Towards the interaction between large solutes in a fluid of small hard spheres
The equivalence of the asymptotic expression for the interaction energy between a pair of large hard spheres in a fluid of small hard spheres that has been obtained by Roth et al. [Phys. Rev. E, 2000, vol. 62, p. 5360] using Fourier transforms and an expression that we have obtained [J. Colloid I...
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| Zitieren: | Towards the interaction between large solutes in a fluid of small hard spheres / D. Henderson, D.T. Wasan, A. Trokhymchuk // Condensed Matter Physics. — 2001. — Т. 4, № 4(28). — С. 779-783. — Бібліогр.: 12 назв. — англ. |
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irk-123456789-1205292017-06-13T03:05:38Z Towards the interaction between large solutes in a fluid of small hard spheres Henderson, D. Wasan, D.T. Trokhymchuk, A. The equivalence of the asymptotic expression for the interaction energy between a pair of large hard spheres in a fluid of small hard spheres that has been obtained by Roth et al. [Phys. Rev. E, 2000, vol. 62, p. 5360] using Fourier transforms and an expression that we have obtained [J. Colloid Int. Sci., 1999, vol. 210, p. 320] using Laplace transforms is pointed out and commented upon. Вказано і прокоментовано еквівалентність асимптотичного виразу для енергії взаємодії між парою великих твердих сфер у флюїді малих твердих сфер, який був отриманий Росом та ін. [Phys. Rev. E, 2000, том 62, с. 5360], використовуючи Фур’є перетворення і виразу, який ми отримали [J. Colloid Int. Sci., 1999, том 210, с. 320], використовуючи перетворення Лапласа. 2001 Article Towards the interaction between large solutes in a fluid of small hard spheres / D. Henderson, D.T. Wasan, A. Trokhymchuk // Condensed Matter Physics. — 2001. — Т. 4, № 4(28). — С. 779-783. — Бібліогр.: 12 назв. — англ. 1607-324X PACS: 82.70.Dd, 61.20.Gy DOI:10.5488/CMP.4.4.779 http://dspace.nbuv.gov.ua/handle/123456789/120529 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The equivalence of the asymptotic expression for the interaction energy
between a pair of large hard spheres in a fluid of small hard spheres that
has been obtained by Roth et al. [Phys. Rev. E, 2000, vol. 62, p. 5360] using
Fourier transforms and an expression that we have obtained [J. Colloid Int.
Sci., 1999, vol. 210, p. 320] using Laplace transforms is pointed out and
commented upon. |
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Article |
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Henderson, D. Wasan, D.T. Trokhymchuk, A. |
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Henderson, D. Wasan, D.T. Trokhymchuk, A. Towards the interaction between large solutes in a fluid of small hard spheres Condensed Matter Physics |
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Henderson, D. Wasan, D.T. Trokhymchuk, A. |
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Henderson, D. |
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Towards the interaction between large solutes in a fluid of small hard spheres |
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Towards the interaction between large solutes in a fluid of small hard spheres |
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Towards the interaction between large solutes in a fluid of small hard spheres |
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Towards the interaction between large solutes in a fluid of small hard spheres |
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Towards the interaction between large solutes in a fluid of small hard spheres |
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towards the interaction between large solutes in a fluid of small hard spheres |
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Інститут фізики конденсованих систем НАН України |
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2001 |
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Towards the interaction between large
solutes in a fluid of small hard spheres / D. Henderson, D.T. Wasan, A. Trokhymchuk // Condensed Matter Physics. — 2001. — Т. 4, № 4(28). — С. 779-783. — Бібліогр.: 12 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT hendersond towardstheinteractionbetweenlargesolutesinafluidofsmallhardspheres AT wasandt towardstheinteractionbetweenlargesolutesinafluidofsmallhardspheres AT trokhymchuka towardstheinteractionbetweenlargesolutesinafluidofsmallhardspheres |
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2025-07-08T18:04:46Z |
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2025-07-08T18:04:46Z |
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1837102953661464576 |
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Condensed Matter Physics, 2001, Vol. 4, No. 4(28), pp. 779–783
Towards the interaction between large
solutes in a fluid of small hard spheres
D.Henderson 1 , D.T.Wasan 2 , A.Trokhymchuk 1,3
1 Department of Chemistry and Biochemistry, Brigham Young University,
Provo, UT 84602, USA
2 Department of Chemical and Environmental Engineering,
Illinois Institute of Technology,
Chicago, IL 60616, USA
3 Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received September 25, 2001
The equivalence of the asymptotic expression for the interaction energy
between a pair of large hard spheres in a fluid of small hard spheres that
has been obtained by Roth et al. [Phys. Rev. E, 2000, vol. 62, p. 5360] using
Fourier transforms and an expression that we have obtained [J. Colloid Int.
Sci., 1999, vol. 210, p. 320] using Laplace transforms is pointed out and
commented upon.
Key words: colloidal suspensions, interaction
PACS: 82.70.Dd, 61.20.Gy
1. Introduction
This paper is dedicated to Jean-Pierre Badiali on the occasion of his 60th birth-
day. Jean-Pierre is a generous friend who has made many important contributions
to the theory of electrolytes and their interfaces. The authors wish him many more
productive, healthy, and happy years.
In their paper on the depletion potential in hard-sphere mixtures, Roth et al. [1]
have reported results for the asymptotic form of the interaction potential between
two large spheres in a fluid of small spheres. Recently [2–4], we have developed simple
analytic expressions for the force and interaction potential for the same system. Our
result is based on the asymptotic form of these functions, as obtained from the zeros
of the Laplace transform solution of the Ornstein-Zernike equation [2,5]. The result
of Roth et al. is obtained from the zeros of the Fourier transform of the Ornstein-
Zernike relation between pair and direct correlation functions [6]. Quite obviously,
c© D.Henderson, D.T.Wasan, A.Trokhymchuk 779
D.Henderson, D.T.Wasan, A.Trokhymchuk
the Laplace and Fourier transform techniques must lead to the same results. Indeed,
the Fourier transform is related to the Laplace transform with an argument that is
complex. The purpose of this note is to draw attention to the equivalence of the two
results and make a few comments and some generalizations.
2. Consideration
Roth et al. [1] comment on the fact that for a binary hard-sphere mixture in the
dilute limit of the large (radii R) spheres, the denominator of the Fourier transform
of the small-small (ss), large-small (ls) and large-large (ll) total correlation functions
is the same and equal to that of the fluid of pure small (diameter σ) spheres,
D̂(q) = 1− ρĉ(q), (2.1)
with ĉ(q) referring to the Fourier transform of the direct correlation function of the
fluid of pure small particles, and ρ is the number density of the small particles.
Henderson [5] had already noted this using the Laplace transforms. Following the
Percus-Yevick (PY) theory, in the limit of infinite dilution of large spheres, the
denominator of the Laplace transforms of all three total correlation functions is
P (s) = L(s) + esS(s), (2.2)
where L(s) and S(s) are two polynomials, first defined by Wertheim [7]. This result
is not as remarkable as Roth et al. feel. Since the large spheres are present in extreme
dilution, the basic behaviour of all the correlations in the system, which is governed
by the pole structure of the Fourier transform or Laplace transform, must be that
for a fluid of small spheres. This is valid in general and presumes that the same
approximation is used throughout.
The pole structure of the total correlation functions is determined by the zeroes
of P (s) or D̂(q). In general, there are an infinite number of solutions for P (s) = 0
or D̂(q) = 0, but the asymptotic behaviour is governed only by the zero with the
smallest imaginary part. Such a zero of P (s) in the form s = κ+iω has been tabulated
as a function of the packing fraction of the small spheres by Perry and Throop [8]
in their calculations of the Percus-Yevick asymptotic form of the total correlation
function for a one-component hard-sphere fluid. The zero of D̂(q) in the form q =
a1 + ia0 has been analyzed by Roth et al. [1]. Although both zeroes have been
parametrized by slightly different analytical functions [1,2], their numerical values
are the same. The resulting expression is an exponentially damped trigonometric
function
hPY
ss
(r) ∼ (Ass/r) cos(ωr + ϕss) exp(−κr), (2.3)
where Ass and ϕss are the amplitude and the phase parameters, respectively. It
has been already pointed [2,6] that asymptotic (2.3) reproduces the Percus-Yevick
solution not only at very large separations but is remarkably accurate down to the
second nearest-neighbour shell.
780
Interaction between large solutes
The decay of the Percus-Yevick total correlations, hPY
ll
(r), between two large
spheres is of the similar form as given by equation (2.3) only with the new amplitude
and phase parameters, All and ϕll. The function hPY
ll
(r) might be used to calculate
the interaction W (r) between two large solutes through the exact relation
βW (x) = − ln[1 + hll(x)], (2.4)
where x = r − R − σ/2. However, it has been argued [9] that if 2R ≫ σ, the
Percus-Yevick theory for total correlations between large spheres being applied in
equation (2.4), i.e. assuming hll = hPY
ll
, leads to the result which does not obey the
experimentally observed law: W ∼ R [10]. A qualitatively correct approximation for
the total correlations between two large spheres which still involves Percus-Yevick
theory has the form [5]
1 + hll(x) = exp[hPY
ll
(x)]. (2.5)
For the potential of mean force between two large spheres this leads to [2]
βW (x) = −hPY
ll
(x), (2.6)
and
βW (x) ∼ −All cos(ωx+ ϕll) exp(−κx). (2.7)
The last result has been obtained by Roth et al. [1] as well, through the linearization
of equation (2.4) aiming to get the asymptotic behaviour. However, they have com-
mented upon the fact that the asymptotic results (2.7) are in good agreement with
density functional theory “at very large separations and, strikingly, at intermediate
separations”. Assuming the validity of the Derjaguin approximation, this is simi-
lar to our own observations [2,11] regarding approximation (2.6), density functional
theory, as well as some other theories. To comment on this, we wish to stress that
the use of approximation (2.6) and expression (2.7) is not merely the result of a
linearization that is reasonable at large separations. Its meaning has much deeper
significance. The fact that expressions (2.5)–(2.7) are accurate, being constructed
from the Percus-Yevick result hPY
ll
, can be seen from their consistency with the ex-
perimental observation [10] that W ∼ R for very large spheres in low concentrations.
The expressions (2.6) and (2.7) for the potential of mean force can be thought
as resulting from the exact relation (2.4) if the considered system will be described
by the set of Ornstein-Zernike equations with the Percus-Yevick approximation for
the small-small and large-small Ornstein-Zernike counterparts together with the use
of the hypernetted chain (HNC) approximation for the large-large Ornstein-Zernike
equation. This may be designated the PY/PY/HNC approximation and is not to be
confused with a full implementation of the HNC approximation (HNC/HNC/HNC)
used by Attard and Patey [12], for example. The latter would not result in analytical
expressions but would satisfy the observation: W ∼ R. In contrast, a PY/PY/PY
procedure together with exact relation (2.4), would yield W ∼ lnR, a result with
an incorrect quantitative form and seriously in error. It is only the combination
PY/PY/HNC with equation (2.4), or equivalently PY/PY/PY with equation (2.5),
that is sensible.
781
D.Henderson, D.T.Wasan, A.Trokhymchuk
3. Conclusions
The force between two large spheres can be obtained by differentiation of the
interaction potential with respect to separation. However, since differentiation in
direct (real) space is related to multiplication by s in Laplace space, a more com-
pact expression and further generalizations can be easier obtained by using the
Laplace transform formalism. In particular, the asymptotic behaviour of the force
and, through the Derjaguin approximation, the decay of the interaction energy per
unit area between two planar surfaces are again determined by the same zero of P (s)
and both have the form of equation (2.7) but with other amplitude and phase coef-
ficients [3,4]. In the case of two planar walls (slit-like film), the important property
is the disjoining pressure which can be obtained by differentiation of the energy per
unit area with respect to separation. From multiplication of the Laplace transform
of the pair correlation function between two large spheres by s2, we again have the
result of the form of equation (2.7) for disjoining pressure only with other amplitude
and phase parameters. In as much as the Fourier transform is related to the Laplace
transform with a complex argument, equivalent results can be obtained from the
procedure of Roth et al.
4. Acknowledgements
DH and AT were supported in part by the NSF (Grant No CHE98-13729). DTW
was supported in part by a DOE grant.
References
1. Roth R., Evans R., Dietrich S. // Phys. Rev. E, 2000, vol. 62, p. 5360.
2. Trokhymchuk A., Henderson D., Wasan D.T. // J. Colloid Interface Sci., 1999,
vol. 210, p. 320.
3. Wasan D.T., Nikolov A.D., Trokhymchuk A., Henderson D. – In: Encyclopedia of
Surface Science. Edited by A. Hubbard. San Diego, Academic Press (in press).
4. Trokhymchuk A., Henderson D, Nikolov A., Wasan D.T. // Langmuir, 2001, vol. 17,
No. 16, p. 4940–4947.
5. Henderson D. // J. Colloid Interface Sci., 1988, vol. 121, p. 486.
6. Evans R., Leote de Carvalho R.J.F., Henderson J.R., Hoyle D.C. // J. Chem. Phys.,
1994, vol. 100, p. 591.
7. Wertheim M.S. // Phys. Rev. Lett., 1963, vol. 10, p. 321.
8. Perry P., Throop G. J. // J. Chem. Phys., 1972, vol. 57, p. 1827.
9. Experimental measurement [10] for the force F indicates that F (x) = ∂W (x)/∂x ∼
R. Since exact relation (2.4), we obtain the conclusion that in the colloidal limit
ln[1 + hll(x)] ∼ R. In particular, at the contact x = 0, the PY result is 1 + hPY
ll
(0) =
(R/σ)3
2
η/(1 − η)2 that would predict W ∼ lnR.
10. Israelachvili J. Intermolecular and Surface Forces. New York, Academic Press, 1992.
11. Henderson D., Sokolowski S., Wasan D.T. // J. Stat. Phys., 1997, vol. 89, p. 243;
J. Phys. Chem., 1998, vol. B102, p. 3009.
782
Interaction between large solutes
12. Attard P., Patey G.N. // J. Chem. Phys., 1990, vol. 92, p. 4970.
До взаємодії між великими сферами у флюїді малих
твердих сфер
Д.Гендерсон 1 , Д.Т.Васан 2 , А.Трохимчук 1,3
1 Факультет хімії і біохімії, Університет Брайхем Янг,
Прово, UT 84602, США
2 Факультет хімічної інженерії
та охорони навколишнього середовища,
Іллінойський технологічний інститут, Чикаго, IL 60616, США
3 Інститут фізики конденсованих систем НАН України,
79011 Львів, вул. Свєнціцького, 1
Отримано 25 вересня 2001 р.
Вказано і прокоментовано еквівалентність асимптотичного виразу
для енергії взаємодії між парою великих твердих сфер у флюїді малих
твердих сфер, який був отриманий Росом та ін. [Phys. Rev. E, 2000,
том 62, с. 5360], використовуючи Фур’є перетворення і виразу, який
ми отримали [J. Colloid Int. Sci., 1999, том 210, с. 320], використову-
ючи перетворення Лапласа.
Ключові слова: колоїдні суспензії, взаємодія
PACS: 82.70.Dd, 61.20.Gy
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