VOLUME 6, NUMBER 5, MAY 2006 © Copyright 2006 by the American Chemical Society
Imaging Nanoscale Spatio-Temporal Thermal Fluctuations P. S. Crider and N. E. Israeloff* Department of Physics, Northeastern UniVersity, Boston, Massachusetts 02115 Received March 12, 2006
ABSTRACT Spatial and temporal fluctuations of the electric polarization were imaged in polymer thin films near the glass transition using electric force microscopy. Below the glass transition the fluctuations are quasi-static, and spatial fluctuations were found to quantitatively agree with predictions for thermal fluctuations. Temporal fluctuations appear near the glass transition. Images of the space−time nanoscale dynamics near the glass transition are produced and analyzed.
Thermal fluctuations measured locally in complex materials can reveal important features of the microscopic dynamics.1 Via the fluctuation-dissipation relation (FDR), local moduli such as rheological properties can be inferred from the fluctuations.2 However, local deviations from the FDR have been predicted for aging glassy systems.3 Recent theory of glasses has also emphasized space-time trajectories of the spontaneous dynamics.4 In this paper we report on quantitatively imaging the nanoscale spatio-temporal thermal fluctuations of the electric polarization, near the glass transition, Tg, in thin polymer films. We use a form of noncontact atomic force microscopy (AFM) in which a cantilever with a sharp metal-coated tip is oscillated at its resonance frequency in ultrahigh vacuum. The AFM tip is biased with voltage, V, relative to a conducting substrate coated with a dielectric sample. The bias provides an electrostatic force between the tip and surface, which shifts the cantilever resonance frequency by δf, which is then measured by means of a precision phase-locked loop (NanoSurf). This technique is a type of electric force micros* Corresponding author. E-mail:
[email protected]. 10.1021/nl060558q CCC: $33.50 Published on Web 04/06/2006
© 2006 American Chemical Society
copy (EFM). An RHK UHV 350 variable temperature AFM system is used. The samples studied are polyvinyl acetate (PVAc), 0.25 µm thick, mol wt 167 000, prepared by spinning by from a toluene solution onto flamed Au coated glass, and annealed in vacuum at Tg ) 308 K for 30 min. The δf is quadratic in V; however, the minimum δf is not necessarily centered on V ) 0. One source of this offset is residual charge on or near the sample surface and its image charge on the tip.5 We have found that by applying a dc bias of 1 V while the tip is parked (in noncontact) over one location for several minutes just below Tg, an induced polarization charge appears, producing a surface potential, Vp, which offsets the applied voltage by ∼200 mV. The residual Vp decays away after the bias is removed. The time scale for relaxation is strongly temperature-dependent near Tg and varies from seconds to hours, similar to the (alpha) glassy relaxation-time.6 This is expected because the strong electric field beneath the tip locally polarizes the dielectric. A residual polarization should persist for an alpha relaxationtime after the bias is removed.
Figure 2. Finite element model of EFM tip (white) and dielectric polymer sample (light blue). The electric fields fall off by a factor of 2 in the colored region below the tip, defining a probed volume.
Figure 1. 600 × 600 nm2 polarization images of PVAc at 303 K. Top: first scan. Bottom: 17 min later, showing some decorrelation. The dark oval at the top left in each image is a topographic feature used as a marker.
The tip electrostatic force can then be calculated from the charging energy as a function of tip height z, U ) 1/2C(z)V2 and F ) -dU/dz. The small shift in cantilever resonance frequency, δf, is used as the imaging feedback parameter. It is proportional to the fractional change in the cantilever spring constant, k, due to the force gradient dF/dz. The electrostatic component of δf is therefore δf )
1 f0 dF 1 f0 d2C (V - Vp)2 )2 k dz 4 k dz2
(1.1)
A lock-in amplifier was used to monitor the response of the δf to an applied sinusoidal bias V ) V0 sin(ωt) at frequencies between 100 and 200 Hz, and V0 ) 1 V. A term in eq 1.1 proportional to V20 will appear as a 2ω lock-in response, V2ω. The offset voltage, Vp, due to polarization, will be seen as a 1ω lock-in signal, Vω, proportional to 2V0Vp. Both V2ω and Vω have the same multiplicative factors; thus, Vp can be obtained directly from Vp ) 1/4VωV0/V2ω. Because V2ω varies very little across the sample under fixed conditions, we measure V2ω once before imaging Vω and extracting Vp. As seen in Figure 1, a polarization (Vp) image of PVAc near the glass transition shows an interesting random structure, which can be resolved down to about 30 nm 888
currently. This structure is reproducible over short time intervals but is quite distinct from the topography. The topography (height) was quite smooth aside from a distinct bump used as a marker. Most striking, the polarization image shows evolution in time at temperatures close to Tg. After 17 min, there is some similarity but considerable loss of correlation. Note that this is 5 K below Tg. Quantifying local polarization precisely is difficult because not all of the parameters are well known. However, for the case of thermally induced polarization we can do quite well in predicting its effect on our measured Vp under certain conditions. The capacitance as a function of height above the surface, C(z), can be modeled using a finite element (Bela) calculation for a realistic tip shape. See Figure 2. Near the glass transition of a polymer, the static dielectric susceptibility, s, has a fast component, ∞, and a slow component due to glassy reorientation of polar molecules, ∆ ) s - ∞. The sample surface to back electrode forms a capacitance whose slow component we designate C∆, in series with the tip-to-sample surface capacitance, whose total capacitance we call Ctip. We find that the nominal tip radius of 10 nm is consistent with the measured δf(z,V) and gives Ctip ) 4.4 × 10-19 F and C∆ ) 2.9 × 10-18 F, and a probed volume with depth ) 7 nm, diameter ) 30 nm, for z ) 7 nm. Any capacitor will have integrated thermal fluctuations (noise) given by the equipartition theorem: ) kBT/ C. There will be very fast fluctuations, which are not detected in the measurements. The slow fluctuations, kT/C∆, will appear as residual surface potential fluctuations but are reduced by a fraction, (∆/s)2, due to the parallel C∞. Thus, the integrated noise can be calculated from known quantities and quantities obtained from the model calculations: < δV2p > )
( )
kBT ∆ C∆ s
2
(1.2)
Equation 1.2 predicts the integrated temporal fluctuations in Vp that would be observed in a single location. However, Nano Lett., Vol. 6, No. 5, 2006
Figure 3. Space-time images of polarization fluctuations at 301.5 K (top) and 305.5 K (bottom).
) 0.024 ( 0.005 V for z ) 7 nm, with a very weak z dependence for z ) 5-15 nm. To quantitatively test this idea, we used repeated onedimensional scans (a horizontal line on Figure 1) to obtain Vp(x,t). We subtract the long-time-averaged (6900 s), assumed to be due to surface or back electrode topography to obtain δVp(x,t) and plot these as space-time fluctuation images, as shown in Figure 3 for two temperatures. These images are striking in that they clearly demonstrate the spatio-temporal aspects of glassy dynamics. We see that correlations along the time axis are distinctly longer at the lower temperatures and hints of possible dynamic heterogeneity. At each time we calculate the rms spatial fluctuation and average this over time (6900 s) to obtain . This was then repeated several times at different locations to obtain better statistics. At 30.5 C, we observed ) 0.023 ( 0.004. At 32.5 C, we found ) 0.028 ( 0.004. Thus, we find very good agreement with the prediction for thermally induced spatial variations in the polarization. We can also measure the time autocorrelation function at each x and then average over x. This global autocorrelation function, C(t) (Figure 4) clearly shows the decrease in the alpha relaxation time with increasing temperature and is similar in shape to the decay of polarization we see after a bias change. Space-time images of polarization fluctuations offer a tremendous opportunity to study nanoscale variations of material properties as well as complex local dynamics. In glassy polymers, dynamic heterogeneity and various correlation functions can be studied and compared with the predictions of different models.3,4 The observed fluctuations agree quantitatively with the FDR. Thus, it might enable the study of local FDR violations3 under nonequilibrium conditions. Acknowledgment. This work was supported by NSF DMR-0205258, NSF NSEC 0425826 Center for High Rate Nanomanufacturing, and ACS PRF 38113-AC7. References
Figure 4. 305.5 K.
Global autocorrelation functions for 301.5 and
by scanning, we are in essence sampling different capacitors. At temperatures just below Tg, the fluctuations are essentially frozen for hundreds of seconds. Thus, we expect the spatial variations in Vp to reflect the full variance as given by eq 1.2, for a fast enough scan. We predict rms fluctuations δVp
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(1) (a) Weeks, E. R.; Weitz, D. A. Phys. ReV. Lett. 2002, 89, 095704. (b) Vidal Russell, E.; Israeloff, N. E. Nature 2000, 408, 695-698. (2) Mason, T. G.; Weitz, D. A. Phys. ReV. Lett. 1995, 74, 1250-1253. (3) Castillo, H. E.; Chamon, C.; Cugliandolo, L. F.; Kennett, M. P. Phys. ReV. Lett. 2002, 88, 237201-237204. (4) Garrahan, J. P.; Chandler, D. Phys. ReV. Lett. 2002, 89, 035704. (5) Terris, B. D.; Stern, J. E.; Rugar, D.; Mamin, H. J. Phys. ReV. Lett. 1989, 63, 2669-2672. (6) Walther, L. E.; Israeloff, N. E.; Vidal Russell, E.; Alvarez Gomariz, H. Phys. ReV. B: Condens. Matter Mater. Phys. 1998, 57, R15112-R15115.
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