SOLID

Crystalline solids (1)Atoms and molecules are arranged in a regular 3 dimensional arraySolids have very definite geometric shapeBasic unit: unit celle.g. metals, diamond, salt, sugar

Remarks:Crystalline structure is energetically stableCrystals have definite melting pointEquivalent planes of atoms are all parallel in a ‘single crystal’Polycrystalline samples consist of tiny crystals (grains) packed together and orientated randomly

Crystalline solids (2)

Amorphous solidsThe particles are assembled in a more disordered way e.g. plastics, glass, wax, chalkRemarks:

No definite melting pointBecomes softer and more mobile when heatedRegarded as liquids with very high viscosity

Mechanical properties of solids (1)

StrengthThe force a material can withstand before breakingIf a material can withstand a large force, it is said to be very strong, otherwise, it is said to be weak

StiffnessThe opposition set up to being distorted by having its shape or size changedA stiff material is very rigid, i.e. it only extends slightly when stretched A material which is not stiff is said to be soft Strength and stiffness are independent properties

Mechanical properties of solids (2)

Ductilitythe ability of a material to be hammered, pressed, bent or stretched into useful shapesA ductile substance is one that can be permanently strained e.g. metalsA material is said to be brittle if it breaks immediately after exceeding the elastic limit e.g. glass, concrete figure

Mechanical properties of solids (3)

Toughnessit relates to how readily a material will crack

Mechanical properties of solids (4)

Stress-strain behaviour (1)Searle apparatus can be used to investigate the extension of a wire when it is subjected to tensionStress

it is the force acting on unit cross-section area Mathematical form:Unit: Nm-2 or Pa

A

F

Further reading

Strainit is the extension of unit length Mathematical form:It has no unit

Stress-strain behaviour (2)

l

e

Further reading

Elastic material it returns to its original size and shape when the deforming force is removedif a material has no tendency to return to its original size or shape on removal of the deforming force, it is said to be plasticwhen the applied load is not too large, a graph of stress versus strain gives a straight line passing through the origin

Stress-strain behaviour (3)

Stress-strain behaviour (4)tensile stress

tensile strain constant

Mathematical form:E =Fl

Ae

E is called the Young modulus of the materialUnit: Nm-2 or Pa

Notes:The value of Young modulus depends on the nature of the material and not on the dimensions of the sample large E a large stress is required to produce a small strain The proportionality between stress and strain is called Hooke’s law

Stress-strain behaviour (5)

Young modulus of some materials:

Stress-strain behaviour (6)

Material E/1010 Pa

Aluminium 6.9

Copper 12.4

Glass 7

Iron 19.7

Steel 20.6

Lead 1.79

Tin 4.69

Polythene about 0.5

Rubber about 0.005

Behaviour of larger stressForce-deformation curveStress-strain curve

Region OP: Hooke’s law is obeyed and P is the proportional limitRegion PL: Hooke’s law is not obeyed but the deformation is still elastic and L is the elastic limit

Note: in some materials, points P and L overlap with each other

Stress-strain behaviour (7)

Beyond L, plastic deformation occurs and permanent deformation resultsAt Y, a sudden remarkable increase in extension occurs and Y is the yield point. The material is undergoing plastic deformationStress at M is the maximum stress the material can withstand without breaking and is called the breaking stress or the strength of the materialThe material eventually breaks at B

Stress-strain behaviour (8)

figure

Determination of Young modulus (1)

Experimental setupProcedure

Test wire should be thin and longmain scale is carried by the reference wire and the vernier is carried by the test wire The two wires, P and Q, should be made of the same material and suspended from the same support

Both wires should be free of ‘kinks’ The diameter of the test wire must be found by a micrometer screw gauge at several places The readings on the vernier are also taken when the load is gradually removed in steps A graph of extension against load is plotted and the gradient is measured

Determination of Young modulus (2)

Equation:

Determination of Young modulus (3)

)/(/

/

FxA

L

Lx

AFE

gradient A

LE

Various stress-strain graphs

Comparison of stress-strain graphs of

a) Strong and weak materialsb) Stiff and soft materialsc) Ductile and brittle materials

Energy of deformation (1)Work in stretching and compressing an object is stored as strain energyWithin elastic limit, strain energy can be recovered completely when the stress is removedStrain energy =

1

2Fe

Strain energy per unit volume

Energy of deformation (2)

1

2

1

2E2

1

2 E

2

Intermolecular force and potential energy (1)

The molecules exert net repulsive force at very small separations otherwise all matters would collapseThe intermolecular forces must be negligible to account for the free movement of molecules in gaseous state

Kinetic theory: molecules are in continual motion at all temperatures above absolute zero, therefore they possess k.e.Molecules exert forces on each other, therefore they possess p.e.The relative magnitude of the kinetic and potential energies determines whether a substance exists in the solid, the liquid or the gaseous state

Intermolecular force and potential energy (2)

Forces between moleculesElectromagnetic in originWhen two molecules approach each other, the charges on each are distorted slightly resulting in attractive intermolecular forceWhen the molecules become so close that the electron clouds begin to overlap, the intermolecular force becomes repulsive

Intermolecular force and potential energy (3)

Model for a solidAtoms are more or less locked in positionPotential energy- separation curve

P.E. increases when the solid is compressed (r decreases) or stretched (r increases)P.E. = 0 when the atoms are infinitely apart

Interatomic force-separation curve (1)

Interatomic force is given by

FdU

dr

Force-separation curve

RemarksMolecules in a solid seem to be in equilibrium on the average It is hard to push molecules in a solid closer together. Therefore as they come closer together the repelling forces must increase. It is also hard to pull the molecules of a solid further apart. So as they move further apart the attracting forces must also increase

Interatomic force-separation curve (2)

A metal rod can be broken by pulling on its ends. However, it cannot be broken by pushing its ends inwards. The repelling forces must, therefore, continue to increase as the molecules come closer together, but the attracting forces must reach a limit

Interatomic force-separation curve (3)

Characteristics about the F(r) curve (1)

Sign convention: F is positive for repulsionAt r = ro, F = 0. Molecules are at equilibrium separationSteeper slope for r < ro than r > ro more difficult to compress When r < ro, the graph continues to rise steeply without limit a solid cannot be broken by compression

When r > ro, the curve shows a maximum value for the attractive force. Hence a solid can be broken by stretching When r is larger than several molecular diameters, F(r) becomes virtually zero

Characteristics about the F(r) curve (2)

Intermolecular force is a short-range force, F(r) is related to high powers of r

F(r) = a/rm - b/rn The first term represents the repulsive component where the second term represents the attractive component

Characteristics about the F(r) curve (3)

Characteristics of U(r) curve (1)

Due to the existence of intermolecular force, a molecule would possess different potential energies at different distances from its neighbour. The molecular potential U(r) is also a function of r Negative slope represents repulsive intermolecular force

At ro, U is a minimum most stable position-Eo = bonding or binding energy which is the minimum energy required to separate the molecules to infinityPoint of inflexion: the separation having the maximum attractive forcer1 represents the maximum molecular separation before the solid breaksF1 represents the breaking force

Characteristics of U(r) curve (2)

figure

Interatomic bonds (1)Ionic (electrostatic) bond

Formed between atoms of elements at opposite sides of the periodic table, e.g. sodium & chlorine A sodium atom has a loosely held outer electron which is readily accepted by a chlorine atom and two oppositely charged ions are formedThe two ions are then bonded by the electrostatic attraction between their unlike charges The ions tend to pack closely together

Covalent bondBonding electron sharing occurs between two or more atoms Highly directional e.g. carbon atoms in diamond

Interatomic bonds (2)

Metallic bondMetal atoms have one or two loosely held outer electrons that are readily lost These free electrons are shared by all atomsThe strong attraction between the ions and electrons constitutes the metallic bond Close packing arrangement

Interatomic bonds (3)

Van der Waals bondWeakExists in all atoms and moleculesArises due to the formation of a weak electric `dipole’ giving rise to an attractive force between opposite ends of such dipoles in neighbouring atoms

Interatomic bonds (4)

Microscopic interpretations of

macroscopic phenomena (1)

Elasticity & Hooke’s lawAtoms have minimum P.E. at the equilibrium position (P.E.-separation curve)When slightly displaced, atoms tend to move back to the equilibrium position at which the atoms attains the most stable state This explains the elastic property of matter

For sufficiently small displacements from the equilibrium point ro, the F-r graph is linear Therefore the force required is directly proportional to the desired extension or compression (Hooke’s law)

Microscopic interpretations of

macroscopic phenomena (2)

Young modulusImagine the atoms to be joined by springs each with a certain force constant k If a solid consists of N atoms on each side, with separation ro between atoms

Microscopic interpretations of

macroscopic phenomena (3)

Ek

ro

Thermal expansionThe PE curve is steeper for compression and flatter for stretching At absolute zero, the atom is essentially stationary at ro When temperature increases, the mean separation between atoms will increase resulting in expansion

Microscopic interpretations of

macroscopic phenomena (4)

Definite shapes for solidsAt low temperatures the kinetic energy of the molecules is low compared with their potential energy The molecules of solids can merely vibrate about fixed positions and are therefore locked into a geometrically ordered array Therefore, a solid has both a fixed volume and a fixed shape

Microscopic interpretations of

macroscopic phenomena (5)

The latent heat of sublimation of a solidConsider a solid in which each atom has n nearest neighboursOne mole of solid contains NA atoms (where NA is the Avogadro constant) and each atom is involved in n bonds The total energy required to separate the atoms in one mole of solid is

Microscopic interpretations of

macroscopic phenomena (6)

)(2

1oA nEN

The latent heat of sublimation of one mole of solid at 0 K is

Microscopic interpretations of

macroscopic phenomena (7)

oAS EnNL2

1

Effect of temperature and time on mechanical

properties (1)The effect of temperature

With increasing temperature, a solid softens and its strength and stiffness decreasesBy suitable heating and cooling, a ductile piece of steel can be changed into a brittle piece. Re-heating and slow cooling can reverse the effect

The effect of time (Fatigue and creep)If a material is repeatedly stressed and unstressed, it becomes weaker and the material may fracture even though the maximum stress applied in any of the cycles does not exceed the breaking stress The failure of a material in this way is called fatigue

Effect of temperature and time on mechanical

properties (2)

Creep is the gradual increase in strain when a material is subjected to stress for a long time (creep curve)It occurs even when the stress is constantIt is most marked at elevated temperatures The turbine blades in jet engines are particularly susceptible to creep Soft metals, e.g. lead, and most plastics, show considerable creep even at room temperature

Effect of temperature and time on mechanical

properties (3)

Theory of plastic deformation (1)

Ductility of metals: due to slipping of the crystal planesIn an ideal crystal all atoms are arranged in a perfectly regular lattice. To make one layer of atoms to slip over the other would require a very large stress because this would involve the breakage of a large number of bonds simultaneously

In a real metal, the force required to cause plastic deformation is much less than the expected value This is due to imperfections formed during manufacture such as missing atoms, extra or foreign atoms Those misplaced atoms are called dislocations

Theory of plastic deformation (2)

When plastic deformation occurs, the crystal planes slip by the movement of dislocationsThe movement of dislocation requires a much smaller force because only one bond is broken at a time

Theory of plastic deformation (3)

Elastic hysteresis of rubberStress-stain curve of a rubber

The curve during unloading does not coincide with the curve during loadingThe extension during unloading is always greater than that during loading with a given stretching forceThe effect is known as hysteresisThe sample returns to its original length when the load is completely removed

Properties of rubber (1)When rubber is stretched it becomes warmer. When the stress is released its temperature falls but it remains a little warmer than it previously was The net increase in the heat content per unit volume of the sample during the stretching-relaxation cycle is equal to the area of the hysteresis loop

Low stiffness It consists of long chain-like molecules with a repetitive unit of (C5H8)n The elasticity of rubber is caused by the coiling of the molecules It does not obey Hooke’s law

Properties of rubber (2)

From the figure, the sample stretches easily at first, but has become very stiff (steep slope) by the time the extension corresponding to the point A has been reached At A the extension is such that the long-chain molecules of the rubber have become fully straightened out

Properties of rubber (3)

Heating a stretched rubber band causes it to contract: The higher temperature produces increased lateral bombardment of the long chain molecules causing them to kink and so shorten The stress/strain ratio increases with increasing temperature

Properties of rubber (4)

Shear modulusShear stress: the stress which changes the shape of a bodyShear strain: the distortion resulted from shear stressShear modulus (modulus of rigidity):

strainshearG

stressshear

Unit: Nm-2 or Pascal

‘Single crystal’

Polycrystalline sample

Ductile & brittle failures

Searle apparatus

Stress

Strain

Stress-strain graph (within proportional region)

Force-deformation curve

Further reading

Stress-strain curve (1)

Further reading

Stress-strain curve (2)

Experimental set up

Extension Vs load graph

Strong and weak materials

Stiff and soft materials

Ductile and brittle materials

Strain energy

W1 = F1 e1

= area of shaded

strip

Strain energy

= area OAB

= 1

2Fe

P.E.- separation curve

ro = separation of atoms at equilibrium position

Eo = minimum P.E. attained

|Eo| = bonding energy

Interatomic force-separation curve

At ro , F = 0

r < ro, F is repulsive

r > ro, F is attractive

F(r) curve

U(r) curve

Closely-packed ions

Close packing arrangement

Dipole

Microscopic view of Hooke’s law

Atoms joined by springs

Thermal expansion

Creep curve

Movement of dislocationsBond BC is broken and is replaced by the bond CA

The process is repeated and the dislocation moves from left to right through the crystal

Hysteresis loop

Shear stress and strain