Module 3: ES

Question 1

Long-Range Electrostatics

(Problems)

Answer 1

• electrostatic potential converges slowly (1/r )
• dipole term is conditionally convergent, depending on whether one does periodic summation or shell summation
  
  • Boundary conditions:

Question 2

Direct Summation

Answer 2

• compute potential using pbc
• requires large n_cut
• bad scaling O(N²)
  
  • number of cells scales with n³_cut

Question 3

Ewald Summation

Answer 3

• separate into short-range and long-range by “adding zero” with Gaussian covering point charge
  
  • short-range in real space
  • long-range in k-space
• both terms now converge relatively quickly
• scaling O(N^3/2)
• requires r_cut, k_cut
  
  • minimum image convention → r_cut < L/2

Question 4

Ewald Summation and P3M Errors

Answer 4

• rms force error depends on choice of Gaussian width α (and cut-offs)
• there exists methods to choose α based on desired accuracy

Question 5

P3M Method

(Overview)

Answer 5

• replace long-range contribution k-space sum with FFT on mesh grid
• scales O(NlogN)

Question 6

P3M Method

(Algorithm)

Answer 6

1. Interpolate charges onto lattice charge density
2. FFT transform lattice charge density to k-space
3. Get potential by solving Poisson equation by multiplication of optimal influence function
4. Get field iva k-space differentiation
5. Inverse FFT transform back to real space
6. Extrapolate field to position of charges
7. Calculate force

Question 7

ELC + P3M

Answer 7

• P3M assumes 3D pbc
• ELC allows for “post-processing” correction for modelling 2D periodic systems using P3M
• requires charge neutral “buffer space” in non-periodic dimension

Question 8

Poiseuille Flow

Answer 8

Flow of incompressible, Newtonian fluid in channel is highest at the center

Question 9

Electro-Osmotic Flow

Answer 9

Flow of liquid in microchannel due to applied voltage

Question 10

Electrophoresis

Answer 10

motion of dispersed particles relative to a fluid under the influence of a spatially uniform electric field

Question 11

Rouse Model

(Overview)

Answer 11

• conformational dynamics of ideal chain
• no excluded volume interaction
• each monomer subject to Langevin Dynamics
• no hydrodynamic interactions
• usually assocaited with free-draining

Question 12

Rouse Model

(Good Performance)

Answer 12

• correctly predicts long-time diffusion for chains shorter than entanglement length

Question 13

Zimm Model

(Overview)

Answer 13

• extension of Rouse model to include hydrodynamic interactions
  
  • uses Oseen matrix

Question 14

Zimm Model

(Good Performance)

Answer 14

• predicts diffusion relative to N ^-ν
  
  • consistent with experiment