The joint Rouse-Zimm theory of the dynamics of polymers in dilute solutions
We propose a theory of the dynamics of polymers in dilute solution, in which the popular Zimm and Rouse models are just the limiting cases of an infinitely large and small draining parameter. The equation of motion for the polymer segments (beads) is solved together with Brinkman’s equation for th...
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irk-123456789-1212882017-06-14T03:08:02Z The joint Rouse-Zimm theory of the dynamics of polymers in dilute solutions Lisy, V. Tothova, J. Zatovsky, A. We propose a theory of the dynamics of polymers in dilute solution, in which the popular Zimm and Rouse models are just the limiting cases of an infinitely large and small draining parameter. The equation of motion for the polymer segments (beads) is solved together with Brinkman’s equation for the solvent velocity that takes into account the presence of other polymer coils in the solution. The equation for the polymer normal modes is obtained and the relevant time correlation functions are found. A tendency to the time-dependent hydrodynamic screening is demonstrated on the diffusion of the polymers as well as on the relaxation of their internal modes. With the growing concentration of the coils in the solution, they both show a transition to the exactly Rouse behaviour. The shear viscosity of the solution, the Huggins coefficient and other quantities are calculated and shown to be notably different from the known results. 2006 Article The joint Rouse-Zimm theory of the dynamics of polymers in dilute solutions / V. Lisy, J. Tothova, A. Zatovsky // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 95-102. — Бібліогр.: 23 назв. — англ. 1607-324X PACS: 36.20.Fz, 36.20.-r, 82.35.Lr, 82.37.-j, 83.80.Rs DOI:10.5488/CMP.9.1.95 http://dspace.nbuv.gov.ua/handle/123456789/121288 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We propose a theory of the dynamics of polymers in dilute solution, in which the popular Zimm and Rouse
models are just the limiting cases of an infinitely large and small draining parameter. The equation of motion
for the polymer segments (beads) is solved together with Brinkman’s equation for the solvent velocity that
takes into account the presence of other polymer coils in the solution. The equation for the polymer normal
modes is obtained and the relevant time correlation functions are found. A tendency to the time-dependent
hydrodynamic screening is demonstrated on the diffusion of the polymers as well as on the relaxation of their
internal modes. With the growing concentration of the coils in the solution, they both show a transition to the
exactly Rouse behaviour. The shear viscosity of the solution, the Huggins coefficient and other quantities are
calculated and shown to be notably different from the known results. |
| format |
Article |
| author |
Lisy, V. Tothova, J. Zatovsky, A. |
| spellingShingle |
Lisy, V. Tothova, J. Zatovsky, A. The joint Rouse-Zimm theory of the dynamics of polymers in dilute solutions Condensed Matter Physics |
| author_facet |
Lisy, V. Tothova, J. Zatovsky, A. |
| author_sort |
Lisy, V. |
| title |
The joint Rouse-Zimm theory of the dynamics of polymers in dilute solutions |
| title_short |
The joint Rouse-Zimm theory of the dynamics of polymers in dilute solutions |
| title_full |
The joint Rouse-Zimm theory of the dynamics of polymers in dilute solutions |
| title_fullStr |
The joint Rouse-Zimm theory of the dynamics of polymers in dilute solutions |
| title_full_unstemmed |
The joint Rouse-Zimm theory of the dynamics of polymers in dilute solutions |
| title_sort |
joint rouse-zimm theory of the dynamics of polymers in dilute solutions |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121288 |
| citation_txt |
The joint Rouse-Zimm theory of the dynamics of polymers in dilute solutions / V. Lisy, J. Tothova, A. Zatovsky // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 95-102. — Бібліогр.: 23 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT lisyv thejointrousezimmtheoryofthedynamicsofpolymersindilutesolutions AT tothovaj thejointrousezimmtheoryofthedynamicsofpolymersindilutesolutions AT zatovskya thejointrousezimmtheoryofthedynamicsofpolymersindilutesolutions AT lisyv jointrousezimmtheoryofthedynamicsofpolymersindilutesolutions AT tothovaj jointrousezimmtheoryofthedynamicsofpolymersindilutesolutions AT zatovskya jointrousezimmtheoryofthedynamicsofpolymersindilutesolutions |
| first_indexed |
2025-07-08T19:32:18Z |
| last_indexed |
2025-07-08T19:32:18Z |
| _version_ |
1837108445116891136 |
| fulltext |
Condensed Matter Physics 2006, Vol. 9, No 1(45), pp. 95–102
The joint Rouse-Zimm theory of the dynamics of
polymers in dilute solutions
V.Lisy1, J.Tothova1, A.Zatovsky2
1 Institute of Physics, P.J.Safarik University,
Jesenna 5, 041 54 Kosice, Slovakia
2 Department of Theoretical Physics,
I.I.Mechnikov Odessa National University,
2, Dvoryanskaya Str., 65026 Odessa, Ukraine
Received July 11, 2005
We propose a theory of the dynamics of polymers in dilute solution, in which the popular Zimm and Rouse
models are just the limiting cases of an infinitely large and small draining parameter. The equation of motion
for the polymer segments (beads) is solved together with Brinkman’s equation for the solvent velocity that
takes into account the presence of other polymer coils in the solution. The equation for the polymer normal
modes is obtained and the relevant time correlation functions are found. A tendency to the time-dependent
hydrodynamic screening is demonstrated on the diffusion of the polymers as well as on the relaxation of their
internal modes. With the growing concentration of the coils in the solution, they both show a transition to the
exactly Rouse behaviour. The shear viscosity of the solution, the Huggins coefficient and other quantities are
calculated and shown to be notably different from the known results.
Key words: polymers, dynamics, dilute solution, rheology
PACS: 36.20.Fz, 36.20.-r, 82.35.Lr, 82.37.-j, 83.80.Rs
1. Introduction
Interest in dilute polymer solutions arises primarily from their importance in the characteriza-
tion of polymers, their interaction with solvent and from a fundamental interest in understanding
macromolecular response to hydrodynamic forces, free from the complications of intermolecular
entanglements. Despite five decades of investigations, these issues are not entirely understood. So,
there is a systematic discrepancy between the dynamic scattering data and the theory [1–3]. The
existing theories give different results for the viscosity of dilute polymer solutions (see, e.g., [4,5]
and the citations therein), the observed monomer motion in single polymer chains cannot be ex-
plained by the available theories [6], the time dependence of hydrodynamic screening in solutions
has not been explained [7], etc. For other problems we refer the reader to the recent review [8].
The aim of this work was to contribute to the solution of some of these problems by developing
a phenomenological bead-spring theory of diffusion of an individual “test” polymer and the relax-
ation of its internal modes in the solution of unentangled polymers. Our approach differs from the
traditional ones in several main points. Firstly, the joint Rouse-Zimm theory is exploited [9]. In
the literature, following de Gennes [10], it is assumed that at θ conditions, the Zimm modes with
the dispersion of relaxation times τpZ ∼ p−3/2 , where p is the mode number, at low frequencies,
should always dominate the Rouse modes with τpR ∼ p−2; we show that these both contributions
to the polymer characteristics should be taken into account. Next, the internal modes are distri-
buted discretely (the assumption of their continuous distribution with respect to p is true only in
a restricted time domain and often leads to incorrect interpretation of experimental data [11]. The
formalism from our theory with the time-dependent hydrodynamics of polymers has been adopted
[9,12] and, finally, the Brinkman’s theory [13,14] for the flow in porous media is used in order to take
into account the effect of other coils on the test polymer. The presented theory has the following
limitations. The considered time scales are t � τR = R2ρ/η, where R is the hydrodynamic radius
c© V.Lisy, J.Tothova, A.Zatovsky 95
V.Lisy, J.Tothova, A.Zatovsky
of the polymer, ρ is the density and η is the viscosity of the solvent. This means that the effects
of hydrodynamic memory (or the viscous aftereffect) are neglected [9,12,15]. The distribution of
the coils in solution is considered to be stationary (this is justified at least for the times t � τD;
the choice of this time scale is possible since always τp � τD, where τD = R2/D is the character-
istic time of the coil diffusion with the diffusion coefficient D). Our theory is also restricted to θ
solvents [4]; generalizations to other cases require the knowledge of the equilibrium distribution of
the segments when the exclude volume interactions are taken into account. As already mentioned,
only solutions of unentangled polymers are considered. We are thus limited to the concentrations
of the chains c < 1/[η], where [η] is the intrinsic viscosity and c is the number of polymers per
unit volume [8]. In spite of all these restrictions and other ones, like those that we do not consider,
i. e., the internal viscosity of polymers and the self-entanglements (the importance and even the
reality of these interactions are uncertain [8]), we believe that the results of our theory could be of
interest. In particular, we have found new expressions for the quantities describing the behaviour
of flexible polymers in solution, such as the diffusion coefficient of the coil, the relaxation times of
the internal modes, the viscosity of the solution, and the Huggins coefficient. These quantities have
been obtained from a generalized Rouse-Zimm equation for the position vectors of the polymer
segments and the Oseen tensor describing the velocity field of the solvent due to perturbation. Fi-
nally, our theory describes in a simple manner the hydrodynamic screening, i.e. the concentration
and time-dependent transition between the Zimm and (as distinct from the previous theories) the
exact Rouse behaviour of the polymer.
2. The Rouse-Zimm-Brinkman theory of polymer dynamics
We choose one coil as a “test” polymer. The equation of motion of its nth segment is
M
d2−→x n(t)
dt2
=
−→
f fr
n +
−→
f ch
n +
−→
f n . (2.1)
Here, −→x n is the position vector of the segment (a spherical bead) from the N ones constituting the
polymer, M is the bead mass,
−→
f ch
n is the force from the neighboring beads along the chain,
−→
f n
is the random force due to the motion of the molecules of solvent, and
−→
f fr
n is the Stokes friction
force on the bead during its motion in the solvent [4,16]:
−→
f fr
n = −ξ
[
d−→x n
dt
−−→v (−→x n)
]
, (2.2)
where −→v denotes the velocity of the solvent in the place of the nth bead due to the motion of
other beads. The friction coefficient on the bead with radius b is ξ = 6πηb. This expression holds
in the case of steady flow and takes into account the hydrodynamic interaction. In a more general
case, with the hydrodynamic memory [9,12,15] the force (2) should be replaced by the Boussinesq
force and equation (1) has to be solved together with the nonstationary hydrodynamic equations
for the macroscopic velocity of the solvent. To take into account the presence of other polymers
in solution, we use the Brinkman’s work [13] (see also [14]) in which a polymer is considered as a
porous medium. In our approach, the whole solution is such a medium while the coils are obstacles
to the solvent flow. Then, in the right hand part of the Navier-Stokes equation, a term −κ2η−→v
has to be added, where κ−2 is the solvent permeability. This term has the sense of an average
value of the force acting on the liquid in the element of volume dV , provided the average number
of polymers in solution per unit volume is c; then κ2η = cf , where f is the friction factor on one
coil. Thus, for an incompressible solvent (∇−→v = 0)we have to solve the equation
ρ
∂−→v
∂t
= −∇p+ η4−→v − κ2η−→v + −→ϕ . (2.3)
Here p is the pressure and −→ϕ is the density of the force from the beads of the studied polymer
acting on the solvent [16],
−→ϕ (−→x ) = −
∑
n
−→
f fr
n (−→x n) δ (−→x −−→x n) . (2.4)
96
Dynamics of Rouse-Zimm polymers
To solve this equation is a difficult problem since the polymer chains are mobile. However, restricti-
ng ourselves to the times much shorter than τD, the concentration c can be assumed constant. The
above equations then describe the motion of one bead in the solvent with the effective action of
other coils on the motion of the solvent flow. This problem can be transformed to that solved
already in [17] (see also [9,15]). The velocity field can be in the Fourier representation in the time
written as follows:
vω
α (−→r ) =
∫
d−→r ′
∑
β
Hω
αβ (−→r −−→r ′)ϕω
β (−→r ′) . (2.5)
Here the analog of the Oseen tensor is
Hω
αβ (−→r ) = Aδαβ +Brαrβr
−2, (2.6)
A = (8πηr)−1
{
e−y − y
[(
1 − e−y
)
y−1
]′′}
,
B = (8πηr)−1
{
e−y + 3y
[(
1 − e−y
)
y−1
]′′}
, (2.7)
y = rχ, χ2 = κ2−iω%/η, and the prime means the differentiation with respect to y. In the particular
case ω = 0 and for permeable solvent, κ = 0, equation (6) coincides with the well-known result of
Zimm [4]. Using this solution, a generalization of the Rouse-Zimm equation can be obtained from
the equation of motion [9]. The preaveraging of the Oseen tensor over the equilibrium Gaussian
distribution of the beads [4,16] gives
〈
Hω
αβnm
〉
0
= δαβh
ω (n−m) , −→r nm ≡ −→xn −−→xm ,
hω (n−m) =
(
6π3|n−m|
)−1/2
(ηa)−1
[
1 −
√
πz exp
(
z2
)
erfc(z)
]
. (2.8)
Here a is the mean square distance between the beads along the chain and z ≡ χa (|n−m|/6)1/2.
Then in the continuum approximation with respect to the variable n the new Rouse-Zimm equation
reads
−iω−→x ω(n) =
1
ξ
[
3kBT
a2
∂2−→x ω(n)
∂n2
+Mω2−→x ω(n) +
−→
f ω(n)
]
+
∫ N
0
dmhω(n−m)
[
3kBT
a2
∂2−→x ω(m)
∂m2
+Mω2−→x ω(m) +
−→
f ω(m)
]
. (2.9)
It is solved with the help of the Fourier transformation (FT) in n, taking into account the
boundary conditions at the ends of the chain [16], ∂−→x /∂n = 0 at n = 0, N : −→x ω(n) = −→y ω
0 +
2
∑
p>1
−→y ω
p cos(πnp/N). The inverse FT then yields
−→y ω
p =
−→
f ω
p
[
−iωΞω
p −Mω2 +Kp
]−1
, (2.10)
where
Ξω
p = ξ
[
1 + (2 − δp0)Nξh
ω
pp
]−1
, Kp = 3kBT
( πp
Na
)2
, p = 0, 1, 2, . . . (2.11)
and the Oseen matrix is [9,15]
hω
pp =
1
πηa
√
3πNp
1 + χp
1 + (1 + χp)
2
, χp ≡
√
N
3πp
χa, p = 1, 2, . . . , (2.12)
hω
00 =
2√
6Nπηa
1
χω
[
1 − 2√
πχω
+
1
χ2
ω
(
1 − expχ2
ωerfcχω
)]
, χω ≡
√
N
6
χa. (2.13)
Using the fluctuation-dissipation theorem, the time correlation functions of the normal modes are
ψp(t) = 〈yαp(0)yαp(t)〉 =
kBT
(2 − δp0)πN
∫ ∞
−∞
dω cosωt
ReΞω
p∣∣−iωΞω
p −Mω2 +Kp
∣∣2 . (2.14)
97
V.Lisy, J.Tothova, A.Zatovsky
2.1. Diffusion of the coil
In the stationary limit ω = 0 so that χ = κ. Then the preaveraged Oseen tensor (6) is
〈
Hω
αβ
〉
0
=
〈
exp(−χr)
r
〉
0
. (2.15)
Thus, the quantity 1/κ (for small κr only) can be considered as a screening length. For an individual
polymer we have (p = 0 in equation (14)) [9]
ψ0(0) − ψ0(t) = Dt (2.16)
with the diffusion coefficient D = DR + DZ (R and Z denote the well-known Rouse and Zimm
limits [4]. Now instead of equation (12) we have h0
00 with χ0 = κRG (RG being the gyration radius),
the diffusion coefficient depends on the concentration of the coils c,
D = DR +DZ(c), (2.17)
(DZ(0) = DZ) and consists of the Rouse (independent of the presence of other polymers) and the
Zimm contributions. The latter one can be expressed in the form
DZ(c) = DZf(c), (2.18)
where f(c) is a “universal” function for every polymer:
f(c) =
3
√
π
4χ0
[
1 − 2√
πχ0
+
1
χ2
0
(
1 − expχ2
0erfcχ0
)]
. (2.19)
The dependence of the permeability on the concentration is estimated as follows. The friction
coefficient in the quantity κ2 = cf/η from equation (3) can be determined using the Einstein
relation D = kBT/f . In such a picture
κ2 =
27
√
π
16
c̃
R2
G
(
1 +
3
4
√
2h
)−1
. (2.20)
Then, the values of κ and χ0 depend on the draining parameter h = 2(3N/π)1/2b/a (if h � 1,
the dynamics is of the Zimm type, for h � 1 we deal with the Rouse polymers). The quantity
c̃ ≡ 4πR3
G
c/3 denotes the number of polymers per volume of a sphere with the radius RG. With
the increase of c the Zimm term decreases and for large c (small permeability κ when χ0 � 1) it
becomes ∼ 1/
√
c,
DZ(c) ≈ 2kBT
πηNa2
1
κ
. (2.21)
The realistic case of small c corresponds to χ0 = κRG � 1 when
DZ(c) = kBTh
0
00(c) = DZ
(
1 − 3
8
√
π
κRG + · · ·
)
. (2.22)
The concentration dependent correction to DZ is thus proportional to
√
c and differs from other
results (compare [18] and citations there, where this correction is ∼ c). The behaviour of a free
polymer depends on the draining parameter h. If h is large, the Zimm polymer (at c = 0) with
growing c should change its behavior to the diffusion with the Rouse coefficient DR.
2.2. Dynamics of internal modes
In the stationary case (ω = 0) and at zero concentration (κ = 0) the diagonal elements of the
Oseen matrix are well known [4]. Now h0
pp from equation (13) depends on c. The internal modes
98
Dynamics of Rouse-Zimm polymers
relax exponentially as in previous theories, ψp(t) ∝ exp(−|t|/τp), but their relaxation rates consist
of the Rouse contribution and the concentration-dependent Zimm part,
1
τp(c)
=
1
τpR
+
1
τpZ(c)
, (2.23)
where τpR and τpZ(0) ≡ τpZ are given in [4,16] and
τpZ(c) =
1
2
1 + (1 + χp)
2
1 + χp
τpZ , (2.24)
which behaves as
τpZ(c) = τpZ
(
1 +
N
6πp
κ2a2 − · · ·
)
(2.25)
if c→ 0, and (although unrealistic), as c→ ∞ one has
τpZ(c) ≈ 1
2
τpZχp =
(
Na2
)2
η
6πkBTp2
κ. (2.26)
Note that for the internal modes, the draining parameter depends on the mode number p: h(p) =
τpR/τpZ = h/
√
p. The “universal” dependence of τpZ(c)/τpZ(0) on χp (24) indicates that with the
growing c every polymer shows a tendency to become the Rouse one.
2.3. Steady state viscosity and the Huggins coefficient
The shear viscosity of the solution can be calculated from the formula [4,8,16]
η(c) = η +
1
2
kBTc
∞∑
p=1
τp(c). (2.27)
Using equation (24), in the Rouse limit we have the familiar result [4] η(c) − η = πN2a2bcη/6. In
the Zimm limit at small concentrations
η(c) − η
η
=
c
2
√
3π
(√
Na
)3
ζ
(
3
2
)[
1 + cNa2RZζ
−1
(
3
2
)
ζ
(
5
2
)
+ · · ·
]
= 0.425c
(
Na2
)3/2
[
1 + 0.140c
(
Na2
)3/2
+ · · ·
]
, (2.28)
where RZ is the Zimm hydrodynamic radius [4] and ζ is the Riemann zeta function. The first term
coincides with the known result [4]. A more general expression for the viscosity, following from
equations (27) and (24), is
η(c) − η
η
=
1
π
N2a2bc
∞∑
p=1
1
p2
(
1 +
2h√
p
1 + χp
1 + (1 + χp)
2
)−1
. (2.29)
At very low concentrations when χp � 1 one has
η(c) − η
η
=
1
π
N2a2bc
∞∑
p=1
1
p2
(
1 +
h√
p
)−1
. (2.30)
Due to the dependence on h, the difference between this and the classical result [4] can be notable.
So, for a polymer with small h, the ratio of the intrinsic viscosity [η]h = limc→0[η(c) − η]/(ηc)
at h < 1 (when the polymer is assumed to be the Rouse one) to that with h = 0 changes as a
function of h from 1 to ≈ 0.55, at h = 0.5 being 30 per cent smaller than in the case of a pure
Rouse polymer. For a very large h, the intrinsic viscosity is [η]h�1 = 3
√
2/πR3
G
ζ(3/2) = 6.253R3
G
.
99
V.Lisy, J.Tothova, A.Zatovsky
Considering the viscosity normalized to this expression, one can find that even for rather large h
the difference from the traditional result for the pure Zimm polymer is significant. So, at h = 10,
it represents some 25 per cent and it is still about 10 per cent even for h as large as 50.
One of the important rheological parameters of polymer solutions is the Huggins coefficient kH.
It can be determined from the general expression for the viscosity (29), using the intrinsic viscosity
[η]h at zero concentration (see equation (30)):
η(c) − η
ηc
= [η]h (1 + kH[η]hc+ · · · ) . (2.31)
We find
kH =
3π
23/2
∞∑
p=1
1
p7/2
(
1 +
h√
p
)−2(
1 +
3
4
√
2h
)−1
[
∞∑
p=1
1
p2
(
1 +
h√
p
)−1
]−2
. (2.32)
Thus, for large h (the Zimm case) one has
kH =
3π
23/2
ζ
(
5
2
)
ζ−2
(
3
2
)
≈ 0.655. (2.33)
This value differs from the literature results, see, e.g., [4] where kH = 0.757 is given; in [19] one
finds kH = 0.6949, and in [20] the calculations give the value 0.3787. The Freed and Edwards
theory [21,22] possesses an intrinsic viscosity, which is inconsistent with the Kirkwood-Riseman
steady-state limit and gives the hydrodynamic screening even for infinitely dilute solutions (the
discussion of this issue has been already given in the paper [5]).
As h → 0 (the Rouse case), kH approaches zero as kH ≈ 2πhζ(7/2)ζ−2(2) and when h grows,
the Huggins coefficient slowly converges to the Zimm limit (33). The difference from this limit is
significant in a broad region of h, e.g., with the maximum ≈ 1.27 of the function kH/kHZimm at
h = 3, and with kH/kHZimm ≈ 1.15 for h = 20.
2.4. Monomer motion
In connection with the unresolved problem of the dynamic nature of hydrodynamic screening
in polymer solutions (see Introduction), it is of special interest to consider the time-dependent
quantities describing the polymer behaviour. Among such quantities, the relaxation modulus, which
determines the shear stress at shear flows can be easily studied since it is given simply by a sum of
exponentials containing the relaxation times from equation (23) [4,8]. Here we shall briefly focus on
the simplest (but observable [6]) motion of the end monomer within a polymer coil and calculate
its mean square displacement (MSD). The MSD part due to internal modes is [4,11]
〈
r2(t)
〉
int
=
4Na2
π2
∞∑
p=1
1
p2
[
1 − exp
(
− t
τp(c)
)]
. (2.34)
As already shown, with growing concentration c, every polymer tends to behave as a Rouse one,
which is due to the decrease of the Zimm contribution to the relaxation rates τ−1
p . The time
dependence of this screening is well displayed considering, e.g., the ratio of the Rouse part of
the MSD (i.e., if the polymer was the pure Rouse one, h = 0) to the total MSD in the joint
Rouse-Zimm model. This function, 〈r2(t)〉int,R/〈r2(t)〉int, depends on the draining parameter h,
the concentration c, and the time. With the growing t, the above relation converges to unity
showing the transition to the Rouse behaviour. For example, at a concentration c̃ = 0.1 and h = 10
we have 〈r2(t)〉int,R/〈r2(t)〉int ≈ 0.75 at t = τ1R, at t = 2τ1R the difference from the Rouse MSD
is only about 10 per cent, and at t = 5τ1R the initially Zimm polymer becomes indistinguishable
from the Rouse one. When the same relation is considered as a function of c̃ for different times,
one sees that the tendency to approach the Rouse limit with the increase of c̃ is more and more
expressed as the time growths. At long times, as expected, the polymer behaves as the Rouse one
already at small concentrations.
100
Dynamics of Rouse-Zimm polymers
3. Conclusion
The behaviour of complex polymer systems that are attractive due to their unusual properties
as well as numerous applications cannot be comprehended without understanding the behaviour
of a single polymer in a liquid, its interaction with the solvent and with other polymers in dilute
solutions. Even in situations when we deal with solutions of flexible and unentangled polymers,
a number of open questions do exist and new “puzzles” appear. In our opinion, some of the
problems are only due to inappropriate use of the existing theories as well as due to a great
influence of simple “universal” laws of polymer behaviour, such as the famous k3 law for the first
cumulant of the dynamic structure factor or the tα laws for the monomer MSD, where α = 1/2
for the Rouse polymer and 2/3 for the Zimm one. A closer look at these laws shows, however, that
their application to real situations is rather restricted and they frequently do not correspond to
experimental conditions. The model developed in the present work is not particularly new. This
model is partially known (when dealing with the single polymer diffusion) since the publication
of the work by Kirkwood and Riseman [23]. As to the internal polymer dynamics, our approach
corresponds to that by Dubois-Violette and de Gennes [10] who, however, have assumed that
the internal modes of the polymers should behave as the Zimm modes (i.e., with the dispersion
∼ p−3/2), thus neglecting the Rouse contribution, initially being present in their theory. Such a
simplification requires quantitative arguments and in many cases it is not substantiated, just like
the assumption of the continuous distribution of the internal modes; we believe that this is clearly
shown in the present work. To take into account the presence of other coils in the solution, we have
used the well-known Debye and Bueche (or Brinkman’s) theory for a porous medium. Again, this
approach has been already used in the polymer physics. However, according to our earlier results
regarding the hydrodynamic theory of the polymer dynamics, we could ultimately build a model
that is capable of predicting new results on the fundamental characteristics of polymer behaviour
in dilute solutions. Some of the quantities characterizing the polymer solutions (viscosity, Huggins
coefficient) could be verified in standard experiments (taking into account the draining parameter).
We have also proposed a description of the time dependence tending to hydrodynamic screening
in dilute polymer solutions; this effect seems to be suitable for computer simulation studies similar
to those in [7].
Acknowledgements
This work was supported by the Scientific Grant Agency of the Slovak Republic.
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