ch10 - Advanced Topics in Equilibrium Statistical Mechanics...

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Advanced Topics in Equilibrium Statistical Mechanics Glenn Fredrickson 10. Electrolyte and Colloidal Solutions A. Systems and Models Thus far, we have restricted our attention to the equilibrium SM of simple ﬂuids and polymers. However, there are many other systems of general interest to chemical engineers: colloidal solutions: 1-1 electrolyte solution: polyelectrolyte solution: We model these systems a bit diFerently, depending on the lengthscales involved. In many cases, it is convenient and accurate to replace the solvent by a continuum , so that we do not have the added computational expense of describing the solvent structuring around the solute . Let’s start with the simplest case of a solution of neutral particles . Imagine that we have a solution of non-charged, neutral molecules or colloids, dissolved in a low molecular weight solvent: 1

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If such systems are dilute in solute , then we might think about using a virial expansion to address the thermodynamic properties. The eFective interactions between solute molecules would correspond to the potential of mean force w ( r ) between two solute molecules separated by some distance r in the pure solvent . Indeed there is a theory known as the McMillan-Mayer theory that makes a rigorous connection between dilute gases and dilute nonelectrolyte solutions: Gases p k B T = ρ + X n =2 B n ( T ) ρ n EOS u ( r ) pair potential in vacuum Dilute Non-electrolyte solutions Π k B T = ρ + X n =2 B n ( T,µ ) ρ n EOS w ( r j µ ): potential of mean force between two solutes at separation r in pure solvent at chemical potential µ . ρ : solute concentration # density) Π: osmotic pressure of the solution The B n are related to w in precisely the same way that B n are related to u , e.g B 2 ( )= 1 2 Z d r [ e βw ( r,µ ) 1] Thus, dilute non-electrolyte solutions are no more diﬃcult to deal with than gases, provided we can make good models of w ( r ; µ ). ±or, large ( 1 µ m) colloidal particles, it is sensible to neglect the granularity of the solvent and model w by a simple hard sphere potential. ±or “ sterically stabilized ” colloids with grafted polymers, we might want to employ a softer repulsive potential with a longer range starting at 2 R 2 . ±inally, for molecular solutes , one might do a MC or MD simulation of 2 solutes in the solvent to explore w ( r ; µ ). Next let’s consider a simple electrolyte solution , e.g. NaCl in water—a so- called 1-1 electrolyte. The crudest description of such a solution is the so-called primitive model : 2
Treat solvent as a continuum with dielectric constant ² . Let “1” be the cation (+) species and “2” be the anion( ) species. Model their interactios as the sum of a hard sphere and coulomb potential: e.g., w 12 ( r, µ )= ½ ,r < 1 2 ( σ 1 + σ 2 ) q 1 ,q 2 ²r ,r> 1 2 ( σ 1 + σ 2 ) Here, σ 1 and σ 2 are the cation and anion hard sphere diameters; q 1 and q 2 are the charges ( q 1 =+ e , q 2 = e for NaCl).

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ch10 - Advanced Topics in Equilibrium Statistical Mechanics...

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