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 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.
Contents
1Non-Structural forces between bodies and surfaces...... 3
1.1 Van der Waals force...... 3
1.2 Electrostatic force between bodies in liquids...... 4
2Structural 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
3Conclusion...... 15
4References...... 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 potentialbetween two molecules or particles, which is usually known as pair potential. It is related to the force between these two molecules or particles by . Since the derivative of gives the force, and hence the work that can be done by the force, is often referred to as the free energy.
Pair potential for two solute molecules in a solvent does not include only the direct solute-solute 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.
1.1Van der Waals force
Van der Waals force is a sum of three different intermolecular forces of electric origin with the same distance dependence , where 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 centered at and a flat surface located at in the limit of small distance , equals
.(1)
Hamaker introduced Hamaker constant which includes all material specific properties, relevant for the interaction, in the form
, (2)
where stands for number density, and 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 , approaching a dependence. This is called the retardation effect. For two molecules in free space retardation effects begin at separations above . In a medium, where the speed of light is slower, retardation effects come in at smaller distances.
1.2Electrostatic 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
(3)
where is the electrostatic potential and the number density of ions of valency at any point between two surfaces. Since only differences in potential are physically meaningful, we may choose at the midplane where also .
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 :
(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 , and counterion density , at any point 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 , potential profile and the field between the surfaces.
Armed with this solutions, it is not too difficult to calculate the pressure at any point 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 surface-solvent 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 surfaces (the radius of curvature ()) of molecularly smooth mica, Figure 4. Multiple beam interferometry was used for measuring the surface separation to and forces were measured by a spring deflexion method with sensitivity of better than .
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 . This is most commonly used liquid in these kinds of experiments for its large 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 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 (divided by the radious of curvature ) plotted against the distance between surfaces.
Since one of the mica surfaces was suspended at the end of a centilaver spring of stiffness , the force could be measured only in regions where . When the gradient of the force exceeds the spring stiffness , 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 ) can be measured directly, while those shown by dashed sections () 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 the average periodicity of the force oscillations is 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 , 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, , in Figure 5, where the Hamaker constant 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.
Fig. 5: a) Liquid density profile at a vapor-liquid interface. is the bulk liquid density; is the scale of molecular ‘roughness’ of the interface. b) Liquid density profile at an isolated solid-liquid interface. is the contact density at the surface. c) Liquid density profile between two hard walls a distance apart. is a function of 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 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 liquid-liquid 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 and when they are at infinite separation:
,(5)
Here and are the density of liquid molecules at each surface at surface separation 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 varies. We see, Figure 6, that will be high only at surface separations that are multiples of but must fall at intermediate separations.