# Microfluid Mechanics/Intermolecular and Surface Forces

## Fundamental interactions

In particle physics, fundamental interactions (sometimes called interactive forces or fundamental forces) are the ways that elementary particles interact with one another. An interaction is fundamental when it cannot be described in terms of other interactions. The four known fundamental interactions, are:

All are non-contact forces. Gravitation is a long range force, as it acts at large distances. Tough electromagnetic forces are short range forces, they might have long range effects: Surface tension and related phenomena are good examples of long range effect electromagnetic forces. Electromagnetic forces (see also Electromagnetism) are causing the intermolecular interactions and, therefore, have to be considered in microfluid mechanics.

## Common types of intermolecular interactions between atoms, ions and molecules

The interactions between fluid molecules, molecules of particles and molecules of surfaces are mainly due to electromagnetic forces. The bonds generated by these interactions are of two types:

As atoms covalently bond (i.e. form chemical bonds), the electron charge distributions of the atoms change completely and merge. Physical bonds occur as a result of small electromagnetic interactions and hold the molecules together in liquids and solids. They can be as strong as covalent bonds. Even the weakest physical binding force is strong enough to hold together all but the smallest atoms and molecules in solids and liquids at STP as well as in colloidal and biological assemblies. These properties, coupled with the long-range nature of physical forces, make them the regulating forces in all phenomena that do not involve chemical reactions [1].

 All molecules are either charged (i.e. ions), polar, or non-polar (which are polarizable). When the thermal energies ${\displaystyle kT}$ of the molecules are more than the orientation dependencies of dipole-dipole and ion-dipole interaction energies, the molecules are not fully aligned and they rotate freely. The intermolecular interactions which are responsible for physical bonding are: Interactions involving charges (charged molecules) and permanent di/multi-poles (polar molecules) Charge-charge interaction(Coulomb interaction) Charge-dipole interaction Charge-induced dipole interaction: The interaction between a charged molecule (ion) and a nonpolar molecule Dipole-dipole interaction between permanent spatially fixed dipoles in molecules Van der Waals interactions London dispersion interaction Dipole-induced dipole interaction (Debye force or induction force): The attractive interaction between a freely rotating multipole on one molecule with an induced (by the former di/multi-pole) multipole on another. Dipole-dipole interaction of freely rotating polar molecules(Keesom interaction or Orientation force): The Boltzmann-averaged interaction between two permanent dipoles. Hydrogen bond: A particularly strong type of directional dipole-dipole interaction. Because of the small size of the --H+ group, it is far stronger than that predicted by the point dipole approximation. Repulsive interactions: Intermolecular interactions causing physical bonds (hydrogen bond is excluded)

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, so one cannot simply add all the pair potentials of a molecule to obtain its net interaction energy with all the other molecules. This is because 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. This effect adds an additional contribution to the total van der Waals interaction energy. The nonadditive property of van der Waals forces is more important in the interactions between large particles and surfaces in a medium[1].

## Total intermolecular pair potentials

 The total pair potential created by intermolecular interactions can be modeled as a function of the distance between two molecules ${\displaystyle \ (r)}$. One mostly used model is the Lennard-Jones pair pottential, which represents the contributions of van der Waals and repulsive interactions (steric effects): ${\displaystyle \displaystyle w(r)=\underbrace {-{\frac {A}{r^{6}}}} _{negative}+\underbrace {\frac {B}{r^{12}}} _{positive}[J]}$ where ${\displaystyle \displaystyle A}$ and ${\displaystyle \displaystyle B}$ are molecule dependent constants and they are equal to ${\displaystyle \ A=4\varepsilon \sigma ^{6}\ \ {\text{and}}\ \ B=4\varepsilon \sigma ^{12}}$ in which ${\displaystyle \ \varepsilon }$ is the depth of the potential well (maximum attraction energy) and ${\displaystyle \ \sigma }$ is the finite distance at which the inter-molecule potential is zero. ${\displaystyle \ \sigma }$ is also called as the collision diameter. Inserting ${\displaystyle \displaystyle A}$ and ${\displaystyle \displaystyle B}$ in the above interaction potential reveals the other form of the Lennard-Jones pair pottential: ${\displaystyle \ w(r)=4\varepsilon \left[-\left({\frac {\sigma }{r}}\right)^{6}+\left({\frac {\sigma }{r}}\right)^{12}\right]}$ The van der Waals potential is the negative contribution, which varies with the inverse 6th power of the distance ${\displaystyle \displaystyle r}$. The intermolecular force is the derivative of the pair potential w.r.t. the distance. ${\displaystyle \displaystyle F(r)=-{\frac {dw(r)}{dr}}=\underbrace {-6{\frac {A}{r^{7}}}} _{attractive}+\underbrace {12{\frac {B}{r^{13}}}} _{repulsive}[N]}$ Hence, it can be seen that the first term is the attractive force, i.e. the van der Waals force. The second term is the repulsive force and it hinders that two atoms and molecules do not exist at the same location at an instant. Note that, force and potential energy are directly related. The potential energy (Lennard-Jones potential) is the negative of the work done by the net electromagnetic force moving the molecule towards another molecule to its given position in space from infinity. Note that for interactions of charged molecules and fixed dipoles, other types of potentials has to be used. Lennard-Jones potential and the intermolecular force derived from it. Selected constants are A=1e-77 [J m6] & B=1e-134 [J m12] which are for Argon.

## Forces between particles and surfaces

The fundamental forces involved are the same as those already described (i.e., electrostatic, van der Waals, solvation forces). However, they can manifest themselves in quite different ways and lead to qualitatively new features when acting between large particles or extended surfaces. Surfaces referred here are between a solid and a fluid (gas or liquid) or between two fluids (gas-liquid or liquid-liquid). All special cases will be explained when needed in specific chapters of this course. However, a detailed account of these forces can be found in the literature [1].

## References

1. Israelachvili, J.N.: Intermolecular and Surface Forces, 3rd Ed., Academic Press, London, 2011.