Curvature Effects in the Analysis of Pendant Bubble Data: Comparison of Numerical Solutions, Asymptotic Arguments, and Data

Modeling of adsorption dynamics at air–liquid interfaces using statistical rate theory (SRT)

“…It is thus necessary to employ numerical methods to solve the system of equations. Different numerical techniques, including integral, finite difference, and finite element methods, have been reported in the literature [11][12][13]19]. In the present work, the method of lines (MOL) is used.…”

“…The lower and upper limits of the maximum surface concentration (Γ ∞ ), the diffusion coefficient (D), the Langmuir constant (b), and the adsorption rate constant, (K a ) are taken from the literature [2,[11][12][13][14][15][16][17]19,22,[25][26][27]. The lower limit of the bulk concentration (C 0 ) is estimated from the minimum surface concentration needed to induce a change in surface tension and the upper limit is the critical micelle concentration (CMC) of the commonly used surfactants (e.g., short-chain alcohols and bolaform).…”

“…Although the rate constants can be calculated from the theory in principle, we estimated these rate constants using reported experimental data and related rate constants of the same system. Table 1 lists the physically realizable range of values for each of the parameters involved in the governing equations [2,[11][12][13][14][15][16][17]19,22,[25][26][27]. For example, the range of the bulk concentrations is considered from a very dilute concentration to the critical micelle concentration of common surfactants.…”

“…An analogous solution was obtained by Lin [13], where adsorption onto a bubble was considered and the bubble shape was approximated as a spherical interface. Ferri et al [12] reported that the geometry of the interface significantly affected the adsorption. Adsorption onto a spherical interface is faster than that onto a planar interface for a bubble, whereas, for a pendant drop, the adsorption onto a spherical interface is slower than that onto a planar interface [13,19].…”

“…The adsorption kinetics is mixed diffusion and transfer controlled when steps (i) and (ii) are equally important. Currently, most of the theoretical models have been developed using the following two mechanisms: diffusion controlled [9][10][11][12] and mixed diffusion and transfer controlled [13][14][15][16][17]. Rearrangements at the surface may consist of reorientation, conformational changes, complex formation, chemical reactions, phase transitions, or formation of three-dimensional structures resembling liquid crystals.…”