon 2,5
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J. Phys. Chem., Vol. 98, No. 46, 1994 11855 T = 2 5 ‘C
: i
of maximum
g ( r ) was a decreasing function of density for pressures less than about 101.8 bar. At higher pressures and densities the height of the first peak in g ( r ) was an increasing function of pressure. At 69.6 bar (closest to the critical point) the function exhibited a long-range tail. It was this increase in g ( r ) for large r which was responsible for the large value of S(0) and the large correlation lengths in the SAXS results. The height of the radial distribution function at short range was reduced at intermediate densities because interactions with other atoms act to reduce the effective attraction between xenon pairs under these conditions. The effective repulsion can be seen by examining the density expansion of the radial distribution function,45 ,.] =
where y ( r ) is the background (or bath) correlation function. As discussed above, each gn(r) gives the effect that n atoms surrounding the central pair have on the probability of finding the pair a distance r apart. The background correlation function provides an aggregate measure of the attractive or repulsive character of the solvent surrounding a pair of atoms separated by a distance r. At low densities the radial distribution function is approximately the Boltzmann factor exp(-u/kT) and y ( r ) PX 1. At intermediate densities the functions g,(r) make important contributions to the distribution function. The function gl is27 g l ( r ) = Jdr,[exp(-
2) 2) - l][exp(-
- 11 (26)
where r = I lrlzl I and 1-3 is moved over the volume of the system. This integral describes how single atoms surrounding a pair of atoms influence the likelihood of finding the pair a distance r apart. The function (J - l)/@ is plotted versus separation distance r in Figure 11 for three different densities. The modified Barker potential was used in the calculation, and the temperature was 25 O C . The three curves correspond to the low-density limit and to pressures of 101.8 and 2187 bar. The
low-density limit shown is equal to g l ( r ) . gl was rigorously calculated using eq 26. At higher densities (J - l)/@was estimated using the HMSA theory (since the g,(r) are in general intractable and the convergence behavior of the series is unknown). These functions were negative for r in the excluded region between the first and second neighbors (between about 4.5 and 7.5 A). The fact that y ( r ) was less than 1 indicates that atoms surrounding each pair reduce the height of g ( r ) over these separation distances below the low-density (Boltzmann factor) limit. As discussed above, separation distances less than about 7.5 A make the predominant contribution to the chemical shift. g l ( r ) was similar to Ly(r) - I]/@ over the separations probed by NMR, except for r less than about 4.5 8, at high densities. Thus, the negative well in gl largely accounts for why the height of the radial distribution function by NMR (as measured by de)was reduced with increasing density at low to intermediate densities. At higher pressures and densities, higher order g, make an increasing contribution to the height of y ( r ) and g ( r ) for r between 3.9 (the approximate core size) and 4.5 A. This range of separations includes the first neighbor atoms. The increase in the magnitude of g ( r ) over this range of separations causes o/@ to be an increasing function of density at high densities. An examination of how different regions of space sunounding a pair of atoms contributes to gl (and therefore to g ( r ) ) is useful, since this allows a mechanical interpretation of why the specific shielding decreases with increasing density at low to intermediate densities. It is the interaction of the coordination spheres surrounding each atom that acts to reduce the height of gl and the radial distribution function at short range. The simple cage model ignores the fact that each atom in the cage has its own cage, and complete filling of all of them is impossible because of their overlap. Shown in Figure 12A is a contour plot of the product of Mayer factors [exp(-u(rl3)/kt) - l][exp(-u(r32)/ kT) - 11 (which appear in the integrand of eq 26 for g l ) for atoms separated by the distance to the potential minimum r,. The Mayer factors are symmetric about the interatomic axis so the plot is made for a plane through the centers. Areas lighter than the gray background are positive and make a positive contribution to gl(rm)and g(rm). Areas that are darker than the background are negative and act to reduce the g(rm). The large
Pfund et al.
11856 J. Phys. Chem., Vol. 98, No. 46, 1994 T=25 OC P = 2187 bar p = 0.02352mole/cm3
40 -
T = 25OC Low density limit = g 1
20 -
0 --
-20 -
I
rcI
v! t
I
Io
1
2
I
a
I
1
v!
2
0, Separation distance r, 8,
I
2
I
If. 7
rl
Figure 11. First order term (gl) in the density expansion of the radial distribution function of xenon (assuming pairwise additivity) at 25 "C; estimates of the same function plus higher order terms at 0.013 98 moVcm3 (approximately 101.8 bar) and at 0.023 52 moVcm3 (approximately 2187 bar).
Figure 12. Overlap of first coordination shells of a pair of xenon atoms separated by (A) 4.45 A, (B) 6.0 A, and (C) 8.8 A. Areas lighter than the background are attractive; those that are darker are repulsive.
circles have radii r, and represent the most probable location of the nearest neighbors of each atom. Solvent atoms located at the intersection of both of these spheres (the cusps at the top and bottom of the figure) exert an attraction on each atom in the pair and act to increase gl(rm). The region with the footballshaped cross section is well within the coordination spheres of both atoms and is a volume from which solvent is completely excluded. As with hard spheres near contact, this volume also acts to increase gl and g (for hard spheres at contact this region has a volume of glhs(d+)= 5n&/12 = B3/B2, where d is the diameter and B2 and B3 are the known second and third virial coefficients of hard spheres). Where the first coordination
sphere of one atom penetrates that of the other, solvent atoms interact attractively with one atom and repulsively with the other. In these regions of space (dark gray to black in Figure 12) the product of Mayer factors are negative and they act to reduce gl. Shown in Figyre 12B is a similar plot for atoms separated by a distance (6 A) intermediate between the first and second neighbor distance. The positive regions are significantly reduced in size; the negative regions are not. The negative overlap regions are not significantly reduced until the atoms are separated by the second neighbor distance, as shown in Figure 12C. At this large distance, the coordination shells (or cages) of the two atoms are separate and both can be filled without
Fluid Structure in Supercritical Xenon mutual interference. In short, an atom in the local environment of another (separation distance r < about 2rm)has part of its first coordination shell (or solvent cage) unavailable for occupancy, due to repulsive overlap of (potential) shell atoms with the core of the second atom.
Conclusions The SAXS results presented here give evidence of the longrange density fluctuations near the critical point discussed by DebenedettL30 These fluctuations reflect an increase in the radial distribution function g(r) at large separation distances r.13 However, the density dependence of fluctuation size provides little information about how the local density at small separations varies with the bulk density. The NMR results demonstrate that the height of the radial distribution function in pure xenon for separations less than the second neighbor distance is a decreasing function of density at low to intermediate densities. The reduction in the height of g(r) is primarily due to the overlap of mutually exclusive regions of the first coordination shells of the pair of atoms separated by distance r. The depression of g(r) from the Boltzmann factor is largest not at the density where the compressibility is maximized but instead at approximately the density where the compressibility factor P/@kTis minimized. A similar density dependence for the solventlsolute g(r) in asymmetric mixtures can be expected. NMR is a useful tool for studying the local solute environment in simple supercritical fluids because of the known, simple, short-range effect of van der Waals interactions on the nuclear shielding constant. Through the combination of techniques we have shown that a short-range solvent depletion together with a long-range enhancement (relative to the bulk density) exists near the critical point of xenon, demonstrating clustering of the Debenedetti type, which is purely long-range.
Acknowledgment. Work at the Pacific Northwest Laboratory (PNL) was supported by the Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the U.S.Department of Energy, under Contract DE-AC0676RLO 1830. The authors acknowledge Richard D. Smith, PNL, for helpful discussions during the course of this work. References and Notes (1) Tanko, J. M.; Blackert, J. F. Science 1994, 263, 203-205. (2) Kim, S.; Johnston, K. P. Ind. Eng. Chem. Res. 1987, 26, 12061213. (3) Shaw, R. W.; Brill, T. B.; Clifford, A. A.; Eckert, C. A.; Frank, E. U. Chem. Eng. News 1991, (Dec. 23) 26-93. (4) Knutson, B. L.; Tomasko, D. L.; Eckert, C. A.; Debenedetti, P. G.; Chialvo, A. A. In Supercritical Fluid Technology, Theoretical and Applied Approaches to Analytical Chemistry; Bright, F. V., McNally, M., Eds.; ACS Symposium Series 488; American Chemical Society: Washington, DC, 1985; pp 60-72. (5) Brennecke, J. F.; Tomasko, D. L.; Eckert, C. A. J . Am. Chem. SOC. 1990, 112, 7692-7700.
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