THE JOURNAL OF CHEMICAL PHYSICS 142, 074303 (2015)
Para-hydrogen and helium cluster size distributions in free jet expansions based on Smoluchowski theory with kernel scaling Oleg Kornilova) and J. Peter Toennies Max-Planck-Institut für Dynamik und Selbstorganisation, Am Fassberg 17, 37077 Göttingen, Germany
(Received 6 November 2014; accepted 26 January 2015; published online 18 February 2015) The size distribution of para-H2 (pH2) clusters produced in free jet expansions at a source temperature of T0 = 29.5 K and pressures of P0 = 0.9–1.96 bars is reported and analyzed according to a cluster growth model based on the Smoluchowski theory with kernel scaling. Good overall agreement is found between the measured and predicted, Nk = A k a e−bk, shape of the distribution. The fit yields values for A and b for values of a derived from simple collision models. The small remaining deviations between measured abundances and theory imply a (pH2)k magic number cluster of k = 13 as has been observed previously by Raman spectroscopy. The predicted linear dependence of b−(a+1) on source gas pressure was verified and used to determine the value of the basic effective agglomeration reaction rate constant. A comparison of the corresponding effective growth cross sections σ11 with results from a similar analysis of He cluster size distributions indicates that the latter are much larger by a factor 6-10. An analysis of the three body recombination rates, the geometric sizes and the fact that the He clusters are liquid independent of their size can explain the larger cross sections found for He. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4907601]
I. INTRODUCTION
Clusters have attracted considerable attention in many areas of physics and chemistry where they serve as model systems for studying finite size effects1,2 as matrices for spectroscopy,3 as catalysts,4 and also as building blocks in nanotechnology and materials science.5 Coalescence growth or agglomeration, the formation of bigger particles by combination of smaller ones, is a ubiquitous process of fundamental importance. The theory of homogeneous nucleation and condensation governing the growth of clusters has a long history that goes back more than a century.6 Early theories of coalescence were based on statistical thermodynamics.7 Although suitable for describing near-equilibrium conditions, i.e., slow growth that occurs in cloud chambers, this approach is not suitable for the fast growth of clusters in free jet supersonic expansions by which clusters are mostly produced in the laboratory today. The extremely rapid cooling in free jet expansions with a rate of between 106 and 1010 K/s8 calls for a kinetic theory which considers the rates of the many individual steps leading from monomers to larger aggregates. Considerable effort has been invested in developing such theories but progress up to recently has been hampered by the unavailability of reliable data on cluster size distributions. The distributions of ions formed by electron impact ionization of neutral clusters, on which early experimental evidence was based, were soon shown to be suspect because of the extensive fragmentation following the ionization process.9 In recent years, several techniques have appeared that provide reliable information on the neutral cluster size distributions. The impact momentum transfer technique of a)Present address: Max-Born-Institute, Max-Born-Straße 2 A, 12489 Berlin,
Germany. 0021-9606/2015/142(7)/074303/11/$30.00
Buck and Meyer10 is suitable for tightly bound clusters with sizes up to about k = 9. Larger clusters can be analyzed using threshold photoionization since extensive fragmentation can often be avoided.11,12 These techniques, however, are not suitable for light quantum fluid clusters made up of pH2 and He. High resolution Raman spectroscopy has recently been used to resolve and follow the growth of small (k < 9) pH2 clusters in the course of the expansion.13,14 For the very fragile He clusters, the only technique available is matterwave diffraction from nanoscale transmission gratings.15,16 This technique utilizes the dependence of the de Broglie wavelengths on the mass (size) of the cluster. Since these cluster beams are nearly monoenergetic, the different sized clusters are diffracted into different angles. The size distributions of pH2 clusters analyzed in the present study were measured using this technique. All coalescence growth is initiated by a fundamental nucleation event, in which either three monomers collide nearly simultaneously (three-body recombination) or two monomers interact dynamically to form a metastable dimer, subsequently stabilized by a third collision partner. The ensuing agglomeration steps involve mostly two-body collisions. As the coalescing partners become larger, with many internal states, the internal states temporarily stabilize the nascent cluster by providing a large number of channels for redistributing the energy released in condensation. Ultimately, some of the excess energy must be removed, either by a third collision or by boiling or evaporating off fragments. Previously, simulations17 have shown that a variety of perturbations to the kinetics of smaller clusters can be imposed without affecting significantly the predicted populations of larger ones. Moreover, because of the large number of parallel kinetic pathways for producing any particular large cluster size, details of earlier nucleation events do not significantly
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influence the final cluster size distributions. This allows significant simplification of the coalescence model. In the model used here, each elementary two-body collision process is governed by a rate constant for the combination of a particle of size p with a particle of size q to form a particle of size p + q = r or decomposition of a particle of size r into particles of sizes q and p = r − q. A great simplification in dealing with the large number of kinetic events was introduced by Marian Smoluchowski,18 who already in 1916 suggested (1) the neglect of decomposition, (2) the neglect of possible changes in the rate constants with progressing condensation, and (3) the assumption that the rate constants for combination reactions depends on the size of the combining particles in a regular way. For the case that all the rate constants Kpq were the same (Kpq = K11), Smoluchowski was able to solve the resulting equations explicitly, obtaining algebraic expressions for the time-dependent distributions where Nk is the number of particles of size k at time t. It has since been possible to derive the asymptotic form of the cluster size distribution for large k when the association rate constants scale with particle numbers. In this interesting case, the particle numbers are multiplied by α and β, respectively, and the rate constant scales as Kαi, β j = α µ βν Kij, where µ and ν are scaling parameters and i < j.19 The derived distributions were Maxwellian, i.e., Nk is proportional to k a e−bk, where the parameters a and b depend on µ and ν, and b also depends on time of coalescence. Despite these simplifications, the Maxwellian form describes very well the results of many kinds of agglomeration processes.19–23 When experimental data are fit to this model, deviations provide information on how well the assumptions of the theory resemble the actual behavior of the system. Deviations from these assumptions may be due to exothermicity, or particularly stable or unstable cluster sizes due to magic numbers, dissociation events, or more complicated forms of the rate constants’ size dependence.17,24 In an earlier study,25 the Maxwellian distribution, a A k e−bk, was demonstrated to describe the results of diffraction measurements of 4He cluster size distributions with up to about 100 atoms over several orders of magnitude, except, of course, for magic-number clusters. Helium is a special case since it has the least dimer exothermicity of all common gases.26 In the present contribution, measurements of the size distributions of (pH2)k clusters with number sizes up to 70 molecules, based on the same diffraction technique, are presented and compared using a theory modified to account for time dependent agglomeration rate constants in free jet expansions. Although the pH2 cluster exothermicities are in general larger by roughly a factor of 30-40 (see Table VI), similarly good agreement is found for pH2 as for He. Deviations of experimental Nk from the Maxwellian distribution indicate that clusters consisting of 13 molecules are particularly stable, as recently reported experimentally13 and predicted by theory.27,28 Using the scaling and the thermodynamic relations describing nozzle beam expansions, the basic agglomeration rate constant K11 for pH2 is calculated. A direct comparison of the new results with the earlier values for He is made possible by converting to the ratio of the effective cross sections, which are independent of the T0 and
J. Chem. Phys. 142, 074303 (2015)
P0 conditions of the particular experiments. The comparison reveals that He has a much larger effective growth cross section. An analysis of the three body recombination rates, the geometric sizes, and the fact that the He clusters are liquid independent of their size can explain the larger cross sections found for He. The procedure reported here is applicable to systems in which monomers of arbitrary size and mass coalesce in the limit of small exothermicity. Thus, it provides an internally consistent approach for systematically studying size distributions of clusters of larger gas molecules. The deviations between the fit and actual data then provide evidence on the effect of the exothermicity and other details of the interactions present in the actual system, which are not accounted for by the theory. II. THEORY
The present model has been described in detail elsewhere24,25 and here only the essential aspects are reviewed. Let Nk (t) be the number of clusters of size k at time t in a volume V , so that the average cluster size is given by Nk (t) k k ⟨k (t)⟩ ≡ . (1) Nk (t) k
The Smoluchowski equation writes the rate of change of each Nk as a combination of second-order reactions,18–23,17,24 i.e., ∞ k−1 d(Nk ) V Nk Ni Ni Nk−i = −V Kki . Ki,k−i dt 2 i=1 V V V V i=1
(2)
Neglecting decomposition and evaporation of the clusters has the advantage that ⟨k(t)⟩ must increase monotonically with t. Essentially, these processes are accounted for in the effective reaction rate constants. Since, as discussed later in detail, the ratio of the effective He and pH2 growth cross sections compare favorably with other available data, it appears that decomposition and evaporation either do not play a significant role or if they do, their effect cancels in comparing the two systems. The time dependence of the Nk (t) is therefore determined by the competition between the increase resulting from the combination of smaller clusters (an i-mer and a jmer) to form new i + j = k-mers and the decrease due to the combination of the k-mers with clusters of any size (a kmer and a j-mer to give a (k+ j)-mer). Initially, N1 decreases with time, while each Nk for k > 1 increases with time and eventually passes through a maximum and decreases as larger clusters are formed. If the initial number of monomers is N1, so that, at time t = 0, Nk (0) = Nδk1, conservation of particles requires ∞
Nk (t) · k = N.
(3)
k
The rate constants Kij (also called Kernels) in the rate equations Eq. (2) are assumed to scale according to Kαi, βj = α µ βν Kij
(4)
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for i ≤ j, where µ and ν are the scaling parameters or scaling exponents which characterize the growth processes. Moreover, we impose the physical restriction29,30 that ν ≤ 1 since the reaction rate of a large j-mer (i.e., j ≫ 1) should not increase faster than its volume, and similarly, µ + ν ≤ 2, since, for clusters of equal size K j, j = j µ+ν K1,1 should not increase faster than j 2. The limiting case µ = ν = 1 corresponds to gelation, a process in which a single cluster containing all the monomers is formed in one step. The scaling and symmetry conditions on Kij imply that Kij = K11i µ j ν for i < j and Kij = K11iν j µ for i > j, where K11 is the base agglomeration rate constant. It should be emphasized that K11 is in fact not the rate constant for dimer formation, but simply the constant that appears in the general formula, Kij = K11iν j µ for i > j. Formation of a dimer from two monomers is a three-body process, whereas the formation of a (k + j)-mer from a k-mer and a j-mer where k and j are much larger than 1 is a two-body process. Under the assumption that Eq. (4) holds the asymptotic (large k, large t) solution to Eq. (2) is of the form Nk (t) = Ak a e−bk,
(5)
where b is time dependent and A is determined by normalization, i.e., ∞
k Nk = A
k=1
∞
k
a+1 −bk
e
= N.
(6)
k=1
The average value of cluster size is then a+1 −bk k e k ⟨k⟩ = a −bk . k e
(7)
k
By substituting Eq. (5) into Eq. (2) one can show that, for large t, a = −(µ + ν)
(8)
and b−(a+1) − b0−(a+1) C(µ, ν) = N K11 (t), a+1 Γ(a + 2)
(9)
where Γ(a + 2) refers to the standard gamma function and b0 is the value of b at t = 0. b0 depends on the equilibrium concentration of dimers in the source which, being negligible in He, is significant in the case of pH2 (see Sec. V). C(µ, ν) appearing in Eq. (9) is given by 1/2 C(µ, ν) =
y −ν (1 − y)−µ dy 0
∞ 1 n+1−ν ( /2) µ(µ + 1) . . . (µ + n − 1) . (10) = n + 1 − ν n! n=0
Since a, as given by Eq. (8), is independent of time and, moreover, since in our experiment, no gelation is observed, the value of a + 1 is positive,24,29 then according to Eq. (9), b decreases with time. Assuming the particle number density in the source, N/V , obeys the ideal gas law, P0 = N k BT0/V , where k B is the Boltzmann constant, T0 is the source temperature, and
P0 the source pressure; Eq. (9) predicts that b−(a+1) should be a linear function of P0. In deriving (5) from (2), it is assumed that the rate constants Kij are independent of time. However, the scaling exponents are derived assuming Kjk is a product of the relative velocity of the particles and their reactive cross sections Kij = νijσij. The relative velocities depend on temperature which, in a supersonic beam, decreases rapidly with the expansion time. Furthermore, the divergence of the beam means, effectively, that the volume V containing the particles increases with time. If the temperature varies according to T = T0 f (t) and the volume varies according to V = V0g(t), the asymptotic solution to the Smoluchowski equation is still given by Eq. (5), except that t is replaced by w(t), where f (t) dt. dw(t) = g(t) Equation (9) is then replaced by b−(a+1) − b−(a+1) P0 C(µ, ν) 0 [w(τ) − w(0)] , = K11 a+1 k BT0 Γ(a + 2) (11) where τ is the coalescence time and a = −(µ + ν) as before. Thus, if b is determined by fitting experimental data on {Nk } for various P0 to the asymptotic Maxwellian form, the quantity Θ, defined in Eq. (12), can be determined from the slope of b−(a+1) vs. P0, ∂b−(a+1) K11 C(µ, ν) [w(τ) − w(0)] ≡ Θ. = (a + 1) ∂P0 k BT0 Γ(a + 2) (12) In a nozzle beam, the variation of temperature with distance ρ is given by31 (
T0 T
)2(
T0 −1 T
) − 12
=
( )2 ( )2 4 √ ρ 3 , 3 ρ∗
(13a)
where T0 is the source temperature, ρ the distance travelled by the gas jet, and ρ∗ = 0.7d, with d the dimension of the nozzle orifice. With several approximations, 2 ( ) 2 3 2 3 (0.7d) T = (13b) √ [u(t)]−4/3 ≡ f (t), T0 4 3 where u(t) is the velocity of the expanding gas. The increase in volume as a result of the expansion is given by
V = d −2(d + u(t))2 ≡ g(t). (14) V0 Using Eqs. (13) and (14), and assuming that the gas has reached its terminal velocity u∞ at the coalescence time τ, it can be shown that32 1.312d . (15) u∞ Since, as discussed, in Sec. III from the measured Θ, K11 can be determined using Eq. (12) with Eq. (15) in place of w(t) =
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[w(τ) − w(0)], provided that the parameter a is known from some model for cluster growth. Several different models are used to estimate the relative velocities relating Kjk to a cross section. In the ballistic model,24,25 the particles are assumed to move in straight lines between collisions. This model is appropriate in the gas phase if all the particles involved in cluster-number-changing collisions are in thermal equilibrium. Assuming a Maxwell distribution, the relative velocities are given by ( )( ) 1 8k BT 1 + , (16) vjk = π m j mk where m j is the mass of a j-mer. Since m j is proportional to j, vjk is proportional to ( j)−1/2 for j
Para-hydrogen and helium cluster size distributions in free jet expansions based on Smoluchowski theory with kernel scaling Oleg Kornilova) and J. Peter Toennies Max-Planck-Institut für Dynamik und Selbstorganisation, Am Fassberg 17, 37077 Göttingen, Germany
(Received 6 November 2014; accepted 26 January 2015; published online 18 February 2015) The size distribution of para-H2 (pH2) clusters produced in free jet expansions at a source temperature of T0 = 29.5 K and pressures of P0 = 0.9–1.96 bars is reported and analyzed according to a cluster growth model based on the Smoluchowski theory with kernel scaling. Good overall agreement is found between the measured and predicted, Nk = A k a e−bk, shape of the distribution. The fit yields values for A and b for values of a derived from simple collision models. The small remaining deviations between measured abundances and theory imply a (pH2)k magic number cluster of k = 13 as has been observed previously by Raman spectroscopy. The predicted linear dependence of b−(a+1) on source gas pressure was verified and used to determine the value of the basic effective agglomeration reaction rate constant. A comparison of the corresponding effective growth cross sections σ11 with results from a similar analysis of He cluster size distributions indicates that the latter are much larger by a factor 6-10. An analysis of the three body recombination rates, the geometric sizes and the fact that the He clusters are liquid independent of their size can explain the larger cross sections found for He. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4907601]
I. INTRODUCTION
Clusters have attracted considerable attention in many areas of physics and chemistry where they serve as model systems for studying finite size effects1,2 as matrices for spectroscopy,3 as catalysts,4 and also as building blocks in nanotechnology and materials science.5 Coalescence growth or agglomeration, the formation of bigger particles by combination of smaller ones, is a ubiquitous process of fundamental importance. The theory of homogeneous nucleation and condensation governing the growth of clusters has a long history that goes back more than a century.6 Early theories of coalescence were based on statistical thermodynamics.7 Although suitable for describing near-equilibrium conditions, i.e., slow growth that occurs in cloud chambers, this approach is not suitable for the fast growth of clusters in free jet supersonic expansions by which clusters are mostly produced in the laboratory today. The extremely rapid cooling in free jet expansions with a rate of between 106 and 1010 K/s8 calls for a kinetic theory which considers the rates of the many individual steps leading from monomers to larger aggregates. Considerable effort has been invested in developing such theories but progress up to recently has been hampered by the unavailability of reliable data on cluster size distributions. The distributions of ions formed by electron impact ionization of neutral clusters, on which early experimental evidence was based, were soon shown to be suspect because of the extensive fragmentation following the ionization process.9 In recent years, several techniques have appeared that provide reliable information on the neutral cluster size distributions. The impact momentum transfer technique of a)Present address: Max-Born-Institute, Max-Born-Straße 2 A, 12489 Berlin,
Germany. 0021-9606/2015/142(7)/074303/11/$30.00
Buck and Meyer10 is suitable for tightly bound clusters with sizes up to about k = 9. Larger clusters can be analyzed using threshold photoionization since extensive fragmentation can often be avoided.11,12 These techniques, however, are not suitable for light quantum fluid clusters made up of pH2 and He. High resolution Raman spectroscopy has recently been used to resolve and follow the growth of small (k < 9) pH2 clusters in the course of the expansion.13,14 For the very fragile He clusters, the only technique available is matterwave diffraction from nanoscale transmission gratings.15,16 This technique utilizes the dependence of the de Broglie wavelengths on the mass (size) of the cluster. Since these cluster beams are nearly monoenergetic, the different sized clusters are diffracted into different angles. The size distributions of pH2 clusters analyzed in the present study were measured using this technique. All coalescence growth is initiated by a fundamental nucleation event, in which either three monomers collide nearly simultaneously (three-body recombination) or two monomers interact dynamically to form a metastable dimer, subsequently stabilized by a third collision partner. The ensuing agglomeration steps involve mostly two-body collisions. As the coalescing partners become larger, with many internal states, the internal states temporarily stabilize the nascent cluster by providing a large number of channels for redistributing the energy released in condensation. Ultimately, some of the excess energy must be removed, either by a third collision or by boiling or evaporating off fragments. Previously, simulations17 have shown that a variety of perturbations to the kinetics of smaller clusters can be imposed without affecting significantly the predicted populations of larger ones. Moreover, because of the large number of parallel kinetic pathways for producing any particular large cluster size, details of earlier nucleation events do not significantly
142, 074303-1
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influence the final cluster size distributions. This allows significant simplification of the coalescence model. In the model used here, each elementary two-body collision process is governed by a rate constant for the combination of a particle of size p with a particle of size q to form a particle of size p + q = r or decomposition of a particle of size r into particles of sizes q and p = r − q. A great simplification in dealing with the large number of kinetic events was introduced by Marian Smoluchowski,18 who already in 1916 suggested (1) the neglect of decomposition, (2) the neglect of possible changes in the rate constants with progressing condensation, and (3) the assumption that the rate constants for combination reactions depends on the size of the combining particles in a regular way. For the case that all the rate constants Kpq were the same (Kpq = K11), Smoluchowski was able to solve the resulting equations explicitly, obtaining algebraic expressions for the time-dependent distributions where Nk is the number of particles of size k at time t. It has since been possible to derive the asymptotic form of the cluster size distribution for large k when the association rate constants scale with particle numbers. In this interesting case, the particle numbers are multiplied by α and β, respectively, and the rate constant scales as Kαi, β j = α µ βν Kij, where µ and ν are scaling parameters and i < j.19 The derived distributions were Maxwellian, i.e., Nk is proportional to k a e−bk, where the parameters a and b depend on µ and ν, and b also depends on time of coalescence. Despite these simplifications, the Maxwellian form describes very well the results of many kinds of agglomeration processes.19–23 When experimental data are fit to this model, deviations provide information on how well the assumptions of the theory resemble the actual behavior of the system. Deviations from these assumptions may be due to exothermicity, or particularly stable or unstable cluster sizes due to magic numbers, dissociation events, or more complicated forms of the rate constants’ size dependence.17,24 In an earlier study,25 the Maxwellian distribution, a A k e−bk, was demonstrated to describe the results of diffraction measurements of 4He cluster size distributions with up to about 100 atoms over several orders of magnitude, except, of course, for magic-number clusters. Helium is a special case since it has the least dimer exothermicity of all common gases.26 In the present contribution, measurements of the size distributions of (pH2)k clusters with number sizes up to 70 molecules, based on the same diffraction technique, are presented and compared using a theory modified to account for time dependent agglomeration rate constants in free jet expansions. Although the pH2 cluster exothermicities are in general larger by roughly a factor of 30-40 (see Table VI), similarly good agreement is found for pH2 as for He. Deviations of experimental Nk from the Maxwellian distribution indicate that clusters consisting of 13 molecules are particularly stable, as recently reported experimentally13 and predicted by theory.27,28 Using the scaling and the thermodynamic relations describing nozzle beam expansions, the basic agglomeration rate constant K11 for pH2 is calculated. A direct comparison of the new results with the earlier values for He is made possible by converting to the ratio of the effective cross sections, which are independent of the T0 and
J. Chem. Phys. 142, 074303 (2015)
P0 conditions of the particular experiments. The comparison reveals that He has a much larger effective growth cross section. An analysis of the three body recombination rates, the geometric sizes, and the fact that the He clusters are liquid independent of their size can explain the larger cross sections found for He. The procedure reported here is applicable to systems in which monomers of arbitrary size and mass coalesce in the limit of small exothermicity. Thus, it provides an internally consistent approach for systematically studying size distributions of clusters of larger gas molecules. The deviations between the fit and actual data then provide evidence on the effect of the exothermicity and other details of the interactions present in the actual system, which are not accounted for by the theory. II. THEORY
The present model has been described in detail elsewhere24,25 and here only the essential aspects are reviewed. Let Nk (t) be the number of clusters of size k at time t in a volume V , so that the average cluster size is given by Nk (t) k k ⟨k (t)⟩ ≡ . (1) Nk (t) k
The Smoluchowski equation writes the rate of change of each Nk as a combination of second-order reactions,18–23,17,24 i.e., ∞ k−1 d(Nk ) V Nk Ni Ni Nk−i = −V Kki . Ki,k−i dt 2 i=1 V V V V i=1
(2)
Neglecting decomposition and evaporation of the clusters has the advantage that ⟨k(t)⟩ must increase monotonically with t. Essentially, these processes are accounted for in the effective reaction rate constants. Since, as discussed later in detail, the ratio of the effective He and pH2 growth cross sections compare favorably with other available data, it appears that decomposition and evaporation either do not play a significant role or if they do, their effect cancels in comparing the two systems. The time dependence of the Nk (t) is therefore determined by the competition between the increase resulting from the combination of smaller clusters (an i-mer and a jmer) to form new i + j = k-mers and the decrease due to the combination of the k-mers with clusters of any size (a kmer and a j-mer to give a (k+ j)-mer). Initially, N1 decreases with time, while each Nk for k > 1 increases with time and eventually passes through a maximum and decreases as larger clusters are formed. If the initial number of monomers is N1, so that, at time t = 0, Nk (0) = Nδk1, conservation of particles requires ∞
Nk (t) · k = N.
(3)
k
The rate constants Kij (also called Kernels) in the rate equations Eq. (2) are assumed to scale according to Kαi, βj = α µ βν Kij
(4)
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for i ≤ j, where µ and ν are the scaling parameters or scaling exponents which characterize the growth processes. Moreover, we impose the physical restriction29,30 that ν ≤ 1 since the reaction rate of a large j-mer (i.e., j ≫ 1) should not increase faster than its volume, and similarly, µ + ν ≤ 2, since, for clusters of equal size K j, j = j µ+ν K1,1 should not increase faster than j 2. The limiting case µ = ν = 1 corresponds to gelation, a process in which a single cluster containing all the monomers is formed in one step. The scaling and symmetry conditions on Kij imply that Kij = K11i µ j ν for i < j and Kij = K11iν j µ for i > j, where K11 is the base agglomeration rate constant. It should be emphasized that K11 is in fact not the rate constant for dimer formation, but simply the constant that appears in the general formula, Kij = K11iν j µ for i > j. Formation of a dimer from two monomers is a three-body process, whereas the formation of a (k + j)-mer from a k-mer and a j-mer where k and j are much larger than 1 is a two-body process. Under the assumption that Eq. (4) holds the asymptotic (large k, large t) solution to Eq. (2) is of the form Nk (t) = Ak a e−bk,
(5)
where b is time dependent and A is determined by normalization, i.e., ∞
k Nk = A
k=1
∞
k
a+1 −bk
e
= N.
(6)
k=1
The average value of cluster size is then a+1 −bk k e k ⟨k⟩ = a −bk . k e
(7)
k
By substituting Eq. (5) into Eq. (2) one can show that, for large t, a = −(µ + ν)
(8)
and b−(a+1) − b0−(a+1) C(µ, ν) = N K11 (t), a+1 Γ(a + 2)
(9)
where Γ(a + 2) refers to the standard gamma function and b0 is the value of b at t = 0. b0 depends on the equilibrium concentration of dimers in the source which, being negligible in He, is significant in the case of pH2 (see Sec. V). C(µ, ν) appearing in Eq. (9) is given by 1/2 C(µ, ν) =
y −ν (1 − y)−µ dy 0
∞ 1 n+1−ν ( /2) µ(µ + 1) . . . (µ + n − 1) . (10) = n + 1 − ν n! n=0
Since a, as given by Eq. (8), is independent of time and, moreover, since in our experiment, no gelation is observed, the value of a + 1 is positive,24,29 then according to Eq. (9), b decreases with time. Assuming the particle number density in the source, N/V , obeys the ideal gas law, P0 = N k BT0/V , where k B is the Boltzmann constant, T0 is the source temperature, and
P0 the source pressure; Eq. (9) predicts that b−(a+1) should be a linear function of P0. In deriving (5) from (2), it is assumed that the rate constants Kij are independent of time. However, the scaling exponents are derived assuming Kjk is a product of the relative velocity of the particles and their reactive cross sections Kij = νijσij. The relative velocities depend on temperature which, in a supersonic beam, decreases rapidly with the expansion time. Furthermore, the divergence of the beam means, effectively, that the volume V containing the particles increases with time. If the temperature varies according to T = T0 f (t) and the volume varies according to V = V0g(t), the asymptotic solution to the Smoluchowski equation is still given by Eq. (5), except that t is replaced by w(t), where f (t) dt. dw(t) = g(t) Equation (9) is then replaced by b−(a+1) − b−(a+1) P0 C(µ, ν) 0 [w(τ) − w(0)] , = K11 a+1 k BT0 Γ(a + 2) (11) where τ is the coalescence time and a = −(µ + ν) as before. Thus, if b is determined by fitting experimental data on {Nk } for various P0 to the asymptotic Maxwellian form, the quantity Θ, defined in Eq. (12), can be determined from the slope of b−(a+1) vs. P0, ∂b−(a+1) K11 C(µ, ν) [w(τ) − w(0)] ≡ Θ. = (a + 1) ∂P0 k BT0 Γ(a + 2) (12) In a nozzle beam, the variation of temperature with distance ρ is given by31 (
T0 T
)2(
T0 −1 T
) − 12
=
( )2 ( )2 4 √ ρ 3 , 3 ρ∗
(13a)
where T0 is the source temperature, ρ the distance travelled by the gas jet, and ρ∗ = 0.7d, with d the dimension of the nozzle orifice. With several approximations, 2 ( ) 2 3 2 3 (0.7d) T = (13b) √ [u(t)]−4/3 ≡ f (t), T0 4 3 where u(t) is the velocity of the expanding gas. The increase in volume as a result of the expansion is given by
V = d −2(d + u(t))2 ≡ g(t). (14) V0 Using Eqs. (13) and (14), and assuming that the gas has reached its terminal velocity u∞ at the coalescence time τ, it can be shown that32 1.312d . (15) u∞ Since, as discussed, in Sec. III from the measured Θ, K11 can be determined using Eq. (12) with Eq. (15) in place of w(t) =
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O. Kornilov and J. P. Toennies
J. Chem. Phys. 142, 074303 (2015)
[w(τ) − w(0)], provided that the parameter a is known from some model for cluster growth. Several different models are used to estimate the relative velocities relating Kjk to a cross section. In the ballistic model,24,25 the particles are assumed to move in straight lines between collisions. This model is appropriate in the gas phase if all the particles involved in cluster-number-changing collisions are in thermal equilibrium. Assuming a Maxwell distribution, the relative velocities are given by ( )( ) 1 8k BT 1 + , (16) vjk = π m j mk where m j is the mass of a j-mer. Since m j is proportional to j, vjk is proportional to ( j)−1/2 for j
