Steady-State Properties of Single-File Systems with Conversion

30 Pages·0.5 MB·English

by  S. V. Nedea

Steady-State Properties of Single-File Systems with Conversion
S.V. Nedea,∗ A.P.J. Jansen,† J.J. Lukkien,‡ and P.A.J. Hilbers§
(Dated: February 2, 2008)
We haveused Monte-Carlo methods and analytical techniquesto investigate theinfluenceof the
characteristic parameters, such as pipe length, diffusion, adsorption, desorption and reaction rate
constantson thesteady-stateproperties ofSingle-File Systemswith areaction. Welooked at cases
2 when all the sites are reactive and when only some of them are reactive. Comparisons between
0 Mean-Field predictions and Monte Carlo simulations for the occupancy profiles and reactivity are
0 made. Substantial differences between Mean-Field and the simulations are found when rates of
2
diffusion are high. Mean-Field results only include Single-File behavior by changing the diffusion

n rate constant, but it effectively allows passing of particles. Reactivity converges to a limit value
a if more reactive sites are added: sites in the middle of the system have little or no effect on the
J kinetics. Occupancyprofilesshowapproximately exponentialbehaviorfrom theendstothemiddle

1 of thesystem.
3
PACSnumbers: 02.70Uu,02.60.-x,05.50.+q,07.05.Tp

]
h
p I. INTRODUCTION ticle can only move to an adjacent site if that site is not
-
occupied.
m
This processofSingle-File diffusion hasdifferent char-
e Molecular sieves are crystalline materials with open
acteristics than ordinary diffusion which affects the na-
h framework structures. Of the almost two billion pounds
ture of both transport and conversion by chemical re-
c ofmolecularsievesproducedinthelastdecade,1.4billion
. actions. For Single-File diffusion, the mean-square dis-
s pounds were used in detergents, 160 millions pounds as
c placement of a particular particle is proportional to the
catalysts and about 70 millions pounds as adsorbents or
i square-rootof time
s desiccants. [1]
y
h Zeolites represent a large fraction of known molecular r2 =2Ft21
h i
p sieves. These are all aluminosilicates with well-defined
[ pore structures. In these crystalline materials,the metal whereFistheSingle-Filemobility.[3]Thisisincontrast

atoms (classically, silicon or aluminum) are surrounded to normal diffusion, where mean-square displacement is
1
v by four oxygen anions to form an approximate tetrahe- directly proportional to time. A variety of approaches
0 dron. These tetrahedrathen stackin regulararrayssuch havebeenusedtodescribethemovementoftheparticles
7 that channels and cages are formed. The possible ways inSingle-FileSystems,mostofthemconcentratedonthe
0 for the stackingto occur is virtuallyunlimited, and hun- role of the Single-File diffusion process.
1 dreds of unique structures are known. [2] Molecular Dynamic(MD) studies of diffusion in zeo-
0 liteshavebecomeincreasinglypopularwiththeadventof
2 Thechannels(orpores)ofzeolitesgenerallyhavecross
powerful computers and improved algorithms. In a MD
0 section somewhat larger than a benzene molecule. Some
simulation the movement is calculated by computing all
/ zeoliteshaveone-dimensionalchannelsparalleltoonean-
s forces exerted upon the individual particles. MD results
c other and no connecting cages large enough for guest
have been found to match experimental observations of
si molecules to cross from one channel to the next. The Single-Filediffusionforsystemswithonetypeofmolecule
y one-dimensional nature leads to extraordinary effects on
withoutconversionandwithveryshortpores.[4, 5,6, 7]
h thekineticpropertiesofthesematerials. Moleculesmove
Because a molecule can move to the right or to the left
p
in a concerted fashion, as they are unable to pass each
: neighboring site only if this site is free, MD simulations
v other in the channels. These structures are modeled
under heavyloadcircumstancesrequirea highcomputa-
i by one-dimensional systems called Single-File Systems
X tionaleffortforparticlesthathardlymove. However,the
where particles are not able to pass each other. A par-
r level of detail provided by MD simulations is not always
a necessary.
Thus, deterministic models are used also but they
are mainly focused on dynamic and steady-state infor-
∗Department of Mathematics and Computing Science, Eindhoven
mation of short pore systems. [8, 9, 10] Several re-
UniversityofTechnology, P.O.Box513,5600MBEindhoven,The
Netherlands.;Electronicaddress: [email protected] searchers [11, 12, 13] used a stochastic approach, i.e.,
†Department of Chemical Engineering, Eindhoven University of Dynamic Monte Carlo(DMC), to determine the proper-
Technology, P.O.Box513,5600MBEindhoven, TheNetherlands. ties of Single-File Systems. In DMC reactions can be
‡Department of Mathematics and Computer Science, Eindhoven included. The rates of the reactions determine the prob-
UniversityofTechnology, P.O.Box513,5600MBEindhoven,The
ability with which different configurations are generated
Netherlands.
§Department of Biomedical Engineering, Eindhoven University of and how fast (at what moment in time) new configura-
Technology, P.O.Box513,5600MBEindhoven, TheNetherlands. tions are generated. The most severe limitation of the
DMC method arises when the reaction types in a model morecorrelationsbetweenneighboringsitesin regionsof
can be partitioned into 2 classes with vastly different re- the systems with high occupancy gradients and less cor-
actionrates. Inthiscase,extremlylargeamountsofcom- relationsinregionswithlowandnooccupancygradients.
puter time are requiredto simulate a reasonablenumber StartingfromWei[8]resultsaboutcorrelationsinSingle-
of chemical reactions. However, in general the system File Systems, Okino and Snurr [10] used a deterministic
can be simulated for much longer times than with MD. model where each site was assumed to have equal activ-
ity towards reaction. Doublet approximation was found
All the previous references put the emphasis on the
tooverpredicttheoccupancyofthesitesandtheincreas-
transportpropertiesofadsorbedmoleculesastheimpor-
ing mobility raised the concentration of reactants in the
tantfactorinseparationandreactionprocessesthattake
pore.
place within zeolites and other shape-selective microp-
orouscatalysts. Ro¨denbeckandK¨arger[9]solvednumer- UsingDMCsimulationswehaveobservedthatevenfor
icallytheprincipaldependenceofsteady-stateproperties infinitelyfastdiffusion,westillhaveSingle-Fileeffectsin
suchasconcentrationprofilesandtheresidencetimedis- the system. Instead of focusing on diffusion at different
tribution of the particles, on the system parameters for occupancies of the system, we therefore concentrate in
sufficiently short pores. In multiple papers, Auerbach et this paper on the reactivity of the system, studying the
al. [14, 15] used Dynamic Monte Carlo to show different reactivity of the system for different sets of kinetic pa-
predictions about Single-File transport and direct mea- rameters, the length of the pipe and the distribution of
surements of intercage hopping ion strongly adsorbing the reactive sites. We analyse the situations when MF
quest-zeolite systems. Saravananand Auerbach [16, 17] gives good results and when MF results deviate strongly
studied a lattice model of self-diffusion in nanopores, to from the DMC simulations. We investigate the effect of
explore the influence of loading, temperature and adsor- thevariousmodelassumptionsmadeaboutdiffusion,ad-
batecouplingonbenzeneself-diffusioninNa-XandNa-Y sorption/desorption, and reaction on the overall behav-
zeolites. They applied Mean-Field(MF) approximation ior of the system. We look at the total loading, loading
for a wide set of parameters, and derived an analyti- with different components, generation of reaction prod-
cal diffusion theory to calculate diffusion coefficients for ucts and occupancies of individual sites as a function of
various loadingsat fixed temperature,denoted as ”diffu- the various parameters of a Single-File System.
sionisotherms”. Theyfoundthatdiffusionisothermscan
In section II we specify our mathematical model for
be segregated into subcritical and supercritical regimes,
diffusion and reaction in zeolites together with the theo-
depending upon the system temperature relative to the
retical background for the analytical and simulation re-
critical temperature of the confined fluid. Supercriti-
sults. In section IIIA we present the various results for
cal systems exhibit three characteristic loading depen-
the simplified modelwithout conversion. InsectionIIIB
dencies of diffusion depending on the degree of degen-
we use MF theory to solve the Master Equation govern-
eracy of the lattice while the subcritical diffusion sys-
ing the system behavior for the case when all the sites
tems are dominated by cluster formation. Coppens and
havethe sameactivity towardsconversion. Similarly the
Bell [18, 19, 20] studied the influence of occupancy and
resultsobtainedusingDMC simulationsarepresentedin
pore network topology on tracer and transport diffusion
section IIIB2 and are compared with MF results. We
inzeolites. They foundthatdiffusioninzeolitesstrongly
pay special attention to the infinitely fast diffusion case
depends on the pore network topology and on the types
andtotheinfluenceofthelengthofthepipeontheover-
and fractions of the different adsorption sites. MF cal-
all behavior of the system. In section IIIC we analyze
culations can quickly estimate the diffusivity, although
again the MF and simulation results but for the case
large deviations from the DMC values occur when long-
when only some of the sites are reactive. The influence
timecorrelationsarepresentathigheroccupancies,when
of the position and number of reactive sites on the reac-
the site distribution is strongly heterogeneous and the
tivity and site occupancy of the system is outlined. The
connectivity of the network low.
last section summarizes our main conclusions.
Few researchers included also reactivity in Single-File
Systems. Tsikoyannis and Wei [8] considered a reac-
tive one-dimensional system with all the sites reactive
in order to get more information about the reactivity
and selectivity in one-dimensional systems. They used a
II. THEORY
Markov pure jump processes approach to model zeolitic
diffusion and reaction as a sequence of elementary jump
events taking place in a finite periodic lattice. Monte In this section we will give the theoretical background
Carlo and approximate analytical solutions to the de- for our analytical and simulation results. First we will
rived Master Equation were developed to examine the specifyourmodelandthenwewillshowthatthedefined
effect of intracrystalline occupancy on the macroscopic system obeys a Master Equation. [21] We will simulate
diffusional behavior of the system. One conclusion was thesystemgovernedbythisMasterEquationusingDMC
that better results using analytical approach can be ob- simulations. The rate equations used for the derivation
tainedcomparedtoDMCsimulationresultsbyincluding of the analytical results are outlined.
A. The Model B. Master Equation
Because we are interested in reaction of molecules in Reaction kinetics is described by a stochastic process.
Single-FileSystems,wecallthesystemwearemodelling, Everyreactionhasamicroscopicrateconstantassociated
Single-File System with conversion. We model a Single- with it that is the probability per unit time that the
File System by a one-dimensional array of sites, each reaction occurs. Stochastic models of physical systems
possibly occupied by a single adsorbate. The sites are can be modelled by a Master Equation. [21]
numbered 1, 2, ... , S. An adsorbate can only move if By α, β, we will indicate a particular configuration of
an adjacent site is unoccupied. The sites could be reac- the system i.e.,a particularwayto distribute adsorbates
tive or unreactive andwe note with Nreac the number of over all the sites. Pα(t) will indicate the probability of
reactive sites. A reactive site is the only place where a findingthesysteminconfigurationαattimetandWαβ is
reaction may take place. the rate constant of the reaction changing configuration
We consider two types of adsorbates, A and B, in β to configuration α.
our model and we denote with X the site occupation The probability of the system being in configuration
of a site, X=( , A, B), which stands for an empty α at time t + dt can be expressed as the sum of two
site, a site occu∗pied by A, or a site occupied by a B, terms. The first term is the probability to find the sys-
respectively. The sites at the ends of the system are tem already in configuration α at time t multiplied by
labeled with m, and the reactive sites are labeled with the probability to stay in this configuration during dt.
r (see figure 1). We restrict ourselves to the following The second term is the probability to find the system in
mono and bi-molecular transitions. some other configuration β at time t multiplied by the
probability to go from β to α during dt.
a) Adsorption and desorption
Adsorption and desorption take place only at the Pα(t+dt)=(1 dt Wβα)Pα(t)+dt WαβPβ(t)
−
two marginal sites i.e., the left and rightmost sites at β β
X X
the ends of the system. (1)
By taking the limit dt 0 this equation reduces to a
→
Master Equation:
A(gas) + m Am
∗ −→
Am A(gas) + m
−→ ∗
Bm −→ B(gas) + ∗m, dPα(t)
= [W P (t) W P (t)]. (2)
αβ β βα α
where m denotes a marginal site. Note that there is no dt −
β
X
B adsorption. B’s are formed only by a reaction.
Analytical results can be derived as follow. The value
b) Diffusion ofapropertyX isaweightedaverageoverthevaluesXα
which is the value of X in configuration α:
In the pipe, particles are allowed to diffuse via hopping
to vacant nearest neighbor sites.
X = P X . (3)
α α
h i
α
X
An + ∗n+1 ←→ ∗n + An+1 From this follows the rate equation
Bn + n+1 n + Bn+1,
∗ ←→ ∗
d X dP
where the subscripts are site indices: n=1, 2, ... , S-1. h i = αX
α
dt dt
α
X
c) Reaction
= [W P W P ]X
αβ β βα α α (4)
−
αβ
An A can transform into a B at a reactive site. X
= W P (X X ).
αβ β α β
−
αβ
X
Ar Br.
−→
The initial state of the system is all that all sites are C. Dynamic Monte Carlo
empty (no particles in the pipe). In this paper we will
only look at steady-state properties and not to the time Because it might be not always possible to solve the
dependenceofthesystempropertiesstartingwithnopar- MasterEquationanalytically,DMC methods allowus to
ticles. simulate the system governed by the Master Equation
desorption
adsorption
m r r m
.......
adsorption
desorption
FIG. 1: Picture of a Single-File System with two typesof adsorbates, A(lighter colored) and B(darker colored). The marginal
sites are labeled with m, and the reactive sites(lighter colored) with r. Adsoption of A and desorption of A and B can take
place only at thetwo marginal sites. AnA can transform into a B only on r labeled sites.
overtime. We simplify the notationofthe MasterEqua- DMC method is not to compute probabilities P (t) ex-
α
tionbydefiningamatrixWcontainingtherateconstants plicitly, but to start with some particular configuration,
W , and a diagonal matrix R by R W , if representative for the initial state of the experiment one
αβ αβ ≡ γ γβ
α = β, and 0 otherwise. If we put the probabilities of wantsto simulate,andthengenerateasequenceofother
the configurations P in a vector P, we canPwrite the configurations with the correct probability. The method
α
Master Equation as generatesatime t′ whenthe firstreactionoccursaccord-
ing to the probability distribution 1 exp[ R t]. At
αα
time t′ a reaction takes place such tha−t a new−configura-
dP = (R W)P. (5) tion α′ is generated by picking it out of all possible new
dt − − configurationsβ withaprobabilityproportionaltoWα′α.
At this point we can proceed by repeating the previous
where R and W are assumed to be time independent.
steps,drawingagainatimeforanewreactionandanew
We also introduce a new matrix Q, Q(t) exp[ Rt].
≡ − configuration.
This matrix is time dependent by definition and we
can rewrite the Master Equation in the integral form
t
P(t)=Q(t)P(0)+ dt′Q(t t′)WP(t′). (6)
−
Z0 III. RESULTS AND DISCUSSION
By substitution we get of the right-hand-side for P(t′)
A. No conversion
P(t)=[Q(t)
t
+ dt′Q(t t′)WQ(t′) We mention in this section various results for the sys-
− tem without conversion. These results can be derived
Z0 (7)
t t′ analytically. Thederivationsarenotdifficult,soforcom-
+ dt′ dt′′Q(t t′)WQ(t′ t′′)WQ(t′′) pleteness we give them in the appendix. We will use the
− −
Z0 Z0 results when we deal with the system with conversion.
+...]P(0).
In a Single-File System without conversion, the rele-
Suppose at t = 0 the system is in configuration α vantprocessestodescribeareadsorption,desorptionand
with probability Pα(0). The probability that, at time diffusion. So, Wαβ is given by
t, the system is still in configuration α is given by
Q (t)P (0) = exp( R t)P (0). This shows that the
αα α αα α
firsttermrepresents−thecontributiontotheprobabilities W =W ∆(ads)+W ∆(des)+W ∆(diff), (8)
αβ ads αβ des αβ diff αβ
when no reaction takes place up to time t. The ma-
trix W determines how the probabilities change when
a reaction takes place. The second term represents the where∆(rx) equals1 ifa reactionoftype “rx”cantrans-
contribution to the probabilities when no reaction takes αβ
form the system from β to α, and equals 0 otherwise.
placebetweentimes0andt′,somereactiontakesplaceat
W , W , W are the rate constants of adsorption,
timet′,andthennoreactiontakesplacebetweent′andt. ads des diff
desorption and diffusion respectively.
The subsequenttermsrepresentcontributionswhentwo,
three, four, etc. reactions take place. The idea of the If we substitute expression (8) into the Master Equa-
tion (2), we get and
2
ddPtα =Wads ∆(αaβds)Pβ −∆(βaαds)Pα hAnAn+1i=(cid:20)WadWs+adWs des(cid:21) , (15)
Xβ h i
Notethatthisprobabilitydoesnotdependonthesite,
+Wdes ∆(αdβes)Pβ −∆(βdαes)Pα (9) all sites have equal probability to be occupied and that
Xβ h i there is no correlation between the occupation of neigh-
+W ∆(diff)P ∆(diff)P . boringsites. Againdiffusiondoesn’tinfluencetheseprop-
diff αβ β − βα α erties. Note also that these expressions are the same as
Xβ h i for a model in which particles are allowed to pass each
Using this expression we can show that when the sys- other.
tem is in steady state then the probability of finding the
system in a certain configuration depends only on the
number of particles in the system. B. All sites reactive
P =q N (10) We look first at the situation with all sites reac-
α α
tive: i.e., conversion of an A into a B particle can
where Nα is the number of pa(cid:0)rtic(cid:1)les in configuration α. take place at any site including the marginal sites. For
The expression for q(N) is: simplicity we consider W =W =W , and also
desA desB des
W =W =W . We will be looking at the total
diffA diffB diff
S N
W W loading(Q),thetotalloadingofA’s(Q ),thetotalload-
q(N)= des ads . (11) A
(cid:20)Wdes+Wads(cid:21) (cid:20)Wdes(cid:21) ingofB’s(QB),thenumberofB’sproducedperunittime
(B ), and how the distribution of A’s and B’s varies
prod
Note that diffusion has here no effect on steady-state from site to site ( A and B ).
n n
h i h i
properties. Note that the total loading of the pipe for the model
Theloadingofthepipe,definedastheaveragenumber with conversion is the same as for the model without
of particles per site, is then conversion
W
1 S W Q= ads . (16)
QA = Np(N)= ads , (12) Wads+Wdes
S W +W
ads des
N=0 The loadings and the production of B’s can easily be
X
derived from the probabilities A and B so we first
n n
where p(N) is the probability that there are N particles h i h i
focus on them. For a non-marginal site we can write
inthesystem. Noteagainthatdiffusiondoesn’tinfluence
the steady-state loading. dhAni =R (A,diff)+R(rx), (17)
The standard deviation, i.e., the fluctuation in the dt n n
number of particles is then:
where R (A,diff) is the rate of diffusion of A from and
n
to site n, and R (rx) is the rate of conversion of A to
2 n
S S
B on site n. The conversion takes place at one site and
√σ2 = N2p(N) Np(N)
vu −" # is therefore easier to handle than the diffusion. Using
uNX=0 NX=0 (13) equation (4) we have
t
W W
ads des
= S. R (rx) =W ∆ (rx)P (A A ), (18)
s(Wdes+Wads)2 n rx αβ β nα− nβ
αβ
X
To determine how the parameters of the system influ- where A = 1 if site n is occupied by an A in configu-
nα
ence the kinetics of the system, we are interested in the ration α and A =0 if not. We have A A =0 if
nα nα nβ
− 6
correlation in the occupancy between neighboring sites. there is an A at site n in configuration β (A =1) that
nβ
We lookatone site occupancyandat twosites occupan- has reactedto a B leading to configurationα (A =0).
nα
cies. We denote by An the probability that an A is at This gives us
h i
site n and with A A the probability to have an A
h n n+1i ′
at site n and one at site n+1. R (rx) = W P = W A , (19)
n rx β rx n
Oneandtwo-siteprobabilitiescanbe derivedfromthe − − h i
β
factthatallconfigurationswiththesamenumberofpar- X
ticleshaveequalprobabilityandtheexpressionsforq(N). where the prime restricts the summation to those β’s
We find with Anβ =1. For the diffusion we similarly get
An = Wads , (14) Rn(A,diff) =Wdiff ∆αβ(A,diff)Pβ(Anα−Anβ). (20)
h i Wads+Wdes Xβ
There are four ways in which A A = 0 and
nα nβ
− 6
∆αβ(A,diff) =0inβ: thereisanAatsitenthatcanmove W W
to site (n 6 1), there is an A at site n that can move to 0= diff des An+1 + An−1 2 An Wrx An ,
(n+1),th−ereisanAatsite(n 1)thatcanmovetosite Wads+Wdes − −
nandthereisanAatsite(n+1−)thatcanmovetositen. 0= WdiffWdes (cid:2)(cid:10)B (cid:11)+(cid:10)B (cid:11) 2(cid:10)B (cid:11)(cid:3)+W (cid:10)A (cid:11),
n+1 n−1 n rx n
Inallcaseswehave∆ (A,diff) =1.Inthefirsttwocases Wads+Wdes −
αβ
W W (cid:2)(cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11)(cid:3) (cid:10) (cid:11)
wAenαhavAenAβn=α1−.TAhneβs=um−m1atainodnoinvetrhβe ilnastthtewfiorswtecahsaevies 0= Waddsiff+Wdedses A2 − A1 −Wrx A1 −Wdes A1
−
restrictedto configurationswithanAatsiten anda va- WadsWdes (cid:2)(cid:10) (cid:11) (cid:10) (cid:11)(cid:3) (cid:10) (cid:11) (cid:10) (cid:11)
+ ,
cantsite(n−1). Thisgivesaterm−Wdiffh∗n−1Ani.The Wads+Wdes
aonthderWcdaiffsehs∗ngAivne+t1eir.mTshe−rWatdeiffehqAuna∗tnio+n1si,thWendiffbheAconm−1e∗sni, 0= WWaddsiff+WWdedses B2 − B1 +Wrx A1 −Wdes B1 ,
W W (cid:2)(cid:10) (cid:11) (cid:10) (cid:11)(cid:3) (cid:10) (cid:11) (cid:10) (cid:11)
dhdAtni =Wdiff[−hAn∗n−1i−h∗n−1Ani+hAn−1∗ni 0= Waddsiff+Wdedses AS−1 − AS −Wrx AS −Wdes AS
W W (cid:2)(cid:10) (cid:11) (cid:10) (cid:11)(cid:3) (cid:10) (cid:11) (cid:10) (cid:11)
+ nAn+1 ] Wrx An . + ads des ,
h∗ i − h i W +W
(21) ads des
W W
diff des
0= B B +W A W B .
S−1 S rx S des S
For B we get similarly W +W − −
n ads des
h i (cid:2)(cid:10) (cid:11) (cid:10) (cid:11)(cid:3) (cid:10) (cid:11) (25(cid:10)) (cid:11)
d B
n
h i =Wdiff[ Bn n−1 n−1Bn + Bn−1 n We haveusedhere the probabilityfor asite to be vacant
dt −h ∗ i−h∗ i h ∗ i
thatwehavedeterminedforthecasewithoutconversion.
+ B ]+W A .
h∗n n+1i rxh ni We note that these equations are identical to the MF
(22)
equations of a system in which the particles can move
independently with a rate constant for diffusion equal
The marginal sites have also adsorption and desorption.
to W W /(W +W ). This means that the MF
Theycanbedealtwithastheconversion. Therateequa- diff des des ads
does not really model the non-passing that characterizes
tions for A are
a Single-File System.
d A The continuum limit of the MF equation is
1
h i =W [ A + A ]+W
diff 1 2 1 2 ads 1
dt −h ∗ i h∗ i h∗ i
∂a/∂t 1 b a ∂2a/∂x2 a
−WdeshA1i−WrxhA1i, ∂b/∂t =D −b 1 a ∂2b/∂x2 +Wrx −a ,
(cid:18) (cid:19) (cid:18) − (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
d AS (26)
h i =W [ A + A ]+W
diff S S−1 S S−1 ads S
dt −h ∗ i h∗ i h∗ i
W A W A , where a=a(x,t) is the probability distribution of A’s (a
des S rx S
− h i− h i similardefinitionholdsforb),andD=W d2,withdthe
(23) diff
distancebetweenneighboringsites(seeappendix). These
and the rate equations for B are the equations that are normally used to describe dif-
fusion in Single-File Systems. [19, 22, 23, 24] As this
d B equation is derived from the MF equations, it has the
1
h i =W [ B + B ] W B +W A ,
diff 1 2 1 2 des 1 rx 1 same drawback; i.e., the Single-File behavior is only in-
dt −h ∗ i h∗ i − h i h i
d B corporated by the reduction of the diffusion, but it does
S
h i =Wdiff[ AS S−1 + SAS−1 ] Wdes B1 effectively allow for passing of particles. This shows up
dt −h ∗ i h∗ i − h i
as so-called counter diffusion of A’s and B’s. [22, 23, 24]
+W A .
rxh Si We see thatequations(25)arelinearandwe cansolve
(24)
them at least numerically. We think however that it is
worthwhile to use an analytical approach. We consider
Note that these coupled sets of differential equations are
the ansatz
exact.
A xn (27)
n
∝
1. Mean Field results inthe steady-stateequa(cid:10)tion(cid:11)s(25)for An .This leadsto
h i
x2 2(1+α)x+1=0, (28)
We will now look at the loadings QA and QB and the −
site occupation probabilities An and Bn . We will with
h i h i
first determine steady-state properties using the (MF)
W W +W
approximation: i.e, we put A = A etc. rx des ads
n n+1 n n+1 α= . (29)
in the rate equations. This gihves∗us i h ih∗ i 2Wdiff Wdes
100 2.8
90 2.6
80 2.4
70 2.2
60 2
50 Bprod1.8
1.6
40
1.4
30
1.2
20
1
10
0.8
0 0 500 1000 1500 2000 2500 3000
0 0.1 0.2 0.3 0.4x0.5 0.6 0.7 0.8 0.9 1 Wads
1
FIG. 2: The characteristic length ∆ as a function of x1. FIG.3: Bprodperunittimeatonemarginalsiteasafunction
ofWadsforS=Nreac=30,Wdes=0.8,Wdiff =2andWrx=0.4.
The quadratic equation yields two solutions x and x set Wads Wdes Wdiff Wrx
1 2
with x = x−1. We have x = x = 1 only when α = 0, a) 0.2 0.8 0.05 0.01
2 1 1 2 b) 0.2 0.8 0.05 0.1
i.e. when W = 0. We will therefore assume α > 0 and
rx c) 0.2 0.8 2 0.1
x <1. Then
1 d) 0.2 0.8 1 2
e) 0.2 0.8 10 2
x =(1+α) α(α+2). (30)
1 − f) 0.8 0.2 0.05 0.01
We can write then the soluption A = a (x )n + g) 0.8 0.2 0.05 0.1
n 1 1
a (x )S+1−n. The symmetry in thhe oiccupancy of the h) 0.8 0.2 2 0.1
2 1 i) 0.8 0.2 1 2
pipe A = A yields a = a = a. So, the gen-
h ni h S+1−ni 1 2 j) 0.8 0.2 10 2
eral solution for the steady state has the form:
A =a(xn+xS+1−n). (31)
n 1 1 TABLE I: The sets of parameters used for thesimulations
The coefficien(cid:10)t a(cid:11)is to be determined from the equa-
tions for the marginal sites in the set of equations(25).
so, the expression for Q is
A
Inthe leftsideofthe systemnissmalland(S+1 n)is
−
large. Because x <1 we can neglect the second term in
1
equation(21)and An x1n.Thismeansthattheprob- a S 2ax (1 x S)
ability of finding an A a∝t site in the left-hand-side of the QA = [x1n+x1S+1−n]= 1 − 1 , (34)
systemisanexpon(cid:10)enti(cid:11)allydecreasingfunctionofthesite S n=1 S 1−x1
X
index. If we write An e−∆n, we find ∆ = 1/ln(x1)
h i ∝ −
for the characteristic length of the decrease. The loga-
Q =Q Q . (35)
B A
rithm makes this length only a slowly varying function −
ofthe rateconstants(see figure 2). WhenWdiff becomes The total production of B’s is
larger, α approaches 0, x approaches 1 and ∆ diverges.
1
Note that this is a MF result. We will see that in the x (1 x S)
1 1
B =W Q S =2aW − . (36)
simulations ∆ remains finite. Also when the conversion prod rx A rx 1 x
1
is slow moreA’s arefound awayfromthe marginalsites. −
The second factor in the expression for α equals the re-
ciprocal of a site being vacant. Low loading leads to a 2. Simulation results
smaller α than high loading. Because of the vacancies
the A’s can penetrate farther into the system before be- We presentnowthe resultsfor differentsets ofparam-
ing converted. For slow conversion or fast diffusion α is eters and we compare them with MF results. Because
small and ∆ can be approximated by we cansee fromequation(36)that largerpipes don’tin-
creasetheproductivityofthesystem,weconsiderforthe
W W
diff des comparisonsoftheresultsasystemsizeS=30. Wehave
∆= . (32)
r Wrx Wdes+Wads considered separately the sets of parameters in Table I.
The sets of parameters from a) to e) are for the cases
The total loading with A’s, Q , is
A oflowloadingandfromf)toj)forthehighloading. The
parametersinthetabledescribethe followingsituations:
S a)andf)forveryslowreactionandslowdiffusion;b)and
1
QA = An (33) g)forslowreactionandslowdiffusion; c)andh)forslow
S h i
n=1 reactionandfastdiffusion;d)andi)forfastreactionand
X
QA Bprod Q For the case Wads we have
set MF Sim MF Sim Sim →∞
2W W
a) 0.0330 0.0318 0.0099 0.0100 0.209 rx des
B = . (37)
prod
b) 0.0149 0.0148 0.0491 0.0472 0.198 W +W
rx des
c) 0.0385 0.0342 0.1156 0.1024 0.204
From the simulations (see figure 3) we see that for high
d) 0.0040 0.0041 0.2449 0.2463 0.200
e) 0.0046 0.0044 0.2767 0.2729 0.201 adsorption rates, Bprod converges to a point and the
f) 0.0798 0.0748 0.0239 0.0235 0.795 corresponding value is equal to the analytical value for
g) 0.0376 0.0373 0.1129 0.1157 0.804 the case adsorption is infinitely fast. The reason for
h) 0.0598 0.0486 0.1796 0.1406 0.802 this is that all the sites are occupied, diffusion is com-
i) 0.0048 0.0049 0.2931 0.2943 0.797 pletely suppressed, and only the marginal sites play a
j) 0.0050 0.0049 0.3013 0.2957 0.801 role. The expression above can be seen as a factor 2
for the two marginal sites, the probability that an A at
the marginal sites is converted to a B before it desorbs
TABLEII: Simulation andMF resultsfor QA andBprod for
W /W +W , and the rate constant for desorption
all thesets of parameters rx rx des
W .
des
The accuracy of the simulation results for Q and
A
B canbederivedbylookingatthetotalloadingQin
prod
Table II. For the total loading Q, the simulation results
can be comparedwith the values of the exact expression
(12). We remark that the largest deviation from the ex-
slow diffusion; e) and j) for fast reaction and fast diffu-
act analytical results is 0.04, so the relative errors are
sion.
around 0.02%.
We can see from Table II that the simulation and MF
The differences between MF and the simulations be-
results match for all the cases except the cases when we
comes especially clear in the limit W . Because
have low reaction rates and fast diffusion for both low diff
→ ∞
this limit makes the system homogeneous in MF we get
and high loading. In these cases MF overestimates the
amount of A’s in the pipe, and consequently overesti- W W
ads rx
mates the B production. In figure 4 we have the site QB = , (38)
W +W W +W
ads des rx des
occupancy with A and B both from the simulations and
MF. We again see that the MF and the simulation re-
W W
sults agree reasonably well, except for low reactionrates Q = ads des . (39)
A
W +W W +W
and fast diffusion. MF overestimates the characteristic ads des rx des
length ∆ and allows A’s to penetrate farther into the
pipe than in the simulations. The reason for this is that
MFdescribesthefactthattheparticlescannotpasseach The first factor in these expressions is the probability
other by reducing the diffusion, but this effectively does thatasiteisoccupied. Thesecondfactorindicatesifthe
allow for passing. The larger ∆ in MF means also a particleis convertedto a B ornotbefore itdesorbs. The
larger Q . As a consequence the B production in MF is simulationsshowthatthe systemshouldnotbe homoge-
A
largerand,becausetheseB’shavetobeabletoleavethe neous at all (see figure 6). The B production increases
pipe via desorption, the probabilities B and B are linearly with S only for the case of infinitely-fast diffu-
1 S
largerinMF.The probabilities A ahnd iA ahrethiere- sion, otherwise it converges to a limiting value.
1 S
h i h i
fore smaller, which means that the MF curves and the
simulation curvesin figure cross eachother, as can actu-
C. Only some of the sites reactive
ally be seen. The behavior of the systemat high loading
and at low loading is about the same, except that ∆ is
smaller at high loading. We consider now the situation that not all the sites
Onemightexpectthatthelargerthenumberofreactive are reactive, and that these reactive sites can be either
sitesthemoreB’swillbeproducedinthepipe. Fromthe uniformly distributed inside the pipe or distributed in
simulations we see that the amount of B’s produced per compact blocks. We will show that the number of reac-
unit time by all reactive sites goes to a limit value when tive sites doesn’t change qualitatively the properties of
the number of reactive sites is increased. In figure 5, the the system. QA, QB and number of B produced for a
marked line represents the B production as a function variable number of reactive sites are compared with the
on the length of the pipe and the dashed line the B pro- previous results.
duction according to MF. For short pipe lengths, the B
production from both MF and simulations increase lin-
early with S, while for higher lengths it converges to a 1. Mean Field
limiting value. The limiting value is higher for MF. This
could be seen also from the Table II. According to MF From the Master Equation it is easy to show that the
there are more B produced in the pipe. total loading is again just the same as in the case when
0.25 a) 0.25 b) 0.25 c)
0.2 0.2 0.2
<A >,<B >nn00.1.15 <A >,<B >nn00.1.15 <A >,<B >nn00.1.15
0.05 0.05 0.05
00 5 10 15 20 25 00 5 10 15 20 25 00 5 10 15 20 25
n n n
0.25 d) 0.25 e)
0.2 0.2
B >n0.15 B >n0.15
<A >,<n 0.1 <A >,<n 0.1
0.05 0.05
0 0
0 5 10 15 20 25 0 5 10 15 20 25
n n
FIG.4: ThesiteoccupancywithA(hAni)andB(hBni)asafunctiononsitenumberforcasesa,b,c,d,ewhenS =Nreac =30.
The continuous line and the corresponding symmetric line represent MF results for hAni and hBni respectively. The dashed
lines represent DMC results for hAni and hBni. hAni is decreasing towards the middleof thepipe while hBni is increasing.
0.12
0.1
0.08
d
o
pr 0.06
B
0.04
0.02
0
5 10 15 20 25 30 35 40 45 50
S
FIG. 5: B production as a function on the length of the pipe for Wads=0.2, Wdes=0.8, Wdiff=2, and Wrx=0.1. The marked
line represents theDMC results and thedashed line representsthe MF results.
0.9
0.8
0.7
0.6
>
B n0.5
<
>, 0.4
A n
< 0.3
0.2
0.1
0
0 5 10 15 20 25
n
FIG. 6: Analytical and simulation results for site occupancy of a system when parameters are: S = Nreac = 30, Wads=0.8,
Wdes=0.2, Wdiff=100 and Wrx=2. The continuous line and the corresponding symmetric line represent the simulation profiles
for site occupancy with A and B particles. The bottom and the upper straight lines represent the analytical results for
occupancy with A and with B particles respectively.
all the sites are reactive. We introduce an extra coef- always be occupied by a particle A and the B’s will not
ficient ∆ in the MF equations to the reaction term. beabletogetoutofthepipe. InMFparticleseffectively
n
∆ = 1 if n is a reactive site and ∆ = 0 if it is not canpasseachother,soBparticlesarethenabletogetout
n n
a reactive site. The steady-state equations are identical ofthepipe. Evenforthecaseoflowloadingwestillhave
to equations (25), except that W should be replaced deviations from MF for fast diffusion and fast reaction.
rx
by W ∆ . The resulting set of equations is linear again In this case MF overestimates A’s for nonreactive sites.
rx n
and it should be possible to solve them numerically. In For fast diffusion and slow reaction, MF underestimates
fact only the probabilities for the marginal and reactive A’s for nonreactive sites and for slow diffusion and slow
sites have to be solved numerically. For the other sites reaction MF overestimates B’s for the reactive sites in
the probabilities can be obtained by simple linear inter- the middle.
polation. That this is correct can be seen because those Figures 7 and 8 show how the probabilities A and
n
h i
sites only have the diffusion term. We can also remove B vary in the pipe. The situations for reactive sites
n
h i
the probabilities for the B’s, because we have from the forming blocks at the ends of the pipe are not shown
model without conversionthat as they are almost the same as when all the sites are
reactive (see figure 4). When the reactive sites are ho-
W
A + B =1 = ads . (40) mogeneously distributed the plots look also very similar
n n n
− ∗ W +W
ads des to the ones with all sites reactive, except that the char-
(cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11)
acteristic length ∆ is larger. A and B look very
The resulting equations for the reactive sites have the h ni h ni
differentwhenthe reactivesites formablockinthe mid-
same form as equation (25) for the non-marginal sites.
dle of the pipe. The MF results show, as predicted, a
We expect therefore that we get anexponential decrease
linear behavior at the nonreactive sites. The MC results
of A on the reactive sites when we move from the
n
h i show,however,anonlinearbehaviorinthe formofS-like
marginal sites to the center of the pipe, and a linear de-
curves. At the reactive sites the behavior is similar to
pendence on n between the unreactive sites.
the situation with all sites reactive with the MC results
showing a more rapid approach to the value at the mid-
dle of the pipe than MF, i.e., smaller ∆. The values at
2. Simulation results
the marginal sites can differ between MC and MF quite
a lot. This reflects the difference in B mentioned be-
Thenumberofreactivesitesisconsideredtovaryfrom prod
fore: adifferentB mustbeaccompaniedbyadifferent
1 to 50% and the reactive sites are distributed either in prod
B desorption at steady-state. As we have already seen
blocks situated near the marginal sites, in the middle of
fromthecasewhenallthesiteswerereactive,B very
the pipe, or homogeneously distributed in the pipe. We prod
rapidly approaches the limiting value when the pipe is
will first compare the MF results with the MC simula-
made longer (see figure 5). Similarly when we start with
tion results for different sets of parameters and then we
few reactive sites and,instead ofincreasingthe lengthof
look at the dependence of B production and total load-
the pipe, we increase the number of reactive sites. The
ing Q onthe number andpositionofreactivesites. For
A loading Q is already almost the same as the value with
the comparisonbetweenMFandMC resultsweconsider B
all sites reactive when only about 10% of all sites are re-
the system size S = 30 and the number of reactive sites
active provided there are reactive sites at or very near
N =10. The sets of parameters used for the specific
reac the marginal sites. If the reactive sites are moved away
situations to be studied are the same as the sets used in
from the ends of the pipe, then the loading Q and the
thecasewithallthesitesreactiveintheprevioussection. B
B production decreases.
We can see from the Tables III and IV that when the
reactive sites are homogeneously distributed or situated
as a block in the middle ofthe pipe, there aresignificant
IV. SUMMARY
differences between MF results and MC results. When
thereactivesitesformblocksnearthemarginalsites,the
resultsarealmostthesameaswhenallsitesarereactive: We have used analytical and simulation techniques to
theMCandtheMFresultsdifferifwehavefastdiffusion study the reactivity in Single-File Systems.
andslowreaction. The sites inthe centerofthe pipe are The MF results show that MF models Single-File be-
not relevant when the sites at the ends of the pipe are havior by changing the diffusion rate constant, but it
reactive. Whenthereactivesitesaresituatedonlyinthe effectively does allow passing of particles.
middle of the pipe, we have deviations for all the sets of Whenallthesitesarereactive,thesimulationandMF
parameters. They are very prominent for the case when results are very similar for all the parameters, except
we have high loading, fast diffusion and fast reaction. for the case when we have low reaction rates and fast
MFstronglyunderestimatesA’sforallnon-reactivesites, diffusion. InthesecasesMFoverestimatestheamountof
butwehavealsoimportantdeviationsforhighloadingin A’sinthepipe. TheamountofBproducedperunittime
thecaseswithfastdiffusion-slowreaction,slowdiffusion- byallreactivesitesgoestoalimitvaluewhenthenumber
fast reaction, and slow diffusion-slow reaction. This is of reactive sites is increased. For high adsorption rates,
happening because for high loading, the end sites will B converges to a point and the corresponding value
prod