1 / 8

Structure Factors in Particulate Systems: Ornstein-Zernike Equation & Approximations

This chapter explores the fundamental concepts of structure factors in particulate systems, focusing on the Ornstein-Zernike (OZ) equation, Percus-Yevick (PY) approximation, and Mean Spherical Approximation (MSA). It delves into the Fourier transforms of inter-particle structure factors and interaction potentials such as hard spheres and screened Coulomb. By implementing closure relations and evaluating scattering cross-sections, we derive essential results that inform modeling in concentrated systems. Understanding these approximations is crucial for predicting behavior in various physical contexts.

yank
Download Presentation

Structure Factors in Particulate Systems: Ornstein-Zernike Equation & Approximations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 32 – STRUCTURE FACTORS FOR PARTICULATE SYSTEMS 32:1. THE ORNSTEIN-ZERNIKE EQUATION 32:2. THE PERCUS-YEVICK APPROXIMATION 32:3. THE MEAN SPHERICAL APPROXIMATION

  2. h(r) c(r) Hard Sphere Screened Coulomb U(r) Square Well r 32:1. THE ORNSTEIN-ZERNIKE EQUATION The OZ equation: Take Fourier transform: Inter-particle structure factor: Interaction potentials:

  3. Hard Sphere U(r) r 32:2. THE PERCUS-YEVICK APPROXIMATION PY closure relation: Hard-sphere interaction potential: PY solution of the OZ equation: . Parameters: Result:

  4. RESULT Inter-particle structure factor: Scattering cross section:

  5. FORM FACTOR P(Q) AND STRUCTURE FACTOR SI(Q)

  6. Screened Coulomb U(r) r 32:3. THE MEAN SPHERICAL APPROXIMATION Screened Coulomb interaction potential: MSA closure relation: MSA solution of the OZ equation: Take Fourier transform… to obtain tedious result… not reproduced here

  7. RESULT Inter-particle structure factor: Scattering cross section:

  8. COMMENTS -- The OZ equation along with one of the closure relations (the PY or the MSA for charged systems) is used to calculate the structure factor for scattering particles. -- Structure factors are needed to mode the cross section for concentrated systems.

More Related