Investigation of ortho-positronium annihilation for porous materials with different geometries and topologies

In this work, we present the results of the ortho-positronium (o-Ps) annihilation lifetimes and nitrogen adsorption measurements for different porous materials and an approach for describing the annihilation of o-Ps in a pore, which results in a surface-volume formula (SVF) for calculating the pore-related o-Ps lifetime. This proposed formula gives the relationship between the o-Ps annihilation rate and the effective pore radius, bulk composition, and pore structure, including pore geometry and topology. The pore-related o-Ps lifetimes of different materials calculated by the SVF are consistent with experimental results for both micro- and mesopores (and macropores) with different geometries and topologies. The SVF is convenient for calculations of pore dimensions for many cases of metal organic frameworks and zeolites. This approach enables us to fully explain the temperature dependence of the o-Ps annihilation lifetime over a wide temperature range, 20–700 K.


Supplementary information S1. The o-Ps energy
When trapped in a pore with an energy of a few electron-volts 1 , o-Ps is scattered multiple times from atoms and molecules at the pore surface, thereby thermalizing and undergoing pick-off annihilation 2,3 with average energy, Eav(τ), τ is average lifetime of o-Ps. By the result of Nagashima et al. 3 , the Eav(τ) has been found to monotonically decrease with o-Ps lifetime 3 . For o-Ps trapped in a pore, an increase of effective pore radius (3V0/S0, here V0 and S0 are pore volume and surface area, respectively) also results in an increase of the o-Ps lifetime. This means that Eav(τ) decreases as effective pore size increases. Therefore, Eav(τ) depends on V0/S0. In other words, Eav(τ) is a decreasing function of effective pore radius 3 . Normally, the value of Eav(τ) is small enough to expand Eav(τ) into 1 st -order Maclaurin series of effective pore radius, R0 (R0 = 3 V0/S0): Differentiating Eq. (1) with respect to the variation of the effective pore radius R0, results in: Note that E av ′ (τ) is found to be approximately proportional to minus Eav(τ) 3 . This results in these following relations: d(E av (τ)) = −μ 1 (T)E av (τ)d(R 0 ) (2) d(E av (τ)) E av (τ) = −μ 1 (T)d(R 0 ) where μ1 is proportionality factor. Integrating differential equation, Eq. (3), gives: Eav(τ) = E0exp(-μ 1 (T)R0) + Eth (4) where E0 are constants in unit of eV. Eth is approximated the value of Eav(τ) with the very large value of R0, E th = lim R 0 →∞ E av (τ).

S2. The derivation for wavefunction and annihilation rate of o-Ps
Supposed o-Ps being trapped in a pore, the o-Ps lifetime monotonically increases with pore volume 5,10 , and decreases with pore surface area 5,8 . However, even with the same value of pore volume, a pore can have different surface areas in different pore shapes (e.g., cubic, and spherical pores). As a result, the lifetimes of the o-Ps states in different pores with the same volume and with different surface areas, are different. Considering the o-Ps states in different pores with different values of pore volumes and surface area, we assume that the lifetimes of o-Ps states in those pores with the same value of V0/S0, have the same value (without taking into account the role of pore surface composition). Based on this assumption, we presented the model for describing o-Ps annihilation in a pore as shown by Eq. (1) and Fig. 1 of main text. In the spherical polar coordinate, the radial function, ψ + (r), of the o-Ps spherical wavefunction can be obtained from deriving the Schrödinger equation with a finite potential, U(r) as presented in main text. The o-Ps radial wavefunctions are different for different regions of a pore. In region I (r < R0 -ΔR1), the radial wavefunction of the o-Ps is approximated by the ground state wave function of o-Ps as follows (see ref. 7 for example): where α is a factor independent on r, k 0 = √4m E av (τ)/ℏ 2 , me is the electron mass, and ℏ is the reduced Planck's constant. For R0 -ΔR1 ≤ r ≤ R0, the radial wavefunction of the o-Ps is approximated as follows: where r1 = r -(R0 -ΔR1), β0 is a factor independent on r, and, k = √4m (U 0 − E av (τ))/ћ 2 . The o-Ps can undergo pick-off annihilations in both regions II and III. For those o-Ps that have not annihilated in region II or moved back to region I for the selfannihilation, they penetrate the region III (R0 ≤ r ≤ R0 + ΔR) with a nonzero probability. The radial wavefunction of o-Ps in region III, ψ 3+ (r), is therefore satisfied this following relation: where coefficient, D 2 , is the fraction of o-Ps entering region III from region II. It is noted that the coefficient, D 2 , arises only for the case of unstable o-Ps (for the case of stable particle such as electron, the wavefunction in region III is the same with that as in region II). We will consider below the coefficient, D 2 , which depends on the pore surface area and volume. Here, we deal with o-Ps, which have no angular momentum, the azimuthal and magnetic quantum numbers, l0 and m0 are all zero, the spherical harmonic 7 , Ylm(θ, φ) = (1/4π) 1/2 ). Neglecting the o-Ps escaping from pores, the pick-off annihilation rate, λ pick−off , can be expressed as follows 5,8 : where r0 is the classical electron radius (nm), and c is the speed of light (nm/ns), ρe0(T) is the average bulk electron density (nm -3 ) in the region III (bulk density of electron), and ρeth(T) is the average electron density (nm -3 ) in region II arisen by the thermal atomic vibrations at the pore surface 9,10 at temperature T. Note that ρe0(T) linearly relates to bulk density of the atoms, ρa0(T), by a factor Zeff, an effective number of electrons 11 , ρe0(T) = Zeffρa0(T). Eq. (9) can be re-written as: where λ 0 (T) = ρ e0 (T)πr 0 2 (ns -1 ) 8,10,11 , being the bulk annihilation rate of o-Ps at temperature, T(K); η(T) is the ratio of ρeth(T)/ρe0(T). The value of k depends on the value of Eav(τ), which in turn, depends on the effective pore radius and temperature. The expression of, λ pick−off , which is obtained by integrating right hand-side of Eq. (10), is as follows: At room temperature, the Eq. (11) becomes: where η0 is the value of η(T) at the room temperature, T = 298 K. The pore related annihilation rate of o-Ps with taking into account the 3γ self-annihilation rate, λ 3 , is expressed as 12 : where κ explained as relative contact density 12 is applied for the correction of the 3γ self-annihilation rate, λ 3 , of o-Ps in the medium. For small void, the value of λ 3 is negligible, while for large pore, is approximately unit 12 . Consider to β 0 2 , provided that the unit of o-Ps annihilation rate in Eq. (11) is ns -1 , the dimension of β 0 2 should be nm -1 . Furthermore, for R0 > ΔR1, substituting r = R0 -ΔR1 into Eq. (7) gives where P(R1) is an expression relevant to the probability of finding o-Ps at position, R1 = R0 -ΔR1. Provided that P(R1) has defined, nonzero value, β 0 2 must be in this following expression: and, therefore: where f1(R1) is expression of R1 having defined and nonzero value when R1 = 0. Note that the greater the pore effective radius, R1, the higher the 3γ self-annihilation rate, the lower the pick-off annihilation rate of o-Ps, therefore, the lower the probability of finding o-Ps at R1, therefore, f1(R1) inversely varies with V1, the volume of sphere with radius R1. Referring to the dimension of β 0 2 , the expression, f1(R1), can be set as below: such that ϰ1 is constant, and the dimensionless factor, f2(R1) = 1, when R1 is very large. Eq. (15) becomes: Provided that f1(R1) has the defined and nonzero value when R1 = 0, the value of f2(R1) must vanish when R1 = 0. It is allowed to assume that f2(R1)/R1 ≈ 1/R0. Therefore: where ϰ = ϰ1/4π, being constant, S0 and V0 are respectively the surface area and volume of a pore with effective radius, R0. Considering the coefficient D 2 , it is noted that an increase in volume results in a decrease in D 2 ; and with a given volume, V0, an increase in the pore surface area, S0, results in an increase in the pick-off annihilation rate in region III. Therefore, D 2 is decreasing function of S/V. In the first order approximation, D 2 is expressed as where q is a coefficient in nm, D 0 = D 2 |S V =0 = 0. Let to assume that the relative contact density, κ = 1, for R0 > ΔR1, the pore related o-Ps annihilation rate, λ o−Ps , is calculated using Eq. (13) as follows: or: At room temperature, the Eq. (20) becomes: and Eq. (21) becomes: where λ0 is the value of λ0(T) at room temperature, T = 298 K. Eq. (18) is meaningful and applicable only for those cases with pore radii greater than ΔR1. For R0 = ΔR1, the fraction of 3γ self-annihilation of the o-Ps can be negligible 12 , almost o-Ps undergo pick-off annihilation with electron of the pore surface. Due to relative contact density, κ, is negligible 12 for these cases, the following condition is satisfied: The condition of Eq. (24) provided that at room temperature, and for the case of a pore with effective pore radius equal ΔR1, the o-Ps annihilation rate is λ0, which in fact, is approximately 2 ns -1 for many materials. For those pores with pore radii smaller than ΔR1, all equations, Eq. (6) -Eq. (12), and Eq. (14) -Eq. (23) are unapplicable. However, this following condition is held: where R2 = R0 + ΔR1, ψ + (r) is the radial part of a normalized wavefunction of the o-Ps in a pore with effective radius smaller than ΔR1. The expression of ψ + (r) may differ from that of the case with pore radius greater than ΔR1 that presented above. In practice, the value of ΔR1 is very small (ΔR1 is about 2.4a0) 13 , and the bound of e + -ecannot exist in the dispace smaller than 8 0.097 nm. It is supposed that o-Ps formed and existed in the region, 0.097 nm -2.4a0, behaves as if it does in the bulk materials, and o-Ps annihilation rates are approximate λ0. It is noteworthy to found that the RTE model also gives similar result. The RTE model can be applied only for those values of the pore-size-related parameter, "a", defined by Gidley et al. 5 and Dull et all. 14 , greater than or equal 0.36 nm. Namely, when, "a" = 0. 36 nm, the pore diameter is zero, and RTE calculation gives the result of o-Ps annihilation rate approximates 2 ns -1 . The similar situation occurs with the TE model 15 . The TE calculation gives the annihilation rate be to 2 ns -1 when pore size approaches zero. These clearly show that at room temperature, the bulk o-Ps annihilation rate is 2 ns -1 interpreted by the TE and RTE models, and that is λ0, in the interpretation of SVF.
Using condition of Eq. (24) and applying the experimental results of o-Ps lifetimes associated with porous samples, MCM-41, ZSM-5, and Al-Mil-53 as shown in Table 1, along with the values of pore volumes, surface areas associated with these porous sample as shown in Table 2 to Eq. (22) and Eq. (23) results in four transcendental equations of four parameters, η0, ϰ, q, and μ, which turn out to be consistent. The calibration of SVF into experimental results accomplished by deriving those equations gives, η0 ≈ 0.12, ϰ = ϰ1/4π = 0.716/4π = 0.057, q ≈ 0.03 nm, μ ≈ 2.5 nm -1 . Notably, the value of, η0 = 0.12, agrees with the theoretical calculation 13,16 , It is noted that for the calculus using radial variable, r, the role of the parameters, k, Eav(τ), and V/S can be clarified as follows. For the SVF model as described by Eq. (1) and Fig. 1 of main text, the pore with volume, V0, and with surface area, S0, and with any shape, is modelled as a spherical pore with pore radius, R0 = 3V0/S0 (the dimension is in nm). The value of R0 is defined as an effective pore radius. The value, 3V0/S0, is measured by gas adsorption method. In spherical coordinate, the radian variable, r, the distance of o-Ps from pore center, defines the radial position of o-Ps. The dimension of r (nm) is the same dimension of R0 =3V0/S0 (nm). The value of r is varied from 0 to (R0 + ΔR). However, r is a variable specifying the radial position of the moving o-Ps, while 3V0/S0 is not variable in the calculus of the integration or differentiation, it is a constant value for a given pore size. The value of Eav(τ) and k are functions of V0/S0 of different pore sizes, but they are constants for a given pore size (it is not function of r). The variable, r, is used to calculate the integration or differentiation, etc. to obtain the analytical expressions of o-Ps annihilation rate for a given pore size. While taking the integral or differentiate, r is considered as variable, but R0, k, Eav(τ), and V0/S0 (of a given pore) are considered as constant. Therefore, in the formula of SVF, there are only values of R0 and V0/S0 related to the o-Ps annihilation rate.

S3. Appendix
The explanation for the RTE calculation results presented in Tables 3, 4, 5. The RTE model 5 use modeling parameters, a, b, c, for calculations of the o-Ps lifetimes. Therefore, in the RTE model, the o-Ps lifetime depends on these modeling parameters, a, b, c, but not directly depends on the pore size. Gidley et al. 5 and Dull et al. 14 have provided the relation of these parameters, a  (or b, or c), and pore radius, a = 2 (R + δRTE), where R is the pore radius (nm), δRTE = 0.18 nm, being the RTE constant, and the connection of these parameter, a (and b, and c), with mean free path, a = l/2 for 1D geometry, a = l = 4 V/S for 2 D geometry, and a = (3/2)l = 6 V/S for 3D cubic geometry. Hence, it is possible to specify different solutions to define the RTE parameters, a, b, c.
In Table 3, the values of these parameters (3D cubic geometry, a = b = c, are determined from the value of mean free path, l = 4 V/S, and, l = (2/3)a. For the RTE calculation in these cases, it is inconsistently for defining the geometry to calculate the RTE parameter, a. For example, for the case of MCM-41, the chosen geometry of 2D seems to be better than that of 3D. However, for the cases of ZSM-5 and Al-Mil-53, the chosen geometry of 3D seems to be better than that of 2D. Note that in Table 3 of main text, in addition to 3D calculation results, we indicate the values of RTE calculation for 2D geometry in bold italic data, for reference.
In Table 4, if one uses the value of side length of cubic pore, am, as RTE parameter, a, the RTE calculation results are much less than the experimental results; and, it is seemly that the relation, a = 4 V/S for the case of 2D geometry, or a = 6 V/S for the case of 3D geometry, are both not suitable solution for the RTE calculation for the cases of MOF-5, MOF-20, and MOF-8. Below is the explanation for this inadequacy.
Consider to the calculation of RTE model for the cases of MOF-5, MOF-20 and MOF-8 (Table 4). According to the definitions of the RTE modeling parameters 5,14 , a, b, c, one can specify the different relations between the RTE parameter, a, and the pore size parameter for a given pore, and one can calculate the different values of these parameters, a (= b = c), and therefore, the different results of o-Ps lifetime can be obtained from RTE calculations for a given pore. This inconsistency is illustrated by different case below: i) Case 1. Assigning the value of the side length, am, to modeling parameters, a, am = a (= b = c), the results of o-Ps lifetimes calculated by RTE model presented in Table S3.1a, which are much less than experimental results. Hence, this solution is not suitable for calculation of parameter, a from the value of cubic side, am. ii) Case 2. Using the equality 5,14 , a (= b = c) = 6 V/S, where, V/S ≈ am/4, is the volume-to-surface area ratio of infinite rectangular prism, with sides, am = bm = side length of pore, and side, cm, is infinite. For this case, the results of o-Ps lifetimes calculated by RTE model are presented in Table S3.1b, which are deferent with experimental results. This solution is not good one for calculation of parameter, a, from the value of cubic side, am. iii) Case 3. Using, a (= b = c) = 2(R + δRTE), where R is pore radius of circumscribed sphere of cubic pore with a side length, am, the o-Ps lifetimes calculated by RTE model are presented in Table S3.1c, relatively differ from experimental results. This solution is not best one for the calculation of parameter, a. iv) Case 4. Using, a (= b = c) = 2 (R0 + δRTE), where R0 is effective pore radius, R0 = 3V/S as described in the case ii), the results of o-Ps lifetimes are presented in Table S3.1d, which are consistent with experimental results. The o-Ps lifetime calculated by the RTE model using the case 4, which is presented in Table S3.1d, are best results by means of the best agreement with the experimental results. It is noted that the assumption that o-Ps can move along the infinite rectangular prism, which result in in the calculations of all SVF, TE, and RTE models consistent with experimental results for micropores of MOF, agrees with discussion of Dutta et al. 17 . Nevertheless, although, the solution iv) is the best one for the RTE calculations of MOF-5, MOF-20, and MOF-8, it is not good solution for the cases of pores presented in Table 5. In Table 5, the RTE parameter, a (= b = c) = 2(R + δRTE), where R is pore radius referred from literature 19,20 . Using the values of a calculated from that relation, the RTE calculation results are not agreed with experimental results of literature 14,19,20 .
The mean square amplitude of normal component of the thermal atomic vibrations. Firstly, we consider temperature dependence of o-Ps lifetime for high temperature range. In this region, we can determine the expression of u1, the mean square amplitude of normal component of the thermal atomic vibrations that approximately depends on the kBT. We apply this result to calculate the o-Ps lifetimes using SVF for different temperatures and compare these results with experimental results published by other authors 10,17 . It is observed that these results of the calculated o-Ps lifetimes agree with experimental results of o-Ps lifetimes measured at vacuum condition and for different high temperatures. Noticeably, for high temperature, the changes of o-Ps lifetimes calculated by SVF, are agreed with those of the o-Ps lifetimes calculated by RTE. However, for low temperature, the application of u1 to calculate the changes of o-Ps lifetimes using SVF are not consistent with experimental results reported by Dutta et al. 10 The o-Ps lifetimes calculated by RTE for low temperature also are inconsistent with data of Dutta et al. 10 Referring to the result of Dutta et al. 10 , we propose another expression, u2, the mean square amplitude of normal component of the atomic vibrations, that approximately depends on the (kBT) 1/2 for the low temperatures. The application of u2 to calculate the o-Ps lifetimes using SVF for different low temperature results in consistency of the SVF calculated o-Ps lifetimes and experimental results of Duta et al. 10 Several values of u1 and u2 are calculated and plotted against (kBT) 1/2 in Fig. S2.   It is noted that the presentation of u1 and u2 is only the intermediate stage, our purpose is to search for the general expression of un, the mean square amplitude of normal component of the thermal atomic vibrations adequate for at all temperature. In general, its analytical expression is not well known, however, it manifests itself that un approximately depends on kBT for the high limit of temperatures, and (kBT) 1/2 for the low limit of temperature. This suggests us to approximate it by polynomial of (kBT) 1/2 variables. The values of parameters related to 3 rd polynomial described in Eq. (10) of main text, a0 = 0.00324 Å 2 , a1 = 0.113 Å 2 (eV) -1/2 , a2 = -0.561 Å 2 (eV) -1 , and a3 = 2.80 Å 2 (eV) -3/2 , are numerically calculated by fitting un into the calculated values of u1 and u2 (Fig. S3. shows the results of this fitting that is simply and reliably to return the fitting coefficients). In Fig. S2, the simulation of un (blue solid line) shows to be well fitted with the values of u1 and u2: the simulation of un differs from u1 and u2 with mean relative deviation of 0.017.
Noticeably, in Fig. 3-6, the RTE simulation uses the RTE parameter, a = (3/2)l, where l = 7 nm and l = 3.33 nm; while, in Fig. 3-5, Dutta simulations 21 are applied for pore radii, Rd = 7. 2 nm and Rd = 2.76 nm. In Fig. 4, SVF1 is simulated by SVF for cylindrical shape with pore aperture, Rc = 5.95 nm, and for high temperatures, T ≥ 273 K, SVF2 is done so for Rc = 2.55 nm, and for low temperature, T ≤ 273 K. In Fig. 5, SVF simulates for cylindrical pore with pore apertures, Rc = 5.95 nm, and 2.55 nm, and for temperature range of 20-700 K. In Fig. 6, SVF simulates for Rc = 5.85 nm and Rc = 2.45 nm, and for temperature varied from 20-700 K, while the Goworek simulation (the red solid line) uses simulated data of Goworek et al. 21 . In the calculations of SVF for different temperatures, the change of the parameter q with temperature, which is simply calculated as, q(T) = q[η0//η(T)], has been taken into account, where, q = 0.03 nm, and η0 = 0.12.

Comparison of Eav(τ) used in SVF and that used by Nagashima et al. 3 .
Eav(τ) defined by SVF is presented in the dependence of the pore radius, which is not the same with that defined by Nagashima et al. 3 , which is in the time dependence. However, because the o-Ps lifetime depends on the pore radius as described by Eq. (23), one can simulate the Eav(τ) in the o-Ps lifetime dependence. In Fig. S4, the simulation of Eav(τ) with E0 being 0.96 eV and μ1 = 2.5 nm -1 /3, and the simulation of (1) ( ) with the value of parameter, bN = 1.3× 10 7 s -1 and with E0 being 0.96 eV, are plotted against o-Ps lifetime. As shown in Fig. S4, Eav(τ) and (1) ( ), which is described in Eq. (10) of ref. 3, are almost approximated. It is shown that using the SVF calculation presented by Eq. (23), the o-Ps lifetime dependence of Eav(τ) is consistent with time dependence of (1) ( ) introduced by Nagashima et al. 3 . This result further supports the validity of the approach and expression of SVF.

The temperature dependence of Eav(τ)
The change of the energy of o-Ps annihilation, Eav(τ), due to the change of temperature from 20 to 700 K, is explained as follows. Consider to o-Ps being annihilation in given pore with effective pore radius, R0. For different sample temperatures, T2 > T0 = 298 K. As presented above, the value of parameter, μ =2.5 nm -1 , as expressed in Eq. , and for R0 > 3.6 nm, the relative change of the o-Ps lifetime being less than 2%. It is similar results can be given for the case, T0 > T2 ≥ 20 K. Therefore, one can use the room temperature value, μ =2. 5 nm -1 , to calculate the o-Ps lifetime for different temperatures, T = 20-700 K with the relative errors less than less than 2% for pore radius greater than 1 nm.
For R0 < 1 nm, it is necessary to take into account the change of parameter, μ, to calculate the o-Ps lifetime of sample temperature, T, varied over range of 20 -700 K.