Theory of linear viscoelasticity of semiflexible rods
in dilute solution
V. Shankar, Matteo Pasquali,a) and David C. Morseb)
Department of Chemical Engineering and Materials Science, University of
Minnesota, 421 Washington Avenue, S. E., Minneapolis, Minnesota
(Received 25 October 2001; final revision received 19 June 2002)
Synopsis
We present a theory of the linear viscoelasticity of dilute solutions of freely draining, inextensible,
semiflexible rods. The theory is developed expanding the polymer contour about a rigid rod
reference state, in a manner that respects the inextensibility of the chain, and is asymptotically exact
in the rodlike limit where the polymer length L is much less than its persistence length L p . In this
limit, the relaxation modulus G(t) exhibits three time regimes: At very early times, less than a time
␶ 储 ⬀ L 8 /L 5p required for the end-to-end length of a chain to relax significantly after a deformation,
the average tension induced in each chain and G(t) both decay as t ⫺3/4. Over a broad range of
intermediate times, ␶ 储 Ⰶ t Ⰶ ␶⬜ , where ␶⬜ ⬀ L 4 /L p is the longest relaxation time for the
transverse bending modes, the end-to-end length decays as t ⫺1/4, while the residual tension
required to drive this relaxation and G(t) both decay as t ⫺5/4. As later times, the stress is dominated
by an entropic orientational stress, giving G(t) ⬀ e ⫺t/ ␶ rod, where ␶ rod ⬀ L 3 is a rotational
diffusion time, as for rigid rods. Predictions for G(t) and G * ( ␻ ) are in excellent agreement with
the results of Brownian dynamics simulations of discretized free draining semiflexible rods for
lengths up to L ⫽ L p , and with linear viscoelastic data for dilute solutions of
poly-␥-benzyl-L-glutamate with L ⬃ L p . © 2002 The Society of Rheology.
关DOI: 10.1122/1.1501927兴
I. INTRODUCTION
While the theoretical problem of predicting linear viscoelastic functions of dilute
polymer solutions was largely solved in the 1950s for the cases of completely flexible
共Gaussian兲 polymers and rigid rods, the corresponding problem for the wormlike chain
model has resisted solution. Here, we present a solution to this problem in the relatively
simple limit of rodlike chains, of length L much less than their persistence length L p .
The theory is asymptotically exact in the limit L Ⰶ L p , and is found 共by comparison to
both simulations and experiment兲 to remain surprisingly accurate for chains of length up
to L ⬃ L p .
The wormlike chain 共WLC兲 model describes the backbone of a polymer as a smooth
contour with a finite elastic resistance to bending, but an infinite resistance to tangential
extension or compression. Solving this model has proved difficult largely because the
dynamical equations for a single such chain are nonlinear, even in the absence of hydrodynamic interactions. The nonlinearity is a result of the constraint of inextensibility,
a兲
Present address: Department of Chemical Engineering, Rice University, 6100 Main St., Houston, TX 77005.
Author to whom correspondence should be addressed; electronic mail: [email protected]
b兲
© 2002 by The Society of Rheology, Inc.
J. Rheol. 46共5兲, 1111-1154 September/October 共2002兲
0148-6055/2002/46共5兲/1111/44/$25.00
1111
1112
SHANKAR, PASQUALI, AND MORSE
which must be enforced by a tension 共i.e., Lagrange multiplier兲 field whose value at each
point on a chain depends upon the conformation of the entire chain. The physical reason
for treating such polymers as inextensible is suggested by a simple model of a wormlike
chain as cylindrical elastic solid of diameter a and Young’s modulus Y: The energy per
unit length required to change the length of such a cylinder by a specified fractional strain
is proportional to Y a 2 , while the energy per length required to bend it into an arc of
radius R is of order Y a 4 /R 2 , which is smaller than the strain energy by a factor of order
(a/R) 2 . This property of thin, weakly curved filaments is familiar to anyone who has
tried to bend and stretch a thread or human hair.
An influential previous theory for the viscoelasticity of solutions of wormlike chains
关Harris and Hearst 共1966兲; Hearst et al. 共1966兲兴 attempted to bypass the difficulties posed
by the inextensibility constraint by introducing a modified Gaussian model in which the
constraint is imposed only in a temporally and spatially average sense, by a constant
Lagrange multiplier tension. This and related Gaussian models yield reasonable approximations for many static properties, and some dynamic properties, but, we find, yield
qualitatively incorrect results for the viscoelasticity of solutions of semiflexible rods,
because they neglect flow-induced constraint forces. We find that, at all frequencies for
which the behavior of semiflexible rods in solution differs substantially from that of true
rigid rods, the polymeric contributions to the stress is dominated by contributions arising
from constraint forces, and that a rigorous treatment of the constraint is thus a necessity.
The absence of an adequate theory has not prevented the accumulation of a body of
experimental data on the viscoelasticity of dilute solutions of wormlike polymers, much
of it by J. Schrag and coworkers 关Warren et al. 共1973兲, Nemoto et al. 共1975兲, Carriere
et al. 共1985兲, 共1993兲兴. Of particular relevance to the present work are measurements of
the linear viscoelasticity of dilute solutions of poly共␥-benzyl-L-glutamate兲 共PBLG兲 with
L ⬃ L p by Warren et al. 共1973兲 and of somewhat shorter chains, at significantly higher
frequencies by Ookubo et al. 共1976兲. The measurements of Warren et al. revealed that the
rigid rod theory of Kirkwood and Auer 共1951兲 adequately describes the behavior of
G * ( ␻ ) for such chains in the low frequency terminal regime, but not at any higher
frequencies. The present paper thus focuses on explaining the viscoelasticity of solutions
of semiflexible rods at relatively high frequencies, where their behavior differs qualitatively from that of truly rigid rods.
A. Model
We consider a dilute solution of monodisperse wormlike polymers of contour length L,
diameter d, and number density c. The conformation of a single chain is parametrized by
a space curve r(s), where s is the arc length measured along the contour of the polymer.
The bending energy of a wormlike chain with a bending rigidity ␬ is given by
Ubend ⫽
1
2
␬
冕
L
0
ds
冏 冏
⳵ u共 s 兲
⳵s
2
,
共1兲
where u(s) ⬅ ⳵ r(s)/ ⳵ s is the local tangent vector, and ⳵ u(s)/ ⳵ s is a curvature vector,
whose magnitude is the inverse of the local radius of curvature. The constraint of local
inextensibility is expressed as a requirement that 兩 ⳵ r(s)/ ⳵ s 兩 ⫽ 1 at each point on the
chain, so that u(s) is a unit vector. The persistence length L p ⬅ ␬ /k B T is the distance
along the chain over which tangent vectors remain correlated in equilibrium. Chains with
L Ⰷ L p thus are 共globally兲 random coils, whereas those with L Ⰶ L p are semiflexible
rods.
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
1113
The Brownian motion of a single free-draining wormlike chain in an imposed mean
flow with velocity gradient ␥˙ ⬅ (“v) T is described by the Langevin equation
␨•
冉
冊
冉 冊
⳵r
⳵4r ⳵
⳵r
T
⫹␩.
⫺ ␥˙ •r ⫽ ⫺ ␬ 4 ⫹
⳵t
⳵s
⳵s
⳵s
共2兲
Here, ␨ is a local friction tensor of the form ␨ ⫽ ␨ 储 uu⫹ ␨⬜ (I⫺uu) where ␨ 储 and ␨⬜ are
friction coefficients for motions parallel and perpendicular to the local tangent u(s),
respectively. The left side of Eq. 共2兲 is a hydrodynamic frictional force. The first term on
the right side is the force arising from the bending energy. The second term on the right
side is a constraint force required to enforce inextensibility, in which T(s,t) is a fluctuating tension, or Lagrange multiplier, field. The last term on the right side is a random
Brownian force with vanishing mean value and a variance 具 ␩(s,t) ␩(s ⬘ ,t ⬘ ) 典
⫽ 2k B T ␨(s,t) ␦ (s⫺s ⬘ ) ␦ (t⫺t ⬘ ). Solutions of Eq. 共2兲 must satisfy boundary conditions
requiring that ⳵ 2 r/ ⳵ s 2 ⫽ ⳵ 3 r/ ⳵ s 3 ⫽ T ⫽ 0 at both chain ends.
This free-draining model ignores the effects of long range hydrodynamic interactions.
The theory of slender body hydrodynamics 关e.g., Batchelor 共1970兲兴 shows that, in the
limit of a rodlike object of length L much greater than its hydrodynamic diameter d, the
effects of hydrodynamic drag can be mimicked, to within logarithmic corrections, by the
use of an anisotropic local friction, with coefficients ␨⬜ ⫽ 2 ␨ 储 ⯝ 4 ␲ ␩ s , where ␩ s is the
solvent viscosity. In addition, the theory predicts a weak logarithmic dependence of the
effective friction coefficient on the characteristic distance for variation of the drag forces
along the chain. For example, the effective friction coefficient for rigid transverse motion
of a rod is approximately 4 ␲ ␩ s /ln(L/d), while that for a bending mode of wavelength
␭ Ⰷ d is approximately 4 ␲ ␩ s /ln(␭/d) 关Granek 共1997兲, Kroy and Frey 共1997兲兴. The
free-draining model of Eq. 共2兲 ignores this logarithmic scale dependence, but does allow
us to retain the factor of 2 difference between ␨ 储 and ␨⬜ when comparing to experiment.
B. Time and length scales
We briefly review the characteristic time and length scales relevant to stress relaxation
in a solution of semiflexible rods, with L Ⰶ L p 关Morse 共1998b兲兴.
The slowest relaxation process in a solution of rods is rotational diffusion. The free
draining model described earlier yields a rotational diffusivity D rot ⫽ 12k B T/( ␨⬜ L 3 ),
and a corresponding terminal relaxation time
␶rod ⫽
␨⬜ L 3
72k B T
共3兲
for the relaxation of flow-induced anisotropies in the distribution of rod orientations.
The longest wavelength bending mode of a semiflexible rod of length L has a decay
time proportional to
␶⬜ ⬅
␨⬜L4
␬
.
共4兲
This time scale can be estimated by dimensional analysis of Eq. 共2兲, by balancing the
frictional force with the bending force. We also define a corresponding time-dependent
length scale
␰⬜共t兲 ⬅
冉冊
␬t
␨⬜
1/4
共5兲
1114
SHANKAR, PASQUALI, AND MORSE
which, for t ⬍ ␶⬜ , is approximately the wavelength of a bending mode with a relaxation
time equal to t. Hereafter, the notation ␰⬜ ( ␻ ) ⬅ ( ␻␨⬜ / ␬ ) ⫺1/4 refers to the same length
scale expressed as a function of a frequency ␻ ⫽ 1/t.
Understanding the longitudinal response of a semiflexible rod is the key to understanding the high-frequency viscoelasticity of a solution of rods. At nonzero temperature, a
semiflexible rod undergoes thermally excited transverse undulations. As a result, its average end-to-end length is thus always less than its full contour length L, and can be
changed slightly by longitudinal forces applied to the chain, without changing the actual
contour length, by suppressing 共for T ⬎ 0兲 or enhancing 共for T ⬍ 0兲 the magnitude of the
thermally excited ‘‘wrinkles.’’ The resulting effective extensibility can be quantified by
an effective longitudinal extension modulus B ⬅ T / 具 E典 measured in a hypothetical experiment in which a uniform infinitesimal tension T is applied to the chain, and results in
an average strain 共i.e., fractional change in end-to-end length兲 具 E典 ⬅ 具 ␦ L 典 /L, where
具 ␦ L 典 is the change in the average end-to-end length.
MacKintosh et al. 共1995兲 calculated an effective static modulus by using equilibrium
statistical mechanics to calculate the longitudinal response to a spatially uniform, static
tension. They obtained
B⬀
␬2
共6兲
kBTL3
with a prefactor that depends upon the boundary conditions imposed on the ends of the
chain. The effective extensibility of the chain arises from the existence of thermal transverse fluctuations; thus, the modulus increases with decreasing temperature T or increasing rigidity ␬. B depends strongly on the chain length, because the calculated compliance
1/B is dominated by the contributions of the longest wavelength bending modes.
Subsequently, both Gittes and MacKintosh 共1998兲 and Morse 共1998b兲 calculated a
frequency-dependent dynamic longitudinal modulus B( ␻ ) ⫽ T( ␻ )/ 具 E( ␻ ) 典 , by considering the longitudinal response of a semiflexible rod to a spatially uniform but temporally
oscillating tension of complex amplitude T共␻兲 at a frequency ␻, and calculating the
amplitude 具 ␦ L( ␻ ) 典 ⫽ L 具 E( ␻ ) 典 of the resulting change in the average end-to-end length.
Both authors found a modulus
lim B 共 ␻ 兲 ⫽
⫺1
␻ Ⰷ ␶⬜
␬2
kBT
冉 冊
2i ␻␨⬜
␬
3/4
共7兲
⫺1
, and confirmed that they recovered the static modulus of
at high frequencies, ␻ Ⰷ ␶⬜
⫺1
. The ␻ 3/4 dependence of B( ␻ ) at high frequencies
Eq. 共6兲 at low frequencies, ␻ Ⰶ ␶⬜
can be qualitatively understood as follows: When a chain is subjected to an oscillatory
⫺1
tension with ␻ Ⰷ ␶⬜
, only bending modes with wavelengths shorter ␰⬜ ( ␻ ), which
have relaxation rates greater than ␻, can respond to the oscillatory tension. Only these
short wavelength modes contribute significantly to the extensibility of the chain. To
⫺1
, the length ␰⬜ ( ␻ ) replaces the
estimate B共␻兲, we can thus assume that, for ␻ Ⰷ ␶⬜
chain length L as the long-wavelength cutoff in Eq. 共6兲. This scaling argument gives
B共␻兲 ⬃
in agreement with Eq. 共7兲.
␬2
3
kBT␰⬜
共␻兲
共8兲
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
1115
The slight longitudinal extensibility of a semiflexible rod gives rise to a nontrivial
longitudinal dynamics 关Morse 共1998b兲, Everaers et al. 共1999兲兴. We have noted previously
关Morse 共1998b兲, Pasquali et al. 共2001兲兴 that the average longitudinal force balance for a
semiflexible rod with a frequency-dependent longitudinal modulus B共␻兲 can be recast in
the form of a modified diffusion equation for the average tension 具T共s,␻兲典, with a
frequency-dependent diffusivity D(␻) ⫽ B(␻)/␨. This yields a kind of anomolous diffusion in which tension and longitudinal strain propagate a distance
␰储共t兲 ⬅
冑
兩 B 共 1/t 兲 兩 t
⫽
␨储
冋冉 冊
3
2 ␨⬜
k B TL 5p
␨储
␨储
t
册
1/8
共9兲
in a time t, giving ␰ 储 (t) ⬀ t 1/8. The notation ␰ 储 ( ␻ ) ⬅ 冑兩 B( ␻ ) 兩 /( ␻␨ 储 ) denotes the same
length as a function of ␻ ⫽ 1/t. Setting ␰ 储 (t) ⫽ L yields a characteristic time scale
␶储 ⬅
冉 冊
␨储
2␨⬜
3
␨ 储 L8
共10兲
kBTL5p
required for strain and tension to diffuse the entire length of the chain, which is also the
time required for the chain length to relax significantly after a sudden deformation.
Equations 共3兲, 共4兲, and 共10兲 yield ratios of the time scales ␶ rod , ␶⬜ , and ␶ 储 :
␶储
␶⬜
⬀
冉冊
L
Lp
4
,
␶⬜
␶rod
⬀
L
Lp
共11兲
.
These three time scales thus become well separated in the rodlike limit L Ⰶ L p , and
form a hierarchy ␶ 储 Ⰶ ␶⬜ Ⰶ ␶ rod . The dynamical response of a solution of such rods is
therefore expected to exhibit three time regimes in an experiment, such as a step strain,
that subjects the system to a sudden perturbation: At early times, t Ⰶ ␶ 储 , the end-to-end
length has insufficient time to relax significantly and so will retain the value imposed on
it by the initial perturbation. At intermediate times, ␶ 储 Ⰶ t Ⰶ ␶⬜ , the longitudinal deformation of the chains has had time to undergo significant 共but not necessarily complete兲
relaxation, but the longest wavelength bending modes have not yet relaxed. At late times,
t ⬎ ␶⬜ , both longitudinal and transverse degrees of freedom are fully relaxed, and the
polymer behaves like a rigid rod. Note that the width of the intermediate regime grows
rapidly with decreasing L/L p , but that the gaps between ␶ 储 , ␶⬜ , and ␶ rod disappear as L
approaches L p from below, causing the intermediate regime to vanish.
C. Stress relaxation
Here we discuss some qualitative features of the relaxation of stress
␴(t) ⫽ G(t) 关 ␥0 ⫹ ␥T0 兴 after a step strain of infinitesimal magnitude ␥0 at t ⫽ 0. In what
follows, the polymer contribution to G(t) in a dilute solution of c chains per unit volume
will be characterized by an intrinsic relaxation modulus, defined by
关G共t兲兴 ⫽ lim
c→0
G共 t 兲⫺␦共 t 兲␩s
c
,
共12兲
which gives the contribution to G(t) per chain and by a corresponding intrinsic dynamic
⫺i ␻ t .
modulus, 关 G * ( ␻ ) 兴 ⫽ (G * ( ␻ )⫺i ␻ ␩ s )/c, where 关 G * ( ␻ ) 兴 ⫽ i ␻ 兰 ⬁
0 dt 关 G(t) 兴 e
We first review the behavior of solutions of true rigid rods 关Kirkwood and Auer
共1951兲, Doi and Edwards 共1986兲, Bird et al. 共1987兲兴. The stress in a dilute solution of thin
rigid rods is given by
1116
SHANKAR, PASQUALI, AND MORSE
␴⫽c
冕
L
0
ds 具 T共 s 兲 uu典 ⫹3ck B T 具 uu⫺I/3典 ,
共13兲
where T共s兲 is the tension at point s along a rod, and u is the unit vector parallel to a rod.
The first term on the right-hand-side 共rhs兲 is a viscous stress arising from the tension
T(s,t) that is required to maintain the constraint of inextensibility, which depends linearly
on the instantaneous rate of strain ␥˙ (t). The second term is the entropic orientational
stress arising from anisotropies in the distribution of rod orientations. For a step strain of
magnitude ␥ 0 at t ⫽ 0, the rate-of-deformation tensor ␥˙ (t) is a delta-function ␥˙ (t)
⫽ ␥0 ␦ (t). This induces a tension
T共 s,t 兲 ⫽ 21 ␨ 储 ␦ 共 t 兲 s 共 L⫺s 兲 ␥0 :uu,
共14兲
in a rod with orientation u, with a ␦-function time dependence, a parabolic dependence on
s, and a magnitude that depends on u. Using this tension to evaluate the viscous tension
stress in Eq. 共13兲, averaging the result over chains of different orientations, and calculating the effect of a step strain upon the distribution of rod orientations to obtain the
entropic orientational stress, yields an intrinsic modulus
关G共t兲兴 ⫽
␨储L3
180
3
␦ 共 t 兲 ⫹ k B Te ⫺t/ ␶ rod,
5
共15兲
with a delta-function viscous contribution, and an exponentially decaying orientational
contribution that decays by rotational diffusion, with the decay time ␶ rod given in Eq. 共3兲.
The behavior of G(t) in a solution of semiflexible rods is expected to be similar to that
predicted for true rigid rods at late times t ⬎ ␶⬜ , when all internal deformations have
had time to relax. This paper thus focuses on predicting linear viscoelastic behavior at
times t Ⰶ ␶⬜ , or corresponding frequencies.
Both Gittes and MacKintosh 共1998兲 and Morse 共1998b兲 have predicted that G * ( ␻ ) in
solutions of wormlike chains should vary asymptotically as G * ( ␻ ) ⬀ (i ␻ ) 3/4 at very
high frequencies, or 共equivalently兲 that G(t) ⬀ t ⫺3/4 at very early times. This prediction
is based on the assumption that, at sufficiently high frequency in an oscillatory flow, or at
sufficiently short times after a step deformation, the frictional coupling between the chain
and the solvent will be strong enough to enforce a nearly affine deformation of the
end-to-end vector of a chain, i.e., the same longitudinal extension or compression as that
experienced by a straight line of ink drawn in the solvent with the orientation of a
particular rod. In an oscillatory flow, this affine strain yields a corresponding oscillatory
tension with a ␻ 3/4 frequency dependence that directly reflects the frequency dependence
of B( ␻ ). The underlying assumption of an affine longitudinal deformation must break
down, however, at frequencies ␻ ⬍ ␶ 储 , or times t ⬎ ␶ 储 , for which the end-to-end length
has sufficient time to significantly relax. This earlier prediction is thus expected to be
valid only in the high frequency or early time regime.
The behavior of G(t) in the intermediate time regime ␶ 储 Ⰶ t Ⰶ ␶⬜ is more subtle. We
showed in an earlier report on this subject 关Pasquali et al. 共2001兲兴 that the stress is
dominated at these intermediate times 共as at early times兲 by a ‘‘tension’’ contribution that
arises from the constraint forces, and that both G(t) and the tensions in individual rods
decay as t ⫺5/4 in this regime. This algebraic decay of the tension at intermediate times is
shown here to be an indirect result of the free relaxation of transverse bending modes
throughout this time regime: The sequential relaxation of bending modes of increasing
wavelength throughout this regime leads to an accordion-like motion, which, as a result
of the inextensibility of the chain contour, is found to yield an average longitudinal
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
1117
strain that relaxes as t ⫺1/4. The residual tension induced by frictional forces that oppose
the longitudinal component of this motion is proportional to the the time derivative of the
strain, and so decays as t ⫺5/4.
At late times, t ⬎ ␶⬜ , the coupled relaxation of the transverse undulations and longitudinal strain is complete, and the macroscopic stress is dominated by a remaining
entropic orientational component analogous to that found for rigid rods. This component
decays exponential with a decay time ␶ rod identical 共for L Ⰶ L p ) to that found for rigid
rods.
The above description applies rigorously only to very stiff chains, with L Ⰶ L p . As L
approaches L p from below, the intermediate time regime disappears, leaving an essentially featureless crossover from a t ⫺3/4 decay of G(t) at early times to an exponential
termain decay. For much longer coil-like chains, with L Ⰷ L p we expect G(t) to crossover as a function of time from a t ⫺3/4 decay at early times to a Rouse-like t ⫺1/2 decay
at later times. This crossover should occur at a time proportional to ␨⬜ L 3p /(k B TL p ),
which is roughly the relaxation time for a bending mode of wavelength L p , after which
the stress relaxation will be controlled by the relaxation of Rouse-like modes with wavelengths greater than L p .
D. Outline
The rest of this paper is organized as follows. In Sec. II, we expand the equation of
motion and related quantities about a rigid rod reference state. In Sec. III we derive an
integrodifferential equation relating the average longitudinal tension and strain fields in a
semiflexible rod of known orientation, and obtain a formal solution of this relation in
terms of a spatially nonlocal longitudinal compliance, or a corresponding nonlocal modulus, which is a generalization of the frequency dependent modulus B共␻兲 discussed earlier.
In this section, we also introduce an analytically tractable ‘‘local compliance approximation’’ that ignores the spatial nonlocality of the relationship between tension and strain,
which yields the simple modified diffusion equation for strain that was used in our
previous work 关Pasquali et al. 共2001兲兴. In Sec. IV, we calculate the nonlocal compliance
⫺1
, the
in an approximation that is valid at intermediate- and high-frequencies, ␻ Ⰷ ␶⬜
results of which show that the local compliance approximation is also valid throughout
the same frequency range. In Sec. V, we use the local compliance approximation to
calculate the stress and strain field along a rod in weak flow field. In Sec. VI, we review
the formal expression of Morse 共1998a兲 for the stress in a solution of wormlike polymers,
and cast this in a form appropriate to a nearly straight rod. In Sec. VII, we use the local
compliance approximation to obtain analytic results for the linear viscoelastic moduli for
⫺1
␻ Ⰷ ␶⬜
. In Sec. VIII, we formulate a more complete theory that is valid in the limit
L Ⰶ L p at arbitrary ␻, but that must be evaluated numerically. In Sec. IX, we present
Brownian dynamics simulations of semiflexible rods, and compare simulation results for
G(t) to the predictions of the full theory. In Sec. X, we compare predictions for G * ( ␻ )
to experimental data for dilute solutions of poly共benzyl-glutamate兲. In Sec. XI, we
present an accurate analytic approximation to the full theory. In Sec. XII, we compare our
theory to that of Harris and Hearst 共1966兲. Conclusions are summarized in Sec. XIII.
The simulations results presented here, and a brief presentation of the local compliance approximation for the tension stress contribution to G(t), have appeared previously
in Pasquali et al. 共2001兲. Similar simulations of G(t) have also been conducted by Dimitrakopoulos et al. 共2001兲. The theoretical treatment of the inextensibility constraint used
1118
SHANKAR, PASQUALI, AND MORSE
here is similar to that used by Liverpool and Maggs 共2001兲 in a recent theoretical study
of dynamical light scattering from long semiflexible filaments.
II. EXPANSION ABOUT A ROD
We expand the dynamical equations about a rotating rodlike reference state. Our
reference state is a straight rod parallel to a unit vector n共t兲 that rotates like a thin
non-Brownian rod in a homogeneous velocity gradient
dn
dt
⫽ 共 I⫺nn兲 • ␥˙ •n.
共16兲
We expand the polymer contour r共s兲 about this line as
r共 s,t 兲 ⫽ 关 s⫹ f 共 s,t 兲兴 n共 t 兲 ⫹h共 s,t 兲 ,
共17兲
where f (s,t) and h(s,t) are the longitudinal and transverse displacements, respectively.
The transverse displacement h(s,t) always remains orthogonal to n(t), and can be expanded as
h共 s,t 兲 ⫽
兺
h ␣ 共 s,t 兲 e␣ 共 t 兲 ,
␣ ⫽ 1,2
共18兲
where e1 (t) and e2 (t) are two unit vectors that are always orthogonal to n(t) and to each
other. Greek subscripts are used hereafter to represent transverse directions, and can take
values 1 and 2. We choose 关following Hinch 共1976兲兴 the time evolution of e1 (t) and e2 (t)
as
de1
dt
de2
⫽ ⫺n共 e1 • ␥˙ •n兲 ,
dt
⫽ ⫺n共 e2 • ␥˙ •n兲 ,
共19兲
so that each transverse basis vector rotates in a plane spanned by itself and n(t) so as to
remain always orthogonal to n(t).
The constraint of inextensibility, which requires that 兩 ⳵ r(s)/ ⳵ s 兩 2 ⫽ 1, can be expanded as
冉冊
⳵f
冉 冊
2
⳵f ⳵ h
⫹2 ⫹
⳵s
⳵s
⳵s
2
⫽ 0.
共20兲
In the limit L/L p Ⰶ 1, where 兩 ⳵ f / ⳵ s 兩 Ⰶ 1, this can be expanded, to leading approximation, as
⳵f
⳵s
⯝⫺
冉 冊
1 ⳵h
2
2 ⳵s
共21兲
.
It is useful to describe longitudinal displacements in terms of a longitudinal strain field
E共 s 兲 ⬅
⳵f
⳵s
⫺
冓冔
⳵f
⳵s
,
共22兲
eq
where 具 ••• 典 eq denotes an average value evaluated in the thermal equilibrium state, i.e.,
with ␥˙ ⫽ 0. Using approximation 共21兲, E共s兲 can be expressed in terms of transverse
displacements as
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
E共 s 兲 ⯝ ⫺
1
2
冋冏 冏 冓 冏 冏 冔 册
⳵h
⳵s
2
⳵h
⫺
1119
2
⳵s
共23兲
,
eq
to leading order in derivatives of h(s).
To expand the equation of motion, we substitute expansion 共17兲 for r(s,t) into Langevin equation 共2兲, while using Eqs. 共16兲 and 共19兲 for the rates of change of n(t), e1 (t),
and e2 (t). By projecting the result onto n(t), e1 (t), and e2 (t), we obtain longitudinal
and transverse components. The transverse component in direction e␣ is
冋
␨⬜
⳵h␣
⳵t
⫺
兺␤ ␥˙ ␣␤h␤
册
⫽ ⫺␬
⳵4h␣
4
⳵s
⫹
⳵
⳵s
冉 冊
T
⳵h␣
⳵s
⫹␩␣ ,
共24兲
where ␥˙ ␣␤ ⬅ e␣ • ␥˙ •e␤ , and ␩ ␣ (s,t) ⬅ e␣ (t)• ␩(s,t). The longitudinal component is
冋
␨储
⳵r储
⳵t
⫺r储␥˙ :nn⫺
兺␣
册
␥˙ : 共 ne␣ ⫹e␣ n兲 h ␣ ⫽ ⫺ ␬
⳵4r 储
⳵s
4
⫹
⳵
⳵s
冉 冊
T
⳵r储
⳵s
⫹␩储 ,
共25兲
where r 储 (s,t) ⬅ s⫹ f (s,t), and ␩ 储 (s,t) ⫽ n(t)• ␩ (s,t). Equations 共24兲 and 共25兲, together with constraint 共21兲, form the equations of motion in the rodlike limit.
III. LONGITUDINAL DYNAMICS
In this section, we derive a formal equation relating the average of the strain and
tension fields along a rod of known orientation. We consider the average of longitudinal
force balance 共25兲 with respect to rapid fluctuations of f, h, and T on a semiflexible rod
of much more slowly varying orientation n, which is regarded as constant for this purpose. This averaging procedure, which will be denoted by the symbol 具•••典 throughout
this section, yields
冋
册
⳵具 f 典
⳵ 4具 f 典 ⳵具T 典
˙
␨储
⫹
⫺s␥ :nn ⫽ ⫺ ␬
.
⳵t
⳵s4
⳵s
共26兲
To derive Eq. 共26兲, we have used the fact that the 具 h ␣ 典 ⫽ 0 in Eq. 共25兲 as a result of the
invariance of Eq. 共24兲 under the symmetry h → ⫺h. Differentiating Eq. 共26兲 with respect to s yields an equivalent relationship
冋
␨储
⳵具E典
⳵t
册
⫺ ␥˙ :nn ⫽ ⫺ ␬
⳵ 4 具 E典
⳵s4
⫹
⳵ 2具T 典
⳵s2
共27兲
in terms of the average strain 具 E(s,t) 典 .
To solve Eq. 共27兲, which relates derivatives of the average strain 具 E(s,t) 典 and the
average tension 具 T(s,t) 典 , we need a second relationship between these two fields. This is
obtained by combining transverse equation of motion 共24兲 with Eq. 共23兲 for the constraint, which relates the longitudinal strain to the derivatives of h ␣ (s), by using the
transverse equation of motion to calculate the linear response of the average 具 兩 ⳵ h/ ⳵ s 兩 2 典
of the quantity that appears on the rhs of Eq. 共23兲 for E(s,t). We thus consider the
solution of transverse Eq. 共24兲 driven by small perturbations arising from the tension field
T(s,t) in the second term on the rhs and also from the velocity gradient tensor ␥ ␣␤ , in
the second term on the left-hand-side 共lhs兲. To describe the dynamical linear response of
the 具 E(s,t) 典 to these perturbations, we must calculate the response of 具 兩 ⳵ h/ ⳵ s 兩 2 典 to a
temporally oscillating velocity gradient and tension field at arbitrary frequency ␻. We
hereafter adopt a convention in which Fourier transforms of all time dependent functions
1120
SHANKAR, PASQUALI, AND MORSE
are indicated by replacing the time argument t by a frequency ␻, so that, e.g., E(s, ␻ )
⬅ 兰 dte ⫺i ␻ t E(s,t). In the limit of small perturbations, we expect the responses of
具 E(s, ␻ ) 典 to the velocity gradient and to the tension to be additive, and so expect to be
able to write the average strain as a sum
具E共 s, ␻ 兲 典 ⫽ ⌰ 共 s, ␻ 兲 ␥ 共 ␻ 兲 :nn⫹
冕
L
0
ds ⬘ ␹ 共 s,s ⬘ ; ␻ 兲 T 共 s ⬘ , ␻ 兲
共28兲
of contributions linear in ␥共␻兲 and T(s ⬘ , ␻ ), with response functions ⌰(s, ␻ ) and
␹ (s,s ⬘ ; ␻ ) that will be calculated in subsequent sections. The function ␹ (s,s ⬘ ; ␻ ) is a
nonlocal longitudinal compliance that gives the average strain induced at point s by a
tension applied at s ⬘ with frequency ␻. The tensor form of the term linear in ␥共␻兲 is
dictated by the fact that this contribution to the strain must be a scalar function of ␥ and
n and linear in ␥, and that ␥共␻兲 must be traceless for an incompressible fluid.
A. Full theory
A closed integrodifferential equation for 具 T(s, ␻ ) 典 can be obtained by Fourier transforming Eq. 共27兲 with respect to time and substituting Eq. 共28兲 for 具 E(s, ␻ ) 典 , while
equating the unspecified tension T(s,t) in linear response relationship 共28兲 with the
average value 具 T(s,t) 典 that appears in Eq. 共27兲. This yields
冋
i␻␨储⫹␬
⳵4
⳵s4
册冕
L
⳵ 2 具 T 共 s, ␻ 兲 典
0
⳵s2
ds⬘␹共s,s⬘;␻兲具T 共 s ⬘ , ␻ 兲 典 ⫺
⫽ i ␻␨ 储 ␥共 ␻ 兲 :nn关 1⫺⌰ 共 s, ␻ 兲兴 .
共29兲
Here and hereafter, n(t) is approximated by its time average over one period of oscillation. The dimensionless response function ⌰(s, ␻ ) is calculated in the Appendix, and is
1
found to approach a maximum value proportional to L/L p in the limit ␻ Ⰷ ␶⬜
. The term
involving ⌰(s, ␻ ) on the rhs of Eq. 共29兲 is thus uniformly smaller than the leading term
on the rhs in the limit L Ⰶ L p of interest, and so can be neglected. Making the derivatives dimensionless by defining ŝ ⬅ s/L and dividing by i ␻␨ 储 then yields
冉
1⫹
1 ␨⬜ ⳵ 4
i␻␶⬜ ␨储 ⳵ŝ4
冊冕
L
1
⳵ 2 具 T 共 ŝ, ␻ 兲 典
0
i ␻␨ 储 L 2
⳵ ŝ 2
ds⬘␹共s,s⬘;␻兲具T 共 s ⬘ , ␻ 兲 典 ⫺
⫽ ␥共 ␻ 兲 :nn.
共30兲
An equivalent expression involving the strain, rather than the tension, can be obtained by
introducing a nonlocal modulus B(s,s ⬘ ; ␻ ), defined such that
冕
L
0
ds⬘B共s,s⬘;␻兲␹共s⬘,s⬙;␻兲 ⫽ ␦共s⫺s⬙兲,
共31兲
i.e., such that B is the functional inverse of ␹. If we consistently neglect the term involving ⌰(s, ␻ ) in Eq. 共28兲, as done to obtain Eq. 共30兲, we can write 具 T(s, ␻ ) 典
⫽ 兰 ds ⬘ B(s,s ⬘ ; ␻ ) 具 E(s ⬘ , ␻ ) 典 , to obtain the equivalent expression
冉
1⫹
1 ␨⬜ ⳵4
i␻␶⬜ ␨储 ⳵ŝ
4
冊
具E共 ŝ, ␺ 兲 典 ⫺
⳵2
1
2
i ␻ 储 L ⳵ ŝ
2
冕
L
0
ds ⬘ B 共 s,s ⬘ , ␻ 兲 具 E共 s ⬘ , ␻ 兲 典 ⫽ ␥共 ␻ 兲 :nn.
共32兲
Equations 共30兲 and 共32兲 are the starting points for the full theory developed in Sec. VIII.
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
1121
B. Local compliance approximation
We find in what follows that Eqs. 共30兲 and 共32兲 can be further simplified in the limit
1
of intermediate and high frequencies, ␻ Ⰷ ␶⬜
. First, we note that the coefficient multiplying the fourth derivative term in Eq. 共32兲 is proportional to (i ␻ ␶⬜ ) ⫺1 , and that, as
result of this small prefactor, is negligible compared to the first term on the left side at all
⫺1
␻ Ⰷ ␶⬜
. Moreover, we find that the nonlocality of the compliance becomes unimpor⫺1
; thus ␹ (s,s ⬘ ; ␻ ) can be approximated at these frequencies
tant at frequencies ␻ Ⰷ ␶⬜
by a frequency dependent but spatially local compliance, of the form
␹共s,s⬘;␻兲 ⯝ ␹共␻兲␦共s⫺s⬘兲.
共33兲
Here ␹ ( ␻ ) ⫽ 1/B( ␻ ), where B共␻兲 is the frequency-dependent modulus given in Eq. 共7兲,
which was obtained by calculating the spatial average strain induced by a spatially uni⫺1
. The resulting local compliance approximation
form tension at frequencies ␻ Ⰷ ␶⬜
共LCA兲 assumes that the average strain and tension are locally proportional, so that
具E共 s, ␻ 兲 典 ⫽ ␹ 共 ␻ 兲 具 T 共 s, ␻ 兲 典 .
共34兲
Substituting this approximation in Eq. 共27兲, and neglecting the fourth derivative term,
yields the LCA longitudinal balance equation
冋
i␻⫺
B共␻兲 ⳵2
␨储 ⳵s2
册
具E共 s, ␻ 兲 典 ⫽ ␥˙ 共 ␻ 兲 :nn,
共35兲
where ␥˙ ( ␻ ) ⬅ i ␻ ␥( ␻ ).
Equation 共35兲 has the form of a diffusion equation, with a frequency-dependent diffusivity D( ␻ ) ⬅ B( ␻ )/ ␨ 储 , and a spatially uniform source term arising from the rate of
straining along direction n. An analytic solution to this diffusion equation, which satisfies
the boundary condition 具 T(s, ␻ ) 典 ⫽ 0 at the chain ends, is presented in Sec. V. At high
frequencies, ␻ Ⰷ ␶ ⫺1
, for which ␰ 储 ( ␻ ) Ⰶ L, the solution of Eq. 共35兲 is found to yield
储
tension and strain fields that vary over lengths of order the longitudinal diffusion length
⫺1
␰ 储 ( ␻ ), as suggested by dimensional analysis. At intermediate frequencies, ␶⬜
Ⰶ␻
Ⰶ ␶ ⫺1
,
for
which
L
Ⰶ
␰
(
␻
),
the
tension
and
strain
are
found
to
vary
smoothly
over
储
储
the entire chain length L.
To justify the LCA, the predicted characteristic distances for spatial variations of
T(s,
␻ ) 典 at each frequency must be compared to the calculated range of the nonlocality
具
of ␹ (s,s ⬘ ; ␻ ): The approximation of ␹ (s,s ⬘ ; ␻ ) by a ␦ function is justified only if the
predicted 具 T(s, ␻ ) 典 varies slowly over distances comparable to the range of values of
兩 s⫺s ⬘ 兩 for which ␹ (s,s ⬘ ; ␻ ) remains significant. In Sec. IV, ␹ (s,s ⬘ ; ␻ ) is calculated in
⫺1
an approximation valid at ␻ Ⰷ ␶⬜
, and it is shown that ␹ (s,s ⬘ ; ␻ ) has a range of
nonlocality proportional to the transverse length ␰⬜ ( ␻ ). Using this result, the validity of
⫺1
the LCA can be justified at all ␻ Ⰷ ␶⬜
by using Eqs. 共4兲 and 共10兲 to confirm that
␰⬜ ( ␻ ) Ⰶ ␰ 储 ( ␻ ) throughout the high-frequency ␻ Ⰷ ␶ ⫺1
regime in which the tension
储
varies over distances of order ␰ 储 ( ␻ ), and that ␰⬜ ( ␻ ) Ⰶ L throughout the intermediate
⫺1
regime ␶ ⫺1
Ⰷ ␻ Ⰷ ␶⬜
in which the tension varies smoothly over the entire chain
储
⫺1
, however, for
length. The LCA fails at frequencies comparable to and less than ␶⬜
which the nonlocality of ␹ (s,s ⬘ ; ␻ ) extends over the full chain length.
1122
SHANKAR, PASQUALI, AND MORSE
IV. NONLOCAL COMPLIANCE AT INTERMEDIATE AND HIGH FREQUENCIES
We now calculate ␹ (s,s ⬘ ; ␻ ) in an approximation appropriate to the limit ␰⬜ ( ␻ )
⫺1
Ⰶ L or, equivalently, ␻ Ⰷ ␶⬜
. In this limit, the calculation of ␹ (s,s ⬘ ; ␻ ) from the
transverse equation of motion becomes insensitive to the finite length of the chain, and so
the calculation can be performed as if the chain were infinite, by using an expression of
h ␣ (s) in a continuous distribution of Fourier modes
h␣共s,t兲 ⫽
冕␲
dq
2
h␣共q,t兲e⫺iqs,
共36兲
with amplitudes h ␣ (q,t) ⫽ 兰 dsh ␣ (s,t)e iqs . The bending energy of Eq. 共1兲 can be expanded to quadrative order in Fourier amplitudes as
Ubend ⫽
冕
1
兺
2 ␣
dq
2␲
␬ q 4兩 h ␣共 q 兲兩 2.
共37兲
Fourier transforming Eq. 共24兲 yields a transformed transverse equation of motion
冉
␨⬜
⳵
⳵t
冊
⫹␬q4 h␣共q兲 ⫽ ␨⬜
兺␤ ␥˙ ␣␤h␤共q兲⫺ 冕 2␲ q共q⫺q1兲T共 q 1 兲 h ␣ 共 q⫺q 1 兲 ⫹ ␩ ␣ 共 q 兲 ,
dq1
共38兲
where T(q,t) and ␩ ␣ (q,t) are the corresponding spatial Fourier transforms of T(s,t) and
␩ ␣ (s,t), respectively, and the random force satisfies 具 ␩ ␣ (q,t) 典 ⫽ 0 and
具 ␩ ␣ (q,t) ␩ ␤ (q ⬘ ,t ⬘ ) 典 ⫽ ␦ ␣␤ 2 ␲ ␦ (q⫹q ⬘ ) ␦ (t⫺t ⬘ )2k B T ␨⬜ .
To calculate the nonlocal compliance, we calculate the linear response of the average
strain 具 E(k,t) 典 to a prescribed tension, where 具•••典 is used in this section to represent an
average over different realizations of the transverse noise. Fourier transforming Eq. 共23兲
for E(s,t), yields
具E共 k,t 兲 典 ⫽
冕
dq
2␲
共39兲
q 共 k⫺q 兲 a 共 q,k;t 兲 ,
where
a共q,k;t兲 ⬅
1
兺 关具h␣共q兲h␣共k⫺q兲典⫺具h␣共q兲h␣共k⫺q兲典eq兴 .
共40兲
2 ␣
The time derivative of a(q,k;t) is calculated by using Eq. 共38兲 to evaluate
⳵a共q,k;t兲
⳵t
⫽
1
兺
2 ␣
冓
⳵h␣共q兲
⳵t
h␣共k⫺q兲⫹h␣共q兲
⳵h␣共k⫺q兲
⳵t
冔
,
共41兲
while setting ␥˙ ␣␤ ⫽ 0 on the first term on the rhs of Eq. 共38兲. To evaluate the rhs of Eq.
共41兲, we approximate averages of the form 具 h ␣ (q,t)h ␤ (q ⬘ ,t) 典 that appear multiplied by
explicit factors of T (⫺q⫺q ⬘ ) by their equilibrium values, for the purpose of calculating
the linear response to T, using the equilibrium variance 具 h ␣ (q 1 )h ␤ (q 2 ) 典 eq
⫽ ␦ ␣␤ 2 ␲ ␦ (q 1 ⫹q 2 )k B T/ ␬ q 41 , which is obtained by applying the equipartition theorem
to Eq. 共37兲. We also use the identify 具 ␩ ␣ (q 1 )h ␤ (q 2 ) 典 ⫽ k B T ␦ ␣␤ 2 ␲ ␦ (q 1 ⫹q 2 ) for
terms involving the random force 关see, for example, Doi and Edwards 共1986兲 p. 112兴.
This yields the differential equation
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
再
␨⬜
⳵
⳵t
冎
⫹␬关q4⫹共k⫺q兲4兴 a共q,k;t兲 ⫽ T (k,t)
冋
q
共 k⫺q 兲
3⫹
1123
册
共 k⫺q 兲 k B T
.
q3
␬
共42兲
Fourier transforming Eq. 共42兲 with respect to time yields a corresponding algebraic
equation
a共q,k;␻兲 ⫽
T 共 k, ␻ 兲
4
4
冋
q
i ␻␨⬜ ⫹ ␬ 关 q ⫹ 共 k⫺q 兲 兴 共 k⫺q 兲
3⫹
册
共 k⫺q 兲 k B T
,
q3
␬
共43兲
for a(q,k; ␻ ) ⬅ 兰 dte ⫺i ␻ t a(q,k;t), where T (k, ␻ ) is the corresponding Fourier transform of T (k,t). Finally, substituting Eq. 共43兲 for a(q,k; ␻ ) into the temporal Fourier
transform of Eq. 共39兲 gives a strain of the form
具E共 k, ␻ 兲 典 ⫽ ␹ 共 k, ␻ 兲 T 共 k, ␻ 兲 ,
where
␹共k,␻兲 ⫽
冕 ␲ ␻␨
␬
kBT dq
冋
1
共44兲
q2
⫹
2 i ⬜⫹␬关q4⫹共k⫺q兲4兴 共k⫺q兲2
共k⫺q兲2
q2
册
共45兲
is a k- and ␻-dependent nonlocal compliance.
Prior results for the response to a spatially uniform tension can be recovered by
evaluating ␹ (k, ␻ ) at k ⫽ 0, which we denote by a function ␹ ( ␻ ) ⬅ ␹ (k ⫽ 0,␻ ). This
yields
␹共␻兲 ⫽
kBT
␬
冕
dq
q ⬃ 1/L
1
2 ␲ 2i ␻␨⬜ ⫹ ␬ q 4
共46兲
.
The lower limit of integration is included as a reminder that the existence of a finite chain
length L imposes a cutoff on the range of allowable wave numbers. At frequencies ␻
⫺1
Ⰷ ␶⬜
, where the calculation given in this section is valid, the integral is insensitive to
this lower cutoff, and yields
lim ␹ 共 ␻ 兲 ⫽
⫺1
␻ Ⰷ ␶⬜
冉 冊
k B T 2i ␻␨⬜
␬2
⫺3/4
␬
共47兲
,
in agreement with results of Gittes and MacKintosh 共1998兲 and Morse 共1998b兲 for
⫺1
B( ␻ ) ⬅ 1/␹ ( ␻ ). For ␻ Ⰶ ␶⬜
, the integral in Eq. 共46兲 instead is controlled by its lower
limit, indicating that the calculation is no longer quantitatively valid, as a result of our use
of a continuous distribution of Fourier modes rather than discrete bending modes to
expand h ␣ (s). However, a scaling relation can be obtained by using a lower cutoff
proportional to 1/L, which yields ␹ ( ␻ ) ⬀ k B TL 3 / ␬ 2 , in agreement with the result of
MacKintosh et al. 共1995兲 for the static compliance.
⫺1
can be made more
The wavenumber dependence of ␹ (k, ␻ ) at frequencies ␻ Ⰷ ␶⬜
transparent by rewriting Eq. 共45兲 in the scaling form
␹共k,␻兲 ⫽ ␹共␻兲F共k␰⬜共␻兲兲,
where ␹共␻兲 is given by Eq. 共46兲, and
F共k̂兲 ⬅ 23/4
冕
dq̂
1
冋
q̂ 2
2 ␲ i⫹q̂ 4 ⫹ 共 k̂⫺q̂ 兲 4 共 k̂⫺q̂ 兲 2
共48兲
⫹
共 k̂⫺q̂ 兲 2
q̂ 2
册
,
共49兲
1124
SHANKAR, PASQUALI, AND MORSE
where q̂ ⬅ q ␰⬜ ( ␻ ) and k̂ ⬅ k ␰⬜ ( ␻ ), and where F(0) ⫽ 1. Carrying out an inverse
spatial transform of Eq. 共48兲 yields a corresponding nonlocal compliance of the form
␹共s⫺s⬘,␻兲 ⫽ ␹共␻兲F̃
冉 冊
s⫺s⬘
␰⬜共␻兲
共50兲
,
where F̃(x) is the inverse Fourier transform of F(k̂). The existence of this scaling form
shows that, at a given frequency ␻, ␹ (s,s ⬘ ; ␻ ) depends only on the ratio 兩 s
⫺s ⬘ 兩 / ␰⬜ ( ␻ ), and thus has a range proportional to ␰⬜ ( ␻ ). Using this fact, the LCA can
⫺1
be shown to be a consistent approximation at all ␻ Ⰷ ␶⬜
because it leads to a tension
具 T (s, ␻ ) 典 that varies slowly over distances of order ␰⬜ ( ␻ ) at these frequencies.
V. TENSION AND STRAIN IN THE LCA
In this section, we use the LCA to calculate the average strain and tension fields along
a rod of known orientation n in a solvent subjected to a weak oscillatory or step strain. To
begin, we rewrite Eq. 共35兲 in dimensionless form as
再
⳵2
1
1⫺
␭2共␻兲 ⳵ŝ2
冎
具E共 s, ␻ 兲 典 ⫽ ␥共 ␻ 兲 :nn,
共51兲
where ŝ ⬅ s/L is dimensionless arc length, and where we have introduced a complex
dimensionless parameter
␭共␻兲 ⬅
冋 册
i␻␨储L2
1/2
B共␻兲
⫽ 共 i ␻ ␶ 储 兲 1/8 ⫽ i 1/8
L
␰ 储共 ␻ 兲
.
共52兲
The parameter ␭共␻兲 characterizes the relative importance of the first term on the lhs of
Eq. 共51兲, which arises from longitudinal drag forces, relative to the second derivative
term, which arises from forces produced by gradients in 具 T (s, ␻ ) 典 . 关This definition of
␭共␻兲 differs by a factor of 2 from that used in Pasquali et al. 共2001兲兴. Equation 共51兲 can
be solved exactly subject to the boundary condition that the average tension 共and hence
the average strain兲 vanishes at the chain ends ŝ ⫽ 0 and ŝ ⫽ 1:
再
具E共 s, ␻ 兲 典 ⫽ 1⫺
cosh关␭共␻兲共ŝ⫺ 21兲兴
cosh关␭共␻兲/2兴
冎
␥共 ␻ 兲 :nn.
共53兲
The corresponding tension is given in the LCA by T (s, ␻ ) 典 ⫽ B( ␻ ) 具 E(s, ␻ ) 典 .
Hereafter we focus on the response to oscillatory strains of magnitude ␥共␻兲, which can
be directly described by Eq. 共53兲, and to step strains ␥(t) ⫽ ␥0 ⌰(t) of magnitude ␥0 at
t ⫽ 0, which must be treated by inverse Fourier transformation of Eq. 共53兲. To treat the
latter problem, we note that such a step strain yields a rate of strain ␥˙ (t) ⫽ ␥0 ␦ (t) whose
Fourier transform ␥˙ ( ␻ ) ⫽ i ␻ ␥( ␻ ) ⫽ ␥0 is independent of ␻, giving Fourier amplitudes
␥共␻兲 ⫽ ␥˙ 共␻兲/共i␻兲 ⫽ ␥0 /(i ␻ ). To clarify the physical content of Eq. 共53兲, it is useful to
consider separately the high-frequency limit where ␭( ␻ ) Ⰷ 1, corresponding to ␻
Ⰷ ␶ ⫺1
, and the intermediate regime where ␭共␻兲 Ⰶ 1, corresponding to ␶ ⫺1
Ⰷ␻
储
储
⫺1
Ⰷ ␶⬜ .
A. High frequencies and short times
We first consider the limiting behavior of the 具 E(s, ␻ ) 典 and 具 T (s, ␻ ) 典 in an oscillatory
flow at frequencies ␻ Ⰷ ␶ ⫺1
, where ␭共␻兲 Ⰷ 1. In this limit, the first term on the left side
储
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
1125
FIG. 1. Magnitude of the average strain 具 E(s, ␻ ) 典 , normalized by the affine strain ␥共␻兲:nn, as a function of
ŝ ⫽ s/L, as calculated from the local compliance approximation 关Sec. V, Eq. 共53兲兴, for ␻ ␶ 储 ⫽ 1010 共left panel兲
and ␻ ␶ 储 ⫽ 10⫺5 共right panel兲.
of Eq. 共51兲, representing the drag force, is much larger than the second derivative term.
Balancing this drag force against the extensional strain on the rhs yields a spatially
uniform average strain
具E共 s, ␻ 兲 典 ⯝ ␥共 ␻ 兲 :nn.
共54兲
This is simply the affine strain that would be experienced by a line of ink drawn in the
fluid. In the LCA, this yields a corresponding spatially uniform tension
具T 共 s, ␻ 兲 典 ⯝ B 共 ␻ 兲 ␥共 ␻ 兲 :nn
共55兲
that, for fixed strain amplitude, grows as ␻ 3/4. Because the boundary conditions require
that T (s, ␻ ) ⫽ 0 at the chain ends, however, there must be a narrow boundary layer near
each chain end where the average strain and tension drop from these uniform values to
zero; there the second derivative term in Eq. 共51兲 must remain important. The thickness
of the boundary layer is given by the distance ␰ 储 ( ␻ ) ⬀ ␻ ⫺1/8 over which tension can
diffuse in a time 1/␻. This behavior is shown in the left panel of Fig. 1, where the
magnitude of the exact solution 共53兲 is plotted at a reduced frequency of ␻ ␶ 储 ⫽ 1010.
The boundary layers remain rather wide even at this extremely high frequency because
␰ 储 ( ␻ ) decays only as ␻ ⫺1/8.
The decay of the tension at correspondingly early times t Ⰶ ␶ 储 after a small step strain
can be obtained by inverse Fourier transformation of Eq. 共55兲, using Fourier amplitudes
␥共␻兲 ⫽ ␥ 0 /(i ␻ ). This yields
具T 共 s,t 兲 典 ⫽
冕
d␻ B共 ␻ 兲
2␲ i␻
e i ␻ t ␥0 :nn
共56兲
everywhere outside of the boundary layers near the chain ends. Noting that B( ␻ )
⬀ (i ␻ ) 3/4, one finds, either by power counting or by evaluating the integral, that T (t)
remains nearly spatially uniform outside the boundary layers near the chain ends, and
decays with time as
T 共 t 兲 ⬀ t ⫺3/4.
共57兲
The corresponding strain 具 E(s,t) 典 is also found to remain nearly uniform outside the
boundary layers, and to approximately retain the value E(s,t) ⫽ ␥0 :nn produced by the
initial affine deformation.
1126
SHANKAR, PASQUALI, AND MORSE
B. Intermediate frequencies and times
We now consider the behavior in oscillatory flow at intermediate frequencies ␶ ⫺1
储
⫺1
Ⰷ ␻ Ⰷ ␶⬜
, for which ␭( ␻ ) Ⰶ 1, but for which the LCA still remains valid. In this
regime the dominant balance in Eq. 共51兲 is between the extensional strain on the rhs and
the second derivative term on the lhs. The limit ␭共␻兲 → 0 can be obtained by setting
B( ␻ ) → ⬁. To leading order in ␭共␻兲, the behavior found in this regime is thus identical
to that found for an inextensible rigid rod. However, we find that the first order correction
to this result must be analyzed to understand adequately the qualitative behavior of the
stress in this regime. We thus expand the tension as a perturbation series
具T 共 s, ␻ 兲 典 ⫽ 具 T0 共 s, ␻ 兲 典 ⫹ 具 T1 共 s, ␻ 兲 典 ⫹•••
共58兲
where each subsequent term is smaller than the previous by a factor the small parameter
关 ␭( ␻ ) 兴 2 , or (i ␻ ␶ 储 ) 1/4, and T0 (s, ␻ ) is the asymptotic behavior obtained by setting
␭共␻兲 → 0, or B共␻兲 → ⬁.
The leading contribution 具 T0 典 can be obtained by ignoring the first term on the lhs of
Eq. 共35兲, while setting 具 T0 (s, ␻ ) 典 ⫽ B( ␻ ) 具 E(s, ␻ ) 典 in the second. This yields a differential equation
⳵2具T0 共 s, ␻ 兲 典
⳵ ŝ 2
⫽ ⫺i ␻␨ 储 L 2 ␥共 ␻ 兲 :nn,
共59兲
whose solution gives
具T0 共 s, ␻ 兲 典 ⫽ 21 i ␻␨ 储 L 2 ŝ 共 1⫺ŝ 兲 ␥共 ␻ 兲 :nn.
共60兲
This leading order tension is identical to that obtained for a rigid rod, which varies
linearly with the rate of strain ␥˙ ( ␻ ) ⫽ i ␻ ␥( ␻ ). In a semiflexible chain, unlike a rigid
rod, this leading order tension induces a nonzero leading order strain 具 E0 (s, ␻ ) 典
⬅ 具 T0 (s, ␻ ) 典 /B( ␻ ). Equation 共8兲 for B( ␻ ) ⬀ (i ␻ ) 3/4 yields a strain
具E0 共 s, ␻ 兲 典 ⫽ 21 共 i ␻ ␶ 储 兲 1/4ŝ 共 1⫺ŝ 兲 ␥共 ␻ 兲 :nn
共61兲
␻ 1/4
that varies as
with frequency, with the same parabolic dependence on ŝ as the
tension. This strain is small compared to the affine strain ␥共␻兲:nn when ␻ Ⰶ ␶ ⫺1
, and
储
.
This
small
parabecome comparable in magnitude to the affine strain when ␻ ⬃ ␶ ⫺1
储
bolic strain is shown in the right panel of Fig. 13, where the magnitude of the exact
solution is plotted for a reduced frequency ␻ ␶ 储 ⫽ 10⫺5 .
Next, we consider the decay of the leading order tension and strain in the intermediate
time regime ␶ 储 Ⰶ t Ⰶ ␶⬜ after a step deformation. The leading order tension, which can
be obtained by inverse Fourier transforming Eq. 共60兲, is identical to that given in Eq. 共14兲
for a rigid rod
具T0 共 s,t 兲 典 ⫽ 21 ␦ 共 t 兲 ␨ 储 L 2 ŝ 共 1⫺ŝ 兲 ␥0 :nn,
共62兲
and has a delta-function time dependence. The ‘‘leading order’’ contribution to 具 T (s,t) 典
in the intermediate time regime, if defined as the inverse Fourier transform of the leading
order contribution to 具 T(s, ␻ ) 典 in the corresponding frequency regime, thus vanishes in
the time domain ␶ 储 Ⰶ t Ⰶ ␶⬜ of interest. However, the associated leading order longitudinal strain 具 E0 (s,t) 典 induced by this tension does not vanish at intermediate times, and
can be calculated by Fourier transforming Eq. 共61兲 for 具 E0 (s, ␻ ) 典 . Evaluating 共or power
counting兲 the resulting Fourier integral, using Fourier components ␥( ␻ ) ⫽ ␥0 /(i ␻ ),
yields a strain
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
1127
具E0 共 s,t 兲 典 ⬃ 共 t/ ␶ 储 兲 ⫺1/4ŝ 共 1⫺ŝ 兲 ␥0 :nn
共63兲
that decays with time as t ⫺1/4. This is the strain induced by the ␦-function impulse of
tension given by Eq. 共62兲. This impulse creates a nonequilibrium distribution of bending
mode amplitudes at the beginning of the intermediate time regime, creating a strain that
relaxes via the free relaxation of bending modes of increasing wavelength throughout the
intermediate time regime. The rate of decay of the longitudinal strain in the intermediate
time regime is thus limited primarily by the rate of free decay of the bending modes,
rather than by resistance to longitudinal motion, and is completed only when the slowest
bending mode relaxes.
Because the leading order approximation for the tension vanishes at times ␶ 储 Ⰶ t
Ⰶ ␶⬜ , the tension in this regime is dominated by the first order correction 具 T1 (s,t) 典 . To
calculate this correction, we again start in the frequency domain, and expand Eq. 共51兲 to
first order in ␭ 2 ( ␻ ), using the zeroth order solution to cancel all zeroth order terms. By
this method, we find that 具 T1 (s, ␻ ) 典 must satisfy
⳵2具T1 共 s, ␻ 兲 典
⫽ i ␻␨ 储 具 E0 共 s, ␻ 兲 典
⳵s2
共64兲
or, equivalently, that
⳵2具T1 共 s,t 兲 典
⳵s
⫽ ␨储
2
⳵ 具 E0 共 s,t 兲 典
共65兲
⳵t
in the time domain. Equations 共64兲 and 共65兲 show that 具 T1 (s,t) 典 is the tension required
to balance the longitudinal frictional force that opposes relaxation of the zeroth order
longitudinal strain 具 E0 (s,t) 典 . This residual tension has a subdominant effect on the relaxation of the strain, but is important because it is found in Sec. VII to yield the
dominant contribution to G(t) at intermediate times. Solving Eqs. 共64兲 and 共65兲 yields
具T1 共 s, ␻ 兲 典 ⫽
冉
冊
共66兲
␥0 :nn
共67兲
ŝ
ŝ 4
3
␨ 储 L 2 共 i ␻ 兲 5/4␶ 1/4
⫺
⫺
␥共 ␻ 兲 :nn
⫹ŝ
储
12
2
2
1
in an oscillatory flow, or
冉
ŝ
ŝ 4
2
2
具T1 共 s,t 兲 典 ⬀ t ⫺5/4 ⫺ ⫹ŝ 3 ⫺
冊
following a step deformation. This residual tension decays as t ⫺5/4 because, by Eq. 共65兲,
the first order tension is proportional to the time derivative of the leading order strain,
which decays as t ⫺1/4.
VI. STRESS TENSOR
The intramolecular polymer contribution to the stress is given, for any discrete model
of beads interacting via an intramolecular potential energy, by the Kramer–Kirkwood
expression 关Doi and Edwards 共1986兲兴:
␴p ⬅
⫺1
V
N
兺
i⫽1
具 Ri Fi 典 ,
共68兲
where Ri is the position of bead i,Fi ⫽ ⫺ ⳵ 关 U⫹k B T ln ⌿兴/⳵Ri is the effective force on
bead i,U( 兵 R其 ) is an intramolecular potential energy, and ⌿共兵R其兲 is a single chain prob-
1128
SHANKAR, PASQUALI, AND MORSE
ability distribution function. An expression for the intramolecular stress in a solution of
wormlike chains has been obtained by Morse 共1998a兲 by applying the Kramers–
Kirkwood formula to a discretized model of a wormlike chain as a chain of N beads
connected by very stiff springs with a preferred distance a between neighboring beads,
with a three-body bending potential that is a discretized version of bending energy 共1兲. By
evaluating the rhs of Eq. 共68兲 for this discrete model, then taking the limit a Ⰶ L p of
continuous, weakly curved chains, and re-expressing the results in terms of a continuous
arc length variable s ⫽ ia, Morse 共1998a兲 obtained a stress
␴p ⫽ ␴curv⫹ ␴tens⫹ ␴ornt⫺ck B TI,
共69兲
where
␴tens ⫽ c
␴curv ⫽ c ␬
冕
L
ds
0
冓
冕
L
0
ds 具 T uu典 ,
冏 冏冔
⳵u ⳵u
⳵u
⫺uu
⳵s ⳵s
⳵s
2
⫹
共70兲
ck B T
a
冕
L
0
ds 具 3uu⫺I典 ,
␴ornt ⫽ 23 ck B T 具 u共 0 兲 u共 0 兲 ⫹u共 L 兲 u共 L 兲 ⫺ 32 I典 ,
共71兲
共72兲
where u ⫽ u(s) and T ⫽ T 共s兲 are the unit tangent and tension for a bond at position s
⫽ ia. The physical meaning of these three stress contributions 共briefly兲 is: The ‘‘tension
stress’’ ␴tens arises from the constraint forces that enforce inextensibility in the chain. The
‘‘curvature stress’’ ␴curv contains both a purely mechanical contribution arising from the
bending forces 关the first term on the rhs of Eq. 共71兲兴 and an entropic contribution arising
from the orientational entropy of the links 共the second term兲. The curvature stress was
shown, using the underlying discrete model, to vanish in a hypothetical partially equilibrated state in which the variance of the curvature at each point on the chain retains its
thermal equilibrium value, or, for a rodlike polymer, in which the distribution of bending
mode amplitudes is equilibrated. The curvature stress thus arises from disturbances of this
equilibrium distribution of bending mode amplitudes. The ‘‘orientational stress’’ ␴ornt is
a residual contribution of the orientational entropy arising from the two end links.
These expressions can be further simplified in the rodlike limit. In this limit, to leading
order, u(s) can be approximated by the global rod orientation n except in terms involving
the curvature ⳵ u/ ⳵ s. This approximation immediately reduces the forms of the orientational and tension stress to those found for a rigid rod. Terms involving the curvature
⳵ u/ ⳵ s can be approximately by noting that ⳵ u/ ⳵ s must be orthogonal to u, because u共s兲
is a unit vector, and thus nearly orthogonal to n: Approximating ⳵ u/ ⳵ s by its projection
onto the plane perpendicular to n yields ⳵ u/ ⳵ s ⯝ ⳵ 2 h/ ⳵ s 2 . In this approximation
␴tens ⫽ c
␴curv ⫽ c ␬
冕
L
0
ds
冓
冕
L
0
ds 具 T nn典 ,
冏 冏冔
⳵2h ⳵2h
⳵2h
⫺nn
⳵s2 ⳵s2
⳵s2
2
⫹
␴ornt ⫽ ck B T 具 3nn⫺I典 .
共73兲
ck B TL
a
具 3nn⫺I典 ,
共74兲
共75兲
In Eqs. 共73兲–共75兲, 具•••典 represents a complete thermal average, over both the rapid fluctuations of h, f, and T, and over overall rod orientation n. Below, we calculate
␴tens , ␴curv , and ␴ornt separately, and express G(t) as a sum
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
1129
G共t兲 ⫽ Gtens共 t 兲 ⫹G curv共 t 兲 ⫹G ornt共 t 兲 ⫺ ␩ s ␦ 共 t 兲 ,
共76兲
of contributions arising from different stress contributions, where, e.g., ␴curv(t)
⫽ G curv(t) 关 ␥0 ⫹ ␥T0 兴 , and similarly expand G * ( ␻ ) and the intrinsic moduli 关 G(t) 兴 and
关 G*(␻) 兴.
VII. VISCOELASTICITY IN THE LCA
Here, the linear viscoelastic response of a solution of rods at intermediate and high
⫺1
, is computed using the LCA for the average tension.
frequencies, ␻ Ⰷ ␶⬜
A. Tension stress
The LCA approximation for the tension stress is obtained by setting the tension in Eq.
共73兲 equal to 具 T(s, ␻ ) 典 ⫽ B( ␻ ) 具 E(s, ␻ ) 典 , while using Eq. 共58兲 for the strain. This yields
a stress
␴tens共 ␻ 兲 ⫽ cL
冕
1
0
再
dŝ B( ␻ ) 1⫺
cosh关␭共␻兲共ŝ⫺1/2兲兴
cosh关␭共␻兲/2兴
冎
␥共 ␻ 兲 : 具 nnnn典 .
共77兲
Here, 具•••典 denotes an average over rod orientations, which can be approximated in a
linear response calculation by an average over an isotropic distribution, using the identity
关Doi and Edwards 共1986兲兴:
具nin jnknl典eq ⫽
1
15
共 ␦ i j ␦ kl ⫹ ␦ ik ␦ l j ⫹ ␦ il ␦ k j 兲
共78兲
for randomly oriented unit vectors. By evaluating the integral with respect to s, and
requiring that the trace of ␥共␻兲 vanish in an incompressible fluid, we obtain a stress of the
form
T
␴tens共 ␻ 兲 ⫽ c 关 G *
tens共 ␻ 兲兴关 ␥共 ␻ 兲 ⫹ ␥ 共 ␻ 兲兴 ,
with
关G*
tens共 ␻ 兲兴 ⫽
1
15
再
LB 共 ␻ 兲 1⫺
tanh关␭共␻兲/2兴
␭ 共 ␻ 兲 /2
冎
共79兲
共80兲
.
Equation 共80兲 has the following limiting behaviors
共1兲 High frequencies and short times: At frequencies ␻ Ⰷ ␶ ⫺1
, for which ␭共␻兲 → ⬁,
储
Eq. 共80兲 reduces to
lim 关 G *
tens共 ␻ 兲兴 ⯝
␻ Ⰷ ␶ ⫺1
储
L
15
B共 ␻ 兲 ⯝
2 3/4
15
k B TLL 2p
冉 冊
i ␻␨⬜
␬
3/4
,
共81兲
3/4
and hence, 关 G *
tens( ␻ ) 兴 ⬀ ␻ . Inverse Fourier transforming this asymptote yields a
modulus
lim 关 G tens共 t 兲兴 ⫽ C 1 k B TLL 2p
t Ⰶ ␶储
冉 冊
␬t
␨⬜
⫺3/4
,
共82兲
where C 1 ⫽ 2 3/4/ 关 15⌫( 14 ) 兴 ⫽ 0.0309.
共2兲 Intermediate frequencies and times: At intermediate frequencies, such that ␶ ⫺1
储
⫺1
Ⰷ ␻ Ⰷ ␶⬜
, where ␭( ␻ ) ⫽ (i ␻ ␶ 储 ) 1/8 Ⰶ 1, Eq. 共80兲 can be expanded in powers of
␭ 2 ( ␻ ). The first two terms of the expansion are
1130
SHANKAR, PASQUALI, AND MORSE
* 共 ␻ 兲兴 ⯝
关Gtens
1
180
i ␻␨ 储 L 3 ⫺
kBT
冉 冊
␨储
1800 23/4 ␨⬜
2
共 i ␻ ␶⬜ 兲 5/4⫹•••.
共83兲
The leading contribution, which arises from the leading order tension, is identical to the
viscous contribution found for true rigid rods, and is purely imaginary. The first correction, which is proportional to (i ␻ ) 5/4, thus dominates the real part of 关 G *
tens( ␻ ) 兴 . We thus
obtain a loss modulus 关 G ⬙tens( ␻ ) 兴 ⬀ ␻ , with the same prefactor as that of rigid rods, but
a storage modulus 关 G ⬘tens( ␻ ) 兴 ⬀ ␻ 5/4. This ␻ 5/4 contribution to 关 G ⬘ ( ␻ ) 兴 is a subdominant contribution to 兩 关 G * 兴 兩 , because it is small compared to 关 G ⬙ ( ␻ ) 兴 , but is much larger
than the ␻-independent storage modulus of 53 k B T per chain found at these frequencies for
rigid rods, and is found to dominate 关 G ⬘ ( ␻ ) 兴 .
Upon transforming this intermediate asymptote to the time domain, the leading order
i ␻ term yields an apparent ␦共t兲 contribution identical to that found for rigid rods, which
does not contribute to G(t) at intermediate times. As a result, 关 G tens(t) 兴 is dominated at
intermediate times by the transform of the (i ␻ ) 5/4 term, which yields
冉 冊冉 冊
␨ 储 2 t ⫺5/4
,
关 G tens共 t 兲兴 ⯝ C 2 k B T
␨⬜
␶⬜
␶ 储 Ⰶ t Ⰶ ␶⬜
lim
共84兲
where C 2 ⫽ 1/关 2 3/47200⌫(3/4) 兴 ⫽ 6.74⫻10⫺5 . This stress arises from the t ⫺5/4 decay
of the tension.
B. Curvature stress
Here, the curvature stress is computed with transverse equation of motion 共24兲, in an
approximation similar to that used to calculate the high-frequency compliance in Sec. IV,
in which finite size effects are ignored and all quantities are expanded in a continuous
distribution of Fourier modes.
We first expand Eq. 共74兲 for the curvature stress in terms of Fourier amplitudes of h.
Because ␴curv vanishes when the distribution of bending mode amplitudes is equilibrated
关Morse 共1998a兲兴, the contribution of ␴curv from a rod with a specified orientation n may
be equated with the difference between the rhs of Eq. 共74兲 and its thermal equilibrium
value. Expanding this difference in Fourier modes yields a contribution
␴curv ⫽ c ␬
冕
dq
2␲
q 4 具 b共 q,t 兲 ⫺nnTr 关 b共 q,t 兲兴 典 n ,
共85兲
where b(q,t) ⬅ 兺 ␣␤ e␣ (t)e␤ (t)b ␣␤ (q,t) is a tensor with components
b␣␤共q,t兲 ⬅ 具h␣共q兲h␤共⫺q兲典⫺具h␣共q兲h␤共⫺q兲典eq .
共86兲
Here, the average 具•••典 in Eq. 共86兲 represents an average over fluctuations of the bending
mode amplitudes for rods of known orientation, while the symbol 具 ••• 典 n in Eq. 共85兲 is
used to indicate an average with respect to rod orientations.
To evaluate Eq. 共85兲, it is convenient to calculate the contributions to b(q,t) and ␴curv
that are induced by the presence of a nonzero value of ␥˙ ␣␤ in Eq. 共38兲 separately from
those induced by the existence of a nonzero tension and then add these two contributions.
共1兲 Flow-induced curvature stress: We first consider the contribution induced directly
by the velocity gradient in Eq. 共38兲, while setting T ⫽ 0. The resulting contribution to
b ␣␤ (q, ␻ ) on a chain of known orientation in an oscillatory flow can be calculated using
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
1131
equation of motion 共38兲 by a method closely analogous to that used to derive Eq. 共43兲 for
a(q,k; ␻ ) in Sec. IV. This yields
b␣␤共q,␻兲 ⫽
i␻␨⬜
kBTL
4
␬q i␻␨⬜⫹2␬q4
关␥␣␤共␻兲⫹␥␤␣共␻兲兴.
共87兲
We then expand b(q, ␻ ) ⬅ 兺 ␣␤ b ␣␤ (q, ␻ )e␣ e␤ , and use the identity 兺 ␣␤ ␥ ␣␤ ( ␻ )e␣ e␤
⫽ (I⫺nn)• ␥( ␻ )•(I⫺nn) to express b(q, ␻ ) explicitly as a function of ␥共␻兲 and n.
After substituting the resulting expression for b(q, ␻ ) into the temporal Fourier transform
of Eq. 共85兲, averaging over random rod orientations, and integrating with respect to q, we
* 兴关 ␥ ( ␻ )⫹ ␥ T ( ␻ ) 兴 , with an intrinsic
obtain a stress of the form ␴curv,t ( ␻ ) ⫽ c 关 G curv,t
modulus
* 共 ␻ 兲兴 ⫽
关Gcurv,t
3k B T
2 3/410
共 i ␻ ␶⬜ 兲 1/4.
共88兲
* ( ␻ ) 兴 , because it arises directly
We refer to this as the ‘‘transverse’’ contribution to 关 G curv
from the components of ␥˙ ( ␻ ) along the directions transverse to n.
共2兲 Tension-induced curvature stress: We next consider the contribution to b(q, ␻ ) and
␴curv( ␻ ) induced by the tension on the rhs of Eq. 共38兲, while setting ␥˙ ␣␤ ⫽ 0. This
calculation is closely analogous to the calculation of a(q,k; ␻ ) in the case k ⫽ 0. On an
infinite chain, b ␣␤ (q, ␻ ) depends only on the q ⫽ 0 component of the tension, as a result
of translational invariance. On a finite chain at intermediate or high frequencies,
b ␣␤ (q, ␻ ) thus depends upon the integral T (q ⫽ 0,␻ ) ⫽ 兰 dsT (s, ␻ ). Another calculation analogous to that leading to Eq. 共43兲, in which the tension in Eq. 共38兲 is equated to
the thermal average 具 T(s, ␻ ) 典 , yields
b␣␤共q,␻兲 ⫽ ⫺␦␣␤
2kBT
1
2
冕
L
␬q i␻␨⬜⫹2␬q
4
0
ds 具T 共 s, ␻ ;n兲 典 ,
共89兲
in which 具 T (s, ␻ ;n) 典 is the average, with respect to rapid transverse fluctuations of the
tension in a rod with known orientation n. Substituting this expression into Eq. 共85兲, and
again evaluating averages with respect to chain orientations using Eq. 共78兲, yields a stress
contribution characterized by an intrinsic modulus
关G*
curv,l共 ␻ 兲兴 ⫽
2
3
L
5/4
Lp
共 i ␻ ␶⬜ 兲 ⫺1/4关 G *
tens共 ␻ 兲兴 .
共90兲
Here 关 G *
tens( ␻ ) 兴 is the tension modulus given in Eq. 共80兲, which also depends on the
integral 兰 ds 具 T(s, ␻ ;n) 典 . Equation 共90兲 has the following limiting behaviors: For ␻
3/4
Ⰷ ␶ ⫺1
, where 关 G *
储
tens( ␻ ) 兴 ⬀ (i ␻ ) ,
* 共 ␻ 兲兴 ⬃ k B T
lim 关 G curv,l
⫺1
␻ Ⰷ ␶储
冉 冊
Lp
L
共 i ␻ ␶⬜ 兲 1/2.
共91兲
⫺1
For ␶ ⫺1
Ⰷ ␻ Ⰷ ␶⬜
, where the dominant contribution to 关 G *
储
tens兴 scales as
(k B TL p /L)(i ␻ ␶⬜ ),
lim
␨储
* 共 ␻ 兲兴 ⬃ k B T 共 i ␻ ␶⬜ 兲 3/4.
关 G curv,l
␨⬜
⫺1
␶ ⫺1
Ⰷ ␻ Ⰷ ␶⬜
储
共92兲
1132
SHANKAR, PASQUALI, AND MORSE
* ( ␻ ) 兴 is termed the ‘‘longitudinal’’ contribution to 关 G curv
* ( ␻ ) 兴 because it arises
关 G curv,l
from the tension, which is proportional to the component ␥˙ 共␻兲:nn of ␥˙ along n. The total
* ( ␻ ) 兴 is given by the sum of 关 G curv,t
* ( ␻ ) 兴 and 关 G curv,l
* (␻)兴.
curvature modulus 关 G curv
C. Results
The total modulus is given by the sum of the tension and curvature contributions
calculated earlier and of an orientational contribution. In the rodlike limit, L Ⰶ L p , the
orientational contribution can be approximated by that of a dilute solution of rigid rods
关G*
ornt共 ␻ 兲兴 ⫽
3
5
kBT
i ␻ ␶ rod
1⫹i ␻ ␶ rod
.
共93兲
For frequencies ␻ ␶ rod Ⰷ 1, this expression for 关 G *
ornt兴 yields a plateau of magnitude
3
k
T
in
the
storage
modulus.
5 B
This orientational component dominates the total storage modulus at low frequencies,
⫺1
␻ Ⰶ ␶⬜
, giving behavior identical to that of rigid rods, but becomes negligible compared to either the tension or curvature components at intermediate and high frequencies.
A comparison of the earlier expressions for the different contributions to G * ( ␻ ) shows
that, throughout intermediate and high frequency regimes, they form a hierarchy
* 共 ␻ 兲 Ⰶ G tens
* 共 ␻ 兲.
G*
ornt共 ␻ 兲 Ⰶ G *
curv,t 共 ␻ 兲 Ⰶ G curv,l
共94兲
⫺1
The tension contribution 关 G *
tens( ␻ ) 兴 dominates the total modulus at all ␻ Ⰷ ␶⬜ . At
⫺1
frequencies ␻ ⬃ ␶⬜ , all four of these contributions to 关 G * ( ␻ ) 兴 become comparable to
k B T, and thus to each other. Though the earlier LCA calculation is valid only at ␻
⫺1
Ⰷ ␶⬜
, the more complete calculation given in the next section shows that both the
curvature and tension contributions to G(t) decay exponentially at t Ⰷ ␶⬜ , with terminal
times of order ␶⬜ , leading to terminal behavior in the corresponding components of
⫺1
G * ( ␻ ) at ␻ Ⰶ ␶⬜
. The resulting time dependence of G tens(t),G curv(t), and G ornt(t)
for chains with L Ⰶ L p is shown schematically in Fig. 2.
VIII. FULL THEORY
⫺1
In order to describe accurately viscoelasticity at frequencies of order ␶⬜
and lower,
the preceding calculation must be extended in two ways. First, because the nonlocality of
␹ (s,s ⬘ ; ␻ ) becomes important at these frequencies, we must abandon the LCA and use
the full nonlocal compliance when solving the average longitudinal force balance of Eq.
共30兲, while respecting the condition that the tension vanish at both chain ends. Second,
when using the transverse equation of motion to calculate both the compliance and the
curvature stress, the transverse displacement field must be expanded in discrete eigenmodes that respect the boundary conditions for the transverse dynamics. To formulate a
full theory, accurate at arbitrary frequency, we first introduce expansions of T and h in
appropriate basis functions.
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
1133
FIG. 2. Schematic showing the asymptotic behaviors of G tens(t),G curv(t), and G ornt(t), in a log–log plot, for
stiff chains with L Ⰶ L p .
A. Longitudinal mode expansion
The tension T 共ŝ兲, which must vanish at both ends of the chain, is expanded in a basis
of sines
T 共 ŝ 兲 ⫽
兺k Tk ␾ k 共 ŝ 兲 ,
␾ k 共 ŝ 兲 ⫽
冑2sin共␭kŝ兲,
共95兲
where ␭ k ⫽ k ␲ for k ⫽ 1,2,3,... . Substituting this expansion into Eq. 共30兲, multiplying
the result by ␾ i (ŝ) and integrating with respect to ŝ yields the expansion of the longitudinal force balance
兺k
冋
i␻␨储␹ik共␻兲⫹
冋 册
␬ ⳵4␹共␻兲
L4
⳵ŝ4
⫹␦ik
ik
␭2i
L2
册
具Tk 典 ⫽ i ␻␨ 储 ␥共 ␻ 兲 :nnFi ,
共96兲
where
␹ik共␻兲 ⫽ L
冋 册
⳵4␹共␻兲
⳵ŝ4
冕 冕
1
0
⫽L
ik
1
dŝ
0
ds⬘␾i共ŝ兲␾k共ŝ⬘兲␹共ŝ,ŝ⬘;␻兲,
冕 冕
1
1
⳵4
0
⳵ŝ4
dŝ
0
dŝ⬘␾i共ŝ兲
and
Fi ⫽
冕
1
0
dŝ ␾ i 共 ŝ 兲 ⫽
再
2 冑2
i␲
0
␹共ŝ,ŝ⬘;␻兲,
i odd
.
i even
Here, ␹ ik ( ␻ ) are the ‘‘matrix elements’’ of ␹ (s,s ⬘ ; ␻ ) in a basis of sines.
共97兲
共98兲
共99兲
1134
SHANKAR, PASQUALI, AND MORSE
B. Transverse mode expansion
The transverse displacement h ␣ (ŝ) is expanded as
h␣共ŝ兲 ⫽
兺j h̃␣,jWj 共 ŝ 兲 ,
共100兲
where W j is the jth normalized eigenfunction of the fourth order eigenvalue problem
关Aragon and Pecora 共1985兲, Kroy and Frey 共1997兲, Wiggins et al. 共1998兲兴:
⳵4W j 共 ŝ 兲
⳵ ŝ
4
⫽ ␣ 4j W j 共 ŝ 兲 ,
共101兲
subject to the homogeneous boundary conditions
⳵2W j
⳵ ŝ 2
⳵ 3 Wj
⫽
⳵ ŝ 3
⫽0
共102兲
at ŝ ⫽ 0 and ŝ ⫽ 1. The eigenfunctions W j are orthogonal, because the differential
operator in 共101兲 is self-adjoint, and are taken to be orthonormal, so that 兰 10 dŝW j Wk
⫽ ␦ jk . The eigenvalue problem has a degenerate trivial eigenvalue ␣ 0 ⫽ 0, with eigenfunctions W ⫽ 1 and W ⫽ 冑12(ŝ⫺1/2) corresponding to rigid translations and rotations, respectively. The nontrivial eigenfunctions, corresponding to bending modes, are
W j 共 ŝ 兲 ⫽ A j 兵 关 sinh共␣ j 兲⫹sin共␣ j 兲兴关sin共␣ jŝ兲⫹sinh共␣ jŝ兲兴
⫹关cos共␣ j 兲⫺cosh共␣ j 兲兴关cos共␣ jŝ兲⫹cosh共␣ jŝ兲兴其,
共103兲
⫺1/2. The eigenvalues ␣ satisfy the
where A j ⫽ 关 12 ⫹ 21 cosh(2␣ j)⫹␣⫺1
j
j cosh(␣ j)sin(␣ j)兴
solvability condition cos(␣ j) ⫽ 1/cosh(␣ j). The first few nontrivial eigenvalues are ␣ 1
⫽ 4.73 ⫽ 3 ␲ /2⫹0.0176, ␣ 2 ⫽ 5 ␲ /2⫺0.000 781 6, and ␣ 3 ⫽ 7 ␲ /2⫹0.000 033 5.
Higher eigenvalues are accurately approximated by setting cos(␣ j) ⫽ 0, which yields
␣ j ⫽ (2 j⫹1) ␲ /2.
The harmonic approximation 共37兲 of the bending energy can be expanded in mode
amplitudes as
Ubend ⫽
1 ␬
2 L
3
兺j ␣ 4j 共 h̃ ␣ , j 兲 2 .
共104兲
The transverse dynamical Eq. 共24兲 can be expanded by substituting expansion 共100兲 for
h ␣ and expansion 共95兲 for T, multiplying the result by W j (s), and integrating with
respect to s; this yields
␨⬜
兺␤
冋
册
1
⳵
␦␣␤ ⫺␥˙ ␣␤ h̃␤,j ⫽ ⫺␬L⫺4␣4j h̃␣,j⫺ 2 兺 HljkTl h̃ ␣ ,k ⫹ ␩˜ ␣ , j ,
⳵t
L lk
共105兲
where
Hljk ⬅
冕
1
0
dŝ ␾l共ŝ兲
⳵W j 共 ŝ 兲 ⳵ Wk 共 ŝ 兲
⳵ ŝ
⳵ ŝ
,
共106兲
and ␩˜ ␣ , j ⫽ 兰 L0 dŝW j (ŝ) ␩ ␣ (ŝ) is a mode amplitude for component ␣ of the transverse
noise ␩⬜ .
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
1135
C. Nonlocal compliance
We now present a calculation of the nonlocal compliance ␹ (s,s ⬘ ; ␻ ) at arbitrary
frequency. The longitudinal strain given in Eq. 共23兲 is expanded as
具E共 ŝ 兲 典 ⫽ ⫺
1
⳵ W j ⳵ Wm
兺
jm
a jm ,
共107兲
兺 关具h̃␣,jh̃␣,m典⫺具h̃␣,jh̃␣,m典eq兴 .
共108兲
L
2
⳵ ŝ
⳵ ŝ
where
a jm ⬅
1
2␣
To calculate the compliance, we derive an expression for da jm (t)/dt in the presence of
an infinitesimal tension, while setting ␥ ␣␤ ( ␻ ) ⫽ 0, by a method closely analogous to
that used to derive Eq. 共42兲 for da(q,k;t)/dt. Here, we use expansion 共105兲 of the
transverse dynamical equation, and evaluate the required thermal averages using the
relations 具 h̃ ␣ , j h̃ ␤ ,k 典 eq ⫽ ␦ ␣␤ ␦ k j k B TL 3 /( ␬␣ 4k ) and 具 ␩˜ ␣ , j h̃ ␤ ,k 典 ⫽ ␦ ␣␤ ␦ jk k B T/L. This
yields the differential equation
再
␨⬜
⳵
⳵t
冎
4
⫹␬L⫺4共␣4j ⫹␣m
兲 a jm共t兲 ⫽ ⫺
kBTL
␬
⫺4
⫺4
共␣m ⫹␣ j 兲
兺l HlmjTl 共 t 兲 ,
共109兲
for a jm (t), or the equivalent algebraic equation
a jm共␻兲 ⫽ ⫺
⫺4
⫹␣⫺4
␣m
j
kBTL
4
␬ i␻␨⬜⫹␬L⫺4共␣4j ⫹␣m
兲
兺l HlmjTl 共 ␻ 兲 ,
共110兲
for its Fourier transform a jm ( ␻ ) ⬅ 兰 dta jm (t)e ⫺i ␻ t . Substituting Eq. 共110兲 for a jm ( ␻ )
into Eq. 共107兲 yields an average strain of the form 具 E(s, ␻ ) 典
⫽ 兰 L0 ds ⬘ ␹ (s,s ⬘ ; ␻ )T(s ⬘ , ␻ ), where
␹共s,s⬘;␻兲 ⫽
⫺4
⫺4
共␣m ⫹␣ j 兲Hljm ⳵W j 共 ŝ 兲 ⳵ Wm 共 ŝ 兲
␾ l 共 ŝ ⬘ 兲
2
4
␬L ljm i␻␨⬜⫹␬L⫺4共␣4j ⫹␣m
⳵ ŝ
兲 ⳵ ŝ
kBT
兺
共111兲
is the desired nonlocal compliance.
The matrix elements ␹ ik ( ␻ ) and 关 ⳵ 4 ␹ / ⳵ s 4 兴 jk defined in Eqs. 共97兲 and 共98兲 can be
obtained by evaluating the defining double integral, using the orthonormality of the
functions ␾ k (ŝ). The result can be expressed in terms of non-dimensional matrix elements ␹¯ ik ( ␻ ) ⬅ i ␻␨⬜ L 2 ␹ ik ( ␻ ) and 关 ⳵ 4 ␹¯ / ⳵ ŝ 4 兴 ik ⬅ i ␻␨⬜ L 2 关 ⳵ 4 ␹ ( ␻ )/ ⳵ ŝ 4 兴 ik , which
are given by
冋 册
⳵4␹¯ 共␻兲
⳵ŝ4
⫺4
⫺4
共␣ j ⫹␣m 兲HijmHkjm
4 ,
Lp jm 1⫹共i␻␶⬜兲⫺1共␣4j ⫹␣m
兲
L
␹¯ ik共␻兲 ⫽
⫽
ik
兺
兺
where
Fijm ⫽
⫺4
⫺4
共␣ j ⫹␣m 兲FijmHkjm
4 ,
Lp j,m 1⫹共i␻␶⬜兲⫺1共␣4j ⫹␣m
兲
L
冕
冉
1
⳵ 4 ⳵W j ⳵ Wm
0
⳵ŝ 4 ⳵ ŝ
dŝ␾i
共112兲
⳵ ŝ
冊
.
共113兲
1136
SHANKAR, PASQUALI, AND MORSE
D. Tension and tension stress
To calculate the spatial Fourier components of the average tension, longitudinal force
balance 共96兲 can be expressed in nondimensional form as a matrix equation
兺k
再
␨储
␨⬜
␹¯ ik共␻兲⫹
1
i␻␶⬜
冋 册
⳵4␹¯ 共␻兲
⳵ŝ4
冎
⫹␭2i ␦ik 具T k* 共 ␻ 兲 典 ⫽ Fi ,
ik
共114兲
where
具T k* 共 ␻ 兲 典 ⬅
具 Tk 共 ␻ ;n兲 典
i ␻␨ 储 L 2 ␥共 ␻ 兲 :nn
共115兲
is the nondimensional Fourier component of the tension, and 具 Tk ( ␻ ;n) 典 is a Fourier
amplitude for the average tension in a chain with a known orientation n. By Eq. 共96兲,
具 Tk ( ␻ ;n) 典 depends linearly on ␥共␻兲:nn. This dependence has been factored out in the
earlier definition, so that the reduced Fourier amplitude 具 T *
k ( ␻ ) 典 is independent of both
n and ␥共␻兲. These reduced Fourier amplitudes are calculated by solving matrix Eq. 共114兲
with a finite number of modes.
* ( ␻ ) is calculated by substituting expansion 共95兲 in Eq. 共73兲
The tension modulus G tens
for the tension stress, while using Eq. 共115兲 to recast the results in terms of the dimensionless Fourier components of the tension. This yields a stress
␴tens共 ␻ 兲 ⫽ ci ␻␨ 储 L 3 ␥共 ␻ 兲 : 具 nnnn典
兺p 具 T *p 共 ␻ 兲 典 Fp .
共116兲
After evaluating the average of 具nnnn典 over rod orientations, as before, we obtain a stress
T
of the form ␴tens( ␻ ) ⫽ c 关 G *
tens( ␻ ) 兴关 ␥( ␻ )⫹ ␥ ( ␻ ) 兴 , with
关G*
tens共 ␻ 兲兴 ⫽
1
15
i ␻␨ 储 L 3
兺p 具 T *p 共 ␻ 兲 典 Fp .
共117兲
E. Curvature stress
Substituting expansion 共100兲 for the transverse displacement in Eq. 共74兲 for the curavture stress, and using the fact that ␴curv vanishes when the bending modes are equilibrated, yields an expansion of the curvature stress as
␴curv ⫽ c ␬ L ⫺3
兺k ␣ 4k 具 bk ⫺nn Tr关 bk 兴 典 n ,
共118兲
where bk (t) ⬅ 兺 ␣␤ e␣ (t)e␤ (t)b k, ␣␤ (t) is a tensor with components
bk,␣␤ ⬅ 具h␣,kh␤,k典⫺具h␣,kh␤,k典eq .
共119兲
The calculation of the components of bk ( ␻ ) in a weak oscillatory flow field is closely
analogous to the calculation of b(q, ␻ ) given in Sec. VII B, except for the use of expansions 共100兲 and 共95兲 for h and T, respectively, rather than Fourier transforms, and the
corresponding use of Eq. 共105兲 for time derivatives of the mode amplitudes. As in Sec.
VII B, we consider separately the contributions to bk and ␴curv arising from the direct
coupling of the velocity gradient to h, and from the tension.
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
1137
共1兲 Flow-induced curvature stress: We calculate the curvature stress that is induced
directly by the velocity gradient tensor by calculating the time derivative of b k, ␣␤ (t),
which setting the tension to zero in Eq. 共105兲. This yields
bk,␣␤共␻兲 ⫽
kBTL3
␬␣4k
i␻␨⬜
i␻␨⬜⫹2␬L⫺4␣4k
关␥␣␤共␻兲⫹␥␤␣共␻兲兴
共120兲
for the Fourier transform of b k, ␣␤ (t). Substituting Eq. 共120兲 in Eq. 共118兲, and evaluating
* ( ␻ ) 兴关 ␥( ␻ )
the orientational averages yields a stress contribution ␴curv,t( ␻ ) ⫽ c 关 G curv
⫹ ␥T ( ␻ ) 兴 , with
* 共 ␻ 兲兴 ⫽
关Gcurv,t
3
5
kBT
兺k i ␻␨
i ␻␨⬜
⬜ ⫹2 ␬ L
⫺4 4 .
␣k
共121兲
共2兲 Tension-induced curvature stress: The contribution to b k, ␣␤ induced by the tension, for ␥共␻兲 ⫽ 0, is given by
bk,␣␤ ⫽ ⫺␦␣␤
2kBTL
1
␬␣4k i␻␨⬜⫹2␬␣4k L⫺4
兺l Hlkk具Tl 共 ␻ 兲 典 .
共122兲
Substituting this in Eq. 共118兲, expressing the tension coefficients in terms of the reduced
coefficients 具 T* ( ␻ ) 典 , and averaging over chain orientations, yields a stress contribution
characterized by an intrinsic modulus
* 共 ␻ 兲兴 ⫽
关Gcurv,l
2
5
kBT
i ␻␨ 储
兺k i ␻␨
⬜ ⫹2 ␬ L
⫺4 4
␣k
兺l H lkk 具 T *l 共 ␻ 兲 典 .
共123兲
F. Results
The total complex modulus of dilute solutions of semiflexible rods, with L Ⰶ L p but
arbitrary frequency, is obtained by adding Eqs. 共117兲, 共121兲, 共123兲, and 共93兲:
关G*共␻兲兴
kBT
⫽
24 ␨ 储
5 ␨⬜
⫹
共 i ␻ ␶ rod兲
兺k 具 Tk* 共 ␻ 兲 典 Fk
2 ␨储
1
兺
5 ␨⬜ k 1⫹2 ␣ 4 共 i ␻ ␶
⬜兲
k
⫹
3
3
兺l 具 T*l 共 ␻ 兲 典 H lkk
1
兺
5 k 1⫹2 ␣ 4 共 i ␻ ␶
k
⫹
⫺1
i ␻ ␶ rod
5 1⫹i ␻ ␶ rod
,
⬜兲
⫺1
共124兲
where 具 T *
k 典 is obtained as a solution to matrix Eq. 共114兲, Fk is defined in Eq. 共99兲, and
H l jm is defined in Eq. 共106兲. The first line in Eq. 共124兲 represents 关 G *
tens( ␻ ) 兴 , the second
* ( ␻ ) 兴 the third 关 G curv,t
* 兴 , and the fourth 关 G *
line represents 关 G curv,l
(
ornt ␻ ) 兴 .
An example of theoretical predictions for various contributions to the storage and loss
modulus, viz. 关 G *
tens兴 , 关 G *
curv,l兴 , 关 G *
curv,t兴 , and 关 G *
ornt兴 , are shown in Fig. 3 for L/L p
⫽ 1/8. Also plotted in this figure are the corresponding asymptotes derived in Sec. VII
for high and intermediate frequency regimes. Note that the high frequency ␻ 3/4 asymptotes of 关 G ⬘ ( ␻ ) 兴 and 关 G ⬙ ( ␻ ) 兴 are approached rather slowly and at very high frequen-
1138
SHANKAR, PASQUALI, AND MORSE
FIG. 3. Predictions of the full theory for the tension 共tens兲, curvature, 共curv,l and curv,t兲, and orientation 共ornt兲
contributions to the intrinsic storage modulus 关 G ⬘ ( ␻ ) 兴 共left panel兲 and intrinsic loss modulus 关 G ⬙ ( ␻ ) 兴 共right
panel兲, divided by k B T, as functions of ␻ ␶ rod for L/L p ⫽ 1/8. Intrinsic moduli are defined in Eq. 共12兲 as
contributions per chain. Dashed lines are the asymptotic power laws predicted for each component at intermediate and high frequencies.
cies, ␻ ␶ rod Ⰷ 109 , corresponding roughly to ␻ Ⰷ ␶ ⫺1
. The slow approach to this as储
ymptote is a result of the slow ␻ ⫺1/8 decrease in the thickness of the boundary layers at
the chain ends. The ␻ 5/4 dependence of 关 G tens
⬘ ( ␻ ) 兴 at intermediate frequencies is clearly
visible for this value of L/L p , over a range ␻ ␶ rod ⬃ 103 – 106 . The quantity 关 G ⬙tens( ␻ ) 兴 ,
⫺1
, is proportional to ␻ at all ␻ Ⰶ ␶ ⫺1
, with a
which dominates 关 G ⬙ ( ␻ ) 兴 at all ␻ Ⰷ ␶ rod
储
constant of proportionality identical to that found for rigid rods. As a result, 关 G ⬙ ( ␻ ) 兴 for
a solution of semiflexible rods closely approaches that of a corresponding rigid rod
. Both the tension contribution and the two curvature contribusolution at all ␻ Ⰶ ␶ ⫺1
储
tions to 关 G * ( ␻ ) 兴 exhibit terminal behavior below ␻ ␶ rod ⬃ 102 , corresponding roughly
⫺1
to frequencies below the relaxation rate ␣ 41 ␶⬜
of the longest wavelength bending mode.
At all lower frequencies, for which the bending modes are relaxed, the overall 关 G ⬘ ( ␻ ) 兴
and 关 G ⬙ ( ␻ ) 兴 both closely mimic the behavior of rigid rods, leading to a plateau of
⫺1
magnitude 关 G ⬘ ( ␻ ) 兴 ⫽ 53 k B T at ␶ rod ⬍ ␻ ⬍ ␣ 41 ␶⬜
, which grows broader as L/L p is
⫺1
decreased, and terminal behavior at ␻ Ⰶ ␶ rod .
Figure 4 shows the evolution of the total predicted storage and loss moduli as L/L p is
varied from 1/8 to 1. As L/L p increases, both the ␻ 5/4 intermediate regime in G ⬘ ( ␻ ) and
the orientational plateau at lower frequencies gradually disappear, while the highfrequency ␻ 3/4 asymptote is approached at lower reduced frequencies for more flexible
chains.
Figure 5 compares the predictions of the full theory to those of the LCA, and to the
analytic approximation of Sec. XI, for L/L p ⫽ 1/8. The LCA agrees well with the full
⫺1
theory at frequencies well above the relaxation rate ␣ 41 ␶⬜
of the longest wavelength
bending mode, but fails at lower frequencies. The conspicuous failure of the LCA at low
frequencies is primarily a result of the fact that the LCA predicts algebraic frequency and
time dependence for the tension and curvature contributions to the modulus even at
⫺1
, due to the use of a continuous distribution of modes with no lower
frequencies ␻ ⬍ ␶⬜
cutoff, which yields a theory with no terminal relaxation time. The full theory instead
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
1139
FIG. 4. The storage 关 G ⬘ ( ␻ ) 兴 and loss 关 G ⬙ ( ␻ ) 兴 moduli vs ␻ ␶ rod obtained from the full theory for different
values of L/L p . The solid and dash-dotted straight lines are the predicted ␻ 3/4 high-frequency asymptote for
G ⬙ and ␻ 5/4 intermediate frequency asymptote for G ⬘ , respectively.
yields normal terminal frequency dependence of all components of G * ( ␻ ) at these
frequencies, and exponential decay in G(t) at corresponding times, due to the finite
length of the chain. Because existing experimental data for dilute solutions of semiflexible rods 共which are discussed in Sec. X兲 are in the regime ␻ ␶ rod ⬍ 104 , we must use the
full theory when making quantitative comparisons to experiments.
IX. BROWNIAN DYNAMICS SIMULATIONS
To test these predictions over a much wider frequency range than those accessible to
current experiments, we have carried out Brownian dynamics simulations of noninteracting, free-draining wormlike chains. We simulate discretized wormlike chains in which
each chain contains N beads with positions Ri for i ⫽ 1,...,N, which act as point sources
of frictional resistance, connected by N⫺1 rods of constant length a. We use a discretized bending energy
1140
SHANKAR, PASQUALI, AND MORSE
FIG. 5. Comparison of predictions for G ⬘ ( ␻ )) and G ⬙ ( ␻ ) as functions of ␻ ␶ rod for L/L p ⫽ 1/8, as obtained
from the full theory 共continuous black lines兲, the local compliance approximation 共dashed black lines兲, and the
analytic approximation of Sec. XI 共dashed gray lines兲.
Ubend ⫽ ⫺
␬
N⫺1
a
i⫽2
兺
ui •ui⫺1 ,
共125兲
where
ui ⬅
Ri⫹1 ⫺Ri
兩 Ri⫹1 ⫺Ri 兩
共126兲
is a unit tangent vector along rod i. For simplicity, we simulate free-draining chains with
isotropic friction at each bead, corresponding to a continuum model with ␨ 储 ⫽ ␨⬜ ⫽ ␨ ,
in the creeping flow limit, where we ignore any inertia of the chain.
The equation of motion for a chain in a flow with velocity gradient ␥˙ is
冋
␨b
dRi
dt
册
tens
rand
⫺ ␥˙ •Ri ⫽ Fi ⫽ Fbend
⫹Fmet
.
i
i ⫹Fi ⫹Fi
共127兲
Here, ␨ b ⫽ ␨ a is a bead friction coefficient, Fbend
⫽ ⫺ ⳵ U bend / ⳵ Ri is the force on bead
i
met
is a
i due to the bending of the chain, Fi is a ‘‘metric force’’ 共discussed below兲, Ftens
i
constraint force that is chosen to impose the constraints of constant rod length 共discussed
below兲, and Frand
is a random Langevin force with vanishing mean and a variance
i
rand
rand
具 Fi (t)F j (t ⬘ ) 典 ⫽ 2k B T ␨ b I␦ i j ␦ (t⫺t ⬘ ) given by the fluctuation dissipation theorem.
The stochastic equation of motion 共127兲 for the bead positions Ri is integrated numerically with a mid-step algorithm 关Grassia and Hinch 共1996兲兴 to generate bead trajectories.
is of the form
The constraint force Ftens
i
Ftens
⬅ Ti ui ⫺Ti⫺1 ui⫺1 ,
i
共128兲
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
1141
where Ti is the tension in bond i. Requiring that dtd Ri⫹1 ⫺Ri 兩 2 ⫽ 0 for each bond in the
chain, and using Eq. 共127兲 to calculate this time derivative, yields a set of linear equations
for the instantaneous tensions
N⫺1
兺
j⫽1
HijT j ⫽ ui • 共 F̃i⫹1 ⫺F̃i 兲 ,
共129兲
rand
˙
⫹Fmet
where F̃i ⬅ Fbend
i
i ⫹Fi ⫹ ␨ b ( ␥•Ri ), and H i j is an N⫻N symmetric, positive
definite, tridiagonal matrix with elements H ii ⫽ 2 and H i j ⫽ ⫺ui •u j for i ⫽ j⫾1.
The metric force Fmet
for such a free draining chain is given by the derivative
i
1
⳵ ln共det H)
⫽ ⫺ kBT
Fmet
i
2
⳵ Ri
共130兲
of the ‘‘metric pesudopotential’’ introduced by Fixman 共1978兲 where det H is the determinant of the matrix with elements H i j introduced above. This metric force must be
included in simulations of free-draining chains with constrained rod lengths, in the midstep
algorithm
used
here,
to
obtain
a
Boltzmann
distribution
exp关⫺Ubend(u1 , . . . ,uN )/k B T 兴 of rod orientations in thermal equilibrium 关Fixman
共1978兲, Hinch 共1994兲兴. The metric forces are computed using the algorithm described by
Pasquali and Morse 共2002兲.
The stress relaxation function G(t) is obtained from equilibrium simulations, with
␥˙ ⫽ 0, by using the Kubo relation that relates G(t) to the autocorrelation function of the
microscopic stress tensor
G共t兲 ⫽
1
kBT
具␴xy共t兲␴xy共0兲典,
共131兲
where ␴ ⬅ ⫺ 兺 i Ri Fi is the contribution of a single chain to the stress tensor. The
Brownian contribution to ␴共t兲 is calculated using the stochastic filtering method of Grassia and Hinch 共1996兲 and Doyle et al. 共1997兲, which avoids including large but temporally uncorrelated contributions to the stress arising from the random force.
To simulate G(t) over a wide range of time scales, we use a technique introduced by
Everaers et al. 共1999兲 and run simulations with different values of N for each value of
L/L p 共where L ⫽ Na), using simulations with relatively coarse-grained chains 共small N兲
to resolve slow relaxation processes 共e.g., rotational diffusion兲 and shorter simulations of
finer-grained chains 共large N兲 to resolve G(t) at shorter times. For each value of N,
meaningful results for G(t) are obtained only at times greater than a time proportional to
the relaxation time ␨ a 4 / ␬ of a bending mode of wavelength a, below which 关 G(t) 兴
saturates to a finite value whose existence is an artifact arising from the use of a discretized model. The initial chain conformations in each simulation are chosen from an
equilibrium Boltzmann distribution. These are generated by an algorithm in which chains
are ‘‘grown’’ from one end, by adding each new rod at an orientation chosen randomly
from a Boltzmann distribution e ⫺ ␬ ui⫹1 •ui /(ak B T) for the bending energy of the joint
between each new rod and the previous one. The use of a pre-equilibrated distribution of
initial states allows one to use data from relatively short simulations of fine-grained
chains without having to equilibrate the system initially. Results for chains with the same
L/L p but different N are collapsed onto master curves of 关 G(t) 兴 vs t/ ␶ rod , where ␶ rod
⫽ ␨ b a 2 N 3 /72k B T for the discretized WLC. An example of this collapse is shown in Fig.
6. Ensuing figures show only the regions of overlap of the results obtained with different
values of N, which reflect the behavior of a continuous wormlike chain.
1142
SHANKAR, PASQUALI, AND MORSE
FIG. 6. Simulation results illustrating collapse of simulation data for 关 G tens(t) 兴 共top curve, ⫻兲, 关 G curv(t) 兴
共middle curve, ⫹兲, and 关 G ornt(t) 兴 共bottom curve, 䊊兲 vs t/ ␶ rod with ␶ rod ⬅ ␨ b N 3 a 2 /(72k B T) for L/L p
⫽ 1/8 and N ⫽ 8, 16, 22, 32, 46, 64, 90, 128. 关 G ornt(t) 兴 is shown only for the few smallest values of N.
We decompose the stress ␴ obtained from our simulations as a sum ␴ ⫽ ␴ornt
⫹ ␴curv⫹ ␴tens⫺k B TI of orientational, curvature, and stress components 关Morse
共1998a兲兴, given by
␴ornt ⬅ 23 k B T 共 u1 u1 ⫹uN⫺1 uN⫺1 ⫺2I/3兲 ,
N
␴curv ⬅ ⫺
兺
i⫽1
共132兲
N⫺1
Ri Fbend
⫹3k B T
i
兺
i⫽1
共 ui ui ⫺I/3兲 ⫺ ␴ornt ,
␴tens ⬅ ␴⫺ ␴ornt⫺ ␴curv⫹k B TI.
共133兲
共134兲
Corresponding, 关 G(t) 兴 ⫽ 关 G ornt(t) 兴 ⫹ 关 G curv(t) 兴 ⫹ 关 G tens(t) 兴 , where 关 G ␣ (t) 兴 , with
␣ ⫽ ‘‘ornt,’’ ‘‘curv’’, or ‘‘tens,’’ describes the decay of the stress component 具 ␴␣ (t) 典
after a small step deformation. These partial intrinsic moduli are calculated from the
Kubo relation
关G␣共t兲兴 ⫽
1
kBT
具␴␣,xy共t兲␴xy共0兲典,
共135兲
which cross correlates components of the single-chain stress with the total.
Figure 7 shows the master curves obtained for the components
关 G tens(t) 兴 , 关 G curv(t) 兴 , and 关 G ornt(t) 兴 for chains with L/L p ⫽ 1/8, 1/4, 1/2, and 1,
respectively. Also shown, as continuous gray lines, are the predictions of the full theory
for these functions. Dashed lines represent the predicted asymptotes for 关 G tens(t) 兴 at
early and intermediate times 关Eqs. 共82兲 and 共84兲兴. Figure 8 shows simulation results and
predictions for the total intrinsic modulus 关 G(t) 兴 共rather than the individual components兲
for L/L p ⫽ 1/8 and L/L p ⫽ 1. The theoretical predictions were obtained by numerically
Fourier transforming the results of the full theory for the tension and curvature components to 关 G * ( ␻ ) 兴 , using isotropic friction coefficients, ␨ 储 ⫽ ␨⬜ ⫽ ␨ b /a, as in the simulations.
Consider first the stiffest chains simulated, with L/L p ⫽ 1/8. For these, we obtain
excellent agreement between theory and simulations over the entire range of time scales
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
1143
FIG. 7. Comparison of theory and simulations for different values of L/L p : Data points 共black symbols兲
represent simulation results for 关 G tens(t) 兴 共top curve, ⫻兲, 关 G curv(t) 兴 共middle curve, ⫹兲, and 关 G ornt(t) 兴 共bottom
curve, 䊊兲, plotted vs t/ ␶ rod with ␶ rod ⬅ ␨ b N 3 a 2 /(72k B T). Continuous gray lines are our theoretical results for
these three components G(t). Short dashed gray lines are predicted asymptotic power laws as indicated.
accessible to simulations. This set of simulations clearly show a broad intermediate
regime where 关 G tens(t) 兴 ⬃ t ⫺5/4, but does not access the t ⫺3/4 decay predicted at extremely early times, which was computationally inaccessible. A clear orientational plateau, with 关 G(t) 兴 ⯝ 关 G ornt(t) 兴 ⯝ 53 k B Te ⫺t/ ␶ rod, is also visible at times approaching
FIG. 8. Comparison between theoretical prediction 共gray lines兲 and the results of Brownian dynamics simulations 共black symbols兲 for the intrinsic modulus 关 G(t) 兴 vs t/ ␶ rod , for L/L p ⫽ 1/8 共left panel兲 and L/L p ⫽ 1
共right panel兲.
1144
SHANKAR, PASQUALI, AND MORSE
␶ rod . As L/L p is increased from 1/8 to 1/4, 1/2, and 1 in the remaining simulations, the
intermediate regime gradually disappears, as ␶ 储 共rapidly兲 and ␶⬜ 共more slowly兲 approach
␶ rod , while the predicted t ⫺3/4 regime moves into the computationally accesible window.
For the most flexible chains shown, with L/L p ⫽ 1, both the intermediate regime and the
orientational plateau are absent, but the simulation results begin to closely approach the
early time t ⫺3/4 asymptote. The predictions of the full theory become noticeably less
accurate with increasing L/L p , as expected for a theory that is based upon an expansion
about a rigid rod reference, but remain remarkably accurate for chains of length up to
L ⫽ L p . For the most flexible chains, with L ⫽ L p , the predictions of the individual
components of 关 G(t) 兴 are noticeably less accurate at times approaching ␶ rod than the
predictions of the total 关 G(t) 兴 , indicating a partial compensation of errors.
Simulations very similar to those discussed above and in our earlier report 关Pasquali
et al. 共2001兲兴 have also been carried out by Dimitrakopoulos et al. 共2001兲. These authors
reported that their simulation results for 关 G(t) 兴 could be adequately fit over a wide range
of intermediate times by a single power-law decay 关 G(t) 兴 ⬀ t ⫺ ␣ , with an apparent
exponent ␣ that varies continuously with L/L p and approaches ␣ ⫽ 5/4 for L/L p Ⰶ 1.
This description is broadly consistent with the simulation data of both groups: For example, the simulation data for 关 G(t) 兴 for L/L p ⫽ 1 in Fig. 8 is fit well by a t ⫺7/8 power
law, as reported by these authors. The theory shows, however, that this is an empirical
description of a broad crossover in both time and L/L p , which can be accurate only at
intermediate values of L/L p , since universal power laws are predicted for L Ⰶ L p and
L Ⰷ L p , and only in an intermediate range of computationally accessible times, since a
universal t ⫺3/4 decay is expected at early times for all L/L p .
X. COMPARISON WITH EXPERIMENTS
In this section, we compare our predictions with the experimental data of Warren et al.
共1973兲 and Ookubo et al. 共1976兲, who carried out linear viscoelastic measurements of
G ⬘ ( ␻ ) and G ⬙ ( ␻ ) for dilute solutions of PBLG in the solvent m-Cresol. The persistence
length of PBLG is approxiamtely 150 nm, though there does not appear to be a consensus
in the literature on the exact value; reported estimates range from 100 to 180 nm. The
average lengths of the chains used in the experiments of Warren et al. 共1973兲 include
L ⫽ 108 and 162 nm, while Ookubo et al. 共1976兲 used chains with L ⫽ 116, 82, and 51
nm. The chain lengths in most of these experiments are thus comparable to the persistence length. Although our theory is constructed so as to be accurate only in the limit
L/L p Ⰶ 1, a comparison between the theory and these experiments seems reasonable in
light of the level of agreement found above between the theory and Brownian dynamics
simulations of chains with comparable values of L/L p .
A. Experiments of Warren et al.
Warren et al. 共1973兲 used a multiple-lumped resonator to measure 关 G * ( ␻ ) 兴 of dilute
solutions of PBLG in m-Cresol with molecular weights ranging from 16⫻10 4 to 57
⫻104 in the frequency range 106 – 6060 Hz and concentration range 0.0015–0.005 g/ml.
They reported intrinsic moduli data, which were obtained by extrapolation to infinite
dilution, for three samples containing chains of length 108, 162, and 387 nm. We compare the theory only to data from the samples with L ⫽ 108 and 162 nm, since L significantly exceeds L p for the third sample. Warren et al. 共1973兲 report a polydispersity index
M w /M n ⫽ 1.234 for these samples.
We take into account polydispersity in the theory by calculating an average
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
1145
FIG. 9. Comparison with the experiments of Warren et al. 共1973兲: Intrinsic storage modulus 关 G ⬘ 兴 /(k B T) and
loss modulus 关 G ⬙ 兴 /(k B T) as a function of reduced frequency ␻ ␶ rod for two different 共average兲 lengths L ⫽ 108
nm, L ⫽ 162 nm. ␶ rod for the experiments is determined from ␲ ␩ s L 3 / 关 18k B T ln(L/d)兴 where d ⫽ 2.5 nm.
Symbols 䊊 共L ⫽ 162 nm兲 and 䊐 共L ⫽ 108 nm兲 are the data from Warren et al. 共1973兲 and the lines are
theoretical predictions for L p ⫽ 130 nm. Also shown as a dash-dotted line is the predicted 关 G ⬘ ( ␻ ) 兴 for true
rigid rods.
G共t兲 ⫽
冕
⬁
0
dL ␯共L兲关G共t;L兲兴,
共136兲
where 关 G(t;L) 兴 is the predicted intrinsic modulus for chains with length L, and ␯ (L)dL
is the number of chains per unit volume with contour length between L and L⫹dL. We
assume a distribution of the form
␯共L兲 ⬀
冉冊
L ␣
exp关⫺L/L0兴,
L0
共137兲
where L 0 is chosen to obtain a weight averaged length equal to the reported value. We
use an exponent ␣ ⫽ 3, which yields a polydispersity index of 1.25 very close to the
reported value.
In Fig. 9, we compare theoretical predictions and the data reported by Warren et al.
共1973兲 共digitized from their Fig. 4兲 for the ratio 关 G * ( ␻ ) 兴 /(k B T) vs ␻ ␶ rod . We have
followed Warren et al. in plotting data for chains with two different lengths 共L ⫽ 108 nm
and L ⫽ 162 nm兲 in a single graph, as in their Fig. 4, because the resulting data very
nearly superpose for these two samples. Theoretical predictions for both reported chain
lengths are calculated using a persistence length L p ⫽ 130 nm, with anisotropic friction
coefficients ␨⬜ ⫽ 2 ␨ 储 . Frequencies in the theoretical curves have been rescaled using
␶ rod ⫽ ␨⬜ L 3 /(72k B T), where L is the length averaged chain length. The experimental
data in this and all other plots have been made dimensionless using the rotational diffusion time ␶ rod ⫽ ␲ ␩ s L 3 / 关 18k B T ln(L/d)兴 predicted by slender-body hydrodynamics,
where L is the reported weight-averaged chain length, using the hydrodynamic chain
diameter of d ⫽ 2.5 nm reported by Warren et al. 共1973兲 and the reported solvent viscosity. Also shown in this figure is the storage modulus predicted by the rigid rod theory
1146
SHANKAR, PASQUALI, AND MORSE
FIG. 10. Effect of variations of L p on the theoretical prediction 共lines兲, and on the comparison with the data
共䊐兲 of Warren et al. 共1973兲: 关 G ⬘ 兴 /(k B T) and 关 G ⬙ 兴 /(k B T) vs ␻ ␶ rod for L ⫽ 108 nm, for L p ⫽ 110, 130, and
150 nm.
共corrected for polydispersity兲 which agrees well with the data in the terminal regime but
clearly fails at higher frequencies. Like the experimental data, our predictions for the two
different chain lengths are only slightly different, despite the 50% difference in L, indicating that both the predicted and measured shape of the curves in this representation
change only slowly with changes in the ratio L/L p for L ⯝ L p . The same point is
evident in Fig. 10, where we show the effect of varying L p from 110 to 150 nm on the
theoretical predictions for L ⫽ 108 nm. Aside from the uncertainty introduced by the
existence of a substantial range of estimated values for the persistence length of PBLG in
the literature, which seems to have a weak effect on our predictions, this comparison with
experiment contains no adjustable parameters.
B. Experiments of Ookubo et al.
Ookubo et al. 共1976兲 used a torsional free decay method to measure linear viscoelastic
measurements of dilute PBLG solutions in m-Cresol at concentrations 0.002–0.05 gm/ml
in a frequency range 2.2⫻103 – 5.25⫻105 Hz, giving a maximum frequency an order of
magnitude larger than that obtained by Warren et al. 共1973兲. These authors, who reported
measurements of the complex viscosity ␩ * ( ␻ ) ⬅ ␩ ⬘ ( ␻ )⫺i ␩ ⬙ ( ␻ ) ⫽ G * ( ␻ )/(i ␻ ), reported that their accuracy for ␩ ⬘ ( ␻ ) ⫽ G ⬙ ( ␻ )/ ␻ is higher than that for ␩ ⬙ ( ␻ )
⫽ G ⬘ ( ␻ )/ ␻ , and that the lack of accuracy of their data for ␩ ⬙ ( ␻ ) made extrapolation to
infinite dilution to impossible for this component. We thus consider the data of Ookubo
et al. 共1976兲 to be less reliable than that of Warren et al. 共1973兲. We nevertheless have
compared the theory with this data because it contains data for both shorter chain lengths
and significantly higher frequencies than those reported by Warren et al. 共1973兲.
Ookubo et al. 共1976兲 carried out measurements on three fractionated samples, with
lengths L ⫽ 116, 82, and 51 nm, and one unfractionated sample, which we do not consider. No value for the polydispersity index is reported, and so, in the absence of a better
estimate, we assume a polydispersity of 1.25 similar to that reported by Warren et al.
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
1147
FIG. 11. Comparison with the experiments of Ookubo et al. 共1976兲 for different values of L: Intrinsic storage
modulus 关 G ⬘ 兴 /(k B T) and loss modulus 关 G ⬙ 兴 /(k B T) as a function of ␻ ␶ rod for three different chain lengths
indicated above, with at several concentrations per chain length. Symbols are the data from Ookubo et al.
共1976兲 and the lines are from our theory with L p ⫽ 130 nm. Unfilled symbols represent storage modulus and
filled symbols represent loss modulus.
Figures 11共a兲–11共c兲 show the values of 关 G ⬘ 兴 /(k B T) and 关 G ⬙ 兴 /(k B T) vs ␻ ␶ rod obtained
for these three chain lengths at all of the reported concentration. The intrinsic moduli in
these graphs are calculated from the digitized plots of Ookubo et al. 共1976兲 for ␩ ⬙ ( ␻ )
and ␩ ⬘ ( ␻ ) by dividing the resulting G ⬘ ( ␻ ) and G ⬙ ( ␻ )⫺i ␻ ␩ s by the actual concentration, rather than by extrapolating to infinite dilution. Values of ␶ rod are calculated, as
before, using reported chain lengths and solvent viscosity of ␩ s ⫽ 0.105 P. The solid
lines are corresponding theoretical predictions for the intrinsic moduli, calculated for
L p ⫽ 130 nm, for these three chain lengths. The lack of a smooth variation of the data
for 关 G ⬘ ( ␻ ) 兴 with concentration explains the authors’ conclusion that this data cannot
support a meaningful extrapolation to infinite dilution. Figures 11共a兲–11共c兲 indicate that
there is nonetheless good agreement between theoretical and experimental results for
1148
SHANKAR, PASQUALI, AND MORSE
FIG. 12. Attempted data collapse of intrinsic storage and loss modulus data reported in Warren et al. 共1973兲
共L ⫽ 108, 162 nm兲 and Ookubo et al. 共1976兲 共L ⫽ 116 nm兲 as a function of ␻ ␶ rod . Symbols denote the data
at different concentrations of Ookubo et al. 共1976兲 and the two different lengths of Warren et al. 共1973兲. All
unfilled symbols represent storage modulus data and filled symbols represent loss modulus data. Solid lines are
our theoretical predictions for L ⫽ 116 nm and L p ⫽ 130 nm. The dotted line is the predicted 关 G ⬘ ( ␻ ) 兴 /(k B T)
for rigid rods.
关 G ⬙ ( ␻ ) 兴 and 共in light of the evident experimental uncertainties兲 reasonable agreement
for 关 G ⬘ ( ␻ ) 兴 at all three chain lengths.
In Fig. 12, we have combined the infinite dilution data of Warren et al. 共1973兲 with
L ⫽ 108 and 162 nm, which have already been shown to nearly superpose, together with
the data of Ookubo et al. at several concentrations for the fraction with L ⫽ 116 nm. In
general, there is good collapse of these three data sets, over a combined frequency range
of nearly four decades, which is substantially wider than that obtained in any single
measurement. Theoretical predictions for L ⫽ 116 nm and L p ⫽ 130 nm agree well with
this data over this entire range.
XI. AN ANALYTIC APPROXIMATION
The LCA results for the stress relaxation modulus G(t) and related quantities such as
the single chain compliance ␹共t兲 agree well with the results of the full theory at short
times, less than the longest transverse relaxation time ␶⬜ / ␣ 41 , but fail at longer times
because the approximation uses a continuous distribution of Fourier modes rather than
discrete set of bending modes, and thus contains no terminal relaxation time. As a result,
the LCA predicts power law decays for the tension and curvature contribution to G(t)
even at times t Ⰷ ␶⬜ , whereas the full theory predicts exponential decay of these components with relaxation times proportional to ␶⬜ / ␣ 41 .
A physically motivated approximation to the full theory can thus be obtained by taking
the LCA results for these components of G(t) and multiplying them by an exponential
cutoff, i.e., by approximating
G␣共t兲 ⯝ G␣,LCA共 t 兲 e ⫺t/ ␶ ␣ ,
共138兲
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
1149
where G ␣ ,LCA(t) is the LCA prediction for component ␣ of G(t), and ␶ ␣ is a time
proportional to ␶⬜ . Corresponding analytic approximations for the components of
G * ( ␻ ) can be obtained by using the following property of Fourier transforms: If
G共t兲 ⫽ F共t兲e⫺t/␶,
共139兲
⫺i ␻ t of G(t) is given by
then the one-sided Fourier transform G̃( ␻ ) ⬅ 兰 ⬁
0 dtG(t)e
G̃共␻兲 ⫽
冕
⬁
0
dt F共t兲e⫺t/␶e⫺i␻t ⫽ F̃共␻⫺i␶⫺1兲,
共140兲
i.e., by the analytic continuation of the one-sided transform F̃( ␻ ) of F(t) to a complex
frequency ␻ ⫺i ␶ ⫺1 . The one-sided Fourier transform of G(t) is given by the ratio
⫺i ␻ t , so this prescription requires us to analytically
G̃( ␻ ) ⬅ G * ( ␻ )/(i ␻ ) ⫽ 兰 ⬁
0 dtG(t)e
continue the LCA predictions for components of G * ( ␻ )/(i ␻ ), rather than of G * ( ␻ ).
We obtain analytic approximations for the tension and curvature components of
* ( ␻ )/i ␻ and
G * ( ␻ ) by evaluating the analytic LCA predictions for the components G tens
* ( ␻ )/i ␻ , and for the ratio G curv,l
* ( ␻ )/G *
G curv,t
tens( ␻ ), at complex frequencies with imagi⫺1
nary parts proportional to ⫺ ␣ 41 ␶⬜
. This yields
* 共 ␻ 兲兴 ⯝
关Gtens
1
15
关G*
curv,l共 ␻ 兲兴 ⯝
2
冋
LB 共 ␻ 兲 1⫺
3
L
5/4
Lp
关G*
curv,t共 ␻ 兲兴 ⯝
␭ 共 ␻ 1 兲 /2
␻1
,
共 i ␻ 2 ␶⬜ 兲 ⫺1/4关 G *
tens共 ␻ 兲兴 ,
3
2
册
tanh关␭共␻1兲/2兴 ␻
3/4
10
共 i ␻ 3 ␶⬜ 兲 1/4
␻
␻3
共141兲
,
where ␭( ␻ 1 ) ⬅ 关 i ␻ 1 ␨ 储 L 2 /B( ␻ 1 ) 兴 1/2 is calculated by evaluating Eq. 共7兲 for B( ␻ ) at a
complex frequency ␻ 1 , and in which ␻ 1 , ␻ 2 , and ␻ 3 are three complex frequencies, of
the form
⫺1
␻i ⬅ ␻⫺iCi␣40␶⬜
共142兲
with i ⫽ 1,.2,3 with different numerical constants C 1 ,C 2 , and C 3 . These numerical
constants are treated as fitting parameters, which are adjusted to optimize the fit of the
resulting approximation to the results of the full theory. The choice C 1 ⫽ 0.14,
C 2 ⫽ 0.72, and C 3 ⫽ 1.26 yields an approximation of the total loss and storage moduli
that differs from the results of full theory by less than 6% at any frequency for
L/L p ⫽ 1/8, 1/4, 1/2, and 1. The approximate storage and loss moduli obtained for
L/L p ⫽ 1/8 with these constants are compared with the results of the full theory in Fig.
5.
XII. RELATION TO THE HARRIS AND HEARST MODEL
We now compare our theoretical predictions for linear viscoelastic moduli to those of
Harris and Hearst 共1966兲 and Hearst et al. 共1966兲 共HH兲, who also attempted to calculate
linear viscoelastic moduli for dilute solutions of wormlike chains. These authors considered a generalized Gaussian approximation for a wormlike chain, in which the equilibrium distribution of contour R共s兲 is controlled by an effective potential energy
1150
SHANKAR, PASQUALI, AND MORSE
U⫽
1
2
冕
L
0
冋冉 冊 冉 冊册
ds ␬
⳵2R
2
⫹T
⳵s2
2
⳵R
⳵s
,
共143兲
in which T is a Lagrange multiplier introduced to impose approximately the constraint of
constant length. In this model, however, T is not treated as a fluctuating field, but as a
constant, which is independent of both s and t. The value of T for a given chain length L
is chosen to yield an equilibrium mean-squared end-to-end distance equal to that of a true
wormlike chain of equal length. Harris and Hearst 共1966兲 find that, in the rodlike limit
L Ⰷ L p of interest here, this criterion yields a value T ⯝ 3k B T/L. While the HH model
can be applied to chains of arbitrary length, we consider only its predictions for the
rodlike limit, L Ⰶ L p .
The dynamical equation used by Harris and Hearst 共1966兲 is identical to our Eq. 共2兲,
except for the crucial fact that HH treat the tension T as a constant, independent of s, t,
and the state of flow. Their model thus does not allow any tension to be induced in the
chain by flow. HH expand all three Cartesian components of the chain contour R共s兲 in
eigenfunctions of the eigenvalue problem
⳵4W j
⳵ ŝ 4
⫺⌳
⳵ 2 Wj
⳵ ŝ 2
⫽ ␣ 4j W j ,
共144兲
with ⌳ ⬅ T L 2 / ␬ , with corresponding boundary conditions
⳵2W j
⳵ ŝ
2
⫽ 0,
⳵ 3 Wj
⳵ ŝ
3
⫺⌳
⳵ Wj
⫽0
⳵ ŝ
共145兲
at both chain ends. Equations 共144兲 and 共145兲 reduce to our Eqs. 共101兲 and 共102兲 in the
limit ⌳ → 0. In the rodlike limit, L Ⰶ L p , where the coefficient ⌳ ⯝ 3L/L p is small,
the presence of the second derivative term in Eq. 共144兲 has a significant effect only on the
smallest eigenvalue ␣ 0 , which vanishes when ⌳ ⫽ 0. We find by a perturbation analysis
of Eqs. 共144兲 and 共145兲 that the presence of a small nonzero ⌳ ⫽ 3L/L p splits the
degeneracy between the two zero modes of our Eq. 共100兲, yielding a vanishing eigenvalue for the translation mode, with W ⬀ const., but producing a small nonzero eigenvalue ␣ 40 ⯝ 36L/L p for the eigenvector W ⬀ (ŝ⫺1/2) that, in Eq. 共100兲, represents rigid
rotations.
In the present notation, the Harris and Hearst 共1966兲 result 关their Eq. 共55兲兴 for the
complex modulus is
⬁
* 共 ␻ 兲兴 ⫽ k B T
关GHH
i␻
兺
⫺1 4 .
␣
i ⫽ 0 i ␻ ⫹2 ␶
⬜
共146兲
i
In the rodlike limit, the eigenvalues ␣ i can be approximated by those obtained with
⌳ ⫽ 0, except for the i ⫽ 0 mode, for which ␣ 40 ⫽ 36L/L p . Comparing the relaxation
⫺1 4
⫺1
␣ 0 of the contribution arising from the i ⫽ 0 mode to the relaxation rate ␶ rod
rate 2 ␶⬜
for the orientational stress in a solution of rods shows that they are equal
⫺1
⫺1 4
␶rod
⫽ 2 ␶⬜
␣0 ⫽
72k B T
␨⬜ L 3
.
共147兲
Separating the contribution of the i ⫽ 0 mode from the remaining sum in Eq. 共146兲 yields
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
* 共 ␻ 兲兴 ⫽ k B T
lim 关 G HH
L Ⰶ Lp
i␻
⬁
兺
⫺1 ⫹k B T
i ␻ ⫹ ␶ rod
i⫽1
i␻
⫺1 4
i ␻ ⫹2 ␶⬜
␣i
1151
.
共148兲
The first term on the rhs of Eq. 共148兲 resembles Eq. 共93兲 for 关 G *
ornt( ␻ ) 兴 , and the remain(
␻
)
.
However,
our Eqs. 共93兲 and
ing sum over i ⭓ 1 resembles Eq. 共121兲 for 关 G *
兴
curv,t
共121兲 each contain a prefactor of 3/5 that is absent from either term on the rhs of Eq.
共148兲; thus, the HH result for 关 G * ( ␻ ) 兴 in the rodlike limit is related to our results by
3
5
* 共 ␻ 兲兴 ⫽ 关 G ornt
* 兴⫹关G*
关GHH
curv,t兴 ,
共149兲
where the two terms on the rhs are given by Eqs. 共93兲 and 共121兲, respectively.
This relationship shows that, in the rodlike limit, the Harris–Hearst model neglects the
two contributions to 关 G * ( ␻ ) 兴 that dominate at intermediate and high frequencies, viz.
* 兴 , which both arise from flow induced tension, while retaining con关G*
tens兴 and 关 G curv,l
* 兴 , which are
tributions that 共aside from a different prefactor兲 resemble 关 G *
curv,t兴 and 关 G ornt
subdominant at these frequencies. As a result, the HH model predicts storage and loss
⫺1
* ( ␻ ) 兴 , and so enormously
moduli that increase as ␻ 1/4 at ␻ Ⰷ ␶⬜
, like our 关 G curv,t
⫺1
⫺1
underestimates 关 G ⬘ ( ␻ ) 兴 at all ␻ Ⰷ ␶⬜ and 关 G ⬙ ( ␻ ) 兴 at all ␻ Ⰷ ␶ rod
, as shown in Fig.
13. The model’s predictions are less egregiously wrong at lower frequencies: At ␻
⫺1
Ⰶ ␶⬜
, where 关 G ⬘ ( ␻ ) 兴 is dominated by the orientational stress, the model correctly
⫺1
, but
predicts a Maxwellian behavior for 关 G ⬘ ( ␻ ) 兴 , with the correct relaxation rate ␶ rod
with a numerical prefactor that is too large by 5/3. Interestingly, the model predicts the
correct intrinsic zero shear viscosity of 关 ␩ 0 兴 ⫽ k B T ␶ rod for a model of rods with isotropic friction, because the use of too large a prefactor for the ‘‘orientational’’ contribution
to ␩ 0 共i.e., the i ⫽ 0 mode兲 is exactly compensated by the absence of a tension contribution. This success is a consequence of the fact that the zero shear viscosity of such a
free-draining model depends only on the polymer’s equilibrium radius of gyration R g
关see Eq. 共16.3-20兲 of Bird et al. 共1987兲兴, and that the model gives the correct limiting
value for R g in the rodlike limit.
XIII. CONCLUDING REMARKS
This paper presents a theory that describes accurately the linear viscoelastic response
of dilute solutions of freely draining semiflexible rods, with lengths L smaller than their
persistence lengths L p , over the whole range of possible frequency and time scales. The
theory treats the inextensible wormlike chain as an effectively extensible rod with an
effective longitudinal compliance that arises from the existence of transverse thermal
fluctuations. A simplified, analytically solvable local compliance approximation, which
ignores the spatial nonlocality of the relationship between the average tension and strain
fields, describes accurately the viscoelastic behavior throughout the intermediate and high
frequency regimes in which the predicted behavior differs significantly from that of rigid
rods.
In the limit of very stiff chains, the theory predicts a stress relaxation modulus G(t)
that decays as t ⫺3/4 at very early times, as found previously by Morse 共1998b兲 and Gittes
and MacKintosh 共1998兲, but that decays as t ⫺5/4 over a range of intermediate times ␶ 储
⬍ t ⬍ ␶⬜ , which broadens rapidly as L/L p decreases. At times larger than the relaxation time ␶⬜ of the longest bending mode, the theory predicts an exponential decay
G(t) ⬀ e ⫺t/ ␶ rod identical to that found for rigid rods. This description is accurate, however, only for very stiff rods: As L approaches L p from below, the times ␶ 储 , ␶⬜ , and ␶ rod
1152
SHANKAR, PASQUALI, AND MORSE
FIG. 13. Comparison of the present theoretical predictions with the theoretical results of Harris and Hearst
共1966兲 共denoted as HH in the figure兲: Intrinsic storage and loss moduli vs ␻ ␶ rod for L/L p ⫽ 1/8.
approach one another, and hence both the intermediate t ⫺5/4 decay and the rigid rod
orientational plateau disappear gradually. An initial t ⫺3/4 decay of G(t) is expected for
any value of L/L p , but only below a time ␶ 储 ⬀ ␶ rod(L/L p ) 5 that remains much lower
than ␶ rod even for L ⬃ L p and that drops with decreasing L/L p to values that, for L
Ⰶ L p , rapidly become inaccessible to either our simulations or experiment. The intermediate time t ⫺5/4 decay in G(t), and the corresponding intermediate ␻ 5/4 frequency
dependence of G ⬘ ( ␻ ), is observable at more easily accessible times and frequencies, but
is well defined only for very stiff chains. In light of the resulting difficulties facing any
attempt compare our asymptotic power law predictions directly to experiment, we wish to
emphasize that the full theory and the analytic approximation of Sec. XI both provide
accurate predictions over much wider ranges of frequency and reduced chain length L/L p
than those provided by asymptotic analysis alone.
In the opposite limit of a dilute solutions of coillike chains, with lengths much larger
than a few persistence lengths, we expect a t ⫺3/4 decay at very early times followed by
a Rouse-like t ⫺1/2 decay at longer times for models of free-draining chains, or a Zimmlike decay for very long chains with hydrodynamic interactions. Thus, with the results of
the present study, theoretical understanding of the linear viscoelasticity of dilute solutions
of freely draining wormlike chains is nearly complete for the whole range of L/L p from
rigid rods (L/L p → 0) to random coils (L/L p Ⰷ 1), except for a small 共in a logarithmic
sense兲 crossover region in which L is somewhat larger than L p , where there must be a
crossover from the behavior described here to a Rouse–Zimm behavior.
Comparison of the theoretical predictions to the results of our Brownian dynamics
simulations of stress relaxation show striking quantitative agreement over roughly nine
orders of magnitude in time. Predictions of G(t) remain reasonably accurate for chains of
length up to L ⫽ L p , despite the expansion about a rigid-rod reference state that is used
throughout the derivation of the theory. We have also compared the theory with the
available linear viscoelastic data for dilute solutions of rodlike poly 共␥-benzyl-Lglutamate兲 in m-Cresol 关Warren et al. 共1973兲, Ookubo et al. 共1976兲兴. These data are for
chains with L/L p ⯝ 0.4– 1.2 at reduced frequencies of ␻ ␶ rod ⯝ 10⫺1 – 104 , which is a
crossover region in which none of the asymptotic power laws are valid. The predictions
of the full theory are nonetheless found to be in good quantitative agreement with these
experiments, with essentially no adjustable parameters.
LINEAR VISCOELASTICITY OF SEMI-FLEXIBLE RODS
1153
ACKNOWLEDGMENTS
This work was supported by NSF Grant No. DMR-9973976 and by the Minnesota
Supercomputing Institute.
APPENDIX
In this appendix, we calculate the response function ⌰(s, ␻ ) defined in Eq. 共28兲,
which describes the direct response of 具 E(s, ␻ ) 典 to the velocity gradient that appears in
transverse dynamical Eq. 共24兲. Here, as in the full theory of Sec. VIII, we use eigenvector
expansion 共100兲 for h ␣ (s), expansion 共105兲 of the transverse dynamical equation, and
expansion 共107兲 of the average strain field. A calculation similar to that used to obtain Eq.
共120兲 shows that the velocity gradient term in Eq. 共105兲 introduces a contribution to
a kl ( ␻ ) given by
akl共␻兲 ⫽ ␦kl
kBTL3
␬␣4k
i␻␨⬜
兺 ␥␣␣共␻兲.
⫺1
i␻␨⬜⫹2␣4k ␶⬜
␣
共A1兲
Substituting this into Eq. 共107兲, and using the expression 兺 ␣ ␥ ␣␣ ( ␻ ) ⫽ ␥( ␻ ):(I⫺nn)
⫽ ⫺ ␥( ␻ ):nn for traceless ␥共␻兲 yields a strain of the form 具 E(s, ␻ ) 典
⫽ ⌰(s, ␻ ) ␥( ␻ ):nn, where
⌰共s,␻兲 ⫽
L
兺
Lp k
冉 冊
⳵Wk
⳵ ŝ
2
1
.
␣ 4k 关 1⫹2 ␣ 4k 共 i ␻ ␶⬜ 兲 ⫺1 兴
共A2兲
The rhs of Eq. 共A2兲 is a factor of L/L p times a dimensionless function of ŝ and ␻ ␶⬜
alone, which is defined by the sum. This sum increases linearly with i ␻ ␶⬜ for ␻ ␶⬜
⫺1
Ⰶ 1, and approaches a finite limit for ␻ Ⰷ ␶⬜
. In the rodlike limit L/L p
Ⰶ 1,⌰(s, ␻ ) is thus always small compared to unity as a result of the overall prefactor
of L/L p . This justifies our neglect of this contribution in the main text, as discussed in
Sec. III.
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