Model for dynamic shear modulus of semiflexible polymer
solutions
F. GITTES, B. SCHNURR, C.F. SCHMIDT, P. D. OLMSTED*, and F.C. MACKINTOSH,
Department of Physics &' Biophysics Research Division, University of Michigan, Ann Arbor, MI 48109, and *Department of Physics, University of Leeds, Leeds, LS2 9JT, United
Kingdom.
Abstract We discuss a dynamical model for the frequency-dependent shear modulus of an
entangled solution of semifexible polymers, based on longitudinal fluctuations in filaments
between entanglement points or crosslinks. The goal is to explain non-Rouse, power-law
scaling of the bulk shear modulus that is found via microscopic rheology of highly entangled
F-actin solutions. This generalizes a previous model for the static modulus. Hydrodynamic
effects, and the validity of a local drag approximation below the scale of the mesh size,
are discussed. We test aspects of our model via a molecular dynamics simulation, and also
present for comparison experimental results from microrheology on F-actin.
Introduction
An entangled or crosslinked network of semiflexible polymers is an interesting physical object,
and is quite distinct from more familiar networks composed of flexible polymers [1, 2]. A
semiflexible network is characterized by two basic quantities. One is the persistance length
of the filaments, fp, which is the decay length of angular correlation along a single filament,
observed in isolation; ip is thus the arc length over which large bends occur through Brownian
motion. The second quantity is the concentration of the network; this is expressed most
concisely by p, the density of length per volume of the medium (in units of length- 2 ), but is
more intuitively given by the mesh size (em) which is an estimate of average distance between
neighboring filaments. This can be estimated by taking f_ = p- 1 / 2 which, if the filaments
were arranged in a cubic structure, would be the width of the cubes.
When the network is dense, so that em << 4, one can view the semiflexible network as a
"haystack" of interpenetrating filaments. Upon reflection, it becomes clear that the general
dynamics of such a structure may be quite subtle. For example, suppose the network is
not cross-linked, so that filament motions are constrained only by other filaments in the
network. An important length scale is then the "entanglement length," 4,, which is the
average length between effective constraints on a filament. We will find below, for example,
that the low-frequency shear modulus of an entangled network, within our model, depends
in a sensitive way on f, as has also been noted by previous theories [3, 4, 5, 6]. Because of
the interpenetrating nature of the stiff filaments, 4 is not known a priori, other than that
it must be larger than the mesh size em. (For dilute, arbitrarily stiff filaments, in fact, one
expects that the entanglement length becomes arbitrarily large). Low-frequency dynamics
are consequently both interesting and non-trivial [5, 6].
Here we discuss the high-frequency dynamics for which, as we will show, the physical
picture can be simplified greatly. We show that a scaling response can be explained by
semiflexible polymer dynamics in the presence of a freely-draining solvent. We will show,
in particular, how network considerations drop out of an estimate for the high-frequency
shear modulus; one obtains a power-law behavior G*(w) cx w3/ 4 , where the proportionality
depends only on single filament bending stiffness (i.e. on the persistence length fp), together
with the spatial density of filaments, p, and their transverse hydrodynamic drag coefficient
C. This w3/ 4 scaling law is distinct from scaling behavior predicted for flexible polymers at
2
high frequency, where the shear modulus obeys power-laws in the range of w1/ (the Rouse
23
model) to w / (the Zimm model) [2]. The discussion presented here describes a theory that
will appear more fully in a separate publication [7].
49
Mat. Res. Soc. Symp. Proc. Vol. 489 ©1998 Materials Research Society
102
a
91
10
0~
E 0
10
G'
.,
10-
10-1
102
101
10°
103
Frequency (Hz)
Figure 1: Real part G' and imaginary part G" of the complex (dynamic) shear modulus
G*(w) for a 1 mg/ml F-actin solution. This shear modulus was deduced from the observed
fluctuation spectrum of a 5-/tm diameter silica bead embedded in the actin solution.
Experiments
A major obstacle in studying semiflexible polymer systems, compared to the flexible-polymer
systems, has been that among the vast range of synthetic polymers available, few possess
a sufficiently large aspect ratio of persistence length (4p) to molecular diameter to display
distinctive semiflexible behavior. Biopolymers, in contrast, can easily attain very large aspect ratios due, paradoxically, to their large cross-sections: the protein F-actin (filamentous
actin), with a diameter of 4-7 nm, has a persistence length of 15-18 micrometers [8, 9, 10],
while microtubules have diameters of 28 nm and persistence lengths of several millimeters
[9].
Initial experimental work on F-actin, using small magnetic beads embedded in concentrated F-actin solutions[1 1], found anomalous dynamics. Using video tracking, the Brownian
motion of an imbedded object was found to obey a subdiffusive power law in time, (X2) (Xt3/4
(rather than the (X2) oc t of ordinary diffusive motion). Subdiffusive behavior was interpreted
within an independent-filament picture, in which bead motion reflected the lateral diffusion
of one, or very few, filaments. For an isolated filament whose length is smaller than the
persistence length, one can easily write a differential equation for the lateral displacement
r±(s) as a function of position s along the contour of the filament:
77i = -K-
,
(1)
where r is the bending rigidity, ( is the transverse drag per unit length, and s is position
along the contour of the filament. One finds from Eq. 1 that the lateral position of a point
on a filament exhibits just this t1/ 4 subdiffusive motion [11], as will also be shown further on.
50
However, given the small mesh sizes of dense and entangled gels of our earlier work [12, 13],
we find it implausible that our experiments should observe bead motion that directly reflects
lateral motion of independent filaments.
In our own experimental work, we have observed the Brownian motion of beads embedded
in a concentrated F-actin solution [12, 13]. For relatively large beads, compared to the mesh
size of the network, at high frequencies (f > 1 Hz), we observed Brownian fluctuation
spectra (X2) ox w-1.75 above a lower frequency of about 1 Hz; such a power law is equivalent
to subdiffusion with (X2) oCt 3 / 4 . When the bead is much larger than the mesh size of the
network, and at frequencies well above 1 Hz, one can show that the network and solvent
should move together[12] so that the surrounding medium is describable as a continuum,
whose frequency-dependent shear modulus we have deduced via the fluctuation-dissipation
theorem and hydrodynamic considerations. In our experiments, the bead was chosen to be
large enough so that scaling behavior emerged in the macroscopic modulus. Figure 1 shows
the real part G' and the imaginary part G" of the dynamic shear modulus G*(w). The
deduced shear modulus in Figure 1 will result in anomalous macroscopic dynamics in the
medium; we wish to explain the macroscopic G*(w) in terms of a microscopic network model.
In the picture to be discussed, a semiflexible polymer network is taken to be a random
array of long stiff chains, that are entangled or crosslinked at intervals f, shorter than the
persistence length of the filaments. Under a virtual macroscopic shear of the network, filaments undergo an affine distortion that must be accommodated by longitudinal compression
or extension of the fluctuating segments between entanglements. A given segment is put
under either compression or tension by the applied shear, depending on its orientation, and
a time-dependent tension is induced in it that contributes to the dynamic shear modulus.
We calculate stress contribution in terms of the dynamic response of an isolated segment in
a viscous solvent. What follows is a streamlined discussion of the model; a more complete
discussion is given in [7]. The long-time average tension within this model is that previously
calculated in Ref. [3]. An independent derivation of the high-frequency shear modulus,
complementary to ours in its methods but equivalent in its conclusions, is given by Morse
[6].
Response function and modulus
The first ingredient to our model is the longitudinal dynamic response of a single filament to
a time-dependent tension. Here we will not give details of the argument [7]. The approach is
based on the linear Eq. 1, from which, if one imposes boundary conditions of laterally constrained ends (with an undulating length f,), one finds dynamical modes for each transverse
direction,
u(s)
=
2E Uqsinqs
teq
(q = n•r/4,n >1),
(2)
4
From Eq. 1, one finds that these modes decay with an inverse time constant of Wq = (r/()q
and have a thermal mean square value (u ) = t.
From these two quantities, one finds
that the correlation function of different modes, with t > 0, is
(Uq(t) uq(O))
fe _exp(w
--- 2q4jp ep
(-at(3
3
while modes of different q are uncorrelated. Now, because of the length constraint, transverse
bending described by the linear equation Eq. 1 implies a total end-to-end distance for the
filament of
6f(t)
-
j ds (Or±/Os)
51
2
102
-,
100
_
10-'
PSD of
projected length
-s
10_1
C 10,
10'-
2
10
10
10
10
10,
10
10
Frequency (natural units)
Figure 2: Solid line: fluctuation spectrum of end-to-end distance, obtained from a dynamical
simulation of a fluctuating filament composed of twenty inextensible segments. Dotted line:
Theoretical expression for the end-to-end distance fluctuation spectrum.
E q 2 (Uq(t) 2 + Vq(t) 2 ).
(4)
eeq
where u and v denote the two transverse directions. In order to construct the fluctuation
spectrum of end-to-end distance, we must construct a correlation function
_(
-
(6)ý 2,
(5)
whose Fourier transform is the desired spectrum. Putting Eq. 4 into Eq. 5, one arrives
at fourth-order averages of the Uq'S and vq's at various times. It is in fact possible to
factorize these averages because Uq and Vqhave an underlying Gaussian distribution [7]. The
fluctuation spectrum that results from this procedure is
(e)
-
1
E
8+
1
(6)
where w,1 is the relaxation rate of the slowest mode, w,
(K/()(T/4) 4 . This fluctuation
spectrum, comparing Eq. 6 with a dynamical simulation, is shown in Figure 2. For the time
and length units used, and a description of the calculation, see [7]. An equivalent result has
been obtained by Granek [14]. The important feature of Eq. 6 at high frequencies is that it
quickly approaches a pure power-law (6 2)% cx w 1.75, a fact that is not hard to deduce from
Eq. 6 by using an integral approximation for the sum.
One can use the fluctuation-dissipation theorem to obtain either from Eq. 6 or, at high
frequencies from the power-law (562 ) cx w- 1.75, a response function a, for end-to-end distance (i.e. if a longitudinal tension r, is applied to this filament, the linear response is
6f,, = cc).
At high frequencies, the response is given by [7]
1
I
52
3/4
(W> Ws).
(7)
10'
10,
D 10'
10-13
10'
100
101
102
10
Relative frequency wo/2o
Figure 3: Real part, G', and minus the imaginary part, -G", of the complex (dynamic)
shear modulus G*(w) given by the theoretical formula, Eq. 10, but using a full expression for
a, that will appear elsewhere (Ref. 12). The scaling regime G*(w) cxW3/4 at high frequency
is about a factor of seven higher than the experimental curve in Fig. 1. The low-frequency
plateau in G'(w) does not appear in the experimental data, probably reflecting a large value
of the entanglement length, 4,.
Equation 7 is the origin of the G*(w) cx w3/ 4 power law that we seek. To see this, we
consider an entangled network modeled as an isotropic and homogeneous distribution of
filaments in space. The force directed along each filament is -rh, where the induced tension
is -r, = 5f,/ose.
Defining p as the length/vol of filaments (the mesh size of the network may
be be defined as f£m - p 1/2), it is not hard to show that the dynamic shear modulus of the
network is given simply by [7]
G (w)
-Pfe/ au
15
(8)
Figure 3 shows the real and imaginary parts of the shear modulus in Eq. 8, using not
just the high-frequency power law given in Eq. 7, but the full response function a, at
all frequencies with the previously stated boundary conditions on the filaments between
entanglement points.
Conclusions
Prior discussion of the motion of particles embedded in an entangled network of semiflexible filaments has used free transverse motion of independent filaments in order to explain
anomomalous time and frequency dependence of diffusion or forced motion of the particles.
Instead, we have described a model [7] in which transverse fluctuations have used indirectly,
by way of an implied longitudinal response, in order to explain a power-law in terms of a
macroscopic dynamic shear modulus of the medium, considered to be homogeneous on a
large scale.
We can compare the amplitude of the power-law modulus, found by inserting Eq. 7 into
Eq. 8, with the experimental G(w) in Figure 1. When this is done [7], the predicted amplitude
53
of the high-frequency modulus in Eqs. 7 and 8, at a concentration of c = 2 mg/mL of F-actin,
is found to be about seven times larger than the observed power-law modulus in Figure 1,
between 10 and 100 Hz. We consider this reasonable agreement. The experimental value
might be depressed by incomplete polymerization or by a high fraction of short filaments
(much shorter than the entanglement length).
Acknowledgements
We thank David C. Morse for discussions of his own work. This work was supported in
part by the Whitaker Foundation and by the National Science Foundation (BIR 95-12699
and DMR 92-57544).
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