Effect of Adsorption on Solute Dispersion: A Microscopic Stochastic

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Effect of Adsorption on Solute Dispersion: A Microscopic Stochastic Approach Dzmitry Hlushkou,†,‡ Fabrice Gritti,§ Georges Guiochon,§ Andreas Seidel-Morgenstern,‡ and Ulrich Tallarek*,† †

Department of Chemistry, Philipps-Universität Marburg, Hans-Meerwein-Strasse, 35032 Marburg, Germany Max-Planck-Institut für Dynamik komplexer technischer Systeme, Sandtorstrasse 1, 39106 Magdeburg, Germany § Department of Chemistry, The University of Tennessee, Knoxville, Tennessee 37996−1600, United States ‡

ABSTRACT: We report on results obtained with a microscopic modeling approach to Taylor−Aris dispersion in a tube coupled with adsorption−desorption processes at its inner surface. The retention factor of an adsorbed solute is constructed by independent adjustment of the adsorption probability and mean adsorption sojourn time. The presented three-dimensional modeling approach can realize any microscopic model of the adsorption kinetics based on a distribution of adsorption sojourn times expressed in analytical or numerical form. We address the impact of retention factor, adsorption probability, and distribution function for adsorption sojourn times on solute dispersion depending on the average flow velocity. The approach is general and validated at all stages (no sorption; sorption with fast interfacial mass transfer; sorption with slow interfacial mass transfer) using available analytical results for transport in Poiseuille flow through simple geometries. Our results demonstrate that the distribution function for adsorption sojourn times is a key parameter affecting dispersion and show that models of advection−diffusion−sorption cannot describe mass transport without specifying microscopic details of the sorption process. In contrast to previous one-dimensional stochastic models, the presented simulation approach can be applied as well to study systems where diffusion is a rate-controlling process for adsorption.

L

solid−liquid interface. If a solute can be adsorbed at the channel wall, this leads to additional solute spreading, because the adsorbed solute molecules cannot be transported with the solvent flow. In 1958, Golay3 extended the Taylor−Aris theory to channels with circular and rectangular cross sections, the walls of which are coated with a thin adsorbing layer. The solute was assumed to partition between flowing fluid (mobile phase) and adsorbing layer (stationary phase) such that, in equilibrium, the amount of solute in the adsorbing layer was related to the amount in the flowing fluid by the constant k′. Golay established a set of mass-balance equations for the mobile phase considering radial and axial diffusion as well as the flow velocity profile. By resolving the partial differential equations using Laplace transformation, he showed that for tubes with circular cross-section D is given by

ateral molecular diffusion of a solute that is transported by a fluid stream with nonuniform velocity affects the rate of local solute motion and results in additional longitudinal dispersion relative to the average fluid velocity. In the 1950s, Taylor1 and Aris2 showed that at long times the dispersion coefficient of an inert (i.e., nonreacting and nonadsorbing) solute in incompressible fluid flow through a channel can be determined from the molecular diffusion coefficient Dm and an additional term that depends, in particular, on the actual geometry of the channel cross-section and the flow velocity profile. For instance, in a circular tube of radius R, the dispersion coefficient D is given by D = Dm +

uav 2R2 48Dm

(1)

where uav is the average fluid velocity in the tube. According to the Taylor−Aris theory, after a time sufficient for diffusion over the channel cross-section (lateral equilibration), an inert solute with initial delta-pulse concentration distribution, injected into the channel under laminar flow conditions, is distributed axially symmetrically about a point that moves with the average speed of solvent flow. Taylor−Aris dispersion is important in numerous fields including, in particular, chemical engineering and chromatography. Frequently, Taylor−Aris dispersion is accompanied by adsorption and desorption processes that take place at the © 2014 American Chemical Society

D=

2 2 Dm 1 + 6k′ + 11k′2 uav R + 3 1 + k′ 48Dm (1 + k′)

(2)

if diffusion in the adsorbing layer is negligible. From a comparison of this expression with eq 1, it can be realized that sorption affects dispersion. In particular, D depends on the Received: January 23, 2014 Accepted: April 6, 2014 Published: April 6, 2014 4463

dx.doi.org/10.1021/ac500309p | Anal. Chem. 2014, 86, 4463−4470

Analytical Chemistry

Article

plate height expression was identical to that obtained by Scott and Fritz.12 Further extensions of the stochastic approach included the assumption that the times the molecules spent in the mobile phase follow a Gaussian probability function,14,15 the modeling of a heterogeneous adsorbing layer,16,17 and the development of a stochastic−dispersive theory.18 All of the aforementioned stochastic approaches address mass transport and transfer in one-dimensional systems; in particular, they do not account for a real (nonuniform) fluid flow pattern. Recently, Levesque et al.19 developed a stochastic approach based on Laplace transformation of the transverse diffusion equation for Poiseuille flow in planar and cylindrical geometries with reversible adsorption at the walls. For flow through a circular tube, their final expression for D is identical to eq 3 by Khan4 following a different mathematical approach, if the retention factor is defined as k′ = 2ka/Rkd, where ka is the adsorption rate constant (m s−1) and the desorption rate constant has the dimension of s−1. Later, Levesque et al.20 extended a kinetic, lattice-Boltzmann approach to account for the effect of sorption in their analysis of pure diffusion and advection−diffusion in Poiseuille flow in simple geometries (a slit channel and an ordered sphere packing). In 2012, Berezhkovskii21 presented an approach based on the axial displacement of an individual particle, which moves in a plane perpendicular to the tube axis along a trajectory, to analyze the Taylor−Aris problem. Averaging of the displacement and its square over realizations of particle trajectories allows one to determine, respectively, the first moment and second central moment of the displacement. These statistical moments were then used to calculate the effective velocity and dispersion coefficient in a circular tube with Poiseuille flow transporting particles, which can reversibly adsorb at the wall as well as drift and diffuse on the wall.22 It was shown in ref 22 that dispersion depends in a complex manner on flow and drift velocities and diffusivities in the fluid and on the wall, as well as on the adsorption and desorption rate constants. Though the stochastic approaches proposed in refs 19−22 involved a parabolic flow profile in open channel systems, they did not account for the effects of a variance in the adsorption sojourn time and assumed that adsorption of solute molecules is characterized only by the value of the desorption rate constant, k d. In 2007, Chen23 developed a two-dimensional statistical model of open-tubular chromatography. He established relationships between the moments of column residence time and the moments of sojourn times in the stationary and mobile phases, which were obtained from the mass-balance principle. This model accounts for a parabolic flow profile in an open tube and for diffusion of solute molecules in the stationary phase. Expressions for up to the fourth moment of the column residence time were developed, and it was shown that the proposed two-dimensional model covers all relative results from previous theories. Specifically, for negligible diffusion of solute molecules in the stationary phase and a uniform velocity field, Chen’s expression for chromatographic plate height is identical to those developed by Fritz and Scott12 and Dondi and Remelli.13 Assuming a parabolic velocity profile and a firstorder kinetics for the sorption process, Chen’s model also reproduces the results obtained with Khan’s model.4 Here, we present a microscopic three-dimensional simulation approach to advective−diffusive transport in an open tube with adsorbing wall. We combine a random-walk particle-tracking method to simulate advective−diffusive transport of molecules

relative amount of solute retained in the stationary phase, as given by the retention factor k′. Importantly, the theory of Golay is based on the assumption of interfacial equilibrium, i.e., an instant interfacial exchange between stationary and mobile phases. In 1962, Khan4 extended Golay’s theoretical model to account for interfacial resistance to mass transfer. He derived an expression similar to eq 2, which contains an additional term corresponding to the interfacial resistance D=

2 2 uav 2da Dm 1 + 6k′ + 11k′2 uav R k′ + + 3 3 1 + k′ 48Dm (1 + k′) (1 + k′) kd

(3)

where da is the thickness of the adsorbing layer and kd is the desorption rate constant (m s−1). Equation 3 assumes that the contribution to D due to interfacial mass transfer resistance is proportional to da and inversely proportional to kd. If kd → ∞, eq 3 implies that the interfacial exchange rate does not affect dispersion. The mass-balance models developed by Golay3 and Khan4 and their extensions found a wide range of applications in fluid dynamics, chemical engineering, and chromatography.5−10 However, an analytical solution of the system of partial differential mass-balance equations can be obtained only for simplified physical systems (e.g., with a neglect of longitudinal diffusion in the adsorbing layer) and by using only macroscopic, not local system parameters (e.g., coefficients characterizing the interfacial mass transfer). For instance, the above approach is inapplicable to describe solute dispersion in systems with a heterogeneous adsorbing surface. Another class of theoretical models developed to study advective−diffusive transport coupled with sorption is based on stochastic approaches. Their idea is to consider a solute as an ensemble of individual molecules and describe molecular transport with solvent flow via random migration interrupted by adsorption−desorption events. Though the molecular level approach has the appeal of making the physical picture very clear, realistic, and easily understandable, the mathematical treatment of the stochastic approaches can be as well rather complicated. In 1955, Giddings and Eyring11 presented the first stochastic model of the chromatographic process. In this model, it was assumed that the solute molecules perform a random number of adsorption−desorption events, characterized by a Poisson distribution, while moving along a chromatographic column. Another assumption was that the linear mobile phase velocity is constant, independent of the coordinates, i.e., the Giddings−Eyring model is inherently one-dimensional only. In 1984, Scott and Fritz12 proposed a stochastic model for chromatography based on the statistical renewal theory. They showed that the variances of the time spent by solute molecules in the adsorbed state and in the mobile phase affect peak broadening. However, this model provided no approach to determine these variances, and the authors assumed that the sojourn times in both the stationary and mobile phases are governed by exponential distributions. Later, Dondi and Remelli13 showed that the characteristic function method can be used to realize many models of chromatography. They established general expressions allowing one to determine three fundamental chromatographic quantities: the plate height, specific asymmetry, and specific excess of a chromatographic peak. These expressions include, in particular, the statistical characteristics of the residence time distributions of solute molecules in the mobile and stationary phases. For the Giddings−Eyring model11 (i.e., with a Poisson distribution for entries in the stationary phase and an exponential distribution for adsorption sojourn times), the developed 4464



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⎛ r2 ⎞ u(r ) = 2uav ⎜1 − 2 ⎟ ⎝ R ⎠

in laminar flow and a stochastic (probabilistic) approach to model sorption at the tube wall. In contrast to all previous stochastic models, both mass transport and the adsorption− desorption kinetics are explicitly simulated at the microscopic (molecular) level in three-dimensional space. Depending on the microscopic parameters of the sorption kinetics, the developed numerical model allows one to simulate systems, the behavior of which can be described by any of the presented macroscopic models: Taylor−Aris (no sorption); Golay (sorption with fast interfacial mass transfer); and Khan (sorption with slow interfacial mass transfer). The results of our three-dimensional microscopic numerical simulations confirm that, besides flow velocity, transverse dimension of the tube, diffusion coefficient, and adsorption/ desorption rate constants, the dispersion coefficient in Poiseuille flow through a circular tube with adsorbing wall depends as well on the distribution of the adsorption sojourn time, i.e., the waiting time for desorption of individual molecules. At the same time, since our numerical model is inherently microscopic, we are able to construct the retention factor k′ through the independent adjustment of the adsorption probability p (which can be attributed to geometrical and energetic restrictions for the adsorption process) and the average adsorption sojourn time (which depends on the mean activation energy for desorption). This approach can realize the same value of k′ by different sets of p and to implement either adsorption controlled (p ≪ 1) or diffusion controlled (p = 1) interfacial mass transfer kinetics, in contrast to all the previous stochastic models11−18 that have been developed with the assumption that diffusion is not a ratecontrolling step for adsorption. Further, as the proposed approach is inherently threedimensional, it can be also employed to study advection− diffusion−sorption mass transfer in real (three-dimensional) chromatographic systems with particulate or monolithic stationary phases, accounting explicitly for heterogeneity of the local fluid velocity. The developed modeling approach can be applied to any distribution of adsorption sojourn times, e.g., based on experimental data, or to model multisite adsorption as well as to simulate systems with irreversible adsorption.

During the simulations, the coordinates of each tracer are monitored, which allows us to determine the statistical moments of the tracer displacements. Particularly, the first moment and the second central moment, respectively, of the longitudinal displacement were calculated as Δx(t ) =

σ 2(t ) =

N

1 N

∑ (xi(t ) − xi(t = 0))

1 N

∑ (Δxi(t ) − Δx(t ))2

i=1

(6)

N i=1

(7)

These two quantities, determined from information at the microscopic level, can be related to the dispersion coefficient and average migration velocity of the solute, respectively, by D(t ) = umig =

1 ∂σ 2(t ) 2 ∂t Δx(t ) t

(8)

(9)

3,4

It was shown that the retention factor k′ can also be determined from the ratio of the migration velocities for unretained (umig,0) and retained (umig,R) samples umig,0 k′ = −1 umig,R (10) If a tracer hits the tube wall, it can be either reflected back into the fluid with probability (1 − p) or adsorbed at the surface with probability p for the adsorption sojourn time τa. p can be less than unity because of spatial and energetic restrictions.4,28−31 The former results from a discontinuous location of adsorption sites and is characterized by just a fraction of the total surface area that is active in this regard. The latter is associated with a nonuniform distribution of sorption energy: Not every collision is energetic enough to overcome the barrier for kinetic transfer between fluid and adsorbing layer. Theoretical and experimental estimates of the adsorption probability p range from unity down to 10−7.4,32−36 The sorption energy also affects the average adsorption sojourn time , a further microscopic characteristic that we use to describe the sorption kinetics. The Frenkel equation36 allows us to relate to the mean adsorption energy Ea

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MODELING APPROACH Advective−diffusive mass transport was simulated by a randomwalk particle-tracking technique.24−27 This approach is based on the evolution of the coordinates of individual and distinguishable point-like tracer “particles”. In this study, we used N = 106 tracers, which initially (at t = 0) are distributed randomly within the tube of radius R. Then, the time evolution of tracer coordinates (changing due to flow and diffusion) is monitored. Advective displacement is determined by the local velocity, and diffusive displacement is determined from the diffusion coefficient. During each time step Δt, the displacement Δl of a tracer is determined as the sum of advective and diffusive contributions Δl = u(r )Δt + δ

(5)

⎛E ⎞ = τ0exp⎜ a ⎟ ⎝ RT ⎠

(11)

where R is the gas constant, T is the temperature, and τ0 is a constant typically about 10−13 s at room temperature.37,38 Results in this study were obtained for constant adsorption sojourn times (τa = const) and τa values following exponential probability distributions30,39 f (τa) =

(4)

where δ is a vector having a random orientation in space and a length governed by a Gaussian distribution with variance 6DmΔt and u(r) is the local velocity along the tube with an amplitude depending on the distance from the tube axis

⎛ τ ⎞ 1 exp⎜ − a ⎟ ⎝ ⎠

(12)

Equation 12 is characterized by the average adsorption sojourn time. Random values of τa (following the distribution function defined by eq 12) were generated with the method of inverse transform sampling40 4465


Analytical Chemistry τa = − ln(1 − η)

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simulation results). Using the reduced velocity, eq 1 can be rewritten in the following simplified form

(13)

where η represents random numbers that are generated with uniform probability density in the range [0,1]. Whereas exponential distribution of τa corresponds to a sorption process described by a first-order kinetic equation, constant adsorption sojourn times imply a constant activation energy for the desorption process. Both microscopic sorption characteristics (p and ) determine the relative amount of solute that is retained (adsorbed) at the tube wall, i.e., the retention factor k′. Importantly, a given value of k′ can be realized by numerous combinations of these parameters: the smaller the adsorption probability p, the larger is the average adsorption sojourn time for a given k′. Experimental results of single-molecule spectroscopic studies used to measure the adsorption time estimate its range from fractions of a millisecond up to seconds, depending on the physiochemical properties of the adsorbent surface and the adsorbed molecule.39,41,42 The developed numerical model of the advective−diffusive transport coupled with sorption at the solid−liquid interface was realized as a parallel code in C language and implemented on Hydra, an IBM iDataPlex system for High Performance Computing at Rechenzentrum Garching (RZG, Garching, Germany). Simulations of a single advection−diffusion− sorption transport scheme (for a given velocity, retention factor, and adsorption probability) took from 5 min up to 4 h at 8 processor cores.

D ν2 =1+ Dm 48

(14)

The data in Figure 1 demonstrate excellent agreement between simulated and theoretical values of D/Dm over the whole velocity range we analyzed, i.e., from ν = 0.25 (diffusion-limited regime) up to ν = 100 (advection-dominated regime). Next, we simulated advective−diffusive transport coupled with adsorption−desorption at the tube wall, assuming exponential distributions of the adsorption sojourn times τa for four retention factors (k′ = 1, 3, 5, and 7) and two adsorption probabilities (p = 1 and 0.01). Because all tracers initially (t = 0) were assumed to be distributed only within the fluid (mobile phase), the dispersion coefficient demonstrates nonmonotonic transient behavior: An initial increase is followed by a decrease and approach of an asymptotic value. In Figure 2, an example of transient dispersion is shown

■

RESULTS AND DISCUSSION To validate the developed numerical model, we initially compared dispersion coefficients obtained by simulations of advective−diffusive transport of unretained tracers (no adsorption, k′ = 0) with those calculated from Taylor−Aris theory. In Figure 1, the simulated and theoretical values of the Figure 2. Dispersion coefficient (ν = 10) normalized by the molecular diffusion coefficient D/Dm vs the normalized time t/. Adsorption probability: p = 0.01 (k′ = 7). Adsorption sojourn times τa were governed by the exponential probability distribution function (eq 12) with = 1.26 s.

(simulated for ν = 10, k′ = 7, and p = 0.01). This nonmonotonic behavior (which is not observed for unretained tracers) is a result of the nonequilibrium initial condition. At t = 0, no tracer is adsorbed on the tube wall and the sorption dynamics is shifted toward the adsorption event. With time, the system approaches equilibrium characterized by the relative amount of adsorbed tracers determined from a given retention factor, and the dispersion coefficient becomes independent of observation time. For our analysis, we used only asymptotic values of D (t → ∞), which were determined as values averaged over time intervals when no significant changes were observed any longer (cf. dashed line in Figure 2). The adsorption of solute on the wall also reduces its migration velocity along the tube by the factor (1 + k′). The dispersion coefficient is often normalized by this factor to facilitate the comparison of systems characterized by different retention factors. In Figure 3, we compare dispersion coefficients normalized by the factor (1 + k′)/Dm, as obtained from simulations with exponential distributions of the adsorption sojourn times for p = 1 (symbols), with values calculated from eq 15 developed according to the Golay theory (lines)

Figure 1. Dispersion coefficient normalized by the molecular diffusion coefficient D/Dm vs the reduced flow velocity ν = uavR/Dm in a cylindrical tube without wall adsorption.

dispersion coefficient normalized by the molecular diffusion coefficient of the solute (D/Dm) are shown as a function of the reduced flow velocity ν = uavR/Dm. The reduced velocity is often used to characterize the ratio of contributions from advection and diffusion to the overall mass transport. In this study, we assume Dm = 2.0 × 10−9 m2 s−1, R = 50 μm, and Δt = 10−7 s, resulting in an average length of the vector δ (eq 4) of 3.33 × 10−8 m (a further reduction of Δt did not affect 4466



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times equal to τa = 1.26, 3.76, 6.28, and 8.79 ms (instead of the earlier exponential distributions with the same values for ). For these constant values of τa, the simulated retention factors were very close (±3%) to k′ = 1, 3, 5, and 7 (guaranteeing conserved retention properties), and the corresponding dispersion coefficients were indistinguishable from those we obtained with the exponential distributions of τa (data not shown in Figure 3). This demonstrates that fast interfacial mass transfer (adsorption−desorption) contributes insignificantly to solute dispersion. In addition, this contribution remains very small, independent from the variations in adsorption sojourn times of the individual solute molecules. A completely different picture developed as we conducted the simulation of systems with a high resistance to interfacial mass exchange (slow adsorption−desorption). This case was realized by a reduction of the adsorption probability to p = 0.01. Assuming again exponential distributions of adsorption sojourn times, values of k′ = 1, 3, 5, and 7 were obtained with = 0.18, 0.54, 0.9, and 1.26 s, respectively. In Figure 4a, we compare the simulation results for these values of (symbols) with theoretical predictions (lines) calculated by eq 15 based on the Golay theory. It is obvious that the Golay model underestimates the dispersion coefficients in these systems with slow sorption. This is indeed unsurprising given the inability of the Golay model to account for the contribution of a finite interfacial mass transfer resistance to dispersion. In addition, we conducted simulations with p = 0.01 for constant adsorption sojourn times τa = 0.18, 0.54, 0.9, and 1.26 s, which allowed us to realize the same set of retention factors, i.e., k′ = 1, 3, 5, and 7. The data in Figure 4b show that the dispersion coefficients decrease compared to those with exponential adsorption sojourn time distributions, Figure 4a, but they are still larger than those for the systems with fast sorption processes (Figure 3). The diversity demonstrated by Figure 4 cannot be explained with a difference in interfacial mass transfer rates: Both systems, either with a constant τa or with τa following exponential distributions, are characterized by the same desorption rates (kd), because they have the same values of ∼ kd−1. Thus, using Khan’s model (eq 3), albeit able to account for finite interfacial mass transfer resistance, they must be characterized still by the same dispersion coefficient.

Figure 3. Normalized dispersion coefficient D(1 + k′)/Dm vs the reduced flow velocity ν = uavR/Dm. Adsorption probability: p = 1. Adsorption sojourn times τa were governed by the exponential probability distribution function (eq 12) with = 0, 1.26, 3.76, 6.28, and 8.79 ms for k′ = 0, 1, 3, 5, and 7, respectively. The simulation results obtained for the constant values of τa = 0, 1.26, 3.76, 6.28, and 8.79 ms for k′ = 0, 1, 3, 5, and 7, respectively, are indistinguishable from those obtained for the exponential distributions of τa.

D(1 + k′) 1 + 6k′ + 11k′2 ν 2 =1+ 48 Dm (1 + k′)2

(15)

as a function of the reduced velocity. Because p = 1, every collision of a tracer with the tube wall results in its adsorption corresponding to local equilibrium between the concentrations of solute molecules in the fluid and adsorbing layer. With this condition, the average adsorption sojourn times were 1.26, 3.76, 6.28, and 8.79 ms for k′ = 1, 3, 5, and 7, respectively. Short average adsorption times correspond to fast exchange and small resistance to interfacial mass transfer. The simulated data in Figure 3 are therefore in good agreement with theoretical predictions from eq 15. To our knowledge, this is the first reported comparison between results predicted by the Golay model and those obtained from three-dimensional, direct numerical simulations of solute dispersion in a cylindrical tube with adsorption at its wall. We conducted a similar set of simulations, but with constant (uniform) adsorption sojourn

Figure 4. Normalized dispersion coefficient D(1 + k′)/Dm vs the reduced flow velocity ν = uavR/Dm. Adsorption probability: p = 0.01. (a) Adsorption sojourn times τa were governed by the exponential probability distribution function (eq 12) with = 0, 0.18, 0.54, 0.9, and 1.26 s for k′ = 0, 1, 3, 5, and 7, respectively. (b) Adsorption sojourn times were uniform and equal to τa = 0, 0.18, 0.54, 0.9, and 1.26 s for k′ = 0, 1, 3, 5, and 7, respectively. 4467



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Figure 2: Because of the nonequilibrium initial distribution of tracers, the system is gradually relaxed to equilibrium. The transient behavior of the adsorption rate observed with constant adsorption times is characterized by decaying oscillations, which have the same origin. However, the constant adsorption time for all tracers results in a synchronization of their adsorption−desorption events, a picture that is blurred by diffusion. Nevertheless, the asymptotic values of the adsorption rate for both types of adsorption time distribution are the same. This observation is in agreement with the definition of the retention factor in the stochastic theory of chromatography, k′ = ka/kd. Our simulations show that the identity of asymptotic values of ka for constant and exponentially distributed adsorption sojourn times is preserved for all reduced velocities and values of p and k′ in this study. For our analysis, we used the asymptotic values of ka determined as time-averaged values after the system has relaxed to equilibrium (steady-state). Similarly, only the asymptotic (steady-state) values of the average migration velocity of retained tracers umig,R were used to determine k′ (eq 10). It should be mentioned that the nonmonotonic behavior of ka observed in Figure 5 becomes smoother with smaller values of k′. In Figure 6a, we compare the normalized values of Dads obtained with exponential distributions of the adsorption sojourn time in two ways: (i) defined as the difference between simulated values of D and corresponding values calculated by eq 15 (symbols) and (ii) calculated by eq 16 (lines). We observe excellent agreement between the results obtained by these two approaches over the whole range of reduced velocities (for all retention factors). This allows one to conclude that Khan’s model4 based on the mass-balance equation and the stochastic approach developed by Levesque et al.19 accurately describe advective−diffusive transport in a tube with interfacial mass transfer resistance due to reversible adsorption at the wall, characterized by an exponential distribution of adsorption sojourn times. However, both models cannot explain a reduction of the dispersion coefficient in the system with uniform distributions of the adsorption sojourn time, as seen in Figure 4b. Equations 3 and 16 do not contain information on the distribution function for the adsorption sojourn times and provide the same Dads for systems with constant and exponentially distributed τa. In Figure 6b, we present normalized values of Dads (symbols) simulated with constant values of τa, plotted on the same scale as in Figure 6a. The contribution to solute dispersion from the adsorption with constant adsorption sojourn times, Figure 6b, is remarkably smaller than for the exponential distributions, Figure 6a. The observed difference can be explained with the stochastic model of chromatography based on the renewal theory, proposed by Scott and Fritz in 1984.12 They developed a simplified one-dimensional model assuming a constant flow velocity through a chromatographic column filled with particles that are coated with a thin adsorbing layer. The authors established an expression to evaluate the contribution to chromatographic peak broadening due to the random sorption process through the statistical moments of sojourn times in the mobile and stationary phases. This contribution to peak broadening is composed of two additive terms originating from the variation in the number of adsorption events for individual molecules and the variation in the adsorption sojourn times. For constant adsorption sojourn times, the second variation is equal to zero. By adapting the original expression

According to Khan’s model, the two contributions to solute dispersion associated with mobile phase resistance to mass transfer (second term in the rhs of eq 3) and interfacial resistance (third term in the rhs of eq 3) are additive. For our subsequent analysis, we defined Dads as the difference between the simulated values of the dispersion coefficient D and those determined using eq 15, i.e., Dads is associated only with the adsorption and desorption kinetics (interfacial resistance). The model of Khan4 includes the thickness of the adsorbing layer (da) as a parameter, but in our system, its thickness is assumed as infinitesimal. Therefore, we used the alternative approach by Levesque et al.19 to evaluate the additional contribution of the adsorption−desorption kinetics to solute dispersion. According to that approach, Dads can be defined as Dads(1 + k′) u 2 = av a Dm Dm

ka kd 2

(1 + ) ka kd

(16) −1

where the adsorption rate ka has the dimension of s . It should be mentioned that an expression mathematically identical to eq 16 can also be obtained from Khan’s model (eq 3), assuming that the value of kd/da used in that model, which has the dimension of s−1, is equal to kd (s−1) used in the model of Levesque et al.19 While running our simulations, we recorded the total number of adsorption events, Na(t), as a function of time. This quantity can be translated into the adsorption rate, ka, which corresponds to the average number of adsorption events per tracer and unit time

Na(t ) (17) Nt Figure 5 shows adsorption rates determined according to eq 17 for constant adsorption sojourn times (τa = 1.26 s) and those ka =

Figure 5. Adsorption rate ka vs the normalized time t/ for constant (τa = 1.26 s) and exponentially distributed ( = 1.26 s) adsorption sojourn times, ν = 20. Adsorption probability: p = 0.01 (k′ = 7).

governed by exponential distribution ( = 1.26 s) as a function of time for k′ = 7, p = 0.01, and ν = 20. At the beginning, the transient behaviors of the adsorption rates for the constant adsorption sojourn times and the exponentially distributed ones are very different. The adsorption rate for the exponential distribution decreases monotonically. This behavior has the same cause as the one noted earlier in a discussion of 4468



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Figure 6. Normalized contribution to the dispersion coefficient Dads(1 + k′)/Dm due to adsorption vs the reduced flow velocity ν = uavR/Dm. Adsorption probability: p = 0.01. (a) Adsorption sojourn times τa were governed by the exponential probability distribution function (eq 12) with = 0, 0.18, 0.54, 0.9, and 1.26 s for k′ = 0, 1, 3, 5, and 7, respectively. (b) Adsorption sojourn times were uniform and equal to τa = 0, 0.18, 0.54, 0.9, and 1.26 s for k′ = 0, 1, 3, 5, and 7, respectively.

from ref 12, the following equation for evaluation of Dads with uniform τa can be obtained12,43 Dads(1 + k′) u 2τ = av a Dm 2Dm

The smaller the Da, the stronger is the role of diffusion relative to that of adsorption in an overall mass transfer process. In our simulations, Da ∼ O(1) for the system with p = 0.01, whereas Da ∼ O(100) with p = 1, implying that diffusion is a ratecontrolling mechanism. To our knowledge, the data in Figure 3 are the first reported results obtained with the stochastic model applied to systems with large Da.

ka kd 2

(1 + ) ka kd

(18)

■

It should be pointed out that expressions identical to eqs 16 and 18 can be as well obtained from the one-dimensional and two-dimensional stochastic models developed by Dondi and Remelli13 and Chen,23 respectively. In Figure 6b, we compare the normalized values of Dads simulated with constant τa (symbols) and corresponding values calculated using eq 18 (lines). Excellent agreement between simulation results and theoretical predictions based on eq 18 is observed. A comparison of Figure 6a,b demonstrates that adequate description of advective−diffusive transport coupled with sorption processes cannot be generally realized without accounting for the intrinsic microscopic details of the adsorption−desorption kinetics. Ignoring this information eliminates from the model relevant characteristics of solute dispersion. At the same time, an inherent feature of all previous onedimensional stochastic models11−18 is a “memory-free” behavior of the solute molecules. It limits the applicability of these models to systems where adsorption probability for a solute molecule is independent of its spatiotemporal trajectory, in particular, of its current position characterized by the local flow velocity. In these systems, diffusion is not a ratecontrolling process for adsorption.11 In many situations, however, diffusive transport of solute molecules affects adsorption. Regarding the kinetics of an individual molecule, in these systems, there is a higher probability for adsorption of a molecule immediately following its desorption. This scenario is realized in our simulations with p = 1 (Figure 3). It is noteworthy that p characterizes adsorption probability as a tracer hits the wall, not the adsorption probability averaged over the tracer ensemble; i.e., p is a conditional probability. The ratio between the rates of diffusion and adsorption processes can be characterized by the Damköhler number (Da) Da =

R2 Dm

CONCLUSIONS We developed a microscopic numerical model combining simulations of advective−diffusive transport with a stochastic approach to the sorption process at the solid−liquid interface. The use of the numerical stochastic approach to simulate the sorption process has allowed us to address microscopic details of interfacial mass transfer and to analyze the influence of the adsorption kinetics on the dispersion coefficient, D. In particular, it has been illustrated that a given retention factor k′ can be constructed by independent adjustment of the adsorption probability and the mean adsorption sojourn time. The results of our simulations reveal that both the randomness of sorption events and variations in the adsorption sojourn times τa of individual molecules affect the value of D. This has been confirmed by a comparison between the simulated dispersion coefficients obtained with constant adsorption sojourn times or exponential adsorption sojourn time distributions, eq 12. It has been demonstrated that the additional contribution to solute dispersion from adsorption can be accurately evaluated with the renewal theory of chromatography, but while the renewal theory requires one to express the statistical moments of the adsorption sojourn time and the sojourn time in the fluid in analytical form, our approach enables the explicit determination of these quantities from the simulations. This allows us in the future to realize a variety of adsorption models, e.g., with heterogeneous adsorption sites, or models based on a description of adsorption sojourn times using single-molecule experimental measurements. Our results confirm that an adequate model of the coupled advection−diffusion−sorption transport scheme needs to account for detailed information on the sorption kinetics, a conclusion obtained already from one-dimensional stochastic models of chromatography.12,13 However, the results presented in this study were derived for the first time by direct threedimensional microscopic simulation of adsorption−desorption

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Article

and advection−diffusion in a nonuniform flow velocity field. Additionally, the proposed approach does not make the assumption that diffusion is not rate-controlling during adsorption, an inherent feature of all previous stochastic models of chromatography. The presented modeling approach holds promise for the analysis of complex separations, e.g., separations of enantiomers on chiral stationary phases, when slow adsorption−desorption processes, different sojourn time distributions, and heterogeneous adsorption behavior come together.

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(30) Giddings, J. C. Dynamics of Chromatography, Part I: Principles and Theory; Marcel Dekker: New York, 1965. (31) McGuffin, V. L.; Krouskop, P. E.; Wu, P. J. Chromatogr., A 1998, 828, 37−50. (32) Clement, C. F.; Kulmala, M.; Vesala, T. J. Aerosol Sci. 1996, 27, 869−882. (33) Takahama, S.; Russell, L. M. J. Geophys. Res. 2011, 116, D02203. (34) Gross, A.; Mikkelsen, K. Adv. Quantum Chem. 2005, 50, 125− 142. (35) De Boer, J. H. The Dynamical Character of Adsorption; Clarendon Press: Oxford, 1968. (36) Frenkel, J. Kinetic Theory of Liquids; Dover: New York, 1955. (37) Felinger, A. J. Chromatogr., A 2006, 1126, 120−128. (38) Benvenuti, C. Molecular surface pumping: The getter pumps. In CERN Accelerator School: Vacuum Technology; Turner, S., Ed.; CERN: Genève, 1999; pp 43−50. (39) Ludes, M. D.; Wirth, M. J. Anal. Chem. 2002, 74, 386−393. (40) Devroye, L. Non-Uniform Random Variate Generation; Springer: New York, 1986. (41) Wirth, M. J.; Swinton, D. J.; Ludes, M. D. J. Phys. Chem. B 2003, 107, 6258−6268. (42) Bemgard, A. K.; Blomberg, L. G. Anal. Chem. 1989, 61, 2165− 2171. (43) Takács, L. Ann. Probab. 1974, 2, 420−431.

AUTHOR INFORMATION

Corresponding Author

*E-mail: tallarek@staff.uni-marburg.de. Phone: +49-(0)642128-25727. Fax: +49-(0)6421-28-27065. Notes

The authors declare no competing financial interest.

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Effect of Adsorption on Solute Dispersion: A Microscopic Stochastic

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