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Thermodynamics of Associating Fluids

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Chains of molecules

• 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 association2. misdirection3. 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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Association limits1. 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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Chains of molecules

• 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 volumein water/R1233zd mixtures

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Origins of SAFT

• 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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Basics of Wertheim’s theory

• 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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Basics of Wertheim’s theory

• 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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Basic model equations• ra = ra(1) + ra(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 solutionof these equations

H-bond interaction

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Basic model equations

the integration is performedover all possible orientationsand molecular separations

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Potential model

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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Potential model

The coulombic interactions are turned off if the dipole center-dipole center RDD distanceis larger than a given cutoff (rC). If s < 0.55 rc only dimers form.

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Calculated fraction of monomers and DU of mixing

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Calculated fraction of monomers as a function of mole fraction and reduced dipole moment

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Excess enthalpy of mixing

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Effect of location of dipole off-center 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

# association sitesin each moleculemole fraction of molecules not bonded at A

sum over all the associating sites

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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 rj

association strength

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Chain term

change in the Helmholtz free energy due to bonding

mi : # 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 carboxylicacids where dimersform

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SAFT theory

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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; thesolvent 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 Xring is the density of “rings” (intramolecular associations)

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Non-homogeneous fluids

• 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

segment i density at position rbulk chemical potential of segment i

external potential felt by segment i at 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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Some results• There are only two type of association

sites: O and H and only unlike associations are allowed:

significant intramolecular interactionat low Ts; enthalpy of bond formationovercomes entropy penalty of forminga 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)

molecules at a hydrophobic surfacemodeled as a smooth hard wall.

solvent: blue

associating end segments: red

non-associating middle segments: green

At high T the chains & the solventwet the surface because H-bonding at thebulk decreases

At low T, moderate wetting, middle segmentsassociate close but not at the surface

*e = 2(higher T)

*e = 6(lower T)

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Inhomogeneous system (DFT)

segment density profile

solvent: blueassociating end segments: rednon-associating middle segments: green