P. ANNEHILTNERAND IRVIN M. KRIEGER
2386
Diffraction of Light by Ordered Suspensions
by P. Anne Hiltner and Irvin M. Krieger Department of Chemistry and the Division of Macromolecular Science, Case Western Reserve University, Cleveland, Ohw (Received December 97,1968)
An experiment is described for applying the Bragg equation to determine lattice parameters in ordered colloidal dispersions. Measurements on electrolyte-free monodisperse latexes show diffraction peaks; upon dilution of the dispersions, the peaks become increasingly sharp and the positions of the maxima shift to indicate increased interparticle spacing. The center-to-center particle separation D, the particle diameter Do,and the ~ 0.74, the value 0.74 being the volume fraction of spheres in a volume fraction 4 obey the relation + ( D / D o )= close-packed array. The variation of peak width with concentration is accounted for quantitatively in terms of the scattering power of homogeneous spheres in a perfect crystalline array. Diffraction peaks were observed at volume fractions as low as 1%, where the particles are four diameters apart. The existence of order in such dilute dispersions is attributed to long-range electrostatic forces.
Concentrated aqueous suspensions of spherical polymer particles can be prepared by emulsion polymerization. Such suspensions appear milk-white in color, and are therefore called “latexes;” they usually contain a wide range of particle sizes. By employing special polymerization conditions, it has been found possible in this laboratory and elsewhere to produce particles with a very narrow siEe distribution. These “monodisperse” latexes exhibit interesting diffraction phenomena both in liquid suspension and as dried films. When the particle diameter falls in the range 150-500 mp and the volume fraction of polymer exceeds 40%, monodisperse suspensions exhibit iridescent colors characteristic of the particle size. Luck, et uL,l demonstrated that the color is due to I .iiJg diffraction of visible light from hexagonally ordered layers of particles stacked parallel to the surface of the container. They state that the three-dimensional lattice is face-centered cubic, although the distance between layers would be the same for hexagonal close packing. Luck and coworkers assumed that the particles in the array were touching, so that the lattice spacing was determined only by the particle diameter. The latex particles are charged, however, and the aqueous phase contains small ions. Each particle is surrounded by an electrical double layer, and this layer should play an important role in the ordering process. This phenomenon, and its effect on the iridescence, was discussed by Vanderhoff, Tausk, and Overbeek,2 who also observed that interparticle spacing increases when deionized latex is diluted. A program was initiated to study this phenomenon quantitatively, exploring the effects of particle size and concentration, electrolyte content, and surface charge. This paper describes an experiment which permits accurate measurement of the diffraction and its application to the investigation of order in electrolyte-free latexes as a function of particle concentration. The Journal of Physical Chemistry
Experimental Section Apparatus. Diffraction measurements were made with a modified General Electric X-ray goniometer table (Model A4954A), whose original function was to measure the angular positions of X-ray diffraction maxima. The main protractor and the sample holder rotate coaxially and are coupled so that the sample moves through precisely one-half the angle through which the protractor moves. The X-ray components were removed from the table and two lengths of triangular aluminum optical bench were mounted, one on the rotatable protractor and one in place of the stationary X-ray tube. This creates a versatile instrument on which optical components can be manipulated without disturbing the angular positioning. For diffraction experiments on latex suspensions, a Bausch and Lomb Model 33-86-02 scanning monochromator with a tungsten lamp was mounted on the protractor. A collimating lens and a variable slit completed the light source arrangement. An RCA 6217 photomultiplier tube, sensitive in the visible wavelength region, was mounted on the stationary optical bench. To utilize the original alignment of the goniometer table, it is essential that the light beam not be refracted before it strikes the surface of the suspension. A special cell was designed, one wall of which is a solid glass halfcylinder (Figure 1). The light beam enters and leaves the cell along radii of the half-cylinder, and hence no refraction occurs at the air-glass interface. The sample holder supplied with the instrument was used, with minor modification to accommodate the cylindrical cell. (1) W. Luck, M. Klier, and H. Wesslau, Ber. Bunsenges. Phys. Chem., 67, 7 5 , 84 (1963); Naturwissenschaften, SO, 486 (1963). (2) J. W. Vanderhoff, H. V. van den Hul, R. J. M. Tausk, and J. G. Th. Overbeek, “The Preparation of Monodisperse Latexes with Well-Characterised Surfaces,” Proceedings of the Conference on Clean Surfaces, Marcel Dekker, New York, N. Y., in press.
2387
DIFFRACTION OF LIGHTBY ORDERED SUSPENSIONS mXo/n, = 2d[1
-
(n,/n,)2 cos2 0 ~ 1 ’ ’ ~
(3)
The distance D between particle centers was calculated from the lattice parameter d, on the assumption that the particles form a close-packed lattice (either face-centered cubic or hexagonal), with hexagonally ordered planes parallel to the interface. Thus d = (2/3)”’D, and eq 3 becomes
kJ-
SUSPENSION
m2X02= (S/3)Dz(ns2 - ne2cos20),
Figure 1. Glass cell for measuring diffraction by latex suspensions (viewed from above).
Since the goniometer table is programmed to scan through angles with the Bragg condition fulfilled, diffraction measurements were made a t fixed wavelengths. However, it is also possible with this equipment to man through wavelengths at fixed angle. A typical series of diffraction scans at fixed wavelengths is shown in Figure 2.
Figure 2. Diffraction scans a t five different wavelengths, = 9.2%, D = 346.7 mp. Intensities are on an arbitrary scale.
(4)
Measurements were made at various wavelengths, and a graph of i o 2 vs. cos2 00 constructed. A typical graph is shown in Figure 3. From the slope and intercept of the best straight line, obtained by least-squares analysis, the spacing D and the mean refractice index na were calculated. This is the calculation procedure used by Luck, et aE. Particle spacings in the range 2501000 mp could be measured in this way.
DO = 171 mp, d,
0
I
I
I
I
0.2
0.4
0.6
0.8
3
cos*eo
Analysis of Data. The relationship among the wavelength h of the radiation, the angle 0 of the diffraction maximum, and the lattice parameter d is given by Bragg’s equation
mX
= 2d sin 0
(1) where m is the diffraction order. If n, is the refractive index of the suspension and Xo the wavelength in air, then X = Xo/n,. Also, since the light is refracted at the glass-suspension interface, Snell’s law must be applied to relate the goniometer angle 00 to the diffraction angle e sin ( n / 2 - eo> = -cos - -eo - n, sin (n/2 - 6) cos 0 n,
(2)
where ng is the refractive index of the glass. With these substitutions, Bragg’s law becomes
Figure 3. Graphical test of modified Bragg relation (eq 4):0, D = 286.0 mp; M, D = 458.6 mp; 0 and 0, D = 712.8 mp, first- and second-order reflections, respectively.
Materials. The method used to prepare the uniform latex studied here has been described in detail elsewhere. Styrene was emulsion-polymerized in an aqueous suspension containing both ionic and nonionic surfactant; the reaction was initiated by free radicals from the thermal decomposition of K2S208initiator. The final latexes were highly iridescent and contained ca. 50% by volume of polymer. Electron microscopic examination showed uniformity indices below 1-01. (The uniformity index is the ratio of the weight-average particle diameter to the number-average diameter.) (3) M. E. Woods, J. S. Dodge, I. M. Krieger, and P. E. Pierce, J . Paint Technol., 40, 627 (1968).
Volume 73, Number 7 July 1969
2388
P. ANNEHILTNERAND IRVIN M, KRIEGER
A method similar to that of van den Hul and Vanderhoff4 was used to remove electrolyte from the latex. One gram of a monobed ion-exchange resin (Amberlite MB-3, from the Rohm and Haas Co.) was added to each 25 g of latex; the resin was removed by filtration after 24 hr. Dilutions were effected with distilled water. Polymer contents of all suspensions were determined by drying accurately weighed samples to constant weight in an oven set at 110". The density of polystyrene was taken as 1.05 g/cc for the conversion of weight fraction into volume fraction.
0.7 1
Results Particle spacings D obtained from the diffraction measurements were always significantly larger than DO, the particle diameter determined by electron microscopy. When the original latex was diluted from 50 vol % polymer, the particle spacing remained unchanged until the concentration fell below 40%; at lower concentrations the suspensions were completely white, and no diffraction peaks were observed. Removal of electrolyte from the most concentrated suspensions did not appreciably affect the particle spacings. Upon dilution of the electrolyte-free latex, however, the diffraction peaks became progressively sharper and were displaced in the direction of larger spacings, as shown in Figure 4. Furthermore, the
Figure 5. Reciprocal reduced volume as a function of the polymer volume fraction: 0, Do = 171.0 mp; A, Do = 205.0 mfi; 0, DO = 256.3 mp; A, DO= 283.0 mfi; 0, DO = 328.0 mp. The solid line represents the equation @vr = 0.74.
obey the equation n, = 1.33 deviation of less than 1%.
+ 0.27 4, with an average
Table I : Particle Separation and Refractive Index of the Ordered Suspension a t Various Volume Fractions (Do= 256.3 mp)
loo@ 45.8 33.8 23.7 18.3 14.0 11.7 9.6 6.3 3.4
Y
F
Figure 4. Diffraction scans a t five different polymer volume fractions, DO = 171.0 mp and X = 450 mr. Intensities are on an arbitrary scale.
diffraction peaks persisted to concentrations as low as 1% polymer by volume, with particle spacings approximately four times as large as the particle diameter. Table I shows D values us. polymer content for a dilution series of an electrolyte-free latex with Do = 256 mF. Figure 5 shows similar data for five different latexes with Do values ranging from 171 to 392 mp. To obtain superposition, a reduced volume vr = D3/DO3 was calculated, and its reciprocal graphed against the volume fraction 4. The straight line represents the equation 4vr = 0.74. Mean refractive indices n, were calculated for each suspension and are also given in Table I. The refractive indices are lineally intermediate between those of water (1.33) and of the pure polymer (1.60) and The Journal
of
Physical Chemistry
285.0 323.5 361.4 398.3 458.6 486.1 507.8 571.9 712.8
1.470 1.434 1.408 1.389 I . 361 1.360 1.356 1.349 1.335
The experimental results clearly show that, upon dilution of electrolyte-free latexes, the particles move apart while retaining their latticelike configuration. For close-packed lattices of uniform spheres, whether face-centered cubic or hexagonal, the volume fraction is 0.74. The correspondence in Figure 5 between the experimental points and the line 4vr = 0.74 strongly implies that the lattice is either face-centered cubic or hexagonal close-packed and that the structure persists throughout the entire suspension. (For electrolytecontaining latexes, on the other hand, the invariance of spacing with dilution over the concentration range where diffraction is observed indicates that regions both of order and disorder are present.) The increase in sharpness with dilution, as shown in Figure 4, should (4) H. J. van den Hul and J. W. Vanderhoff, J. Colloid Interjac. Sci., 28, 336 (1968).
2389
DIFFRACTION OF LIGHTBY ORDERED SUSPENSIONS therefore be attributed to the decrease in turbidity rather than to increasing perfection of crystallinity. I n the concentrated suspensions, only a few planes near the interface contribute to diffraction, whereas in dilute suspensions the increased transparency permits reflections from deeper planes to sharpen the diffraction peaks. Quantitative expressions for the peak shape which have been previously derived for X-ray diffractions can be applied here with minor modifications. For perfect crystals of identical nonabsorbing atoms, the ratio A , of the amplitude of the diffracted beam to that of the incident beam at an angle 0' close to the Bragg angle 0 is given by q = -E
& &2
- 42
(5)
Here e = (2n)-lhd(0 - 0') cos 0, and the scattering coefficient 4 for a plane of atoms is proportional to the scattering factor f, the polarizability of the scatterers CY, and the number of atoms per unit volume N . q =
Ndhf 4a2a ____ sin 0 h2
Although derived for reflection of X-rays, eq 5 is applicable to diffraction by any ordered array of particles if the scattering coefficient is modified appropriately. For homogeneous spheres of diameter Do, f is calculated using the Rayleigh-Gans approximation and a = (n21)003/24,where n is the refractive index of the particles relative to the medium. The half-width A&,, of the peak is then given by A&,, = 0.0827(n2 - I.)[
sin x
- x coz] sin 2e
(7)
where x = (3/2)'/%Do/D. Experimental values of the peak width and those calculated from eq 7 are compared in Table 11. The close agreement indicates that the peak shape is satisfactorily described by eq 5; this fact supports the use of a perfectly crystalline model to describe diffraction by these highly ordered suspensions. A particularly striking result of this study is the demonstration that colloidal spheres in suspension can maintain themselves in a regular lattice structure, even when the interparticle spacing is several diameters. The contrast between the behavior of electrolyte-containing and electrolyte-free suspensions indicates that
Table 11: Experimental and Calculated Values of the Peak Width a t Half the Maximum Intensity (DO= 206.4 mp and AO = 500 mfi) Exptl
D,
loo+
mp
6, deg
47.1 32.0 24.1 11.7 9.0 5.3
235 .O 267.1 298.4 381.1 417.8 496.4
60.5 53.0 46.8 35.7 32.6 27.1
A@/2,
deg
Calod A@/%, de@;
10.2 9.4 8.4 4.6 4.0 3.2
8.84 8.60 8.27 4.66 3.82 2.80
the explanation lies in the theory of the electrical double layer. The original latex particles are charged due to bound charges from the initiator fragments and to adsorption of ionic surfactant. I n addition, the original latex contains sufficient electrolyte to provide a screening double layer, so that the effective range of interparticle forces is reduced to a fraction of a particle diameter. When the latexes are dialyzed or treated with ion-exchange resin, both the free electrolyte and the ionic surfactant are removed, but the bound charges and their free counterions remain. The concentration of available ions is far too low to provide an effective screen for the interparticle coulombic forces of repulsion. The range of forces is therefore so long that appreciable repulsion is felt even when the particles are separated by several diameters. It is this long-range repulsive force to which the existence of order in dilute suspensions is attributed. Further study is in progress to investigate the effect of ionic strength, particle size, and surface charge density on the lattice parameters. An additional focus for further study is the transformation between the iridescent form of the suspension and the nondiff racting form, which is an order-disorder transition in a system where the jnterparticle potential can be experimentally varied and theoretically calculated.
Acknowledgment. The work reported in this paper was supported by a grant from the Public Health Service. ( 5 ) Derivations can be found in most monographs on X-ray diffraction, for example, R. W. James, "The Optical Principles of the Diffraction of X-Rays," Cornel1 University Press, Ithaca, N. Y . , 1948, Chapter 11.
Volume 76, Number 7 July 1969
