1. NC STATE UNIVERSITY CHE597B / CH795N Multi-Scale Modeling of Fluids and Soft Matter Instructors: Stefan Franzen and Keith E. Gubbins Lecture 7: Structure and Distribution Functions

2. NC STATE UNIVERSITY Radial Distribution Function Distribution and correlation functions are useful because they tell us the molecular structure of the material The radial distribution function, g(r), measures the probability of finding 2 molecules a distance r apart. (Figure taken from Tester and Modell, Thermodynamics and its applications, 3 rd edition, Prentice Hall, New Jersey (1997)

3. NC STATE UNIVERSITY Radial Distribution Function: Fluids Example: homogeneous, isotropic fluid (from Tester and Modell, Thermodynamics and its applications, 3 rd edition, Prentice Hall, New Jersey (1997) Radial distribution functions for gas and liquid phases of water, and their relationship to molecular-level visualizations

4. NC STATE UNIVERSITY Radial Distribution Function: Crystals Example: crystalline solid (from Tester and Modell, Thermodynamics and its applications, 3 rd edition, Prentice Hall, New Jersey (1997) Radial distribution function for a solid phase of water, and its relationship to molecular-level visualizations

5. NC STATE UNIVERSITY Anisotropic Liquids: Liquid Crystals Example: homogeneous, anisotropic fluid → liquid crystals (adapted from Leach, Molecular modelling, principles and applications, Longman, London (1996)

6. NC STATE UNIVERSITY - The distinguishing characteristic of the liquid crystalline state is the tendency of the molecules (mesogens) to point along a common axis, called the director. - This is in contrast to molecules in the isotropic liquid phase, which have no intrinsic order. - In the solid state, molecules are highly ordered and have little translational freedom. - The characteristic orientational order of the liquid crystal state is between the traditional solid and liquid phases, and this is the origin of the term mesogenic state, used synonymously with liquid crystal state. Note the average alignment of the molecules for each phase in the following diagrams. Liquid Crystals Example: homogeneous, anisotropic fluid → liquid crystals: (taken from http://plc.cwru.edu/tutorial/enhanced/files/lindex.html)

7. NC STATE UNIVERSITY Liquid Crystals Example: homogeneous, anisotropic fluid → liquid crystals (taken from http://plc.cwru.edu/tutorial/enhanced/files/lindex.html) nematic: no long-range translational order, long-range orientational order smectic: long-range translational order (one or two dimensions), long-range orientational order (taken from F. Barmes, PhD Thesis, Sheffield Hallam University, June 2003)

8. NC STATE UNIVERSITY Liquid Crystals: Smectic Phases Example: homogeneous, anisotropic fluid → liquid crystals (taken from F. Barmes, PhD Thesis, Sheffield Hallam University, June 2003)

9. NC STATE UNIVERSITY Three-Body Correlations Example: 3-body correlations (e.g., allotropic forms of carbon) (taken from X. Bourrat, in Sciences of carbon materials, ed. H. Marsh and F. Rodríguez-Reinoso, Publicaciones de la Universidad de Alicante (2000)

10. NC STATE UNIVERSITY Graphite Example: 3-body correlations (e.g., allotropic forms of carbon) (from T. J. Bandosz et al., in Chemistry and physics of carbon vol. 28, ed. L. R. Radovic, Marcel Dekker, New York (2003)

11. NC STATE UNIVERSITY Scattering and Diffraction Experiments Diffraction experiment: SourceSample  Detector ’ -Source can be neutrons, x-rays, light, electrons, from reactor, accelerator, etc. - Static approximation: = ’ - Neutrons scatter off nuclei - x-ray scatter off electron cloud

12. NC STATE UNIVERSITY Structure Factor for Molecular Liquids References: K. E. Gubbins et al., Mol. Phys. 25, 1353 (1973); P. A. Egelstaff, An introduction to the liquid state, 2 nd edn., Clarendon Press, Oxford (1992) Properties of three kinds of radiation RadiationScattering center Size of scattering center relative to size of atom ElectromagneticElectron density1 ElectronsCharge density 1 for electrons 0 for nucleus NeutronsNucleus 0 (from Egelstaff, “An introduction to the liquid state”)

13. NC STATE UNIVERSITY Scattering and Diffraction Experiments SourceSample  Detector ’ -Source can be neutrons, x-rays, light, electrons, from reactor, accelerator, etc. - Static approximation: = ’ - Neutrons scatter off nuclei - x-ray scatter off electron cloud Theory: k 0 A i riri B  k’k’ A free particle at 0 would scatter a wave of amplitude b i (constant) Consider a particle i at r i. To get scattered amplitude, calculate phase difference Experiment:

14. NC STATE UNIVERSITY Scattering Cross-Section Path difference ( PD ) Phase difference Amplitude of wave scattered by i Total scattered amplitude for N scatterers and scattering cross section (intensity)(1) For neutrons, the sum over k is over all nuclei; for x-rays or light, over all electron clouds k 0 A i riri B  k’k’

15. NC STATE UNIVERSITY Structure Factor for Molecular Liquids We now consider neutron diffraction and write the sum over nuclei as a sum over molecules i and nuclei  within molecules or (2) b i  = “scattering length” for nucleus  in molecule i r i  j  = r i  – r j   vector from nucleus  in molecule i to nucleus  in j = average over grand canonical ensemble + average over nuclear isotope and spin states

16. NC STATE UNIVERSITY Structure Factor for Molecular Liquids We first carry out averaging over nuclear states: if i = j,  otherwise Carrying out the average over nuclear states in the first term: if i = j,  otherwise

17. NC STATE UNIVERSITY Structure Factor for Molecular Liquids (3) where incoherent c.s. (4) = coherent c.s. (5) where S(q) is the structure factor, (6)

18. NC STATE UNIVERSITY Structure Factor in Terms of Distribution Functions It is convenient to separate S(q) into intramolecular and intermolecular parts, due to scattering interference for two centers within a single molecule ( S intra ) and two centers in different molecules ( S inter ), respectively ci = center of molecule i We write r i  j  as. Substituting this in (6) and separating i = j and i ≠ j terms: ci cj R ij r ci  r cj  ii jj rijrij (7)

19. NC STATE UNIVERSITY Structure Factor in Terms of Distribution Functions where (8) (9) S inter is entirely due to intermolecular correlations, while S intra is solely due to intramolecular effects.

20. NC STATE UNIVERSITY Intramolecular Structure Factor If we assume rigid molecules, S intra is easily evaluated: Note: (10)

21. NC STATE UNIVERSITY Structure Factor in Terms of Distribution Functions To evaluate (9), we note that each term will give the same result on integration. Thus, (11) where

22. NC STATE UNIVERSITY Structure Factor in Terms of Distribution Functions Thus (12) This is often written as (13) where the “forward scattering” has been subtracted, (14)

23. NC STATE UNIVERSITY Limiting cases of S(q) For the atomic case, (13) reduces to (15) and (16) As q → 0, (17) where  = isothermal compressibility As q→ , (18)

24. NC STATE UNIVERSITY Specific and Generic Distribution Functions probability molecule 1 (or specific molecule) is in molecule 2 is in around while simultaneously around, and so on up to molecule N, which is is around specific N-body distribution function Since molecules of the same species are indistinguishable, we are more interested in the generic distribution function probability a molecule (any molecule) is in, another is in, …, and another is in

25. NC STATE UNIVERSITY Specific and Generic Distribution Functions The number of ways of assigning N identical molecules to N locations is N !, so (54) P and f are normalized differently

26. NC STATE UNIVERSITY If there are no correlations among the molecules, i.e., no intermolecular forces (e.g., ideal gas), P and f can be factored into 1-body distribution functions Specific and Generic Distribution Functions (55) where and are probability densities for finding a molecule in about, irrespective of where the other (N – 1) molecules are.

27. NC STATE UNIVERSITY Reduced Distribution Functions In practice, we rarely need the N-body distribution function. Mostly we will be concerned with the 1- and 2-body (and occasionally 3-body) distribution functions. The h-body distribution functions are defined by probability density for finding specific molecules 1, 2, … h in the configuration, irrespective of the positions and orientations of the other (N – h) molecules is proportional to the probability density for observing any set of h molecules in around, regardless of where the other (N – h) molecules are

28. NC STATE UNIVERSITY Reduced Distribution Functions or (56) Here, = the number of ways of choosing h molecules from N

29. NC STATE UNIVERSITY and are normalized according to: Reduced Distribution Functions (57)

30. NC STATE UNIVERSITY Correlation Functions For an ideal gas, the reduced distribution functions factor according to eqn. (55). For real fluids and solids we can write (58) where = the h-body correlation function For ideal gases,. For real substances the departure of from 1 is a direct measure of the correlation between molecules due to intermolecular forces

31. NC STATE UNIVERSITY Centers Correlation Function Sometimes (e.g. in theory of solutions, compressibility) we are only interested in the distribution functions and correlation functions for molecular centers, regardless of molecular orientations. Then where means an unweighted average over the molecular orientations. For the 2-body case (homogeneous) radial distribution function

32. NC STATE UNIVERSITY First Order Distribution Functions In general, (a) Isotropic, homogeneous fluid: is independent of both r 1 and  1 so (60) Here, = 4  (linear) = 8  2 (nonlinear)

33. NC STATE UNIVERSITY First Order Distribution Functions (b) Anisotropic, homogeneous fluid (e.g., nematic liquid crystal): depends on   but not r 1. In the case of the nematic, or for uniaxial molecule Define ; then

34. NC STATE UNIVERSITY First Order Distribution Functions (c) Anisotropic, inhomogeneous fluid (e.g., gas-liquid or liquid-solid interface): Interface lies in the xy plane. Fluid is inhomogeneous in the z direction. Also anisotropic as molecules will have a preferred alignment For linear molecules

35. NC STATE UNIVERSITY Two-Body Correlation Functions For an isotropic, homogeneous fluid,, so from (58): (61) where denotes an ensemble averaging over and, and is the Dirac delta function, and The   function form is usually used in the theory of radiation scattering In (61), (nonlinear) (linear)

36. NC STATE UNIVERSITY Two-Body Correlation Functions

37. NC STATE UNIVERSITY Site-Site Correlation Functions  are sites on molecules 1 and 2 probability density for finding  site in one molecule at distance r   from the  site in a different molecule

38. NC STATE UNIVERSITY Site-Site Correlation Functions (63) where the averaging is over all values of   and    keeping r  fixed The site-site correlation functions arise naturally in the theory of: - Radiation scattering - Dielectric constant - Kerr constant - Isothermal compressibility For other properties the site-site correlation functions arise only if the potential is of site-site type.

39. NC STATE UNIVERSITY Site-Site Correlation Functions

40. NC STATE UNIVERSITY

41. Site-Site Correlation Functions

42. NC STATE UNIVERSITY Discotic Molecules Example: homogeneous, anisotropic fluid → liquid crystals - Discotic molecules typically have a core composed of aromatic rings connected in a circular arrangement from which alkyl chains extend radially (see figure 1.4) (taken from F. Barmes, PhD Thesis, Sheffield Hallam University, June 2003)

43. NC STATE UNIVERSITY - Discotic molecules form discotic, nematic and columnar phases. - Several types of the latter exist (see figure 1.5) namely disordered, ordered and tilted, and for each of these there can be three column arrangements (hexagonal, rectangular and oblique) Motivation Example: homogeneous, anisotropic fluid → liquid crystals (taken from F. Barmes, PhD Thesis, Sheffield Hallam University, June 2003)

44. NC STATE UNIVERSITY Three Body Correlations Example: 3-body correlations (e.g., allotropic forms of carbon) (taken from X. Bourrat, in Sciences of carbon materials, ed. H. Marsh and F. Rodríguez- Reinoso, Publicaciones de la Universidad de Alicante (2000)

45. NC STATE UNIVERSITY Diamond Example: 3-body correlations (e.g., allotropic forms of carbon) (from T. J. Bandosz et al., in Chemistry and physics of carbon vol. 28, ed. L. R. Radovic, Marcel Dekker, New York (2003)

46. NC STATE UNIVERSITY Three Body Correlations Example: 3-body correlations (e.g., allotropic forms of carbon) (from T. J. Bandosz et al., in Chemistry and physics of carbon vol. 28, ed. L. R. Radovic, Marcel Dekker, New York (2003)

47. NC STATE UNIVERSITY Correlation Functions y – function. Occasionally we shall use the “y – function”, sometimes called the “cavity correlation function”, (59) In particular, for h = 2 We show later that for dilute gases (second virial regime),. This is the correlation between two isolated molecules, i.e., the direct correlation between molecules 1 and 2. Thus, y(12) can be thought of as the indirect correlation between 1 and 2, due to effects of molecules 3, 4, etc. y(12) arises in treating hard body molecules.

48. NC STATE UNIVERSITY First Order Distribution Functions

49. NC STATE UNIVERSITY Total and Direct Correlation Function: h and c The total correlation function is and tends to 0 as. It measures the total effect of molecule 1 on molecule 2, and is of longer range than u (12). The total correlation between molecules 1 and 2 can be thought of as the sum of a direct and an indirect part: Total correlation h(12) = direct correlation + c(12) (1 acting on 2 directly) indirect correlation (1 influences molecules 3, 4, etc., which in turn influence 2)

50. NC STATE UNIVERSITY Total and Direct Correlation Function: h and c The direct correlation function is defined by the Ornstein - Zernike (OZ) equation (1914): (62) To see the physical significance of the indirect part, use (62) to eliminate h under the integral: =

51. NC STATE UNIVERSITY Total and Direct Correlation Function: h and c For a dilute gas, the indirect parts vanish and