XORPHOLOGY OF COLLOIDAL GOLD
Sept. 5, 1964
cannot be enlarged to describe the behavior of monomer-dimer-n-mer associations. When applying the theory to the nonideal situations it has to be recognized t h a t data in very dilute solutions are required. With the introduction of the Rayleigh methods and of 30-mm. double sector cells for the experiment, sufficiently greater precision in such dilute systems is promised and one is encouraged to essay such experiments. Incidentally, it is of interest that the theory developed by Steiner for ideal associating systems requires that the equilibrium constants be evaluated from limits taken at infinite dilution ; this restriction is now relaxed. A main objective is the determination of the quantity L = KZ - BM1. The equation of Steiner, subject to its restriction, provides it through his equation
Herefl is the weight fraction of the monomer. I t can be calculated by means of our eq. 16. Plotting the quotient against the quantity fit, the intercept, 4Kz/ M1, and the limiting slope at infinite dilution, 9K3/M12, are available. For nonideal systems undergoing the monomer-dimer association Adams and Fujita' already have shown that this intercept is the desired quantity Kz - BM1. I t has been proposed, for quite different purposes to be sure, t h a t the micelles formed in solutions of paraffin-chain salts may serve as convenient models for proteins. Problems of the kind we have discussed may lead one to be able to state just how far the approach used by Debye in his study of micelle formation
3461
can be applied in protein physical chemistry. -4s a longer range objective it has been our purpose to indicate new means to converge on problems of this kind. For ideal associating systems there exists another and simpler manner for testing the association mechanism instead of using eq. 14 and 14a. Using eq. 5, with all yi = 1 for an ideal solution, and eq. 11 one obtains
This may be rearranged to give
A plot of the quantity on the left-hand side of eq. 53 against ec gives a horizontal line for a monomer-dimer association and an inclined line for a monomer-dimertrimer or monomer-trimer association. We can distinguish between a monomer-dimer-trimer and a monomer-trimer association with a second plot; we note t h a t
3c - (1/2AMlr)dc/dr = 2fia caeml
+ fZae"
(54)
Here a plot of the left-hand term of eq. 54 against el' gives a horizontal line for a monomer-trimer association and an inclined straight line for a monomer-dimer or monomer-dimer-trimer association. Acknowledgment.-We wish to thank Drs. J . H. Elliott, F. E. LaBar, D. L. Filmer, and V. J. MacCosham for their comments and interest in this work.
THE DEPARTMENT O F CHEMISTRY, WILLIAXMARSHRICE UNIVERSITY, HOUSTON, TEXAS, AND T H E DEPARTMENT OF PHYSICS, TULANE UNIVERSITY, N E W ORLEANS, LOUISIANA]
[CONTRIBUTION FROM
Morphology of Colloidal Gold-A
Comparative Study
BY W. 0 . MILLIGAN'~ A N D R. H. MORRISS'~ RECEIVEDMARCH 2, 1864
A comparative study of the morphology of colloidal gold was made using light absorption and scattering, X-ray diffraction line broadening, and electron microscopy. .A series of colloidal gold samples ranging in average particle "diameter" from 10 to 400 A. and exhibiting a wide range of crystallite morphology was prepared and examined by each method. Extinction coefficients, absolute light scattering coefficients, and depolarization factors were measured for wave lengths between 4000 and 6500 A. Using the theory of Mie for the interaction of light with conducting spheres and the extension of this theory by Gans t o ellipsoids of various axial ratios, the approximate shape of the crystallites was calculated. The pure diffraction line broadening of the (200), ( 1111, (2201, and (311 ) reflections was measured, and by applying the Scherrer equation, the mean crystallite dimension along [ l O O ] , [ 1111, [ 1101, and [311] was calculated. The morphology a s calculated from light absorption and scattering and X-ray diffraction was found t o be in fairly close agreement with the electron microscopic findings for most colloidal gold samples. However, each technique, including electron microscopy, was found to have limitations, in particular particle size ranges.
I. Introduction Beginning with the slit ultramicroscope which was developed shortly after 1900, several indirect methods have been employed to study the morphology of colloidal particles. One method is that of light absorption and scattering. In 1908 >lieza presented a theoretical treatment of the interaction of light with (1) ( a ) D e p a r t m e n t of C h e m i s t r y , William M a r s h Rice University, present address T e x a s Christian University, F o r t W o r t h , T e x a s , (b) Dep a r t m e n t of Physics, T u l a n e Univerqity
small, conducting spheres On the basis of Maxwell's electromagnetic theory, Mie calculated the true absorption and the intensity and polarization properties of light scattered by spherical particles of varying size in terms of the macroscopic optical properties of the metal In 1912 G a m z b extended the calculations of Mie by generalizing the particle shape to prolate and oblate ellipsoids of revolution Whereas Mie's cal(2) ( a ) G Mie, A n n P h y s i k , 9 6 , 377 (1908) (1912)
( b ) R Gans. ibrd
87, 833
,'Mi2
W 0. MILLIGAN A N D K H MORRISS
culations apply to particles smaller than the wave length of light, Gans' calculations are limited to particles smaller than approximately one-tenth the wave length of light. I n 1028 Lange3 made a detailed investigation of the absorption and scattering of light by gold hydrosols ovcr the wave length range G!5(J-iO(lO A. The absorption measurements were in good agreement with hlie's theory indicating the particles to be approximately spherical in shape. However, the data obtained for the degree of polarization of the transversely scattered lig.ht were in better agreement with C h i s ' theory, thereby indicating ellipsoidal morphology. ,. I ( J explain this seeming contradiction, Lange postulated that a large majority of particles in a gold hydrosol are spherical in shape which manifest themselves in alxorptiun measurements, arid a lesser number of particles which are in the form of ellipsoids of revolution account for the polarization properties of the scattered light. In 1 0 3 i Krishnan4 measured the absorption and polarization of a group of gold hydrosols over the wave length range :3O(JO-il)OO A. Froin both sets of data, he found the particles t o behave optically as prolate ellipsoids with an axial ratio of 0.73. Sivarajan" measured the scattering intensities for a group of gold hydrosols siinilar to those of Krishnan and likewise obtained good agreement with Gans' theory lor prolate ellipsoids of axial ratio (1. i 3 . Scherrer" in 191s showed that the mean dimension, D , of the crystallites composing. a crystalline powder is related to the pure diffraction line broadening, /3, b y the equation
where k ' is a constant factor which is related to the shape of the crystallites and the manner in which 11 and 3 arc defined. If D is referred to as D h k i and is the effective crystallite thickness in a direction is deperpendicular to the ( i l k / ) planes and /3 = fined as the half-maximum line breadth, i t has been shown6,.'that k' has a value of approximately 0.9 and is independent of the crystal shape. The Scherrer equation then becomes
By measuring several values of pji2 corresponding to different ( h k / ) reflections, one can obtain some indication of the crystal shape characteristics. In 1938 Jones' carried out an X-ray diffraction line broadening experiment o n a sample of colloidal gold and found the average dimensions of the particles to he 703 X 176 x I i ( j .%. which would indicate the particles to be rod shaped. The development of the electron microscope in the late 10:31's offered a new. direct means of studying the morphology of colloidal particles. Beischer and i:i) H l,ange, Z p h y s i i i ' h p m , 132, I f1!12§, (4) R S Krishnan Pvoc I n d z a ? j -lcaii .sa 5 , 94 ( 1 8 3 7 ~ ,,j, S K S i v a r a j a n $ L i d , 37. 418 '1%33' ~ i j ,P S c h e r r e r , .\achi k g ! Grs li'lsr Go!:titgeii, 2 , 9§ :I9181 (71 I , H r a g ~ ', T h e C r ) s t a l l i n e S t a t e , 4 C;eneral S u r v e y " Vol I , G Bell a n d S U ~ , , ~ . t d I.rindon 1:$1
