## Existence of condensed phase in lattice gas model

**R****en**** UK A D****atta**

*Centre o f Adva?iced S t u d y i n A p p l i e d IJa lh e ^n a iic s *

92, *A c h a r y a F r a f u lk i C h a n d ra F o o d , C a lm tfa -lO O O O d . *
*{Received* 6 *A u g m t* 1970, *revised* 27 *Sepiem er* 1970}

**Indian J. Phya.**

**5lA, **

371-381 (1977)
A throe dimonsioiinl Jaithiu gas niadol is cunsidorod, for investigations o f tUermodyiiainic j)i'opoj‘lios of an assembly of ])artioles, eaeli ol tlu^m eomaudiiig a rigid volume of exeliisioii. Dnlta’s Ud tieo model (1953) for gases has been taken as Uie basis, and a(tempts have Ix'on made lo dodue.e the oquatioii of state. The possibility oi'existence of condenscHl ])hase in th(^ lattice gus model has beiai cxa-mii»ed by considering dif»- tributiou of particles in lattice points and by comparing the eonjff c ftBibi- lity ol two extreme cases. For simplicity, short iang<^ forces oi'^cry small range have been considered hoif^.

**1 . ** **I****n****T R O J )U O T IO N**

In 1953 it was shown by Dutta that tFe properties of tJu^ ri‘al gases and th(‘ ()istri- biition of the ions in the strong electrolytes in solutions might b(‘ dednccnl b.y cpiasi lattice theoiy. [n the said <piasi-Iatti(;e theory the partic^b's (mr>lecules, atoms, ions etc,) are distributed in a lattice. The distribution forjunla, deducerl in this way for ions in strong eloetj’olytes in solutions has heim used in the th(;ury of strong oloetrolytos. Later maii}'^ author ((vg. Riudle 1909, h’ish('r 1972 etc.) have worked on tins mod(d and t lu^y^ rcfcj’ the struct ure as lalt ice gas. Ruclle (1969) called the latti(:o gas as a system eonstitut.(ul with hard cores which usually required that the Euclidian dist ance of each pair of part ic^b^s j cmains largcw than some constant (say,7i?) and so the coufigunitioii sj)aee is taken to 3n-dim(MiLBioiuil with lattice points. Fishej- (1972) also has taken the abt^ve stnicturi^ ol latt.ice gas in one dimension.

In the above work. It is found that the lattice ihoojy is conveiiiojit for treating the matter in all the usual states i.(^. solid, liauid gas. Alfemxhs liavo been made to investigate the condensed phases and the phase transitions from the lattice gas model by the metliod of clusters (Temporley 1954, Jkoda 1974) and others (Neumann 1974). Neumann claimed that he s(jlvcd the cooj[)orativ(i problem for a lattice gas on a simple cubic lattice by a system of iwo coupled, transcendental equations, derived by a combinatorial medhod. Actually he used the combinatorial method only tr> calculate the weight function which he had introduced in the partition function,

**371**

**372** Kenuka Batta

Here wo show that by usual combinatorial method followed by maximising the thermodynamic probability and by the use of Boltzmann principle, the matter in gaseous pliase and also in condensed phased, with the simple lattice gas model may be treated. A phase is identified as gaseous jdiase when the compressibility is comparatively high, wJioreas the same is identified as a condensed phase wh^n lh(i compressibility is very low, as good as incompressible. The pliase transitioji has not b(^en considered and is hojKid to be studied in future. Only the existenw^

of gaseous phase and condonsod phase, in llu^ sense explained above, have been demonsti'alod by applying combinatoric method Avith usual calculations.

Case i. *R m l gas i n lattice n u d e t*

Lattice structure is convenient for treating real gas and in this st^ction we have obtained the nature of the gas and have also calculated its compressibilily.

which has been compared with the eomprossibility obtained in Case ]J.

*D e s c rip tio n o f the m odel*

For investigation of certain ju'operties of lattice gases it is essential to (Jonsidei- the distributi(>n of the particles in a Euclidian space under the unilateral restric

tion * ^{2a}* < . It is easy to show that (Ruell 1969), it is practically equi

valent to the distribution of particles in threes dimensional lattice points. In
simple language tihe above distribution is (iquivalent to tlu^ distribution of particles
on the lattice points and this model is goiiora-lly referred to as lattice gas model.'
First Ave divide the whole space in a containei- in three dimensional lattice in such
a Avay that- the Euelidian distance betAvoon tw^o points of the lattice must be
equaL-to oi' greater tlum sonic^ coust-aut-, say *2a* and also our assumption is that
each lattice point may be occupied by only one particle. A pair of particles has
only binding energy * 2 w* if 2a

**<**jat—

**<**2a/ «2-

**"I'I**

**ilcii**

**,**

**aac**assume that the particles of only t wo typos of w^hieh some are single particles and some are paii’

part.icles are distributed among tlte lattice points. 8iuglo particles arc those type
of particles ai‘(>uiid which nearest neighbours arc‘. vacant and so they ImA^c no
potential tuiergy, have oaly kinetic energy. Each oi the particles ol a paj*ticlc-
pair around which nearest eolls are vacant has potential energy * ^{iv}* along with its
kinetic energy. Since foi* a single particle six nearest positions must bo vacant,
the effective exclusion volume for those type ol particles is 6 ^ ~ 3 ^

(.he effective exclusion volume of particle pair is 6^ -^(2a)^j. Lot the elfec-
tive exclusion volume of a particle-pair in prexence oi a single particle and of a
single particle in prosoiico of a piirticle pair be w^hero < 1 and *b i**2* “

*h i+ h n .* These facts are clear from figure 1, Avhich is the intersection of the lattice
structure by a plane passing through central lattice,

**Condensed p h ase in lattice gas model**

*M a th e m a tic a l Calctdatioiiff*

Let the total volume be F, and so the total number of lattice points are

*^* where *b* = *%ra^.* Let the total number of particles bo JV, the number of single
particles be and the number of particle-pairs V)0 munbor
of particles with kinetic energy *e{.* Lot *E* be t he total energy of the system.

*V i f T .* 1. OiHbrilmtion f)f Kiuglo particles anrl partirln-paiTK in latticr poinis.

The thennodynamii- piobal)ili(.y when the sinslo parl.icles ure distrihuted first and then the. particle pairs are distributed is

**H'la - -**
*N ,*

**i l }**

**i l }**

^{{ }

^{l ±}

^{N \}**Oi!** ^{. . . }** ^{(}**1

**)**

Similarly, th(' thormodyiiamio probability when the partirle-paiis an^ distributed first and then the single particles are di.stributod is

**( i ) '**

**( i ) '**

^{m}*n a i\*

Then **th e en tro p y **S **o f the system (Cf. D u tta **1953, **m ixtu re o f gases) is**

**... (**2**)**

,<?= |-(log Fr„+logTT„) ... (3)

**3 7 4** **Benuka Datta**

Usinf! iSlirlin>?’M lormula in oq. (]) and (2) and ihon maximising Ike expression (3) iind(!i- ilie following conditiona

**Sajei+2JYjM> — **

^{E}wo got.

*a i = e‘~^*

*1*

**\** ^{h ,}**)**

**.**

*. . .* (4.1)
... (4.2)
... (4.3)

... (5)

**= e - * V •••(6)**

**iV > ..**

*-2 h i(2 v + 2 u w )* ... (7)

**^ e**

Avlu^rn /?,//, *V* arc Lagrange’s undorL^nninecl mnitipliers. According to onr nnidcl
wo can assume

**V *** ’* F ' '

(6) and (7) avo got

- ) e - v * ^{.. .}* 18)

j ^{^’^}*^{2v~2/io)}* ... (9)

Using c<is. (5), (8), and (0) in eq (D) wo have

*S - h* i */ l E - ***Nj **^{log } *V* ^{^ } ^{log [ } ^{) }^{+ N}^{ log AT+}

+ i _{1}( ^ . ) ' - ( ( ) -

+ ( - F - a .6„i\^ \ / r - a A a ^ i

*\* *1* 1 *h*

**+ ( I = 5 ^ a j l o g**

**Condensed phase in lattice gas model **

**3 7 5**

log j | j

From the thormodynamic relation

*T * *\ d E t v*

**(**

### 10

**)**

it follows

Wo got from oqs. (5) and (4.2)
*II*** ^**

*k T*

A = log { ^5,3 (W !T )> /= }

### ( **11** )

**. . . ** **(**12^{)}

Finally, oqs. (8) and (9) loads to tho following (quadratic oqnation I’or e'’:

**/ F \ ** **.**

*< )* ^{+ Q}**e - - i r . o**

tho positive solution of whicJi is (retaining upto second order of small quantity)

### r ; )

" '1### -

*2w*

**an d so ** **v = log ** **j — log iV — .log | l —** **« K 2 ' j ( U H in g ( ] l) )**

Fi’ora oqs. (8) and (9) *N g* and become (using oqs. (11))
*2w*

" 1 ^{i - f .}

*N b ^*
*V*

*m b . * *■*

. — .* e

*2w*

*~ K T*

F

**(**

**14**

**)**

'I’Koti imttiuK (II), (12), (13) ** ^{and }**(14)

**ecj. (10)**

^{in }**th e e n tro p y becomes**

*k*I -) Wj loK A ' -1ok ( l - I ' . + lo g (

**376** Benuka Datta

*'Sw*

*IcT* *( > v*

**. fi ■ **

**kT**** )los f —)** **<-1** **6, ** **r ** **/ “ Ift, / &2 ■**

**- [!— i v ( i - ** **.**

**2«j**

**Lft, ** **1 ** **ft., ** **F**

**-JV^ 1** **26, ** **7V6, **

**b , ■ V**

*2ir*

**1****, ** **26, ** **JV6, **

**^ ** **fta ■ 7**

i2w’

**- o r ) ]**

**>o ** ^{ho }^{\}

*N b , *

**fta ** **1'**

**2«/'**

**Tr j — F _ a iftia ^ /]**

**- 0*2 ** **^2 ** **^**

**26, iVft, **
**ft., ■ ** **r***2ir*

**X r " l- 'l’**** )** *I),*

*~h.,*** ■ ** **]■**

*2w*
**. r r /**

*w*

**r**** " jx lo R** * ^{V}* a|6,

**. ** *he* **« ( i - j**26, *N h ,*

*b.*

■ r
**2'****m****’**

**- , r )**

**l!<r**

*V* **r L h , ** *b, * *b , ' * *y*** • ** **! ' ^ U , ** **ft., ** *y*

**')]**

*2ir*

f JC**1 ft**, 2 6, ^{2w}* _{k T}* iV (1 - .

*2w*

A^fti , - 7 2

J Lft_{1} ft. ft. * ^{V}* \ 62 F

< l"!t 1 ^ Jtj612 ft. A'^^ft, ^{2}w>

" ^{) : T}*n* *U* 26, A7j, e“ 7 2 ' ) l \

" lft. ^>1 62 F * ^{\}* ft, •'

_{^ }-'J J

. /; , v{ 1, * *C '^* ) - 1"KA' - l o g / 26, JV6,

1 ' ft', ■ V' . e' f ci rj
**f l oK ( ^ J ^ , - ( W - 7 - ) » ' “) ) | + iV lo R J V - J V ( l -**

*Gond&Med phase in lattice gas model*

- l i

**2****m****'**

**•)i**

*f* ^ _{N b ,}

*h , * *■* r

*2 i r * *' 2 i r*

f K ^1^12* _{N}* I(,

^{2 b , }

^{N b ,}

^{.}

^{,}

^{-}

^{'}

^{S}

^{\}

^{_}** ’■‘1. , - S i*

^{k}/ ^{'^2} I J

L ^{^^2} ^{bn}^{\}^{[ }^{b ^ ' }^{V}

*2uf*

### 377

**k ) ** **k**

) ■■ [
r \ **k )**

**k**

**11**

’ />,
*h * *k -* **I'** ^{k l'}

[/^X **i»2 ** T I Li^] *^>}* I **J**

] 111. ^{1. , ;}

^1^12 ^{m b . }* ^{-}* 2'//'

**A:T**_{— N 1}_{1}- ^{•ib,}

*Vb,* /q *l)n* T _{V}*b..*

loK r I’

Ux ' ^{b\ 'bn '}^{1}' ^{il}

*2w*

*k T* ( . -

**2vr**

## J J

'Flui t.lioriiio(l\iiann'f‘ r(‘lati(ni

7’ / ,2 i_ \

**“ ** **\ 5 F j**

(Hi)

loads to the oquatior) of stato of tho systoni (I’C'taiiiing iq> (o ll)(‘ second oid( r l<‘nnh
**: : l )**

*N b , * *N b ^*
**ot —- ** **as**

whore

, a ^{N k T}

*\ n*** ■**

**•J ** **1**

**a _ ***a ( T )*** =- ***k N % * T* **KT -**

9 1 **-w**

*/i = l i { T ) ^ .* **.J V V K2'**

**... (17)**

)

... (IS)

### 378 Benuka Batta

Tho oquaiioix *o f* state (17) is the same form as the van der Waal eqixation of
state of real gas. The onl^^ difference is that in van der Wall’s equation *a* aiid */J *

are eoiistaiit, but here a and /? aj c funcions o f temperature and tho binding energy

*o f* parti do-pairs.

Trom (5q. (17) the compressibility for the above system is given by 1 / \

*V \ d p*

**1**

*N J c T V ~* *2 a*

**> 2**

... (19)

Taking * ^{oc}* — 1.42 X 10^ atoms xem^,

**f i***^*c.c. for a gram mole at N.T.P. of a van der Wall’s gas (e.g. Argon) (Cf. Saha & Srivastava 1965) wo liave estimated the valuer of the exclusion volume

*h*of a ])artide ajid the binding energy

*w*of a 2)air-partide. The vahui o f

*h ^*10'^^ eju'^ which is actually happend to be (jf prop(»r order. We have also calculated the comx)ressibility by tho formula (19) and it is shown that the coinxiressibility .T] ~ 10~® dynes cm~^ which actually occurs for a gas.

Case II : *L a ttic e gas m odel i n condensed ph a se*

Here wo have oonsidored the general phyBi(;al j)rop(utios along with tho com- X>rossibility of tho system. CoJUi)aring the compressibility of this case with tliat o f the former (iaso wo can see that it is much smaller than that obtained in Case 1.

TUoJi, w'o can call this system a condensed phasic

*D e s c rip tio n o f the m odel*

In this case we also divide tho wdxole Bjiaco in a container into tliroe dimon- sional lattice structure as tjxplained in the formo)‘ case. It is also assnmed that each lattice sites can be occupied hy only one particle. For this case our assump

tion is that most, of all tlm lattice sites arc occupied bj^ particles and so tlxo number of unoccupied lattice sites are very small as (ionij)ared to that of occupied lattice sites. It is dear that jxossibilit.ics of coming of two unoccupied colls (say, holes) in tlu‘ two neighbou4ing silos is very small. Tho effective exclusion volume

of a hole is ^ (2a)“ j , which is the same as that in the former case. Those are dear from the figure 2. which is tho intersection of the three dimensional lattice model by a plane through the central lattice.

*M a th e m a tic a l C a lc u la tio n s*

First wx^ assume all tho lattice points are filltKl up by the particles. There

fore, if be tho binding eneigy of two neighbouring x>artidos, each jiartide has potential energy (i/c. 'Fhen we distribute among them few pseudo-particles with potential energy — 1 2?c, sudi iliat they coiiiributo potential energy to the partides at their own positions so that they (say, holes) have potential

energy zero and also contribure potential energy — u? to the six neighbouring
**Condensed p h a s e 'in lattice gas m odel**

### 379

particles.

O J

o o o o g o o

### O O Q Q O O O ( 2 ® j D O O

a w o o

o o o

**Fig, 2. Distribution of holos amongst particles.**

Let *V* bo the voliimo the container and so the total number of lattice points
are *V jh ,* where *h —* 4/3(7ra”). Lot *N* bo the number of particles, *N* *h* be the numbor
of holes. Let * ^{Cm}* be the number of partieles with kinetic energy

^{tjjn. }*be the total energy of the system. Then tlici thermodynamic probability is*

^{E}*i b i ) ' * *N\*

n **Cm'-**

*w*

Using Stirling’s approximation and taking logarithm (jf (20) wo have

log PT = log i V^l ogi VH— log

*N* ^ *^mVm*

(**20**)

Maximising log *W* under the etmditions
**Scm - ***N*
*m*

**“i . Otc—iV ^ .1 2 ic + S ***CmVm*** “ ***E *

*^ * *m*

**.. ** **(****21**^{)}

**. . . (****2 2****.****1****) **

**... (22 ^{.}2^{) }**

... (23)

we have Cj» = ^

**where A', fi' are Lagrange’s undetermined multipliers.**

Since F^{/6}** is the total number of lattice points and N is the total number of particle.s ****and Nff IB the number of boles, we have another condition :**

**(24)**

### 380 **R e n u k a D a t t a**

**fjasorting eqs. (23), (24) in eq. (21) and using the relation i&f ** **A; log IF, we have **
**the entropy,**

*8*** = ** **(***l 2 N w - 6 .*** ^ . w ) +2ff log ***N +*

+ ( 4 ^ ) ‘« * ( ^ ) 4 ( ^ ■ + T

- I ( - T ) - ( -

**... (26)**

**As before we get**

**, ** ***2'**

**= - X** **... (20)**

**and ** **A' = log ** **(27rmfe2r)*'*|** **... (27)**

**finally, the entropy is**

**F ;**
6
S in o o

**(28)**

**i>- = « - y “ ‘ [ ** **(2™ W** **} - i** ** ( l2 J f« ,- 6 . t » ) +**

T h u s , t h e e q u a t io n o f s t a t e fo r t h is s y s t e m is

**(29)**

(30)

**Condensed p h ase in lattice gas model**

### 381

From this equation of state we get the compressibility o f this system. Since

*N t i* are very small compared to the total number of lattice points, thoiefojo,
retaining up to the first order term * o f *-'^*^i.e, 1— the compressibility oi'
this system is

*K - - - * *{ * *\* - L -

* ' F \ ^{d p I}* ^{t}* ""

*(31) Since 1---- oompaiing the expreasions (19) and (31) we see that the com-*

^{N k T}*N b*

prosBibility in Case [ is much larger than that in Case H Using the values ol

*h* and to estimated from Case I and taking *p* ^ 50 atoms, *T * *-* “ 200”C (Since
critical iiressuie and temperature are 48 atoms and ™12 2°c rospoctiveJv for Argon)
in (30) we calculate and then by (31) we calculate the eompressibility for this
system. The compressibility ~ 10'^’’ dynes cm--*. One will be inclined to
interpret it as a condensed phase.

3. CONOLUniNG BEMARKH

We have discussed two extreme eases only to show that tiic lattice gas model is suitable for investigations of matter in the gaseous phase and in the condensed phase, It is to be noted that in our calculation in Case 11 we have got the com

pressibility very low even lower than the usually accepted value of condensed phase. In this connection it is to be noted that there are many intermediate cases where existence of particle-triplots, -quadruplets etc. may be frequent, but they are not considered here. It is naturally expected that in one of these cas(>s quantitative agreement (at least in the order) may be obtained in the case (if condensed phase. Moreover, when these intK^nnediatc possibility would he in

vestigated a clear insight about the phase' transition may be obtained.

**A****cknowledgement**

The author is grateful to Professor M. Dutta, CVntie of Advanced Study in Applied Mathematics and Professor-in-Charge, S. N. Pose Institute of Physical Since for his kind help and valuable suggestions in preparing this paper.

References

**Dutta, M. 1953 **

*P roc. Nat. Innt, Sc. India,*

** 16, ISH.**

**Fiaher, M. E. 1972 **

*Cornmun, Math. Phy&.*

** 26, 0.**

**Ikeda, K. and Nisizima, K. 1076 **

*P rog. Thep. Phya.*

** 53, 32t.**

**Noumaxui, H. P, 1974 **

*Z. N aurforach*

** 29a, 65.**

**Buelle, D. 1969 **

**Statistical Mechanics : Rigorous Results****W. A. Benjamin, Inc., New York. **

**Saha, M. N. and Srivastava, B. N. 1906 **

**A Treatise on Heat,****the Indian Preas Private Ltd.,**

**Allahabad.**

**Tomperley, H. N. V. 1969 **

**Proc, Ph^s. Soc.**