# Figure the notations and are the chemical potential

Figure 1 shows the schematic diagram of T shaped microfluidic reactor, which has two intel with 100 µm diameter and 1.5 mm length. Two electrodes are placed at 200 µm downstream of the T junction and the distance between these two electrode is 100 µm. Two different liquid phases for two inlets are assumed to be Newtonian, immiscible, incompressible, leaky dielectric, isothermal. The following governing equations are described below to calculate the motion of the phases inside microfluidic reactor,

(1)

(2)

Here, denotes 1, 2 for oil phase and water phase respectively. The over dot symbol defines the time derivative. The notations, anddenote density, velocity, pressure of phase respectively and defines acceleration due to gravity. The symbol is used to describe hydrodynamic stress tensor, which can be written as the constitutive relation for Newtonian fluids, and is used for Maxwell stress tensor for the applied electric field, which can be written as . Here and denote dynamic viscosity and electric field. , surface tension force is defined as where the notations and are the chemical potential and phase field parameter respectively.

The irrotational AC electric field (without the presence of magnetic field),is applied across the microfluidic channel, which can be further expressed as , is the electric field potential function. Substituting this relation for in the Gauss’s law for both pure dielectric and leaky dielectric leads to the Laplace equation, , where andare dielectric constant and conductivity respectively. For leaky dielectric fluids, the electrostatic force is calculated as,

(3)

Here, is the charge density and this can be further expressed as .

The interface of two phases is tracked using the phase field method,

(4)

Here, defines the mobility of the interface. The values of for oil phase and water phase are 1 and -1 respectively. The values of in between 1 and -1 defines the interface of the two phases., chemical potential can be written in terms of free energy functional, ,

(5)

(6)

Here, andrepresent the volume of the liquid phase domains and total free energy density respectively. is defined as the summation of surface energy and bulk energy or double well potential where mixing energy density, is expressed as , in which and are diffused interface thickness and interfacial tension. The interfacial parameters,,, are represented as the function of ,

(7)

Here, can be any of the above said interfacial parameters.

A pair of chemical species in phases inside a microfluidic reactor are governed by following Advection-Diffusion-Reaction equation,

(8)

In this equation, the subscript ‘i’ denotes Benzyl Chloride ( ) (i = A) in oil phase and Sodium Phenolate ( ) (i = B) in water phase. denotes concentration vector, denotes the source term due to reaction with respect to ith species, is the total flux, which can be represented as the summation of diffusion flux () and advective flux (). So . It can be further rearranged as . denotes the diffusivity of the species. Now replacing the total flux into equation (8) and after the plugging the equation (1) to get the following equation,

(9)

The reaction between the species is given by,

Here, the reaction is assumed to be irreversible and isothermal. The products are Benzyl Phenyl Ether () (i = C) and Sodium Chloride () (i = D). Therefore, the concentration of products are denoted by and respectively. Here, Tetrabutylammonium Iodide as phase transfer catalyst (PTC) is used, which makes the reaction rate faster54, 55. So the species balance equations are given below,

, , and . These can be replaced as , , and . So Each reaction rate coefficient follows the Arrhenius equation,

(10)

Here, denotes the activation energy, is the gas constant, is pre-exponential factor or frequency factor anddenotes the temperature.