Modifying Fragility and Collective Motion in Polymer Melts with Nanoparticles

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by  Francis W. Starr

Modifying Fragility and Collective Motion in Polymer Melts with Nanoparticles
Francis W. Starr
Department of Physics, Wesleyan University, Middletown, CT 06459, USA
Jack F. Douglas
Polymers Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
(Dated: Submitted 31 August 2010)
Weinvestigatetheimpactofnanoparticles(NP)onthefragilityandcooperativestring-likemotion
inamodelglass-formingpolymermeltbymoleculardynamicssimulation. TheNPcausesignificant
changes to both the fragility and the average length of string-like motion, where the effect depends
on the NP-polymer interaction and the NP concentration. We interpret these changes via the
Adam-Gibbs(AG)theory,assumingthestringscanbeidentifiedwiththe“cooperativelyrearranging
1
regions”ofAG.Ourfindingsindicatefragilityisprimarilyameasureofthetemperaturedependence
1
of the cooperativity of molecular motion.
0
2

n The addition of a small concentration of nanoparti- of CRR, and we confirm this relationship.
a
cles (NP) to glass-forming (GF) polymer materials can Our findings are based on equilibrium molecular dy-
J
lead to large property changes that are difficult to com- namics simulations of a nanoparticle surrounded by a
8
2 prehend by extension of the effects of macroscopic filler densepolymermelt,aswellassimulationsofapuremelt
additives. Depending on system details, changes may be for comparison purposes. We utilize periodic boundary

] rationalized by the large surface-to-volume ratio of NP, conditions so that our results represent an ideal, uni-
t
f chain bridging [1–3], or NP self-assembly into extended form dispersion of NP. The polymers are modeled by a
o
structures[4,5]. Changesintheglasstransitiontemper- well-studied bead-spring model [14], but with the cut-
s
t. ature Tg have been emphasized, and both experimental off distance between pairs extended to include attractive
a and theoretical studies indicate that attractive or repul- Lennard-Jones (LJ) interactions. All monomer pairs in-
m
sive(non-attractive)polymer-NPinteractionstendtoin- teract via a LJ potential, and bonded monomers along a
d- crease or decrease Tg, respectively. Correspondingly, the chain are connected via a FENE anharmonic spring po-
n interfacial polymer layer around the NP shows a slowing tential. The NP consists of 356 Lennard-Jones particles
o down (increased Tg) or an acceleration of dynamics (de- bonded to form an icosahedral NP; the facet size of the
c creased Tg), providing a molecular interpretation [6–9]. NP roughly equals the equilibrium end-to-end distance
[
Unfortunately, Tg changes provide only a limited un- for a chain of 20 monomers. Details of the simulation

1 derstanding of how NP affect the properties of GF poly- protocol and our model potentials can be found in the
v mer melts. We also expect that the T dependence of supplemental material and in ref. [8].
7
dynamical properties approaching T , or the “fragility” We simulate systems with 100, 200, or 400 chains of
6 g
of glass formation [10], will be altered. Fragility changes M = 20 monomers each (for totals of N = 2000, 4000,
5
5 have been argued for on theoretical grounds [11], based and 8000 monomers) to address the effect of varying the
. on the finding that changes in the molecular packing in NPvolumefraction. Underconstantpressureconditions,
1
0 the glass state (T < Tg) should also alter the fragility the addition of nanoparticles can give rise to a change in
1 of glass formation. The NP we study should be particu- the overall melt density. A slight change in density can
1 larly effective at modifying molecular scale packing and cause a significant change in the dynamic properties rel-
v: motion, since their size is roughly commensurate with ative to the pure melt. In order to probe only changes
Xi the heterogeneity scale of fluids near Tg [12]. causedbytheinteractionsbetweentheNPandthepoly-
Inthisletter,weaddresshowpolymer-NPinteractions mer melt, we have matched the density of monomers far
r
a and NP concentration affect the fragility of glass forma- from the NP with that of the pure polymer melt [8].
tion, and how this effect relates to cooperative motion. To quantify changes in the nanocomposite dynamics,
We demonstrate that the changes in the relaxation time we evaluate the effect of φ and the polymer-NP interac-
τ can be related to changes in the average size L of the tions on τ, measured from the relaxation of the coherent
string-like cooperative motion of monomers. The Adam- intermediate scattering function (see supplementary in-
Gibbs (AG) theory [13], predicts a specific relationship formation). The effects of interactions on τ and T for
g
between between size of hypothetical “cooperatively re- some φ were presented in ref. [8]; here we provide addi-
arrangingregions”(CRR)andstructuralrelaxationtime. tional simulation data and focus our analysis on fragility
If L corresponds the size of the CRR, we find that the and cooperative motion. As expected, Fig. 1 shows that
AGrelationholdsforallconcentrationsφandinteraction attractive polymer-NP interactions slow the relaxation
types considered. Additionally, the AG theory predicts (τ becomes larger), while non-attractive polymer-NP in-
that fragility is sensitive to the T dependence of the size teractions give rise to an increased rate of relaxation (τ
2
102 m
0.3 D -1
1.6
Attractive T /T
g c
Tg0.02.52 Non-Attractive gility1.4 TL-0T /TdgL/dT
pure101 gerf 0.150 0.05 f 0.1 0.15 0.2 ve Fra1.2 Attractivge MFroargeile
t / t Lar Attractive, f = 0.151 elati 1
R
100 Pure 0.8 Non-Attractive
Stronger
(a)
f Non-Attractive, f = 0.151
0.6
0 0.05 0.1 0.15
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Concentration f
Temperature T
300 2.4
(b)
Fragility
FIG. 1: Structural relaxation time τ as a function of T for 2
200
tethraaecchtciφovrenreoisnrpmteoarnaldiczitenidognbTsygiτnapcsurreeaafsfoeurnτtchtaeinopndurToefg,mcowenlhtc.ielenTtnhraoetniin-oasntettφras.hctoAiwvtes- m100 Non- Pure Attractive 1.6E¥
interactions decrease τ and Tg. In both cases, the effect is Attractive Activation Energy
morepronouncedwithincreasingconcentration. Theconcen- 1.2
trationsareφ=0.0426(red(cid:13)andorange(cid:52)),0.0817(green
(cid:50) and brown (cid:47)) and 0.151 (blue (cid:51) and violet (cid:53)). The pure 00..1155 00..22 00..2255 00..33 00..3355
meltisindicatedinblack. ThefitsoftheVFTrelationtothe TT
gg
data deviate by at most 0.5 %.
FIG. 2: (a) Fragility dependence on concentration φ rela-
tive to the pure melt. We consider five different measures
becomes smaller). The effect of φ is more clearly seen of fragility, which are discussed in the text. All measures of
fragilityshowthesamequalitativetrend: namely,thesystem
by rescaling τ by the value τ in the pure melt, which
pure with attractive polymer-NP interactions becomes more frag-
shows that τ can be altered by a factor of more than an
ile,whilethesystemwithnon-attractiveinteractionsbecomes
order of magnitude on cooling. The effect of the NP is
lessfragile(stronger). Thelastmeasure,relatedtothestring
more pronounced at low T. lengthL,isdiscussedlaterinthetext. (b)Demonstrationof
theproportionalitybetweenT andfragilitym,aswellasthe
g
Wenextexaminehowthesechangesinτ affectT and high-T activationenergyE . Thedifferentcolorsofsymbols
g ∞
fragility. For reference, the inset of Fig. 1 confirms that indicate the nature of the polymer-NP interactions.
T increaseswhenthereisattraction,anddecreaseswith
g
non-attractive interactions. To estimate T , we fit the
g dence of τ near T , namely [15],
data using the Vogel-Fulcher-Tammann (VFT) expres- g
(cid:12)
sion [10] τ ∝ exp(D/[T/T0−1]). T0 is an extrapolated d(lnτ) (cid:12)
m= (cid:12) . (1)
divergence T of τ, while D provides one measure of the d(T /T)(cid:12)
fragility. We use the VFT fit to estimate T based on g Tg
g
the condition that τ(T ) = 100 s (the canonical defini- For strong GF systems, the rate of change of τ with re-
g
tionofthelaboratoryglasstransition[10]),assumingthe spect to T is smaller than that of fragile systems; hence
one time unit in standard LJ reduced units corresponds m is larger for more fragile GF fluids. We estimate m
to1ps(reducedunitsdefinedinsupplementaryinforma- using our VFT fit. Fragility can also be estimated by
tion). the ratios T0/Tg or Tg/Tc. We estimate Tc using the
power-law form τ ∼ (T/T −1)−γ in an appropriate T
c
Since there is no single agreed upon measure of range(seesupplementaryinformationforfittingdetails).
fragility, weconsiderseveraldifferentmeasurestoensure T /T and T /T are larger in more fragile systems [11].
0 g g c
consistency. First, as indicated above, the parameter D Wesummarizetheresultsforthevariousfragilitymet-
fromaVFTfittoτ iswidelyutilized;specifically,alarger rics in Fig. 2(a), where we find that – for all definitions
value of D indicates a stronger (less fragile) GF fluid so – attractive polymer-NP interactions lead to more frag-
that D−1 increases with increasing fragility. The most ile glass formation as a function of φ; conversely, non-
common definition of fragility is based on the T depen- attractivepolymer-NPinteractionsleadtostrongerglass
3
formation. These changes in fragility mirror the changes
1.15
in Tg (fig. 1 inset). In particular, fig. 2(b) shows that erφ 2
mpoly∝meTrgi,c cgolansssistfeonrtmewristh[1e6x]paenridmeanntaallyttircencdaslcuinlatpiounres 1.1 Larg L1.8
based on the entropy theory of glass formation [17]. Fig- 1.6
ure 2(b) also shows that the high-T activation energy
E is roughly proportional to T . We discuss the impli- ure1.05 1.40.4 0.5 0.6 0.7 0.8 0.9 1
∞ g Lp T
cations of these scaling relationships below.
Our findings for fragility changes are consistent with L / 1
experimental studies of polymer-NP systems. Bansal et Lar Attractiveφ = 0.043
al.[6]foundthatdispersionsofNPhavingrepulsiveinter- ger Attractiveφ = 0.082
0.95 φ Attractiveφ = 0.151
actionscausedTg todecrease,accompaniedbyanappre- Non-Attractiveφ = 0.043
ciablebroadeningoftheglasstransitionregion,indicative Non-Attractiveφ = 0.082
Non-Attractiveφ = 0.151
ofincreasedstrength(decreasedfragility)ofglassforma- 0.9
0.4 0.5 0.6 0.7 0.8 0.9 1
tion. For fullerenes dispersed in polystyrene, Cabral and
TemperatureT
co-workers[18]reportedbehaviorexpectedforattractive
polymer-NP interactions, namely an increase in T , ac-
g
companiedbyanincreasedfragility. Forsmallφ,negligi- FIG. 3: T dependence of L for different φ normalized by L
ble changes in the fragility have been reported [19], also for the pure melt. Note the parallelism to fig. 1. The inset
consistentwithoursmallφresults. Ourresultsarelikely also shows a snapshot of an example string of length L=9.
not applicable when the NP-polymer interactions are so
strong that non-equilibrium effects (leading to “bound”
ilar in all cases.
polymer) [20] or phase separation may dominate.
While the changes in L reflect the changes in τ for
We next examine how the NP interactions impact the
any given T, how do the T dependence of these changes
heterogeneity of molecular motions and how this relates
compare? In other words, can L be used to predict the
to the observed changes in T and fragility. Both small-
g fragilitychangesinfig.2(a)? Toanswerthisquestion,we
molecule and polymeric liquids exhibit pronounced spa-
areguidedbyAGtheory,whichproposesthatrelaxation
tialcorrelationsinmobility,commonlyreferredtoas“dy-
in GF liquids is dominated by cooperatively rearranging
namical heterogeneity” [12]. In particular, the most mo-
regions (CRR). Specifically, AG argue that τ is related
bile atoms or molecules tend the cluster on a time scale
to the average number of particles in the CRR z by a
after the “breaking of the cage”, but before the primary
generalized Arrhenius relation [23],
relaxation τ. These clusters of mobile molecules can be
further dissected into “strings” involving particles mov- τ =τ exp(zE /T). (2)
∞ ∞
ingroughlyco-linearly[21];thesestructuresappeartobe
For high T, cooperativity is minimal, so z ≈ 1, and E
themostbasicunitsofcooperativerelaxation. Thechar- ∞
(a constant) can be identified with the high-T activation
acteristic size of both the mobile-particle clusters and
energyofthefluid. Accordingly,theT-dependentactiva-
strings grow as a fluid is cooled toward T . Notably,
g
tion energy E(T)=z(T)E should govern the fragility.
thestring-likecollectivemotionisnotstronglycorrelated ∞
Thestringsareanaturalcandidatetodescribetheab-
withchainconnectivity[22],soitshouldnotbeconfused
stract CRR of AG. Fig. 4 shows that eq. (2) with z ∝L
with reptative motion.
provides an excellent prediction for τ, consistent with
We evaluate the string size L(T) following the proce-
identifying L with the CRR of AG [24]. Therefore, the
dures developed in ref. [21], which are slightly modified
non-Arrhenius behavior of τ can be viewed as a conse-
forthespecificcaseofthebead-springpolymerwestudy,
quence of the increase in L on cooling. Accordingly, L
as discussed in ref. [22] (see supplementary material for
must encode the fragility of glass formation. In particu-
the technical details). Figure 3 shows L for all T and φ,
lar, combining eqs. (1) and (2) implies a direct relation
where we compare systems by normalizing by the value
between m and L,
of L of the pure melt. Given that L grows as τ grows
oncooling,weexpectthattheattractiveNPinteractions (cid:2) (cid:3)
m=(E /T ) L(T )−T dL/dT| . (3)
(which increase τ relative to the pure melt) should cause ∞ g g g Tg
Ltoincreaserelativetothepuremelt,andvice-versafor Since E and T are proportional (fig. 2(b)), the prefac-
∞ g
non-attractive NP interactions. Indeed, the variation of torE /T isroughlyconstantforallφ,andthusdoesnot
∞ g
L is consistent with these expectations, and we conclude impact the nanocomposite fragility relative to the pure
thatthetheattractiveNPinteractionscauseanincrease melt. Therefore, the fragility changes measured by m on
in the degree of correlated molecular motion for fixed T, addingNPshouldresultfromchangestoLandT dL/dT
while non-attractive NP interactions cause a decrease in at T . To test this, we extrapolate L to T by assum-
g g
L. NotethattheT dependenceofLisqualitativelysim- ingconsistencybetweeneq.(2)andtheVFTexpression.
4
u2/Æu2æ
0
104 104
103
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g tivated mass transport, it is natural to apply their ar-
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guments to a mass diffusion relaxation time. However,
molecular mass polymer polymer materials [16], implies the decoupling relation between structural α-relaxation
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