8.2 Methodology
8.2.3 Extended Dynamically Corrected Transition State
Several publications showed that the self-diffusion of molecules adsorbed in zeolite materials can be accurately computed by the approach of extended dy-namically corrected transition state theory109,110,116,117 (extended DCTST).
As the self-diffusion coefficient, DS, is, in general, a function of loading for a given adsorbate–zeolite system at a given temperature, extended DCTST provides a valuable means to discuss the loading dependence of DS on the basis of those two factors that comprise the theory:
1. Free-energy contribution (static property). Usually, one of the Cartesian directions is identified as the transport coordinate (histor-ically: reaction coordinate), q, that measures the progress of a jump event from one cage, qA, through the window (i.e., transition state
‡), q‡, toward the target cage, qB; note that q ≡ z in the following.
Free-energy profiles along the transport coordinate, F(q), are calcu-lated from residence histograms of a tagged adsorbate molecule, as obtained from the simulations. Finally, the relative conditional prob-ability, P(q‡)|q(0)<q‡, to find the molecule on top of the barrier (i.e., in the window region) provided that is has been located in cage A before the hop attempt [i.e.,q(0)< q‡] can be computed according to:
P(
where the integration limits highlight that cage A is defined by the encompassing transition states ’‡ and ‡ (cf., Section 4.4).
2. Flux through dividing surface (dynamic property). The idealized TST flux through the dividing surface at q‡ is approximated by kinetic gas
8.2 Methodology 125
theory so that the jump frequency from A to B, kTSTA→B, reads:
kA→BTST =
+kBT 2πm ·P(
q‡)''
q(0)<q‡. (8.2)
kB is Boltzmann’s constant, T the absolute temperature, and m the mass of the bead(s), or atom(s) involved in the transport coordinate (in this chapter: center-of-mass of entire molecule). Spurious crossings are accounted for by computing the reactive flux correlation function110–112 (RFCF),κ(t): whereq(0) and ˙q(0) denote the initial position and velocity of the mo-lecule, respectively. H is the Heaviside function [H(x) = 1 for x≥0 and H(x) = 0 otherwise] and #δ(x) the Dirac delta function [#δ=∞ for x= 0 and#δ= 0 otherwise]. Starting configurations for the RFCFs were generated using an MD-based approach (BOLAS122 and EPS123). Oth-erwise, the procedure for the RFCF simulations is the same as in Ref-erence 110. The plateau of the transient RFCF,κ, yields the (velocity-wise averaged) fraction of hop attempts from the barrier toward target cage B. It is referred to as the transmission coefficient or dynamical correction factor and may attain values between zero and unity only.
The methodology described above is also known as the Bennett–Chandler approach.111,112 It has often been used in order to understand diffusion in nanopores at the limit of infinite dilution; see, for example, the numerous references in Reference 110. The key to extending DCTST to diffusion at finite loadings is the computation of effective hopping rates of a single tagged molecule. Surrounding adsorbate molecules are viewed as an additional ex-ternal field to the tagged molecule, and naturally fluctuating cage occupan-cies are crucial to the hopping rate computed.116 In fact, this viewpoint is similar to what Chandler112 anticipated for the isomerization of n-butane:
the rate constant would strongly depend on the solvent density that exerts an external field to n-butane.
The self-diffusion coefficient, DS, can be calculated on basis of DCTST by:109,110
DS= 1
2d·κ·kTSTA→B·l2cage, (8.4) whereddenotes the dimensionality of the pore system (here: d= 1) and lcage
the separation of hopping sites in the zeolite structure (here: lcage= 4.242 ˚A).
126 8 In-Depth Analysis of Interface Dynamics Transition state theory has also been used to characterize equilibrium trans-port at crystal surfaces.9,182 However, the dynamic correction (spurious cross-ings) in such crystal surface transport situations have so-far been neglected.
Critical Crystal Thickness
In order to assess the relative importance of surface transport effects, Arya et al. pointed out that any such assessment “must include an estimate of the critical crystal dimension beyond which the barrier resistance becomes insignificant”.172They, therefore, introduced a critical ratio of the two lengths involved, that is, of the pore length to the length of the pore exit region, and, on the basis of a simple activated transport model, provided a good estimate for this ratio. In a similar manner, and also using a simplified equilibrium model, Newsome and Sholl8,176defined a critical crystal thickness, δcrit. Using the example of adsorption, the model of the authors finally reduced to the following equation:176
where Rads and Rintra denote the adsorption resistance and intracrystalline transport resistance, respectively, DC is the corrected diffusivity, α the sur-face permeability rating the sursur-face transport, δ the crystal thickness, and cfeed is the concentration inside the zeolite in equilibrium withpfeed, the latter of which being the gas-phase pressure outside the crystal. Note that News-ome and Sholl defined α as the derivative of the flux density with respect to pressure, α≡dj/dp.8
This work exploits the computation of DCTST fluxes for a relative assess-ment of surface transport effects. The molar flux density in terms of DCTST is given by:
The computation of concentration profiles, c(q), takes place in parallel to the computation of residence histograms. The two quantities and the free-energy profile stand in a direct relationship {P(q) ∝ c(q) ∝ exp[−βF(q)]}. Note that the simulations of this work showed that computing TST fluxes (i.e., Equation 8.6 and setting κ= 1) is exactly equivalent to computing one-way fluxes.8 This indicates that the temperature distributions along the transport coordinate were homogeneous enough to apply kinetic gas theory for estimating the ideal flux through the dividing surfaces.
8.2 Methodology 127 Because a transport resistance is inversely proportional to the corres-ponding flux, the definition of a critical crystal thickness used in the present work can, in analogy to the work by Newsome and Sholl,8 be interpreted as equating the transport resistance at the external surface (or rather in the boundary layer between gas and zeolite) with the intracrystalline transport resistance:
δcrit= 2·lcage· jzeol‡ 11/jsurf‡ + 1/jgas‡
2−1. (8.7)
In essence, this length is the minimal crystal dimension in tracer-exchange experiments for which the surface transport resistance can, in a good ap-proximation, be neglected in the evaluation model, considering the usually encountered high experimental measurement uncertainties (cf., Chapter 7).
Most previous studies viewed the transport through the boundary layer as a single-step process. That is, a molecule that is initially located in the last zeolite cage hops out onto the external surface and is then considered to have left the zeolite. Equation 8.7 highlights however that a two-step boundary permeation process forms the basis of the critical crystal thickness of this work. That is, this work considers also the slow hopping from the external surface adsorption layer to the gas phase. Therefore, a single-step critical thickness, δ1stepcrit , neglecting the second step is calculated along with the two-step δcrit from Equation 8.7 to rate the importance of the second step:
δ1stepcrit = 2·lcage· jzeol‡
jsurf‡ . (8.8)
128 8 In-Depth Analysis of Interface Dynamics