Fluctuation Theoretical and Turbidimetric Studies of Fast Mixing Processes in Liquids by Dynamic Volume Pulse Techniques. Part 1. Fluctuation Theory

Abstract

AbstractFast optimum mixing processes in open, binary ideal liquid mixtures are theoretically treated as a superposition of a temporal set of noninterfering concentration fluctuations. For periodically generated volume pulses the relations of the quasi‐stationary mixing equilibrium as well as the temporal course of the corresponding concentration fluctuations are deduced. It is shown that the ”optimum differential mixing time ϑ” is a characteristic elementary time quantum. It also corresponds to the “waiting time θ*” which elapses during the equilibration of a sudden compression of the local mixture concentration: ϑ = ϑ*.The investigation of the time course of concentration fluctuations (ε = γ‐&γmacr; ; γ = volume fraction of constituent 1) results in an estimation of ϑ. A statistical‐thermodynamic fluctuation theory is used for the calculation of ϑ by means of a liquid model. The mean differential mixing time 〈ϑ〉 corresponds to the quotient of the “average course time (Verlaufszeit)” (〈ϑϵ〉 = 1.5 · 10−5sec) and the “fluctuation probabilityP(ε)”: 〈ϑ〉 = 〈ϑϵ/P(ϵ) · 10−2sec. The mean square relative deviation is given by 〈ε2/(&γmacr;)2= 〈(γ ‐ &γmacr;)2〉/(&γmacr;)2= 1/n̄1≈︁ 10−22(n̄1≈︁ 1022= average number of molecules of constituent 1 per cm3).It is shown that the elementary time quantum ϑ may also be interpreted as a relaxation time. Such an approach permits some understanding of the molecular events which proceed during fast mixing processes e.g. dependence on temperature and viscosity. Optical observation i.e. turbidity measurement is discussed on the basis of coupling fast mixing processes to polymer precipitation as an indicator reaction.