Amplificación mediante PCR 1 Oligonucleótidos

Chapter 2 explains the theory behind molecular dynamic simulations and clar- ifies the linking of the EDIP and ZBL potentials. The Kinchin and Pease model for hard sphere collisions is explained. A literature review into previous stud- ies within this field is presented.

The methodology employed to simulate radiation damage cascades in graphite, diamond, glassy carbon, high and low density amorphous carbons is described in Chapter 3. A description of neutron collisions is also presented along side a discussion about the physical properties of cascades in graphite and carbon materials.

Chapter 4 considers the qualitative behaviour of cascades in graphite along with the presentation of quantitative results. Results are obtained from; ther- mal spike simulations, collision cascade and defects analysis. Collision cas- cades are simulated over 20 initial directions with initial energies between 25 and 2000 eV. Results obtained from cascades with a high initial temperature are also documented.

Cascade simulations in diamond, glassy carbon and high and low density amorphous carbons are presented in Chapter 5. Results include thermal spike simulations, collision cascades and defect analysis.

Chapter 6 links graphite cascades with historical models in the literature and discusses how fundamentally different radiation damage in graphite is com- pared to other carbon materials. Quantitative and qualitative data is presented to further highlight this. A plan for future work is also presented.

Chapter

2

Theory

Molecular dynamics (MD) is a computational technique used for computing the equilibrium and transport properties of a classical many-body system [50]. Given the initial co-ordinates and velocities of an ensemble of particles that interact in an explicit manner, under certain conditions, this method integrates the equations of motion numerically, providing new sets of co-ordinates and velocities at each time step. The obtained particles trajectories allow one to calculate various system properties as statistical averages.

The simulation of radiation damage using MD has a long history, extending back to the first ever publication in 1960 of an MD simulation involving fo- cused collision sequences in copper [51][52]. A great number of radiation cascade simulations were performed over the following decades facilitated by the development of the embedded atom method for metals and Buckingham- type potentials for ionic solids and oxides [1][2]. Whilst there have been vast MD simulations achieved within metals and oxides, there have been hardly any MD simulations in graphite. Further developments of MD potentials have been found to be essential to accurately simulate radiation damage in graphite. MD simulations are in many respects very similar to real experiments. It is for this reason that it is useful to employ MD simulations before conducting experimental research in order to gain a greater understanding of the effects to be expected in experiments. Simulations can be used to explain experiments, to reveal processes, develop intuition and suggest directions of research. The MD simulations described in Chapters 4 and 5 have been used to simulate the effect that neutron irradiation has on graphite and carbon allotropes. The results have been used to pinpoint areas of damage, analyse defects and to demonstrate the effect the irradiation has on the rest of a cell.

2.1

Equations of Motion

The MD simulation method is based on Newton’s Second Law, equation 2.1. The knowledge of the force acting on atoms allows the acceleration of each atom in the system to be calculated. Integration of the equations of motion then yields a trajectory that describes the positions, velocities and acceleration of the particles as they vary with time. The trajectory obtained as a direct result of integration allows average values of a systems properties to be determined [53].

Fi =miai (2.1)

whereFi is the total force exerted on particle i (i = 1, 2, 3 ... n) and mi is the

mass of particlei anda_i is the acceleration of particlei.

The force can also be expressed as the gradient of the potential energy, equa- tion 2.2.

F_i =_−∇iV (2.2)

whereV is the potential energy of the system and is assumed to depend only on the particle positions.

Combining equations 2.1 and 2.2 produces:

−dV

dri

=mi

d2r_i

dt2 (2.3)

To calculate the trajectory of an atom, one only needs the initial positions of the atoms, an initial distribution of velocities and the acceleration, which is determined by the gradient of the potential energy function. The Verlet algorithm is employed to integrate equation 2.3 and is described in section 2.3. To measure an observable quantity in an MD simulation, the observable func- tion must be able to be expressed as a function of the positions and momenta of all the particles in a system. A prime example of an observable quantity in MD simulations is temperature. In a classical, many-body system, a reliable definition of the temperatures makes use of the equipartition of energy over all degrees of freedom that enter quadratically in the Hamiltonian of the sys- tem. The theorem of equipartition of energy states that molecules in thermal

equilibrium have the same average energy associated with each independent degree of freedom of their motion and the average kinetic energy is given in equation 2.4 [50]. This alters depending on the degrees of freedom, for ex- ample, in polyatomic molecules where there are 3N degrees of freedom and in diatomic molecules where there are 2N degrees of freedom, the thermal energy will be distributed between rotations and vibrations of the molecule. The equipartition theorem begins to break down if the system is not in thermal equilibrium. This is because when a system reached thermal equilibrium, each sub-system is guaranteed to have the energy which is attributed to it by the equipartition theorem. 1 2mv 2 α = 1 2kBT (2.4)

Wheremis the mass, vα is the velocity of the particle whereα labels the trans- lational degree of freedom,kB is the Boltzmann constant and T is the temper-

ature.

Equation 2.4 can be employed as an operational definition of the temperature. However, in practice, the total kinetic energy of the system would be measured and divided by the number of degrees of freedom. During an MD simulation, the total energy of a system fluctuates as does the instantaneous temperature (equation 2.5). These fluctuations occur as a direct result of the finite size of the cell. The relative fluctuations in the temperature will be of the order 1/p

Nf

and asNf is typically of the order of 102- 103, the statistical fluctuations in the

temperature are of the order of 5-10% (whereNf is the number of fluctuations).

To get an accurate estimate of the temperature, one should average over many fluctuations [50]. T(t) = N

∑