Effect of geometric confinement on the structure of polymers

The adsorption of a polymer chain onto a surface is driven by the competition between enthalpic and entropic interactions: the gain in enthalpy due to the binding of monomers from the chain onto the surface and the associated conformational entropic cost associated with this adsorption.^[Reference Degennes and Pincus^39, Reference Rubinstein and Colby^40] As a result, the “packing” of segments of chains is not spatially uniform throughout a polymer film. At a hard, atomically smooth substrate, the average monomer density is higher than in the bulk; simulations reveal that the amplitude of the monomer density profile oscillates away from the substrate, with the amplitude decreasing gradually into its bulk value.^[Reference Yoon, Vacatello and Smith^41–Reference Daoulas, Harmandaris and Mavrantzas^44] Such simulations further suggest a “layering” of the segments at the substrate, including the center of mass of the chain. Specifically, the component of the radius of gyration of a chain parallel to the wall, 〈R _xy〉, is larger than 〈R _z〉, that of the radius of gyration component normal to the wall.

At an unconstrained or free surface, simulations of linear-chain polymers reveal that the local segmental “packing” is lower than the bulk.^[Reference Daoulas, Harmandaris and Mavrantzas^44–Reference Mansfield and Theodorou^46] Experimental studies using positron annihilation lifetime spectroscopy (PALS) reveal the presence of increased free volume at the free surface in relation to the bulk.^[Reference Glynos, Frieberg, Oh, Liu, Gidley and Green^30, Reference Glynos, Frieberg, Chremos, Sakellariou, Gidley and Green^31] This is experimental evidence that supports the premise that the chain segments near a free surface possess larger configurational freedom than the bulk. Simulations also show that the sharpness, or breadth, of this interface increases with decreasing temperature.^[Reference Daoulas, Harmandaris and Mavrantzas^44–Reference Mansfield and Theodorou^46] In typical polymer systems, the length scales of the breadth of such an interface are of the order of nanometers. To this end, we schematically illustrate in Fig. 2 the local packing density of a supported polymer film, confined between a free surface and a hard interface. In this figure, the overall film thickness is h; h _f is the length-scale from the free surface where there would be increased configurational freedom. The length-scale δ, of the order of a nanometer, is the thickness from the substrate where the chain segment orientations are restricted and there is a preferential orientation of the segments in the plane of the film. The distance h _s + δ is the length-scale (~ few nanometers) over which the density of the film is perturbed due to the substrate. In addition to changes in the density at an interface, the average molecular weight between entanglements near an interface differs from the bulk and this can have important implications on mechanical properties, including interfacial adhesion and fracture.^[Reference Sussman, Tung, Winey, Schweizer and Riggleman^47] Other physical properties, such as the time-dependent displacements of chain segments in thin films, would differ from the bulk^[Reference Ye, Wiener, Tyagi, Uhrig, Orski, Soles, Vogt and Simmons^48–Reference Soles, Douglas and Wu^50]

Figure 2.

Schematic illustrates variations in the local density of the segmental “packing” at distances away from the interfaces.

Generally polymer segmental dynamics at polymer/substrate interfaces are comparatively slow;^[Reference Soles, Douglas and Wu^50–Reference Soles, Douglas, Wu, Peng and Gidley^53] this sluggishness is believed to be due to the long time-scales (compared with bulk relaxations) associated with adsorption/desorption of segments from the “walls.”^[Reference Yelash, Virnau, Binder and Paul^54] Stronger interactions with the walls lead to longer desorption times, thereby leaving the molecules in a metastable state, and hence slower dynamics. In fact, broad band dielectric spectroscopy (BDS) studies reveal a reduction of the dielectric strength (Δε) indicating a portion of the film is “dead” when the chains are permanently adsorbed onto a substrate and the dipoles unable to reorient.^[Reference Napolitano and Wubbenhorst^52] This effect, permanently adsorbed chain segments, is polymer/substrate specific and typically occurs after heating the film for sufficiently long periods of time. Simulations reveal that the dynamics at “walls” are anisotropic; the dynamics of the segments normal and parallel to the substrate differ in magnitude.^[Reference Peter, Meyer and Baschnagel^55, Reference Hanakata, Douglas and Starr^56]

With regard to free, unconstrained, surfaces it has been demonstrated by experiments and by simulations that the polymer chain dynamics are fast relative to the bulk.^[Reference Keddie, Jones and Cory^1–Reference Long and Lequeux^10, Reference Chai, Salez, Mcgraw, Benzaquen, Dalnoki-Veress, Raphaël and Forrest^57–Reference Torres, Nealey and De Pablo^64] The dynamics near the free surface are enhanced due to excess configurational freedom of the chain segments. The excess configurational freedom would be responsible for a reduction of the effective friction coefficients, which dictate the time scales of the center-of-mass diffusion of chains near a surface, leading to an increase in translational diffusion coefficient of a chain.^[Reference Lange, Judeinstein, Franz, Hartmann-Azanza, Ok, Steinhart and Saalwächter^65] Later it will be shown that changes in physical properties of polymers occur over much longer length-scales than the length scales that characterize changes in their segmental densities at interfaces.

Wetting of star-shaped macromolecules

Differences between the wetting properties of star-shaped macromolecules and their linear-chain analogs are profound. To understand these differences it is instructive to recall basic ideas about the wetting of a non-volatile liquid droplet on a substrate.^[Reference Bonn, Eggers, Indekeu, Meunier and Rolley^66–Reference Young^70] The wetting of a liquid on a substrate is dictated by a combination of short- and long-range intermolecular forces: within a specific distance of the line of contact the shape of the droplet is distorted by the intermolecular forces. The molecular length associated with the long-range van der Waals interactions is defined as:

(1)$$\alpha = \sqrt { - A/G\pi \gamma}, $$

where A is the Hamaker constant, and γ the liquid/vapor surface tension of the liquid. The macroscopic angle of contact θ _eq, defined away from the line of contact, is dictated by Young's equation, which relates it with interfacial energies:

(2)$$\gamma \cos {\theta _{{\rm eq}}} = {\gamma _{{\rm SV}}} - {\gamma _{{\rm SL}}},$$

where γ _SV is the surface tension, and γ _SL is the interfacial tension between the liquid and the substrate.

Recently we showed that the macroscopic contact angle θ _eq for an 8-arm star-shaped polystyrene (PS) macromolecule on silicon oxide substrates (SiO_x) is approximately one order of magnitude smaller than linear-chain PS on the same substrate.^[Reference Glynos, Chremos, Frieberg, Sakellariou and Green^28, Reference Glynos, Frieberg and Green^29] The adsorption of a macromolecular segment onto a substrate is largely dictated by a competition between the gain of enthalpy due to binding monomers to the substrate, and the conformational entropic cost of adsorption. The enhanced wetting of the star-shaped macromolecules is associated with the fact that these branched polymeric molecules experience smaller losses in configurational entropy when adsorbed to interfaces compared with their linear analogs.^[Reference Striolo and Prausnitz^71–Reference Chremos, Camp, Glynos and Koutsos^75] These smaller losses of configurational entropy of the star-shaped macromolecules, in relation to their linear chain analogs of the same total number of monomers, may be considered to be an entropic attraction of the star-shaped molecules to the substrate.

The data in Fig. 3 describe the fascinating wetting characteristics of the star-shaped macromolecules.^[Reference Glynos, Chremos, Frieberg, Sakellariou and Green^28] The error bars this figure represent were determined from measurements of the contact angles of tens of droplets. It is shown that the macroscopic contact angle rapidly decreases with increasing functionality of the star, before exhibiting a minimum at f = 16. For f > 16, θ _eq increases with f until it becomes nearly constant in the high functionality limit. The values are smaller than that of linear chains.

Figure 3.

Equilibrium contact angles, θ _eq, are plotted as a function of f. Reproduced (adapted) with permission from Ref. Reference Glynos, Chremos, Frieberg, Sakellariou and Green28, copyright 2014 American Chemical Society.

The wetting behavior of star-shaped molecules in relation to their linear-chain analogs may not be fully understood in terms of the surface tension of the molecules; effects associated with the entropic attraction must be considered. The surface tension for star-shaped polymers is reasonably well described by:

(3)$${\gamma ^{\hskip1.5pt\rm s}}({M_{\rm n}}) = {\gamma ^{\hskip1.5pt\rm s}}(\infty ) + \rho RT\displaystyle{{\,f{\hskip-0.9pt{U}^{\rm e}} + {U^{\rm b}}} \over {M_{\rm n}^{\rm s}}}, $$

where γ(∞) is the surface tension of the theoretical infinite number average molecular mass polymer, ρ is the bulk density, R is the universal gas constant, T is the temperature, M _n is the number average molecular mass, and U ^e and U ^b represent the effective attractive, or repulsive, interactions between the interface and the chain ends, and branch points, respectively.^[Reference Minnikanti and Archer^72, Reference Qian, Minnikanti, Sauer, Dee and Archer^73] Because this relationship predicts a monotonic decrease of the surface tension with increasing functionality, the non-monotonic behavior of θ _eq cannot be explained solely in terms of changes in the surface tension. As discussed below, the wetting properties of star-shaped polymers stem primarily from the strong surface adsorption behavior in relation to linear chains of otherwise identical chemical structure.

Molecular dynamics (MD) simulations of a bead-spring model of polymer chains show that the ratio of the perpendicular and parallel components of the radii of gyration, ${\langle R_{\rm g}^2 \rangle _z}/{\langle R_{\rm g}^2 \rangle _{xy}}$, of macromolecules adsorbed to an interface depends on the functionality of the polymer [Fig. 4(a)].^[Reference Glynos, Chremos, Frieberg, Sakellariou and Green^28] These MD simulations indicate that ${\langle R_{\rm g}^2 \rangle _z}/{\langle R_{\rm g}^2 \rangle _{xy}}$ increases with increasing f, but exhibits a maximum for f ≈ 8. Beyond this maximum, f > 8, the ratio decreases and reaches a constant value in the high f limit. The densities of adsorbed chains, ρ _ads (the number of adsorbed chains per unit area), extracted from the MD simulations and experimental measurements are plotted in Fig. 4(b). These data indicate that dependence of ρ _ads on f is consistent with the ${\langle R_{\rm g}^2 \rangle _z}/{\langle R_{\rm g}^2 \rangle _{xy}}$ [Fig. 4(a)] versus f trends.

Figure 4.

(a) The ratio of the parallel and perpendicular components of the mean-square radii of gyration 〈R _g²〉_z/〈R _g²〉_xy of the adsorbed polymers is plotted as a function of ƒ. Reproduced (adapted) with permission from Ref. Reference Glynos, Chremos, Frieberg, Sakellariou and Green28, copyright 2014 American Chemical Society. (b) The number of adsorbed chains per unit area (density), ρ _ads, is shown as a function of ƒ, obtained from the experiment and from the simulations. Reproduced (adapted) with permission from Ref. Reference Glynos, Chremos, Frieberg, Sakellariou and Green28, copyright 2014 American Chemical Society.

The maxima of the anisotropy of orientation of the molecules and of the density [data in Figs. 4(a) and 4(b)] are due to the same two competing mechanisms. The entropic attraction of a star-shaped molecule to the substrate increases as the functionality ƒ of the molecule increases. Additionally with this increase of ƒ, entropic intermolecular repulsions between the arms of a molecule extending from the branch point necessarily increases; this leads to stretching of the arms (i.e., a reduction of the entropy per chain). This intermolecular repulsion has the effect of reducing any potential “crowding” of the molecules at a surface. In other words, the entropic restrictions imposed by increasing f lead to a decreasing compliance of the star-shaped polymers, which suppresses their ability to deform on the surface when adsorbed. Hence, the angle of contact decreases.

This is the first time that such significant differences in the wetting behavior are achieved by simply changing the architecture of the molecule, without changing the chemistry. The practical implications are that the wetting at an interface may be “tailored” via changes in the functionality and the length of the arms of the macromolecule.

Glass transition of thin films

This topic is introduced by recalling that the reduction of the temperature of substances, from small molecule liquids to network glass melts and polymer melts, is accompanied by a significant increase in the structural relaxation time. Practically, at a sufficiently low temperature where the structural relaxation time becomes longer than the observation time, or duration, that the cooling rate permits, the system becomes “frozen,” this temperature denotes the glass transition. Additionally virtually any substance, if cooled at a sufficiently rapid rate, would not crystallize due to a lack of time for the constituents to form an ordered structure. It is important to note that by virtue of their molecular structures, atactic polymers for example, will not crystallize and therefore exhibits a T _g, regardless of cooling rate. Glass-forming melts solidify, with the absence the formation of long-range order, at a lower temperature than the crystallization temperature. The glass-transition temperature, T _g, occurs over a very narrow range of temperatures and is cooling-rate dependent. With these brief comments, we discuss this phenomenon with regard to the behavior of thin polymer films.

It is a known fact that the average glass-transition temperature of a sufficiently thin polymer film differs from the bulk.^[Reference Baschnagel and Varnik^8, Reference Pye and Roth^63, Reference Forrest, Dalnokiveress and Dutcher^76–Reference Ediger and Forrest^79] This effect is often observed when the polymer film is of the order of tens of nanometers, whether or not it is supported by a substrate. In the case of freely standing linear-chain polymer films, the average T _g decreases with decreasing h. When the polymer films are supported by non-wetting surfaces, such as linear-chain PS supported by SiO_x, the average T _g decreases with decreasing h (see Fig. 5).^[Reference Keddie, Jones and Cory^2, Reference Kawana and Jones^80] However, when the interactions between the polymer and the substrate are sufficiently strong (i.e., polar interactions), the average T _g increases with decreasing h. Two examples of this are poly(methyl methacrylate) (PMMA) films supported by SiO_x and PS supported by gold substrates.^[Reference Keddie, Jones and Cory^1, Reference Priestley, Mundra, Barnett, Broadbelt and Torkelson^81–Reference Pham and Green^84] The important point is that the average glass-transition temperature of a thin polymer film is influenced by the polymer and by the nature of the interactions between the polymer segments and the external interfaces, as further discussed below.

Figure 5.

(a) T _g, plotted as a function of depth, for linear-chain PS and for star-shaped PS (f = 8, M _arm = 10 and 25 kg/mol.); these data were obtained using depth-profile PALS. Reproduced (adapted) with permission from Ref. Reference Frieberg, Glynos and Green32, copyright 2012 American Physical Society and Ref. Reference Glynos, Frieberg, Oh, Liu, Gidley and Green30, copyright 2011 American Physical Society. (b) The relative length scales of the effects of the monomer density profiles and of the T _g are depicted here.

It has been suggested that due to the lower segmental configurational freedom of chains at a free surface, the T _g at the free surface would be lower than the bulk.^[Reference De Gennes^58] Experiments show that polymer segments, within a distance of a few nanometers of the free surface, remain mobile at temperatures below the bulk T _g.^[Reference Chai, Salez, Mcgraw, Benzaquen, Dalnoki-Veress, Raphaël and Forrest^57, Reference Ediger and Forrest^59–Reference Paeng, Swallen and Ediger^62, Reference Forrest^85] Commensurate with this observation, depth-profile PALS measurements demonstrate that the glass-transition temperature at the free surface of linear polystyrene is lower than the bulk; these data are shown in Fig. 5(a) (squares).^[Reference Glynos, Frieberg, Oh, Liu, Gidley and Green^30, Reference Frieberg, Glynos and Green^32] The T _g becomes equal to the bulk at a distance of tens of nanometers into the film. Other experiments also lead to conclusions that the T _g across the film is not uniform.^[Reference Yang, Fujii, Lee, Lam and Tsui^4, Reference Ellison and Torkelson^60] For the case of PS/SiO_x where the interactions between the PS chain segments and SiO_x are non-wetting, and the T _g at the free surface is lower than the bulk, the average T _gs of such PS films are lower than the bulk value. Generally, when the T _g at the free surface is lower than the bulk and the polymer/substrate interactions are sufficiently weak, the average T _gS of sufficiently thin polymer films are lower than the bulk. However, if the polymer substrate interactions are sufficiently strong (e.g., hydrogen bonding), and responsible for an enhancement of the local T _g at the substrate, then the average T _g of a sufficiently thin film may be larger than the bulk value. The PS/Au and PMMA/SiO_x systems are excellent examples.

It is important to recognize that the length-scale ξ _ρ over which the “packing” of segments is influenced by the proximity to the interface is of the order of a nanometer.^[Reference Baschnagel and Varnik^8, Reference Torres, Nealey and De Pablo^64, Reference Shavit and Riggleman^86] However, the length-scale ξ _glass over which the glass-transition temperature deviates from the bulk is at least one order of magnitude larger (Fig. 5). This behavior has not been fully captured using simulations. However, a recent analytical theory provided the first explanation for the differences between the length-scales associated with segmental packing and those longer distances associated with changes in the glass transition.^[Reference Mirigian and Schweizer^87] The displacements of polymer segments are associated with long-range collective elastic distortion processes, occurring over tens of nanometers. Near a surface the effects are different from the bulk due to the “reduced neighbors” and subtle differences in the “packing” of segments. For this reason, the near surface and the bulk vitrification temperatures, and their associated length scales, are different.

Figure 6.

(a)The glass-transition temperatures are plotted as a function of film thickness for linear-chain PS, supported by SiO_x, and for star-shaped PS molecules of varying functionalities. The molecular mass per arm of each macromolecule is ≈10 kg/mol. Reproduced (adapted) with permission, from Ref. Reference Glynos, Frieberg, Chremos, Sakellariou, Gidley and Green31, copyright 2015 American Chemical Society, and Ref. Reference Glynos, Frieberg, Oh, Liu, Gidley and Green30, copyright 2011 American Physical Society. (b) The deviation of T _g from the bulk for a h = 30 nm thick PS film on SiO_x as a function of the number of arms. Reproduced with permission (Ref. Reference Glynos, Frieberg, Chremos, Sakellariou, Gidley and Green31), copyright 2015 American Chemical Society.

Since the star-shaped macromolecules experience lower entropic penalties than their linear-chain analogs when adsorbed at an interface, it would be reasonable that their thin-film glass-transition temperatures would differ from linear chains, as suggested by the data in Fig. 5(a).^[Reference Glynos, Chremos, Frieberg, Sakellariou and Green^28, Reference Glynos, Frieberg and Green^29, Reference Striolo and Prausnitz^71–Reference Chremos, Camp, Glynos and Koutsos^75] To this end the h-dependencies of the average T _gs of several different star-shaped PS thin films supported by SiO_x substrates are reported in Fig. 6. In this figure, M _arm was fixed and the functionalities are varied, 3 < f < 64. For cases where f = 2, 3, and 4 the average T _gs decrease with decreasing h. This behavior is similar to linear-chain PS, where T _g decreases with decreasing h.^[Reference Glynos, Frieberg, Oh, Liu, Gidley and Green^30] However, for molecules possessing functionalities in the range 4 < f < 32, the average T _gs increase with decreasing h. This remarkable shift in the trend, based on the functionality, exhibited by vitrification of the films is more clearly illustrated in Fig. 6(b). In this figure, the T _gs of different h = 30 nm thick films of molecules of varying f relative to the bulk T _gs are plotted. It is apparent that the T _g behavior of the star-shaped polymers is similar to that of linear chains when the arms are sufficiently long (M _arm is sufficiently high); here the effect of the functionality is not important.^[Reference Glynos, Frieberg, Oh, Liu, Gidley and Green^30] Simulations reveal an enhancement of the segmental densities of short-arm star-shaped molecules at the external interfaces, in relation to the bulk. The enhancement at the free interface is less pronounced than at the substrate, which is not unexpected. These results provide a rationale for the shifting trends of the T _g versus h behavior.

We now discuss the increase of the T _g with decreasing h, for f > 4. PALS measurements show that the free surface T _g for the 8-arm star-shaped films is higher than the bulk (Fig. 5). Our MD simulations reveal that at the free surface, the packing density of the 8-arm star-shaped macromolecules is higher than the bulk. Additionally, a strong positional correlation between the star-shaped molecules (spatial ordering of the cores) near the free surface occurs.^[Reference Glynos, Frieberg, Chremos, Sakellariou, Gidley and Green^31] As described in the previous section, the star-shaped molecules in this regime exhibit a stronger tendency to adsorb to the substrate. This results in a higher packing density and thus a larger local T _g at the substrate interface. Because the T _gs at the free surface and at the substrate are larger than the bulk for the 8-arm star with M _arm = 10 kg/mol., the average T _g of this system increases with the decreasing film thickness [Fig. 6(a)].^[Reference Glynos, Frieberg, Oh, Liu, Gidley and Green^30] When arms are sufficiently long, M _arm > 25 kg/mol., the thickness-dependent T _g behavior is virtually identical to that of linear-chain PS. These results confirm the notion that the thickness dependence of the T _g may be understood in terms of the interactions between the polymer segments with its external interfaces. The interactions are enhanced due to the entropic attraction of the chains to the interfaces with increasing f. As described in the previous section this enhanced entropic attraction is offset by an entropic effect associated with crowding, and an associated loss of mechanical compliance, of the molecules as f further increases. This behavior is largely responsible for the maximum in Fig. 6.

The glass transition of star-shaped PS polymer films on SiO_x may be summarized in terms of a “Diagram of States” where the deviation of T _g from the bulk is illustrated as a function of f and M _arm in Fig. 7. When the number of arms is sufficiently large and the arms are sufficiently short, the star-shaped molecules “pack” like soft-colloids; the average glass transition temperatures of these films are equal to that of the bulk. It has also been reported in that bulk star-shaped molecules with high f and low M _arm exhibit behavior akin to that of soft-colloids.^[Reference Vlassopoulos, Fytas and Cloitre^36–Reference Pearson and Helfand^38] In this regime the molecules self-assemble to form ordered structures and uniform ordered layers across the film. This is illustrated in Fig. 8 with an atomic force microscopy (AFM) image, line-scan and schematic. This organization is consistent with the simulations, which suggest evidence of ordering of the cores of the molecules at the interfaces, for the molecules with large f.

Figure 7.

The relative glass-transition temperature, ΔT _g = T _g(h = 30 nm) – T _g (bulk), is shown here for star-shaped polymers of varying functionalities and arm lengths. Gray region (black squares) represents ΔT _g < 0, blue region (blue triangles) represents ΔT _g > 0, and the red region (red circles) represents ΔT _g = 0. Reproduced with permission from Ref. Reference Glynos, Frieberg, Chremos, Sakellariou, Gidley and Green31, copyright 2015 American Chemical Society.

Figure 8.

An AFM image of a thin-film star-shaped macromolecule of functionality f = 64 (and M _arm ~ 9 kg/mol.) is shown. The thickness of each layer is represented by the line scan; a schematic representation of the organization of two layers of molecules is shown.

The decreased compressibility of the polymers, with increasing f and decreasing M _arm would have the effect of suppressing the influence of the interface on wetting and adsorption, and therefore changes in the glass transition as shown. In the “linear-chain” region of Fig. 7, the average T _g of thin films is lower than the bulk. The transition between the linear-chain region and the soft colloidal-like behavior occurs over limited values of f and M _arm. This is this region that represents the maximum in Fig. 6(b).

The primary message of this section is that the average T _g of a thin polymer film is dependent on the thickness of the film, the macromolecular architecture and the external interfaces. These data in Fig. 7 illustrate the fact that the glass-transition temperatures of films may be “tailored” simply by changing the macromolecular architecture. The glass-transition temperature is not constant throughout the film; it is a function of the proximity to an interface. Secondly, the architecture of the molecule and the interactions between the molecule and the interfaces have a significant impact on the magnitude of the local T _g and hence the average T _g of the film. The primary implication of these results is that when a thin film is cooled to lower temperatures, different regions within the thin film, based on their proximities to the interfaces, fall out of equilibrium at different temperatures. This has important implications on processing and on properties such as physical aging, which we discuss in the next section.

Physical aging of thin films

The term physical aging refers to a system undergoing slow dynamics as the system approaches equilibrium, due to frustration associated with transitions between the metastable states (configurations).^[Reference Struik^88] This phenomenon is ubiquitous, occurring in diverse systems: network glasses (i.e., SiO₂),^[Reference Zhao, Simon and Mckenna^89] spin glasses,^[Reference Bouchaud^90] colloids,^[Reference Bouchaud^90] metallic glass,^[Reference Wang^91] and polymer glasses.^[Reference Struik^88] To understand this phenomenon in polymers we begin by recalling that as the temperature of a glass-forming liquid is decreased, the structural relaxation time, and the viscosity, increases in a non-linear manner. The temperature dependence of the viscosity is well described by the Vogel–Fulcher–Tammann (VFT)^[Reference Fulcher^92–Reference Vogel^94] equation [also equivalently the Williams–Landel–Ferry equation (WLF)]:^[Reference Williams, Landel and Ferry^95]

(4)$$\eta (T) = Aexp\left( {\displaystyle{B \over {T - {T_\infty}}}} \right),$$

where T _∞ is the Vogel temperature, which is related to T _g, and B is a parameter associated with thermally activated processes. Adam and Gibbs, in 1965, were the first to propose the existence of cooperativity between the dynamics of the constituents of the material and the effect of cooperativity on the overall dynamics. Specifically, there exists a length-scale, ξ _coop, which characterizes a distance over which the dynamics are highly cooperative. They, and now others, argue that it is the increase of ξ _coop with decreasing T that is responsible for the increase in the activation energy and hence the non-linear increase of time scales of the relaxation dynamics, with decreasing T.^[Reference Adam and Gibbs^96]

It is currently understood that the dynamics of liquids are spatially and temporally heterogeneous.^[Reference Ediger^97] Specifially, the dynamics of local regions (size ~ nm at high T) vary spatially; in some local regions the dynamics are very fast compared with the average and for others the dynamics are slow compared with the average. The positions of these local regions are correlated temporally in space over particular length-scales ξ _het of the order of nanometers.^[Reference Long and Lequeux^10, Reference Adam and Gibbs^96–Reference Starr, Douglas and Sastry^99] It is suggested by dynamic heterogeneity models that density fluctuations throughout the system are responsible for the strong spatial variations of the dynamics. It is the percolation of the slow domains that is responsible for the glass transition, below which the system becomes a glass.

When a substance rapidly cooled into the temperature regime T < T _g, the physical properties of the material become time-dependent in this non-ergodic state, as the system attempts to reach equilibrium.^[Reference Priestley, Ellison, Broadbelt and Torkelson^5, Reference Struik^88, Reference Hutchinson^100–Reference Mckenna^103] Time-dependent properties include: enthalpy, entropy, the compliance, gas permeability/selectivity, density, index of refraction, and dielectric susceptibility. In polymers, the physical aging process is characterized by three regimes: an initial plateau, an intermediate time-dependent stage, and an equilibrium plateau. The aging rate:^[Reference Baker, Rittigstein, Torkelson and Roth^104]

(5)$$\beta = \displaystyle{1 \over{{h_\infty}}} \displaystyle{{dh} \over {d\left[ {ln({t_{{\rm age}}})}\right]}},$$

is obtained from the slope of the lines shown in Fig. 8(a).^[Reference Frieberg, Glynos, Sakellariou and Green^33] The data shown in this figure represent the intermediate, time-dependent, aging stage. It is well established that the aging rate is determined by the difference between the glass-transition temperature of the material T _g and the aging temperature T _age: T _g–T _age = ΔT _a.^[Reference Hutchinson^100–Reference Mckenna^103] For small values of ΔT _a the aging rate is fast and increases with increasing ΔT _a; however, the aging rate reaches a maximum and then decreases with increasing ΔT _a. This maximum is the result of two competing processes: the driving force for aging increases with decreasing ΔT _a, but the thermal energy decreases as T decreases. Experimentally, the time scale associated with achieving equilibrium could range from minutes to decades. In our case, we investigated the time dependent changes in the film thickness of a supported polymer film after it is quenched to a temperature T _age.

The aging rates of star-shaped and linear-chain polymers differ: whereas the aging rate of linear-chain polymers is independent of the length of the chain,^[Reference Hutchinson^100–Reference Mckenna^103] the rates of star-shaped polymers depend on the functionalities and on the arm lengths of the polymers.^[Reference Frieberg, Glynos and Green^32, Reference Frieberg, Glynos, Sakellariou and Green^33] It is shown in Fig. 9(a) that as the functionality of the star is increased the aging rate decreases at a constant temperature relative to T _g.^[Reference Frieberg, Glynos, Sakellariou and Green^33] However, when M _arm is sufficiently large the star-shaped polymers exhibit an aging rate similar to linear-chain PS [Fig. 9(b)].^[Reference Frieberg, Glynos, Sakellariou and Green^33] It should be intuitive, therefore, that as the length of the arms is increased, or the number of arms is decreased, the reduction of the aging rate associated with the branching point is suppressed.

Figure 9.

(a) The time-dependent change in film thickness for h = 1000 nm thick films of three different PSs: linear-chain and star-shaped (f = 8, 16). Reproduced (adapted) with permission from Ref. Reference Frieberg, Glynos, Sakellariou and Green33, copyright 2012 American Chemical Society. (b) The physical aging rate is plotted as a function of f and M _arm at a temperature which is 35 °C below the T _g for the h = 1000 nm thick films. The black squares represent linear PS, the red circles represent 8-arm star PS and green triangles represent 16 arm star PS. Reproduced (adapted) with permission from Ref. Reference Frieberg, Glynos, Sakellariou and Green33, copyright 2012 American Chemical Society.

It has been noted that the glass-transition temperature of a thin film is not a constant value, but varies as a function of distance from an interface. With regard to aging, the local aging rates of a film are dictated by the local T _gs throughout the depth of the film. This is illustrated in Figs. 10(a) and 10(b), where it is shown how the aging rate changes across thin films of linear-chain PS [Fig. 10(a)].^[Reference Frieberg, Glynos and Green^32] MD simulations of the aging of freely standing linear-chain polymer films yield similar results; the aging rate is lowest near the free surface and increases with increasing distance, reaching a maximum, in the interior of the film.^[Reference Shavit and Riggleman^105] Note that in Fig. 10(a), the film is supported and T _g at the substrate is approximately equal to the bulk T _g for this linear-chain PS/SiO_x system, as discussed earlier. Hence, the aging rate necessarily reaches a plateau, as shown.^[Reference Frieberg, Glynos and Green^32] The aging rates for the 8-arm star-shaped PS [Fig. 10(b)] are also depth-dependent and also dependent on the difference between the aging temperature and the local T _g, i.e., the T _g at that depth.^[Reference Frieberg, Glynos and Green^32] The other important result of this model is shown in Fig. 9(c), which shows the lower T _g surface layer for the linear chain increases with increasing T, for temperatures below T _g. This is the liquid-like surface layer to which we referred in the section on the T _g of thin films; its thickness is dictated by T _g–T. These data in Fig. 10(c) are consistent, both in magnitude and the trend with temperature, with independent experiments by other researchers.^[Reference Paeng, Swallen and Ediger^62, Reference Paeng and Ediger^106] When the outer surface layer has a higher T _g than the bulk, the surface layer thickness decreases (opposite trend) over the same temperature range.

Figure 10.

(a) The aging rate is shown here for linear-chain PS as a function of distance from the free surface, for a h = 50 nm thick film; (b) similar data are shown for the 8-arm star shaped PS. (c) The thicknesses of the surface layers are shown here at a function of the difference between the bulk T _g and the aging temperatures for linear-chain and star-shaped polymers. Reproduced (adapted) with permission from Ref. Reference Frieberg, Glynos and Green32, copyright 2012 American Physical Society.

We conclude this section by reiterating the following observations. The aging of a thin film is dictated by the local T _gs near the interfaces. The average aging rates of star-shaped macromolecules are lower than that of their linear-chain analogs. This has important implications regarding the use of macromolecules of different architectures for thin-film membrane applications as well as a host of thin-film packaging applications.

Future outlook

Star-shaped polymers exhibit different physical properties from their linear-chain analogs when confined to thin-film geometries. The key differences include the following. (i) The surface wettability of star-shaped polymers may be “tailored” in a controllable manner by varying the functionality of a star-shaped polymer and the arm length. (ii) The substrate and free surface T _gs of the star-shaped polymers are dependent on the functionality and molecular mass per arm. (iii) Star-shaped polymers age more slowly than linear-chain polymers.

The practical implications of such differences are important for a variety of applications. The mechanical properties of films are sensitive to film thickness as shown by MD simulations,^[Reference Bohme and De Pablo^107, Reference Yoshimoto, Jain, Nealey and De Pablo^108] finite-element analysis,^[Reference Clifford and Seah^109] nanoindentation experiments,^[Reference Clifford and Seah^110] thin-film buckling measurements,^[Reference Stafford, Harrison, Beers, Karim, Amis, Vanlandingham, Kim, Volksen, Miller and Simonyi^13, Reference Stafford, Vogt, Harrison, Julthongpiput and Huang^111, Reference Torres, Stafford and Vogt^112] and Brillouin light scattering. The thickness dependence is associated with the fact that near the free surface the local T _g is lower than the bulk T _g, which suppresses the average T _g of the film. The magnitude of the modulus depends on the difference between the bulk T _g and the ambient temperature at which the measurements are conducted. Star-shaped polymers would offer an important alternative to linear-chain polymers, because their interfacial properties are dictated by f and M _arm. Another obvious application would be adhesion to various types of interfaces, due to their enhanced wettability. Additionally, the star-shaped polymers may be suitable for membrane applications for gas selectivity and/or separation. Since the star-shaped macromolecules age at appreciably slower rates than the linear chains, they offer important advantages with regard to the stability and reliability of pore sizes in polymer membranes.