Table Of ContentInduced Orientational Effects in Relaxation of Polymer Melts
J. M. Deutsch and J. H. Pixley
Department of Physics, University of California, Santa Cruz, California 95064
(Dated: January 25, 2009)
Westudystressrelaxationinbi-disperseentangledpolymersolutions. Shorterchainsembeddedin
amajorityoflongeronesareknowntobeorientedbycouplingtothem. Weanalyzethemechanism
forthisbothbycomputersimulationandtheoretically. Weshowthattheresultscanbeunderstood
intermsof stressfluctuationsinapolymermelt andchain screening. Stressfluctuationsare frozen
on the relaxation time of the longer chains, and these will induce strong orientational couplings in
theshorter chains.
9
0 PACSnumbers:
0
2
Polymer molecules in concentrated solution or melts which leads to a small compressibility and chain screen-
n
a haveveryunusualvisco-elasticproperties[1]andarethe ing,thatis,idealchainstatisticsonalargescale[2]. The
J subject of much research [2, 3]. After shearing such a maintenance of screening leads to an increase in free en-
5 system of long chains, the modulus has a long plateau ergy when the chain moves out of a tube [7, 8, 9]. To
2 that depends strongly on chain length, yet the height of maintain constant density, this produces a stress field
the plateau appears almost independent of it. whose strength grows linearly with the new tube. This
] The most successful model in explaining the experi- implies that the motion will be activated as the original
t
f mental data is the “reptation” model of de Gennes [4]. tube is vacated. The stress is relieved by other chains
o
Theinitialworkconcentratedonthecaseofapolymerin moving into or out-of the stressed regions.
s
t. a backgroundnetwork such as a gel which is most easily Thismechanismcauseschainstofollowthestresstrails
a understood, but since then, reptation has had many ap- ofotherchainsbecauseasachainleavesatube, itleaves
m
plicationstopolymersolutions. Theideaisthattopolog- a vacancy whichis attractiveto other chains. The mem-
- ical effects restrict the movement of a polymer to a tube ory for this stress mechanism will decay with a longer
d
n by virtue of entanglements thereby suppressing trans- relaxation time than the time for the chain to vacate a
o verse motion for scales larger than the “tube diameter”. tube. This was proposed as an explanation for the dif-
c However chain ends can move freely forming new tube. ference between diffusion and stress relaxation[10].
[ A polymer then has a curvilinear diffusion coefficient in-
The effects of interaction of chains can perhaps most
versely proportional to its length L, so that the time to
1 easily be seen by the fact that they induce partial align-
3
form a new tube L . This is also expected to be the
v ∝ ment with each other. Many experiments have shown
0 relaxation time in a polymer melt. that there is strong induced orientational coupling of
3 Tocomparethis toexperimentaldataonthe viscosity,
smaller chains under stretching of a system containing
9 we recall that the elastic modulus G(t) is related to the
mainly longer ones, for example in bi-disperse polymer
3 viscosity η through
melts[11]wheretheratiooftheorientationofthesmaller
.
1
∞ chainstothe longerhostchainswasfoundtobeǫ=0.45
0
η G(t)dt. (1) which is quite large. It is the purpose of this work to
9 ∝Z0 show that this phenomenon can at least partially be ac-
0
: Because the plateau is independent of L, this gives a counted for by the same mechanism responsible for the
v viscosity L3. The experimental data in the entan- many-body effects related to chain screening mentioned
Xi gled regim∝e fits better with L3.4. The explanation for above. Earlier work looking at this by simulation tech-
r this discrepancy has yielded a number of theoretical ex- niquescouldreproducesimilarorientationalresponsesby
a planations. The two main reasons are both legitimate means of a direct simulation of these systems [12]. This
extensions of de Gennes original idea. First, finite size work should help to clarify the origin of this effect theo-
effects [5, 6], such as tube length fluctuations, yield a retically.
largerslope for this quantity. Second,many body effects We start by performing computer simulations similar
that give rise to chain screening [7, 8, 9] require modi- to those done earlieronmanybody effects for long poly-
fication of the original single chain model. Researchers mer chains [7, 8]. Consider two chain lengths L1 and
have been able to fit experimental data separately using L2. We will assume reptation dynamics [13] for moving
either idea and it is clear that in fact both are real and chains. Chains live on a cubic lattice in a B B B
× ×
experimentally contribute. box. Ineachstep,the headortailmonomerofachainis
Thatmanybodyeffectsmustbeimportantcanbeseen chosenat random, and then an attempt is made to reat-
by considering what happens if the chains were all inde- tach it, in a random orientation, to the other end. It is
pendent. Then density fluctuations of the system would rejected only if the new lattice position is occupied. We
become very large and the system would be very com- assumeacurvilineardiffusioncoefficient 1/L ,i=1,2,
i
∝
pressible. Arealsystemhasexcludedvolumeinteractions which is expected from reptation [4]. Translated to a
2
Monte-Carlo simulation, we say that the probability of length L2 = 256 and one chain of length L1 = 64, with
choosing a chain is 1/L . As discussed above, this a number density of 0.58. We monitor two kinds of
i
∝ ≈
system had been analyzed numerically in considerable quantities. First we monitor l(t), the average amount of
detail [8] and shows activated motion in agreement with theoriginaltubeleftattimet. Wealsomeasurethebire-
the theoreticalpredictions [8, 9] giving a relaxationtime fringencewhichisthedeviationofpolymerlinksfroman
exp((L/L∗)2/3) for very long chains, where L∗ is a isotropic state. We define this as
∼
constant that depends on the density of the system. In
arealmelt itwouldmoredirectlydepend onthe plateau n 2nz nx ny (2)
≡ − −
modulus of the melt.
wheren ,n ,andn arethe fractionofmonomersgoing
For the simulation to be relevant it must be in the x y z
in the x, y, and z directions respectively.
same universality class for statics and dynamics as that
In the Doi Edwards model, the number of entangle-
of a real melt. However the lattice model does not al-
ment points still remaining after time t is a measure of
lowmonomerstosimultaneouslyrearrangethemselvesas
thestressofthesystem. Henceintheirmodelthe length
monomers reptate to new sites. In a real system, such
of original tube l(t) is proportional to the elastic modu-
localrearrangementwill happenas is requiredby the in-
lus. However if there are collective effects, even after a
compressibility of the system. A consequence of this is
chainhasleftitsoriginaltube, itcanstillhavearesidual
that stress will be createdby these rearrangementslead-
orientationalbias. Thereforeweshouldmonitormoredi-
ing to an increase in free energy that turns out to be lo-
rectly, the birefringence of the system, which for small
cal [8]. This local increase in the free energy is precisely
forcing will be linear will the stress of the system.
what one simulates with reptation dynamics. Therefore
we do expect the same essential physical effects in these
lattice simulations as well. 1
Tounderstandmemoryeffectsinsuchsystems,wepre-
pare the simulation in a sheared state, where the orien- 0.8
tation of each monomer is now anisotropic. We do this
by biasing the Monte-Carlo algorithm so that it is more 0.6
likely to make steps in the z-direction rather than the )
(t
othertwo. Thesystemisequilibratedinthisstatebylet- ), l 0.4
ting it evolve for many relaxation times. At this point, (t
r
the effects of shearing are turned off and directions of 0.2
motion are chosen to be isotropic.
0
1 -0.2
0.9 0 100 200 300 400 500
0.8 t
0.7
FIG. 2: (Color Online) The residual birefringence r(t) and
0.6
theaverageamountofchainleft inatubel(t)areshown. As
n 0.5 is clear, thetwo curves closely follow each other.
0.4
0.3
0.2 In Fig. 1 we plot the decay of the birefringence sepa-
0.1 rately for the longer chains (upper plot) and the shorter
chain (middle plot). The lower plot is l(t), the amount
0
0 100 200 300 400 500 of chain left in the original tube.
In the original reptation picture, one expects that the
t
birefringence should be proportional to the chain left in
the tube. Clearly for the shorter chains, this is not the
FIG.1: (ColorOnline)Thedecayoftheaveragebirefringence
case. Thebirefringencequicklydecaysinitially,following
for a simulation using reptational motion in a polymer melt
with two different chain lengths. The top plot, L2 = 256, l(t)quiteclosely,howeveritappearstoleveloffatanon-
and the middle plot L1 = 64. The lower plot is the average zero value. After this is decays, but at a much longer
amount of L1 length chain left in thetubeat time t. time scale. We will analyze what mechanism gives rise
to such a striking effect.
In the initially biased preparation of the system, links
To illustrate the effect, weshallanalyzewhathappens aremorelikelytobeorientedinthez direction,therefore
in a polydisperse system made up predominantlyof long as the polymer chain leaves its tube, it will see a biased
chains L2 with a minority of shorter chains L1. For ex- environment. Inthe extremecaseofanenvironmentori-
ample, we consider a box of size B =16 with 9 chains of ented as rods in the z direction, the chain will also be
3
biased in the same direction as well [14]. In the case of state has been obtained by equilibrating the chains so
smaller orientational biases, it is then expected to have thattheirdirectionsarebiasedinthez direction. Thisis
an effect which is linear in the average birefringence of not identical to a real experiment because lattice chains
the chains. If the surrounding chains are much longer cannot be continuously deformed due to a lack of con-
than the shorter one, it will see an almost constant ori- tinuous translationalsymmetry. A test of the validity of
entational bias as it moves. Evenafter it has completely such a procedure is to ask if we are in the small defor-
leftitsoriginaltubeitwillpossessaresidualorientational mation, linear regime. We therefore test to determine if
bias. This will then slowly decay with time at the same the relaxation of the system is linear in the initial bire-
rate as the bias of the longer chains. fringence. We calculatedthe integralof n(t), normalized
Therefore we should modify the non-interacting rep- by its initial value, for two different initial biases, and
tational hypothesis by saying that the orientational bias found that indeed this was the case. For example, we
of the shorter chains is the amount of chain left in the considered a melt of chains of length 64 with a 0.58 of
tube at time t plus the bias due to the environment. thelattice sitesoccupied. Whenthe biaswasreducedby
Thisispreciselythesameassumptionmadeinreptation- 38%, this integral only changed by 2.3%.
basedtreatmentsofthiseffectmadepreviously[15]Sowe Second, as mentioned above, there are no multi-
candefinethe“residualbirefringence”duetotheshorter monomer moves with reptation dynamics, and so neigh-
chains as boringchainsare unable to rearrangethemselvesaround
new tube being formed. This causes an increase in the
r(t) n1(t) ǫn2(t) (3) local chain density of a region. In reality, what happens
≡ −
instead is that the system is almost incompressible, so
where ǫ is the fraction of orientational coupling induced
that stress and strain are created by the introduction of
by the longer chains, as has been measured experimen-
new monomers into a region. These effects in equilib-
tally [11]. After subtracting out the background bias
rium have been analyzed in detail previously [8, 9] and
from the longer chains, we are left with the bias from
give rise to the same behavior as in the simulations.
chains that have yet to leave their tube. We therefore
In the simulation described here, the reason that the
expect r(t) l(t). To test if this is the case, we find the
∝ polymer chain acquires an anisotropy is due to the in-
valueofǫthatbesteliminatesthelongertimedecay,and
fluence of the surrounding medium. The surrounding
then plot r(t) and l(t) as shown in Fig. 2. It is clear
mediumisinhomogeneousandthedensityofthemedium
thatthe twocurvesfit eachotherquite well. Inthis case
is frozen in on the time scale of a relaxation time of the
ǫ=0.14.
majority(longer) chains. The anisotropyof density fluc-
Thereforewehavegainedagoodunderstandingofhow
tuations cause the polymer to orient in the same direc-
orientation decays in this polydisperse polymer lattice
tion. In a realmelt, the analogousquantities to consider
simulation. Shorter chains continue to have substantial
arestressfluctuationsaswasanalyzedbyRubinsteinand
anisotropy even after completely leaving their original
Semenov[9]whenthesystemwasisotropic. Wewillcon-
environments by the influence of the surrounding longer
sider the effect of these fluctuations in a stressed poly-
chains, that still have not relaxed to an unbiased state. mer melt. The network stress σ(r) will be frozen in for
This is in accord with earlier experimental and theo-
timescales less than the relaxation time. When the end
retical work on these systems. This orientational bias is
ofachainentersacertainregion,thenetworkwilldeform
aneffectidenticaltowhatisseeninmoremicroscopicac- by an amount ∆u(r). The excess free energy is then
curate simulation on effectively shorter chains [12]. The
degree of orientational coupling is given by the factor ǫ,
which given the density of this system, appears to be F ∆u(r)σ(r)d3r. (4)
∼Z
consistent with the value of coupling found in the more
microscopic approach [12]. The typical energy of a fluctuation of order an entan-
Now we turn to the relation between this lattice sim- glement length √N , must obey Boltzmann statistics
e
ulation and real polymer melts. There are two idealiza- and is therefore∝of order k T [9]. At the same time, the
B
tions of the simulation that need to be addressed that energy of such a fluctuation in a volume V N3/2 is
e
stem from a lack of continuous translational symmetry ∝
for lattice models. The initial conditions from which the Vσ2
system relaxes, and the inability of surrounding chains W (5)
∼ K
to simultaneously rearrange themselves so as to accom-
modate new chain. whereK is the (longitudinal)elastic modulus ofthe net-
First, in a real experiment, the elastic modulus is de- work and is k T/N [2].
B e
∝
fined as the stress relaxation observed after a small and The increasein free energy in placing additional chain
sudden deformation of the material. After such a de- in a new regionis responsible for the exponential depen-
formation, the system is in a nonequilibrium state, and denceofrelaxationtimeonchainlengthashasbeenpre-
the polymer chains will then move to restore themselves viously analyzed [8, 9]. However what we are interested
to equilibrium statistics. The simulation here is started inhereisthefluctuationintheadditionalfreeenergythat
in a deformed state and is then relaxed. The deformed mustbepayedbyplacinganentanglementlengthofchain
4
inanewregion. Theaveragedeformationu N /V and Thisargumentcanbeturnedaround: the factthatwe
e
Using Eqs. 4 and 5, this gives [16], δF T/∼N1/4. musthavecomplete screeningforarbitrarilylongchains,
e
Wecanthereforethinkabouttheelast∼icfluctuationsas and that screening will be imperfect for polymers con-
causingarandompotentialwithanR.M.S. N−1/4that finedtoatube,impliesthatthepolymerchainmustmove
e
thechaininteractswith. Wenowaskwhath∝appenswhen in a weak random potential whose strength is such as to
the network is deformed. Suppose an affine deformation givecompletecancellationofthesecondvirialcoefficient.
eis appliedthatisoforder1. The randompotentialwill In other words,the assumptionof reptation, plus the re-
now also be deformed, just as the density had been de- quirementofscreening,givesrisetotherandompotential
formedinthesimulations. Soweexpectananisotropyin with the strength given above.
the potentialalsooforderk TN−1/4. Thereforea chain After stressing the system, the random potential will
B e changesothat the correlationfunction is v (r r′). The
segmentoflengthNe willnolongerexploreitssurround- s −
ings isotropically, but have a bias of order 1/Ne1/4 in the fporromximofattheleypvot=entvi(aΛlrs)howuhledrenoΛwisbtehaenaiffisontreotpriacnasfnodrmapa--
z direction. Therefore the birefringence of the chains in s
tion of the unsheared coordinates to the sheared state.
this new region will, more generally be eN−1/4.
e In the direction of maximum extension, the range of v
∼ s
To understand this more quantitatively, write the cor-
has now increased, and perpendicular to this, the range
relation function for the random potential φ(r) as
has decreased. This clearly leads to an alignment of the
chains in the direction of maximum extension. To cal-
φ(r)φ(r′) =v(r r′). (6)
h i − culate the local anisotropy quantitatively is not possible
without additional information. The exact form of fluc-
For a single chain in this environment, the annealed and
tuation in Eq. 6 depends on the correlations of entan-
quenchedaveragesyieldthesameresults[17]. Averaging
glements in these systems. This, at the moment, is un-
over realizations of the random potential leads a weak
known. Relatedtothis,ishowtheimperfectscreeningof
attraction of the polymer to itself, which interacts with
itself via the potential v(r). chains in tubes, discussed above, is modified by stretch-
ing. Thisisnecessarytodeterminechainstatisticsinthe
At first,this weak attractionwouldseemto be in con-
stretched state.
tradiction with the screening theorem [2] that says for
long chains, the statistics of individual chains are ideal. The ǫ factor in Eq. 3 is Ne−1/4. This is a substan-
∝
Howeverweareconsideringapotentialthatisstaticonly tial amount of orientational bias on the scale of a tube
on a time scale much less than the reptation time. On diameter. In a melt one would also expect that chain
thattimescale,thecurvilinearmotionofthetubeissmall packing would also induce further alignment locally due
and we can consider what happens if we stop chain ends to local chain packing and indeed an ǫ = 0.45 has been
completely from reptating. Because of entanglements, reported experimentally. However in concentrated and
chain motion is confined to a tube and this means that semidilutesolutions,suchlocalpackingeffectswouldnot
we do not expect complete screening in this case. Chain be expected to play a large role. In this case, we expect
of length less than N can move quite freely, but above that the mechanism discussed here would dominate.
e
thatentanglementsreducethenumberofdegreesoffree-
Inconclusionwehaveshownthatinducedorientational
dom. Thereforeweexpectthatthescreeningofexcluded
coupling of chains in a concentrated polymer solution
volume is not complete, and the screening should scale
can be understood as a consequence of the many body
as it would for a melt of chains of length N . Therefore
e interactions of a polymer system undergoing reptation.
the excluded volume parameter 1/N [2].
e Reptationalmotionfreezesindegreesoffreedomcausing
∝
On the scale of blobs of length of N , this leads to a
e incomplete screening that must be balanced by a ran-
2 −1/2
net repulsive potential ∼ (1/Ne)Ne/V ∼ Ne , with dom potential causing chain attraction. This random
a range of a tube diameter. This is exactly the same potential has its origin in elastic fluctuations of the sys-
scale as the scale of the effective attractive potential in tem[9]. When the systemis stretched,the correlationin
Eq. 6, because the height of v is N−1/2 also with the therandompotentialnolongerscreensinteractionscom-
e
∼
rangeofatubediameter. Thereforeoneexpectsthatthis pletely and causes the chain to elongate in the direction
weak attraction is canceled out by the excluded volume of stretch. This was analyzed directly by computer sim-
repulsion weakened by screening, in order to have ideal ulationusingreptationdynamicsanddemonstratedthat
chains over large distances. this orientational bias is substantial.
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