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• Many molecules are not spherical, but could be represented as consisting of connected spherical segments: chains
• Concept of chains of hard-spheres allows the development of EOS to represent polymers of complexes
(amines, alcohols, acids) .
• Statistical mechanics makes possible to model assemblies of spherical segments forming chains
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Hydrogen bonds and associating fluids
• Hydrogen bonds are intermolecular forces existing between a hydrogen atom and an electronegative atom such as oxygen, nitrogen or fluorine.
• The hydrogen bonding is considered as a chemical strength or chemical equivalent, whose intensity is several orders of magnitude greater than the physical forces and an order of magnitude lower than that of chemical bonds.
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Hydrogen bonds and associating fluids
• For example, the interaction energy of van der
Waals forces for methane is 0.6 kJ / mol , while that of the H- bond of water is 22.3 kJ / mol , and that of an OH chemical bond is 465 kJ / mol .
• H- bonds allow the formation of complex molecules and polymers, to which they impart specific behaviors.
• For example, in the absence of H- bonding, the boiling temperature of water at atmospheric pressure would be 80 °C instead of 100 °C, and its melting point - 110 °C instead of 0 °C.
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H- bonds in the water molecule
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H- bonds and associating fluids
• Associating fluids can be described as fluids having an ability to form H- bonds .
• Associating fluid molecules combine to form long chains of polymers or dimers.
• The intermolecular forces involved in these fluids are intermediate between the dispersion forces or weak electrostatic interactions , and the forces characteristic of chemical reactions forming the molecules.
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H- bonds and associating fluids
• Associating fluids exhibit one or more association sites, each of them being characterized by a potential placed near the perimeter of the molecule.
• Associating interactions depend on the distance and orientation of the molecules.
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H- bonds and associating fluids
• Example of spherical segments equipped with association sites A and
B.
The two spheres can form AB dimer bonds only if the distance and orientation of the sites are favorable :
1. molecules too far– no association
2. misdirection
3. association achieved
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Possible association schemes
1. two sites cannot associate with a third;
2. a site of the molecule i cannot associate simultaneously on two sites of j;
3. double association is not permitted
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1. a complex molecule is modeled as consisting of four segments of hard spheres;
2. the interaction forces between two remote molecules are represented;
3. an associating link between two sites links two chains.
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• The association function via a square well potential can be characterized by two values, the energy of association (depth of the well) and a parameter characterizing the volume of association
(bound to the well width).
• In addition to these values, the association scheme must be specified, that is to say the number of sites and their type.
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EOS based on associating fluid theory
• Based on these concepts, new theories derived from statistical mechanics lead to the development of the Statistical Associating Fluid
Theory (SAFT) or Cubic Plus Association (CPA)
EOS able to accurately represent the thermodynamic properties of fluids, be they associating or not, complex or polar molecules.
• These new equations of state can be used to model, with a small number of parameters, some refrigerants whose molecules are usually polar but not necessarily associating
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Results: liquid volume of water in mixtures water/R1233zd the CPA model, in dots and dashes is excellent thanks to the association term
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Water molecule association sites the CPA model considers that the water molecule has four association sites among which two electron acceptor protons C and D
(so-called scheme 4C)
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Results:R1233zd liquid volume in water/R1233zd mixtures
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• theory for mixtures in which molecular association occurs, and application to binary mixtures of components A and
B in which AB dimers are formed, but there is no AA or BB dimerization.
• Based on Wertheim theory of cluster expansion
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• Cluster expansion done in terms of two densities: the equilibrium monomer density and the (initial) overall number density. In so doing, we are guaranteed the correct low density limit.
• Assumption: the repulsive core interactions restrict the highly orientationally dependent H- bond to only dimer formation
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• The resulting equations are applicable to a pure fluid with short-range, highly orientationally dependent attractive forces and hard repulsive cores, such that only dimers can form. Chapman et al extended the theory to binary mixtures of components A and B
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• r a
= r a
(1) + r a
(2)
• overall number density of component a is the sum of the equilibrium density of monomers + the equilibrium density of dimers
• Helmholtz free energy of the associating mixture minus that of the reference mixture
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the monomer fraction results from the solution of these equations
H-bond interaction
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the integration is performed over all possible orientations and molecular separations
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Components A and B have the same hard sphere diameter s and both have off-center point charge dipoles of equal magnitude and opposite sign.
Like pairs of molecules interact as hard spheres. Unlike pairs interact as hard spheres with a sum of coulombic interactions.
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The coulombic interactions are turned off if the dipole center-dipole center RDD distance is larger than a given cutoff (rC). If s
< 0.55 r c only dimers form.
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Calculated fraction of monomers and D U of mixing
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Calculated fraction of monomers as a function of mole fraction and reduced dipole moment
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Effect of location of dipole offcenter on excess properties
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Comparison of excess enthalpy with experiments
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Comparison of excess Gibbs free energy with experiments
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Comparison of equilibrium constant with ideal solution
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Basics for the SAFT equation of state residual Helmholtz free energy segment-segment interactions (LJ) covalent segment-segment bonds forming a chain specific interactions, i.e H-bonding
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The association term for pure components sum over all the associating sites mole fraction of molecules not bonded at A
# association sites in each molecule
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the integral can be approximated as: that depends on the segment diameter d and in the rdf of the segment g(d)
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approximating the segments as HS:
Carnahan-Starling with m= # of segments in a chain
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The association term for mixtures linear with respect to mole fractions mole fraction of i molecules not bonded at site A
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molar density r j association strength
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Chain term change in the Helmholtz free energy due to bonding m i
: # of spherical segments in molecule i
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Compressibility factor for a pure HS fluid with one attractive site
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Compressibility factor for pure chains of various lengths
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Liquid density and vapor pressure of n-octane
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Here the reference fluid is the LJ potential
Simulations: association interactions:
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association interactions: model carboxylic acids where dimers form
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Helmholtz free energy of the molecular chain g(r) is estimated from: this model has been shown to represent very well properties of alcohols and water
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More recent work on SAFT important in proteins, carboxylic acids, glycols, ethers
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Potential Model
Chains: flexible linear chains of tangentially bonded spheres with the end segments having 2 H-bonding association sites: one e acceptor and one e donor.
Water: single sphere with 4 association sites: 2e donors and 2 e acceptor sites
Effective pair potential between unbonded segments:
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Theory
--a mixture of m chain segments and one solvent segment
--chains are formed by forcing segments to bond at specific sites
--chain ending segments have multiple association sites and one chain forming site; the chain middle segments have 2 chain-forming sites; the solvent segment has various association sites
--specifying which sites can bond, chains of defined length can be formed
--the Helmholtz free energy is expressed as a function of the density of the segments, where X ring is the density of “rings” (intramolecular associations)
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• A fluid can be non-homogeneous of an external field (could be electric, magnetic, or the presence of a surface
(wall) that modifies the structure of the fluid in its vicinity
• In a homogenous fluid (no external field) we can use SAFT as explained before
• In a non-homogeneous fluid, we use a different approach, known as molecular density functional fluid
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Molecular density functional theory
• At equilibrium, there is a minimum in the grand potential:
• The grand potential W is related to A external potential felt by segment i at r bulk chemical potential of segment i segment i density at position r
Equations can be solved for the equilibrium density profile given an expression for A
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Expression for the intrinsic A
• Given our model,
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Functional derivatives
Once these equations are solved we get the equilibrium density profile r (r)
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• There are only two type of association sites:
O and H and only unlike associations are allowed: significant intramolecular interaction at low Ts; enthalpy of bond formation overcomes entropy penalty of forming a ring
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effect of intramolecular association dotted: no intramolecular association solid: w/intramolecular association symbols: simulations
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Inhomogeneous system (DFT) e*
= 2
(higher T) e*
= 6
(lower T) molecules at a hydrophobic surface modeled as a smooth hard wall.
solvent: blue associating end segments: red non-associating middle segments: green
At high T the chains & the solvent wet the surface because H-bonding at the bulk decreases
At low T, moderate wetting, middle segments associate close but not at the surface
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segment density profile solvent: blue associating end segments: red non-associating middle segments: green
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