Surface Restricted Diffusion in Chromatography

As mentioned above, the contribution of surface diffusion for an analyte is in the form of a molecular diffusion coefficient and is related to the total variance due to longitudinal molecular diffusion (!_!",! _{!) by:}6

!!"! ,! =!2!!!!!!+2!!!!!′!! Equation 3-4 where ! is the obstruction factor, related to the tortuosity of the path hindering diffusion, and !!is the solute diffusion coefficient. The subscripts ! and ! are used to designate contributions from the mobile and stationary phases. !_!",!! _{is the sum of the parallel contributions of diffusion in the mobile}

and stationary phases. The contributions from the mobile and stationary phases will sum to an overall effective diffusion coefficient that a solute band is experiencing (!_!"") during a time (!), simplifying Equation 3-4.7

!!",!! =2!!""! Equation 3-5

diffusion to the effective diffusion coefficient.7 !!""= !_!!_! 1+!′+ !′!_!!_! 1+!′ Equation 3-6

Using Equation 3-6, if !′ increases, then the contribution from !!!! begins to dominate !!"" thus illustrating how it could become an issue if surface diffusion is high under gradient initial conditions.

The most widely accepted model for surface diffusion in reversed phase chromatography is the surface restricted diffusion model, developed by Miyabe and Guiochon.8,9 This model assumes that stationary phase diffusion is analogous to molecular diffusion in the mobile phase, but now occurring in the potential field of the stationary phase. The process of diffusion through the mobile phase is considered an activated process.8

!!=!!,!!exp!( −!_!

!" ) Equation 3-7

Here !!,! is the frequency factor of molecular diffusion in the mobile phase and !! is the activation energy of molecular diffusion. !_! is related to the process of hole making in the mobile phase, which is required for diffusion. !! is correlated to the evaporation energy of the solvent system. !! can also be described in terms of absolute rate theory:10

!_!=!!_!"##!!_!"## Equation 3-8

where !!"## is the distance between two equilibrium positions and !!"## is the rate constant of diffusion.8 !_!"## = 1 !_!!/! ! !!! 2!!_! exp!( −!! !" ) Equation 3-9 Here, !! is the free volume of the solvent, which is related to the solvent molar volume and

evaporation energy, !_! is the Boltzmann constant, and !_! is the molecular weight of the solute. The combination of Equations 3-7, 3-8, and 3-9 allow estimation of !!,! by:8

!!,!!= !_!"##! !_!!/! ! !_!! 2!!! !/! Equation 3-10

The values of !!"##, !!, and !! are dependent on the solute and solvent system used, but values of

!!,! generally fall in the range of 3 x 10-3 to 2 x 10-2 cm2/s for reversed phase LC.

As mentioned previously, the surface restricted diffusion model assumes that stationary diffusion is similar to mobile phase diffusion, but now in the potential field of the surface.9

!!=!!+!β(−!!")! Equation 3-11 Here !! is the required activation energy for surface diffusion and ! is some fraction of the isosteric

heat of adsorption (!!"). Equation 3-11 provides insight into the energetic processes that must occur

for stationary phase diffusion to occur.11 _{First a hole must be made for the solute, assumed here to be}

equivalent to a hole making process in the mobile phase (!!), but in the potential field of the surface. Then the solute must break a certain energy barrier related to its retention (β(−!_!")), in order to hop into the hole. The process of hole making and hopping steps in the potential field of adsorption is at the core of the surface restricted diffusion model and as such, !! takes on a similar form as Equation

3-7.8

!_!=!_!,!exp −!!

!" =!!,!exp

−(!!+!β −!!" )

!" Equation 3-12

The values for !_!,! (surface diffusion frequency factor) and !_!,! are expected to be similar, but

!!"##, !!, and !! cannot be estimated accurately in the potential field of the surface thus making determination of !_!,! difficult.

By dividing Equation 3-12 by 3-7, the ratio of !!

!! can be determined. 8 !! !_!= !!,! !_!,! exp !!" !" ! Equation 3-13 Using the van’t Hoff equation, the isosteric heat of adsorption can be related to the partition

equilibrium constant (!) by:

!=!!exp

−!_!"

!" Equation 3-14

Equations 3-13 and 3-14 yields the relationships between !! !! and !. !! !_! = !!,! !_!,! !! ! ! Equation 3-15 Since !∝ !′, !!

!! will decrease with increasing retention of a solute. This trend is important as it

should limit the amount of broadening that can occur over the course of a gradient separation since analytes tend to have a high !′ until the mobile phase reaches a strength needed to elute the peak.

The surface restricted diffusion model has been shown to hold true for a number of reversed phase chromatography systems including C18 bonded particles as well as monolithic columns.9,11,12

To the best of my knowledge, the study of surface diffusion on porous graphitic carbon stationary phases has not been previously investigated. As our interest is to utilize long capillary columns with long gradient run times, it is necessary for us to understand the surface diffusion properties of the PGC particles.

3.2 Experimental Methods 3.2.1 Chemicals

Methanol (MeOH, HPLC grade), acetone (HPLC grade), potassium chloride (KCl) and L- ascorbic acid were purchased from Fisher Scientific (Fair Lawn, NJ). Deionized water was obtained using a Barnstead Nanopure ultrapure water system (Dubuque, IA). Formic acid (FA), mandelic acid (MA), hippuric acid (HA), 2-methylhippuric acid (2-MHA), 3-methylhippuric acid (3-MHA), 4- methylhippuric acid (4-MHA), potassium ferrocyanide, and potassium ferricyanide were purchased from Sigma Aldrich (St. Louis, MO).