1. --Experimental determinations of radial distribution functions --Potential of Mean Force 1

2. Radial distribution function from experiments A diffraction experiment uses radiation of a wavelength < the molecular size For example X-Ray scattering ( = 0.01 to 10 Å) or neutron beams ( = 1 to 10 Å ) How does it work? Electrons of an atom or molecule do the scattering in X-Ray (needs an X-Ray generator) ; while in neutron scattering the nucleus of the atom is the scattering center of neutrons (needs a neutron beam source) 2

3. How does it work? A liquid is subjected to a monochromatic beam (fixed wavelength) that has been collimated so all rays are parallel and in phase 3

4. Reflection scattering experiment The scattered radiation is measured as a function of the scattered angle 4

5. Scattering is defined by the vector s In X-Ray, the electrons are the scattering sites, and the scattering cross section is related to the Fourier transform of the electron density: can be calculated and it is known for most atoms 5

6. the experiment measures the intensity of the scattered radiation at each scattered angle difference of the path length of the two scattered rays is given by a distance x2 –x1 amplitude of scattered radiation at the angle that corresponds to s 6

7. Intensity of the scattered radiation N is the # of scatters in the target region of the liquid that is subjected to the beam (not known); so is normalized to the scattering of atoms without interference 7

8. the final diffraction equation is 8 or total structure function

9. Intensity of scattered radiation for liquid Ar at T= -125 o C and 0.982 g/cc 9 90% confidence interval

10. Structure function H(s) from the X-ray diffraction data 10

11. Radial distribution function computed from the experimental H(s) 11 uncertainty shown by the hatched region

12. Potential of Mean Force 12 at low densities --w is the effective potential between two particles modified by the presence of all other particles; --range of w (r12) much longer than that of u(r12); --w12 is a function of density and temperature (unlike u(r12))

13. more formal expression of the PMF 13 force between two atoms in vacuum as they move apart in a fluid

14. relation of PMF to g(r) 14

15. Relation of PMF to g(r) 15

16. physical interpretation of PMF 16 since w(r 12,T,  ) is the integral of the force over the distance, it is also the work done to bring two particles together from an infinite separation in a dense fluid to a separation r. Since this work is done at N, V, and T then w(r) is the Helmholtz free energy of the process

17. g(r) and w(r) for the Hard spheres fluid: --Range of PMF is longer than range of u 17

18. -- Note that the PMF shows attractive potential when the HS is purely repulsive, why? 18

19. 2 atoms sufficiently close to each other; on each collision with surrounding atoms, the force is indicated by arrows; there is a region shielded from collisions with other atoms. Due to this imbalance there is a net attractive force between the two atoms at these distances 19

20. Behavior of polymers, colloids, or proteins in solution one possibility of obtaining the rdf for these systems (besides molecular simulations) is to develop an expression for he PMF and then fit the parameters to experimental data, for example osmotic P or precipitation data Since the solvent molecule is small compared to the macromolecules, solvent is treated as a continuum 20

21. Example: precipitation of globular proteins in aqueous solution induced by a polymer PMF model: 21 Attractive term: Van der Waals, for example 12-6 LJ: H is called the Hamaker constant

22. Example: precipitation of globular proteins in aqueous solution induced by a polymer For the electrostatic term: charge-charge, charge-dipole, charge- induced dipole if the molecules are charged, but modified by the presence of the solvent: 22

23. 23 Hamaker electrostatic pmf for this set of parameters, the potential is attractive for r/  > 1.5

24. 24 for this set of parameters, the potential is always repulsive

25. 25 for this set of parameters, the potential is attractive for r/  > 1.1

26. 26 for this set of parameters, the potential is attractive for r/  > 1.56

27. Example: precipitation of globular proteins in aqueous solution induced by a polymer An attractive force arises due to the exclusion fo the polymer from the region between two macromolecules; this is added as an osmotic term: the osmotic term depends on the size of the polymer 27

28. Osmotic pressure and PMF for colloidal and protein solutions The Virial EOS was derived for a dilute concentration of atoms in vapor phase, i.e., space between atoms is vacuum. Another Virial EOS can be derived considering the solvent as a continuum fluid where the molecules are floating. The solvent is chracterized by T, P, dielectric constant, and chemical potential . 28

29. Osmotic pressure and PMF for colloidal and protein solutions Considering the addition of solute to the solution at constant T and chemical potential of the solvent, . As solute is added, the equilibrium pressure above the solution increases to keep the chemical potential constant (that is changing due to the addition of the solute) 29

30. Osmotic pressure and PMF for colloidal and protein solutions 30 At moderate solute concentrations,

31. Osmotic pressure and PMF for colloidal and protein solutions It is important to understand the difference of the B2 in gases vs. B2 for solutions; in the first case B2 depends on u(r) but B2 of solutions depends on w(r, , T) The B2 values can be obtained from osmotic pressure measurements. If the values are negative (positive), the net force is attractive (repulsive). The sign of the 2 nd osmotic Virial corfficient gives hints regarding whether the protein is going to precipitate (crystallize) or not. 31