## Motivation

After we had successfully synthesis quantum dots by JM109, we detect the excitation spectrum of our quantum dots. The result is in Fig.1 and look at the the fluorescence spectrum, we can find out that there are two peaks in the Fig.1.

## Possible Factor

Since the excitation wavelength is approximately proportional to the cubic of the diameter of quantum dots. We would have a deduction which is that the quantum dots we synthesis have at least two different size.^{[1]}

## Reference research

However, the two different size of quantum dots in our result is not a good thing due to the fact that we would like to have uniformity of quantum dots' size in manufacturing. In other words, we want the size of quantum dots to be about the same. Thus, we want to figure out what lead to the different size of quantum dots?

According to reference 2, we realize that there are two pathway for JM109 to synthesis quantum dots, extrcellular and intracellular. The extracellular pathway take place at the plasma membrance and the nucleation is supported by membrane protein. Owing to the unstable condition in extracellular pathway, the fluctuation of enviroment would cause the shape of quantum dots not regulate. On the other hand, the intracellular pathway happen in cytoplasma, which is relatively more stable. That is, the enviroment is more stable, which implies the shape of quantum dots would be similar, hexagonal.

Although there are many research and hypothesis about the extracellular and intracellular pathway in different species, we still not know the exact pathways in JM109. Yet, without the exact pathway, we can still do a simulation to help find out the cause of difference size of quantum dots in extracellular and intracellular.

## Model

To simulate the nucleation process, we would introduce three common used model, Phase Field model, Phase Field Model, and Molecular Dynamic Model.

^{[4]}

Phase Field model is a macroscopic mean field model. This model can not see the separation of two phase while we can not see the microstructure of crystal. Since the size of quantum dots is about 2-10 nanometer, the length scale of phase field model is about micrometer. We would not used this model. The molecular dynamic model is a mircoscopic model which is dominated by Schrodinger Equation. We need to know the potential measured by experiment so that we can do the simulation. What's more, the simulation can only do a short time scale and the atom in this simulation is kept moving even in equilibrium, which means that it is difficult for us to determine whether it is in equilibrium. Hence, we choose the phase field model to do the simulation for three reason: First, it is a mean field model which means it averages the non-local region versus time. In equilibrium, the size and shape of crystal would remain the same. Second, it is the mesoscopic model. We can see the microstructure of the crystal and the time scale is diffusive time scale. Third, pfc model unlike pf model can be connected with classical physics model.(DFT model). Next, we will introduce the detail of phase field crystal model.

## PFC model

PFC model is a morphology model that can cause period patterns emerge, which are used in nano-crystal study. The free energy of phase field crystal model can be written as below:

where q0 is the length of reciprocal lattice vector and rho is the density that has theoritical solution:

When we minimized the free energy, the wave number would tend to be q0 in k space while the cubic term of rho cause the formation of triangular crystal. In other words, the stable solution is to form triangular crystal with lattice constant 2*pi/q0.This model can be connected with classical model, density function theory(DFT).

Density functional theory is divided free energy into two parts, ideal free energy and excess free energy:

Based on statistic physics, ideal free energy can be written as

which describe the local free energy arised from an entropically driven ideal gas or fluid. The excess free energy can be written as below

then we can expand ideal free energy with Taylor series into

Next, we define the excess free energy in operator form

We can find out that the whole free energy can be written as

which is similar to PFC model. K. R. Elder [5] describe the first peak correlation function with quadratic term in k space which is the original PFC model.

^{[5]}

## Simulation

Since all nucleation process is quite the same except the constraints, we can simplify the intracellular pathway into a homogeneous nucleation process while the extracellular pathway is heterogeneous nucleation.

## Heterogeneous Nucleation (Extracellular)

The simulation of intracelluar pathway is to simulate the homogeneous process. In this part, we place a nuclei in the middle of solution, which is showed in Fig.7.

Since we want to know what influence the nucleation speed, the parameter we have are mean density and temperature. The mean density corresponding to the concentration of Cd. The closer the mean density to the constant phase, the lower the concentration of Cd.

We first use rho 0 =0.2807 to simulate the crystal growth and record the energy change. The crystal growing process and its corresponding free energy change is showed in Fig.9.

With a closer look of the free energy versus time in Fig.9, we can divide the process into three: The first is that the free energy decreases so slowly that the energy versus time looks horizontal. The second is that the free energy decreases obviously. In this process, the nuclei growth bigger and bigger into crystal. The third is that the free energy reach the minimum. It becomes stable.

In first part, we can find out that there is an inflection point. It means that before the inflection point, the energy decreases slower and slower while after the inflection point, the free energy decreases faster and faster. It is crucial for us since it can be used to evaluate the nucleation speed. Then we change different rho 0 from 0.2907 to 0.2007 to see the time it needs to reach the inflection point. The result is plotted in Fig.11.

## Conclusion

Look at the Fig.11, we find out that the inflection point and the mean density had the exponential growth relationship. In other words, the nucleation time of different Cd concentration is exponential decay. Since JM109 does not have specialized Cd channel, the Cd concentration of intracellular and extracellular are quite difference- intracellular concentration would be far lower than extracellular. These concentration difference leads to different nucleation speed, which cause different size of quantum dots. That is the simplest way to make the size of quantum dots uniformity is to introduce a Cd channel protein, mnth, to balance the extracellular Cd concentration and intracellular Cd concentration.

## References

- W. William Yu, Lianhua Qu, Wenzhuo Guo, and Xiaogang Peng. Experimental Determination of the Extinction Coefficient of CdTe, CdSe, and CdS Nanocrystals.
*Chem. Mater. 2003, 15, 2854-2860* - K. E. Marusak,a Y. Feng,a C. F. Eben,b S. T. Payne,b Y. Cao,b L. Youb and S. Zauscher. Cadmium sulphide quantum dots with tunable electronic properties by bacterial precipitation.
*RSC Adavnce Issue 80, 2016, Issue in Progress.* - Carla Gallardo-Benavente,1 Ornella Carrión, Jonathan D. Todd, Joana C. Pieretti, Amedea B. Seabra, Nelson Durán, Olga Rubilar, José M. Pérez-Donoso,,* and Andrés Quiroz. Biosynthesis of CdS Quantum Dots Mediated by Volatile Sulfur Compounds Released by Antarctic Pseudomonas fragi.
*Front Microbiol. 2019; 10: 1866.* - Author links open overlay panelLászlóGránásyabGyula I.TóthcJames A.WarrendFrigyesPodmaniczkyaGyörgyTegzeaLászlóRátkaiaTamásPusztaia. Phase-field modeling of crystal nucleation in undercooled liquids – A review.
*Progress in Materials Science Volume 106, December 2019, 100569* - K. R. Elder and Martin Grant.Modeling Elastic and Plastic Deformations in Non-Equilibrium Processing Using Phase Field Crystals.
*Physical Review E · December 2004.*