Journal of Non-Crystalline Solids 235±237 (1998) 717±722
Simulation of polymers in dilute solution under elongational
¯ow
J.G. Hern
andez Cifre, J. Garcõa de la Torre
*
Departamento de Quõmica Fõsica, Facultad de Quõmica, Universidad de Murcia, Campus de Espinardo, 30071 Murcia, Spain
Abstract
The behavior of polymer chains, modeled as chains of ®nitely extensible non-linear elastic springs, in solutions undergoing elongational ¯ow, has been studied by Brownian dynamics simulation. The coil±stretch transition is observed
_ exceeds some critical _c that is determined, in steady-state condition, as a function of
when the elongational rate, ,
chain length. We have simulated the time dependence of polymer dimensions when _ is increased from below to above
_c , and then decreased to the initial value. In the coil-to-stretch transition there is an induction time, required to observe
the onset of the transition, which varies among the chains in a sample. However, the stretch-to-coil transition follows
the same path for all the molecules. The observable properties (sample averages) show a hysteresis cycle that is intensi®ed by the e€ect of hydrodynamic interaction (HI). Ó 1998 Elsevier Science B.V. All rights reserved.
1. Introduction
When a dilute polymer solution is subjected to
elongational ¯ow, with a ¯ow rate that exceeds
some critical _c , the solution properties show a
change. This fact is due to a transition of the polymer chains from the coil conformation to a
stretched state. The so-called coil-stretch transition
was predicted by DeGennes [1] and experimentally
observed by Keller, Odell and coworkers [2,3] and
by Fuller, Leal and coworkers [4,5]. Both the laboratory realization and the theoretical description
of the phenomenon are dicult, and therefore various aspects still remain obscure. On the theoretical side, it is generally accepted that the
DeGennes' theory predicts and qualitatively describes the transition, but some of its important aspects have been reasonably questioned, by Bird
*
Corresponding author. Tel.: +34-68 364 148; e-mail:
[email protected].
0022-3093/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved.
PII: S 0 0 2 2 - 3 0 9 3 ( 9 8 ) 0 0 5 1 8 - 3
et al. [6], particularly the time-evolution of the
polymer properties.
In addition to essentially experimental and theoretical work, a third method is computer simulation.
Our group has pioneered the Brownian dynamics
simulation of polymer solutions under ¯ow [7],
where non-averaged, ¯uctuating hydrodynamic interaction (HI) is included (as it should be). We have
already employed this technique for studying polymers in elongational ¯ow [8±12]. In the present
work, we concentrate on the time-resolved response
of polymer conformation and properties when the
_ is greater than and then less
elongational rate, ,
than _c . In the simulation we observe the time-evolution of each chain in the sample, which allows a
deeper understanding of the coil±stretch transition.
2. Methods
We consider a dilute polymer solution subjected
to an elongational ¯ow with a velocity ®eld
718
J.G. Hern
andez Cifre, J. Garcõa de la Torre / Journal of Non-Crystalline Solids 235±237 (1998) 717±722
1
1
_
_
vz ˆ ÿ z:
…1†
vy ˆ ÿ y;
2
2
This ®eld is produced by the opposing jets device
pioneered by Keller, Odell and coworkers [2,3].
The polymer molecules are modeled as beadand-spring chains with FENE (®nitely extensible
non-linear elastic) springs, for which the spring
force is given by [13]
_
vx ˆ x;
Fˆÿ
1ÿ
H
Q
2 Q;
ter with an asterisk). Dimensions, energy and force
are reduced by b, kT , and kT =b, respectively, and
reduced time is given by
t ˆ t=…fb2 =kT †
…3†
and the elongational rate is reduced as a reciprocal
of time
_ 2 =kT †:
_ ˆ …fb
…2†
Qmax
where Q is the spring vector and Qmax is the maxQ
1; Eq. (2) reduces
imum spring length. At Qmax
to the force law of the Hookean (Gaussian) spring,
F ˆ ) HQ, where the spring constant is
H ˆ 3kT =b2 , and b2 ˆ hQ2 i0 is the mean square
spring length in the limit of low elongation (in
our case, in the absence of ¯ow). For some purposes, we alternatively used the simplest model,
namely the Rouse chain, with Hookean springs.
The dynamics of the polymer chains is simulated using the Brownian dynamics algorithm of
Ermak and McCammon [14] as modi®ed by Iniesta and Garcõa de la Torre [15]. The position of
the beads, ri , i ˆ 1; . . . ; N ; after a time step, Dt,
are calculated from the initial positions r0i , the
spring forces Fi , i ˆ 1; . . . ; N ÿ 1; formulated in
Eq. (2) (with Qj ˆ rj‡1 ÿ rj ), the di€usion tensor
Dij and the ¯ow velocities at the bead centers, given by Eq. (1). The diagonal components of the diffusion tensor are given by Dii ˆ …kT =fi †I where fi
is the bead friction coecient. An important aspect of our simulation is to ascertain the e€ect of
HI and for this purpose the di€usion tensor is expressed in terms of the Oseen tensor, Tij as
Dij ˆ kT Tij [16]. If HI is neglected, Dij ˆ 0. For
the bead friction we use a Stokes coecient
f ˆ 6pg0 r; where r ˆ 0:257b, which corresponds
to a HI parameter, h ˆ 0:25. It should be remarked that Brownian dynamics simulation of
long chains with HI are time consuming: CPU
time is approximately proportional to N 3 and, furthermore, longer chains require longer trajectories,
with a larger number of simulation steps.
For the Brownian dynamics procedure and the
presentation of results, it is convenient to employ
reduced, dimensionless quantities (denoted hereaf-
3. Results
3.1. Steady-state properties. Critical elongational
rate
The critical elongational rate, _c , is determined
by means of experiments in which polymer properties are monitored in steady-state regime with a
_ The experiments are carried out for varigiven .
_ For low values of ,
_ the polymer properties
ous .
remain very close to those in quiescent solution.
However, when _ reaches the critical value, _c ,
the polymer properties change dramatically. Thus,
_c can be found in a series of experiments in which
its value is bracketed by experiments in which the
polymer either stretches or remains as a coil.
The results for _c are plotted vs. N in Fig. 1. A
least-squares ®t gives the result _c ˆ …14:1
1:1†N ÿ…1:550:03† . This is in agreement with the scaling law observed for polymer chains in theta solvents, _c / M ÿ3=2 [3,5]. The same procedure was
indeed employed in the determination of _c for
Gaussian chains [8], with a very similar result. In
the present paper we are mainly concerned with
the time-dependent behavior, as described below,
so that the _c vs. N relationship are used just for
the determination of _c .
3.2. Time-dependent behavior
To describe the properly dynamic kinetic aspects of the coil±stretch and stretch±coil transitions, we devised and carried out some computer
experiments in which the time dependence of polymer properties was followed.
A simple but illustrative situation is that of a
_ above _c . Rather than
pulse of elongational rate, ,
J.G. Hernandez Cifre, J. Garcõa de la Torre / Journal of Non-Crystalline Solids 235±237 (1998) 717±722
719
Fig. 1. Dimensionless critical elongational rate, _c , plotted vs.
chain length, N . Results for FENE chains with Qmax ˆ 10b.
The straight line is the least-squares ®t, with slope ÿ1:55 0:03.
an inception and cessation of ¯ow from and to
_ ˆ 0, the simulation consisted ®rst of an `equilibration' period for 100 units of time at a ¯ow rate
_low ' 0:8_c . This was followed by an instantaneous jump to a ¯ow rate above 20% above _c ,
_high ' 1:2_c , which was kept constant for 400 time
units (t.u.), with a ®nal jump down to the initial
_low , which was maintained for 400 more t.u.
This ¯ow protocol was applied to each molecule in a sample of Nmol molecules, which were
generated initially in random-coil conformation.
The properties of each individual chain were monitored as a function of time, and for each time the
sample averages of the properties were evaluated.
The most illustrative and relevant property is the
square radius of gyration, S 2 . For extensional
¯ows, S 2 is preferred to the end-to-end distance,
since the latter has low values not only for random coils, but also for extended hairpin-like conformations that may be found in ¯ows of this
type.
Fig. 2 shows the individual square radius of gyrations, S 2 , for each chain in the sample, as a function of time in two runs, with and without HI. For
the ®rst, short equilibration interval at _low , the individual S 2 values show ¯uctuations similar to
those in quiescent solution, around the no-¯ow
value, hS02 i ' …N ÿ 1†=6 ' 3 (Note that these val-
Fig. 2. Results for the up-and-down experiment. The arrow indicates the time (t ˆ 100) at which _ is increased from _low to
_high . Values of …S 2 † of individual chains are plotted vs. time.
Data: Nmol ˆ 20, N ˆ 20, Qmax ˆ 10b. (a) Without HI,
_c ˆ 0:072, _low ˆ 0:065, _high ˆ 0:100. (b) With HI, _c ˆ 0:133,
_low ˆ 0:110, _high ˆ 0:160.
ues seem to be very close to zero in Fig. 2 and
the following ®gures, as a consequence of the
range that has to be covered by the ordinate axis).
The absence of polymer deformation at ¯ow rates
quite close to _c but less than it, is a known feature
of elongational ¯ow.
After the jump to _high , we can observe the
sudden coil-to-stretch transition of the individual
chains. A salient feature is that there is a variability, from molecule to molecule, in the instants at which the transition occurs. Sooner or
later, most chains reach a stretched conformation
in which S 2 ¯uctuates around a well de®ned
value.
When _ is instantaneously decreased to the
original value, _low , S 2 decays to the initial value,
but the di€erence with the rise step is that the decay is immediate and simultaneous for all the
720
J.G. Hern
andez Cifre, J. Garcõa de la Torre / Journal of Non-Crystalline Solids 235±237 (1998) 717±722
molecules in the sample. There is no lag time, and
the decay trace is nearly the same for all the molecules. The time required for an individual molecule to experience the stretch-to-coil transition is
similar to that of the coil-to-stretch (measured
from the onset); they are both several times the
longest relaxation time but close to it in order of
magnitude.
As commented in the introduction, a key aspect
in the coil±stretch transition is the role of HI.
Comparing Fig. 2(a) and (b), we see that the behavior is qualitatively similar in both the rise
and decay intervals. However, one notes that the
transitions in the HI case are steeper and sharper,
and that the distribution of the induction times for
the coil-to-stretch transit is somehow broader with
HI (incidentally, 2 of the 20 chains failed to
switch).
We also carried out computer experiments with
a more elaborate ¯ow program in which _ was increased and then decreased, between values of _
below and above _c . This was done in a step-bystep manner, with step duration much longer than
the longest relaxation time of the chains. The elongational rate went from a value _low < _c to
_high > _c in steps of 100 t.u., except for the highest
step at _high , which was maintained for 400 t.u. The
downward run back to _low was symmetric with the
upward run.
The results of this simulation are plotted in
Fig. 3. As in the previous experiment, we note that
the coil-to-stretch transition of the individual molecules occurs at quite separate instants and there_ On the other hand, the traces
fore at di€erent .
described by all the molecules in the downwards
run are analogous: S 2 begins to decrease immediately after _ is decreased, without any induction
time.
In laboratory experiments, what one actually
observes are sample averages. Thus, we calculated
the mean hS 2 i over the molecules in the sample at
each step, with a further averaging over various
observations at the same _ (i.e. along a step in
the ¯ow program). The resulting variation of
hS 2 i with _ is plotted in Fig. 4. In the upward
run, an increase in hS 2 i over hS 2 i0 is noted when
_ suciently exceeds _c . The hS 2 i value of the
stretched state is reached and in the downward
Fig. 3. (a) Flow program of the stepwise simulation, for the noHI case, with _low ˆ 0:600 and _high ˆ 0:100. The dashed line
marks the critical value, _c ˆ 0:072. For the simulation with
HI the ¯ow program is similar, with _low ˆ 0:110, _high ˆ 0:160
and _c ˆ 0:133. (b.c) Time evolution of the individual S 2 values.
Chain data are as in Fig. 2.
run the hS 2 i vs. _ curve follows a di€erent path,
which gives rise to a typical hysteresis cycle.
In regard to the e€ects of HI, we ®rst note in
Fig. 3 that the induction times for the coil-tostretch transition are longer and more broadly distributed in the simulation with HI than without
HI. The time decay of S 2 in the no-HI case is
roughly linear and follows the same trend as the
_ For the HI case the initial part of
decrease of .
the decay is slower, although at the end of the simulations, at _low , all the chains return to the coil
conformation. These di€erences are clearly manifested in the hysteresis cycle (Fig. 4) which show
a more pronounced e€ect (a larger cycle area) in
the simulation including HI.
J.G. Hern
andez Cifre, J. Garcõa de la Torre / Journal of Non-Crystalline Solids 235±237 (1998) 717±722
721
induction times that vary greatly from chain to
chain, as in our computer experiments (see our
Figs. 2 and 3). The variability is attributed to the
rich diversity of conformations of what we generically call the random coil state of ¯exible polymer
chains [17]. For some speci®c conformations, the
unraveling and stretching in the extensional ¯ow
may be easier for some chains than for others. Actually, in the laboratory experiments, a few di€erent unraveling pathways were detected, thanks to
the direct visualization of the intermediate conformations. Such di€erent mechanisms give extension
vs. residence time traces which display qualitative
di€erences. On the other hand, all the chains in
our simulations have the same behavior. There
may be various reasons for this similarity, perhaps
related to the simplicity of our chain model (relatively short FENE chains without excluded volume, or to the special propensity of DNA for
some speci®c conformations). Hopefully, more detailed simulations along the line initiated in this
paper, may be helpful in clarifying such questions.
Fig. 4. Variation of hS 2 i with _ along the upward and downward runs, showing the hysteresis cycles, with and without HI.
4. Discussion
The dynamic coil-stretch transition, as simulated in our time-resolved computer experiment,
have some resemblance to the predictions of the
classical DeGennes' paper [1]. Thus, we observe
a delay in the response of the molecules after _
_ and the plots of the properties
has surpassed ,
vs. _c display a hysteresis cycle, as predicted in that
theory. Furthermore, we note other features that
had not been explicitly described in previous work.
Particularly, we note that the stretch-to-coil and
coil-to-stretch transitions di€er in aspect (particularly in their time dependence), which suggests different mechanisms, both being more complex than
the transition above a barrier of free energy, as in
the DeGennes' description.
In a very recent work Perkins et al. [17] have visualized in the laboratory the coil-to-stretch transition of individual DNA molecules, observing
Acknowledgements
This work has been supported by grant PB961106 from the Direcci
on General de Investigaci
on
Cientõ®ca y Tecnica. We also acknowledge support
from the Consejerõa de Educaci
on, Comunidad
Aut
onoma de la Regi
on de Murcia, through grant
FI-CON96/9, and a predoctoral fellowship awarded to J.G.H.C.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
P.G. DeGennes, J. Chem. Phys. 60 (1974) 5030.
D.P. Pope, A. Keller, Colloid Polym. Sci. 256 (1978) 751.
A. Keller, J.A. Odell, Colloid Polym. Sci. 263 (1985) 181.
G.G. Fuller, L.G. Leal, Rheol. Acta 19 (1980) 580.
C.A. Cathey, G.G. Fuller, J. Non-Newtonian Fluid Mech.
34 (1990) 63.
J.M. Wiest, L.E. Wedgewood, R.B. Bird, J. Chem. Phys. 90
(1989) 587.
F.G. Diaz, J. Garcõa de la Torre, J.J. Freire, Polymer 30
(1989) 259.
J.J. L
opez Cascales, J. Garcõa de la Torre, J. Chem. Phys.
95 (1991) 9384.
J.J. L
opez Cascales, J. Garcõa de la Torre, J. Chem. Phys.
97 (1992) 4549.
722
J.G. Hern
andez Cifre, J. Garcõa de la Torre / Journal of Non-Crystalline Solids 235±237 (1998) 717±722
[10] J.J. L
opez Cascales, J. Garcõa de la Torre, J. Non-Cryst.
Solids 172±174 (1994) 823.
[11] K.D. Knudsen, J.G. Hern
andez Cifre, J. Garcõa de la
Torre, Macromolecules 29 (1996) 3603.
[12] K.D. Knudsen, J.G. Hern
andez Cifre, J. Garcõa de la
Torre, Colloid Polym. Sci. 1997, in press.
[13] R.B. Bird, C.F. Curtiss, R.C. Amstrong, O. Hassager,
Dynamics of Polymeric Liquids: Kinetic Theory, 2nd ed.,
vol. 2, Wiley, New York, 1987.
[14] D.L. Ermak, J.A. McCammon, J. Chem. Phys. 69 (1978)
1352.
[15] A. Iniesta, J. Garcõa de la Torre, J. Chem. Phys. 92 (1990)
2015.
[16] H. Yamakawa, Modern Theory of Polymer Solutions,
Harper and Row, New York, 1971.
[17] T.T. Perkins, D.E. Smith, S. Chu, Science 276 (1997) 2016.