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Team:Paris Saclay/Modeling
From 2014.igem.org
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We use the following phenomenological law suggest by Adolphe Fick in 1855: | We use the following phenomenological law suggest by Adolphe Fick in 1855: | ||
| - | '' In an homogeneous and isotropic environment, containing particles distributed inhomogeneously,appears spontaneously a volumetric flow density vector particle $\overrightarrow{J}(M,t) $. In any point $M$ in space, this vector is proportional to the gradient of the particle density $n(M,t)$. Mathematicaly, this relationship take the form: \[ \overrightarrow{J}(M,t) = - D \times \overrightarrow{grad} n(M,t) \qquad (1) \] where $D$ is the diffusion coefficient.'' | + | <p> |
| + | '' In an homogeneous and isotropic environment, containing particles distributed inhomogeneously,appears spontaneously a volumetric flow density vector particle $\overrightarrow{J}(M,t) $. In any point $M$ in space, this vector is proportional to the gradient of the particle density $n(M,t)$. Mathematicaly, this relationship take the form: \[ \overrightarrow{J}(M,t) = - D \times \overrightarrow{grad} n(M,t) \qquad (1) \] where $D$ is the diffusion coefficient.'' | ||
| + | </p> | ||
| - | Referring to the article '''[1]''', the diffusion coefficient of | + | Referring to the article '''[1]''', the diffusion coefficient of oxygen in agarose is $ D = 0{,}256 \times 10^{-8} m^2 s^{-1} $. |
* To simplify the problem, we consider that the diffusion of oxygen particle occurs only in one direction. So $\overrightarrow{J}(M,t) = J(x,t) \overrightarrow{e}_x $. | * To simplify the problem, we consider that the diffusion of oxygen particle occurs only in one direction. So $\overrightarrow{J}(M,t) = J(x,t) \overrightarrow{e}_x $. | ||
| - | * | + | * Spatial variations in the density of particles are connected to spatial variations of the vector $\overrightarrow{J}(M,t)$ by '''the material's equation of conservation''' in presence of volume distribution of particle source $\sigma (x,t)$ (device which injects or subtracted particles to the system) : |
\[ \frac{\partial n}{\partial t} (x,t) = - \frac{\partial J}{\partial x} (x,t) + \sigma (x,t) \qquad (2) \] | \[ \frac{\partial n}{\partial t} (x,t) = - \frac{\partial J}{\partial x} (x,t) + \sigma (x,t) \qquad (2) \] | ||
By replacing $(2)$ in $(1)$, we obtain the following '''equation of diffusion''' | By replacing $(2)$ in $(1)$, we obtain the following '''equation of diffusion''' | ||
\[ \forall t, \forall x, \bigg( \frac{\partial}{\partial t} - D \frac{\partial^2}{\partial x^2} \bigg) n(x,t) = \sigma (x,t) \qquad (3) .\] | \[ \forall t, \forall x, \bigg( \frac{\partial}{\partial t} - D \frac{\partial^2}{\partial x^2} \bigg) n(x,t) = \sigma (x,t) \qquad (3) .\] | ||
| + | |||
| + | As our lemon is exposed to the ambient air, we stay in steady state where the source $ \sigma (x,t) $ is equal to $N_0$ the quantity of $O_2$ in the air. | ||
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'''[1]''' A.C. Hulst, H.J.H. Hens, R.M. Buitelaar and J. Tramper, ''Determination of the effective diffusion coefficient of oxygen in gel materials in relation to gel concentration'', Biotechnology Techniques Vol 3 No 3 199-204 (1989). | '''[1]''' A.C. Hulst, H.J.H. Hens, R.M. Buitelaar and J. Tramper, ''Determination of the effective diffusion coefficient of oxygen in gel materials in relation to gel concentration'', Biotechnology Techniques Vol 3 No 3 199-204 (1989). | ||
| + | |||
'''[2]''' Vincent Renvoizé, ''Physique PC-PC*'', Cap Prepas, Pearson Education, 2010. | '''[2]''' Vincent Renvoizé, ''Physique PC-PC*'', Cap Prepas, Pearson Education, 2010. | ||
{{Team:Paris_Saclay/default_footer}} | {{Team:Paris_Saclay/default_footer}} | ||
Revision as of 11:24, 9 August 2014
Modeling
To realise our artwork, we use an agarose gel to obtain the shape of a lemon. To push the resemblance to the extreme, we wish to have a crust in the edge of the lemon when we seperate it. In fact, we build bacteria who produce yellow/green color in presence of oxygen. Thus we must evaluate the penetration of the oxygen in the gel !
We use the following phenomenological law suggest by Adolphe Fick in 1855:
In an homogeneous and isotropic environment, containing particles distributed inhomogeneously,appears spontaneously a volumetric flow density vector particle $\overrightarrow{J}(M,t) $. In any point $M$ in space, this vector is proportional to the gradient of the particle density $n(M,t)$. Mathematicaly, this relationship take the form: \[ \overrightarrow{J}(M,t) = - D \times \overrightarrow{grad} n(M,t) \qquad (1) \] where $D$ is the diffusion coefficient.
Referring to the article [1], the diffusion coefficient of oxygen in agarose is $ D = 0{,}256 \times 10^{-8} m^2 s^{-1} $.
- To simplify the problem, we consider that the diffusion of oxygen particle occurs only in one direction. So $\overrightarrow{J}(M,t) = J(x,t) \overrightarrow{e}_x $.
- Spatial variations in the density of particles are connected to spatial variations of the vector $\overrightarrow{J}(M,t)$ by the material's equation of conservation in presence of volume distribution of particle source $\sigma (x,t)$ (device which injects or subtracted particles to the system) :
\[ \frac{\partial n}{\partial t} (x,t) = - \frac{\partial J}{\partial x} (x,t) + \sigma (x,t) \qquad (2) \]
By replacing $(2)$ in $(1)$, we obtain the following equation of diffusion \[ \forall t, \forall x, \bigg( \frac{\partial}{\partial t} - D \frac{\partial^2}{\partial x^2} \bigg) n(x,t) = \sigma (x,t) \qquad (3) .\]
As our lemon is exposed to the ambient air, we stay in steady state where the source $ \sigma (x,t) $ is equal to $N_0$ the quantity of $O_2$ in the air.
[1] A.C. Hulst, H.J.H. Hens, R.M. Buitelaar and J. Tramper, Determination of the effective diffusion coefficient of oxygen in gel materials in relation to gel concentration, Biotechnology Techniques Vol 3 No 3 199-204 (1989).
[2] Vincent Renvoizé, Physique PC-PC*, Cap Prepas, Pearson Education, 2010.
