Marangoni instability induced by a mono-enzymatic interfacial reaction. Effect of pH on the reaction rate

Abstract

The dynamical behaviour of a Newtonian reacting interface separating two homogeneous incompressible and semi-infinite Newtonian fluids is considered. The temperature is constant and uniform, electrical effects are not taken into account and the system is maintained in non-equilibrium conditions. Indeed, surfactants cross the interface where they are free to react and to diffuse. The surface tension depends on the interfacial concentrations of surfactants through an interfacial equation of state. The interfacial chemi-diffusion, adsorption–desorption and convective processes determine the surface tension and its variations. This provides a link between chemical and mechanical effects. Indeed, according to the Marangoni equation, a surface tension gradient is compensated for by the mechanical forces acting on the interface, especially the viscous tractions of the bulk phases. The Scriven theory of interfacial hydrodynamics is the starting point for a linear analysis of interfacial mechano-chemical stability. Our purpose is to obtain the conditions for the occurrence of surface motions and deformations in a system initially at rest and characterized by a flat interface. In the present treatment, the surface chemi-diffusion steps, the adsorption–desorption and the surface convection processes are the only relaxation processes responsible for the instability. We consider a mono-enzymatic Michaelian surface reaction catalysing the deprotonation of a substrate. In addition to a monoprotonated active form, the enzyme exists in unprotonated and biprotonated inactive forms. The reaction rate is then a function of the pH of the reacting medium. Consequently, the reaction is auto-exalted for a range of values of pH > pH(where the peculiar value pH corresponds to the optimum enzyme activity), while for pH < pH, it is auto-inhibited. We show that aperiodic (time) hydrodynamic and chemical instabilities are possible in the non-autocatalytic conditions pH < pH when the chemical and convective processes are coupled. Quantitative and qualitative stability criteria are reported. The stability behaviour differs depending on whether the enzyme forms are soluble in the bulk phases or not. Indeed, when they are soluble, the instability occurs for a finite range of non-vanishing values of the wavenumber characterizing one normal mode of perturbation. Furthermore, the viscous effects then play an expected stabilizing role since increasing but finite values of the viscosity coefficients completely remove the instability. Finally, the smallest value of the chemical constraint compatible with a state of neutral stability (transitory régimes between aperiodically stable and unstable situations), i.e. the critical constraint, corresponds to a non-vanishing and finite value of the wavenumber. On the other hand, when the enzymes are insoluble in the bulk phases, the instability occurs for an infinite range of wavelengths. Furthermore, the instability is only removed for infinite values of the viscosity coefficients. Finally, the critical constraint now corresponds to an infinite wavelength. The results obtained are discussed and interpreted by invoking the Marangoni effect caused by the mechano-chemical processes considered.