J. Phys. Chem. 1981, 85,2013-2021
2013
Solvent Effect on the Dispersion (Hamaker-London) Coefficient from Third-Order Perturbation Theory V. L. Vliker,”+ G. P.
Uyeno,+ and W. 0. McMlllan*
Unlversw of Calllwnia, Los Angeles. California 90024 (Received: October 9, 1980; In Final Form: Merch 26, 1981)
An effective pair potential is developed for the (unretarded) dispersion interaction between two multiatom solute particles-e.g., two macromolecules or colloidal particles-in a solvent medium, analogous to the Hamaker-London potential between two such particles in a vacuum. This requires taking into account not only the usual London dispersion interactions between all pairs of atoms respectively in the two particles, but also the “mediation” of these pairwise interactions by third atoms in the solvent medium or in the particles themselves. This is accomplished through application of the third-order perturbation treatment developed by previous workers, together with (approximate) summation of the contributions of all third atoms. The resulting equations are cast in a form that permits evaluation of effective Hamaker-London coefficients in solution. Theoretical Coefficients calculated for aqueous solutions of, respectively, acetone, polystyrene, and bovine serum albumin are in good agreement with the dispersion interaction coefficients estimated by other methods. In comparison with the Hamaker-London dispersion coefficients in vacuo, those in aqueous solution for these systems are smaller by factors ranging from 10 to 100. 1. Background a n d Purpose Many problems of fundamental importance in physical chemistry, colloid science, biochemistry, molecular biology, and medicine involve the interactions of multiatom particles, or macromolecules, in various media. These interactions pervade such wide-ranging phenomena as osmotic pressure, colloid coagulation, gene replication, and immunology, to cite but a few examples. It is the purpose of this paper to adapt recent significant advances in the understanding of long-range interatomic interactions to such macromolecular systems. In particular, we shall show that the modifications introduced by the quantum-mechanical treatment of third-atom “mediation” of the London dispersion interaction between macromolecules in aqueous solution can, with good approximation, be represented in the traditional HamakerLondon pair-potential form, with alteration of only the numerical coefficient. The traditional “microscopic” or “discrete atom” description of those intermolecular interactions commonly associated with the name of van der Waals proceeds by summing the several contributions to the intermolecular pair potential arising from dipole-dipole, dipole-induced dipole, and induced dipole-induced dipole (dispersion) interactions.’ By contrast, the “macroscopic” or “field” approach introduced by the Russian school adapts the theory of fluctuations of the electromagnetic (EM) field to yield an expression for the interaction energy in terms of the frequency dependence of the (complex) EM permeabilities of the interacting particles and the solvent mediurne2 At the macromolecular concentrations that prevail in many natural phenomena, it is the long-range attraction that dominates-or that, at least, encourages the intermolecular proximity a t which other, shorter-range forces are brought into play. In both the microscopic and macroscopic approaches this long-range attraction derives principally from dispersion. A recent study of the interactions between albumin protein molecules in electrolyte solution suggests that dispersion is the dominant source of long-range attraction also between biological macromolecule~.~ ‘Chemical, Nuclear, and Thermal Engineering Department. t Department of Chemistry. 0022-3654/81/2085-2013$01.25/0
In its current state of development, the macroscopic approach treats each medium (particle or solvent) as though continuous rather than atomistic, but is otherwise formally rigorous in that it takes into account all manybody interactions at all electromagnetic frequencies. Major difficulties encountered in applying this approach to colloidal particles and biological macromolecules include the need for permeability data over a wide frequency range (far-ultraviolet to microwave), and an unwieldy mathematical complexity for problems of even the simplest geometry (e.g., two spheres of equal size). This complexity impedes a straightforward incorporation with other contributions to the interparticle potential (such as screened monopole interactions) for the purpose of synthesizing a composite potential function. Moreover, the assumption of continuous media enters in such an essential way as to obscure application to small interparticle separations where the solvent molecular structure becomes imp~rtant.~J’ Accordingly, the considerations in this paper will be limited to an extension of the traditional microscopic approach. 2. Preview of Coming Attractions
Table I offers a perspective on the evolution of the quantum-mechanical theory of intermolecular electronic correlations (dispersion interactions) and the place of the present application in that development. London’s initial treatment of the dispersion interaction between hydrogen-like atoms in vacuo was followed by his extension of the theory to more complex Slater and Kirkwood8 used a variational method to derive an expression (1)E. J. W.Verwey and J. Th. G. Overbeek,“Theory of the Stability of Lyophobic Colloids”, Elsevier, New York, 1948,Chapter VI. (2)J. Mahanty and B. W. Ninham, “Dispersion Forces”, Academic Press, New York, 1976. Macroscopic theory is based on earlier works: I. E. Dzyaloshinskii,E. M. Lifshitz,and L. P. Pitaevskii, Adu. Phys., 10,165 (1961);E. M. Lifshitz, Sou. Phys.-JETP (Engl. Trawl.),2,73 (1956);S. M. Rytov, “Theory of Electric Fluctuations and Thermal Radiation”, Academy of Sciences Press, Moscow, 1953. (3)V. L. Vilker, C. K. Colton, and K. A. Smith, J. Colloid Interface Sci., 79,548 (1981). (4)B. W.Ninham, J . Phys. Chern. 84,1423 (1980). (5)D. Y.C. Chan, D. J. Mitchell, B. W. Ninham, and B. A. Pailthorpe, J . Colloid Interface Sci., 68, 462 (1979). (6)F. London, Z . Phys., 60, 245 (1930). (7)R. Eisenschitz and F. London, Z . Phys., 60,491 (1930);F.London, Trans. Faraday Soc., 33, 8 (1937). (8)J. C. Slater and J. G. Kirkwood, Phys. Reu., 37, 682 (1931).
0 1981 American Chemical Society
2014
The Journal of Physical Chemktry, Vol. 85, No. 14, 1981
TABLE I:
Vilker et al.
Evolution of Microscopic Theories f o r Dispersion Interactions interaction system
ref
Two-Body Interactions t w o a t o m s in vacuo L O n d ~ n , Slnter ~ . ~ a n d Kirkwood,8 K i r k ~ o o d Mueller’O ,~ t w o spherical molecules in vacuo considering o n l y interactions Hamaker’l between pairs o f c o n s t i t u e n t a t o m s t w o spherical molecules immersed in a solvent with refererice Hamaker’* state a t infinite separation Three-Body Interactions Axilrod and Teller,14 M ~ t o , ’Midzuno ~ and Kihara,16 Bade and Kirkwood,17 Kestner a n d S i n a n o g l ~ ’ ~ * ’ ~ t w o a t o m s in solvent with third-atom corrections Sinanoglu20 t w o spherical molecules in solvent with third-atom corrections this work three a t o m s in vacuo
that is similar to London’s result except for the inclusion of a factor that accounts for contributions from outer-shell electrons. Kirkwoodggave an elegant derivation of atomic polarizability by the variation method, which yielded a remarkably accurate relationship with the diamagnetic susceptibility. Mueller’O combined Kirkwood’s result with the London dispersion interaction to give an attractive intermolecular pair potential in somewhat improved agreement with experiment. Later, the effect of the finite propagation speed of light (i.e., retardation) on the dispersion interaction was introduced by Casimir and Polder.I1 Employing the London atom dispersion interaction, HamakeP extended the interatomic dispersion interaction in a vacuum to the case of two spherical multiatom particles having different radii, each composed of a single atomic species. Then, assuming that the dispersion interaction between all such multiatom particles is independent of the surrounding medium, e.g., whether vacuum or solvent Yamaker calculated the difference in the London di;,,ersion energy between two spherical particles a t the finite intercenter separation r and a t infinite separation. This is just the particle pair potential in solution using infinite separation as the reference state of zero energy. It has long been recognized that the London interatomic even for two dispersion potential is quite appr~ximate,’~ atoms in a vacuum. But beginning with the introduction of third-order perturbation corrections by Axilrod and Teller,14 the importance of the contributions of third atoms-whether of either particle or the surrounding medium-in “mediating” the interatomic potentials has become increasingly appreciated.1520 The most extensive investigations of the third- and higher-order perturbation terms are those of Sinanoglu and co-workers.1g20 As recently as 1976, Nir21observed that these third-atom contributions had not been applied to the case of interacting multiatom particles. The present paper attempts to fill this gap. In carrying out the (approximate) summation of Sinanoglu’s third-order atom-pair corrections to the dispersion energy for two (equal) spherical particles (or macromolecules), we apply Sinanoglu’s results much (9)J. G.Kirkwood, Phys. Z., 33,57 (1932). (10)A. Mueller, Proc. R. SOC.London, Ser. A , 154,624 (1936). (11) H.B. G. Casimir and D. Polder, Phys. Rev., 73,360 (1948). (12) H.C. Hamaker, Physica, 4 , 1058 (1937). (13)K. S. Pitzer, “Quantum Chemistry”, Prentice-Hd, New York, 1953,p 201. (14)B. M. Axilrod and E. Teller, J. Chem. Phys., 11, 299 (1943). (15)Y.Muto, P ~ o cPhys.-Math. . SOC.Jpn., 17,629 (1943). (16)Y. Midzuno and T. Kihara, J. Phys. SOC.Jpn., 11, 1045 (1956). (17)W.L.Bade and J. G. Kirkwood, J. Chem. Phys., 27,1284(1957). (18)N. R. Kestner and 0. Sinanoglu, Discuss. Faraday Soc., 40,266 (1965). (19)N. R.Kestner and 0. Sinanoglu, J. Chem. Phys., 38,1730(1963). (20) 0.Sinanoglu in “IntermolecularForces”, J. 0. Hirschfelder, Ed., Wiley-Interscience, New York, 1967,p 283. (21)S. Nir, Prog. Surf. Sci., 8, 1 (1976).
Flgure 1. Geometry for pair interaction between muttlatom particles I and J. Volume element A7, contains local number density n,’(i,) of atoms of species a in particle I , and volume element AT contains local number density nbJ(?)of atoms of species b in partfcle J .
as Hamaker applied those of London. A correction term to the Hamaker-London pair coefficient is derived which is a function of parameters obtainable from optical frequency refractive index measurements and the Lorenz-Lorentz equation used in the classical microscopic approach. Those interactions occurring at frequencies outside the ultraviolet-visible range and which are included in the macroscopic approach are not included in our result. Nir21has shown that interactions at these frequencies do not contribute significantly to the dispersion energy. The range of particle separation (between nearest approach of the particle surfaces) for which our results apply is -20-1000 A. The upper limit is approximately the separation below which it is reasonable to ignore retardation effects, while interactions a t separations less than -20 would require consideration of solvent structure effects. 3. The Hamaker-London Coefficient To set the stage and establish the notation for the developments to follow, we first recapitulate the essential features of Hamaker’s seminal paper12 of 1937, which addressed the calculation of the pair interaction energy between two multiatom particles embedded in a liquid me-
Solvent Effect on the Dispersion Coefficient
The Journal of Physical Chemistry, Vol. 85, No. 14, 1981 2015
dium based upon the corresponding interactions in a vacuum. With reference to Figure 1, the two particles I and J are composed of a variety of atoms of species designated by a (or b) having local number densities of n,'(Fi) and nbJ(ij)a t respective positions ii and Fj. We denote by (7) Here, the angular bracket symbolizes the average obtained by using as weighting factor the 5u,l of eq 5a
the usual London dispersion energy between two such atoms having mean excitation energies22 6, and 6b and (static) polarizabilities a, and a b and separated by distance - , rij = ri - rj (2)
-
If we employ volume elements Asi that are large compared to atomic volumes yet small compared to the total volume of the particle, the number of atoms of species a in Ari a t Fi is given approximately by n,'(Fi)Ari. In summing eq 1over all atom-pair interactions between particles Z and J, Hamaker replaced the finite volume elements by their infinitesimal counterparts, thus treating the atom number densities as continuous functions. With this approximation the total dispersion energy between particles I and J is given by
(3) Assuming that the atom number densities are reasonably uniform, the n,'(ii) may be replaced by their position-independent average values fit,and thus be removed from the integrand:
(6'/2)1
=
Es0'/2c,'/d
(8)
0
An alternative averaging process applied to the summation in eq 4 serves to establish contact with Hamaker's notation. Denoting by ii' the total mean atom number density En,' in particle I, and using the definition of %ab in eq 1, we obtain cC%ab?%,'?ibJ = fi'fiJ( 8) a b
so that
-AIJg(Z,J geometry) 6%) Here, AIj is the (multispecies generalization of the) familiar Hamaker coefficient
AIj I r2fi1fiJ( 8) (10) and the dimensionless geometric factor g is defined by
Although Hamaker evaluated the integral (eq 11)for two spherical particles of different radii, we will limit our consideration to the case of identical radii. With this specialization, Hamaker's integration yields
(4) For later numerical evaluation it will prove convenient to have the summand in eq 4 be separable into factors belonging respectively to the two particles I and J. To this end we first define the specific polarizability of atoms of species a using the atom number density for the particle in question: cOI
-I na
(54
The corresponding average total specific polarizability of particle I is thus
'E = Clt,laa
(5b)
0
We now invoke the well-known theorem that, for a set of quantities-such as the mean excitation energies Gi--that cluster in a narrow band with only small fractional deviations ti, the arithmetic mean exceeds the geometric mean by terms of second order in the deviations:
or With this substitution, eq 4 becomes
where s is the interparticle center-to-center distance R scaled to the particle radius a: s = R/a (13) It is easy to verify that for s >> 2 the geometric factor (eq 12) reduces correctly to the London limit from eq 11: g(s >> 2) 16/(9s6) (14) The second step of Hamaker's development is illustrated in Figure 2. Solute particles 1and 2 are represented by solid circles, and equivalent volumes of solvent 3 by dashed circles. Here the objective is to evaluate the energy change which attends bringing a pair of solute particles immersed in the solvent medium from infinite separation to distance r12. Spherical geometry is employed in Figure 2 for clarity only; particles 1 and 2 may be of any shape in the figure so long as each 1-3 or 2-3 interaction is calculated by using a solvent volume which is of shape and size identical with the particle replaced. As indicated in the figure, this change in distance involves changes in interactions also with and between spherical solvent "particles" of volume identical with that of the solute particles. The resultant pair potential energy, v',",',is the sum of these steps:
-
q!J) = v12 + v33 - v13 - v23 or, when 1 and 2 are the same material
~~
(22)
T.Kihara, Adu. Chem. Phys., 1, 267 (1958).
V# = VI,
+ V3, - 2VI3
(154
(15b)
Vilker et ai.
The Journal of Physical Chemistty, Vol. 85,No. 14, 1981
2016
1.
Break 1-3 and 2-3 interactions, with the 1-2 pairs at infinite separation.
I
-10
I
---I-lOO
2. Break 3-3 interaction at r 1 2 separation.
OD
3. Form 1-2 interaction at r 1 2 separation.
I
-10
1
1
---
t
;lo
\
\ 0
P
Figure 3. (a) Three atoms in (scaled) bicentric and cylindrical coordinates; center of atom i at z = 0, center of atom j at z = 1, and center of atom k a t cylindrical coordinate z,p and arbitrary azimuth 9 . (b) Geometric factor (1 3 cos 8, cos 8) cos 8k)/r,/3r,k3rk,3 as a function of atom k position.
+
4. Form 3-3 and 3-3 interactions at separation r I 2 ,
with pairs at infinite separation. //
[\
-\
\
/- -\
/
3+r12-3
'- / &'
\ /
\.-A
/e-.
3 + r\l 2 + - 3 /
I
,
\ \.-/
j
Ne-\
\
-.'
\ .
+*v33 Figure 2. Energy changes accompanying the process of bringing spherical molecules 1 and 2 of equal size (solid circles) from infinite separation to separatlon r,2 In solvent 3. Dashed circles are volumes of solvent equivalent in volume to molecules 1 and 2.
From eq 9b and 15b, Hamaker12 showed that solution phase interaction coefficients, Ai;), could be expressed in terms of the corresponding in vacuo coefficients: 23
A\!) = All + A 3 3 -
(16)
4. Dispersion Energy of Three Atoms in Vacuo The dispersion interaction energy udisP among three atoms in vacuo, carried to the third order in perturbation theory, includes the sum of the three pair interactions of the second-order (London) type of eq 1plus a three-atom correction term, Ui;k: UdkP = u i j + u j k + u k i + U i j k (17)
As discussed in detail by Kestner and Sinanogl~,'~ of the 27 terms that enter the quantum-mechanical calculation of ui;k all but three types vanish. Each such type corresponds to a correlation between the fluctuating electronic dipole moments of a particular pair of atoms (e.g., those numbered i and j , respectively, in particles I and J) indirectly coupled or "mediated" by the third atom (numbered k ) . This mediation takes three forms wherein (i) the field of atom i (the source), fluctuating with its characteristic frequency, induces a correlated dipole moment in atom k , which in turn-now acting as a source but with its own characteristic frequency-induces a correlated dipole moment in atom j , (ii) the inverse process occurs, in which the roles of i and j are interchanged, and (iii) atom (23) This expression for 34';) i s often simplified by using the approximation for AIS= (34113433)1/ddue to D.Berthelot, C. R. Hebd. Seances Acad. Sei., 126, 1703 (1898). The result 3415,'= (AIl1I2 - 34s1/2)2 can lead to large dilferences from the result of eq 16 when All and 34, are nearly equal and 3415,'i s small.
k acts as a field source that induces correlated dipole moments simultaneously in both atoms i and j . Since any one of the three atoms can serve as mediator, there are altogether nine three-atom interactions that make up the third-order perturbation correction. Kestner and Sinanoglulgintroduced a clever operational method that considerably shortens the derivation of the expression for U i j k in terms of the mean atomic excitation energies ( s i , 6;, 6 k ) and polarizabilities ( c y i , cyj, a$: uijk =
-3 2
+ 6; + 8 k ) a i a j a k (6i + 6j)(6j + 6 k ) ( 6 k + s i ) 6i6j6k(6i
[
1
+ 3 cos 8 i cos 8; cos 8 k r,.3r. 3r 11 ~k ki
,
1
(18) The vertex angles ( e i , e k , e,) are defined in the bicentric coordinate system of Figure 3a, as are also the scaled interatomic distances Tij = R, ?;k = R?, ?k = Rs' to be employed below. The angular factor in eq 18 shows that both the interatomic distances and the relative geometry of the three atom locations affect this three-atom energy. Figure 3b is a plot of the geometric factor for arbitrary azimuth Q, (to which eq 18 is invariant), scaled to unit distance between atoms i and j , and with atom k off-axis a radial distance p at altitude z. Evidently Ui;k can assume either positive or negative values depending on the geometric configuration.
5. Two-Atom Interaction Mediated by Solvent Third Atom In the three-atom dispersion energy, udisP, of eq 17 it is clear that the third-atom term U i j k belongs jointly to all three atoms rather than to any particular pair. If it were our objective to calculate, for example, the total dispersion contribution to the cohesive energy of a large ensemble of molecules, it would be necessary to sum eq 17 over all atoms, with due care to avoid redundant counting. This problem has been treated by Sinanoglu.20 In seeking the solution analogue of the Hamaker-London particle-pair interaction, however, we need to focus on how the London dispersion interaction ui, between two atoms i and j (ultimately to be regarded as belonging respectively to the two solute molecules I and J)is modified through medi-
The Journal of Physical Chemistry, Vol. 85, No. 14, 1981 2017
Solvent Effect on the Dispersion Coefficient
ation by a third atom belonging either to the solvent or to one of the solute molecules. For brevity we denote by region X the total solvent volume exclusive of the volume regions I and J and denote by k a particular atom of the solvent located at position ?k within region x. The Hamaker process of Figure 2 does not involve second-order London pair interactions Ukk' within the solvent (i.e., the solvent cohesive energy), nor between the solute molecules and the solvent u j k or u k i (energy of solution). These several types of pair interactions do not enter-not because they are negligible but rather because they do not change in the Hamaker process, and thus cancel out in taking the energy difference between final (separation r ) and initial (separation m ) states. As we shall see, this has the effect of assigning the third-atom modification U i j k (almost) entirely to the i-j interaction u i j . The resulting modified atom-pair dispersion interaction u,,* is thus given by u..* 11 = u1J. . + X u llk .. (19)
To this end, we first note that the third-atom summation was obtained in closed form by Kestner and Sinanoglu for the case of two spherical atoms embedded in a homogeneous solvent that extends up to the atom "surfaces". In our notation, the totality of this region is constituted of the sum of regions J , d, and X (i.e., all occupied by solvent) with the exception of the volumes occupied by the two atoms i and j themselves. This suggests that in each of the four potentials (eq 20) we construct the sum over K = J , d, and X and examine the terms left over. We thus find, for example
and similarly for the other three potentials of eq 20. When added together according to the prescription of eq 22, the correction terms in brackets sum to
k
For clarity, when the mediating third atom is in volume region X,it will bear index k;when in I, index i'( # i), and similarly for J. A prime on either index or summation will signify omission of any term bearing two identical indexes.
6. Two Spherical Solute Particles with Third-Atom Mediation To obtain the totality of atom-pair interactions between the two solute molecules, one must sum eq 19 over all atoms respectively in regions I and J. With reference to eq 15a, the Hamaker process of Figure 2 involves atom-pair interactions not only between the two spherical solute particles themselves, but also between one particle and an equivalent sphere of solvent occupying the volume region vacated by the other particle, as well as between two such equivalent spheres of solvent. These equivalent solvent spheres, occupying respectively the particle volume regions I and J, will be designated by the corresponding script capitals J and 6. With this understanding, the summation of eq 19 yields four (third-order-corrected (*)) terms entering the analogue of the Hamaker formula of eq 15a: vIJ*=
(uij i€I j € J
+
Uijk
k€X
+ y2
' uiji' i'€I
+ 72
uijy)
j'€J
(204
vJJ*= i E J I,e (uij + EJ k€X
Uijk
+ '/z
UijiI i'E3
+ 72 j ' E J
uijj')
(20b)
(204 The only subtlety in this development is the appearance of the factors l/z wherever two of the three atoms occur in the same volume region, I , J, J , or d. This factor is necessary to avoid redundant counting, as is immediately evident, for example, in the double summation:
c c
i>i'
Uij?
both E I
=
y2
c c
all i # i'
uiji'
(21)
We now wish to combine the four potentials, eq 20, in the analogue of the Hamaker eq 15a:
v&)*=
VI,*
+ v33* - VIS*
- v2,*
(22)
With reference to the detailed form of U i j k in eq 18, it can be seen that, if the solute particles and the solvent medium have closely similar optical properties (mean excitation energies and specific polarizabilities), the sum of differences (expression 24) would vanish. It is well-known that the effective mean excitation energies lie in a quite narrow band. Moreover, the specific polarizabilities of the aqueous medium and the several solute molecules that we later use to test the theory are also closely similar. Even for the bovine serum albumin (BSA) macromolecules the specific polarizability is close to that of the saline water solution in which it is commonly dispersed, especially considering that the BSA molecule imbibes and carries with it a significant volume of the solvent. One might think that optical similarity would make the Hamaker sum, eq 15a or 22, itself tend toward zero. Indeed, for just this reason the pair potential k$) in solution proves to be much smaller than its vacuum counterpart Vl2.
It should be emphasized that the various terms of expression 24 represent the differences generated by substituting an atom of the equivalent solvent sphere for an atom of the corresponding solute molecule in the role of the third (mediating) atom in the third-order-perturbation correction. Since it is only to this role of third atom mediator that this substitution applies, physical intuition argues that this can scarcely make a significant difference. In this sense, the sum (expression 24) is one of those corrections to a correction that are traditionally neglected. When to these reasons is added the observation that neglect of expression 24 renders mathematically tractable an otherwise totally intractable problem, such provisional neglect highly commends itself on the basis of simple practicality. In what follows we shall thus omit the bracketed terms of eq 23 for each of the four interaction potentials, noting that such omission may be justifiable, if a t all, only by virtue of approximate cancellations in expression 24 that result upon taking the Hamaker sum, eq 22. With this approximation we are left with four simplified particle pair interaction energies of the form
Vilker et al.
The Journal of Physical Chemistfy, Vol. 85, No. 14, 1981
2010
Here, the region K is occupied exclusively by solvent and extends everywhere outside the radii of atoms i and j . 7. Passage to Continuous Media
In eq 25, the double sum of uij has already been carried out in eq 1-7. We thus proceed directly to the triple sum of uijk as given in eq 18. To this end we employ the bicentric coordinates of Figure 3 r.. 11 = R
rjk = Rr
rki = Rs
(26)
and follow the notation of Kestner and Sinanoglu in introducing 0 for
e = COS ei COS e, =
(
COS
s2+1-r2 2s
The averages of the roots of the mean excitation energies can be simplified, in good approximation, by inverting the order of the averaging and root operations so that (61/n)I
)(
&33*
)(
r2+s2-1
2rs
)
(27) As before, we proceed by employing continuous atom number densities within each volume region. Then, if our interest is restricted to particle separations R sufficiently large to justify ignoring the molecular structure of the solvent medium (e.g., in aqueous media, >-20 A between nearest surfaces), the summations may be replaced by integrations-again excepting the sum over discrete species a, b, and c. The steps closely parallel those of section 3 and lead directly to
E
6
p
(33)
In evaluating these AIJ* coefficients for the several terms that enter the Hamaker sum, eq 22, the simplest case is the interaction of two solvent spheres, Jq33, for which the mean excitation energy 6 is the same for all three regions, I , J, and K
ek r2+1-s2 2r
(&;in)
=
&33(1
- 2TCK)
(34)
In this and the equations immediately following, the notation $ has been retained in preference to its equivalent &(3) for simplicity. When particles I and J are identical, we approximate eq 32 by All* = All(1 - 2a&K(63/61)1/3)
(35)
and when K is the same as either I or J &31* = &13* = A31(1- 2a#(63/6,)’/‘)
(36)
The correspondence between corrected and uncorrected Hamaker coefficients is thus given approximately by A($)*= A\$)-
+ A33- 2Jq13(63/61)1/6](37)
When all ratios (63/61)’/“ are close to unity, the corrected Hamaker coefficient reduces to the simple form A!?)’ = A(?)[l- 2a$]
Under the stipulated proscription against close approach of the two particles, Sinanoglumhas evaluated the bicentric double integral to give
At this point, we again approximate the arithmetic mean by the geometric mean, eq 6a, to simplify eq 28: 3
???5
+ a b + 6,) + &)(ab + 6c)(6c + 6,)
6a6b6c(6a
(6,
X
If we now factor out of this expression the quantity VZJ, eq 7, we find
We thus finally arrive at the connection between the Hamaker coefficients AzJ* corrected for third-atom mediation, in terms of the original Hamaker-London coefficients introduced in eq 9:
(38)
8. Modified Hamaker-London Coefficients a n d Comparisons The results of the previous section will now be used to calculate effective Hamaker-London coefficients for aqueous solutions of, respectively, acetone (a prototypical small organic molecule), polystyrene (synthetic polymer particle), and bovine serum albumin (BSA, a globular biological macromolecule). Literature values for the static polarizabilities and characteristic frequencies of water, styrene, and BSA are summarized in Table 11. For acetone these parameters were calculated by the method discussed by Gregoryz4 using the Lorenz-Lorentz equation applied to literature data for the acetone refractive index as a function of optical frequency.25 These data are used to calculate the Hamaker-London coefficients for water shown in Table I11 and for the materials in aqueous solution shown in Table IV. The calculated in vacuo coefficients (A33 for water in Table I11 or All for pure materials in Table IV) are evaluated from eq 10 with (48) from eq 7 and 33 (also see ref 12, 24, and 29): (48) = y46,’/2d6:/2d
(39)
(24)J. Gregory, Adu. Colloid Interface Sci., 2, 396 (1969). (25)I. Timmermans, “Physico-Chemical Constanta of Pure Organic Compounds”, Vol. 1, Elsevier, New York, 1950,p 356;Val. 2, 1965,p 272. (26)M. Andersen, L. R. Painter, and S. Nir, Biopolymers, 13, 1261 (1974). (27)J. R. Brown, “Albumin Structure, Function and Uses”, W. M. Rosenoer, M. Oratz, and M. A. Rothschild, Eds., Pergamon Press, New York, 1977,p 1. (28)P. Putzeys and J. Broeteaux, Bull. SOC. Chirn. Biol., 18. 1681 (1936). (29)J. Visser, Adu. Colloid Interface Sci., 3 , 331 (1972).
The Journal of Physical Chemistry, Vol. 85, No. 14, 1981 2010
Solvent Effect on the Dispersion Coefficient
TABLE 11: Data Used for Calculating Dispersion (Hamaker-London) Coefficients material/molecular unit
( M ) , g mol-’
mol wt
mol vol (l/ii),aA 3
static polarizability (e),A 3
( u 0 ) , 1015 s-’
char freq
waterlwater m ~ l e c u l e ’ ~ acetonelacetone m o l e c ~ l e ’ ~ polystyrene/styrene monomer2‘ bovine serum albumin/BSA moleculez6
18.0 58.1 104.1 66100b
29.9 103.5 164.5 81890
1.433 5.41 12.79 63OOc
3.35 3.20 2.62 3.06
a Calculated from l/ii = p N / M where the density p is 1.0, 0.932, 1 . 0 5 , and 1 . 3 5 g c m - 3 for water, acetone, polystyrene, Reference 27. Andersen, Painter, and Nir26also report a lower polarizability ( 5 6 0 0 A 3 ) ; we and BSA, respectively. choose the higher value shown since these data extrapolate t o the correct refractive index for solid BSA, q D = 1 . 5 9 (see ref 28).
in the table are reduced from their in vacuo counterparts by factors of 8-70. Three-atom corrections provide a The total further reduction to the values shown for Ai?)’. effect of the solvent is to reduce the in vacuo HamakerA 33* 34 Lif 34 0 3 dl 13 London coefficient by factors ranging from 10 to 100. (eq 3 4 ) (lit.) (es 10) (lit.) Acetone was one of several organic substances for which 2.64 4.3gb 1.7-30b the in vacuo Hamaker-London coefficient was evaluated 3.78 3.5c 2.60‘ by Croucher and Hair30 using a corresponding-states principle. Since this theory was based on the F o w k e P J ( = 0 . 0 6 2 5 e V , or 2.5 kT). All A in units of expression for the dispersion contribution to the surface Reference 29. Reference 21. free energy, which in turn assumes pairwise additivity, the with 6 = hvo where h is Planck’s constant and I = J = 3 Croucher and Hair coefficient should be equivalent to our (water) or I = J = 1 (pure material of Table IV). The All. Their value of 4.2 X J compares well with our coefficients corrected for the effect of third-body interAll. Radke and PrausnitP estimated that water reduces or by eq 35 (All*). actions are evaluated by eq 34 (A33*) the dispersion energy between acetone molecules by a Effective Hamaker-London coefficients in solution (Ai:)) factor of 10 from the in vacuo value. This estimate was and these effective coefficients corrected for solvent made by comparing potential energy parameters obtained are evaluated for Table IV third-atom interactions (AI!)*) from dilute-gas second virial coefficient data with those by eq 16 and 37, respectively. obtained from dilute-solution osmotic second virial coefComparisons between our calculated microscopic coefficient data. By comparing Ai:)’with All in Table IV, we ficients and similar dispersion interaction coefficients from would predict a factor of 100 reduction of the dispersion the literature are shown in the tables. VisserZ9has proenergy. We also note that for acetone-water, since .Al1 vided a critical review through 1972 of the estimated and = A33,the approximate formz3of eq 16 leads to a much measured microscopic and macroscopic coefficients. We smaller value of the effective coefficient Ai:)than the value do not repeat his assessment of the methods involved in given in Table IV. obtaining the tabulated values except where these asPolystyrene lattices have been used extensively in exsessments are helpful in making comparisons between perimental investigations of colloid theories. Visserm revalues in Tables I11 and IV. The Lifshitz-van der Waals ports a wide range of estimates and measurements for All, coefficient determined from macroscopic theory is labeled 5.6 X 10-%54.9 X J, although the upper limit of this for pure materials or A!& for materials in aqueous J if the Moelwyn-Hughes range is lowered to 16.8 X solution. Since these coefficients automatically include methods for calculating the London coefficient is excluded. many-body effects, they ought to be comparable with our (The expression for B of Moelwyn-Hughes is reported to three-atom corrected coefficients, A * or for pure be in error.29) Our value of All is close to the median of materials in vacuo, or with our A$ for materials imthis range. Lifshitz-van der Waals coefficients of 6.4 X mersed in water. and 7.9 X J are reported by Visser% and = 1.7 X 10-20-30 For water in Table 111, the range P a r ~ e g i a nrespectively. ,~~ These are larger than our All* X J has been reported by Visser.% Our calculated and again may reflect contributions to the dispersion invalue of 3.78 X J is in good agreement with estimates teractions at other frequencies. For interactions in water, made by using comparable refractive-index data. We show is 0.1 the range of reported effective coefficients,% Ai?), that the third-atom correction gives = 2.64 X X 10-m-ll X J. Visser estimates the best Lifshitz-van J, which is about one-third smaller than the uncorrected J, while Parsegian31 der Waals coefficient to be 0.35 X coefficient. In comparison, Visser reports the Lifshitz-van calculated a coefficient for the interaction of polystyrene der Waals coefficient for water to be 4.38 X loTMJ. The to -0.3 slabs in water which decreased from 1.3 X larger Lifshitz-van der Waals coefficient may be due to X J as slab separation increased. This decrease is the inclusion of contributions by dispersion at frequencies thought to be a consequence of declining influence of other than vo and contributions due to dipole-dipole and high-frequency interactions a t increasing separations. dipole-induced dipole forces. On the other hand, Nir2* Table IV shows that A$:)* is near the median of this range gives values of the unretarded microscopic dispersion for ~ f ! ~ coefficient (3.5 X J) and the Lifshitz-van der Waals dispersion coefficient (2.6 X J) that agree very well (30)M.D.Croucher and M.L. Hair, J.Phys. Chem., 81,1631(1977). with our values for A,,and respectively. (31)V.A. Parsegian in “Physical Chemistry: Enriching Topica from When the pure materials listed in Table IV are imColloid and Interface Science”, H. van Olphen and K. T. Mysels, Eds., mersed in water, the calculated coefficients All, A{!) and , Theorex, La Jolla, CA, 1975,p 27. (32)S. N.Srivastava, Z.Phys. Chem. ( L i e p i g ) ,233,237 (1966). A{?’”show that the largest effect reducing the in vacuo (33)F. M.Fowkes, Ind. Eng. Chem., 56, 40 (1964). coefficient results from accounting for the energy change (34)C. J. Radke and J. M. Prausnitz, J.Chem. Phys., 67,714 (1972). from infinite solute molecule separation, i.e., the Hamaker (35)E.A. Moelwyn-Hughes,“Physical Chemistry”,2nd ed.,Pergamon process in Figure 2. The resulting values of Ai:)shown Press, London, 1961,p 393.
TABLE 111: Hamaker-London Coefficients for Water Showing Effects o f Third-Atom Corrections and Comparisons with Literature Estimates o f Dispersion Interaction Coef ficientsa
-
-
acetone polystyrene
4.3
7.77
4.2b 5.6-54.9‘
2.99 5.23
BSA
8.87
1.04-46f
6.11
Figure 4. Conceptual routes for modifying the Hamaker-London Coefficient to include the effect of the medium on the dispersion interaction. The upper route involves modification of the net pairwise by; one of several proposed simple functions of the coefficient 4’ dielectric constant. The lower route, followed in the present work, modifies each interaction by correcting for the effect of solvent third atoms on the dispersion interaction.
Hamaker-London coefficients for bovine serum albumin in vacuo and in water have been investigated b SrivasAll = 46 X J and Ai)= 0.12 t a ~ a He . ~ calculated ~ X J from surface-tension data, and All = 1.04 X 10-20-2.60 X J and A{:)= 1.4 X 10-20-2.0 X J from approximate amino acid composition and assuming additivity of the amino acid molar refractivities. He also measured A$ = 5 X J indirectly from studies of the aggregation of albumin-coated hydrocarbon emulsion droplets. NirZ1evaluated the effective (unretarded) Hamaker-London coefficient, Ai?), to be 0.69 X J, and the (unretarded) Lifshitz-van der Waals coefficient, A&, to be 0.48 X J. Althou h these values are lower than our calculated values for All and Ai:)*(probably because of slight differences in the interpretation of the literature agrees very optical dataB), our calculated ratio A{:)*/A& well with that of Nir.
8)
9. Discussion
It has long been suspected that the dispersion energy between two particles embedded in a solvent medium is not correctly given by the Hamaker-London treatment. In response to this vague uneasiness, several authors have attempted to modify the Hamaker coefficients given by eq 16 to allow for the effect of the medium on the propagation of the phenomenon underlying the dispersion interaction. Since this interaction is between electric dipoles, it is not surprising that the earlier correction factors invoked simple functions of the refractive index or dielectric constant. These attempts, for which no theoretical basis has ever been established, may be represented diagrammatically as the upper (counterclockwise)route of Figure 4. Followin this reasoning, various authors have suggested that AI:)be divided by the high-frequency dielectric constant of the medium (1.77 for Setlow and Pollards suggested division by the square of this dielectric constant since the field of one atom induces a moment in the second atom whose field must be transmitted back to ~
~
~~~
~~
~~~
6.4c 7.9d
0.06 0.71
1.07
0.1-1 1c 0.12-5‘ O.6gR
0.04 0.41 0.72
0.3 5‘ 0.3-1.3dre 0.4V
the first atom to effect a correlation. V i ~ s e suggested r~~ that Ai:)be multiplied by 1.6 for interactions in water. Dispersion interactions, however, are unlike interactions between charges or static dipoles embedded in a dielectric medium since the quantum-mechanical perturbation process (which gives rise to dispersion interactions) is strictly an energy rather than a free energy. The correct approach, as emphasized by Kestner and Sinanoglu,l9 is to take account of the third-atom influence on the perturbation process between all solute-solute, solute-solvent, and solvent-solvent atom pairs. This approach, adopted herein, is represented by the lower route of Figure 4. The results, eq 34-37, are in the desirable form of effective Hamaker-London coefficients which require no additional data for evaluation beyond that required for the uncorrected coefficients. Comparisons made in the previous section between the effective Hamaker-London coefficients and the Lifshitzvan der Waals coefficients show close agreement. This is true despite the facts that .A{:)* comes from taking the differences between quantities which are not very different from one another and that &E) possibly includes contributions from forces at frequencies other than vo and from dipole-dipole and dipoleinduced dipole interactions. The good agreement suggests that corrections to the effective Hamaker-London coefficient for higher-order three-atom perturbations or corrections involving four or more atoms will not make significant changes to A$;)*. Kestner and Sinanoglu estimated that fourth- and fifth-order perturbation terms would alter the three-atom correction to the pair interaction energy by less than 30%. One aspect of the general problem of third-atom effects on the dispersion energy between two spherical molecules in a solvent (the last entry of Table I) has also been investigated by Renne and N i j b ~ e r . ~They ~ derived an expression for the (unretarded) van der Waals interaction energy between an atom, represented by an isotropic harmonic oscillator, and two dielectric spheres composed of the same atomic species, yielding an interaction energy equivalent to our V33*. Their result Aii* = & i i [ l - T C ~ ( F ~ / F ~ ) ] (40) is similar to our eq 34 above. The ratio of functions F3/Fl depends on s, the scaled separation between the centers of the spheres: F3/F1varies from 0.99 to 0.09 as s changes from 2.001 to 4.0. This indicates a smaller third-atom effect on the pair interaction than that predicted by eq 34, and that the third-atom effect decreases with increasing separation. These authors did not investigate the more general problem of third atoms which were different from the constituent atoms of the spheres, so they did not derive expressions corresponding to our eq 35-37. It should be emphasized that the results of the present work yield a modified Hamaker-London coefficient dif-
~
(36) R. B. Setlow and E. G. Pollard, “Molecular Biophysics”, Addison-Wesley, Reading, MA,1962, p 170.
(37) M.J. Renne and B. R. A. Nijboer, Chem. Phys. Lett., 6, 601 (1970).
J. Phys. Chem. 1981, 85, 2021-2026
fering from its unmodified counterpart by a constant factor, Le., invariant with distance of separation. Acknowledgment. We are most grateful to Professor Bruno H. Zimm, Department of Chemistry, University of
2021
California, San Diego, who called to our attention the work of 0. Sinanoglu, and to the William and Mary Beedle Fellowships Fund (awarded to G.P.U.), the UCLA Academic Senate Research Committee, and the Department of Chemistry for financial support.
Electron Trapping in Glassy Normal Alcohols. A Pulse Radiolysis Study at Temperatures Down to 6 K G. V. Buxton,' J. Kroh,' and G. A. Salmon The Universiry of Leeds, C&rklge Radiation Research Centre, CookrMge Hospital, Leeds LS 16 606, United Kingdom (Received: November 10, 1980: In Final Form: February 18, 1981)
End-of-pulse spectra of the trapped electron (e,) have been measured in glassy methanol + 5 mol % water, ethanol, 1-propanol, 1-propanol + 5 mol % water, 1-butanol, and 1-pentanol in the temperature range 6-115 K. The spectra depend on the temperature and the alcohol, and it is inferred from the kinetic behavior that they represent the initial distributions of e;. The results are explained in terms of a multitrap (small-polaron) model according to which electrons are trapped by OH groups and by at least two other, shallower traps associated with the alkane moiety. It is concluded that trapping by the shallow traps is more efficient at low temperatures and trapping by the OH groups is more efficient at high temperatures. Spectral changes with time are ascribed to migration of e; from shallow to deep traps.
Introduction The mechanism by which excess electrons, generated by ionizing radiation or photoionization, are trapped in liquids and amorphous and crystalline solids continues to attract attention from experimentalists and theoreticians alike.2 On the experimental front effort is being increasingly directed toward obtaining information about the optical absorption spectrum of the trapped electron, e t , in the very early stages of its existence, since this is most relevant to the actual trapping process. The most recent techniques employed include pulse radiolysis at cryogenic temperatures down to 6 KS5 with submicrosecond time resolution, both pulse radiolysis and flash photolysis at ambient temperature with subnanosecond6and even subpi~osecond~ resolution, and steady-state radiolysis at 4.2* and 1.6 Ka9 An important conclusion arising from such work is that the initial trapping process results in the formation of more than one kind of trapped electron, e.g., in crystalline ice,1° certain aqueous glasses.11J2and liquid6and glassf alcohols. (1)Senior Visiting Fellow from Sept 1978 to March 1979. On leave from Institute of Applied Radiation Chemistry, Technical University of Lodz, Lodz, Poland. (2)see, for example, J. Phys. Chem., Colloq. Weyl, 5th, 84 (1980). (3)(a) G. V. Buxton, K. Kawabata, and G. A. Salmon, Chern. Phys. Lett., 60,48(1978); (b) K. Kawabata, G. V. Buxton, and G. A. Salmon, ibid., 64,487(1979);(c) G.V. Buxton, J. Kroh, and G. A. Salmon, ibid., 68,554(1979); (d) G. V. Buxton and G. A. Salmon, ibid., 73,304 (1980). (4)N. V. Klassen and G. G. Teather, J. Phys. Chem., 83,326 (1979). (5)J. W. van Leeuven, L. H. Straver, and H. Nauta, J.Phys. Chem., 83,3008 (1979). (6)(a) G. A. Kenney-Wallace and C. D. Jonah, Chem. Phys. Lett., 39, 596 (1976); (b) ibid., 47,362 (1977). (7)J. M.Wiesenfeld, and E. P. Ippen, Chem. Phys. Lett., 73,47(1980). (8)M. Ogasawara, K.Shimizu, K. Ycahida, J. Kroh, and H. Yoshida, Chem. Phys. Lett., 64,43 (1979). (9)G. Dolivo and L. Kevan, J. Chem. Phys., 70,2599 (1979). (IO) G.V. Buxton, H. A. Gillis, and N. V. Klassen, Can. J.Chem., 55, 2385 (1977). (11)G. V. Burton, H.A. Gillis, and N. V. Klaasen, Can. J. Chern., 54, 367 (1976). (12)T.Q.Nguyen, D. C. Walker, and H. A. Gillis, J.Chern. Phys., 69, 1038 (1978). 0022-3854/81/2085-2021$01.25/0
Subsequent time-dependent spectral shifts have been variously attributed to trap deepening through dipole relaxation9J3or migration from shallower to deeper t r a p ~ . ' ~ J ~ For alcohols it has been suggested16that there are two kinds of trapping sites, one associated with the hydroxyl group and the other with the alkyl group. Alcohols, therefore, are appealing systems in which to investigate electron trapping since the size of the alkyl group can be varied systematically and hence the relative importance of the OH and alkyl moieties in the trapping process determined. In addition, subsequent interconversion between different forms of trapped electrons can be followed. In this paper we present results of a pulse radiolysis study of the normal aliphatic alcohols from methanol to pentanol in the temperature range 6-115 K. These show that the initial trapping is dependent on the alcohol and the temperature. The results for 1-propanol glass, which were presented in a preliminary report,3Eshowed that the end-of-pulse spectrum at 6 K is totally different from previously reported spectra obtained by pulse radiolysis at 77 K or steady-state radiolysis a t liquid-helium temperatures.
Experimental Section Methanol and ethanol (BDH Aristar) and 1-propanol (BDH Analar) were used as received. 1-Butanol (BDH Laboratory Reagent) and 1-pentanol (Hopkin and Williams Analar) were refluxed over sodium borohydride (10 g L-l) for 24 h and fractionally distilled, and the middle fraction comprising 75% of the total was collected. Glassy samples were made by injecting argon-purged alcohol into (13)L.Kevan, J. Chem. Phys., 56,838 (1972). (14)J. H.Baxendale and P. H. G. Sharpe, Chem. Phys. Lett., 39,401 (1976). (15)R. L. Bush and K. Funabashi, J. Chem. SOC.,Faraday Trans. 2, 73, 174 (1977). (16)T . Shida, S. Iwata, and T. Watanabe, J. Phys. Chern., 76,3683 (1972).
0 1981 American Chemical Society
