As a model for molecular traffic control (MTC) we investigate the diffusion
of hard core particles in crossed single-file systems. We consider a square
lattice of single-files being connected to external reservoirs. The (vertical)
alpha-channels, carrying only A-particles, are connected to reservoirs with
constant density ra. B-particles move along the (horizontal) beta-channels,
which are connected to reservoirs of density rB. We allow the irreversible
transition A to B at intersections. We are interested in the stationary density
profile in the alpha- and beta- channels, which is the distribution of the
occupation probabilities over the lattice. We calculate the stationary currents
of the system and show that for sufficiently long channels the currents (as a
function of the reservoir densities) show in the limit of large transition
rates non analytic behavior. The results obtained by direct solution of the
master equation are verified by kinetic Monte Carlo simulations.Comment: 11 page

A. Brzank

E. G. Derouane

F. Spitzer

G. M. Schütz

G. Schütz

H. Spohn

J. Kärger

P. Bräuer

English

arXiv

Crossref

Molecular traffic control in single-file networks with fast catalysts

As a model for molecular traffic control we investigate the diffusion of hard core particles in crossed single-file systems. We consider a square lattice of single-files being connected to external reservoirs. The (vertical) alpha channels, carrying only A particles, are connected to reservoirs with constant density rho(A). B particles move along the (horizontal) beta channels, which are connected to reservoirs of density rho(B). We allow the irreversible transition A-->B at intersections. We are interested in the stationary density profile in the alpha and beta channels, which is the distribution of the occupation probabilities over the lattice. We calculate the stationary currents of the system and show that for sufficiently long channels the currents (as a function of the reservoir densities) show in the limit of large transition rates nonanalytic behavior. The results obtained by direct solution of the master equation are verified by kinetic Monte Carlo simulations

Brzank, A.

Schütz, G. M.

Bräuer, P.

Kärger, J.

Juelich Shared Electronic Resources

PHYSICAL REVIEW E 69, 031102 ~2004!Molecular traffic control in single-file networks with fast catalystsA. Brzank,1 G. M. Schu¨tz,2 P. Bra¨uer,1 and J. Ka¨rger11Fakulta¨t fu¨r Physik und Geowissenschaft, Universita¨t Leipzig, Abt. Grenzfla¨chenphysik, Linne´strasse 5, D-04103 Leipzig, Germany2Institut fu¨r Festko¨rperforschung, Forschungszentrum Ju¨lich, D-52425 Ju¨lich, Germany~Received 6 October 2003; published 9 March 2004!As a model for molecular traffic control we investigate the diffusion of hard core particles in crossedsingle-file systems. We consider a square lattice of single-files being connected to external reservoirs. The~vertical! a channels, carrying only A particles, are connected to reservoirs with constant density rA . Bparticles move along the ~horizontal! b channels, which are connected to reservoirs of density rB . We allowthe irreversible transition A→B at intersections. We are interested in the stationary density profile in the a andb channels, which is the distribution of the occupation probabilities over the lattice. We calculate the stationarycurrents of the system and show that for sufficiently long channels the currents ~as a function of the reservoirdensities! show in the limit of large transition rates nonanalytic behavior. The results obtained by directsolution of the master equation are verified by kinetic Monte Carlo simulations.DOI: 10.1103/PhysRevE.69.031102 PACS number~s!: 82.75.2z, 05.50.1qI. INTRODUCTIONThe concept of molecular traffic control was introduced inthe beginning of the 1980s @1,2#. It is postulated that theeffective reactivity of microporous catalysts may be en-hanced by directing the reactant and product molecules alongdifferent pathways. The so-called molecular traffic control~MTC! effect has recently been verified by Monte Carlosimulations @3#. For a better insight into the restrictions andlimitations of the MTC effect it is desirable to understand theconditions which determine the actual flow of particles. Thisquestion is closely related to the problem of calculating thestationary density profile of the system. From a theoreticalpoint of view the MTC system can be modeled as an array ofseveral interacting symmetric exclusion processes ~SEP! @4#.The stationary properties of the SEP @5# within a channel, inparticular, the linear density profile with the stationary par-ticle current j}1/L , where L denotes the number of latticesites between intersections, together with the transition rulesat the catalytic sites lead to an interesting and unexpectedbehavior of the composed system.II. DESCRIPTION OF THE SYSTEMWe consider 2N21 channels of type a and b. Slightlygeneralizing the setup of @6# we assume the a~b! channels tobe connected to reservoirs of constant densities such that theentrances of the respective channels have fixed particle den-sities rA(B) . There should be no interaction other than hardcore repulsion which forbids double occupancy of the latticesites. D is the elementary ~attempt! rate of hopping betweenlattice sites and it is assumed to be the same for both speciesof particles. Therefore we have a network of interacting one-dimensional symmetric exclusion processes ~SEP!. From thiswe can expect linear density profiles between two intersec-tions @4#, the slope being inversely proportional to the num-ber of lattice sites L . L is the number of lattice sites betweentwo neighboring intersections or intersection and the adja-cent boundary site. See Fig. 1.At the intersections we place a catalyst that changes irre-1063-651X/2004/69~3!/031102~5!/$22.50 69 0311versibly A particles into B particles. In a discrete-time settingthis mechanism can be specified by saying that per time unitDt an A particle is converted into a B particle without havinga jump attempt. In a continuous-time description this prob-ability reduces to the reaction rate c which is the reactionprobability per time unit Dt .In our present calculations we assume L@1 and c largeenough so it is unlikely that an A particle that has entered anintersection point will leave the intersection before havingbeen converted into a B particle. Therefore in the steady statethe bulk of the network contains almost no A particles. Onlythe first and the last b channels and the first segment of the achannels are of interest. Due to reflection symmetry it issufficient to consider only a half-system and finally the sys-tem reduces to Fig. 2.III. THE GENERAL CASELet us label the lattice sites for the rth a-channel segmentfrom 1 to L and denote the catalytic site by I5L11 and theFIG. 1. MTC system with three a channels ~vertical! and threeb channels ~horizontal! and L53 lattice sites between adjacent in-tersections.©2004 The American Physical Society02-1Brzank et al. PHYSICAL REVIEW E 69, 031102 ~2004!FIG. 2. Reduced MTC system. The B particle density of the rth intersection is labled with r˜r .boundary site by 0. We use a similar labeling for the rth bchannel ~bulksites 1 to L , intersections by I50) since it willbe clear which one we refer to. Because of the exclusioninteraction the local occupation numbers akr and bkr for A andB particles can only take values 0,1. We also define the ‘‘va-cancy occupation number’’ vkr. In the a channel we havevkr512akr, in the b channel vkr512bkr, and on the catalyticsite v Ir512bIr2aIr. The expected local densities ^air& for Aparticles and ^bir& for B particles satisfy the following set ofequations:ddt ^a1r &5D~^a2r &22^a1r &1rA!, ~3.1!ddt ^air&5D~^ai11r &1^ai21r &22^air&!, 2<i<L21~3.2!ddt ^aLr &5D~^aL21r &2^aLr &1^aIrvLr &2^aLr v Ir&!, ~3.3!ddt ^aIr&5D~^aLr v Ir&2^aIrvLr &!2c^aIr& , ~3.4!ddt ^bIr&[D~^bLr21v Ir&2^bIrvLr21&1^b1r v Ir&2^bIrv1r &!1c^aIr&,~3.5!ddt ^b1r &5D~^b2r &2^b1r &1^bIrv1r &2^b1r v Ir&!, ~3.6!ddt ^bir&5D~^bi11r &1^bi21r &22^bir&!, 2<i<L21~3.7!ddt ^bLr &5D~^bL21r &2^bLr &1^bIr11vLr &2^bLr v Ir11&!,~3.8!ddt ^bLN&5D~^bL21N &22^bLN&1rB!. ~3.9!In these equations 0<r<N21, notice that ^bL11N &[rB and^a0r &[rA . The linear nature of the bulk equations ~3.2!, ~3.7!results from a cancellation of joint probabilities ^airai11r &,^birbi11r & due to the SU~2! symmetry of the symmetric ex-clusion process @7#. For a generic channel they may be writ-ten in the standard form of a lattice continuity equation forthe local density03110ddt rn5 jn212 jn ~3.10!with the densities rnA5^an& or rnB5^bn&, respectively, at siten and the corresponding bulk currentsjnA5D~^an21&2^an&!, ~3.11!jnB5D~^bn21&2^bn&! ~3.12!between bonds (n ,n11).For the stationary solution that we wish to study the timederivative vanishes. Because of particle conservation in thebulk the currents jnA , jnB do not depend on the lattice site nwhich implies linear density profiles. For notational simplic-ity we will drop the n and add the information about thechannel we are talking about instead.For cÞ0 we derive from Eqs. ~3.3!–~3.6! for the rthchannelj rA5c^aIr& , ~3.13!j rB5 j r21B 1c^aIr& , j0B52 j0B1c^aI0& . ~3.14!@In the absence of catalysis (c50) one would have an equi-librium state with all currents equal to zero.# Equations~3.13!, ~3.14! imply conservation of currentsj rB5 j r21B 1 j rA , 1<r<N21 ~3.15!j0B512 j0A. ~3.16!For futher analysis we neglect correlations between the oc-cupancy of a catalytic site and its three neighboring sites.This mean field approximation is motivated by exact resultsfor the correlations in the stationary state of the symmetricexclusion process from which it is known @8# that nearest-neighbor correlations in the vicinity of the boundary of asystem of size L are of order 1/L2 and hence small under theassumption L@1 made here. Within the mean field we there-fore replace joint probabilities ^xy& by the product ^x&^y&.Equation ~3.11! together with Eq. ~3.3! and Eq. ~3.12! to-gether with Eq. ~3.6! then become two equations for thecurrents, depending only on the local densities of the border-ing catalytic sites r˜rA, r˜rB, r˜r11B,j rA5Dr˜rA1rA~ r˜rB21 !L~ r˜rB21 !21, 0<r<N21 ~3.17!2-2MOLECULAR TRAFFIC CONTROL IN SINGLE-FILE . . . PHYSICAL REVIEW E 69, 031102 ~2004!j rB5Dr˜rB1 r˜r11B ~ r˜rA21 !L~12 r˜rA!11, 0<r<N21. ~3.18!For further discussion we neglect the densities of A particleson the intersections which comes from the model assumptionthat an A particle is very likely to be converted into a Bparticle before returning to the a channel. Equations ~3.17!and ~3.18!, together with Eq. ~3.15!, finally leads to a set ofequations for the unknown densities r˜rB[r˜r ,FIG. 3. Particle current obtained by Monte Carlo simulation ~3!vs theoretical current ~dots!. Lattice N51, L5250, rB50.7. Thedeviations are mainly due to finite-size corrections.03110r˜r2 r˜r11L11 5r˜r212 r˜rL11 1rA~ r˜r21 !L~ r˜r21 !21, 0<r<N21~3.19!r˜02 r˜1L11 512rA~ r˜021 !L~ r˜021 !21. ~3.20!These equations are quadratic in r˜r and to leading order inthe system size one has the solutionsFIG. 4. For a lattice N51, rA50.6, rB50.7 one finds r˜51. Forincreasing L the density of the intersection obtained by simulationsapproaches the theoretical value.r˜r5H 112 ~ r˜r111 r˜r211rA!, r˜05 r˜1112 rA , 1<r<N21.~3.21!FIG. 5. Normalized theoretical currents of the a channels ~left! and b channels ~right! in a lattice of N53. The first intersection r˜0saturates when rA529 , r˜1 becomes one for rA513 .2-3Brzank et al. PHYSICAL REVIEW E 69, 031102 ~2004!Before giving an explicit expression for the local densities atthe intersections we would like to discuss which of thesesolutions the system selects. We note that owing to exclusionr˜r<1. Owing to particle conservation in the bulk of the sys-tem all currents are non-negative. This implies an increase ofthe densities r˜r for decreasing r . Therefore all solutions withr˜n51 and r˜m,1 for m,n need to be discarded. The actu-ally selected solution is determined by the boundary densi-ties rA and rB . For small rA each a channel is in itsmaximal-current state since the probability of finding A par-ticles on the catalytic sites is almost zero. Each b channelcan support the accumulated currents that enter through the achannels. However, this implies a rather high density of Bparticles at the first catalytic site. Then, with an adiabaticFIG. 6. Densities r˜0 , r˜1 , r˜2 of B particles obtained by MonteCarlo simulation ~3! vs theoretical densities ~dots!. Lattice N53,L5250, rB50.03110increase of rA there is a critical value at which this densityreaches 1. Increasing rA further eventually leads to the sec-ond catalytic site saturating. Then there is no current any-more in the first part of the b channel. This goes on until thelast catalytic site also saturates and the system reaches asituation where the current is limited by the capacity of thelast segment of the b channel; only in this channel is there anactual flow of particles.To quantify this we note again that for small rA none ofthe catalytic sites is saturated and the recursion relation~3.21! is solved byr˜r512 ~N22r2!rA1rB . ~3.22!Notice that r˜r<1 which implies the range0<rA,212rBN2 . ~3.23!Increasing rA to such an extent that the densities of the firsts intersections become 1 we haver˜r51N2s F ~N2r !1~r2s !rB1 r2s2 ~N2r !~N2s !rAG ,r5s11,....,N21 ~3.24!with the corresponding rangerA,2~12rB!~N2s21 !~N2s ! . ~3.25!IV. SIMULATIONS AND RESULTSWe used dynamic Monte Carlo simulations to verify theresults obtained in the previous section. The algorithm usesrandom sequential update. Statistical errors are of order 0.1%and therefore not additionally marked on the graphs. We cal-culated the stationary profile for a system of N51 a chan-nels and L5250 and a system of N53 a channels and L5250. See Fig. 3. In the first case we find the two solutionsj5H D2 rAL , 0<rA<222rBD12rBL ,222rB,rA<1~4.1!which implies that the system selects the smaller of the twovalues ~4.1!,j5minH D2 rAL [ jmaxA ,D 12rBL [ jmaxB J . ~4.2!This has an intuitive physical meaning. In each channel thesystem tries to maximize its current, but the actually selectedcurrent is limited by the smaller one of the two maximalcurrents. To understand the origin of this selection principlewe employ the picture developed in the preceding section.Notice that the stationary probability of finding an A particle2-4MOLECULAR TRAFFIC CONTROL IN SINGLE-FILE . . . PHYSICAL REVIEW E 69, 031102 ~2004!on the catalytic site is almost zero. Now let us assume thatfirst rA , i.e., the reservoir density of A particles, is verysmall. For small rA the system tries to sustain the current j5 jmaxA which is possible as long as the b channel can supportthis current. This is the case if the density of B particles atsite N required to generate this current in the b channel isless than 1. Notice that in this regime the density of A par-ticles both on site I and on the last site of the a channel isnearly zero. Now we assume rA to be increased adiabati-cally, i.e., so slowly that the system reaches stationarity be-fore a further increase occurs. On the catalytic site the Bdensity r˜ increases in order to sustain the enhanced station-ary current. If rA becomes so large that jmaxA 5jmaxB one hasr˜51. Then the current is limited by the capacity of the bchannel and increasing rA further does not increase the cur-rent. Instead the last site L in the a channel acquires a finitedensity of A particles.For increasing L one expects the simulations to approachthe theoretical densities. Figure 4 shows the calculations fora lattice of N51, rA50.6, and rB50.7 with the theoreticaldensity at the intersection r˜51.We consider the system of N53 a channels. For simpli-fication we assume rB to be zero. Figure 5 shows the theo-retical currents of the a channels and b channels as a func-tion of rA . The first intersection r˜0 saturates when rA5 29 ;r˜1 becomes one for rA5 13 . For the same lattice Fig. 6 com-pares the densities obtained by simulation with the theoreti-cal values.03110V. CONCLUSIONWe discussed the stationary occupation probabilities ofthe molecular traffic control system with fast catalysts. Ourmain results are the obtained intersection densities ~3.22!–~3.25!, calculated in the limit of large file length. They aresufficient for the derivation of the density profile as a func-tion of the reservoir densities rA ,rB due to linear densityprofiles of the channel segments. As an intriguing result weshowed that an increase of the reservoir gradient does notnecessarily mean an increase of the currents inside the sys-tem. In fact, due to saturating intersection sites certain chan-nels show ~Fig. 4! an actual decrease of the current and fi-nally become zero. Since the mean field approximation isessential in our calculation the results are not valid for sys-tems of short file length. The stationary behavior of thesesystems remains an open question. The main ingredient inour approach is current conservation inside channels. Hencea similar analysis may be performed for more realistic modelsystems with density-dependent currents.The nonanalyticity of the current is not a critical phenom-enon in the usual sense of being associated with a divergentcorrelation length. It results from a nonequilibrium transportphenomenon, viz., the saturation of channels with the maxi-mal current they can support. We expect this phenomenon tooccur also in three dimensions. A quantitative investigationin three dimensions by means of Monte Carlo simulation isin progress.@1# E. G. Derouane, and Z. Gabelica, J. Catal. 65, 486 ~1980!.@2# E. G. Derouane, Appl. Catal., A 115, N2 ~1994!.@3# P. Bra¨uer, A. Brzank, and J. Ka¨rger, J. Phys. Chem. B 107,1821 ~2003!.@4# F. Spitzer, Adv. Math. 5, 246 ~1970!.@5# H. Spohn, J. Phys. A 16, 4275 ~1983!.@6# J. Ka¨rger, P. Bra¨uer, and H. Pfeifer, Z. Phys. Chem. ~Munich!214, 1707 ~2000!.@7# G. Schu¨tz and S. Sandow, Phys. Rev. E 49, 2726 ~1994!.@8# G. M. Schu¨tz, in Phase Transitions and Critical Phenomena,edited by C. Domb and J. Lebowitz ~Academic Press, London,2000!.2-5

Molecular traffic control in single-file networks with fast catalysts

Abstract

Similar works

Full text

Available Versions

Crossref

Juelich Shared Electronic Resources