concentrations
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft
op gesag van de Rector Magnificus, Prof. Ir. K. C. A. M. Luyben,
voorzitter van het College voor Promoties, in het openbaar te verdedigen op dinsdag, 10 april, 2012 om 12:30 uur
door
TOMASZ MICHAŁ TRZECIAK
Master of Science
Warsaw University of Technology geboren te
Prof. Dr. A. Schmidt-Ott Copromotor:
Dr. J. C. M. Marijnissen
Samenstelling promotiecomissie: Rector Magnificus voorzitter
Prof. Dr. A. Schmidt-Ott Technische Universiteit Delft, promotor Dr. J. C. M. Marijnissen Technische Universiteit Delft, copromotor Prof. Dr.-Ing. H. J. Schmid Universiteit Paderborn
Prof. Dr.-Ing. E. Kruis Universiteit Duisburg-Essen Prof. C. M. Sorensen Kansas State University Prof. Dr. R. F. Mudde Technische Universiteit Delft Dr. G. Biskos Technische Universiteit Delft
Prof. Dr. S. J. Picken Technische Universiteit Delft, reservelid
Prof. Dr. A. Podgórski, Warsaw University of Technology, heeft als begeleider in belangrijke mate aan de totstandkoming van het proefschrift bijgedragen.
ISBN 978-94-6191-251-0
Copyright © 2012 by Tomasz Michał Trzeciak
All rights reserved. The author encourages the communication of scien-tific contents and explicitly allows reproduction for scienscien-tific purposes, provided the proper citation of the source.
concentrations
Thesis
presented for the degree of doctor at Delft University of Technology under authority of the Rector Magnificus,
Prof. Ir. K. C. A. M. Luyben, Chairman of the Board of Doctorates,
to be defended in public in the presence of a committee Tuesday, 10thof April, 2012 at 12:30
by
TOMASZ MICHAŁ TRZECIAK
Master of Science
Warsaw University of Technology born at
Prof. Dr. A. Schmidt-Ott Co-supervisor:
Dr. J. C. M. Marijnissen
Composition of Examination Committee: Rector Magnificus Chairman
Prof. Dr. A. Schmidt-Ott Technische Universiteit Delft, promotor Dr. J. C. M. Marijnissen Technische Universiteit Delft, copromotor Prof. Dr.-Ing. H. J. Schmid Universiteit Paderborn
Prof. Dr.-Ing. E. Kruis Universiteit Duisburg-Essen Prof. C. M. Sorensen Kansas State University Prof. Dr. R. F. Mudde Technische Universiteit Delft Dr. G. Biskos Technische Universiteit Delft
Prof. Dr. S. J. Picken Technische Universiteit Delft, reserve member
Prof. Dr. A. Podgórski, Warsaw University of Technology, has provided substantial guidance and support in the preparation of this thesis.
ISBN 978-94-6191-251-0
Copyright © 2012 by Tomasz Michał Trzeciak
All rights reserved. The author encourages the communication of scien-tific contents and explicitly allows reproduction for scienscien-tific purposes, provided the proper citation of the source.
Behorend bij het proefschift
“Brownian coagulation at high particle concentrations”
van
TOMASZ MICHAŁ TRZECIAK
1. Een promovendus wordt verondersteld een zelf-geleidend projectiel te zijn: het doel wordt gegeven en de baan moet onderweg gevonden worden.
2. Met de huidige tendens om wetenschappelijk onderzoek meer ondernemingsge-richt te maken—gepland, geondernemingsge-richt, met een duidelijke impact en opbrengsten— moet niet vergeten worden dat de meest interessante ontdekkingen vaak onge-pland en bij toeval worden gedaan. Het is belangrijk om exploratie ruimte in het onderzoek open te houden.
3. Het verschil tussen theorie en praktijk is in theorie veel kleiner dan het in de praktijk is (toegeschreven aan Steve Crocker).
4. Dingen moeten zo eenvoudig mogelijk gemaakt worden, maar niet eenvoudiger (Albert Einstein).
5. Bij het zoeken naar eenvoud moet complexiteit als eerste worden aangepakt. 6. Omdat de gecompliceerdheid van modellen continu toeneemt in het nastreven
van steeds betrouwbaardere voorstellingen van de echte wereld, worden op een gegeven moment modellen zelf, in plaats van de echte wereld, het doel van de studie.
7. De waarde van een nieuw idee wordt vaak beoordeeld naar de realisatie in plaats van haar verdiensten. Zo kan zelfs het beste idee ten prooi vallen aan een slechte uitvoering.
8. Het enige wat nog slechter is dan generalisatie uit één voorbeeld is generaliseren zonder voorbeeld (Bob Scheifler & Jim Gettys: “X Window System”).
9. Ben er zeker van dat de hersenen in gang zijn gebracht voordat de mond wordt ingeschakeld (vermoedelijk een uithangbord op de Air Force Academy in Colo-rado).
Deze stellingen woerden verdedigbaar geacht als zodanig goedgekeurd door de promotor, Prof. Dr. A. Schmidt-Ott.
Adjunct to the thesis
“Brownian coagulation at high particle concentrations”
by
TOMASZ MICHAŁ TRZECIAK
1. A promovendus is expected to be like a self-guided missile: the target is given and the trajectory has to be found on the way.
2. With the current trend to make scientific research more enterprise-like—planned, focused, with clear impact and deliverables—it should not be forgotten that the most interesting discoveries are often accidental and not a part of the plan at all. It is important in research to leave room to simply explore.
3. The difference between theory and practice is much smaller in theory than it is in practice (attributed to Steve Crocker).
4. Things should be made as simple as possible, but not simpler (Albert Einstein). 5. In quest for simplicity, complexity has to be handled first.
6. As models’ complexity constantly increases in pursuit of ever more faithful representation of the real world, at some point models themselves, rather than the real world, become the object of study.
7. The value of a new idea is often assessed by its realization instead of its merits. That’s how even the best idea can fall victim to poor execution.
8. The only thing worse than generalizing from one example is generalizing from no examples at all (Bob Scheifler & Jim Gettys: “X Window System”).
9. Be sure brain is in gear before engaging mouth (supposedly a sign at the Air Force Academy in Colorado).
These propositions are considered defendable and as such have been approved by the supervisor, Prof. Dr. A. Schmidt-Ott.
The process of Brownian coagulation, whereby particles are brought together by thermal motion and grow by collisions, is one of the most fundamental processes influencing the final properties of particulate mat-ter in a variety of technically important systems. It is of importance in colloids, emulsions, flocculation, air pollution, soot formation, materials manufacture and growth of interstellar dust, to name a few of its appli-cations. With continuous progress in particulate matter processing there is a constant trend to increase particle loadings in technical applications for efficiency reasons. The classical coagulation theory, which assumes that the collision kernel is independent from the particle concentration, is well proven for the diluted particle systems, however, this basic as-sumption does not necessarily hold any more once the particle system becomes highly concentrated. In this dissertation the consequences of coagulation at high concentrations are investigated theoretically, by computer simulations and by comparison with available experimental data.
In Chap. 1the classical theory of Brownian coagulation is critically reviewed and forms the background to the rest of the work.
In Chap. 2a novel simulation algorithm is developed to determine the Brownian coagulation kernels based on direct simulation of parti-cle motion and collisions, while at the same time maintaining partiparti-cle size and concentration as constant. The basic premise of the proposed method is to use first principle Langevin dynamics simulation to track the motion of particles placed in a cubic box endowed with periodic bound-ary conditions until a collision event occurs. The number of particles and their size is kept constant during the simulation by redistributing the collided particles back into the simulation box. The redistribution step requires special care to avoid introducing spurious effects and to
preserve statistical independence of collisions. This is accomplished by choosing the positions of the redistributed particles in such a way, so as to preserve the nearest neighbour distance distribution naturally developing during the simulation. The more technical details of efficient practical implementation of this redistribution algorithm are relegated to AppendixA.
In Chap.3the new simulation method is used to investigate Brownian coagulation at high particle volume fractions and over the entire size spectrum from the free-molecule to continuum regime. It is shown that the coagulation kernel is in general concentration dependent and that the classical concentration-independent values are recovered only in the limit of vanishing particle volumetric concentration. In the free-molecule regime (ballistic motion), the concentration effect is of a moderate mag-nitude, while the contrary is true in the opposite limit of the continuum regime (diffusional motion), where much stronger dependence on the particle concentration is observed. For conditions intermediate between the free-molecule and the continuum regime all data reduce to one master curve if plotted versus a ratio of coagulation kernels calculated for the two limiting cases. This master curve is well approximated by the classical interpolation formulae for the transition regime provided that the limiting coagulation kernels in this formula are corrected for the concentration dependence.
The focus of Chap. 4 shifts from the geometrical crowding effects towards the coagulation at particle number densities that are large in comparison to that of the background gas. With an increasing rarefaction of the suspending gas, while keeping the particle concentration fixed, there will be a point, when the particle relaxation time becomes longer than the characteristic coagulation time. At that point the average veloc-ity of particles will decrease due to mutual inelastic collisions and will fall below the value predicted by the energy equipartition theorem for there is insufficient time between the collisions to restore the thermal equilibrium. This reduced particle velocity will in turn lead to a lower coagulation rate and this effect is shown through first principle Langevin dynamics simulations. A model quantifying the effect of the coagulation rate suppression is formulated by introducing the concept of a
thermal-to extend the classical coagulation theory thermal-to this new non-equilibrium regime of coagulation. A closed-form, analytical formula for the coagula-tion kernel in this regime is obtained and its excellent agreement with the first principle simulations is also demonstrated.
In the final Chap.5the problem of accelerating kinetics in sol-gel tran-sition is analysed. This process can be considered to be universal with respect to the particle volume fraction (or more specifically, to the free volume available for particle motion). A simple monodisperse model of agglomeration with the coagulation kernel dependent on the volumetric concentration is formulated and used to find a non-dimensional transfor-mation that is required to obtain a universal solution for the kinetics of this process. By exploring this concept of free-volume universality of co-agulation further, a coco-agulation kernel enhancement function established for a hard-sphere system is applied to the problem of agglomeration through an appropriate definition of the agglomerate volume fraction. It is shown through comparison with computational modelling and ex-perimental data that the kinetics of agglomerate growth are adequately captured in this model before the agglomerates become interdigitated. The apparent increase in coagulation kernel homogeneity observed in cluster-cluster agglomeration, but not in coalescent coagulation, is iden-tified as originating from the increase of the effective volume fraction of growing agglomerates stemming from their fractal-like structure.
Summary xi
1. Brownian coagulation – classical theory 1
1.1. Free-molecule regime . . . 2
1.2. Continuum regime (Smoluchowski theory) . . . 3
1.3. Transition regime . . . 4
1.3.1. Particle mean free path . . . 4
1.3.2. Flux matching method . . . 6
1.3.3. Fokker-Planck equation . . . 9
2. Brownian Dynamics algorithm 15 2.1. Particle motion. . . 16
2.2. Time stepping algorithms . . . 17
2.3. Calculation of coagulation kernel – basic algorithm . . . . 20
2.4. Particle redistribution . . . 22
3. Coagulation kernels in concentrated systems 27 3.1. Free-molecule regime . . . 27
3.2. Continuum regime . . . 29
3.3. Transition regime . . . 36
3.4. Conclusions. . . 37
4. Coagulation under thermal non-equilibrium 41 4.1. Brownian motion and energy dissipation in collisions . . . 42
4.2. Coagulation kernel in the non-equilibrium regime . . . 46
4.3. Coagulation rate of equal size particles . . . 49 4.4. Overall coagulation rate of polydisperse particle population 51 4.5. Non-equilibrium coagulation via kinetic Monte Carlo method 55
4.6. Discussion . . . 60 4.7. Conclusions. . . 63
5. Monodisperse model for sol-gel transition 67
5.1. Agglomeration and fractal-like structure . . . 67 5.2. Universal master curve for gelation kinetics . . . 70 5.3. Comparison with simulations of cluster-cluster
agglomer-ation . . . 74 5.4. Comparison with experimental data. . . 81 5.5. Conclusions. . . 84
A. Particle redistribution algorithm 87
A.1. Sampling from NNDD during redistribution step . . . 87 A.2. Nonparametric density estimation with kernel smoothing 88 A.3. Bandwidth selection . . . 89 A.4. Boundary bias and boundary correction . . . 91 A.5. Fast computation of kernel smoothing estimates . . . 92
B. Calculation of the thermalization number 97
Summary in Dutch 99
Acknowledgement 103
Curriculum Vitae 105
Brownian coagulation – classical
theory
Introduction
Coagulation is a physical process by which particles grow through inelas-tic collisions with and attachment to other parinelas-ticles. There are various phenomena that can bring particles into contact with each other: Brown-ian motion, gravitational settling, turbulent flow, shear flow, inter-particle interactions (e.g., electrostatic, van der Waals), and others. Which of these phenomena will play the dominant role depends on physicochem-ical properties of particles, suspending fluid and flow conditions. For particles in the submicron range the coagulation process is usually de-termined by a random, thermally induced particle motion and therefore this process is often termed Brownian coagulation.
One can distinguish two limiting cases of Brownian coagulation: the free-molecule coagulation and the continuum coagulation. The first one occurs when the drag exerted by fluid on suspended particles is small and these particles are decelerated only very slowly. In effect, particles move along straight trajectories over distances much larger than their size and behave in a similar fashion to gas molecules. In the latter case the converse is true and the direction of particle motion changes over a length scale that is a small fraction of the particle size. This leads to particle transport and collisions through random walk that is characteristic for the continuum diffusion process.
1.1. Free-molecule regime
Particle behaviour in the free-molecule regime can be described using the gas kinetic theory. According to this theory the collision rate between two particles a and b, with radii ra and rband having concentrations Ca and Cb, can be expressed as (Hidy and Brock,1970):
Rab= Z dσab Z dva Z dvb f (va) f (vb) va− vb = = π(ra+ rb)2 p (
v
a)2+ (v
b)2CaCb (1.1) where f (v) = C m2πkp BT 3 2 exp −m2kp(v · v) BT (1.2) is the Maxwellian distribution of velocities with the meanv
:v
= È 8kBT πmp (1.3) and variancev
2:v
2= 3kBT mp (1.4)kBis the Boltzmann constant, T – absolute temperature, mp – particle mass andRdσab= σab= π(ra+ rb)2is the collision cross-section.
The coagulation kernel for the free-molecule regime Kfmis calculated from the collision rate as:
Kfm= CRab
aCb = π(ra+ rb) 2p(
v
a)2+ (
v
b)2 (1.5) For equal-sized particles the above expression becomes:Kfm,mono= 16r2 p p πkBT/mp= 8 p 3rpkBT/ρp (1.6) where ρpis the particle density. Thus, even in the monodisperse case the free-molecule coagulation kernel remains dependent on the particle size.
1.2. Continuum regime (Smoluchowski theory)
In the continuum regime collisions between particles occur as a result of Brownian motion. The rate of this process is diffusion-limited and can be found by considering a stationary central absorber in an infinite medium with suspended particles. This central absorber forms a perfect sink, therefore at the collision distance r0 particle concentration must vanish. At large distances particle concentration is assumed to approach a constant value of C∞and it is further assumed that the concentration is constant at the very beginning of the process throughout the entire medium. The concentration profile at any given time can be found by solving the diffusion equation:∂ C ∂ t = 1 r2 ∂ ∂ r r2D∂ C ∂ r (1.7)
where D = kBT/ζ is the diffusion coefficient, ζ = 3πµdp/Cslip – drag coefficient, µ – fluid viscosity, dp– particle diameter and
Cslip= 1 + Kn 1.2310 + 0.4695 exp −1.1783Kn (1.8)
is the Cunningham slip correction factor (Hutchins et al.,1995). The Knudsen number Kn is defined in the next section, where its relevance to the coagulation problems is briefly discussed. Equation (1.7) is com-pleted with initial and boundary conditions:
C = C∞, t = 0, r > r0 (1.9a)
C = C∞, t > 0, r → ∞ (1.9b)
C = 0, t > 0, r = r0 (1.9c) The solution to (1.7) has the following form:
C C∞ = 1 − r0 r 1 − erf r − r0 2pDt (1.10)
Using the above result particle deposition flux onto the central collector is calculated as: J = − 4πr2D∂ C ∂ r r=r0 = −4πr0DC∞ 1 +pr0 πDt (1.11)
From this equation the coagulation kernel for the continuum regime Kco is obtained as: Kco(t) = |J|/C∞= 4πr0D 1 +pr0 πDt (1.12)
Although the above equation was derived for a stationary collector, it is equally applicable to coagulation of two particles in relative motion, if only a substitution r0= ra+ rband D = Da+ Db is made in order to account for the relative motion of these two particles. Furthermore, in the long time limit stationary conditions are reached, i.e., one can take
∂ C/∂ t = 0, in which case the well known classic result is obtained: Kco= 4π(ra+ rb)(Da+ Db) (1.13) For the special case of equal-sized particles that are large enough to neglect the slip correction factor (Cslip = 1) the coagulation kernel becomes independent of particle properties and equal to:
Kconst= 8kBT
3µ (1.14)
1.3. Transition regime
1.3.1. Particle mean free path
The transition regime spans between the free-molecule regimes and the continuum. The Knudsen number Kn = λg/rp, which is defined as the ratio of the gas mean free path λgto the particle radius rpis often used to discriminate between these two regimes of coagulation.
However, the Knudsen number characterises transition from contin-uum to rarefied conditions in fluid flows rather than in particle coagula-tion. In the latter context it is the particle Knudsen number Knp= λp/rp,
defined as the ratio of the particle mean free path λp to the particle radius, that plays this role (Fuchs,1980; Dahneke,1983). In contrast to the gas Knudsen number, the particle Knudsen number unambiguously characterises the transition of particle motion from ballistic to random walk and reduces the number of non-dimensional groups required in description of the coagulation process (Molski,1984).
The actual definition of λp used in the literature differs among various authors. Fuchs (1964) uses the particle stopping distance ¯
v
/β, i.e., theaverage distance travelled by the particle before its motion is completely diverted into a direction perpendicular to the initial one. Here, the reciprocal of the particle relaxation time is given by the mass specific friction coefficient:
β = mζ
p (1.15)
Friedlander (2000) chooses D
v
2−1/2 for λp (note: the original expression in Friedlander’s book contains typographical errors), while Dahneke (1983) arrives at 2D/ ¯
v
by requiring consistency between the kinetic gas theory and the continuum description of particle diffusion.In fact, all of the above definitions are equivalent to each other and differ only by a constant, multiplicative factor. It is not clear, there-fore, which one should form the base for the particle Knudsen number definition. Furthermore, in the case of coagulation, transport proper-ties for relative particle motion are of interest rather than those for a single particle motion. Hence, the relative mean thermal velocity
v
ab=(v
a)2+(v
b)21/2, the relative diffusion coefficient Dab= Da+Db, and the particle collision radius rab= ra+ rb, should be used accordingly instead of their single particle equivalents.It is straightforward to show that, regardless of which of the above expressions is adopted for λp, using the quantities for the relative par-ticle motion leads to an expression for the parpar-ticle Knudsen number that is also proportional to the ratio of coagulation kernels (or
equiv-alently coagulation rates) calculated according to the continuum and free-molecule theories. Therefore, it seems appropriate to use this ratio for the definition of Knp, especially in light of the aforementioned lack of consensus on any other definition. However, to avoid any further confusion and noting that the particle Knudsen number can be useful in characterisation of other particulate processes than coagulation, we follow Fuchs and define Knpas:
Knp= λp rp = ¯
v
β rp (1.16)and we reserve a separate symbol κ for the coagulation kernel ratio:
κ =KKco
fm (1.17)
For equal sized particles we have the following relation between these two numbers:
Knp= 2 p
2
π κ (1.18)
In the remainder of this work it is the dimensionless group κ that is chosen to discriminate between the free-molecule and the continuum regimes in the coagulation process for its simple meaning and unambigu-ous interpretation.
1.3.2. Flux matching method
There are two main approaches employed in the theoretical derivation of coagulation kernels in the transition regime: the flux matching method and the solution of the Fokker-Planck equation. The former one is attributed to Fuchs (1934); Fuchs (1964) who assumed that particle transport from the suspending fluid to a central absorber happens in two distinct stages: (I) diffusion from the surrounding medium into the vicinity of the absorber and (II) free ballistic motion close to the central collector. These two regions are assumed to be separated by a dividing
θmax
l r0
I II
Figure 1.1: Geometry of the collision model in the flux matching method
surface at some distance l as shown in Fig.1.1, where the geometry of the problem is depicted.
For distances larger than l the diffusion theory is assumed to be applicable. The particle flux at the dividing surface can be calculated by solving the stationary version (i.e., with ∂ C/∂ t = 0) of the diffusion equation (1.7) with boundary conditions C(r = l) = C0and C(r → ∞) =
C∞: JD= − 4πr2D∂ C ∂ r r=l= −4πDl(C∞− C 0) = = −KcoC∞ l r0 1 −CC0 ∞ (1.19) On the other hand, the particle flux at the interface between (I) and (II) is given from the kinetic gas theory as:
JG= −4πl2 Z 2π 0 dϕ Z θmax 0 dθ Z ∞ 0 d
v v
cos θ f (v
, θ, ϕ) (1.20)where the Maxwell distribution of velocities in spherical coordinates is defined as: f (
v
, θ, ϕ) = C0 m 2πkBT 3 2v
2e−mpv 2 2kBT sin θ (1.21)The collision radius is denoted as before by r0and
v
cos θ is the particle velocity component perpendicular to the dividing surface. The integra-tion limit θmax= arcsin(r0/l) marks the angular range for which ballistic collision occurs. By integrating equation (1.20) one arrives at:JG= −πr2
0
v
¯C0= −KfmC0 (1.22) By requirement of mass conservation at the dividing surface it follows that both of these fluxes must be equal JD = JG = J. Using this rela-tion and equarela-tions (1.19) and (1.22) one can deduce that the particle concentration at the interface is:C0 C∞ = 1 +r 2 0
v
¯ 4l D −1 (1.23)The coagulation kernel for the transition regime Ktrcan now be obtained as:
Ktr= |J|/C∞=
Kco
r0/l + Kco/Kfm (1.24)
The last unspecified quantity in the above equation is the location of the regime separation surface, which we can take without any loss of generality as l = rab+ δab, with rab = r0 = ra+ rb. Probably the simplest choice for the distance between the collision surface and the regime dividing surface is δab= 0, i.e., to perform flux matching directly at the collision surface, which leads to the harmonic mean coagulation kernel1:
Kharm=K−1 fm + Kco−1
−1
(1.25)
1Even though this name is customarily used in the literature, the actual definition
Fuchs (1964) proposed: δp= h (dp+ λp)3− (dp2+ λ2p)3/2 i /(3dpλp) − dp (1.26) with λp= ¯
v
/β for a single particle and for a pair of particles suggested: δab= δ2a+ δ2b1/2
. This seems somewhat arbitrary, since in light of the earlier discussion about definition of the particle mean free path, one should rather use quantities defined for the relative particle motion. This route was taken by Dahneke (1983), who arrived at δab = 2Dab/
v
ab by requirement of consistency between the kinetic gas theory and the continuum description of particle diffusion and he obtained the following coagulation kernel for the transition regime:KDahn= Kco2 + κ(2 + κ)2 + κ (1.27)
The above equation, despite its simplicity, stands in remarkably good agreement with the asymptotic solution of the Fokker-Planck equation presented in the next section.
1.3.3. Fokker-Planck equation
The flux matching method is a simplification. In reality, the character of particle motion does not change abruptly from ballistic to random walk, but rather gradually. A theoretically more sound approach to calculate the coagulation kernel in the transition regime is to solve the Fokker-Planck equation. For motion of a single particle this equation takes the form: ∂ Ψ ∂ t + v · ∇xΨ = ∇v· βv−F (ex t) m Ψ +β km ∇BT 2vΨ (1.28) where Ψ = Ψ(x, v) is the joint probability distribution for particle position and velocity, F(ex t) is a vector of external forces acting on the particle and indices x and
v
refer to differentiation with respect to the particle position and velocity, respectively. For relative motion of equal-sized0.1 1 0.1 1 κ γ( κ) harmonic Dahneke Fuchs Sahni, trunc. Sahni, asympt.
Figure 1.2: Comparison of formulae for Brownian coagulation kernels in the transition regime.
particles the above equation remains still valid provided that the reduced mass mab= (m−1a + m−1b )−1and the reduced friction coefficient ζab= (ζ−1a + ζ−1b )−1is used. However, in case of different sized particles this can be done only as an approximation, since the relative and collective motions of a particle pair are coupled together (Sahni,1983b).
Solution to the Fokker-Planck equation in the form of the Burnett function expansion was proposed by Sahni (1983b), who also obtained two closed-form approximations for the coagulation kernel. The first one neglects all but the most significant terms of the expansion to give:
Ktrun= Kco 1 + κ +κ 2 h π κ2 − 1 + π κ2 erfpπ κ i (1.29)
Table 1.1: Transition regime Brownian coagulation kernels.
Name Transition regime formula γ(κ)
harmonic (1 + κ)−1 Dahneke 2 + κ 2 + κ(2 + κ) Fuchs κ + 1 +p∆2a+ ∆2b −1−1 ∆p= (2+Knp) 3−(4+Kn2 p)3/2 12Knp − 1 Sahni, Eq. (1.29) n1 + κ +κ 2 h π κ2 − 1 + π κ2 erfpπ κ io−1 Sahni, Eq. (1.30) n1 +κ 2 h 1 + expπ κ2 erfcpπ κ io−1
condition of particle number conservation (Sahni,1983a):
Kasym= Kco 1 +κ 2 h 1 + expπ κ2 erfcpπ κ i (1.30)
Despite further efforts to provide even more accurate solutions by retaining additional expansion terms (Kumar and Menon, 1985) or by using an integral formulation (Monvel-Berthier and Dita,1991), to the best of authors’ knowledge no practical method has been devised to calculate the coagulation kernel to an arbitrary numerical accuracy. Therefore, the above formulas of Sahni still remain perhaps the most rigorous closed-form solutions to the problem of coagulation in the transition regime.
Nevertheless, with the exception of the harmonic average, the differ-ences between different approximations to the coagulation kernel remain fairly small as can be seen in Fig.1.2. For example Dahneke’s and Fuchs’ interpolation formulas and equation (1.29) are within 2, 5 and 8 percent of Sahni’s asymptotic result (1.30), respectively. Therefore, they can be regarded as largely equivalent for practical applications and the choice of which one to use can be made based on pragmatic considerations. A very
popular choice is Fuchs’ formula, however, our general recommendation goes towards Dahneke’s formula for its sound theoretical motivation, ex-cellent agreement with Sahni’s asymptotic result and particularly simple form that makes it amenable in analytical treatment of some coagulation problems (Otto et al.,1999).
In what follows it will be convenient to express the coagulation kernel in the transition regime through a general formula:
Ktr= Kcoγ(κ) = Kcoγ KcoKfm (1.31) where γ(κ) is an interpolation function satisfying conditions γ(κ → 0) → 1 and γ(κ → ∞) → κ−1. Alternatively, one could also take
Ktr= Kfmγ0(κ) with γ0(κ) = κ−1γ(κ), which is just a reformulation of (1.31). The expressions for γ(κ) proposed by various authors have been gathered for convenience in Tab.1.1.
References
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Friedlander, S. K. (2000), Smoke, Dust, and Haze: Fundamentals of Aerosol
Dynamics, 2nd ed., Oxford: Oxford university Press.
Fuchs, N. A. (1934), “Theory of coagulation”, Z. Phys. Chem. 171 (3), 199–208.
Fuchs, N. A. (1964), The Mechanics of Aerosols, Oxford: Pergamon Press. Fuchs, N. A. (1980), “On the Brownian coagulation of aerosols”, J. Colloid
Interface Sci. 73 (1), 248–249.
Hidy, G. M. and J. R. Brock (1970), The Dynamics of Aerocolloidal Systems, Oxford: Pergamon Press.
Hutchins, D. K., M. H. Harper, and R. L. Felder (1995), “Slip correction measurements for solid spherical particles by modulated dynamic light-scattering”, Aerosol Sci. Technol. 22 (2), 202–218.
Kumar, V. and S. V. G. Menon (1985), “Condensation rate on a black sphere via Fokker-Planck equation”, J. Chem. Phys. 82 (2), 917–920.
Molski, A. (1984), “On the use of the Knudsen number and aerosol Knudsen number in the kinetics of Brownian coagulation of aerosols”,
Colloid Polym. Sci. 262 (5), 403–405.
Monvel-Berthier, A. B. de and P. Dita (1991), “Brownian motion near an absorbing sphere”, J. Stat. Phys. 62 (3), 729–736.
Otto, E., H. Fissan, S. H. Park, and K. W. Lee (1999), “The log-normal size distribution theory of Brownian aerosol coagulation for the en-tire particle size range: Part II - Analytical solution using Dahneke’s coagulation kernel”, J. Aerosol Sci. 30 (1), 17–34.
Patterson, R. I., J. Singh, M. Balthasar, M. Kraft, and W. Wagner (2006), “Extending stochastic soot simulation to higher pressures”, Combust.
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Sahni, D. C. (1983a), “An exact solution of Fokker-Planck equation and Brownian coagulation in the transition regime”, J. Colloid Interface Sci. 91 (2), 418–429.
Sahni, D. C. (1983b), “Comment on exact Brownian coagulation”, J.
Brownian Dynamics algorithm for
calculation of coagulation kernels
Introduction
Classical theory of coagulation described in Chap. 1 is based on the analysis of two-particle systems in isolation and therefore cannot be easily applied to many-particle systems. While formal extensions to this theory are possible, in general they involve many-particle distributions (cf. Molski, 1988) and analytical treatment required to obtain some
useful results becomes challenging to say the least.
Alternatively one can resort to direct simulation of particles’ motion and interactions to obtain a very detailed picture of system’s behaviour on a micro scale. A unique challenge in this approach is how to extract the relevant information. In the case of Brownian coagulation, kernels are typically determined in the same way as from experimental data—by examining how concentration or average particle size changes over time (Urbina-Villalba et al., 2004). This method, however, poses the same problems as experiments, i.e., how to infer parametric dependence of the kernel on, e.g., particle concentration, while this parameter changes during simulation. For this reason a novel simulation algorithm is pro-posed here that does away with this kind of limitations by keeping the coagulating system under constant conditions with regard to particle size, concentration and volume fraction.
2.1. Particle motion
Particle thermal motion in quiescent fluid is governed by the stochastic Langevin equations:
md
v
id t = −ζ
v
i+ Fi(ex t)+ Fi(B) (2.1) wherev
i, i = 1, 2, 3, are the Cartesian components of the particle velocity,F(ex t)
i – external force acting on the particle (including inter-particle interactions) and F(B)
i denotes random Brownian force.
The solution to this stochastic equation has a form of joint probability distribution for particle velocity and position (Chandrasekhar,1943):
Ψ(
v
i, xi) = 1 2πσviσxi p 1 − ρ2 i × × exp − vi−vi σvi 2 −2ρi(vi−vi)(xi−xi) σviσxi + xi−xi σxi 2 21 − ρ2 i (2.2)where expected values
v
iand xifor velocity and position are respectively:v
i=v
0ie−β t+ F(ex t) 0i mβ 1 − e−β t (2.3a) xi= x0i−v
0i β 1 − e−β t+ F0i(ex t) mβ t −β11 − e−β t (2.3b) where the index 0 refers to the initial values of the quantities at timeρi in equation (2.2) are given by: σ2vi =
v
i−v
i2= kBT m 1 − e−2β t (2.4a) σ2x i = xi− xi 2 = kmβBT 2β t − 3 + 4e−β t− e−2β t (2.4b) ρi= vi−vi σvi xi−xi σxi = 1 − e−β t2 Æ 1 − e−2β t 2β t − 3 + 4e−β t− e−2β t (2.4c)2.2. Time stepping algorithms
The solution to the Langevin equation presented in the previous section is strictly valid only for point source in infinite medium, i.e., it gives prob-ability of finding a particle with a given velocity and at a given location after some time t in function of its initial position and velocity. In many-particle system one would have to account for possible collisions (sticky or otherwise) among different particles, which would be equivalent to solving many-particle version of the Fokker-Planck equation.
Unfortunately, such a problem is hardly tractable analytically. Never-theless, over sufficiently short periods of time the solution for a single free particle constitutes a good approximation to the behaviour of each individual particle in the many-particle system and thus it is often used as a basis for numerical integration of the Langevin equation.
One such algorithm was proposed by Podgórski (2001). In this algo-rithm one first updates particle velocity and then uses this updated value in calculation of particle displacement as follows:
v
i=v
i+ σviGvi (2.5a) xi= xi+ ρi σxi σvi(v
i−v
i) + Æ 1 − ρ2 iσxiGxi (2.5b) where Gxi and Gvi are random numbers drawn from the standard normal distribution.Another popular algorithm was proposed by Ermak and Buckholz (1980):
v
i =v
i+ σviGvi (2.6a) xi = x0i+ 1 βv
i+v
0i− 2F(ex t) 0i mβ ! tanh β t 2 + +2F (ex t) 0i mβ t + σxi|viGxi (2.6b) The particle velocity is updated in the same way as in Podgórski’s algo-rithm but particle displacement is drawn from the conditional distribu-tion:Ψ(x|v) = Ψ(x, v) ÂZ
Ψ(x, v)dx (2.7)
with standard deviation:
σxi|vi = 2kBT mβ β t− 2 tanh β t 2 (2.8) In fact both of these algorithms are mathematically equivalent since they sample from the same underlying probability distribution. This has also been confirmed by our numerical simulations. For the most part Podgórski’s algorithm is used except for very large Knudsen numbers, in which case e−β t→ 0, and large round-off errors occur on finite precision computing machines, especially in the evaluation of the correlation coefficients (2.4c). Since the algorithm of Ermak and Buckholtz does not require computation of the correlation coefficients ρi, it is better suited in this situation.
The choice of integration time step to use in simulation should guaran-tee sufficient level of accuracy on the one hand and acceptable computa-tional time on the other hand. The influence of time step on calculated values of coagulation kernels is illustrated in Fig. 2.1(coagulation of 10 µm unit density particles at 5 % concentration and normal conditions). Filled symbol signifies the result for step size typically employed in our simulations.
10−1 100 101 102 103 1.5 2 2.5 β∆t Kco (φ ) (8π D dp )
Figure 2.1: Dependence of the coagulation kernel on time step size (dp= 10 µm, φ = 5 %).
For large time steps coagulation kernel is clearly underestimated. The reason for this is that particles are checked for collisions only at discrete times in between the steps, therefore there is a possibility that some collisions go undetected. This is more pronounced in the continuum regime, where taking large steps means that particle’s Brownian trajec-tories will be poorly resolved reducing the neighbourhood explored by a particle and thus reducing also the chance of encounter and collision with another particle.
To control the magnitude of introduced error, integration step is deter-mined based on the requirement that in one step particles move only a small fraction (taken as 1% in our simulations) of their diameter. The mean square displacement of the particle can be found by using (2.3b) and averaging over all initial velocities (see also Chandrasekhar,1943, Eq. 175):
This equation can be solved to find the time step given the mean square particle displacement:
β t = A + 1 + W−e−A−1 (2.10) where
A = x(t) − x0 2 mβ 2
6kBT (2.11)
and the Lambert W function is defined through an implicit relation
z = W(z)eW (z).
The above equation can be approximated as:
β t≈pA(A+ 2) (2.12)
with an accuracy better than 7 %. For the purpose of the integration time step selection this approximation is more than sufficient and it has a benefit of being computationally more convenient, since the evaluation of the implicit Lambert W function in not required.
2.3. Calculation of coagulation kernel – basic
algorithm
This section presents the basic algorithm used for calculation of coagula-tion kernels from the simulated trajectories and collisions of Brownian particles. The basic logic of the proposed algorithm is as follows:
1. Choose simulation parameters: fluid viscosity µ and temperature T, particle diameter dpand density ρp, number of simulated particles
Np and their volumetric concentration φ and maximal number of collisions Ncoll.
2. Determine size of the simulation box Lbox= dp3 q
πNp
6φ.
3. Generate initial positions of particles by scattering them randomly in a non-overlapping manner inside the simulation box.
4. Generate initial velocities of particles from the Boltzmann distribu-tion.
5. Proceed with the next time step and update particle positions and velocities.
6. Apply periodic boundary conditions to particles that left the simu-lation box to avoid particle loss during simusimu-lation.
7. Detect and count collisions among particles.
8. Select randomly one particle from each collision pair and redis-tribute it back into the simulation box to ensure that particle size and concentration remain constant during the entire simulation. 9. If the number of collisions is less than Ncollgo to step5, otherwise
record the total time of the coagulation process, ttot, and calculate the coagulation kernel from (2.13).
K = Ntcoll
tot
2L3 box
Np(Np− 1) (2.13)
The above equation can be derived in the following way. A given pair of particles will on average collide with K/L3
boxfrequency. Since there are Np(Np− 1)/2 unique pairs in the system of Np particles, there will be Ncoll= KNp(Np− 1)/2L3boxcollisions during the time ttot. From this equation (2.13) follows immediately.
Physical conditions of the coagulation process are specified by the parameters µ, T, rp, ρp and φ. The number of simulated particles, Np, determines the size of the coagulating system but does not influence the statistical accuracy of the calculated coagulation kernel, which is controlled by the maximal number of collisions Ncoll. Except for the redistribution step8, which is discussed in the next section, the imple-mentation of the above algorithm is fairly straightforward and therefore will not be discussed in more detail.
0 5 10 15 20 0.8 1 1.2 1.4 1.6
Recollision delay [# of collisions]
R
ecollision
frequency
Random redist. Matched NNDD
Figure 2.2: Correlations in successive collisions (dp= 10 µm, φ = 10 %).
2.4. Particle redistribution
The redistribution step8in our algorithm is the most error prone. On the one hand it is necessary if particle size, concentration and volume fraction are to be kept constant during the simulation. On the other hand, since it does not have a clear counterpart in the real process of coagulation, there is a risk of introducing some artificial effects.
Our initial approach to particle redistribution was to make the most obvious choice – to generate the new particle positions randomly with uniform probability in the entire simulation box but without overlaps with other particles and to draw particle velocities from the Maxwellian distribution as dictated by thermal equilibrium with the suspending fluid (Trzeciak et al.,2004). Closer examination of the results obtained using this approach revealed, however, that such redistribution leads to a certain degree of correlation between successive collisions.
This is demonstrated in Fig.2.2, where normalized frequency of parti-cle recollision is shown for coagulation of 10 µm spheres at 10 %
concen-0 1 2 3 4 5 0 0.5 1 1.5 t/τc Probability density Random redistr. Matched NNDD Exponential
Figure 2.3: Distribution of collision times (dp= 10 µm, φ = 10 %).
tration in air at standard conditions standard conditions (1 atm, 20◦C). Those values of recollision frequency where obtained by counting reap-pearances of the same particle as a collision partner after a specified number of collisions, which should be roughly similar to the autocorrela-tion of particle recollision probability. In the absence of any correlaautocorrela-tion this probability is N2
p and independent of when the particle has last
collided1and this value is used as a normalization constant.
It is clearly visible that for the random redistribution case there is a positive correlation for particle recollision meaning that the same particle is more likely to collide again than any other particle in the system. This, in effect, leads to a non-Poisson statistics of collision times (see Fig. 2.3) and overall overestimation of the coagulation kernels, since the collision frequency is higher on average than it would be for uncorrelated collisions.
1There areNp(Np−1)
2 possible collision pairs and any given particle is present in Np− 1
of them, hence the value of 2
1 1.2 1.4 1.6 1.8 2 0 1 2 3 r/dp NNDD Sampled Random redistr. Matched NNDD
Figure 2.4: Nearest neighbour distance distributions (dp= 10 µm, φ = 10 %).
To understand the origin of the above mentioned correlations one has to realize that in general the relative arrangement of particles in the system is not random on average, since this would imply a flat concentration profile (pair distribution function) around each particle. Only in the free-molecule regime this is true. In the continuum and transition regime random particle arrangement is present only at the very start of the simulation (due to the chosen initial conditions) and after a short period of time non-uniform profiles are developed (see Fig.3.3). For this reason higher collision frequency is observed at the very beginning of the simulation, before the stable concentration profiles develop. This effect can be readily explained with the help of equation (1.12), which shows that the coagulation kernel is higher for small times before it approaches the steady state value. The time scale of this approach in the diluted system is d2
p/D, while for concentrated one the inter-particle gap should be used as a relevant length scale if it is comparable or smaller than the particle diameter. In principle, this initial
period should be discarded from the simulation if steady-state value of the kernel is of interest. However, in practice we found out that its influence was well below the statistical spread of our simulation data and therefore no provision has been made in this regard.
In the case of particle redistribution it is much more difficult to tackle this problem. One can expect that incorporation of some information about locally developed profiles should be beneficial when generating new particle positions. The nearest neighbour distance distribution (NNDD) can be used as a sensitive measure of a close range particle arrangement. In Fig.2.4the NNDD sampled during simulation is shown in comparison with the one obtained for randomly redistributed particle. Clearly, there is a large mismatch between the two of them and randomly redistributed particles have nearest neighbour much closer on average than other particles, which results in a higher collision frequency. It is possible, however, to redistribute particles in accordance to the sampled NNDD by using the standard rejection-acceptance method. The results of such a procedure are shown in the same figure. In this way one can effec-tively remove correlations from successive collisions as shown in Fig.2.2 and recover the Poisson statistics of collision times (cf. Fig.2.3). More details about procedure used to match NNNDs are given in AppendixA. Another assumption inherent in the simple redistribution approach is that the velocities of the redistributed particles remain in thermal equilibrium with the suspending fluid. This assumption is reasonable as long as the particle relaxation time is much shorter than the inter-particle collision time. Even though the kinetic energy of inter-particle motion is dissipated during each collision, there will be sufficient time in be-tween collisions to restore the thermal equilibrium in the system. This condition, however, does not always hold. Indeed, in the free molecule regime under increased concentration conditions collisions can become so frequent that thermal equilibrium cannot be assumed any more. It is straightforward to account for this effect in the simulation by simply calculating the post-collision particle velocities from the law of momen-tum conservation, rather than generating them from the Maxwellian distribution.
References
Chandrasekhar, S. (1943), “Stochastic problems in physics and astron-omy”, Rev. Modern Phys. 15 (1), 1–89.
Ermak, D. L. and H. Buckholz (1980), “Numerical integration of the Langevin equation: Monte Carlo simulation”, J. Comput. Phys. 35 (2), 169–182.
Molski, A. (1988), “A source term formalism for the reactive Fokker-Planck dynamics”, Chem. Phys. Lett. 148 (6), 562–566.
Podgórski, A. (2001), “Brownian dynamics - II. Algorithms for stochastic simulations of a solid spherical aerosol particle motion near a solid wall”, J. Aerosol Sci. 32 (Suppl. 1), 713–714.
Trzeciak, T. M., A. Podgórski, J. W. M. van Erven, and J. C. M. Marijnissen (2004), “Aerosol coagulation in dense systems - langevin dynamics simulations”, J. Aerosol Sci. 35.Suppl. 2, 963–964.
Urbina-Villalba, G., J. Toro-Mendoza, A. Lozsan, and M. Garcia-Sucre (2004), “Effect of the volume fraction on the average flocculation rate”,
Coagulation kernels in
concentrated systems
Introduction
A number of computer experiments have been done using the algorithm described in the previous chapter. Unless specified otherwise simulations were done for coagulation of unit density (1 g cm−3) particles in air at normal conditions (1 atm, 20◦C). However, the key results are presented in the dimensionless form, making them applicable to other conditions. The total number of collisions after which a simulation is terminated has been fixed at Ncoll= 5000 and the number of simulated particles Np has been taken between 16 and 128, without any noticeable influence on the obtained values of the coagulation kernel. Size and concentration of aerosol particles have been varied over several orders of magnitude to cover a wide range of conditions, from free-molecule to continuum and from diluted to highly concentrated ones.
3.1. Free-molecule regime
In Fig.3.1the concentration enhancement factor of the coagulation ker-nel ηfm= Kfm/Kfm,0, which is the kernel normalized by its low
concentra-tion limit, is shown as a funcconcentra-tion of particle volumetric concentraconcentra-tion φ for simulations in the free-molecule regime. The normalization constant
Kfm,0= Kfm(φ = 0) denotes the value obtained for a diluted system as given by equation (1.5).
0 0.05 0.1 0.15 0.2 0.25 1 1.5 2 φ ηfm = Kfm Kfm,0
Figure 3.1: Coagulation enhancement factor in the free-molecule regime: ( ) Brownian coagulation simulations, ( ) hard-sphere gas simulations (elastic collisions), solid line – fitted curve (3.1), dashed line – Chapman-Enskog theory (3.2).
A moderate influence of particle volume fraction on the coagulation kernel is observed, which has been fitted with the equation:
ηfm= Kfm/Kfm,0= 1 + 10.5φ2 (3.1) This relation is included as a solid line in Fig. 3.1. For the highest concentration examined (26 %), the coagulation kernel is increased by approximately a factor of 1.7.
A similar effect of the collision rate enhancement due to particle crowd-ing is also known to occur for hard-sphere gases Noije and Ernst,2001. In Fig.3.1the collision enhancement factor obtained from simulations of a hard-sphere gas are shown for comparison. For these simulations the same algorithm was used as for the coagulation but with the redistri-bution step replaced by elastic rebounding of particles. An expression for the collision rate enhancement for this case can be obtained from the
Chapman-Enskog theory (Silva and Liu,2008):
ηHS=
1 −12φ
(1 − φ)3 (3.2)
A very good agreement of the simulation results can be observed with the above theoretical relation, however, from comparison of the hard-sphere gas collision rate to the coagulation rate it follows that the two are equivalent only in the limit of vanishing concentration (φ → 0). With increasing concentration they start to differ not only in magnitude but also show different behaviour. In particular, the linear scaling present at low concentrations in equation (3.2) seems to be absent from the coagulation enhancement.
It is important to note here, however, that not all assumptions of our simulation model hold for the coagulation case in the free-molecule regime. Specifically, the requirement of thermal equilibrium between particles and the background gas should be lifted for the ballistic coagu-lation, because for sufficiently rarefied gas conditions the coagulation process becomes faster than the thermalization of the particle motion and one has to account for the kinetic energy loss due to inelastic collisions. Although this lack of thermalization turns out to have a dominant effect on the coagulation kernel, the enhancement due to particle crowding plays a role too and therefore it is useful to analyse it separately. We will return to these problems in the next chapter and analyse in detail what happens under such non-thermal conditions.
3.2. Continuum regime
The influence of particle concentration on the coagulation kernel in the continuum regime is shown in Fig.3.2. The first observation to be made is that, contrary to the free-molecule case, the coagulation kernel in the continuum regime becomes strongly dependent on the particle volumetric concentration. For low concentrations it increases steeply as shown in the inset of Fig.3.2, then follows roughly a linear growth, and finally escalates again.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 5 10 15 20 25 30 35 φ ηco = Kco Kco,0 0 0.02 0.04 0.06 1 2 3 4
Figure 3.2: Coagulation enhancement factor in the continuum regime: ( ) uniform particle redistribution (this work), ( ) matched NNDDs (this work), ( ) Heine and Pratsinis (2007), ( ) Urbina-Villalba et al. (2004), ( ) estimated from Mountain et al. (1986, Fig. 6), ( ) estimated from Rottereau et al. (2004, Fig. 8), solid line – equation (3.7), dashed line – fitted cor-rection (3.9), dotted line – correlation (3.8). Inset shows the close-up for lower concentrations.
Such a behaviour can be explained with a Brinkman-like model of particle diffusion around a central absorber immersed in a reactive medium (Bonnecaze and Brady,1991):
1 r2 d d r r2D ab dC d r = keff C − C∞ (3.3) with the usual boundary conditions C(r = rab) = 0 and C(r → ∞) = C∞. In this approach the particles move from infinity through a reactive medium with an effective reaction constant keff towards the central absorber. The reaction constant keffaccounts for the presence of other particles in the system and is yet to be defined. The solution to this equation is: C C∞ = 1 − rab r exp È keffr2 ab Dab 1 −rr ab (3.4)
The coagulation kernel is determined, as in the purely diffusive case, from the particle flux at the collision surface:
Kco(φ) = 4πr 2 ab C∞ Dab dC d r r=rab = Kco,0 1 + s 4πkeffrab3 Kco,0 (3.5)
As a first order approximation the effective reaction constant can be defined using the coagulation kernel for diluted conditions, i.e., keff=
Kco,0C∞. After substitution in (3.5) this leads to a coagulation kernel concentration correction for monodisperse particles:
ηco= Kco/Kco,0= 1 +p24φ (3.6) By requirement of self-consistency it can be argued, however, that one should take keff= Kco(φ)C∞. After substitution in (3.5) this gives rise to an implicit relation for the coagulation kernel Kco(φ). This relation can be solved analytically and for equal sized particles the concentration enhancement factor to the coagulation kernel is obtained as:
The above formula is plotted as a solid line in Fig. 3.2and it coin-cides quite well with the simulation results up to about 10 % volume fraction. At lower concentrations this expression reduces to the square root dependence of (3.6). At higher concentrations this model is prob-ably not sufficient due to the omission of particle ordering and caging effects. These effects are manifested as increasingly pronounced maxima and minima on the pair distribution function g(r) = C(r)C∞, which is a measure of the mean concentration of one particle around another (Fig.3.3). At these conditions each particle is tightly surrounded by sev-eral other neighbouring particles in a cage-like structure, thus inhibiting particles’ ability to move past each other without collision.
Our simulation results are in very close agreement with recent data of Heine and Pratsinis (2007) from Brownian dynamics simulation of coalescent coagulation, which are also reproduced in Fig. 3.2. For an overall enhancement of the coagulation kernel they proposed the following correlation (dotted line in Fig.3.2):
ηHeine= Kco/Kco,0= ςco
1 + a1
1 − φ(− log10φ)a2
(3.8) where the fitted parameters are equal to a1= 2.5 and a2 = −2.7. The polydispersity enhancement factor for a dilute system ςco= 1.0734 was obtained from the moments of the self-similar particle size distribution (Pratsinis,2010) using the data from Vemury et al. (1994). In compari-son to equation (3.7) this correlation provides a much better fit to the simulation data. However, one should keep in mind that there are no adjustable parameters in (3.7), as this equation was derived from a physically motivated model and thus goes beyond a mere correlation offering additional insight into the nature of concentration effects. In particular, a square root dependency is predicted for low particle volume fractions (φ ® 0.001), a feature that is absent from (3.8).
It is interesting to note that for higher concentrations the data of Heine and Pratsinis (2007) are slightly, but consistently, below our results. This is somewhat surprising, since for diluted systems an increase rather than decrease of the coagulation rate is observed due to population polydispersity (Fuchs,1964), cf. the polydispersity enhancement factor
1 2 3 4 5 0 0.5 1 1.5 r/dp g( r) 1% 4% 10% 20% Vol. fraction
Figure 3.3: Pair distribution function at several different particle volume fractions as indicated in the graph. More pronounced lo-cal minima and maxima for higher volume fractions signify increasing short-range order of particle arrangement.
ςcoin equation (3.8). This suggests that at these higher concentrations
the polydispersity effect is offset by changes in the functional form of the coagulation kernel for different sized particles.
For very high concentrations a run-away of the coagulation kernel can be expected as the particle configuration approaches a close-packed arrangement. This region could not be probed effectively with our simulations, however, and thus we do not feel compelled to suggest any particular functional form for it. For this reason only a polynomial correction ∆ηco= −4φ +600φ3(with the less significant quadratic term being omitted) is added to (3.7) to provide a more accurate prediction at higher concentrations:
ηco= 1 + 12φ +p12φ(2 + 12φ) − 4φ + 600φ3 (3.9) We also compare our results with the coagulation enhancement factor
estimated from data reported by Mountain et al. (1986) and Rottereau et al. (2004) for their simulations of Brownian agglomeration. These estimates were obtained from the plots of time evolution of particle concentration in the former case and the average particle size in the latter case. A very good agreement can be observed for the lower concentrations. For higher concentrations the data from Rottereau et al. (2004) suggest much larger values of the coagulation kernel than the ones obtained in this work. This discrepancy can be explained in part by the fact that in our simulations particle volume fraction is kept constant, while in case of agglomeration it effectively increases as agglomerates span space disproportionately to their mass. The other reason is that the initial condition of random (uniform) particle arrangement was used in the simulations of that work. The influence of the initial correlation between the particle positions will increase with particle volumetric concentration and will carry over to the estimate of the coagulation kernel based on the initial rate of the particle concentration change. This is corroborated by much closer agreement of these data with our results from simulations employing uniform particle redistribution rather than obeying naturally developing nearest neighbour distance distribution (NNDD). A further support to the proposed explanation is lent by the data reported in Urbina-Villalba et al. (2004), which were obtained from the population half-life of instantaneously coalescing particles. They lie between our results for simulations with and without matching the particle NNDD.
Enhancement of agglomeration kinetics was previously reported also by Fry et al. (2002) for their Brownian dynamics simulations and was later corroborated experimentally by Dhaubhadel et al. (2009). The enhancement was presented in terms of the degree of coagulation kernel homogeneity, which was shown to increase in time as the agglomeration proceeds towards gelation. It is difficult to compare these findings quantitatively with the results presented here without considering the time evolution of the agglomerating system and we will address this aspect in Chap.5.
Finally let us make a remark on the practical relevance of the results presented in this section. Considering the range of volumetric
concentra-10−2 10−1 100 101 102 100 101 Kn Ktr (8k B T / 3µ ) 2 % 5 % 10 % 15 %
Figure 3.4: Coagulation kernel in the transition regime: symbols – Brow-nian dynamics simulations for concentrations as indicated in the graph, solid line – Dahneke’s transition regime inter-polation formula, dashed lines – Dahneke’s formula with concentration correction.
tion at which enhancement of coagulation kernel becomes appreciable our findings have probably most bearing for colloidal systems, where such conditions are not uncommon. For aerosols typical values of volume fraction are below 1 × 10−4, therefore they are too small to produce any substantial deviations from the infinite dilution baseline. However, in the case of particle agglomeration the effective volume fraction increases over the course of the coagulation process as mentioned above. For some conditions this increase can be so high that it can even lead to gelation (Sorensen et al.,1998; Kim et al., 2006) and therefore concentration enhancement of the coagulation kernel should not be disregarded in such cases.
Another aspect of coagulation in dense systems that warrants some further attention is the influence of initial conditions on the kinetics
10−1 100 101 102 103 10−3 10−2 10−1 100 κ = KcoKfm γ = Ktr Kco 2 % 5 % 10 % 15 %
Figure 3.5: Coagulation kernel in the transition regime – master curve: symbols – Brownian dynamics simulations for concentrations as indicated in the graph, solid line – Dahneke’s transition regime interpolation formula.
of the coagulation process. To the best of our knowledge this problem has not received much attention to date, however, as evidenced by the results and analyses presented in this section, for extremely high concentrations the starting particle arrangement can play a significant role in determining the initial kinetics of the process. Therefore it might prove important to take this effect into consideration for successful modelling of experimental data.
3.3. Transition regime
Coagulation kernels calculated from Brownian dynamics simulations in the transition regime for several particle volumetric concentrations are shown in Fig. 3.4. The normalization constant of the coagulation
