Temperature dependence of some properties of star-branched polymers in confined space. A Monte Carlo study
A simulation study of star-branched molecules with 3 arms was performed in a space confined between two parallel repulsive walls, which were at the distance d apart. The simulations were performed for chains on a simple cubic lattice. The chains consisted of N = 49 , 100 and 199 beads. During th...
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| Cite this: | Temperature dependence of some properties of star-branched polymers in confined space. A Monte Carlo study / P. Romiszowski // Condensed Matter Physics. — 2002. — Т. 5, № 1(29). — С. 5-14. — Бібліогр.: 16 назв. — англ. |
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irk-123456789-1205742017-06-13T03:05:08Z Temperature dependence of some properties of star-branched polymers in confined space. A Monte Carlo study Romiszowski, P. A simulation study of star-branched molecules with 3 arms was performed in a space confined between two parallel repulsive walls, which were at the distance d apart. The simulations were performed for chains on a simple cubic lattice. The chains consisted of N = 49 , 100 and 199 beads. During the simulations the solvent quality has been varied by means of changing the segment-segment interactions. Also the thickness of a layer between the walls has been changed. The results show that there is a coil-to-globule transition. The transition temperature does not depend on the confinement. However, the scaling exponents of the radius of gyration of the chains are sensitive to value d . За допомогою копм’ютерного моделювання досліджено зірково-гіллясті молекули з 3 -х ланцюгів у просторі між двома паралельними відштовхувальними стінками, розташованими на відстані d одна від другої. Моделювання проводилося для ланцюгів на простій кубічній гратці. Ланцюги складалися з N = 49 , 100 і 199 ланок. Якість розчинника під час моделювання регулювалася за допомогою зміни взаємодії між сегментами. Також змінювалася товщина шару між стінками. Результати свідчать про те, що відбувається перехід полімерний клубок–глобула. Температура переходу не залежить від обмеження простору. Однак, показники скейлінгу радіусу гірації чутливі до величини d . 2002 Article Temperature dependence of some properties of star-branched polymers in confined space. A Monte Carlo study / P. Romiszowski // Condensed Matter Physics. — 2002. — Т. 5, № 1(29). — С. 5-14. — Бібліогр.: 16 назв. — англ. 1607-324X PACS: 02.70.Lq, 36.20.Ey, 61.25.Hq, 07.05.Tp DOI:10.5488/CMP.5.1.5 http://dspace.nbuv.gov.ua/handle/123456789/120574 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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A simulation study of star-branched molecules with 3 arms was performed
in a space confined between two parallel repulsive walls, which were at the
distance d apart. The simulations were performed for chains on a simple
cubic lattice. The chains consisted of N = 49 , 100 and 199 beads. During
the simulations the solvent quality has been varied by means of changing
the segment-segment interactions. Also the thickness of a layer between
the walls has been changed. The results show that there is a coil-to-globule
transition. The transition temperature does not depend on the confinement.
However, the scaling exponents of the radius of gyration of the chains are
sensitive to value d . |
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| author |
Romiszowski, P. |
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Romiszowski, P. Temperature dependence of some properties of star-branched polymers in confined space. A Monte Carlo study Condensed Matter Physics |
| author_facet |
Romiszowski, P. |
| author_sort |
Romiszowski, P. |
| title |
Temperature dependence of some properties of star-branched polymers in confined space. A Monte Carlo study |
| title_short |
Temperature dependence of some properties of star-branched polymers in confined space. A Monte Carlo study |
| title_full |
Temperature dependence of some properties of star-branched polymers in confined space. A Monte Carlo study |
| title_fullStr |
Temperature dependence of some properties of star-branched polymers in confined space. A Monte Carlo study |
| title_full_unstemmed |
Temperature dependence of some properties of star-branched polymers in confined space. A Monte Carlo study |
| title_sort |
temperature dependence of some properties of star-branched polymers in confined space. a monte carlo study |
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Інститут фізики конденсованих систем НАН України |
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2002 |
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http://dspace.nbuv.gov.ua/handle/123456789/120574 |
| citation_txt |
Temperature dependence of some
properties of star-branched polymers in
confined space. A Monte Carlo study / P. Romiszowski // Condensed Matter Physics. — 2002. — Т. 5, № 1(29). — С. 5-14. — Бібліогр.: 16 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT romiszowskip temperaturedependenceofsomepropertiesofstarbranchedpolymersinconfinedspaceamontecarlostudy |
| first_indexed |
2025-07-08T18:09:27Z |
| last_indexed |
2025-07-08T18:09:27Z |
| _version_ |
1837103232582680576 |
| fulltext |
Condensed Matter Physics, 2002, Vol. 5, No. 1(29), pp. 5–14
Temperature dependence of some
properties of star-branched polymers in
confined space. A Monte Carlo study
P.Romiszowski∗
Department of Chemistry, University of Warsaw,
02–093 Warszawa, Poland
Received August 30, 2001
A simulation study of star-branched molecules with 3 arms was performed
in a space confined between two parallel repulsive walls, which were at the
distance d apart. The simulations were performed for chains on a simple
cubic lattice. The chains consisted of N = 49 , 100 and 199 beads. During
the simulations the solvent quality has been varied by means of changing
the segment-segment interactions. Also the thickness of a layer between
the walls has been changed. The results show that there is a coil-to-globule
transition. The transition temperature does not depend on the confinement.
However, the scaling exponents of the radius of gyration of the chains are
sensitive to value d .
Key words: Monte Carlo simulation, confined space, branched polymers,
coil-globule transition
PACS: 02.70.Lq, 36.20.Ey, 61.25.Hq, 07.05.Tp
1. Introduction
For the recent 20 years, the problem of the properties of polymeric systems that
are observed in a special environment has attracted the attention of numerous re-
searchers. The modern technology of polymers as well as their application required
the solution of many basic problems connected with the specific geometrical con-
finement. Many polymeric systems used as composites, microlayers or thin films
exhibit quite different properties compared to the melt or solution systems in un-
restricted 3D space. The presence of geometrical constraints causes a considerable
shrinking of the conformational space of the system. The space confinement most
frequently used in literature was a slit [1–9], which served as a tool for investiga-
tions of the properties of linear polymers in thin layers, microstructures and finally
in biosystems. Moreover, the cylindrical space confining the molecules was applied
∗E-mail: prom@chem.uw.edu.pl
c© P.Romiszowski 5
P.Romiszowski
in simulations [10]. Some papers were devoted to the static and dynamic properties
of a polymer melt in confinement [11,12]. All mentioned authors have reported the
system anisotropy which effected the specific properties of the systems.
The properties of star branched polymers in unrestricted space have recently
been studied by many authors. They paid attention to both the static and dynamic
properties of the systems under consideration. The star-branch molecules appear to
be nice stepping-stone model objects for studying the systems in which the topology
of the molecule plays an important role. The simplest star-branched polymers are
the molecules with f = 3 arms of equal length. As it has been shown, such objects
sometimes exhibit the properties different from their linear counterparts [13]. The
differences were mostly caused by the non-uniform density of stars as well as by the
interactions between the arms.
The confinement of the molecule changes the properties of the molecule which
does not behave as a free chain – the geometrical obstacles change the configuration
space in comparison with the unrestricted conditions. Also, the interactions with
the walls confining the object deform the molecule. As a result of these phenomena,
one observes considerable effects of the presence of confinement in static and dy-
namic properties of the system. An interesting result concerning the coil-to-globule
transition has recently been presented by Sotta et al. [10] that carried out the sim-
ulations of a grafted polymer chain confined in a tube. They found out that the
coil-globule transition is continuous. As they showed, the chain structure was a suc-
cession of blobs within which the transition took place independently of the overall
chain length.
In this study, the influence of temperature (solvent quality) of the system is
investigated, especially with respect to the size of the molecules located between
the two impenetrable walls. Also, the coil-to-globule transition of the system has
been derived from the simulations. The simulations were performed for a series of
different separations between the walls in order to see the effect of the size of the
slit. On the other hand, the solvent quality was varied as well as the chain length.
Here the three different chain lengths (the next one doubled the previous length)
were used, which enabled one to obtain the scaling of the system.
2. The model
The model system considered in these simulations consisted of a Monte Carlo
box with periodic boundary conditions in xy plane. The walls perpendicular to z
axis were impenetrable for the chain located between them. The scheme of the sys-
tem is shown in figure 1. The impenetrable (purely repulsive) walls were separated
at a distance d, which was varied in a series of simulations. The simulations were
started from d = 3, i.e. the distance between the walls was 3 lattice units, which
is the distance large enough to enable all the micromodifications of the chain in 3
dimensions provided that the ergodycity of the system is present. The variation of
the distance between the walls d enabled one to study the effect of deformation of
the chains in two regimes: first (d much less than the mean square radius of gyra-
6
Star-branched polymers in confined space
G
�
]
\
[
Figure 1. Schematic representation of the chain located between the walls used
in the simulations.
tion without any confinement 〈S2〉0 in which the chain is highly squeezed, but still
all possible micromodifications of the chain segments can be executed; the second
regime is at d larger than 〈S2〉0, but still the chain “feels” the confinement. The
chains were simulated on a simple cubic lattice with excluded volume effect present,
which means that the double occupation of the same lattice knot was forbidden.
The molecules consisted of f = 3 arms of equal length n = 17, 34 and 67 emanating
from one point, called a branching point. The total number of beads in a molecule
was N = f(n−1)+1, in this case the chain length was N = 49, 100 and 199 beads,
respectively. In order to study the properties of the system along with the change
of the solvent quality, a contact potential of the mean force V = ε has been intro-
duced. The potential has been in effect for the pairs of nonbonded beads separated
by a distance of 1 lattice unit; otherwise, for longer distance V = 0. No other local
potentials were present in the system. Since it has been assumed that the attrac-
tive forces are present in the system, the interaction parameter ξ = ε/kT has been
negative. The reference state (ε = 0) means that the solvent-segment and segment-
solvent interactions are identical. For ε < 0, the segment-segment interactions are
more preferable than segment-solvent interactions and one observes gradual changes
the of the properties of the system. The changes in ε mean the changes in solvent
quality – the more negative ε, the worse is the solvent quality. On the other hand,
the changes in solvent quality can be interpreted as the changes of the temperature
T of the system without changing the solvent. In this paper this method will be
used, as a measure of the temperature T ∗ = |kT/ε| = 1/|ξ| is applied.
3. Results and discussion
The simulations were started at given T ∗ and d = 3 from a flat random struc-
ture located between the walls. Then, the equilibration run was performed and the
simulation was started. During the simulation run, the trajectory was recorded –
it consisted of 3000 to 5000 snapshots of the system taken in equal time intervals.
Having finished the simulation, the distance between the walls was increased and
the new procedure was repeated. The last chain conformation served as a starting
chain for the new equilibration run and so on. After the series of simulations, the
trajectories were analyzed and the data were extracted. It was the aim of this study
to examine the coil-to-globule transition in the confined system as well as to obtain
the scaling dependencies.
7
P.Romiszowski
The initial configuration of a model chain in a given simulation run underwent
a series of local micromodifications during which the positions of some beads (seg-
ments) were randomly changed. As it has been shown, there exists a set of ele-
mentary micromodifications which allows the chain to move to any point in the
phase space. These are the following: (i) two-bond chain end motion, (ii) two-bond
kink motion, (iii) three-bond kink motion, (iv) three-bond 90◦ crankshaft motion,
(v) branching point collective, i.e. accompanied by the move of one of the nearest
bonded bead, motion. A detailed description of the above elementary motions was
given elsewhere [14]. Since the total energy of the system was the sum of all pairwise
segment-segment contacts (the contact was assumed when the distance between the
non-bonded beads was equal to one lattice unit, i.e. 1), the micromodifications of the
chain usually changed the total configurational energy of the molecule along with
the changes of a number of contacts between the segments. In the simulation, the
number of these contacts has been monitored continuously. The Metropolis sampling
was applied in order to efficiently sample the configurational space and to calculate
the temperature dependencies of the system. The transition of the system from an
“old” to a “new” configuration was effective unconditionally when the number of
contacts did not decrease (i.e. the configurational energy remained constant or was
increased). If the number of segment-segment contacts decreased during the micro-
modification, then the probability of the acceptance of such a move was computed
according to the relation P = exp(−∆E/kT ) (where ∆E was the difference between
the “new” and “old” conformation) and then the change of the chain conformation
was executed with this probability. By performing the series of simulations at a
wide range of reduced temperature T ∗ (from T ∗ = 10 to 1) one could examine the
properties of the chains both in a good and in a poor solvent regime.
The changes in the solvent quality strongly influence the dimensions of the chains,
which are usually characterized by the mean square radius of gyration 〈S 2〉 - this
parameter was calculated for each simulation condition (d and T ∗). In order to
study the dependence of chain dimensions on the distance between the plates d for
different solvent quality, we have plotted the results in figures 2–4 which are done for
different chain lengths. The x-axis of these figures represents the ratio d/2〈S 2
0
〉
1/2
T .
The doubled mean square root of the radius of gyration with no confinement present
〈S2
0
〉T taken at a given temperature T ∗ , is roughly equal to the diameter of the coiled
macromolecule. Calculating the ratio of this quantity with respect to the distance
between the wall, enables one to estimate the proportion between the size of the
molecule and d, which results in the degree of “squeezing” by the repulsive plates
confining the chain. The y-axis of the plots is the ratio of the observed value of the
mean square radius of gyration to that obtained without the confinement at the
given solvent quality (temperature T ∗ of the system). Therefore, we can normalize
the effect of the interaction of the chain with the walls independently of the size
of the molecule. In fact, the curves obtained for different temperature values of
interaction parameter ξ form a family of similar plots. Each of these curves exhibits
the minimum which is located close to the abscissa value 1. However, one can notice
that the position of this minimum changes toward the larger values of abscissa
8
Star-branched polymers in confined space
Figure 2. The dependence of the radius of gyration of the chain on the distance
between the walls at various solvent conditions (values of interaction parameter
ξ are given in the inset) – N = 49. See text for details.
Figure 3. The dependence of the radius of gyration of the chain on the distance
between the walls at various solvent conditions (values of interaction parameter
ξ are given in the inset) – N = 100. See text for details.
9
P.Romiszowski
Figure 4. The dependence of the radius of gyration of the chain on the distance
between the walls at various solvent conditions (values of interaction parameter
ξ are given in the inset) – N = 199. See text for details.
along with the decrease of ξ, which is equivalent to the decrease of T ∗. Also, one can
observe that the depth of the minimum on curves is less pronounced as the quality
of the solvent worsens. For a bad solvent regime, the minimum vanishes, which is in
agreement with the work by Cordeiro et al. [5]
In each figure, the data for an a thermal solution (good solvent regime) are given.
The presence of the minimum on each curve has also been observed by other authors.
The explanation of this phenomenon can be as follows: as the distance between the
walls decreases, the molecule becomes more flat since the z-axis of the system is
limited by d. Therefore, the z-component of the radius of gyration decreases. On
the other hand, during the “squeezing” process the xy-components of 〈S2〉 increase.
Since 〈S2〉 is the sum of these two components, the minimum have to appear on the
plot. The location of this minimum in the vicinity of abscissa value 1 is obvious, since
at d close to the diameter of the molecule, the walls start to interact considerably
with the chain and the evident “squeezing” of the molecule starts as d decreases. As
the quality of the solvent worsens, this effect takes place at larger values of d. The
partially collapsed chain forms a more dense structure and, therefore, it starts to
interact with the walls when the distance between the plates is larger than in case
of a good solvent.
Since the simulations were made for different values of T ∗, this enabled one to
study the collapse of the chain from a coil to globular state. In order to do so, the
energy fluctuations during the simulations were calculated, which enabled one to
10
Star-branched polymers in confined space
Figure 5. Plots of heat capacity Cv as a function of reduced temperature T ∗ for
different distances between the walls – N = 49.
Figure 6. Plots of heat capacity Cv as a function of reduced temperature T ∗ for
different distances between the walls – N = 100.
11
P.Romiszowski
Figure 7. Plots of heat capacity Cv as a function of reduced temperature T ∗ for
different distances between the walls – N = 199.
plot the heat capacity Cv according to the fluctuation theorem:
Cv/k = 〈E2〉 − 〈E〉2,
where E is the total configurational energy of the molecule averaged over the large
number (∼ 106) of configurations. Plotting Cv as a function of the temperature of the
system shows whether or not there is a phase transition of the system. In figures 5–7,
the plots of Cv against the reduced temperature T ∗ are shown for different values
of d. Each of the curves exhibits the maximum which signs the position of a coil-to-
globule transition temperature. The transition temperature Tc estimated from the
plots is 1.1, 1.7 and 2.0 forN = 49, 100 and 199, respectively. These values are almost
the same as those obtained for star-branched molecules with no confinement (1.0
for N = 49 and 2.0 for N = 199) [15]. As one can notice, there are no considerable
differences in the course of the plots for various values of d, which means that even
great changes in the environment of the chain does not influence its phase transition
behaviour. This effect is interesting because along with the decrease of the slit size
d one approaches the 2D regime. On the other hand, even at the smallest value of
d = 3, all the mechanisms of changing the conformation were present and from the
standpoint of the system characteristics, one deals with the objects featuring the
same mechanism of conformational changes.
The scaling of dimensions with the number of beads in a molecule gives one the
very important information about the expansion of the chain in a solvent – in a
good solvent regime the dimensions are proportional to (N −1)γ with γ = 6/5, at Θ
conditions γ = 1 and for a globular state it reaches the value 2/3. These relations are
valid for a linear chain, but also hold in a case if star-branched polymers in a limit of
infinite N [16]. In this work, scaling relations were extracted for three cases: a good
12
Star-branched polymers in confined space
Table 1. Values of scaling exponents γ for different solvent regimes for small
distances between the walls d.
ξ d = 3 d = 5 d = 7 d = 10
−0.1 1.30± 0.01 1.24± 0.05 1.14± 0.05 1.05± 0.01
−0.4 1.00± 0.02 0.83± 0.03 0.72± 0.07 0.68± 0.06
−1.0 0.78± 0.03 0.59± 0.04 0.57± 0.01 0.55± 0.01
solvent regime (ξ = −0.1), the vicinity of the coil-to-globule transition temperature
(ξ = −0.4) and finally, a bad solvent regime (ξ = −1.0). The scaling exponents of
the relation 〈S2〉 ∼ (N − 1)γ were calculated for different values of the separations
of the plates d. The results are given in table 1.
As one can see, the decrease of the distance between the walls d increases the
scaling exponent γ. At a good solvent regime, the scaling strongly depends on d
giving the values much higher than those observed for chains without confinement
– this is in agreement with the results of Milchev and Binder [3]. At considerably
small values of d, the system approaches the 2D case, for which the scaling exponent
γ = 3/2 in a good solvent, which is in agreement with the tendency shown in Table
1 (value 1.30). The scaling for the collapsed globule (ξ = −1) is closer to that
from the theoretical predictions of value 2/3. Discussing this problem, one should
remember that the repulsive effect of the walls introduces a strong anisotropy of the
molecule and,thus, the further detailed investigations of the shape of the confined
star-branched molecules in a slit would complete the study.
References
1. Sanat K.K., Vacatello M., Yoon D.Y Off lattice Monte Carlo simulations of polymer
melts confined between two plates. // J. Chem. Phys., 1988, vol. 89, p. 5206–5215.
2. van Vliet J.H., ten Brinke G. Orientation and shape of flexible polymers in a slit. //
J. Chem. Phys., 1990, vol. 93, p. 1436–1441.
3. Milchev A., Binder K. Dynamics of polymer chains confined in slit-like pores. //
J. Phys. II France, 1996, vol. 6, p. 21–31.
4. Milchev A., Binder K. A polymer chain trapped between two parallel repulsive walls:
A Monte Carlo test of scaling behaviour. // Eur. Phys. J. B, 1998, vol. 3, p. 477–484.
5. Cordeiro C.E., Molisana M., Thirumalai D. Shape of confined chains. // J. Phys. II
France, 1997, vol. 7, p. 433–447.
6. Van Giessen A.E., Szleifer I. Monte Carlo simulations of chain molecules in confined
environments. // J. Chem. Phys., 1995, vol. 102, p. 9069–9076.
7. Cifra P., Bleha T. Anisotropy in the dimensional and elastic parameters of confined
macromolecules. // Macromol. Theory Simul., 1999, vol. 8, p. 603–610.
8. Kuznetsov D., Balazs A.C. Phase behaviour of end-functionalized polymers confined
between two surfaces. // J. Chem. Phys., 2000, vol. 113, p. 2479–2483.
13
P.Romiszowski
9. Sommer J.-U., Hoffmann A., Blumen A. Block copolymer films between neutral walls:
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10. Sotta P., Lesne A., Victor J.M. The coil-globule transition for a polymer chain confined
in a tube: a Monte Carlo simulation. // J. Chem. Phys., 2000, vol. 113, p. 6966–6973.
11. Pakula T. Computer simulation of polymers in thin layers. I. Polymer melt between
neutral walls – static properties. // J. Chem. Phys., 1991, vol. 95, p. 4685–4690.
12. Yethiraj A. Monte Carlo simulation of confined semiflexible polymer melts. //
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13. Sikorski A., Romiszowski P. Motion of star-branched vs linear polymer: A Monte Carlo
study. // J. Chem. Phys., 1996, vol. 104, p. 8703–8712.
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mol. Theory Simul., 1993, vol. 2, p. 309–318.
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polymers: a computer simulation study. // J. Chem. Phys., 1998, vol. 109, p. 2912–
2920.
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Температурна залежність деяких властивостей
зірково-гіллястих полімерів в обмеженому
просторі. Вивчення методом Монте Карло
П.Ромішовскі
Кафедра хімії, університет м. Варшави,
02–093 Варшава, Польща
Отримано 30 серпня 2001 р.
За допомогою копм’ютерного моделювання досліджено зірково-гіл-
лясті молекули з 3 -х ланцюгів у просторі між двома паралельними
відштовхувальними стінками, розташованими на відстані d одна від
другої. Моделювання проводилося для ланцюгів на простій кубічній
гратці. Ланцюги складалися з N = 49 , 100 і 199 ланок. Якість роз-
чинника під час моделювання регулювалася за допомогою зміни вза-
ємодії між сегментами. Також змінювалася товщина шару між стінка-
ми. Результати свідчать про те, що відбувається перехід полімерний
клубок–глобула. Температура переходу не залежить від обмеження
простору. Однак, показники скейлінгу радіусу гірації чутливі до вели-
чини d .
Ключові слова: моделювання методом Монте Карло, обмежений
простір, гіллясті полімери, перехід полімерний клубок–глобула
PACS: 02.70.Lq, 36.20.Ey, 61.25.Hq, 07.05.Tp
14
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