Minimal model of relaxation in an associating fluid: viscoelastic and dielectric relaxations in equilibrium polymer solutions.

Abstract

Cluster formation and disintegration greatly complicate the description of relaxation processes in complex fluids. We systematically contrast the viscoelastic and dielectric properties for models of equilibrium polymers whose thermodynamic properties have previously been established. In particular, the monomer-mediated model allows chain growth to proceed only by monomer addition, while the scission-recombination model enables all particles to associate democratically, so that chain scission and fusion occur at the interior segments as well as at chain ends. The minimal models neglect hydrodynamic and entanglement interactions and are designed to explore systematically the competition between chemical reaction and internal chain relaxation and how this coupling modifies the dynamics from that of a polydisperse solution of Rouse chains with fixed lengths (i.e., "frozen" chains). As expected, the stress relaxation is nearly single exponential when the assembly-disassembly reaction is fast on the time scale of structural chain rearrangements, while multiexponential or nearly stretched exponential relaxation is obtained when this reaction rate is slow compared to the broad relaxation spectrum of almost unperturbed, nearly "dead" chains of intrinsically polydisperse equilibrium polymer solutions. More generally, a complicated intermediate behavior emerges from the interplay between the chemical kinetic events and internal chain motions.

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Minimal model of relaxation in an associating fluid: Viscoelastic
and dielectric relaxations in equilibrium polymer solutions
Evgeny B. Stukalina?and Karl F. Freed
The James Franck Institute, University of Chicago, Chicago, Illinois 60637
?Received 7 September 2006; accepted 6 October 2006; published online 13 November 2006?
Cluster formation and disintegration greatly complicate the description of relaxation processes in
complex fluids. We systematically contrast the viscoelastic and dielectric properties for models of
equilibrium polymers whose thermodynamic properties have previously been established. In
particular, the monomer-mediated model allows chain growth to proceed only by monomer addition,
while the scission-recombination model enables all particles to associate democratically, so that
chain scission and fusion occur at the interior segments as well as at chain ends. The minimal
models neglect hydrodynamic and entanglement interactions and are designed to explore
systematically the competition between chemical reaction and internal chain relaxation and how this
coupling modifies the dynamics from that of a polydisperse solution of Rouse chains with fixed
lengths ?i.e., “frozen” chains?. As expected, the stress relaxation is nearly single exponential when
the assembly-disassembly reaction is fast on the time scale of structural chain rearrangements, while
multiexponential or nearly stretched exponential relaxation is obtained when this reaction rate is
slow compared to the broad relaxation spectrum of almost unperturbed, nearly “dead” chains of
intrinsically polydisperse equilibrium polymer solutions. More generally, a complicated
intermediate behavior emerges from the interplay between the chemical kinetic events and internal
chain motions. © 2006 American Institute of Physics. ?DOI: 10.1063/1.2378648?
I. INTRODUCTION
Molecular systems may self-assemble to form complex
equilibrium structures that are subject to the directional in-
teraction encoded in the intermolecular potentials of the as-
sembling particles.1,2Beyond the basic problem of under-
standing the thermodynamic characteristics for this type of
transition, there is a basic need for understanding the dy-
namical properties of associating fluids from a fundamental
viewpoint.
Equilibrium polymerization is a common physical pro-
cess in which monomers or n-mers form linear chains at
equilibrium. The chains are “dynamic” in the sense that the
polymers are constantly forming and disintegrating. These
associating systems are also termed “living” polymer sys-
tems in contrast to systems where the mass distribution is
arrested on experimentally accessible time scales and conse-
quently where macroscopic properties are determined by an
average over the frozen polydisperse mass distribution. Liv-
ing polymer systems exhibit rather rich and nontrivial equi-
librium and dynamic properties that vary greatly with the
degree of aggregation. These properties include the molecu-
lar size distribution, the morphologies of the aggregates, and
dynamic properties, such as the viscoelasticity, self-
diffusion, and dielectric permittivity. Examples of reversible
equilibrium aggregating systems include flexible linear mac-
romolecules such as poly-?-methylstyrene,3,4giant micelles
comprised of surfactant molecules,5some inorganic com-
pounds ?sulfur6and selenium7?, and protein filaments3,8?ac-
tin filaments and microtubules?. Association is known to de-
termine the stability of micellar systems, ionic fluids, and
nanoparticle dispersions. Aggregating systems provide im-
portant candidates for synthesis and molecular manipulation
to produce complex structures with new properties.3,9How-
ever, our understanding of the impact of self-assembly on the
macroscopic dynamic properties of aggregating systems is
still quite limited.
Thermodynamic theory considers the time average prop-
erties of reversibly aggregating systems, such as the average
number L of monomers in the chains, the fractions ?1of
monomers in the associated state, and the temperature depen-
dence of the time average of these quantities. The rate at
which these aggregation-dissociation processes occur is evi-
dently important for determining the dynamical properties of
these fluids. Recent simulations have established the impor-
tance of the persistence of association in understanding the
dynamic properties of aggregating polymer solutions at
equilibrium,10and this phenomenon requires greater illumi-
nation. Cates and co-workers have recently addressed similar
issues based on a model that incorporates phenomenological
reptation concepts with a particular model for the reaction
kinetics. Contact is made with this work below, and common
features are found along with distinct differences.
Equilibrium polymerization actually connotes a family
of models involving different constraints on the association-
dissociation dynamics. The general classification scheme has
been introduced by Tobolsky and Eisenberg, who pioneered
the theory of equilibrium polymerization.11These thermody-
namic models include the I model in which monomers only
add to the chain ends in the presence of an “initiator” species
a?Electronic mail: stukalin@uchicago.edu
THE JOURNAL OF CHEMICAL PHYSICS 125, 184905 ?2006?
0021-9606/2006/125?18?/184905/16/$23.00© 2006 American Institute of Physics
125, 184905-1
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Page 2

that lowers the free-energy barrier for bond formation. We
consider one model of an associating fluid that is similar to
the I model in the sense that single monomers can add or
dissociate at chain ends. This monomer-mediated12?MM?
reaction model differs from the I model only in the absence
of a chemical initiator. Another basic thermodynamic model
is the FA or free-association model where all particles can
associate and dissociate into chains without any constraint
except the condition of thermodynamic equilibrium. Thus,
the reaction kinetics of the FA model are identical to what
hasbeen called thescission-recombination
mechanism.5,12The third basic model, the activated associa-
tion model, requires that the monomers must be thermally
“activated” before particles can be associated. This later situ-
ation occurs in sulfur where S8sulfur rings must break open
due to thermal motions before chain formation can occur.13
The present work confines itself to the extreme cases of the
SR and MM models, which are identical or similar to the FA
and I models, respectively, and are representations of the
rather broad range of behaviors seen in equilibrium models.
The present paper describes initial steps in this direction
by systematically treating the stress and dielectric relaxation
in solutions of equilibrium polymers where hydrodynamic
and entanglement interactions are neglected, interactions that
become important phenomenologically in dilute and in con-
centrated high molecular weight solutions, respectively.
However, the general trends that are described are believed
to apply also to fluids having these more complex interac-
tions. The main focus here is on the particular problem of
how the cluster formation-disintegration process and the time
scales governing the clustering dynamics modify the stress
and dielectric relaxation processes from those of solutions of
clusters having a fixed polydisperse mass distribution.
We provide a theoretical description of the influence on
the dynamics of reversible aggregating systems of the cou-
pling between chemical relaxation, due to chemical reactions
that result in a change in chemical species, and internal re-
laxation, that arises from the structural rearrangements
within chains of fixed lengths. The analysis uses a discrete
chain molecular model and explicitly considers the vis-
coelastic properties and the dielectric relaxation. The general
theory is applicable to a wide range of dynamic models since
the relaxation is described in terms of the relaxation time
distribution. However, because our focus is on understanding
general trends emerging from this coupling, many illustra-
tions of the general formulation use the analytically simplest
Rouse model for the internal chain dynamics and two distinct
kinetic models for the association/dissociation that corre-
spond at equilibrium to the simplest FA and I models.14As-
suming the independence of the chemical and structural re-
laxation processes, we derive analytical expressions for the
viscoelastic and dielectric functions for an equilibrium sys-
tem composed of self-associating linear macromolecules as a
functions of polymer density, temperature, chain length, ki-
netic rate constants for the polymerization process, and inter-
action and geometric parameters of the bead-spring model,
such as the monomeric friction coefficient and the mean-
square end-to-end distance. In contrast to the independence
of equilibrium properties to the reaction kinetics within the
?SR?
SR and MM models,14dynamic properties strongly depend
on the mechanism of chemical relaxation that is dictated by
the kinetics of self-association and disintegration.
Section II considers the stress relaxation function and
dielectric permittivity in the presence of the reversible break-
ing and restoration of chains, focusing on the competition
between chemical and structural relaxation processes. Calcu-
lations illustrate the equilibrium properties of polydisperse
systems and the influence of chemical relaxation for two dif-
ferent mechanisms of polymerization corresponding to the
SR and MM models. Section III presents the derivation of
explicit general expressions and illustrations of general
trends for the dynamical properties of living polymer sys-
tems, such as the complex viscosity and dielectric permittiv-
ity. Dynamic properties of equilibrium polymer systems are
compared for MM model and SR model reaction kinetics.
The models are analyzed both for slow and fast chemical
relaxation dynamics in Sec. IV, and a comparison with the
relaxation model of Cates is illustrated for Rouse-type chain
dynamics. The Appendix presents the derivation of closed
form analytical expressions for the contributions from indi-
vidual chains to the viscoelastic and dielectric functions of
living polymers with Rouse dynamics as well as those for the
viscoelasticity of living and unbreakable chains in the repta-
tion model.
II. STRESS RELAXATION FOR EQUILIBRIUM
POLYMERS
In the absence of chemical relaxation ?i.e., for “frozen”
chains?, the dynamics is described by models that are often
represented in terms of a set of relaxation times ?pfor the
“modes” p of relaxation. We consider this type of general
expression for the complex frequency dependent viscosity
and dielectric permitivity, but numerical calculations are il-
lustrated for the simplest bead-spring Rouse model,15–17
which describes dynamics in melts and concentrated solu-
tions of nonentangled polymers. The Rouse model is used
because of its greater analytical tractability and the need for
a minimum number of parameters. As is well known, the
Rouse model ignores the influence of hydrodynamic interac-
tions and entanglements.
We consider a polydisperse system, where the distribu-
tion of chain lengths is determined by chemical equilibrium
for the polymerization process and by the overall concentra-
tion ? of the solute. If the chemical relaxation is extremely
slow on the experimental time scale, the reversible breaking
and restoration of chains only results in a redistribution of
mass between the aggregates, but the overall mass distribu-
tion is fixed by the condition of chemical equilibrium.18The
simplest model for the chemical relaxation process posits
that each association and dissociation event involves only
one chemical bond in the linear chain and is characterized by
two mean-field rate constants kAand kDfor chain growth and
decomposition, respectively. Both kAand kDare assumed, for
simplicity, to be independent of chain length and configura-
tion.
This minimal model is consistent with at least two alter-
native kineticmechanisms
aggregation process.5,12,19,20The first is a monomer-mediated
forthepolymerization/
184905-2E. B. Stukalin and K. F. FreedJ. Chem. Phys. 125, 184905 ?2006?
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Page 3

kinetic scheme12?MM model? in which each chain can grow
through a bimolecular reaction with a monomer or the chain
can lose a monomer at the chain end. Only one end is con-
sidered to be the active site of a macromolecule. The second
alternative mechanism is the scission-recombination scheme
?SR model?.5,19We consider the specific example of the SR
model in which a chain may break with equal probability
into two shorter chains at any site along its length, or two
chains can recombine with one another to form a larger
chain. These two reaction kinetic models can be described as
cN? cN−1+ c1, MM model,
?1?
cN? cN?+ cN?,SR model,
?2?
where N=N?+N?
The equilibrium chain length distribution has been ob-
tained analytically for the model of Eq. ?2? after treating the
chain length N as continuous parameter.5,21The equilibrium
number density for chains of length N is obtained as the
exponential distribution,
cN=?
L2exp?− N/L?,
?3?
where the average polymer length is L=??Keq/2?1/2with
Keq=kA/kDthe equilibrium constant for the addition of a
monomer and ?=?NNcNis the total monomer density. The
monomer-mediated polymerization reaction scheme also
produces an exponential equilibrium distribution when L is
not very small.20However, here the distribution is found by
treating N as a discrete variable,12
Keq?1 + 2?Keq−?1 + 4?Keq
2?Keq
cN=
1
?
N
??
L2exp?− N/L?,
?4?
where L=2?Keq??1+4?Keq−1?−1???Keq?1/2with Keqde-
fined as above. Using a discrete representation, both mecha-
nisms are found to yield the same equilibrium chain length
distribution cN=c1qN−1, where q=1−1/L. Note that this dis-
tribution does not apply for the activated A model and the
living I polymerization model with initiator present because
additional thermal activation and chemical initiation steps
can alter the variation of L with ? and the sharpness of the
polymerization transition. For living polymerization systems,
the average polymer length L scales nearly linear with ?
above the “critical polymerization concentration” defined as
the value of ?*at which no polymers are present in a solu-
tion ?L=1? at a fixed concentration of chemical initiator. This
behavior contrasts with that of the free-association model
where the polymerization transition is extremely broad and L
varies as ?1/2.14When the living polymerization is chemi-
cally initiated, the equilibrium mass distribution NcN, which
is relevant for averaging the dynamic functions, is given by
NcN
bution NcNdescribed in Eq. ?4? for L large enough. Also the
relation between the “exponential” factor q and the average
polymer length L is different.14
liv=?N+2?CqN+2, and only slightly differs from the distri-
A. Dynamic properties of polydisperse systems of
frozen chains
The polymer contribution to the relaxation modulus for a
system of frozen chains with exponential polydispersity is
evaluated by summing over the normal modes for the collec-
tive motions and by averaging over the equilibrium polymer
distribution,15,22
G?t? =
1
L2?
N=1
+?
NqN−1GN?t?
=
?RT
M0L2?
N=1
+?
?
p=1
N−1
qN−1exp?− 2t/?p?.
?5?
Here
modulus for chains of fixed length N, M and M0are the
molecular weights of a polymer and monomer, respectively,
? is the mass density of polymers, and the ?p?p=1,...,N
−1? are the normal mode relaxation times. Attention here is
restricted to a minimal model for the dynamics of associating
fluids. In order to elucidate general trends arising from the
coupling between particle association-dissociation and chain
relaxation, this section illustrates these trends for the sim-
plest Rouse model where the ?pare known analytically as
12kBTsin−2??p
GN?t?=?RT/M?p=1
N−1exp?−2t/?p?
is the relaxation
?p=
?0b2
2N?.
?6?
The prefactor L−2=?N=1
malization condition for averaging over the equilibrium dis-
tribution. Section IV describes an application in which a rep-
tation model is used for the ?p, while other models for the
chain relaxation times may be substituted in Eq. ?5?.
The Rouse model does not apply for the monomer con-
tribution to Eq. ?5?, so it is necessary to specify this quantity.
For simplicity, the monomer contribution to the viscoelastic
function is taken as identical to that of a solvent molecule,
and the contribution of monomers and solvent is omitted
from the dynamical properties to focus on the response of the
aggregates. It is also evident that short chains are not Gauss-
ian, and strictly speaking the Rouse formula of Eq. ?6? is
inapplicable for small N. However, the equilibrium weight
density distribution M0NcNis maximum at the average poly-
mer length L. Hence, if L is not small, the error in using Eq.
?6? is minimal because most of the contributions to the re-
laxation modulus arise from chains whose lengths are near
the maximum of the weight density distribution. Again, these
approximations are consistent with our focus on general
trends, and more detailed models are readily considered by
substituting the appropriate ?p.
The viscoelastic functions are obtained similarly. For ex-
ample, the zero-shear viscosity ?0,Nfor chains of fixed length
N in the Rouse model is
+?NqN−1in Eq. ?5? arises from the nor-
?0,N=?NA?0b2
36M
?N2− 1? ?NA?0b2
36M0
?N,
?7?
where NAis Avogadro’s number, ?0is the monomer friction
coefficient, b is the effective length of a segment, and M0is
the molecular weight of a monomer. Averaging over the ex-
184905-3Viscoelastic and dielectric relaxations J. Chem. Phys. 125, 184905 ?2006?
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Page 4

ponential chain length distribution ?this time expressed with
a continuous distribution? leads to
?0= ?−1?
0
?
NcN?0,NdN = 2?0,L,
?8?
where L is the average chain length.
B. Associating fluids with reversible chemical
dynamics
When the chemical relaxation induced by the bond
breakage and reformation reactions is not slow on the time
scales of measurements, the effect of the chemical dynamics
cannot be described solely through an average over the equi-
librium distribution of chain lengths as in Eq. ?5?. Stress
relaxation may thus occur either via diffusive motion of in-
dividual chains or due to an “evaporation-condensation”
mechanism arising from mass transfer between chains that
takes place by virtue of the association-dissociation events.
We assume that each normal mode p of chain motion
may relax either due to the dynamics of a chain of fixed
length or because of a chemical reaction that changes the
chain length. Both of these relaxation processes are taken to
be independent, implying that the effective relaxation rate
1/?efffor stress relaxation may be estimated as if these pro-
cesses occur in parallel, i.e., as the harmonic average 1/?eff
=1/?p?+1/?c, where 1/?p?=2/?pis the rate of structural relax-
ation for mode p in an unbreakable chain ?see Eq. ?5??. When
?cis comparable with the longest structural relaxation times
?p, the efficient mechanism for stress relaxation is deter-
mined by a coupling between the two types of system rear-
rangements. Given this assumption of independence, the
shear relaxation modulus Gc?t? is described by
Gc?t? =
1
L2?
N=1
+?
NqN−1Gc,N?t?
=
?RT
M0L2?
N=1
+?
?
p=1
N−1
qN−1exp?− 2t/?p?exp?− t/?c?,
?9?
where ?cis the characteristic time for the chemical reaction
?1? or ?2? and where ?cdepends on N in a manner to be
determined below. Equation ?9? clearly behaves properly in
the limit that ?cbecomes infinite and the chains have fixed
lengths.
The stress relaxation for an equilibrium polydisperse
system with very rapid, monomer-mediated reaction kinetics,
?c??pfor all p, is exponential over the whole time scale t.
Polydispersity does not destroy the exponential dependence
since the lifetime ?cof chains is practically independent of
their lengths ?see next section?. The stress relaxation for an
equilibrium polydisperse system with scission-recombination
reaction kinetics is almost exponential for small ?c. Polydis-
persity slightly influences the stress relaxation decay func-
tion since the lifetime ?c depends on chain length ??c
?N−1? ?Ref. 23? ?see below?. Thus, as ?cchanges from large
to small values, the relaxation spectrum changes from non-
exponential to exponential. The complex viscosity ?c
*=?c?
−i?c? is related to the shear relaxation modulus and can be
obtained through Fourier-Laplace transformation as ?c
=1/?i???0
*???
?exp?−i?t?Gc?t?dt.
C. Dielectric relaxation
The complex dielectric permittivity ?*=??−i?? of equi-
librium associating systems is found similarly. We consider
polymers whose dipoles are parallel to the chain contour
?A-type?, whereupon the dielectric relaxation is determined
by the first derivative of the normalized time correlation
function for the end-to-end vector R?t?,15,24,25
NqN−1????
???R?t?R?0??
?*??? = ??−
1
L2?
N=1
?R2??dt?.
+?
0
?
exp?− i?t?d
dt
?10?
Here, ??is the high frequency limit of the dielectric con-
stant, and ?? is the relaxation strength which for A-type
chains is related to the mean-square end-to-end distance
?R2?, the molecular weight M, and the solute concentration
?,24,25
?? =?4?NA?2F
3kBT????R2?
M?,
?11?
where ? is the dipole moment per unit contour length, and F
is the ratio of internal and external electric fields.
For a monodisperse system following Rouse dynamics,
the time correlation function in Eq. ?10? is given by
?R?t?R?0??
?R2?
=
8
?2?
p=1
N−11
p2exp?− t/?p??p odd?.
?12?
In the presence of the reversible breaking-restoration of
chains, the time correlation function is modified in a similar
fashion as the stress relaxation, and the Rouse model yields
the contribution from chains of fixed length N as
??R?t?R?0??
c
p=1
?R2??
=
8
?2?
N−11
p2exp?− t/?p?
?exp?− t/?c??p odd?.
?13?
Substituting Eq. ?13? into Eq. ?10? yields our description for
polydisperse equilibrium aggregating systems, namely,
?*??? = ??+
1
L2?
N=1
+?
NqN−1??
??
8
?2?
p=1
N−11
p2
?p
−1+ ?c
−1+ ?c
−1
?p
−1+ i??
?p odd?.
?14?
D. Chain scission relaxation times
The reaction schemes in Eqs. ?1? and ?2? produce infinite
sets of coupled nonlinear kinetic equations ?see below? with
a broad range of characteristic relaxation times. Thus, a num-
ber of different approaches exist for estimating the charac-
teristic times of reactions for these aggregating systems. This
184905-4 E. B. Stukalin and K. F. FreedJ. Chem. Phys. 125, 184905 ?2006?
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Page 5

difference depends, in part, on the manner in which the sys-
tem may be perturbed, e.g., by a transition to nonequilibrium
state following a c-jump or T-jump process, i.e., by sudden
changes in concentration or temperature, respectively, or by
an equilibrium fluctuation that implies a small instantaneous
perturbation. The mean lifetime ?cis defined as the charac-
teristic time necessary for the system to relax after the per-
turbation is applied. The calculation of characteristic times
for chemical relaxation becomes simplified for systems close
to equilibrium where the chemical reaction dynamics is de-
scribed by a system of coupled linear ordinary differential
equations. Cates and co-workers have estimated relaxation
times for both concentration and temperature jumps in the
scission-recombination model,21while others have consid-
ered the relaxation based on a simplified treatment of the rate
equations.12,23We focus on the response to concentration
fluctuations as more indicative of the dynamics in equilib-
rium aggregating systems, especially when considering the
relaxation for chains of particular sizes since this type of
relaxation process enables defining a size dependent relax-
ation time.
The rate equations for the MM model reaction scheme
can be expressed as the system of nonlinear differential
equations,
dc1
dt
= − 2kAc1
2− kAc1?
j=2
?
cj+ 2kDc2+ kD?
j=3
?
cj,
?15?
dcj
dt= − ?kAc1+ kD?cj+ kAc1cj−1+ kDcj+1,
j ? 1,
where kAis the association rate constant, kDis the dissocia-
tion rate constant, and Keq=kA/kDis the equilibrium con-
stant. We omit an initial initiation step, for simplicity, in
order to have the minimum number of adjustable parameters.
There is no difficulty in extending the analysis to more
elaborate kinetic models involving activation or initiation
steps.14The factors of two in the first of equation ?15? arise
because the reaction of two monomers to produce a dimer
consumes two chains of size j=1, while the decomposition
of a dimer yields two monomers. The equilibrium distribu-
tion over cluster sizes may readily be determined by setting
the left hand side of Eqs. ?15? to zero, leading to cj,0
=c1,0qj−1, where q=kAc1,0/kD?1, and cj,0and c1,0are equi-
librium concentrations of j-mers and monomers, respec-
tively. The absolute values of the equilibrium concentrations
cj,0are determined by the total polymer density through the
mass conservation condition ?=?jcj.
The dynamical response to small perturbations in the
equilibrium concentrations of each cluster is described by the
deviations ?cj?t?=cj?t?−cj,0. Substituting the preceding ex-
pression for ci?t? into Eqs. ?15?, using the properties of the
equilibrium solution, and also assuming that the perturba-
tions ?cj?t? are small, Eq. ?15? reduces to a set of linearized
equations governing the chemical dynamics near equilib-
rium,
1
kD
d?1
dt
= − ?4q + q2/?1 − q???1+ ?2 − q??2+ ?1 − q??
j=3
?
?j,
?16?
1
kD
d?j
dt
= − ?1 + q??j+ q?j−1+ ?j+1+ qj−1?1 − q??1.
Numerical solutions of Eq. ?16? can only be obtained after
truncating the equations by introducing a maximal cluster
size Nm, such that the equilibrium concentration cNm,0is neg-
ligible for clusters of size Nmand greater. Then, the rate
equations for ?j?j=1,...,Nm?, consistent with the mass con-
servation condition, reduce to
1
kD
d?1
dt
= − ?4q + q2?1 − qNm−2?/?1 − q???1+ ?2 − q??2
+ ?1 − q??
j=3
Nm
?j,
1
kD
d?j
dt
= − ?1 + q??j+ q?j−1+ ?j+1+ qj−1?1 − q??1,
?17?
1
kD
d?Nm
dt
= q?Nm−1− ?Nm+ qNm−1?1.
Equations ?16? are the limiting case of Eqs. ?17? when Nm
approaches infinity.
In general, the solution to the finite system of Eqs. ?17?
can be represented in terms of the fundamental solutions as
?j?t? =?
i=1
Nm
aiuijexp?− ?it?,
?18?
where uij is the jth component the ith eigenvector ui
=?ui1,...,uiNm? for the linearized system of equations, −?i
are the corresponding eigenvalues that must be nonpositive,
and aiare constants that depend on the initial conditions for
the concentration fluctuations. Due to the constraint of mass
conservation, one eigenvalue vanishes, say, ?Nm=0, and the
corresponding coefficient must be set as aNm?0 too, in order
that the system is fully relaxed back to equilibrium in the
large time limit.
We define a mean lifetime ?cfor each individual cluster
ck?1?k?Nm? by introducing a local perturbation ?j?0? as
the initial concentration fluctuation, such that the initial
changes differ from zero only for those “reactants” involved
into the rate equation for clusters of a specific size k, namely,
for three “adjacent” clusters ?k, ?k±1?0 and the monomers
?1?0, while all other initial deviations from equilibrium
concentrations areassumed
?1,k,k±1?. The mass conservation condition for this choice
of initial perturbations implies that only three of the ?j?0? are
independent, i.e., mass conservation implies k?k?0?+?k
−1??k−1?0?+?k+1??k+1?0?+?1?0?=0. Given the freedom of
choice for k different from either 1 or Nm, we set ?1?0?=0
and take ?k−1=?k+1=−?k/2, so that the response only in-
volves the single parameter ?k. For k=1, we set ?2=−?1/2,
while for k=Nm, ?1=?Nm−1=−?Nmis chosen. The truncated
to vanish,
?j?0?=0?j
184905-5Viscoelastic and dielectric relaxationsJ. Chem. Phys. 125, 184905 ?2006?
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Page 6

linear set of ordinary differential equations may be solved
numerically for each k using these initial conditions defining
the local concentration fluctuations. The solutions are used to
define characteristic relaxation times.
The characteristic time constant ?c?k? for each cluster k
and for the above choices of zero time perturbations ?j?0? is
naturally determined as a linear combination of reciprocal
eigenvalues −?i
?k?0??
0
−1,
?c?k? =
1
?
?k?t?dt =
1
?k?0??
i=1
Nm−1
uki?i
−1aik.
?19?
Thus, a matrix ?ccan be defined whose diagonal elements
are the ?c?k?,
??c?k?? = UT???−1A,
?20?
where UT=?uij?Tis the transpose of the eigenvector matrix,
???−1=??i
reciprocal of the eigenvalues, and A=?aij? is a matrix that
may be determined from the t=0 conditions for the kth set of
concentration fluctuations for the clusters cj?j=1,...,Nm?
when the concentration of the cluster of a size k is perturbed
locally. The initial conditions imply that ?j
and hence, the factor of ?k?0? cancels between numerator and
denominator of Eq. ?19?, making ?c?k? independent of ?k?0?.
Explicit numerical calculation of Eq. ?18? for the char-
acteristic relaxation times following small concentration
fluctuations shows the ?c?k? to be almost independent of the
cluster size k and weakly dependent on the total polymer
density. Only the monomers relax more quickly than the
other species. The cutoff Nmhas been chosen in the numeri-
cal calculations, such that the equilibrium properties for in-
finite Nmare reproduced with good accuracy. Beginning
from dimers j?2, the lifetimes for all cluster sizes are ap-
proximated quite well by the constant ?c?j?=?2kD?−1, a result
consistent with a rough analysis of the rate equations by
Milchev.12Indeed, neglecting the last two terms in Eq. ?15?
containing cj±1produces the simplified rate equation dcj/dt
=−kD?1+q?cj, which yields the characteristic time 1/?c
=kD?1+q?=kD?2−1/L?.12For sufficiently large L, Milchev’s
estimate is close to that from our numerical analysis of
1/?c?j??2kD. This relaxation time is explained by the fact
that “cluster” reactions are local and may occur only at the
chain ends, so ?c?j? is independent of chain length.
It appears that an analytical solution to the rate equations
for the scission-recombination mechanism can only be found
in the continuum limit. Characteristic times for scission-
recombination reactions have been obtained in an elegant
analysis by Cates and co-workers for the time evolution fol-
lowing a small c-jump perturbation from equilibrium,21,26,27
−1? is a diagonal matrix whose elements are the
k?0?=?i=1
Nmujiaik,
?c=
1
kD?2L + N?.
?21?
This result for the lifetime ?cis again as expected since a
chain of size N has N possible reaction sites for bond break-
age and hence has N-times larger probability to break than a
chain in the monomer-mediated kinetic model. Thus, a quali-
tatively different chain length dependence of the relaxation
times emerges for the MM and SR aggregation mechanisms.
The weak dependence or practical independence on N for the
MM model contrasts with the inverse proportionality to N in
the SR model.
III. VISCOELASIC PROPERTIES AND DIELECTRIC
RELAXATION
The complex viscosity is readily derived from the relax-
ation modulus in terms of the set of relaxation times ?p. One
benefit of using the Rouse model for illustrating the theory is
the fact that the expression for the complex viscosity may be
evaluated in closed form using contour integration ?see Ap-
pendix for details?. Hence, calculations for equilibrium poly-
merization systems can be performed with a single summa-
tion, rather than the double summation required when using,
for example, a Zimm model for the chain relaxation times.
The viscoelastic functions are affected by chemical dynam-
ics, and the resultant functions are derived for the Rouse and
reptation models. All expressions for the reptation model are
presented in the Appendix, while the viscoelastic functions
for Rouse dynamics are detailed in the present section.
The final closed form expression for the contribution to
the complex viscosity from chains of particular length N fol-
lowing Rouse-type dynamics but affected by the reaction ki-
netics is obtained from the Appendix as
24M0?
??N+ i?r?1 + ?N+ i?r
2N?
?N+ i?r
?c,N
*??r? =?NA?0b2
coth?2N arcsinh??N+ i?r?
−
1
1
+
1
1 + ?N+ i?r??,
?22?
where ?r=?? is the reduced frequency, ?N=??c
mensionless characteristic time for the chemical reaction,
?0b2
24kBT=?2
−1is the di-
? =
8?0,
?23?
and ?0is the Rouse relaxation time constant. Alternatively,
?Ncan be expressed using the ratio of the longest structural
relaxation time ?R=?0N2to ?c, both N dependent,
?2
8N2
?c
?N?
?R
.
?24?
The final expression for the complex viscosity involves
anaverage over the
=L−2?N=1
For frozen chains, ?N=0, and the frequency dependent
viscosity reduces to that for the Rouse model. It is interesting
to compare the expression for ?N
phenomenological model of Blizard,28who considers mol-
ecules as springs moving in a viscous medium with “spring
constant” 3kBT/qb2?q is the number of subunits? and intro-
duces a constant viscous coupling per unit length to the sur-
rounding medium ?equivalent to the introduction of a mono-
meric friction coefficient ?0?.28The frequency dependent
viscosity ?*=G*/?i?? in this representation is given by
*??? = ?C1/M????iC2M2/?? coth?iC2M2? − 1?,
massdistribution
?c
*??r?
?NqN−1?c,N
*??r?.
*??? in Eq. ?22? with the
?N
?25?
where C1and C2are unspecified constants. Equations ?22?
and ?25? with ?N=0 are similar but not identical. In the low
184905-6E. B. Stukalin and K. F. Freed J. Chem. Phys. 125, 184905 ?2006?
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Page 7

frequency limit, the two formulas are equivalent, and the
direct comparison of these equations leads to the following
parameter mapping: C1=?RT/2 and C2=?0b2/6M0kBT. At
high frequencies, the Blizard equation yields the limiting real
and imaginary complex viscosities ??=??=1/2C1?C2/??1/2
as equal, while the asymptotic limits obtained using the
Rouse model are different ?see below?. However, within a
relatively wide range of ?, the continuous “Rouse-type”
Blizard model and our analytical representation for the dis-
crete version of the classical bead-spring model are in close
agreement. Differences appear only for very high frequen-
cies.
Separating the viscoelastic function into real and imagi-
nary parts leads to
48M0?f??z+? + f??z−?
N?
?N
48M0?i?f??z+? − f??z−??
N?
?N
where z±are given by
?c,N
? ??r? =?NA?0b2
−1
?N
2+ ?r
2+
1 + ?N
?1 + ?N?2+ ?r
2??,
?26?
?c,N
? ??r? =?NA?0b2
−?r
1
2+ ?r
2+
1
?1 + ?N?2+ ?r
2??,
?27?
z±= ?N± i?r,
?28?
and the functions
f??z±? =coth?2N arcsinh?z±?
?z±?1 + z±
.
?29?
The average over the equilibrium chain length distribu-
tion ?see Eqs. ?8? and ?9?? can be evaluated numerically, but
some estimates are derived here. It is instuctive to compare
the zero-shear viscosity ?c,0of a polydisperse system with an
exponential mass distribution for both kinetic reaction
mechanisms with the “normalized” contribution ?c,0,Lfrom a
monodisperse system of chains with length L. For ?c→?,
the chemical dynamics is irrelevant, and ?0?2?0,L?see Eq.
?8??. When ?c→0, the average zero-shear viscosity of the
polydisperse system is
?c,0=
1
L2?
N=1
+?
NqN−1?c,0,N=
?RT
M0L2?
N=1
+?
qN−1?N − 1??c. ?30?
Because ?c is practically N independent for monomer-
mediated kinetics, ?c,0=?RT/M0?1−1/L??c, while for the
scission-recombinationscheme,
=?RT/M0?1−1/L???c,L. Thus, ?c,0=??c,0,L, with ??1 for
monomer-mediated dynamics, and ??0.83 for scission-
recombination reaction kinetics ?the last number depends
slightly on L?.
Figure 1 presents logarithmic plots for the real and
imaginary parts of the complex viscosity as a functions of
the dimensionless frequency ?R=?0???8/?2??rfor three
polydisperse systems characterized by the same average
chain lengths L but different chemical dynamics. The plots
?c?N−1
and
?c,0
are normalized by the zero-frequency shear viscosity for un-
breakable chains with the same polydispersity. The three
curves correspond to ?a? frozen chains, i.e., ?N=0, ?b?
monomer-mediated reaction kinetics with ?N=?2?0kD/4, and
?c?
thescission-recombination
=?2?0kD?2L+N?/8. Identical dissociation rate constants “per
site” kDand structural relaxation times ?Rare used for the
two reacting systems. Chemical relaxation leads to faster
stress relaxation, diminishes the viscoelasticity for lower fre-
quencies, and shifts the peak in the imaginary part to higher
frequencies. The “tail” of the imaginary part of the viscoelas-
tic function for high frequencies ?0??1 is unaffected by the
chemical dynamics as demonstrated in Fig. 1?b? and as also
deduced by analysis of Eq. ?27?. However, in general, cases
?b? and ?c? exhibit a quite different ? dependence to the
complex viscosity. For monomer-mediated polymerization,
the zero-shear viscosity is higher, while the real part of ?*???
depends more strongly on frequency than for the scission-
recombination mechanism, provided the kinetic rate con-
stants per site are the same. However, if both kinetic rate
constants kAand kDare scaled to make the zero-frequency
shear viscosities for cases ?b? and ?c? coincide, the whole ?
curves become superimposed, with the relative difference not
mechanismwith
?N
FIG. 1. Normalized dynamic viscosities for three equilibrium polymerizing
systems with identical average chain lengths L=100 as a function of the
dimensionless frequency ?R=?0?. The curves correspond to unbreakable
chains ?UCs? and to living polymers with monomer-mediated ?MM? mass
exchanges and with scission-recombination ?SR? reaction kinetics. For two
last cases, ?0kD=0.01. Real components of the shear viscosity ?? are dis-
played in part ?A?, while imaginary components ?? are presented in part ?B?.
184905-7Viscoelastic and dielectric relaxations J. Chem. Phys. 125, 184905 ?2006?
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Page 8

exceeding 10%. In order to achieve this superposition, the
rate constants in Fig. 1 have to be scaled down for the
scission-recombination model, or, equivalently, they are both
scaled up for monomer-mediated kinetics by a factor of
?190. These results imply that it is difficult to distinguish
between these two mechanisms only from an analysis of the
frequency dependence of the viscoelastic function without
independent information on the reaction rates.
The complex dielectric permittivity for Rouse model
chains may be found in a similar fashion. However, the con-
tour integration used for evaluating the finite discrete sum for
the Fourier-Laplace transform of the normalized decay func-
tion could not completely be resolved analytically, and two
additional integrals NJNand NJN? remain to be calculated
numerically. When N is sufficiently large, these integrals
yield small corrections since both scale as N−1. Here the
expressions for JNand JN? with even N are presented, while
the Appendix provides the case of odd N. The complex per-
mittivity is evaluated from averaging over the mass distribu-
tion
?c,N
*??? − ??,N= ???N
*???,
?31?
where the reduced frequency dependent part for even N is
?N
*?? ˆr? =
?ˆN
?ˆN+ i? ˆr
+ i? ˆr?
tanh?N arcsinh??ˆN+ i? ˆr?
N??ˆN+ i? ˆr?1 + ?ˆN+ i? ˆrarcsinh2??ˆN+ i? ˆr
+8N
?2JN??−8N
?2JN,
?32?
where ?ˆN=2?N, and ? ˆr=2?r. The coefficient of 2 arises from
the difference in the structural relaxation times ?pin expres-
sions for viscoelastic and dielectric functions of the same
factor.
The real and imaginary components yield the contribu-
tions to the dynamic dielectric constant and dielectric loss,
?N
*??r?=?N? −i?N?,
?N??? ˆr? =
?ˆN
2+ ? ˆr
2
?ˆN
2+i? ˆr
2?f??z ˆ+? − f??z ˆ−?? −8N
?2JN,
?33?
?N??? ˆr? =
?ˆN? ˆr
?ˆN
2+ ? ˆr
2−? ˆr
2?f??z ˆ+? + f??z ˆ−?? −8N? ˆr
?2JN?,
?34?
where z ˆ±=?ˆN+i? ˆr, the functions f??z ˆ±? have the same struc-
ture as the first term in square brackets in Eq. ?32?,
tanh?N arcsinh?z ˆ±?
N?z ˆ±?1 + z ˆ±arcsinh2?z ˆ±,
f??z ˆ±? =
?35?
and the ?- and N-dependent correction terms JNand JN? are
obtained as the integrals ?N even?,
JN=?
0
?x2+ N2?2
fN
JN? =?
0
?x2+ N2?2
fN
?x tanh??x/2?
fN
2?x? + ? ˆr
2?x?dx
2,
?36?
?x tanh??x/2?
fN?x?dx
2?x? + ? ˆr
2,
?37?
with fN?x?=cosh2??x/2N?+?ˆN. Equations for unbreakable
chains, i.e., with infinite lifetimes, are generated by setting
?ˆN=0 and averaging over the mass distribution. Naturally,
the first term in Eq. ?32? vanishes in that case.
The frequency dependent dielectric permittivity exhibits
similar trends as for the viscoelastic properties. As is seen
from the normalized dielectric loss spectrum E?=??/?? ?see
Fig. 2?, more “rapid” chemical dynamics produces a nar-
rower peak, and the maximum shifts towards higher frequen-
cies because of the smaller effective stress relaxation time.
When the chains are sufficiently long, the corrections JNand
JN? become negligible, and the closed form analytical formula
for dielectric relaxation is reliable. The “worst” approxima-
tion in neglecting JNand JN? is for the model with ?ˆN=0
?unbreakable chains?. Figure 3 presents the ratios of absolute
values of the complex dielectric permittivity for different fre-
quencies and chain lengths in the Rouse model as evaluated
using Eq. ?32? with the correction terms JNand JN? neglected
and as calculated without this approximation by performing
the direct summation over all the modes ?see Appendix, Eq.
?A8??. The relative error increases with frequency. However,
within the frequency range of 0??R?0.1, the absolute
FIG. 2. Normalized dielectric loss E? as a function of dimensionless fre-
quency ?Rfor three systems with equal average chain lengths L=100 and
different relaxation dynamics. The three systems are unbreakable chains
?UCs? and living polymers follow the monomer-mediated ?MM? and
scission-recombination ?SR? reaction kinetics. For living polymers, ?0kD
=0.01.
184905-8E. B. Stukalin and K. F. FreedJ. Chem. Phys. 125, 184905 ?2006?
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Page 9

value of the dielectric permittivity changes by approximately
two orders of magnitude for chains of length N=100 and
approximately three orders of magnitude for chains with N
=1000, while the relative overestimation of ??*??R?? is at
most only 1.16.
IV. SELECTED APPLICATIONS
A. Brief comparison to model by Cates and co-workers
The model by Cates and co-workers focuses on the par-
ticular dynamic regime in which the average time ?break
?1/?kDN? for a chain of length N to break into two pieces is
lower than the characteristic time of stress relaxation ?relfor
unbreakable “representative” chains of the same length. The
characteristic stress relaxation time ? is generally affected by
the scission-recombination reaction kinetics within the
model of Cates and co-workers, and ? is identified with the
waiting time for a break in the chain to occur within some
characteristic arc distance n.29The contour distance n, which
is usefully “utilized” for the stress relaxation, is estimated
from scaling arguments. These arguments assume that the
relaxation time ?relfor a section of chain n ?say, the Rouse
time ?0n2when Rouse-type dynamics are considered? is
comparable to ?break. The overall relaxation time ? in the
regime ?break??relis then given by
? ? ?breakNl2/?r2??break??,
?38?
where ?r2??break?? is the mean-square displacement of a
monomer on an unbreakable chain of length N in a time
?break?l is a subunit length?.
In order to compare some predictions between our model
and that of Cates and co-workers, attention is restricted to the
scission-recombination polymerization mechanism in con-
junction with the Rouse model described above, and it suf-
fices to compare predictions for contributions from chains of
length N. The zero frequency shear viscosity within our
theory immediately follows from Eq. ?22? as
?0,N=?NA?0b2
24M0?
2N?
coth?2N arcsinh??N?
??N?1 + ?N
1 + ?N??
−
1
1
?N
+
1
? G0,N?,
?39?
where G0,N=Gmax,N
modulus,
8?
2N?
?
???=?RT?1−N−1?/M0 is the plateau
N − 1??
??N?1 + ?N
1 + ?N??,
? =?2
?0
coth?2N arcsinh??N?
−
1
1
?N
+
1
?40?
?0=?0b2/3?2kBT is the “Rouse constant,” and
?N=?2?0kD?2L + N?
8
.
?41?
Note that if ?c→0, the stress relaxation is exponential with
time constant ?→?c, and the average zero-shear viscosity
?c,0is given by Eq. ?30?.
The model of Cates and co-workers predicts that if the
chain motion is also Rouse-type and if ?break??R, the viscos-
ity obeys the scaling relation,6
?0,N? G0,N??R?break?1/2= ?G0,N?N?0/kD?1/2,
where ?R??0N2is the Rouse time of an unbreakable chain of
length N, G0,Nis the plateau modulus, ? is a constant of the
order of unity which cannot be calculated exactly, and ?0is
defined above. Equations ?39?–?42? imply that the zero fre-
quency shear viscosity in both the model of Cates and co-
workers and our model can be represented as a function of
?42?
FIG. 4. Comparison between the effective stress relaxation times for poly-
mer systems following Rouse dynamics evaluated using the present theory
and the model of Cates and co-workers for various values of the parameter
?0kDand the chain lengths N ?see text?. The chemical reaction dynamics is
assumed to proceed through the reversible scission-recombination ?SR?
model mechanism. The middle “plateau” regime delineates the range of
kinetic parameters over which there is general agreement between both
theories, while the departure from this plateau for large ?0kDdemonstrates
the difference between the models in the ??1 regime. The departure from
the plateau level for small ?0kDarises because the square root coupling
expression of Eq. ?42? in the model of Cates and co-workers is not appli-
cable when ??1.
FIG. 3. The ratio of absolute values of the dielectric permittivity in the
Rouse model as evaluated using Eq. ?32? neglecting the correction terms JN
and JN? and as calculated by direct summation over the normal modes as
function of dimensionless frequency ?Rfor different chain lengths.
184905-9 Viscoelastic and dielectric relaxationsJ. Chem. Phys. 125, 184905 ?2006?
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Page 10

the universal dimensionless parameter ?0kDsince Eq. ?42?
can be rewritten as ?0,N=?G0,N?0?N/?0kD?1/2.
The direct comparison between two approaches may be
effected for ?0,N/G0,N?0at fixed ?, ?0, T by varying both kD
and kAin parallel while maintaining as constant the mean
length L?N. Figure 4 compares the two approaches for the
dependence of the ratio of values for ?0,N/G0,N?0from two
models on the kinetic rate constants. Each curve corresponds
to chains of fixed length N. A range of dissociation rate con-
stants exists in which the two theories agree qualitatively
apart from a factor of the order of unity. This range becomes
larger for longer chains, and the agreement only occurs
within the limits in which the model of Cates and co-workers
is applicable, i.e., for ?=?c/?R???0kDN3?−1?1. The ratio
between the values of ?0,N/G0,Nfor the two models is
roughly a constant within some range of ?0kD, being close to
0.5. The difference between the approaches occurs for small
kD?and kA? where the model of Cates and co-workers is not
applicable because his “scaling” model does not reduce to a
description of frozen, unbreakable, polydisperse polymers in
the limit of vanishing kDas in our theory and as in the gen-
eral formulation of Cates and co-workers ?see below?. A dif-
ference between the theories also emerges for very large val-
ues of the kinetic rate constants. This is illustrated for ?0kD
varying between 10−6and 10−3, where the effective stress
relaxation times in the present model and in the model of
Cates and co-workers change in a parallel fashion for “inter-
mediate” sized chains ?N=100?, while the model of Cates
and co-workers is inapplicable over most of the range for
“short” chains ???1?. A departure between our model and
that of Cates and co-workers is evident for very rapid reac-
tion kinetics ???1? of “long” chains. Applying the scaling
theory of Cates and co-workers type to Rouse-type dynamics
suggests that the coupling ????R?break?1/2is valid in the
small ?c??breakregime, while our approximation suggests
that the asymptotic time constant ? approaches the lifetime ?c
as ?c→0.
Granek and Cates26also derive an analytical approxima-
tion for the stress relaxation function for living polymers in
the framework of a Poisson renewal model. Their expression
yields a relation ?in Laplace transform space? between the
stress relaxation function ?rm?t?=Grm?t?/Grm,0 for living
polymers in a chemically reversible system and the stress
relaxation function ?N?t?=GN?t?/G0,Nfor unbreakable chains
of length N. Here Grm,0and G0,Nare the plateau moduli for
living polymer and unbreakable polymers of length N, re-
spectively. The Poisson model relies on the assumption that
no correlation exists in the dynamics between renewal inter-
vals and that during one renewal interval. Hence, the stress
relaxation is taken as independent of what transpired within
the previous interval. The complex viscosity of the polymer
system in the renewal model is
?rm
*?i?? =
?0
?dNNcN?c
?dNNcN?N
*?i? + ?c
−1G0,N
−1?
1 − ?0
−1?N
*?i? + ?c
−1?,
?43?
where ?N
unbreakable chains, ?N
age over the mass distribution is explicitly indicated. Thus,
*?s? denotes the complex viscosity for a system of
*?s?=?0
?dte−stG0,N?N?t? and the aver-
Grm?t? for a living polymer system is obtained from a mul-
tiple convolution of Gc?t? from Eq. ?9? ?see Eq. ?43? above?.
The transform Grm
rational fraction of Gc
quantities Gc?t? and its transform Gc
may readily be converted into those for the renewal model.
The renewal model is now used to describe viscoelastic
properties of a polymer system where a dominant mecha-
nism for the stress relaxation is chain reptation. The stress
relaxation function for the reptation model is
L2?
0
p2?
0
*??? emerges from the renewal model as a
*???, so we focus attention on the basic
*???. Hence, our results
??t? =
1
?
N exp?− N/L??N?t?dN
=
8
L2?2?
p=1
?
1
?
N exp?− N/L?
?exp?− tp2/?rep?dN
?p odd?,
?44?
where ?rep=? ¯rep?N/L?3is the reptation time of a chain of
length N and ? ¯repis that of a chain with the average length L.
In the limit of small ?c, the sum over p can be replaced by an
integral. After averaging over the chain length distribution
?exponential as usual?, the effective stress relaxation time ?
??c??=0? for the living polymers in the SR model with
?c=?break/?2+N/L? is given by ?=???break? ¯rep?1/2with ?
?0.38.26Use of the full expression for ?N?t? in Eq. ?44? for
unbreakable reptative chains is essential to obtain this par-
ticular square root dependence of ? on the scission time
?break. Note that both our model and the renewal model pro-
duce the correct result for infinitely long ?c. The viscosity for
a system of unbreakable chains, with an identical chain
length distribution as the living polymer system, is obtained
in this limit as expected from our theory.
If the polymer dynamics is Rouse-type and if ?c??0, our
model and the renewal approximation are in agreement be-
cause the time scales in both expressions are combined using
the harmonic mean of ?pand ?c. However, as ?cbecomes
smaller ?say, less than ?0?, the difference between the models
is evident since the denominator in Eq. ?43? is distinct from
unity and diminishes as ?cgets smaller.
Cates suggests that for Rouse dynamics, the assumption
of independent stress relaxation in different renewal intervals
fails because the low frequency modes are more affected by
chain length “jumps” in renewal events than the “local,” high
frequency modes. We believe that it is still an open question
as to what is the proper limit for the stress relaxation when
the chemical relaxation is very fast and as to how the effec-
tive time ? for this process depends on ?c. The prediction of
our model emerges from Eqs. ?30? and ?40? in the limit of
?N
the transform ?N???=G0,N?N???/i? of Eq. ?44? and the
transform ?c,N???, i.e., the viscoelastic function as modified
by chemical relaxation processes for the reptation model.
−1→0. The Appendix presents a closed form resolution of
B. On qualitative nature of relaxation in associating
fluids
When the chain lifetime is very large for unbreakable
reptative chains, i.e., ?c→?, the stress relaxation for mono-
184905-10 E. B. Stukalin and K. F. Freed J. Chem. Phys. 125, 184905 ?2006?
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Page 11

disperse systems is nonexponential, as found for the Rouse
model. On the other hand, averaging over the exponential
equilibrium distribution in Eq. ?5? in the limit of very slow,
nearly frozen chemical dynamics does not produce qualita-
tively new features in the time dependence of G?t?. A
stretched exponential approximation G?t??e−at?
over a limited range of times. The calculated ? is altered if
the sum over p in Eq. ?5? is truncated to retain only the
slowest p=1 mode ?a description that has been used as a
crude model of stress relaxation in glass forming liquids?.30
The stress relaxation becomes exponential at very large times
t. The lifetime for the long time exponential decay of mono-
disperse systems is proportional to N2, as expected. When
the chemical dynamics becomes faster, the relaxation spec-
trum for a polydisperse system changes from rather disperse
to nearly exponential for all times.
The evolution of the stress relaxation spectrum as the
rate of assembly-disassembly kinetics switches from being
slow to fast may be followed in the frequency domain as
well. The analysis of this frequency dependence is conve-
niently represented in terms of Cole-Cole plots, in which, for
instance, the imaginary part of the complex modulus G????
is presented as a function of the real part G????, or analo-
gously, the imaginary component of the dielectric permittiv-
ity ????? is plotted versus the real component ????? in order
to analyze the dynamics of viscoelastic and dielectric relax-
ation, respectively.31,32Systems with a single exponential de-
cay yield perfectly semicircular Cole-Cole plots, characteris-
tic of those for a Maxwell model. In general, deviations from
semicircular plots appear and are indicative of nonexponen-
tial stress relaxation.
Figure 5 presents a redrawing of the data from Fig. 1 in
form of Cole-Cole plots for the normalized complex modu-
lus G*???/G0of a set of Rouse model polymer systems with
different reaction kinetics but with identical equilibrium
macromolecular distributions. Frozen Rouse chains ?un-
breakable chains ?UC?? yield a nonsymmetric Cole-Cole
plot, while the plots become close to semicircular for
fits G?t?
scission-recombination model living chains ?SR? that can
break and recombine freely. The plot for MM reaction kinet-
ics lies intermediate. However, for sufficiently rapid chemi-
cal dynamics ?large values of kDand kA?, the Cole-Cole plot
again approaches a semicircle. Thus, based on an analysis of
the shape of the Cole-Cole plot for a reversibly aggregating
system, the relaxation spectrum may be deduced as either
nearly exponential, which is characteristic of that for a living
polymer system with an effective interplay between the ki-
netic mechanism of viscoelastic relaxation and stress relax-
ation due to chain motions, or as quite disperse, indicative of
“unbreakable” macromolecules with negligible mass redistri-
bution.
To demonstrate the general trends, Fig. 6 presents the
frequency dependence of the real and imaginary components
ofthe normalizedcomplex
G????/G0 for polydisperse system following scission-
recombination dynamics for systems with average chain
length L=100. The relaxation spectrum shifts to higher fre-
quencies as the characteristic times for the scission-
recombination reaction ?cdecrease and the chemical relax-
modulus
G????/G0
and
FIG. 5. Cole-Cole plots for Rouse model polymer systems with L=100 and
with different reversible dynamics: unbreakable chains ?UCs?, monomer-
mediated polymerization ?MM?, and scission-recombination ?SR? reaction
kinetics. For both systems of living polymer chains ?MM and SR?, the
parameter ?0kDis equal to 0.01.
FIG. 6. The normalized complex modulus G*as function of dimensionless
frequency ?R=?0? for equilibrium polydisperse systems with scission-
recombination ?SR? model reaction kinetics and average chain length L
=100. Real components G? are displayed in part ?A? while imaginary com-
ponents G? are presented in part ?B?. The parameters ?c/?0are in legends,
where ?cis the mean lifetime of an average chain of length L and ?0is the
Rouse constant ?see text?. The solid curves correspond to unbreakable
chains with ?c→?.
184905-11Viscoelastic and dielectric relaxationsJ. Chem. Phys. 125, 184905 ?2006?
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Page 12

ation becomes faster. It is interesting to contrast the Cole-
Cole plot for unbreakable chains, whose dominant stress
relaxation mechanism is reptation, with those whose relax-
ation follows a Rouse model. The high frequency part of the
Cole-Cole plot for a purely reptative mechanism passes
through the point ?1,0? with an asymptotic of slope −1,31
while this part of the plot for Rouse-type dynamics is close
to a semicircular shape.
Cole-Cole plots for the normalized dielectric permittivity
are presented in Fig. 7 for the three different reaction models.
The characteristic times ?cfor the chemical reactions ?1? and
?2? in the Cole-Cole plots in Figs. 5 and 7 lie within the
range of structural relaxation times ?pfor the representative
chains with N=100. Indeed, for monomer-mediated kinetics,
?c/?p?p2/?kD?0N2??1 for p?10 since kD?0=0.01 in the
figures, whilethecharacteristic
recombination reaction is comparable with the “shortest”
structural relaxation time ?0since ?c/?0?1/?kD?0N??1. The
choice of parameters ??0kDand L? leads to Cole-Cole plots
for the dielectric relaxation of living polymer systems with
both SR and MM reaction schemes that are symmetric and
close to semicircular, while only the system with scission-
recombination dynamics exhibits a semicircular viscoelastic
Cole-Cole diagram. However, as noted above, parameters
can be chosen to superimpose the viscoelastic ?as well as the
two dielectric? functions for scission-recombination and
monomer-mediated kinetics. This identity is achieved for
monodisperse system using kD,SR?kD,MM/N.
timeforscission-
V. CONCLUSIONS
The influence of reversible chemical dynamics in linear
macromolecules on their viscoelastic functions and dielectric
relaxation is studied by analyzing two different reaction
mechanisms that correspond to monomer-mediated ?MM?
model and scission-recombination ?SR? model polymeriza-
tions. Analytical expressions for the complex viscosity and
the dielectric permittivity for Rouse chains are derived in
closed form within the bead-spring model for chains of fixed
length N. The general analysis for extremely slow chemical
relaxation in equilibrium reversibly aggregating systems
demonstrates that the only influence on the dynamic proper-
ties of the system is associated with the polydisperse macro-
molecular distribution, and the viscoelastic and dielectric
functions correspond in that limit to those for a polydisperse
system of “unbreakable” chains. When the chain scission
times are comparable to times within spectrum of the struc-
tural relaxation times or when the chemical dynamics is
rapid on the time scales of internal chain motions, the inter-
play between the chemical reaction events and the structural
rearrangements increases the stress relaxation rate, often con-
siderably, and the relaxation effectively becomes a single ex-
ponential decay.
The present approach is compared with that of Cates and
co-workers for Rouse dynamics using a different model for
the coupling between the kinetic and “regular” ?e.g., Rouse-
type or reptative? mechanisms. Within the domain of appli-
cability of the approximations of Cates and co-workers, both
models produce similar results for quite long chains. How-
ever, the predictions of the models are different for very
rapid chemical dynamics where experimental data are unava-
iable and numerical simulation results are lacking.
By proper choice of the chemical reaction rate constants
for both reaction mechanisms, the respective viscoelastic and
dielectric functions can be brought into nearly full coinci-
dence over the whole range of frequencies for equilibrium
aggregating system that follows monomer-mediated and
scission-recombination kinetics.
A closed form expression is obtained for the contribu-
tions from chains of length N to the viscoelasticity of reptat-
ing polymers that are affected by the reaction kinetics. An
analysis of the expression derived for the viscoelasticity in-
dicates that for infinitely long ?c, the theory recovers the
viscoelastic function of unbreakable reptative chains as re-
quired, while when the chain scission times are finite, the
stress relaxation is influenced by an interplay of single chain
dynamics and chemical relaxation.
ACKNOWLEDGMENTS
This research is supported, in part, by NSF Grant No.
CHE-0416017 and PRF Grant No. 41728-AC7. The authors
thank J. F. Douglas for valuable discussions.
APPENDIX: EVALUATION OF FINITE
TRIGONOMETRIC SERIES BY CONTOUR
INTEGRATION
1. The bead-spring model with reversible chemical
dynamics: Viscoelastic function
The calculation of dynamic properties, such as viscosity
and dielectric permittivity in our model requires the evalua
FIG. 7. The dependence of the imaginary component of the normalized
dielectric permittivity E? on its real part E? for equilibrium polymerization
systems with L=100: unbreakable chains ?UCs? and living polymers with
monomer-mediated ?MM? and scission-recombination ?SR? kinetics. The
living chains are described using ?0kD=0.01.
184905-12E. B. Stukalin and K. F. Freed J. Chem. Phys. 125, 184905 ?2006?
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Page 13

tion of finite series involving trigonometric functions. We use
contour integration to resolve these series into closed
form.33,34
Consider the contour C in the complex plane z as shown
in Fig. 8 where ? is an arbitrary constant between 0 and 1
while ? is taken to approach infinity. Next, we exactly evalu-
ate the finite sum necessary for calculating the contribution
?c,N
*??? from chains of length N to the complex viscosity,
?c,N
*??? =?RT
M?
p=1
N−1
1
?−1sin2??p/2N? + ?c
−1+ i?
=??RT
M?
p=1
N−1
1
sin2??p/2N? + ?N+ i?r
=?NA?0b2
24M
SN,
?A1?
with the prefactor ? defined above Eq. ?23?. To evaluate SN,
consider the contour integral
IN=?
C
cot??z?dz
sin2??z/2N? + ?N+ i?r
=?
C
FN?z?dz,
?A2?
with C of Fig. 8. The integrations along the horizontal por-
tions C2and C4vanish as ?→+? since sin??z/2N?→+?
for ?z?→+?. After noting that cot?z? and sin2?z? both have a
period of ?, it is seen that integrands for the integrals along
C1and C3are both equal to cot????/?sin2???/2N?+?N
+i?r? ??=?+i?, ? varies?, but they are traversed in opposite
directions. Hence, these two integrals compensate each other,
and the integral INvanishes,
?
C
cot??z?
sin2??z/2N? + ?N+ i?r
= 0.
?A3?
On the other hand, FN?z? ?for N?2? has simple poles at
z=0,±1,±2,...,±?N−1?,−N
=?2Ni/??arcsinh??N+i?r, all lying within the contour C.
Application of the residual theorem to Eq. ?A3? yields after
rearrangement,
and
z=±z0
with
z0
?
p=1
N−1
Res?F?z?,z = ± p? + Res?F?z?,z = − N?
+ Res?F?z?,z = 0? + Res?F?z?,z = ± z0? = 0.
?A4?
For p=0,±1,...,±N, the residues are evaluated as
Res?F?z?,z = ± p? =
1
??sin2??p/2N? + ?N+ i?r?,
?A5?
while
Res?F?z?,z = ± z0? = −N coth?2N arcsinh??N+ i?r?
???N+ i?r?1 + ?N+ i?r
.
?A6?
Hence, after substitution into Eq. ?A4?, we have the exact
result
?2/??SN+
1
???N+ i?r?+
−2N coth?2N arcsinh??N+ i?r?
???N+ i?r?1 + ?N+ i?r
1
??1 + ?N+ i?r?
= 0.
?A7?
Substituting SNinto Eq. ?A1? and using the relation M
=NM0produce Eq. ?22?. The separation of complex viscos-
ity into the real and imaginary components may be checked
by direct substitution, but they may be evaluated separately
by contour integration beginning from the expressions for
?c,N
?
and ?c,N
?
in the form of separate series.
2. The bead-spring model with reversible chemical
dynamics: Dielectric relaxation
The average over internal chain modes for the dielectric
permittivity constant as a function of frequency can be de-
rived similarly. However, contour integration does not pro-
duce a closed form since the integral of the auxiliary func-
tion over the contour is distinct from zero and is not known
in closed form. In spite of this, a quite good closed form
approximation can be derived. Consider the case of even N
first. Define the integration contour in the same way as in
Fig. 8 but with ?=0. We need to calculate
?N
*??? =?c,N
*??? − ??,N
??
=
8
?2?
p=1
N−11
p2
?1/2??−1sin2??p/2N? + ?c
?1/2??−1sin2??p/2N? + ?c
−1
−1+ i?
=
8
?2?
p=1
N−11
p2
sin2??p/2N? + ?ˆN
sin2??p/2N? + ?ˆN+ i? ˆr
=
8
?2SN?
?A8?
FIG. 8. Integration contour used for analytically evaluating the finite series
for the viscoelastic function in the Rouse model. Here ? is an arbitrary
positive constant less than unity and ? is taken to approach infinity. The
circled points are simple poles of the auxiliary contour function ?see text?.
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Page 14

by summing over odd p. Here ?ˆN=2?N, and ? ˆr=2?rfollow-
ing from the definition of Eqs. ?9? and ?13?. To evaluate SN?,
apply the contour integral
IN? =?
C?
sin2??z/2N? + ?ˆN+ i? ˆr
=?
C?
tan??z/2?dz
z2
sin2??z/2N? + ?ˆN
FN??z?dz,
?A9?
with C? of Fig. 8 and with ?=0. The integrations along
horizontal parts C2? and C4? vanish as ???→+? since
?z2?→+?. The integrals along vertical lines C1? and C3? are
equal to each other. To show this, recall that tan?z? and
sin2?z? both have a period of ?. Hence, the integrands for the
integrals along C1? and C3? have the integration variables
z1,3=±N+i?, ?N is an even number, ? is a variable? and are
given by
FN??z1,3? =tan??z1,3/2?
z1,3
2
sin2??z1,3/2N? + ?ˆN
sin2??z1,3/2N? + ?ˆN+ i? ˆr
=i tanh???/2?
?i? ± N?2
cosh2???/2N? + ?ˆN
cosh2???/2N? + ?ˆN+ i? ˆr
,
?A10?
where the plus sign before N applies for FN??z1?, while the
minus sign is for FN??z3?. After simple transformations, we
find
FN??z1? = ?N2− ?2− 2i?N?i?N????,
?A11?
FN??z3? = ?N2− ?2+ 2i?N?i?N????,
where
?N???? =tanh???/2?
?N2+ ?2?2
fN
2??? − ifN???? ˆr
fN
2??? + ? ˆr
2
,
?A12?
with fN???=cosh2???/2N?+?ˆN. Recall that in the course of
integration the contour parts C1? and C3? are traversed in op-
posite directions. Because both parts contribute equally to
the contour integral, it becomes
IN? =?
C?
−?
FN?dz = 4iN?
+?
??N????d?,
?A13?
where dz1,3=id?.
On the other hand, FN??z??N?2? has simple poles at z
=0,±1,±3,...,±?N−1? ?odd numbers? and z?=±z0? with
z0?=?2Ni/??arcsinh??ˆN+i? ˆrall lying within the contour C?.
Application of the residual theorem to Eq. ?A13? yields after
rearrangement,
?
p=1
N
Res?FN??z?,z = ± p:odd?
+ Res?FN??z?,z = 0? + Res?FN??z?,z = ± z0??
2?i?
C?
=
1
FN??z?dz.
?A14?
The residues for p=±1,...,±N are
Res?FN??z?,z = ± p:odd? = −2
?
sin2??p/2N? + ?ˆN
p2?sin2??p/2N? + ?ˆN+ i? ˆr?
,
?A15?
for p=0 the residue is
Res?FN??z?,z = 0? =
??ˆN
2??ˆN+ i? ˆr?
,
?A16?
while
Res?FN??z?,z = ± z0??
=
i?? ˆrtanh?2N arcsinh??ˆN+ i? ˆr?
4N??ˆN+ i? ˆr?1 + ?ˆN+ i? ˆrarcsinh2??ˆN+ i? ˆr
. ?A17?
After substituting into Eq. ?A14?, we have
− ?4/??SN? +
??ˆN
2??ˆN+ i? ˆr?
+
i?? ˆrtanh?2N arcsinh??ˆN+ i? ˆr?
2N??ˆN+ i? ˆr?1 + ?ˆN+ i? ˆrarcsinh2??ˆN+ i? ˆr
=2N
??
−?
+?
??N????d?.
?A18?
Inserting SN? into Eq. ?A8? and redefining the interal IN? in terms of the pair of integrals JN, JN? for real and complex components,
Eq. ?32? is obtained. Note, that due to symmetry, the integral IN? is transformed to the limits from 0 to +? for JN, and JN?, so
a coefficient of 2 appears in the final formulas.
Now consider the case of odd N. The vertical lines of the new integration contour C? are to be shifted parallel to the
corresponding lines of contour C? for even N by one unit towards the center ?recall the auxillary function FN??z? has poles at
odd N?. Analogously, the integration along horizontal parts vanishes, while integration along N−1+i? and −N+1+i? again
results in equal contributions ?as numerical analysis verifies?. The sum of these contributions equals the contour integral IN? and
is given by twice the integral along C1?. After some algebra, we obtain
IN? = 2?
C?
−?
FN−1
? dz = − 2?
+?
??N − 1?2− ?2− 2i?N − 1????N????d? = − 2JN?,
?A19?
with
184905-14E. B. Stukalin and K. F. Freed J. Chem. Phys. 125, 184905 ?2006?
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Page 15

?N???? =
tanh???/2?
??N − 1?2+ ?2?2
?x + ?ˆN?2+ y?y + ? ˆr? − i? ˆr?x + ?ˆN?
?x + ?ˆN?2+ ?y + ? ˆr?2
,
?A20?
where
=1?2sin??/N?sinh???/N?. Analogous to Eq. ?A14? for odd
N, IN? is
??ˆN
2??ˆN+ i? ˆr?
i?? ˆrtanh?2N arcsinh??ˆN+ i? ˆr?
2N??ˆN+ i? ˆr?1 + ?ˆN+ i? ˆrarcsinh2??ˆN+ i? ˆr
x=1?2?1+cos??/N?cosh???/N??
and
y
− ?4/??SN? +
+
=
1
2?iIN?,
?A21?
and hence
?N
*?? ˆr? =
?ˆN
?ˆN+ i? ˆr
+
i? ˆrtanh?2N arcsinh??ˆN+ i? ˆr?
N??ˆN+ i? ˆr?1 + ?ˆN+ i? ˆrarcsinh2??ˆN+ i? ˆr
−2i
?2JN?,
?A22?
where JN? is presented in Eqs. ?A19? and ?A20?.
3. Reptation model: Viscoelastic functions
The contribution to the complex viscosity from chains of
fixed length N for the reptation model are affected by chemi-
cal relaxation,
?c,N
*??? =
8
?2G0,N?
p=1
?
1
p2?p2?rep
−1+ ?c
−1+ i??
=
8
?2G0,N?rep?
p=1
?
1
p2?p2+ ?rep?c
−1+ i?rep???p odd?
=
8
?2G0,N?repSrep.
?A23?
Define the integration contour as in Fig. 8 with ? is taken to
approach infinity. Replace the fixed number N by a variable
integer m, which also approaches infinity, and use the inte-
gration contour Cinf. We evaluate Srepfrom the contour inte-
gral,
I =?
Cinf
tan??z/2?dz
z2?z2+ ?rep?c
−1+ i?rep??=?
Cinf
F?z?dz.
?A24?
The integrations along the horizontal and the vertical por-
tions of Cinfvanish as ?→+? and/or as m+?→+? since
the auxiliary function F?z? approaches zero.
On the other hand, F?z? has simple poles at z
=0,±1,±3,...,±p,...
?odd numbers?
and
z
=±i??rep?c
cation of the residual theorem to Eq. ?A24? yields after rear-
rangement,
−1+i?rep?, all lying within the contour Cinf: Appli-
?
p=1
?
Res?F?z?,z = ± p? + Res?F?z?,z = 0? + Res?F?z?,z
= ± i??rep?c
−1+ i?rep?? = 0.
?A25?
The residues for p=±1,...,±N are
Res?F?z?,z = ± p:odd? = −2
?
1
p2?p2+ ?rep?c
−1+ i?rep??,
?A26?
for p=0 the residue is
Res?F?z?,z = 0? =?
2
1
?rep?c
−1+ i?rep?,
?A27?
while the last two residues are
Res?F?z?,z = ± i??rep?c
−1+ i?rep??
tanh??/2??rep?c
??rep?c
= −1
2
−1+ i?rep??
−1+ i?rep??3/2
.
?A28?
Hence, after substitution into Eq. ?A25?, we have the exact
result for the sum
− ?4/??Srep+?
2
1
?rep?c
−1+ i?rep?
−tanh??/2??rep?c
??rep?c
−1+ i?rep??
−1+ i?rep??3/2
= 0.
?A29?
Substituting Srepinto Eq. ?A23? produces the expression for
the contribution from chains of length N to the complex vis-
cosity for living polymers whose dynamics is described by
the reptation model with chemical relaxation. Thus, the con-
tribution to the viscoelastic function from chains of length N
is
*??? = G0,N?
?c
?
?c,N
1
−1+ i?−2
tanh??/2??rep?c
?rep
−1+ i?rep??
1/2??c
−1+ i??3/2 ?.
?A30?
The average viscosity is given by integration over the equi-
librium mass distribution, namely,
*??? =?
0
?c
?
dNNcN?c,N
*???,
?A31?
with N-dependent characteristic times for chemical reaction
of assembly-disassembly ?cand structural rearrangement of
the chain ?rep, respectively. The dependence of ?con the
chain length is considered in Sec. II D, while the scaling of
the reptation time ?repwith N is discussed briefly in the com-
ments after Eq. ?44?.
For infinitely long ?c→?, we recover the viscosity for
the reptation model in the absence of chemical relaxation.
The contribution from chains of fixed length N is obtained in
this limit as
184905-15Viscoelastic and dielectric relaxationsJ. Chem. Phys. 125, 184905 ?2006?
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