Scaling theory and computer simulation of star polymers in good solvents
The scaling theories and the results of the renormalization-group ε = 4−d expansion ( d is the spatial dimensionality) as well as the computer simulations such as Monte Carlo simulations are extensively reviewed for star polymers with very long flexible arms of equal length in a dilute solution...
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Інститут фізики конденсованих систем НАН України
2002
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| Цитувати: | Scaling theory and computer simulation of star polymers in good solvents / K. Ohno // Condensed Matter Physics. — 2002. — Т. 5, № 1(29). — С. 15-36. — Бібліогр.: 65 назв. — англ. |
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irk-123456789-1205782017-06-13T03:05:19Z Scaling theory and computer simulation of star polymers in good solvents Ohno, K. The scaling theories and the results of the renormalization-group ε = 4−d expansion ( d is the spatial dimensionality) as well as the computer simulations such as Monte Carlo simulations are extensively reviewed for star polymers with very long flexible arms of equal length in a dilute solution of the good solvent limit, with a close connection to general polymer networks. In particular, the asymptotic behaviour of the conformational and entropic quantities in the long chain limit is discussed in detail in terms of the polymer-magnetism analogy. Discussions are given not only for static properties such as the distribution functions and the osmotic pressure or entropy but also for dynamic properties such as the relaxation time and the intrinsic viscosity of star polymers. Проведено огляд теорій скейлінгу і ренормгрупового ε = 4 − d розкладу ( d – вимірність простору) а також комп’ютерного моделювання зіркових полімерів (і полімерних сіток), що складаються з довгих гнучких ланцюгів однакової довжини і знаходяться в розведеному розчині в границі доброго розчинника. Зокрема, в термінах аналогії полімер-магнетик детально обговорюється асимптотична поведінка конформаційних і ентропійних величин в границі довгих ланцюгів. Розглянуто не лише статичні характеристики, такі, як функції розподілу і осмотичний тиск чи ентропія, але і динамічні властивості, такі як час релаксації і власна в’язкість зіркових полімерів. 2002 Article Scaling theory and computer simulation of star polymers in good solvents / K. Ohno // Condensed Matter Physics. — 2002. — Т. 5, № 1(29). — С. 15-36. — Бібліогр.: 65 назв. — англ. 1607-324X PACS: 05.10.Cc, 05.10.Ln, 61.41.+e, 82.35.Gh, 82.35.Lr, 82.70.Uv DOI:10.5488/CMP.5.1.15 http://dspace.nbuv.gov.ua/handle/123456789/120578 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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The scaling theories and the results of the renormalization-group ε = 4−d
expansion ( d is the spatial dimensionality) as well as the computer simulations
such as Monte Carlo simulations are extensively reviewed for star
polymers with very long flexible arms of equal length in a dilute solution
of the good solvent limit, with a close connection to general polymer networks.
In particular, the asymptotic behaviour of the conformational and
entropic quantities in the long chain limit is discussed in detail in terms of
the polymer-magnetism analogy. Discussions are given not only for static
properties such as the distribution functions and the osmotic pressure or
entropy but also for dynamic properties such as the relaxation time and the
intrinsic viscosity of star polymers. |
| format |
Article |
| author |
Ohno, K. |
| spellingShingle |
Ohno, K. Scaling theory and computer simulation of star polymers in good solvents Condensed Matter Physics |
| author_facet |
Ohno, K. |
| author_sort |
Ohno, K. |
| title |
Scaling theory and computer simulation of star polymers in good solvents |
| title_short |
Scaling theory and computer simulation of star polymers in good solvents |
| title_full |
Scaling theory and computer simulation of star polymers in good solvents |
| title_fullStr |
Scaling theory and computer simulation of star polymers in good solvents |
| title_full_unstemmed |
Scaling theory and computer simulation of star polymers in good solvents |
| title_sort |
scaling theory and computer simulation of star polymers in good solvents |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2002 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120578 |
| citation_txt |
Scaling theory and computer
simulation of star polymers in good
solvents / K. Ohno // Condensed Matter Physics. — 2002. — Т. 5, № 1(29). — С. 15-36. — Бібліогр.: 65 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT ohnok scalingtheoryandcomputersimulationofstarpolymersingoodsolvents |
| first_indexed |
2025-07-08T18:09:52Z |
| last_indexed |
2025-07-08T18:09:52Z |
| _version_ |
1837103257991774208 |
| fulltext |
Condensed Matter Physics, 2002, Vol. 5, No. 1(29), pp. 15–36
Scaling theory and computer
simulation of star polymers in good
solvents
K.Ohno
Department of Physics, Faculty of Engineering,
Yokohama National University,
79-5 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan
Received December 7, 2001
The scaling theories and the results of the renormalization-group ε = 4−d
expansion ( d is the spatial dimensionality) as well as the computer simu-
lations such as Monte Carlo simulations are extensively reviewed for star
polymers with very long flexible arms of equal length in a dilute solution
of the good solvent limit, with a close connection to general polymer net-
works. In particular, the asymptotic behaviour of the conformational and
entropic quantities in the long chain limit is discussed in detail in terms of
the polymer-magnetism analogy. Discussions are given not only for static
properties such as the distribution functions and the osmotic pressure or
entropy but also for dynamic properties such as the relaxation time and the
intrinsic viscosity of star polymers.
Key words: renormalization group, Monte Carlo simulation, total number
of configurations, virial coefficient, relaxation time, hydrodynamic effect
PACS: 05.10.Cc, 05.10.Ln, 61.41.+e, 82.35.Gh, 82.35.Lr, 82.70.Uv
1. Introduction
The complicated statistics of polymer networks [1] having ring [2], star [3], and
other branched topologies in solution have received a continuous attention over a long
time. The statistics of polymer networks resolved in a good solvent in the dilute limit
can be measured in terms of monomer distribution, end-to-end distance distribution,
entropy, and so on. On the other hand, typical dynamic properties of a dilute solution
of polymer networks in a good solvent are as follows: relaxation time, diffusion
constant, viscosity, sedimentation coefficient, hydrodynamic radius, and so on. They
can be measured experimentally by means of optical, neutron-diffraction and other
techniques. They are theoretically estimated by means of computer simulations and
other numerical techniques (see [4,5] for review and see also [6–14] for star polymers)
as well as using sophisticated analyses such as the exact results in two dimensions
c© K.Ohno 15
K.Ohno
[15,16], scaling theory [2,17–19] and renormalization-group (RG) techniques (for star
polymers see [20–25] as well as [17,19], and for other polymer networks see [15,26]).
All these quantities and techniques are of course not special for polymer networks
but have been rather common for linear polymers. However, the resulting behaviour
is not always similar to that of linear polymers. As an example, a ring polymer
having a very simple polymer architecture, which is characterized by a closed loop
without branch, is known to have the statistics in a dilute solution different from that
of an open chain [2]. Another extreme example of polymer networks is a polymer
gel, which is a huge network of many flexible linear chains, and its statistics and
dynamics have attracted interest [2].
A star polymer is composed of many long arm chains starting from a center unit.
Still it has a simple enough structure but its statistics and dynamics involve a lot of
important ingredients of more general polymer networks. In fact, it has been found
in two [15,16] and arbitrary dimensions [15,17,18] that the behaviour of the total
number of configurations of star polymers determines the behaviour of general poly-
mer networks, The study of star polymers is also interesting in other respects since
it has a close relationship to the subject of micellar and other polymeric surfactant
systems [27–30], and has a potential to bring widespread applications.
In the past two decades, the great progress in the synthesis of highly qualified
monodisperse polymer networks [31–38] has stimulated many experimental and the-
oretical studies of star polymers. More recently, star polymers with a huge number of
arms has also been synthesized in a controlled fashion [39]. The experimental stud-
ies of star polymers have stimulated theoretical studies to be more closely tied up
with the experiments. For example, the virial coefficients [40–42], relaxation times
[43,44] and hydrodynamic effects [45–48] have been more recently investigated with
a powerful tool of computer simulations and some of their results have been success-
fully compared with the experiments [31–37,49,50] (for a recent review, see [51]).
The effect of walls or other confining geometries has also found interest [52–54] in
connection , for example, with surface critical phenomena (see [55] for a review).
The center-absorbed star polymers are also related to the polymer brush [56] when
the interchain distances on the substrate become short enough. In this article, the
study of star polymers in good solvents in a dilute regime will be reviewed from the
theoretical standpoint.
2. Lattice model and magnetic analogy of polymer networks
The theory of linear polymers in a good solvent has been very successful as a
result of the connection established by de Gennes [2] and des Cloizeaux [57] between
the polymer statistics and the critical phenomena of an n-component classical spin
model in the n → 0 limit. The discussions presented in this section can be seen
mostly in [17–19].
16
Scaling theory and computer simulation
2.1. Lattice chain model and n -vector model
Let us first start from a lattice chain model and assume that there are f chains
of length l1, l2, ..., lf on a regular lattice. To be specific, let us use a d-dimensional
hypercubic lattice with a unit lattice constant (d = 3 for usual purposes). Each
segment has a unit length and can rotate only by 90◦. If the position of a joint of
segments at a contour distance si along the ith lattice chain is designated by r(si),
the Hamiltonian of this lattice chain system can be written as
H = a
f
∑
i=1
li−1
∑
si=1
{[r(si + 1)− r(si)]
2 − 1}+ b
f
∑
i=1
f
∑
j=1
li−1
∑
si=1
lj−1
∑
sj=1
δr(si)r(sj), (1)
where δr(si)r(sj) means Kronecker’s delta and the parameters a and b are chosen to
be 0<a<b. Then, we see that the energy is b−a(>0), zero, a(>0), 2a(>0), 3a(>0),
..., respectively, when the distance between two adjacent joint points is 0, 1,
√
2,√
3, 2 and so on. And, if two joint points, which are on two different chains or on
the same chain but apart from each other along the chain contour, meet at the
same point, the energy is increased by b(>0). Therefore, keeping the conformations
with zero energy only, we obtain f chains which do not pass the same lattice point
twice, i.e., satisfying the self-avoiding condition. They are called self-avoiding walks
(SAWs) on a lattice. We want to count the number of all such configurations with a
given set of fixed chain-end points, Oi = r(si = 1) and Pi = r(si = li), i = 1, 2, ..., f .
This is possible when we estimate the “restricted” partition function
N (O1P1, O2P2, ..., OfPf ; l1, l2, ..., lf ) = lim
T→0
f
∏
i=1
li−1
∏
si=2
∑
{r(si)}
exp
[
− H
kBT
]
(2)
of this Hamiltonian at T = 0. (The meaning of “restricted” is that the sum with
respect to joint points {r(si)} over the lattice points is taken with fixed chain ends
at Oi = r(si = 1) and Pi = r(si = li), i = 1, 2, ..., f .) That is, the total number
of configurations of f -chains on a lattice with fixed end points is exactly given by
equation (2) at T = 0. From the complete knowledge on the total number of con-
figurations, one can calculate various averages of geometrical quantities of polymer
networks.
Physical polymer networks contain loops and branch points, while the f linear
chains introduced above are disconnected. However, by requiring some of the end
points of linear chains to be in close proximity of each other, one can obtain desired
structures of the considered network. If O1, O2, ..., Of are put as the nearest neigh-
bours of each other while the positions of P1, P2, ..., Pf are not restricted, an f -arm
star geometry is realized. In fact, such a proximity constraint has the same physical
effect as putting in suitable chemical crosslinks (functional units) to form the net-
work. In what follows, we will consider a general polymer topology G and treat each
crosslink, at which several chains meet at close vicinities, as a single branch point
as a whole.
17
K.Ohno
Total number of configurations of a polymer network with a fixed topology G is
NG(l1, l2, ..., lf) =
1
N
∑
end and branch points
N (O1P1, O2P2, ..., OfPf ; l1, l2, ..., lf ), (3)
where the summation with respect to end and branch points is taken over all N
lattice points. The prefactor 1/N is necessary because there are N translationally
identical configurations which should not be distinguished. It is often convenient
to introduce a slightly different definition for the total number of configurations in
which the total length of chains is fixed at L:
NG(L) =
∞
∑
l1=0
∞
∑
l2=0
· · ·
∞
∑
lf=0
NG(l1, l2, ..., lf)δl1+l2+...+lf ,L . (4)
Now that the lattice chain model is defined completely, one can demonstrate its
relation to the lattice spin model called the n-vector model. To this end, we define
the generating function for the total number of configurations as
ZG(K) =
∞
∑
L=0
NG(L)K
L. (5)
This generating function (5) is proved to be equal exactly to a nonlinear suscepti-
bility (in the limit n→ 0) of the n-vector model:
ZG(K) = χG(K), (n→ 0). (6)
The n-vector model is defined by the Hamiltonian
H = −JH = −kBTKH, K =
J
kB
, (7)
H =
∑
〈ij〉
Si · Sj . (8)
In equation (8), the summation runs over all the nearest-neighbour pairs, ij, on a
lattice, and each spin S i at each lattice point i has n components
Si = (S
(1)
i , S
(2)
i , ..., S
(n)
i )
and fixed length |Si|2 = n. This Hamiltonian describes the Ising model, the planar
(XY) model and the classical Heisenberg model, respectively, for n = 1, 2 and 3.
The relevant nonlinear susceptibility χG(K) can be constructed as follows: con-
sider the system of f linear chains which do not have contacts with each other and
suppose that the polymer network has nk, k-functional units. The k-functional unit
around point P consists of g neighbouring points P1, P2, ..., Pk, at which each linear
chain starts. Then we are requested to introduce the kth order composite operator
Ψ
(m1,m2,...,mk)
k =
∑
P
S
(m1)
P1
S
(m2)
P2
...S
(mk)
Pk
(9)
18
Scaling theory and computer simulation
for each g-functional unit constituting the network with the topology G. The sum
with respect to P is performed with a fixed vertex structure, because the neigh-
bouring points P1, P2, ..., Pk do not change their configuration around P . Then, for
a given topology G of polymer networks, the nonlinear susceptibility is given by
χG(K) =
1
N
〈
∏
k
[Ψ
(m1,m2,...,mk)
k Ψ
(m1′ ,m2′ ,...,mk′)
k ...Ψ
(m1′′ ,m2′′ ,...,mk′′)
k ]
〉
. (10)
One should recall that ith chain carries ith spin component, so that the same com-
ponent appears just twice in the brackets on superscript of Ψ’s in (10).
When the nonlinear susceptibility χG(K) is calculated with the renormalization
group (RG) theory, the average of the product of composite operators becomes, after
renormalization, a linear combination of the original function and other functions
which are related to simpler topologies obtained from the original topology G by
shrinking some arms and have the same or lower canonical dimensions. If we are
interested in the networks having the same chain length l, we should discard all these
terms which occur due to additive renormalization. That is, it is only necessary to
identify the renormalization factor associated with the topology G itself. We will call
this part of the renormalized nonlinear susceptibility the essential part.
2.2. Relation to critical phenomena
The equivalence between polymer and magnetic systems enables us to study
conformational and entropic properties of polymer networks of long flexible chains.
From the knowledge of the n-vector model, we expect that the nonlinear suscep-
tibility (10) generally exhibits a singular behaviour like
χG(K) ∼ tγ̂G . (11)
Here t is the reduced temperature defined by µK = e−t with µ = 1/Kc = kBT/J ;
µ is a parameter proportional to the critical temperature of the n-vector model in
the limit n → 0. Note that t = 0 corresponds to the critical point K = Kc and
t ∼ 1− µK holds only near the critical point.
If we replace the summation with respect to L in (5) by an integration and use
µK = e−t, the relation (6) becomes a form of Laplace transformations: χG(K) ∼
∫∞
0
dL[NG(L)/µ
L]e−tL, where a symbol ∼ is used to indicate its validity only in the
scaling limit, L → ∞. Because the nonlinear susceptibility is expected to have a
power-law singularity (11) near t ∼ 0, an inverse-Laplace-transformation gives
NG(L) ∼ Lγ̂G−1µL. (12)
Thus, the exponent γ̂G for the total number of configurations of networks with
a fixed total length L is found to be identical to the exponent of the nonlinear
susceptibility (11). In enumerations of SAWs on a lattice, the constant µ is sometimes
called the effective coordination number. Note that, in the case of simple random
walks, µ is equal to the coordination number of a lattice (2d for the d-dimensional
19
K.Ohno
hypercubic lattice) and γ̂G is equal to 1. In contrast, in the case of SAWs, µ is slightly
smaller than the coordination number minus 1, the number of possible directions of
elongating a chain end by one segment without folding backward onto itself. This
mapping shows that µ is independent of the topology of the polymer network.
As a more important case, when the length of every single chain is all the same
and given by l, the total number of configurations NG(l, l, ..., l) behaves differently
from the NG(L) with a fixed total chain length L. The behaviour of this NG(l, ..., l)
may be explored by recognizing that, if all chain lengths l1, ..., lf are of the same
order, then NG(l1, ..., lf ) behaves like (l1 + · · ·+ lf)
γ̂G−1µl1+···+lf . Then, since such a
region dominates in the sum of (4), we obtain
NG(L) ∼ LγG−1µL
∞
∑
l1=0
∞
∑
l2=0
· · ·
∞
∑
lf=0
δl1+l2+···+lf ,L ∼ LγG+f−2µL. (13)
The comparison between (12) and (13) yields
γG = γ̂G + 1− f. (14)
Let us turn our attention to the end-to-end distribution of a general polymer
network G. We consider the total number of configurations NG(Oi, Pj;L) which is
defined by the NG(L) summed up with respect to all ends and branch points except
Oi and Pj. It depends on spatial coordinates only through the distance rij between Oi
and Pj . Then, introducing the reduced temperature and the Laplace transformation,
and applying the magnetic analogy for this function, we are led to the local nonlinear
susceptibility χG(Oi, Pj;K) expressed as ∼
∫∞
0
dL[NG(Oi, Pj;L)/µ
L]e−tL. From the
scaling theory of spin systems, we expect that it obeys a scaling form
χG(Oi, Pj;K) ∼ 1
r
d−2+ηG
ij
ϕG
( rij
t−ν
)
, (15)
where ν is the correlation-length exponent; and ηG is the anomalous dimension and
is related to γ̂G via γ̂G = ν(2 − ηG). Then, an inverse-Laplace-transformation gives
NG(Oi, Pj;L) ∼ (µL/Lr
d−2+ηG
ij )ϕ̃G(rijL
−L). Finally, dividing it by NG(L), we find
the end-to-end distribution function for fixed topology G and fixed total length L,
pG(rij) =
NG(Oi, Pj;L)
NG(L)
∼ 1
Lγ̂Gr
d−2+ηG
ij
φ̂G(rijL
−L) =
1
rdij
φG(rijL
−ν), (16)
similar to the single chain problems. The distribution function for general polymer
networks where all chains have the same length l is also expected to have the same
form as (16). From (16), we get to the mean square end-to-end distance
〈r2ij〉 ∼ L2ν , (17)
which also behaves like a single polymer chain. Thus, we expect that if a flexible
polymer network made of connecting very long chains is desolved in a good solvent,
its gyration radius is characterized by the same exponent ν as for a single self-
avoiding walk (SAW) (0.588 for d = 3) irrespective of the network structure [2,17].
20
Scaling theory and computer simulation
2.3. Scaling theory of general polymer networks
Although the gyration radius exponent ν of an arbitrary polymer network is
the same as that of a single self-avoiding walk (SAW), the exponent for the total
number of configuration is generally different from that of the SAW and depends on
the topology of the network G.
To see this, we will discuss a phenomenological scaling theory of general polymer
networks. We assume that the gth-order composite field hg, which is conjugate to
the gth-order composite operator Ψg, scales as t
∆g irrespective of their components.
That is, we assume that the free energy of the system has a scaling form
F ∼ t2−αΦ
(
h1
t∆1
,
h2
t∆2
, ...
)
, (18)
where α is the specific heat exponent of the spin system (α ∼ 0.23 for d = 3) and is
related to ν by the hyperscaling relation α = 2− νd
The nonlinear susceptibility with composite operators is generally not multiplica-
tively renormalizable but mixed with other functions which have the same or lower
canonical dimension [58]. Correspondingly, in the configurations of the network G
with a fixed total length L, many simpler topologies are realized by shrinking some
of the linear chains in G. In order to discuss the configurations with all chain lengths
being the same, one should discard all such simpler (and less singular) terms and
preserve only the singularity associated with the topology G (the essential part).
The corresponding exponent γG is obtained from the exponent γ̂G of the essential
part associated with the topology G of the nonlinear susceptibility via equation (14).
We have the essential part of the nonlinear susceptibility behaving as
χG(K) ∼ − 1
N
[
∏
g
∂ng
∂h
ng
g
]
F
∣
∣
∣
∣
h=0
∼ t−γ̂G (19)
with γ̂G = α − 2 +
∑
g ng∆g. Then using the relation (14), the exponent γG for the
total number of configurations of f chains having the same length l is given by
γG = α− 1− f +
∞
∑
g=1
ng∆g. (20)
This expression was first obtained by Duplantier [15] who used the two-dimensional
exact analysis and the renormalization-group (RG) approach. The present derivation
is based on a phenomenological scaling argument given by Ohno and Binder [17].
If we consider a comb polymer composed of g = (f −1)/2 3-functional units and
g + 2 side branches, its exponent is given by γcomb(g) = γ + g[γ(3) − γ] from the
scaling relations (20) and (21). This expression is verified to O(ε) in the RG ε = 4−d
expansion, if we use γ(3), which was calculated specifically for star polymers to O(ε)
by Miyake and Freed [20,21] [see also (22) in the next section], and gcomb(g), which
was calculated specifically for comb polymers to O(ε) by Vlahos and Kosmas [26].
21
K.Ohno
2.4. Scaling behaviour of star polymers
Let us discuss the scaling behaviour of star polymers. First of all, the exponent
γG ≡ γ(f) for an f -arm star polymer is expressed by the scaling relation (20) as
γ(f) = α− 1 +
f
2
(γ − α) + ∆f . (21)
Thus, combining (20) and (21), one can express γG for an arbitrary polymer network
G in terms of well known exponents γ, ν and α, and star polymer exponents γ(f).
Now we mention the result of the RG ε expansion for the star polymer expo-
nents. For the exponent γ(f), the first-order term in ε was determined in 1983 by a
pioneering work of Miyake and Freed [20,21]. The expression valid up to the second
order in ε was obtained by Ohno and Binder [17] as follows:
γ(f) = 1 + (γ − 1)
[
f − f(f − 1)
2
]
+ f(f − 1)(f − 2)A(f), (22)
A(f) =
1
64
ε2 +O(ε3); (23)
A(f) is a regular function of f . Its value at O(ε3) was identified by Duplantier [24].
The expansion is, however, an asymptotic expansion and does not converge even at
small ε and f [25]. The large f behaviour of γ(f) was predicted by Ohno [23] to be
γ(f) ∼ −f d/(d−1) for arbitrary dimensions, 2 6 d < 4. The γ(f) was also evaluated
by Monte Carlo techniques [7,11,13] (see section 3.1).
It is possible to discuss additional scaling relations which relate the contact
exponents of a linear chain to the star-polymer exponents. Using the scaling relations
(20) and (21), Duplantier and Saleur [59] and Ohno and Binder [17] obtained
γ(3) = 2γ − 1− νθ1, γ(4) = 2γ − 1− νθ2, (24)
where θ1 denotes the contact exponent characterizing the short distance behaviour
(s.d.b.) between one end and one interior point of a linear chain; and θ2 characterizes
the s.d.b. between two interior points of a linear chain. The scaling relations (24)
between contact and star-polymer exponents are fulfilled if we use (22) with (23)
for γ(f) and the contact exponents calculated up to O(ε2) by des Cloizeaux [60].
For a star polymer, the mean distance RC of an arbitrary monomer j from the
center, the radius of gyration of the total polymer Rgyr, and the mean center-end
distance RCE are predicted by Daoud and Cotton [3] to behave as [see also (17)]
RC ∼ Rgyr ∼ RCE ∼ fσlν , σ =
1− ν
d− 1
. (25)
The short distance behaviours of the monomer density distribution function
ρ(r) =
1
rd−1/ν
ψ
(
r
RC
)
, (26)
22
Scaling theory and computer simulation
and the center-end distribution function
g(rCE) =
1
Rd
CE
φ
(
rCE
RCE
)
, (27)
which are normalized as
∫
drCEg(rCE) = 1 and
∫
drρ(r) = L, are given by [18]
ψ(x) = const, φ(x) ∼ xθ(f), (for x≪ 1) (28)
θ(f) = [γ − γ(f + 1) + γ(f)− 1]/ν; (29)
γ being the configuration number exponent of a free linear polymer. Equations (26)–
(29) refer to free star polymers only; a more general form is discussed in [61].
2.5. Star polymers and polymer networks in semi-infinite geo metry
The scaling theory for general polymer networks can be generalized to the case
where some end or branch points are grafted on a flat surface [16–19]. In the last two
sections, we saw that the exponent γG for a complicated network can be expressed
as a linear combination of only these star polymer exponents γ(f), and as the well-
known exponents of single linear polymer chains. Similarly, the exponent γ
G
for
a polymer network G which has nh h-functional units free, n′
h h-functional units
grafted at the surface and totally f linear polymers with the same length obeys the
scaling relation
γG = α− 1− f + ν +
∞
∑
h=1
[nh∆h + n′
h∆
′
h]. (30)
This scaling relation was first obtained for two dimensions by Duplantier and Saleur
[16] and then for arbitrary dimensions by Ohno and Binder [17].
For center-absorbed stars γs(f) is used for γG , and for stars which have one, two,
. . . ends of arms grafted at the wall, γ1(f), γ11(f), etc., are used. From (30), γs(f),
γ1(f), gll(f), ... are expressed, by assuming that g11...1(f) has g subscripts 1, as
γs(f) = α− 1 + ν +
f
2
(γ − α) + ∆′
f , (31)
γ11...1(f) = γ(f) + ν + g(γ11 − γ1). (32)
By the RG ε expansion, the exponent γs(f) was found to be [18]
γs(f) = 1 + (γ1 − 1)f − (γ1,1 + ν)
f(f − 1)
2
+ f(f − 1)(f − 2)B(f), (33)
where B(f) = cε2+O(ε3) and c is an unknown constant. Colby et al. [10] and Shida
et al. [14] performed Monte Carlo simulations to evaluate γ s(f) (see section 3.1).
These exponents take different numbers depending upon whether the surface
is repulsive, attractive, or “marginal” with respect to the monomers forming the
arms of the star [18]. Here we call a surface “marginal” if the system is right at
23
K.Ohno
the adsorption transition where for l → ∞ the chain configuration changes from d-
dimensional to (d− 1)-dimensional due to the attractive monomer-wall interaction.
Here, we draw attention to other properties of star polymers such as distribution
functions [18,19]. In the semi-infinite geometry, we also expect RC ∼ Rgyr ∼ RCE ∼
lν as (25) (ν = 3/4 in two dimensions and ν = 0.588 in three dimensions). In the case
of center-absorbed stars, a dependence on the distances parallel and perpendicular
to the surface (r||, z) appears. The monomer density profile behaves as
ρ(r||, z) = L (RC)
−d ψs
(
r||
RC
,
z
RC
)
, (34)
with the short distance behaviour of the scaling function ψs(x, y)
ψs(x, 0) ∼ x−d+λ(f), ψs(0, y) ∼ y−d+1/ν . (35)
Equation (35) contains a new exponent λ(f) which depends on f . This λ(f) should
coincide with 1/ν in the limit f → ∞ where the Daoud–Cotton theory [3] or the cone
picture [23] is applicable. The value of λ(f) was determined with the ε expansion
by Ohno and Binder [19] and is given by (f − 1)ε/4 +O(ε2); it was also evaluated
by means of Monte Carlo simulations by Shida et al. [14] (see section 3.1).
The center-end distribution function gs(r
CE
|| , zE) behaves as
gs(r
CE
|| , zE) = (RCE)
−dφs
(
rCE
||
RCE
,
zE
RCE
)
, (36)
where the scaling function φ(x, y) has the following short distance behaviour:
φs(x, 0) ∼ xθ||(f), φs(0, y) ∼ yθ⊥(f), (37)
θ||(f) = [γ1 − γs(f + 1) + γs(f)− 1]/ν, (38)
θ⊥(f) = [γ − γs(f + 1) + γs(f)− 1]/ν. (39)
Here the exponents γ1, (γ11) refer to the number of configuration of a linear chain
polymer with one end, (two ends) grafted at a surface [54,55]. These relations were
tested for f = 2−15 stars by means of Monte Carlo simulations by Shida et al. [14].
Equations (34)–(39) hold not only for a repulsive wall but also for a “marginal
wall”, where the adsorption transition from d-dimensional to (d−1)-dimensional be-
haviours takes place. For a marginal wall, the exclusion effect against polymers by
the hard wall is cancelled by a sufficient strength of the attractive wall interaction.
For slightly stronger attractive walls, polymers are adsorbed on the wall and exhibit
(d − 1)-dimensional behaviour. In the polymer-magnetism analogy, this behaviour
corresponds to the surface transition above the bulk Tc of a magnetic system where
the coupling constant at the surface (Ks) exceeds that in the bulk (K). The poly-
mer problems near a marginal wall correspond in magnetic analogy to the critical
behaviour at a “special transition” or a “surface-bulk” (SB) multicritical point [55].
The scaling relation (30) for networks with grafted chains also applies to the ad-
sorption transition if ∆′
h is replaced by ∆
SB
h . For example, ∆
SB
2 is no more given by
24
Scaling theory and computer simulation
−ν but by ϕ, which is the “crossover exponent” [54,55]. The γ
G
obeys the scaling
relation
γ
SB
G
= α− 1− f + ν +
∞
∑
h=1
[nh∆h + n′
h∆
SB
h ]. (40)
∆h and ∆
SB
h are related, respectively, to the exponents of star polymers via (21) and
γ
SB
s (f) = α− 1 + ν +
f
2
(γ − α) + ∆
SB
f . (41)
Note that these equations (40),(41) have exactly the same form as the repulsive case
[see equations (30),(31)], if we replace ∆
SB
h by ∆′
h. First several ∆’s are expressed
with only the exponents of a linear polymer:
∆1 = 1 + (γ − α)/2, ∆2 = 1, ∆′
1 = (d− 1)ν/2 + γ11/2, ∆′
2 = −ν,
∆
SB
1 = (d− 1)ν/2 + γ
SB
11 /2, ∆
SB
2 = ϕ.
For the attractive surface, equations (34)–(39) need modification, since the char-
acteristic lengths parallel and perpendicular to the surface differ and are given by
RCE,|| ∼ RC,|| ∼ Rgyr,|| ∼ lν
(d−1)
, l → ∞, (42)
RCE,⊥ ∼ RC,⊥ ∼ Rgyr,⊥ ∼ |c|−1 ∼ const, l → ∞. (43)
Instead of equations (34) and (35), we have
ρ(r||, z) = L (RC,||)
−(d−1)(RC,⊥)
−1ψs
(
r||
RC,||
,
z
RC,⊥
)
, (44)
with ψs(x, 0) ∼ x−(d−1)+1/ν(d−1)
. That is, the asymptotic properties of a star at an
attractive wall are the same as those of a star in a (d − 1)-dimensional geometry.
Similar to (44), we have for the center-end distribution function
gs(r
CE
|| , zE) = (RCE,||)
−(d−1)(RCE,⊥)
−1φs
(
rCE
||
RCE,||
,
zE
RCE,⊥
)
, (45)
with φs(x, 0) ∼ xθ
(d−1)
(f), and θ
(d−1)
(f) given by an equation similar to (29) but with
all exponents taking their (d−1)-dimensional values. Note that equations (42)–(45)
are valid only for d > 2, since for d = 2 only two arms of a star polymer would be
adsorbed on the surface, and the configuration of the remaining arms would be just
that of a star with f − 2 arms at a repulsive wall.
At the adsorption transition, γs(f) and γ1 in equations (34)–(39) must be re-
placed by γ
SB
s (f) and γ
SB
11 . Using the RG ε (= 4 − d) expansion, Ohno and Binder
[18] found
γ
SB
s (f) = 1 + (γ
SB
1 − 1)f + (ϕ− γ
SB
11 )
f(f − 1)
2
+ f(f − 1)(f − 2)C(f). (46)
In equation (46), C(f) is a polynomial of both ε and f and of order ε2, ϕ is the
crossover exponent, and γ
SB
1 , (γ
SB
11 ) are the multicritical values of the conformation
number exponents γ1, (γ11) of linear polymers with one, (two) ends at the surface.
25
K.Ohno
3. Numerical simulations
Using various computer simulation techniques, one can simulate star polymers
in solvents. Using the molecular dynamics method, Grest et al. [8] have investigated
gyration radius of star polymers with up to 50 arms. Grest et al. also investigated
relaxation times of star polymers [43]. To investigate the total number of configu-
rations, it is necessary to count the probability of elongating arms (with sufficiently
long chains to get to the scaling regime). For this purpose, Monte Carlo techniques
have been used successfully. Off-lattice Monte Carlo simulations have been per-
formed by Rey et al. [45], Freire et al. [47] and Rubio and Freire [41]. Lattice Monte
Carlo simulations of star polymers with up to 6 arms have been done by Lipson et
al. [6] and Wilkinson et al. [7,46]. Batouris and Kremer [11] used a biased sampling
method in their Monte Carlo simulations. Ohno and Binder [12,13] and Shida et al.
[48] have applied an efficient enrichment algorithm to star polymers which enables
one to treat star polymers with relatively large number of arms (∼ 32). Colby et al.
[10] and Shida et al. [14] investigated grafted star polymers in a semi-infinite geome-
try. There is also an application of static Monte Carlo techniques to the investigation
of a relaxation time by Ohno et al. [44].
3.1. Enrichment algorithm
In the enrichment algorithm for star polymers [12–14,44,48], we generate mono-
disperse f -arm stars with arm length l + 1 from those with arm length l which is
shorter by one segment, using a standard Monte Carlo technique. The number of
arms, f , is fixed throughout the computation. On a simple cubic lattice, for example,
we have five ways of elongating one end of an arm by one segment, because the 6th
direction makes the arm fold backwards on itself. The success ratio is given by
(µ/5)f = (0.93706)f (the effective coordination number of the simple cubic lattice
is µ = 4.6853 [62]). Then, one-step larger star polymers are generated by elongating
every arm by one segment. At each step, the self-avoiding condition is tested; unless
this condition is fulfilled, generated configurations are simply discarded.
If we had to consider all possible realizations at the l+1st step fromM l distinct
realizations at the lth step, we would have to make all 5fMl trials. However, doing
only mMl trials (m ≪ 5f) which are much less than the full 5fMl trials, we can
collect a sufficient number of samples which are statistically isomorphic to the full
realizations. That is, we generate only a limited number of samples M l+1 from Ml
samples by a Monte Carlo method. This process can be iterated when Ml ≈Ml+1 ≈
· · ·. The conditionMl+1/Ml > 1 is satisfied when we choosem > (5/µ)f = (1.0672)f .
In practice, it is necessary to increaseMl gradually as l increases, because one has to
avoid an unphysical “bias” caused from this iteration. However, even for f as large
as f = 18 a rather small value of m(≈ 10) results. (For small l, it is better to work
with somewhat larger values of m.) This enrichment algorithm for star polymers
significantly reduces computing time, since it becomes quite efficient asymptotically
for very long arms. In contrast, the conformation of shorter stars, typically with
l = 1 or 2, are more rapidly counted by the direct SAW algorithm.
26
Scaling theory and computer simulation
Table 1. The values of the exponent γ(f) for f -arm star polymers with f =
3− 6, 12, 18, 32 estimated by Monte Carlo simulations.
f 3 4 5 6
[7] 1.05± 0.03 0.88± 0.03 0.55± 0.05 0.20± 0.05
[11] 1.089± 0.001 0.879± 0.001 0.567± 0.002 0.16± 0.01
f 12 18 32 6
[13] −3.4± 0.3 −8.9± 0.2 −29± 2 0.18± 0.05
Table 2. The values of the exponents γ(f) and λ(f) for f -arm star polymers with
f = 2 − 6, 8, 10, 12, 15 estimated by lattice enrichment Monte Carlo simulations
[14].
f 2 3 4 5 6 8 10 12 15
γs(f) 0.10 −0.52 −1.21 −2.03 −2.92 −4.9 −7.1 −9.4 −13
λ(f) 0.8 0.9 0.9 1.0 1.l 1.1 1.3 1.4 1.5
The total number of configurations of f -arm stars with each arm length equal
to l as its successive ratio Nl+1/Nl can be identified as 5fMl+1/mMl. If we consider
alternatively (Nl+1/Nl)
1/f , we have, from Nl ∼ µfllγ(f)−1,
(Nl+1
Nl
)
1
f
=
(
5fMl+1
mMl
)
1
f
= µ
{
1 +
γ(f)− 1
fl
[
1 +
γ(f)− 1− f
2fl
]
+O(
1
l3
)
}
. (47)
Since the quadratic correction becomes negligible for l ≫ |γ(f)−1−f |/(2f), the con-
figuration number exponent γ(f) can be determined from a plot of (5fMl+1/mMl)
1/f
versus 1/fl: if the numerical data fall on a straight line with the (known) intersec-
tion µf on the ordinate axis, the slope of this straight line yields [γ(f) − 1]µ. The
enrichment algorithm was recently applied to the semi-dilute regime of linear chain
solutions [63], although its extension to star polymer solutions has not been done
yet.
Using the enrichment algorithm, Ohno [13] generated 6-, 12-, 18- and 32-arm
star polymers with an octahedral core as center unit. The resulting exponent γ(f)
is listed in table 1 together with earlier works for up to 6-arm stars by Wilkinson et
al. [7] and Batoulis and Kremer [11]. The exponents γs(f) and λ(f) [see (35) for its
definition] for center-adsorbed stars in a semi-infinite geometry were evaluated by
Shida et al. [14] (first several numbers for γs(f) were also estimated by Colby et al.
[10]). They are listed in table 2. As was pointed out by Ohno and Binder [18], λ(f)
is an increasing function of f and approaches 1/ν ≈ 1.70 as f increases.
The data also allow a significant study of linear dimensions such as the mean
square distance from the center, RC, and the mean square distance RCE between
27
K.Ohno
the end (E) of an arm and the center (C). All results [6,7,11–13] are consistent in
high accuracy with the Daoud-Cotton prediction (25) for the f and l dependences.
3.2. Virial coefficient
Entropic properties of star polymers in good solvents are closely related to the
total number of configurations. Let us consider the osmotic pressure Π of a solution
of monodisperse star polymers, which have f arms, each consisting of l segments.
In a dilute solution, Π is expressed in a power series of the concentration c as
Π
NAkBT
=
c
M
+ A2c
2 + A3c
3 + ..., (48)
where NA stands for Avogadro’s number, andM = flm means the molecular weight
of one star polymer (m is the molecular weight of one segment). This series is
usually called the virial expansion, and A i (i = 2, 3, ...) is referred to as the ith virial
coefficient of the solution. Available information about the virial coefficients is largely
limited to A2, because of the difficulty in estimating A3 and higher coefficients.
Consider two star polymers in a solution, and write the ith joint point between
two adjacent segments of the first (second) star as ri (σi) and the center of the first
(second) star as r0 (σ0). In the good solvent limit, the two body interaction u(r i−σj)
is given by (α/β)δ(ri − σj) with β = 1/kBT and α(>0) being the excluded-volume
parameter. Then, the second virial coefficient A2 can be expressed as
A2 = − NA
2NM2
∑
{r}
∑
{σ}
P{r}P{σ}
{
L
∏
i=0
L
∏
j=0
[1− δri,σj
]− 1
}
, (49)
where N is the volume of the solution (i.e., the number of lattice points), and P{r} is
the one-body distribution function of a star polymer normalized as
∑
{r} P{r} = N
(the {r}-sum is taken by r0 as well). The total number of configurations N (l, D) of
two star polymers apart at a vector distance D = r0 − σ0 is expressed as
N (l, D) = N (l)2
∑
{r}
∑
{σ}
P{r}P{σ}δr0,σ0+D
L
∏
i=0
L
∏
j=0
[1− δri,σj
], (50)
where N (l) denotes the total number of configurations of an isolated star polymer.
Then, we find the following relation between A2 and N (l, D):
A2M
2
NA
= −1
2
∑
D
{g(l, D)− 1}, g(l, D) =
N (l, D)
N (l,∞)
. (51)
Here we used N (l)2 = N (l,∞) which guarantees no interference at D = ∞. The
function g(l, D) in equation (51) is the pair distribution function in the dilute limit.
It approaches zero for smallD as Da2 (with a2>0) and goes to unity for large D. It is
28
Scaling theory and computer simulation
related to the effective interstar potential U(l, D) via g(l, D) ∝ exp[−U(l, D)/kBT ].
Then, we obtain the logarithmic dependence of the effective interstar potential
U(l, D) ∼ −a2kBT logD, (52)
which was first predicted by Witten and Pincus [29].
The penetration function Ψ, which is a combination of the second virial coefficient
and the mean square radius of gyration as
Ψ =
A2M
2
4π3/2NAR2
gyr〉3/2
, (53)
is often used instead of A2 itself, since it is known to be a universal quantity [37].
Ohno et al. [40] and Shida et al. [42] performed Monte Carlo simulations based on
the enrichment algorithm to obtain the effective interstar potential and the second
virial coefficient of star polymers. Rubio and Freire [41] evaluated the second virial
coefficient of star polymers by using off-lattice Monte Carlo simulations.
In the enrichment algorithm, the successive ratios N (l, D)/N (l − 1, D) of the
total number of configurations are obtained automatically. Multiplying these ratios
from l(≫10) to l=10 yields N(l, D)/N(9, D). Then, g(l, D) in (51) is given by
g(l, D) =
N(l, D)
N(9, D)
N(9,∞)
N(l,∞)
=
N(l,D)
N(l−1,D)
N(l−1,D)
N(l−2,D)
N(l−2,D)
N(l−3,D)
· · · N(11,D)
N(10,D)
N(10,D)
N(9,D)
N(l,∞)
N(l−1,∞)
N(l−1,∞)
N(l−2,∞)
N(l−2,∞)
N(l−3,∞)
· · · N(11,∞)
N(10,∞)
N(10,∞)
N(9,∞)
. (54)
In the second equality, we used N (9, D) = N(9,∞) = N 2(9) for sufficiently large
D. (In fact this relation holds for all D satisfying D > 18.) Then the substitution of
equation (54) into equation (51) yields A2 which can be evaluated from the subse-
quent ratios of the total number of configurations. The values for the gyration radius
Rgyr, the second virial coefficient A2, the penetration function Ψ and the coefficient
of the interstar potential a2 obtained by Ohno et al. [40] are summarized in table 3.
The results were successfully compared with the RG ε expansion by Douglas and
Freed [22], the prediction by Witten and Pincus [29] and the experiments for Ψ by
Douglas et al. [37], Roovers et al. [49] and Okumoto et al. [50].
Table 3. The estimated values [40] of the mean-square radius of gyration
R2
gyr, the second virial coefficient A2M
2/NA, the penetration function Ψ =
(A2M
2)/(4π3/2NAR
3
gyr) and the coefficient of the repulsive potential a2.
f 3 4 5 6
R2
gyr 91.38 103.08 111.14 118.11
A2M
2/NA 6735.78 10622.13 13760.61 18534.96
Ψ 0.35 0.46 0.53 0.64
a2 2.04 2.43 3.28 3.50
29
K.Ohno
3.3. Relaxation time
In the presence of excluded volume interaction, each arm is approximately re-
stricted in its configuration to a cone the angle of which vanishes proportional to
1/r1/2 as f → ∞. For this situation, Grest et al. [43] suggested the existence of
several relaxation times; the autocorrelation time τB of the distance between a core
and an arm end is associated with the size of the largest blob: ξmax ∝ R(f)−1/2,
which contains lBmax ∝ ξ
1/ν
max ∝ lf−1/2ν effective monomers. Then Grest et al. [43]
proposed
τB ∝ ξ2maxlBmax ∝ l1+2νf−(1+2ν)/2, f ≫ 1. (55)
For a relaxation time for the shape fluctuation of a star polymer, where a densi-
ty fluctuation has to diffuse a distance of the radius R of the star polymer, they
proposed a formula (“el” stands for “elastic”)
τel ∝ τB(R/ξmax)
2 ∝ l1+2νf 1−[(1+2ν)/2] ≈ l2.176f−0.088. (56)
Grest et al. [43] performed molecular dynamics simulations of multi-arm star poly-
mers and confirmed these relations as a function of f . Equation (56) was also con-
firmed by Ohno et al. [44] who used the enrichment Monte Carlo simulation com-
bined with the Kramers potential method.
3.4. Hydrodynamic interactions
In the calculation of hydrodynamic properties, one often uses a rigid-body ap-
proximation, in which polymer chains in the flow are assumed to move as if they
were rigid molecules, i.e., their equilibrium conformations are presurved in the flow.
This leads to the Kirkwood–Riseman equation [64] for each polymer molecule,
1
6πη0a
F i +
∑
i 6=j
T ijF j − ui = −v
0
i . (57)
Here, η0 is the viscosity of the fluid, a is the hydrodynamic radius of each segment,
F i = (Fix, Fiy, Fiz) the frictional force exerted on the ith segment, and T ij is the
Oseen tensor defined by
T ij =
1
8πη0rij
{
I +
rijrij
r2ij
}
. (58)
Here rij is the distance between the ith and jth segments.
From these equations, the hydrodynamic radius RH can be calculated by setting
the angular velocity at zero, and the intrinsic viscosity η can be calculated by setting
both the total force and total moment to be equal to zero. Such a treatment with
Monte Carlo simulations was first given for a linear chain in 1956 by Zimm [65]
and therefore is nowadays called a Zimm model. The Monte Carlo investigation for
star polymers was first performed in 1986–1987 by Rey et al. [45] and then in 1988
30
Scaling theory and computer simulation
by Wilkinson et al. [46], and recently by Shida et al. [48]. Table 4 lists the results
by Shida et al. [48] of the factors for the intrinsic viscosity and the hydrodynamic
radius,
η = B(f)Lν , RH = C(f)Lν , (59)
as well as the Flory viscosity factor
Φ ≡ [η]M/(6R2
gyr)
3/2 (60)
and the hydrodynamic factor
ρ ≡ Rgyr/RH. (61)
The data of 10−23Φ mol−1 for f = 2 and 12 obtained by Shida et al. [48] are close
to those (1.80 and 5.18 for f = 2 and f = 12, respectively) obtained by Freire et
al. [47] who used simulations without the rigid-body approximation. This implies
that the approximation does not make much of an error on viscosity values. Table 4
also contains the values obtained from the empirical formulae for g η ≡ B(f)/B(2)
and gH ≡ C(f)/C(2), which were proposed by Douglas et al. [37] by fitting to their
experimental data. As is seen in this table, the agreement between the Monte Carlo
results and the empirical formulae is excellent.
Table 4. Monte Carlo results [48] of the factors for the intrinsic viscosity, B(f) and
gη ≡ B(f)/B(2), and for the hydrodynamic radius, C(f) and gH ≡ C(f)/C(2).
The values obtained from empirical formulae by Douglas et al. [37] are also listed
in the columns indicated by “Emp.”. Results [48] of the Flory viscosity factor
Φ and the hydrodynamic factor ρ defined, respectively, by (60) and (61) are
compared with experimental values [31–37] (the columns indicated by “Exp.”).
f 2 3 4 6 8 12 18
B(f) 1.10 0.95 0.80 0.61 0.49 0.37 —
gη — 0.86 0.73 0.55 0.45 0.34 0.25
Emp. ([37]) — 0.83 0.71 0.56 0.46 0.33 0.22
C(f) 0.41 0.38 0.37 0.35 0.33 0.30 0.27
gH — 0.94 0.92 0.86 0.82 0.73 0.66
Emp. ([37]) — 0.96 0.92 0.86 0.80 0.72 0.63
10−23Φ mol−1 2.0 2.3 2.9 3.7 4.3 5.8 7.2
Exp. ([31–37]) — 2.6 3.1 3.3–3.9 4.1 5.5–6.1 5.8
ρ 1.36 1.26 1.16 1.04 0.97 0.88 0.80
Exp. ([31–37]) — — 1.09 1.01 — 0.92 0.88
31
K.Ohno
References
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K.Ohno
Теорія скейлінгу і комп’ютерне моделювання
зіркових полімерів в добрих розчинниках
К.Оно
Кафедра фізики, технологічний факультет,
Національний університет м. Йокогама,
79-5 Токівадаі, Годогая-ку, Йокогама 240-8501, Японія
Отримано 7 грудня 2001 р.
Проведено огляд теорій скейлінгу і ренормгрупового ε = 4 − d роз-
кладу ( d – вимірність простору) а також комп’ютерного моделюван-
ня зіркових полімерів (і полімерних сіток), що складаються з дов-
гих гнучких ланцюгів однакової довжини і знаходяться в розведеному
розчині в границі доброго розчинника. Зокрема, в термінах аналогії
полімер-магнетик детально обговорюється асимптотична поведін-
ка конформаційних і ентропійних величин в границі довгих ланцюгів.
Розглянуто не лише статичні характеристики, такі, як функції розпо-
ділу і осмотичний тиск чи ентропія, але і динамічні властивості, такі як
час релаксації і власна в’язкість зіркових полімерів.
Ключові слова: ренормалізаційна група, моделювання методом
Монте Карло, загальна кількість конфігурацій, віріальний
коефіцієнт, час релаксації, гідродинамічний ефект
PACS: 05.10.Cc, 05.10.Ln, 61.41.+e, 82.35.Gh, 82.35.Lr, 82.70.Uv
36
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