PHY1039 Properties of Matter Macroscopic (Bulk) Properties: Thermal Expansivity, Elasticity and Viscosity 20 & 23 February, 2012 Lectures 5 and 6

Thermal Expansivity,  + dT V o T o +F+F + F +F+F 3-D +F+F VoVo +F+F +F+F + dV T + dT Constant P (dV and dT usually have the same sign) 1-D +F+F +F+F LoLo A ToTo T o + dT L o +dL +F+F +F+F Linear Expansivity,  Constant F (dL is usually the same sign as dT) F

Potential Energy of a Harmonic Oscillator Stretching or compressing the spring raises the potential energy. Extension = r – r o roro uouo K is a spring constant Figure from “Understanding Properties of Matter” by M. de Podesta At equilibrium, the spring length (atomic spacing) is r o

r = r 0 ; Potential energy is at minimum. Kinetic energy is maximum. Potential energy is at maximum. Kinetic energy is minimum (or zero for an instant) Atomic Origins of Thermal Expansion: Anharmonic Potential Thermal energy is the sum of the kinetic and potential energies. roro Increasing T raises the thermal energy. r

 L/L)*100% T  increases slightly with temperature. Thermal Expansivity of Metals and Ceramics Substance Linear expansivity,  (K -1 ) (room T) Invar steel1 x Pyrex glass3 x Steel11 x Aluminium24 x Ice51 x Water*6 x Mercury*6 x Steel SiC * Deduced from  (   3  )  liquid >>  solid

C.A. Kennedy, M.A. White, Solid State Communications 134, (2005) 271. Negative Thermal Expansivity The volume of these materials decreases when they are heated! Science, 319, 8 February (2008) p Low T High T Low T High T

VoVo F 3-D T V o +dV +dF+dF +d F T Bulk Modulus, K (dV is usually negative when dP is positive) Constant T 1-D LoLo A T Young’s Modulus, Y +d F L o +dL +d F T Constant T (dL is usually positive when d F is positive) +F+F + F +F+F Initial pressure could be atmospheric pressure. Increased pressure: dP

P-V Relation in an Ideal Gas Volume, V Pressure, P

Potential Energy, u, for Pair of Molecules Separation between molecules (r/  ) r  Potential Energy for a Pair of Non-Charged Molecules Equilibrium spacing at a temperature of absolute zero, when there is no kinetic energy. Figure from “Understanding Properties of Matter” by M. de Podesta

- + Relating Molecular Level to the Macro-scale Properties Considering the atomic/molecular level, the slope of this curve around the equilibrium point describes mathematically how the force will vary with distance. Compression Tension Figure from “Understanding Properties of Matter” by M. de Podesta u r/ 

Strain,  Applied Stress,  Elastic (Young’s) Modulus, Y Length, L x Brittle solids will fracture Y Stress: Strain: LoLo +F+F +F+F L A T

Young’s and Bulk Moduli of Common Solids and Liquids MaterialY (GPa)K (GPa) Polypropylene2 Polystyrene3 Lead167.7 Flax58-- Aluminium7070 Tooth enamel83-- Brass9061 Copper Iron Steel Tungsten Carbon Nanotubes~ Diamond Mercury--27 Water--200 Air

A0A0 F F L dLdL L b b dbdb Poisson’s Ratio Poisson’s ratio = Therefore, usually is positive. Solids become thinner when pulled in tension. b usually decreases when L increases. If non-compressible (constant V), then = 0.5.

Auxetic Materials have a Negative Poisson’s Ratio! /action/index.html

Summary of Bulk Properties Property Volume expansivity (3-D) Equation of State f(P,V,T) =0 FormulaSI Units K -1 Linear expansivity (1-D) f( F,L,T) =0  K -1 Isothermal Bulk modulus (3-D) f(P,V,T) =0 Pa = Nm -2 Young’s modulus (1-D) f( F,L,T) =0  Pa = Nm -2 Isothermal compressibility (3-D) f(P,V,T) =0 Pa -1 = N -1 m 2

A A y F xx There is a velocity gradient (v/y) normal to the area. The viscosity  relates the shear stress,  s, to the velocity gradient. The top plane moves at a constant velocity, v, in response to a shear stress: v  has S.I. units of Pa s. Definition of Viscosity Viscosity describes the resistance to flow of a fluid.

Inverse Dependence of the Viscosity of Liquids on Temperature Thermal energy is needed for molecules to “hop” over their neighbours. Viscosity of liquids increases with pressure, because molecules are less able to move when they are packed together more densely.

Temperature Dependence of Viscosity Flow is thermally- activated. Viscosity is exponentially dependent on 1/T

Viscosity, , of an Ideal Gas Viscosity varies as T ½ but is independent of P.

Figure from “Understanding Properties of Matter” by M. de Podesta