In this Letter we show that in the rheology of electrostatically assembled soft materials, salt concentration plays a similar role as temperature for polymer melts, and as strain rate for soft solids. We rescale linear and nonlinear rheological data of a set of model electrostatic complexes at different salt concentrations to access a range of time scales that is otherwise inaccessible. This provides new insights into the relaxation mechanisms of electrostatic complexes, which we rationalize in terms of a microscopic mechanism underlying salt-enhanced activated processe

Cohen Stuart, M.A.

Gucht, J., van der

Lemmers, M.

Sprakel, J.H.B.

Spruijt, E.

English

Name not available

Relaxation dynamics at different time scales in electrostatic complexes: Time-salt superposition

Wageningen Yield 

Relaxation Dynamics at Different Time Scales in Electrostatic Complexes:
Time-Salt Superposition
Evan Spruijt,1,* Joris Sprakel,2 Marc Lemmers,1 Martien A. Cohen Stuart,1 and Jasper van der Gucht1
1Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6,
6703 HB Wageningen, The Netherlands
2School of Engineering and Applied Sciences, Harvard University, 9 Oxford Street, Cambridge, 02138 Massachusetts, USA
(Received 15 June 2010; published 8 November 2010)
In this Letter we show that in the rheology of electrostatically assembled soft materials, salt
concentration plays a similar role as temperature for polymer melts, and as strain rate for soft solids.
We rescale linear and nonlinear rheological data of a set of model electrostatic complexes at different salt
concentrations to access a range of time scales that is otherwise inaccessible. This provides new insights
into the relaxation mechanisms of electrostatic complexes, which we rationalize in terms of a microscopic
mechanism underlying salt-enhanced activated processes.
DOI: 10.1103/PhysRevLett.105.208301 PACS numbers: 83.80.Rs, 41.20.Cv, 82.35.Rs, 83.60.Bc
Electrostatically driven assemblies are ubiquitous in
natural and man-made soft materials. In nature, attractive
electrostatic forces play an important role in the conden-
sation of DNAwith histones [1], in the binding of periph-
eral proteins to membranes [2] and in the exceptional
thermal stability of enzymes in thermophilic bacteria [3].
Nature’s approaches for electrostatic assembly have been
mimicked using synthetic building blocks to create pho-
tonic materials [4], self-healing gels [5], and multilayered
capsules of various sizes and possible cargos [6]. Despite
the plethora of phenomena where electrostatic attractions
are relevant, it remains poorly understood how environ-
mental factors, such as ionic strength, influence the dy-
namics of these complexes. Moreover, there are no clear
methods available to experimentally analyze these effects.
In this Letter, we propose a new approach to study the
dynamics of electrostatic complexes. We show that prob-
ing the rheological behavior of a model electrostatic com-
plex at different ionic strengths is equivalent to probing the
behavior at different time scales. This allows for super-
imposing rheological data at different salt concentrations,
which together with the proposed model leads to a new
way to systematically investigate the effects of temperature
and salt concentration on relaxation behavior and mechani-
cal properties of these complexes.
Superposition is used as a tool in rheology to probe
dynamical processes in a wide variety of soft materials,
thereby extending our understanding of their dynamics.
For example, time-temperature superposition [7] relies on
the acceleration of all activated processes at high tempera-
tures, enabling probing of longer effective time scales at
high temperatures. Analogously, strain-rate-frequency su-
perposition allows one to probe low-frequency structural
relaxation of soft solids, such as concentrated suspensions
and foams, by enhancement of activated processes by
external force [8]. In this Letter, we apply a similar ap-
proach to study the dynamics of a yet underrepresented
class of soft materials: electrostatic complexes. In addition
to the model materials presented here, our approach could
easily be extended to other electrostatic complexes, both
natural and man made, to help elucidate unresolved issues
in the dynamics of electrostatic complexes.
We use complexes of two equally long and oppositely
charged polyelectrolytes, namely, poly(acrylic acid), a pol-
yanion (effective pKa 5.0–5.5 for csalt ¼ 2:0–0:10 M), and
poly(N;N-dimethylaminoethyl methacrylate), a polycation
(effective pKa 8.0–7.5). The phase behavior of these com-
plexes in aqueousmonovalent electrolyte solutions has been
worked out experimentally [9]. We use polyelectrolytes
with a degree of polymerization of 20, 50, 150, or 510
(PDI< 1:2) at pH 6.5 where the polycation and polyanion
have equal charge densities. When mixed together, the
mixture phase separates into a dilute phase and a dense,
viscoelastic phase that contains about 70 wt % water. Our
polyelectrolyte complexes are prepared at different salt
concentrations but always at 1:1 stoichiometry of polymeric
charges. In addition, another set of polyelectrolytes, synthe-
sized by atom transfer radical polymerization, with a degree
of polymerization of 100 (PDI< 1:1) is used [10]. These
polyelectrolytes contain nopH-dependent charges, but per-
manently charged sulfonate and quarternary ammonium
groups. Complexes with these polyelectrolytes are made
in the same way as described above. All rheological mea-
surements are carried out on an Anton Paar MCR501 series
stress-controlled rheometer using a cone-plate geometry.
The frequency sweeps shown in Fig. 1 are typical ex-
amples for our model polyelectrolyte complexes in a re-
gime of linear deformation. The rheological response of
polyelectrolyte complexes to an oscillatory strain depends
strongly on salt concentration. At low salt concentration
the response is similar to that of soft solids [8], whereas the
response is liquidlike at high salt concentration. These
observations are in line with results on biological electro-
static complexes [11].
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Salt is known to accelerate dynamic processes in poly-
electrolyte complexes. This leads to faster equilibration and
eventually to hysteresis-free cycles of compression and
disruption of polyelectrolyte complexes, as was shown
using atomic force microscopy [12]. In well-known super-
position methods in rheology, like time-temperature super-
position and strain-rate-frequency superposition, such
acceleration of dynamic processes provides the basis for
probing dynamic processes on a range of time scales that is
otherwise experimentally inaccessible. Couldwe use analo-
gous rescaling of measurements shown in Fig. 1 in order to
get more insight into the dynamics of these polyelectrolyte
complexes as well? Figure 2(a) shows that the frequency
sweeps for different salt concentrations can indeed be
rescaled onto a single mastercurve using salt concentration
dependent shift factors for the frequency, such that
!scaled ¼ !c, and for the moduli, such that G0scaled ¼
G0=Gc and G00scaled ¼ G00=Gc. Shift factors c and Gc are
chosen such that the crossover between storage and loss
modulus, which is related to a typical relaxation time in
many rheological models, occurs at a rescaled frequency
and modulus of unity. Figure 2(a) shows that the relevant
dynamic processes in these polyelectrolyte complexes re-
spond similarly to salt concentration, which enables us to
probe these complexes over seven decades in frequency.
The limits of the accessible dynamic range are dictated
by a combination of instrumental and thermodynamic fac-
tors. The upper limit is given by the maximum accessible
frequency of the rheometer and a theoretical minimum salt
concentration close to zero. The lower limit is given by the
critical salt concentration, beyond which polyelectrolyte
complexes no longer form. In this case, we use a creep
test to probe the lowest frequencies and hence longest
time scales. Data of creep compliance as a function of
time are converted into frequency-dependent storage and
loss moduli using a direct mathematical procedure [13].
If the apparent relaxation times, c, and moduli, Gc, are
to provide a meaningful measure for the relevant dynamic
processes in electrostatic complexes, we expect that they
can be used to rescale nonlinear flow experiments as well.
Indeed, Fig. 2(b) shows that both linear and nonlinear
regimes of strain rate sweeps at various salt concentrations
can be rescaled using the same shift factors as for Fig. 2(a).
The flow curves all start with a Newtonian regime, fol-
lowed by a nonlinear regime of strong shear thinning,
which becomes more pronounced at low salt concentra-
tions. This regime is characterized by large intermittent
stress fluctuations, reminiscent of other associative net-
works [14], for which the steady state is taken as the
average.
As shown in the inset of Fig. 2(b), a similar rescaling can
also be applied to polyelectrolytes of different chain
10−1
101
10−2 100 102
10−5
10−3
10−1
101
10−5 10−3 10−1 101 103
G
' /
G
c 
,
 
 
 
G
'' /
G
c
N = 510
ω τ
c
10−6
10−3
100
10−6 10−2 102
σ
 /G
c
γ.τ
c
1 Ν = 510
150
100
50
20
10−6
10−4
10−2
100
10−6 10−4 10−2 100 102
σ
 /G
c
γ.τ
c
1
a
b
N = 510
FIG. 2 (color online). (a) Rescaled frequency sweep of a N ¼
510 polyelectrolyte complex at  ¼ 2%. Storage moduli are
shown using circles and crosses, loss moduli using triangles
and pluses. Filled symbols are converted from a creep test.
Solid lines are drawn to guide the eye. (b) Rescaled flow curves
for N ¼ 510 polyelectrolyte complexes. The inset shows re-
scaled flow curves for N ¼ 20 (4), 50 (), 100 (), 150 (h),
and 510 () polyelectrolyte complexes. For reasons of clarity,
one per five data points is shown in the inset.
101
103
100 101 102 103
ω  /   rad s−1
103
104
101
103
104
105
G
' ,
G
'' /
 P
a
0.50 M
0.75 M
1.05 M
1.20 M
FIG. 1 (color online). Frequency sweeps of a N ¼ 510 poly-
electrolyte complex at  ¼ 2% and salt concentrations as in-
dicated in the figure, showing storage () and loss (4) moduli.
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lengths, and even to a different set of polyelectrolytes
(N ¼ 100). A close inspection of the curves reveals that
the onset of the nonlinear regime is not exactly the same for
all chain lengths. This can be explained by the fact that
polymer concentrations in these complexes are not com-
pletely independent of chain length [9], and entanglements
become important for longer polymers.
Our observation that both linear and nonlinear rheolog-
ical data can be superimposed by using two salt-dependent
shift factors, c and Gc, suggests that a simple mechanism
underlies the dynamics of electrostatic complexes. Figure 3
shows that there is a strong dependence of the apparent
relaxation time, c and a much weaker dependence of the
modulus, Gc, on salt concentration for all chain lengths.
The depicted dependences of c and Gc are closely
related to the stress relaxation mechanism inside poly-
electrolyte complexes. These complexes can be seen as
concentrated two-component polymer solutions. Stress
relaxation occurs by movement of chains, which is hin-
dered by associations in the form of ion-ion pairs between
two oppositely charged polyelectrolytes (sticky points).
This suggests that the dynamics of chains with ‘‘sticky
points’’ should apply, for which [15]
c  Np0ðcsaltÞ (1)
where N is the polymer chain length, p is the polymer
volume fraction and 0 is an effective friction coefficient,
which depends on salt concentration. The magnitude of 
depends on the regime of relaxation: for Rouse-like relaxa-
tion  ¼ 2, for reptation  ¼ 3:4. The magnitude of 
depends somewhat on solvent quality, but is typically close
to unity [15]. The breaking of association points is related
to the dissociation rate of ion-ion pairs. We assume this is
an activated process, governed by an energy barrier, Ea
0ðcsaltÞ ¼ 1!0 exp

EaðcsaltÞ
kBT

: (2)
Reorganization of associations encompasses the break-
ing of polymeric ion-ion pairs, separating each two ionic
groups, thereby surrounding them with monovalent coun-
terions, and reforming polymeric ion-ion pairs at different
positions. This route contains an energy barrier that can be
estimated as the difference between the electrical free
energy of four separated ionic groups of charge e in
an electrolyte solution, and the Coulombic energy of
two ion-ion pairs in contact: Ea  Ecorr  ECoul. The elec-
trical correlation energy is approximated as Ecorr 
2e2=4		0 / ﬃﬃﬃﬃﬃﬃﬃﬃcsaltp [16], where  is the inverse
Debye length. The Coulombic energy can be written as
ECoul ¼ e2=2		0d, where d is the contact distance of
charged groups in an ion-ion pair. Combining this with
Eqs. (1) and (2), we find
c / 1!0N
p exp½aðTÞ ﬃﬃﬃﬃﬃﬃﬃﬃcsaltp þ bðTÞ (3)
where csalt is the ionic strength in mol=L for a 1:1 electro-
lyte, aðTÞ ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2000NAp e3=2ð		0kBTÞ3=2 and bðTÞ ¼
e2=2		0kBTd.
Equation (3) predicts that the activation barrier depends
on the square root of salt concentration, leading to a
stretched exponential dependence of c on csalt. We have
previously found that also p depends weakly on csalt [9].
This dependence is, however, significantly weaker than
the variations in c shown in Fig. 3. The relaxation times
presented in Fig. 3(a) have therefore been fitted to
Eq. (3), using only an adjustable prefactor {A ¼ !10 N
p exp½bðTÞg, and a fixed value of 11 for aðTÞ. We find an
excellent agreement between our data and the presented
model. Moreover, the value of aðTÞ we find is quantita-
tively in agreement with the theoretical prediction of
11.3 for 	 ¼ 45 and T ¼ 293 K, which is a reasonable
assumption for concentrated polyelectrolyte complexes
[17]. Finally, the prefactors, A, scale according to N,
with  ¼ 2:5, as expected.
A key assumption in our conceptual picture is that an
activated dissociation of ion-ion pairs governs the relaxa-
tion dynamics of electrostatic complexes. Consequently, c
should be a function of temperature as well. This is verified
in a time-temperature superposition experiment, where we
determine relaxation times as a function of temperature
[Fig. 3(a), inset]. Assuming that 	 is independent of T [17],
we expect c to scale with expðT1Þ or expðT1:5Þ as
the two limiting cases [Eq. (3))]. The fits in the inset of
Fig. 3(a) for expðT1Þ (solid lines) and expðT1:5Þ (dashed
lines), show that it is experimentally not possible to dis-
tinguish between these limiting cases for aqueous systems.
10−3
10−1
101
103
τ c
/s
N = 510
150
100
20
50
104
105
0 0.5 1 1.5
c
salt    / M
G
c
 
 
 
/P
a
a
b
10−3
10−1
3 3.2 3.4 3.6
c
salt
 = 1.2 M
c
salt
 = 0.85 M
(1000 / T )  / K−1
τc /  s
FIG. 3 (color online). (a) Relaxation times and (b) moduli shift
factors, for rheological data of polyelectrolyte complexes with
N ¼ 20 (4), 50 (), 100 (), 150 (h), and 510 (). The solid
lines in (a) are fits to Eq. (3); the inset shows relaxation times for
different temperatures at two indicated salt concentrations.
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In contrast to the relaxation times, the moduli are found
to depend only weakly on csalt. This is simply due to the
coupling between salt concentration and overall polymer
density, as was shown in [9]. We find a linear relation
between the modulus and p=N (data not shown), as
expected for unentangled polymer solutions in which
each chain contributes kBT to the modulus [15].
The presented analysis of the dynamics in polyelectro-
lyte complexes provides a general tool to study dynamics
in other soft materials that are assembled by electrostatic
complexation. To verify this generality, we carry out the
same approach on a transient network formed by telechelic
polymers [5]. This network is made from ABA triblock
copolymers with charged end blocks that are complexed
with oppositely charged homopolymers. Electrostatic
complexes form the nodes of this network, while the
neutral and soluble B blocks form the links between the
nodes. We find that frequency sweeps of these networks
can be rescaled onto a single mastercurve as well, as shown
in Fig. 4. Using superposition we can access over eight
decades in frequency and obtain the typical relaxation
times at different salt concentrations.
The dynamics of electrostatically assembled materials is
strongly dependent on salt concentration. A detailed under-
standing of these effects requires an effective method to
probe the widely varying dynamic behavior of electrostatic
assemblies at different salt concentrations. By rescaling
rheological data, one can probe linear and nonlinear me-
chanical properties of these materials over a wide range of,
otherwise inaccessible, time scales and study the salt de-
pendence of relaxation mechanisms. This approach offers
new ways to elucidate unresolved issues in the dynamics of
electrostatic complexes: for example, those concerning the
formation mechanism of electrostatic complexes [18] and
those concerning the dynamics of biological complexes,
such as the DNA-histone complex, at rest or under me-
chanical and chemical stresses during cell division [19].
The authors gratefully acknowledge Frans Leermakers
for fruitful discussions. E. S. and J. S. acknowledge the
Netherlands Organisation for Scientific Research (NWO)
for financial support.
*evan.spruijt@wur.nl
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10−4
10−2
100
0 0.5 1 1.5 2
c
salt   / M
τ
c
 / s
10−4
10−2
100
10−5 10−3 10−1 101 103
G
' /
G
c
 
 
,
 
 
 G
'' /
G
c
ω τ
c
FIG. 4 (color online). Rescaled frequency sweeps of a tele-
chelic network of PSPMA28PEO230PSPMA28 and PAH171 at
 ¼ 5%. Storage moduli are shown using circles; loss moduli
using diamonds. The inset shows c as a function of csalt.
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Relaxation dynamics at different time scales in electrostatic complexes: Time-salt superposition

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