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Copyright Brian J. Kirby. With questions, contact Prof. Kirby here. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.

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Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

H.1 Thermodynamics of intermolecular potentials [atomistic modeling top]

Rarely-interacting atoms are straightforward to model thermodynamically. That these atoms rarely interact is the basic premise of the ideal gas law and the ideal solution approximation. For example, theBoltzmann approximation for ions in an electrolyte consists of treating the ions as if they do not interact with each other, but rather a continuum electrical field. The solvent molecules (typically water) are ignored. The only effect of the solvent molecules is to dictate theelectrical permittivity ε of the mean field. The Boltzmann approximation leads to the simple conclusion that the energy of an atom is given exclusively by the singlet energy e1(microfluidics textbook nanofluidics textbook Brian Kirby Cornell i)—namely, that
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(H.1)

which was presented earlier as Equation 9.1, fully describes the potential energy landscape that dictates atom distributions. This is asinglet potential, meaning that the potential is a function of position, but not a function of the location of or properties of other atoms. The potential energy of a system owing to singlet energies has the mathematical form E = ie1(microfluidics textbook nanofluidics textbook Brian Kirby Cornell i), where E is the potential energy, i denotes atom i, e1 is the singlet energy, and microfluidics textbook nanofluidics textbook Brian Kirby Cornell i is the position vector of atom i. These terms carry the effects of external fields on the atoms, but do not account for atom-atom interactions. For example, if walls are at x = 0 and x = L, e1 would be zero far from the walls, but would become large as x approached 0 or L.

Unfortunately, as systems become more dense, we must consider the interaction of systems. This leads to pair potentials, which are a function of two atoms and their relative location, triplet potentials, which are a function of the relative positions of sets of three atoms, and so on. The following sections discusspair potentials in some detail. As compared to a Boltzmann model, considering these pair potentials in detail leads to vastly different solutions for the distribution of atoms, as shown in Figure H.1. We describe these in the coming sections with reference to the multipolar theory in Appendix F.

Pair potentialsimply energies that are given by the interactions between pairs of atoms. These energies have the mathematical form E = j>iie2(microfluidics textbook nanofluidics textbook Brian Kirby Cornell i,microfluidics textbook nanofluidics textbook Brian Kirby Cornell j). The sum denoted by j>i, in this context, means that the energy from the interactions between molecules i and j are counted only once. The most fundamental challenge and limitation of molecular dynamics is the definition and evaluation of the pair potential e2. In particular, pair potentialscan be precisely defined only if a large amount of data (including fluid properties, quantum modeling, and spectroscopic data) is used. They can be rather precisely defined for simple atoms (e.g., liquid Argon), but they are difficult to define for more complex structures, and are more difficult to define for water. Even once a pair potential formula is agreed upon, evaluating the potential numerically is computationally expensive. If all pair potentials in a system with N molecules is considered, order N2 evaluations are required. Given that N is typically large, this is onerous and approximate techniques must be used. When the number of evaluations is reduced with some sort of engineering approximation (say, by only calculating energies for molecules that are reasonably close to each other), the true pair potential no longer conserves momentum and energy, and typically must be replaced with an effective pairpotential (or some other approximation) that corrects for these limitations. Finally, the pair potentials are typically corrected for tripletand higher-order energies.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure H.1: Continuum vs. atomistic depiction of ion concentration near a wall.


H.1.1 Monopole pair potentials

The simplest pair potential considers the electrostatic interaction between the charges on atoms. Recall that the electrostatic potential energy in a vacuum between two atoms with point charges q1 and q2 is
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(H.2)

where Δr12 is the length of the vector between the two locations microfluidics textbook nanofluidics textbook Brian Kirby Cornell 1 and microfluidics textbook nanofluidics textbook Brian Kirby Cornell 2. The force is simply the derivative of the potential with respect to Δr12:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(H.3)

If the atoms are no longer in a vacuum, we must approximate the interactions of the atoms with all of the other atoms (i.e., the solvent atoms) somehow. Here, we often make a continuum approximation that the medium in which the points exist can be described by a single value (the electric permittivity) which describes how the potential is attenuated by the dielectric screening (see Section 5.1.2):
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(H.4)

The only difference is that ε0 has been replaced by ε, the electrical permittivity of the medium.

H.1.2 Spherically-symmetric multipole pair potentials

All atomic interaction potentials that are not described by point charges (i.e., monopolar interactions) correspond to distributions of charge (i.e., multipoles). The forces caused by electrostatic interactions between electron orbitals of molecules are typically described by a time-averaged potential that accounts for all of the multipolar electrostatic interactions. We consider only spherically-symmetric potentials, since most models for water involve a spherically-symmetric pair potential combined with monopole pair potentials to account for the partial charge on each of the atoms.

Hard sphere potential

A strong, short-ranged repulsion prevents multiple atoms from existing in the same location at the same time—this stems from the Pauli exclusion principle as applied to the electron orbitals. A simple way to handle this is to treat the atoms as hard spheres. From the standpoint of pair potentials, this corresponds to
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(H.5)

where d12 is the effective hard-sphere diameter of the two-body system. Hard-sphere systems are simple to study and exhibit some semblance of reality, but fail to predict a number of simple phenomena, such as a liquid-gas transition.

Lennard-Jones potential

Real atoms always show long-range Coulomb attraction known as Van der Waals attraction that scales inversely with the sixth power of the distance. The short-range repulsive potential is large but neither infinite nor discontinuous, and it has been deemed convenient to approximate this with a Δr12-12 dependence. There is no real physical source for the scaling of this term, but it works well and liquid properties are not a strong function of the functional form used for the repulsion term. More importantly, it is computationally more efficient to calculate an r-12 term since the r-12 term is the square of the r-6 term. Thus, the most common form of pair potential has historically been a Lennard-Jones potential (alternately called an LJ potential or LJ 6-12 potential), with general form

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

where εLJ is the depthof the potential well and σLJ is the pointat which the pair potential is zero. This potential is straightforward to define mathematically, efficient computationally, and roughly approximates real pair potentials for spherically-symmetric atoms. An example is shown in Figure H.2. The value of σLJ for Argon atoms is approximately 3.4 Å, and the value of εLJ∕kB for Argon atoms is approximately 121 K.

The depth of the potential well naturally leads to a means for normalizing the temperature. We define a reference temperature εLJ∕kB, which is the temperature at which kBT is equal to the well depth. Then a nondimensional temperature T * can be defined as

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

See Appendix E for further discussion on nondimensionalization. At T * smaller than one, we expect the intermolecular potentials to strongly affect the atom distributions. At T * much larger than one, we expect the intermolecular potentials to not affect the atom distributions much at all.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure H.2: The Lennard-Jones potential.


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Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

Copyright Brian J. Kirby. Please contact Prof. Kirby here with questions or corrections. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.


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