MOLECULAR PHYSICS, 1997, VOL. 92, NO. 2, 265± 270
Some exact results for isolated hard-disc chain
and ring molecules
By MARK P. TAYLOR
Department of Chemistry, Dartmouth College, Hanover,
New Hampshire 03755, USA
( Received 28 January 1997; revised version accepted 9 April 1997 )
In this paper we study two-dimensional ¯ exible interaction-site molecules composed of n
tangent hard discs. In particular, we obtain exact results for the n-site probability function
for isolated tangent-hard-disc chains and rings with n< 6. All con® gurational properties of the
chain and ring n-mers can be determined from this probability function. We present analytical
and `exact’ numerical results for the mean-square site± site separations, mean-square radii of
gyration, average shape and two-site intramolecular distribution functions.
1. Introduction
A useful model in the study of molecular and polymeric ¯ uids is that of the ¯ exible interaction-site molecule. Such a molecule is composed of `simple-¯ uid’
monomers which, in the simplest case, are connected
by completely ¯ exible universal joints. The structure of
an interaction-site molecule can be described in terms of
a set of spherically symmetric site± site distribution functions. These functions are analogous to the pair correlation function used to describe simple ¯ uids [1] and, in
principle, can be computed using the integral equation
techniques of liquid-state theory [2± 5].
In this paper we present a number of exact results for
isolated two-dimensional hard-core interaction-site
molecules. Such results are useful for validating both
simulation algorithms and the above-mentioned integral
equation theories. Also, results for hard-core systems
provide a basis for the development of perturbation
theories for non-hard core potentials. Here we study
¯ exible chain and ring molecules composed of n tangent
hard discs with n< 6. We obtain the exact n-site probability function for these molecules from which all con® gurational properties can be determined. We give exact
results for the mean-square site± site separations, meansquare radii of gyration, average shape and two-site
distribution functions. When possible we have obtained
analytical results; otherwise exact numerical results are
presented.
2. Theory
In this work we study two-dimensional ¯ exible interaction-site molecules composed of n hard discs of diameter s connected by universal joints of bond length s .
The probability that an isolated tangent-hard-disc n-mer
0026± 8976/97 $12 . 00
Ñ
chain will be found in a speci® c con® guration is written
in terms of the coordinates of the n hard-disc sites as
chain
Pn
n- 2
1
( r1 , . . . , rn ) =
´
Z chain
n
Õ
n- 1
a =1
Õ Õ
n
exp
i= 1 j = i+ 2
s( ra ,a
+1
[- b
u( rij )
),
]
( 1)
chain
where Z n
is the single-chain partition function,
b = 1 /kB T , rij = |ri - rj |, s( r) = d ( r - s ) /2p s is the intramolecular distribution function between bonded
sites, and
¥ , r <s ,
( 2)
b u( r) =
0, r > s ,
{
is the hard-disc potential. This formal de® nition for the
n-site con® gurational probability function can be greatly
simpli® ed by changing from the ® xed Cartesian coordinates {ri } to the set of intramolecular coordinates {la , µk }
where la is the bond length between sites ( a , a + 1) and
µk is the angle created by sites ( k - 1, k, k + 1) [6]. The
n- 1
Jacobian of this transformation is simply
a = 1 la =
n- 1
s
. By locating site 1 at the origin and integrating
out the n - 1 ® xed bond lengths la , the n-mer chain
con® gurational probability function can be expressed
in terms of the n - 2 bond angles as
P
( µ2 , . . . , µn- 1 )
P chain
n
=
where µk
ì
î
í
1
, no hard-core overlaps,
( 2p ) n- 2 Z chain
n
0,
( 3)
otherwise
can be determined via
Î [p /3, 5p /3] and Z chain
n
1997 Taylor & Francis Ltd.
266
M. P. Taylor
Table 1. Allowed bond angle ranges for tangent-hard-disc nmer chains. Entries are restricted to the µ2 Î p /3, p range
since for µ2 Î p , 5p /3 the n-mer probability function can
chain
be determined via the identity Pn ( µ2 , . . . , µn- 1 ) =
)
(
µ
µ
P chain
2
.
.
.
2
.
For
angles
outside the
p
p
- 2, , - n- 1
n
allowed ranges the probability functions are identically
zero. The limiting angle g ( µ2 , µ3 , x) is given by equation
(23).
Table 2. Allowed interior bond angle ranges and multiplicity
factor h for tangent-hard-disc n-mer rings. For angles outside the allowed ranges the probability functions are identically zero. The limiting angles for the case of n = 6 are
de® ned as b 1 = ( 3p - µ2 ) / 2 - cos - 1 ( cos µ2 ) / 2 sin ( µ2 / 2) ,
b 2 = g ( 2p / 3, µ2 - p / 3,1), g 1 = g ( µ2 ,µ3 ,1), g 2 = g ( µ2 - p /3,
µ3 , 1) , and g 3 = g ( µ2 , µ3 , 2) where the angle g is given by
equation (23).
n
n
[
3
4
5
]
µ2
[p /3, p ]
[p /3, 2p /3]
[2p /3, p ]
[p /3, 2p /3]
[2p /3, p ]
[
µ3
]
µ4
[p - µ2, 5p /3]
[p /3, 5p /3]
[p - µ2, 4p /3 - µ2]
[4p /3 - µ2, 4p /3]
[4p /3, 5p /3]
[p /3, 4p /3 - µ2]
[4p /3 - µ2, 2p /3]
[2p /3, 4p /3]
[4p /3, 5p /3]
[g
[p
[p
[g
[p
[p
[p
]
µ3 , 1) , 5p / 3
/ 3, 5p / 3
/ 3, 3p - µ3
( µ2 , µ3 , 1), 5p / 3
- µ3, 5p /3
/ 3, 5p / 3
/ 3, 3p - µ3
( µ2 ,
]
]
]
]
]
]
[
µ2
µ3
[
[p /3, 2p
[2p /3, p
[p /3, 2p
]
3
/ ] [p - µ2 , 4p /3 - µ2 ]
] [p /3, 4p /3 - µ2]
/3] [p - µ2 , 4p /3 - µ2 ]
[b 1, 4p /3 - µ2]
[4p /3 - µ2, 5p /3 - µ2]
[4p /3 - µ2, p ]
[2p /3, p ] [p /3, 4p /3 - µ2]
[b 1, 4p /3 - µ2]
[4p /3 - µ2, 2p /3]
[4p /3 - µ2, b 2]
[2p /3, 5p /3 - µ2]
[p , 4p /3] [p /3, 5p /3 - µ2]
]
µ4
h
4 p / 3, 2p /3
1
5
1
1
6
[g 1, g 1 + p /3]
[g 1 + p /3, g 3]
[p /3, 2p - µ2 - µ3]
[2p - µ2 - µ3, g 3 ]
[g 1, g 1 + p /3]
[g 1 + p /3, g 3]
[p - µ3, g 2]
[g 2, g 3]
[p /3, g 2]
[p - µ3, g 2]
1
2
1
2
1
2
1
2
1
1
the normalization condition
ò ò
( µ2 , . . . , µn- 1 ) = 1.
´´´ dµ2 ´´´ dµn- 1 Pchain
n
( 4)
Thus the speci® cation of the full n-site con® gurational
probability function amounts to determining the physically permitted ranges of the n - 2 bond angles. Results
for tangent-hard-disc n-mer chains with n = 3, 4 and 5
are given in table 1. The n-site chain probability function
is invariant with respect to the exchange {µ2 , . . . , µn- 1} ®
{2p - µ2 , . . . , 2p - µn- 1 } which implies that the average
of any of the bond angles is simply k µi l = p .
A tangent-hard-disc n-mer ring is simply a special case
of the n-mer chain with sites 1 and n in contact. The ring
n-site probability function can thus be written as
ring
P n ( r1 , . . . , rn ) = 2p s
ring
chain
2 Zn
ring
Zn
s( r1n )P n
chain
( r1 , . . . , rn ) ,
( 5)
ring
where Z n is the single ring partition function and P n
has been de® ned such that the ratio of the ring to the
chain partition functions is equal to the probability that
an n-mer chain will be in a closed ring conformation (i.e.
chain
chain
chain
Z ring
= 2p w 1n ( s ), where w 1n
is the end-ton /Z n
end distribution function of an n-mer chain). Transforming from ® xed Cartesian to intramolecular coordinates, as above, the n-mer ring probability function can
be expressed in terms of n - 2 adjacent interior bond
angles. Further simpli® cation is achieved by using the
ring closure constraint to integrate out the angle µn- 1 .
Thus the n-mer ring con® gurational probability function
can be written in terms of n - 3 adjacent interior bond
angles as
ring
P n ( µ2 , . . . , µn- 2 )
ì
=
í
î
hn ( µ2 , . . . , µn- 2 )
, no hard-core overlaps,
( 2p ) n- 2 Z ring
n
0,
otherwise,
ring
( 6)
where Z n is given by the analogue of equation (4) and
the weight factor hn arises from the integration over µn- 1
via the d ( r1n - s ) constraint. This weight factor is
related to the probability of closing an n-mer ring once
the positions of n - 1 sites have been speci® ed and is
given by hn = 2h /|sin µn | where the multiplicity factor
h ( = 1 or 2) is the number of ways of placing the nth
site to form a closed ring and the ring closing angle is
2
2
-1
µn = cos ( 1 - r 1,n- 1 /2s ) . This latter quantity can be
expressed in terms of the n - 3 speci® ed bond angles
using equation (8) (de® ned below). Thus speci® cation
ring
of P n amounts to determining the physically permitted
ranges of the n - 3 bond angles of an ( n - 1)-mer harddisc chain such that ring closure, via the addition of an
nth site, is possible. Results for tangent-hard-disc n-mer
rings with n = 4, 5 and 6 are given in table 2. The n-site
ring probability function is translationally invariant
with respect to the labelling of sites. Since the sum of
the n internal angles of the hard-disc n-mer ring is constrained to be ( n - 2)p , this invariance principle implies
that the average of any of these equivalent angles is
267
Hard-disc chain and ring molecules
simply k µi l = ( n - 2)p /n (which is the internal angle of
an n-sided regular polygon ).
All con® gurational properties of a tangent-hard-disc
n-mer chain or ring can be expressed in terms of
averages over the above n-site con® gurational probability functions. For example, the mean-square site±
site separations for an n-mer chain are given by
k r 2ij l
=
ò ò
. . . dµ2 . . . dµn- 1 r 2ij ( µi+ 1 , . . . , µj- 1 )
( µ2 , . . . , µn- 1 ),
´ Pchain
n
( 7)
where the squared distance between any two nonbonded sites i and j ( i < j ) can be determined via recursive application of the following formula:
r ij ( µi+ 1 , . . . , µj - 1) = s
2
2
2
å
´
j- 1
Figure 1. End-to-end probability functions 2p xw1n ( x) for
tangent-hard-disc n-mer chains with n = 3, 4, 5 and 6 as
indicated. The n = 3 function diverges at x = 2 while the
n = 4 function vanishes for x > 3.
2
+ r i,j- 1 + 2s
( - 1) j- k cos
k = i+ 1
(å )
j- 1
( 8)
µl .
l= k
Similarly, dimensionless site± site distribution functions
for an n-mer chain are given by the following integral
transform:
chain
w ij
( r) =
s
2
2p r
ò ò
[
. . . dµ2 . . . dµn- 1 d r - rij ( µi+ 1 , . . . , µj- 1 )
( µ2 , . . . , µn- 1 ).
´ Pchain
n
k
n
Z chain
n
3
4
5
6
2/3
5/12
0. 254 94
0. 154 28
k
r 213 l /s
2
2. 826 99
2. 904 77
2. 912 62
2. 912 44
Table 4.
2
n
1
k r2ij l
n 2 i<j
å
( 10)
and the average shape (i.e. deviation from circular symmetry) can be expressed in terms of the ratio of the
2
2
2
principle orthogonal components k L 1 l and k L 2 l of k Rg l ,
2
2
2
2
where k Rg l = k L 1 l + k L 2 l and the L i components are
determined by diagonalizing the ring or chain inertia
tensor [7]. A number of analytic results for the above
quantities for n-mer chains and rings with n< 6 are given
in the following sections. Exact numerical results are
summarized in tables 3 and 4 and two-site distribution
functions are shown in ® gures 1 and 2.
]
( 9)
Analogous expressions for n-mer rings, and for other
con® gurational properties (e.g. bond correlation functions and three-site distribution functions), are readily
obtained. The average size of an n-mer chain or ring is
given by the mean-square radius of gyration
Table 3.
Rg l =
Exact con® gurational properties for tangent-hard-disc n-mer chains.
k
r 224 l /s
2
2. 904 77
2. 992 23
3. 002 43
k r 214 l
/s
k r 225 l
2
5. 295 87
5. 474 37
5. 503 15
/s
k
2
5. 474 37
5. 677 49
r 215 l / s
k
2
8. 273 31
8. 566 85
r 216 l /s
2
11. 674 55
k
R2g l / s
2
k
0. 536 33
0. 881 59
1. 281 58
1. 731 16
Exact con® gurational properties for tangent-hard-disc n-mer rings.
n
Z ring
n
k µi l /p
3
4
5
6
2 /3 1/2 p
ln 3 / 2p 2
0. 015 744
0. 005 777 6
1/3
1/2
3/5
2/3
k r 2i,i+ 2 l
/s
2
1
2
2. 539 52
2. 795 37
k
r 2i,i+ 3 l / s
2
1
2. 539 52
3. 713 60
k
R2g l / s
2
1/3
1/2
0. 707 90
0. 942 03
k
L 22 l / k L 21 l
1
( ln 3) /ln 2 - 1
0. 519 17
0. 521 34
L 22 l / k L 21 l
0. 138 31
0. 137 37
0. 139 83
0. 140 19
268
M. P. Taylor
k
R2g l
s
(
5
3
+
8 10p
=
2
3 1 /2 +
p
3
),
( 16)
and the following analytic expressions can be derived for
the site± site distribution functions:
ìïïï
íïï
w13 ( x) =
ï
î
and
3
F( u 1 , q) - F( u 0 , q) ,
5p x
6
p
F , q - 2F( u 0 , q)
5p 3 x1 /2
2
ìïïï
ïïï
íïï
ï
3 1 /2
w14 ( x) =
Figure 2. Two-site probability functions 2p xwi,i+ n/2 ( x) for
tangent-hard-disc n-mer rings with n = 4, 5 and 6 as indicated. The n = 4 and 5 functions vanish for x > 3 1/2 and
x > 2 respectively.
3.
Results for tangent-hard-disc chains
3.1. The 3-mer chain
The 3-mer chain probability distribution function
P3 ( µ2 ) is given by equation (3) with Z3 = 23 and
µ2 Î p /3, 5p /3 . The mean-square end-to-end distance
and mean-square radius of gyration are given by
[
]
k r213 l
= 2+
k
=
2
s
R2g l
s
2
3
3 /2
,
2p
( 11)
1
4 3 /2
+
,
9 6p
( 12)
respectively and the end-to-end distribution function is
w13 ( x) =
3
1
,
2p 2 x ( 4 - x2 ) 1 /2
1<
x<
2,
( 13)
where x = r /s . This function is zero for x outside the
speci® ed range.
3.2. The 4-mer chain
The 4-mer chain probability distribution function
P4 ( µ2 , µ3 ) is given by equation (3) with Z4 = 125 and the
allowed ranges of ( µ2 , µ3 ) are given in table 1. The meansquare site± site separations and mean-square radius of
gyration are given by
k r213 l
2
s
k r214 l
s
2
6
= 2+
5p
= 3+
(
(
3
1 /2
+
p
2
12 1 /2 4
3 +
5p
p
),
),
( 14)
( 15)
4 1 + ( 3 /p ) sin - 1 ( x /2)
, 1 < x < 31 /2,
( 4 - x 2 ) 1 /2
5p 2 x
8
1
3 1 /2 < x < 2.
,
5p 2 x ( 4 - x2 ) 1 /2
( 17)
ïïï
ïï
î
6
5p x
[
]
[( )
3 1 /2 F
( , ),
p
m
, q) =
ò
],
2<
u m
0
3
sin - 1
[
zm = cos
q=
p
(
3
x+ 1
4
( ) ],
)
1-
1
( x + 1)( 3
x
x)
x2
2
1 /2
3,
( 18)
4( 1 - zm ) - 2( x - 1) 2
( 1 - zm )( x + 1)( 3 - x)
(
x<
da
( 1 - q2 sin 2 a ) 1 /2
+ m cos -
2,
x<
and
is an elliptic integral of the ® rst kind with
u m=
3 1 /2 ,
x<
31 /2 <
q
2
where x = r /s , w14 ( 1) = ( 3 ln 3) /5p
F( u
1<
)
( 19)
1 /2
,
( 20)
( 21)
( 22)
.
The above site± site functions vanish for x outside the
speci® ed ranges.
3.3. The 5-mer chain
The allowed ranges of bond angles for the tangenthard-disc 5-mer chain are given in table 1. The limiting
angle g ( µ2 , µ3 , x) can be written in terms of the site± site
separations as follows:
g ( µ2 , µ3 , x) = cos - 1
(
+ cos -
s
1
(
2
s
+ r 14 - r 13
2s r14
2
2
+
2
r 214
-
2s r14
)
x 2s
2
),
( 23)
where the distances r13 and r14 are given in terms of the
angles µ2 and µ3 via equation (8). Owing to the complicated expression for the limiting angle g ( µ2 , µ3 , 1), simple
analytic expressions are not easily obtained for the
con® gurational properties of the 5-mer chain. Here we
settle for semianalytic formulas involving integrals
269
Hard-disc chain and ring molecules
[
]
over g ( µ2 , µ3 , 1) or cos g ( µ2 , µ3 , 1) which are readily
evaluated via numerical quadrature. The resulting
numerically `exact’ results are given in table 3. Exact
site± site distribution functions can be obtained via
numerical evaluation of equation (9). Results for the
end-to-end distribution function w15 ( r) are shown in
® gure 1.
3.4. The 6-mer chain
Determination of the allowed bond angle ranges for
the tangent-hard-disc 6-mer chain is straightforward,
although admittedly rather tedious. The full results are
too lengthy to include here. We do include, in table 3,
the numerically exact results for the 6-mer chain partition function, mean-square site± site separations and
mean-square radius of gyration. Results for the
end-to-end distribution function w16 ( r) are shown in
® gure 1.
4.
Results for tangent-hard-disc rings
4.1. The 4-mer ring
The 4-mer ring con® gurational probability function
ring
ring
P 4 ( µi ) is given by equation (6) with Z 4 =
( ln 3) /2p 2 , h4 = 2 /sin µi , and µi Î p /3, 2p /3 . The
exact mean-square site± site separation and mean2
2
2
square radius of gyration are k r i,i+ 2 l = 2s and k Rg l =
2
2
s /2 respectively and the principal components of k Rg l
2
2
2
2
2
are k L 1 l = ( s ln 2) /2 ln 3 and k L 2 l = s /2 - k L 1 l . The
site± site distribution function is
[
wi,i+ 2 ( x) =
2
1
,
2(
p ln 3 x 4 - x2 )
1<
]
x<
3 1 /2 ,
( 24)
where x = r /s . This function is zero for x outside the
speci® ed range.
4.2. The 5-mer ring
The physically allowed con® gurations of a tangenthard-disc 5-mer ring are given in table 2 and the assoring
ciated con® gurational probability function P 5 ( µ2 , µ3 )
is given by equation (6) with h = 1 and µ5 =
1
cos - cos µ2 + cos µ3 - cos ( µ2 + µ3 ) - 12 . The singlering
ring partition function Z 5 , as well as the meansquare site± site separations and mean-square radius of
gyration have been determined via numerical quadrature and the numerically exact results are given in
table 4. It is possible to derive an analytic expression
for the site± site distribution function for the 5-mer
ring which is given as follows:
[
]
5p
ìïïï
wi,i+ 2 ( x) =
íïï
ïï
ïïï
ïïï
ïïï
î
[
]
chain
( x)wchain
x < 3 1 /2 ,
14 ( x) , 1 <
ring w 13
18Z 5
(u
)
1
chain
( x) F 10/2, q , 3 1 /2 < x < 2,
w
2 ring 13
x
3p Z 5
1
x = 2,
,
ring
4p 4 Z 5 3 1 /2
( 25)
where x = r /s , F( u 0 , q) is given by equation (19), and
w chain
is the end-to-end distribution function for a tan1n
gent-hard-disc n-mer chain. The above function vanishes
for x outside the speci® ed range.
4.3. The 6-mer ring
The physically allowed ranges of adjacent internal
angles for a tangent-hard-disc 6-mer ring are given in
table 2 and the associated con® gurational probability
ring
function P 6 ( µ2 , µ3 , µ4 ) is given by equation (6). The
multiplicity factor h associated with each range of
angles is also included in table 2 and numerically exact
ring
results for the 6-mer ring partition function Z 6 , as well
as the mean-square site± site separations and meansquare radius of gyration are given in table 4. Analytic
expressions for the site± site distribution functions for
the 6-mer ring have been derived only for a limited
range of site± site separations. These are given as follows:
wi,i+ 2 ( x) =
wi,i+ 3 ( x) =
2p
1
ring 2
Z6
[Z
chain chain
w 13
3
( x)Z chain
w chain
5
15 ( x) ],
2p 1 chain chain
w 14 ( x)
ring 2 Z 4
Z6
[
1<
x<
3
1 /2
, ( 26)
] , 1<
x<
3
1 /2
, ( 27)
2
where x = r /s and w chain
is the end-to-end distribution
1n
function for a tangent-hard-disc n-mer chain. The 6-mer
ring functions wi,i+ 2 ( x) and wi,i+ 3 ( x) vanish for x > 2
1
and x > 1 + 3 /2 respectively. Numerical results for the
entire `end-to-end’ function wi,i+ 3 ( x) are shown in ® g. 2.
5. Discussion
In this paper we have presented exact results for the nsite probability function for tangent-hard-disc n-mer
chains and rings with n < 6. From this probability function all con® gurational properties of the hard-disc chain
and ring molecules can be determined. We have presented analytical and exact numerical results for meansquare site± site separations, mean-square radii of gyration, average shape and two-site distribution functions.
Calculation of the full n-site probability functions for
hard-disc chains and rings is the continuum analogue to
an exact enumeration of self-avoiding walks and polygons on a lattice. The tables of allowed bond angle
270
Hard-disc chain and ring molecules
ranges given here for n-mer chains and rings correspond
to a catalogue of all possible ( n - 1)-step lattice walks
and n-sided lattice polygons. Our exact calculations have
been limited to fairly small values of n owing to the
extremely large con® guration space available to chains
and rings in the continuum.
The approach used here to study isolated hard-disc
molecules can be directly extended to study isolated
sticky-disc molecules (i.e. molecules composed of sites
interacting via an attractive contact or Baxter [8] potential). By determining the subset of allowed bond-angle
ranges for a hard-disk n-mer chain or ring in which nonbonded discs are in contact, one can construct the exact
n-site probability function for a sticky-disc n-mer chain
or ring. For n < 6 the partition function of such a
sticky-disc molecule is a well behaved function of temperature (i.e. inverse `stickiness’ parameter) and, with
decreasing temperature, such molecules are expected to
undergo a collapse transition [9,10]. In contrast, the
partition function of a sticky-disc n-mer chain or ring
with n > 7 possesses singular contributions and thus
such molecules cannot be treated as well de® ned thermodynamic systems. This pathological behaviour is a feature of the Baxter sticky potential which was ® rst
pointed out by Stell [11]. Stell has proposed an alternate
form of the sticky potential [11] which avoids these
problems and thus may prove useful in the study of
sticky-disc polymers. Work on such systems is currently
in progress.
This work was supported in part by the National
Science Foundation (grant No. DMR-9424086 to
J. E. G. Lipson).
References
[1] Hansen, J. P., and Mc Donald , I. R., 1986, Theory of
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