Lecture 2: Forces and Potentials

What did we cover in the last lecture? Microscopic and nanoscale forces are important in a number of areas of nanoscience, nanotechnology and biology These forces start to become important on the sub micron length scale The relevant energy scale on these small length scales is the thermal energy scale

In this lecture… Chapter 3 1) The relation ship between force and potential 2) Recap of electrostatic interactions 3) Ionic interactions 4) The range of interactions and solubility of ions 5) Covalent interactions and bonding 6) Metallic bonding

Relationship between force and potential The force, F, on an object can be related to the gradient in its potential energy, U, with respect to distance, x, by the formula This is an extremely important formula in physics and we will use it a lot in this module!!

The ‘sign’ of the potential When the potential energy is negative the interaction between two bodies is favourable Potential energy U < 0 When the potential energy is positive, interactions are unfavourable Potential energy U > 0

The sign of the force Positive forces are repulsive |F| > 0 Negative forces are attractive |F| < 0

Strong inter- atomic and intermolecular interactions I: Coulomb Forces You have already met Coulomb interactions in the first year module ‘Newton to Einstein’ Some atoms and molecules may acquire a net charge and will interact via electrostatic forces r +6e

Electrostatic potential The potential energy of two molecules having charges q 1 and q 2 that are separated by a distance, x, is given by  o =8.85 x C 2 N -1 m -1  =relative permittivity Inverse power law dependence (x -n ) is typical of many different types of potential

Electrostatic forces The force acting between two point molecules q 1 and q 2 separated by a distance, x, is given by The sign of the force depends upon the signs of the charges If q 1 q 2 < 0 force is attractive (negative sign) If q 1 q 2 > 0 force is repulsive (positive sign)

Problem 1 Two protons (mass m=1.67x kg and charge q=+1.6x C) are fixed in place at the coordinates (0, 1) nm and (0, -1) nm. A third proton is fired in the negative x direction from the position (10, 0) nm with a speed v a) Calculate the closest distance of approach between the particles if v =14,850 ms -1 b) Sketch the form of the potential energy of the particle as a function of the distance along the x axis

The range of an interaction Thermal motion of atoms and molecules tends to disrupt the interactions between them At ‘small’ distances interactions between atoms and molecules are strong enough to overcome the effects of thermal motion At ‘large’ distances interactions between atoms and molecules become too weak to overcome the effects of thermal motion What do we mean by small and large, strong and weak? How do we define the range?

Can we be more quantitative? We can estimate the range of an interaction by comparing the thermal energy to the size (magnitude) of the potential energy of two molecules Small and large ranges are nebulous terms and depend upon the nature and strength of the interaction between two atoms/molecules When |U(x)|  kT thermal motion disrupts the interactions When |U(x)| > kT the atoms/molecules still feel the interactions between them

Range of electrostatic interactions We can obtain an estimate of the range X range of the electrostatic interaction between two atoms/molecules such that Rearranging gives For x ≥ X range electrostatic interactions become unimportant x < X range electrostatic interactions start to influence atoms/molecules In water (  =80) this has a special name- ‘Bjerrum length’

Problem 2 Calculate the range of the electrostatic interaction (at room temperature) between two positively charged ions having a charge of 1.6 x C in a) vacuum (  =1), b) water (  =80) and c) toluene (  =2.38)

Ionic Crystals Electrostatic interactions are responsible for the formation of ionic crystals For example, consider NaCl Each Cl- has 6 nearest neighbour Na + ions at a distance r 12 next-nearest neighbour Cl - ions at √2r 8 next-next-nearest neighbour Na + ions at √3r r

Cohesive energy of ionic crystals We can calculate the (cohesive) energy per ion holding the crystal together by summing up the contributions to the potential energy from all neighbours Where M is called the Madelung constant (M=1.748 for NaCl) e is the electronic charge (1.6 x C)

Problem 3 Calculate the cohesive energy of a NaCl crystal in a) vacuum, b) water and c) toluene if the nearest neighbour spacing between Na + and Cl - ions is r=0.276 nm. Use your answers to parts b) and c) to explain why ionic crystals dissolve in water, but not in organic solvents.

The Born (or ‘self’) energy When considering the solubility of ions and charged particles in different media, it is important to consider how much energy is associated with placing the ion/particle in the medium (even when it is not interacting with other particles) This energy is the energy required to bring the constituents of the ion/particle from infinity and construct an ion/particle of radius a. For an ion/particle with a charge of +ze this energy is given by See OHP

Strong inter- atomic and intermolecular interactions II: Covalent Interactions Covalent bonds are highly directional and are chemical in origin (e.g. pi bond created by overlap of d electron orbitals) These interactions originate from the sharing of valence electrons to facilitate the filling of electronic shells within atoms → stable structures The directionality of the bonds is caused by the mutual repulsion of the electrons in different bond orbitals

Properties of covalent bonds Covalent forces (bonds) operate over short distances nm The energies associated with the formation of these bonds are typically 100 – 300 kT per bond You will learn more about these in the 3 rd year Solid State module p32

Strong inter- atomic and intermolecular interactions III: Metallic bonding Sharing of electrons between metal atoms gives rise to another form of bonding However, in metallic bonds, the electrons become delocalised throughout the material in such a way that they are shared between many nuclei The ‘sea’ of delocalised electrons helps to screen the repulsion between neighbouring nuclei

Properties of metallic bonds Optical – The optical properties of metals are determined by how the ‘sea’ of electrons responds to the oscillating electric and magnetic fields associated with incident light Electrical and Thermal – Delocalised electrons are easier to move, so metals are good conductors of heat and electricity Again you will learn more about these in future modules! Metallic bonds are comparable in strength to covalent bonds (U= kT per bond) The delocalisation of electrons can be used to explain many/all of the properties of metals

The force on an object is the gradient in potential energy The range of an interaction is the distance at which thermal effects start to dominate Summary of key concepts Electrostatics can be used to explain the interactions in ionic solids. The cohesive energy and Born energy of ions/particles can be used to explain why they are more soluble in water than other solvents The energies associated with the formation/breaking of strong covalent, ionic and metallic bonds are Covalent U ~ kT Ionic U ~ kT Metallic U ~ kT