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CHAPTER 5

Processes and Kinetics of Cluster Growth and Decay

5.1 Processes of Growth and Evaporation of Clusters in Parent Gas

The simplest way to support the equilibrium between clusters and the parent gas proceeds through processes of evaporation of atoms from the cluster surface and attachment of atoms to the cluster. The rates of these processes are connected by the principle of detailed balance, and hence the equilibrium between a cluster and parent vapor is similar to that between a bulk surface and this vapor. The equilibrium vapor pressure at a given temperature T of the surface and vapor is equal to the saturated vapor pressure Psat. The temperature dependence of this value is

Psat '" exp (-~) , where eo is the atom binding energy for the surface. Evidently, the same dependence is valid for clusters, so that the equilibrium vapor pressure, when the rates of cluster evaporation and atom attachment are equal for a cluster consisting of n atoms, depends on the temperature as

Pn '" exp ( - ~ ) ,

where en is the atom binding energy for this cluster. From this it follows that

( en - eo) Pn = Psat exp - --T- (5-1)

First we assume clusters to be similar to bulk particles when the cluster binding energy en is a monotonic function of the number of atoms. Indeed, the atom binding energy en includes the surface energy; hence en < eo, and en drops with a decrease

B. M. Smirnov, Clusters and Small Particles© Springer-Verlag New York, Inc. 2000

5.1 Processes of Growth and Evaporation of Clusters in Parent Gas 145

in cluster size. Let the vapor pressure be p > Psar. which gives for large clusters P > Pn· This means that the rate of evaporation of a large cluster is smaller than the rate of attachment of atoms to this cluster, and such a cluster will grow at a given vapor pressure and temperature. On the other hand, for small clusters we have P < Pn, and small clusters evaporate under these conditions. The cluster size at which the rate of cluster evaporation is equal to the rate of atom attachment to a cluster is called the critical size. At this cluster size we have P = Pn. The concept of the critical size holds a central place in the kinetics of cluster growth in the parent vapor.

Note the character of evolution of the size distribution of clusters when the equilibrium results from processes of cluster evaporation and atom attachment. If clusters are inserted in a fixed volume, they evaporate, and the pressure of free atoms increases. If the vapor pressure P exceeds the value Pn at which the initial cluster size is equal to the critical one at a given size n, then clusters grow. Small clusters evaporate. Hence, any size distribution of clusters in the parent vapor is not stationary owing to processes of cluster evaporation and atom attachment. As a result of these processes, the average cluster size increases, and their number density decreases. The process of the growth of large clusters and evaporation of small clusters proceeds through free atoms, even if the number density of free atoms is constant in time.

Problem 5.1. Determine the rates of cluster evaporation and attachment of atoms to the cluster surface assuming the cluster surface to be similar to the macroscopic one.

Let us first consider processes of evaporation and attachment of atoms to a macroscopic surface and assume the parameters of the cluster surface to be identical to that of the bulk surface. The attachment flux of gaseous atoms to a macroscopic surface is the product of three factors

jar = J T . N~, 21rm

(5-2)

so that the first factor is the average velocity of atoms in the direction perpendicular to the surface, m is the atomic mass, N is the atom number density, ~ is the probability of an atom joining with the surface after their contact. The flux of evaporating atoms is given by the fonnula

jev = C exp( -eo / T)

where eo is the binding energy for atoms of a bulk surface, and the parameter C depends weakly on the temperature and is determined by properties of the surface. If the atom number density is equal to the number density of saturated vapor Nsar at this temperature, the attachment flux becomes equal to the evaporation flux:

jev = jar = J 2:m Nsar(T)~,

146 5. Processes and Kinetics of Cluster Growth and Decay

where Nsar(T) = No exp( -eol T). Thus for the factor C in the fonnula for the evaporation flux we have

C = J T No~ 2nm

Within the framework of the liquid drop model of the cluster, we have the expression (5-l) for the flux of attaching atoms. If we transit from a macroscopic surface to the corresponding cluster surface, we can use the above expression for the evaporation flux with a change of the atom binding energy eo for the bulk surface by the cohesive energy of cluster atoms en . Then the fonnula for the evaporation flux obtains the fonn

(5-3)

Problem 5.2. Write the balance equation for the cluster size taking into account the processes of cluster evaporation and atom attachment.

Clusters of different sizes are found in an equilibrium with a gas or vapor consisting of the same atoms as a result of processes

(5-4)

Then evolution of the cluster size with an initial size n is given by the balance equation

dn 2. . dt = 4nr (jar - lev),

where r is the cluster radius which is connected with its size by fonnula (1-1), and jar, jev are the fluxes of atom attachment and cluster evaporation correspondingly.

On the basis offonnulas (5-1), (5-3) we have

dn 2r;E [ ( en-eo)] - = 4nr --;. N - Nsar(T)exp ---dt 2nm T

(5-5)

From this it follows that clusters are divided in two groups, so that small clusters evaporate with time, while large clusters grow as a result of attachment of atoms. The boundary cluster size is called the critical cluster size and is detennined by the relation

__ Tln[Nsat(T)] en eo - N (5-6)

Since en < eo, the critical cluster size and the above regime of cluster evolution exist only for a supersaturated vapor N > Nsat(T) .

Problem 5.3. Write the balance equation for the number density of clusters of different sizes that results from the equilibrium between clusters and their vapor.

Let us introduce the rates of cluster evaporation and attachment atoms to a cluster and account for them in the kinetic equation for the number density of clusters of a


given size. This kinetic equation for the number density fn of clusters containing n atoms has the form

ofn at = Nkn-tfn-I - Nknfn - vnfn + vn+tfn+1' (5-7)

where N is the number density of free atoms of this kind, kn is the rate constant of atom attachment to a cluster consisting of n atoms, and Vn is the frequency of atom evaporation for a cluster of n atoms.

The expressions for rate constants of a large cluster within the framework of the cluster liquid drop can be obtained on the basis offormulas (5-1) and (5-3). Indeed, we have for a large cluster with macroscopic properties of the surface, as it occurs for the liquid drop model,

jat . rrr2 = knN,

where r is the cluster radius, which is determined by formula (1-1), and the attachment flux is given by formula (5-1). Hence, for the rate constant kn of atom attachment to a cluster consisting of n » 1 atoms we obtain

kn = kol;n2/3,

where

ko = rr (3m )2/3 J 8T = 1.93TI/2ml / 6p-2/3 (5-8) 4rrp rrm

Here m is the atomic mass, and p is the bulk density. The principle of detailed balance that was used for the deduction of formula (5-3) for a macroscopic cluster yields for the rate Vn+1 of evaporation of an atom from a cluster consisting (n + 1) atoms

( 8n+I-8o) Vn+1 = knNsat(T)1; exp - T (5-9)

Thus the kinetic equation for clusters whose equilibrium with the parent vapor is maintained by processes (5-4) has the form

0!rn = Nkol; [fn-I(n - li/3 - f nn2/3]-

kol; NsateEo/T [fn(n - li/3 exp ( - ~) - fn+ln2/3 exp ( - 8;1)] (5-10)

Problem 5.4. Show that the flux of ions from the surface of a metallic cluster of a mean charge is small compared to the flux of atoms.

The flux of evaporating atoms from the cluster surface is determined by formula (5-3) and depends on the cluster temperature T as j ""' exp( -En/ T), where En

is the binding energy of a surface atom of a cluster consisting of n atoms. At high temperatures clusters are charged, and according to formula (1-27), a typical cluster charge Z is estimated as Z ""' r T / e2, where r is the cluster radius and e is the electron charge. Let us consider a large cluster whose charge is high, Z » 1. If


an ion is released from the surface of a large cluster, it obtains an additional energy of order Ze2 / r because of the Coulomb interaction between the cluster and the removed ion. This gives that the ion binding energy with the cluster surface is equal to en = Ze2 / r, and the ratio of the ion flux jion from the cluster surface to that of the atoms jat is estimated as

jion Z (ze2 ) - '" -exp -jat n2/ 3 rT

Above we account for the number of ions Z on the cluster surface and the number of surface atoms of order n2/3, and Z « n2/3. For the mean cluster charge Z '" r T / ~ this ratio is

jion Z -.- '" n l / 3 ' lat

i.e., the flux of evaporating atoms is high compared to that of ions.

Problem 5.5. Determine the size distribution jUnction of clusters that are under thermodynamic equilibrium with the parent vapor on the basis of the balance of processes (5-4) of cluster evaporation and attachment of atoms to the cluster surface.

In the case of equilibrium between clusters containing n - 1 and n atoms, the rates of decay and formation of clusters must be identical, i.e.,

Injev = In-Ijat,

where In is the number density of clusters containing n atoms. From this it follows that the relation between equilibrium number densities of clusters of neighboring sizes is given by

In-I N _ ( en-eo) -- - Nsatexp --- , In T

(5-11)

As is seen, this formula has the form of the Saha distribution. This distribution is established during a typical time of attachment of one atom Tn '" (Nkon2/3~)-1 or a typical time of evaporation of one atom Tn '" (Nsatkon2/3 ~)-I exp[(en - eo)/ T»).

Problem 5.6. Determine the cluster critical size when it is sufficiently large and the equilibrium between clusters and the parent vapor is established by means of processes (5-4) of cluster evaporation and attachment of atoms to the cluster surface.

Let us use an asymptotic formula (3-15) for the total binding energy of a cluster or small particle that includes the cluster surface energy:

(5-12)

where A is the specific surface energy of the cluster. Taking into account these terms in the cluster energy, we assume the specific surface energy A to be independent


of the cluster size n for n » 1. Then the binding energy of a surface atom is

dEn l1e en = dn = eo - n1/3 ' (5-13)

where l1e = ~A. Substituting the relation (5-13) into formula (5-11), one can represent it in the form

in (l1e ) -- =Sexp ---in-I n l/3 T

(5-14)

where

(5-15)

is the vapor supersaturation degree. Condensation of atoms takes place at S > I if the vapor density exceeds its saturation value for a given temperature. Then from formula (5-14) it follows that the cluster number density as a function of its size has a minimum at the critical number of cluster atoms (5--6), which accounting for formula (5-13) has the form

ncr = (T~SY (5-16)

The concept of critical size means that for large clusters whose size exceeds the critical one, the probability of cluster growth exceeds the probability of evaporation, and such clusters grow, whereas small clusters with n < ncr evaporate.

Problem 5.7. Prove that the generation of clusters can only proceed under nonequilibrium state between clusters and the parent vapor.

According to formula (5-14), the number density of clusters as a function of their size has a minimum at the critical size of clusters. From this follows an important conclusion. Clusters are an intermediate phase of matter between gaseous and condensed phases. According to formula (5-14), the majority of atoms of a supersaturated vapor S > I consisting of a gas and a condensed phase under thermodynamic equilibrium are found in the condensed phase. On the other hand, in a nonsaturated vapor S < I the majority of atoms are found in the gaseous phase. This means that clusters include a small portion of the atoms of a system under thermodynamic equilibrium. But transition from the gaseous to the condensed phase in a volume proceeds through the formation and growth of clusters. From this one can conclude that a high portion of atoms in clusters corresponds to unequilibrium conditions. Hence, methods of generation of clusters are based on fast nucleation of vapors or gases in a volume at the stage of violating their thermodynamic equilibrium with a condensed phase.

For this reason, generators of clusters or small particles when they are formed from a supersaturated vapor use expansion of the vapor in a region oflow pressure. In the course of expansion, the vapor temperature decreases and the vapor comes to be supersaturated, which leads to the formation and growth of clusters or small


2 3 4

7

111111111 1

FIGURE 5.1. Scheme of generation and deposition of clusters by free jet expansion for "ion cluster beam" method. I-camera; 2- heater; 3- nozzle; 4- skimmers; 5 - crossing electron beam; 6 - accelerator of cluster ions; ·7 - cluster beam. A metallic vapor is formed in the camera as a result of heating of the metal. The vapor is transformed into clusters by means of free jet expansion, and formed clusters are ionized by an electron beam.

1

6

'5 11111 7

8 ,

FIGURE 5.2. Scheme oflaser generation of metallic cluster ions. I - laser beam, 2 - a rod - target; 3 - flow of a buffer gas; 4 - beam of a buffer gas and clusters after a nozzle, 5 -skimmers, 6 - crossing electron beam, 7 - ion optics and accelerator, 8 - final beam. Atoms of a metallic vapor are formed as a result oflaser irradiation of a metallic rod. Further atoms are transformed into clusters as a result of expansion in a buffer gas and expansion of the formed mixture in a vacuum.

particles. But the time for the process is not sufficient for the transformation of clusters in a condensed system, so that almost all the atoms of the incident vapor belong to clusters at the end of the process.

Some schemes of cluster generators use a free jet expansion. Fig. 5.1 gives a version of this scheme when a vapor or its mixture with a buffer gas expands in a vacuum. In the other version of this scheme, a vapor is emitted in a flow of a buffer gas, and furthermore this mixture expands through a jet. In particular, Fig. 5.2 gives such a scheme of cluster generation when an atomic vapor results from evaporation of a surface under the action of a laser beam or a beam of fast atomic particles (electrons, ions, atoms). Then heating of the surface leads to evaporation of atoms, and a forming beam of atoms expands into the surrounding space. Cooling of this

5.2 Kinetics of Cluster Evolution in Parent Gas 151

beam in the course of its expansion causes nucleation of evaporated atoms and formation of clusters.

Problem 5.8. Determine the size distribution function near magic numbers of cluster atoms.

From the kinetic equation (5-10) follows the relation (5-11) in the equilibrium case, when the left-hand side of equation (5-10) is zero. This relation has the form

fn (en_eo) -f, =Sexp -T '

n-I

where S is the supersaturation degree (5-15) of the vapor. This equilibrium is established during a typical time of attachment of one atom t'n 'V (Nkon2/3~)-1 or a typical time of evaporation of one atom t'n 'V (Nsatkon2/3~)-1 exp[(en - eo)jT)] [notation is given in formulas (5-1), (5-2)].

Letn be a magic number, i.e., en > en±I.Takingen-en+1» Tanden-en-I» T, we get fnlfn-I » fn+1Ifn, i.e., fn » J fn-dn+l. By analogy we have fn-I « J fnfn-2. Next, assuming len+1 - en+21 'V T, we have /;+1 'V fnfn+2, . f, /.3/2 / /.1/2 A . Co. th ·d . f, I.e., n+2« n+1 n-I· ssummg uom e symmetry consl erabons n-I 'V

fn+ I, we obtain from this fn » fn± 1 > fn±2. Thus, the size distribution function of clusters has a maximum at the magic numbers of cluster atoms.

5.2 Kinetics of Cluster Evolution in Parent Gas

The size distribution function of clusters located in the parent vapor is not stationary when it is governed by processes (5-4). Hence, the kinetics of the distribution function are of importance in this case. The evolution of a cluster in the parent vapor consists in an increase or decrease of its size depending on the initial size and the vapor pressure. So, small clusters evaporate and large clusters grow in the parent vapor if the vapor pressure is constant in the course of this process. Under this condition, the evolution of clusters can also depend on other factors. The behavior of a charged large cluster or a bulk particle differs from the behavior of a neutral cluster of the same size because of other character of their interaction with an individual me atom in the elementary processes (5-4). Next, the process of cluster evolution in a buffer gas depends on the ratio of the cluster radius and the mean free path of vapor atoms in the buffer gas. If this ratio is large, collisions of free atoms of the parent vapor with atoms of the buffer gas is of importance, because a slow transport of atoms in a buffer gas due to their diffusion delays processes (5-4). Below we consider the growth and evaporation of clusters in the parent and a buffer gas under different conditions.

Problem 5.9. Determine the free enthalpy of a small particle, and find from this the critical size for particles located in the parent vapor.


The free enthalpy of a drop particle is a sum of two terms that account for the space and surface energies:

G = n(cg - co) + 4JTr2y,

where n is the number of atoms in the particle, which is connected with the particle's radius r by the relation (1-1); Co is the binding energy of internal atoms of the particle; Cg is an atom's energy in the surrounding gas, y is the surface tension. We have separated the space and surface atomic energies for the particle under consideration. Note that we take the free enthalpy of the gas to be G = O. Below we find the equilibrium conditions for the system of particle and gas. At given values of the temperature and pressure, this condition requires the maximum of the free enthalpy and has the form d G I d r = o.

Use the expression for the saturation vapor pressure above a bulk surface

Psat = Po exp( -colT),

where Po is a parameter, and T is the gaseous temperature. Let us introduce the degree of vapor supersaturation (5-15)

S = pi Psat(T) = N I Nsat(T),

where N is the number density of atoms in the gas, Nsat(T) is the number density of atoms for a saturated vapor at a given temperature. From this we have

Psat Co In - = -- + const,

P T so that the particle free enthalpy using formula (1-1) obtains the form

4JTp G = 4JTr2y - -r3(T In S + const)

3m

Let us use that the equilibrium condition d G I dr = 0 for a plane surface has the form S = 1 according to the definition of the pressure of a saturated gas. In this limiting case one can neglect by the surface energy, so that this condition gives const = O. Thus, the particle free enthalpy is given by the expression

G=4JTr2y- 4JTPr3TInS 3m

(5-17)

The condition of maximum free enthalpy dGldr = 0 leads to the following expression for the critical radius:

2ym r ---

cr - pT InS

This expression is identical to formula (5-16)

Problem 5.10. Determine the critical size for a charged particle located in the parent vapor.


Let us calculate the part of the free enthalpy of a particle !1 G that corresponds to the particle's charge. We have

00

!1G = f eE2 dR 8rr

o Here R is the coordinate of a point in the frame of reference where the origin is located at the particle's center and it is the point of ion location, E is the electric field strength, and e is the dielectric constant of matter, which we assume to be unity outside the particle and to be a constant inside it. Then the electric field strength is equal to e/(eR2) inside the particle and e/ R2 outside it. From this it follows for the considering part of the free enthalpy of the particle that

!1G = ~ (l-~) +const,

where cons t does not depend on the particle's radius. Taking this term into account, we obtain for the free enthalpy of the particle instead of (5-17)

2 4rrp 3 e2 ( 1) G =4rrr y - --r TlnS+ - 1- - +const, 3m 2r e

The equilibrium condition, corresponding to the maximum of the free enthalpy, has the form dG /dr = 0 and leads to the following equation for the critical radius

2 4rr p 2 e2 ( 1 ) 8rrr y--r TlnS-- 1-- =0 cr m cr 2rcr e (5-18)

Problem 5.11. Determine the supersaturation degree when the growth of a charged particle in the parent gas takes place for any particle sizes.

Rewrite equation (5-19) in the form

InS = 2ym [1 _ e2 (1 _ ~)] rcrpT 16rrr~ry e

As a function of the critical radius, this expression has a maximum at

which corresponds to

_[ e2 ( 1)]1/4 ro - -- 1- - , 4rry e

my lnSmax =--

ropT

If InS exceeds this value, the above equation for In S has no solution, and a particle of any size grows.

Problem 5.12. Determine the typical time of evaporation of a small neutral particle whose size is smaller than the critical radius for the considered pressure of the parent vapor.


The rate of evaporation from the particle's surface has the form (5-3)

lev = 47rr2 jev = 4;rrr2 J 2;m Nsa,(T)~ exp (T~~/3 ) = 4;rrr2 j T Nsa,(T)~ exp (2ym)

2;rrm pTr (5-19)

This gives the balance equation for the number atoms in an evaporating particle:

dn 4;rrpr2 dr 2{l; [1 (2ym) ] - = --- =4;rrr --N~ -exp -- -1 , dt m dt 2;rrm S pTr

or

dr = ~ rrm Nsa,(T)~ [exp (2ym) - exp (~)J dt p Y 2;; pTr pTrcr

Let us consider the limiting cases. If r cr - r « r, one can expand the expression in the parentheses, and the balance equation is

dr = ~jTm N~. 2ym (~_~) = rcr - r, dt p 2;rr pT r rcr ro

where

-1 1 rrm 2ym ro = pY 2;;N~ . pTr~r

From this we find the total time of particle evaporation:

rcr rev = ro In --

rcr - r

In the limiting case 2;:; «r « r cr we have for the evaporation time

_1 = ~ . ~JTm N~ . 4ym rev r2 p 2;rr pT

In the limiting case r « 2m y / (T p) one can neglect the second term in the balance equation, and we have for the total evaporation time

rev = ro. r: . exp (_ 2m y ) rcr Tpr

Problem 5.13. Determine the typical time offormation of a drop of a given size resultingfrom ion growth in the parent vapor at a high degree of supersaturation.

Under the given conditions one can neglect the evaporation processes. Then the balance equation for the number n of particle atoms has the form

dn 4;rrr2 p dr 2 . 2{l; -d = -- . - = 4;rrr Jal = 4;rrr --N~,

t m dt 2;rrm


where we used formulas (1-1) and (5-1). Solving this equation, we have for the particle radius, which it at time t,

r = !....jTm NE, . t P 2n

Note that we have assumed that the particle's growth does not change the number density N of free' atoms.

Problem 5.14. Analyze the character of cluster evolution in the case where the binding energy of the cluster atoms is a stepwise jUnction of the number of cluster atoms n with maxima at magic numbers.

The critical cluster size is introduced through the relation

( en-eo) S=exp --T- ,

which follows from the kinetic equation (5-11), if we assume in = in-I. If en is a monotonic function of n, this relation gives one value for the cluster's critical size ncr. In particular, if the value en is given by formula (5-13), the critical size ncr is determined by formula (5-16). In this case the critical size divides the cluster sizes n into two ranges. Clusters with n > ncr grow, and clusters with n < ncr evaporate. Correspondingly, the cluster flux in the space n is directed to large sizes for n > ncr and to zero for n < ncr, and the flux is zero at the critical cluster size.

In the case where en is a stepwise function of n, the range where the cluster grows is not continuous. For such n according to the above relation we have

( en-eo) S> exp ---T- , (5-20)

The same conclusion corresponds to the size range where clusters evaporate. Assume that in the size range where clusters evaporate relation (5-20) is satisfied only at several magic numbers. Then the evolution of clusters in the course of their evaporation has the following character at a constant number density offree atoms. When the cluster reaches a magic number of atoms, it attaches an atom, and the forming cluster evaporates. This process repeats several times, and the probability Pn of emitting one atom for a cluster consisting of n atoms is given by

1 (en-eo) - = 1 + S exp -- , Pn T

Analogously, the total time of cluster evaporation is in this case given by

f dn [ (en - eo)] rev = ~ 1 + S exp --T- ,

where Vn is the rate of evaporation of a cluster of n atoms. In this case one can see the increase in the evaporation time compared to the case

where Vn is a continuous function of n. If the numbers of cluster atoms for which the criterion (5-20) is fulfilled or is violated are distributed randomly, the character


of the cluster evolution has a more complex character. In this case the diffusion character of the motion in n -space takes place, and each step of this diffusion is not random, but is determined by the competition of processes of cluster evaporation and cluster growth for a given cluster size. Thus each step is determined by the probability Pn , which is not a continuous function of n.

Problem 5.15. Clusters of an identical size n are located in a supersaturated vapor and criterion (5-20) is satisfied for them. Analyze the character of the size evolution of clusters if q free atoms correspond to each cluster and the variation in the number of cluster atoms as a result of this eVolution is relatively small. Assume that the atomic cohesive energy for a cluster is determined by formula (5-13).

Neglecting the size fluctuations during the cluster growth, we find clusters to be of identical sizes in the course of their growth. The cluster growth is accompanied by a decrease of the number density of free atoms, and it stops when the size of clusters corresponds to the critical size at a given number density of free atoms. If at the end of the process q' free atoms correspond to one cluster, the condition of the cluster critical size has the form

q' ( D.s ) -S = exp q T(n + q - q,)1/3

where q is the initial number of free atoms per cluster. Because the relative variation of cluster atoms is small, i.e. n » q - q', from this relation it follows that

, q (D.S) q = S exp Tn1/3

Problem 5.16. Determine the typical time of particle evaporation in a dense buffer gas for the diffosion motion of atoms of the parent gas in a buffer gas.

The concentration c of atoms of the parent vapor in a buffer gas is,small. Because of processes of evaporation and attachment of atoms to the particle, the atomic concentration varies near the particle. Let us denote their concentration far from the particle by coo. The flux of attachment atoms to the particle's surface is given by

j = -DNVc,

Since atoms are not consumed in the volume, we have J = 4n R2 j = const, where R is the distance from the particle. This is the equation for the concentration of atoms of the parent vapor in a volume

2 de J = -4nR D

dR'

and the solution of this equation is

J c(R) = Coo + ---

4nDNR


The total flow of atoms toward the particle is

S- So 1 = lat - lev = lev --S;-'

where lev is given by formula (5-20), S = N / Nsat is the supersaturation degree [formula (5-13)], and So is the supersaturation degree if the particle's radius is equal to the critical radius. In the case of a dense buffer gas we have lat ~ lev » l.

Hence, near the particle's surface S = So, i.e. the number density of atoms of the parent vapor reaches such a value when the particle radius becomes to equal to the critical radius. Then the concentration of free atoms of the parent gas on a distance R from the particle's center is c(R) = Coo + (c* - coo)r/ R, and the rate of particle evaporation is given by

lev = 4rr DNr(c* - Coo),

where c* is the concentration of parent atoms in the buffer gas at which the particle's radius is equal to the critical radius. Assuming c* » Coo, we obtain the balance equation for the evaporation of the particle:

dn 4rrr2p dr (2ym) - = -- . - = 4rrrDNsat (T)exp -dt m dt pTr

Taking r »2ym/(pT), we get from this for the evaporation time

r2p 't'ev=-----

2mDNsat(T) (5-21)

From the above relation we obtain for the atom concentration near the particle surface

and the parameter (1--44), which characterizes the competition between the kinetic and diffusion regimes of the process, is given by

Thus we have

a = J T . ~r 2rrm D

Coo + ac* Co = 1 +a '

and the criterion of the diffusion regime of evaporation corresponds to Coo « Co.

Under the condition Coo « c* this corresponds to

(;f ~r a= --·-»1 2rrm D

(5-22)

It is the criterion of the diffusion character of particle evaporation. This analysis is valid if the particle's radius is large compared to the mean free path ). of free parent atoms in the buffer gas, because we use the diffusion character of motion of


atoms near the particle's surface. Using the estimation for the diffusion coefficient

of atoms D '" Aj"f, we obtain

~r Ol '" -

A

Thus, the diffusion regime of cluster growth and evaporation (Ol » I) takes place for a large particle radius

A r»-~

(5-23)

Problem 5.17. Determine the typical time of a particle's evaporation in a buffer gas for the diffusion regime of the process if the number density of atoms of the buffer gas is restricted.

We found in the previous Problem that the concentration of atoms of a parent gas near the particle surface is equal to c*, which corresponds to the number density of atoms of the parent gas at which the particle's radius is equal to the critical radius. Evidently, C* .::: 1. If the number density of atoms of the buffer gas is less than the number density of the parent gas atoms which makes the particle radius by the critical one, it is necessary to change the formulas of the previous Problem. Replacing in this case the number density of the parent gas atoms near the particle's surface by the number density of atoms of buffer gas, i.e., Co = I, we transform the formula for the evaporation time to the form

r2p rev = 2mDN'

where N is the number density of atoms of the buffer gas.

Problem 5.1S. Determine the typical time for the drop growth on the nuclei of condensation in a buffer gas in the case of the diffusion regime of the process.

The concentration of attaching atoms at a distance R from the particle for the diffusion regime of the growth process is c(R) = coo(l - r / R), so that the rate of attachment is given by

Jar = 4Jl'r2 DN IVcl = 4Jl'rDNcoo , (5-24)

and from the balance equation for the number n of particle atoms dn/dt = Jar we get that the law of evolution of the particle's radius as a result of this process is

r = JmDNcoot 2p

Problem 5.19. Determine the typical time of drop evaporation in a dense buffer gas taking into account both the kinetic and diffusion regimes of this process.

Comparing formula (5-21) of Problem 5.16 for the evaporation time in the diffusion regime of evaporation with the corresponding formula of Problem 5.12


for the kinetic regime of this process, we obtain for the ratio of these times

g;T ym cr. = -~SRo/D, where Ro =-,

Hm pT (5-25)

and the size Ro is the typical size of the particle's growth and evaporation. For this particle size the saturation vapor pressure for a particle becomes different from that of a plane surface.

Note that in the limiting cases we chose ,the minimum time of two versions according to the above formulas. Hence, combining both regimes of particle evaporation, one can represent the evaporation rate as the sum of the above rates:

_1 = 2mD~sal(T) . (1 + cr.) rev r p

Problem 5.20. During expansion of a buffer gas with an admixture of a nucleating vapor, a small portion of the atoms of the vapor is converted into clusters because of the three-body character of formation of diatomic molecules. Assuming the average size of clusters at the end of the process to be large, determine the size distribution of clusters within the framework of the liquid drop model for clusters.

Diatomic molecules are formed as a result of the three-body process

2A+M -+ A2+M

where A is an atom of the nucleating vapor, and M is an atom of the buffer gas. Correspondingly, the number density of diatomic molecules N2 is determined by the balance equation

dN2 = KNbN2, dt

where K is the rate constant of this three-body process, Nb is the number density of atoms of the buffer gas, and N is the number density of the nucleating gas. We assume that starting [ram diatomic molecules, the cluster growth process proceeds as a result of pair processes (5--4), so that the number of formed diatomic molecules is the number of condensation nuclei. Next, we assume that the expansion process leads to a strong drop in temperature, and at typical temperatures of cluster growth one can neglect the evaporation of clusters. The number density of atoms of a buffer gas and atoms of a vapor vary in time by the law Nb '"" exp(-t/rex ), N '"" exp(-t/rex ), where rex is the typical expansion time. The total number density of forming molecules and clusters at time t follows from the above balance equation:

N2 = ~ [1 - exp (- :e:)] KNbN2rex,

where Nb, N are the initial number densities of the corresponding atoms. From this it follows that the regime of cluster growth requires the criterion

(5-26a)


Under this condition, the number of cluster atoms n within the framework of the liquid drop model for a cluster is described by the equation

dn 2. - = 47fr Jal, dt

where the flux of attaching atoms to a cluster jal is given by formula (5-2). Then, using formula (1-1) for the cluster radius r, we obtain this equation in the form

dn = k n2/ 3 N dt 0 ,

where ko is given by formula (5-8). Accounting for N ~ exp( -tITex) and solving this equation, we get for the maximum number of atoms in clusters that result from nucleation of diatomic molecules formed at t = 0 that

_ (ka NTex)3 nmax - 3 '

where the atom number density corresponds to the initial time. Hence, the regime of cluster growth corresponds to the criterion

(5-26b)

As it is seen, the validity of criteria (5-26aa) and (5-26ab) requires the condition

Nb« kolK

Typical values of the rate constants are ko ~ 1O- ll em 3 Is, K ~ 1O-33em6 Is, so that this regime of nucleation is valid if N b « 1 022 em - 3 .

Let us use the connection between the cluster size n at the end of the process and the time t of formation of diatomic molecules that are nuclei of condensation for these clusters. Clusters have the following size if they result from diatomic molecules that are formed at a moment t:

( kaNTex) (3t) ( 3t) n = --- exp -- = nmaxexp -- , 3 rex rex

where N is the number density of vapor atoms at the initial time. Next, the number density of nucleation centers that are formed at time t satisfies the equation

dN2 = K NbN2 exp (_2:) dt Tex

Because the cluster size n relates definitively to the time of formation of the condensation nuclei, we have from these equations for the number density in of clusters of a size n at the end of the process

din Nlol

dn nmax

where NIOI = K NbN2Tex 13 is the total number density of condensation nuclei at the end of the process, and it is normalized to the initial time. It follows from this that the value *' does not depend on n. In addition, the total number density

5.3 Processes of Association of Two Small Particles 161

of atoms in clusters at the end of the process is equal to i Ntotnmax exp( -t / Lex).

Because we assume that the majority of atoms remain free at the end of the process, the following condition must be satisfied instead of (5-26a) (Ntotnmax « N):

KN N (k N )3 I k-3/ 4 -1/4N-1/ 4 b Lex 0 Lex « or N Lex« 0 K b (5-26c)

Under these conditions, we have in = nNtot / nmax , and nin is the number density of atoms that are contained in clusters of size n. Then the average size of clusters fonned is

and the total number density of atoms in clusters is

1 ~ nin = 2: Ntotnmax « N

5.3 Processes of Association of Two Small Particles

We consider the elementary process of association of two clusters or bulk particles within the framework of the model of liquid drops for clusters. Then the contact of two clusters in the course of their collision leads to the joining of clusters into a larger liquid drop. The cross section of this process is equal to Jr (rl + r2)2 , where r1, r2 are radii of joining neutral clusters, because for interacting clusters the distance of closest approach of joining clusters is ro = rl + r2. This is the basis for the analysis of the kinetics of cluster evolution.

We use the liquid drop model of a cluster for association processes when additional factors influence the character of this process. Among these factors, the influence of an external electric field is of importance for the association process. Indeed, the induced dipole moment of a cluster or bulk particle under the action of an electric field is proportional to the number of cluster atoms. Hence, the interaction between induced dipole momenta of joining clusters can be essential even in weak electric fields for large clusters. This interaction can detennine the character of cluster association in a buffer gas and the kinetics of cluster growth.

If the cluster's size is large compared to the mean free path of gaseous atoms or molecules, the motion of clusters is detennined by their diffusion in the gas. This process can detennine the character of approach of clusters in the course of their association, so that the diffusion movement of associating clusters influences on parameters of their association and the kinetics of cluster growth.

Problem 5.21. Determine the rate constant of association of two small particles as a jUnction of their sizes within the framework of the liquid drop model.

Taking joining particles to be liquid and spherical ones, we assume within the framework of the liquid drop model that each contact of such particles leads to


their union. This assumption gives the following expression for the rate constant of the uniting of two particles containing nl and n2 numbers of atoms:

2/*T 1/3 1/3 2 .jnl + n2 ![fT 2 k(nl,n2)=1f(rl+r2) -=ko(nl +n2 ) r;;-:-;:;:: ,ko = -· 1frw 1fJ.L "nln2 1fm

(5-27) Here ko is given by formula (5-8), rl, r2 are the radii of the particles containing nl and n2 atoms correspondingly; m is the atomic mass; p is the bulk density; (3m/41fp)I/3 is the Wigner-Seits radius in accordance with formula (1-1); and J.L is the reduced mass of the colliding particles. It follows from this formula that the rate constant of association of two large neutral particles of the same order of sizes n, '"" n2 '"" n is of order k(n" n2) '"" kon l/6 .

Problem 5.22. Determine the rate constant of the joining of two liquid particles in a dense buffer gas if the sizes of the joining particles are larger than the mean free path of the gaseous atoms.

Under these conditions, the approach of the particles results from their diffusion motion. If we fix one particle, the flux of particles of the second type toward its surface is given by

aN j = -D aR'

where D is the diffusion coefficient, R is distance from the test particle, and N is the number density of particles of the second type. Assuming that these particles are absorbed by the test particle, we obtain N = 0 at R = r, where r is the sum of the radii of the associating particles. Because gaseous particles are consumed on the surface of the test particle, we have J = 41f R2 j = const. From the solution of this equation with the above boundary condition we obtain N = N o( I - r / R), where No is the number density of particles far from the absorbing center. From this it follows the Smoluchowski formula (1-24) for the number of associating particles per unit time J = 41f Dr No, and the rate constant of this process is given by

kas = 41fDr

In this formula D is the diffusion coefficient which is responsible for the approach of the particles. In particular, according to the nature of the diffusion motion, for the relative distance between particles we have

R2 = (R, - R2)2 = 6Dt

if at the beginning the particles are located together. Here t is time, R I, R2 are the coordinates of the associating particles, and a bar means averaging over time. Let us express the diffusion coefficient D of the relative motion of particles through the diffusion coefficients D I, D2 of these particles in a gas. By definition, we have

RI = 6Dlt, ~ = 6D2t, so that

R2 = (RI - R2)2 = Ri + R~ = 6(DI + D2)t


We take into account that the motion of the particles is independent, so that RI R2 = RI . R2 = O. Thus, for the rate constant of association of two particles we get

kdi/ = 4Jl'(DI + D2)(rl + r2)

Using formula (1-15) for the diffusion coefficient of a large particle in a dense gas D = T /(6Jl'rTJ), where r is the particle's radius, and TJ is the gas viscosity, for the coefficient of association of two large particles in a dense gas we get

kdi/ = 2T (~ + ~) (rl + r2) 3TJ rl r2

(5-28)

Problem 5.23. Determine the rate constant of association of spherical charged particles of opposite charges in a dense buffer gas if the sizes of joining particles are larger than the mean free path of gaseous atoms.

Denote the charges of the particles by ±q. A force F = q2 / R2 acts between particles located at a distance R, so that the particles move toward each other with velocities [see formula (1-14)]

q2 q2 v+ = 6Jl'r +TJR2' v_ = 6Jl'r _TJR2 •

where TJ is the viscosity coefficient of the gas, and r +. r _ are the radii of the corresponding particles. From this we obtain the flow of negative particles toward a test positive particle:

2 2q2 L = 4Jl'R (v+ + v_)N_ = -N_.

3rTJ

where N _ is the number density of negatively charged particles, and the effective radius of the particles is equal to r = 1/( 1 / r + + 1/ r _). The balance equation for the number density of positively charged particles is by definition

dN+ -- = -J_N+ = -krecN_N+

dt

From this we get the recombination coefficient krec for two oppositely charged particles:

2q2 2q2 (1 1 ) krec = 3rTJ = 3;J' r+ + ~

In particular, if charged particles have charges +Z+e and -Z_e, we have

2Z+Z_e2 (1 1 ) krec = 3 . - +-

TJ r+ r_ (5-29)

Problem 5.24. Compare the rate constants of association of two neutral liquid particles and two oppositely charged particles in a buffer gas.


According to formulas (5-28), (5-29), the ratio of the rate constants of association of the particles is given by

krec Z2 e2 -=--, kdif Tr

where r = r) +r2, and Z is the charge of each particle expressed in electron charges. In particular, if Z = I and T = 300K, this ratio equals unity at r = O.061L.

Problem 5.25. Determine the rate constant of association of charged and neutral small particles as a result of polarization interaction between them.

The interaction potential of a charged particle with charge Ze and a neutral particle at a distance R between them is given by

CiZ2e2 U(R)=---

2R4 '

where q is the particle's charge, and Ci is the polarizability of the neutral particle. This causes the motion of particles toward each other, and according to formula (1-13) their relative velocity is

F (1 1 ) CiZ2e2 ( 1 1) 1 v = 61f11 r) + r2 = 31fT} r) + r2 RS'

where r), r2 are the radii of the particles. Solving the equation v = dR/dt, we obtain the time of association of particles r if at the beginning they are located at a distance R

~ = 2CiZ2e2 (~+~) ~ r 1f11 r) r2 R6

Let us determine the average time of association for a charged particle that is located in a gas of neutral particles. If p neutral particles are located in a spherical volume Vo whose center is a charged particle, the probability d P that the location of the nearest neutral particle is at distance R from the charged one in the volume interval d V is

dP = p dV . (1 _ ~)P , Vo Vo

where the volume V is contained in a sphere of radius R. Using the number density of neutral particles N = p / Vo and taking p » 1, we obtain

dP = NdVe-NV

This distribution function gives for the average time of the charged particle with - 2

the nearest neutral particle (V2 = (41f R3 /3) = 2/ N 2)

r=!rdP= 911ref , 161fCiZ2e2N2


where the effective radius of particles is ref = (.1 + .1 )-1 . Introducing the effective '1 T2

rate constant of the process as kpol = 1/(N'f), we get

kpol = 16rraZ2e'lN (5-30) 9TJref

In particular, in the case of identical radii of associate particles rl = r2 = 2r ef and for a metallic neutral particle (a = ri) this formula gives

32rrr2 Z2 e'2 N kpol = ~TJ

Note that the effective rate constant of this pair process depends on the number density of particles.

Let us compare the effective rate constant of association in this case (5-30) with the rate of association of large neutral particles (5-28). For identical radii of colliding particles formula (5-28) gives kas = 8T I(3TJ). Therefore, the mechanism of association due to the polarization interaction is stronger than the association of neutral particles in a dense gas of large neutral particles if the following criterion is valid:

T N»Z222 e r

Problem 5.26. Determine the rate of association of two neutral spherical particles in a uniform electric field as a result of interaction of induced dipole momenta. The direction of the electric field coincides with the direction of the principal axis of the polarizability tensor for each particle.

The interaction potential of two dipoles at large distance R =Rn (n is the unit vector along R) between them is

U = [DID2 - 3(Dln)(D2n)] R3

Since the dipole momenta of particles Dl, D2 are induced by the electric field, their values are DI = al E, D2 = a2E, where ai, a2 is the polarizability of the corresponding particle. Thus the interaction potential of the particles is

ala2 E2 2 U = - R3 . (3 cos () - 1),

where () is the angle between vectors R and E. As it is seen, attraction of the particles takes place for a small interval of angles 0 < () < arccos(lIJ3), but just this interval is responsible for the association of the particles. The normal and tangential components of the force between particles owing to the interaction of the induced momenta are

3ala2E2 2 3ala2E2 Fn = R4 . (3 cos () - 1), Fr = R4 . sin2(}

For simplicity, below we neglect the tangential force for small angles (). Then this angle does not vary during particle collision, and repeating the operations of


the previous Problem, for the time of particle approach if at the beginning the distance between them is R we obtain

21(TjR5ref r- ~--~~--~--~

- 5ala2E2(3 cos2 () - 1)'

where the effective radius of the particles is ref = (l/rl + l/r2)-I. For determination of an average association time we take a volume around the

particle in the center of mass frame of reference in such a way that the time of approach of the center is the same for all points of the surface. Then if Ro is the distance from this surface for () = 0, the distance for other angles cos() > 1/..(3 is equal to Ro(3 cos2 () - 1)1/5. This elemental volume is given by

R. 1

V = f R2dR f 21(dcos()· [(3cos2 () - 1)/2]3/5 = 0.518R~ o 1/.J3

Thus, averaging over the initial relative positions of the particles, we get

_ 2.82Wef r= ,

ala2E2N5/3

and the effective rate constant of the process is

k . - _1 __ 0 3 4ala2E2N2/3 md - _ - . 5 (5-31)

Nr Tjref

Problem 5.27. Compare the rate constants of association of two neutral spherical particles as a result of diffusion motion in a buffer gas and due to the interaction of induced dipole momenta under the action of an external electric field.

Comparing formulas (5-28) and (5-31) for identical radii of particles rl = r2, we get for small metallic particles (al = a2 = r?)

k. E2 N 2/3r5 md = 0.27 I

kdif T

i.e., association in an external electric field due to interaction of induced electric momenta is preferable at high electric field strengths and large number densities of particles. In particular, let us make numerical estimations for water drops in the atmospheric air. Take a high degree of humidity so that the water content is 0.1 g / g (gram of water per gram of air), which corresponds to the density of water drops under normal conditions 1.3 . 10-4 g / em3 , or it corresponds to the value of the parameter N r? = 3 . lO-5. Then from the above formula it follows that the above ratio of rate constants equals to unity at the electric field strength which is given by the relation

( rl )3/2 E=Eo -

a

If we take the parameter a = IlL, we get Eo = 3.8k V / em. As it is seen, this mechanism requires high values of the electric field strength.


Problem 5.2S. Show that the association of a small spherical particle with a chain aggregate in an external electric field proceeds near the poles of the aggregate.

According to formula (2-13) and Fig. 2.1, a spherical particle is attracted to a cylindrical one if it is located near its poles, and repulsion takes place if the spherical particle approaches the trunk of the cylindrical particle. Hence, the growth of the cylindrical particle resulting from the joining of spherical particles proceeds near their poles, i.e., it leads to an increase in the length of the cylindrical particle. Note that this takes place only for a large length of the cylindrical particle, so that its axis is directed parallel to the external electric field.

Problem 5.29. Determine the rate of association of a small spherical particle with a large chain aggregate as a result of interaction of induced charges in an uniform electric field.

Let us model the chain aggregate by a cylindrical metallic particle. If it is sufficiently large, the axis of this cylindrical particle is directed along the electric field (see Problem 2.9). Then the induced electric charge per unit length of the aggregate is a(z) = Cz, where the z axis is directed along the aggregate axis, and the origin is located at the middle of the aggregate. Taking the aggregate length to be

/

2/, we have -I S z S I. Hence, the aggregate dipole moment is D2 = J Cz2dz, -/

i.e., C = 3D2/(2z3) = 3a2E 1(213), where the aggregate polarizability a2 along its axis was found in Problem 2.6 and is given bya2 = 13 1[3ln(llr»), and r is the aggregate radius. The interaction potential of a charge e and dipole moment D is given by U(R) = eDol R2, where R is the distance between these objects, and 0 is the unit vector along R. We have R = J (z - Z,)2 + p2, where z' is the charge coordinate along the polar axis, and z, p are coordinates of the dipole. From this we obtain for the interaction potential of the induced charges of the cylindrical particle if the induced dipole moment of the spherical particle is determined according to the formulas of Problem 2.6 that

/ /

f D]o 3D2 f D]o U(R) = Czdz R2 = 2[3 zdz R2

-/ -/

In particular, in the limiting case R » 1 it follows that the interaction poteotial of the two dipole momenta is given by

( D]D2 - 3(D]0)(D20) U R) = ---...,---R3

Let us consider the other limiting case 1 » R in the attraction region near the end of a cylindrical particle, i.e., z > I. Then we get the following expression for the interaction potential:

/ /

U(R) = 32~2 f z' dz'(D]o)1 R2 = 3~~~2 f z'(z' - z)dz'l R3 -/ -/


In this limiting case the interaction potential is detennined by the region of the cylindrical particle near its end. This allows us to replace z' under the integral by l, so that we get

U(R ) = _ 3D)D2 o 2l2Ro '

where Ro = J Z2 + p2 is the distance of the spherical particle from the end of the cylindrical one, and this fonnula is valid for z > l. From this for the force that acts on the spherical particle from the cylindrical one we get

3D)D2 3£r)£r2E2 r3lE2 F=----D=- 0=- 0

2[2 R~ 2[2 R~ 2R~ In(l I r)

Now we use the method of Problem 5.25. Take the cylindrical particle.to be motionless. Then the velocity v of the motion of the spherical particle to the end of the cylindrical particle follows from the equation

r3lE2

F = 6nrv1/ = - 2R~ In(llr) '

so that for the velocity of a spherical particle we have

dRo F r2lE2 v=-=--=

dt 6nl1r 12n11R~ In(llr)

Solution of this equation gives

4n11R~ In(llr) r = ---,-:-:'--:-'-'--'-

r2lE2

for the time of attachment of the spherical particle to the end of the cylindrical particle if the initial distance from this particle is Ro. Averaging this time, as to Problem 5.25, by using the probability of a given distance of the nearest spherical particle from the end of the cylindrical one, we get

_ 311 In(l/r) r = ---:-~-'-

r2lE2 N '

where N is the number density of spherical particles. This expression is valid if the following criterion is fulfilled:

Let us compare this with the result of Problem 5.26, where we get for the mean time of joining of two spherical particles of identical radii r the expression T = 1.41/ l(r5 E2 N 5f 3), and the criterion that spherical particles attach to cylindrical ones under these conditions has the fonn

N5f3r 3 (I) Ncy/ » -l- In -;. ,

5.4 Kinetics of Association of Small Particles 169

where N, Ncyl are the number densities of spherical and cylindrical particles. Since spherical particles form a gas in a buffer gas, we have N r3 « 1. One can see that at the corresponding number density of cylindrical particles the rate of attachment of a test spherical particle to the ends of cylindrical particles exceeds remarkably that of its association with a spherical particle.

Let us estimate the rate of the growth of a cylindrical particle in a gas of spherical ones. From the equationdl/dt '" r /fwe estimate the typical time.1 of the growth of a cylindrical particle with the accuracy up to logarithmic terms to be

TJ ·1 '" r3 E2N

Note that this does not depend on the length I of the cylindrical particle.

5.4 Kinetics of Association of Small Particles

Along with processes (5-4) of atom attachment and cluster evaporation, which govern the size evolution of clusters in the parent vapor, the coagulation process of cluster association can be of importance for the kinetics of cluster growth. This process proceeds according to the scheme

(5-32)

and results in the joining of colliding clusters. A reversible process with respect to (5-32) that leads to fragmentation ofa cluster is hampered if the cluster is located in a gas, because this process requires high energy. Hence, the irreversible process (5-32) takes place in a gas. This leads to cluster growth and decrease of the cluster number density.

In the case of joining of two clusters, the binding energy of atoms in the forming cluster exceeds the total initial binding energy of the atoms in the clusters. Hence, heat extraction results from the process (5-32). The energy excess of this process can be consumed in the release of atoms, i.e., the coagulation process is accompanied by the formation of free atoms. In addition, this heat release can influence the heat regime of the cluster growth process. In particular, the standard method of generation of cluster beams consists in a free jet expansion of an atomic vapor. As a result of vapor expansion, the gaseous temperature decreases, and the vapor becomes supersaturated. Then the vapor is transformed into clusters, and an atomic beam is converted into a cluster beam. In this process the energy of gas compression is used for its cooling in the course of expansion. But the energy of gas compression per atom usually does not exceed the thermal atomic energy, while the binding energy of cluster atoms per one atom exceeds remarkably the typical thermal energy of an atom. Hence, the process of formation and growth of clusters as a result of vapor expansion influences the heat regime of the expansion process. In particular, the total transformation of atoms of an expanding vapor into clusters is impossible without a buffer gas.


Problem 5.30. Determine the growth of the mean particle radius in a system of liquid small particles that are joined in more large drops as a result of their contact. Assume the diffusion character of motion of particles in a buffer gas and the typical size of particles to be larger than the mean free path of gaseous atoms.

The process of particle growth proceeds according to the scheme (5-32) and leads to the joining of colliding particles. Use formula (5-28) for the rate constant of joining of two liquid particles in a buffer gas in the case of the diffusion character of the particle motion in a buffer gas. Because of a weak dependence of the rate constant on the particle radii, below we assume the rate constant of the joining of the particles kdiJ to be independent of the radii of the particles.

Let us introduce the number density fn of particles consisting of n atoms. Assume the buffer gas to be motionless. Then we have that the total number density of particle atoms No = .En nfn is conserved during the evolution ofliquid particles. We introduce the average size of particles:

.En2 fn I n= n = -N Ln2 fn

.Enfn 0 n (5-33)

n

If the process of particle growth proceeds according to the scheme (5-32), the kinetic equation for the distribution function fn of particles on the number of atoms has the form:

and the factor 1/2 accounting for the process of collision of particles consisting of m and n - m atoms is taken into account twice. Considering large particles n » 1, it is convenient to replace the summation by integration, so that the kinetic equation takes the form

(5-34)

The kinetic equation (5-34) can be used for the analysis of particle evolution. In particular, multiplying this equation by n and integrating over n, we obtain

00 00 00 00 n

:t f nfndn = -kdiJ f nfndn f fm dm + ~kdiJ f ndn f fn-mfm dm = 0 o 0 0 0 0

00

This means that the total number density of bound atoms No = J nfndn is cono

served during particle growth. Let us multiply the kinetic equation by n2 and


integrate over dn. Then we obtain 00 00 00

d fan! 2 f dt nfndn = dt = -kdif n fndn fm dm o 0 0

00 n

+ ~kdif f n2dn f fn-mfmdm = kdifNo o 0

From this it follows that the average particle size varies in time by the law

n = kdifNot, (5-35)

if we assume at the beginning the average particle size to be small.

Problem 5.31. Under the conditions of the previous Problem, find the size distribution jUnction of liquid particles.

For this goal we solve the kinetic equation (5-34). Let us introduce the concentration of particles of a given size Cn = fnNo, where No is the total number density of bound atoms in particles. The normalization condition for the particle concentrations has the form L nCn = 1, and the kinetic equation (5-34) is

n

00 n

aCn f 1 f ~ = -Cn cmdm + "2 cn-mcmdm (5-36)

o 0

where the reduced time is r = Nokdift. Let us take the solution of this equation in the form

C = ~ exp (_ 2n) n n2 n' (5-37)

00

This form of Cn satisfies the normalization condition J ncndn = 1 and gives the o

average value of the cluster size n in accordance with (5-35). Then for the left-hand side of equation (5-36) we have

aCn 2cn dn 2ncn dn -=---+-ar n dr nZ dr

The right-hand side of equation (5-36) gives

-2cn/n + 2ncn/n2

One can see that (5-37) is the solution of equation (5-36) ifn = r in accordance with the result of the previous Problem, i.e.,

4 2n Cn = -exp(--) r2 r

Evidently, the solution (5-37) is valid for large r if the initial size distribution of the particles is not essential.


Problem 5.32. A buffer gas containing liquid small particles expands in a vacuum. Taking into account nucleation of small particles according to the scheme (5-32), determine the mean size of small particles at the end of the process. Assume the diffosion character of the motion of particles in a buffer gas, and take the typical size of particles to be larger than the mean free path of gaseous atoms.

Under these conditions, the nonnalization condition takes the fonn L nfn = n

No exp( -t /rex ), where rex is the typical expansion time. Correspondingly, the kinetic equation (5-36) is transfonned to the fonn:

(5-38)

The character of the process is the following. In the course of expansion the rates of collision of particles decrease, and at the end of the process the buffer gas becomes to be rare, so that collisions of particles cease. From this one can estimate the mean size of the particles at the end of the process to be fi 'V Nokdifrex.

For a more accurate detennination of the mean particle size at the end of the expansion process let us multiply equation (5-38) by n2 and integrate the result over dn. Using the nonnalization condition L nCn = exp( -t /rex ), we obtain

according to (5-33) n

00 f n2cndn = fie- tlr"

o

Then the above operations lead to the equation

dfi - = N kd·fe-tlr" dt 0 I ,

The solution of this equation for t --* 00 is

(5-39)

Problem 5.33. Within the framework of the liquid drop model, write the kinetic equation for the cluster growth as a result of processes (5-32) in a motionless buffer gas if the typical cluster size is small compared to the mean free path of gaseous atoms. Determine the character of growth of the average cluster size.

The rate constant of the processes (5-31) is given by fonnula (5-27),


where ko is given by formula (5-8) and does not depend on the sizes of the colliding clusters. Correspondingly, instead of equation (5-38) we have in this case

00 n

oCn / 1 / a:; = -Nocn k(n, m)cmdm + 2No k(n - m, m)cn-mcmdm, (5-40)

o 0

where No is the total number density of the atoms inside the clusters. Multiplying equation (5-40) by n and integrating over dn, we get for a motionless buffer gas

00 00 00 d / /(n+m) / dt ncndn = -No 2 dn dmk(n, m)cncm

o 0 0 00 n

+ ~No / dn / nk(n - m, m)cn-mcmdm o 0

Here we write the first term of the right-hand side of the equation in symmetric form. Introducing the variables x = n, y = m in the first term of the right-hand side of the equation and the variables x = n - m, y = m in the second term, we obtain

00 which gives f ncndn = 1.

o

00 ~/nCndn = 0, dt

o

Multiplication of equation (5-40) by n2 and integration over dn leads to the following relation:

d /00 2 /00 (n2 + m2) /00 dt n cndn = -No 2 dn dmk(n, m)cncm

o 0 0 00 n

+ ~No / dn / n2k(n - m, m)cn-mcmdm, o 0

Using the variables x = n, Y = m in the first term of the right-hand side of the equation and variables x = n - m, Y = m in the second term, we transform it to the form


00

Above we use the relation for a motionless gas f ncndn = I. Because of the o

weak dependence k(x, y), one can use in the right-hand side of the relation the expressions (5-37) for cx , cy of Problem 5.30 that corresponds to the case k = const. Then we obtain

where

00 00 /x+Y J = 16 f xdx f ydy exp(-2x - 2y)(x 1/3 + yl/3)2 V ~ = 5.54

o 0

From this it follows the law of the cluster growth:

n = 6.3(koNot)1.2 (5-41)

Problem 5.34. Under the conditions of the previous Problem, determine the average size of clusters whose growth proceeds according to the scheme (5-32) and is accompanied by expansion of a buffer gas with clusters in a vacuum. Assume the typical cluster size to be small compared to the meanfree path of gaseous atoms.

In this case we obtain instead of the kinetic equation (5-40)

00 n

aCn Cn f I f - = ---Nocn k(n,m)cmdm+-No k(n-m,m)cn_mcmdm, at rex 2 (5-42)

o 0

where the first term takes into account the expansion of the buffer gas with the clusters. Repeating the operations of the previous Problem, we transform it to integral form. Using the result of the previous Problem, we get

dn - = koNonl/6Je-I / Tex

dt

Taking J = 5.54, from this we obtain the average size of the clusters at the end of the expansion process:

n = 6.3(ko No rex )1.2 (5-43)

This corresponds to formula (5-41) of the previous Problem.

Problem 5.35. Determine the number of released atoms resulting from joining two large clusters.

Take the binding energy of a large cluster consisting of n atoms in accordance with formula (3-15) E = eon - An2/3. As it is seen, the process (5-32) leads to the energy release

(5-44)

5.4 Kinetics of Association of SmaIl Particles 175

In particular, for m = n/2, when this function of m at a given n has maximum, formula (5-44) gives

!l.Emax = O.25An2/3

As it is seen, this released energy is created by the surface cluster energy. Hence, it is small compared to the total binding energy of the cluster atoms, but it can exceed the binding energy of one surface atom.

Release of this energy leads to an increase in the temperature of the formed cluster compared to that of the joining clusters, which can cause the release of surface atoms. Note that an excited cluster can emit only one atom, because evaporation of molecules and fragments is characterized by a small probability. Hence, an excited cluster resulting from the joining of two clusters can subsequently emit several atoms, and we have a more complicated process compared to (5-32), which proceeds according to the scheme

An-m + Am -+ An- q + qA, (5-45)

where A is an atom. Assume that in the end of the process (5-45) the temperature of a formed cluster

is equal to the temperature of colliding clusters. This means that the released energy is spent on the liberation of atoms, so that the number of released atoms is given by

q = !l.E (dE/dn)-1 ,

where dE/dn is the binding energy of the surface atoms. In the case ofa large cluster we take dE/dn = Eo - 2A/(3nl/3) ~ Eo, so that the maximum number of released atoms as a result of the formation of a cluster containing n atoms is given by

qmax = 0.25An2/3/Eo (5-46)

In particular, for a large fcc-cluster with a short-range interaction of atoms we have qmax = 0.32n2/3, for a large icosahedral cluster with a short-range interaction of atoms this formula gives qmax = O.28n2/3, and for a large icosahedral cluster with the Lennard-Jones interaction of atoms it follows from this formula that qmax = 0.40n2/3.

Thus, this effect of cluster heating resulting from th joining of clusters can be responsible for the formation of free atoms in an expanding nucleating vapor, but the number of released atoms is small compared to the number of atoms in a formed cluster.

Problem 5.36. Determine the variation of temperature of an expanding buffer gas with an admixture of a nucleating vapor if this gas expands through a jet.

A jet expansion is an adiabatic process that is accompanied by transitions of energy between different degrees of freedom of an expanding gas. For the analysis of this process let us extract a gas volume V in which are located nb atoms of a


buffer gas and nv atoms of a vapor, and these atoms are initially free. The variation of the total energy of this volume is

dE =dQ+ pdV,

where Q is the thennal energy of particles in this volume, and p is the gaseous pressure. Assuming the concentration of vapor atoms in the buffer gas to be small, we neglect the thermal energy of vapor atoms. Then the variation of the thermal energy is

3 dQ = -nbdT - L Ekdnkt

2 k

where dT is the temperature variation, nk is the number of clusters consisting of k atoms and located in a given volume, and Ek is the total binding energy of the atoms in a cluster containing k atoms. Introducing the number density of a buffer gas Nb and using its definition Nb = nb/V, we obtain pdV = -nbTdNb/Nb, where we use the equation of the gas state p = N b T. On the basis of the adiabatic character of the expansion process dE = 0, we obtain, accounting for the above relations,

dE = dQ + pdV = ~nbdT - nb TdNb - L Ekdnk = ° 2 Nb k

(5-47)

Neglecting the nucleation process in equation (5-47), we get the adiabatic law of expansion of a monatomic gas

Nb ~ T3/2

Let us assume that the flow of a buffer gas conserves cylindrical symmetry during a free jet expansion. Denote the beam radius by R and take into account that the atom flux J = T( R2 N bU is conserved, where the drift velocity of atoms U

is assumed to be independent of the temperature. Then we have Nb/ No = R;/ R2, where No, Ro are the initial values of the beam parameters. Substituting this in equation (5-47), we get

3 -nbdT - 2nbTdR/ R - L Ekdnk = ° 2 k

Now let us assume that initially all vapor atoms are free and that condensation takes place in a narrow temperature range near T*. The solution of this equation under these conditions has the form

( Ro)4/3 (2ee) T = To Ii" exp 3T* (5-48)

where c = nv/nb is the concentration of vapor atoms, and E is the cluster binding energy per atom.

From this it follows that the nucleation process does not influence the character of gas expansion if the criterion


is fulfilled. In the opposite case, the thennal effect of the nucleation process stops this process, so that only a portion of the free atoms can fonn clusters. For this reason, a buffer gas is required for the total condensation of an expanding vapor.

Problem 5.37. Determine the maximum concentration of atoms informed clusters as a result of free expansion of a pure vapor. Assume that the formation of clusters proceeds in a narrow range of temperatures near T*.

For this goal we use equation (5-47), which takes the following fonn under the conditions considered:

3 dN "2 ndT -nT N - LEkdnk = 0,

k

where n is the number of atoms in the expanding volume, and N, T are the number density of atoms and the temperature. Let the initial values of these parameters be No, To, and at the beginning of clusterization let these values be N *, T*. During the first stage of expansion, when clusterization is absent, the vapor state is governed by the equation

3 dN "2dT -TN =0,

so that N "" T3/2 , and at the beginning of clusterization we have

N* = No(T*1 To)3/2

At the next stage of the process dT = 0, and we obtain

dNIN + LEkdndn = ° k

The clusterization process starts at temperature T* when the fonnation of diatomic molecules, nuclei of condensation, is possible. Hence, this temperature can be estimated from the relation N2(T*) "" N, where N2(T*) is the number density of molecules at thennodynamic equilibrium. Let us assume that the clusters fonned are large, so that the binding energy per atom is close to that of bulk, which we denote by So. Then the above equation of the heat balance takes the fonn

3 "2ndT - nTdN IN - sondc = 0,

where c is the concentration of bound atoms, i.e., the ratio of the number of bound atoms to the total number of atoms in a given volume. Let us consider the clusterization process under conditions (5-26a) of Problem 5.19 when the criterion

K N 2-rex « 1

is fulfilled. Here K is the rate constant of the three body process 3A ~ A2 + A, and -rex is the expansion time. If the clusterization process is limited by heat release, it proceeds at constant temperature and leads to the maximum concentration of bound atoms at the end of the process, which according to the solution of this


equation is given by

where N f is the final number density of free atoms when the clusterization process finishes. According to the formulas of Problem 5.20, this follows from the relation

koNfTexn2j:X "-' 1,

where nmax is a typical number of cluster atoms at the end of the process, which is given by

nmax "-' (koN*Tex )3 "-' (koNoTex)3(T*/To)9/2

Thus the maximum concentration of bound atoms at the end of the expansion process is given by

with accuracy up to numerical factor under logarithm. In reality T* / £0 "-' 0.1, so that the real maximum degree of clusterization in a pure vapor is of order 10% .

Problem 5.38. Compare the rates of cluster growth on the basis of processes (5-4) and (5-32).

The case where the cluster growth is determined by the attachment of free atoms to nuclei of condensation and where a pair process of cluster growth starts from diatomic molecules is considered in Problem 5.20. Then the average cluster size is

(5-49a)

where N is the initial number density of nucleating atoms, and this result is valid under the condition N » No, where No is the total number density of bound atoms in clusters. In the other limiting case, where the cluster growth is determined by the cluster coagulation (5-32), the average cluster size is given by formula (5-43),

(5-49b)

where No is the initial total number density of bound atoms. This formula is valid if the total concentration of bound atoms does not vary in the course of the expansion of gases, i.e., No » N. Though these formulas correspond to different regimes of expansion, one can see from comparison of the corresponding values Ii that formula (5-49a) is valid if the expansion time Tex is small compared to the typical condensation time, i.e., the time during which all the vapor can be transformed into clusters. Formula (5-49b) corresponds to the opposite relation between these times. Thus, in the first case we have N » No, and the cluster growth is determined by the process (5-4). In the opposite case, if the typical condensation time is smaller than the expansion time, we have N « No

5.5 Growth Processes in Gases Involving Fractal Systems 179

during a typical time of cluster growth that proceeds according to the processes (5-32).

5.5 Growth Processes in Gases Involving Fractal Systems

Fractal aggregates are porous systems consisting of elemental compact particles. Bounds between elemental particles are formed in the areas of contact. Pores inside fractal aggregates are determined by the character of their formation, and the relative volume of pores increases with an increase in aggregate size. For simplicity, we consider a fractal aggregate consisting of identical elemental particles of radius a and mass mo' Characterizing a fractal aggregate by its radius R and mass M, we have the following connection of these parameters with the parameters of elemental particles:

(5-50)

where the parameter fJ is called the fractal dimensionality of the aggregate. The fractal dimensionality is the principal characteristic of a fractal system. For

three dimensional space we have f3 < 3, i.e., fractal systems are not compact. The fractal dimensionality is determined by the character of the aggregate's growth. Below we consider two mechanisms of this process that corresponds to different growth models.

Diffusion limited aggregation (DLA) takes place when elemental particles with a diffusion character of their motion in a buffer gas attach to a motionless aggregate. The fractal dimensionality of this aggregate is given by f3 = 2.45 for three dimensional space (Witten T.A. and Sander L.M. Phys.Rev.Lett. 47, 1400,1981). Cluster-cluster diffusion limited aggregation (CCA) takes place when aggregates join to form a more larger aggregate as a result of their contact and the motion of the aggregates in the buffer gas has diffusion character. The fractal dimensionality for the CCA mechanism of aggregation in three dimensional space is f3 = 1.77 (Meakin P. Phys. Rev. Lett. 51, 1119, 1983)

Fractal aggregates are rare systems of a low density, and their density decreases with increase of the aggregate size. This determines the specifics of the growth of aggregates from the elemental particles located in the buffer gas.

Problem 5.39. Small spherical particles of radius a are associated on the surface of an aggregate, and this process is accompanied by diffUsion motion of elemental particles in a buffer gas. The probability of a particle sticking as a result of contact with the aggregate surface is unity. Determine the rate of the aggregate growth.


According to fonnula (5-50), we have for the number of small elemental particles contained in the fractal aggregate of a radius R

(5-51)

The rate of the aggregate growth is given by the Smoluchowski fonnula (1-24):

dn - = 4rrRD, dt

where D is the diffusion coefficient of the aggregate in the buffer gas, which is expressed through the aggregate radius by fonnula (1-16). From the above relations it follows that

( f3 1 )11<t1-1) R = R~-I + 4rr D-j-afJ t ,

where Ro is the initial aggregate radius. This fonnula gives in the limit of large times R '" t 1/(fJ-1». A typical value of the fractal dimensionality for this process in the three-dimensional case is f3 = 2.45, so that the law of the growth of the large aggregate is R '" to.69 for the DLA regime of the aggregation process.

Note that an aggregate of radius R collects particles during its growth from a gaseous ball of radius

From this it follows that the number density of small particles in a gas drops if

R' = (_3_)1 /3 R(fJ-3)/3a-fJ/3 > I R 4rrN

This proceeds up to aggregate radius

R = (_3_) 1/(3-fJ) a-fJ/(3-fJ) c 4rr N '

(5-52)

when the atom number density in the fractal aggregate coincides with the number density of free atoms in collects particles gas. Here Rc is the correlation radius of this system. In particular, for the case of diffusion limited aggregation f3 = 2.45 this fonnula has the fonn

Rc = 0.074N-1.82a -4.45

At larger radii the density of small particles in the aggregate becomes lower than that in a gas.

Problem 5.40. This mechanism of the growth of fractal aggregates consists in joining of fractal aggregates during their collisions (CCA-mechanism - clustercluster diffusion limited aggregation). Assuming the diffusion character of motion offractal aggregates in a gas, determine a typical time of the aggregate growth up to


a size when aggregates consist of n particles on average. The fractal dimensionality of the forming aggregates is fJ.

Using the rate constant (5-28) for the aggregation process, assume sizes of joining aggregates to be close, so that the rate constant (5-28) of aggregate association does not depend on the radii of colliding aggregates:

8T kdif =-,

371

where 71 is the viscosity coefficient of the buffer gas. Then the evolution of the aggregate radius is analogous to that of Problem 5.30, and the average number of particles in a forming fractal aggregate is determined by formula (5-35):

n = kdifNot.

where No is the initial number density of small particles constituting the fractal aggregate.

On the basis of formula (5-51), from this for the evolution of the mean radius of a fractal aggregate we have

( 8T )1/fJ R = a ""3ri Not (5-53)

For the aggregation of solid particles in a dense gas that corresponds to the CCA aggregation mechanism and is characterized by the fractal dimensionality fJ = 1.7 - 1.8, formula (5-53) gives

(8T )O.57±O.02

R =a -Not 371

Problem 5.41. Small spherical particles join in fractal aggregates on the basis of the CCA-mechanism. These particles are located in a buffer gas, and their initial number density is No At the end of the process all the particles form a bound system similar to an aerogel that possesses all the gaseous volume. Find the time of formation of this system.

The aggregate growth finishes when the number density of particles in the fractal aggregate coincides with the initial number density No of small particles in the buffer gas. Then the system of free small particles in the buffer gas is transformed into an aerogel- a bulk system of bound particles. Fig. 5.3 shows an element of an aerogel as a porous bulk system of bound particles. According to formula (5-52), the aggregate radius that corresponds to formation of an aerogel is

R = -- a-fJ /(3-fJ) ( 3 )1/(3-fJ )

c 41rN •

and the typical time of transition from free particles in a buffer gas to an aerogel through formation of fractal aggregates is given by

371 ( 3 ) 1/(3-fJ) _ 4-~ t = - - (N afJ) 3-~

8T 41r 0 (5-54)


I 10nm I

FIGURE 5.3. The internal structure of an aerogel consisting of small silica particles.

Taking f3 = 1.77, a typical fractal dimensionality for fractal aggregates resulting from the CCA mechanism of this process, we transform formula (5-54) into the form

3'1 ( 3 )0.75 (1.77)-1.75 0.13'1 t = - - Noa = ---:-:::::-'-:0-::-:-

8T 4;rr T N;·75a I.01 (5-55)

Note the peculiarities of the formed system of bound particles, an aerogel. It is uniform for large sizes R » Rc and has a fractal structure for R « Rc. This means that if we cut off a sphere of a radius R from the aerogel, it will have fractal properties when R « Rc. The value Rc is called the correlation radius of the aerogel.

Problem 5.42. The growth of a fractal aggregate results from its fall under influence of gravity in a buffer gas containing small particles that attach to the fractal aggregate. Find the typical time of transformation of the fractal aggregate into an aerogel.

Consider the character of the growth of the falling aggregate when the gravitational force determines its motion. The aggregate growth proceeds due to two mechanisms. First, a falling aggregate collects small particles that are located in its path. Second, particles go to the surface of the falling aggregate as a result of diffusion in the buffer gas and attach to the aggregate. Then the balance equation for the number of particles n constituting the aggregate is given by the equation

dn dt = VI + V2,

where VI, V2 are the rates of the corresponding mechanism of aggregation. The rate of the first process is

VI = ;rr R2 vN,

where R is the aggregate radius, v is the velocity of gravitational fall, and N is the number density of particles in the buffer gas. The velocity of the falling


aggregate can be determined by analogy with formula (1-18) from the equality of the gravitational force P = mgn of the aggregate and the resistive force F = 61f RI1 v, where m is the mass of an individual particle, g is the free fall acceleration, n is the number of elemental particles of the aggregate, and 11 is the viscosity coefficient of the buffer gas. Note that for simplicity we consider the aggregate as a spherical symmetric system, while this process of aggregation stretches it in the vertical direction. From this it follows that for the drop velocity we have

mgRP-I v - ~--;;-- 61fT}aP •

so that the rate of this process is given by

mgRP+I N mgNan1+I/P VI = =

611aP 611

The rate of the second aggregation process is given by the Smoluchowski formula (1-24)

V2 = 41fDRN = 41fDNanl / P•

where D is the diffusion coefficient of individual particles in the buffer gas. Thus, the balance equation for the number of particles of the aggregate has the form

dn - = mgNan1+1/P /(6T}) + 41f DNanl / P = Nk l n1+I/P + Nk2n1/P• dt

where kl = mga/(611), k2 = 41f Da. As it is seen, the aggregation process accelerates when the first mechanism of aggregation becomes the principal one. This means a finite time of transformation of the aggregate in the aerogel which is given by

(5-56)

I-! ! The main contribution to the integral over time is determined by N kl ~ k; sin j

aggregate sizes n '" kd k l • We suppose that the number of particles in the aerogel formed exceeds remarkably this value. For a typical fractal dimensionality of this aggregate fJ = 2.45 this formula takes the form

3.3 t = --;:-:,:-;:-=

NkY·59kg.41 (5-57)

Problem 5.43. The character offormation of an aerogel in a buffer gas containing small aggregating particles consists in the joining of fractal aggregates as a result of their collision and their fall in a gravitational field. Determine the time of aerogel formation.

This process is analogous to that of the previous Problem, so that

VI = 1f (RI + Rd IVI - v21 Na• V2 = 41f(DI + D2)(RI + R2)Na•


where N a is the number density of aggregates at a given time, and other subscripts mean the number of colliding aggregates. Let us use the simple size distribution function of aggregates f ....., exp( -n/n), where n is the average aggregate size. Averaging over aggregate sizes on the basis of this distribution function, we get the balance equation for the mean number of aggregate particles:

dn -I+I/tl - = 1.4Nokin + (1l + 2)Nok2• dt

where No = nNa is the initial number density of particles. kl = mga/(6T}), and k2 = 2T /(3T}).

From this for the time of aerogel formation we get

1l{3

= [(,8 + l)No sin tl:1 . (1.4kJ)tlI<.8+1)[(1l + 2)k2]i/(tl+I)]

For a typical fractal dimensionality for the CCA mechanism of aggregation we take {3 = 1.77, so that this formula takes the form

_ N-Ik-o.64k-o.36 t - 0 I 2 (5-58)

Problem 5.44. Determine the rate of cluster-cluster aggregation in a uniform electric field due to the interaction of induced dipole momenta of joining aggregates. Compare it with the rate of the growth of fractal aggregates according to cluster-cluster aggregation in the absence of an electric field.

We assume fractal aggregates to be similar to spherical metallic particles. Then the effective rate constant of their joining is given by formula (5-30):

(XI (X2 E2 N2/3 kef ....., TJR •

where (XI, (X2 are the polarizabilities of joining aggregates, E is the electric field strength, N is the number density of aggregates, and TJ is the viscosity coefficient of the buffer gas in which the process proceeds. Considering aggregates as metallic particles, we have (XI ....., (X2 ....., R3, where R is the typical radius of aggregates at a given time.

Let us use the definition of the effective rate constant of the process

dn dt = kefNn.

where n is the average number of elemental particles constituting a fractal aggregate. We have N ....., No/n, where No is the total number density of elemental particles in aggregates. On the basis offormula (5-51) we obtain

~: ....., R5 E2 N;/3 /TJ ....., a5 E2 N;!3n 5/ tl - 2/3 /T}.


where a is a typical radius of an elemental particle, and f3 is the fractal dimensionality of the aggregates.

Comparing this with the rate of cluster-cluster aggregation in the absence of electric fields (problem 5-37), we obtain for the ratio of typical times of these processes

iCCA E2 N 2/ 3 5 5/fJ-2/3 --"-'- 0 an ,

iel T (5-59)

where iel is the typical time of aggregation resulting from interaction of induced dipole momenta, iCCA is a typical time of diffusion limited aggregation in the absence of electric fields (Problem 5.40). As it is seen, aggregate growth due to interaction of induced dipole momenta is essential for large aggregates. In particular, taking the fractal dimensionality of an aggregate f3 = 1.77, we have the following criterion for the mechanism of aggregation due to interaction of induced dipole momenta to determine the rate of the aggregate growth:

TI/2 E » (5-60)

N;/3 a5/2n 1.1

Problem 5.45. Determine the rate of the growth of a fractal fiber by means of attachment of fractal aggregates to its ends as a result of interaction of induced dipole momenta of joining aggregates in a uniform electric field.

The aggregation process proceeds according to the scheme of the previous Problem until an induced dipole moment of the forming aggregate D is relatively small, i.e., DE « T, where E is the electric field strength, and T is a typical thermal energy. Under this condition, a forming fractal aggregate can be considered spherical. If the opposite criterion is fulfilled, a forming fractal aggregate resembles a cylindrical particle whose axis is directed along the electric field. We call this particle a fractal fiber (see Fig. 0.3). In this case spherical aggregates attach to a fractal fiber near its ends, as considered in Problems 5.28 and 5.29.

In calculating the effective rate constant of the process of the growth of a fractal fiber we note an analogy of this problem with Problem 5.29. Thus, one can use the result of Problem 5.29 for determining the effective time of attachment of a spherical particle to a cylindrical one. In this case a typical time il of the growth of a fractal fiber as a result of attachment of fractal aggregates to the ends of the fractal fiber follows from Problem 5.29:

il "-' 17/(r3E2N)

Note one more peculiarity of the growth of a fractal fiber compared to a continuous cylindrical particle. We assume that the conductivity of a former fiber is great enough that the electric potential of its surface is constant over the fiber when it is located in an external electric field. But the formation of fast bonds between fractal aggregates inside a fractal fiber requires time, so that this scenario can be valid only for a slow process.

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