Fluid-substrate interactions; structure of fluids inside pores

2 Question: where are the goats?

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 Integral equation theories  Classical density functional theory  They have in common the introduction of intermolecular potential functions to describe the system 7

8  If a glass tube with a bore as small as the width of a hair is dipped into water then the liquid rises in the tube to a height greater than that at which stands outside. Rise is 3cm in a tube with a bore of 1mm. Molecular theory of capillarity, J. S. Rowlinson and B. Widom, Oxford, 1982

9 L F Liquid film Force needed to balance the tension in the 2-sided film is proportional to L F = 2  L Note that the work: F  x =  A A tension per unit length has the same units that a surface energy per unit area Tension/length Surface energy/area

10 plpl pgpg Laplace’s equation: p l – p g = 2  /R internal pressure (based on cohesive energy) first indication of molecular forces   sg =  ls +  lg cos  Young’s equation

11 k = cos  = (  sg –  ls )/  lg -1 < k < 1  k= -1  ls large and >0; liq. does not wet solid solid liquid weak liq-solid forces or strong liq-liq or sol-sol

12 if -1 < k < 0  ls not too large; Hg on glass  = ~140 o strong forces in the liquid if 0 < k <1 solid is unwetted or partially wetted solid is wetted if k = 1  = 0 o liquid wets completely (water/clean glass) water/glass forces >> water/water

13 l z one phase above and one phase below the plane (line) Helmholtz free energy

 An “external” potential Another fluid molecule (like or unlike) A solid “wall” Electric, magnetic fields… 14

 classical mechanics (Laplace, Young)  classical thermodynamics (Gibbs)  statistical mechanics 15

Statistical Mechanics of Fluids n A classical, isotropic, one-component, monoatomic fluid. n A closed system, for which N, V and T are constant (the Canonical Ensemble). Each particle i has a potential energy U i. n The probability of locating particle 1 at dr 1, etc. is n The probability that 1 is at dr 1 … and n is at dr n irrespective of the configuration of the other particles is n The probability that any particle is at dr 1 … and n is at dr n irrespective of the configuration of the other particles is

N-particle distribution function n If the distances between n particles increase the correlation between the particles decreases. n In the limit of |r i -r j |  the n-particle probability density can be factorized into the product of single-particle probability densities. Measures the extent to which the fluid structure deviates from randomness

Radial Distribution Function n In particular g (2) (r 1,r 2 ) is important since it can be measured via neutron or X-ray diffraction n g (2) (r 1,r 2 ) = g(r 12 ) = g(r)

Radial Distribution Function n g(r 12 ) = g(r) is known as the radial distribution function n it is the factor which multiplies the bulk density to give the local density around a particle If the medium is isotropic then 4  r 2  g(r)dr is the number of particles between r and r+dr around the central particle

Correlation Functions n Pair Correlation Function, h(r 12 ), is a measure of the total influence particle 1 has on particle 2 h(r 12 ) = g(r 12 ) - 1 n Direct Correlation Function, c(r 12 ), arises from the direct interactions between particle 1 and particle 2

Ornstein-Zernike (OZ) Equation n In 1914 Ornstein and Zernike proposed a division of h(r 12 ) into a direct and indirect part. n The former is c(r 12 ), direct two-body interactions. n The latter arises from interactions between particle 1 and a third particle which then interacts with particle 2 directly or indirectly via collisions with other particles. Averaged over all the positions of particle 3 and weighted by the density.

Thermodynamic Functions from g(r) n If you assume that the particles are acting through central pair forces (the total potential energy of the system is pairwise additive),, then you can calculate pressure, chemical potential, energy, etc. of the system. n For an isotropic fluid

Classical density functional theory The thermodynamic grand potential   T, V) is a functional of the one particle distribution function,  (r) So we have  =  (r)] n The equilibrium density profile is obtained by minimizing this functional

Intrinsic Helmholtz free energy Chemical potential External potential  (r) is the fluid number density at position r free energy functional inhomogeneous hard-sphere fluid

25   (r) is the pair distribution function mean field approximation for attractive forces

26 Repulsive part = ideal gas +excess part (Tarazona, Mol. Phys. 1984) ideal gas part is directly a functional of  r) excess part is a functional of a smoothed density: weighting function

27 Intrinsic Helmholtz free energy Chemical potential External potential  (r) is the fluid number density at position r

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30 Tan and Gubbins, JPC, 1992

31 enthalpy and volume differences between adsorbed phase and vapor phase measures the strength of the forces between adsorbate and substrate Q  Q heat of vaporization

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 Molecular simulations (Gibbs ensemble) Useful to determine phase equilibria, adsorption isotherms, heats of adsorption  Molecular simulations (Grand canonical ensemble) Useful to determine adsorption isotherms, heats of adsorption, density profiles  Classical density functional theory Useful to determine density profiles, pore size distribution, adsorption isotherms, heats of adsorption 33

34 T 1 = T 2 P 1 = P 2  1 =  2 Gibbs …which is the condition for phase coexistence in a one-component system. Given two phases 1 and 2, they will be at equilibrium in all points where:

35 gasliquid achieve equilibrium by coupling them

36 Overall system: NVT ensemble N = N 1 + N 2 V = V 1 + V 2 T 1 = T 2 distribute N 1 particles change the volume V 1 displace the particles

37 Particle displacement Volume change  equal P Particle exchange  equal μ 3 different kinds of trial moves:

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39 Panagiotopoulos, Mol. Phys., 1987

40 Panagiotopoulos, Mol. Phys., 1987

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