Kinetic Behavior of Aggregation-Fragmentation Process with Annihilation

Comun. Thcor. Plys. (Beijing, (Ihnina) 37 (2002) pp 207 302@ Intornational Acrademnic PublisthersVol. 37. No. 3. March 15. 2002Kinetic Behavior of Aggrcgation- Fragmentation Proccss with Annihilation"KE Jan-llong and LIN Zhen-Quan04 ADepartment of Plysics and Flectrouic Information Science. Wernzho Norrnal (College, Wenuzhou 325027. Zhrjiang Province.China(Received May 30, 2001; Revised Angust 16, 2001)A bstract The kinetic bchavior of an aggrcgation-lragrnent ation-annihilation systeu with two litinc. sprecies in stud-icd. We propuse: t.hat the aggregation reaction occurs only bet ween two clustcrs of the sune sprcies, and the ireversibleannihilation reaction occurs onuly betwecn two clusters of different spccies, mcarnwhile there exists the frugmientatioureaction of a clustcr into two snaller clusters for cither specis. Based u1l the mral ficld theory, we investigate therate cquations of the process with cunstant reaction rates and obtain the asymptotic dcscriptions of the cluster- sassdistribution. In the casc of the sume initial corncentrations of two speics, the scaling descriptions for the clustcr mnassdistributions of the two specics are found to break down completely. It is also obscrved that the kinctic belaviors ofdistiuct species arc quite counplicated for the case of difterent initial concentratiors of the two species. The clusters oflarger initial conccutration specics (heavy species) possess pecnliar scaling propertics, while the clusler-mass distributionof light specics has not scaling behavior. The exponents describing the scaling bchavior for hcavy species strongly dependton its fragnentation rate and initial nnouomer concentrations of two kinds of rcactants.PACS numbers: 05.45.-aKey words: kinetic behavior, aggregation fragmcntation-annihilation process. scaling description, rale equa-tion1 Introductionously;KA(1,j)Aggregation processes are inportalnt phenomena inA.+A,一A.:J(.j) .many fields, such as physics, chemistry and biology. Con-siderable interest has been focused on aggregation,"-5[ andKg(1J])annihilation'16-1川and fragmentation processes,![1.111 andB.+B,一Br+,，their interactionL3-18! since 1970. Krapivsky stndied theJg(i,J)irreversible two-species aggregation process (aggregation and irreversible annihilation reactions between the clus-without fragmentation) with annihilation annd found thatters of diferent species areits long tine asymptotic behevior of cluster-mass distri-A.+B;+j气B,bution bas scaling nature in the case that two speciesandhave the samne initial cluster concentrations.141 But thereA;tj+ B,"Ajare few studics concerning processes in which aggregation,fragmentation and anihilation cxist simultaneously and (Here, irreversible aruihilation reaction betwe theinteract arnong themsclves. Most of them are only desane quantities of monomers of two different species re-voted to the aggregation with fragmentation or aggrega- sults in inertness and withdrawal fron the aggregation-tion and annihilation. In fact, those reversible aggregationannibilation proceses), where A; denotes a cluster con.proceses of aggregation- fragmentation with anihilation sisting of the i-mers of species A, and similarly for B.are of grcat pactial sgifcance. For example, in a two- The rate of coalescencr reaction betwcen the chusters A,speis bilogical systen with costuet monomers A and Ai is equal to0 Ka(ij), while that bewen the clusand B, such as leucocyte and germ, the fragmentation比ters B; and Bj is KB(i,j). JA(i, i) denotes the reactionactions of the clusters always 0ccur simultancously whilerate of A2+j fragmentating into two smaller clusters A:they bond with the other clusters of the sarne species. Onand A, and Jg(i,j) for fragmentation reaction rate ofthe other hand, the clusters of diferent specics will perishBx+j. The reaction rate of annihilation between A. andB+, is equal to的and that between A:tj and R equalstogether after they meet each other.J'。， The present investigation is based on the mcan- fieldIn this work, we investigate the coumpetitions among thcory.中国煤化工ion procceds withaaggregation, fragmentation and annihilation processes of rate pr_entration. Thus thetWo specics; A and B. We assume that the aggregation mean-tYHCN M H Gatial Anchationofoccurs between two clusters of the sane species with the the reactant densities and therefore applies to the case ofreverse processes of fragmentation coexisting simultane- the spatial dimension d of the system greater than or egual●The projet supported by Zhejiang Provincial Natnral Science Foundation of Chia under Grant No.19905020%KE Jian-liong and LIN Zhcen.QuanVol 3710 the upper critical dintension d,. When d < du fiuctna- procruss give1 by Krapivsky! ald writr the corspound.tions il the drusities of reactants ilay lead 10 dintension- ing Sinolucluowski rate: cquatioms for | his ig;rt gation-deprndent kinetic behavior in the loug; time limit. For fraguneut at io-amnihila! ion system ius follows:sinuplicity. we su that in (our sysiten the spatial di-mnension d is greater than de.on-六E,n之,Ii order to investigaie thoroughly the evolut ion behav-上k1=1iors of reversible agreationannihilation systen, we as-su that the reaction rates of aggregation; annihilation+1( 2 apb,-us,and fragmentation are different constants. We have de-p=g-kj=lrived the asyumptotic solutions for the cluster-mass (clus-ter concentration) distributions. T'he rcsults show that the(k+.J，(la)evolntions of the two clusters behave as 1nu scaling in thesame initial cases of two spccies, and in different initialdbs。它的cases the cluster-mnas distribution of the heavy initial.mass species possesses scaling behavior. while that of theJ=1light initial-nass specics does not scale. And generally,the fragmentation may lead to breakdown of the standard+川( E b.oao-bo二川)scaling belavior of the cluster- Imass distributions.μ=g+kThe paper is organized as follows. In Sec. 2, we des-cribe a two-component reversible aggrcgation- annihilation-五2 b.+2Eb. (Ib)j=Imodel, and give the corresponding rate equatious withconstant reaction rates on the basis of mean-field theory.We sssi1 that tbere only exist A- and B-clusters ofIn Sec. 3, we derive the asymptotic solutions of the cluster-monomers and their cuncentratiors are equal如Au andmass dstributious of the two speies and then discuss their Bo al l= 0. rspectively. Then the initial conditious arecharacteristics. A brief summary is given in Sec.4.ar(0) = Ao8k1.b;(0)= Bo61. (2)2 Model of Reversible Aggregation ProcessesThe rate equations can be solved with the help ofwith Annihilationansatz in Ref. [14]. We asme ak and bk have the fol-In our investigation, the theoretical approach of lowing fornsaggregation-fragmnentation-annihilation processes is baseda&()= A(t)a^-1(), be(t)= B()b*-+(t).(3)on the ratc equations according to the mean-field assumpwhere A(t), a(t), B(t) anld b(t) are functious dependent ontion. Fluctuations in densities of reactants are ignored andtime. Substitnting Eq. (3) into Eqs (1) and consideringclusters are considered to be homogencous-distributed atthe identity of all the reaction rate equations of auggre~-all time throughout the wlole procs. The concentra-gates, we can transform the rate equations (1) into thetions of A- and B-clusters of the h mers are denoted as akfollowing differential equationsand bk, respectively. For simplicily, we consider a modelwith coustant reaction rates, and set all the rates of an-da_ A(4a)nihilation processes equal to J. The aggregation rates,di= 2K(i,j), and fragmentation rates,小(i, j), of A-clusters aredAAuaAequal to 1 aud小while those of B- clusters equal to I andHt一+JAB("--)+小。(4b)-aJ2, respectively. In our reversible aggregation systerm, Ak4bran be produced by the following reactions(4c)diA+g+B,→Ax，Ak-.+A.→AkdB1B2bBanddt=1-bt JAB(-二)+2 (4d)Ant)乌Ak+A;Correspondingly, the init ial conditions are rewritten心while the reactionsa=0，A= Ao,da_ AnAk+B,→Ak-j (or By-k whilej> k),dt= 2Ak+A;→Ak+j and Ak二 Ai+ Ak-r中国煤化工=0.(52)result: in decrease of concentration of Ak, and similarlyMHCNMH GIBofor Bk. All the rates of aggregation, fragmentation or an-nihilation reactions are considered to be directly propor-b=0B- Bo,dt 2tional to the concentrations of their rcactants. Wr gen-cralize the rate equations of the aggregation-annihilationdB= -IB-JAoBo. at t=0.(5b)atNo.3Kinetic Bechavior of Aggregation hragmcntaton Process wIl H Anmihilatacn2993 The Cluster- Mass Distribution Solutions of wherteSpecies A and B2C+1-V.2 + 4J.CDiferential equations (1b) and (4d! can be easily2Ctransformed into tle followiug equationsc,-2C +1+ VFR+4JiCd[4-)|(1-0)2_ B (1-a(1-b)(6a)C3=5V只? +4Ch.dt .1(1-6)21-abSubstituting Fq. (}0) into Eq. (8a), we obtain the asymup-[(L -.6)2B- b)23(1-a)21-ab(1-a)(1- b6)(6b)totic solution of F1(0) alt long tine,F(1)≈(11)1- exp(-C4t)The system reaches its equiibriuu slate at t→oc andwherethe conditions can be given as follows:C=号(+4Ch-1)da_ Aan一2一性-0、(7a)It shows that F(t) is approxinatcly equal t0) C in thelongtine limit and thr above approximation, Fi(t)≥ C,1AA2is reasonable.at-1-ab~ 1- b)From Eq. (9b), o1C can fiud the asyinptotic solution of+小;aAn =0,(7b)b(l),b(t)≈C3(e-z2t72_e Ct).(12)db_(7c)ICj=18-J2号=0,(13)JV升+4Ch-11-h1B2ll1-b\1-ab~ 1-a)The solutions for A(t) and B(t) can be derived frunEq. (8a),bB+12=0.(7d)A()≈_C_ (1-C1)-(1-C2)e Ca'12(140)1- e-CAti1- e-Cst. llere we may set An > Bo (A is the heavy speciesCe-cst(1- Cse-2t/2 + Cse* Cat)2with larger initial concentration) for the initial conditionB(t)≥- 1- eet(14b)without loss of any generality: and thus there must not beThus we obtain the asyinptotic solutions for the mass dis-B-clusters at the end because of the annihilation process.tributions of A- and B-clusters,Prom Eqs (7), we find b= B =0and A = Ja at the equi-C(1-C1)-(1- C2)e-csty2librium state for the asymmetric system. It implics thata()≈- e"-cr.1- e-Gatb, B→0,a→cand A→JIc (c is a constant greatertban zero) in the long-time limit. Thus equations (6) can[Ci一C2e -Cs1k-I(!5a)be rewritten as1- (-CstACe-C4(1-Cge-21/2 + Cse Cst)2hi=(-a)=(1-0)]2C.(8a)br(t)≌1- e-CdF(1 -a)(1-6) .d1=-JF(F-C)1- ab(8b)x |Cs(e(15b)When J1 = 0, one can find the Smoluchuwski solutionwhereC-Ao - Bo, and F(t) is a function of tine and for ireversible aggregation process frou E4. (15a) (se.equations (4) can be recast to the following equationsRefs [川} and [141) .da4C(Ct- 2)k-1=号气(t-o)-只,(9a)uk(l)≌Cryx+1- ≤4(C12) 'ex(-x), (16)Ib_ Fr-Cwhich is valid in the scaling regiondi<(1-6)2- b(9b)中国煤化工1)= fnite.It is obvious that F(t)→C at long time and thus we ob- This shYHC N M H Gibution of A- speciestain the asynptotic solution of a(t) in the long-ime limit belhaves the sraling law in the forml4by substituting Fi≌C into Eq. (9a),ak()≌1"φ[k/S(t)|，5(t)xt"， (17)a(t)≌CL-Czexp(-C3l)where t.he function S(t) is used to denote the characteristic1 - exp(-Cat)mass.300KE Jian-long and LIN Zhen-Quan\ol. 3.The results suggest that the evolution behavior of the can be rwritten asmass distribution of B-clusters is nonscaling: and the Inassof B-clusters decreases rapidly with time. But the 0V0-N(t)= 2 a(t) x{g()j"1[()!".(25a)k=1lution behavior of the mass distribution of A-clusters ismore complicated and depends on the value of fragmnenta-M(1)-厂kas(t)x lg<(t] 1f()“。(25b)}tion rate J. The evolution of coucentratiou of A-clustersk-lcomcs in a scaling form in the loug time limit when the The rlatins among the exponents arefragmnentation process dloes not cxist. In the scaling reX'=w'-'μl'=w'-2x'.(26)gion. the average llass of clusters increases indefinitely.The total nunber Na(t) and the total nass MB(t)of When f()二g(t)三t. one can rasily find the relationsA-clusters can also be written in the power-law fornbetwcen (w, z,入川) anrd (r': ur',z'.X.”,Na(l)= 2 ax(t)xt-^;(18a)λ=r' +w'-z'.μ=r' +u'-2z'. (27)Ma(t)=2 kae(l)xt-".(18b)From Eqs (21) and (23), we can obtain all the rxponentsof A-cluster,r' =0,w' =0,The four exponents (u, z,入,山arc universally used to describe the scaling nature of aggregation annihilation pro-2=号V?+4(Ao-B0)厅，cesscs at long tine. From Eqs (17) and (18), one can findthe relations alwng the four scaling exponentsx=-3V2 +4(Ao - Bo)d,λ=w一z,μ二w- 2z.(19)ul =-J2 + 4(Ao- Ba)J.(28)In the case of JI二0, we obtain the four exponents ofGenerally, whent 》1 and k》1, we can obtain theA-clusters from Eqs (16) and (19),fllwing equation (29) from Eq. (15a),w=2，z=1， λ=1， μ=0.ax(t)≌C(1 - C)}C* exp(-y).(29)Further, when J > 0 and 4CJ《1, we obtain an w hereapproximate solution of Eq. (15a) for the large clusters(C2- C1)kexp( -Czt) .(30)(k》1),Cae(t)≌C(1- C1)2 exp(-x);(21)It is evident that whent十x, (C1 - Cre" C3*)/(1-e-Cst)→Ci < 1, and ax(t) will decay along with k whenwherek》1. But equation (29) is valid in the scaling regiouy= (Cz-C) exp(- -Cgt).(22) ke-Cst ≌finite. The Isual scaling description for theC1mass distribution of A-cluster breaks down partly whenThus we obtain the total number and the total mass ofthe fragmentation rate is not cqual to zero, and only ifC1≌1, can distribution of A-cluster mass be approxi-A clusters,mately described by the universal scaling form of Fq. (24).C(1 - C1)PC1N(t)≌C2-C1exp(C31).(23a)We may further modify the above scaling description (24)intoMa(I)≌C(G.- C3)2 xp[2C2x).(23b)a(t)= Ch"*g()-"If(t)- " [/S()，(Gr-C1)pS(t)x[f()]"、f(t)>0，(31)The result shows that the distribution of A-clusters has arather pecuiar scaling carateristic in this cae. In order where h denotes a constant and 00,(24)MHCNMHGwhere g(t) and f(t) are the peculiar functions of time,i" =V3 + 4(Ao- Ba)4;such as e', logt, 2', etc.'The total number and the total mass of the clustersX"=0，μl"=0,(32)No.3.Kinetic Belavior of Aggregation-Fragmentation Process with Auililation301which are different from those of Eq. (28).fllows:Finally, we liscuss the distributions of the total 1u1ber and total mass of A- and B-clusters in the case ofa(1)≈2e+1(号).(37a)diferent initial concentrations. N(1) and M(t) of A- andB-clusters in the long time limit can be obtained fromb(t)= 2ee 12(号),(37b) .Eqs (25) as fllows:where E;(t) is an exponential integral function,Na()=立。k()==l-aE(0)=-/°τd1.(38)It can be partially intgrated and we then derive the so-≥一C (1-C1)- (1- C2)e-Cot(33a)lutions of a(t) and b(t) in the fllowing explicit formus:- e-Cs1- e-Cgt(2-1(i-11!MA(t)=》kae(t)=(1-a)≈i-e-Gs: (33)a()≈}+Z((00)].(39a)k=l0()≥Z -2-1(i-(39b)Na()=〉之b()=1-b(o,")I.k-1In the long time case, a(t) and b(t) can be simplfied as~ Ce-Cia(1-Cge-J2t/2 +Cge-Cst)(33c)1- e-Csta(t)≌元t-'，(40a)BCe-CstMp(t)= 2 kbe(t)= ;(33d)b(t)≌(40b)k二1(1-6)2一1-c CAEJJIt is shown that both the total number and total massThe solutions of A(t) and B(t) can then be given as fol-of A-clusters dominatc over thosc of B-clusters in the long-lows:time limit, which is natural for the case of initial concen-A(t)=F2(1-a)2≈40J-°ε-，(4)trations Ao > Bo. The evolution behaviors of A- and(J.Jht)3B-clusters strongly depend on the annihilation rate J andtheir fragmentation rates Ji and J2, but the aggregationB()= Fr(I_)p≌2111-1(J.J2)3rate I of B-clusters has very little effect on the evolutionThe Inass distributions of A- and B-clusters ran br given,behavior. Moreover, the initial condition of the systemalso plays an important role in the evolution behaviors ofae(t)≌J(JJl- 1)2J(42a)the reversible aggregation- annihilation system.(J+)+2(JJ0)xNow we discuss the special case of the sarme initial corl-bx{t)≌J2(JJ2t- I)2Ik-1(42b)centrations Aη= Bo. Equations (8) can be rewritten as(JJ20)*+2-宁(示A(34a)and the total nunber and total Inass of A- aad B-clustersF2-(1-0)2 (1-6)2.aredFz_J1p2(1-a)(1_-)(34b)Na()-〉ax()≌JJL- !(43a)dt1- ab12.11211where F2(t) is a function of time. Then we rewrite Eqs (9)as follows:Ng()= Eo(≤出⊥(43b) .J2J2t2=1dt- 21-a)2 -号a,(35a)Ma(t)=E kae(t)≌jt',(43c)Hi-2(1-)2-号(35b) .21Fron Eqs (7), the equilibrium state of the system withMg()= 2 kbe()≈.(43d)the sarne initial concentrations of two species is found tobea=A=b= B=0. Thusit is obvious thata《1 Equal中国煤化工that the usual scal-andb《1 at long time, and we derive the zero-order ing (distribution for ag-asymptotic solution of F2(t) from Eq. (34b),gregatMYHC N M H Gror our agregation-F2()≈=r'.(36)fraguentatiou-anuihilation systern in the case of the sarueinitial concentrations of two species. Both the mass con-Subsitnting Fq (36) into Eqs (35), we obtain the asymp- centratious of A- and B-clusters decay quickly with tinme,tortic solutions of a(t) and b6(t) in the long-time limit as and the total number and total mass of the two species.302KE Jian-Hng and LIN Zlu:n Quasl .V'nl. 故decroase with tinc. lo0. The masu distribution crucially the decreasing rates of the total nuber aund total Imnas ofdeprnds on the reaction rates I.小J2 aund J. Making the clusters 1ainly deprend on the vadue of the auihiltiona cumparison bctween Eqs (42a) and (42b). we findl that renctiom rate J.when I > J2/1. the heavy clusters of B species dorur4 Summaryinate over those of A-species and the light clusters areon the contrary. and the total number of A-clusters isWe have studicdl a reversible aggregation auililationlarger than that of B-clusters. And it is quire the reverse model with two kinds 0f distinct. specinsou the bha-forJ < J2/1. The total mass of Aclwters is always sis of thr manfield thery. Considering the costatidentical with that of B-clusters tlroughout the reversiblereaction-rate mordel, we analyze the kinutic belhaviors ofaggregation-annilhilation prooss, which is apparently trmethe aggregation fragment anmmihilatin systemn. Iufor the sarue initial conditions Ao= Bo.the case of the salne initial conrentrat ions of two species,Finally, we investigate the synmetrical case betweenwe have found that th: scaling descriptions for ihe masA- and B-clusters of I = I, 小二J2 and the sane ini-distributions brcak down. The total mber and totaltial concentrations Ao = Bo. Then we lhave a = b andmass of the clusters dec ay rapidly with tinte in the long-. A = B. Let Ce(t) denotes the concentrations of the -time limit, and their decay rates depend on the values ofmers of specics A or B, i.e, ce(t) = ax(t) = be(t). thethe reaction rates. Further, for different intitial Cucentrarasyimnplotic solution of the cluster mass distribution cantions of two species we obscrve the completely diferentevolution behaviors of the lwo distinct species. Fir the ir~then be derived from Eqs (42):reversible aggregation anrihilation proress, the mas dis-cx(t)≌J{JJt)-*.tribution of heavy specirs can be described by a standardIn the case of the same initial conditions, it was found scaling form. But a nore universal scalitng description ofby Krapivsky that the cluster. -nass distibution has scal-mass distribut ion of heavy species has bee1 found in ge)-ing natureat人二0,!14| but our solntious show that the eral proccss of aggregation- fraguentatiou with unihila-scaling description for the cluster- mass distribution breaks tion. It is found in our model that whether the standlarddown completely when小> 0.scaling description of mass distribution exists or breaksThe total number N(l) and the total mass M(t) of the dowr strongly depends on the fragmentation rate andclusters can be given,the initial concentrations of the two species. But for thelight species, the evolution behavior lndergoes a 1oIIaN(1)= ,(45a). ing way; and the mass distribution of the elusters alway>ae广decays rapidly with time.M(t)=(45b) AcknowledgmentsOne of the authors, LIN Zhen-Quan, is grateful toEqnations (44) and (45) show that in the synmetrical Prof. YANG Than-Ru and Dr. ZH0U Zhi Xin for helpfulcase, c&(t); N{t) and M(t) decay quickly with timc, and discussions.References[9] I. Ispolatov and P.L. Krapivsky, Phys. Rev E53 (1996)3154.[川RL. Drake, Topic of Current Areosol Research, eds. G.M.|10] B. Bonnier and R. Brownl, Phys. Rev. E55 (1997) 6661.Hidy and J.R. Brook, Pergarnon, New York (1972).2| M. Branmon and J.I. Lebowitz, Plyx. Rev. lett 61 (1988) 川E.D. McGrady and R.M. Zif, Phys. Rev. Lett. 58 (1987)2397.892.[3} P.L. Krapivsky and E. Ben-Naim. Phys. Rev. E53 (1996) [12] Z. Cheng and s. Redner. J. Phys. A23 (1990) 1233.291.(13} F. Family, P. Meakin and J. l)eutch. Phys. Rev. Lett. 57[4! P.IL. Krapivsky and S. Rudner, Plhys. Rev. 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