Article pubs.acs.org/Macromolecules
Fragility-Confinement Effects: Apparent Universality as a Function of Scaled Thickness in Films of Freely Deposited, Linear Polymer and Its Absence in Densely Grafted Brushes Tian Lan† and John M. Torkelson*,†,‡ †
Department of Materials Science and Engineering and ‡Department of Chemical and Biological Engineering, Northwestern University, Evanston, Illinois 60208, United States S Supporting Information *
ABSTRACT: Ellipsometry measurements as a function of cooling rate are used to study nanoscale confinement effects on dynamic fragility (kinetic fragility), m, in supported films of freely deposited, linear polymer. Polymers include neat polystyrene (PS), neat polycarbonate (PC), and PS + 2 wt % 1,10-bis(1-pyrene)decane (BPD) as small-molecule diluent; in each case, the substrate/polymer interface lacks significant attractive interactions. In terms of both the length scale at which confinement effects become evident and the percentage reduction in m from its bulk value, the magnitude of the mconfinement effect increases with increasing bulk polymer system m. Additionally, for films of linear polymer lacking significant attractive interactions with the substrate surface, m-confinement effects are evident at larger onset thicknesses than those commonly reported in the literature for the glass transition temperature (Tg)-confinement effect. Evans et al. [Macromolecules 2013, 46, 6091] found that the Tg-confinement effect in related films exhibits a universal nature as a function of scaled thickness. Fragility-confinement effects of films of freely deposited, linear polymer chains exhibit a similar universal nature as a function of scaled thickness using shift factors consistent with those used by Evans et al. However, when PS is confined in a dense brush with one end of each chain covalently attached to the substrate surface, both m and Tg are independent of brush thickness. The strong correlation of fragility-confinement and Tg-confinement effects has important implications for understanding the fundamental natures of both the Tg-confinement effect and the glass transition itself.
1. INTRODUCTION m=
Over the past two decades, intense research efforts have been focused on the effects of nanoscale confinement on glassforming polymers.1−30 Many studies have reported that confinement can lead to large deviations from bulk response in a variety of physical properties and behaviors, including the glass transition temperature (Tg),1−23 physical aging rate,24−27 surface viscosity,28 elastic modulus,29 diffusivity,30 etc. As measured by (pseudo)thermodynamic methods including differential scanning calorimetry (DSC), ellipsometry, X-ray reflectivity, and fluorescence, among others, both free-standing films of linear polymer and supported films of freely deposited, linear polymer lacking substantial attractive interactions with the substrate can exhibit a decrease in Tg with decreasing nanoscale thickness.1−4,10,12,21−23 The decreases can be significant, with Tg reductions of as much as 50−80 K.2,21 Dynamic fragility (also called kinetic fragility) has also attracted major research interest over the past two decades.21,31−47 As described by Angell, “fragility is defined in terms of the deviation of the relaxation time temperature dependence from simple Arrhenius behavior.”32 The fragility index, m, can be expressed as follows:32 © XXXX American Chemical Society
d log τα dTg /T
T = Tg
(1)
where τα is the α-relaxation time and T is temperature. Upon being cooled from the rubbery or liquid state to the glassy state, glass-formers which exhibit a nearly Arrhenius T-dependence of τα are “strong” (with relatively small m values), whereas those exhibiting a highly non-Arrhenius T-dependence (most often described equivalently by the Vogel−Fulcher−Tammann equation or Williams−Landel−Ferry equation) with a steep variation of τα are “fragile” (with relatively large m values).31−38 The non-Arrhenius behavior indicates substantial cooperativity during relaxation. Fragility is reported to be intimately related to various properties of glass-formers, including the nonexponentiality of the segmental relaxation function,36 the molecular weight (MW) dependence of polymer Tg,37 the experimentally determined self-concentration parameter in the Lodge− McLeish model for miscible blends,38,48 the kinetics of cold Received: November 17, 2015 Revised: February 1, 2016
A
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or Krytox oil and smaller decreases for films spun cast directly onto the flash DSC chip.66) Priestley and co-workers68,69 used DSC to study the fragility of polymer spherical particles, with diameters confined from ∼120 to ∼630 nm, under isobaric or isochoric conditions, with the isochoric condition achieved by capping particles with silica shells. Although the silica-capped polymer particles exhibited an invariant Tg with decreasing diameter, in sharp contrast to uncapped particles, substantial decreases were reported in fragility of the silica-capped particles as diameter decreased from ∼400 to ∼120 nm.68 Arabeche et al.70 also used DSC to measure the fragility of PC/PMMA alternating multilayer films prepared by multilayer coextrusion. As the individual layer thickness decreased from 16 μm to 12 nm, m decreased in PC layers but remained invariant in PMMA layers within error. Here, we report a study73 of the fragility-confinement effect in films of linear polymers supported on Si/SiOx substrate as a function of polymer species, small-molecule diluent addition, and covalent attachment of chain ends to the substrate. In three cases, the polymers are freely deposited by spin coating; in one case, the polymers are dense brushes with one end of each chain covalently attached to the substrate surface.74 As with a recent report by Glor et al.,75 we use cooling-rate-dependent measurements of Tg by spectroscopic ellipsometry to determine m as a function of confinement in supported polymer films. (We note that a 2005 study by Fakhraai and Forrest76 is a seminal precursor to ref 75 and the current study73 as it demonstrated both a strong cooling rate dependence of the Tgconfinement effect in thin PS films and interpreted the results in terms of activation energy of the α-relaxation which is related to fragility.) The supported films consist of freely deposited linear polymers, including neat PS, PS doped with 2 wt % of a small-molecule diluent (1,10-bis(1-pyrene)decane (BPD)), and neat PC, all of which lack significant attractive interactions with the Si/SiOx substrate surface. Dense PS brushes that are covalently attached to the substrate are also studied. Fragility values measured in bulklike films via ellipsometry agree well with those measured by DSC. With confinement, major decreases in m are observed in films of more highly fragile neat PS and neat PC, whereas a muted decrease is observed in films of PS + 2 wt % BPD. In contrast, in dense PS brushes m is invariant with confinement down to a thickness of 27 nm. These fragility-confinement effects correlate well with previous findings which show that the effect of confinement on the average Tg across the film thickness is substantial in confined films of freely deposited, linear PS and PC and muted in films of linear PS with several wt % small-molecule diluents11,50 but is absent in dense PS brushes.74 Using shift factors (correlated with bulk m values) consistent with those used by Evans et al.,21 who demonstrated universality in the Tg-confinement effect of polymer films, we find that a similar universal nature of the fragility-confinement effect is evident in thin films of freely deposited, linear polymer lacking attractive interactions with the substrate.
crystallization,39 Poisson’s ratio,40 and elasticity.37,41 Recently, Evans et al.21 studied films of seven linear polymer systems lacking attractive interactions with the substrate and reported that the strength of the Tg-confinement effect was related to bulk polymer fragility.49 For example, polymers with high bulk m values, such as poly(vinyl chloride) and polysulfone, exhibited much larger Tg-confinement effects than polymers with much lower bulk m values, such as polystyrene (PS) doped with 2 or 4 wt % dioctyl phthalate. They concluded that fragility plays a key role in defining the magnitude of Tg reduction and the length scale associated with effects of confinement on Tg. The fragility of bulk polymer correlates with chain backbone stiffness and the stiffness of side groups relative to the backbone.47 Fragility can also be affected by the presence of additives, including (anti)plasticizers21,50 and nanoparticles.51,52 The fragility index of bulk polymer has been measured by various techniques, including but not limited to dielectric relaxation spectroscopy (DRS),53−58 rheology,46,59,60 dilatometry,61 photon correlation spectroscopy,62 and DSC.21,63−65 However, few techniques have been used to characterize m in confined polymers, with DRS53−58 and DSC26,66−70 being the two main methods up to now. Using DRS, Fukao and Miyamoto53 found that m decreased from ∼120 to 95 as thickness decreased from ∼100 to 10−20 nm in single-layer supported PS films. Recently, Fukao and co-workers studied stacked ultrathin films of poly(2-chlorostyrene)55 and poly(methyl methacrylate) (PMMA)56 by DRS and also observed a substantial depression of m with confinement. A more dramatic decrease of fragility was reported via DRS in free-standing PS films by Napolitano and Wübbenhorst.54 They found that m decreased from ∼150 in high MW, bulk PS to ∼75 in a 40 nm thick free-standing film. The m-confinement effect has also been studied by simulations. Riggleman et al.50 showed via molecular dynamics simulation that nanoscale confinement reduced the relative fragility of a polymer. They also reported that a reduction in the m-confinement effect occurs upon incorporating antiplasticizing additives in polymer films. Recently, Marvin et al.71 examined the fragility index as a function of polymer film thickness and location inside the film. They indicated that polymer systems “exhibiting an interfacial suppression in Tg under confinement should always exhibit a suppression in fragility”71 whereas “interfacially enhanced Tg could potentially exhibit an interfacially-driven enhancement in overall fragility”.71 Direct measurement of dynamic fragility in bulk has been done via DSC60−62,72 by relating m with the activation energy for enthalpy relaxation, with the latter determined by measurements as a function of cooling rate. Fragility measurements by DSC have also been done in nanoconfined polymers. Simon and co-workers used DSC to investigate confinement effects on fragility in stacked ultrathin PS films; they reported that m = 98 for PS films with individual layer thickness of 38 nm, substantially lower than the value of 144 they reported for bulk PS.26 Simon and co-workers66,67 recently studied dynamic fragility via flash DSC in single-layer ultrathin films of polycarbonate (PC) and PS. Both polymers exhibited decreases in m with decreasing nanoscale thickness. (However, for reasons that were not clear at the time of the publication of those studies, their “result for the bulk polycarbonate of m = 102 is lower than generally reported in the literature”67 and the extent of the decrease in m with confinement of PS films was strongly affected by the substrate on which the films were supported: larger decreases were observed on Apiezon grease
2. EXPERIMENTAL SECTION An anionically polymerized PS standard (nominal MW = 900 kg/mol, Mw/Mn = 1.06, as reported by the supplier) was from Pressure Chemical and used as received. Bisphenol-A-polycarbonate (Mn = 17.3 kg/mol, Mw/Mn = 1.7, as reported by the supplier) was from Scientific Polymer Products and washed carefully twice (to remove low-MW additives in the commercial sample) prior to use by dissolving in 1,1,2trichloroethane (Chem Cruz) and precipitating in methanol. 1,10B
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Macromolecules Bis(1-pyrene)decane (Molecular Probes; MW = 543 g/mol) was used as received. Bulk Tg values were measured via DSC (Mettler-Toledo 822e) using ∼3 mg samples. The Tg values (onset method) were determined at a heating rate of 10 K/min after annealing the samples above Tg and cooling to 298 K at 40 K/min (see Table 1). The fragility index of bulk
described in ref 74, with the only difference being that the silica wafer was exposed to aminosilane vapor for 1 h instead of 0.5 h. As described in ref 74, the grafting density was determined to be 0.3−0.4 chain/ nm2. Spectroscopic ellipsometry (M-2000D, J.A. Woollam Co., Inc.) was used to determine Tg by monitoring thickness as a function of temperature (T). Measurements were done in air at 420−1000 nm wavelength range (incident angle = 65°). A Cauchy layer model, which is composed of a silicon substrate with a 2 nm thick native SiO2 layer and a polymer film on top,10 was fitted to the data to interpret film thickness. Before measurement, films were annealed at T > Tg + 28 K on the T-controlled sample stage (Instec) for 5 min. They were then cooled through the transition region at rates ranging from 20.0 to 0.2 K/min against liquid nitrogen flow with sampling time ranging from 2.5 to 40 s depending on cooling rate. The T-controller was calibrated against an indium film (see Supporting Information for details).
Table 1. Tg, Molecular Weight, Dispersity, and Fragility polymer PS PS + 2 wt % BPD PC dense PS brushes a
Mn (kg/mol)
Mw/Mn
900a
1.06a
Tgb (K) 374 370
17.3a
1.70a
418
45−164
1.15−1.78
373−374
b
mc (bulk, ellipsometry) 159 ± 5 (±1592) 133 ± 2 (±1592) 206 ± 3 (±1592) 168 ± 8 (±1592)
3. RESULTS AND DISCUSSION 3.1. Cooling Rate Dependence of Tg in Films of PS, PS + 2 wt % BPD, and PC Determined by Ellipsometry. To analyze Tg in thin polymer films via ellipsometry, the most widely used method is to fit the thickness (h)−T data in the glassy and rubbery states with two lines, with the intersection defining Tg. As shown in Figure 1a, with a cooling rate of 0.2 K/min, a 771 nm thick PS film exhibits Tg = 372 K, whereas a 28 nm thick film exhibits Tg = 363 K. To better quantify the transition region and determine Tg, we employed the thermal expansivity analysis method first used by Kawana and Jones3 and later by others.10,74,79,80 The T-dependent expansivity, α(T), is computed by numerically differentiating h(T) with respect to T:
c
Reported by suppliers. Onset Tg measured by DSC. The average fragility values and the uncertainty (one standard deviation) were reported by ellipsometry in multiple bulklike samples. Specifically, in neat PS films, samples with thicknesses larger than 220 nm were used; in PS + BPD, samples with thicknesses larger than 90 nm were used; in neat PC films, samples with thicknesses larger than 300 nm were used; in dense PS brushes, samples with thicknesses larger than 50 nm were used. For a more detailed discussion regarding the uncertainty of ±15, see ref 92. PS + 2 wt % BPD was also measured by DSC via the cooling rate dependence of fictive temperature using rates ranging from 1 to 40 K/ min followed by heating at 20 K/min; the fictive temperature was evaluated using the Richardson method.77 Thin polymer films were prepared by spin-coating solutions onto silica wafers bearing intact oxide layers at speeds of 1500−2000 rpm. Concentrations of polymer solutions and spin speeds were adjusted to achieve a desired film thickness.78 Polystyrene and PS + 2 wt % BPD were dissolved in toluene, and PC was dissolved in 1,1,2-trichloroethane. After spin coating, neat PS and PC films were annealed at a temperature exceeding Tg,bulk + 10 K for 2 h in vacuum. Films of PS + 2 wt % BPD were annealed under ambient pressure rather than vacuum to avoid BPD sublimation. Dense polymer brushes were synthesized on silica wafers (WaferNet) using ARGET ATRP as
α(T ) =
h(T + ΔT /2) − h(T − ΔT /2) h(T0) × ΔT
(2)
Here T0 is a reference temperature which we chose to be Tg,bulk as measured by DSC, and ΔT is the differentiation region with a typical value taken to be 5.0 K.3,79 The calculated expansivity is smoothed using an adjacent-average method (shown as red curves in Figures 1b and 1c) following ref 79. As shown in Figures 1b and 1c, which illustrate α(T) derived from
Figure 1. Spectroscopic ellipsometry characterization of Tg in thin films of neat PS. (a) T-dependent normalized film thickness for (bottom) 771 nm thick film and (top) 28 nm thick film deposited on silicon wafer with a native oxide layer, cooled at 0.2 K/min. For both curves, thickness is normalized by the value measured at 374 K, and the data of 28 nm thick film have been shifted vertically for clarity. (b) T-dependent thermal expansivity for (bottom) 771 nm thick film and (top) 28 nm thick film, cooled at 0.2 K/min. (c) T-dependent thermal expansivity for (bottom) 771 nm thick film and (top) 28 nm thick film, cooled at 20.0 K/min. The data for the 28 nm thick film in (b) and (c) have been shifted vertically by 0.0006 K−1 for clarity. The red curves in (b) and (c) represent the smoothed data of thermal expansivity. In the case of (b) and (c), Tmid values are taken as Tg. C
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Figure 2. Spectroscopic ellipsometry characterization of Tg in thin films of PS + 2 wt % BPD. (a) T-dependent normalized film thickness for (bottom) 418 nm thick film and (top) 27 nm thick film deposited on silicon wafer with a native oxide layer, cooled at 0.2 K/min. For both curves, thickness is normalized by the value measured at 370 K, and the data of 27 nm thick film have been shifted vertically for clarity. Inset of (a) shows the chemical structure of the small-molecule diluent of BPD. (b) T-dependent thermal expansivity for (bottom) 418 nm thick film and (top) 27 nm thick film, cooled at 0.2 K/min. (c) T-dependent thermal expansivity for (bottom) 418 nm thick and (top) 27 nm thick film, cooled at 20.0 K/min. The data for the 27 nm thick film in (b) and (c) have been shifted vertically by 0.0006 K−1 for clarity. The red curves in (b) and (c) represent the smoothed data of thermal expansivity.
Figure 3. Spectroscopic ellipsometry characterization of Tg in thin films of neat PC. (a) T-dependent normalized film thickness for (bottom) 635 nm thick film and (top) 28 nm thick film deposited on silicon wafer with a native oxide layer, cooled at 0.2 K/min. For both curves, thickness is normalized by the value measured at 418 K, and the data of 28 nm thick film have been shifted vertically for clarity. (b) T-dependent thermal expansivity for (bottom) 635 nm thick film and (top) 28 nm thick film, cooled at 0.2 K/min. (c) The T-dependent thermal expansivity for (bottom) 635 nm thick and (top) 28 nm thick film, cooled at 20.0 K/min. The data for the 28 nm thick film in (b) and (c) have been shifted vertically by 0.0006 K−1 for clarity. The red curves in (b) and (c) represent the smoothed data of thermal expansivity.
T− and a 3 K increase in T+ relative to the bulklike film, resulting in only a 3 K decrease in Tmid. Taking Tg = Tmid, there is a major reduction in the Tg-confinement effect with measurements at increasingly higher cooling rate. This result is in accord with other studies on the effect of cooling rate on the Tg-confinement effect in spin-coated PS films.66,75,76,81 Similar studies were done in films of PS + 2 wt % BPD. Figure 2 shows measurements for a 418 nm thick, bulklike film and a 27 nm thick ultrathin film. When analyzed by extrapolating the T-dependent thicknesses in the glassy and rubbery states (Figure 2a, 0.2 K/min cooling rate), the 27 nm thick film of PS + 2 wt % BPD shows a smaller reduction of Tg (compared to bulk) than the 28 nm thick film of neat PS. The Tg may be better characterized with thermal expansivity analysis (Figures 2b and 2c). When cooled at 0.2 K/min, the 418 nm thick film of PS + 2 wt % BPD has Tmid = 368 K; a 27 nm thick film of PS + 2 wt % BPD has Tmid = 364.5 K, i.e., Tg − Tg,bulk =
measurements at 0.2 and 20.0 K/min cooling rates, there is a nearly steplike change in α in both 771 nm thick and 28 nm thick films. There is a nearly T-independent αrubbery, a much smaller αglassy, and a transition region between the rubbery and glassy states characteristic of a glass transition breadth, all of which can be fitted with three straight lines.10,74,79,80 The two intersections represent the onset (as measured upon cooling) and end point of the glass transition region and are denoted as T+ and T−. The midpoint between T+ and T− is defined as Tmid (Tmid = 0.500(T+ + T−)), which can be taken as an ellipsometrically determined Tg.3,10,74,79,80 When cooled at 0.2 K/min (Figure 1b), the 28 nm thick PS film exhibits a 17 K decrease in T− and 1 K decrease in T+ relative to the 771 nm thick bulklike film, resulting in a 9 K decrease in Tmid from 372 K in the thicker film to 363 K in the thinner film. When cooled at 20.0 K/min (Figure 1c), the 28 nm thick PS film shows a 9 K decrease in D
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Figure 4. Comparison of thickness dependent Tg values measured by ellipsometry for films of (a) neat PS, (b) PS + 2 wt % BPD, and (c) neat PC, which were measured at different cooling rates: (square) 20.0 K/min, (circle) 10.0 K/min, (triangle) 5.0 K/min, (diamond) 2.0 K/min, (star) 1.0 K/ min, and (pentagon) 0.2 K/min. The uncertainty associated with each Tg value is ±0.5 K for films thicker than 30 nm and ±1.0 K for films thinner than 30 nm.
increases from ∼30 to ∼60 nm as cooling rate decreases from 20.0 to 0.2 K/min. Similarly, Figure 4c shows that honset for neat PC films increases from ∼50 to ∼100 nm as cooling rate decreases from 20.0 to 0.2 K/min. The qualitative effect of cooling rate on honset may be explained as follows. As shown in Figure 4, Tg,bulk is a function of cooling rate, taking on higher values at higher cooling rates. This result is consistent with the idea that the glass transition occurs when the (average) α-relaxation time associated with cooperative segmental mobility, τα, equals the experimental time scale for relaxation to an equilibrium state. The experimental time scale decreases with increasing cooling rate, and thus both τα(Tg) and the extent of cooperativity at Tg decrease with increasing cooling rate. The Tg-confinement effect may be understood to reflect the extent to which the free surface of the film perturbs (reduces) the requirement for cooperativity of segmental mobility not only at the free surface but also by propagating the perturbation into the film interior. It is expected that the free surface will reduce the requirement for cooperativity to a greater extent, i.e., cause a greater perturbation, when there is a greater level of cooperativity associated with Tg, e.g., when Tg occurs at lower temperature due to a lower cooling rate. Thus, at lower cooling rate, the Tgconfinement effect will be evident over a broader range of thickness with a higher value of honset. We note that the results shown in Figure 4 are qualitatively consistent with computational studies that have reported that the length scale over which a film interface modifies τα in the film interior grows on cooling.82−86 As a consequence of their computational results, Lang and Simmons effectively argued that in the neighborhood of glass formation “the interfacial dynamic length scale is determined by the size of cooperatively rearranging regions (CRRs), which facilitate propagation of interfacial dynamics farther into the film than would be expected on the basis of a free volume layer model alone.”85 Starr and Douglas and co-workers51,87,88 showed that this scale is related to the characteristic scale of stringlike cooperative motion.86 In particular, Hanakata et al.86 indicated that because the length of the interfacial mobility gradient of films is inversely related to their configurational entropy,89 and because string length grows linearly with inverse configurational entropy,87 then to a good approximation the length scale of the mobility gradient at the free surface will vary linearly with
−3.5 K, which is a substantially smaller reduction than the Tg − Tg,bulk = −9 K in a 28 nm thick neat PS film. When cooled at 20.0 K/min, Tmid = 373.5 K in the 418 nm thick film and 371.5 K in the 27 nm thick film, i.e., Tg − Tg,bulk = −2 K. In comparison, Tg − Tg,bulk = −3 K in a 28 nm thick neat PS film. Thus, in comparison with the results in Figure 1, Figure 2 shows that a muted Tg-confinement effect is observed in films of PS containing 2 wt % of a low-MW diluent, independent of cooling rate. This muted Tg-confinement effect agrees with results by Ellison et al.,11 who determined via fluorescence measurement (at ∼1 K/min average cooling rate) that the Tgconfinement effect in PS films is suppressed or eliminated by small-molecule diluent addition: a 20 nm thick neat PS film exhibits Tg − Tg,bulk = −25 K, whereas Tg − Tg,bulk = −5 K and Tg − Tg,bulk = 0 K in 20 nm thick PS films with 2 or 4 wt % dioctyl phthalate addition, respectively. Related observations have been made by simulation.50 Figure 3 shows ellipsometry measurements in 635 nm thick, bulklike, and 28 nm thick ultrathin PC films, which were cooled at 0.2 and 20.0 K/min from the rubbery state to the glassy state. When cooled at 0.2 K/min (Figure 3b), the 28 nm thick PC film exhibits a 26 K decrease in T−, a 3 K decrease in T+, and a 14.5 K decrease in Tmid relative to the 635 nm thick film. When cooled at 20.0 K/min (Figure 3c), the 28 nm thick PC film exhibits a 15 K decrease in T−, a 1 K decrease in T +, and an 8 K decrease in Tmid relative to the 635 nm thick film. Similar to results for neat PS, the use of a slower cooling rate results in an enhanced Tg-confinement effect in neat PC. Overall, the Tg reduction is larger in PC than in PS, in agreement with results in ref 21. (Small quantitative difference in Tg reductions measured via ellipsometry in this study relative to those measured by fluorescence in ref 21 are expected based on the sensitivity of each technique to Tg. Such small differences have been addressed in ref 79.) Figure 4 shows the ellipsometrically determined Tg values for supported films of PS, PS + 2 wt % BPD, and PC as a function of thickness and cooling rate. Six different cooling rates were used which cover 2 orders of magnitude from 0.2 to 20.0 K/ min. In agreement with results of the seminal work by Fakhraai and Forrest76 and other studies,66,75,81 there is a significant effect of cooling rate on the Tg-confinement effect. If we take Tg − Tg,bulk = −2.5 K to define an onset thickness (honset) for this effect, then Figure 4a shows that honset for neat PS films E
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Figure 5. (a) Determination of dynamic fragility by spectroscopic ellipsometry for (square) 771 nm thick film and (triangle) 28 nm thick film of neat PS: m = 163 for the 771 nm thick film, and m = 69 for the 28 nm thick film. (b) Determination of dynamic fragility by spectroscopic ellipsometry for (square) 418 nm thick film and (star) 27 nm thick film of PS + 2 wt % BPD: m = 134 for the 418 nm thick film, and m = 108 for the 27 nm thick film. (c) Determination of dynamic fragility by spectroscopic ellipsometry for (square) 635 nm thick film and (circle) 28 nm thick film of neat PC: m = 208 for the 635 nm thick PC film, and m = 75 for the 28 nm thick PC film.
average string size. Such a direct relationship between average string size or average CRR size and the length scale of the interfacial mobility gradient (or, equivalently, an interfacial Tg gradient) indicates that measurements of the type shown in Figure 4 or related measurements yielding the length scale of the interfacial Tg gradient as a function of experimental time (or cooling rate) may provide a means to probe directly the Tdependence of the length scale of cooperative relaxation present in bulk polymers. Thus, the results in Figure 4 have important implications for understanding the fundamental nature of the glass transition in bulk glass formers. Future study is warranted to undertake this important experimental challenge. 3.2. Dynamic Fragility in Films of PS, PS + 2 wt % BPD, and PC Determined by Ellipsometry. As discussed by Robertson et al.63 in 2000 and Wang et al.64 in 2002, the DSC method used to determine dynamic fragility is “a modification of the fictive temperature method used by Moynihan and coworkers90,91 to obtain the activation energy (of glasses) for enthalpy relaxation.”64 In ref 64, the cooling rate dependence of Tg was represented as follows: ⎛ Eg ⎞ ⎟⎟ Q = Q 0 exp⎜⎜ − ⎝ RTg ⎠
By plotting data from Figure 4 for a given polymer at a given thickness according to eq 6, both the slope and the intercept of the fitted line yield dynamic fragility m. Figure 5a shows such plots for representative thick and ultrathin PS films. The average m value measured in bulklike films of PS is 159 (see Table 1). The level of uncertainty reported in Table 1 for m (bulk PS) is ±5, which represents ±1 standard deviation of the average m value resulting from five PS films with thicknesses of 228, 267, 430, 771, and 1000 nm. (Investigator-to-investigator differences in m value determinations employing Tg values obtained from ellipsometry may be larger than the reported standard deviation of ±5, which is why we provide a second estimate of error in Table 1 of ±15; see ref 92.) The bulk m value for PS of ∼159 is in accord with those reported in the literature for high MW PS measured by DSC (144,26 146,46 ∼150,69 ∼155,63 160 ± 15,94 and 180 ± 1565). Figure 5b shows related plots for representative thin and ultrathin PS + 2 wt % BPD films. As shown in Table 1, the average m value measured in bulk films of PS + 2 wt % BPD by ellipsometry is 133 ± 2 (or ±1592), which also agrees well with the m value of 136 ± 6 that we determined by DSC. It is noteworthy that the addition of 2 wt % BPD to PS leads to a 4 K reduction in Tg,bulk (measured by DSC and ellipsometry), which is a very small, ∼1% reduction relative to Tg,bulk of neat PS. In contrast, bulk fragility of neat PS is reduced by more than 10% upon addition of 2 wt % BPD. Thus, with regard to percentage change, m is more sensitive than Tg,bulk to the presence of low-MW additives than Tg,bulk. Figure 5c shows related plots for representative thick and ultrathin PC films. As shown in Table 1, the average m value measured in bulklike films of PC by ellipsometry is 206 ± 3 (or ±1592). This value is in reasonable agreement with a DSC-measured m value of 234 ± 25 reported in ref 94 by Evans et al.49 Figure 6 summarizes the m-confinement effects obtained in the three polymer systems studied here. Figure 6 also includes m values that we calculated from results reported by Fakhraai and Forrest76 for 6, 11, and 24 nm thick neat PS films (see ref 95 for details of the calculation). All three systems of films of freely deposited polymer chains exhibit a reduction in m upon confinement. The larger m-confinement effect observed in more highly fragile polymers also agrees with the trend that more highly fragile, linear polymers show a larger Tgconfinement effect.21 In turn, this suggests a strong correlation
(3)
Here Q is the cooling rate, Q0 is a reference cooling rate, Eg is the activation energy for enthalpy relaxation at Tg, and R is the gas constant. Comparing a cooling rate Q with a standard rate Qs, eq 3 can be written as follows: ⎛ ⎛Q ⎞ Eg ⎛ 1 Eg T ⎞ 1⎞ ⎜⎜ ⎜⎜1 − g,s ⎟⎟ log⎜⎜ ⎟⎟ = − ⎟⎟ = Tg ⎠ 2.303RTg,s ⎝ Tg ⎠ ⎝ Q s ⎠ 2.303R ⎝ Tg,s (4) 59,64,68
The dynamic fragility m can be expressed by
m=
Eg 2.303RTg
(5)
Thus ⎛ ⎛Q ⎞ T Tg,s ⎞ ⎟⎟ = m − m g,s log⎜⎜ ⎟⎟ = m⎜⎜1 − Tg Tg ⎠ ⎝Qs ⎠ ⎝
(6) F
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It is also important to note that the length scale associated with the m-confinement effect is substantially larger than those normally reported in the literature for the Tg-confinement effect. For example, a typical onset thickness reported for the Tg-confinement effect in supported PS films is 40−50 nm,4,21 a factor of 3−4 smaller than that determined here for fragility. A similar difference is observed in the length scales associated with the Tg-confinement effect21 and the m-confinement effect in neat PC. How may these different length scales of m- and Tgconfinement effects in a given polymer be resolved? At least two factors may contribute to this apparent difference. First, as evident from Figure 4 and other studies,66,75,76,81 the Tgconfinement effect depends on cooling rate. In the research literature, Tg-confinement effects are commonly reported at a single cooling rate, often at ∼1 K/min. Because fragility at a given film thickness is determined from the cooling rate dependence of Tg, it is possible that the longer apparent range of m-confinement effects may reflect, at least in part, the longer range of the Tg-confinement effect at lower cooling rate. (In fact, if there were no cooling rate dependence of the Tgconfinement effect, then application of eq 6 to determine m would result in no apparent effect of confinement on fragility.) A second possible contributing factor to this difference in apparent length scales of confinement effects is related to how confinement changes the breadth of the α-relaxation distribution. If the fragility value for a given polymer system reflects its extent of dynamic cooperativity71 and the breadth of its αrelaxation distribution, then confinement results in a reduction of that breadth. Near Tg, the breadth of the α-relaxation distribution in bulk polymer covers many orders of magnitude in time.96,97 Upon confinement, fragility decreases and the αrelaxation distribution narrows. In the case of PS, if the narrowing of the relaxation distribution occurs predominantly or almost exclusively on the short-time side with confinement down to 50 nm thickness, then little effect of confinement in Tg may be observed over those length scales. This is because the value of Tg reflects the long-time side of the α-relaxation distribution.96,97 (The apparent decoupling reported in the literature of the well-documented Tg-confinement effect characterized by (pseudo)thermodynamic measurement of a transition temperature and the (near) absence of confinement effects on cooperative segmental mobility98 may complicate this hypothesis.) Future study is warranted to address how the effects of confinement on different properties or behavior are related to the m-confinement effect. As described elsewhere,47,99,100 dynamic fragility is associated with the packing efficiency of glass-forming polymers. With nanoscale confinement, a larger fraction of chain segments present in freely deposited polymer films are adjacent to the free surface. With access to the free surface, these chain segments have smaller packing frustration, which results in a reduced average dynamic fragility with decreasing thickness. In addition, fragility may be reduced by confinement effects in the absence of free surfaces, i.e., via intrinsic size effects. The fragility-confinement effect in PS films shown in Figure 6 agrees qualitatively with that measured by Fukao and Miyamoto53 in supported PS films by DRS; they observed that fragility decreased from ∼120 to ∼90 as thickness decreased from above 100 to 20−30 nm. (The quantitative differences between our results and those in ref 53 may arise from different techniques. In ref 53, the use of DRS involved placing a metal electrode layer on top of the thin films. This metal capping
Figure 6. Dynamic fragility measured as a function of film thickness for neat PS (open circle), PS + 2 wt % BPD (diamond), and neat PC (square). Also included are fragility values (half-open circle) that we calculated from results reported by Fakhraai and Forrest76 for 6, 11, and 24 nm thick neat PS films (see ref 95 for details of the calculation). The uncertainties associated with measured fragility values for bulklike films are listed in Table 1; for confined films, the uncertainties are estimated to be less than ±10.
between the susceptibility of Tg to be perturbed by confinement and the susceptibility of fragility to be likewise perturbed. Neat PC, which exhibits the highest bulk fragility of the three linear polymer systems, shows the largest m-confinement effect in terms of both the length scale at which the confinement effect is evident, Tg to obtain bone-dry polymer. (50) Riggleman, R.; Yoshimoto, K.; Douglas, J.; de Pablo, J. Influence of Confinement on the Fragility of Antiplasticized and Pure Polymer Films. Phys. Rev. Lett. 2006, 97, 045502. (51) Starr, F. W.; Douglas, J. F. Modifying Fragility and Collective Motion in Polymer Melts with Nanoparticles. Phys. Rev. Lett. 2011, 106, 115702. (52) Sanz, A.; Wong, H. C.; Nedoma, A. J.; Douglas, J. F.; Cabral, J. T. Influence of C60 Fullerenes on the Glass Formation of Polystyrene. Polymer 2015, 68, 47−56. (53) Fukao, K.; Miyamoto, Y. Slow Dynamics near Glass Transitions in Thin Polymer Films. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2001, 64, 011803. (54) Napolitano, S.; Wübbenhorst, M. Structural Relaxation and Dynamic Fragility of Freely Standing Polymer Films. Polymer 2010, 51, 5309−5312. (55) Hayashi, T.; Fukao, K. Segmental and Local Dynamics of Stacked Thin Films of Poly(methyl methacrylate). Phys. Rev. E 2014, 89, 022602.
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DOI: 10.1021/acs.macromol.5b02489 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules
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DOI: 10.1021/acs.macromol.5b02489 Macromolecules XXXX, XXX, XXX−XXX
