Original Paper orma, 5, 9 7, Molecular Dynamic Simulation of Nonequilibrium Effect ociated with Thermally ctivated Exothermic Reaction Jerzy GORECKI and Joanna Natalia GORECK Intitute of Phyical Chemitry, Polih cademy of Science, Kaprzaka 445, PL--4 Waraw, Poland Intitute of Phyic, Polih cademy of Science, l. Lotnikow 46, PL--668 Waraw, Poland E-mail: grecki@ichf.edu.pl, gorec@ifpan.edu.pl (Received May 6, ; ccepted June 6, ) Keyword: Reaction, Nonequilibrium Effect, Molecular Dynamic btract. In thi paper we dicu the influence of nonequilibrium effect on the rate of a thermally activated, exothermic reaction. The ytem with a binary proce B B{ energy } i conidered a an example, on which we compare molecular dynamic imulation with a imple phenomenology baed on the aumption that a nonequilibrium tate can be characterized by many time dependent temperature. good agreement between reult of both method i oberved. We have found that the rate contant are changed ignificantly by the nonequilibrium effect, which affect ytem evolution.. Introduction In order to predict the time evolution of a chemical ytem we uually apply the tandard chemical kinetic baed on the ma action law. In uch approach the rate contant i jut a number, which link the concentration of reactant with the reaction rate. t the microcopic level of decription the rate contant for an elementary reaction i related to two factor: the reaction cro ection σ (Ε i ) for a reactive encounter between reactant at a particular quantum tate Ε i and the rate uch tate appear in the ytem p(e i ). In thi notation the rate contant k i the average of σ over all tate: k = σ ( Ei) p( Ei). Ei () Let u aume that at the beginning all reactant are equilibrated. In the cae of an activated proce the energetic barrier ha to be croed and the reaction cro ection increae with the energy of interacting molecule (STILLER, 989). Therefore the high energy part of reactant ditribution function p will be trongly diturbed by the reaction, becaue the correponding tate can be eaily tranformed into product, and the rate contant calculated from () hall differ from it initial, equilibrium value. the reult, the time evolution of the ytem will differ from the one calculated uing the equilibrium rate (cf. ig. ). 9
J. GORECKI and J. N. GORECK ig.. The concentration of product B a a function of time for the ytem characterized by = and u = 4 ( = 5). The dahed line MD imulation, the thick olid line claical phenomenology (Eq. (4) and (5)); the thin olid line Eq. (6) (8). The influence of nonequilibrium effect on the rate contant ha been invetigated for more than 5 year. Mot of the theoretical conideration were concerned with the thermally activated, thermoneutral chemical procee (PRIGOGINE and XHOUET, 949; PYUN and ROSS, 964; SHIZGL and NPIER, 996; NOWKOWSKI, 998). The number of reult for thermally activated egzothermic reaction i limited (ee for example PRIGOGINE and MHIEU, 95; XYSTRIS and DHLER, 978; BRS and MLEK MNSOUR, 989, 997; GORECKI and GORECK, ). In thi paper we tudy the nonequilibrium effect for the reaction: B Benergy () uing molecular dynamic (MD) imulation. We alo how that a very imple phenomenology give a good agreement with imulation. We think that it may be eaily applied to etimate the trength of nonequilibrium effect in other ytem with egzothermic reaction.. Reult Let u aume that the activation energy for the direct proce of () ( B B) i E and the reaction heat U >. Therefore the activation energy of the revere reaction
Molecular Dynamic Simulation of Nonequilibrium Effect (B B ) i E = E U and the reaction heat U. Our MD imulation are performed for the implet cae in which the molecule of both and B are repreented by identical hard phere (the ame mae an diameter). Now the only form of molecule energy i the kinetic one. One can define σ * for a thermally activated proce uing the lineof-center model for reactive colliion (PRESENT, 959). ccording to thi model a colliion between reactant molecule in which the energy of the relative motion along the line of center calculated in the center of ma reference ytem exceed E may lead to reaction. The probability that reaction occur for a colliion atifying the energetic condition i called a teric factor ( ). The line-of-center model for reactive colliion can be naturally adopted in imulation of exothermal procee. When a reactive colliion of two particle occur the reaction heat i releaed a the additional kinetic energy of outgoing B particle. In cae of reactive colliion of two B particle the energy of product i decreaed by the value of reaction heat in the ame way. or a Maxwellian ditribution of velocitie at the temperature T the reaction rate k for line-of-center model equal (GORECKI, 99): E k = ( T) η k T () where η(t) i the frequency of colliion between phere. Our imulation tart with all phere marked a the reactant and randomly ditributed in the ytem with velocitie choen from the Maxwellian ditribution correponding to the initial temperature T (t = ). We follow ytem evolution by checking which colliion are reactive and if o changing the chemical identity of phere and their kinetic energy. The periodic boundary condition are applied o the total number of particle i conerved and the ytem i adiabatic. In a ingle imulation run one obtain the number fraction of particle () and the average energie per particle for (E a ) and B(E b ) and for the ytem a a whole (E ) a function of time. The outcome of a ingle imulation program trongly depend on the initial condition, therefore we preent the data, which are averaged over more than of individual run. They have been obtained for 5 phere and the packing fraction of the ytem wa.. We have retricted the imulation to the initial tage of reaction becaue in thi region the nonequilibrium effect are the mot pronounced. uming that reaction () proceed in a cloed ytem, the tandard kinetic equation decribing ytem temperature and concentration of are: B = ( ), (4) = ( ) u (5)
J. GORECKI and J. N. GORECK where i the number fraction of molecule and, u,, are the ytem temperature T, reaction heat and the activation energie caled to the initial temperature of the ytem repectively ( = T (t)t (t = ), u = Uk B T (t = ), i = E i k B T (t = )). Equation (4) and (5) are written in the time cale in which η(t (t = )) =. The accuracy of uch decription for reaction () can be eaily teted if one compare it with the reult of molecular dynamic imulation and for example for = and u = 4 the outcome i rather diappointing (compare the thick olid and the dahed line in ig. ). What i the origin of the dicrepancy? igure how reult of MD imulation (dahed line) for average energie of and molecule of the ytem treated a a whole caled by k B T (t = ) a a function of the number fraction of molecule for = and two value of u (u =.5 and u =.5). It i convenient to ue (t) intead of time a the meaure of reaction progre. One can notice that the difference between the average energy of particle and the average energy per particle of the ytem are ignificant, what indicate that the energy ditribution of particle differ from the energy ditribution of the whole ytem. t the initial tage of the reaction E a decreae becaue the mot energetic particle are conumed by the direct reaction. Next it tart to increae a the reult of the energy exchange in non-reactive colliion with B particle and becaue highly energetic particle of B are tranformed into in the revere reaction. ig.. The average energy per a particle of reagent (the thin line) and per an average particle of the ytem (the thick line) caled to k B T a a function of. The reult are preented for = and two value of u. The dahed line how reult of MD imulation(long dahed u =.5, hort dahed u =.5), the thin olid line are olution of Eq. (6) (8).
Molecular Dynamic Simulation of Nonequilibrium Effect igure how the average energy per particle of B and per a particle of the ytem treated a a whole for the ame et of parameter a in ig.. t the initial tage of proce the average energy of B particle i much higher than E becaue only the mot energetic particle are tranformed into B. The difference between E b and E i more important than thi between E a and E. Now it i eay to lain the difference between MD imulation and Eq. (4) and (5) hown in ig.. The tudied reaction i very fat o the energy exchange between reagent i not ufficient to keep the ame energy ditribution for all of them. The rate contant for the direct proce hould be maller than conidered in Eq. (4) and (5) becaue E a () < E (). On the other hand E b () > E () o the revere reaction i fater. Both effect work in the ame direction, o finally the actual rate of product creation i much lower than predicted by the tandard kinetic. Let u notice that the effect i qualitatively imilar to the one oberved for coupled reaction (GORENSEK and KOSTIN, 984, 985; CUKROWSKI et al., 99; GORECKI and HNZKI, 994): a fat reaction with low activation energy ( ) prepare a nonequilibrium ditribution of reactant which ignificantly change the rate of another proce with a higher activation energy. The implet decription of the nonequilibrium effect can be given if one aume that the ytem i characterized by two different, time dependent temperature: the reactant ha a temperature (t)t (t = ) and the whole ytem i characterized by the temperature (t)t (t = ). NOWKOWSKI and GORECKI (996) demontrated that imilar phenomenology give an accurate decription of the time evolution for nonequilibrium ytem with thermoneutral reaction. Conidering all energy exchange procee between colliding particle one obtain the following equation: ig.. The average energy per a particle of B and per an average particle of the ytem caled to k B T a a function of. Notation a in ig..
4 J. GORECKI and J. N. GORECK =, (6) = ( ) 7 4 7 4 7 4 ( ) ( ) ( ) ( ) u 7, ( )
Molecular Dynamic Simulation of Nonequilibrium Effect 5 = u. ( 8) The reult calculated on the bai of Eq. (6) (8) (thin olid line on ig. 5) are in a good agreement with imulation. Such agreement ha been alo obtained for the other tudied value of, and u. In molecular dynamic imulation the nonequilibrium rate contant for the direct and revere reaction (k d and k r repectively) a function of can be calculated from the average time <δt> () the ytem pend in the tate characterized by a given. In order to ee clearly the nonequilibrium effect it i convenient to cale the rate contant for a given by the value calculated from the rrheniu law uing temperature (). Such caled rate contant for the direct proce obtained from imulation (the hort dahed line) i preented in ig. 4. The thin long dahed line mark tatitical error of the imulation data ( = and u =.5). igure 5 how caled rate contant of the revere reaction k r a a function of for = and two value of reaction heat (u =.5 (long dahed line) and u ig. 4. The rate contant for the direct reaction a a function of for = and u =.5( = 4.5) caled to it equilibrium value. The hort dahed line MD imulation reult; the long dahed line etimate it tatitical error; the olid line olution of Eq. (6) (8).
6 J. GORECKI and J. N. GORECK ig. 5. The rate contant for the revere reaction a a function of for = and two value of u: u =.5 ( =.5) and u =.5 ( = 4.5). The dahed line are reult of MD imulation, The olid line come from olution of Eq. (6) (8). =.5 (hort dahed line)). ected the influence of nonequilibrium effect on the revere reaction i more pronounced. In our opinion the mall dicrepancy between imulation and phenomenology i mainly caued by hot pot of product B which make the ytem inhomogeneou. lthough the revere proce help to retore ytem homogeneity for a large reaction heat the activation energy of the revere proce i high and o the invere reaction i low and inefficient. The rate contant i calculated from MD reult auming that the ytem i homogeneou and it hould be lower if one conider a higher effective concentration of B in the pot. Thu it i not urpriing that it (ee ig. 5) overetimate theory.. Concluion In thi paper we preented reult of direct meaurement of the nonequilibrium reaction rate contant in an adiabatic ytem with thermally activated reverible exothermic reaction (). The meaurement were performed on the bai of molecular dynamic imulation for the reactive hard phere. We oberve that the rate contant for the direct reaction i decreaed with repect to the correponding equilibrium value and the decreae i imilar a for the thermoneutral proce with the ame activation energy. The rate contant for the revere reaction i increaed by nonequilibrium effect and it value may be order of magnitude higher than the correponding equilibrium value.
Molecular Dynamic Simulation of Nonequilibrium Effect 7 We compared the reult of MD imulation with imple phenomenology which aume that the ytem a a whole and reactant are decribed by ditribution function with time dependent temperature. The agreement between phenomenology and imulation i very good o uch imple decription of nonequilibrium effect may help to predict the time evolution of many reactive ytem. or example the nonequilibrium effect can have a ignificant influence on ytem exhibiting thermochemical ocillation (GRY et al., 988). REERENCES BRS,. and MLEK MNSOUR, M. (989) Phy. Rev. Lett., 6, 79. BRS,. and MLEK MNSOUR, M. (997) dv. Phy.Chem.,, 9. CUKROWSKI,. S., KWCZYNSKI,. L., POPIELWSKI, J., STILLER, W. and SCHMID, R. (99) Chem. Phy., 59, 9. GORECKI, J. (99) J. Chem. Phy., 98, 4. GORECKI, J. and GORECK, J. N. () Chem. Phy. Lett, 9, 7. GORECKI, J. and HNZKI, I. (994) Chem. Phy., 8, 9. GORENSEK, M. B. and KOSTIN, M. D. (984) J. Chem. Phy., 8, 77. GORENSEK, M. B. and KOSTIN, M. D. (985) J. Chem. Phy., 8, 8. GRY, P., Kay, S. R. and SCOTT, S. K. (988) Proc. Roy. Soc., 46,. NOWKOWSKI, B. (998) J. Chem. Phy., 9, 4. NOWKOWSKI, B. and GORECKI, J. (996) cta Phyica Polonica, B7, 895. PRESENT, R. D. (959) J. Chem. Phy.,, 747. PRIGOGINE, I. and MHIEU, M. (95) Phyica, 6, 5. PRIGOGINE, I. and XHOUET, E. (949) Phyica, 5, 9. PYUN, C. W. and ROSS, J. (964) J. Chem. Phy., 44, 87. SHIZGL, B. and NPIER, D. G. (996) Phyica,, 5. STILLER, W. (989) rrheniu Equation and Non-Equilibrium Kinetic, Teubner-Texte zur Phyik, Band, BS B. G. Teubner, Leipzig. XYSTRIS, N. and DHLER, J. S. (978) J. Chem. Phy., 68, 87.