# Mathematical Modelling of the Effects of Plasma Treatment on the Diffusivity of Biofilm
## 2 Materials and Methods  
### Development of a Mathematical Model of CHX Diffusion in a Biofilm  
The mathematical development of CHX diffusion in a biofilm is based on [PERSON] (1975) and [PERSON] (2017) [33]. The Figure 1 shows the physical characteristics of the biofilm. In this figure, S is the surface area of the biofilm normal to the CHX diffusion and \(\delta\) is the thickness of the biofilm. Figure 1: Biofilm model on a titanium surface. Considering a thin slice in the biofilm (from \(x\) to \(x+\Delta x\)), the accumulation of CHX at time \(t\) is given by \(SAx\frac{\partial C}{\partial t}\). Now, from the mass-balance equation, \(accumulation=in-out+generation-consumption\), and using [PERSON]'s law to describe CHX diffusion we obtain,  
\[S\Delta x\frac{\partial C}{\partial t}=D_{e}S\frac{\partial C}{\partial x} \mid_{(x+\Delta x)}-D_{e}S\frac{\partial C}{\partial x}\mid_{(x)} \tag{1}\]  
where \(D_{e}\) is the effective diffusivity of CHX in the biofilm and C is the biofilm concentration of CHX. It should be noted that in Equation (1), CHX generation is zero and CHX consumption is zero, because CHX degradation is assumed to be negligible during the treatment time as CHX is known to be a very stable molecule. The lifetime of CHX is in the order of days and even weeks [34; 35]. Equation (1) then becomes,  
\[\frac{\partial C}{\partial t}=D_{e}\ \frac{\partial^{2}C}{\partial x^{2}} \tag{2}\]  
The initial and boundary conditions for Equation (2) are,  
\[ \begin{array}{l}\text{Initial condition};\ t=0,\ C=0\\ \text{Boundary condition 1}:\ \text{at}\ x=0,\ \frac{\partial C}{\partial x}=0\\ \text{Boundary condition 2}:\ \text{at}\ x=\delta,\ C=C_{b}\end{array}  \tag{3}\]  
where \(C_{b}\) is the bulk concentration of the CHX. From [PERSON] (1975), the solution to Equations (2) and (3) is given by,  
\[C(x,t)=c_{b}\left[1-\frac{4}{\pi}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}. \exp\ \left[\frac{-D_{e}(2n+1)^{2}\pi^{2}t}{4\xi^{2}}\right].\cos\left[\frac{(2n+1) \ \pi\ x}{2\delta}\right]\right] \tag{4}\]  
From the definition of effective diffusivity,  
\[D_{e}=\frac{\varepsilon D_{AB}}{\tau} \tag{5}\]  
where \(D_{AB}\) is the diffusivity of CHX in the media assumed to have the same properties as water, \(\tau\) is the tortuosity, and \(\varepsilon\) is the porosity. From [PERSON] 2017, and using the molecular weight (MW) of CHX as 505 g/mole, we can estimate \(D_{AB}\) in water at 37 \({}^{\circ}\)C,  
\[ \begin{array}{ll}D_{AB}&=1.013\times 10^{-4}(MW)^{-0.46}\\ &=5.782\times 10^{-6}\text{cm}^{2}/\text{s}\end{array} \]  
We can then adjust this value of \(D_{AB}\) at 20 \({}^{\circ}\)C using the Stokes-Einstein equation ([PERSON] 2017),  
\[ \begin{array}{ll}D_{AB}&=5.782\times 10^{-6}\times\frac{293}{310}\times \frac{0.691}{1.002}\\ &=3.77\times 10^{-6}\ \text{cm}^{2}/\text{s}\end{array} \]  
where 0.691 and 1.002 are the viscosities of water at 310 K and 293 K, respectively. Now consider the cell balance within the biofilm, we let  
\[\overline{X}=\frac{Number\ of\ cells}{Volume\ of\ biofilm} \tag{6}\]  
where \(\overline{X}\) is the cell concentration based on the total film volume assuming that the cells remain stationary, that is, the cells do not move or diffuse. The death rate of the cells in the biofilm is proportional to the total amount of cells. The proportionality constant is the specific death rate \((\mu_{d})\),\[\frac{d\overline{X}}{dt}=\mu_{d}\ \overline{X}\quad\text{and},\mu_{d}=k_{d}\ C(x,t) \tag{7}\]  
The specific death rate constant, \(\mu_{d}\) is assumed to depend on the local concentration of CHX, which is given by Equation (4), and a death rate constant \(k_{d}\).