Interactions between star polymers: High-order calculations of the scaling exponents
The scaling behavior of star polymers can be calculated in the m → 0 limit of an m -component spin system with an additional composite operator. The resulting scaling exponents describe the effective interaction of such polymer stars, i.e. objects whose behavior interpolates between that of poly...
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Інститут фізики конденсованих систем НАН України
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| Cite this: | Interactions between star polymers: High-order calculations of the scaling exponents / V. Schulte-Frohlinde , Yu. Holovatch , C. von Ferber, A. Blumen // Condensed Matter Physics. — 2003. — Т. 6, № 4(36). — С. 703-711. — Бібліогр.: 29 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1207692025-06-03T16:30:09Z Interactions between star polymers: High-order calculations of the scaling exponents Взаємодія між зірковими полімерами: обчислення скейлінгових показників у вищих порядках теорії збурень Schulte-Frohlinde, V. Holovatch, Yu. von Ferber, C. Blumen, A. The scaling behavior of star polymers can be calculated in the m → 0 limit of an m -component spin system with an additional composite operator. The resulting scaling exponents describe the effective interaction of such polymer stars, i.e. objects whose behavior interpolates between that of polymer coils and that of hard sphere colloidal particles. We extend the existing renormalization group calculations from the third to the fourth order. Аналіз масштабної (скейлінгової) поведінки зіркових полімерів можна проводити на основі границі m → 0 m -компонентної спінової системи з додатковим композитним оператором. Скейлінгові показники, які при цьому отримуються, описують ефективну взаємодію полімерних зірок: об’єктів, поведінка яких інтерполює між поведінкою полімерних клубків та колоїдних частинок. Ми доповнюємо існуючі ренормгрупові обчислення третього порядку теорії збурень четвертим. We thank the Deutsche Forschungsgemeinschaft (SFB 428), the Fonds der Chemischen Industrie and the Bundesministerium fur ¨ Bildung und Forschung (BMBF) for support. 2003 Article Interactions between star polymers: High-order calculations of the scaling exponents / V. Schulte-Frohlinde , Yu. Holovatch , C. von Ferber, A. Blumen // Condensed Matter Physics. — 2003. — Т. 6, № 4(36). — С. 703-711. — Бібліогр.: 29 назв. — англ. 1607-324X PACS: 64.60.Ak, 61.41.+e, 64.60.Fr, 11.10.Gh DOI:10.5488/CMP.6.4.703 https://nasplib.isofts.kiev.ua/handle/123456789/120769 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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| description |
The scaling behavior of star polymers can be calculated in the m → 0
limit of an m -component spin system with an additional composite operator.
The resulting scaling exponents describe the effective interaction
of such polymer stars, i.e. objects whose behavior interpolates between
that of polymer coils and that of hard sphere colloidal particles. We extend
the existing renormalization group calculations from the third to the fourth
order. |
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Article |
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Schulte-Frohlinde, V. Holovatch, Yu. von Ferber, C. Blumen, A. |
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Schulte-Frohlinde, V. Holovatch, Yu. von Ferber, C. Blumen, A. Interactions between star polymers: High-order calculations of the scaling exponents Condensed Matter Physics |
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Schulte-Frohlinde, V. Holovatch, Yu. von Ferber, C. Blumen, A. |
| author_sort |
Schulte-Frohlinde, V. |
| title |
Interactions between star polymers: High-order calculations of the scaling exponents |
| title_short |
Interactions between star polymers: High-order calculations of the scaling exponents |
| title_full |
Interactions between star polymers: High-order calculations of the scaling exponents |
| title_fullStr |
Interactions between star polymers: High-order calculations of the scaling exponents |
| title_full_unstemmed |
Interactions between star polymers: High-order calculations of the scaling exponents |
| title_sort |
interactions between star polymers: high-order calculations of the scaling exponents |
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Інститут фізики конденсованих систем НАН України |
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2003 |
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https://nasplib.isofts.kiev.ua/handle/123456789/120769 |
| citation_txt |
Interactions between star polymers:
High-order calculations of the scaling
exponents / V. Schulte-Frohlinde , Yu. Holovatch , C. von Ferber, A. Blumen // Condensed Matter Physics. — 2003. — Т. 6, № 4(36). — С. 703-711. — Бібліогр.: 29 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT schultefrohlindev interactionsbetweenstarpolymershighordercalculationsofthescalingexponents AT holovatchyu interactionsbetweenstarpolymershighordercalculationsofthescalingexponents AT vonferberc interactionsbetweenstarpolymershighordercalculationsofthescalingexponents AT blumena interactionsbetweenstarpolymershighordercalculationsofthescalingexponents AT schultefrohlindev vzaêmodíâmížzírkovimipolímeramiobčislennâskejlíngovihpokaznikívuviŝihporâdkahteoríízburenʹ AT holovatchyu vzaêmodíâmížzírkovimipolímeramiobčislennâskejlíngovihpokaznikívuviŝihporâdkahteoríízburenʹ AT vonferberc vzaêmodíâmížzírkovimipolímeramiobčislennâskejlíngovihpokaznikívuviŝihporâdkahteoríízburenʹ AT blumena vzaêmodíâmížzírkovimipolímeramiobčislennâskejlíngovihpokaznikívuviŝihporâdkahteoríízburenʹ |
| first_indexed |
2025-11-26T19:06:40Z |
| last_indexed |
2025-11-26T19:06:40Z |
| _version_ |
1849881006686863360 |
| fulltext |
Condensed Matter Physics, 2003, Vol. 6, No. 4(36), pp. 703–711
Interactions between star polymers:
High-order calculations of the scaling
exponents
V.Schulte-Frohlinde 1 , Yu.Holovatch 2,3 , C. von Ferber 1 ,
A.Blumen 1
1 Theoretische Polymerphysik, Universität Freiburg,
Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany
2 Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
3 Ivan Franko National University of Lviv,
12 Drahomanov Str., 79005 Lviv, Ukraine
Received September 23, 2003
The scaling behavior of star polymers can be calculated in the m → 0
limit of an m -component spin system with an additional composite op-
erator. The resulting scaling exponents describe the effective interaction
of such polymer stars, i.e. objects whose behavior interpolates between
that of polymer coils and that of hard sphere colloidal particles. We extend
the existing renormalization group calculations from the third to the fourth
order.
Key words: polymers, renormalization-group, fractals, critical exponents
PACS: 64.60.Ak, 61.41.+e, 64.60.Fr, 11.10.Gh
Star polymers which consist of linear polymer chains linked at one end are the
simplest branched objects. They act as colloidal particles [1,2]; when increasing
the number of arms their behavior changes from that of linear polymers to that
of polymeric micelles [3]. From the scaling properties of star polymers, one may
derive their effective interactions. The partition function of two star polymers with
f arms each and a distance r between their cores obeys a power law governed by
the so-called contact exponent Θff [4]
Z
(2)
ff (r) ∼ rΘff . (1)
The logarithm of the partition function is the free energy, F
(2)
ff = −kBT lnZ
(2)
ff .
Differentiating F
(2)
ff with respect to r gives the mean force Ff (r) between two star
c© V.Schulte-Frohlinde, Yu.Holovatch, C. von Ferber, A.Blumen 703
V.Schulte-Frohlinde et al.
polymers [4–6]:
1
kBT
Ff (r) =
Θff
r
. (2)
The mean force between the cores of two star polymers is experimentally acces-
sible and determines the properties of star polymer solutions. Estimates for the
dependence of the contact exponent Θff on the number of arms f have been found
by scaling arguments and by using the so-called cone approximation [7,8], giving
Θff ∼ f 3/2. Making use of the known results [5] for Θff for f = 1 and f = 2 also
fixes the prefactor. Thus [1]
Θff ≈
5
18
f 3/2. (3)
The contact exponents Θff are related to a family of exponents for single star
polymers [4,9,10]. We denote by Z (∗f) the partition function of a single star polymer
with f chains, each consisting of N monomers. Z (∗f) obeys the following scaling
form [4,11]:
Z(∗f) ∼ eµNfNγf−1, (4)
with the star configuration exponent γf and a connectivity constant eµ. We note that
the exponents γf are directly accessible through Monte Carlo (MC) simulations [12–
14]. In particular, γ2 = γ1 is the configuration exponent of a polymer chain.
Now, the γf can be expressed in terms of additional exponents ηf which appear
naturally in the field theory that we develop below. The relations between these
exponents are [15]
γ1 = γ2 = 1 − νη2 (5)
and
γf = 1 + ν(ηf − fη2), f > 2. (6)
In equations (5) and (6) ν is the usual Flory exponent, which relates the mean square
end-to-end distance R2 of the polymer chain to its number of monomers N through
R2 ∼ N2ν .
The star exponents γf and ηf are universal; they depend on f and on the space
dimension d only [15]. The exponents ηf allow to express Θff in a simple way [4,16]:
Θff ′ = ηf + ηf ′ − ηf+f ′ . (7)
So far, ηf (or equivalently γf) were evaluated perturbatively in expansions up to
the third order both in ε = 4 − d [4,8–11,15,17] and in fixed dimension d = 3 [16].
Here, we calculate ηf up to the order ε4.
It is well known that the scaling exponents of polymer chains may generally
be obtained from the m → 0 limit of O(m)-symmetric m-vector magnets [18–20].
The same is true for the star exponents, but the underlying field theory has to
be extended to represent stars. Our calculations are based on the field-theoretical
renormalization group (RG) approach in the minimal subtraction scheme [20–22].
We follow the formalism of [16] and refer the reader to [23] for a more detailed
account. The field-theoretical description of a star polymer may be performed in
704
Interactions between star polymers
terms of the Edwards model of continuous chains [24], generalized in order to describe
a set of f chains. The configuration of a linear polymer, say a, is given by a path
ra(s) in d-dimensional space IRd parameterized by a variable 0 6 s 6 Sa where Sa
is given by Nl2, l being the length of a monomer. Denoting the excluded volume
interaction by u, the Hamiltonian H for f chains is given by [15]
1
kBT
H(ra, {Ss}) =
f
∑
a=1
∫ Sa
0
ds
(
dra(s)
2ds
)2
+
1
6
f
∑
a,b=1
u
∫
drρa(r)ρb(r), (8)
with the densities ρa(r) =
∫ Sa
0
ds δd(r− ra(s)). In this formalism the partition func-
tion of f polymer chains is calculated as a functional integral:
Z(f){Sa} =
∫
D[ra(s)] exp
{
−
1
kBT
H(ra, {Sa})
}
. (9)
Here, the symbol D[ra(s)] includes the normalization such that Z (∗f){Sa} = 1 if
u ≡ 0. In order to have a well-defined bare theory a cutoff s0 is introduced such that
all simultaneous integrals of any variables s and s′ on the same chain are cut off by
|s − s′| > s0. The f polymers can now be used to form a star by constraining them
to have a common starting point. The corresponding partition function is:
Z(∗f){Sa} =
∫
D[ra] exp
{
−
1
kBT
H(ra, {Sa})
} f
∏
a=2
δd(ra(0) − r1(0)). (10)
The continuous chain model of equations (9) and (10) can be mapped onto a
corresponding field theory by a Laplace transform from the variables Sa to the
conjugate chemical potentials (“mass variables”) µa [15]:
Z̃(∗f){µa} =
∫
∞
0
(
∏
b
dSbe
−µbSb
)
Z(∗f){Sa}. (11)
The Laplace-transformed partition function Z̃(∗f){µa} is expressed as the m = 0
limit of the functional integral over vector fields φa, a = 1, . . . , f with m components
φα
a , α = 1, . . . , m :
Z̃(∗f){µb} =
∫
D[φa(r)] exp[−L{φb, µb}]|m=0. (12)
The Landau-Ginzburg-Wilson Lagrangian L of f interacting fields φb, each with m
components, reads
L{φb, µb} =
1
2
f
∑
a=1
∫
dr
[
µaφ
2
a + (∇φa(r))
2
]
+
1
4!
f
∑
a,a′=1
u
∫
drφ2
a(r)φ
2
a′(r), (13)
where φ2
a =
∑m
α=1(φ
α
a )2. The limit m = 0 in equation (12) can be understood as a
selection rule for the diagrams that contribute to the perturbation theory expansions.
705
V.Schulte-Frohlinde et al.
The one particle irreducible (1PI) vertex functions Γ(n)(qi) of this theory are defined
by:
δ
(
∑
qi
)
Γ(n)
a1...an
(qi) =
∫
eiqi·ridr1 . . .drn〈φa1
(r1) . . . φan
(rn)〉L1PI,m=0. (14)
The average 〈· · ·〉 in equation (14) is taken with respect to the Lagrangian in equa-
tion (13), keeping only the contributions which correspond to one-particle-irreducible
graphs and which have non-vanishing tensor factors in the limit m = 0.
The vertex part of the Laplace transform of equation (10) is given by [15]:
δ(p +
∑
qi)Γ
(∗f)(p,q1 . . .qf ) =
=
∫
ei(p·r0+qi·ri)dr0dr1 . . .drf 〈φ1(r0) . . . φf (r0)φ1(r1) . . . φf(rf)〉
L
1PI,m=0. (15)
Thus, one obtains the vertex function Γ(∗f) by insertion of the composite operator
∏
a φa into the 1PI vertex function with f external legs. The scaling dimension of
this operator defines the star exponent of the corresponding polymer.
Now, we sketch the RG analysis of the field theory in equation (13). The initial
expressions for the calculations are the bare vertex functions (∂/∂k2)Γ(2)(u), Γ(4)(u),
and Γ(∗f)(u). Ultraviolet divergences occur when the bare vertex functions are eval-
uated naively [20,22]. The polymer limit m = 0 leads to essential simplifications of
the renormalization: each field φa and mass µa renormalizes as if the other fields
were absent. Since the theory is renormalizable, we can collect all the divergences in
the so-called renormalization factors Z and define a finite theory of the same struc-
ture as the original one by renormalizing the parameters. Here we use dimensional
regularization in which divergences are expressed as poles in ε, defined by d = 4− ε.
We render the theory finite by using the minimal subtraction scheme in which
only pole terms in ε are subtracted. The renormalization is carried out with the
help of the KR̄-operation [22], which recursively subtracts all pole terms of lower
order. The expressions for the Z-factors are given as power series in the renormalized
coupling constant g:
Zφa
(g) = KR̄Γ(2)
aa (g), Zg(g) = KR̄Γ(4)(g), (16)
where the renormalized coupling g is defined by:
u = κεZ−2
φa
Zgg. (17)
The parameter κ sets the scale of the external momenta in the renormalization
procedure.
In order to renormalize the star vertex functions the renormalization factors Z∗f
are introduced by:
Z
f/2
φa
Z∗f = KR̄Γ(∗f)(g). (18)
The renormalized couplings g and renormalizing Z-factors depend on the scale
parameter κ. This dependence defines the RG functions by the following relations:
κ
d
dκ
g = β(g), κ
d
dκ
ln Zφa
= ηφa
(g), κ
d
dκ
ln Z∗f = ηf (g). (19)
706
Interactions between star polymers
Figure 1. The contributing fourth order diagrams which have a φ5-vertex as star
vertex.
The function ηφa
(g) describes the pair correlation critical exponent [see equa-
tion (16)], while the functions ηf (g) correspond to the set of exponents for poly-
mer stars introduced in equation (5).
We have constructed explicit expressions for the β and η functions up to four
loops. For the renormalized propagators and couplings of equations (16) and (17)
we use the expansions for the corresponding magnetic spin system which are dia-
grammatic expansions with four-point interactions only, and apply the appropriate
combinatorics in the limit m → 0. For equation (18), we have to establish all terms
containing an f -point interaction for the star vertex. At the four loop level this
also includes a five-point interaction which does not appear among the four-point
interactions in the spin system. Therefore, we have to calculate and renormalize six
diagrams which contain a five-point interaction on top of their four-point interac-
tions and which are shown in figure 1. It turns out that the integrations of five of
these diagrams are identical to those of the diagrams which contain only four point
interactions and which have been calculated for the spin system. We calculated the
renormalized part of the last diagram in figure 1 by applying to it the so-called infra-
red rearrangement [25] and the R̄∗-operation [26]. The details and explicit forms for
the β and γ-functions are given in a separate publication together with a study of
the RG flow and the fixed points; there we have generalized the whole approach to
cover the copolymer stars as well [23].
The asymptotic values of the scaling exponents were evaluated at the fixed point
of the β-function. We find the following expansions for the polymer star exponents:
ηf = f(f − 1)
{
−
ε
8
+
ε2
256
[8(f − 2) − 9] + ε3[a31 + (f − 2)(a32 + (f − 3)a33)]
+ ε4[a41 + (f − 2)(a42 + (f − 3)(a43 + (f − 4)a44))]
}
. (20)
Here
a31 = 33ζ3/512 − 49/4096,
a32 = 9/512 − 7ζ3/128,
a33 = −1/64,
a41 = 477ζ3/16384 + (99ζ4 − 465ζ5)/2048 + 47/262144,
a42 = (95ζ5 − 21ζ4)/512 − 153ζ3/4096 + 133/32768,
a43 = (65ζ3 + 75ζ5 + 1)/2048,
a44 = 21/2048
and ζk = ζ(k) are the values of the Riemann-ζ-function. Equation (20) recovers the
ε3 results of [15,16].
707
V.Schulte-Frohlinde et al.
-12
-10
-8
-6
-4
-2
0
2
0 1 2 3 4 5 6 7 8
η f
f
-12
-10
-8
-6
-4
-2
0
2
0 1 2 3 4 5 6 7 8
η f
f
(a) (b)
Figure 2. Star exponent ηf for d = 2. Lines: exact result [9]. (a) Conformal
resummation; open squares: fourth order (ε4), open triangles: third order (ε3),
(b) Padé approximants; open squares: ε4 – Padé(2,2) approximant, open triangles:
ε3 – Padé(2,1) approximant, full diamonds: ε4 – Padé(3,2) approximant.
As is well known, the RG perturbative expansions are asymptotic at best [27]
and appropriate resummation techniques should be applied in order to extract re-
liable quantitative information from them [20,22]. Here, we apply a resummation
procedure based on a conformal mapping, which is widely used in the analysis of
perturbative RG expansions [18,19]. The procedure relies on the fact that the series
for the β-functions of the φ4-theory are asymptotic and Borel-summable. In our case
we assume that the perturbation series for ηf is of the form
ηf =
∑
k>1
Af,k , Af,k ∼ k!(âf)k, k � 1, (21)
where â = 3/8 as found for the β-function [18]. For comparison we also apply
simple Padé approximations. First, we report the results for the star exponents ηf .
In figure 2 we give the third (∼ ε3) and fourth (∼ ε4) order results of the resummed
ε-expansion for d = 2, together with the exact result [9] for the exponent ηf , which
reads:
ηf (d = 2) = −
(f − 1)(9f + 4)
48
. (22)
Our results coincide with the exact result at small values of f . The expansion co-
efficients in the series for the star exponents grow with increasing f and become,
therefore, less reliable for larger values of f . Nonetheless, for all f the ε4 contribution
is closer to the exact data than the third order approximation.
From figure 2(b), where we compare different Padé approximants for ηf with
the exact result, we conclude that better agreement is found for the non-diagonal
approximants (2,1) and (3,2), which both have a quadratic high f behavior. This
seems to indicate that the weights of the contributions of the different ε-orders are
not optimally chosen in our Borel resummation. A reason for this might be that
the growth of the combinatorial factors in the expansion of the star exponents is
stronger than assumed in equation (21).
708
Interactions between star polymers
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
0 2 4 6 8 10 12
γ f
f
(a)
0
1
2
3
4
5
6
7
8
9
0 2 4 6 8 10
Θ
ff
f
(a) (b)
Figure 3. Star exponent γf (a) and contact exponent Θff (b) at d = 3. (a) Open
squares: fourth order (ε4), open triangles up: third order (ε3), open triangles
down: third order RG calculated at fixed dimension (d=3) [16], full triangles up:
Monte Carlo simulations (MC) [13], full triangles down: MC [12], full diamonds:
MC [14]. (b) Open squares: fourth order (ε4), open triangles: third order (ε3), line:
cone approximation Θff = 5
18f3/2. The values for Θff are found via equation (7)
after resumming the series for the exponents ηf .
In figure 3(a) we compare the resummed ε3 and ε4 expansions for the d = 3
star exponents. The exponents γf were obtained from the d = 3 data for ηf [28]
via equation (5), with ν = 0.588. These results are confronted with those of MC
simulations [12,13] and with the resummed three-loop pseudo-ε expansion obtained
within the massive RG scheme at fixed d = 3 [16]. The agreement gets worse for
larger f .
In figure 3(b) we plot the results for the exponent Θff , obtained through our
method, which we compare to equation (3) established in the cone approximation.
Two obvious conclusions from figure 3(b) are: (i) for small f , where the perturbative
expansions are known to give precise results, the cone approximation gives a reliable
description; (ii) the fourth order expansion confirms the third order results in this
respect.
In summary, we obtained in d = 2 and d = 3 the scaling exponents for a polymer
star with an arbitrary number f of arms. We confronted our results in d = 3 with
those of other approaches, based on Monte Carlo simulations [12,13,29], on the fixed
dimension RG technique [16], and on the cone approximation [10]. In d = 2, we com-
pared our findings to the exact expressions [9]. Our perturbative approach to fourth
order in ε confirms the results previously obtained using lower order perturbation
theory methods. A more careful study of the asymptotic behavior of our series might
permit a refinement of the resummation in order to extract better numerical values
for the star-star interaction at larger numbers f of arms.
709
V.Schulte-Frohlinde et al.
Acknowledgements
We thank the Deutsche Forschungsgemeinschaft (SFB 428), the Fonds der Che-
mischen Industrie and the Bundesministerium für Bildung und Forschung (BMBF)
for support.
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Взаємодія між зірковими полімерами: обчислення
скейлінгових показників у вищих порядках теорії
збурень
В.Шулте-Фролінде 1 , Ю.Головач 2,3 , К. фон Фербер 1 ,
А.Блюмен 1
1 Теоретична фізика полімерів, Університет Фрайбург,
Німеччина, D-79104 Фрайбург, Герман-Гердер-Штрассе, 3
2 Інститут фізики конденсованих систем НАН України,
79011 Львів, вул. Свєнціцького, 1
3 Львівський національний університет ім. Івана Франка,
79005 Львів, вул. Драгоманова, 12
Отримано 23 вересня 2003 р.
Аналіз масштабної (скейлінгової) поведінки зіркових полімерів мож-
на проводити на основі границі m → 0 m -компонентної спінової
системи з додатковим композитним оператором. Скейлінгові по-
казники, які при цьому отримуються, описують ефективну взаємо-
дію полімерних зірок: об’єктів, поведінка яких інтерполює між пове-
дінкою полімерних клубків та колоїдних частинок. Ми доповнюємо
існуючі ренормгрупові обчислення третього порядку теорії збурень
четвертим.
Ключові слова: полімери, ренормалізаційна група, фрактали,
критичні показники
PACS: 64.60.Ak, 61.41.+e, 64.60.Fr, 11.10.Gh
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