Nonequilibrium Distribution Function Theory of Many-Particle Effects in the Reversible Reactions of the Type A+B C+B

Transcription

1 986 Bull Korean Chem Soc 5, Vol 6, No Jinuk Lee et al Nonuilibrium Ditribution Function Theory of Many-Particle Effect in the Reverible Reaction of the Type AB CB Jinuk Lee, Jeik Uhm, Woojin Lee, Sangyoub Lee, * and Jaeyoung Sung,* Department of Chemitry, Seoul National Univerity, Seoul 5-747, KoreaUG * angyoub@nuackr Department of Chemitry, Chung-Ang Univerity, Seoul , Korea * jaeyoung@cauackr Received October 7, 5 We tudy the relaxation kinetic of reverible reaction of the type A B : C B by applying the manyparticle kernel theory, which we have developed to invetigate many-particle ect on general diffuioninfluenced reaction It i hown that for the target model, where A and C molecule are immobile and their interconverion i induced by the encounter with the B molecule that are preent in much exce, the manyparticle kernel theory give a reult that coincide with the known exact reult Key Word : Reverible diffuion-influenced reaction, Many-particle ect, Nonuilibrium ditribution function theory Introduction For the lat two decade, a great deal of ort ha been directed to invetigate the many-particle ect on the kinetic of reverible diffuion-influenced reaction -8 In reverible reaction, reactant molecule that were extinguihed by the forward reaction are regenerated by the backward reaction In general, the regenerated reactant would ee a non-uilibrium ditribution of reactant partner that depend on it previou reaction hitory Accordingly, the regenerated reactant would have a time-dependent probability of encountering reaction partner Thi fact make it difficult to develop a rigorou theory for the kinetic of reverible reaction By now, there are everal theoretical framework dealing with the kinetic of reverible reaction Among thee, reduced ditribution function (RDF) approach ha been one of the mot popular one, -4,9-,7 which tart from a hierarchical et of evolution uation for the RDF of the reactant molecule Explicit olution to the et of kinetic uation i then obtained by introducing a truncation approximation called the dynamic uperpoition approximation (SA),,9-,7 Due to it implicity and flexibility, the SA-baed RDF approach ha been applied to many type of reaction,,,9-,7,9, one of which i reverible aociationdiociation reaction, A B : C,,9- For thi type of reaction, the prediction of the SA-baed RDF theory agree well with that of Brownian dynamic (BD) imulation, 8 in the aymptotic time region a well a at hort time However, it wa revealed by Yang, Lee, and Shin (YLS) that for other imple reverible reaction of the type A B : C B, the SA-baed RDF theory how large dicrepancy from the exact reult 6 In addition, Gopich and Burhtein reported that the quantum yield for the latter type of reaction calculated by the SA-baed RDF theory differ much from the exact one 5 Intead of the SA-baed RDF theory, YLS preented another theoretical framework in which memory uation for the one-particle reactant denity field i obtained by applying Mori projection operator technique to formal mater kinetic uation decribing complete many-particle reaction dynamic Then the memory function appearing in the reulting memory uation wa analyzed by fully renormalized kinetic theory developed by Mazenko 3 The reult of their theory wa in better agreement with the exact reult than that of the SA-baed RDF theory, for the reaction of the type A B : C B Alo for the reverible reaction of the type A B : C, the YLS theory wa hown to provide a better qualitative explanation on the BD imulation 8 reult than the SA-baed RDF theory did, except at long time; 7 the YLS theory predict an incorrect amplitude of the aymptotic decay curve More recently, Gopich, Kipriyanov, and Doktorov applied the modified integral encounter theory (MET) to the reverible reaction of the type A B : C B 4 In MET, a formal olution to the mater kinetic uation i manipulated with the denity expanion of the propagator and ophiticated diagrammatic technique In the work, it wa hown that the MET and the YLS theory have the comparable accuracy However, unfortunately, even the MET and the YLS theory become inaccurate a the reactant denity increae, becaue low reactant denity approximation are addreed in both theorie: the reult of the former predict a lower relaxation and that of the latter predict a fater relaxation than the exact reult And neither of them reduce correctly to the exact Smoluchowki reult 4 in the irreverible limit Therefore, there wa acute need for developing more rigorou theory for the reverible reaction kinetic Recently, we reported an advanced RDF theory that doe not rely on the dynamic SA 3,4 In the work, it wa hown that the RDF formalim give a formally exact non-markovian rate uation and exact expreion for the time-dependent reactant number denitie The rate kernel appearing in the rate uation and the reactant number denity expreion involve a many-particle kernel (MPK) that carrie the information on the mean-field dynamical influence of urrounding reactant molecule on the reaction between a pair of

2 Many-Particle Effect in Diffuion-Influenced Reaction Bull Korean Chem Soc 5, Vol 6, No 987 reactant molecule The expreion for the many-particle kernel were determined in a ytematic manner by conidering the evolution uation for the three-particle RDF Thi new olution procedure in the RDF formalim ha been widely cited a the MPK theory in enuing literature 5,6 When it wa applied to the reverible aociation reaction, A B : C, 3 the reult of the MPK theory were better than any previouly reported theory and howed almot perfect agreement with the BD imulation 8 for the whole time range and for any et of reaction parameter In the preent work, we tudy the relaxation kinetic of reverible reaction of the type A B : C B by applying the MPK theory It i hown that the triking ucce of the MPK theory i not limited only to the reaction of the type A B : C: the MPK theory turn out to give the known exact reult, when it i applied to the reaction of the type A B : C B Theory The obervable quantitie are time-dependent number denitie of A and C molecule denoted by a( and c(, repectively On the other hand, the number denity of B molecule aumed to be preent in great exce of A and C molecule would remain ectively contant and i denoted by C B For the ake of implicity, we conider the reaction ytem where the pherical A and C molecule with radiu σ are immobile while the B molecule are mobile point particle and mutually independent The chemical reaction between A (or C) and B molecule i aumed to occur when B molecule come into contact with A (or C) molecule Only for the jut mentioned imple reaction model, the exact analytic reult i available According to reaction cheme, we can write down the kinetic uation for the time-dependent reactant number denity a d d ----a() t ----c() t κ () dt dt f ( σ, ( σ, Here, κ f and κ r repreent the inherent rate parameter for the forward and the revere bimolecular reaction The firt and the econd term decribe the change in the timeevolution of the number denitie due to the forward and the revere bimolecular reaction, repectively The twoparticle RDF (r, in the firt term repreent the averaged product of the number denitie of A and B molecule at the two location eparated by r, and the other two-particle RDF (r, in the econd term ha the imilar meaning The evolution uation for thee function are, in turn, given by d ----C dt AB L( r) δ ( r σ) σ d f σ, σ, () d ----C dt CB L( r) δ ( r σ) [ κ σ d f κ r κ f σ, κ r σ, L(r) in Eq () and (3) denote the operator decribing the thermal motion of B molecule In d patial dimenion, it explicit expreion i given a L( r)f( r) D B r d ( d/dr)r d e U ( d/dr)e U f( r), where D B i the diffuion contant of B molecule and U(r) i the potential of mean force (in unit of thermal energy) between A (or C) and B molecule The econd term on the right ide of Eq () and (3) take account of the change in the two-particle RDF due to the forward and the backward bimolecular reaction A B : C B i the metric factor that depend on the dimenionality, d, of the reaction ytem; it ual for 4π, for d 3, π for d, and for d For the impler onedimenional ytem where B molecule reide at one ide of an A or C molecule, The third and the fourth term decribe the change due to the competitive reaction of a third B molecule with the A and C molecule whoe pair dynamic with the B molecule at a ditance r i of primary concern Thee term contain the three-particle RDF, (r,r', and (r,r',; (r,r', denote the averaged product of the number denity of A molecule and the number denitie of B molecule at the two location eparated from the A by r and r', and (r,r', ha the imilar phyical meaning In turn, the evolution uation for thee three-particle RDF are given by ---- C t ABB r, [ L( r) L( r ) r, δ ( r σ) σ d f r, r, δ -- ( r σ) σ d f r, r, B r σ,, B r σ,, ---- C t CBB r, [ L( r) L( r ) r, δ( r σ) σ d f r, r, δ( r σ) -- σ d f r, r, κ f B r σ,, κ r B r σ,, Each term in the right ide of Eq (4) and (5) ha a imilar phyical meaning with the correponding term in Eq () and (3) In analogou manner, we can write down the evolution uation for the higher-order RDF up to N-th order, where N i the number of B molecule in reaction (3) (4) (5)

3 988 Bull Korean Chem Soc 5, Vol 6, No Jinuk Lee et al ytem For treating the macrocopic ytem where N i order of Avogadro number, we introduce a truncation approximation, intead of olving the whole et of hierarchical uation Before doing thi, let u looor the formally exact olution The method of olution can be delineated more clearly in the Laplace domain Denoting the Laplace tranform of a function f( by f ( [ d te f() t, we can ˆ t rewrite Eq () a â( a [ ĉ( c (6) Ĉ AB ( σ, Ĉ CB ( σ, αˆ ( where we have defined an auxiliary function αˆ ( a and c denote the initial value of a( and c( Next by defining a many-particle kernel according to the uation αˆ ( C B ξˆ g r we can obtain the formally exact formula for the twoparticle RDF a from Eq () and (3) repectively In Eq (8) and (9), ΔĈ AB Ĉ AB a( C B and ΔĈ CB Ĉ CB ĉ( C B, where g(r) denote the uilibrium pair correlation function obeying L(r)g(r) The initial pair correlation of A-B or C-B pair are aumed to be given by g(r) o that ) a C B and ) c C B Then, by ubtituting Eq (8) and (9) into Eq (6), we get a rate uation a where [ Ĉ ABB σ, Ĉ CBB r, σ, ΔĈ AB ΔĈ CB kˆ f σ d, αˆ ( δ( r σ) C L r B ξˆ r, αˆ L r δ( r σ) C σ d B ξˆ αˆ ( Ĉ AB ( σ, Ĉ CB ( σ, kˆ r kˆ f( â( C B kˆ r( ĉ( C B k r κ f ; Fˆ κ r F( ( r σ) C σ d B ξˆ r σ (7) (8) (9) () () () for â( and ĉ( from Eq (6) and (): where â( ( A)a ( C)c ĉ( Ŷ C ( A)a Ŷ C ( C)c Ŷ C Ŷ C ( A) [ kˆ r( C B Q ( C) kˆ r( C B Q ( A) kˆ f( C B Q ( C) [ kˆ f( C B Q (5) (6) (7) (8) (9) () with Q [ kˆ f( C B kˆ r( C B Here, i the probability that an A molecule at time i found till a A molecule at a later time t, while Y C i the probability that an A molecule at time i found to be C molecule at a later time t Y C ( tc) and Y C have the imilar meaning An exact reciprocal relation hold between ( tc) and Y C By comparing Eq (8) and (9), we get Ŷ C ( A) ( C) kˆ f K kˆ r( () In addition, we can alo get the exact generalized ma action law, ie Y C ( tc)k K Y C ( tc) () (3) Note that we have not yet made any approximation and the reult obtained up to now are exact However, a in Eq () and (3), the complicated many-particle ect are incorporated into the rate kernel kˆ f( and kˆ r( through the many-particle kernel ξˆ that i not yet determined To determine the many-particle kernel ξˆ a defined in Eq (7), we hould evaluate Ĉ ABB rˆ, and Ĉ CBB rˆ, From Eq (4) and (5), we can derive Ĉ ABB r, â( C B g( r ) αˆ ( C B δ r σ L r [ g( r ) ξˆ ( r, L( r ) σ d αˆ ( C B L r δ -- ( r σ) [ ξˆ L( r ) σ d k r Here, and denote uilibrium forward and backward reaction rate contant given by κ f g( σ) and κ r g( σ), repectively Note that the forward and the backward rate kernel kˆ f ( and kˆ r( defined in Eq () atify the detailed balance condition kˆ f ( kˆ r( k r K exactly, irrepective of the pecific form of evolution operator L(r) and the many-particle kernel ξˆ r In term (, ) of the rate kernel, we can obtain the following expreion αˆ ( C B [g r L( r ) ξˆ ξˆ ( r, Ĉ CBB r, ĉ( C B g( r ) αˆ ( C B L r ξˆ ( r, g( r )ξˆ δ( r σ) [ g( r ) ξˆ ( r, L( r ) σ d (4)

4 Many-Particle Effect in Diffuion-Influenced Reaction Bull Korean Chem Soc 5, Vol 6, No 989 C B αˆ δ( r σ) [ L( r ) σ d ξˆ C B [g r αˆ L( r ) ξˆ ξˆ ( r, (5) with the ue of the following approximation to truncate the hierarchy at the level of three-particle reaction dynamic: κ f Ĉ ABBB (6) Then by ubtituting Eq (6), (8), (9), (4) and (5) into Eq (7), we can obtain the following integral uation for ξˆ : (7) Then, according to the appendix of Ref 4, Fˆ ( defined in Eq (3) in term of many-particle kernel atifying Eq (7) can be approximately given by ( Fˆ ( (8) ( Here, ( i the urvival probability of A molecule undergoing the irreverible reaction AB B with the ective uilibrium forward rate contant given by k r The exact expreion for ( i given by (9) where i the time-dependent rate coicient that can be derived from the Smoluchowki approach for the irreverible reaction The expreion for are given by Ω( k t t D ) (d) (3) ξˆ ( r, g( r )ξˆ r σ,, Ĉ CBBBB r σ,, [ ξˆ [ κ f Ĉ ABB ( r σ,, C B g r Ĉ CBBB ( r σ,, α( C B g r ξˆ [ ξˆ [ g( r ) ξˆ ( r, δ ( r σ) ξˆ σ d r, κ f L r κ r C B [ δ ξˆ ( r σ) ( r, L( r ) σ d δ( r σ) -- ξˆ ( σ d r, ξˆ ( r, ξˆ C B ( )C B exp C B dτ ( τ) [ k π -- dx exp( x t t D ) x [ ( x) kj ( x) [ xy ( x) ky ( x) xj (d) (3) { kω[ ( k) t t D } k (d3) (3) d where k σ D DB, t σ D D /D B, Ω( x) (x )erfc(x), and J n, and Y n are the nth order Beel function of the firt and the econd kind, repectively 7 The ective contact ditance σ D are defined by σ D drexp[ U( r) r (33) σ When the ect of potential of mean force i negligible, U( r), the expreion for given by Eq (3)-(3) become exact We now have the expreion for the timedependence of the reactant number denitie in the Laplace domain, given by Eq (5)-() with the rate kernel in Eq () and (8) Thee Laplace domain reult can be converted numerically to the time domain with the ue of a numerical invere-laplace tranform routine 8 Reult and Dicuion A can be verified from Eq (5)-(), the uilibrium value of a( and c( are k r ( a c )/( )[ a and ( a c )/( )[ c, repectively, and the reactant number denitie relax to the uilibrium value according to the following relaxation law Δâ Δa Δĉ( Δc( ) [ ( ( C B (34) where Δa a a and Δc c c When the expreion of the rate kernel given by Eq () and (8) are ubtituted into Eq (34), the reult can be put down in the time domain a () Δa t Δa Δc() t Y Δc( ) A () t (35) In fact, for the imple reverible reaction of the type A B : C B, there exit an eay way to prove the relaxation predicted by Eq (9) and (35) i exact one 5 However, none of general purpoe many-particle theorie for reverible reaction ucceeded in recover the exact reult for thi type of reaction In the preent theory, the error in the truncation approximation, Eq (6) that wa introduced to derive the uation for ξˆ in a cloed form i exactly cancelled by that in Eq (8) which relate the diffuion ect function Fˆ ( to the irreverible urvival probability Fortunately, thi type of error cancellation alo occur in parallel when MPK theory i applied to other type of reaction 3,4 Acknowledgment Thi work wa upported by a grant from Korea Reearch Foundation (KRF-4-5- C34)

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