E. T. ADAMS,J R . ,
3454 [CONTRIBUTIOX FROM
THE
AND
J. W. WILLIAMS
voi. XG
DEPARTMENT OF CHEMISTRY, THEUNIVERSITY OF WISCONSIN, MADISONti, WISCOXSIN]
Sedimentation Equilibrium in Reacting Systems. 11. Extensions of the Theory to Several Types of Association Phenomena',2 B Y E. T. ADAMS,J R . , 3 a
AND
J.
w. \lTILLIAMS3b
RECEIVEDAPRIL 15, 1964 Prior theoretical treatments for the study of protein a n d detergent associations in systems a t equilibrium, such a s osmotic pressure, light scattering, and sedimentation equilibrium, and in systems during transport, t o include chromatography, electrophoresis, a n d velocity sedimentation, are t o be found in the literature For the most part i t is idealized behaviors which are described. Here are considered extensions of t h e sedimentation equilibrium approach, t o both a description of t h e centrifugal behavior in the presence of such reactions, and an indication a s t o how t h e effects of solution nonideality m a y be included in the essential working equations without limiting the discussion t o the simplest monomer-dimer case.
I. Introduction T h e measurement and interpretation of the amount of scattering at the several angles as a beam of light passes through a solution of macromolecules may provide useful inforniation about the size, shape, and configuration of the solutes. Studies of this kind, made possible largely by the pioneering inyestigations of D e b ~ e have , ~ been used by him and later by others with certain detergent solutions in order to determine the size of the micelles, or more precisely, the number of primary units of which they are composed. In addition Debye has provided a theory descriptive of the formation of the micelle, with its structure representing an equilibrium between the repulsive long-range Coulomb forces due to the charges concentrated in the "heads" of the detergent molecules and the shortrange attractive van der Waals forces which are operative when the hydrocarbon portions of the molecules are forced from the surrounding water into the core of the structure.s According to this exposition one would expect a substantially homogeneous population of micelles. Here sedimentation equilibrium experiments can give valuable information, including an estimate of the nuiiiber of monomers associating to form the given niicelle, and an indication a t least of any appreciable polydispersity. Descriptions of experiments of this kind are now just beginning to make their appearance6 and methods of interpretation of the data are being developed. Association phenotnena are also quite common in protein systems ; representative references are here g i v e ~ i . ~ - 'In ~ certain cases they appear to be quite ( I ) T h i s investigation was supported b y G r a n t ~ i n - a i d AI~0464.5-01, h'atir,n;tl I n s t i t u t e s of H e a l t h (1) I'ur more detailed discussions of certain of these problems reference is made t o t h e P h I). dissertation of E. T A , J r . , University o f Wisconsin, 1962 C'/ also. I< T A d a m s , Jr., Federofcon P r o c . , 13, 215 (19641 ( 3 ) ( a ) Physical C h e m i s t r y Ilivision, Hercules Research C e n t e r . Wilmingtlm, 11.~1. 198B'J. (h) t o whom reprint requests should b e a d dressed
( 4 ) I' I l e b y e , J . P h y s . Colloid C'hem., 61, 18 (1947). ( 5 ) P . Ilehye. A n n N Y . Acad. Sci., 61, 575 (1949). J P h y s . Colloid ( ' h r v t , 63, 1 (1949) ( t i ) IC W A n a c k e r , R M R u s h , a n d J S J o h n s o n , J P h y s . C h e m . , 68, 81 (1061)
(7) R T. A d a m s , J r , a n d H. F u j i t a , "Ultracentrifugal Analysis in T h e o r y a n d I'.xperiment," J W Williams. Ed.. Academic Press, I n c , New York, N Y 1!46;i, p 119. (8) A . l'iselius, %. nhysik. Chem.. 114, 449 (1926). ( ! I ) P I) Jeffrey a n d J H C o a t e s , N o i u r e , 197, 1104 (19631. 1 1 O j hl 1. 1,udwip. hl J . H u n t e r , a n d J I>, O n r l e y , A b s t r a c t s . 1 3 7 t h Saticrnal M e e t i n g o f t h e American Chemical S o c i e t y , Cleveland, Ohio, .4pril i 1 1 . 1!IOO f 1 1 ) I' C,. S q u i i e a n d C H 1.i. J . A m . Chem S O L , 8 3 . 3521 (1961) ( 1 ' 2 hl. S N. Rat) a n d G Kegeles i b i d , 8 0 , 5724 (1958)
involved, for they are believed to undergo monomerdimer-n-mer associations instead of the forinally simpler monomer-n-mer reactions. In others, the association number may be two, three, four, eight, etc., to correspond to a number of basic building blocks to form the particular protein. Presumably such numbers may be derived either from X-ray analysis of crystals or by such experiments as osmotic pressure, light scattering, and sedimentation, either equilibrium or transport, with solutions. I n a previous report from this laboratory,' there was presented a mathematical analysis to describe the combined sedimentation and chemical equilibriuni i n an associating solute system of the monotner-dimer type. Here, the attempt is to be made to extend the treatment to systems in which the solute is matlc up of monomer-n-mer, or monorner-diiner~iz-~iierin chemical equilibria with each other. As examples of these types of interaction detailed consideration will be given to monomer-dimer and monomer-n-mer equilibria, including micelle formation. Some additional remarks will be directed to those cases which contain several complexes in equilibrium with mo110mer. X short preliminary account of the work has already appeared.2 In i t , the final sentence should read : For nionomer-dimer-trimer equilibria it appears and 2K:, one can only evaluate K J i l - ls'~-2f~2
11. Basic Equations In the simplest of the associating systems, a monomer and its dimer form the solute species in a single solvent component. Such species are designated by the subscripts 1 and 2, or in general i. Although we propose detailed considerations of the monomer -dimer and the monomer-n-mer cases, a more complicated association, t h a t of monoIiier~diIner-trimer, will be used as reprcsentative system in order to set down a few basic equat,ions. I t is represented as where PIrepresents the macromolecular species undergoing the indicated reaction. In Sedimentation equilibrium experiments with such a solute system, especially with protein associations, two assumptions are usually made: one is that the partial specific volumes of the associating species are equal, and the other (13)
V.
W
F. H a r r i n g t o n , a n d U S. H a r t l e y . T l i c 24 (IY~A). (14) E I, S m i t h , J R . K i m m e l . a n d I ) M B r o w n , .l f9:ol < hc,m 2 0 7 , 5 3 3 (10.54) ( 1 5 ) J I,, B e t h u n e , J A m Chem S o c , 86, 4 0 4 7 111363:. (16) K T o w n e n d , R J W i n t e r b o t t o m . and S S T i m a s h e l f . , b i d , 8 2 , 3161 (1960) JMassey,
C U S S I O I S F Q Y ~ ~ Q.SOC., Y
ao,
Sept. 5, 1964
SEDIMENTATION
EQUILIBRIUM IN REACTING SYSTEMS
requires t h a t the refractive index increments are equal. The equilibrium constants K2 and K3 are defined by the equations KZ =
(:)($)
2;K3 =
= a2
=
(:3)($)
a3
3455
vicinity of .infinite dilution. With the aid of the basic equations we write for the combined sedimentation and chemical equilibrium
(2)
a13
with the concentrations, c, being expressed in grams per unit volume of solution, and the activity coefficients, y, measure nonideality on this concentration scale. On a molar basis, the conditions for chemical equilibrium are 2Pl
= P2
3Pi
= Pa
(3)
whereA = (1 - vp)u2/RT. For the monomer, the sedimentation equilibrium is described by the formula
In solution clr = claeA M 1 ( r 2 -
Wl = Pm
The chemical potentials, p i , are given in the units cal./ mole. The condition for Sedimentation equilibrium for the i t h reactant is dp, - Miw2rdr = 0
(4)
Thus for the constituent P1(constant temperature)
In this expression fil = chemical potential per mole of monomer, M1 = monomer molecular weight, w = angular velocity of the rotor, r = radial distance from the center of rotation, fil = partial specific volume of the monomer, c1 = concentration of monomer at any radial position (g. per dl.), and d P / d r = d r p , where p is the density of the solution. The total solute concentration, in the monomerdimer-trimer system which has been selected is
a9
= Claed~
(9)
The quantity a represents the radial distance to the meniscus in the cell, and $1 = AMl(r2 - u2). Here the concentrations a t any radial position, c,, and the concentration a t the meniscus, c,, along with the radial distances r and a themselves, have been used as limits of integration. T h e equation could be generalized by using any chosen radial position, r$, with the corresponding concentration, c,,, instead of the values at the meniscus, provided t h a t a 2 r* 5 b. The radial position b is t h a t at the bottom of the cell. For the dimer, trimer, etc., and in the same way we have cir
=
K2~la2e261 = ciae2+l, etc.
(10)
Thus, for the system in discussion
(11) =
T h e weight-average molecular weight for this associating system (as in all others) varies with the total solute concentration; we adopt the symbol Mwc,) to indicate this fact. Here
claed'
+ 2 K 2 ~ ~ +~ 3K3cla3e3*' ~ e ~ ~ '
Dividing both sides of this equation by c,, the total concentration of macromolecular component a t the meniscus, we have
3
C cirMi i = l Mw(c)(=-~wr) =
3
C
Cir $ = l
=
c,
((c,
+ 2K2clr2(e) + Yz
The quantities, f l a , etc., represent the weight fractions of the several associating species, monomer, etc., at the meniscus. Dividing through by e@", we define a q u m tity
7
The symbol cl, represents the monomer concentration at position r in the cell.
111. The Ideal Case A. Monomer-Dimer-Trimer Equilibria.-Ideal associating systems are defined as those in which excess chemical and electrical potentials are assumed to be absent. Thus, the theoretical working equations may be said to provide data to allow one to test for the ideality of solution behavior as well as for the association mechanism at appreciable solute concentrations, instead of being required to make evaluations in the
= fla
+ 2f2,e" + 3f3,e2+'
Since the sum of the weight fractions is one fla
=
1
- fia
- f3a
and $ - 1 = fia(2e"
- 1) -k f3a(3e2m'- 1)
(14)
Thus, a plot of (* - l)/(2c*' - 1) us. (3e2+' - 1 ): (2eml - 1) may be used to distinguish between a monomer-dimer association, and either a rnonome r trimer or a monomer-dimer-trimer system. This is
3456
E. T. ADAMS,JR.,
AND
J. W. WILLIAMS
Vol. 86
I
+ - I
I
2
3exp(e+,) eexp(+J
.
-I -I
3exp(eq)
Fig. 1.-Plot based on eq. 14: Curve 1 would be given by a monomer-dimer association. The ordinate of this line has the value fz0. Curve 2 could be given by a monomer-dimer-trimer association or a niononier-trimer association. The value of the abscissa a t the meniscus is 2.
-I
Fig. 2.-Plot based on eq. 14a: Curve 1 would be given by a monomer-trimer system only. The ordinate in this horizontal line has the value f3.. Curve 2 would be representative only of a monomer-dimer-trimer association. T h e value o f the abscissa a t the meniscus is 1 / 2
demonstrated in Fig. 1, in which it will be seen t h a t for the monomer--dimer case the plot is a horizontal straight line, b u t in either of the other situations the line is inclined. Now, in order to distinguish between a monomerdimer-trimer and a monomer-trimer association, we rewrite eq. 14 as
C
In Fig. 2 is shown a plot of ($ - 1)/(3e2+' - 1) US. (2em' - 1)/(3e2*' - 1). For the monomer--trimer association a horizontal straight line results, but for the more complicated situation with monomer-dimertrimer in equilibrium an inclined straight line would be found. Having tested for the niechanism of the association by the methods given above for the more common sedimentation equilibrium experiment which makes use of schlieren optics to record the total concentration gradient of the redistributed components over the cell, the Iiayleigh optical system now can be employed to evaluate the quantities cia, K2cla2, etc. Again, referring to this system, there are obtained from a generalized form of eq. 4 the set of equations CY,
=
la
e A M ~ ( r ~-2 a t )
+
Fig. 3.-Plot of Ki(app)v s . c : The quantity K,c.,,, is calculated from eq. 15. For ideal solutions it is independent of c, the total concentration of the macromolecular component. For nonideal systems concentration dependence will be exhibited, a possible form of which is indicated.
Then, remembering that
or log
~
~
~
~
~ -~ a 2~)
~ 1 = )
log K ,
+
%
log
~1
(17)
the value of n, the degree of aggregation, is available from a simple plot. For this case, the equation corresponding t o (14) would read
K ~ ~ ~ ~ z ~ ~ A -. a*) M I+ ( ~ I *
K
-
(C
A
M
I
(
~
I
~
$
-
1
=
f,,(ne"'
- 1)
Thus, a plot of $ - 1 v s . (nem'- 1) will provide an inclined line if only one species is present. If the 2 ZAMt(rg2 - as) + cr3 = eAMl(n2 - aZ) n-mer is polydisperse upward curvature of the line will K2Cla e la 3 3 A M i ( r 2 2 - 0%) be found. This plot is more sensitive to polydispersity K3Cla e than is the plot based on eq. 1 7 ; however, the use of c, = C l o KzC,,2 K3Cla3 this equation is required for the estimate of the value n. of For ideal solutions the apparent and thermodynamic equilibrium constants are equal, thus a plot of K z ( ~ ~ ~ ) IV. The Nonideal Case or KB(~,,,]) vs. c should give a horizontal straight line, A. Monomer-Dimer-Trimer Equilibria.-b'hen the Fig. 3 . For the nonideal case, soon to be considered, equations are extended to include thermodynamic concentration dependence is involved. nonideality effects one restricts a t once the concentraB. Monomer-n-Mer Equilibria.--With associations tion range over which observations can be interpreted. of the monomer-%-mer type such as occur in detergent The expression for the chemical potential becomes solutions, the weight fraction, f,of monomer and thereRT In yici fii = fore the concentration of monomer, 61, can be obtained b y the application of an equation of Seiner,'' namely The quantity In y , can be expanded in powers of c to give
+
+ +
(15)
+
+ O(c2) In y i = B.Wzc + O(c2) In y i
( 1 7 ) R . F. Steiner, Arrh. Biochem. BiophyJ., 39, 333 11952); 49, 400 (1854)
=
BMlc
SEDIMENTATION EQUILIBRIUM IN REACTING SYSTEMS
Sept. 5 , 1964
an equation is obtained which defines the apparent weight fraction of monomer ; thus
or in general In y i = iBMlc
3457
+ O(c2)
only for very dilute systems. From eq. 4 there can be obtained for the solution as a whole the result
since
(The symbol c1 is here used to represent clr, the concentration of monomer at any radial distance, 7, in the cell.) With the definition of M w ( c )and some rearrangement, we find 1
wheref = cl/c. It should be noted that the integrand in eq. 22 is not an indeterminate form at infinite dilution of the macromolecular component (c = 0) ; thus i t is readily (or monomershown for a monomer-dimer-trimer dimer+-mer) association t h a t
1
Equation 23 also gives -KZ for a monomer4imer association ; however, for a monomer-n-mer association (n I 3) eq. 23 has the value zero. Combining eq. 5 and 22 we have
where
6
- 1)/
c = (Kz
- BM1)cl/c
+
and 1 dc - r dr
=
cMa(l
-
Lo2
vp)
-
2
RT
s2/c
+ . . . ,)
(24)
From eq. 19 it will be noted t h a t
Also and
-
c-0
-- B
MI
C
(2Oa)
(’cleBMIC - ’)
= {(Kz
- BMl)
+ ( [ K I - 2KzBM1 +
Furthermore and and
It follows as well t h a t Thus, by plotting { (l/M1 - ( l / M a ) } / c against c or l/Ma against c a relation between Kz and B M I is obtained. Defining L = Kz - B M I ,we have
Kz
=
L
+ BMI
(21)
The quantity L or its analogs often appear in equations descriptive of the associating system. For example it has been shown’ to be involved in the equations of Steiner.” In the next paragraphs we develop additional equations which involve the quantities Kz, KX, and AMI. The first step in the process is t o define the apparent weight fraction of monomer, .fa. Going back t o Steiner’s relation, eq. 16, and using the apparent molecular weight, Ma,which is experimentally available,
Several other equations may be derived from eq. 23; thus In f a
+ ( K z - BMl)c
=
E. T. ADAMS,J R . ,
3458
AND
There are a variety of additional relationships which can be derived in similar manner. Thus, making use of a definition
BMl - KzBMlcl - . . .
.}
(26)
i t is found t h a t
lim c-0
(2)
= -X(Kz -
BMl)
+
(2K3 - 2Kz2 - KzBMi)
(27)
In eq. 26 and 27 the quantity X can take on the values
B M l , K z , - M ~ / C M and ~ ~ ~so~on. , So, when X K z ,eq. 27 becomes
=
J. W. WILLIAMS
VOl. 86
This provides no new information. Similar results are obtained with other combinations. The unfavorable result shows that these equations (30, 31, and 32, for example) are not mathematically independent Thus, unless additional and independent relations which involve the quantities IT2, K 3 , and H.M1 can be developed, all one can expect to evaluate from nonideal systems of the monomer-dimer-trimer (or nmer) type are the quantities L = Kz - H.Vl and Q = 2K3 - 3K22 B. The Monomer-Dimer Equilibrium.--Where a simple mondmer-dimer equilibrium exists, one may apply eq. %Oar20b, 24a, 24c, and 24d, to obtain L = K z - BMI. Having confidence t h a t the monomer-dimer equilibrium is the only association reaction present, and representing the activity coefficient for each associating species in the conventional way, as has been indicated, the following equation can be applied7 (34)
=
c-0
2K3 - 3K22G !AI
(28)
X = XI
It should also be noted that Ai any value of c, eq. 34 is a cubic equation in K z ; it can be solved by standard algebraic methods. Alternatively, one can use the following equations to obtain K Zand cl. Using eq. 5 , 19, 21, and 23, it can be shown t h a t lim
(2
=
- fa)
-
K22
{f*
cM,, - c M ~= KZC~'- B12!Ta~Wm~c~~2
-B c*
2
- KS (30)
=
c =
~1
+ KzclZ
(36a)
K2 =
(C
- c1)/cl2
(36b)
Then, with eq. 21,36a, and 36b we obtain = c1yc
{(GI-
and
lirn c
While i t would appear t h a t we have sufficient equations to evaluate K z , Ks, and B M l , we should examine what occurs when we substitute eq. 28 into eq. 30. We find
- K3
2
(33)
-
CI) --
c1)
-
LcI'} ( 2 ~ cl)cMa ( 3 6 ~ )
At any given value of c, eq. 3Gc is cubic in the quantity cl, and the quantity KZmay be obtained from eq. 3Gb, once c1 is known. As already suggested, Steiner's equations are valid. Equation 28 or 29 will give KZ2for this system; thus there are several checks available. Of course, the previous results' could be applied, b u t this newer approach eliminates any necessity for calculating Kapp. C. The Monomer-n-Mer Equilibrium.-The monomer-n-mer equilibrium can be represented as ? L P Z P,
(n 2 3)
Assuming t h a t the activity coefiicients for the associating species can be represented in the usual fashion, we write
However, for the monomer-n-mer quantity .Mzc(c)becomes J
-
(35)
or
c12(cMa - CMl)
c-0
+ 2Kzcl')~
For the monomer-dimer association, eq. 5 reduces to
+ K.J3Ml
+
Kzci2 - UMaMl(cl
n 2
-
L2 -
2
equilibrium the
Sept. 5, 1964
SEDIMENTATION
EQUILIBRIUM IN REACTING SYSTEMS
3459
+
Eisenberglgb = i/2(1 - (iZcM3/Mlc3) , .), 0 = (c3/M3) - (rZc/M1), and c3 = concentration (g./ dl.) of BX. The quantity z is a factor such t h a t 0 6 i 6 1. For a neutral macromolecule, i = 0 ; when a strong electrolyte is involved, i = 1. For macromolecules which behave as weak electrolytes the value of the quantity i lies between zero and one. Remembering t h a t the concentration for a monornerdimer equilibrium can be expressed by eq. 36a and the concentration gradient by an equation of the same form as eq. 18, we obtain for the sedimentation equilibrium of a ' charged, associating monomer-dimer equilibrium
Thus
Again, the equations of Steiner are valid. Thus, eq. 16 applies and we have as before, f a = f e B M l c . = AMl(C1 2K2c12) - (Cl - 2KZC12) x Since we can evaluate the quantity BM1 from eq. 38, we can readily estimate f and hence c1. Knowing M a and B M 1 , we obtain Mto(c) from (19) and the definition of Ma. I t is apparent that if B M 1 , c, and M m ( c ) or are available, one can obtain n and K , from eq. 17. D. Charged Macromolecules and Further Remarks (404 on Nonideal Systems.-Proteins and detergent micelles are usually charged macromolecules, and sedimentation Here, B* = BM1 f(z)/Ml and c M W ( J M 1 = (cl equilibrium studies are generally carried out in buffers, 2K2c12). It should be noted t h a t the second term in often along with additional supporting electrolyte. f ( z ) is usually much, much smaller than the main term, By using the Scatchard-Bregman l8 or the Casassaso t h a t the second term may be omitted, or one may Eisenberglg definitions of components, one may make an approximation by assuming reduce the sedimentation equilibrium equation t o that of a two-component system. dc3 Ms(1 - fi@) These definitions of components have evolved from dc - M 1 ( l - fiPX, - r Z ( B X ) P ) the LammZ0theoretical description of the sedimentation equilibrium of dissolved charged macromolecules It is here taken for granted t h a t the activity coefin the presence of supporting electrolyte. ficients ( L e . , the excess electrochemical potential) for We consider a charged, associating macromolecule each associating species can be expressed a s P X , and a supporting electrolyte BX. The associating species are defined as PX, - r Z ( B X ) for the monomer, In yi = iB*Mlc 0(c2) (40b) [PX, - I'Z(BX)]2 for the dimer, and so on. The It now follows from this convention t h a t the data for sedimentation equilibrium equation for the monomer charged, associating system can be analyzed in the becomes manner already indicated in sections A, B, and C. AMld(r2) = d In c1 zf(z)dc BMldc (39) E. Other Activity Coefficient Treatments.-To this point use has been made of a single virial coefficient. Here Additional terms can be included but they make the equations (and calculations) much more complicated. - 2r)(1 - r) r dc Thus, if the activity coefficient for each associating species can be represented by the series
+
+
N
+
~
+
+
+
+ +
M1(e
+g)
(s)] +
In yi
=
iBIMlc
+
iB2Mlc2
[I - 2r - ( g ) ] d c 3 )
O(C3),
+ i
= 1, 2 ,
3, . . .
. ..
(41)
we still have C =
where z = charge on the monomer, Ml = molecular weight of monomer [PX, - rZ(BX)], ;If3 = molecular weight of BX, c = total concentration (g./dl.) of macromolecular component a t any radial position in the ultracentrifuge cell, A = (1 - fipx2 - r Z ( B X ) P ) w 2 / 2 R r , r = membrane distribution parameter of Casassa and (18) G Scatchard and J. Bregman, J . A m . Chem. Soc., 81, 6095 (1959). (18) (a) E . F. Casassa and H Eisenberg, J . P h y s . Chem., 64, 7 5 3 ( 1 9 6 0 ) ; ( b ) H . Eisenberg and E . F . Casassa, J Polymer S i . , 47, 29 (1960). ( 2 0 ) 0 . Lamm, Arkiw Kern:, .Wineral Geol., 1'78, N o . 2 5 (1943-1944).
C1
+
KzCIZ
for a monomer-dimer-trimer follows that
+ K3C13 association.
I t also
E. T. ADAMS,JR.,
3460
AND
Equations 20a, 24a, 24c, and 24d are also valid in this instance. There may be systems which are nonideal in more complicated ways than can be described by the conventions for the activity coefficient we have used. In this event, the activity coefficients may be described as
In yl for the monomer. tion
BIMlc
=
+ BzMlc2
(44)
Now for a monomer-dimer associa-
Pz
2P and
Thus
On expansion
Then 21.
[l
+
a
(F) J L
y2 =
c +
......
c = o
I=
Hence
K2c12(1
+ +. (YC
.
. .)
I t can then be shown that
or
J. W. WILLIAMS
Vol. 86
trimer (or n-mer) associations, respectively, but the Steiner equations now become much more complicated.
V. Discussion An ever increasing number of dissociation-association reactions are being studied in the ultracentrifuge, along with other methods. Involved may be proteins, surfactants, and other molecules in solution. As of now the experimental results of such investigations, either restricted to the sedimentation methods or with the inclusion of the other approaches, are at considerable variance. Thus, as an immediate objective i t has seemed worthwhile to attempt to put the theory of the combined chemical and sedimentation equilibrium into improved form for later experimental programs, eventually to eliminate some of these inconsistencies. There is, of course, the more common sedimentation transport experiment for the purpose in discussion. Gilbert and associates21 have been concerned with the modification of the form of the boundary gradient curve as the combined transport and reaction process proceeds. In order to simplify some of the mathematical problems which have arisen certain rather restrictive assumptions have been required, but the theoretical description does provide for a monomodal boundary for a monomer-dimer system and a bimodal one in the case of a monomer-n-mer (n > 2) reaction, as observed experimentally. Also the more classical combination of sedimentation velocity and diffusion experiments has been used to study associations both in protein and in surfactant systems, with the object being to evaluate the association number. For the paraffin-chain salt case the necessary theory has been recently set down and developed at considerable length by MijnlieffZ2; i t will be strictly correct only if the micellar system which forms is monodisperse. We have elected to look into the possibilities of the sedimentation equilibrium experiment for several reasons. I t has the more sound theoretical foundation, with i t solution nonideality effects may be more readily taken into account, and presumably the experiments may be carried out at lower solute concentrations. Such experiments have not been popular in the past because of the relatively long time to attain equilibrium, but the situation in this respect is now changed because of the re-introduction of the use of short solution columns, already described in the Svedberg and Pedersen monograph,23by van Holde and B a l d ~ i n , ~ ' and by Yphantis.z5 In the earlier report' there was presented an elaborate mathematical analysis for a monomer-dimer system a t sedimentation equilibrium, one which included the effects of solution nonideality. Here, additional methods have been developed for the study of this system, and extensions have been provided whereby monomer-n-mer (n > 2) systems are included. Further, i t appears t h a t the theory in its present state (21) ( a ) G . A Gilbert, Discussions F a r a d a y Sac., 8 0 , 68 ( 1 9 5 5 ) ; P r o c . R o y . S O L .( L o n d o n ) , A160, 377 ( 1 9 5 9 ) ; A163, 420 (1959); ( b ) G . 4 . Gilbert
Blc
+ 2B2c2
(51a)
Equations 20b and 2Oc are still true for the monomerdimer (and n-mer) association and for the monomer-
and R . C . 1.1. Jenkins, "Ultracentrifugal Analysis in Theory and Experiment," J W . Williams. Ed , Academic Press. I n c . , N e w York, N. Y . , 1963, p p . 59, 73 ( 2 2 ) P . J Mijnlieff, "Ultracentrifugal Analysis in Theory and Experi1963, m e n t , " J . W Williams, E d . , Academic Press, I n c , N e w Y o r k , K Y . . p. 81. ( 2 3 ) T . Svedberg and K . 0. Pedersen, "The Ultracentrifuge," Clarendon Press, Oxford, 1940, p p 5 7 , 305. ( 2 4 ) K . E. van Holde and R . L . Baldwin, J . Phys. C h e m . , 61, 734 ( 1 9 5 8 ) . ( 2 5 ) D . A Yphantis, A n n A , Y . A c a d . S c i . , 8 8 , 586 ( 1 9 6 0 ) .
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 Texas 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, Tulane 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
