Colloidal suspensions confined to a film: local structure and film stability
This paper summarizes recent experimental and theoretical research conducted in our laboratories on understanding the properties of colloidal suspensions confined to a film. The results of statistical mechanics modelling to explain some experiments on thinning liquid films formed from concentrate...
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| Zitieren: | Colloidal suspensions confined to a film: local structure and film stability / D.T. Wasan, A. Nikolov, A. Trokhymchuk, D. Henderson // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 361-374. — Бібліогр.: 13 назв. — англ. |
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irk-123456789-1204422017-06-13T03:04:17Z Colloidal suspensions confined to a film: local structure and film stability Wasan, D.T. Nikolov, A. Trokhymchuk, A. Henderson, D. This paper summarizes recent experimental and theoretical research conducted in our laboratories on understanding the properties of colloidal suspensions confined to a film. The results of statistical mechanics modelling to explain some experiments on thinning liquid films formed from concentrated mono- and bidisperse colloidal suspensions are reported. The effect of colloidal particle size bidispersity on the local density distribution and film stability is discussed in detail. Стаття узагальнює останнi експериментальнi та теоретичнi дослiдження, проведенi в наших лабораторiях, щодо пояснення властивостей колоїдних суспензiй в обмежених об’ємах, зокрема колоїдних плiвок. У статтi представлено результати статистико-механiчного моделювання експериментальних спостережень процесу звуження плiвок, утворених одно- та двокомпонентними колоїдними суспензiями. Детально обговорюється вплив розмiрiв колоїдних частинок на розподiл локальної густини у плiвках та їх стабiльнiсть. 2001 Article Colloidal suspensions confined to a film: local structure and film stability / D.T. Wasan, A. Nikolov, A. Trokhymchuk, D. Henderson // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 361-374. — Бібліогр.: 13 назв. — англ. 1607-324X PACS: 82.70.Dd, 64.60.Cn, 64.75.+g, 65.50.+m DOI:10.5488/CMP.4.2.361 http://dspace.nbuv.gov.ua/handle/123456789/120442 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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This paper summarizes recent experimental and theoretical research conducted
in our laboratories on understanding the properties of colloidal suspensions
confined to a film. The results of statistical mechanics modelling
to explain some experiments on thinning liquid films formed from concentrated
mono- and bidisperse colloidal suspensions are reported. The effect
of colloidal particle size bidispersity on the local density distribution and
film stability is discussed in detail. |
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Article |
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Wasan, D.T. Nikolov, A. Trokhymchuk, A. Henderson, D. |
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Wasan, D.T. Nikolov, A. Trokhymchuk, A. Henderson, D. Colloidal suspensions confined to a film: local structure and film stability Condensed Matter Physics |
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Wasan, D.T. Nikolov, A. Trokhymchuk, A. Henderson, D. |
| author_sort |
Wasan, D.T. |
| title |
Colloidal suspensions confined to a film: local structure and film stability |
| title_short |
Colloidal suspensions confined to a film: local structure and film stability |
| title_full |
Colloidal suspensions confined to a film: local structure and film stability |
| title_fullStr |
Colloidal suspensions confined to a film: local structure and film stability |
| title_full_unstemmed |
Colloidal suspensions confined to a film: local structure and film stability |
| title_sort |
colloidal suspensions confined to a film: local structure and film stability |
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Інститут фізики конденсованих систем НАН України |
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2001 |
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http://dspace.nbuv.gov.ua/handle/123456789/120442 |
| citation_txt |
Colloidal suspensions confined to a
film: local structure and film stability / D.T. Wasan, A. Nikolov, A. Trokhymchuk, D. Henderson // Condensed Matter Physics. — 2001. — Т. 4, № 2(26). — С. 361-374. — Бібліогр.: 13 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT wasandt colloidalsuspensionsconfinedtoafilmlocalstructureandfilmstability AT nikolova colloidalsuspensionsconfinedtoafilmlocalstructureandfilmstability AT trokhymchuka colloidalsuspensionsconfinedtoafilmlocalstructureandfilmstability AT hendersond colloidalsuspensionsconfinedtoafilmlocalstructureandfilmstability |
| first_indexed |
2025-07-08T17:53:27Z |
| last_indexed |
2025-07-08T17:53:27Z |
| _version_ |
1837102227150340096 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 2(26), pp. 361–374
Colloidal suspensions confined to a
film: local structure and film stability
D.T.Wasan 1 , A.Nikolov 1 , A.Trokhymchuk 2,3 , D.Henderson 3
1 Department of Chemical and Environmental Engineering,
Illinois Institute of Technology,
Chicago, IL 60616, USA
2 Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
3 Department of Chemistry and Biochemistry, Brigham Young University,
Provo, UT 84602, USA
Received January 26, 2001
This paper summarizes recent experimental and theoretical research con-
ducted in our laboratories on understanding the properties of colloidal sus-
pensions confined to a film. The results of statistical mechanics modelling
to explain some experiments on thinning liquid films formed from concen-
trated mono- and bidisperse colloidal suspensions are reported. The effect
of colloidal particle size bidispersity on the local density distribution and
film stability is discussed in detail.
Key words: colloidal suspensions, disjoining pressure, film stability
PACS: 82.70.Dd, 64.60.Cn, 64.75.+g, 65.50.+m
1. Introduction
Much work has been done during the last century to investigate the properties of
aqueous and organic solvents confined by dispersed nanosized colloidal particles. The
last decade has shown an increased interest to the understanding of the nature of the
processes carried out where the systems in which the colloidal particles themselves
are under the confinement of macrosurfaces such as the walls of a container or
the outer surfaces of a macroscopic bodies [liquid droplets, gas bubbles etc.] are
immersed in a colloidal suspension. This is the case of numerous products such
as paints, inks, food and beverages, coatings, emulsions and foams. The properties
of colloidal suspensions near a single wall and the stability of the films formed
from concentrated colloidal suspensions become essential for the processes such as
sedimentation, lubrication, paper making, the mechanism for pollutant removal from
solid surfaces, the wetting of solids by micellar solutions, etc. However, the present
c© D.T.Wasan, A.Nikolov, A.Trokhymchuk, D.Henderson 361
D.T.Wasan et al.
understanding of the nature of the properties of fluid films formed from molecular
solvents and, particularly, from supramolecular fluids, is still far from complete.
A number of experimental techniques have been used to investigate concentrat-
ed colloidal suspensions. A new era of research towards understanding the nature of
the properties of the liquid films formed from aqueous latex suspensions, surfactant
micellar solutions, and microemulsions was opened when two of us discovered the
layering phenomena within the film and layer-by-layer thinning of such an ordered
intra-film structure [1]. Our further experiments using the differential and common
interference methods, low-angle transmitted light and Kossel diffraction techniques,
digital optical imaging, and video microscopy have clearly established the presence
of structural forces arising from the particle layering and the in-layer particle struc-
turing in colloidal suspensions confined between the film surfaces. These important
experimental findings revealed the role of the so-called non-DLVO entropic forces
such as oscillatory structural and depletion forces and thus stimulated the theoretical
modelling and the studies of confined colloidal dispersions.
A big advantage of the statistical mechanics approach is that once an adequate
model is formulated and an appropriate level of approximate description is estab-
lished, a broad set of properties [including thermodynamics, structure and dynam-
ics], their origin and interdependence may be examined simultaneously within the
framework of the same [theoretical] “experiment”. The most popular and widely
used statistical mechanics model of colloidal suspensions is called the “primitive”
model. This has a long history and is grounded in the fact that a small size ratio
of solvent molecule to colloidal particle [typically 10−1 − 10−4] could allow one to
neglect the discrete nature of the solvent. Due to this, it is assumed that the solvent
molecules and/or small colloidal particles 1 are invisible and considered to be a con-
tinuum [the DLVO level of description of colloidal systems] on the length and the
time scales of the large colloidal particles. However, the recent development of ex-
perimental and theoretical approaches and the results so obtained cast some doubts
about the universality of this model. This has been discussed already [2] for the
dynamics of colloidal suspensions and illustrates that the continuum has a strong
effect via Stokesian friction and hydrodynamic interactions, but it is still believed
that such “primitive” modelling has no influence on the static properties. At the
same time the Surfaces Forces Apparatus measurements [3] indicate that even the
crossed cylinders [infinitely large colloids] “recognize” the molecular scale and force
acting between them has oscillatory profile with periodicity of about the diameter of
solvent molecules. Statistical mechanics modellings [4,5] performed for aqueous and
organic solutions with the solvent and solute taken into account explicitly confirmed
these experimental observations.
We hope that even such brief insights into the state of the art of the confined
colloidal suspensions sufficiently highlight the applied and basic impact of the sub-
ject and shows the existing problems and the prospects of future development. The
particular goal of the present publication is two-fold. First, we wish to show an im-
1Colloidal dispersions invariably contain polydispersed particles with size ranges from several
percent [polymerization processes] to hundreds of percent [emulsification processes].
362
Colloidal suspensions confined to a film
portant role the non-DLVO [entropic] structural forces play in properties of confined
colloidal suspensions. Originally, this has been revealed and indicated by experimen-
tal investigations. We wish to show the link between statistical mechanical modelling
and observations. Second, we argue that an adequate modelling of confined colloidal
suspensions cannot be based on the “primitive” level. To do this, we apply the
bimodal highly size-asymmetric hard-sphere model to a confined colloidal suspen-
sions and compared with the “primitive” [one-component] model. In particular, we
report the results of microscopic statistical mechanics modelling for the local densi-
ty, disjoining pressure and interaction energy of the film formed from concentrated
monodisperse and bidisperse colloidal suspension.
2. Observation of the film stratification
To observe the behaviour of thin films formed from colloidal suspensions, an
original method has been developed [1]. According to this technique, the first step
is to form a biconcave drop of a liquid inside a cylindrical tube with a hydrophilic
inner wall. Next, the liquid is slowly sucked out of the drop through a capillary
orifice in the tube wall, creating a horizontal flat film [of radius from about 5×10−3
to 10−1 cm] encircled by a biconcave liquid meniscus. The temperature is maintained
constant by enclosing the glass assembly in a thermostat. Monochromatic light from
the bottom of the glass cell is incident on the film surface and the light, reflected
from a small portion of the thinning film area is conducted through a fiber optic
probe and photomultiplier to electrometer and, finally, it is recorded in the form
of photocurrent as a function of time. The film thickness, H , at any instant of
time, can be estimated from the intensity of the photocurrent using the expression
derived by Lyklema et al. [6]. The described method is a general one, appropriate for
studying the films formed from micellar surfactant solutions, suspensions of latex,
silica particles, and other colloidal dispersions.
Using a film formed from a micellar surfactant solution [micelle diameter, about
10 nm] as an example, the following phenomena have been observed. As soon as the
film forms, it begins to decrease in thickness. After it becomes thinner than about
ten micelle layers, the film thickness starts to change in steps [see the left panel of
figure 1]. The film first rests for a few seconds in a metastable state with a uniform
thickness. Then, one or more spots [with a smaller thickness than the remaining
part of the film] appear and gradually increase in size. Soon, the spots cover the
entire film and the film rests for several seconds in a new metastable state. Then,
even darker spots appear and, after their expansion, a subsequent metastable state
ensues. This process continues until the film finally reaches a stable state [of the
thickness about one micelle layer] with no more step-wise changes. The calculated
height of the steps is shown in the same figure and we can see that the magnitude is
approximately constant for all steps [about 10.6 nm], which corresponds to a micelle
diameter.
The experimental data for stratifying films formed from monodispersed latex
and silica hydrosol colloidal solutions, have been obtained as well. We find that the
363
D.T.Wasan et al.
0 10 20 30 40 50
-8
-4
0
4
8
12
16
10.1nm
10.1nm
10.1nm
particle volume fraction 34%
particle diameter 10nm
film thickness, [nm]
e
ne
rg
y
pe
r
un
it
ar
ea
, [
10
11
kT
/c
m
2 ]
Figure 1. Left panel: Interferogram of a film formed from a micellar surfactant
solution. Right panel: Film interaction energy for the film formed from a monodis-
perse suspension of hard-sphere particles as calculated from equation 3.8 [solid
line] and predicted [7] by Monte Carlo simulations [open squares].
process of film thinning occurs with the same stepwise thickness transitions. The
sizes of the transitions are uniform and are close to the effective diameters of the
colloidal particles. The number of thickness transitions increases with the particle
volume fraction. All this shows that film stratification is a universal phenomenon.
We suggest that this stepwise film thinning results from the ordered structures of
the colloidal particles that are formed within the film. The stepwise transitions are
nothing more than layer-by-layer squeezing of these structures from the middle area
of the film.
A direct proof of the presence of microstructures within thin films is of paramount
importance. We have performed some preliminary experimental and theoretical in-
vestigations of particle structuring inside the thinning film. Experimental studies
have been conducted using the low-angle transmitted light diffraction techniques
with a laser wavelength of 632.8 nm. A vertical microscopic film of the diameter
of 2.5 cm was formed from 40 vol% latex suspension. On the left panel of figure 2
we present the diffraction pattern obtained. The pattern symmetry corresponds to
a predominantly 2D hexagonal packing structure, i.e. colloid crystal-like in-layer
structure. This hexagonal structure appeared only at film thicknesses corresponding
to one and two particle layers. No such in-layer hexagonal structuring was observed
for films with three particle layers.
364
Colloidal suspensions confined to a film
3. Statistical mechanics modelling of the film stratificatio n
The experimental set-up described in a previous section can be imagined theoret-
ically as a huge reservoir filled with a colloidal suspension that is a size-asymmetric
two-component hard-sphere mixture of the small (s) and large (l) particles. The
small component of this mixture has a number density, ρb
s, hard-core diameter, d,
and its volume fraction is φs = πρbsd
3/6. The large component has a number density,
ρbl , hard-core diameter, D, and its volume fraction is φ l = πρblD
3/6. This is a ho-
mogeneous suspension and we refer to this as the bulk (b) suspension. To proceed,
we immerse two identical freely moveable flat walls, w, located in the XY plane at
z = 0 and z = H , in a middle part of the reservoir. To describe the interaction of
the suspension species with the walls, we used a hard-core particle-wall potential,
ϕiw(z), in a form which means that the closest approach to the wall surface occurs
when the geometric centers of fluid particles are located at z = d/2 [s component]
and at z = D/2 [l component]. In such a reference frame, H is the actual thickness
of the film, i.e. thickness H/D = 1 corresponds to one large particle layer.
The main statistical mechanics tool which we applied in the present study to
obtain theoretical results is based on the integral equation (IE) theory approach,
although some Monte Carlo (MC) computer simulations data [to prove our theoret-
ical approximations] are reported as well. To adjust the IE approach to the model
set-up, we start from the Ornstein-Zernike (OZ) relation between the total and di-
rect correlation functions written for the three-component hard-sphere mixture of
s, l and w species. The wall spheres have number density, ρb
w, and diameter, 2R.
The two flat walls appear in the theory as a result of the so-called “wall limit” when
the number density of the w species tends to zero [ρbw → 0] whereas their diameter
tends to infinity [2R → ∞]. This leads us to the set of the three OZ equations of
the form:
hij(r12)− cij(r12) =
∑
n=S,L
ρbn
∫
hin(r13)cnj(r32)dr3, (3.1)
hiw(r12)− ciw(r12) =
∑
n=S,L
ρbn
∫
hin(r13)cnj(r32)dr3, (3.2)
hww(r12)− cww(r12) =
∑
n=S,L
ρbn
∫
hwn(r13)cnw(r32)dr3, (3.3)
where h and c are the pair and direct correlation functions, respectively.
Equation (3.1) is just the OZ equation for the suspension in a bulk region. The
solution of equation (3.2) gives us the normalized local density distributions of the
particles confined by two walls, i.e. in a film region,
ρ⋆i (H, z) ≡
ρi(H, z)
ρbi
= 1 + hiw(H, z) . (3.4)
Finally, the solution of equation (3.3) can be used to obtain the effective interaction,
v(H), between wall solutes through the exact relation
βv(H) = − ln[hww(H) + 1] . (3.5)
365
D.T.Wasan et al.
1 2 3 4 5 6
-1.2
-0.8
-0.4
0.0
0.4
0.8
φ = 0.45
φ = 0.25
φ = 0.35
film thickness, H/D
fi
lm
in
te
ra
ct
io
n
en
er
gy
, [
kT
/D
2 ]
Figure 2. Left panel: Diffraction pattern from a vertical foam film formed from
40 vol% latex suspension. Right panel: Film interaction energy for the film formed
from monodisperse suspension of hard-sphere particles at three different bulk
volume fractions. The dashed curves indicate the exponential decay of the form
±W0e
−κH .
The question of what is an appropriate approximation for use in solving equa-
tions (3.1)–(3.3) is both controversial and unresolved. Although, the evidence [8]
indicates that the Percus-Yevick (PY) closure is reasonably reliable for the problem
of a thermal colloidal suspensions and molecularly smooth substrates, care must be
taken in each particular case. Namely, it has been argued [9] that if 2R ≫ D, the PY
theory for total correlation function between a pair of wall particles when applied
in equation (3.5), i.e. assuming hww = hPY
ww, leads to the result which does not obey
the experimentally observed law: v ∼ R [3]. A qualitatively correct approximation
for the total correlations between two walls has the form [4]
hww(H) + 1 = exp[hPY
ww(H)] . (3.6)
For the effective interaction between two walls this leads [5]
βv(H) = −hPY
ww(H) . (3.7)
The energy per unit area between two flat walls can be obtained by differentiation
of the interaction potential with respect to separation, i.e. W (H) = ∂v(H)/∂H .
Using the analytic PY result for the Laplace transform L[hPY
ww(r)], we derived a
simple equation to calculate the energy of the slit-like film formed frommonodisperse
suspension in the form [10]
W (H) = −P [D −H ]− 2σ , 0 < H < D
(3.8)
= W0 cos(ωH + ϕ)e−κH +W1e
−δ(H−D) , H > D ,
where P is the Carnahan-Starling result [11] for a suspension bulk pressure, σ is
the scaled particle theory result [12] for the surface tension near a single wall, and
W0,W1, ϕ, κ and δ are density dependent parameters [10]. This equation has been
366
Colloidal suspensions confined to a film
developed in a way to satisfy the known and exact statistical mechanics results and
has been tested against computer simulation data.
The right panel of figure 1 displays the profile of energy per unit area between
film surfaces for the film formed from monodisperse suspension as predicted by
equation (3.8). The conditions, i.e. volume fraction and particle diameter are close
to that of an experiment on film stratification in micellar surfactant solution. We can
see that the periodicity of the energy oscillations is about one particle diameter, i.e.
the same as the size of the steps in the experiments on film thinning [see left panel of
the same figure 1]. Below we will find the same periodicity for the density oscillations
in the film formed from monodisperse suspensions. This is a strong evidence that this
oscillatory behaviour of energy is caused by the formation of well-defined particle
layers inside the film and that the film will become thinner in a step-wise manner
with each step being about the particle diameter.
To proceed, on the right panel of figure 2 we summarize the theoretical predic-
tions for the effect of the particle volume fraction on both the decay length and
the periodicity of oscillations of the film energy W (H), as scaled with the particle
diameter. It is noted, that the decay length, 1/κD, increases upon increasing the
volume fraction, while the periodicity of oscillations, 2π/ωD, has the opposite trend.
When the particle concentration in a bulk region corresponds to around a 30% vol-
ume fraction, both the decay length and periodicity of oscillation are close to the
particle hard-core diameter. This theoretical result is consistent with experimental
findings. In general, the dependence of both the decay length and periodicity of
oscillation suggest that upon increasing colloidal particle volume fraction, the 2D
layered structure becomes a 3D array structure.
4. Local density distribution within the stratifying film
Local density variations are an essential feature of the suspensions under con-
finement. A suspension far from a confining substrate, i.e. in the bulk region, is
homogeneous and the particle local density is constant throughout. For film confine-
ment, the local density variations depend additionally on the separation between the
confining surfaces, i.e. the film thickness. In the present discussion we are focused on
the case of a thick film and will pay main attention to the changes in local density
for close vicinity of the film surfaces.
Two sets of theoretical “experiments” have been carried out to study the local
density distributions near the film surfaces. In each we started from a bulk monodis-
perse suspension of only the large or only the small particles with volume fractions,
φl = 0.20 and φs = 0.15, respectively. The size ratio of particle diameters, d : D,
was chosen to be 1 : 10 and was maintained in all our calculations. A diameter of
large particles, D, is used for scaling throughout. Since both surfaces of the film are
identical, we will display only half of the film region. Such results for the normalized
local densities2, ρ⋆i (z), are shown in figures 3 and 4.
2Since we are discussing the thick film, we omit the film thickness, H , from the notation of
normalized local density given by equation (3.4).
367
D.T.Wasan et al.
0
1
2
3
surface layer
1.27
L
a)
monodisperse
φ
L
= 0.20
0
1
2
φ
S
= 0.05φ
L
= 0.20
bidisperse
1.25
b)0.86
S
L
S
L
0 1 2 3
0
1
2
distance from the surface, z/D
l o
c
a
l
d
e
n
s
i
t y
d
i s
t
r
i b
u
t
i o
n
φ
S
= 0.10φ
L
= 0.20
bidisperse
1.25
c)
0.84
0.5 1.0
0
3
6
9
L
0.5 1.0
0
3
6
9
L
0
1
2
3
surface layer
S
a)
0.13
monodisperse
φ
S
= 0.15
0
1
2
φ
L
= 0.01φ
S
= 0.15
bidisperse
b)
0.13 0.58
S
L
S
L
0 1 2 3
0
1
2
distance from the surface, z/D
l o
c
a
l
d
e
n
s
i
t y
d
i
s
t r
i
b
u
t i
o
n
φ
L
= 0.05φ
S
= 0.15
bidisperse
c)
0.13 0.58
0.5 1.0
0
3
6
9
L
0.5 1.0
0
3
6
9
L
Figure 3. Left panel: Normalized local density profiles of a bidisperse suspension
near a film surface. The volume fraction of the large particles is fixed at φ l =
0.20 and volume fraction of the small particles changes. The open circles in part
(a) correspond to MC data. The arrows indicate the positions of corresponding
density minima. Right panel: The same as in left panel, but volume fraction of
the small particles is fixed at φs = 0.15 and volume fraction of the large particles
changes.
The initial fragments [parts (a)] of the left and right panels of figure 3 present
the local density distributions of pure suspensions of the large and of the small
particles at bulk volume fractions φ l = 0.20 and φs = 0.15, respectively. Such
monodisperse adsorbates are found to form a layer structure near the film surfaces:
the first [or surface] layer and the second layer are separated by well-defined minima
at z = 1.27D [for a film from the suspension of large particles] and z = 0.13D [in the
case of small particles]. The maxima of the surface layers are centered at z = 0.5D
and z = 0.05D, respectively, i.e. on the film surfaces. The surface layers are diffuse,
i.e. not well-ordered and, as a consequence, their thickness is about 1 1
2
diameter of the
constituent particles. These density distributions reflect the well-known [3] particle
layering phenomena in a monodisperse film. The PY approximation reproduces this
structure rather well as can be seen from the comparison with computer simulations
data presented on part (a) of the left panel as well.
The transformations of the local density of the large particles, caused by changing
the composition of the bulk suspension from φ l = 0.20;φs = 0 to φl = 0.20;φs =
0.10, are displayed on the remaining fragments of the left panel of figure 3. When
the bulk volume fraction of the small particles rises to φ s = 0.05 [part (b)], the film
surface contact density of the large particles is jumped over the bulk value by a
factor of up to approximately 5 and becomes two times higher than in the case of
a monodisperse suspension [part (a)]. Further increasing of the volume fraction of
the small species in a bulk region to φs = 0.10 [part (c)] causes more large particles
368
Colloidal suspensions confined to a film
0 1 2 3
0
1
2
3
distance from the surface, z/D
surface layer
2nd layer
2nd monolayer
surface monolayer
lo
ca
l d
en
si
ty
d
is
tr
ib
ut
io
n φ
S
= 0.15
φ
L
= 0.20
bidisperse suspensionL
S
Figure 4. Left panel: The same as in figure 4, but the volume fractions of both the
large and the small particles are fixed at φl = 0.20 and φs = 0.15, respectively.
Right panel: Pictorial two-dimensional interpretation of bidisperse suspension
ordering near a film surface.
to be adsorbed on the film surface giving rise to the formation of a thin surface
sublayer, which we recognize below [see figure 4] as a surface monolayer of the large
particles. A close inspection of ρ⋆
l (z) near the film surface reveals that, indeed, the
amount of large species located directly on the surface is increased, however, their
number density in the entire surface layer3 has not been affected and remains that
for the monodisperse case [part (a)]. At the same time, when the volume fraction
of the small species increases to φs = 0.15 [figure 4], the first maximum of ρ⋆
l (z)
totaly splits into a well-defined monolayer located directly on the film surface and
one or two [not so pronounced but clearly seen] thin sublayers. Meanwhile, with an
increase of the concentration of the small particles in a bulk region, also the second
layer of the large particles near a film surface becomes more pronounced. As for the
small particles, they fill the remaining free space on and in the vicinity of the film
surface extended up to 8–9 small particle diameters4. The nearly monotonic shape
and slow decay of ρ⋆s(z) in this region indicate a high packing of the small particles
[the local density is higher than in the bulk region] but without evidence for particle
structuring with respect to the wall, except their surface layer.
Finally, the remaining fragments of the right side of figure 3 present the evolution
of the local density of the small species near a film surface due to changing the
suspension composition in a bulk region from φs = 0.15;φl = 0 to φs = 0.15;φl =
0.05. The addition of only 1% of the large particles [φ l = 0.01] to the bulk region
3The integral over separation from the wall, z = 0, to the first minimum of the density profile
at z = 1.25D.
4This follows from the position of the first minima of the small particle density profiles, which
is located at z ≈ 0.8D.
369
D.T.Wasan et al.
does not affect the local density of the small particles near the film surface [part
(b)] but clearly indicates the formation of a surface localized monolayer of the large
particles being almost identical to that we observe in figure 4 at a high concentration
of the large particles [φl = 0.20] in the bulk region. We turn attention to this fact,
since no layering phenomena are observed when a monodisperse suspension of only
the large particles at a low bulk volume fraction, such as φ l = 0.01, is used. This
is a strong argument of the prominent role the small particle background play in
bidisperse suspensions [or solvent molecules in a monodisperse suspensions] to create
the conditions of a strong and pure entropic attraction within the large particles and
between the large particles and film surface. A well-defined second sublayer of the
large particle is also seen in the same part (b), i.e. the large particles are adsorbed on
a surface layer of the small species. Further increase of the concentration of the large
particles in a bulk region to φl = 0.05 [part (c)] affects only this second sublayer,
which becomes less organized, indicating the destruction [by the surface-attracted
large particles] of the surface layering of the small species, that has been formed in
their pure suspension [part (a)]. In such an environment, the small particles prefer
to be adsorbed on the surface of the large spheres and fill the cavities made by the
large spheres and the film surface, forming an effective surface coverage. A further
increase of the volume fraction of the large particles in a bulk region brings the film
to the state shown in figure 4 with φl = 0.20.
5. Film disjoining pressure and interaction energy
The properties of a major interest for the study of film stability are the film
disjoining pressure and the energy of interaction between film surfaces. According
to the definition [3], the disjoining pressure, Π(H), is a normal pressure inside the
film, measured relative to the pressure outside the film, i.e. to the pressure in a bulk
region:
Π(H) = PN(H)− PB . (5.1)
The normal pressure, P N(H), is the pressure exerted on the inner side of film surfaces
[in the direction perpendicular to the surfaces] by the film particles. This can be
evaluated from the local density profiles, ρi(H, z), and as a function of film thickness
is given by
βPN(H) = −β
∫ H
0
(
∂Usw(z,H)
∂z
ρs(H, z) +
∂Ulw(z,H)
∂z
ρl(H, z)
)
dz . (5.2)
The bulk pressure, P B, is one of the parameters that determine the thermodynamic
state of the suspension in a bulk region. This pressure is the limiting value of the
normal pressure in the case of infinitely thick film, i.e. PB ≡ PN(H → ∞). In
particular, to calculate the bulk pressure we used a surface separation of ten large
particle layers. This provides us with a well-defined homogeneous region at the
middle of the film. On the other hand, according to Derjaguin formalism [3], the
disjoining pressure can also be measured by displacing one of the film surfaces by
370
Colloidal suspensions confined to a film
a distance ∆H . Then, the energy [work] per unit area to bring the surfaces from
infinity to the separation H , can be calculated as
W (H) =
∫
∞
H
dH ′Π(H ′) . (5.3)
The equation (5.3) is an important expression because it links the calculated prop-
erties, i.e. local density distributions and normal pressure with the experimental
measurements of the energy of the interaction between film surfaces. We also wish
to remind that colloidal suspensions in the present study are modeled by hard-
sphere-like particles, i.e. suspensions are a thermal, and both the disjoining pressure
and the energy of interaction between film surfaces are of pure entropic origin. They
result only from a distinct structural ordering of colloidal particles in a bulk and
film regions discussed above.
Figure 5 displays the profiles of the resulting structural disjoining pressure and
structural interaction energy between film surfaces as functions of a film thickness
for monodisperse and bidisperse suspensions. A sharp peak in the disjoining pressure
exerted by a monodisperse film near a thickness of one particle diameter is apparent:
as the film thickness is reduced to about one particle diameter, it becomes very
difficult to squeeze the particles in the last layer out of the film. This force is purely
steric. Thus, the disjoining pressure rises to a high value that corresponds to the
repulsive stabilization energy barrier. Once the film thickness is reduced to a value
slightly smaller than one large particle diameter, there will be no particles inside the
film, the steric force disappears, and the disjoining pressure drops to zero [or even
less] leading to a depletion attraction in the energy profile and to film floculation.
Qualitatively similar but quantitatively quite different, i.e. a more sharp and twice
higher peak is observed for a film formed from bidisperse suspension. Surprisingly,
when the film thickness is reduced to a value slightly smaller than one large particle
diameter, the pressure inside the film does not drop significantly and the resulting
disjoining pressure is still positive and large [in our particular case it is the same
order as stabilization disjoining pressure in a monodisperse film at the thickness of
one large particle diameter]. Consequently, the “depletion” energy is repulsive and
the film remains stable.
This will cause different trends in the film stratification process. In particular,
the film thicknesses that correspond to mechanical equilibrium [Π(H) = 0] will be
different. The detailed analysis of the film energy and disjoining pressure profiles
vs film thickness reveals that mechanical equilibrium between the film formed from
monodisperse suspension and the monodisperse bulk reservoir with volume fraction,
φ = 0.35, is observed at film thicknesses, H = 3.2D; 2.2D; 1.2D while in the case of
a bidisperse film and bidisperse bulk reservoir with the same volume fraction, i.e.
φ = φl + φs = 0.35 it is reached at H = 2.85D; 1.95D; 1.13D and additionally at
H = 1.02D. These thicknesses correspond to a minima of film energy and will define
the metastable film thicknesses that could be measured experimentally by light
interferometry. Note, that not all zeroes of disjoining pressure will correspond to a
metastable film thicknesses. In general, the film of thicknesses H will be metastable
if a condition: ∂Π/∂H < 0 is fulfilled.
371
D.T.Wasan et al.
1 2 3 4
-6
0
6
12 particle volume fraction 35%
bidisperse suspension
monodisperse suspension
di
sj
oi
ni
ng
p
re
ss
ur
e,
Π
D
3 /k
T
film thickness, H/D
1 2 3 4
2
0
2
4
monodisperse suspension
bidisperse suspension
particle volume fraction 35%
st
ru
ct
ur
al
fi
lm
e
ne
rg
y,
W
D
2 /k
T
film thickness, H/D
Figure 5. Structural disjoining pressure [left panel] and structural interaction
energy between film surfaces [right panel] of a films formed from monodisperse
[dashed line] and bidisperse [solid line] suspensions. The particle volume fractions
in both cases are 35%.
6. Conclusions
A model film formed from a bimodal hard-sphere-like suspension with particle
size ratio 1 : 10 confined between two structurless hard walls has been investigated.
We have confirmed the conclusions, obtained from the observation, that the presence
of the small particles becomes crucial and has a strong impact on the ordering of
the large species near the planar wall. The most striking prediction from the present
study is concerned with the formation a surface-localized monolayer of the large
particles. Perhaps we should note a very evident correlation between the bidisperse
composition, i.e. φl and φs, when the surface monolayer of the large particles be-
comes well-defined [the dip developed in ρ⋆
l (z) at z = 0.58D is well-defined], for the
bidisperse compositions at which a surface crystalline phase was observed [13].
As the number density of the large particles in the monolayer is increased, a
question is raised: What is the lateral structure of the large particles on and near
the film surface? The singlet level of integral equation theory, that we employed in
the present study, does not give information about the in-layer correlation functions
and, obviously, we cannot definitely answer the above question. Nevertheless, we still
can obtain some information from the local density variations near the film surface.
In particular, the estimated number density of the large species in a monolayer is
≈ 2 per surface area of ten large particles. Thus, it is very unlikely that crystalline
order can be formed within the monolayer of the large particles. At the same time,
the surface coverage provided by the small particles in the vicinity of the wall is quite
high [≈ 240 small particles per surface area of one large particle for the composition
conditions corresponding to figure 4]. In conjunction with the strong large particle-
film surface entropic attraction, such a density distribution of the small particles
372
Colloidal suspensions confined to a film
will limit the large species movement on the wall forming an effective atomically
structured or corrugation-like coverage of the film [adsorbent] surface. This allows
us to speculate that the remaining large particles, that are adsorbed on such a
structured substrate, will grow into the crystallites in the direction normal to the film
surface that are observed experimentally. In this case the formation of the surface
monolayer of the large particles, revealed by the present calculations, might be the
necessary condition and physical reason that can explain why surface crystallization
and surface phase separation foreshadows phase separation in the bulk bidispersion.
The structural disjoining pressure and structural energy of a bidisperse film as
the functions of film thickness have oscillatory decay profiles but the periodicity
and phase of oscillations are more complex then for a film formed from a monodis-
perse suspension. When the film is thick, the periodicity of the oscillations is of the
order of one large particle diameter, as it for a monodisperse film, but the phases
of oscillations are almost opposite, i.e. metastable thicknesses for monodisperse and
bidisperse films are different. Due to the adsorption of the small particles on the film
surfaces and on the surfaces of the large particles as well, at the film thicknesses
about one large particle layer, the pressure exerted by the film and the film energy
start to oscillate with a periodicity of the small particle diameter. This results in a
novel [with respect to the film formed from a monodisperse suspension] energy min-
ima and zero of disjoining pressure which have been observed at the film thickness
only slightly larger than one layer of the large particles, which is rather stable due
to the very low decrease in film energy and very high increase in disjoining pressure.
7. Acknowledgements
AT and DH were supported in part by NSF (Grant No. CHE98–13729). DTW
and AN were supported in a part by a DOE grant.
References
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2 that would
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Плiвки з колоїдних суспензiй: локальна структура та
стабiльнiсть плiвок
Д.Т.Васан 1 , А.Ніколов 1 , А.Трохимчук 2,3 , Д.Гендерсон 3
1 Факультет хімічної інженерії та охорони навколишнього
середовища,
Іллінойський технологічний інститут, Чикаго, IL 60616, США
2 Інститут фізики конденсованих систем НАН України,
79011 Львів, вул. Свєнціцького, 1
3 Факультет хімії та біохімії, Університет Брайхем Янг,
Прово, UT 84602, США
Отримано 26 січня 2001 р.
Стаття узагальнює останнi експериментальнi та теоретичнi дослi-
дження, проведенi в наших лабораторiях, щодо пояснення власти-
востей колоїдних суспензiй в обмежених об’ємах, зокрема колоїдних
плiвок. У статтi представлено результати статистико-механiчного
моделювання експериментальних спостережень процесу звуження
плiвок, утворених одно- та двокомпонентними колоїдними суспен-
зiями. Детально обговорюється вплив розмiрiв колоїдних частинок
на розподiл локальної густини у плiвках та їх стабiльнiсть.
Ключові слова: колоїднi суспензiї, розклинюючий тиск,
стабiльнiсть плiвок
PACS: 82.70.Dd, 64.60.Cn, 64.75.+g, 65.50.+m
374
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