A synthetic non-ionic amphiphile assembles into aggregates of a single fixed size monomers

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Soft condensed matter · Soft condensed matter

Forces, Machines and Signals, Ch Charges in salt water and the assembly of amphiphiles

Problem

A synthetic non-ionic amphiphile assembles into aggregates of a single fixed size N=50N=50 monomers. At equilibrium the chemical potential of a monomer is the same whether it is free in solution or inside an aggregate: μ1∘+kBTln⁡X1=μN∘N+kBTNln⁡ ⁣(XNN),\mu_1^\circ + k_BT\ln X_1 = \frac{\mu_N^\circ}{N} + \frac{k_BT}{N}\ln\!\left(\frac{X_N}{N}\right), where X1X_1 is the mole fraction of free monomers, XNX_N is the total mole fraction of monomers bound in NN-mer aggregates, and Δμ∘≡μ1∘−μN∘/N\Delta\mu^\circ \equiv \mu_1^\circ - \mu_N^\circ/N is the standard free energy gained, per monomer, on moving from free solution into an aggregate. ((a)) Rearrange the equilibrium condition to show that XN=N[X1eΔμ∘/kBT]N.X_N = N\left[X_1e^{\Delta\mu^\circ/k_BT}\right]^N. ((b)) Explain, using the size of NN, why XNX_N stays negligible until X1X_1 approaches a sharply defined value X1∗=e−Δμ∘/kBTX_1^{\ast}=e^{-\Delta\mu^\circ/k_BT}, and why the free monomer concentration stays pinned near X1∗X_1^{\ast} once the total concentration exceeds it. This value of X1∗X_1^{\ast} is the critical micelle concentration (CMC) expressed as a mole fraction. ((c)) For this amphiphile Δμ∘=11.0 kBT\Delta\mu^\circ = 11.0\,k_BT at 298 K298\,\mathrm{K}. Taking the molar concentration of pure water as 55.5 mol L−155.5\,\mathrm{mol\,L^{-1}}, so that a mole fraction XX corresponds to a molar concentration of 55.5X mol L−155.5X\,\mathrm{mol\,L^{-1}} in a dilute aqueous solution, compute the predicted CMC in mM\mathrm{mM}.

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