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Copyright Brian J. Kirby. With questions, contact Prof. Kirby here. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.

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Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

H.7 Exercises [atomistic modeling top]

  1. Write a MATLAB routine to solve the Ornstein-Zernike equation with hypernetted chain closure to find the radial distribution function for a homogeneous Lennard-Jones fluid.

    Proceed as follows:

    1. Use an iterative technique that, in turn, uses the hypernetted chain closure (Equation H.20) to solve for ftc and the Ornstein-Zernike equation (Equation H.19) to solve for fdc.
    2. Start by setting ftc(r) = fdc(r) = 0 on a domain that ranges from r = 0 to r = 512σ.
    3. In each step, define a new ftc by using the hypernetted chain relation:
      microfluidics textbook nanofluidics textbook Brian Kirby Cornell
      (H.32)

      Note that e1(r) in this case is the Lennard-Jones potential.

    4. In each step, define a new fdc by Fourier transforming ftc and fdc, applying the Fourier-transformed Ornstein-Zernike equation to get a new microfluidics textbook nanofluidics textbook Brian Kirby Cornell, and inverse Fourier-transforming microfluidics textbook nanofluidics textbook Brian Kirby Cornell to get a new fdc. We do this because the Fourier-transformed Ornstein-Zernike equation is much easier to deal with (the spatial integral becomes a product when Fourier-transformed):
      microfluidics textbook nanofluidics textbook Brian Kirby Cornell
      (H.33)

      Here k is the frequency variable and microfluidics textbook nanofluidics textbook Brian Kirby Cornell  is the Fourier transform of f. This can be rearranged to give
      microfluidics textbook nanofluidics textbook Brian Kirby Cornell
      (H.34)

      So we Fourier-transform ftc and fdc to get microfluidics textbook nanofluidics textbook Brian Kirby Cornell and microfluidics textbook nanofluidics textbook Brian Kirby Cornell, apply Equation H.34, and then transform microfluidics textbook nanofluidics textbook Brian Kirby Cornell back.

    5. The two previous steps are repeated until the solutions for ftc and fdc are no longer changing. Some attention to numerical stability is needed, especially if ρ* is high and T * is low.

    Plot your results for nine cases as follows: three values of ρ* (0.1, 0.4, 0.8) and three values of T * (0.5, 1.0, 1.5).

  2. Given the results from Exercise H.1, calculate the potential of mean force that atoms see in this case. Plot the emf for the nine cases from Exercise H.1.
  3. Calculate the magnitude of the dipole moment (in Debye) for a water molecule given the geometry of the SPC model.

  4. Consider hard spheres of radius a.
    1. What is the closest approach of the centers of two spheres?
    2. What is the doublet potential for the interaction between the two spheres?
    3. Calculate the excluded volume. How does the excluded volume compare with the volume of one of the spheres?
  5. Show that the nonintegrability of the Coulomb pair potential guarantees that physical systems must be overall electroneutral.
  6. Using the equation for monopole interaction potentials, explain why sodium chloride might be expected to be a crystalline solid when dry but dissolves when exposed to water.

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Jump To: [Kinematics] [Couette/Poiseuille Flow] [Fluid Circuits] [Mixing] [Electrodynamics] [Electroosmosis] [Potential Flow] [Stokes Flow] [Debye Layer] [Zeta Potential] [Species Transport] [Separations] [Particle Electrophoresis] [DNA] [Nanofluidics] [Induced-Charge Effects] [DEP] [Solution Chemistry]

Copyright Brian J. Kirby. Please contact Prof. Kirby here with questions or corrections. This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.

This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby here. Click here for the most recent version of the errata for the print version.


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