Lecture 15: Aggregation What did we cover in the last lecture? Hydrogen bonds and hydrophobic interactions are stronger than simple dispersion interactions (~4-30 kJmol-1 vs ~1kJmol-1) They have more specificity than dispersion forces and have directionality. They can be used to build nanoscale structures The hydrophobic effect arises due to D 2 S changes in the entropy of water F 1 exp o o surrounding nonpolar molecules. In this lecture… 1) Aggregation 2) Why do aggregates form? 3) What determines the shape of an aggregate? 4) 1D aggregation – rods 5) 2D aggregation – sheets 6) 3D aggregation - spheres Further reading Intermolecular and Surface Forces, J. Israelachvili, chapter 16 Aggregation and phase separation We have seen that particles and surfaces can interact via many different physical interactions. The subtle balance between these interactions determines whether the particles/surfaces will remain separated in solution. When the interactions between particles favour the attraction of particles, they will tend to aggregate If these interactions become really strong, they will eventually drive phase separation in the system. Why do aggregates form? When the energy per particle/molecule inside an aggregate is less than the energy of a free molecule, then aggregates will start to form The average free energy per particle/molecule is often referred to as the chemical potential and is given the symbol, m As we will see the chemical potential depends upon the shape of the aggregates and the number of particles in the aggregate (N) 1 Dimensional aggregates (rods) Consider a linear aggregate where all the N particles/molecules are joined together in a line. Number of particles, N If each of the (N-1) ‘bonds’ in the aggregate has an energy of -u relative to the unbonded state. The total energy of the aggregate becomes Number of ‘bonds’, N-1 U agg Nm N 1u N 1u m 1 N or 1 u N 2D aggregation R The case of 2D aggregation is slightly more complicated. As a disk-like aggregate grows the area of the surface on which the particles attach increases with radius of the disk. The production of this excess surface area has an energy penalty associated with it. t Number of particles, N m mbulk 2 t 1 2 1 2 A consideration of the bulk N and surface contributions of u is molecular/particle volume and is the energy per molecule gives excess surface energy of unbonded (see OHP) particles around circumference 3D aggregation Similarly for 3D aggregation we consider volume and bulk terms and we obtain (See OHP- or leave as exercise for reader) 2 3 3 1 m mbulk 4 1 4 N3 R Number of particles, N What happens as the aggregation number increases? Rods 1 u m 1 u mbulk N N Disks m mbulk 2 t N Spheres/droplets 1 2 1 2 3 4 m mbulk 4 2 3 1 N 1 3 In all cases the average energy per particle/molecule decreases with increasing aggregation number N (tending to –mbulk in an infinite aggregate) Aggregates will tend to grow once formed Critical aggregation number There is a critical aggregation number Nc below which aggregates will not grow Aggregate growth is only favourable when m < 0 In each case, we have competing surface and bulk terms 2 3 3 1 mbulk m surface 1 4 3 N m mbulk 4 surfaceterm When msurface ≥ |mbulk |, m ≥ 0 and aggregation becomes unfavourable Aggregates with N< Nc will therefore tend to ‘dissolve’ Calculation of Nc When m = 0 the aggregation number is N=Nc 1 u 0 1 Nc For 1D aggregates or Nc =1 i.e. linear aggregates will have a tendency to grow and form large structures For 3D aggregates Protein fibrils 300nm 2 3 3 1 0 mbulk 4 1 4 N3 c 3 or 4 3 N c mbulk 4 2 How does Nc influence aggregation? A critical cluster size has to form in order for the aggregates to grow 4 N c mbulk 3 3 4 This means that the local molecule/particle concentration has to be increased so that an aggregate nucleus can be formed This can be caused by the presence of foreign surfaces (nucleating agents) or by local fluctuations in composition (due to e.g. thermal fluctuations) 2 Problem Calculate the critical aggregation number for a spherical aggregate made of molecules of a material with a surface energy of 15 mJm-2, if the bulk cohesion energy per molecule is 15 x 10-20 J and the molecule has a radius of 1.5 nm Calculate the radius above which aggregates of this molecule will start to grow Summary of key concepts When the free energy per molecule/particle inside an aggregate becomes less than that of a free molecule/particle, aggregates will start to form The form of the free energy per molecule (chemical potential) depends 2 upon the shape of the aggregate and 3 3 1 m mbulk 4 1 the aggregation number 4 N3 A critical cluster size sometimes has to 3 2 4 3 form before growth can occur. This is N c needed to overcome the contributions mbulk 4 due to the excess surface energy of the aggregate.