Ring polymers in confined geometries
The investigation of a dilute solution of phantom ideal ring polymers and ring polymers with excluded volume
 interactions (EVI) in a good solvent confined in a slit geometry of two parallel repulsive walls and in a solution
 of colloidal particles of big size was performed. Taking i...
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| Zitieren: | Ring polymers in confined geometries / Z. Usatenko, J. Halun, P. Kuterba // Condensed Matter Physics. — 2016. — Т. 19, № 4. — С. 43602: 1–6. — Бібліогр.: 28 назв. — англ. |
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| citation_txt | Ring polymers in confined geometries / Z. Usatenko, J. Halun, P. Kuterba // Condensed Matter Physics. — 2016. — Т. 19, № 4. — С. 43602: 1–6. — Бібліогр.: 28 назв. — англ. |
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| description | The investigation of a dilute solution of phantom ideal ring polymers and ring polymers with excluded volume
interactions (EVI) in a good solvent confined in a slit geometry of two parallel repulsive walls and in a solution
of colloidal particles of big size was performed. Taking into account the correspondence between the field
theoretical φ⁴ O(n)-vector model in the limit n → 0 and the behaviour of long-flexible polymers in a good
solvent, the correspondent depletion forces and the forces which exert phantom ideal ring polymers and ring
polymers with EVI on the walls were obtained in the framework of the massive field theory approach at fixed
space dimensions d=3 up to one-loop order. Besides, taking into account the Derjaguin approximation, the
depletion forces between big colloidal particle and a wall and in the case of two big colloidal particles were
calculated. The obtained results indicate that phantom ideal ring polymers and ring polymers with EVI due to
the complexity of chain topology and for entropical re
Проведено дослiдження розведених розчинiв iдеальних кiльцевих та кiльцевих полiмерiв з ефектами
виключеного об’єму в доброму розчиннику, обмежених в щiлинi двох паралельних вiдштовхуючих поверхонь, а також у розчинi колоїдних частинок великого розмiру. Беручи до уваги вiдповiднiсть мiж теоретикопольовою φ⁴ O(n)-векторною моделлю в границi n → 0 i поведiнкою довгих гнучких полiмерiв
в доброму розчиннику, було отримано з використанням масивної теорiї поля при фiксованiй вимiрностi
простору d = 3 в рамках однопетлевого наближення вiдповiднi сили збiднення i сили, якi дiють мiж iдеальними кiльцевими та кiльцевими полiмерами з ефектами виключеного об’єму i стiнками щiлини. Окрiм
того, беручи до уваги наближення Дєрягiна, розраховано сили збiднення мiж колоїдною частинкою i стiнкою, а також мiж двома колоїдними частинками великого розмiру. Отриманi результати вказують на те,
що кiльцевi полiмери через складнiсть топологiї i з ентропiйних причин демонструють цiлком iншу поведiнку в обмежених середовищах нiж лiнiйнi полiмери.
|
| first_indexed | 2025-12-07T16:57:26Z |
| format | Article |
| fulltext |
Condensed Matter Physics, 2016, Vol. 19, No 4, 43602: 1–6
DOI: 10.5488/CMP.19.43602
http://www.icmp.lviv.ua/journal
Ring polymers in confined geometries
Z. Usatenko1, J. Halun2, P. Kuterba2
1 Institute of Physics, Cracow University of Technology, 30-084 Cracow, Poland
2 Institute of Physics, Jagiellonian University, 30-348 Cracow, Poland
Received June 25, 2016, in final form September 6, 2016
The investigation of a dilute solution of phantom ideal ring polymers and ring polymers with excluded volume
interactions (EVI) in a good solvent confined in a slit geometry of two parallel repulsive walls and in a solution
of colloidal particles of big size was performed. Taking into account the correspondence between the field
theoretical φ4 O(n)-vector model in the limit n → 0 and the behaviour of long-flexible polymers in a good
solvent, the correspondent depletion forces and the forces which exert phantom ideal ring polymers and ring
polymers with EVI on the walls were obtained in the framework of the massive field theory approach at fixed
space dimensions d=3 up to one-loop order. Besides, taking into account the Derjaguin approximation, the
depletion forces between big colloidal particle and a wall and in the case of two big colloidal particles were
calculated. The obtained results indicate that phantom ideal ring polymers and ring polymers with EVI due to
the complexity of chain topology and for entropical reasons demonstrate a completely different behaviour in
confined geometries compared with linear polymers.
Key words: colloidal systems,critical phenomena, polymers, phase transitions
PACS: 64.70.pv, 61.25.he, 67.30.ej, 68.35.Rh, 64.70.qd
As it was shown in a series of the atomic force spectroscopy (AFM) experiments [1, 2], biopolymers
such as DNA very often present a ring topology. Such a situation takes place, for example, in the case of
Escherichia coli (E.coli) bacteria with a chromosome which is not a linear polymer, but has a ring topol-
ogy [3]. The biopolymers of DNA of some viruses such as bacteriophages λ that infect bacteria oscillate
between linear and ring topology [4, 5]. The physical effects arising from confinement and chain topology
play a significant role in the shaping of individual chromosomes and in the process of their segregation,
especially in the case of elongated bacterial cells [6]. The behaviour of linear ideal and real polymers
with excluded volume interaction (EVI) in a good solvent confined in a slit of two parallel repulsive [7, 8],
inert or mixed walls is well understood [8]. Unfortunately, the physics of confined ideal ring polymers
and ring polymers with EVI effects is still unclear. Ring polymers with specified knot type were chem-
ically synthesized a long time ago [9]. Ring topology of polymers influences the statistical mechanical
properties of these polymers, for example the scaling properties [10, 11] and shape [2, 12, 13] because it
restrains the accessible phase space. An interesting point which was confirmed by numerical studies in
[14] is that longer ring polymers are usually knotted with higher frequency and complexity. In [15], it was
established that ring polymers with more complex knots are more compact and have a smaller radius of
gyration and this decreases their ability to spread out under confinement. The results of Monte Carlo
simulations performed in [16] suggest that the knotted ring polymers will exert higher entropic forces on
the walls of the confining slit than the unknotted or linear polymers. In [16] it was stated that the knotted
ring polymers expanded as the width of the slit increased in contrast to the behaviour of unknotted (or
linear) polymers whose size showed a plateau after a certain width of slit was reached. The entropic force
exerted on the walls arising from confinement to a slit of a knotted ring polymer was calculated using
a bead-spring model by Matthews et al. in [5]. It was found [5] that in the case of a narrow slit, more
complex knot types in a ring polymer exert higher forces on the confining walls of the slit in comparison
to unknotted polymers of the same length, and for the relatively wide slits, the opposite situation takes
© Z. Usatenko, J. Halun, P. Kuterba, 2016 43602-1
http://dx.doi.org/10.5488/CMP.19.43602
http://www.icmp.lviv.ua/journal
Z. Usatenko, J. Halun, P. Kuterba
place. Confining ring polymer to a slab results in a loss of configurational entropy and leads to the appear-
ance of a repulsive force which depends on the entanglements between two walls of the confining slab,
as it was shown in the framework of a new numerical approach based on the generalized atmospheric
sampling (GAS) algorithm for lattice knots in [17]. Thus, at the moment most of the papers devoted to the
investigation of the behaviour of ring polymers compressed in confined geometries like slit or squeezed
by a force in a slab of two parallel walls deal with numerical methods and present analytical results are
incomplete. The above mentioned arguments stimulate us to apply one of the powerful analytical meth-
ods referred to as themassive field theory approach in a fixed space dimensions d < 4 (see [18, 19]) for the
investigation of ring polymers confined to a slit geometry of two parallel walls or immersed in a solution
of big mesoscopic colloidal particles of different size. This method, as it was shown in the series of papers
[8, 20, 21], provides a good agreement with the experimental data and with the results of Monte Carlo
simulations. We consider a dilute polymer solution, where different polymers do not overlap and the be-
haviour of such polymer solution can be described by a single polymer. As it is known, taking into account
the polymer-magnet analogy developed by de Gennes [22, 23], the scaling properties of long-flexible poly-
mers in the limit of an infinite number of steps N may be derived from a formal n → 0 limit of the field
theoretical φ4 O(n)-vector model at its critical point. In this case, the 1/N value plays the role of a crit-
ical parameter analogous to the reduced critical temperature in magnetic systems. Besides, we assume
that the surfaces of the confining slit are impenetrable. It means that the correspondent potential U (z̃) of
the interaction between the monomers of a polymer chain and a wall tends to infinity U (z̃) →∞ when
the distance z̃ between a wall and polymer is less than monomer size l . The deviation from the adsorp-
tion threshold [c ∝ (T −Ta)/Ta] (where Ta is adsorption temperature) changes the sign at the transition
between the adsorbed (the so-called normal transition, c < 0) and the nonadsorbed state (ordinary tran-
sition, c > 0) [24, 25] and it plays the role of a second critical parameter. The value c corresponds to the
adsorption energy divided by kBT (or the surface enhancement in field theoretical treatment). The ad-
sorption threshold for long-flexible polymers takes place, where 1/N → 0 and c → 0. As wasmentioned by
de Gennes [22, 23], the partition function Z (x̃, x̃
′) of a single polymer with two ends fixed at x̃ and x̃
′ is con-
nected with the two-point correlation function G(2)(x̃, x̃
′) = 〈~φ(x̃)~φ(x̃
′)〉 in φ4 O(n)-vector model for field
~φ(x̃) with the components φi (x̃), i = 1, . . . ,n [and x̃ = (r̃, z̃)] via the µ2
0
→ L0 inverse Laplace transform:
Z (x̃, x̃
′; N , v0) =IL µ2
0
→L0
(〈~φ(x̃)~φ(x̃
′)〉|n→0) in the limit, where the number n of components tends to zero.
The conjugate Laplace variable L0 has the dimension of length squared and is proportional to the total
number of monomers N which form the polymer. The effective Ginzburg-Landau-Wilson Hamiltonian
describing the system in semi-infinite (i = 1) or confined geometry of two parallel walls (i = 1,2) is [24]:
H [~φ,µ0] =
∫
d
d x̃
[
1
2
(
∇~φ
)2 + µ0
2
2
~φ2 + v0
4!
(
~φ2
)2
]
+
2
∑
i=1
ci0
2
∫
d
d−1r̃~φ2
, (1)
where the conjugate chemical potential µ0 is the “bare mass” in field-theoretical treatment, v0 is the
“bare coupling constant” which characterizes the strength of EVI. In the case of slit geometry, the walls
are located at the distance L one from another in z̃-direction in such way that the surface of the bot-
tom wall is located at z̃ = 0 and the surface of the upper wall is located at z̃ = L. Each of the two sur-
faces is characterized by a certain surface enhancement ci0
, where i = 1,2. In the case when the ends
of polymer x̃ and x̃
′ in partition function Z (x̃, x̃
′; N , v0) coincide, such partition function corresponds
to the partition function of a phantom ring polymer, i.e., a ring polymer where we perform the sum-
mation over all possible knot structures. The fundamental two-point correlation function of the Gaus-
sian theory corresponding to the effective Ginzburg-Landau-Wilson Hamiltonian (1) in a mixed p̃, z̃ rep-
resentation is: G(2)
i j
(p̃, p̃
′; z̃, z̃ ′) = (2π)d−1δi j δ(p̃+ p̃
′)G̃∥(p̃; z̃ , z̃ ′;µ0,c10
,c20
,L), where the free propagator
G̃∥(p̃; z̃, z̃ ′;µ0,c10
,c20
,L) of the model (1) in the case of ring polymer with coinciding ends x̃ = x̃
′ = x can be
obtained by analogy as it was done in [8] for linear polymer. It should be taken into account that we have
ring polymer of length L0 in a slit geometry of two parallel walls with fixed position of one monomer
at point x = (r, z) by analogy as it was proposed in [25] for description of ring polymer adsorption. Such
representation assumes that the respective correlation function for ring polymer depends on x = (r, z),
e.g., on the position of some monomer which can be the first and the last monomer in the ring. Here,
p̃, p̃
′ are the values of parallel momentum associated with d − 1 translationally invariant directions in
the system. The interaction between the polymer and the walls is implemented by the Dirichlet-Dirichlet
43602-2
Ring polymers in confined geometries
boundary conditions (D-D b.c.) (see [7, 8, 24]): c1 →+∞ , c2 →+∞ or ~φ(r,0) = ~φ(r,L) = 0. We consider the
dilute solution of phantom ideal ring polymers and ring polymers with EVI immersed in a slit geometry of
two parallel repulsive walls and permit the exchange of polymer coils between the slit and the reservoir.
The polymer solution in the slit is in equilibrium contact with an equivalent solution in the reservoir
outside of the slit. We follow the thermodynamic description of the problem as it was given in [7, 8] and
perform the calculation in the framework of the grand canonical ensemble where the chemical poten-
tial µ is fixed. As it was shown in [7], the free energy of interaction between the walls in such a grand
canonical ensemble is defined as the difference of the free energy of an ensemble where the separation
of the walls is fixed at a finite distance L and where the walls are separated infinitely far from each other:
δFR = −kBT N ln
{
Z
R
∥ (L)/[Z R
∥ (L →∞)]
}
, where N is the total number of polymer coils in the solution
and T is the temperature. The Z
R
∥ (L) value is the partition function of one ring polymer located in a
volume V containing two walls at a distance L. The correspondent reduced free energy of interaction δ f
per unit area A = 1 for the case of ring polymer confined in a slit geometry of two parallel walls after
performing Fourier transform in the direction parallel to the surfaces and integration over dd−1r (which
gives us the delta function δ(p) and leads to p = 0 in the respective correlation functions after integration
over dd−1p) may be written in the form (see [8]):
δ f R = δFR
npkBT
= L−
L
∫
0
dz
Ẑ
R
I (z)
Ẑb
+
∞
∫
0
dz
[
Ẑ
R
HS1
(z)
Ẑb
−1
]
+
∞
∫
0
dz
[
Ẑ
R
HS2
(z)
Ẑb
−1
]
. (2)
Here, np = N /V is the polymer density in the bulk solution, and Ẑb = IL µ2
0
→R2
x /2
(
1
2µ0
)
. The functions
Ẑ
R
I (z) and Ẑ
R
HSi
(z) are equal to:
Ẑ
R
I (z) =IL µ2
0
→L0
G(2)
(p = 0; z, z)
∣
∣
n→0
, and Ẑ
R
HSi
(z)=IL µ2
0
→L0
G(2)
HSi
(p = 0; z, z)
∣
∣
n→0
,
where G(2)(p = 0; z, z) and G(2)
HSi
(p = 0; z, z) with i = 1,2 are correlation functions of the model equation (1)
describing the system in a slit geometry of two parallel walls and in semi-infinite geometry, respectively
(see [8, 20]). Dividing the reduced free energy of interaction δ f R by another relevant length scale, for exam-
ple, the size of the polymer in bulk, e.g. Rx (where R2
g =χ2
d
R2
x /2, and χd is a universal numerical prefactor
depending on the dimension d of the system (see [25],[26]) and Rg is the radius of gyration), yields the
universal scaling function for the depletion interaction potential ΘR(y) = δ f R/Rx , where y = L/Rx . The
resulting scaling function for the depletion force between the two walls induced by the polymer solution
is denoted as (see [8]):
Γ
R
(y) =−d(δ f R)
dL
=−dΘR(y)
dy
. (3)
As it is known [25], the resulting force exerted on the surfaces of a confining slit by polymer is equal to
the resulting depletion forcewith the opposite sign: K R = d(δ f R)/dL.
Let us consider at the beginning the case of phantom ideal ring polymer under Θ-solvent condition
trapped in the slit geometry of two parallel repulsive walls. Taking into account that in the case of wide
slit region we have y & 1 for the scaling function of the depletion force ΓR we obtain:
Γ
R,id
DD
(y) = 2e
−2y2
−8y2
e
−2y2
. (4)
The asymptotic solution for the depletion force in the narrow slit region, where y ≪ 1, simply becomes:
Γ
R,id
DD, narr
≈−1. The obtained results for the depletion force Γ
R,id
DD
are presented in figure 1 (a) by blue lines
with triangles for awide slit region and by green lines with triangles for a narrow slit region, respectively.
It should be mentioned that the quantity Γ is normalized to the overall polymer density np. Thus, the
above result simply indicates that the force is entirely induced by free chains surrounding the slit, or,
in other words, by the full bulk osmotic pressure from the outside of the slit. We can state that in the
case of very narrow slit with two repulsive walls, the ring polymers would pay a very high entropy to
stay in the slit or even enter it. As the next step, let us consider the dilute solution of ring polymers
with EVI in a good solvent immersed in a slit geometry of two parallel repulsive walls. As it is known
[22, 23], in a good solvent the effects of the EVI between monomers play a crucial role so that the polymer
43602-3
Z. Usatenko, J. Halun, P. Kuterba
0,0 0,4 0,8 1,2 1,6 2,0
-1,0
-0,8
-0,6
-0,4
-0,2
0,0
a
R
D
D
y
y=L/R
x
ideal ring (wide)
real ring with
EVI (wide)
ideal ring (narrow)
real ring with
EVI (narrow)
8,0 10,0 12,0 14,0 16,0 18,0 20,0
0,0
0,2
0,4
0,6
0,8
1,0
1,2
b
K
R
D
D
(L
)
L[l]
real, R
g
=6.9
real, R
g
=7.28
real, R
g
=7.78
ideal, R
g
=6.9
ideal, R
g
=7.28
ideal, R
g
=7.78
Figure 1. (Color online) (a) The scaling functions Γ
R
DD
(y) for phantom ideal ring and ring polymer with
EVI in a good solvent immersed between two repulsive walls; (b) The functions K R(L) for phantom ideal
ring polymer and ring polymer with EVI in a good solvent immersed between two repulsive walls for
different values of the radius of gyration: Rg(121) = 6.9±0.01[l ] (black lines and black lines with squares);
Rg(91) = 7.28±0.01[l ] (red dashed lines and red lines with dots); Rg(61) = 7.78±0.01[l ] (green dashed lines
and green lines with triangles).
coils occupy a large space compared to the case of ideal polymers. The calculations of the correspondent
partition functions Ẑ
R
I
(z) and Ẑ
R
HSi
(z), which permit to obtain the reduced free energy of interaction δ f R
in equation (2), are connected with the calculations of the correspondent correlation functions G(2)(p =
0; z, z) and G(2)
HSi
(p = 0; z, z) with i = 1,2 where the terms describing the EVI effects are taken into account
via using perturbation treatment in the framework of the massive field theory approach in a fixed space
dimensions d = 3 up to one-loop order approximation. In order to make the theory UV finite in RG sense
directly in d = 3 space dimensions, we perform a standard mass renormalization µ2
0
= µ2 −δµ2
0
and the
coupling constant renormalization v0 = µv of the above mentioned correlation functions by analogy as
it was proposed by Parisi [19]. Besides, the surface enhancement renormalization ci0
= ci +δci of the
correspondent correlation functions in the case of D-D b.c. reduces to an additive renormalization as
it took place in the case of semi-infinite geometry [20] and slit geometry (see [8]). The correspondent
expression for the scaling function of the depletion force between two repulsive walls in the case of wide
slit region y & 1 is:
Γ
R, real
DD
(y) = 2e
−2y2
−8y2
e
−2y2
−
ṽ
2
e
−2y2
[
1
y2
+2
p
πψ(0)
(
−
1
2
)
−D −4yB(y)
]
, (5)
where D = 1
4
[
10−2ln 2−2ψ(0)(− 1
2
)+
p
πψ(0)( 1
2
)−2γE
]
, B(y) =
[
2y
p
πψ(0)(− 1
2
)− 1
y
]
and γE = 0.577 is the
Euler’s constant, ψ(0)(z) is the digamma function, erfc(z) = 1− erf(z) is the complementary error func-
tion. Besides, the v = bn ṽ value was introduced with bn = 6
n+8
(4π)3/2
Γ(1/2)
and the calculations are performed
at the correspondent fixed point ṽ∗ = 1 in the limit n → 0. In the region of a very narrow slit y ≪ 1
the depletion force is: Γ
R, real
DD, narr
≈− 1. The results of calculations for Γ
R, real
DD
are presented in figure 1 (a)
by the red line with circles in a wide slit region and by the black line with open circles in a narrow slit
region, respectively. As it is easy to see from figure 1 (a), the obtained results for the scaling function for
the depletion force in the case of ring polymer immersed between two repulsive walls are characterized
by a completely different behaviour compared to the case of linear polymer (see [8]). The phantom ring
polymer, due to the complexity of chain topology and for entropical reasons, tends to escape from the
space between two repulsive walls and it leads to the attractive depletion force between the confining
walls. Besides, we observe [see figure 1 (a)] that the absolute value of the scaling function for the deple-
tion force for phantom ring polymers in the wide slit region is smaller than for linear polymer chains
(see [8]) and decreases as the width of the slit increases. In figure 1 (b) we present the results for the en-
tropically induced force K R (as functions of L for different values of radius of gyration Rg) which exerts
phantom ideal ring polymer and ring polymer with EVI on the confining two repulsive walls. It should
be mentioned that calculations, presented in figure 1 (b) were performed for different values of radius of
43602-4
Ring polymers in confined geometries
gyration [5]: Rg(121) = 6.9±0.01[l ], Rg(91) = 7.28±0.01[l ], and Rg(61) = 7.78±0.01[l ] which correspond to
the ring polymers with different knot types: 121;91;61. Here, Cp is a standard notation [27], where C de-
notes the minimum number of crossings in any projection on a plane and p is used in order to distinguish
the knot types with the same C . It should be mentioned that the region of validity of the obtained results
for the entropically induced forces K R for phantom ideal ring polymers [see figure 1 (b)] is defined by
the value of y = L/Rx & 1 with Rx =
p
2Rg. We observed that in the wide slit region, ring polymers with
less complex knot types (with bigger radius of gyration) in a ring topology exert higher forces on the
confining walls [see figure 1 (b)] at the same length L. In figure 2 (a), we present the ratio K R
DD/K lin
DD of
force for ring polymer chains and corresponding force for linear polymer chains in a slit geometry of
two parallel repulsive walls as function of the distance L between the walls for different values of the
radius of gyration Rg. Our results for entropically induced force K R are in agreement with the previous
results obtained by Matthews et al. in [5] using a bead-spring model. The difference between our results
and the results obtained by Matthews et al. arises due to the fact that in [5] calculations were performed
for relatively short polymer chains with polymer length of order N ∼ 300 units. Taking into account the
Derjaguin approximation [28], which describes the sphere of the big colloidal particle by a superposition
of immersed plates with local distance from the wall (or from other particle), we performed calculations
of the depletion force −d[ΦR
DD(ỹ)/(npkBT )]/dỹ between colloidal particle and a wall (or between two col-
loidal particles) in a dilute solution of ring polymers. It should be mentioned that ΦR
DD(ỹ)/(npkBT ) is
equal to 2πR̃R2
x
∫∞
ỹ dyΘR(y) with ỹ = a/
p
2Rg (where a is the minimal distance between the sphere and
the wall) and R̃ = R for the case of big colloidal particle of radius R ≫ L (and R ≫ Rg) near the wall and
R̃ = R1R2/(R1 +R2) for the case of two colloidal particles with different radius R1 ,R2 (when Ri ≫ L and
Ri ≫ Rg, i = 1,2). The results of calculations are presented in figure 2 (b) and indicate that the absolute
value of the depletion forcewhich arises between two colloidal particles is smaller than between particle
and wall. Besides, if the number of coils N can be fixed, then the total free energy of the polymer solution
within the slit in the canonical ensemble can be obtained from our results for the grand canonical free
energy via the Legendre transform by analogy as it took place in [7, 8] for linear polymers. The further in-
vestigation of solution of ring polymers with EVI immersed in a slit geometry of two inert walls or mixed
walls as well as the investigation of polymer-colloid interactions for different sorts of colloids is the task
of great interest which is under consideration.
8,0 10,0 12,0 14,0
0,4
0,6
0,8
1,0
1,2
a
K
R
D
D
(L
)/K
lin
D
D
(L
)
L[l]
ideal, Rg=6.9
ideal, Rg=7.28
ideal, Rg=7.78
0,8 1,0 1,2 1,4 1,6 1,8
-0,08
-0,06
-0,04
-0,02
0,00
b
ideal ring, SW
real ring, SW
ideal ring, SS
real ring, SS-d
(
D
D
(y
)/n
pk BT)
/d
y
y=a/R
x
Figure 2. (Color online) (a) The ratio K R(L)/K lin(L) as function of the distance L (in l units) between
the walls for different values of Rg; (b) The functions −d[ΦR
DD
(ỹ)/(npkBT )]/dỹ for phantom ideal ring
polymer and ring polymer with EVI in a good solvent immersed between colloidal particle and wall as
well as between two colloidal particles.
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Кiльцевi полiмери в обмежених середовищах
З. Усатенко1, Й. Халюн2, П. Кутерба2
1 Iнститут Фiзики, Кракiвський Полiтехнiчний Унiверситет, 30-084 Кракiв, Польща
2 Iнститут Фiзики, Ягеллонський Унiверситет, 30-348 Кракiв, Польща
Проведено дослiдження розведених розчинiв iдеальних кiльцевих та кiльцевих полiмерiв з ефектами
виключеного об’єму в доброму розчиннику, обмежених в щiлинi двох паралельних вiдштовхуючих по-
верхонь, а також у розчинi колоїдних частинок великого розмiру. Беручи до уваги вiдповiднiсть мiж те-
оретикопольовою φ4 O(n)-векторною моделлю в границi n → 0 i поведiнкою довгих гнучких полiмерiв
в доброму розчиннику, було отримано з використанням масивної теорiї поля при фiксованiй вимiрностi
простору d = 3 в рамках однопетлевого наближення вiдповiднi сили збiднення i сили, якi дiють мiж iде-
альними кiльцевими та кiльцевими полiмерами з ефектами виключеного об’єму i стiнками щiлини. Окрiм
того, беручи до уваги наближення Дєрягiна, розраховано сили збiднення мiж колоїдною частинкою i стiн-
кою, а також мiж двома колоїдними частинками великого розмiру. Отриманi результати вказують на те,
що кiльцевi полiмери через складнiсть топологiї i з ентропiйних причин демонструють цiлком iншу пове-
дiнку в обмежених середовищах нiж лiнiйнi полiмери.
Ключовi слова: колоїднi системи, критичнi явища, полiмери, фазовi переходи
43602-6
http://dx.doi.org/10.1073/pnas.032095099
http://dx.doi.org/10.1080/00268976.2011.556094
http://dx.doi.org/10.1073/pnas.0605305103
http://dx.doi.org/10.1103/PhysRevE.63.041803
http://dx.doi.org/10.1103/PhysRevE.80.041802
http://dx.doi.org/10.1002/anie.198901891
http://dx.doi.org/10.1073/pnas.0330884100
http://dx.doi.org/10.1103/PhysRevLett.98.058102
http://dx.doi.org/10.1103/PhysRevLett.99.198102
http://dx.doi.org/10.1021/ma801389c
http://dx.doi.org/10.1088/0305-4470/23/15/028
http://dx.doi.org/10.1103/PhysRevLett.73.3317
http://dx.doi.org/10.1088/1742-5468/2007/03/P03001
http://dx.doi.org/10.1088/1742-5468/2012/09/P09004
http://dx.doi.org/10.1007/BF01014429
http://dx.doi.org/10.1016/S0550-3213(98)00489-1
http://dx.doi.org/10.1063/1.3529426
http://dx.doi.org/10.1016/0375-9601(72)90149-1
http://dx.doi.org/10.1103/RevModPhys.79.611
http://dx.doi.org/10.1007/BF01433225
|
| id | nasplib_isofts_kiev_ua-123456789-156549 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T16:57:26Z |
| publishDate | 2016 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Usatenko, Z. Halun, J. Kuterba, P. 2019-06-18T16:26:21Z 2019-06-18T16:26:21Z 2016 Ring polymers in confined geometries / Z. Usatenko, J. Halun, P. Kuterba // Condensed Matter Physics. — 2016. — Т. 19, № 4. — С. 43602: 1–6. — Бібліогр.: 28 назв. — англ. 1607-324X PACS: 64.70.pv, 61.25.he, 67.30.ej, 68.35.Rh, 64.70.qd DOI:10.5488/CMP.19.43602 arXiv:1608.03117 https://nasplib.isofts.kiev.ua/handle/123456789/156549 The investigation of a dilute solution of phantom ideal ring polymers and ring polymers with excluded volume
 interactions (EVI) in a good solvent confined in a slit geometry of two parallel repulsive walls and in a solution
 of colloidal particles of big size was performed. Taking into account the correspondence between the field
 theoretical φ⁴ O(n)-vector model in the limit n → 0 and the behaviour of long-flexible polymers in a good
 solvent, the correspondent depletion forces and the forces which exert phantom ideal ring polymers and ring
 polymers with EVI on the walls were obtained in the framework of the massive field theory approach at fixed
 space dimensions d=3 up to one-loop order. Besides, taking into account the Derjaguin approximation, the
 depletion forces between big colloidal particle and a wall and in the case of two big colloidal particles were
 calculated. The obtained results indicate that phantom ideal ring polymers and ring polymers with EVI due to
 the complexity of chain topology and for entropical re Проведено дослiдження розведених розчинiв iдеальних кiльцевих та кiльцевих полiмерiв з ефектами
 виключеного об’єму в доброму розчиннику, обмежених в щiлинi двох паралельних вiдштовхуючих поверхонь, а також у розчинi колоїдних частинок великого розмiру. Беручи до уваги вiдповiднiсть мiж теоретикопольовою φ⁴ O(n)-векторною моделлю в границi n → 0 i поведiнкою довгих гнучких полiмерiв
 в доброму розчиннику, було отримано з використанням масивної теорiї поля при фiксованiй вимiрностi
 простору d = 3 в рамках однопетлевого наближення вiдповiднi сили збiднення i сили, якi дiють мiж iдеальними кiльцевими та кiльцевими полiмерами з ефектами виключеного об’єму i стiнками щiлини. Окрiм
 того, беручи до уваги наближення Дєрягiна, розраховано сили збiднення мiж колоїдною частинкою i стiнкою, а також мiж двома колоїдними частинками великого розмiру. Отриманi результати вказують на те,
 що кiльцевi полiмери через складнiсть топологiї i з ентропiйних причин демонструють цiлком iншу поведiнку в обмежених середовищах нiж лiнiйнi полiмери. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Ring polymers in confined geometries Кiльцевi полiмери в обмежених середовищах Article published earlier |
| spellingShingle | Ring polymers in confined geometries Usatenko, Z. Halun, J. Kuterba, P. |
| title | Ring polymers in confined geometries |
| title_alt | Кiльцевi полiмери в обмежених середовищах |
| title_full | Ring polymers in confined geometries |
| title_fullStr | Ring polymers in confined geometries |
| title_full_unstemmed | Ring polymers in confined geometries |
| title_short | Ring polymers in confined geometries |
| title_sort | ring polymers in confined geometries |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/156549 |
| work_keys_str_mv | AT usatenkoz ringpolymersinconfinedgeometries AT halunj ringpolymersinconfinedgeometries AT kuterbap ringpolymersinconfinedgeometries AT usatenkoz kilʹcevipolimerivobmeženihseredoviŝah AT halunj kilʹcevipolimerivobmeženihseredoviŝah AT kuterbap kilʹcevipolimerivobmeženihseredoviŝah |