Site Blocking Effect on Diffusion-Mediated Reactions in Porous Media

Article pubs.acs.org/JPCC

Site Blocking Effect on Diffusion-Mediated Reactions in Porous Media Kazuhiko Seki,*,† Aditya Ballal,‡ and M. Tachiya*,† †

National Institute of Advanced Industrial Science and Technology (AIST), AIST Tsukuba Central 5, Higashi 1-1-1, Tsukuba, Ibaraki, Japan, 305-8565 ‡ Indian Institute of Technology Bombay, Powai, Mumbai Maharashtra, India, 400 076 ABSTRACT: Inspired by recent experiments on quenching kinetics of excited probes in zeolites, we study the site blocking effect on the survival probability of an immobile probe by reaction with mobile reactants. Reactants perform random walks on a diamond lattice under the constraint of prohibiting double occupancy. Reaction takes place when reactants occupy the nearest neighbor sites of the probe. Theoretical expressions are assessed by Monte Carlo simulations. By analyzing the published experimental data with a simple theoretical expression, we show that our model is relevant to the study of the quenching of excited anthracene by nitromethane in dehydrated zeolite Y.

1. INTRODUCTION The quenching kinetics of an excited probe by reaction to the surrounding molecules in porous materials is often very different from that in the solution phase.1,2 In solution, reactants diffuse freely, whereas in porous media, they can migrate only to interconnecting neighboring pores. Moreover, the number of reactants in a pore is restricted by steric interaction, and the quenching kinetics depends more strongly on the quencher concentration than without restriction.1,2 Although diffusion-controlled reactions have been studied in great detail, most of the theories developed so far assume free diffusion of reactants at low concentrations.2−6 Random walk theory which takes into account lattice interconnectivity has been also developed in the dilute limit.1,7,8 In the early theory, the reaction is assumed to take place at the site where the probe is located.7 Recently, the theory has been improved to take into account reaction at the neighboring lattice sites.8 The theory has been applied to study highly nonexponential quenching kinetics of an excited anthracene by aromatic molecules in zeolites.8−11 Inspired by the recent experiments, we study the effect of prohibiting the double occupancy on the survival probability of the immobile probe by reaction with mobile reactants. In previous theories developed to study quenching kinetics in zeolites, this effect is ignored. In experiments, energy transfer between excited probe molecules and quenchers in faujasite-type zeolite (zeolite Y) was investigated. The porous network of zeolite Y has a diamond-like structure, and each pore can be occupied by a restricted number of molecules.8−13 In the most restrictive case, the pore can be occupied by at most one molecule. In this case, reactants perform random walks on the lattice, avoiding the probe site and each other.14−16 There is some evidence that reaction takes place when reactants occupy one of the nearest © 2012 American Chemical Society

neighbor pores connecting to the pore occupied by the probe.12,13,17 The assumption of excluding the probe site can be justified even if the steric interaction between the reactant and the probe is small as long as we are interested in the time scale larger than the spontaneous quenching of the probe by the reactant simultaneously occupying the same pore with the probe. The intrapore reaction rate should be orders of magnitude larger than the interpore reaction rate.8−13 It should be also noticed that the transition to the probe site can be blocked by the probe. Even if it is not blocked, the long-term decay of the survival probability can be reasonably examined by setting the effective reaction rate at the nearest neighbor sites of the probe by prohibiting the probe site to be occupied by the reactant. The diamond structure is not only important as a practical lattice structure of zeolite pores but also interesting for its small coordination number in three-dimensional space. The influence of prohibiting double occupancy can be particularly large in this case. In this paper, we assume that reactants perform random walks on the diamond lattice under the condition that each lattice site can be occupied by at most one molecule. In this way, steric interaction among reactants and the probe is fully taken into account. The interpore migration of a reactant is blocked if the site is occupied by the other molecule of any kind. This effect is sometimes referred to as the site blocking effect. It is increasingly important at high reactant concentration. Received: July 28, 2012 Revised: September 18, 2012 Published: September 19, 2012 22086

dx.doi.org/10.1021/jp307481p | J. Phys. Chem. C 2012, 116, 22086−22093

The Journal of Physical Chemistry C

Article

where c is the concentration of reactants (average occupation probability) and

Quenching of an immobile probe by mobile reactants is classified as the target reaction. Recently, the general theory of the target reaction has been developed by taking into account the site blocking effect among reactants.18 In the theory, the site blocking effect was taken into account by applying the method developed to study tracer diffusion by a vacancy mechanism.19−24 It has been shown that the decay of the survival probability is accelerated by the site blocking.18,25 The probability that reactants occupy the reactive sites is increased as the number of sites occupied by reactants is increased by prohibiting the double occupancy.18,25 Reactants move cooperatively for the reaction with the target. Stated differently, the survival probability obtained by allowing double occupancy is an upper bound of that obtained by prohibiting double occupancy.18,25−32 It is a general consequence of the site blocking effect regardless of the lattice structure and arrangement of the reactive sites. The recent theory based on the theory of tracer diffusion involves several approximations. If the reaction takes place when the reactant occupies the probe site, it gives the lower bound of the survival probability, and the lower bound is very close to the Monte Carlo simulation results on the three-dimensional regular lattice.18,25 The approximate theory has not been compared to simulation results for more realistic systems, where the steric interaction between reactants and the probe is present in addition to that between reactants. The theoretical results will be assessed by the results of Monte Carlo simulations on the diamond lattice. We show that the approximate theory is indeed the lower bound and reproduces the results of Monte Carlo simulations in the reaction-controlled limit. However, it deviates from the results of Monte Carlo simulations at long times in the diffusion-controlled limit where the reaction takes place from the multiple sites surrounding the probe site. We present an alternative simple theoretical expression reproducing Monte Carlo simulation results in the diffusion-controlled limit when the site blocking effect is important. This expression is used to estimate the diffusion coefficient and to justify the uniform loading of quenchers from the published experimental data. We show that the theoretical model is relevant to the study of the quenching of excited anthracene by nitromethane in dehydrated zeolite Y. In Section 2, the exact theory allowing double occupancy of reactants is reformulated by using the Green’s function. The site blocking effect by the probe is exactly taken into account in this approach. In Section 3, the approximate theory is reformulated by taking into account the site blocking effect among reactants and the probe. In Section 4, the theoretical results are compared to the results of Monte Carlo simulation. The simple analytical expression applicable in the diffusion-controlled limit is presented in Section 4. The published experimental data are analyzed in Section 5. Section 6 is devoted to conclusions, and the derivation of the Green’s function excluding the origin occupied by the probe is presented in the Appendix.

F (t ) =

(2)

ri ⃗

where P(ri⃗ ,t) is the occupation probability of the site ri⃗ by a reactant at time t. Pd(t) should be valid in the dilute limit for any value of k0. The Laplace transform of eq 2 becomes ζ

F (̂ s) =

∑ [1/s − P(̂ ri ⃗ , s)] = (k 0/s) ∑ P(̂ li⃗ , s) ri ⃗

i=1

(3)

where ζ = 4 is the coordination number for the diamond lattice and li denotes the nearest neighbor vector. In eq 3, P̂ (li⃗ ,s) can be obtained using the Green’s functions of the diamond lattice excluding the origin occupied by the probe. In this section, we present only the formulation of the Green’s function, and the detailed derivation is given in the Appendix. We denote the sublattice occupied by the immobile probe as the sublattice B. The other sublattice is denoted by the sublattice A. In the absence of reaction, the Green’s function G(A)(ri⃗ ,rj⃗ ,t) denotes the probability that the reactant is at site ri⃗ of the sublattice A at time t when it starts from rj⃗ of the sublattice A. Similarly, the Green’s function G(B) (ri⃗ ,rj⃗ ,t) denotes the probability that the reactant is at site ri⃗ of the sublattice B at time t when it starts from rj⃗ of the sublattice A. The calculation of P̂(li⃗ ,s) requires the Green’s functions starting from the nearest neighbor sites of the origin located on the sublattice A. The Green’s function of the reactants on the lattice excluding the origin occupied by the probe is obtained by solving ∂ (A) G ( ri ⃗ , rj⃗ , t ) = ∂t

ζ

∑ Γ (1 − δo ⃗ , r ⃗− l ⃗ )[G(B)( ri ⃗ − lk⃗ , rj⃗ , t ) k=1

ζ

i

k

− G(A)( ri ⃗ , rj⃗ , t )] ∂ (B) G ( ri ⃗ , rj⃗ , t ) = ∂t

(4)

ζ

∑ Γ (1 − δo ⃗ , r ⃗)[G(A)( ri ⃗ + lk⃗ , rj⃗ , t ) k=1

ζ

i

− G(B)( ri ⃗ , rj⃗ , t )]

(5)

where Γ is the hopping frequency. Since the initial configuration of reactants is uniform over all sites except the site occupied by the probe and the reaction rates at all the nearest neighbor sites of the probe are the same, the transient occupancy probability of these sites will also be the same. By denoting Ĝ S(s) = Ĝ (A)(lj⃗ ,lj⃗ ,s), Ĝ D(s) = Ĝ (A)(li⃗ ,lj⃗ ,s) for i ≠ j, we have P(̂ li ⃗ , s) = (1/s) − k 0(ĜS(s) + 3Ĝ D(s))P(̂ li ⃗ , s)

2. SURVIVAL PROBABILITY IN THE DILUTE LIMIT We consider the situation that the immobile probe is located at the origin of one of the sublattices of the diamond lattice. Reactants migrate on the diamond lattice excluding the origin occupied by the probe. Reaction is possible when the reactant occupies one of the nearest neighbor sites of the probe. The reaction rate is denoted by k0. In the absence of the site blocking effect among the reactants, the survival probability can be expressed as4 Pd(t ) = exp( −cF(t ))

∑ [1 − P( ri ⃗ , t )]

(6)

where Ĝ S(s) and Ĝ D(s) are obtained from eq A-15. By substituting eq 6, eq 3 is expressed as F (̂ s) =

4k 0

1 ̂ s 1 + k 0(GS(s) + 3Ĝ D(s)) 2

(7)

After the inverse Laplace transform of eq 7 using Ĝ S(s) and Ĝ D(s) obtained from eq A-15, the survival probability valid in the dilute limit can be obtained from eq 1.

(1) 22087



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⃗ ⃗ ̂ ̂ (A) ⃗ ⃗ where Ĝ E,S(s) = Ĝ (A) E (lj,lj,s) and GE,D(s) = GE (li,lj,s) with (A) i ≠ j. Ĝ E (li⃗ ,lj⃗ ,s) is obtained by solving simultaneously eq 5 with all the Green’s functions changed into those with the subscript E and

3. SURVIVAL PROBABILITY UNDER THE SITE BLOCKING EFFECT In the presence of the site blocking effect, the survival probability of the immobile probe was recently derived approximately in the Laplace domain.18 The expression was derived under general conditions and can be applied to the following case. Each lattice site can be empty or occupied by one of the N reactants migrating among M sites excluding the site occupied by the probe. When the reactant occupies one of the nearest neighbor sites of the probe site, reaction is possible with the rate k0. By taking the thermodynamic limit under the fixed concentration, c = N/M, the Laplace transform of the survival probability is approximately given for the diamond lattice as

∂ (A) G E ( ri ⃗ , rj⃗ , t ) = ∂t

k=1

i

ζ

k

− G E(A)( ri ⃗ , rj⃗ , t )] − (1 − 2c)k 0 ζ

∑ δ r ⃗ , l ⃗ GE(A)( ri ⃗ , rj⃗ , t ) i k

k=1

(9)

⃗ where the initial condition is given by G(A) E (ri⃗ ,lj,0) = δr⃗i,lj⃗ . A closed set of equations is obtained from eq 5 and eq 9, and the solution can be expressed as

P(̂ s) = 1/[s + 4ck 0 − c(1 − c)k 02(4Ĝ E,S(s + 4ck 0) + 12Ĝ E,D(s + 4ck 0))]

ζ

∑ Γ (1 − δo ⃗ , r ⃗− l ⃗ )[GE(B)( ri ⃗ − lk⃗ , rj⃗ , t )

(8)

⎛ Ĝ (s) ⎞ ⎛1 + (1 − 2c)k Ĝ (s) ⎞−1⎛ Ĝ (s) ⎞ ̂ D(s) 3(1 2 c ) k G − E,S 0 S 0 ⎜ ⎟=⎜ ⎟ ⎜ S ⎟ ⎟ ⎟ ⎜ ⎜ ̂ ⎟ ⎜ ⎝G E,D(s)⎠ ⎝ (1 − 2c)k 0Ĝ D(s) 1 + (1 − 2c)k 0(ĜS(s) + 2Ĝ D(s))⎠ ⎝Ĝ D(s)⎠

By substituting eq 10 into eq 8, we obtain the survival probability in the time domain after the inverse Laplace transform of eq 8.18 The survival probability obtained in this section takes into account the fact that any lattice site can accept at most one molecule. Double occupancy of reactants is prohibited, and the probe site cannot be accessed by the reactants. The higher-order correlations resulting from the site blocking effect are taken into account by Padé approximation.18 The accuracy is assessed by Monte Carlo simulations in the subsequent section.

(10)

concentration. In the reaction-controlled limit, the reactant concentration around the probe is not driven much by the concentration gradient induced by the reaction. In the weak nonequilibrium condition, the site blocking effect on the reactant distribution is not large. Although the effect is not large, the agreement between the theoretical results of eq 8 and the numerical simulation results is excellent. In Figure 1, circles represent the results in the static limit given by Pstat(t ) = [1 − c + c exp(−k 0t )]ζ

4. COMPARISON TO NUMERICAL SIMULATION We start the Monte Carlo simulations by randomly placing the reactants at N out of total of M lattice sites. For simplicity, we place the probe site at the center of the lattice. The random walk to the probe site and the occupied lattice sites is prohibited. The rate of a random walk is Γ/4 where Γ is the jump frequency. The reaction rate of the probe with a reactant at one of the nearest neighboring sites is k0. The total rate of all the possible events can be written as ktot = KΓ/4 + Rk0, where K is the number of possible random walks and R is the number of reactants on nearest neighbors of the probe site. We generate a waiting time of an event to occur as Δt = −ln r/ktot, where r is a uniform random number, and the simulation is proceeded by time Δt. Then we randomly select an event that will actually take place with the probability given by the ratio of its rate to ktot.25,33 The simulation is continued until the reaction takes place. The calculations are repeated for at least 104 independent runs, and we obtain a histogram of decay times. By dividing the histogram by the total number of runs, we obtain the survival probability of the probe.25 In Figure 1, we show the results for Γ/k0 = 10.0. Γ/k0 ≫ 1 can be regarded as the reaction-controlled limit. In this limit, eq 8 for the survival probability obtained by prohibiting the double occupancy reproduces numerical simulation results very well for high reactant concentration, c = 0.5. The approximation used to derive eq 8 is justified in this limit. It should also be noticed that the theoretical results obtained without taking into account the constraint on double occupancy, eq 1, do not deviate much from simulation results even though reactant concentration is as high as c = 0.5. The deviation decreases with decreasing reactant

(11)

Figure 1. Time dependence of the survival probability. Diamond lattice, Γ/k0 = 10.0 and c = 0.5. (Red) dots are obtained by the Monte Carlo simulations. The solid line is obtained by the inverse Laplace transform of eq 8 with eq 10 and eq A-15. The solid line overlaps with (red) dots. The dashed line represents the result obtained by substituting the inverse Laplace transform of eq 7 with eq A-15 into eq 1, where double occupancy is allowed. Circles represent the results of static approximation, eq 11. 22088



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where ζ is the coordination number and ζ = 4 for the diamond lattice. The initial decay is static. However, in the reactioncontrolled limit, the majority of the decay cannot be described by the static approximation. By decreasing the hopping frequency, the site blocking effect on the decay of the survival probability increases. In Figure 2, we

Figure 3. Time dependence of the survival probability. Diamond lattice, Γ/k0 = 0.1. c = 0.1, c = 0.3, and c = 0.5 from top to bottom. (Red) dots are obtained by the Monte Carlo simulations. The solid lines are obtained by the inverse Laplace transform of eq 8 with eq 10 and eq A-15. The dashed lines represent the results obtained by substituting the inverse Laplace transform of eq 7 with eq A-15 into eq 1, where double occupancy is allowed. Circles represent the results of static approximation, eq 11. Filled squares are obtained from eq 12.

Figure 2. Time dependence of the survival probability. Diamond lattice, Γ/k0 = 1.0. c = 0.01, c = 0.1, and c = 0.3 from top to bottom. (Red) dots are obtained by the Monte Carlo simulations. The solid lines are obtained by the inverse Laplace transform of eq 8 with eq 10 and eq A-15. The dashed lines represent the results obtained by substituting the inverse Laplace transform of eq 7 with eq A-15 into eq 1, where double occupancy is allowed. Circles represent the results of static approximation, eq 11.

the results of Γ/k0 = 0.1. The large influence of prohibiting the double occupancy can be seen in the diffusion-controlled limit. A significant part of the decay follows the static law given by eq 11. The decay also follows the results obtained by eq 8. Equation 8 provides the lower bounds, and the simulation results approach the results of eq 1 at long times. When the reactants are allowed to migrate to the probe site and the reaction occurs only from the probe site, the results obtained by the same approximate method as those of eq 8 reproduce the simulation results very well at all times even in the diffusion-controlled limit (not shown). Significant deviation between the theoretical results of eq 8 and the simulation results is seen at long times when the reaction takes place from the nearest neighbor sites of the probe at relatively small reactant concentrations, c ≤ 0.3. The results ignoring the site blocking effect, eq 1, reproduce the simulation results when c ≤ 0.1, and the deviation can be seen when c ≥ 0.3. To find a reasonable simple analytical formula in the diffusioncontrolled limit at high reactant concentrations, we note that to a large extent the survival probability decays by the static quenching in this limit and take into account the decay by migration of reactants to reactive sites by assuming at most one hopping event. When the nearest neighbor sites of the probe are initially empty, the transition to the empty site should take place at least once to place the reactant at the reactive site. If the reaction rate is much larger than the transition rate, the reaction takes place before further transition of the reactant away from the reactive site. As long as the reactants initially occupying the reactive sites

show the results of Γ/k0 = 1.0. By prohibiting the double occupancy, the decay of the survival probability is considerably accelerated when the reactant concentration is c = 0.3. The theory in eq 8 approximately takes into account the site blocking effect. The results are the lower bounds of the survival probability. When the migration of reactants to the probe site is allowed and the reaction takes place only from the probe site, the corresponding theoretical results reproduce simulation results almost perfectly (not shown). However, it turned out that the approximation is not that satisfactory when reaction takes place from the neighboring sites of the probe. When the hopping rate is comparable to the reaction rate, the initial decay, k0t < 1, follows the static law. At intermediate times, the decay is partly described by eq 8, where the site blocking effect is approximately taken into account. At long times, the simulation results asymptotically approach the results of eq 1. The theoretical results obtained by eq 1 are exact when the double occupancy is allowed. It should be stressed that the lower bound is obtained from eq 8 and the upper bound is obtained from eq 1. When the hopping rate is smaller than the reaction rate, the survival probability is diffusion-controlled. In Figure 3, we show 22089



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react immediately with the probe and the initially empty reactive sites are populated by a single hop of reactants followed by immediate reaction, the kinetics can be described by a simple analytical function. The transition of a reactant to an empty site can be regarded as being equivalent to the transition of a vacancy to the site occupied by the reactant before the transition; that is, the occupied site is exchanged by the empty site. The average exchange rate is given by c(1 − c)Γ/ζ,34 where ζ is the coordination number. By noticing that only ζ − 1 out of ζ nearest neighbor sites can be occupied by reactants since one of the sites is occupied by the probe, quenching by hopping can be included into eq 11 as ⎤ζ ⎡ ⎧ ζ−1 ⎫ Pa(t ) = ⎢(1 − c)exp⎨− c(1 − c) Γt ⎬ + c exp(− k 0t )⎥ ⎩ ⎭ ζ ⎦ ⎣ (12)

where ζ = 4 for the diamond lattice. Equation 12 should be a good approximation to the exact results in the diffusion-controlled limit when the reactant concentration is high. As shown in Figure 3, the results of eq 12 reproduce the simulation results remarkably well when c ≥ 0.1. Although eq 12 is applicable only in the diffusioncontrolled limit, it is quite accurate and useful for its simplicity.

5. ANALYSIS OF EXPERIMENTAL DATA In the previous section, we found that the simple analytical expression reproduces the simulation results when c ≥ 0.1 in the diffusion-controlled limit. In this section, we analyze the published experimental quenching data in the diffusion-controlled limit.13 The static quenching of the first singlet excited state of anthracene takes place by quenchers (nitromethane) in pores adjacent to the one occupied by the excited anthracene, and the quenching mechanism could be by an electron tunneling as suggested previously.13 Experimental data clearly show the static quenching followed by the long time decay. The static decay component depends strongly on the loading when the loading is equal to or less than 0.25. The loading level of 0.25 suggests the static quenching by a quencher at one of four interconnected pores of the pore occupied by the probe. The static quenching occurs on the time scale shorter than 0.3 [ns]. The slow decay component extends at least over several nanoseconds. The decay is very slow compared to the static quenching, and we assume that at most one hopping event is involved in the observed time range. When k0 ≫ Γ and in the presence of the natural decay of the probe molecule, eq 12 can be rewritten as Pa(t ) = (1 − c)ζ exp[−t /τ0 − c(1 − c)(ζ − 1)Γt ]

Figure 4. Quenching of the excited anthracene by nitromethane in dehydrated, sodium-exchanged zeolite Y. The data are taken from Figure 3 of ref 13 and are shown by circles. The loadings of nitromethane are 0, 0.064, 0.13, 0.25, and 0.51 molecules/pore from top to bottom. The upper most line is the exponential fit with the time constant of the natural decay being 3.7 [ns]. The other (red) lines represent the results of global fitting using eq 13 with the fixed value of Γ. Inset shows the concentrations obtained from the global fitting against the loadings. The line in the inset shows the results when the concentrations coincide with the loadings.

global fitting. They are c = 0.067, 0.18, 0.38, and 0.49 as shown in the inset of Figure 4. The concentrations are almost equal to the loadings obtained by assuming homogeneous distribution of quenchers in zeolites. We conclude that the uniform loading in zeolites is achieved, and the probability of double occupancy can be ignored for the system examined.

6. CONCLUSIONS We have investigated the site blocking effect on the survival probability of an immobile probe by reaction with mobile reactants. The reactants perform random walks on a diamond lattice except the probe site under the constraint of prohibiting double occupancy. Reaction takes place when reactants occupy the nearest neighbor sites of the probe. Under the site blocking effect, the upper bound of the survival probability is obtained from the exact theoretical results where the double occupancy is allowed. The analytical expression in the Laplace domain is obtained by using the Green’s function with excluded origin. The lower bound of the survival probability is obtained by applying recently developed approximate theory. In the reaction-controlled limit, the approximate theoretical results are shown to be very close to the results obtained by the Monte Carlo simulations. In this limit, the site blocking effect on the decay of the survival probability turned out to be small compared to that in the diffusion-controlled limit. Even though

(13)

where τ0 is the time constant of the natural decay of the probe molecule and ζ = 4. The experimental data of ref 13 are analyzed by using eq 13 in Figure 4. From the experimental data, Pa(t) is obtained by normalizing the fluorescence intensity, I(t), with I0 obtained by fitting the data without quenchers to the exponential decay law, I0 exp(−t/τ0). By analyzing the data without quenchers, we obtain τ0. The value of τ0 is used, and Γ = 108 [1/s] is found by global fitting to the data with quenchers of various concentrations. The nearest neighbor distance, l, is ( 3 )/4 of the unit cell length, 24.78 [Å]. The diffusion constant is expressed as l2Γ/(2d) with d = 3. By substituting all values, we find the value of the diffusion constant as ∼2 × 10−11 [m2/s]. The value is larger than the diffusion constants of aromatic molecules in the zeolite, that is, 10−12 ∼ 10−15 [m2/s] at room temperature, but is still reasonably small.12,13 Meanwhile, the concentrations are estimated from the 22090



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reactant concentration is high, c = 0.5, so that the hopping transition is frequently blocked, the deviation of reactant distribution from the equilibrium distribution is small due to the small intrinsic reaction rate at reactive sites compared to the hopping rate. The equilibrium distribution of reactants is uniform regardless of the site blocking effect. The survival probability reflects the reactant distribution. The small deviation from the equilibrium distribution by the site blocking effect implies that the effect on the survival probability is also small. Although the effect is not significant, the approximate theory reproduces the results of the Monte Carlo simulation for the whole time range. However, the approximate theoretical results are merely lower bound and deviate from the Monte Carlo simulation results at long times in the diffusion-controlled limit. At long times, the results of Monte Carlo simulation asymptotically approach the exact results where the double occupancy is allowed. The lower bound obtained from the approximate theory asymptotically approaches the upper bound only under the reaction limited condition. It should be reminded that the approximate theoretical results reproduce the results of Monte Carlo simulations very well when the reaction takes place at the probe site even in the diffusion-controlled limit.25 When the reactive site is located at the probe site, the reactant that escaped from the reactive site should return to the same reactive site to react, while when multiple reactive sites are present, the reactant that escaped from the reactive site may visit the other reactive site to react. In the diffusion-controlled limit, the approximate theory reproduces the simulation results for the former case, but noticeable deviation is seen at long times in the latter case. An alternative simple expression, eq 12, is shown to reproduce the results of Monte Carlo simulation at all times when the site blocking effect is important. When the nearest neighbor sites of the probe are reactive and the reactants migrate on the diamond lattice excluding the probe site, the influence of site blocking on the survival probability is large at concentrations, c ≥ 0.3, and can be ignored for c ≤ 0.1, even in the diffusion-controlled limit. In recent experiments, the energy transfer from an excited probe to surrounding reactants migrating in interconnected pores within the diamond structure of zeolite is analyzed by using the theory where double occupancy is allowed.8,9 In most of the experiments, the concentration of reactants was below 0.1.8,9 Here we show theoretical ways to investigate the survival probability of the probe even when the concentration of reactants is high. The method is applied to estimate the diffusion coefficient and to justify the uniform loading of quenchers by using the published data with high loadings.13 The results of data analysis indicate that the double occupancy in the pore is unlikely for the system investigated.

In summary, we show that the survival probability obtained by the previously developed theory is the lower bound and reproduces the results of Monte Carlo simulation for the whole time range in the reaction-limited condition. In the diffusioncontrolled limit, the approximate theoretical results are the lower bound but fail to reproduce the simulation results at long times. At long times, the simulation results approach to the upper limit of the results obtained by allowing double occupancy. In the diffusion-controlled limit, an alternative expression given by eq 12 accurately reproduces the results of Monte Carlo simulation for the whole time range, in particular when the concentration is high, c ≥ 0.1.

■

APPENDIX We solve eq 4 and eq 5 by introducing the lattice Fourier transform G(A)(k ⃗ , t ) =

∑ exp(ik ⃗· ri ⃗)G(A)( ri ⃗ , rj⃗ , t ) (A-1)

ri ⃗

⃗ is defined similarly. By applying the Fourier− and G (k,t) Laplace transform to eqs 4 and 5 we obtain (B)

⎞ ⎛ (A) ⎛ 1 −ψ (s)D(k)⎞⎜Ĝ (k ⃗ , s)⎟ ⎜⎜ ⎟⎟ ⎟ ⎜ 1 ⎝−ψ (s)D*(k) ⎠⎜⎝Ĝ(B)(k ⃗ , s)⎟⎠ ⎛ φ(s)exp(ik ⃗· r ) + ψ (s)F ⎞ j⃗ 1⎟ =⎜ ⎜ ⎟ −ψ (s)F2 ⎝ ⎠

(A-2)

where the Laplace transform of the waiting time distribution and that of the remaining probability distribution are given by

ψ (s) = Γ/(s + Γ)

(A-3)

φ(s) = [1 − ψ (s)]/s

(A-4)

respectively. In eq A-4, D(k) is defined by D(k) =

1 ζ

ζ

∑ exp(ik ⃗· lm⃗ )

(A-5)

m=1

and F1 and F2 are given by F1 =

F2 =

1 ζ 1 ζ

ζ

(A)

∑ Ĝ

( lm⃗ , rj⃗ , s)

(A-6)

m=1 ζ

(A)

∑ exp(ik ⃗· lm⃗ )Ĝ

( lm⃗ , rj⃗ , s)

m=1

(A-7)

respectively. Equation A-2 can be transformed into

⎛ ̂(A) ⃗ ⎞ ⎛ ψ (s)D(k)⎞⎛⎜ φ(s)exp(ik ⃗· rj⃗) + ψ (s)F1⎞⎟ 1 1 ⎜G (k , s)⎟ ⎜ ⎟⎟ = ⎜⎜ (B) ⎟⎟ 1 − ψ (s)2 λ(k) ⎜ ψ (s)D*(k) ⎟ 1 ⎝ ⎠⎜⎝ ⃗ −ψ (s)F2 ̂ ⎠ G ( k , s ) ⎝ ⎠

(A-8)

λ(k) = [1 + cos(kx)cos(k y) + cos(k y)cos(kz)

where the structure factor is defined by 2

λ (k ) = | D (k )| For the diamond lattice, the structure factor is given by

+ cos(kz)cos(kx)]/4 (A-9)

(A-10)

By introducing the abbreviation Ĝ (A)(ri⃗ ,s) = Ĝ (A)(ri⃗ ,rj⃗ ,s), we obtain from eq A-8

35−39

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(A) 1 Ĝ ( ri ⃗ , s) = φ(s)U ( ri ⃗ − rj⃗ , ψ (s)) + ψ (s) ζ

where the origin on the sublattice B is not excluded. We denote one of the four nearest neighbor sites of the probe site as li⃗ and introduce the abbreviation, Ĝ S(s) = Ĝ (A)(lj⃗ ,lj⃗ ,s) and Ĝ D(s) = Ĝ (A)(li⃗ ,lj⃗ ,s) with i ≠ j. By further denoting

ζ

∑ U ( ri ⃗ − lk⃗ , ψ (s))Ĝ(A)( lk⃗ , s) − ψ (s)2 k=1

1 ζ2

ζ

ζ

∑ ∑ U ( ri ⃗ − lk⃗ , ψ (s))Ĝ(A)( lm⃗ , s)

US(ψ ) = U (0, ψ )

(A-13)

UD(ψ ) = U ( li ⃗ − lj⃗ , ψ ), for i ≠ j

(A-14)

k=1 m=1

(A-11)

In the above, the lattice Green’s function of the sublattice A is expressed as 1 U ( ri ⃗ , ψ ) = (2π )d

π

π

exp( −ik ⃗· ri ⃗)

∫−π ··· ∫−π ddk 1 − ψ 2λ(k)

(A-12)

we obtain

ψ ψ ⎛ ⎞−1 − ψ + − ζ − ψ + ζ − U h U h 1 ( ) ( 1) ( ) ( 1) S D ⎛ Ĝ ⎞ ⎜ ⎟ ⎛ φ U (ψ ) ⎞ ζ ζ ⎜ S⎟ = ⎜ ⎟ ⎜ S ⎟ ⎜ ⎟ ⎜ ̂ ⎟ ⎜ ψ ψ ψ ⎝G D ⎠ ⎜ − UD(ψ ) + h 1 − (ζ − 2)UD(ψ ) − US(ψ ) + h(ζ − 1)⎟⎟ ⎝ φUD(ψ )⎠ ζ ζ ⎝ ζ ⎠

■

where (s) indicating s-dependence is omitted and we define h=

ψ (s)2 [(ζ − 1)UD(ψ (s)) + US(ψ (s))] ζ2

(A-16)

(A-17)

where K(k) is the complete elliptic integral of the first kind and A(ψ ) = (4 − ψ 2)1/2 − (1 − ψ 2)1/2

(A-18)

B(ψ ) = (1/2) − [ψ 2(4 − ψ 2)1/2 + (2 − ψ 2) (1 − ψ 2)1/2 ]/4

(A-19)

UD(ψ) is expressed using US(ψ) as8 UD(ψ ) = [4ψ −2(US(ψ ) − 1) − US(ψ )]/3

(A-20)

In eq A-15, the Green’s functions excluding the probe site are given in the Laplace domain by those including the site occupied by the probe.

■

REFERENCES

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We need the Green’s functions starting from and terminating on one of the nearest neighbor sites of the probe denoted by Ĝ S and Ĝ D in the Laplace domain. These can be obtained from eq A-15, where the origin on the sublattice B is excluded. US(ψ) in eq A-15 is the lattice Green’s function including the origin on the sublattice B and is known8,35−39 ⎛2 ⎞2 US(ψ ) = A(ψ )⎜ K[B(ψ )]⎟ ⎝π ⎠

(A-15)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This research was conducted under Agreement for Collaboration between IIT Bombay and Nanosystem Research Institute, AIST. The authors thank Dr. A. V. Barzykin for careful reading of the manuscript and useful comments. 22092



Article

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Site Blocking Effect on Diffusion-Mediated Reactions in Porous Media

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