Shape and size of a nonionic surfactant micelle. Triton X-100 in

Mar 1, 1980 - Chemical Reviews 2005 105 (9), 3197-3219. Abstract | Full Text HTML ... A Cloud Point Study on the Micellar Growth of an Amphiphilic Dru...
0 downloads 9 Views 1MB Size
Download PDF
J. Phys. Chem. 1980, 84,599-607

599

Shape and Size of a Nonionic Surfactant Micelle. Triton X-100 in Aqueous Solution H. Hasko Paradies" Department of Chemistry, Cornel1 University, Ithaca, New York 14853, and Fachrichtung Blochemie der Pflanzen, Frele Universital Berlin, D-1000 Berlin 33, West Germany (Received March 30, 1979; Revised Manuscript Received December 18, 1979) Publication costs assisted by Deutsche Forschungsgemeinschafi

Small-angle X-ray scattering experiments on aqueous solutions of the nonionic surfactant Triton X-100 at 20 and 30 "C in the concentration range 5-10 mg/mL were performed in order to determine size and degree of hydration. The partial specific volume determined independently by density measurements and by the contrast variation method was found to be 0.9125 mL 8-l at 20 "C, and a molecular weight of 95 700 dalton was assigned to this monodisperse solution of Triton X-100 (pH 8.0). A higher molecular weight of 150000 dalton was found at 30 "C at pH 8.0 in 0.01 M Tris-HCI; this particle had a radius of gyration of 39.2 A, a volume of 4.42 X lo5 A3,and a degree of hydration of 1.21 g of water per gram of Triton X-100. The radius of gyration of the 95700 dalton particle was found to be 29.3 A; the volume was 3.35 X lo5 A3 with a degree of hydration of 1.18 g of water per gram of Triton X-100. The largest dimensions of the Triton X-100 particle were determined from the correlation functions and were 105 A at 20 "C and 119 8, at 30 "C. For both particles the scattering curves are more consistent with an oblate ellipsoid of revolution rather than a prolate equivalent. From small-angle X-ray scattering experiments at various electron densities of the solvent the mean-square electron fluctuation was determined from which a high electron density and a low electron density region within the Triton X-100 particle can be proposed. If a two-level step function is assumed, the radius of gyration of the high electron density region is 18 A for the 95 700 dalton particle and 22.7 A for the 150000 dalton molecule, indicating that the hydrocarbon core of the 95 700 dalton particle has a diameter of 48.0 A, whereas the hydrocarbon core of the 150 000 dalton particle is asymmetric with half axes of a = 23.4 A and b = 46.8 A. The average curvature of the external surface area of the Triton X-100 micelle at 293 K is considerably smaller (28 A) than the average radius obtained from the radius of gyration, indicating that the external surface is convoluted. Laser correlation spectroscopy from aqueous solutions of Triton X-100 at 20 and 30 "C in the concentration range 5-10 mg/mL was carried out in addition to small-angleX-ray scattering experiments. The micellar size at 20 "C corresponds to an effective hydrodynamic radius of 41.8 A, and that at 30 "C of 55.7 A, with translational diffusion coefficients (20 " C ) and (4.15 f 0.06) X cm2 s-l (30 "C). of (4.92 f 0.08) X

Introduction Triton X-100 has been widely used as a nonionic surfactant for solubilizing membrane bound enzymes and for stabilizing this class of enzymes in solution. Triton X-100 is a polydisperse preparation of p-(1,1,3,3-tetramethylbutyl)phenoxypoly(oxyethyleneglycol) containing an average size of 9.5 oxyethylene units per molecule. The molecular weight of Triton X-100 micelles in aqueous solution was first determined in 1954l by using light scattering techniques i3nd has since been measured several time~.~J Estimates of the degree of hydration have been given by correlation of the micellar molecular weight and hydrodynamic measurements such as intrinsic viscosity and actual frictional ratio from sedimentation velocity experiments. However, it is not easy to separate shape and hydration from these measurements since all shape calculations are influenced by the degree of hydration. Triton X-100 micelles are considered spherical or almost spherical in shape4+ and sufficiently bound water is considered to fit the hydrodynamic parameters. The degree of hydration and the influence of water upon the shape of the Triton X-100 micelle still remains unresolved. Recently a note upon size, shape, and hydration of this nonionic surfactant a ~ p e a r e drevealing ,~ that if the hydrophobic core and the whole micelle of Triton X-100 are spherical, than several oxyethylene groups must be embedded in the hydrodynamic core and, in case this does not occur, then the hydrophobic core and the whole Triton

* Address correspondence to this author at the Freie Universitat Berlin address. 0022-3654/80/2084-0599$0 1.OO/O

X-100 micelle cannot be spherical.8 Hydrodynamic methods such as sedimentation velocity, intrinsic viscosity, flow birefringence, dielectric dispersion, and fluorescence depolarization give little information about the overall geometry and structure of a macromolecule, e.g., Triton X-100, above the critical micelle concentration (cmc), beyond a very general description of the degree of asymmetry of the molecule. The most powerful technique available at present for probing the geometry of the structure of macromolecules in solution is smallangle X-ray scattering which makes it possible to determine directly a number of characteristic parameters for the molecule, e.g., radius of gyration, volume, surface area, and the size of the particle in solution. This paper describes the first small-angle X-ray scattering study of Triton X-100 at different temperatures and ionic strengths in order to measure molecular weight, shape, and hydration of this nonionic surfactant. The X-ray scattering study is performed on the detergent at different electron density contrast by varying the electron density of the solvent by addition of glycerol or thallium nitrate. This paper will attempt to detect, on the basis of small-angle X-ray scattering experiments and by using a different contrast between the Triton X-100 micelles and the applied solvent, the hydrophobic core when using the assumption of a two-step density level within the molecule. Furthermore, quasi-elastic light scattering measurements of aqueous solutions of Triton X-100 were performed which allow one to measure the diffusion coefficient. Since the diffusion coefficient is related to the hydrodynamic radius of the nonionic surfactant through the StokesEinstein relation the size and shape of the Triton X-100 @ 1980 American Chemical Society

600

The Journal of Physical Chemistw, Vol. 84, No. 6, 1980

Paradies

TABLE I: Molecular Weights, M,, Sedimentation Constants, So2o9,, Intrinsic Viscosities, and Partial Specific Volumes of Micellar Triton X-100

no.

a

1 0 - 7 0 ~w,~ ~

solvent

1 0.05 M Tris-HC1,pH 8.0 2 0.01 M Tris-HC1, pH 8.0 3 0.05 M NaOAc, pH 5.0 4 0.01 M Na,HPO,, pH 8.0 5 0.01 M NaH,PO,, pH 7.0 6 0.05 M Tris-SO,, pH 8.0 7 0.01 M Tris-PO,, pH 8.0 8 0.05 M Tris-HC1, pH 7.0 9 0.05 M NaOAc, pH 5.5 10 0.01 M Na,HPO,, pH 7.0 11 0.01 M NaH,PO, Determined by sedimentation equilibrium.

T,K 293 293 293 293 293 303 303 303 303 303 303

103

~ , a

98.5 95.0 85.0 86.0 85.2 150.0 150.0 151.0 148.0 151.0 150.0

S020,,

1.69 1.70 1.68 1.58 1.67 1.98 1.98 1.97 1.96 1.96 1.95

cm* 5.40 i. 0.05 5.40 i 0.05 5.35 i. 0.05 5.41 ~t.0.05 5.40 i 0.05 5.38 i. 0.02 5.39 i. 0.03 5.40 i. 0.05 5.37 f 0.05 5.37 f 0.04 5.38 i 0.05

-

u 2 , b mL g - ' 0.9122 0.9123 0.9122 0.9121 0.9123 0.9105 0.9103 0.9103 0.9104 0.9104 0.9104

Determined according to eq 1.

micelles can be determined independently from X-ray scattering measurements. In conjunction with X-ray scattering measurements this investigation may have a substantial application in resolving the controversies with regard to shape, hydration, and micellar structure, e.g., external surface area, of this typical nonionic detergent micelle.

Materials and Methods Reagents and Solutions. Triton X-100 was purchased from Sigma Chemical Go., St. Louis, Mo. All other chemicals were of analytical grade and from Mallinckrodt, St. Louis, Mo. Triton X-100 micelles were dissolved in water with the additives listed in Table I and at the particular temperature given in that table. Purification of Triton X-100 micelles was performed on a BioGel A 1.5-m column (1.5 X 100 cm), where the material eluted as one single Gaussian peak in 0.01 M Tris-HC1, pH 8.0, T = 293 K, as well as by sucrose density gradient centrifugation (13-35 w t %/vol) at 18OOOOg for 70 h in a Beckman ultracentrifuge (L2 65 B). Analytical Ultracentrifugation and Viscosity. Molecular weight measurements also were based on sedimentation equilibrium measurements which were performed on a Beckman Model E analytical centrifuge by using Rayleigh interference opticsgin order to test for the absence of size heterogeneity. Measurements of the intrinsic viscosity were made by using a Cannon viscosimeter (Pa, USA) with a flow time for water at 25 " C of 290 s.l0 Densimetry. Partial specific volumes of the Triton X-100 were measured by using an Anton Paar precision densimeter, Model DMA 02C (Graz, Austria). Tempera" C at 20 "C and f1.40 X ture constantly was f1.2 X "C at 30 "C due to thermostat performance. Concentrations of Triton X-100 in the aqueous solutions were determined by dry weight measurements on a microbalance (Mettler) and by measuring absorbance at 240 or 261 nm due to light scattering. Moreover, identical resulta were obtained when dye absorption methods were used to estimate micellar concentration. Any changes in the partial specific volumes with increasing glycerol concentration can be detected by a plot of d - clo/c, = (1 + a ) - do(^, + abH,o) (1) where d and do are the density of the solution and solvent, respectively, in g/mL, a is the amount of water associated with the particle in g/mL, u2 the partial specific volume of Triton X-100, c, the volume concentration of the nonionic surfactant Triton X-100, and OH& the partial specific volume of water. Inelastic Light Scattering. The diffusion measurements were performed with an argon-helium laser operating at

6358.3 A. The laser beam was focused to a diameter of 100 nm before entering the sample cell which was mounted in the center of a thermostat bath (f0.05 "C at 20 or 30 "C). The goniometer axis of the cell holder was centered in the middle of the bath by a pair of x-y positioners.1° The scattered light was imaged onto a pinhole in front of a photomultiplier tube, the output of which was amplified and passed through a channel correlation analyzer that calculated the autocorrelation function of the photocurrentag The instrument was aligned by measuring the homodyne spectrum from a sample of polystyrene latex of known radius of 450 A, and the determined experimental diffusion coefficient was compared to the calculated one according to the Stokes-Einstein relation ( D = (4.65 f 0.03) X lo-* cm2 s-l), The decay rate, r, for monodisperse solutions of particle scatterers is equal to twice the product of the translational diffusion coefficient, D,and the square of the scattering vector, K = (4rn/&) sin (8/2), where n is the refractive index, 0 the scattering angle, Xo the wavelength in vacuo. The hydrodynamic radius, R H , of the scattering particle is defined through the StokesEinstein relation

Measurements of aqueous Triton X-100 micelles were performed at scattering angles between 40 and 135". The various output devices were used in conjunction with the correlation analyzer. Ultimately, all data were universally analyzed by means of a PDP 11/34 computer by using a nonlinear least-squares procedureall Small-Angle X-ray Scattering Measurements. The X-ray beam, coming from an Elliot rotating anode (GX 13), and focused from a 5 mm X 50 pm line source from the X-ray generator to a single glass mirror according to Franks.12 The focused nickel-filtered radiation passed the sample which was positioned in a thermally controlled holder midway between the mirror and the detector, which in turn was separated by a 35-cm vacuum path from the detector. The detector was a position-sensitive proportional detector (Tennelec, Oak Ridge, USA, PSD 100) with an x-y positioner and operated at 75 psi with argon as the inert gas. The line focused at the detector was 60 pm full-width at half-maximum which yielded a measured line of approximately 250 pm full-width at half-maximumwhen the detector resolution is included. The X-ray beam is flushed with helium from the mirror to the detector, except a short region around the sample holder. The position linearity of the detector was found to vary less than 1.5% over its 8.5 cm length, and the efficiency varied less than 10% over the whole width of the detector. The sample cell was a thin-walled glass capillary of 0.8-1.5 mm in

Shape and Size of a

Nonionic Micelle

diameter which was kept at constant temperature (h0.5 "C). In each experiment the scattered intensity has been normalized with respect to the energy of the primary beaml8J4which was measured after attenuation by properly calibrated Ni filters. The blank scattering from the instrument was subtracted from the observed intensity, e.g., buffer solutions and buffer solutions including Triton X-100, The solvent measurements were performed periodically throughout the measurements of the Triton X-100 solutions, and no difference within the statistical error was observed. The data were stored on a multichannel analyzer (Tracor, Northern, N 1705) and subsequently stored on magnetic tape. Calculations, background subtractions, etc. were perforrned on a PDP 11/30 computer. Collimation distortions due to the length of the incident beam and the width of the detector window were converted according to a program of Taylor and Schmidt,15kindly provided by Dr. Schmidt. Data analysis and processing are similar to those described recently.16J7

Results The objectives of the experiments presented here are to measure the degree of hydration of Triton X-100 micelles, assuming they are homodisperse, the shape, and possibly to detect a high (hydrocarbon) and a low density region (the solvent-permeated region) of this nonionic surfactant, as well as the morphology of the external surface area. Analytical Ultracentrifugation, Intrinsic Viscosity, Molecular Sieve Chromatography, and Inelastic Light Scattering. In order to detect any effects of detergent concentration on molecular weight and heterogeneity in molecular weight distribution, gel permeation chromatography on BioGel A 1.5 m (1.5 X 100 cm), sedimentation equilibrium measurements in the analytical ultracentrifuge, and sedimentation velocity measurements were performed. No changes were found in [77] (small in the globular range) and inspection of sedimentation equilibrium plots of this detergent in 0.01 M Tris-HC1, pH 8.0, a t 293 K were linear, indicating no change in weight-average molecular weight with Triton X-100 concentration as long as the ionic strength of the solvent was in the range of 0.01-0.1, and the temperature and pH were kept constant. Table I lists the molecular weights determined by analytical ultracentrifugation at the two temperatures, different ionic strength, and pH. The formation of larger aggregates dependti markedly on temperature as well as on the number of oxyethylene units. Gel permeation chromatography results in an equilibrium constant Kd for distribution of Triton X-100 in 0.01 M Tris-HC1, pH 8.0, between the gel pores, and the nonionic detergent remains the same, independent of the initial concentration of the detergent, consistent with the sedimentation equilibrium results as long as the temperature is constant throughout the measurernents. Calibration of the gel chromatography column with globular proteins of known molecular weights and Stokes' radius was performed and the Kd value for Triton X-100 at 20 "C was determined as 0.72, corresponding to ;I Stokes' radius of 40 A, while at 30 "C Kd = 0.56, equivalent to a Stokes' radius of 55 A. Diffusion Coefficients. The values for Do20,wobtained at different pH, ionic strength, and the two temperatures are listed in Table 11. The reproducibility of D,,, was within f l %for the same sample preparation and within 3.5% of the values listed in Table I1 for different sample preparations under identical conditions. The angular dependence of the autocorrelation function was studied for Triton X-100 (5 mg/mL) in 0.01 M Tris-HC1, pH 8.0, a t 20 and 30 "C. By varying the scattering angle from 40

The Journal of Physical Chemistry, Vol. 84, No. 6, 1980 601

TABLE 11: Translational Diffusion Coefficients, Radii of Gyration, and Molecular Weights Determined by Small-Angle X-ray Scattering

,,

10-700,~

R,, A Rg,a A R,,b A cm2 s - ? 4.96 j: 0.08 1 98.0 29.5 i 0.04 29.3 32.1 29.4 f 0.04 29.4 32.2 4.92 i 0.05 2 95.7 3 85.9 29.5 i 0.04 29.4 32.3 4.89 j: 0.08 4 86.5 29.5 j: 0.03 29.5 32.3 4.92 i 0.05 5 87.2 2 9 . 6 i 0.03 29.5 4.92 j: 0.05 32.4 4.12 i 0.04 3 9 . 2 i 0.05 39.5 6 151.0 42.8 7 150.8 39.3 k 0.05 39.6 42.9 4.13 i 0.05 4.15h 0.05 39.4h 0.05 39.6 8 151.0 43.0 9 149.0 39.2 j: 0.05 39.5 4.10 h 0.03 42.8 10 149.5 4.12 j: 0 . 0 3 39.5 f 0.05 39.7 43.1 4.13 i 0.03 11 151.3 39.3 i 0.05 39.6 42.9 a Calculated from inelastic light scattering measurements according t o R , = k.BT/(6nq,~0,,,,J2). Calculated from Do,,,,according to R , = k g T / ( 6 n ~ ~ , ( 5 / 3 ) " 2 0 0 , 0 , , ) . no.

103M,

to 135" the value of P was increased by a factor of 12.1. Over this range of scattering angles the deduced values of D were found to be independent of scattering angles within experimental error. If the results given in Table TI for D020,w are used to calculate the effective hydrodynamic radius accordin to eq 2, values of RH = 41.8 A at 20 "C and R H = 55.7 at 30 "C are obtained, corresponding to radii of gyration of R8 = 32.2 and 42.9 A, respectively, according to the relation R = (3/5)lI2RH,or for a circle and Rgdisk= 39.5 8, (30 "C) (disk) Rgdiak= 29.3 A (20 according to R - R H 1/(2)1/2.The latter values are in good agreement witkihe values obtained by small-angle X-ray scattering (Tables I1 and 111). Furthermore, a plot of the apparent translational diffusion coefficient vs. a, the volume fraction of Triton X-100 at 20 and 30 "C, is linear and can be fitted to a straight line satisfying the equations D = 4.90-5.26a (20 "C) and D = 4.15-4.92a (30 "C). Determinations of the linear frictional volume fraction coefficient,lRK,, yielded values of 12.25 at 20 "C and 11.92 at 30 "C, considerablyhigher than for a hard-sphere model which ranges from 6.55 to 7.20.18 The volume fractions calculated from Kfare 2.094 mL/g at 20 "C and 7.120 mL/g at 30 "C. Small-Angle X-ray Scattering. First inspection of the scattering curves, plotted according to Guinier,le at different Triton X-100 concentrations show straight lines with no curvature (Figure 1). It is not straightforward, as normally seen for protein or ribonucleic acid solutions, to extrapolate I(h,po)/c,to h = 0 since the scattering curves, even when extrapolated to zero concentration, show a maximum at low scattering angles. I(h,po)is the experimental intensity curve of the solute corrected for collimation distortions, with h = 4*/X sin '6 the scattering vector and po the density of the solvent in e/A3. c, is the concentration as measured by the ratio of the numbers of electrons of solute to solution. As is seen in Figure 1,at finite concentrations a shallow maximum is observed; the position, however, approaches h = 0 as the concentration decreases. At c, = 0 one obtains a straight line whose slope and intercept on the ordinate lead to the determination of the mass and of the radius of gyration. Furthermore, determinations of R at very low contrast, where very large absolute values of h, are obtained, were performed according to

d

"6)

with H ( x ) the correlation function H ( x ) = (1/2r2).

602

Paradks

The Journal of Physical Chemistry, Val. 84, No. 6, 1980

C 3.5

5.0

' 8.0 6.5

I

I

1.0

2.0

h2

Flgure

1.

X

I

J

lo4

1.0

2.0 h2X

3.0

4.0

lo4

Guinier plots of the experimental scattering curves at 20 and 30 O C . c is the concentration of Triton X-100 in mg/mL.

.fo"KI(h) h2[sin ( h x ) / ( h x ) ]dh and K a constant. The procedure has the advantage that the radius of gyration is calculated from the experimental (smeared) data points by using the whole scattering curve. P(R) is the distance distribution function which can be calculated from the correlation function H ( x ) by multiplying by r2. The advantage of applying the P(R) function is that in dilute systems a value of D can be defined for which the relation applies P(R) = 0 for R >> D, or H ( x ) = 0 for x >> 2R. From this it can be seen that the function P(R)differs from zero only in a certain region of real space, namely, between 0 and D, with D the maximum dimension of the particle (see below). The statistical error in the determined radius of gyration was propagated from the counting statistics in the fitting procedure. This also yields the intensity at zero including statistical error. Values of scattering angle (Io), Io/c, when c is the Triton X-100 concentration, were calculated and varied less than 6.5%, revealing that aggregation effects were not present as well as dissociation. Since the intensity of the primary beam was normalized, the recorded intensity curves were put on an absolute scale,13J4hence the molecular weights can be determined, which are listed in Table 11. The other experimentally determined parameters from small-angle X-ray scattering experiments of this Triton X-100 micelle are listed in Table 11. Since the normalized intensity at zero scattering angle in(O,po) is defined as

I

/

-30-

Figure 2. Plot of (in(O,p,,)/cJ"* vs. po at 20 (0)and 30

CAI -P O W

with \k the partial specific volume of the detergent in A3/e, po the electron density of the solvent in e/A3, and p1 the average electron density of the Triton X-100 micelle; a plot of (in(O,po)/ce)l/zof eq 6 vs. (po - p l ) yields a straight line (Figure 2), giving Vl/m1/2 from the slope. Therefore the

(A).

7

1.0

with z the thickness of the sample which has been measured with a microscope in electrons/cm2, q = (7.9 X X2 where X = 1.54 A is the wavelength of irradiation, and Eo the energy of the incident beam determined by a set of calibrated Ni filters. I(O,po) is the scattering magnitude at zero scattering angle corrected for collimation distortions. cVeis the volume concentration in electrons per A3 of solution, V1 the volume of the Triton X-100 particle, and m the number of electrons per particle according to in(0) m= (6)

O C

ld do (g/cm3)

Flgure 3. Determination of the partial specific volumes according to eq 1 at 20 ( 0 )and 30 O C (A).

volume and p1 from the intercept at po = p1 can be obtained (Table 111). Because densimetry measurementa of Triton X-100 at various concentrations of glycerol added to the aqueous solution of the detergent did not show any departure from a straight line according to eq l (Figure 3), we can assume that the partial specific volume of the detergent does not change; hence the number of electrons, m, can be determined with confidence (Table 11), along with the molecular volume. The independent estimate of the hydrodynamic volume can be made through those conducted densimetry measurements, since the quantity ( V2 + VH8) (eq 1)is the volume V of the hydrated nonionic + detergent per gram of unsolvated particles. Since avH,) = V,(0.602)M;1, V1 was calculated as 3.41 X lo6 A3

(v2

The Journal of Physical Chemistry, Vol. 84, No. 6, 1980 603

Shape and Size of a Nonionic Micelle

TABLE 111: Morphological Parameters for Triton X-100 Obtained by Small-Angle X-ray Scattering in 0.01 M Tris-HC1, pH 8.0, a t Two Temperatures

---

T = 293 K R,, a Rc, a 1 0 3 ~ ~ 103mp 105v,, 8 3 R, A Rs, a R,, '4

29.46 i 0.03 28.1 ?: 0.1 95.7 i 0.5 50.64 3.35 38.2 40.6 42.4 a, a - ' 0.068 u 2 , mL g - ' 0.9125 i 0.0002 0.2550 Pi, e / a w, g of H 2 0 / g of Triton X-100 1.18 1 0 4 p i V , ,e 8.54 io-yxjq)2, 2.58 Rd, 17.6 K , '4 25.0 lc, a 100 1,, '4 147 a,, A 2 1.68 X l o 2 p = b/a. axiail ratio from 3 V ' 2 t (b/a)2 ___ 1.58 4i7RV3 5 s = 2 + (b/a)' 1.56 ?g 5 R,', e/A 5.7 1

T = 303 K 39.2 i 0.05 31.2 ?I 0.2 150.0 i. 1.4 79.41 4.32 50.9 67.6 57.8 0.065 0.9103 i 0.0005 0.2051 1.21 8.86 15.5 20.1 30.2 120 154 1.82 X 10' 1.72

a02

0.01

0.03

~p ( e / I 3 )

Flgure 4. Dependence of the radius of gyration upon A p at 20 (0) and 30 "C (A) for Triton X-100 in 0.02 M Tris-HCI, pH 8.0. a

1 0.1

20°C

I 05

1.0

1.5

21

10-4

h2

1.82 7.09

a pi is the buoyant density in e/A3;R, R v , and R s are the radii of the spheres whose volume, surface area, and radius of gyration are V, S, and R,; the dispersion of these three values is an indication for asymmetry in shape. piV1 is the number of electrons of one solvated Triton X-100 particle. (ZG)2is the square average of electron density at buoyancy. R X 2(e/.&)is the second moment of one particle at buoyancy or the mean square distance of fluctuation from the center shape. R , is the radius of gyration of the cross section; Rd is the radius of gyration of the thickness. l,, l,, and a, are the coherence length, the length of inhomogeneity, and the coherence area, respectively, all three parameters determined from the correlation function. o( is the ratio of external surface to volume occupied by one solute particle in solution; m is the number o f electrons associated with a particle having the hydrodynamic volume V,.

(20 " C ) and 4.32 X lo5 A3 (30 "C). The value of pi, the average electron density at buoyancy, was determined to be pi = 0.2545 e/A3 which is in fair agreement with the value found from densimetry of pI = 0.2480 e/A3. This reveals a degree of hydration of 1.15 g/mL when applying eq 1 for Triton X-100 of molecular weight 95 700 (aggregation number 153) and a mean electron density of Triton X-100 at 20 "C in 0.01 M Tris-HC1 of 0.3448 e/A3, comparable witlh the mean electron density of 0.3500 e/A3 calculated according to (pi) = (m/M)(1/U2)0.602 (e/A3) with m/M = 0.5292 (Table 11) and from the chemical composition. Dependence of the Radius of Gyration upon po. According to Luzzati et a1.20and Stuhrmann21a linear dependence of the apparent radius of gyration upon the electron density contrast should exist if the sample under investigation is an ideal solution of identical particles (monodisperse) and the scattering density distribution of the particle is independent of the scattering density of the solvent. Since we know p 1 from the plots of Figures 2 and 3, we can determine Ap = p o - pl, so the radii of gyration of Triton X-100 at different solvent densities ( p o ) can be plotted as R,2(Ap) vs. A p (Figure 4). The experimental points fit a straight line from which we obtain the radius of gyration of the excluded volume of Triton X-100 from

I

I

I:

0.1

30°C

a5

15

1.0

2n x

h2

Figure 5. Guinier plot of the cross-sectional scattering curve for Triton X-100 at 20 (A) and 30 "C (B).

the slope, and the intercept yields the average square distance from the center of gravity of Vl(R)of the detergent (Table 111). The sign of the second moment i8 negative ((rn2),/v= -104 A2, Table 111),indicating that the main scattering mass is near the center rather than close to the surface of the particle. The radius of gyration obtained from the slope of Figure 4 was determined to be 29.8 (20 "C)an 39.3 8, (30 "C). The determined radii of gyration correspond to radii of 38.2 and 50.9 A, respectively, when assuming a spherical overall shape. These results bear some importance in considerations regarding particle shape, since a spherical Triton X-100 particle would have the smallest dimension which is inconsistent with the results obtained from the correlation function and the pair distribution function (P(R))where the maximum dimensions (see below) yielded values of 105 and 120 8, at 20 and 30 "C, respectively. This comparison thus shows that both Triton X-100 micelles at 20 and at 30 "C must be asymmetric and not spherical. The suspected asymmetry from spherical shape of the Triton X-100 micelles at both temperatures can be verified by determining the radii of gyration of the cross section and the thickness by plotting the quantities I(h)hvs. h2 or I(h)h2vs. h2,respectively22 (Figure 5). The values obtained for the radii of gyration of the cross sections are R , = 28.1 8, (20 "C) and R, = 31.2 A (30 "C) and those for the thickness are Rd = 17.6 8, (20 "C)and Rd = 20.1 8, (30 "C) (see Table 111). The values obtained result in a thickness of 33.7 8, (20 "C) and 40.5 8, (30 "C)and a total length of 104 8, at 20 "C and 125 8, at 30 "C for the Triton X-100 micelles which are consistent

604

The Journal of Physical Chemistry, Vol. 84, No. 6, 1980

Paradies

TABLE IV: High and Low Electron Density Regions of the Triton X-100 Micelle at Two Different TemDeratures

7

parameter

T = 293 K

T = 303 K

P H , e/A3

0.3786 0.3700 0.17 54 0.1820 0.146 0.25 0.754 0.750 5.8 x 104 1.08 x 105 VH, A 3 VL, A 3 25.2 x 104 3.24 x 105 nH, l o 4e 2.19 2.79 nL, lo4 e 4.4 5.15 18.5 22.8 Rg,H, A 24.0 29.6 Ro,H, a W H ,g of H,O/g of Triton X-100 0.15 0.18 w1,,g of H,O/g of Triton X-100 1.00 1.05 a The subscripts H and L refer t o the intrinsic high and low electron density region of the Triton X-100 particles. Rg,p is the radius of gyration of the high electron density region, Ro,H the hydrodynamic radius calculated from R g , or ~ from VH assuming a sphere which is equivalent to a hydrodynamic particle having the volume VH and radius RH. n~ and n L refer to the number of electrons of unhydrated regions within the Triton X-100 particle. @ H and c $ ~are the volume fractions of high and low electron density of Triton X-100. P L , e/A3 @Hi @ Li

Flgure 6. A plot of Q(po) vs. Ap-' for Triton X-100 at 20 (0)and 30 OC (A). The intercept wRh the abscissa gives App and on the ordinate 10 v,.

with the values obtained from the correlation and pair distribution functions. Determination of the Specific Inner Surface Area of Triton X-100. The asymptotic trend of the small-angle X-ray scattering curve for Triton X-100 at different glycerol concentrations follows Porod's l a ~ and ~ ~ can9be~ ~ Triton X-100, can be determined from a plot of Q(po) vs. analyzed according to Luzzati20 by a plot of h41(h)vs. h4 A P - ~where Ap = p o - p1 (Figure 6). This plot for Triton which yielded a straight line for this nonionic detergent; X-100 is a straight line, indicating that the molecule is the specific inner surface area and the volume of Triton independent of the glycerol concentration and homogeX-100 can be determined. Thus the slope of a plot of 141(h,po) vs. I(h)h4will give the constant term which has was obtained neous in density. From the intercept to be subtracted from I(h,po)at higher h values,16and the and was found to be 3.3 X lom3e2/A*. This indicates that intercept will give the specific surface. The value for the mean density fluctuation of Triton X-100 at buoyancy thespecific surface is CY = 0.0675 A-1, which is somewhat higher is Api = 5.75 X e/A3. This value is comparable to the than the value obtained from the characteristic function value obtained from the intercept of the abscissa of a plot (H(R))according to Debye and B u e ~ h e P, ~o~r ~ d and ,~~ of Rg2G vs. which was 5.80 X e A3, taking the Kirste and P ~ r o dwhich , ~ ~ was found to be CY = 0.0669 A-' mean electron density as ( pl) = 0.3448 e/ I!3. With respect from the initial slope of H(R). The development of H(R) to accuracy it should be noted that H(O,po) whose value in a power series results not only in a determination of the involves the asymptotic extrapolation of I(h,po),is probably specific inner surface area of Triton X-100 but also of the less accurate than I(O,po) which is defined by the extrapayerage curvature of the molecule which was found to be olation of I(h,po)to h = 0. K = 28.2 A from the slope of H"(R) at the origin. This Considering the sign of the intercept on the ordinate of value for the average curvature is much smaller than the is the plotof R, ,pp.2Ap vs. Ap, which is > 0, and since value obtained from the radius of gyration (see Table 111)) known from two mdependent methods, we found it feasible which was Ro = 38.3 A,assuming a particle having a degree to calculate a high and a low electron density region for of hydration of w = 1.15 mL/g and the determined volume the Triton X-100 particle, assuming a two-step density V = 3.35 X lo3A3. I t must be inferred that the external function within this Triton X-100 micelle which is verified surface of the Triton X-100 particle of molecular weight by the plot of h41(h)vs. h. Since the average electron 700 is to some extent convoluted and highly hydrated. 95 density of the nonionic surfactant is p1 = 0.3448 e/A3, the The volume which was determined from the charachigh electron density region is calculated to be pH = 0.3786 teristic function was found to be V = 3.40 X lo5A3, in fair e/A3 and the value for the low electron density region is agreement with the value found from the plot [i,(O,po)/ pL = 0.1754 e/A3, assuming a Triton X-100 particle of vs. Ap (Figure 2) of 3.35 X lo5 A3.Calculation of the molecular weight 95 700 consisting of two phases of difdegree of hydration is now possible and a value of 1.15 ferent densities pH and pL. Note the average electron mL/g is obtained when we apply the determined partial density of the -dry Triton X-100 is 0.365 e/A3. Since 4 ~ p specific volume of V2 = 0.9125 mL g-' (Table 111). The + & Q ~ = ~ IApt1 and &ApH + ~ L A P = L 0, we can calculate value determined from densimetry experiments according ) the volume fractions of the high (PH) and low ( p ~ density to eq 1 was 1.08 g/mL (g of HzO/g of Triton X-loo), whereas Robson and Dennis7 calculated a value of 1.18 (g of the surfactant because p L , p ~ and , are known. The of H20/g of surfactant) on geometrical considerations for values are listed in Table IV. As can be seen from Table an oblate ellipsoid of revolution with axial ratio of 1.9. IV, the volume of the high electron density part is rather Since the scattering curves are known from h = 0 to small in comparison to the total volume, namely, only 25 %. infinity and are on an absolute scale, a plot of eq 7, the Assuming a pseudospherical high electron density part within the Triton X-100 molecule, the pseudospherical radius of this region would be 24.0 8, which is equivalent to a radius of gyration of 18.5 A. Measurements at 303 K. Measurements also were conducted at elevated temperatures, especially at 30 "C, (7) resulting in a higher molecular weight, a different translational diffusion coefficient, and a different Stokes' radius square average of electron density fluctuation Gi2of

Gi2

Zi

~

~

The Journal of Physical Chemistiy, Vol. 84, No. 6, 1980 005

Shape and Size of a Nonionic Micelle

IA

depend on the contrast (Ap) in the same way as the radius of gyration or the characteristic scattering function (I(h,po)) = Zs t Zvs Ap + Zv(Ap)2. Fourier transformation (eq 3) of I(h,po)directly yields the distance distribution functions

Ap=O.O4 e / i 3

D(R) = &(R)

IB

C

n 0, x

Ap-0.04

ell3

Ap.O.01

,/i3

3

-2 2 d

1

R

(8)

Figure 7. Distance disbibution function D$R) and D,(R) for Trlton X-100 at different contrast: (a) Ap = 0.04 e/A at 20 (-) and at 30 "C (----) for D@); (b) Ds(R)at Ap = 0.04 e/A3 at 20 (-) and 30 "C (- --); (c) DdR) at Ap = 0.01 e/A3 at 20 (-) and at 30 OC (-----).

--

from gel permeation chromatography on BioGel A 1.5 m. The radius of gyration obtained from a Guinier plot yielded a vdue of 39.2 f 1.2 A, comparable to the value of 39.4 A obtained according to eq 5. Furthermore, sedimentation equilibrium plots were linear, indicating no change in weight-average molecular weight with concentration which reached maximal values of about 7.8-10.0 mg/mL, sirriilar to the measurements at 20 "C. This shows that the micelles have a relatively narrow size distribution in which number and weight averages are closely similar. The best value for the molecular weight in 0.01 M K2HP04, pH 7.0, containing 0.001 M NaCl at 30 "C was 150000 f 2500 (Table I). The molecular weight, the volume, and the specific inner surface area of this micelle form obtained from small-angle X-ray scattering experiments are listed in Table 111. In contrast to the micelle of molecular weight 95700, the 150000 form was very sensitive to changes in the electron density of the buffer because at Ap = 0.05 e/A3 the solutions were no longer monodisperse. Therefore, thallium nitrate was used instead of glycerol since only small amounts of T1N03 are needed to raise the electron density of the solvent. The values obtained are listed in Tables 1-111. From the distance distribution function we obtained for the largest dimension of this particular micelle a value of D(R) = 119 A (Figure 7) which is considerably larger than that of 105 A for the 95700 Triton X-100 micelle. The distance distribution functions for Triton X-100

+ D,s(R)Ap + Dv(R)(AP)~

(8)

with Zs the scattering function reflecting the internal structure, Iv the scattering function of the excluded volume, and lVs the mixed term. &(I?), which is the transform of Zv(h), is always positive, whereas Dvs(R)and &(@ can be positive or negative. The transforms of the scattering curves are shown in parts b and c of Figure 7 at different contrast for Triton X-100 at 20 "C which are obtained by decomposition of the scattering function Z(h,po). Dv(R) and DS(R) are pair distribution functions at infinite and zero contrast (Ap = 0), respectively, which can be analyzed independently and result in a separate description of surface structure and internal structure. The most important information from these calculations, calculated by a least-squares algorithm taking the error limits of the experimental data into account, is that the D,(R) function is practically identical with the experimental results at Ap = 0.01 e/A3 and agrees reasonably with the theoretical function of an obla& ellipsoid of revolution with dimensions of a = 31.8 A and b = 50.2 A. However, the distance distribution function, Dv(R), from the excluded volume deviates from the oblate model (Ap = 0.01 e/A3) no only at larger scattering angles but also at medium scattering angles. Although it can be difficult to decide whether the differences are significant at larger scattering angles, because in this region of low intensity the experimental errors can be of comparable magnitude, the deviation in the medium region of scattering angles is significant. DV(R)at high contrast (Ap = 0.06 e/A3) of the Triton X-100 micelle at 20 "C supports the disklike structure, whereas at lower contrast (Ap = 0.01 e/A3) the maximum of Dv(R) is shifted toward a smaller value. This is consistent with the data obtained from H(R) yielding the average curvature of the external surface area (25 A) and the distribution of chords,16,26indicating that deep convolutions of the external surface have to be present in the Triton X-100 micelle. Furthermore, the anisotropy increased from 1.05 to 1.34 for the 150000micelle particle, indicating a more asymmetric structure than for the 95 700 molecular weight micelle of Triton X-100. Considering a volume of 4.31 X lo5 A3 and an axial ratio of 1.72 in the case of an oblate ellipsoid of revolution, the hydrodynamic scattering equivalent will have dimensions of half axes a = 32.8 A and b = 56.2 A which suits the scattering curve quite well (Figure 8). Moreover, considering a prolate ellipsoid of revolution with an axial ratio of 0.56 with equivalent dimensions of a = 68.9 A and b = 38.2 A, a volume of V = 7.6 X lo6 A3 is obtained, which is not comparable with the experimental volume obtained from ~ po plot or from the correlation function. a ( ~ ( O ) / C J ~ /vs. Therefore, the most reliable description of this 150 000 micelle of Triton X-100 is that of an oblate ellipsoid of revolution with half axes of a = 32.8 A and b = 56.2 A, which is similar in general shape to the 95 700 Triton X-100 micelle but with different dimensions, namely, a = 31.8 A and b = 50.2 A (Figure 8). For this asymmetry, volume, and radius of gyration of the high density region in this 150000 particle of Triton X-100 with Mhigh= 3.18 X lo4 e, the asymmetry of this region can be calculated to be equivalent to an oblate ellipsoid of revolution with half axes of a = 24.4 A and b = 46.8 A, revealing a degree of hydration of 0.21 g of water per gram electron dense region. In comparison to the 95 700 particle, the 150 000 dalton

808

The Journal of Physical Chemistry, Vol. 84, No. 6, 1980

log h

Paradies

log h

Flgure 8. Theoretical (----) and experimental (-) Scattering curves for various models of Triton X-100 at 20 and 30 "C,all normalized to the radius of gyration of R, = 29.5 A (20 "C)and R , = 39.2 A (30 OC); Vis the axial ratios for various ellipsoids of revolution which are equlvalent in scattering.

molecule is more asymmetric than the nonionic surfactant at 20 "C in 0.01 M Tris-HC1, pH 8.0.

Discussion The major concern of this investigation was heterogeneities, changes of the micellar structure upon addition of glycerol or thallium nitrate, since sucrose affects the micellar structure considerably, as well as the partial specific volume. Therefore, the main conclusions of this investigation are dependent upon the assumption that the Triton X-100 micelles at both temperatures are monodisperse and that their internal structure is independent of the additives TlN03 or glycerol. However, these assumptions are met by other independent methods: (i) sedimentation and gel filtration behavior do not show any indication of polydispersity, (ii) the molecular weights determined by small-angle X-ray scattering are in agreement with other methods reported in this paper and published work by ~ t h e r s(iii) , ~the ~ ~linear ~ ~ relationship ~ ~ between R, app2Ap vs. Ap is consistent with the assumption that the internal structure is independent of the solvent density. Furthermore, the presence of some morphological heterogeneities does not have serious consequences on the X-ray scattering analysis but density heterogeneities may have more serious effects. The buoyant density and the mean square electronic distribution at buoyancy, @, which has been evaluated from the linearity of the plot of Q(po) vs. A P - ~at higher and lower solvent densities of pl, is determined by pycnometric measurements indicating that the accuracy is sufficiently reliable to disregard possible serious errors of pl. Furthermore, the isopycnic density is very close to the density of water so that I(O,po) is mostly measured at po > pl; consequently, the plot Q(po) vs. &?(Figure 6) is quite insensitive to pl. Therefore, this value was closer according to the isopycnic density. In this plot the accuracy of I(O,po) is higher than the integral since it involves the asymptotic extrapolation of I(h,po)and the evaluation of the constant term according to Luzzati et alarn Since Porod's law23is obeyed, which does not necessarily mean a step boundary has to be involved, a possible assumption of a two-step level electron density for Triton X-100 can be proposed. All further calculations which are listed in Tables I11 and IV rely on this assumption.

The demonstration of the sizes of the hydrophilic and hydrophobic portions of the Triton X-100 molecule requires the use of the additives glycerol or T1N03. Polydispersity in size, shape, and molecular weight has the effect of blurring the fringes. However, the presence of the well-defined fringes in the scattering curve (Figure 8) indicates that morphological polydispersity, even in the presence of TlN03 or glycerol, is not too large. Consequences with regard to polydispersity upon the additives on eq 5 and 7 are to replace some of the determined parameters by average values over the particle population. Therefore, the form of the equations is preserved and will still be valid if the micelles were found not to be homodisperse. Furthermore, the plots of Q(po) vs. (AP)-~and RZAp vs. (Ap) are not very adequate tests for morphological homogeneity. Shape and Degree of Hydration. Hydrodynamic, NMR, and theoretical studies27indicate that Triton X-100 micelles in aqueous solution are very hydrated particles whose shapes are not far from spherical with low aggregation numbers, but asymmetric at higher aggregation numbers. Moreover, the ratios of the radius of gyration and Mi3is 0.65 for Triton X-100 at 20 "C and 0.72 at 30 "C. These are not typical values for hard-sphere particles which are in the range of 0.45-0.50. The certain asymmetry for the micelles at both temperatures is furthermore substantiated by the pregence of a thickness factor (Figure 6, Table 111) and a radius of gyration of the cross section. Both values obtained are more consistent with an oblate ellipsoid of revolution rather than a prolate ellipsoid of revolution (Figure 8). Inferences with respect to shape of the Triton X-100 micelles can be made by comparing the experimental scattering curves with theoretical ones that are equivalent in scattering. The presence of a secondary maximum at h = 22.4 X A-1 and a minimum at h = 15.4 X lo-' A-1 in the experimental scattering curve for the micelle of molecular weight 95 700 indicates that the molecules are not very asymmetric or that their shape, e.g., external surface, is complex. By comparison with theoretical scattering functions of ellipsoids of revolution, experimental values, all normalized to the same radius of gyration (29.5 A), are in reasonable agreement with an oblate ellipsoid of revolution with half axes a = 31.8 A and b =

Shape and Size of a Nonionic Micelle

50.2 A, but do not fit the maximum and minimum at the same angles as found experimentally. This may be due to the fact that the molecule has a complex external shape, e.g., protuberances, folding of polyoxyethylene chains, and possibly a nonuniform internal structure (see Tables I11 and IV). The same is valid for the Triton X-100 micelle at 30 "C, but the fit of the experimental scattering curve with the theoretical one is better than for the molecule at 20 "C, although the agreement of the maximum and minimum at the same scattering angles is not perfect. Since the sign of the quantity (m2),/uis negative (-104 and -139 A2) or the intercept of the ordinate of Figure 4 is positive,2829 the high electron density part is located inside the Triton X-100 particle having an electron density of 0.3786 e/A3, corresponding to a density of 0.87 g for this region. Furthermore, the degree of hydration is based on the measured hydrodynamic volume obtained by small-angle X-ray scattering measurements and subtracted from the volume occupied by the detergent molecules using the measureid ii2 value and is in agreement with recent Considerations from Robson and Dennis7 on theoretical grounds from known data compiled in the literature, but without consideration of the convoluted external surface area. Moreover, the hydrodynamic volume fits an oblate ellipsoid of revolution which is consistent with the experimental scattering curve (Figure 8). Furthermore, the determined average curvature of the external surface of both micelles is smaller than the average hydrodynamic radius for both particles (R, = (ab2)'I3),indicating that the external surface area is convoluted. This is further substantiated b:y calculation of the distribution of chords,26 revealing that the overall shape of the 95 700 dalton particle is "quasi-spherical" with wrinkles and spikes distributed over its surface, whereas for the particle at 30 "C the asymmetry is more pronounced, revealing two curvatures of kl = 25.4 8, and k2 = 43.0 8, with k = (klk2)ll3= 30.2 8, obtained from the correlation function experimentally, which are considerably smaller than the overall shape with dimensions of a = 32.8 8, and b = 56.2 A. Furthermore, by raising the low density regions with glycerol of ?'1N03 the magnitude of the distribution of chords is much lower than it would be expected for a compact sphiere; this indicates that the hydrophobic core is not compact, and a fraction of the interface (high-low density region) is due to internal compartimentation, consistent with the results obtained from Dv(R) and Ds(R) (Figure 7) at different contrast &. The polydispersity of the oxyethylene chains could be the cause of the nonuniform distribution around the micelle, allowing the overall shape to be closer to a spherical one. Furthermore, Robson and Dennis30 observed a single peak and a similar Stokes' radius on gel chromatography (BioGel A-5 m) at 20 " C , but at higher temperature (40 "C)they fourid heterogeneity and polydispersity of Triton

The Journal of Physical Chemistry, Vol. 84,

No. 6, 1980 807

X-100 micelles. No heterogeneity and polydispersity were found at 20 and 30 "C for Triton X-100 micelles at pH 5.5 in 0.05 M sodium acetate or 0.01 M Tris-HC1, pH 8.0, buffer in the present study.

Conclusion From this work it can be concluded that the high electron density region of the Triton X-100 micelles has a radius of about 25 A for the 95 700 particle and 29.9 A for the 150 000 particle, which indicates that approximately 38 monomers are the entire high electron density region resulting in a degree of hydration of about 0.29 g of H20/g of Triton X-100 in the high density region. Thus for a whole micelle, the hydrophobic region would be 25-27 A and the hydrophilic region would be about 25 A, resulting in an overall radius of 51 A. Acknowledgment. The author is indebted to the anonymous referees for their suggestions and advice with respeat to the presentation of this work.

References and Notes (1) L. M. Krlshner and W. D. Hubbard, J. Phys. Chem., 58, 1163 (1954). (2) K. Kuriyama, Kolloid. Z., 181, 144 (1962). 13) C. J. Biazelle and D. 8. Millar, Biophys. Chem., 3, 355 (1975). S. Yedaar. Y. Barenbiz. and V. G. CooDer. Biochem. B i o s. . ~_h ~Acta, 363, s i (i974). M. Corti and V. Degiorgio, Opt. Commun., 14, 358 (1975). C. Tanford, "The Hydrophobic Effect Formation of Micelles and Biological Membranes", Wiley, New York, 1973, pp 74-80. R. J. Robson and E. A. Dennis, J. Phys. Chem., 81, 1075 (1977). C. Tanford. J . Phvs. Chem.. 78. 2469 (1974). H. H. Paradies, Bidchem. Biophys. Res. Comhun., 88, 810 (1979); J. Biol. Chem., 245, 7495 (1979). H. H. Paradies and W. Vettermann, Arch. Biochem. Blo~hys., 194, . . 88 (1979). P. R. Berington, "Data Reduction and Error Analysis for the Physical Sciences", McGraw-Hill, New York, 1969, pp 220-222. A. Franks, Br. J. Appl. Phys., 9, 349 (1954). V. Luzzati, Acta Crystallogr., 13, 939 (1960). H. H. Paradies, W. Vettermann, and G. Werz, Protoplasm, 92, 43 (1977); H. H. Paradies and W. Vettermann, Arch. Biochem. Bkphys., 191, 169 (1978). T. R. Taylor and P. W. SchmMt, J. Appl. Crystallogr., 2, 153 (1969). H. H. Paradies, J. Zimmermann, and U. D. Schmidt, J. Bbl. Chem., 253, 8972 (1978). H. H. Paradies and W. Vettermann, Arch. Biochem. Biophys., W l , 169 (1978). H. Z. Cummins and E. R. Pike, Ed., "Photon Correlation and Light Beating Spectroscopy", Plenum Press, London, 1979, p 441. A. Guinier, Ann. Phys., 12, 161 (1939). V. Luzzati, A. Tardieu, L. Mateu, and H. B. Stuhrmann, J. Mol. Blol., 101, 115 (1976). H. B. Stuhrmann, Z.Phys. Chem. (FrankfMamMafi),72, 185(1970). G. Porod, Acta Phys. Aust., 2, 255 (1948). G. Porod, Kollold. Z.,124, 73 (1951). P. Debye and H. M. Bueche, J . Appl. Phys., 20, 518 (1949). R. Kirste and G. Porod, Kolloid. Z.,184, 1 (1962). J. Mering and D. Tschoubar, J. Appl. Cvsfallogr., 1, 153 (1968). F. Podo, A. Ray, and G. NBmethy, J. Am. Chem. Soc., 95, 6164 (1973). D. K. Carpenter and W. L. Mattice, Biopolymers, 18, 67 (1977). W. L. Mattice and D. K. Carpenter, Biopolymers, 18, 81 (1977). R. J. Robson and E. A. Dennis, Biochim. Blophys. Acta, 573, 489 (1979).