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Observing the twinkling fractal nature of the glass transition
Journal of Non-Crystalline Solids 357 (2011) 311–319
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Journal of Non-Crystalline Solids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j n o n c r y s o l
Observing the twinkling fractal nature of the glass transition J.F. Stanzione III a, K.E. Strawhecker b, R.P. Wool a,⁎ a b
Department of Chemical Engineering and Center for Composite Materials, University of Delaware, Newark, DE 19716, United States Army Research Laboratory, Aberdeen Proving Ground, MD 21005, United States
a r t i c l e
i n f o
Article history: Received 9 April 2010 Received in revised form 16 June 2010 Available online 25 August 2010 Keywords: Glass transition; Dynamic fractal networks; Percolation; Solid-state structure; Atomic force microscopy
a b s t r a c t A fundamental understanding of the nature and structure of the glass transition in amorphous materials is currently seen as a major unsolved problem in solid-state physics. A new conceptual approach to understanding the glass transition temperature (Tg) of glass-forming liquids called the twinkling fractal theory (TFT) has been proposed in order to solve this problem. The main idea underlying the TFT is the development of dynamic rigid percolating solid fractal structures near Tg, which are said to be in dynamic equilibrium with the surrounding liquid. This idea is coupled with the concept of the Boltzmann population of excited vibrational states in the anharmonic intermolecular potential between atoms in the energy landscape. Solid and liquid clusters interchange or “twinkle” at a cluster size dependent frequency ωTF, which is controlled by the population of intermolecular oscillators in excited energy levels. The solid-to-liquid cluster transitions are in accord with the Orbach vibrational density of states for a particular fractal cluster g(ω) ~ ωdf − 1, where the fracton dimension df = 4/3. To an observer, these clusters would appear to be “twinkling.” In this paper, experimental evidence supporting the TFT is presented. The twinkling fractal characteristics of amorphous, atactic polystyrene have been captured via atomic force microscopy (AFM). Successive two-dimensional height AFM images reveal that the percolated solid fractal clusters exist for longer time scales at lower temperatures and have lifetimes that are cluster size dependent. The computed fractal dimensions, ≈ 1.88, are shown to be in excellent agreement with the theory of the fractal nature of percolating clusters in accord with the TFT. The twinkling dynamics of polystyrene within its glass transition region are captured with timelapse one-dimensional AFM phase images. The autocorrelation cluster relaxation function was found to behave as C(t) ~ t− 4/3 and the cluster lifetimes τ versus width R were found to be in excellent agreement with the TFT via τ ~ R1.42. This paper provides compelling new experimental evidence for the twinkling fractal nature of the glass transition. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The heterogeneous state of matter that exists near the glass transition in an amorphous material is ubiquitous; however, the true nature and structure of this transition is still considered a critical unsolved problem [1]. Recently, a new theory of the glass transition has been proposed by Wool, entitled the twinkling fractal theory (TFT) [2,3]. In addition, recent atomic force microscopy (AFM) images of polystyrene published by Wool and Campanella and those shown in Fig. 1 have captured the percolating twinkling solid fractal structure near the glass transition temperature (Tg), proposed by the TFT [4]. Based on the TFT, fractal hard and soft regions exist which are quite dynamic and which appear and disappear with time. The AFM techniques utilized in this work allows one to obtain sequential “snap-shots” of the fractal structure near Tg to produce twodimensional (2-D) spatial images (Fig. 1) as well as obtain consecutive
⁎ Corresponding author. Tel.: +1 302 831 3312; fax: +1 302 831 1048. E-mail address: [email protected] (R.P. Wool). 0022-3093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2010.06.041
time-lapse, one-dimensional (1-D) spatial images that permit one to visualize and quantify the complex twinkling fractal dynamics, shown later. According to the TFT, the heterogeneous state of matter that exists near the glass transition in amorphous materials evolves from a dynamic fractal solid-like structure in equilibrium with its liquid [2]. The dynamics of the heterogeneity is driven by the Boltzmann population of vibrational energy levels of clusters oscillating in the anharmonic intermolecular potentials associated with the energy landscape [2]. The anharmonic potentials on the intermolecular coordinates create a temperature-dependent solid PS and liquid PL fractions, which dynamically interchange due to the Boltzmann population of vibrational energy levels. When an oscillator bond length thermally expands to the inflection point in the potential function, its force constant decreases toward zero and a thermal fluctuation may send it into the liquid state, which is the origin of the solid-to-liquid twinkling process in the TFT. As Tg is approached from above, the twinkling solid fraction grows to about 50% by volume and percolates rigidity. The onset of long-range connected fractal structures begins at the scalar percolation threshold pcs, which occurs at
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Fig. 1. AFM tapping-mode X–Y height images of the percolating twinkling solid fractal structure in amorphous polystyrene at 298 K (images a–c) and 358 K (images d–f).
T* N Tg and is determined by T*/Tg = (1 − pcs)/(1 − pc) where the rigidity percolation threshold pc ≈ 0.5. Typical values of the scalar and rigidity percolation thresholds give T*/Tg ≈ 1.2 [2]. Above T*, small non-connected clusters of solid material can still form but are relatively short-lived since cluster lifetimes τ is a function of size R as τ ~ R1.42. Fig. 1 shows the percolating twinkling solid fractal structure of polystyrene at 298 K (images a–c) and at 358 K (images d–f) in three sequential AFM X–Y height images at each temperature. As can be seen in Fig. 1, overall, the topological percolating solid fractal structure at 298 K twinkles less frequently than the structure at 358 K. According to the TFT, the percolating fractal twinkles with a frequency spectrum as solid and liquid clusters interchange with frequency ωTF ~ ωexp(− ΔE/kT) [2–4]. This inverse relationship between twinkling frequency and temperature is qualitatively in agreement with the images in Fig. 1. The vibrational density of states g(ω) for a fractal is determined to be g(ω) ~ ωdf − 1, where df = 4/3 [5,6] and the temperature-dependent activation energy behaves as ΔE ~ (T *2 − T2) [2]. Approximately, the TFT predicts that the spatiotemporal thermo-fluctuation autocorrelation relaxation function C(t) behaves as an inverse power law C(t) ~ t− 1/3 for “short” times or high frequencies and C(t) ~ t− 4/3 for “long” times or low frequencies. These predictions were found to be in fair agreement with nanoscale dielectric AFM experiments on poly(vinyl acetate) near Tg by Crider and Israeloff [7] and with tapping-mode AFM studies obtained by Stanzione and co-workers shown in a recent publication by Wool and Campanella [4,8]. The twinkling fractal mechanism is exceedingly complex, which is most likely the source of much discussion, controversy, and debate on the nature of the glass transition as reviewed by Angell et al. [1] and recent international discussions on relaxations in complex systems [9]. For example, in the TFT, there is no characteristic relaxation time or length scale associated with Tg since the fractal vibrational density of states is continuous over all length scales, from molecular to micro, with twinkling frequencies that typically range from 1012 to 10− 5 Hz and potentially even slower. Furthermore, the characteristic break in slope of the volume–temperature curve near Tg is a consequence of the formation of a fractal percolation structure
that is kinetically driven and not a thermodynamic second-order phase transition [4]. In this article, we explicitly show experimental evidence of the twinkling fractal nature of the glass transition through the utilization of an isothermal 2-D spatio tapping-mode AFM technique and an isothermal 1-D spatio-temporal tapping-mode AFM technique. Both AFM methods have captured the fractal percolating structure of a glass-forming amorphous material (polystyrene) in its glass transition region. In addition, the relaxation behavior predicted by the TFT is shown to be in agreement with the experimental data. 2. Twinkling fractal theory The TFT defines the nature and structure of glass formation by fusing together the Adam–Gibbs concept of cooperatively rearranging regions (CRRs) [10], with dynamic heterogeneity, and fractal structure. The concept of dynamic heterogeneity has been employed to explain glass formation, in particular, the non-exponential behavior of relaxation, since the work of Angell et al. in the early 1970s and, according to Böhmer, refers to a system where “it is possible to select a dynamically distinguishable subensemble by experiment or computer simulation [11].” Since then, extensive evidence of the existence of heterogeneity near Tg with length scales from a few nanometers to ~100 nm have been reported in the literature [11–13]. Significant experimental evidence for the existence of dynamic heterogeneities by such experimental techniques as multidimensional NMR, nonresonant spectral hole burning, time resolved optical spectroscopy, photobleaching, dynamic light scattering, and electric force microscopy have been reported [11–13]. Of particular interest and influence on the TFT is the recent theoretical work of Bakai and Fischer, Steveonson et al., and Baljon et al. Bakai and Fischer reported that an array of fractal aggregations of fluctuons, known as Fischer clusters, forms as a liquid is cooled [14]. Stevenson et al. have redefined and reported CRRs near Tg with the incorporation of the concept of fractal strings [15]. Lastly, Baljon et al. proposed a bead-spring theory whereby slow moving “beads” percolate near Tg [16]. Despite all the insight into the glass transition, there lacks a theory that spans all amorphous materials and effectively and collectively incorporates the
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above concepts of cooperativity, dynamic heterogeneity, and fractal structure. In addition, molecular dynamics simulations of the amorphous state of materials have supported the concept of the formation of percolated clusters, some of which exhibit fractal-like structures, within the glass transition region. Medvedev et al. first observed the argument that glass formation is caused by the formation of solid-like percolation clusters made of high-density configurations in molecular dynamics simulations of liquid, supercooled, and quenched states of rubidium that utilized the Voronoi–Delanuay method [17]. Fractal percolated clusters were later observed by Evteev et al. and later included in a review by Ojovan [18–20]. In molecular dynamic simulations of pure, supercooled iron, Evteev et al. reported that the structural organization of the amorphous phase is believed to be a result of the formation and subsequent growth of mutually penetrating icosahedrons that contain atoms at the vertices and at the centers [18,19]. The main idea underlying the TFT is the development of percolating solid fractal structures near Tg, which are said to be in dynamic equilibrium with the surrounding liquid. This concept is derived from the Boltzmann population of excited vibrational states in the anharmonic intermolecular potential between atoms in the energy landscape. Solid-to-liquid transitions occur at a frequency ωTF, which is governed by the Boltzmann population of intermolecular oscillators in excited energy levels in accord with the Orbach vibrational density of states for a particular fractal cluster g(ω) ~ ωdf − 1, where the fracton dimension df = 4/3 [5]. To an observer, these clusters would appear to be “twinkling.” The solid and liquid clusters interchange with frequency ωTF ~ ωexp(− ΔE/kT) in which the activation energy is temperature-dependent and the number of clusters with frequency ω in a fractal is given by the vibrational density of states g(ω) ~ ωdf − 1. Clusters with frequency ω relax in time via c(t) = g(ω)exp(− ωTFt). Thus, for all clusters, the spatio-temporal fluctuation behavior associated with the twinkling fractal process involves a temperature-dependent global autocorrelation relaxation function C(t,T), as [2]:
=
ω=1 t
C ðt; T Þ =
∫ ωo
df −1
ω
19 8 0 < = −β T *2 −T 2 A dω exp −ωt exp@ : ; kT
ð1Þ
where ω is the vibrational frequency with ωo being the maximum (~ 1012 Hz), t is the observation time, β is a universal constant (~ 0.3 cal/mol K2 or 2.00 × 1024 J K− 2), k is Boltzmann's constant, and T* is a critical temperature defined with respect to Tg as T* ≈ 1.2Tg [2–4]. C(t) is mathematically equivalent to the infinite sum of weighted exponentials represented by the empirical Kolrausch–Williams–Watts stretched exponential function C(t) ~ exp [−(t/τ)β] in which τ is a characteristic time and β ≈ 0.5 is the stretching term. Typically τ and β are dependent on the weighting functions, which are given by the fractal cluster vibrational density of states, g(ω) ~ ωdf − 1, in the TFT. As shown later and as predicted by Eq. (1), there is no characteristic relaxation time at Tg but rather a continuous distribution of relaxation times spanning many decades associated with the many different fractal cluster length scales. According to Wool, Eq. (1) can be approximated for “short” times and high frequencies as C(t) ~ t – 1/3 and “intermediate/long” times or low frequencies as C(t) ~ t – 4/3 [2,4]. The exact analytical solution to the integral in Eq. (1) is given as follows: C ðt; T Þ =
ωo1 = 3 1 1 1 expð−αtωo Þ + Γ ; αtωo −Γ ; α 4 = 3 3 3 3αt 3ðαt Þ 1 − 4 = 3 expð−αÞ αt
ð2Þ
313
where β 2 2 α = exp − T * −T kT
ð3Þ
and where Γ(1/3,x) is defined as the “upper” incomplete gamma function. Thus, at a given rate or observation time t and temperature T, the function C(t,T) describes how many of the solid clusters of varying frequency ω could relax to a liquid in a given observation time interval. If the rate of observation or testing rate γ increases, then this time interval decreases since the observation time t behaves as t ~ 1/γ and fewer clusters will have relaxed. Thus, it is necessary to increase the temperature T to generate the same level of relaxation at a higher rate. This is the essential physics of the rate dependence of Tg as developed in Wool and Campanella [4]. The double exponential structure of C(t,T) in Eq. (1) provides a rapid slowing down of the twinkling dynamics when T b Tg but there is no “freeze-out” temperature To in the denominator which would cause the viscosity of a material to approach infinity at To as suggested by the empirical Vogel–Fulcher– Tammann (VFT) relation discussed in reference [1]. Note that when T * ≫ T, the activation energy term in Eqs. (2) and (3) become more Arrhenius-like or strong, and when T is of order T*, the behavior is more VFT-like or fragile, as discussed by Angell [1]. Most polymers are expected to behave in a fragile manner near Tg. Since T * is described by T*/Tg = (1 − pcs)/(1 − pc), materials with a low scalar percolation threshold, such as inorganic glasses, will appear stronger than polymers with higher pcs values. Further analysis of the behavior of Eq. (2) with respect to temperature T and observation time t is provided below. The TFT states that the solid clusters and surrounding liquid dynamically coexist in approximately equal amounts near Tg. A solid is defined as a cluster of atoms whose pair-wise vibrational interactions occur through average bond distances Xij that are less than the critical Lindemann inflection point Xc in their local anharmonic potential U(Xij) in the energy landscape [2]. An oscillator in the solid-state transitions to the liquid state when a thermal fluctuation causes it to expand beyond Xc. Moreover, the vibrational interactions of the solid clusters provide the mechanism of “communication” between the cooperatively rearranging regions discussed in the traditional Adam–Gibbs theory [10]. This is illustrated in Fig. 2 where Ps and PL represent the solid and liquid fractions, respectively. This concept parallels Lindemann's theory of melting [21]. Instead of crystallization, glass formation occurs due to molecular configurational irregularities, which promote the fractal structure formation and leads to viscous retardation of nucleation and crystallization.
Fig. 2. U as a function of Xij for pair-wise vibrational interactions of atoms in an amorphous material where the average bond distance is proportional to temperature. The inflection point Xc denotes the phase transition between the solid PS (Xij b Xc) and liquid PL (Xij N Xc).
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As Tg is approached from above, dynamic solid fractal clusters begin to form. Percolation of the solid clusters increases to a rigid, or vector, percolation threshold pc of ~0.5 at Tg. This formation of a rigid, frustrated fractal structure is said to dictate the structure of the nonequilibrium glass at T b Tg and explain the continuous change in slope on a V–T diagram. This alternative TFT approach of explaining the second-order behavior of amorphous materials replaces the free volume theory, which was originally proposed by Eyring and further developed by Fox and Flory and others [22–24]. In the TFT, the solid fraction PS behaves with temperature as PS ≈ 1 − [(1 − pc) T/Tg] with PL = 1 − PS, which satisfies continuity. In addition, these equations state that PS and PL are continuous through Tg, as are the modulus [25,26] and compressibility [27]. The onset of long-range connected solid fractal clusters begins at the scalar percolation threshold pcs which occurs at T* ≈ 1.2Tg. This is in agreement with the modecoupling theory TC as well as Goldstein's Tx [28,29]. Since g(ω) is continuous from very low to very high frequencies, the complex twinkling dynamics existing near Tg produces a continuous relaxation function, Eq. (1), with many different length scales and times associated with the fractal clusters. Therefore, the TFT is mainly a kinetic theory of the glass transition such that it is universal to all amorphous materials. The twinkling fractal process is not a subtle effect but rather is a strong physical effect such that about 50% of the material near Tg is not only solid but is jumping back and forth into the liquid state with precise dynamics. Therefore, it should be readily observable and quantifiable by relatively simple transparent experiments, as we demonstrate herein with time-lapse AFM on an amorphous solid near Tg. 3. Experimental 3.1. Materials Commercially purchased, amorphous, atactic polystyrene (PS) (MW = 194,000 g/mol and PDI = 1.07 confirmed by gel permeation chromatography; Tg = 375 K confirmed on a differential scanning calorimeter with a heating rate of 3 K/min) was utilized. PS has been chosen as the glass former to study due to its relatively low Tg (~373 K), compared to most ceramics (e.g. Pyrex, Tg = 848 K) and is closer to room temperature (TR) than most smaller molecules (e.g. glycerol, Tg = 183 K) [2]. In addition, PS has been widely used and discussed in the literature as the benchmark material for phenomenological testing. 3.2. Sample preparation PS films were prepared by spin coating from toluene (HPLC grade) (10 wt.% PS) at 1500 rpm onto gold-plated glass slides, which have been heat treated, rinsed in toluene, and subsequently dried under compressed nitrogen flow prior to coating. Film thickness (1150 nm) and roughness (b1.0 nm) were determined by AFM at TR. 3.3. AFM techniques A Veeco Dimension V scanning probe microscope in tapping-mode with a Veeco DMHC-A250 heater/cooler and AFM probes (Veeco FESPs) were used in c.a. 45% relative humidity air to image the dynamic heterogeneity of PS at TR and at isothermal temperatures close to 373 K. The system was allowed to thermally equilibrate prior to imaging in order to minimize thermal drift. The slow scan axis (y-axis) was disabled to generate 1-D space-time images, while the slow scan axis was enabled to generate the normal 2-D spatial images. For both methods, the scan length in the x-direction and scan rate were 700 nm and 1 Hz, respectively, with the XY closed loop turned on and the Z-limit set to 1.0 μm. A uniform amplitude setpoint was used throughout the experiments to ensure comparability. For
generating the 1-D images, each scan traversed the same 700 nm in the x-direction generating 512 space data points (i.e.: 512 pixels) per second for a total of 512 s (i.e.: 512 lines in the y-direction) per captured image. The retrace height and phase data were employed in the following analysis. In order to spatially resolve regions of different stiffness (liquid versus solid fractions), AFM phase images were analyzed. The phase signal from tapping-mode AFM represents the lag between the driving oscillation signal and the resultant oscillation of the cantilever (simple harmonic oscillator.) This lag is zeroed out when the probe tip is unaffected by the surface forces (in air). When the probe tip experiences surface forces, a phase lag is introduced. For “repulsive” amplitude setpoints, the phase lag is often dominated by material stiffness, which is especially true in the case of hydrophobic materials such as polystyrene. When this condition is met, hard regions appear bright (high) in the phase signal (small lag) and softer regions, because of their greater dissipation, appear dark (low) in the phase signal (large lag.) Thus, the phase signal provides a spatially resolved map of hard and soft regions (or small and large dissipation) across the scan area. All AFM images shown have been 1st ordered flattened and lowpass filtered in the AFM software for viewing purposes and qualitative inspections. Raw AFM height and phase data was used as the basis in the quantitative analysis. 3.4. AFM X–Y height images with fractal dimension calculations 3.4.1. Statistical analysis The consecutive AFM X–Y height images in Fig. 1 were quantitatively compared by first converting the raw RGB images to grayscale and then computing a 2-D correlation coefficient between the first and second images and the first and third images. The form of the 2-D correlation coefficient utilized is: Σm Σn Amn −A Bmn −B ffi: r = sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Σm Σn Bmn −B Σm Σn Amn −A
ð4Þ
In Eq. (4), Amn and Bmn are data corresponding to the first and subsequent image, respectively, where A ̅ and B ̅ are the mean values of the images. 3.4.2. Fractal dimension The fractal dimensions Df of the images in Fig. 1 were computed and compared by first converting the RGB images to grayscale and then implementing an approximate box counting method. 3.5. Statistical autocorrelation function Prior to implementing statistical techniques to the AFM X–t phase data, each image, exported as raw ASCII (least significant bit) data from the AFM software, was 1st ordered flattened followed by a 1 × 3 boxcar average filtration to each x position in order to reduce any high frequency noise that was present within the data. In order to quantitatively compare the experimentally captured relaxation behavior of the twinkling events with the TFT C(t,T), a statistical autocorrelation function C(t) of the following form was applied to the AFM X–t phase data: C ðt Þk =
ck co
where ck =
1 N−k ∑ ðx − xÞ xt N t=1 t
+ k− x
:
ð5Þ
Eq. (5) is the Box–Jenkins estimated autocorrelation function [30]. An autocorrelation function reveals how the correlation between any two values of a data series changes as their separation changes and is
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4. Results 4.1. AFM X–Y height images
Fig. 3. Plot of the 2-D correlation coefficient as a function of time for computed coefficients between the first image with itself, the first image with the second (t = 8.53 min), and the first image with the third (t = 17.06 min) for T = 298 K (black ●) and T = 358 K (red ▲).
used mathematically to extract repeating patterns in random processes [30]. Eq. (5) is used to describe the behavior of a strictly stochastic stationary process, which is assumed to adequately describe the dynamics of the twinkling fractals. In Eq. (5), co and ck are the values of the autocorrelation function at the zeroth and kth time lag, respectively, with N being the total number of oscillations. xt and x ̅ are the numerical values of the observation at lag t and the mean, respectively. In order to calculate a global autocorrelation, the C(t)k is calculated at each x position and then averaged over all positions with accompanying standard deviations. According to Box and Jenkins, useful estimates of the autocorrelation function can be calculated for k = 0,1,…, K where K is not larger than ~ N/4 [30].
4.1.1. 2-D correlation coefficient Fig. 1 shows successive AFM X–Y height images of dimensions 700 nm × 700 nm for T = 298 K (a–c, top) and T = 358 K (d–f, bottom). Each image was captured 8.53 min apart from each other. Thermal drift was estimated to be less than 1.0 nm/min. The percolating solid fractal cluster that exists at T b Tg is believed to be revealed in these height images from a 2-D topographical viewpoint. Fig. 3 shows the results of the 2-D correlation coefficient analysis (Eq. (4)) between the consecutive images where the 2-D correlation coefficient r is plotted as a function of time between captured images for T = 298 K (black circles) and T = 358 K (red triangles). The closer the correlation coefficient values are to one, the more correlated the images are with each other. As expected, the room temperature results (298 K) with a much slower twinkling rate show a smaller change with time compared to the higher temperature results (358 K).
4.1.2. Fractal dimension The fractal dimensions Df for the 298 K and 358 K X–Y height images in Fig. 1 were calculated from the digital data to be 1.8777 ± 0.0005 and 1.8534 ± 0.0091, respectively. From the theory of the fractal nature of percolating clusters, Df = 1.89 in 2-D. Typical fractal dimensions of scalar percolation clusters are about 1.75 but one expects the vector percolation fractal dimension to be higher due to the additional mass required to propagate rigidity at the vector percolation threshold near Tg. The observation of fractal structure in amorphous polystyrene is in accord with observations of the fractal structure in amorphous metals, where Df = 2.31 was reported from neutron and X-ray scattering experiments below Tg [31]. Recently, Ojovan examined solid–liquid bond transitions in SiO2 and GeO2 oxides and reported percolation behavior of the solid regions with fractal dimensions Df = 2.55 [32–34].
Fig. 4. 1-D AFM X–t height (top) and phase (bottom) images at 298 K (left), 358 K (middle) and 383 K (right) obtained by repeated scans along a single line (slow scan axis disabled). PS films with thicknesses of 1146 ± 14 nm (358 K and 383 K) and 600 nm (298 K) were used.
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4.2. Single AFM X–t images
4.3. Consecutive AFM X–t phase images
AFM “movies” or many rapid snapshots versus time of the twinkling process to analyze the dynamics would be ideal. To mimic this ideal situation in part, as shown in Fig. 4, 1-D space-time (X–t) height (top row) and phase images (bottom row) of the surface of PS at 298 K (T ≪ Tg), 358 K (T b Tg), and 383 K (T N Tg) were obtained. It is noted that the surface Tg of PS is expected to be less than the bulk Tg due to relaxed constraints to motion [35]; however, all analysis presented herein is with respect to the bulk Tg of polystyrene. The height images in Fig. 4 (top) show the roughness of the PS surface as a function of time. The phase images display elasticity differences as a function of time and thus, can show the solid fractal clusters twinkling at a particular observation length of 700 nm and time scales of 1.0 Hz.
Fig. 5 shows the consecutive AFM X–t phase images, which will be used to compare with the TFT simulations and the theoretical relaxation function C(t,T). In Fig. 5, the spatial scan length of 700 nm is located on the abscissa with time on the ordinate. AFM X–t phase data (1st order flattened and 1 × 3 boxcar average filtered) were combined in a sequential manner to obtain continuous data sets. Different amounts of data were required to produce experimental relaxation results that were comparable to the TFT. As can be seen in Fig. 5, the amount of experimental data required increases as temperature decreases, which correlates well with the TFT. The exact relationship between the number of data points used to obtain comparable results is currently under investigation. 4.3.1. Cluster lifetimes τ By closely inspecting the images in Fig. 5, one can estimate the cluster widths and lifetimes of the dynamic heterogeneities as shown in Fig. 6. Generally, one sees that the wider the cluster, the longer will be its lifetime or “streak height” in Fig. 5. In Fig. 6, the logarithm of cluster lifetimes is plotted as a function of the logarithm of the cluster widths R or radii in nanometers. The logarithmic scale was used in order to compare the experimental data to the theory of dynamic fractal networks from Orbach et al., where τ ~ RDf/df [5,6]. Df is the fractal dimension = 1.89 in 2-D and df = fracton dimension = 4/3 such that Df/df = 1.42, which is the predicted theoretical slope on a logarithmic scale. Fig. 6 shows that the experimental data, though highly scattered as expected for fractal cluster sampling, is in good accord with the theoretical prediction of τ ~ RDf/df. 4.4. Relaxation behavior C(t,T) The theoretical relaxation function C(t,T) in Eq. (2) was calculated and is plotted in Fig. 7 as a function of time on a log–log scale for a time range that spans four orders of magnitude, with temperatures of 298 K, 358 K, and 383 K. The time range from 1 to 1 × 104 s was chosen in Fig. 7 so that the theory can be directly compared to the experimental data shown in Fig. 5. For each temperature T, the observation time t was swept from 10− 14 s to 1010 s with 1000 data points between each order of magnitude. The universal constants β, k, and ωo in Eqs. (2) and (3) were fixed at 2.0 × 10− 24 J K− 2, 1.38 × 1023 J K− 1, and 1012 Hz,
Fig. 5. 1-D X–t AFM phase images at 298 K (left), 358 K (middle), and 383 K (right) obtained by repeated scans along a single line (slow scan axis disabled). PS films with thicknesses of 1146 ± 14 nm (358 K and 383 K) and 600 nm (298 K) were used. Single 512 × 512 data point images were generated during each experiment and were connected together to produce a continuous single image.
Fig. 6. Log–log plot of the cluster lifetimes as a function of cluster widths for a sample of twinkling fractal clusters present in the AFM X–t phase images for temperatures of 358 K (red ▲), and 383 K (blue ■); included is a slope of 1.42 (black line) that represents Df/df ~ 1.42 from fractal theory.
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Fig. 7. Log–log plot of the TFT global autocorrelation function C as a function of t and T for temperatures of 298 K (black), 358 K (red), and 383 K (blue). The noise in the beginning of the 298 K curve is attributed to the instability of the function with respect to α ~ 10− 24 when T = 298 K, which greatly suppresses the frequency probability of twinkling events for glassy PS.
respectively. With respect to polystyrene, the characteristic temperatures Tg = 373 K and T* = 448 K were used in Eq. (3). Fig. 7 shows that at low temperatures the relaxation slopes are close to zero and at higher temperatures increase at short times, on an absolute scale, with values near 0.4 that eventually reach an asymptotic value of 1.33 at longer times. This result is reflective of the fracton dimension df = 1.33 in the fractal vibrational density of states g(ω) ~ ω1.33. Fig. 8 shows the experimental autocorrelation function C(t) versus time plotted on a log–log scale for short times t = 1–10 s (Fig. 8a) and longer times t = 10–100 s (Fig. 8b) for all three experimental temperatures. Linear best-fit curves to the experimental data are also included along with slope and best-fit R2 values. The linear fits were applied to experimental data up to the point where the C(t)k curve on a linear scale first intersects the ordinate and is above the 95% confidence level threshold of the statistical calculation (383 K curve fit an exception to the latter criterion). The slopes in Fig. 8 show values near −1/3 at low temperatures and short times, eventually increasing towards −4/3 and higher values −1.56 at longer times and higher temperatures, which is consistent with the TFT predictions. At much longer times when the vibrational phonon wavelengths no longer “see” the fractal structure, the slope is expected to approach the Debye limit of − 2. 5. Discussion 5.1. AFM X–Y height images In qualitatively comparing the time-lapse AFM X–Y height images in Fig. 1, there exists greater similarity between the features in the low temperature (298 K) images compared to the higher temperature (358 K) images. This reveals that the percolating solid fractal cluster exists for much longer time scales at lower temperatures. Furthermore, in comparing the quantitative results of the 2-D correlation coefficients seen in Fig. 3, the correlation coefficient decreases significantly less and more slowly for the 298 K images than the 358 K suggesting that the 298 K images are more similar and are well correlated. These results confirm what one visually concludes when qualitatively analyzing the time-lapse images in Fig. 1 and that the fractal structures twinkle more slowly at lower temperatures as expected. With regards to the fractal dimensions Df,, the computed values are in excellent agreement with the theory of the fractal nature of
Fig. 8. Log–log plot of the statistical autocorrelation function C(t)k as a function of lags t between t = 1–10 s (a) and t = 10–100 s (b) for temperatures of 298 K (black ●), 358 K (red ▲), and 383 K (blue ■) with corresponding linear best-fit curves for 298 K (solid black line), 358 K (dashed red line), and 383 K (dotted blue line). The slopes corresponding to the linear best-fit curves with respective R2 values are given in the upper right box.
percolating clusters and are in accord with experimentally observed 2-D fractal dimensions for dynamic uniform fractals in emulsions reported by Ozhovan [5,6,36]. This result suggests that indeed percolated solid fractal clusters develop upon cooling amorphous materials through the glass transition and, with the visual aids of Figs. 1, 4, and 7, that dynamic heterogeneity as well as cooperativity is prevalent in glassy materials. 5.2. Qualitative inspection of the single AFM X–t images As can be seen in Fig. 4, the spatio-temporal aspects of PS clearly exist at all three temperatures even at 298 K, which is 75 K below the bulk Tg value. Focusing on the phase images, correlations along the time axis are distinctly longer at 298 K compared to the temperatures closer to the bulk Tg. When qualitatively inspecting the 298 K images and even though PS at room temperature is well within the glassy state, relative to the bulk Tg of PS, clusters seem to be twinkling at a rate inversely proportional to their radii, which correlates with the TFT and supports the results shown in Fig. 6. In addition, the TFT states that a rapid slowing down of the twinkling dynamics occurs when T b Tg
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where vector percolation of the solid fractal clusters dominates the physical behavior of the material, even though a considerable amount of “liquid” still exists, pL ≈ 0.40, at 298 K for PS. The existence of the liquid fraction that is confined by the percolated solid fractal clusters is captured in the 298 K images, particularly, in the phase image as regions that twinkle relatively frequently but do not drastically alter the topological profile of the height image as compared to the higher temperatures. Large solid clusters will have a very low jump rate at the lower temperatures and we suspect that the observed twinkling process becomes dominated by the nanoscale clusters with lifetimes τ/aT ~ RDf/df, where the shift factor aT = α in Eq. (3). In comparing the height and phase images, it is reassuring to observe that the phase images are noticeably different from the height data, particularly at 358 K and 383 K, indicating that indeed potential dynamic heterogeneity, cooperativity, and fractal structure are being captured and not just the observation of topological phenomena. 5.3. Consecutive AFM X–t phase images By qualitatively comparing the X–t images in Fig. 5, one can observe that the twinkling events tend to increase in frequency with increasing temperature as the larger solid clusters can participate in the twinkling process. Even though the 298 K image seems to be twinkling rather frequently at a small cluster scale as expected, the overall rigidity of the PS remains indicative of the dominance of the percolated solid fractal cluster. The frequent twinkles captured at 298 K are believed to be very small solid fractal clusters of the overall solid cluster hopping in and out of the liquid state in a dynamic fashion in order to maintain equilibrium. In the 358 K and 383 K images and similar to the 383 K image in Fig. 4, multiple white (harder) regions are present and are believed to be liquid regions that have twinkled to stable solid fractal clusters. The widths of these solid clusters are roughly 12 nm, which is within range of reported values for the characteristic lengths of dynamic heterogeneities [11]. By selecting a representative sample of clusters from both the 358 K and 383 K images and measuring their respective lifetimes and widths, one can show that on average, twinkling events are more frequent at higher temperatures with respect to the observation time and the cluster size. The solid-to-liquid twinkling frequencies are β given from Eq. (1) by ω(T) = ω exp − kT T *2 −T 2 , where ω is the lowest fundamental frequency of the solid cluster involving the majority of atoms cooperatively in its eigenvector. Thus, in general, the twinkling frequency increases with temperature for clusters of a given radius R. This is shown in Fig. 6 where the cluster lifetimes decrease with decreasing cluster sizes and, in general, the cluster sizes increase with decreasing temperature. In addition, the slope of the experimental data is in excellent agreement with fractal theory as well as with the TFT predictions. 5.4. Relaxation behavior As can be seen in Fig. 7, the global relaxation of twinkling events at 298 K requires many orders of magnitude more than at temperatures in proximity to the Tg of PS. The noise in the beginning of the 298 K curve is attributed to the instability of the function due to α ~ 10− 24 when T = 298 K, which greatly suppresses the frequency probability of significant twinkling events that effectively influence the macroscale physical properties as well as the global relaxation of glassy PS. Therefore, the slope of C(t,T = 298 K) for t = 1.0 to 1.0 × 104 s is ~ 0. As T approaches Tg (T = 358 K), the probability of twinkling events increases, thereby increasing the twinkling frequency and ultimately reducing the relaxation time of the twinkling events, which, in turn, shifts “longer” times closer to the observation time t. This causes the slope of C(t,T = 358 K) for t = 1.0 to 1.0 × 104 s to be equal to −0.43 for t b 10 s, − 1.29 for 10 b t b 100 s and − 1.33 for t N 100 s. Thus,
significant twinkling events should be observable within the experimental AFM scan rate. Focusing on the 358 K AFM X–t phase images in Figs. 4 and 5, one can see the dynamic heterogeneity as multiple clusters transition from the softer phase (darker color) to the harder phase (lighter color) and vice versa. The twinkling behavior is believed to be captured in these images and well reflected in the autocorrelation relaxation function with C(t) ~ t− 4/3. When T = 383 K, which exceeds Tg, the probability of solid–liquid twinkling events increases further, the fractal structure loses its percolated rigidity, and only scalar percolation dominates the physical behavior of the now rubbery material. Due to this increased flexibility, the twinkling frequencies and, in turn, the relaxation time of the twinkling events decrease to the point that prior “longer” times now occur three orders of magnitude less than the observation time t. This causes the slope of C(t,T = 383 K) for t = 1.0 to 1.0 × 104 s to be equal to a constant of −1.33. Therefore, significant twinkling events should still be observable but at an increased rate compared to T = 358 K. This is indeed what is believed to be captured in the 383 K AFM X–t phase images in Figs. 4 and 5. In comparing the 358 K and 383 K phase images, the dynamic heterogeneity is twinkling at a faster rate and the fractal structure is less percolated in the 383 K phase image than in the 358 K image. An interesting feature of the 383 K images in Figs. 4 and 5 is the existence of two rather long-lived white (harder) regions. These regions are speculated to be potential liquid regions that have twinkled to solid fractal clusters of significantly shorter vibrational interactions with consequently slower relaxation dynamics, considerable cooperativity, and relatively rigid fractal structures. Again, the widths of these solid clusters are roughly 12 nm, which is within range of reported values for the characteristic lengths of dynamic heterogeneities [11]. In comparing the experimental relaxation behavior results shown in Fig. 8 with the TFT predictions, the slopes for 358 K and 383 K are in fair agreement with the TFT. The low R2 values for the 383 K curve fits are the result of the autocorrelation falling below its respective 95% confidence bound value of 0.044 (−1.355 on a log scale) in a relatively short amount of lag time. The slopes for 298 K, on the other hand, do not correlate well with the TFT, where one predicts a slope of ~0 over the observation time. Based on this, the statistical autocorrelation seems to be very sensitive to small differences. The statistical autocorrelation seems to be picking up the frequent twinkles of the surrounding liquid but is unable to capture the robustness of the percolated solid fractal cluster. This observation may also explain why the 358 K and 383 K slopes are slightly higher than what is predicted by the TFT. In addition and even though significant time was provided in order to achieve thermal equilibrium, the AFM technique contains intrinsic and unavoidable thermal drift, which can be seen to some extent in the 358 K images in Figs. 4 and 5 and estimated to be less than 1.0 nm/min. Also, even though setpoints of approximately 90– 95% were used, possible AFM tip induced plastic deformation, instead of elastic deformation, may still occur, depending on the surface sensitivity of the material being scanned. Overall, from both a qualitative and quantitative reasoning, the AFM images of polystyrene do capture the twinkling phenomena of this amorphous material in its glass transition region with experimental relaxation behavior in fair agreement with that predicted by the TFT. 6. Conclusions Atomic force microscopy has been utilized to physically observe the twinkling fractal nature of amorphous materials predicted by the TFT. Specifically, the twinkling fractal characteristics of amorphous, atactic polystyrene have been captured in both height and phases images. Successive 2-D height images revealed that the percolated solid fractal clusters exist for longer time scales at lower temperatures. The computed fractal dimensions, Df ≈ 1.88, are shown to be in excellent agreement with the theory of the fractal nature of
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percolating clusters predicted by the TFT. The twinkling dynamics of polystyrene within its glass transition region were captured in timelapse one-dimensional AFM phase images with the physically observed cluster lifetimes and widths in agreement with the TFT and the theory of dynamic fractal networks. A comparison of the twinkling relaxation behavior of the TFT with the physically observed relaxation behavior provided robust support of the TFT. Acknowledgments The authors are grateful to the U.S. Army Research Laboratory for financial support under the Army Materials Center of Excellence Program, contract W911NF-06-2-011. The author J.F. Stanzione III is also grateful for generous mentorship of Dr. Kenneth Strawhecker and Dr. David Webb from the ARL-APG. References [1] C.A. Angell, K.L. Ngai, G.B. McKenna, P.F. McMillan, S.W. Martin, J. Appl. Phys. 88 (2000) 3113. [2] R.P. Wool, J. Polym. Sci. Pol. Phys. 46 (2008) 2765. [3] R.P. Wool, Soft Matter 4 (2008) 400. [4] R.P. Wool, A. Campanella, J. Polym. Sci. Pol. Phys. 47 (2009) 2578. [5] R. Orbach, Science 231 (1986) 814. [6] T. Nakayama, K. Yakubo, R.L. Orbach, Rev. Mod. Phys. 66 (1994) 381. [7] P.S. Crider, N.E. Israeloff, Nano Lett. 6 (2006) 887. [8] R.P. Wool, J.F. Stanzione III, K.E. Strawhecker, Data presented at the APS March meeting., in, 2009. [9] G. Ruocco, K.L. Ngai, In Proceedings of the 6th International Discussions on Relaxation in Complex Systems, Rome, Italy, 2009.
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[10] G. Adam, J.H. Gibbs, J. Chem. Phys. 43 (1965) 139. [11] H. Sillescu, J. Non-Cryst. Solids 243 (1999) 81. [12] U. Tracht, M. Wilhelm, A. Heuer, H. Feng, K. Schmidt-Rohr, H.W. Spiess, Phys. Rev. Lett. 81 (1998) 2727. [13] P.G. Debenedetti, F.H. Stillinger, Nature 410 (2001) 259. [14] A.S. Bakai, E.W. Fischer, J. Chem. Phys. 120 (2004) 5235. [15] J.D. Stevenson, J. Schmalian, P.G. Wolynes, Nat. Phys. 2 (2006) 268. [16] A.R.C. Baljon, J. Billen, R. Khare, Phys. Rev. Lett. 93 (2004) 255701-1. [17] N.N. Medvedev, A. Geiger, W. Brostow, J. Chem. Phys. 93 (1990) 8337. [18] A.V. Evteev, A.T. Kosilov, E.V. Levchenko, J. Exp. Theor. Phys. 99 (2004) 522. [19] A.V. Evteev, A.T. Kosilov, E.V. Levchenko, O.B. Logachev, Phys. Solid State 48 (2006) 815. [20] M.I. Ojovan, Entropy 10 (2008) 334. [21] F.A. Lindemann, Phys. Z. 11 (1910) 609. [22] H. Eyring, J. Chem. Phys. 4 (1936) 283. [23] T.G. Fox, P.J. Flory, J. Appl. Phys. 21 (1950) 581. [24] R. Simha, J. Chem. Phys. 37 (1962) 1003. [25] R.P. Wool, X.S. Sun, Bio-based Polymers and Composites, Elsevier Academic Press, Amsterdam, 2005 Boston. [26] A. Campanella, J.J. La Scala, R.P. Wool, J. Appl. Polym. Sci. (in press), doi:10.1002/ app.32810. [27] L.A. Utracki, J. Polym. Sci. Pol. Phys. 46 (2008) 2504. [28] M. Goldstein, J. Chem. Phys. 51 (1969) 3728. [29] A. Cavagna, Phys. Rep. 476 (2009) 51. [30] G.E.P. Box, G.M. Jenkins, Time Series Analysis: Forecasting and Control, Rev. ed. Holden-Day, San Francisco, 1976. [31] D. Ma, A.D. Stoica, X.L. Wang, Nat. Mat. 8 (2009) 30. [32] M.I. Ojovan, J. Exp. Theor. Phys. Let. 79 (2004) 632. [33] M.I. Ojovan, W.E. Lee, J. Phys.: Condensed Matter 18 (2006) 11507. [34] M.I. Ojovan, Twinkling Fractals and Glass Transition in Amorphous Oxides, Paper presented at the International Congress on Glass, Bahia, Brazil, Sept 20, 2010, 2010. [35] J.L. Keddie, R.A.L. Jones, R.A. Cory, Europhys. Lett. 27 (1994) 59. [36] M.I. Ozhovan, J. Exp. Theor. Phys. 77 (1993) 939.
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Journal of Non-Crystalline Solids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j n o n c r y s o l
Observing the twinkling fractal nature of the glass transition J.F. Stanzione III a, K.E. Strawhecker b, R.P. Wool a,⁎ a b
Department of Chemical Engineering and Center for Composite Materials, University of Delaware, Newark, DE 19716, United States Army Research Laboratory, Aberdeen Proving Ground, MD 21005, United States
a r t i c l e
i n f o
Article history: Received 9 April 2010 Received in revised form 16 June 2010 Available online 25 August 2010 Keywords: Glass transition; Dynamic fractal networks; Percolation; Solid-state structure; Atomic force microscopy
a b s t r a c t A fundamental understanding of the nature and structure of the glass transition in amorphous materials is currently seen as a major unsolved problem in solid-state physics. A new conceptual approach to understanding the glass transition temperature (Tg) of glass-forming liquids called the twinkling fractal theory (TFT) has been proposed in order to solve this problem. The main idea underlying the TFT is the development of dynamic rigid percolating solid fractal structures near Tg, which are said to be in dynamic equilibrium with the surrounding liquid. This idea is coupled with the concept of the Boltzmann population of excited vibrational states in the anharmonic intermolecular potential between atoms in the energy landscape. Solid and liquid clusters interchange or “twinkle” at a cluster size dependent frequency ωTF, which is controlled by the population of intermolecular oscillators in excited energy levels. The solid-to-liquid cluster transitions are in accord with the Orbach vibrational density of states for a particular fractal cluster g(ω) ~ ωdf − 1, where the fracton dimension df = 4/3. To an observer, these clusters would appear to be “twinkling.” In this paper, experimental evidence supporting the TFT is presented. The twinkling fractal characteristics of amorphous, atactic polystyrene have been captured via atomic force microscopy (AFM). Successive two-dimensional height AFM images reveal that the percolated solid fractal clusters exist for longer time scales at lower temperatures and have lifetimes that are cluster size dependent. The computed fractal dimensions, ≈ 1.88, are shown to be in excellent agreement with the theory of the fractal nature of percolating clusters in accord with the TFT. The twinkling dynamics of polystyrene within its glass transition region are captured with timelapse one-dimensional AFM phase images. The autocorrelation cluster relaxation function was found to behave as C(t) ~ t− 4/3 and the cluster lifetimes τ versus width R were found to be in excellent agreement with the TFT via τ ~ R1.42. This paper provides compelling new experimental evidence for the twinkling fractal nature of the glass transition. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The heterogeneous state of matter that exists near the glass transition in an amorphous material is ubiquitous; however, the true nature and structure of this transition is still considered a critical unsolved problem [1]. Recently, a new theory of the glass transition has been proposed by Wool, entitled the twinkling fractal theory (TFT) [2,3]. In addition, recent atomic force microscopy (AFM) images of polystyrene published by Wool and Campanella and those shown in Fig. 1 have captured the percolating twinkling solid fractal structure near the glass transition temperature (Tg), proposed by the TFT [4]. Based on the TFT, fractal hard and soft regions exist which are quite dynamic and which appear and disappear with time. The AFM techniques utilized in this work allows one to obtain sequential “snap-shots” of the fractal structure near Tg to produce twodimensional (2-D) spatial images (Fig. 1) as well as obtain consecutive
⁎ Corresponding author. Tel.: +1 302 831 3312; fax: +1 302 831 1048. E-mail address: [email protected] (R.P. Wool). 0022-3093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2010.06.041
time-lapse, one-dimensional (1-D) spatial images that permit one to visualize and quantify the complex twinkling fractal dynamics, shown later. According to the TFT, the heterogeneous state of matter that exists near the glass transition in amorphous materials evolves from a dynamic fractal solid-like structure in equilibrium with its liquid [2]. The dynamics of the heterogeneity is driven by the Boltzmann population of vibrational energy levels of clusters oscillating in the anharmonic intermolecular potentials associated with the energy landscape [2]. The anharmonic potentials on the intermolecular coordinates create a temperature-dependent solid PS and liquid PL fractions, which dynamically interchange due to the Boltzmann population of vibrational energy levels. When an oscillator bond length thermally expands to the inflection point in the potential function, its force constant decreases toward zero and a thermal fluctuation may send it into the liquid state, which is the origin of the solid-to-liquid twinkling process in the TFT. As Tg is approached from above, the twinkling solid fraction grows to about 50% by volume and percolates rigidity. The onset of long-range connected fractal structures begins at the scalar percolation threshold pcs, which occurs at
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Fig. 1. AFM tapping-mode X–Y height images of the percolating twinkling solid fractal structure in amorphous polystyrene at 298 K (images a–c) and 358 K (images d–f).
T* N Tg and is determined by T*/Tg = (1 − pcs)/(1 − pc) where the rigidity percolation threshold pc ≈ 0.5. Typical values of the scalar and rigidity percolation thresholds give T*/Tg ≈ 1.2 [2]. Above T*, small non-connected clusters of solid material can still form but are relatively short-lived since cluster lifetimes τ is a function of size R as τ ~ R1.42. Fig. 1 shows the percolating twinkling solid fractal structure of polystyrene at 298 K (images a–c) and at 358 K (images d–f) in three sequential AFM X–Y height images at each temperature. As can be seen in Fig. 1, overall, the topological percolating solid fractal structure at 298 K twinkles less frequently than the structure at 358 K. According to the TFT, the percolating fractal twinkles with a frequency spectrum as solid and liquid clusters interchange with frequency ωTF ~ ωexp(− ΔE/kT) [2–4]. This inverse relationship between twinkling frequency and temperature is qualitatively in agreement with the images in Fig. 1. The vibrational density of states g(ω) for a fractal is determined to be g(ω) ~ ωdf − 1, where df = 4/3 [5,6] and the temperature-dependent activation energy behaves as ΔE ~ (T *2 − T2) [2]. Approximately, the TFT predicts that the spatiotemporal thermo-fluctuation autocorrelation relaxation function C(t) behaves as an inverse power law C(t) ~ t− 1/3 for “short” times or high frequencies and C(t) ~ t− 4/3 for “long” times or low frequencies. These predictions were found to be in fair agreement with nanoscale dielectric AFM experiments on poly(vinyl acetate) near Tg by Crider and Israeloff [7] and with tapping-mode AFM studies obtained by Stanzione and co-workers shown in a recent publication by Wool and Campanella [4,8]. The twinkling fractal mechanism is exceedingly complex, which is most likely the source of much discussion, controversy, and debate on the nature of the glass transition as reviewed by Angell et al. [1] and recent international discussions on relaxations in complex systems [9]. For example, in the TFT, there is no characteristic relaxation time or length scale associated with Tg since the fractal vibrational density of states is continuous over all length scales, from molecular to micro, with twinkling frequencies that typically range from 1012 to 10− 5 Hz and potentially even slower. Furthermore, the characteristic break in slope of the volume–temperature curve near Tg is a consequence of the formation of a fractal percolation structure
that is kinetically driven and not a thermodynamic second-order phase transition [4]. In this article, we explicitly show experimental evidence of the twinkling fractal nature of the glass transition through the utilization of an isothermal 2-D spatio tapping-mode AFM technique and an isothermal 1-D spatio-temporal tapping-mode AFM technique. Both AFM methods have captured the fractal percolating structure of a glass-forming amorphous material (polystyrene) in its glass transition region. In addition, the relaxation behavior predicted by the TFT is shown to be in agreement with the experimental data. 2. Twinkling fractal theory The TFT defines the nature and structure of glass formation by fusing together the Adam–Gibbs concept of cooperatively rearranging regions (CRRs) [10], with dynamic heterogeneity, and fractal structure. The concept of dynamic heterogeneity has been employed to explain glass formation, in particular, the non-exponential behavior of relaxation, since the work of Angell et al. in the early 1970s and, according to Böhmer, refers to a system where “it is possible to select a dynamically distinguishable subensemble by experiment or computer simulation [11].” Since then, extensive evidence of the existence of heterogeneity near Tg with length scales from a few nanometers to ~100 nm have been reported in the literature [11–13]. Significant experimental evidence for the existence of dynamic heterogeneities by such experimental techniques as multidimensional NMR, nonresonant spectral hole burning, time resolved optical spectroscopy, photobleaching, dynamic light scattering, and electric force microscopy have been reported [11–13]. Of particular interest and influence on the TFT is the recent theoretical work of Bakai and Fischer, Steveonson et al., and Baljon et al. Bakai and Fischer reported that an array of fractal aggregations of fluctuons, known as Fischer clusters, forms as a liquid is cooled [14]. Stevenson et al. have redefined and reported CRRs near Tg with the incorporation of the concept of fractal strings [15]. Lastly, Baljon et al. proposed a bead-spring theory whereby slow moving “beads” percolate near Tg [16]. Despite all the insight into the glass transition, there lacks a theory that spans all amorphous materials and effectively and collectively incorporates the
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above concepts of cooperativity, dynamic heterogeneity, and fractal structure. In addition, molecular dynamics simulations of the amorphous state of materials have supported the concept of the formation of percolated clusters, some of which exhibit fractal-like structures, within the glass transition region. Medvedev et al. first observed the argument that glass formation is caused by the formation of solid-like percolation clusters made of high-density configurations in molecular dynamics simulations of liquid, supercooled, and quenched states of rubidium that utilized the Voronoi–Delanuay method [17]. Fractal percolated clusters were later observed by Evteev et al. and later included in a review by Ojovan [18–20]. In molecular dynamic simulations of pure, supercooled iron, Evteev et al. reported that the structural organization of the amorphous phase is believed to be a result of the formation and subsequent growth of mutually penetrating icosahedrons that contain atoms at the vertices and at the centers [18,19]. The main idea underlying the TFT is the development of percolating solid fractal structures near Tg, which are said to be in dynamic equilibrium with the surrounding liquid. This concept is derived from the Boltzmann population of excited vibrational states in the anharmonic intermolecular potential between atoms in the energy landscape. Solid-to-liquid transitions occur at a frequency ωTF, which is governed by the Boltzmann population of intermolecular oscillators in excited energy levels in accord with the Orbach vibrational density of states for a particular fractal cluster g(ω) ~ ωdf − 1, where the fracton dimension df = 4/3 [5]. To an observer, these clusters would appear to be “twinkling.” The solid and liquid clusters interchange with frequency ωTF ~ ωexp(− ΔE/kT) in which the activation energy is temperature-dependent and the number of clusters with frequency ω in a fractal is given by the vibrational density of states g(ω) ~ ωdf − 1. Clusters with frequency ω relax in time via c(t) = g(ω)exp(− ωTFt). Thus, for all clusters, the spatio-temporal fluctuation behavior associated with the twinkling fractal process involves a temperature-dependent global autocorrelation relaxation function C(t,T), as [2]:
=
ω=1 t
C ðt; T Þ =
∫ ωo
df −1
ω
19 8 0 < = −β T *2 −T 2 A dω exp −ωt exp@ : ; kT
ð1Þ
where ω is the vibrational frequency with ωo being the maximum (~ 1012 Hz), t is the observation time, β is a universal constant (~ 0.3 cal/mol K2 or 2.00 × 1024 J K− 2), k is Boltzmann's constant, and T* is a critical temperature defined with respect to Tg as T* ≈ 1.2Tg [2–4]. C(t) is mathematically equivalent to the infinite sum of weighted exponentials represented by the empirical Kolrausch–Williams–Watts stretched exponential function C(t) ~ exp [−(t/τ)β] in which τ is a characteristic time and β ≈ 0.5 is the stretching term. Typically τ and β are dependent on the weighting functions, which are given by the fractal cluster vibrational density of states, g(ω) ~ ωdf − 1, in the TFT. As shown later and as predicted by Eq. (1), there is no characteristic relaxation time at Tg but rather a continuous distribution of relaxation times spanning many decades associated with the many different fractal cluster length scales. According to Wool, Eq. (1) can be approximated for “short” times and high frequencies as C(t) ~ t – 1/3 and “intermediate/long” times or low frequencies as C(t) ~ t – 4/3 [2,4]. The exact analytical solution to the integral in Eq. (1) is given as follows: C ðt; T Þ =
ωo1 = 3 1 1 1 expð−αtωo Þ + Γ ; αtωo −Γ ; α 4 = 3 3 3 3αt 3ðαt Þ 1 − 4 = 3 expð−αÞ αt
ð2Þ
313
where β 2 2 α = exp − T * −T kT
ð3Þ
and where Γ(1/3,x) is defined as the “upper” incomplete gamma function. Thus, at a given rate or observation time t and temperature T, the function C(t,T) describes how many of the solid clusters of varying frequency ω could relax to a liquid in a given observation time interval. If the rate of observation or testing rate γ increases, then this time interval decreases since the observation time t behaves as t ~ 1/γ and fewer clusters will have relaxed. Thus, it is necessary to increase the temperature T to generate the same level of relaxation at a higher rate. This is the essential physics of the rate dependence of Tg as developed in Wool and Campanella [4]. The double exponential structure of C(t,T) in Eq. (1) provides a rapid slowing down of the twinkling dynamics when T b Tg but there is no “freeze-out” temperature To in the denominator which would cause the viscosity of a material to approach infinity at To as suggested by the empirical Vogel–Fulcher– Tammann (VFT) relation discussed in reference [1]. Note that when T * ≫ T, the activation energy term in Eqs. (2) and (3) become more Arrhenius-like or strong, and when T is of order T*, the behavior is more VFT-like or fragile, as discussed by Angell [1]. Most polymers are expected to behave in a fragile manner near Tg. Since T * is described by T*/Tg = (1 − pcs)/(1 − pc), materials with a low scalar percolation threshold, such as inorganic glasses, will appear stronger than polymers with higher pcs values. Further analysis of the behavior of Eq. (2) with respect to temperature T and observation time t is provided below. The TFT states that the solid clusters and surrounding liquid dynamically coexist in approximately equal amounts near Tg. A solid is defined as a cluster of atoms whose pair-wise vibrational interactions occur through average bond distances Xij that are less than the critical Lindemann inflection point Xc in their local anharmonic potential U(Xij) in the energy landscape [2]. An oscillator in the solid-state transitions to the liquid state when a thermal fluctuation causes it to expand beyond Xc. Moreover, the vibrational interactions of the solid clusters provide the mechanism of “communication” between the cooperatively rearranging regions discussed in the traditional Adam–Gibbs theory [10]. This is illustrated in Fig. 2 where Ps and PL represent the solid and liquid fractions, respectively. This concept parallels Lindemann's theory of melting [21]. Instead of crystallization, glass formation occurs due to molecular configurational irregularities, which promote the fractal structure formation and leads to viscous retardation of nucleation and crystallization.
Fig. 2. U as a function of Xij for pair-wise vibrational interactions of atoms in an amorphous material where the average bond distance is proportional to temperature. The inflection point Xc denotes the phase transition between the solid PS (Xij b Xc) and liquid PL (Xij N Xc).
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As Tg is approached from above, dynamic solid fractal clusters begin to form. Percolation of the solid clusters increases to a rigid, or vector, percolation threshold pc of ~0.5 at Tg. This formation of a rigid, frustrated fractal structure is said to dictate the structure of the nonequilibrium glass at T b Tg and explain the continuous change in slope on a V–T diagram. This alternative TFT approach of explaining the second-order behavior of amorphous materials replaces the free volume theory, which was originally proposed by Eyring and further developed by Fox and Flory and others [22–24]. In the TFT, the solid fraction PS behaves with temperature as PS ≈ 1 − [(1 − pc) T/Tg] with PL = 1 − PS, which satisfies continuity. In addition, these equations state that PS and PL are continuous through Tg, as are the modulus [25,26] and compressibility [27]. The onset of long-range connected solid fractal clusters begins at the scalar percolation threshold pcs which occurs at T* ≈ 1.2Tg. This is in agreement with the modecoupling theory TC as well as Goldstein's Tx [28,29]. Since g(ω) is continuous from very low to very high frequencies, the complex twinkling dynamics existing near Tg produces a continuous relaxation function, Eq. (1), with many different length scales and times associated with the fractal clusters. Therefore, the TFT is mainly a kinetic theory of the glass transition such that it is universal to all amorphous materials. The twinkling fractal process is not a subtle effect but rather is a strong physical effect such that about 50% of the material near Tg is not only solid but is jumping back and forth into the liquid state with precise dynamics. Therefore, it should be readily observable and quantifiable by relatively simple transparent experiments, as we demonstrate herein with time-lapse AFM on an amorphous solid near Tg. 3. Experimental 3.1. Materials Commercially purchased, amorphous, atactic polystyrene (PS) (MW = 194,000 g/mol and PDI = 1.07 confirmed by gel permeation chromatography; Tg = 375 K confirmed on a differential scanning calorimeter with a heating rate of 3 K/min) was utilized. PS has been chosen as the glass former to study due to its relatively low Tg (~373 K), compared to most ceramics (e.g. Pyrex, Tg = 848 K) and is closer to room temperature (TR) than most smaller molecules (e.g. glycerol, Tg = 183 K) [2]. In addition, PS has been widely used and discussed in the literature as the benchmark material for phenomenological testing. 3.2. Sample preparation PS films were prepared by spin coating from toluene (HPLC grade) (10 wt.% PS) at 1500 rpm onto gold-plated glass slides, which have been heat treated, rinsed in toluene, and subsequently dried under compressed nitrogen flow prior to coating. Film thickness (1150 nm) and roughness (b1.0 nm) were determined by AFM at TR. 3.3. AFM techniques A Veeco Dimension V scanning probe microscope in tapping-mode with a Veeco DMHC-A250 heater/cooler and AFM probes (Veeco FESPs) were used in c.a. 45% relative humidity air to image the dynamic heterogeneity of PS at TR and at isothermal temperatures close to 373 K. The system was allowed to thermally equilibrate prior to imaging in order to minimize thermal drift. The slow scan axis (y-axis) was disabled to generate 1-D space-time images, while the slow scan axis was enabled to generate the normal 2-D spatial images. For both methods, the scan length in the x-direction and scan rate were 700 nm and 1 Hz, respectively, with the XY closed loop turned on and the Z-limit set to 1.0 μm. A uniform amplitude setpoint was used throughout the experiments to ensure comparability. For
generating the 1-D images, each scan traversed the same 700 nm in the x-direction generating 512 space data points (i.e.: 512 pixels) per second for a total of 512 s (i.e.: 512 lines in the y-direction) per captured image. The retrace height and phase data were employed in the following analysis. In order to spatially resolve regions of different stiffness (liquid versus solid fractions), AFM phase images were analyzed. The phase signal from tapping-mode AFM represents the lag between the driving oscillation signal and the resultant oscillation of the cantilever (simple harmonic oscillator.) This lag is zeroed out when the probe tip is unaffected by the surface forces (in air). When the probe tip experiences surface forces, a phase lag is introduced. For “repulsive” amplitude setpoints, the phase lag is often dominated by material stiffness, which is especially true in the case of hydrophobic materials such as polystyrene. When this condition is met, hard regions appear bright (high) in the phase signal (small lag) and softer regions, because of their greater dissipation, appear dark (low) in the phase signal (large lag.) Thus, the phase signal provides a spatially resolved map of hard and soft regions (or small and large dissipation) across the scan area. All AFM images shown have been 1st ordered flattened and lowpass filtered in the AFM software for viewing purposes and qualitative inspections. Raw AFM height and phase data was used as the basis in the quantitative analysis. 3.4. AFM X–Y height images with fractal dimension calculations 3.4.1. Statistical analysis The consecutive AFM X–Y height images in Fig. 1 were quantitatively compared by first converting the raw RGB images to grayscale and then computing a 2-D correlation coefficient between the first and second images and the first and third images. The form of the 2-D correlation coefficient utilized is: Σm Σn Amn −A Bmn −B ffi: r = sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Σm Σn Bmn −B Σm Σn Amn −A
ð4Þ
In Eq. (4), Amn and Bmn are data corresponding to the first and subsequent image, respectively, where A ̅ and B ̅ are the mean values of the images. 3.4.2. Fractal dimension The fractal dimensions Df of the images in Fig. 1 were computed and compared by first converting the RGB images to grayscale and then implementing an approximate box counting method. 3.5. Statistical autocorrelation function Prior to implementing statistical techniques to the AFM X–t phase data, each image, exported as raw ASCII (least significant bit) data from the AFM software, was 1st ordered flattened followed by a 1 × 3 boxcar average filtration to each x position in order to reduce any high frequency noise that was present within the data. In order to quantitatively compare the experimentally captured relaxation behavior of the twinkling events with the TFT C(t,T), a statistical autocorrelation function C(t) of the following form was applied to the AFM X–t phase data: C ðt Þk =
ck co
where ck =
1 N−k ∑ ðx − xÞ xt N t=1 t
+ k− x
:
ð5Þ
Eq. (5) is the Box–Jenkins estimated autocorrelation function [30]. An autocorrelation function reveals how the correlation between any two values of a data series changes as their separation changes and is
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4. Results 4.1. AFM X–Y height images
Fig. 3. Plot of the 2-D correlation coefficient as a function of time for computed coefficients between the first image with itself, the first image with the second (t = 8.53 min), and the first image with the third (t = 17.06 min) for T = 298 K (black ●) and T = 358 K (red ▲).
used mathematically to extract repeating patterns in random processes [30]. Eq. (5) is used to describe the behavior of a strictly stochastic stationary process, which is assumed to adequately describe the dynamics of the twinkling fractals. In Eq. (5), co and ck are the values of the autocorrelation function at the zeroth and kth time lag, respectively, with N being the total number of oscillations. xt and x ̅ are the numerical values of the observation at lag t and the mean, respectively. In order to calculate a global autocorrelation, the C(t)k is calculated at each x position and then averaged over all positions with accompanying standard deviations. According to Box and Jenkins, useful estimates of the autocorrelation function can be calculated for k = 0,1,…, K where K is not larger than ~ N/4 [30].
4.1.1. 2-D correlation coefficient Fig. 1 shows successive AFM X–Y height images of dimensions 700 nm × 700 nm for T = 298 K (a–c, top) and T = 358 K (d–f, bottom). Each image was captured 8.53 min apart from each other. Thermal drift was estimated to be less than 1.0 nm/min. The percolating solid fractal cluster that exists at T b Tg is believed to be revealed in these height images from a 2-D topographical viewpoint. Fig. 3 shows the results of the 2-D correlation coefficient analysis (Eq. (4)) between the consecutive images where the 2-D correlation coefficient r is plotted as a function of time between captured images for T = 298 K (black circles) and T = 358 K (red triangles). The closer the correlation coefficient values are to one, the more correlated the images are with each other. As expected, the room temperature results (298 K) with a much slower twinkling rate show a smaller change with time compared to the higher temperature results (358 K).
4.1.2. Fractal dimension The fractal dimensions Df for the 298 K and 358 K X–Y height images in Fig. 1 were calculated from the digital data to be 1.8777 ± 0.0005 and 1.8534 ± 0.0091, respectively. From the theory of the fractal nature of percolating clusters, Df = 1.89 in 2-D. Typical fractal dimensions of scalar percolation clusters are about 1.75 but one expects the vector percolation fractal dimension to be higher due to the additional mass required to propagate rigidity at the vector percolation threshold near Tg. The observation of fractal structure in amorphous polystyrene is in accord with observations of the fractal structure in amorphous metals, where Df = 2.31 was reported from neutron and X-ray scattering experiments below Tg [31]. Recently, Ojovan examined solid–liquid bond transitions in SiO2 and GeO2 oxides and reported percolation behavior of the solid regions with fractal dimensions Df = 2.55 [32–34].
Fig. 4. 1-D AFM X–t height (top) and phase (bottom) images at 298 K (left), 358 K (middle) and 383 K (right) obtained by repeated scans along a single line (slow scan axis disabled). PS films with thicknesses of 1146 ± 14 nm (358 K and 383 K) and 600 nm (298 K) were used.
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4.2. Single AFM X–t images
4.3. Consecutive AFM X–t phase images
AFM “movies” or many rapid snapshots versus time of the twinkling process to analyze the dynamics would be ideal. To mimic this ideal situation in part, as shown in Fig. 4, 1-D space-time (X–t) height (top row) and phase images (bottom row) of the surface of PS at 298 K (T ≪ Tg), 358 K (T b Tg), and 383 K (T N Tg) were obtained. It is noted that the surface Tg of PS is expected to be less than the bulk Tg due to relaxed constraints to motion [35]; however, all analysis presented herein is with respect to the bulk Tg of polystyrene. The height images in Fig. 4 (top) show the roughness of the PS surface as a function of time. The phase images display elasticity differences as a function of time and thus, can show the solid fractal clusters twinkling at a particular observation length of 700 nm and time scales of 1.0 Hz.
Fig. 5 shows the consecutive AFM X–t phase images, which will be used to compare with the TFT simulations and the theoretical relaxation function C(t,T). In Fig. 5, the spatial scan length of 700 nm is located on the abscissa with time on the ordinate. AFM X–t phase data (1st order flattened and 1 × 3 boxcar average filtered) were combined in a sequential manner to obtain continuous data sets. Different amounts of data were required to produce experimental relaxation results that were comparable to the TFT. As can be seen in Fig. 5, the amount of experimental data required increases as temperature decreases, which correlates well with the TFT. The exact relationship between the number of data points used to obtain comparable results is currently under investigation. 4.3.1. Cluster lifetimes τ By closely inspecting the images in Fig. 5, one can estimate the cluster widths and lifetimes of the dynamic heterogeneities as shown in Fig. 6. Generally, one sees that the wider the cluster, the longer will be its lifetime or “streak height” in Fig. 5. In Fig. 6, the logarithm of cluster lifetimes is plotted as a function of the logarithm of the cluster widths R or radii in nanometers. The logarithmic scale was used in order to compare the experimental data to the theory of dynamic fractal networks from Orbach et al., where τ ~ RDf/df [5,6]. Df is the fractal dimension = 1.89 in 2-D and df = fracton dimension = 4/3 such that Df/df = 1.42, which is the predicted theoretical slope on a logarithmic scale. Fig. 6 shows that the experimental data, though highly scattered as expected for fractal cluster sampling, is in good accord with the theoretical prediction of τ ~ RDf/df. 4.4. Relaxation behavior C(t,T) The theoretical relaxation function C(t,T) in Eq. (2) was calculated and is plotted in Fig. 7 as a function of time on a log–log scale for a time range that spans four orders of magnitude, with temperatures of 298 K, 358 K, and 383 K. The time range from 1 to 1 × 104 s was chosen in Fig. 7 so that the theory can be directly compared to the experimental data shown in Fig. 5. For each temperature T, the observation time t was swept from 10− 14 s to 1010 s with 1000 data points between each order of magnitude. The universal constants β, k, and ωo in Eqs. (2) and (3) were fixed at 2.0 × 10− 24 J K− 2, 1.38 × 1023 J K− 1, and 1012 Hz,
Fig. 5. 1-D X–t AFM phase images at 298 K (left), 358 K (middle), and 383 K (right) obtained by repeated scans along a single line (slow scan axis disabled). PS films with thicknesses of 1146 ± 14 nm (358 K and 383 K) and 600 nm (298 K) were used. Single 512 × 512 data point images were generated during each experiment and were connected together to produce a continuous single image.
Fig. 6. Log–log plot of the cluster lifetimes as a function of cluster widths for a sample of twinkling fractal clusters present in the AFM X–t phase images for temperatures of 358 K (red ▲), and 383 K (blue ■); included is a slope of 1.42 (black line) that represents Df/df ~ 1.42 from fractal theory.
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Fig. 7. Log–log plot of the TFT global autocorrelation function C as a function of t and T for temperatures of 298 K (black), 358 K (red), and 383 K (blue). The noise in the beginning of the 298 K curve is attributed to the instability of the function with respect to α ~ 10− 24 when T = 298 K, which greatly suppresses the frequency probability of twinkling events for glassy PS.
respectively. With respect to polystyrene, the characteristic temperatures Tg = 373 K and T* = 448 K were used in Eq. (3). Fig. 7 shows that at low temperatures the relaxation slopes are close to zero and at higher temperatures increase at short times, on an absolute scale, with values near 0.4 that eventually reach an asymptotic value of 1.33 at longer times. This result is reflective of the fracton dimension df = 1.33 in the fractal vibrational density of states g(ω) ~ ω1.33. Fig. 8 shows the experimental autocorrelation function C(t) versus time plotted on a log–log scale for short times t = 1–10 s (Fig. 8a) and longer times t = 10–100 s (Fig. 8b) for all three experimental temperatures. Linear best-fit curves to the experimental data are also included along with slope and best-fit R2 values. The linear fits were applied to experimental data up to the point where the C(t)k curve on a linear scale first intersects the ordinate and is above the 95% confidence level threshold of the statistical calculation (383 K curve fit an exception to the latter criterion). The slopes in Fig. 8 show values near −1/3 at low temperatures and short times, eventually increasing towards −4/3 and higher values −1.56 at longer times and higher temperatures, which is consistent with the TFT predictions. At much longer times when the vibrational phonon wavelengths no longer “see” the fractal structure, the slope is expected to approach the Debye limit of − 2. 5. Discussion 5.1. AFM X–Y height images In qualitatively comparing the time-lapse AFM X–Y height images in Fig. 1, there exists greater similarity between the features in the low temperature (298 K) images compared to the higher temperature (358 K) images. This reveals that the percolating solid fractal cluster exists for much longer time scales at lower temperatures. Furthermore, in comparing the quantitative results of the 2-D correlation coefficients seen in Fig. 3, the correlation coefficient decreases significantly less and more slowly for the 298 K images than the 358 K suggesting that the 298 K images are more similar and are well correlated. These results confirm what one visually concludes when qualitatively analyzing the time-lapse images in Fig. 1 and that the fractal structures twinkle more slowly at lower temperatures as expected. With regards to the fractal dimensions Df,, the computed values are in excellent agreement with the theory of the fractal nature of
Fig. 8. Log–log plot of the statistical autocorrelation function C(t)k as a function of lags t between t = 1–10 s (a) and t = 10–100 s (b) for temperatures of 298 K (black ●), 358 K (red ▲), and 383 K (blue ■) with corresponding linear best-fit curves for 298 K (solid black line), 358 K (dashed red line), and 383 K (dotted blue line). The slopes corresponding to the linear best-fit curves with respective R2 values are given in the upper right box.
percolating clusters and are in accord with experimentally observed 2-D fractal dimensions for dynamic uniform fractals in emulsions reported by Ozhovan [5,6,36]. This result suggests that indeed percolated solid fractal clusters develop upon cooling amorphous materials through the glass transition and, with the visual aids of Figs. 1, 4, and 7, that dynamic heterogeneity as well as cooperativity is prevalent in glassy materials. 5.2. Qualitative inspection of the single AFM X–t images As can be seen in Fig. 4, the spatio-temporal aspects of PS clearly exist at all three temperatures even at 298 K, which is 75 K below the bulk Tg value. Focusing on the phase images, correlations along the time axis are distinctly longer at 298 K compared to the temperatures closer to the bulk Tg. When qualitatively inspecting the 298 K images and even though PS at room temperature is well within the glassy state, relative to the bulk Tg of PS, clusters seem to be twinkling at a rate inversely proportional to their radii, which correlates with the TFT and supports the results shown in Fig. 6. In addition, the TFT states that a rapid slowing down of the twinkling dynamics occurs when T b Tg
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where vector percolation of the solid fractal clusters dominates the physical behavior of the material, even though a considerable amount of “liquid” still exists, pL ≈ 0.40, at 298 K for PS. The existence of the liquid fraction that is confined by the percolated solid fractal clusters is captured in the 298 K images, particularly, in the phase image as regions that twinkle relatively frequently but do not drastically alter the topological profile of the height image as compared to the higher temperatures. Large solid clusters will have a very low jump rate at the lower temperatures and we suspect that the observed twinkling process becomes dominated by the nanoscale clusters with lifetimes τ/aT ~ RDf/df, where the shift factor aT = α in Eq. (3). In comparing the height and phase images, it is reassuring to observe that the phase images are noticeably different from the height data, particularly at 358 K and 383 K, indicating that indeed potential dynamic heterogeneity, cooperativity, and fractal structure are being captured and not just the observation of topological phenomena. 5.3. Consecutive AFM X–t phase images By qualitatively comparing the X–t images in Fig. 5, one can observe that the twinkling events tend to increase in frequency with increasing temperature as the larger solid clusters can participate in the twinkling process. Even though the 298 K image seems to be twinkling rather frequently at a small cluster scale as expected, the overall rigidity of the PS remains indicative of the dominance of the percolated solid fractal cluster. The frequent twinkles captured at 298 K are believed to be very small solid fractal clusters of the overall solid cluster hopping in and out of the liquid state in a dynamic fashion in order to maintain equilibrium. In the 358 K and 383 K images and similar to the 383 K image in Fig. 4, multiple white (harder) regions are present and are believed to be liquid regions that have twinkled to stable solid fractal clusters. The widths of these solid clusters are roughly 12 nm, which is within range of reported values for the characteristic lengths of dynamic heterogeneities [11]. By selecting a representative sample of clusters from both the 358 K and 383 K images and measuring their respective lifetimes and widths, one can show that on average, twinkling events are more frequent at higher temperatures with respect to the observation time and the cluster size. The solid-to-liquid twinkling frequencies are β given from Eq. (1) by ω(T) = ω exp − kT T *2 −T 2 , where ω is the lowest fundamental frequency of the solid cluster involving the majority of atoms cooperatively in its eigenvector. Thus, in general, the twinkling frequency increases with temperature for clusters of a given radius R. This is shown in Fig. 6 where the cluster lifetimes decrease with decreasing cluster sizes and, in general, the cluster sizes increase with decreasing temperature. In addition, the slope of the experimental data is in excellent agreement with fractal theory as well as with the TFT predictions. 5.4. Relaxation behavior As can be seen in Fig. 7, the global relaxation of twinkling events at 298 K requires many orders of magnitude more than at temperatures in proximity to the Tg of PS. The noise in the beginning of the 298 K curve is attributed to the instability of the function due to α ~ 10− 24 when T = 298 K, which greatly suppresses the frequency probability of significant twinkling events that effectively influence the macroscale physical properties as well as the global relaxation of glassy PS. Therefore, the slope of C(t,T = 298 K) for t = 1.0 to 1.0 × 104 s is ~ 0. As T approaches Tg (T = 358 K), the probability of twinkling events increases, thereby increasing the twinkling frequency and ultimately reducing the relaxation time of the twinkling events, which, in turn, shifts “longer” times closer to the observation time t. This causes the slope of C(t,T = 358 K) for t = 1.0 to 1.0 × 104 s to be equal to −0.43 for t b 10 s, − 1.29 for 10 b t b 100 s and − 1.33 for t N 100 s. Thus,
significant twinkling events should be observable within the experimental AFM scan rate. Focusing on the 358 K AFM X–t phase images in Figs. 4 and 5, one can see the dynamic heterogeneity as multiple clusters transition from the softer phase (darker color) to the harder phase (lighter color) and vice versa. The twinkling behavior is believed to be captured in these images and well reflected in the autocorrelation relaxation function with C(t) ~ t− 4/3. When T = 383 K, which exceeds Tg, the probability of solid–liquid twinkling events increases further, the fractal structure loses its percolated rigidity, and only scalar percolation dominates the physical behavior of the now rubbery material. Due to this increased flexibility, the twinkling frequencies and, in turn, the relaxation time of the twinkling events decrease to the point that prior “longer” times now occur three orders of magnitude less than the observation time t. This causes the slope of C(t,T = 383 K) for t = 1.0 to 1.0 × 104 s to be equal to a constant of −1.33. Therefore, significant twinkling events should still be observable but at an increased rate compared to T = 358 K. This is indeed what is believed to be captured in the 383 K AFM X–t phase images in Figs. 4 and 5. In comparing the 358 K and 383 K phase images, the dynamic heterogeneity is twinkling at a faster rate and the fractal structure is less percolated in the 383 K phase image than in the 358 K image. An interesting feature of the 383 K images in Figs. 4 and 5 is the existence of two rather long-lived white (harder) regions. These regions are speculated to be potential liquid regions that have twinkled to solid fractal clusters of significantly shorter vibrational interactions with consequently slower relaxation dynamics, considerable cooperativity, and relatively rigid fractal structures. Again, the widths of these solid clusters are roughly 12 nm, which is within range of reported values for the characteristic lengths of dynamic heterogeneities [11]. In comparing the experimental relaxation behavior results shown in Fig. 8 with the TFT predictions, the slopes for 358 K and 383 K are in fair agreement with the TFT. The low R2 values for the 383 K curve fits are the result of the autocorrelation falling below its respective 95% confidence bound value of 0.044 (−1.355 on a log scale) in a relatively short amount of lag time. The slopes for 298 K, on the other hand, do not correlate well with the TFT, where one predicts a slope of ~0 over the observation time. Based on this, the statistical autocorrelation seems to be very sensitive to small differences. The statistical autocorrelation seems to be picking up the frequent twinkles of the surrounding liquid but is unable to capture the robustness of the percolated solid fractal cluster. This observation may also explain why the 358 K and 383 K slopes are slightly higher than what is predicted by the TFT. In addition and even though significant time was provided in order to achieve thermal equilibrium, the AFM technique contains intrinsic and unavoidable thermal drift, which can be seen to some extent in the 358 K images in Figs. 4 and 5 and estimated to be less than 1.0 nm/min. Also, even though setpoints of approximately 90– 95% were used, possible AFM tip induced plastic deformation, instead of elastic deformation, may still occur, depending on the surface sensitivity of the material being scanned. Overall, from both a qualitative and quantitative reasoning, the AFM images of polystyrene do capture the twinkling phenomena of this amorphous material in its glass transition region with experimental relaxation behavior in fair agreement with that predicted by the TFT. 6. Conclusions Atomic force microscopy has been utilized to physically observe the twinkling fractal nature of amorphous materials predicted by the TFT. Specifically, the twinkling fractal characteristics of amorphous, atactic polystyrene have been captured in both height and phases images. Successive 2-D height images revealed that the percolated solid fractal clusters exist for longer time scales at lower temperatures. The computed fractal dimensions, Df ≈ 1.88, are shown to be in excellent agreement with the theory of the fractal nature of
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percolating clusters predicted by the TFT. The twinkling dynamics of polystyrene within its glass transition region were captured in timelapse one-dimensional AFM phase images with the physically observed cluster lifetimes and widths in agreement with the TFT and the theory of dynamic fractal networks. A comparison of the twinkling relaxation behavior of the TFT with the physically observed relaxation behavior provided robust support of the TFT. Acknowledgments The authors are grateful to the U.S. Army Research Laboratory for financial support under the Army Materials Center of Excellence Program, contract W911NF-06-2-011. The author J.F. Stanzione III is also grateful for generous mentorship of Dr. Kenneth Strawhecker and Dr. David Webb from the ARL-APG. References [1] C.A. Angell, K.L. Ngai, G.B. McKenna, P.F. McMillan, S.W. Martin, J. Appl. Phys. 88 (2000) 3113. [2] R.P. Wool, J. Polym. Sci. Pol. Phys. 46 (2008) 2765. [3] R.P. Wool, Soft Matter 4 (2008) 400. [4] R.P. Wool, A. Campanella, J. Polym. Sci. Pol. Phys. 47 (2009) 2578. [5] R. Orbach, Science 231 (1986) 814. [6] T. Nakayama, K. Yakubo, R.L. Orbach, Rev. Mod. Phys. 66 (1994) 381. [7] P.S. Crider, N.E. Israeloff, Nano Lett. 6 (2006) 887. [8] R.P. Wool, J.F. Stanzione III, K.E. Strawhecker, Data presented at the APS March meeting., in, 2009. [9] G. Ruocco, K.L. Ngai, In Proceedings of the 6th International Discussions on Relaxation in Complex Systems, Rome, Italy, 2009.
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