Download When two surfaces or particles approach closer than a few

STRUCTURAL FORCES
Anže Koselj
Faculty of Mathematics and Physics, Ljubljana
Advisor: Igor Muševič
April 2002
Abstract
Direct measurements have furnished conclusive evidence for the existence of structural
forces arising from a surface-induced modification of liquid structure. Such forces are
characterized by decaying oscillations with a periodicity equal to the molecular size, and their
measurable range is typically 5  10 molecular diameters. In this range the amplitude of the
oscillations is significant compared to the van der Waals force predicted by continuum
theory, especially in the last few molecular diameters where it is considerably larger. These
observations are of particular relevance to colloid science and biology, where it has been
customary to consider the solvent between interacting surfaces of particles as a structureless
medium.
1
Contents
1
Non-Structural forces between bodies and surfaces. . . . . . . . 3
1.1 Van der Waals force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Electrostatic force between bodies in liquids. . . . . . . . . . . . . . . . . . . . . . . . . . 4
2
Structural forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Measurements. . . . . .. . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Origin of main type of solvation force: the oscillatory force. . . . . . . . . . . . . . . 7
2.3 Properties of solvation forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11
2.4 Computational models of solvation forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.1 Hard sphere model
2.4.2 Lennard-Jones model
3
Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
16
1 Non-Structural forces between bodies and surfaces
In considering the forces between two molecules or particles in liquids, i.e. solvents, several
effects are involved that do not arise when the interaction occurs in free space. This is
because an interaction in a medium always involves many solvent molecules, i.e. it is
essentially a many-body problem.
At the most basic molecular level, we have the interaction potential w(r ) between two
molecules or particles, which is usually known as pair potential. It is related to the force
between these two molecules or particles by F  dw(r ) / dr . Since the derivative of
w(r ) gives the force, and hence the work that can be done by the force, w(r ) is often
referred to as the free energy.
Pair potential for two solute molecules in a solvent does not include only the direct solutesolute interaction energy but also any changes in the solute–solvent and solvent-solvent
interaction energies as the two solute molecules approach each other. This approach can
only be done by displacing solvent molecules from its path. The net force therefore depends
also on the attraction between the solute and the solvent molecules. Solute molecules
perturb the local structure of solvent molecules, Figure 1. If the free energy associated with
this perturbation varies with the distance between the two dissolved molecules, it produces
an additional solvation or structural force between them.
Structural force is the main objective of this seminar. In the first section we will shortly note
continuum theory that does not entail any structure of the solvent, we will discuss the most
important experimental and theoretical properties of solvation forces in section 2 and give
some final remarks in conclusion.
Fig 1: b) Reordering of solvent molecules by solute molecules.
1.1 Van der Waals force
Van der Waals force is a sum of three different intermolecular forces of electric origin with the
same distance dependence 1 / r 6 , where r is the distance between atoms or molecules. The
contributions are Boltzmann averaged force between two permanent dipoles, usually referred
to as Keesom interaction, interaction between dipole and induced dipole, usually referred to
as Debye interaction and the most important part, the dispersion interaction, which is due to
the fluctuations of electric field and is always present. The last contribution, known also as
the London force, due to London who first calculated it, or charge fluctuation force, has
special meaning because it acts between all molecules and atoms without exceptions.
The pioneering work on the van der Waals forces between macroscopic bodies was done by
Hamaker. He had assumed pairwise additivity of intermolecular van der Waals forces and
summed the pair interactions to obtain the total interaction energy between bodies.
Summation is of course done via volume integral and the net van der Waals interaction
energy in the case of 'interaction' between a sphere of radius R centered at (0,0, R ) and a
flat surface located at ( x, y , d ) in the limit of small distance ( R  d ) , equals
3
W (d )  
 2 C N 2 R
6d
.
(1)
Hamaker introduced Hamaker constant which includes all material specific properties,
relevant for the interaction, in the form
A   2 C N 1  N 2 ,
(2)
where  N stands for number density, and C is a numerical factor.
Hamaker made a rough approximation. Unlike gravitational and Coulomb forces, van der
Waals forces are not generally pairwise additive: the force between any two molecules is
affected by the presence of other molecules nearby as the field emanating from any one
molecule reaches a second molecule both directly and by ‘reflection’ from other molecules
since they, too, are polarized by the field. The second thing that comes into play at
appreciable distances is retardation. Time taken for the electric field of the first atom to reach
with the period of the fluctuating dipole itself. When this happens the field returns to find that
the direction of the instantaneous dipole of the first atom atom is now different from the
original and less favorably disposed to an attractive interaction. Thus, with increasing
separation the dispersion energy between two atoms begins to dacay even faster than
 1 / r 6 , approaching a  1 / r 7 dependence. This is called the retardation effect. For two
molecules in free space retardation effects begin at separations above 5nm . In a medium,
where the speed of light is slower, retardation effects come in at smaller distances.
1.2 Electrostatic force between bodies in liquids
Van der Waals force between similar particles in a medium is always attractive and if only
van der Waals forces were operating, we might expect all dissolved particles to stick together
eventually and precipitate out of solution as a mass of solid material. Fortunately this does
not happen, because particles suspended in a solvent are usually charged and repulsive
electrostatic force can prevent them to coagulate.
The charging of a surface in a liquid can come about in two ways: by the adsorption of ions
from the solution on to the surface or by the dissociation of surface groups. However, when
considering the electrostatic force between macroscopic bodies, the condition of
electroneutrality comes into play. Charged surface attracts counterions from the solution
(ions of opposite sign of charge with respect to the surface) and they counterbalance the
surface charge, satisfying the electroneutrality condition, Figure 2.
We must first consider some fundamental equations to describe the counterion distribution
between two charged surfaces in a solution. For the case when only counterions are present
in solution, the chemical potential of any ion may be written as
  ze  kT log 
(3)
where  is the electrostatic potential and  the number density of ions of valency z at any
point x between two surfaces. Since only differences in potential are physically meaningful,
we may choose  0  0 at the midplane ( x  0) where also    0 .
From the requirement that the chemical potential is uniform throughout the gap, Eq. 3 gives
us the expected Boltzman distribution of counterions at any point x :
   0 e  ze / kT
(4)
4
Fig 2: Left: Charged surface attracts counterions from the solution (ions of opposite sign of charge with respect
to the surface) and they counterbalance the surface charge, satisfying the electroneutrality condition. Right:
Graphical presentation of the solution of Poisson-Boltzman equation.
One further important fundamental equation is required, namely the Poisson equation. This
equation, combined with the Boltzman distribution, Eq. 4, when solved, gives the potential
 , electric field E   / x , and counterion density  , at any point x in the gap between
the two surfaces, Figure 2.
The distribution of ions is the most important source of the electrostatic interaction between
the charged surfaces in a liquid since the electric field between the plates originates from
their distribution. If there were no ions, there would be no field in the gap between equally
charged plates, as their fields would exactly cancel out. When the counterions are introduced
into the intervening region they do not experience an attractive electrostatic force towards
each surface. The reason why the counterions build up at each surface is simply because of
their mutual repulsion. The repulsive electrostatic interaction between the counterions and
their entropy of mixing alone determine their concentration profile  (x ) , potential profile
 (x) and the field E between the surfaces.
Armed with this solutions, it is not too difficult to calculate the pressure P at any point x
between the two surfaces, [1]. It can be split into two parts: attractive electrostatic field
contribution and repulsive entropic contribution. The entropic repulsion dominates and the
net effect is therefore repulsive.
The combination of electrostatic double layer interaction and the van der Waals interaction
plays the most important role in colloid and parts of the surface science (adhesion
phenomena, etc.). The theory covering both forces together has been developed and is
called DLVO theory after Drejaguin, Landau, Verwey and Overbeek.
2 Structural forces
When two surfaces or particles approach closer than a few nanometers, continuum theories
of attractive van der Waals and repulsive double-layer force often fail to describe their
complete interaction. This is either because this continuum theory breaks down or because
other non-DLVO forces, as for example solvation or structural force, come into play. These
additional forces can be monotonically repulsive, monotonically attractive or oscillatory, and
they can be much stronger than either of the two DLVO forces at small separations. Such
oscillatory forces have a mainly geometric origin while monotonic forces are due to surfacesolvent interactions.
2.1 Measurements
On the experimental side, first measurements of such short-range oscillatory forces between
two solid surfaces arising from structure in the intervening liquid were done by Israelachvili
and his co-workers in 1980. The experimental technique in this study allowed direct
measurements of the forces between two curved surfaces immersed in a liquid as a function
of the distance between them. Forces were measured between two crossed cylindrical
5
surfaces (the radius of curvature ( R  1 cm )) of molecularly smooth mica, Figure 4. Multiple
beam interferometry was used for measuring the surface separation to  0.1  0.2nm and
forces were measured by a spring deflexion method with sensitivity of better than 10 7 N .
Fig 3: Surface Forces Apparatus (SFA) Israelachvili used to measure short-range solvation forces. The distance
between the two surfaces is controlled by use of a three-stage mechanism of increasing sensitivity: upper rod,
lower rod and, finally, a piezoelectric crystal tube. The later control is used for positioning to 0.1nm. After all, the
separation between the two surfaces is measured by use of multiple beam interference.
The force is measured by expanding or contracting the piezoelectric tube by a known amount and then measuring
optically how much the two surfaces have actually moved; any difference in the two values when multiplied by the
stiffness of the force-measuring spring gives the force difference between the initial and final positions.
The same technique used in measurements of long-range van der Waals forces in vacuum
and electrostatic double-layer forces gives results that are in very good agreement with
underlying theory. Liquid used in this study was OMCTS, octamethylcyclotetrasiloxane
CH 3 2 SiO . This is most commonly used liquid in these kinds of experiments for its large

4
size (molecular diameter  1nm), nearly spherical molecular shape and inertness.
Fig 4: Experimental results of measurements of force F as a function of separation D between two curved mica
surfaces of radius R. The arrows at P and Q, indicating jumps from unstable to stable positions, have a slope of
6
K/R=10^4N/m^2. For comparison the theoretically expected continuum van der Waals force is also shown. The
inset shows the same measurement on a reduced scale.
Figure 4 shows room temperature measurements of force F (divided by the radious of
curvature R ) plotted against the distance D between surfaces.
Since one of the mica surfaces was suspended at the end of a centilaver spring of stiffness
K  10 2 N / m , the force could be measured only in regions where F / D  K . When the
gradient of the force F / D exceeds the spring stiffness K , instabilities occur and the
surface jump to a separation at which they are stable, e.g. inward jumps from force
maximum, or outward jumps from a force minimum. These jumps are analogous to those
occurring between two magnets when one of them, suspended from a spring, is brought
towards the other, or when the magnets are separated. By measuring such jumps the force
maxima and minima can be accurately located. The forces in between (where F / D  K )
can be measured directly, while those shown by dashed sections ( F / D  K ) are
experimentally inaccessible without the use of much stiffer spring which would reduce the
measuring sensitivity.
The results of Figure 4 show spatially decaying oscillatory forces extending up to about ten
molecular diameters. The decay appears to be rapid over the first few molecular diameters,
then it becomes more gradual. At surface separations above 3nm the average periodicity of
the force oscillations is 1.05  0.05nm which correlates well with the mean diameter of this
nearly spherical (oblate spheroid) molecule. However, the first three or four oscillations have
a smaller periodicity of 0.8  0.1nm , suggesting that molecules in the first layer or two on
each surface are more rigidly bound with their short axes perpendicular to the surfaces.
The only force we would expect to measure between these surfaces if the intervening liquid
had no structure would be the purely attractive van der Waals force given by the continuum
DLVO theory. For comparison we have included this force, F / R   A / 6 D 2 , in Figure 5,
where the Hamaker constant A  1.4  10 20 J has been calculated from experimental data.
This highlights the strength of the structural force, especially at small separations.
2.2 Origin of main type of solvation force: the oscillatory force
To understand how solvation forces arise between two surfaces we must first consider the
way solvent molecules order themselves at an isolated surface. After that we can consider
how this ordering changes in the presence of a second surface, and how this determines the
short-range interaction between these two surfaces in the liquid. The structuring of solvent
molecules at a surface is in principle no different from that occurring around a small solute
molecule, or even around another identical solvent molecule, which is determined primarily
by the geometry of molecules and their ability to pack around a constraining boundary.
At a solid-liquid interface, attractive interactions between the wall and liquid molecules and
the geometric constraining effect of the hard wall on these molecules force them to order (or
structure) into quasi-discrete layers, Figure 5. This structuring shows itself through an
oscillatory density profile extending several molecular diameters into the liquid.
7
  is the bulk liquid density;  is the scale of
molecular ‘roughness’ of the interface. b) Liquid density profile at an isolated solid-liquid interface.  s () is the
contact density at the surface. c) Liquid density profile between two hard walls a distance D apart.  s (D) is a
Fig. 5: a) Liquid density profile at a vapor-liquid interface.
function of
D as illustrated in Figure 3.
Effect of two constraining solid surfaces is even more dramatic. Here even in the absence of
any attractive wall-liquid interaction, geometric considerations alone dictate accommodation
of liquid molecules between the two walls, and the variation of this ordering with separation
D gives rise to the solvation force between the two surfaces. In case of spherical molecules
confined between two hard, smooth surfaces solvation force is usually a decaying oscillatory
function of distance. For molecules with asymmetric shapes solvation force has most often
monotonically repulsive or attractive component.
Likewise, if the confining surfaces are themselves not well-ordered but rough or fluid-like, the
oscillations will be smoothed out and the resulting solvation force will be monotonic. As an
example, no structuring is expected when one or both of the interacting surfaces is liquidliquid or liquid-vapour interface.
Numerous experimental measurements have shown that the solvation pressure between two
surfaces in a liquid media or between two solute molecules is proportional to the difference in
liquid density at each surface when surfaces are at distance D and when they are at infinite
separation:
P( D)   s ( D)   s () ,
(5)
Here  s (D) and  s () are the density of liquid molecules at each surface at surface
separation D and  respectively. Thus, a solvation force arises once there is a change in
the liquid density at the surfaces as they approach each other. For two inert hard walls this is
brought about by changes in the molecular packing as D varies. We see, Figure 6, that
 s (D) will be high only at surface separations that are multiples of  but must fall at
intermediate separations.
8
Fig 6: Top: The molecular ordering and density of liquid molecules in contact with surface changes as the
separation D changes. Bottom: Corresponding solvation pressure as qiven by Eq. 5.
At large separations the solvation pressure falls to zero as  s (D) approaches the value for
isolated surfaces  s () . Resulting solvation pressure has the same oscillatory distance
dependence as the density profile with the periodicity roughly equal to  and range of a few
molecular diameters.
In the limit of very small separations, as the last layer of solvent molecules is finally
squeezed out, we have  s ( D  0)  0 and the solvation pressure approaches a finite value
given by
P( D  0)  kT s ()
(6)
That means that the force at contact is negative, i.e., attractive or adhesive.
A number of theoretical studies and computer simulations of various confined liquids have
invariably led to an oscillatory solvation force at surface separations below a few molecular
diameters. In first approximation these may be described by an exponentially decaying cosfunction of the form
P( D)  kT s () cos( 2D /  )e  D / 
(7)
where both the oscillatory period and the characteristic decay length of the envelope are
close to  . By integrating Eq. 7 it is simple to show that the solvation force contribution to
the interfacial energy of two surfaces is
i 
kT s ()
8 2
(8)
9
Since  s () should be approximately equal to the bulk liquid density, we may write
 s ()  2 /  3 , which further simplifies the above to
i 
2kT
8 
2
2

0.02kT
(9)
2
For Hamaker constants smaller than about A  0.2kT , i.e., for A  10 21 J , we expect the
oscillatory solvation force to dominate the adhesion of two surfaces across a liquid medium,
as van der Waals contribution to the interfacial energy is given by
 i  3 A / 2 2 2  0.1A /  2 .
It is important to appreciate that solvation forces do not arise simply because liquid
molecules tend to structure into semi-ordered layers at surfaces. They arise because of the
disruption or change of this ordering during the approach of second surface, as in Eq 5. If
there were no change, there would be no solvation force. Effects of tendency toward
structuring at an isolated surface and solvation force are two different effects, but they are of
course related: the greater the tendency, the greater the solvation force between two such
surfaces.
The most simple case of spherical molecules between two smooth surfaces is well
understood. Real systems are often much more complex: the liquid molecules are usually
non-spherical, and the surfaces are not smooth but corrugated at the atomic level.
Any strongly attractive interaction between a surface and the liquid molecules adjacent to
denser packing of molecules at the walls and thus to higher  s values and a more repulsive
but still oscillatory force (Figure 7, upper curve). On the other hand, if the surface-liquid
interaction is much weaker than the liquid-liquid interaction the oscillatory force tends to be
overall more attractive (Figure 7, lower curve). Such complex force-laws often show both a
short-range oscillatory part and long-ranged monotonic part, the latter arising from more
complex interactions like H-bond correlation interactions and long-range polarization. A
theoretical understanding of the interactions of ‘complex’ liquids is not yet available.
Repulsive structural forces appear to arise whenever water molecules strongly bind to
surfaces containing hydrophilic groups, i.e., certain ionic, zwitterionic, or H-bonding groups,
and their strength depends on the energy needed to disrupt the hydrogen-bonding network
and to dehydrate two surfaces as they approach each other. Their effective range,
determined experimentally, is about 3  5nm , which is about twice the range of the
oscillatory solvation force in water.
A hydrophobic surface is one that is inert to water in the sense that it cannot bind to water
molecules via ionic or hydrogen bonds. The entropically unfavoured water is ejected into the
bulk when surfaces coming together thereby reducing the total free energy of the system.
Similarly, ‘hydrophobic’ forces are of long-range, decaying heavily in the range 0  10nm
while more slowly farther out.
10
Fig 7: Oscillatory solvation force superimposed on a monotonic solvation force. This type of interaction often
arises in aqueous solutions where hydrogen-bond correlation effects can give rise to an additional monotonically
decaying ‘hydration’ force. For hydrophilic surfaces the monotonic component is repulsive (upper dashed curve),
whereas for hydrophobic surfaces it is attractive (lower dashed curve). For simpler liquids there are no such
monotonic components (middle curve).
2.3 Properties of solvation forces
There are new and new experimental measurement techniques and experiments associated
with solvation forces resulting in many important data about them. Let us summarize the
main features of solvation forces gathered through measurements.
 In liquids whose molecules are roughly spherical and fairly rigid, the periodicity of the
oscillatory force is equal to the mean molecular diameter  .

The peak-to-peak amplitudes of the oscillations show a roughly exponential decay with
distance with a characteristic decay length of 1.2 to 1.7  .

The oscillatory force can exceed the van der Waals force at separations below five to
ten molecular diameters, and for simple liquids, merges with the continuum van der
Waals or DLVO force at large separations.

Oscillatory solvation forces are not strongly temperature dependent

The structure of the confining surfaces is just as important as the nature of the liquid for
determining the solvation forces. Between two surfaces that are completely smooth the
liquid molecules will be induced to order into layers, but there will be no lateral ordering
within the layers. In other words, there will be positional ordering normal but not parallel
to the surfaces. However, if the surfaces have a periodic lattice, this will induce ordering
parallel to the surfaces as well, and the oscillatory force will now also depend on the
structure of the surface lattices.
On the other hand, for surfaces that are randomly rough, the oscillatory force becomes
smoothed out and disappears altogether, to be replaced by a purely monotonic
solvation force. This occurs even if the liquid molecules themselves are perfectly
capable of ordering into layers.
11
To summarize, to have an oscillatory solvation force, the liquid molecules must be able
to be correlated over a reasonably long range. This requires that both the liquid
molecules and the surfaces have a high degree of order. If either is missing, so will the
oscillations. A roughness of only a few angstroms is often sufficient to eliminate any
oscillatory component of a force law.
2.4 Computational models of solvation forces
This section describes the hard-sphere and the Lennard-Jones fluid, which are used to
model the mica-OMCTS system. The surfaces are modeled as structureless continua, which
is appropriate for the molecularly smooth mica used in these experiments.
The density of OMCTS at standard temperature and pressure is 0.956 g / mL , and its
molecular weight is 296.62 g / mol . This gives it a number density of 0.00194 A 3 . The
calculated Hamaker constant for the mica-OMCTS system is 1.23  10 20 J and that for micaair is 1 10 19 J .
Physics behind this task is rather complicated. We know we can calculate most of the
thermodynamic functions by knowing the particle pair correlation functions. To actually
calculate them we split the potential between two molecules to the direct part and to the
indirect part, which is due to nearby molecules, Figure 8.
Fig 8: The intermolecular interaction potential is split to the direct part and to the indirect part, which is due to
nearby molecules.
2.4.1 Hard sphere model
One considers a binary mixture of macrospheres (bodies between which forces are to be
calculated) and hard spheres (solvent), the former at infinite dilution. The geometry here is
different (force between two spheres) from that mentioned previously in seminar, that is two
crossed cylinders, but that does not affect the interpretation of the results.
The solvent hard spheres are specified by the pair potential

u11 (r )  
0
rd
(10)
rd
The solute macrospheres are cavities which interact with the solvent by the pair potential

u 01 (r )  
0
r  (D  d ) / 2
(11)
r  (D  d ) / 2
There is no direct interaction between the macrospheres in the region beyond contact,
u00 (r )  0, r  D . d and D are radii of solvent molecule and macrosphere.
12
There is only one free parameter for the hard-sphere fluid, namely, the diameter of the
molecule.
2.4.2 Lennard-Jones model
The OMCTS molecules are modeled as a Lennard-Jones atoms, and the mica is modeled as
a structureless macrosphere composed of Lennard-Jones atoms. The Lennard-Jones pair
potential between atoms of species i and j has the form
uij (r ) 
 i j
r
12
i  j

(12)
r6
The r 6 term is the usual dispersion attraction, and the  i are related to the atomic
polarizabilities and ionisation strengths. The r 12 term accounts for the short-range repulsion
between atoms.
The macrosphere-OMCTS pair potential is derived by integrating the Lenard-Jones pair
potential over a sphere of radius R . One has (with t  (r 2  s 2  2rs cos )1 / 2 )
R

V ( 6) (r; R)  2  dss 2  d sin  t 6 
0
0
4
r3
,r  R
3 (r 2  R 2 ) 3
(13)
and
R

0
0
V (12) (r; R)  2  dss 2  d sin  t 12 
2 4R3 (15r 6  63r 4 R 2  45r 2 R 4  5R6 )
, rR
90
( r 2  R 2 )9
(14)
Only the dispersion interaction between the macrospheres needs to be considered, and one
has
R

W ( 6) (r; R)  2  dss 2  d sin  V ( 6) 
0
0
 2  4 R 2 (r 2  2 R 2 )
6  r 2 (r 2  4 R 2 )
 ln
r2 
, r  2 R (15)
r 2  4 R 2 
In view of the above results, the pair potentials actually used in the Lennard-Jones model
are, for the solvent-solvent
u11 (r ) 
12
r 12

12
(16)
r6
for the macrosphere-solvent

u 01 (r )  
2
2
(12)
(6)
(  0 01  11 )V (r ; R)  (  0  0 1  1 1 )V (r; R)
rR
rR
(17)
and for the macrosphere-macrosphere

u00 (r )  
2
( 6)
 (  0  0  1 1 ) W (r; R)
r  2R
r  2R
13
(18)
Here  0 is the density of the mica continuum comprising the macrosphere, and 1 is the
density of OMCTS. The macrosphere potentials are the sum of the interactions due to the
mica elements in the continuum, less the sum of the interactions due to OMCTS molecules
over the same spherical region. It may be seen that there are five parameters in the above:
1 ,1 , 1 ,  0 0 and  0  0 .
The number density of OMCTS is 1  1.94 10 27 m 3 . The strength of the short-range
repulsion in OMCTS, 1 , will be regarded as a free parameter, which essentially determines
the size of the OMCTS molecule. For simplicity, the quantity  0 0 will be equated to 11 .
From the comparison between Hamaker approximation for the van der Waals force between
two semiinfinite half-spaces and pressure between Lennard-Jones half-spaces we can
deduce that  0  0  1.0  10 10 J and 1 1  6.54 10 11 J . The only adjustable parameter in
the Lennard-Jones model is the size of the OMCTS molecule.
Computational algorithms for interaction energy using our model potentials are far beyond
this seminar. Here we will present only the results.
Fig 9 Interaction free energy per unit area between mica surfaces in OMCTS. The calculated force in the model
hard-sphere fluid (dashed line) and in the L-J fluid (solid line) are compared with experimental results. In each
case the separation has been scaled to give unit period for the oscillatory force.
Figure 9 compares the smoothed experimental data from several runs, with the results of
theoretical calculations for the hard-sphere and for the Lennard-Jones fluids. The
separations have been scaled so that the oscillations have unit period. Hence, the most
important point of comparison is the amplitude of the force and its rate of decay. Model
parameter, the size of OMTCS molecule, was varied to get the best fit. It may be seen that
there is good agreement between theory and experiment for the depth of the three minima
and the height of the maxima between h  6 and h  8 (contact is at h  0 ).
The amplitude of the measured force is fitted equally well by both theoretical models when
plotted with unit period. The oscillations in OMCTS have a period measured to be
0.82  0.02nm . The theoretical curves correspond to periods of 0.74nm for the hard-sphere
and 0.78nm for the Lennard-Jones fluid. On this basis the Lennard-Jones potential is better
model for OMCTS than is the hard-sphere potential. Indeed, even though it obviously grossly
14
simplifies the real intermolecular potential, the Lennard-Jones pair potential evidently retains
those essential elements of reality that give rise to the measured oscillatory forces.
Figure 10 displays the magnitude of the measured and the calculated forces as determined
from the extrema on a logarithmic plot. It is noticeable that the data are roughly linear, which
indicates that the force law corresponds to an exponentially damped sinusoid. The calculated
decay length of theoretical curves is 0.78nm . Experimental data are fitted well by the same
decay length.
Fig 10: Magnitude of the extrema of the free energy plotted on a logarithmic scale against the surface separation,
squares are maxima, diamonds are minima. Crossed symbols are from different measurement method but they
give the same force law. The solid curve joins the consecutive extrema for the theoretical hard-sphere fluid, and
the dashed curve is for Lennard-Jones fluid.
3 Conclusion
Direct measurements have furnished conclusive evidence for the existence of structural
forces arising from a surface-induced modification of liquid structure. Such forces are
characterized by decaying oscillations with a periodicity equal to the molecular size, and their
measurable range is typically 5  10 molecular diameters. In this range the amplitude of the
oscillations is significant compared to the van der Waals force predicted by continuum
theory, especially in the last few molecular diameters where it is considerably larger. These
observations are of particular relevance to colloid science and biology, where it has been
customary to consider the solvent between interacting surfaces of particles as a structureless
medium.
We have also shown that the magnitude of the measured surface force in the case of
OMTCS molecule is well described by a hard-sphere and a Lennard-Jones model where the
single parameter is the size of the molecule. For the case of the hard-sphere fluid, the
consequent period of the oscillation is 0.74nm , while for the Lennard-Jones fluid it is
0.78nm . Quantitative agreement between both theoretical results and experimental
oscillatory force described here is quite satisfying.
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4 References
[1] J. N. Israelachvili, Intermolecular and Surface Forces, (Academic Press, 1991)
[2] K. Kočevar, Study of Ordering and Forces in Liquid Crystal-Solid Interfaces, PhD
thesis, (University of Ljubljana, Faculty of mathematics and physics, 2001)
[2] R. G. Horn, J. N. Israelachvili, 'Direct Measurement of Forces Due to Solvent Structure',
Chem. Phys. Lett 71, 192-194 (1980)
[3] P. Attard, J. L. Parker, 'Oscillatory Solvation Forces: A Comparison of Theory and
Experiment', J. Phys. Chem 96, 5086-5093 (1992)
[4] H. K. Christenson, 'Interactions between Hydrocarbon Surfaces in a Nonpolar Liquid:
Effect of Surface Properties on Solvation Forces', J. Phys. Chem. 90, 4-6 (1986)
[5] P. Tarazona, 'A Model for Density Oscillations in Liquids between Solid Walls', Mol.
Phys. 56, 557-572 (1985)
[6] M. Dijkstra, 'Structure and Solvation Forces in Confined Films of Alkanes', Thin Solid
Films 30, 14-20 (1998)
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