Master Equation Analysis of Thermal and Nonthermal Microwave

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Master Equation Analysis of Thermal and Non-Thermal Microwave Effects Jianyi Ma J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b03941 • Publication Date (Web): 30 Sep 2016 Downloaded from http://pubs.acs.org on October 5, 2016

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The Journal of Physical Chemistry

Submitted to J. Phys. Chem. A, 06/14/2016

Master Equation Analysis of Thermal and Non-thermal Microwave Effects Jianyi Ma Institute of Atomic and Molecular Physics, Sichuan University, Chengdu, Sichuan 610065, China

Master equation is a successful model to describe the conventional heating reaction, it is expanded to capture the "microwave effect" in this work. The work equation of "microwave effect" included master equation presents the direct heating, indirect heating and non-thermal effect about the microwave field. The modified master equation provides a clear physics picture to the non-thermal microwave effect: 1) the absorption and the emission of the microwave which is dominated by the transition dipole moment between two corresponding states and the intensity of the microwave field provides a new path to change the reaction rate constants. 2) In the strong microwave field, the distribution of internal states of the molecules will deviate from the equilibrium distribution, and the system temperature defined in the conventional heating reaction is no longer available. According to the general form of "microwave effect" included master equation, a two states model for unimolecular dissociation is proposed and is used to discuss the microwave non-thermal effect particularly. The average rate constants can be increased up to 2400 times for some given cases without the temperature changed in the two states model. Additionally, the simulation of a model system was executed using our State Specified Master Equation package. Three important conclusions can be obtained in present work: 1) A reasonable definition of the non-thermal microwave effect is given in the work equation of "microwave effect" included master equation 2) Non-thermal microwave effect possibly exist theoretically. 3) The reaction rate constants perhaps be changed obviously by the microwave field for the non-RRKM and the mode-specified reactions.

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Introduction Gedye published the first article on microwave assisted syntheses in household microwave ovens since 1986,1 a steadily growing interest in this research field over the past few decades. Now, microwave activation is a popular unconventional technique in organic chemistry comparing with the conventional heating approach. Microwave radiation consists in electromagnetic waves in the energy range of 0.01-1 cm-1(～0.00001 eV). The energy between the materials–microwave interactions is too low to induce any chemical activation (～0.2eV for hydrogen bond, ～5eV for covalent bond and ～7.5eV for ionic bond).2 Hence, according to the traditional opinion, electromagnetic interactions plays a key role on the absorption of microwave by materials, and the molecular polarity is important for microwave activation.3 A large number of microwave activation reactions have been reported in organic synthesis.4 Microwave-assisted organic synthesis is characterised by some features which cannot be reproduced by conventional heating, such as higher yields, milder reaction conditions, shorter reaction times. Even reactions that do not occur by conventional heating can be performed using microwaves.5 These features in microwave activation reactions cannot be explained by the effect of rapid heating alone, so-called ‘‘microwave effect’’ is proposed by many authors. Microwave effect in chemical reactions is a combination of the thermal effect and non-thermal effects, i.e., overheating, hot spots and selective heating, and non-thermal effects of the highly polarizing field.6 Although several theories have been postulated and also some predictive models have been published, the existence of non-thermal effects is still a controversial issue.

4a, 7

There are few main reasons leading to this phenomenon:5b 1)

The microwave effect is reaction type independent. 2) The experimental details and the microwave systems used are usually insufficiently described and other research groups have trouble reproducing the results. 3) No reaction exists that only proceeds in the microwave field, the microwave effect always couples with the heating processes. 4) There is no well defined thermal and non-thermal microwave effects. Non-thermal effects have been envisaged to have many origins.8 However, same to the thermal effects, non-thermal effects must arise also from interactions between


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the microwave field and the material. So microwave heating interferes with possible non-thermal effects and it cannot be easily separated in real experimental studies. On the other hand, a suitable theoretical recipe can help us to avoid these four issues mentioned above which exist in the experimental studies due to the temperature and the rate constants can be calculated effectively by theoretical model. For a given temperature, the differences of reaction rate between some of the microwave activation and the conventional heating reactions close to 103 times.2 This huge difference should be captured by recent high accuracy theoretical models of the reaction rate calculations, such as RRKM,9 Semi-Classical Transition State Theory (SCTST)10 and Master Equation.11 Especially, the predicted rate constants by 1D and 2D master equation for many reactions are in reasonable agreement with the experiments. Indeed, master equation is a successful model to describe the conventional heating reaction, how to understand the "microwave effect" base on the model of the master equation is the aim of this present paper. Theory I. The Master Equation. There are many versions of the master equation model, in the present paper we focus on the master equation for thermal dissociation.12 The popular two-dimensional master equation for the irreversible dissociation of a molecule immersed in a solvent is 13 ∞ dN ( E , J , t ) = Z ∑ ∫ [P ( E , J ; E ′, J ′) N ( E ′, J ′, t ) − P ( E ′, J ′; E , J ) N ( E , J , t )]dE ′ 0 dt J′ − k ( E, J ) N (E, J , t )

(1)

where N ( E , J , t ) dE is the number density of molecules with total energy which belongs to ( E , E + dE ) and total angular momentum quantum number which equals to J , t presents the time, Z is the collision rate of the molecule with the bath molecule, P ( E , J ; E ′, J ′) is the transfer probability from the states between ( E ′, E ′ + dE ) with J ′ to the states ( E , E + dE ) with J and k ( E , J ) is the



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unimolecular rate coefficient which is calculated theoretically using ab initio data. Letting N ( E , J , t ) = n(t ) × x ( E , J , t ) , where x ( E , J , t ) is the normalized population distribution, we have

dN ( E , J , t ) dn(t ) dx( E , J , t ) = x( E , J , t ) + n(t ) dt dt dt

（2 ）

If we consider that the population distribution is in steady distribution, i.e., dx ( E , J , t ) / dt = 0 , and dn (t ) / dt = − k (T , p ) . Using Eq.(1) and simplifying, one

obtains ∞

− k (T , p) x( E , J ) = Z ∑ ∫ P ( E , J ; E ′, J ′) x( E ′, J ′)dE ′ J′

0

（3）

− Zx ( E , J ) − k ( E , J ) x( E , J ) Sum over the contribution of the index J, following the definitions of ̅ and in Ref 13. Eq.(3) becomes ∞

− k (T , p ) x ( E ) = Z ∫ P ( E , E ′) x ( E ′)dE ′ − Zx ( E ) − k ( E ) x ( E ) 0

(4)

The 2D and 1D master equations (Eq.(3) or Eq.(4)) can be rewritten as matrix equations GX = − k (T , p ) X

(5)

where the vector contains the steady-state energy level populations and G is a real, symmetric matrix. Then, k (T , p ) can be calculated as the largest eigenvalue of G.13 Sometimes, a steady-state energy level population does not exist for a given system, the time dependent equation, Eq.(1), must be used to get the rate constant. Gillespie shown a stochastic method gives the solution to the ordinary differential equations as Eq.(1).14 Gillespie's algorithm gives a recipe for finding the duration of the time step and selecting the transition among the possible states. The duration of the recent state is chosen by using the uniform random number r1,

τ = − ln(r1 ) / kTOT

(6)

where kTOT = ∑i=1 ki is the total transition rate constant. The transition step is M

selected in the M paths by using a second random number r2,


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m−1

∑k i =1

M

i

< r2 kTOT < ∑ ki

(7)

i =m

This operation is repeated step-by-step until the system reaches the final state.

II. Microwave effect By a classical description, microwave enhanced chemistry comes from the efficiency of interactions of molecules with waves by "microwave dielectric heating" effects. This efficiency depends on the ability of molecules to absorb microwave radiation and convert the energy into heat. Polar molecules change their orientation by absorbing microwave and convert the energy into heat by intermolecular interaction. In the position of quantum mechanics, the absorption efficiency of microwave depends on the absorption rate constant15 2π v v 2 (8) |µ mk ⋅E | δ (ω mk − ω ) h2 v where h is Planck’s constant, E is the intensity of the electronic field, ωmk Wkm =

relates to the energy difference between states k and m, ω is the frequency of the

r v microwave, µ mk = ϕ m µ ϕ k

r is the transition moment and µ is the dipole moment

operator. ω is a small number for microwave, so the microwave absorption occurs between the near degenerate states only. We need to reconsider the unimolecular irreversible dissociation process and add the microwave effect reasonably. A energy grained master equation model for unimolecular reaction is shown in Figure 1. If this unimolecular dissociation reaction occurs in the microwave field, some variables, such as E0, k ( E , J ) and collision dependent transition moments

ϕm H ′ ϕk

will be changed by the surrounding

electric field. While a new energy transfer mechanism depending on the microwave

r radiation of ϕ m µ ϕ k needs to be added. (1) E0 depends on the electron structure of the molecule, while the microwave electric field is usually much less than the internal field of the molecule, so E0 should be an unchanged parameter in the master equation.



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(2) For k ( E , J ) , the unimolecular rate coefficient, it is usually estimated by RRKM or SCTST methods. Actually, k ( E , J ) is a average rate for the states between E and E + dE with J . We know that there are numerous states of the molecule in the given energy range, only part of these states can actually undergo conversion to products, this effect is described by a statistical factor in RRKM theory. Microwave radiation provides a new pathway to make the non-reactive states join the reaction, the statistical factor should be modified in RRKM theory, so k ( E , J ) will be changed by microwave effect. (3) Collision dependent transition moments

ϕm H ′ ϕk

depends on the

movement behavior of the environment molecules, and microwave field usually acts on the environment molecules and changes the collision dependent transition moments (relates to P ( E , J ; E ′, J ′) ) indirectly. (4) Microwave radiation leads to the energy transfer between the near degeneration states and changes the states distributions of the active molecules. For the pure thermal process, the collision transfer probability function must satisfy the detailed balancing principle when the system is at equilibrium, and the transfer probability function is taken as a form of exponential usually. Different from the collision energy transfer, the absorption and emission probability of microwave is close to a δ function, and it is far from the thermal equilibrium between a stronger microwave field and the normal reaction system according to the black body radiation theory. So a strong microwave field may break up the equilibrium distribution of the internal states of a molecule, then the temperature of a non-equilibrium system is not a reliable parameter. Base on above four cases, the two-dimensional master equation for the irreversible dissociation of a molecule immersed in a solvent and a microwave environment can be changed as


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∞

[

]

dN ( E , J , t ) = Z ∑ ∫ P MW ( E , J ; E ′, J ′) N ( E ′, J ′, t ) − P MW ( E ′, J ′; E , J ) N ( E , J , t ) dE ′ dt J′ 0 ∞

+ ∑ ∫ [W ( E , J ; E ′, J ′) N ( E ′, J ′, t ) − W ( E ′, J ′; E , J ) N ( E , J , t )]dE ′ J′ 0

− k MW ( E , J , t ) N ( E , J , t )

(9)

∞

= Z ∑ ∫ [Pw( E , J ; E ′, J ′) N ( E ′, J ′, t ) − Pw( E ′, J ′; E , J ) N ( E , J , t )]dE ′ J′ 0

− k MW ( E , J , t ) N ( E , J , t ) where Pw( E, J ; E′, J ′) = P MW ( E, J ; E′, J ′) + W ( E , J ; E′, J ′) / Z , P MW ( E , J ; E ′, J ′) is the transfer probability from the states between ( E ′, E ′ + dE ) with J ′ to the states ( E , E + dE ) with J in the microwave environment, same as the normal transfer

probability function , it can be obtained by QCT (Quasi-Classical Trajectory method) simulation in microwave field.11a W ( E , J ; E ′, J ′) is absorption rate constant of the microwave, it can be calculated by Eq.(8) in principle. And k MW ( E, J , t ) is the unimolecular rate coefficient in the microwave environment, it can be estimated by RRKM or SCTST methods with microwave effect.16 The corresponding one dimensional version is ∞

− k (T , p ) x ( E ) = Z ∫ [ P MW ( E , E ′)x ( E ′) − P MW ( E ′, E ) x ( E )]dE ′ 0

∞

+ ∫ [W ( E , E ′)x ( E ′) − W ( E ′, E ) x ( E )]dE ′ − k MW ( E ) x ( E )

(10)

0

III. Two states model For a real system, we can solve the time dependent linear equations of Eq. (9) by standard mathematical methods, then the contributions of the three terms on the right hand of that equation can be considered automatically. It must be system dependent for which term is a determinate one. According to the equation, the temperature, pressure and the intensity of the microwave field can change the importance of each term. The first two terms relate to the thermal effect, and the non-thermal effect is included in the last term. Here, a simplest two states model is proposed to discuss the behavior of the last term.



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For the simplest two states model, the work equation of Eq.(10) will be changed into a discrete form,

 dx (1)  dt = Z [ P12 x (2) − P21 x (1)] + [W12 x (2) − W21 x (1)] − k1 x (1)  dx (2)  = Z [ P21 x (1) − P12 x (2)] + [W21 x (1) − W12 x (2)] − k 2 x (2)  dt where Z is the collision frequency, Pji

is the transition density, W ji

(11)

is the

microwave absorption and emission rate constant, k1 and k2 are the rate constants of unimolecular dissociation. At a certain concentration, for a given molecule, the collision frequency

is proportional to the average speed of the molecule, i.e.

Z = Z 0 T . The transition density Pji adopt so-called exponential down model in

present paper,17

ρ i Pji e − βE = ρ j Pij e i

− βE j

(12)

where β = 1 / kT with Boltzmann constant of k and temperature of T , ρ j is the state density of energy level of j and E j is the energy of j states. For this two states model, the energy difference between these two states is microwave available, it must be very small, then the expression of transition density can be reduced as P21 / P12 = ρ 2 / ρ1 . The two states model adopted to consider the microwave effect is

shown in Fig. 2 for clear. The internal states of the molecule are divided into two sets by the dissociation rate constants (fast and slow states) to replace by the energy levels. The state densities of these two sets are ρ1 and ρ2. If we suppose that the microwave absorption and emission rate constants between two quantum states are exactly same, then W21 / W12 = ρ 2 / ρ1 . Although P and W take the simplest forms here, it does not affect to describe the microwave effect reasonably. It is obvious, in Eq.(10), the first term on the right hand of the equation relates to the collision energy transfer, the second term relates to the direct absorption and emission of the microwave, the last term relates to the dissociation process. The heating process which relates to the first and second terms on the right hand of Eq.(10) is not included in this two states model (energy difference for the two sets of the


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internal states is too small to change the temperature of the system) due to there is no controversy in existing knowledge of microwave effect, this simple two states model just focus on that the system has gotten a stable temperature and the population of the reactive states is large to get obvious reaction. This model is constructed base on the one dimension version of the master equation. Actually, due to the rotational excitation is very important for a polar molecule in the microwave field, the angular momentum should be important for the real microwave processes. But a one dimension model is enough for capturing all of the features for master equation description of the microwave effect.

IV. Model system Two states model is too simple to get a complete picture for microwave reactions. It is necessary to find an experiment available system to test this model shown in Eq.(9) or Eq.(10), but the reported systems in microwave chemical synthesis are too complex to execute the corresponding theoretical simulations. For a possible reaction, the crossed molecular beam experiment in a strong microwave field should be an ideal approach to study the microwave effect, although I can't find any suitable experimental work recently too. Before any usable and available experiment being reported, a reasonable model system will help us to understand the microwave effect one step ahead. In present model, the events of collision energy transfer, dissociation process, absorption and emission of microwave are involved, these processes had been discussed everywhere.12a, 13, 18 The collision energy transfer process occurs in a time scale of 10-11 s, and it is temperature, pressure and concentration dependent. The collision frequency of Z in Eq. (10) is set as Z = Z 0 T to consider to temperature effect directly. The transfer probability function is presented by a bi-exponential model 14c as following, MW PDown ( E , E ′) = e − ( E ′− E ) /( C1 + C2 E′)，E ′ > E ,



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MW MW PUp ( E , E ′) / PDown ( E ′, E ) =

ρ E ' −( E ′− E ) / kT . e ρE

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(13)

where C1 and C2 are system dependent parameters. ρ E is the state density at the energy of E, it takes the form of ρ E = 100.07

E

which is close to the state density

distribution of the small molecule of HOCO.11b The subscript "Down" and "Up" relate to the energy transferred in a downward and upward directions respectively. The energy dependent unimolecular dissociation rate constant k ( E ) can be calculated by many reliable method,9, satisfies k ( E ) = 100.22

E −15

19

we only suppose that the rate constant

for our model system, and thus distribution is close to

HOCO molecule too.11b Additional, because the fast and slow dissociation states at a same energy grid E may provide a different dissociation mechanism in the microwave field, we divide the states in a energy grid into a fast and a slow sets respectively. The state densities and the rate constants of the fast and slow sets satisfy a strong condition of

ρ Ef + ρ Es = ρ E ρ Ef / ρ Es = Rρ ρ Ef k f ( E ) + ρ Es k s ( E ) = k ( E )

.

(14)

k f ( E ) / k s ( E ) = Rk Here, the density ratio Rρ and the rate constant ratio Rk are taken as the input parameters to decide every variables in Eq. (14), and the superscripts 'f' and 's' denote the fast and slow states. The absorption of a photon of radiation in a moderate field intensity environment occurs very rapidly, on the order of 10-15 s.18 the de-excitation process in microwave field for normal vibrational-rotational energy level should have similar rate comparing with the absorption events due to the state densities for the downward and upward directions are close. In present paper, we use a Lorentzian function to describe the absorption and emission rate constant


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W ( E , E ′) = W0 ρ E ′

1

γ

π (| E - E ′ | -ω0 ) 2 + γ 2

,

(15)

where ω0 is the microwave frequency, and γ denotes the damping constant. ρ E ′ means the rate constant proportional to the state density of the final state. W0 relates to the intensity of the microwave field. The numerical simulation for this model is executed using our State Specified Master Equation package (SSME) which implements the one dimension work equation of Eq.(10) base on Gillespie's algorithm. The relative parameters in the model system are listed in Table 1.

Discussions I. The two states model described by Eq.(11) contains many different processes. Firstly, in our two states model, the internal states of the molecule are divided into two sets of fast one and slow one according to their dissociation rate. Actually, the energy difference between any two of these states is less than the energy of the microwave. If the energies of these states are much higher than the reaction barrier and the energy transfer among the different vibrational modes is very fast, the dissociation rate constants of the each state are very close. For this case we can suppose that k1 = k2, then the Eq.(11) is easy to be solved, the unimolecular dissociation rate constant equal to k1 due to the collision energy transfer and the microwave processes change the internal states distribution only. It is clearly, microwave is useless except heat the system. For the real systems, there are numerous states of the molecule in the given energy range, only part of these states can actually undergo conversion to products. It is reasonable to divide these states into slow and fast sets respectively. Corresponding, in our two states model, the average dissociation rates of the slow and fast sets are k1 and k2 with k1 > k 2 , W21 >> k 2 and ZP12 = ZP21 ≈ 0 . Because the energy difference between the slow and fast dissociation states is very small in the two states model, the temperature is unchanged almost, while the unimolecular dissociation rate constants can change by few hundred times as shown in Fig. 3, thus change of the rate constant comes from microwave non-thermal effects completely. In Eq.(16), k21 and k12 is independent each other and they can take any values. If Pji and Wji adopt the reduced model, i.e., P21 / P12 = ρ 2 / ρ1 and W21 / W12 = ρ 2 / ρ1 , then we have


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k 21 / k12 = ρ 2 / ρ1

(17)

In Eq.11, the density ratio of the slow and fast states is taken as a parameter. For getting the solution of the corresponding equation using Gillespie's algorithm, we suppose that the initial distribution of the slow and fast states proportional to their density ratio ρ1 / ρ 2 firstly. The average rate constants for the unimolecular dissociation processes when the density ratio between the slow and fast states R=

ρ1 / ρ 2 and k21 take different values using the condition of 10000×k1=k2 =1000 are shown in Fig. 4. The average rate constants k are monotonically increasing about the transition rate between the slow states set and the fast states set under the different density ratios of R, When the density ratio R takes a small value of 0.001, it means the number of the fast states is 1000 times larger than the slow ones, most of the molecules dissociate quickly, the real rate constants should be close to the fast rate of k2 =1000. But the calculated average rate constants is less than k2 when k21 is very small, it due to the slow components of the system slow down the average value. For the density ratios of R=0.001, 0.01, 0.1, 1.0, 10, 100 and 1000, the average rate constants increase about 10, 100, 840, 2400, 830, 100 and 10 times respectively when k12 change from 0.1 to 10000. At a given temperature and pressure, the rate constants of the collision energy transfer processes will take a fixed value, k12 can be adjusted by microwave processes. Base on our reduced two states model, the average rate constants of the unimolecular dissociation is very sensitive to k12. Same thing, the average rate constants of the unimolecular dissociation should be sensitive to the intensity of the microwave field. In our two states model, the rate constants about both of the collision energy transfer and the microwave processes take the simplest models. These models include a strong condition of P21 / P12 = W21 / W12 = ρ 2 / ρ1 . In above discussion, we get the conclusion: When both of the rate constants of collision energy transfer and the microwave processes are much larger than the fast dissociation rate constant of k2, the unimolecular dissociation rate constant equals to ( k12 k1 + k 21k 2 ) /( k12 + k 21 ) . It means



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that when the system takes the case of P21 / P12 > W21 / W12 , microwave processes can slow down the unimolecular dissociation rate constant. Additionally, this condition of P21 / P12 = W21 / W12 = ρ 2 / ρ1 in the reduced two states model comes from the principle

of detailed balance for collision processes and the hypothesis of equal probability transition for the microwave processes, the real system which interacts with a strong microwave field must be far from equilibrium state, more complex phenomena and mechanisms are possible, it will be discussed in the future. There are a number of internal states for the real molecules, two states model is too simple to describe the real case, but the real system must contain similar behaviors shown in the two states model. According to above discussion, the two states model has captured the working mechanism of microwave non-thermal effect successfully. Generally, master equation considers as an efficient approach to estimate the rate constant, the microwave effect should be captured by Eq.(9). There are three different terms on the right hand in Eq.(9), we can investigate the behavior of each term by setting different conditions. At low pressure limit, the collision frequency Z is negligible. If we suppose the unimolecular rate constant is a RRKM one, i.e. k MW ( E , J ) = k ( E , J ) . The microwave effect is completely included in the second term on the right hand of Eq.(9). Molecules change their states by absorbing or emitting the microwave, and we know that stimulated excitation and emission rate is much larger than the spontaneous emission usually. If we ignore the spontaneous emission and furthermore suppose that the absorption rate is a constant, i.e., W ( E , J ; E ′, J ′) = C . Due to the density of states increases with increasing energy, the contribution of the high energy levels will increase in the states distribution under the microwave field, then the temperature will increase correspondingly. This case belongs to the specific microwave effect, the microwave heat the molecule directly. When C≪ , the molecule will dissociate before absorbing or emitting the microwave, there is no obvious microwave effect. When C≫ , the molecule will absorb or emit the microwave many times before


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dissociation. This conclusion is in agreement with the comment mentioned by Lewis: "slower reacting systems tend to show a greater effect under microwave radiation than faster reacting ones".20 For the non-RRKM and the mode-specified reactions,21 the rate constants may change obviously when the vibrational mode or the rotational states changed, microwave radiation can join thus processes in principle.22 So if the unimolecular rate constant is a non-RRKM or the mode-specified reactions, since the microwave radiation provides a new pathway to make the non-reactive states join the reaction, normally k MW ( E , J ) or the effective rate constants is larger than k ( E , J ) and depends on the intensity of the microwave field. The case is just described in the two states model. II. The model system governed by Eq. (10) provides a complete description for microwave involved unimolecular dissociation, and all of terms containing in Eq. (10) compete with each other. According to the typical time scales of the collision frequency, absorption and emission of a photon of radiation in a moderate field intensity environment and the unimolecular dissociation processes, the model system is built up using the dynamics parameters listed in Table 1. The dynamics properties are obtained by executing our SSME code. In Figure 5a, both of Rρ and Rk are set as 1, it means that all of the quantum states in each energy grid have the same average dissociation rate constants. At the same temperature of 400K, the simulated dissociation rate constants are obviously dependent on the field strength of W0 , the simulated rate constants are 3.81×105, 3.87×105, 1.39×106 and 1.01×107 molecule/s when W0 takes 0.0, 104, 106 and 107 respectively. And the simulated rate constant is 4.46× 106 molecule/s when the temperature takes 500K without microwave field. The simulation rate constants are shown in Figure 5b for the case of Rρ = 100 and Rk = 1000 , the corresponding simulated rate constants are 2.56×105, 2.66×105,



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1.66 × 106 and 1.50 × 107 molecule/s when W0 takes 0.0, 104, 106 and 107 respectively. And the simulated rate constant is 3.42× 106 molecule/s when the temperature takes 500K and W0 = 0 . According to the conclusion of the two states model, the microwave effect will become significant when the microwave absorption rate is much greater than the collision frequency. At 400K, the collision frequency is about 2×1011/molecule/s, while the absorption and emission rate constant is about 1017/molecule/s when W0 = 107 , absorption rate constant is much larger than collision frequency too. We know that both of the collision frequency and the absorption rate constants of microwave are adjustable easily, these parameters set in the model system are achievable in the real cases. Comparing Figure 5a with 5b, we can see that the microwave effect is more sensitive for thus model system which contains fast and slow states in the same energy grid, but there is no fundamentally difference is found. When W0 = 104 , the absorption and emission rate constant of microwave is about 500 times larger than the collision frequency, but the dissociation rate constant increases about 4 percents comparing with the case of W0 = 0 , this is due to that the states density of the slow dissociation set is larger than the fast one by 100 times ( Rρ = 100 ), and the condition of Eq, (14) for Rρ and Rk is very strong. The distribution of the normalized number density of the internal states of the model system for different cases are shown in Figure 6, where the fast and the slow states' contributions are summed over. The restrictions of Rρ and Rk shown in Eq. (14) are really strong, the distributions of the normalized number density shown in Figure 6a and 6b are really close, although Rρ and Rk are much different for two panels. In Figure 6, the normalized number density distributions of the internal states of the model system at 400K and 500K without microwave field are Boltzmann distribution. The corresponding contributions for W0 = 0 and W0 = 104 at 400K are


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very close, and there has been long tails for W0 = 106 and W0 = 107 cases. If the microwave field is not too strong ( W0 = 104 ), thermal collision determines the state distribution, the dissociation rate constants changes not so much even for the slow and fast states considered case. In the strong microwave field, the distribution of internal states of the molecules will deviate from the equilibrium distribution, and the system temperature is no longer available. This non-equilibrium distribution of the molecular internal states is impossible in conventional heating approach, it should be a kind of the non-thermal or non-equilibrium thermal microwave effect, and this effect has not been described in present two states model too. In the simulation, there are 2000 energy grids are adopted to describe the internal states of the model system, and each energy grid is divided into the fast and slow sets furthermore, so there are 4000 state grids totally. Every sampling point must start from someone state grid, jump between the state grids driven by collision and microwave, dissociate to the product state through someone internal state finally. In Figure 7, dissociation energy states distributions for every trial sampling in the simulation when rate constant ratio Rk=1000 and state density ratio Rρ=100 are plotted, and different microwave field intensities of W0= 0 and 107 are used in panels (a) and (b) respectively. Additionally, black and red points are used to denote the fast and slow channels. For the case of W0= 0, there are 2644 and 2356 sampling points dissociate from fast and slow channels respectively, and the slow channels have higher internal energy. When W0 is taken as 107, the fast and slow points are 3787 and 1213 in a total of 5000 trail points, more fast channels are involved in a strong microwave field. Indeed, microwave interaction provides a new mechanism to drive the slow states to the fast states and speeds up the dissociation rate constants, and this effect speeds up the rate constant not so much comparing with the effect of redistribution of the internal states. The model system displayed a more complete dynamics processes for the microwave involved dissociation reaction. The distribution of internal states of the molecules will deviate from the equilibrium distribution and the dissociation channels



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contains more fast dissociation channels in a strong microwave field. These two features are impossible processes in the conventional heating approach, they are apparent microwave effect. The reported microwave reaction experiments involve complex solvent environment, unstable temperature condition and uneven microwave field. It is too hard to get a reliable simulation for these experimental systems, and it is not the aim of this paper to consider these real systems.

Summary For a given temperature, the differences of reaction rate between some of the microwave activation and the conventional heating reactions close to 103 times. In this work, we used a high accuracy theoretical model of master equation to capture the "microwave effect". Theoretical analysis shows that microwave field can affect the reactions by following ways: 1) New rapid reaction channels are opened by the absorption and emission of the microwave. 2) The distribution of internal states of the molecules will deviate from the equilibrium distribution in a strong microwave field. 3) Microwave heats the system directly or indirectly. Based on the general form of "microwave effect" included master equation, a two states model for unimolecular dissociation is used to discuss the microwave non-thermal effect particularly. In a microwave energy comparable energy range, the internal states of the molecule are divided into slow and fast sets according to the dissociation rate constants respectively, the distribution of the internal states depend on the collision energy transfer and the microwave processes. The average rate constants for the two states system can be increased many times for some given cases without the temperature changed in the two states model. At constant temperature, the two state model shows that a new energy transfer mechanism depends on the

r microwave radiation of ϕ m µ ϕ k

which can change the reaction rate constants

significantly. While the simulation result for the model system shows that the dissociation rate constants can be increased significantly when a strong microwave field is added. The theoretical analysis in this work shows that the non-thermal


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microwave effect possibly exists, and the reaction rate constants can be changed obviously by the microwave field for the non-RRKM and the mode-specified reactions. We know that Arrhenius plot is always used to explain the microwave effect,4a non-thermal microwave effect can't change the pre-exponential factor and the barrier high in Arrhenius formulation directly, it just changes the statistical behavior of the internal states according to our theoretical model. Base on some experimental results, authors get a conclusion of that non-thermal microwave effects do not exist.7a At the same time, there are many more claims to the existence of these effects in organic chemistry.4a, 5b In this work, the formula and the simulations show that the non-thermal microwave effects possibly exist theoretically, specially for the non-RRKM and the mode-specified reactions. In general, few equilibrium thermodynamic parameters can be used to describe the rate constants of the conventional heating reactions, and the temperature of the system, i.e. the thermal average of the molecular energy, is a reliable and stable variable. In a strong microwave field, the distribution of internal states of the molecules will deviate from the equilibrium distribution and the fast reaction channels play a more important role. And the thermal effect expressed by the system temperature is not enough to determine the rate constants, so the concept of the microwave non-thermal or the non-equilibrium thermal effect is necessary to be proposed.

Acknowledgements: JM acknowledges the National Natural Science Foundation of China (91441107, 21303110) for support. JM also thanks Prof. Xiaoqing Yang and Dr. Boyd Swing for useful discussion.

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Gedye, R.; Smith, F.; Westaway, K.; Ali, H.; Baldisera, L.; Laberge, L.; Rousell, J., The use of

microwave ovens for rapid organic synthesis. Tetrahedron Letters 1986, 27 (3), 279-282.



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2.

Perreux, L.; Loupy, A.; Petit, A., Nonthermal Effects of Microwaves in Organic Synthesis. In

Microwaves in Organic Synthesis, Wiley-VCH Verlag GmbH: 2012; Vol. 1, pp 127-207. 3.

(a) Kalhori, S.; Minaev, B.; Stone-Elander, S.; Elander, N., Quantum Chemical Model of an SN2

Reaction in a Microwave Field. The Journal of Physical Chemistry A 2002, 106 (37), 8516-8524; (b) Gabriel, C.; Gabriel, S.; H. Grant, E.; S. J. Halstead, B.; Michael P. Mingos, D., Dielectric parameters relevant to microwave dielectric heating. Chemical Society Reviews 1998, 27 (3), 213-224. 4.

(a) de la Hoz, A.; Diaz-Ortiz, A.; Moreno, A., Microwaves in organic synthesis. Thermal and

non-thermal microwave effects. Chemical Society Reviews 2005, 34 (2), 164-178; (b) Kappe, C. O., Controlled Microwave Heating in Modern Organic Synthesis. Angewandte Chemie International Edition 2004, 43 (46), 6250-6284. 5.

(a) Kaiser, N.-F. K.; Bremberg, U.; Larhed, M.; Moberg, C.; Hallberg, A., Fast, Convenient, and

Efficient Molybdenum-Catalyzed Asymmetric Allylic Alkylation under Noninert Conditions: An Example of Microwave-Promoted Fast Chemistry. Angewandte Chemie 2000, 112 (20), 3741-3744; (b) Nuchter, M.; Ondruschka, B.; Bonrath, W.; Gum, A., Microwave assisted synthesis - a critical technology overview. Green Chemistry 2004, 6 (3), 128-141. 6.

Yang, L.; Huang, K.; Yang, X., Dielectric Properties of N,N-Dimethylformamide Aqueous

Solutions in External Electromagnetic Fields by Molecular Dynamics Simulation. The Journal of Physical Chemistry A 2010, 114 (2), 1185-1190. 7.

(a) Kappe, C. O.; Pieber, B.; Dallinger, D., Microwave Effects in Organic Synthesis: Myth or

Reality? Angewandte Chemie International Edition 2013, 52 (4), 1088-1094; (b) Nuchter, M.; Muller, U.; Ondruschka, B.; Tied, A.; Lautenschlager, W., Feasibility of chemical reactions in microwave field. Chem. Ing. Tech. 2002, 74 (7), 910-920; (c) Laurent, R.; Laporterie, A.; Dubac, J.; Berlan, J.; Lefeuvre, S.; Audhuy, M., Specific activation by microwaves: myth or reality? The Journal of Organic Chemistry 1992, 57 (26), 7099-7102; (d) Kanno, M.; Nakamura, K.; Kanai, E.; Hoki, K.; Kono, H.; Tanaka, M., Theoretical Verification of Nonthermal Microwave Effects on Intramolecular Reactions. The Journal of Physical Chemistry A 2012, 116 (9), 2177-2183. 8.

(a) Berlan, J.; Giboreau, P.; Lefeuvre, S.; Marchand, C., Synthese organique sous champ

microondes : premier exemple d'activation specifique en phase homogene. Tetrahedron Letters 1991, 32 (21), 2363-2366; (b) Garbacia, S.; Desai, B.; Lavastre, O.; Kappe, C. O., Microwave-Assisted Ring-Closing Metathesis Revisited. On the Question of the Nonthermal Microwave Effect. The Journal of Organic Chemistry 2003, 68 (23), 9136-9139; (c) Chokichiro, S.; Tomohiro, K.; Kimihiro, O., Nonthermal Influence of Microwave Power on Chemical Reactions. Japanese Journal of Applied Physics 1996, 35 (1R), 316; (d) Miklavc, A., Strong Acceleration of Chemical Reactions Occurring Through the Effects of Rotational Excitation on Collision Geometry. ChemPhysChem 2001, 2 (8-9), 552-555. 9.

Baer, T.; Hase, W. L., Unimolecular Reaction Dynamics. Oxford: New York, 1996.

10. Miller, W. H.; Hernandez, R.; Handy, N. C.; Jayatilaka, D.; Willetts, A., Ab initio calculation of anharmonic constants for a transition state, with application to semiclassical transition state tunneling probabilities. Chemical Physics Letters 1990, 172 (1), 62-68. 11. (a) Jasper, A. W.; Pelzer, K. M.; Miller, J. A.; Kamarchik, E.; Harding, L. B.; Klippenstein, S. J., Predictive a priori pressure-dependent kinetics. Science 2014, 346 (6214), 1212-1215; (b) Weston, R. E.; Nguyen, T. L.; Stanton, J. F.; Barker, J. R., HO + CO Reaction Rates and H/D Kinetic Isotope Effects: Master Equation Models with ab Initio SCTST Rate Constants. The Journal of Physical Chemistry A 2013, 117 (5), 821-835.


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12. (a) Barker, J. R., Energy transfer in master equation simulations: A new approach. International Journal of Chemical Kinetics 2009, 41 (12), 748-763; (b) Jeffrey, S. J.; Gates, K. E.; Smith, S. C., Full Iterative Solution of the Two-Dimensional Master Equation for Thermal Unimolecular Reactions. The Journal of Physical Chemistry 1996, 100 (17), 7090-7096. 13. Miller, J. A.; Klippenstein, S. J.; Raffy, C., Solution of Some One- and Two-Dimensional Master Equation Models for Thermal Dissociation: The Dissociation of Methane in the Low-Pressure Limit. The Journal of Physical Chemistry A 2002, 106 (19), 4904-4913. 14. (a) Gillespie, D. T., A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. Journal of Computational Physics 1976, 22 (4), 403-434; (b) Gillespie, D. T., Monte Carlo simulation of random walks with residence time dependent transition probability rates. Journal of Computational Physics 1978, 28 (3), 395-407; (c) Barker, J. R., Multiple-Well, multiple-path unimolecular reaction systems. I. MultiWell computer program suite. International Journal of Chemical Kinetics 2001, 33 (4), 232-245. 15. Liang, K.-K.; Chang, R.; Hayashi, M.; Lin, S. H., Principles of molecular spectroscopy and photochemistry. Zhong Xing Ke Yan: 2001. 16. (a) Lourderaj, U.; Hase, W. L., Theoretical and Computational Studies of Non-RRKM Unimolecular Dynamics. The Journal of Physical Chemistry A 2009, 113 (11), 2236-2253; (b) Berne, B. J.; Borkovec, M.; Straub, J. E., Classical and modern methods in reaction rate theory. The Journal of Physical Chemistry 1988, 92 (13), 3711-3725. 17. Penner, A. P.; Forst, W., Analytic solution of relaxation in a system with exponential transition probabilities. The Journal of Chemical Physics 1977, 67 (11), 5296-5307. 18. Linsebigler, A. L.; Lu, G.; Yates, J. T., Photocatalysis on TiO2 Surfaces: Principles, Mechanisms, and Selected Results. Chemical Reviews 1995, 95 (3), 735-758. 19. Miller, W. H.; Hernandez, R.; Moore, C. B.; Polik, W. F., A transition state theory-based statistical distribution of unimolecular decay rates with application to unimolecular decompoisition of formaldehyde. J. Chem. Phys. 1990, 93, 5657. 20. Lewis, D. A., Microwave Processing of Polymers - An Overview. MRS Online Proceedings Library Archive 1992, 269, 21 (11 pages). 21. (a) Wang, F.; Lin, J.-S.; Liu, K., Steric Control of the Reaction of CH Stretch–Excited CHD3 with Chlorine Atom. Science 2011, 331 (6019), 900-903; (b) Li, J.; Jiang, B.; Song, H.; Ma, J.; Zhao, B.; Dawes, R.; Guo, H., From ab Initio Potential Energy Surfaces to State-Resolved Reactivities: X + H2O ↔ HX + OH [X = F, Cl, and O(3P)] Reactions. The Journal of Physical Chemistry A 2015, 119 (20), 4667-4687. 22. Leitner, D. M.; Levine, B.; Quenneville, J.; Martínez, T. J.; Wolynes, P. G., Quantum Energy Flow and trans-Stilbene Photoisomerization: an Example of a Non-RRKM Reaction. The Journal of Physical Chemistry A 2003, 107 (49), 10706-10716.



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Table 1. Dynamics parameters used in the model system. ( E in cm-1) Parameters

Formulations

Values

Collision frequency

Z = Z0 T

Z 0 = 10 9

Transfer

MW PDown ( E , E ′) = e − ( E ′− E ) /( C1 + C2 E′)，E ′ > E

C1 = 100, C2 = 0.015

probability MW MW PUp ( E , E ′) / PDown ( E ′, E ) =

Absorption and emission rate

W ( E , E ′) = W0 ρ E ′

γ π (| E - E ′ | -ω0 ) 2 + γ 2 1

constant density ratio and rate constant ratio

T = 400 K and 500 K

ρ E ' − ( E ′− E ) / kT e ρE

W0 = 0， 10 4， 10 6 and 10 7

ω 0 = 8cm -1，γ = 0.12cm -1

Rρ = 1 and Rk = 1 Rρ = 100 and Rk = 1000

Energy grid

Grid number=2000, ∆E = 30cm -1

Trial number

5000


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Figure captions Figure 1. Representation of energy grained master equation model for unimolecular reaction. E0 is the critical energy below which classical reaction cannot occur. When the state energy less than EL, the energy difference between two adjacent states larger than the energy of microwave radiation, energy transfer realized by intermolecular collision. When the state energy greater than EL, the rotation vibrational states are too close to absorb the microwave photon, energy transfer realized by both intermolecular collision and microwave absorption.

ϕm H ′ ϕk

and

r

ϕm µ ϕk

present the

collision dependent and microwave absorption dependent transition moments respectively. TS presents the transition state of the unimolecular reaction, k ( E , J ) is the unimolecular rate coefficient. Furthermore, the energy levels of the molecule is presented by different colors. Orange states are the reactive states, blue states are the non-reactive states and they can reach the reactive states by both collision or microwave absorption processes, green states are the non-reactive states and they can reach the reactive states by microwave absorption process only.

Figure 2. Two states system for unimolecular reaction in microwave field. The internal states of the molecule are divided into two sets by dissociation rate constants (fast and slow states). The state densities of these two sets are ρ1 and ρ2, and the rate constants are k1 and k2 respectively. Additionally, we suppose that the rate constants of collision energy transfer and the microwave processes are P12, P21, W12 and W21 as shown in this figure.

Figure 3. The stable solution of the unimolecular dissociation rate constants described by Eq.13 when k21 and k12 take different values using the condition of 10000×k1=k2 =1000.

Figure 4. The average rate constants for the unimolecular dissociation processes



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when the density ratio between the slow and fast states R= ρ1 / ρ 2 and k21 take different values using the condition of 10000×k1=k2 =1000.

Figure 5. The dissociation rate constants for the model system at different temperatures (T=400K and 500K), microwave field intensities (W0=0, 104, 106 and 107), rate constant ratio Rk and state density ratio Rρ for the fast and slow state sets.

Figure 6. The distribution of the normalized number density of the internal states of the model system at different temperatures (T=400K and 500K), microwave field intensities (W0=0, 104, 106 and 107), rate constant ratio Rk and state density ratio

Rρ for the fast and slow state sets.

Figure 7. Dissociation energy states distribution for every trial sampling in the simulation when rate constant ratio Rk=1000 and state density ratio Rρ=100. Different microwave field intensities of W0=107 and 0 are used in panels (a) and (b) respectively, while the black and red points in the figures denote the fast and slow channels.


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Figure 1.



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Figure 2.


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Figure 3.



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Figure 4.


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Figure 5.



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Figure 6.


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Figure 7.



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TOC graphic


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Master Equation Analysis of Thermal and Nonthermal Microwave

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