## Surface Thermodynamic Model

The surface thermodynamic model predicts that the system free energy is minimized at equilibrium and adhesion between two surfaces will occur.23,24 Consequently, bio-flocculation will be thermodynamically favored if the process itself causes the system free energy to decrease. Ignoring electrostatic interactions and other specific binding, two hydrophobic surfaces approaching at short distances will result in the surrounding bound water layers to overlap with the eventual displacement of the bound water into the bulk water. This would lead to a decrease in the interfacial free energy and thus bioflocculation.

The interfacial free energy of the interaction between two identical bacterial cells (B), immersed in liquid (L) can be described as follows:

AGflocculation = -2YBL (19.1)

where AGflocculation is the interfacial free energy of floc formation and ybl is the interfacial tension for the bacteria (B)-liquid (L) interface. If the total free energy of a system is reduced (AGflocculation < 0) by cell interactions, then bioflocculation will be thermodynamically favored.23,25,26 Neumann et al.26 have demonstrated that ybl is a function of ybv and ylv (where ybv and ylv stand for the interfacial tension of bacteria-vapor and liquid-vapor, respectively), and developed an equation-of-state to describe the relationship among yBL, ybv, and ylv:

In conjunction with Young's equation:

A third equation is yielded as:

Based on Equation (19.4), the surface tension (ybv) of sludge flocs is determined by measuring the contact angle of a liquid with known surface tension (ylv). The change in the interfacial free energy of the system, AGflocculation, is then calculated from Equations (19.1) to (19.4).27

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