Chapter 11
Nanoscale versus Macroscale Friction in Polymers and Small-Molecule Liquids: Anthracene Rotation in PIB and PDMS 1,2
1,
Mark M. Somoza and Mark A. Berg * 1
Department of Chemistry and Biochemistry, University of South Carolina, Columbia, SC 29208 Current address: Institut für Physikalische und Theoretische Chemie, Technische Universität München, Lichtenbergstrasse 4, 85748 Garching, Germany 2
As an object becomes smaller than the size of a polymer molecule, the friction it experiences in a polymer melt may be altered from the friction on a macroscopic object. These mesoscopic dynamics are unique to polymeric solvents. The mechanisms causing this effect and the transition from small— molecule to polymer-like behavior have been explored by using the rotation time of anthracene as a measure of nanoscale friction in liquids. The viscosity experienced by anthracene molecules is determined as a function of polymer chain length over the range from the dimer to the entangled polymer. The nanoviscosities in poly(dimethylsiloxane) (PDMS) and poly(isobutylene) (PIB) develop very differently as a function of chain length, despite similar static structures of the polymers. These results are attributed to higher torsional barriers in PIB than in P D M S . We suggest that a dynamic length scale is important in determining the friction experienced by mesoscopic particles in polymers.
© 2003 American Chemical Society
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Introduction For macroscopic objects moving in simple fluids, continuum hydrodynamics is highly successful. With the recognition of viscoelastic and high shear effects, motion in polymeric fluids can also be treated with hydrodynamics. However, as the size of a moving object becomes sub-micron, the continuum assumption of hydrodynamics comes into question. Nevertheless, the Stokes-Einstein-Debye (SED) model, which applies continuum hydrodynamics to molecular-sized objects, has considerable success in small-molecule fluids [see (1) and references in (2)\. However, for large-molecule, polymeric fluids, hydrodynamics is expected to breakdown for nanoscale objects. In the studies summarized here (2-5), we look at the breakdown of the SED model as the molecular size of the fluid increases from the small-molecule to the polymeric regime. The aim is to identify the length scale associated with the breakdown of hydrodynamics and to understand the mechanisms governing the motion of nanoscale objects in polymers. Based on our results, we suggest that torsional relaxations of the polymer backbone are the dominant mechanism for facilitating the motion of small objects. The important length scale defining "small" is a dynamic one determined by the average distance between torsional relaxations that occur within the characteristic time of the object's motion. As our probe of small object motion, we measure the rotation time of a solute molecule. In general, hydrodynamics predicts that the rotation time r of a particle due to Brownian motion is determined by the ordinary viscosity //«> of the solvent, r
where k is the Boltzmann constant, Τ is the temperature and is the hydrodynamic volume of the particle (/). The particle shape and boundary conditions determine the parameter λ. The time % is added phenomenologically to correct for deviations at very low viscosity (6). For a nanometer-sized object, this equation may not hold. However, the object still experiences a friction, which can be expressed as a viscosity by inverting eq 1,
^nm=17r(^- o)7
= ^ . This relationship is expected to hold i f the solvent molecules are sufficiently small relative to the object. Solute rotation in either small-molecule solvents or polymers is a wellstudied subject. More extensive references to this literature can be found in references 2, 8 and 9. The new features of our studies are: (1) data over an extensive molecular-weight range that connect studies in small-molecule solvents to those in polymeric solvents, (2) the detailed comparison of two polymers with similar static structure, but different torsional dynamics, and (3) the careful calibration of the hydrodynamic behavior of the probe object, a feature that allows the measurement of the absolute values of the nanoviscosity. Most of the n m
(b)
(a) 0.78 nm
Hag CH | H g ^ H 3
3
Λ 1.18 nm
3
H C CH 3
3
-|0.54 nm
ft
Λ / χ ÎFQ.1
0.25 nm
( & macroviscosity continues to increase with chain length. The surprising aspect of this length is that it is quite long. The S E D model holds even when the polymer molecules are 2.5-3 times larger than the anthracene molecule, judging either by length or by volume. Because the S E D model assumes a continuous fluid, the simple expectation is that this model would already be failing when the polymer molecules are the same size as the anthracene molecule. e
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Figure 3. Macroviscosity (open) and nanoviscosity (solid) of three polymers versus the number of backbone bonds. See textfor fits. (Reproducedfrom reference 5. Copyright 2003 American Chemical Society.)
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Above ^SED? the nanoviscosity also increases with chain length. However, the increase is much weaker than for the macroviscosity, and it levels off for long polymers. Because the rotation of a solute molecule must be a property of a local region near the solute, we expected the nanoviscosity to reach an asymptotic value in the infinite polymer limit. What was not anticipated, was that the limit would only be reached long after isEDThe reason for the slow approach to the limiting value is suggested by empirical fits to the nanoviscosity data
which are shown as solid curves in Figure 3. Because the density of chain ends scales as 1/i, the l/£ dependence suggests that the slow growth to the asymptotic value is related to the loss of end effects. In principle, the nanoviscosity is a more fundamental property than the macroviscosity, and knowledge of the nanoviscosity should allow the macroviscosity to be predicted. Polymer viscosity below the entanglement length has generally been described by the Rouse model (16). In this model, the macroviscosity is predicted from the segmental friction ζ. The nonlinear behavior of PIB macroviscosity has been attributed to the length dependence of this segmental friction. A natural hypothesis is that the nanoviscosity and segmental friction are closely related. We assume that they are proportional to one another, ζ(£) oc η^ί). Coupling this idea with the Rouse model's linear dependence of the macroviscosity on the chain length gives η^κίχη^Ι).
(4)
As normally formulated, the Rouse model does not attempt to treat the smallmolecule limit properly. We know that in this limit, the SED model holds
We have made a simple interpolation between these two regimes
(6) W )
1
0
* &e nanoviscosity increases weakly toward the long polymer value. The nanoviscosity data can be fit with the same model used for PIB (eq 3, solid curve in Figure 3c), suggesting that the elimination of end effects is again the reason for the slow convergence to the asymptotic value. Below the entanglement length, the macroviscosity can be predicted from the nanoviscosity by our Rouse-like model (eq 6, dashed curve in Figure 3c). The nanoviscosity is unaffected by entanglement. Although the static structures of PIB and P D M S are quite similar, the torsional kinetics are very different. In PIB, there is substantial steric interaction between the methyl groups during torsional transitions. A s a result, the energy barrier between gauche and trans conformations is high compared with thermal energies (6 kcal/mol = 10 kT at 300 K ) . P D M S has more open bond angles and longer bond lengths, which combine to almost eliminate the steric interaction between methyl groups. Consequently, torsional barriers in P D M S are negligible (
