Soft matter, PB

Length scales

Specific surface area (total surface area ÷ total mass) is proportional to 1/R
- Colloid particles have a much higher specific surface area than bulk droplets
- Colloids particles are therefore much higher in energy, as extra surface area costs energy (ΔG = γΔA)
Yield stress is proportional to 1/R³
- Colloid particles have fewer bonds per unit volume, resulting in much lower yield stresses

Van der Waals forces are quantum mechanical in origin
An instantaneous dipole in an atom or molecule (1) can induce a dipole in a nearby atom or molecule (2)
The strength of the induced dipole is proportional to the polarizability of atom or molecule 2

\mu _{\mbox{ind}}=\alpha \mathbf {E}

The energy of a van der Waals attraction is proportional to 1/r⁶
- VdW attractions are much stronger at short range – they rapidly diminish as the separation, r, between the two interacting atoms or molecules increases

Molecules in dielectrics – attractions depends on excess polarizability
- If the difference between the dielectric constant of molecules and of the medium is large, vdW attractions between molecules are strong
- If the dielectric constants of the molecules and the medium are similar, vdW attractions between molecules are weak
The energy of vdW attractions in a medium is proportional to (ε₂−ε₁)² and to 1/r⁶
Such energy is usually expressed in terms of the Hamaker constant, A₂₁

A_{21}\sim \pi ^{2}\hbar \omega \left(\epsilon _{2}-\epsilon _{1}\right)^{2}

VdW forces between identical molecules are always attractive
The Hamaker constant for vdW attraction (A₂₁) between molecules in a medium depends on the Hamaker constants for the molecules (A₂) and the medium (A₁) in a vacuum:

A_{21}=\left({\sqrt {A_{2}}}-{\sqrt {A_{1}}}\right)^{2}

In the limit of large separation h between two colloid particles each of radius R,

U_{d}(h)=-{\frac {A_{21}R}{12h}}