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Modelling of water clusters nucleations with the Monte-Carlo methods

Yu.I.Khlopkov1,2, B.V.Egorov1, A.Yu.Khlopkov2*, Zay Yar Myo Myint2

1Central Aerohydrodynamic Institute, (RUSSIA)2Moscow Institute of Physic and Technology (State University), (RUSSIA)

E-mail : [email protected]

Clusters;Monte-Carlo method;

Thermodynamics of naturalgas;

Quantum views;Molecular collisions.

KEYWORDSABSTRACT

Modeling and research of clusters structure will give mankind an alternativeenergy source, creation of chips in the size in some nanometer, devices ofmemory with superdense record of the information, integrated quantumschemes, will allow to find out their influence on an environment. The mostdifficult case is to simulate the collision between clusters. The purpose ofthis paper is to present the idea of method to simulate the collision betweenclusters. 2014 Trade Science Inc. - INDIA

INTRODUCTION

Formation process of condensation�s initial nucle-uses is one of the most difficult and incompletely deter-minate problems in new phase formation theory. Thesenucleuses are able to have a small size and consist onlyof some molecules (Figure 1), called �clusters� ormicrocluster. Microcluster is the smallest units includedfrom two to hundred and more atoms. Modern quan-tum-chemical methods let to find an inertia moment,frequencies of intermolecular and intermolecular oscil-lations in the clusters and their dissociation energy. Onthe basis of these calculations it�s possible to calculateall the thermodynamic and also kinetic characteristicsof the clusters. The clustered method underlies in ther-modynamic of natural gas components.

Modeling and research of clusters structure will givemankind an alternative energy source, creation of chipsin the size in some nanometer, devices of memory withsuperdense record of the information, integrated quan-tum schemes, will allow to find out their influence on an

environment (destruction of an ozone cloud, change ofa climate of Earth).

Clathrates - the structures similar to an ice in whichvarious gases �visitors� contain in the crystal cells gen-erated by molecules of water �owner�, which actuallyalso are clusters. Clathrates structures are formed onthe basis of various substances like football coordina-tion of atoms: gas hydrates � �Clathrates ice�. Moreimportant is hydrate of methane. Stocks of power re-sources on the Earth (coal, gas, oil) will suffice, at mod-ern rates of their use, on different sources, on 50 - 100years. There is a problem of use of alternative sourcesof fuel. To such concern clathrates of methane whichstocks in ground adjournment of the World Ocean andin permafrost are practically inexhaustible and make notless than 250 billion cubic meters. In recalculation ontraditional kinds of fuel it more than twice exceeds quan-tity available on a planet. Special danger is representedwith warming a climate which can start the mechanismof allocation of methane from crystal-hydrate ocean withpernicious consequences for all alive on the Earth.

An Indian Journal

Volume 8 Issue 7

Nano Science and Nano TechnologyNano Science and Nano Technology

NSNTAIJ, 8(7), 2014 [298-305]

ISSN : 0974 - 7494

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A.Yu.Khlopkov et al. 299

Full PaperNSNTAIJ, 8(7) 2014

Nano Science and Nano Technology

An Indian Journal

MONTE-CARLO SIMULATION

The idea of method of clusters dynamics at nucle-ation is follow. We can see that the volume gas consistsof monomers and n-mers clusters. This system is in equi-librium. The most difficult case is to simulate the collisionbetween clusters. Let us consider the problem of nucle-ation of new phase. In the initial moment of time systemconsists only from monomers then we choose a little timeÄt0. During this time molecules-monomers collide eachother. The law of collision depends on the sort of mol-ecules. It may be solid sphere � the most simple case,the pseudo maxwell molecules, Lennard-Jones law andother. If molecules have complex structure it is necessaryto consider internal free degrees change: rotational andvibration energy As we will see later at the processes ofnucleation there is very important the quantum proper-ties of molecules, besides we must know the criteria ofconglutination. At usually it is some threshold energy EpOn next time step Ät we have already a mixture frommonomers and dimmers. Now we must know law of theinteraction between monomers and dimmers. Besides wemust know the threshold energy of conglutination mono-mer-dimmer and dimmer-dimmer. Moreover we mustknow the criteria of jarring. On the n-th time step we willhave a mixture from n kind of clusters from monomer,dimmer to n-mers. And we must have models of interac-tion of different clusters with each other and criteria ofconglutination and jarring of molecules. Below the calcu-

lation of model problem of nucleation of vapour frommonomers to dimmers � n-mers etc. until creation ofnew phase (Figure 2 a,b).

For establishment of low of interaction of differentsubstances it is necessary to consider physics-chemicalproperties of the substance. As the most important prob-lem we consider property of water clusters.

Calculation of the speed of liquid clusters nucle-ation according to a classical theory

Nucleation classical theory is originated as far back

Figure 1

Figure 2a

Figure 2b

Modelling of water clusters nucleations with the Monte-Carlo methods300



An Indian Journal

as 30th and it�s based on two main approximations:Szilard�s model and liquid-drop model. It�s supposedthat oversaturated steam consists of the mixture of asteam of monomers and clusters, contained differentnumber of molecules, in Szilard�s model. Clusters cangrow and evaporate, adding or losing only one mol-ecule at the same time. In the context of this model it�spossible to calculate the compact expression for thepermanent isometric nucleation�s speed.

1 10 * 1

2 2

â

á1

â

ggi

g i i

nJ

(1.1)

in which n1 - monomers� concentration (it�s supposed

that monomers� steam doesn�t become weak in thecondensation process), â

i - speed of molecule joining

to cluster, gi - evaporation speed, g

*- nucleus�s critical

size. Using then liquid drop model, i.e. supposing thatclusters are spherical and similar, one can get the ex-pression for nucleation speed:

322

0 1 2

2ã 16ð ã

ð 3(ln )J an BExp

m S kT

(1.2)

in which a - adhesion factor, v - one liquid moleculevolume, ã- surface tension, S = P/P

- supersaturation,

B - Varnard�s correction.2 / 3

34ð ã

4ðlnB Exp S

kT

(1.3)

that arises at the consecutive kinetic method and is ab-sent on the calculation from classical thermodynami-cally method, e.g. on calculation of work that is neces-sary for nucleus (size g) creation. In the latter case weget Zeldovitch�Frenkel�s formula:

322

0 1 2

2ã 16ð ã

ð 3(ln )J an Exp

m S kT

(1.4)

In classical theory pressure and critical nucleus� sizeare joined by Calvin-Gibbs� formula:

*

2ãln

g

P

P r kT

(1.5)

Or

*

2ãln Cr

g

Sr kT

(1.6)

in which *gr - critical nucleus� size

*

1/ 3

*3

4ðg

gr

(1.7)

From this32 / 3

*

4ð8 ã

3 ñ ln Cr

mg

kT S

(1.8)

For water �3 / 2 3 / 2

1.2 ì ã ì ãlg 0.242

ñ ñlgCrS

T Tk

(1.9)

Surface tension for small-sized water clusters wascalculated to high accuracy in the experiment[1] (Figure

Figure 3 : Surface tensions dependence on cluster�s size




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3). The results were made according to calculation ofcluster�s size with exact coordinates of atoms in cluster.From this it follows that real surface tension of small-sized clusters is 30-50% less than for liquid usual drops.

g* - In this case equals order 3 for T = 230-250 K

(considered temperature), while g*

~ 5 forabovementioned temperatures on the classical theory

Then we can rewrite the formula (1.1) as:

*

1 1

1

2

â

1 óg

gg

nJ

(1.10)

In which

*2

2

( 1) ( ) ( )á

óâ

g

gj j

gj j

g f g f j

ExpkT

(1.11)

Function f(g) is determinate as:

( ) ln gf g kT

(1.12)

For f(g) function was used a dependence calcu-lated with molecular dynamic method in experiment[2].

According to nucleation classical theory the evapo-ration speed of clusters with g molecules is written downas:

ãá

gg g

SS Exp

kT

(1.13)

In which2 / 3

2 / 334ð

4gS g

- surface square of cluster that

consists of g � molecules, v � one molecule�s (mono-mer) volume, and then

2 / 31/ 3

1

2 34ð

3 4ðg g gS S S g

- surface change due

to evaporation of one molecule

EAExp

kT

- evaporation speed from unit of

area of flat limit of division/section, E - evaporation

heat from plane.In nucleation classical theory nucleuses speed

growth âg is expressed through number of collision with

monomers in time unit:

1â2ðg g

kTaS n

m (1.14)

where a � adhesion factor, that is indeterminate valueand can be from 0.01 to 1.

On the basis of abovementioned data nucleationspeed for liquid-drop model with Varnard�s amendment(1.2), (Figure 4), without Varnard�s amendment (1.4),(Figure 5), nucleation speed received on the basis ofexperiment of molecular dynamic at different tempera-tures and g

*, (1.10) (Figure 4, 5). Findings were com-

pared with last experimental data[3].Evidently, classical method for nucleation theory

doesn�t make satisfactory results. Further, quantum-me-chanical method will be used and demonstrated for cal-culation of nucleation speed.

Calculation of water clusters nucleation speed ac-cording to quantum-mechanical theory

Let�s examine a system included gas-phase mol-

Figure 4 : Nucleation speed according to classical theory(liquid-drop model without Varnard�s amendment)

Figure 5 : Nucleation speed according to classical theory(experiment of molecular dynamic, for g

* = 3)




An Indian Journal

ecules À1 of condensed matter (monomers) and mo-

lecular units An (clusters) included two or more mol-

ecules: n 2. Cluster An with dimension n has energy of

dissociation (rupture of connection) En, three moments

of inertia IA(n), I

B(n), I

C (n), and also 3nl-6 oscillation

degrees of freedom, from which (3l � 6)n belong tointramolecular oscillations, and a (6n � 6) � to inter-molecular oscillations (l � number of atoms in mono-mer).

First, let�s examine a pared-down model ofclusterization and suppose that a clusters growth is aresult of joining a single monomer of a molecule-asso-ciation to them. And the clusters� destruction is a resultof intermediate stage � activated complex formation anddecay[4].

Omitting a stage of activated complex and takinginto account only a reaction final product, successionof elementary processes of formation and destructionof clusters A

n can be written down in the summary form:

( )aK n 1 1n nA A M A M

(2.1)

( 1)aK n

Here M � any molecule or gas-phase cluster; Ka(n) È

Kd(n+1) � constants of direct and inverse reactions�

speeds, Ka(n) � a constant of speed of a monomer

joining to a cluster An, and K

d(n+1) � a constant of

speed of a cluster dissociation An+1

to a cluster An and a

monomer.For concentration N

n clusters with A

n form on the

basis of (2.1) one can write down an equation of theirchange with time in the form of[4]:

1n

n n

dNJ J

dt (2.2)

1 1( ) ( 1)n n a n dJ N N MK n N MK n (2.3)

Here Jn � clusters flow in a space of number of mono-

mers in them, e.g. number of clusters passing on in aunit time from n point to n+1 point.

In the terms of thermodynamic balance the speedsof direct and inverse reactions K

a(n), K

d(n+1) are con-

nected through a balance constant:

1 1

1 1

( )

( 1)a n n

n p

d n n

K n N PK kT K kT

K n N N P P

(2.4)

Here Pi =

iP,

i = N

i/N - a partial pressure and molar

part of component i;

1

1

np

n

PK

P P - Balance constant of reaction (2.1), ex-

pressed in partial pressure terms of components ([Kp]

= 1 atm-1).Calculation of a speed constant of cluster dissocia-

tion is based on monomolecular reactions theoryRRKM (of Rise-Ramsperger-Kassel-Marcus)[5], hereyou can see the final result:

1 0

1 2 0 20

( ) exp( / )exp( / )

1 ( ) / [ ]vr

E

vrE

d

aE

P E E kT dELQ E kT

KhQ Q k E E k M

(2.5)

Where,*

0E E E

10 *

01 0

( ) ( )( )

vr

E

a vrE

LQk E E P E

hQ N E E

Here L � number of paths, that are equal in terms ofenergy (for example, L = 2 for dissociation of H

2O to

H and OH), Q1 and Q

1+ - static sums for gyration in

molecule and activated complex accordingly (it wasused Q

1+/Q

1 = 1 in calculations), Q

2 � total (oscillation

and rotary) static sum for dissociate molecule,

( )vr

E

vrE

P E

- number of vibrational and rotary condi-

tion of activated complex in interval of energies [0,E+), N - density of oscillation and rotary reagent lev-els. Condition number of activated complexes anddensity of energy of reagent small-sized moleculeswere calculated by direct converting (for that all thenormal oscillations� frequencies were divided into 9groups). For large clusters Vitten-Rabinovitch�s ana-lytic approximation were used[5]. Cluster K

a(n) for-

mation constant was determinate through dissociationconstant K

d(n+1) and balance constant K

n according

to (2.4).In quasi-equilibrium terms dN

n/dt = 0, so J doesn�t

depend on n, as according to (2.2) Jn = J

n-1 for any n.

In this case equation (2.3) becomes

0 1 1( ) ( 1)n a n dJ N N MK n N MK n (2.6)

Totality of equations (2.6) represented investigatedheterogeneous linear system relatively N

n in balance

terms, when speeds of any direct and inverse processesare balanced, system (2.6) comes to similar equations

0 0 01 10 ( ) ( 1)n a n dN N K n N K n

(2.7)

Here 0nN - quasi-balanced distribution function in the

absence of flow (J0 = 0).




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The solution of (2.7) is:

01

2

( )

( 1)

nn a

nj d

K jN N

K j

(2.8)

Let�s put a degree of steam S supersaturation in thefollowing way:

10

1eq

NS

N (2.9)

Here 01N - numerical density of monomers, 0

1N -balanced numerical density of monomers above theclusters with maximum large size.

11 1

01

( ) 1lim lim ( )

( )d n

leq pn n

a n

K Q QN K n kT

K Q P V

,

63 3/ 2

71 1

1

( )5.6757 10 (1) (1) (1) 1 exp exp ,

ó

disleq a b c

T A EN I I I

T kT

(2.10)

in which Edis

() - separation energy of monomer frommaximum large-sized cluster, �average statistical� dis-criminatory oscillation temperature of cluster with n -

n size is terminated with following correlation:

6 6( )6 6

1

1 1n i

nnn

T T

i

e e

, (2.11)

Quasi-equilibrium distribution (2.8) becomes:

0 0 3 / 2 6( 1)

1 exp ( 1) ( 1) ln 1 exp( / ) nn

EN N n n S n T

kT R V

1 -10 0 3 / 2 6( 1)

2 1

ó

ó

nj j j

j j j j

R V

kT R V

(2.12)

Here - separation energy of monomer from infinitelylarge-sized cluster, � energy of dissociation of dimer, -parameter[4].

In experiment[4] expression for quasi-equilibrium dis-tribution of the clusters with n monomers were got:

0 0 á

1 exp ( 1) ( 1) ln ,n

EN N n n S

kT

(2.13)

it is fully congruent to equation that was got in approxi-mation of liquid-drop model in classical theory of con-densation, if

2 21( 1) ( ) 4ðó( - )dis dis n nE n E r r

and = 2/3,Where r

n, - radius and surface tension.

Equation (2.12) is differing from classical and fromequation (2.13) of the experiment[4] because of the evi-dent calculation of structure and energy characteristics

of intermolecular quantum motions in n-size cluster.Statistic sum of ideal gas with internal freedom de-

grees, consisted of N indistinguishable identical particles� n-sized clusters, is[6]:

1

!N

n nZ QN

So, for one mol of n � sized clusters

3 / 2 4 3 / 2

77 1 11ln( ) ln 1.4949 10 (2 1) exp

ó

nn sn A e

n n

M T n RZ N g S

P V kT

6 6

1 1

( )2ln( ) ln 1.4949 10 (2 1) exp

n

n sn s

kU n

P V kT

(2.14)

where NA - Avogadro constant, M

1 - monomer�s mass,

ge - degeneration of electronic level of cluster, S

1 -

monomer�s nuclear spin.Knowledge of Z

n gives a possibility to calculate all

the necessary thermodynamic functions for one mol ofn -sized clusters[6]. In quasi-equilibrium approximationincrease of Gibbs� free energy can be reduced to thefollowing:

3 / 200 1 1 1, 1

1 1

ó ( )ln exp 1 exp

ó 1n n n dis

n nn n n

P n R V E nG RT S

P n R V kT T

63 / 2

1 1 1

1 1

ó ( )ln exp 1 exp

ó 1n n n dis

n n n

P n R V E n

P n R V kT T

, (2.15)

Here, R � universal gas constant.

Figure 6 : Nucleation speed according to classical theory(experiment of molecular dynamic, for different g

*)




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Figure 7 : Nucleation speed accordingly quantum-mechani-cal theory for different T (217, 239, 244, K)

Solution of dissimilar system (2.6) in quasi-equilib-rium approximation is calculated as following[7]:

0n n nN C N (2.16)

in which - quasi-equilibrium distribution of clusters inaccording to the sizes in the absence of flow (J

0 = 0).

Solution of dissimilar system (2.6) in quasi-station-ary case J

0 0 is converted to:

* 10

0 01 1

11

( )

n

n nj j a

N N JN N K j

(2.17)

Where

*1

0 01 1

1

( )

n

j j a

JN N K j

(2.18)

When constants of clusters association and dissocia-tion are known, quasi- equilibrium solution can be cal-culated and then - using (2.18), number of nucleuses involume unit, passing from n point (n-sized cluster) ton+1 unit (n+1-sized cluster) on in time unit, is calcu-lated for the system with density of monomers � N

1.

As we see in (2.17), numerical density of clustersin stationary state is less than corresponding numericaldensity in quasi-equilibrium state. Decrease of numeri-cal density of clusters in stationary state in comparisonwith quasi-equilibrium state is so much larger than its nsize is larger.

For calculation of J0 a computer program of the

experiment[1] was used. As it�s at the Figure 7, whenusing this quantum-mechanical method, we get a resultthat is very similar to an experimental result. And it con-firms the necessity of using this method.

Further, quantum-mechanical method for thermo-dynamic functions of water clusters will be demon-strated.

CONCLUSIONS

Comparison of two theories � classical and quan-tum-mechanical - for calculation of water nucleationspeed was carried out. The main problem in nucleationspeed determination in classical theory of condensa-tion, as it was discovered in a lot of authors� works,comes to necessity of describing with maximum preci-sion the surface tension , which makes its contributionon the speed. Classical liquid-drop model can�t givethe exact value for small particles, and so a diver-gence with experiment appears. E.g. concept is los-ing meaning for small clusters. Calculations of energiesand heat capacities of different clusters were made atdifferent temperatures and pressures and their contri-bution to the total system�s heat capacity were esti-mated.

As a result we have following: it�s necessary to takeinto account the clusters� contribution into general char-acteristics of the system for the correct description ofthermodynamic functions of a matter, in particular forwater and its steams. In particular, heat capacity ofwater�s monomer greatly differs from water�s heat ca-pacity taking into account dimers�, threemers� and tet-ramers� contribution.

It�s necessary to attract quantum-mechanical per-formances for investigation of clusters� characteristicsand their influence to the environment.

Moreover, development of clusters� theory helps amastering of the technology of mining - potential gen-erator in future.

Besides progress theory clusters help better mas-ter technology gas-hydrate methane production poten-tials source of energy in the future.

The reported study was partially supported byRFBR, research project No. 11-07-00300-a.

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Kinetics forming of ultra light fraction neutral andcharge clasters in gas-dynamics streams of aircraft.Chemical Physics, 23(4), 28-46 (2004).

[2] S.M.Tompson, K.E.Gubbins, J.P.R.B.Walton,R.A.R.Chantry, J.S.Rowlinson; A moleculardynamics study of liquid drops, J.Chem.Phys., 81(1),530-542 (1984).

[3] David J.DiMattio; A dissertation presented to thefaculty of the graduate school of the university ofmissouri-rolla. In partial fulfillment of therequirements for the degree, (1999).

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[5] P.Robinson, K.K.Holbrook; Monomolecularreactions, Moscow: Mir, 380 (1975).

[6] O.N.Poltorak; Thermodimanic in physical chemistry,Moscow: Higher School, 319 (1991).

[7] M.Volmer; Classical Nucleation Theory: Kinetics,Dresden: Steinkopff, 220 (1939).

[8] B.V.Yegorov, Y.E.Markatchev; Dissipative transportcoefficients are clustered gas, Processing of MIPT,Moscow, 44-47 (2009).

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