Notes on Microphysics

1. Introduction

We live in a world that is composed of elements. We are large, say, relatively. So we define ourworld as macro world, even though we are micro in the eye of galaxy or even the earth − theyare macro also. We will speak about the micro world, the world smaller then we normallyregard. The micro world, however, is nothing strange. It is just part of our world at which weare looking smaller scale. Thus comes the scale. We have micro robots of several , theyare still quite large in the context we are discussing, the difference of physical phenomenon isnot very much different. However, when the scale is continuously getting down, we will seethe capillary phenomenon - liquid level in a small capillary vessel could be high than the leveloutside of it. Continuously going down, van der Waals force, electric static force and surface

FIGURE 1. The Scale

cm3

Notes on Microphysics 2

tension play more and more important roles in the determination the behaviours of microscopicthings. In the mean while, inertial force, gravitational force become less dominant. Thosechanges of dominant physical quantities between different scales is called scaling effect.

The scaling effect makes the micro world different from the macro world we get used to. Foran example shown in Fig. 3, grasping an object and releasing it, an operation we do many timea day, could be difficult in the micro world - the object is difficult to release because of theadhesive forces: static electric, van der Waals and surface tension forces.

To understand the scaling effects and how they affect the micro world operation, it is importantto study the physics. The physics is well established for most aspects of our macro world, andmany work have been done in the difference field under different disciplines for the microworld, such as in molecular biology, colloid science etc. In the following sections, we willreview some aspects of intermolecular forces and their effects in a larger scale - the microworld. Intermolecular forces directly contribute to the forces we are interested. For example,Coulomb forces contribute to the static electric forces; dipole interactions contribute to the vander Waals forces and surface tension forces. Based on the understanding of those forces,further details of the micro operation can be studied.

2. Intermolecular Forces

It is well know that there are four forces in nature: strong interaction, weak interaction,electromagnetic interaction and gravitational interaction. The strong and weak interactions actbetween neutrons, protons, electrons and other elementary particles, their range of actions isless than . The electromagnetic and gravitational interactions act between atoms andmolecules as well as between elementary particles, their range of action is from subatomic topractically infinite distance - from physics in daily life to inter galaxy interaction.

The forces we are most interested, in this contexts, are intermolecular forces which shape thephysics of micro world. Essentially, all intermolecular forces are electrostatic in origin. This isindicated by the Hellman-Feynman theorem, which states that once the spatial distribution of

FIGURE 2. Scaling effect FIGURE 3. Scaling effect in micromanipulation.

10 5– nm

Notes on Microphysics 3

the electron clouds has been determined by solving the Schrödinger equation, theintermolecular forces may be calculated on the basis of classical electrostatics. However, it israther difficult to solve the Schrödinger equation exactly, even for two the interactions ofhydrogen atoms in vacuum. Thus, it is useful to classify intermolecular interactions intocategories despite they come from the same origin:

• Purely electrostatic in origin

such as interactions between charges, permanent dipoles and quadrupoles, etc.

• Polarization forces arisen from the dipole moments

such as interactions in solvent medium etc.

• Quantum mechanical in nature

such as covalent or chemical bonding; repulsive steric or exchange interactions, etc.

• Other interactions

such as van der Waals force, magnetic force, etc.

2.1 The Effective Range of Intermolecular Forces

The effective range of intermolecular interactions are quite short, rarely exceeds 100 . Thereason of this is that the potentials and forces of intermolecular interactions decay very fast. Ifwe consider the total energy of one molecule with all the other molecules in its surrounding,we have:

(1)

where is the diameter of the molecules, is the size of the surrounding we consider and is the interaction potential of two molecules having the general form:

(2)

Thus, we get:

(3)

(4)

If , the second term in (3) will be greater and the contribution from more distantmolecules will dominate over the nearby molecules, i.e., the size of the system must be takeninto account. The gravitational force has and systems of inter galaxy distant could stillinteract each other. Thus, the potentials of intermolecular forces do have and this is thereason that properties of material is generally not affected by its volume.

nm

E L( ) w r( )ρ4πr2 rd

σ

L

∫=

σ Lw r( )

w r( ) C

rn----–=

E L( ) 4πCρ–

n 3–( )σn 3–----------------------------- 1

σL---

n 3–

–=

4πCρ–

n 3–( )σn 3–-----------------------------= for n 3 and L σ»>

n 3<

n 1=n 3>

Notes on Microphysics 4

Common types of interactions between atoms, ions and molecules in vacuum is listed in Fig. 4.Their interactions energy decay generally according to the rule except the charge-chargeinteraction which will be discussed later.

Despite the short effective range of intermolecular forces, they determine the macroscopicphenomena such as the height of rise of liquid in capillary channel, the breakup of water flowinto droplets, the shape of spider’s web and the shapes of liquid droplets on surface (Fig. 5).The reason underlying is that surface tension forces are the same as those that hold moleculestogether in solids and liquids, and in both cases they arise from interactions acting over veryshort distances.

2.2 Intermolecular Interactions

Covalent Interaction

Covalent forces are forces that tightly bind the atoms together within the molecule. Theireffective range is about . The interatomic bonds formed by the covalent forces arecalled covalent bonds. The covalent bonds are strong in strength which are in the range of

FIGURE 4. Common types of interactions between atoms, ions and molecules in vacuum.

FIGURE 5. Long range effects produced by short-range forces.

n 3>

0.1 0.2nm–

Notes on Microphysics 5

( ). Depending on the position an atom or elementoccupies in the periodic table, it can have a certain number of covalent bonds with other atoms.The number is know as the atomic valency. Another property of covalent bonds is thedirectionality - they are directed at specific angles relative to each other. The directionality ofcovalent bonds determines the way the multi-valent atoms coordinates themselves inmolecules or in crystalline solids to from an ordered three-dimensional lattice.

Coulomb Forces

The Coulomb force, or charge-charge interaction, acts between two charged atoms (or ions).Coulomb force is the strongest physical forces we are considering, which is stronger even thanmost chemical binding forces. The free energy for Coulomb interaction between two charges

and is:

(5)

where is the permittivity of free space, is the relativepermittivity or dielectric constant of the medium and is the distance between the twocharges.

The Coulomb force is given by:

(6)

where is the elementary ionic charge, and are ionic valencies.

The electric field at a distance away from a charge is defined by

(7)

Equation (5) indicates the potential of Coulomb interaction decay at speed of , which is thesame as gravitational force and has a very long effective range. This apparently break the rulestated in Section 2.1 - all intermolecular force laws must fall with distance faster than or

for energy. In reality, the positive ions are always surrounded by negative ions in latticeor solution, which makes the electric field be screened and thus decays more rapidly awayfrom them than from an isolated ion. This results all Coulomb interactions between ioniccrystals, charged surfaces and dissolved ions much shorter range.

Polar Molecule Interactions

A electric dipole has two poles and , and has net charge of zero. A molecular may havea permanent dipole. Such a molecule is called polar molecule. The dipoles of some moleculesdepend on their environment and can change substantially when they are transformed from onemedium to another. A dipolar ion is a dipole having different positive and negative charges.The dipole moment of a polar molecule is defined as:

100 300 kT/bond– 200 800 kJ/mol–

Q1 Q2

w r( )Q1Q2

4πε0εr-----------------=

ε0 8.854 1012– C2J 1– m 1–×= εr

F r( ) dw r( )dr

--------------–Q1Q2

4πε0εr2-------------------

z1z2e2

4πε0εr2-------------------= = =

e 1.602 1019–× C= z1 z2

r Q

E r( ) Q

4πε0εr2-------------------= Vm 1–

1 r⁄

1 r4⁄1 r3⁄

+q q–

Notes on Microphysics 6

(8)

where is the distance between the two charges and .

As shown in Fig. 6, the energy of interaction between an ion and a polar molecule can becomputed by:

(9)

let and , we have

(10)

where is the electric field of the charge acting on the dipole:

(11)

Polarization of Molecules

Polarization means that the dipole moments induced in molecules by the electric fieldsemanating from nearby molecules. The macroscopic dielectric constant of a medium is nomore than a reflection of the way the molecules of the medium are polarized by the localelectric field. Principally, all atoms and molecules are polarizable. Their polarizability isdefined according to the strength of the induced dipole moment acquired in a field E:

(12)

FIGURE 6. Ion-dipole interactions

u ql=

l +q q–

w r( ) Qq4πε0ε-------------- 1

AB------- 1

AC--------––=

AB r l θ 2⁄cos–= AC r l θ 2⁄cos+=

w r θ,( ) uE r( ) θcos–=

E r( )

E r( ) Q4πε0εr2-------------------=

αuind

uind αE=

Notes on Microphysics 7

The Fig. 7 shows a one-electron atom with and without external field. When no external fieldexist, . When the atom is in a external field the electron’s orbit is shifted by adistance from the positive nucleus so that the induced dipole moments is ,where the polarizability of this case is .

London’s Forces or Dispersion Forces

The London’s forces or dispersion forces are caused by a momentary dipole moment betweenatoms. The moment exist even for atoms which do not contain a permanent polarization.Dispersion forces are quantum mechanical in origin and amenable to a host of theoreticaltreatment of varying complexity. It may be understood intuitively as follows: for a non-polaratom such as helium, the time average of its dipole moment is zero, yet at any instant thereexists a finite dipole moment given by the instantaneous positions of the electrons about thenuclear protons. This instantaneous dipole generates an electric field that polarizes any nearbyneutral atom, including a dipole to an instantaneous attractive force between the two atoms, thetime average of this force is finite.

The main features of dispersion force includes:

• long range force, can be effective from large distances ( ) down to interatomicspacing (about )

• may be repulsive or attractive, and in general does not follow a simple power law

FIGURE 7. Induced dipole in a one-electron atom.

FIGURE 8. Two atoms affect each other.

uind 0= El uind α0E le= =

α0 4πε0Rε3=

l

l+e +e

-e

-e

u = l e

> 10nm0.2nm

Notes on Microphysics 8

• not only bring molecules together but also tend to mutually align or orient them, though thisorienting effect is usually weak

• the dispersion interaction of two bodies is affected by the presence of other bodies nearby(non-additivity of an interaction)

van der Waals Forces

van der Waals force originates from dipole-dipole interactions, it is contributed by threeimportant parts: Keeson-orientation, Debye-induction and the dispersion force. The energy ofall those three parts decays at the rate of .

The Keeson interaction or orientation interaction is the angle averaged interaction betweentwo permanent dipoles, which has energy:

(13)

where .

The Debye interaction or induction interaction is the angle averaged interaction betweendipole and dipole-induce dipole, which has energy:

(14)

for two different molecules each possessing a permanent dipole moment and , andpolarizabilities and , the net dipole-induce dipole energy is:

(15)

The London’s forces or dispersion forces contribute the third and probably the most importantpart of van der Waals forces. They are always present. In the contrast, the other types of forcesmay or may not present depending on the properties of the molecule. The interaction energybetween two identical atoms or molecules based on London’s equation is:

(16)

where is the Planck constant, the orbiting frequency of the electronand is the first ionization potential.

for two dissimilar atoms:

(17)

Some properties of van der Waals forces:

1 r6⁄

w r( )u1

2u22

3 4πε0ε( )2kTr6-------------------------------------–= for kT

u1u2

4πε0ε3----------------->

1kT 4.114 1014– erg× 4.114 1021– J× at 298K(~25°C)= =

w r( )u2α0

4πε0ε( )2r6---------------------------–=

u1 u2

α01 α02

w r( )u1

2α02 u22α01+

4πε0ε( )2r6----------------------------------–=

w r( )Cdisp–

r6--------------- 3

4---

α02hv

4πε0( )2r6-----------------------–

34---

α02I

4πε0( )2r6-----------------------–= = =

h 6.626 1034–× J s= vI hv=

w r( ) 32---

α01α02

4πε0( )2r6-----------------------

hv1v2

v1 v2+( )---------------------–

32---

α01α02

4πε0( )2r6-----------------------

I1I2

I1 I2+( )-------------------–= =

Notes on Microphysics 9

• Non-additivity: unlike gravitational and Coulomb forces, van der Waals forces are notgenerally pairwise additive - the force between any two molecules is affected by thepresence of other molecules nearby. This is because the field emanating from any onemolecule reaches a second molecule both directly and by reflection from other moleculessince they are also polarized by the field. This property normally reduce a small amount ofthe total energy (less than 20%).

• Retardation effects: when two atoms are distance apart, the time taken for the electric fieldof the first atom to reach the second and return can become comparable with the period ofthe fluctuating dipole itself. For example, the distance travelled by light during one rotationof a Bohr atom electron is about . When this happens the field returns to find that thedirection of the instantaneous dipole of the first atom is now different from the original andless favourably disposed to an attractive interaction. Thus, with increasing separation thedispersion energy between two atoms begins to decay even faster then andapproaching a dependence at . This phenomenon is called retardationeffect. The retardation effects are more obvious in medium where the speed of light isslower.

100nm

1 r6⁄–1 r7⁄– r 100nm>