IACS Institutional Repository

Das, Sumit R

English

Holograms of Cosmological Singularities

Mukhopadhyay, Subir

Ray, Koushik

Branes in Hearts with Perverse Sheaves

Forthcoming Publications

Front Contents

Ojha, K S

Gopal, R

Laser Produced Spectrum of B2Σ+-X2Σ+ and B2Σ+-A2 Π Systems of SiN Molecule

Verma, Pavan Kumar

Srivastava, Rajat

Khanna, K N

ndiiii' J P h y s 80 (8), 837-839 (2006)
Q V K iotc
The specific heat of linear tangent hard sphere chains
Pavdn Kumar Verma, Rajai Srivastava and K N Khanna* 
Dcpailmciil o f Physics, VS S H ro llcg c , Kiinpui-2UR U26. Indu
Emuil knkhaniiJ@ieiJillmail cum
R c t c i v c d 2 N i t v e m b e r 200S_ n a c p t c d 2 2 J u n e 2 0 0 6
sii,i(i Tht spcLiliL heal ( C \ ) o f lincai U ngcnl hard sphere chains of varying length is dcleim incd by em ploying W criheini’s fust oidei 
mill,Hum ihmiy ll is shown that Ihe eqiialion o f slale obtained by W eilhciin's first older pcituibalion iheoiy piedicis the same values ol specilic 
,is ihosL ohiaiiiLd ihiough the equation ol slale obtained by Monte Carlo compulei simulation
uu in ls langLMU haid spheie chains, chain lengths Monte Cnilo simulation
( s Nos 11 IS Md, II IS Hs, 05 10 Ln
dl csuhlislied model to study ehatn-like molecular system 
le m which the molecules are modeled as chains formed by 
iiicdecl sphciical segments This model is commonly used 
upicsciU pidymci’s thermodynamics pioperties and their 
JSC hchiivioi Cham molecules ol tangent segments can be 
sidiMccI lully llcxiblc unless bending and loistonal potentials 
L‘\pliLiily incorporated In these models, the pair potential 
'U‘ui monomcis thal form this chain is given by a spheiical 
icitiial Wertheim 11-4J and Chapman, Jackson and Gubbins 
I I'Mesented independently a very successful theory to study 
‘ ihcimodynamic properties of hard core fluids interacting 
I stuiii uinge attractive forces Wertheim’s formulation also 
Ids qiiiic good description for Lennard Jone (LJ) chains [6- 
sciiure well chains [8J, Yukawa chains [9], vapour - liquid 
Jihhiium [lOJ. critical properties [11] and fluid - solid 
'osiciice [ 12-14] Thus, it is possible to derive an equation of
1i^ )i a chain of freely joined tangent hard segments using 
H ihermodynarnic information of monomer reference fluid, 
'' iheory is commonly known as the first order thermodynamic 
nmbation theory (TPT 1) By employing TPT 1, the 
P'oximate equation of state of the chain fluid can be derived 
'^iiowing the equation of state of the monomer fluid and its 
" <-f'i relation function at contact As we have considered the 
-iiud segments of the chain, hard sphere reference system 
quite useful
'^‘Pumling Aulhur
Despite the extensive number of studies devoted to the 
Wertheim’s theory, relatively little attention has been paid to 
the thermodynamic pi opei tics such as specific heat ol the chain 
molecules To the best of our knowledge, no work has been 
done in this direction In the present work, we have carried out 
a simple numeiical calculations by employing a simple toi mula 
of specilic heat depending upon the equation ol state We have 
also carried out the numerical calculations of the specific heal 
from equation of state obtained by Monte Carlo simulations 
and compared both theoretical results and those obtained from 
Monte Carlo results We find a good agreement between the 
two values
Let us consider the spherical monomer particles N“‘' within 
a certain volume Vat temperatuie T These spherical particles 
interact through a spherical pair potential u(0 We shall denote 
this fluid as the refeience fluid and labeled by supersciipl 
reference Let us also consider a fully flexible chains of ni 
monomers such that N = The Helmholtz free energy of
the reference fluid can be divided into an ideal and a residual 
part as follows [12] ■
_______aM t^vsidual = ln (p"^cr ') - l +
N “'f KT' 
(i)
©2(X)61 ACS
838 Pavan Kumar Verma, Rajat Srivastava and K N Kfianna
where is the number density of the reference fluid and a 
IS the thermal de Broglie wavelength
From the Helmholtz free energy, we can derive the expression 
of the equation ol state as
yjhmn ^ Z{r}) ^ mZ'"'' - ( /« - ] ) 1 + 1^ - 0 .577-
( I - 77) ( I - O577) (2)
(I- w '
(3)
 ^ H (a) (4)
t-y. D 
~ 2 ^
dT)
|77Z(77)1 (5)
Now,
d
{r]Z) = m 
dr] dr]
can be evaluated from cq (2) as 
d 77(1 + 77 + 77^
- ( m - I)
dr]
( l - r ; ) ' 
77(1 + 77- 0577* ) '
(l-77)(l-OJ77) (6)
= 77i/| - (m - 1) /t
and
/. =
(1 “ 77)(i + 2 t7 4 377* - 477’) + 3(77 
( i-j? )"
+ 17' +V -r)
(l-7j)(l-05n)(l + 2r)-15^-) + (rj + r;- -057)')(lj-7;)
Thus, the Cp of the chain system can be determined 
employing eqs (2) and (5)
Recently. Malanoski and Mon.son [19J have P''eseniei
For flexible chain molecules, the reference fluid is that ol 
hard spheres in the fluid pha.se and described by Carnahaiv 
Starling Equation of state 115| as
where 77 is the volume traction of the hard spheie defined as 
77 = [njb)p''^^ , the ( a ) can be obtained by employing
Virial route to the pressure as [16]
Now, we know that the sj^ccitic heat of a system of hard 
spheres ol diameter a can be expressed as [17, 18]
where D IS the dimension (in the picsent case D=3) and Z(r]) is 
the equation of state of the hard sphere system We consider 
Z(77) IS the equation of state of chain system which can be 
obtained by employing Wertheim's first order theory 
6{r]Zh
extensive calculations of the equation of state of freels joinej 
chains of tangents hard spheres using Monte Carlo compmu 
simulation, as
f (Q’ + C,ij + C.r)- + C,r;’ + C,r|'‘)
where C ,^ C, are the coefficients toi the ijim
function fits of the equations of state ot the fluid at ditierw 
density ranges and are given in Table 2 of the Ref [ 19| \W\jr 
determine that
^ (T jZ ) = [(Q - t;)(c ,; + 2C|>7 + 3C;7j- + 4C,7?’ +5C jt)'|
+3(C„' + Qr,^ + + C.r,* + C4r|’ )]/(C„ - n)'
Thus, we can determine the values ot C /K^ for chain lluiii- 
employing Monte Carlo equations ol state (7) in eq (i)
In the present wc^ rk, Wertheim’s theory of the flexible 
molecules, called as pearl necklace model has been cmploud 
determine the specific heal (Cy ot the chain molecules ol (litioo 
chain lengths We assume that the simple formulation ol il^; 
.specific heat (C^) of the system ot hard spheres will lem.iin vaiii, 
for the chain molecules by approximating the equation ol m,ii; 
of chain molecules instead of equation ol state ot hard splicc' 
We shall compare the results obtained from the Wciihcmi' 
theory with the simulation results of Malanoski and 
[ 19] for the pearl necklace model Malanoski and Monson!!' 
have provided empirical fits to their simulation result.s oMli« 
equation of state for the fluid phase of chain molecules vuih/' 
ranging from /w = 3 up to m = 8 Therefore, obtained Imir 
these empirical fits are considered here as simulation re^uli' 
A comparison between theoretical and simulation value"’ 
CJ NKg has been shown in table 1 for chain lengths 111=^ -I'
Table 1 Specific heat (CJNK^^) of the pearl-neck luce model m 1111.11““ 
phase from Werlheim's theory and the simulation data
n
m= 3 ni =4 III= 5 HI
TPTT P T MC T P T MC T P T MC
0 20 1 06 1 04 0 86 0 84 0 74 0 68 0 (I6l'i
0 25 1 186 1 14 0 98 0 97 0 86 0 81 0 77
0 30 1 35 1 32 1 14 I 14 1 02 0 98 0 93
[)W]
0 35 1 56 1 56 1 35 1 35 1 23 1 20 1 14
0 40 1 85 1 86 1 63 1 62 1 50 1 48 1 41
0 45 2 20 2 23 I 99 1 96 1 86 1 83 1 77
The specific heat o f linear tangent hard sphere chains 839
( \Vc find a a g r e e m e n t b e t w e e n t h e t h e o r e t ic a l and 
jlitions v a lu es H o w e v e r , th e s m a l l d i s c r e p a n c y at lo w 
increases w ith in c r e a s in g c h a in le n g th T h is m a y b e 
n, [he c h a r a c te r is t ic o t th e f ir s t o r d e r t h e r m o d y n a m ic 
ion iheory, M o r e im p r o v e d t h e o r ie s s u c h a s D im e i 
t'luc fluid theory |20] a n d G e n e r a l iz e d F lo r y D im e r th e o ry 
improve the r e su lts fo r lo n g c h a in m o le c u le s .
lowledgment
(K S and K N ) are th a n k fu l to C S T . L u c k n o w for
[Kul assistance
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8659 (1999)
[10] E J Bias, A (laliiKlo and ( ' Vega Mol PIm 101 449 (2003)
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6212 (1995)

The Specific Heat of Linear Tangent Hard Sphere Chains

Ghosh, Dipak

Dev, Argha

Sarkar, Samabrata

Haldar, Prabir Kumar

Dynamical Azimuthal Fluctuation of Target Fragments in Forward and Backward Hemispheres in case of 32S - AgBr Interactions

Mandal, Gautam

Quantization of Gravitons, Gravity and Sound Waves

Sengupta, Soumitra

Observable Consequences of Gauge Invariant Electromagnetic-Torsion Coupling - A Departure from Einstemian Gravity

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Holograms of Cosmological Singularities

Branes in Hearts with Perverse Sheaves

Forthcoming Publications

Front Contents

Laser Produced Spectrum of B<SUP>2</SUP>Σ<SUP>+</SUP>-X<SUP>2</SUP>Σ<SUP>+</SUP> and B<SUP>2</SUP>Σ<SUP>+</SUP>-A<SUP>2</SUP> Π Systems of SiN Molecule

The Specific Heat of Linear Tangent Hard Sphere Chains

Forthcoming Publications

Dynamical Azimuthal Fluctuation of Target Fragments in Forward and Backward Hemispheres in case of <SUP>32</SUP>S - AgBr Interactions

Quantization of Gravitons, Gravity and Sound Waves

Observable Consequences of Gauge Invariant Electromagnetic-Torsion Coupling - A Departure from Einstemian Gravity

6,371

8,071