NOTES +5
I
-5t
5.9 6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 Fig. 1.-Zeta-potential in millivolts (open circles), and surface conductivity in ohm-' (closed circles) against
PH. were made as described by Buchanan and Heyman. Electrolytes were HC1 and KOH, and AIS was calculated for each plug packing following measurements of C and &. The surface conductivity (ha) was calculated by ha = ( A / S ) K where K is the difference between the conductivit of the electrolyte solution when filling the pores of the pLg, and the bulk conductivity of the solution.
Results and Discussion Figure 1 shows both surface conductance (A,) and zeta-potential (calculated by the HelmholtzSmoluchowski equation) plotted against pH. Table I gives values of zeta and X. a t higher pH's. TABLE I PH 8.7 10.3 10.7
Zeta-potential (mv.)
-17.9 -22.8 -22.2
Surface conductivity (ohm -I),
x
109
2.35 2.74
2.67
Generally, AIS was of the order of 5-6 X The surface conductivity calculated from the zetapotentials is lower than the measured values, being of the order of 10-lo ohm-1 rather than ohm-'. On the other hand, the surface conductivity calculated from the exchange capacity (6.3 meq./g. of air-dried resin) is much greater than the measured value, being of the order of lo-' ohm-I. One can conclude that the Stern layer ions a t least, and probably also a swollen gel layer, contribute t o the surface conductance of this resin. Nevertheless, there is correspondence between measured zeta and A, in that the surface conductance shows a minimum a t approximately the isoelectric point. Rutgers and De Smet' also observed a minimum of surface conductance at the zero of zeta-potential when using thorium nitrate solutions on a glass surface. (7) A. J. Rutgers and M. De Smet, Nat. Bur. Standards (J.S.) Circ. No. 524, 263, 1953.
T H E LENNARD-JONES POTENTIAL FOR SPHERICAL MACROMOLECULES1 BY ANDREWG.DEROCCO~ The Harrison M . Randall Laboratory of Physics, University of Michigan, A n n Arbor, Michigan Received M a y 16, 1058
De Boers and Hamaker4 mem t o have been of the
responsible for the earliest ealealwtiQns
Vol. 62
intermolecular potential between spheres of uniform composition. More recently Atoji and Lipscomb,6 and Isihara and Koyamas have performed similar computations. The results of Atoji and Lipscomb depend on a series approximation valid for molecular distances fairly large in comparison t o molecular radii, while those of Isihara and Koyama appear to be valid a t all distances of approach. In this note we shall demonstrate that the original method of Hamaker may be employed t o obtain results equivalent t o those of Isihara and Koyama but having the feature of a notation more easily associated with recognizable molecular parameters. In general the forces between macromolecules may be of several kinds, and for spherical macromolecules having static charge distributions, the interaction energy a t values of pH well-removed from the isoionic points of the molecules may be described very nicely by a simple coulombic potential. Another related energy develops when a molecule has sites to which, for example, protons may be attached, and is determined by the excess of available sites over the average number of bound protons. Such a condition leads t o a great many configurations which differ only slightly in free energy, and the consequent mobility of the protons over the sites results in an induced polarization leading a t fairly long ranges t o a potential decreasing as R-2. This potential was described by Kirkwood and Shumaker' and shown by them t o play an important role in the interaction of proteins a t the isoionic points of the molecules, where, certainly, structure sensitive electrostatic forces play n serious role. In addition t o the energies mentioned above, a macromolecule also exhibits a considerably weaker kind of interaction, perhaps best described as a kind of Lennard-Jones potential. We mean by this, that part of the interaction energy for macromolecules which is analogous to the use of a Lennard-Jones (12:6) potential for, say, argon atoms. Clearly some kind of integration process is needed if we are t o go from the 12:6 potential to its corresponding form for a macromolecule and one such procedure is described below, but before passing over t o this consideration it should be emphasized that we are dealing with a strictly special case, uix., a spherical macromolecule for which any static or fluctuating charge distributions are completely ignored. We are dealing with weakly interacting macrospheres or a small part of the interaction for the usual case. Consider two macrospheres of radii R1 and Rz, having atomic distribution functions PI and p2 (taken as constants) and separated by an intermolecular distance R. We generalize the Hamaker formalism and write (1) Aided by a grant from the American Cancer Society. (2) Department of Chemistry, University of Michigan, Ann Arbor,
Michigan. (3) J. H. De Boer, Trans. Faraday Soc., 82, 10 (1936). (4) H. C. Hamaker, Physicu, 4, 1058 (1947). (5) M. Atoji and W. N. Lipscomb, J . Chem. Phys. 21, 1480 (1953). ( 6 ) A, Isihara and R. Koyama, J . Phys. Soc. Japan, 12, 32 (1957). (7) J. G. Kirkwood and J. B. Shumaker, Proc. Null. Acad. Sci., 3 8 , 863 (1952).
NOTES
July, 1958
891
Case 1 (y = l).-This represents the interaction between spheres of identical radii. In general (1)
where .fi(s)
=
RiZ -
(Z
- s ) ~ , ~ z (= z ) Rz2 - (R
-z
) ~
The function p(s) represents the potential energy between atoms or small groupings of atoms (quasimolecules) in the two macrospheres. Equation 1 may be obtained in the following manner. Consider small volumes dVl in sphere 1 and dV2in sphere 2 separated by a distance s and interacting according to a potential cp(s). If p1 and pz represent the atomic density distribution functions, then pldVl and p2dVz represent the total number of atoms in the two differential volumes. It is clear, then, that N R ) = Jv2 p d V z
Iv, PI
co(s)dVi
(2)
Equation 1 results from eq. 2 when the indicated integrations are expressed in terms of the radii and the intermolecular separation of the macrospheres. I n order to compute @(R)for two spheres, we need only to specify a form for p(s) and, in keeping with the objectives of this note, we chose for p(s) the Lennard-Jones potential in the form d s ) = vo[(so/s)'a - 2(So/S)61
(3)
where so is the equilibrium separation and po the concomitant energy. If eq. 3 is inserted into eq. 1 two terms are obtained in @ ( R which ) correspond to the repulsive and attractive terms in p(s). The attractive terms in p(s) go as s - ~ , corresponding to the London dispersion forces and the indicated integrations were first performed (in a somewhat different style) by Hamaker to obtain
where r1 = R1 -? R z and rz = R1 - Rz. We shall find it instructive if a t this point we introduce certain reduced variables and express @ d i s ( R ) in terms of these variables. First we define the distance of closest separation by d = R r l ; next the variables x and y are defined by x = d/Di,
y = Dz/Di
(5)
If we arbitrarily call D 1the diameter of the smaller sphere, then x measures d in terms of the smaller sphere. The ratio of the molecular sizes, y, is bounded by 1 5 y 5 05. Introduction of eq. 5 into eq. 4 leads to the expression Y +x2+xl/+x+y
Y @,lis=
'
-H[:cz+x?/+z
+
where we define H as 'she Hamaker constant and give it the value H = (rr2p~p2p~soe/6).The quantity within the brackets represents a reduced potential whose limiting cases are of interest. We write adis
=
-H**dia
(Z'Y)
(7)
@*dis(z,1) =
[m 1 f x
m 1
)
+2
1 xn +Ll ]
(8)
If the spheres now approach very closely, Le., x > R1, and this is equivalent to saying that a sphere of radius RI "looks at" a flat surface. Certainly @*dis(X,m)=
-q
LX + 1 -
1
x+1+21nx+l
(10)
and for close approach **dis(X,
OD)
1 ;
(11)
From eq. 9 and 11 it is noted that a plane surface attracts a sphere twice as strongly as would an equivaIent sphere. All of the preceding remarks are valid for only the dispersion potential and were first, noted by HamakerU2We proceed now to discuss the terms representing the repulsive part of the potential, which terms are overshadowed until very close distances of approach. If we were to imagine these macromolecules as hard macrospheres, then the potential would become infinite at the cutoff value of (Dl D 2 ) / 2 . We treat here the case where the spheres are somewhat deformable. The calculation of the potential energy of repulsion, @rrep(R), follows directly from eq. 1 upon insertion of the Lennard-Jones repulsive potential q ( s ) = q o s ~ l ~ s - Equation ~~. 1may be evaluated by repeated integration-by-parts to yield
+
1
(R
1
- r1)6 ( R + r d b
(R
-
Equation 12 may also be expressed in terms of the reduced variables introduced in eq. 6 . To do so we require the quantities
+ +
+ + +
= (2R1)"x" (R = (2R1)"(2 y l)n, (R (R rZ)n = (2R1)"(x l)n,(R - r# = (2Ri)"(x y)" R/Ri = ( 2 x y l), r1 = Rl(y l), rz = -Rl(y - 1) (13)
+ +
+
+
If the expressions of (13) are introduced into eq. 12 we obtain
892
NOTES
,
or @rep
=H
(3 -
@*rep(z,y)
It is immediately clear from eq. 14 that if D1is at all large, then @rep is very small until z (or d) becomes itself m i t e small. . ~..-. Again two limiting cases become clear. Case 1 (y = l).-For identical spheres we have
Vol. 62
where both molecules have a radius RO and the reduced distance R' is defined by R' = R / R Q . Equation 21 corresponds to eq. 3.3 of Isihara and Koyarna,B who have shown that the minimum value for the energy can be computedfrom eq. 21 and reads @min
3
- 71 601/n n2 pa~o'Rop~ 1 + 3 60-'/e
(3
~
$1 +I{m +'& 1
1
-
{-&6
+2
(z 115
+
If now$
