Forces and the Lennard-Jones 6-12 Potential

F = - dV / dr

• Force is zero at r=a0

… stable equilibrium

Surface tension γ = force per unit length parallel to surface = energy required to create 1m2 of new surface … application to capillary action: h=2γ / ρgr

Elastic Moduli: Young’s modulus revision … to be continued today, along with Thermal Expansion

Stress = Modulus x Strain

Shear Modulus or Rigidity, G Describes response to tangential force producing a shear … Shear stress = F/A Shear strain = Δx/l = tanθ [G is very small a liquid (“no rigidity”)

… related to viscosity]

Bulk Modulus, K Describes change in volume due to application of a uniform pressure … Bulk stress = P Bulk strain = ΔV / V K = -V (dP/dV) [at fixed T] [Comparable for solids & liquids Compressibility = 1/K]

•  Typically, Young’s modulus and Bulk modulus are similar. Shear modulus is a factor 2-3 smaller.

•  Similar physics underlies all three moduli We will take Young’s modulus as an example.

Fractional expansion, ΔL/L related to temperature change ΔT by coefficient of linear expansion, α

Question: A metal rod has length 2m and linear expansion coeffcient α=2x10-5 K-1. By how much does its length increase for an increase in temperature from 10oC to 20oC?

Notes:

•  Usually length changes are small (~0.1%). •  At larger extensions, the solid ultimately melts! •  For a change in volume ΔV/V = 3α ΔT

α~10-5 K-1 for metals

If the length of a body is held fixed as we heat it, change in temperature induces a tension, F.

To calculate F: Compute amount, ΔL, by which body would expand if we allow it

Then calculate the (mechanical) stress required to compress an unconstrained rod of length L+ΔL back to its original length L

Thermal stress:

Thermal Stress can be very large indeed:

Long structures (e.g. railway lines) may buckle in hot weather if they are badly designed.

If we build a structure or equipment out of more than one material (e.g. glass and metal), care must be taken over difference between thermal expansion coefficients

…considered increase in separation between neighbouring atoms in 2 rows for a force F …

… Young’s modulus (and also the other moduli) can be obtained from the gradient of the force versus distance curve at the equilibrium separation, r=a0

Since F = - dU/dr, Young’s Modulus is also related to the curvature of the curve of potential v separation at the equilibrium point

This is easily caculated using the Lennard-Jones 6-12 model of the potential:

For typical ε, a0 values, Y~1011 Pa good agreement with data

Hooke’s law works, because the force v separation curve is linear to good approximation for small deviations from r=a0

If we pull the atoms apart with very strong forces, then: 1)  Linear approximation fails

Hooke’s law breaks down 2) Ultimately we reach the

maximum attractive (-ve) force Fmax between the atoms.

•  If we pull with a force F >= Fmax , applied continuously, the atoms will separate completely … Material “Fails” or “Breaks” •  Max sustainable stress of material obtained by applying Fmax over area of 1 molecule: Breaking Stress = Fmax / a0

2

Theoretical Breaking Force Fmax occurs when dF/dr = 0

Using the Lennard-Jones 6-12 Potential:

occurs when

… leads to a breaking stress, which can be expressed as a multiple of Young’s modulus:

Question: What is theoretical breaking force for thin steel wire of cross-sec A = 1mm2 and Young’s modulus Y = 2 x 1011 Pa

•  “Brittle fractures” take place along naturally occuring lines of weakness, usually initiated by a surface crack. •  All realistic materials have these •  Not understood until well into 20th century, with some disastrous consequences!

Bad news for material strength

Good news for cutting glass, bricks, paving

slabs …

Stress at tip of crack increased by factor

… depends on shape, not size

… huge effect if r is ~ atomic size and d is macroscopic!