A semiflexible polymer chain under geometrical restrictions: Only bulk behaviour and no surface adsorption
We analyse the conformational behaviour of a linear semiflexible homo-polymer chain confined by two geometrical constraints under a good solvent condition in two dimensions. The constraints are stair shaped impenetrable surfaces. The impenetrable surfaces are lines in a two dimensional space. The in...
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irk-123456789-1535262019-06-15T01:26:08Z A semiflexible polymer chain under geometrical restrictions: Only bulk behaviour and no surface adsorption Mishra, P.K. We analyse the conformational behaviour of a linear semiflexible homo-polymer chain confined by two geometrical constraints under a good solvent condition in two dimensions. The constraints are stair shaped impenetrable surfaces. The impenetrable surfaces are lines in a two dimensional space. The infinitely long polymer chain is confined in between such two (A and B) surfaces. A lattice model of a fully directed self-avoiding walk is used to calculate the exact expression of the partition function, when the chain has attractive interaction with one or both the constraints. It has been found that under the proposed model, the chain shows only a bulk behaviour. In other words, there is no possibility of adsorption of the chain due to restrictions imposed on the walks of the chain. Ми аналiзуємо конформацiйну поведiнку лiнiйного напiвгнучкого гомополiмерного ланцюга пiд дiєю двох геометричних обмежень в умовах доброго розчинника у двовимiрному просторi. Обмеження представляють собою непроникнi поверхнi схiдчастої форми. Непроникнi поверхнi є лiнiями у двовимiрному просторi. Нескiнченно довгий полiмерний ланцюг є обмежений двома поверхнями (A and B). Для розрахунку точного виразу статистичної суми використовується ґраткова модель повнiстю напрямлених блукань без самоперетинiв, якщо ланцюг має притягальну взаємодiю з одною чи двома поверхнями. В рамках запропонованої моделi отримано лише об’ємну поведiнку ланцюга. Iншими словами, жодної можливостi для адсорбцiї ланцюга пiд дiєю обмежень на блукання, не спостерiгається. 2014 Article A semiflexible polymer chain under geometrical restrictions: Only bulk behaviour and no surface adsorption / P.K. Mishra // Condensed Matter Physics. — 2014. — Т. 17, № 2. — С. 23001:1-9. — Бібліогр.: 34 назв. — англ. 1607-324X arXiv:1010.2840 DOI:10.5488/CMP.17.23001 PACS: 05.70.Fh, 64.60 Ak, 05.50.+q, 68.18.Jk, 36.20.-r http://dspace.nbuv.gov.ua/handle/123456789/153526 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We analyse the conformational behaviour of a linear semiflexible homo-polymer chain confined by two geometrical constraints under a good solvent condition in two dimensions. The constraints are stair shaped impenetrable surfaces. The impenetrable surfaces are lines in a two dimensional space. The infinitely long polymer chain is confined in between such two (A and B) surfaces. A lattice model of a fully directed self-avoiding walk is used to calculate the exact expression of the partition function, when the chain has attractive interaction with one or both the constraints. It has been found that under the proposed model, the chain shows only a bulk behaviour. In other words, there is no possibility of adsorption of the chain due to restrictions imposed on the walks of the chain. |
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Mishra, P.K. A semiflexible polymer chain under geometrical restrictions: Only bulk behaviour and no surface adsorption Condensed Matter Physics |
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Mishra, P.K. |
| author_sort |
Mishra, P.K. |
| title |
A semiflexible polymer chain under geometrical restrictions: Only bulk behaviour and no surface adsorption |
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A semiflexible polymer chain under geometrical restrictions: Only bulk behaviour and no surface adsorption |
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A semiflexible polymer chain under geometrical restrictions: Only bulk behaviour and no surface adsorption |
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A semiflexible polymer chain under geometrical restrictions: Only bulk behaviour and no surface adsorption |
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A semiflexible polymer chain under geometrical restrictions: Only bulk behaviour and no surface adsorption |
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semiflexible polymer chain under geometrical restrictions: only bulk behaviour and no surface adsorption |
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Інститут фізики конденсованих систем НАН України |
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2014 |
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| citation_txt |
A semiflexible polymer chain under geometrical
restrictions: Only bulk behaviour and no surface
adsorption / P.K. Mishra // Condensed Matter Physics. — 2014. — Т. 17, № 2. — С. 23001:1-9. — Бібліогр.: 34 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT mishrapk asemiflexiblepolymerchainundergeometricalrestrictionsonlybulkbehaviourandnosurfaceadsorption AT mishrapk semiflexiblepolymerchainundergeometricalrestrictionsonlybulkbehaviourandnosurfaceadsorption |
| first_indexed |
2025-07-14T04:38:53Z |
| last_indexed |
2025-07-14T04:38:53Z |
| _version_ |
1837595816115568640 |
| fulltext |
Condensed Matter Physics, 2014, Vol. 17, No 2, 23001: 1–9
DOI: 10.5488/CMP.17.23001
http://www.icmp.lviv.ua/journal
A semiflexible polymer chain under geometrical
restrictions: Only bulk behaviour and no surface
adsorption
P.K. Mishra∗
Department of Physics, DSB Campus, Kumaun University, Nainital-263 002 (Uttarakhand), India
Received January 20, 2014, in final form March 3, 2014
We analyse the conformational behaviour of a linear semiflexible homo-polymer chain confined by two geomet-
rical constraints under a good solvent condition in two dimensions. The constraints are stair shaped impenetra-
ble surfaces. The impenetrable surfaces are lines in a two dimensional space. The infinitely long polymer chain
is confined in between such two (A and B ) surfaces. A lattice model of a fully directed self-avoiding walk is used
to calculate the exact expression of the partition function, when the chain has attractive interaction with one or
both the constraints. It has been found that under the proposed model, the chain shows only a bulk behaviour.
In other words, there is no possibility of adsorption of the chain due to restrictions imposed on the walks of the
chain.
Key words: polymer adsorption, bulk behaviour, geometrical constraints, exact results
PACS: 05.70.Fh, 64.60 Ak, 05.50.+q, 68.18.Jk, 36.20.-r
1. Introduction
Biopolymers (DNA and proteins) are soft objects and, therefore, suchmolecules can be easily squeezed
into the spaces that are much smaller than the natural size of the molecules. For example, actin filaments
in eukaryotic cell or protein encapsulated in Ecoli [1–3] are the examples of confined molecules that may
serve as the basis for understanding molecular processes occurring in the living cells. The conforma-
tional properties of single bio-polymers have attracted considerable attention in recent years due to the
development of single molecule based experiments [4–9]. The entropy of a molecule having an excluded
volume interaction gets modified due to the presence of geometrical restrictions. Therefore, geometrical
constraints can modify conformational properties and the adsorption desorption transition behaviour of
the confined polymer molecules.
The behaviour of a linear and flexible polymer molecule under good solvent condition, confined to
different geometries, has been studied for the past few years [10–17]. Theoretical investigations of a
semiflexible polymer chain under confined geometry also find considerable attention in recent years,
see [18–25] and references quoted therein. For example, Whittington and his coworkers [10–17] used di-
rected self-avoiding walk models to study the behaviour of a flexible polymer chain confined between
two parallel walls on a square lattice and calculated the force diagram for a surface interacting polymer
chain. Rensburg et al. [17] performed numerical studies using an isotropic self-avoiding walk model and
showed that the force diagram obtained for surface interacting polymer chains confined in between two
parallel plates have a qualitatively similar phase diagram obtained by Brak et al. [10–13] for a directed
self-avoiding walk model of the problem.
However, in the present investigation, we consider an infinitely long-linear semiflexible polymer
chain confined in between one dimensional two stair shaped impenetrable surfaces (geometrical con-
straints) under good solvent conditions and we discuss the conformational behaviour of the chain. Such
∗
pkmishrabhu@gmail.com
© P.K. Mishra, 2014 23001-1
http://dx.doi.org/10.5488/CMP.17.23001
http://www.icmp.lviv.ua/journal
P.K. Mishra
an investigation may be useful to understand the behaviour of a macromolecule near a membrane as
well as the behaviour of DNA in micro-arrays and electrophoresis.
To analyze the conformational behaviour of such semiflexible chains we have chosen a fully directed
self-avoiding walk model introduced by Privmann and coworkers [26, 27] and have used a generating
function technique to solve the model analytically for different values of the spacing between the con-
straints. The result so obtained is used to discuss the possibility of an adsorption phase transition be-
haviour of the polymer chain on the stair shaped geometrical constraints. Since the constraint is an attrac-
tive surface, it contributes an energy εs (< 0) for each step of the fully directed self-avoidingwalk touching
the constraint. This leads to an increased probability defined by a Boltzmannweightω= exp(−εs/kBT ) of
stepping on the constraint (εs < 0 or ω> 1, T is temperature and kB is the Boltzmann constant). The poly-
mer chain gets adsorbed on the constraint at an appropriate value of ω or εs. Therefore, the transition
between an adsorbed to a desorbed phase is marked by a critical value of adsorption energy εs or ωc.
In this paper, we analytically solve the fully directed self-avoiding walk model to calculate the exact
expression of the partition function of the chain when the chain has an attractive interaction either with
one or both of the geometrical constraints. The results so obtained are compared with the case when the
adsorption of a semiflexible polymer chain occurs on a flat surface [28–34].
The paper is organized as follows: In section 2, a square-lattice model of fully directed self-avoiding
walk is described for an infinitely long and linear semiflexible homo-polymer chain confined in between
the constraints for a given value of spacing between the constraints. In subsection 2.1, we discuss the
possibility of an adsorption transition of the polymer chain when constraint A has an attractive interac-
tion with the semiflexible polymer chain. Subsection 2.2 is devoted to a discussion of the adsorption of
a semiflexible polymer chain on the constraint B . While in subsection 2.3 the expression of the partition
function of the polymer chain is obtained for the case when the chain has an attractive interaction with
both the constraints. Finally, in section 3 we summarize and discuss the results obtained.
2. Model and method
A model of fully directed self-avoiding walks [26, 27] on a square lattice is used to investigate the
possibility of an adsorption transition of an infinitely long linear semiflexible homopolymer chain on
geometrical constraints, when the chain is confined in between two impenetrable stair shaped surfaces
under a good solvent condition (as shown schematically infigure 1). The directedwalkmodel is restrictive
in the sense that the angle of bending has a unique value, that is 90◦ for a square lattice and the directivity
of the walk amounts to a certain degree of stiffness in the walks of the chain because different directions
of the space are not treated equally. Since the directed self-avoidingwalkmodel can be solved analytically,
it gives exact values of the partition function of the polymer chain. We consider a fully directed self-
avoiding walk (FDSAW)model. Therefore, the walker is allowed to take steps along+x, and+y directions
on a square lattice in between the constraints.
The walks of the chain start from a point O located on the impenetrable surface-A, and the walker
moves in the space in between the two surfaces [as we have shown schematically in figure 1, a walk of
the polymer chains confined in between two surfaces for a value of separation n(= 3) between them].
The stiffness of the chain is accounted for by associating a Boltzmann weight with the bending energy
for each turn in the walk of the polymer chain. The stiffness weight is k= exp(−βεb); where β= 1/kBT is
the inverse of the temperature, εb(> 0) is the energy associated with each bend in the walk of the chain,
kB is the Boltzmann constant and T is temperature. For k = 1 or εb = 0, the chain is said to be flexible and
for 0 < k < 1 or 0 < εb <∞ the polymer chain is said to be semiflexible. However, when εb →∞ or k → 0,
the chain has the shape of a rigid rod.
The partition function of a surface interacting semiflexible polymer chain can be written as follows:
Z (ω,k) =∑N=∞
N=0
∑
all walks of N steps
g Nω
Ns kNb , (2.1)
where Nb is the total number of bends in a walk of N steps (monomers), Ns is number of monomers in a
N step walk (Nb É N −1 and Ns É N ), lying on the surface, g is the step fugacity of each monomer of the
chain, and ω is the Boltzmann weight of the monomer-surface attraction energy.
23001-2
A semiflexible polymer chain under geometrical restrictions
Figure 1. This figure shows a walk of an infinitely long linear semiflexible polymer chain confined in
between two constraints (impenetrable stair-shaped surface). All walks of the chain start from a point O
on the constraint. We show three different cases viz. (i), (ii) and (iii) having separation (n) between the
constraints along the axis three monomers (steps). The separation between the constraints are defined
on the basis of how many steps a walker can successively move at maximum along any of the +x or +y
directions. In the case 1 (i), the constraint A has an attractive interaction with the monomers of the chain,
in 1 (ii) only constraint B has an attractive interaction with the monomers of the chain while in 1 (iii)
both constraints are shown to have an attractive interaction with the monomers of the polymer chain.
2.1. A semiflexible polymer chain interacting with constraint A
The partition function of an infinitely long linear semiflexible polymer chain confined in between the
constraints (as shown schematically in figure 1 (i) and having an attractive interaction with the constraint
A can be calculated using the generating function technique. The components (as shown in figure 2)
of the partition function Z A
3 (k,ω1) (here we have used the suffix three because in figure 1 (i) case, the
maximum step that a walker can move successively in one particular direction is three and ω1 is the
Boltzmann weight of the attraction energy between monomers, and thus the constraint A) of the chain
can be written as follows:
X A
1 = s1 +ks1Y A
3 , (2.2)
where s1 =ω1g .
X A
2 = g + g
(
X A
1 +kY A
2
)
, (2.3)
X A
3 = g + g
(
X A
2 +kY A
1
)
, (2.4)
Y A
1 = g +kg X A
3 , (2.5)
Y A
2 = g + g
(
k X A
2 +Y A
1
)
, (2.6)
and
Y A
3 = s1 + s1
(
k X A
1 +Y A
2
)
. (2.7)
On solving equations (2.2)–(2.7), we find the expression for X A
1 (k,ω1) and Y A
2 (k,ω1). In obtaining the
expression for X A
1 (k,ω1) and Y A
2 (k,ω1), we have solved a matrix of 2n ×2n (n = 3), for the present case
i.e., figure 1 (i). Thus, we have an exact expression of the partition function for an infinitely long linear
semiflexible polymer chain confined between the constraints and having an attractive interaction with
the constraint A [as shown in 1 (i)]. This is written as follows:
Z A
3 (k,ω1) = X A
1 (k,ω1)+Y A
2 (k,ω1) =−u1 +u2 +u3 +2k4s2
1 g 4 −2k5s2
1 g 4
−1+k6s2
1 g 4 +u4
, (2.8)
23001-3
P.K. Mishra
X
Y
k
k
X2
X3
X4
X1
1
Y
g
g
X2
3
Y1
Y1
Y2
Y3
Y4
Y5 2
X5
B
A
s
1 s
1
(i)
(ii)
(iii)
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
Figure 2. The components of the partition function is shown graphically in this figure. Term X A
m (3 É
m É n) indicates the sum of Boltzmann weight of all the walks having the first step along +x direction
and suffix n indicates maximum number of steps that a walker can successively take along +x direction.
Similarly, we have defined Y A
m , where the first step of the walker is along +y direction. In this figure, (ii)
and (iii) graphically represents the recursion relation for equations (2.2) and (2.6), respectively.
where
u1 = s1 −ks2
1 − g −ks2
1 g +k2s2
1 g − g 2 −kg 2 −ks1g 2 +2k2s1g 2 −ks2
1 g 2,
u2 =−k2s2
1 g 2 +3k3s2
1 g 2 −kg 3 +k2g 3 −k2s2
1 g 3 +2k3s2
1 g 3 −k4s2g 3,
u3 =−kg 4 +k3g 4 −ks1g 4 +k2s1g 4 +k3s1g 4 −k4s1g 4 −2k2s2
1 g 4 +2k3s2
1 g 4,
and
u4 =−k4 [
g 4 +2s2
1
(
g 2 + g 4)]+k2 [
g 2 (
2+ g 2)+ s2
1
(
1+ g 2 + g 4)] .
From the singularity of the partition function, Z A
3 (k,ω1), we obtain the critical value of the Boltz-
mann’s weight for the monomer-constraint A attraction energy,
ωc1 =
√
1−2k2g 2 −k2g 4 +k4g 4√
k2g 2 +k2g 4 −2k4g 4 +k2g 6 −2k4g 6 +k6g 6
.
This is required for the adsorption of an infinitely long linear semiflexible polymer chain on the con-
straint A. We obtain the value of ωc1 = 1, when we substitute the value of gc in the expression of ωc1
corresponding to all possible values of k[= exp(−βεb)] or the bending energy εb for which an infinitely
long linear semiflexible polymer chain can be polymerized in between the constraints. It shows the ex-
istence of only one singularity gc of the partition function equation (2.8) and it corresponds to the bulk
behaviour of the chain. There is no possibility of an adsorption transition of the chain on constraint A.
2.2. A semiflexible polymer chain interacting with constraint B
The partition function of an infinitely long linear semiflexible polymer chain confined in between
the constraints [as shown schematically in figure 1 (ii)] and having an attractive interaction with the
constraint B is calculated following themethod discussed in the above subsection. The components of the
partition function Z B
3 (k,ω2) (where ω2 is Boltzmann weight of attraction energy between the monomers
of the chain and the constraint B ) of the chain can be written as follows:
X B
1 = g +kg Y B
3 , (2.9)
X B
2 = g + g
(
X B
1 +kY B
2
)
, (2.10)
X B
3 = s2 + s2
(
X B
2 +kY B
1
)
, (2.11)
23001-4
A semiflexible polymer chain under geometrical restrictions
where s2 =ω2g .
Y B
1 = s2 +ks2X B
3 , (2.12)
Y B
2 = g + g
(
k X B
2 +Y B
1
)
, (2.13)
and
Y B
3 = g + g
(
k X B
1 +Y B
2
)
. (2.14)
On solving equations (2.9)–2.14), we find an expression for X B
1 (k,ω2) and Y B
2 (k,ω2). In obtaining the
expression for X B
1 (k,ω2) and Y B
2 (k,ω2), we have to solve a matrix of 2n×2n [n = 3, for figure 1 (ii) case].
Thus, we obtain an exact expression of the partition function for an infinitely long linear semiflexible
polymer chain confined between the constraints and having an attractive interaction with the constraint
B [as shown in figure 1 (ii)] which is as follows:
Z B
3 (k,ω2) = X B
1 (k,ω2)+Y B
2 (k,ω2) =−−g (s2(1+kg 2 −k2g 2)+2u5 +ks2
2u6)
−1+k6s2
2 g 4 +u7
, (2.15)
where
u5 = 1+k2(−1+ g )g 2 −k3g 3 +kg (1+ g ),
u6 = 1+ g + g 2 −2k3(−1+ g )g 2 +2k4g 3 −k2g
(
2+3g +3g 2)+2k
(−1+ g 3) ,
u7 =−k4 [
g 4 +2s2
2
(
g 2 + g 4)]+k2 [
g 2 (
2+ g 2)+ s2
2
(
1+ g 2 + g 4)] .
From the singularity of the partition function, Z B
3 (k,ω2), we obtain a critical value for the monomer-
constraint B attraction energy,
ωc2 =
√
1−2k2g 2 −k2g 4 +k4g 4√
k2g 2 +k2g 4 −2k4g 4 +k2g 6 −2k4g 6 +k6g 6
=ωc1,
required for adsorption of an infinitely long linear semiflexible polymer chain on the constraint B . In
this case too, we find ωc2 = 1, for all possible values of the bending energy or stiffness of the semiflexible
polymer chain and further there is no possibility for the existence of a new singularity of the partition
function i.e. equation (2.15). Therefore, the adsorption of the chain on constraint B is impossible.
2.3. A semiflexible polymer chain interacting with both the constraints A and B
The partition function of an infinitely long linear semiflexible polymer chain confined in between
the constraints [as shown schematically in figure 1 (iii)] and having an attractive interaction with both
the constraints (A and B ) is calculated following the method discussed in the above subsections. The
components of the partition function Z C
3 (k,ω3,ω4) of the chain can be written as follows:
X C
1 = s3 +ks3Y C
3 , (2.16)
where s3 =ω3g .
X C
2 = g + g
(
X C
1 +kY C
2
)
, (2.17)
X C
3 = s4 + s4
(
X C
2 +kY C
1
)
, (2.18)
here, s4 =ω4g .
Y C
1 = s4 +ks4X C
3 , (2.19)
Y C
2 = g + g
(
k X C
2 +Y C
1
)
, (2.20)
and
Y C
3 = s3 + s3(k X C
1 +Y C
2 ). (2.21)
23001-5
P.K. Mishra
On solving equations (2.16)–(2.21), we get the expression for X C
1 (k,ω3,ω4) and Y C
2 (k,ω3,ω4). In ob-
taining the expression for X C
1 (k,ω3,ω4) and Y C
2 (k,ω3,ω4), we have solved a matrix of 2n ×2n [n = 3, for
figure 1 (iii) case]. Thus, we have an exact expression for the partition function of an infinitely long linear
semiflexible polymer chain confined between the constraints and having an attractive interaction with
the constraints [as shown in figure 1 (iii)]. This is written as follows:
Z C
3 (k,ω3,ω4) = X C
1 (k,ω3,ω4)+Y C
2 (k,ω3,ω4) =−−(g u8 + s3u9)+u10 +u11
−1+k6s2
3 s2
4 g 2 +u12 +u13
, (2.22)
where
u8 = 1+ s4 +kg − (−1+k)ks2
4(1+ g +kg ),
u9 = 1−k2s2
4 g 2 +k4s2
4 g 2 +k
(
1+ s2
4
)
g 2 −k2 [
g 2s2
4
(
1+ g 2)] ,
u10 = ks2
3
[
1+ g + s4g k2s2
4
(
1−2g
)
g +2k4s2
4 g 2 +kg
(−1− s4 +2g
)]
,
u11 = ks2
3
{
kg s2
4(1+2g )−k2 [
2g 2 + s2
4
(
1+2g +2g 2)]} ,
u12 = k2 {
g 2 + s2
4
(
1+ g 2)+ s2
3
[
1+ (
1+ s2
4
)
g 2]} ,
and
u13 =−k4 {
s2
4 g 2 + s2
3
[
g 2 + s2
4
(
1+2g 2)]} .
From the singularity of the partition function, Z C
3 (k,ω3,ω4), we obtain a critical value of the
monomer-constraint attraction energy,
ωc3 =
√
1−k2g 2 −k2g 2ω2
4 −k2g 4ω2
4 +k4g 4ω2
4√
k2g 2 +k2g 4 −k4g 4 −k4g 6ω2
4 −2k4g 6ω2
4 +k2g 6ω2
4 +k6g 6ω2
4
, (2.23)
required for the adsorption of an infinitely long linear semiflexible polymer chain on the constraints A,
when both the constraints have an attractive interaction with the chain.
On substitution of the value of ω4 in equation (2.23) to get the value of ωc3 = 1,
ωc4 =
√
1−2k2g 2 −k2g 4 +k4g 4√
k2g 2 +k2g 4 −2k4g 4 +k2g 6 −2k4g 6 +k6g 6
=ωc2 .
The method discussed above can be used for different values of n. The size of the matrix needed
to solve for the partition function of the chain confined in between the constraints is 2n ×2n. We have
calculated the exact expressions of the partition function for n (3 É n É 19).
We have found that the adsorption transition point of an infinitely long linear semiflexible polymer
chain on the constraint A, B and simultaneously on both the constraints A and B has the value of unity.
The equation (2.23) has only a singularity that corresponds to the polymerization of an infinitely long
linear homopolymer chain in between the constraints. Therefore, there is no possibility of the adsorption-
desorption phase transition in the proposed model. This fact is true for the chosen values of k or the
bending energy (as checked for 3 É n É 19) forwhich an infinitely long polymer chain can be polymerized
in between the constraints.
2.4. General expressions of the recursion relations
In this subsection, we should like to express the recursion relations with the least possible number
of equations. This method is useful in solving a matrix of n ×n rather than 2n ×2n as discussed in the
subsections 2.1–2.3. For instance, equations (2.16)–(2.21) can be written as follows:
W n
1 = s3 +ks2
3 +kg s3
3 +·· ·+kg n−2s2
3 s4 +k2s2
3W n
1
+k2g s2
3W n
2 +k2g 2s2
3W n
3 +·· ·+k2g n−2s2
3 s4W n
n , (2.24)
W n
m = g +kg 2 +·· ·+kg n+1−m s4 + gW n
m−1 +k2g 2W n
m
+k2g 3W n
m+1 +·· ·+k2g n+1−mW n
n−1 +k2g n+1−m s4W n
n , (2.25)
23001-6
A semiflexible polymer chain under geometrical restrictions
where (1 < m < n) andW n
m = 0, whenm < 1.
W n
n = s4 +ks2
4 + s4W n
n−1 +k2s4W n
n . (2.26)
The equations (2.24)–(2.26) can be used to express recursion relations, X C
n (for all values of the chosen
n), and mutual exchange of s3 with s4 will result in the recursion relations Y C
n for the chosen values of n.
The partition function of the chain can now be written as follows:
Z C
n (k,ω3,ω4) =W n
1 +W n
n−1, (2.27)
whereW n
1 is the sum of the Boltzmann weights of all walks starting from a pointO lying on the constrant
A and having the first step along +x direction, whileW n
n−1 is the sum of the Boltzmann weights of all the
walks starting from pointO and with the first step along +y direction.
However, substituting s4 = g and s3 = s1, we have recursion relations and a partition function for
the case 1 (i), as shown in figure 1 and when we substitute s3 = g and s4 = s2, recursion relations and
partition function for the case 1 (ii) of figure 1 were found by us.
If the constraints are assumed to be neutral, the recursion relations can bewritten for any given value
of n as follows:
W n
m = g +kg 2 +kg 3 +·· ·+kg n+2−m + gWm−1
+k2g 2Wm +k2g 3Wm+1 +·· ·+k2g n+2−mW n
n , (2.28)
where 1 É m É n andW n
0 = 0.
3. Summary and conclusions
We have considered an infinitely long linear semiflexible homopolymer chain confined in between
two impenetrable stair shaped surfaces (constraint) in two dimensions under good solvent condition. We
have used a fully directed self-avoiding walk model to study the adsorption phase transition behaviour of
the polymer chain on any of the two constraints (A and B ) and simultaneous adsorption of the polymer
chain on both the constraints (A and B ). The generating function technique is used to solve themodel ana-
lytically and an exact expression of the partition function of the surface interacting semiflexible polymer
chain is obtained for different values of spacing (3 É n É 19) between the constraints.
We find in the case 1 (i), 1 (ii) and 1 (iii) that the bulk behaviour of the polymer chain occurs on the
constraints for the values ωc1 = ωc2 = ωc3 = 1 for all possible values of k or the bending energy of the
chain for which an infinitely long linear semiflexible polymer chain can be polymerized in between the
constraints. The critical value of ω is unity for all cases considered and for different values of spacing
between the constraints (3 É n É 19). This result is obvious because the walks of the chain are directed
along the constraint(s), therefore, the partition function of the chain is dominated by the walks lying on
the constraints, and the bulk behaviour is observed on the constraints. We have shown the results for a
few values of n = 3, 7, 11, 16 in the table 1 for the case 1 (i), when the chain interacts with the constraint
A. The chain is grafted to the constraint A for the case 1 (i), 1 (ii) and 1 (iii), as shown in figure 1. An
infinitely long linear chain is polymerized in between the two constraints A and B , when g = gc.
However, in the case of adsorption of an infinitely long linear semiflexible polymer chain on a flat
surface, the adsorption transition point is found to depend on the bending energy or stiffness of the chain.
In this case, the partition function of the surface interacting chain has two singularities. One singularity
corresponds to the bulk behaviour i.e., polymerization of an infinitely long linear chain and the other
singularity corresponds to adsorption transition of the chain on the surface [28–34].
We have also expressed general expressions of the recursion relations, when the chain has an attrac-
tive interaction with any or both the constraints and when the constraints are assumed to be neutral. It
has been found that polymerization of an infinitely long flexible polymer chain is not possible for sepa-
rations (n) between constraints 3, 6 and 8. In the case of n = 3, the imaginary part of the critical value
of step fugacity is negligible. However, for other values of separation between the constraints, i.e., n = 6
and 8 the imaginary part in the critical value of step fugacity is reasonable and cannot be ignored. We
plan to discuss these issues in a another paper to be submitted elsewhere in due time.
23001-7
P.K. Mishra
Table 1. This table shows the values of gc and sc(=ωc1gc) for different values of separation (n) between
the constrains, for the case 1 (i), as shown in figure 1. The value of sc = gc indicates that ωc1 = 1.
n = 3 n = 7 n = 11 n = 16
k gc sc gc sc gc sc gc sc
0.1 − − − − − − 0.99303 0.99303
0.2 − − − − 0.92215 0.92215 0.88185 0.88185
0.3 − − 0.89545 0.89545 0.83077 0.83077 0.80227 0.80227
0.4 − − 0.80962 0.80962 0.76023 0.76023 0.73867 0.73867
0.5 0.93879 0.93879 0.74186 0.74186 0.70259 0.70259 0.68558 0.68558
0.6 0.84709 0.84709 0.68613 0.68613 0.65400 0.65400 0.64017 0.64017
0.7 0.77358 0.77358 0.63907 0.63907 0.61222 0.61222 0.60072 0.60072
0.8 0.71293 0.71293 0.59857 0.59857 0.57576 0.57576 0.56603 0.56603
0.9 0.66182 0.66182 0.56325 0.56325 0.54359 0.54359 0.53524 0.53524
1.0 0.61803 0.61803 0.53208 0.53208 0.51496 0.51496 0.50771 0.50771
Acknowledgements
The financial support received from Department of Science and Technology, New Delhi (SR/FTP/PS-
122/2010) thankfully acknowledged. The author also would like to thank Professor D. Dhar, TIFR, Mumbai
(India) and the anonymous referee for useful corrections in the earlier version of the manuscript.
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Напiвгнучкий полiмерний ланцюг пiд дiєю геометричних
обмежень: лише об’ємна поведiнка i жодної поверхневої
адсорбцiї
П.К.Мiшра
Фiзичний факультет, Кумаонський унiверситет, м. Найнiтал (Уттаракханд), Iндiя
Ми аналiзуємо конформацiйну поведiнку лiнiйного напiвгнучкого гомополiмерного ланцюга пiд дiєю
двох геометричних обмежень в умовах доброго розчинника у двовимiрному просторi. Обмеження пред-
ставляють собою непроникнi поверхнi схiдчастої форми. Непроникнi поверхнi є лiнiями у двовимiрно-
му просторi. Нескiнченно довгий полiмерний ланцюг є обмежений двома поверхнями (A and B ). Для
розрахунку точного виразу статистичної суми використовується ґраткова модель повнiстю напрямлених
блукань без самоперетинiв, якщо ланцюг має притягальну взаємодiю з одною чи двома поверхнями.
В рамках запропонованої моделi отримано лише об’ємну поведiнку ланцюга. Iншими словами, жодної
можливостi для адсорбцiї ланцюга пiд дiєю обмежень на блукання, не спостерiгається.
Ключовi слова: полiмерна адсорбцiя, об’ємна поведiнка, геометричнi обмеження, точнi результати
23001-9
http://dx.doi.org/10.1209/0295-5075/78/38001
http://dx.doi.org/10.1103/PhysRevE.75.050902
http://dx.doi.org/10.1021/jp806126r
http://dx.doi.org/10.1063/1.3271830
http://dx.doi.org/10.1140/epje/i2010-10626-y
http://dx.doi.org/10.1063/1.454626
http://dx.doi.org/10.1016/S0378-4371(02)01993-3
http://dx.doi.org/10.1088/0953-8984/22/15/155103
http://dx.doi.org/10.1080/01411590903537588
http://dx.doi.org/10.1080/01411594.2010.534657
http://dx.doi.org/10.1021/ma000493s
Introduction
Model and method
A semiflexible polymer chain interacting with constraint A
A semiflexible polymer chain interacting with constraint B
A semiflexible polymer chain interacting with both the constraints A and B
General expressions of the recursion relations
Summary and conclusions
|