Side-group size effects on interfaces and glass formation in supported polymer thin films

It is widely appreciated that interfaces with substrates and free surfaces can significantly affect the dynamics and mechanical properties of polymers under confinement at nanoscale, which is important for applications such as nanoelectronics,¹ nanocomposites,² and biomedical devices.³ Nanostructured polymer systems can exhibit dramatic shifts in glass formation dynamics compared to the bulk state, which is commonly quantified by changes in their glass transition temperature, $Tg$⁠.^4–10 For freely standing or supported films that weakly interact with a substrate, the dynamics of polymers are enhanced due to a mobile “liquid-like” layer formed near the free surface, which generally causes a reduction in $Tg$⁠.^{4,6,8,11–13} On the other hand, for polymers confined by a strongly attractive substrate, their dynamics are generally suppressed, leading to an increase in $Tg$⁠.^14–17 For polymer films with the presence of both free surface and substrate-film interface—a common setup for many polymeric nano-devices—their effects often compete, causing a change in $Tg$ that depends not only on the film thickness but also on many other factors, such as adhesive interactions and existence of an additional underlayer.^14,18–22 Despite progress, there is still a lack of fundamental understanding of how molecular characteristics arising from monomer structure influence the glass transition of polymer thin films.

To understand the molecular mechanisms behind the glass formation of confined polymers, it is important to understand their properties in bulk. Recent studies have already proved that molecular features and characteristics are indeed important in determining bulk glass formation dynamics. For instance, there has been growing evidence suggesting that the fragility of glass formation is strongly associated with molecular features, such as chain stiffness and architecture.^23,24 That is, polymers with higher chain stiffness that cause more packing frustration generally exhibit higher bulk fragility and $Tg$⁠. These findings are consistent with the predictions of molecular simulations and theories, such as the general entropy theory (GET) of glass formation^25,26 and the elastically collective nonlinear Langevin equation (ECNLE) theory.^27,28 If fragility is considered to be a manifestation of the molecular structure of a polymer, an important question to address is how the fragility and molecular structure relate to $Tg$-confinement phenomena in nanoscale polymer films.

Several studies have aimed to draw a connection between this molecular picture and the free surface confinement effects on the glass transition of polymer thin films. For example, many investigations have suggested that polymers with greater fragility exhibit a larger decrease in $Tg$ in thin films.^12,29,30 Assuming that the effect of free surface is dominant over the substrate, Torkelson and co-workers have argued that the extent to which the free surface relieves the packing frustration experienced by polymer chains near the surface can be correlated to the polymer’s fragility in the bulk.¹² However, the relationship does not appear to hold in other polymer systems. Contrasting observations have been made both experimentally³¹ and computationally,³² where polymers with increased backbone stiffness (and thus larger fragility) exhibit reduced free surface confinement effects on film $Tg$ reduction.

The molecular picture of confinement phenomena is even more lacking for systems under the influence of a strongly attractive substrate. Previous studies have made strides in identifying the effect of substrate features, such as substrate roughness and interfacial interaction strengths, on the overall $Tg$ shift of the supported film.^{14,21,33–35} However, an in-depth investigation of the effect of molecular characteristics on the effect of the attractive substrate remains to be explored. If the correlation between molecular structure and the free-surface induced confinement effect is assumed to be valid, we can hypothesize that molecular structure should play an important role in determining the $Tg$ changes near the substrate-film interface. The present work aims to investigate this possibility computationally.

Relating molecular features such as monomer structure to glass formation processes in a supported polymer film requires us to separate the free surface and substrate effects in thin films. Here, we use coarse-grained (CG) molecular dynamics (MD) simulations to quantify how side-group size (i.e., volume or bulkiness) plays a role on the $Tg$-confinement effects in supported polymer thin films. We employ our recently developed two-bead-per-monomer (one backbone bead and one side-group bead) CG model of poly(methyl methacrylate) (PMMA), which has been parameterized from all-atomistic (AA) simulations to match the dynamic and mechanical properties and density.³⁶ To understand the role of side-group size in glass transition of polymer films, we generalize our model and systematically change the van der Waals radius of the side-group CG bead by varying the Lennard-Jones (LJ) parameter σ. Our study starts from an investigation of bulk glass formation properties, and then we systematically explore how the side-group size affects the $Tg$-confinement phenomena in polymer thin films supported by the substrate. Our simulations reveal that the side-group size has a prominent effect on the interfacial confinement behaviors of supported polymer thin films through its influence on interfacial packing, wetting, and relaxation.

We adopt an atomistically informed 2-bead type CG model of PMMA, where the backbone bead “A” incorporates the methacrylate group (C₄O₂H₅) and the side-group bead “B” includes the methyl group (CH₃) (Fig. 1(a)). The CG model was originally developed for simulating methacrylate-based polymers spanning both small and bulky side-groups, employing a universal backbone chain structure but a distinct side-group. Here, to study the effect of side-group volume, we simply focus on changing the side-group van der Waals radius. As discussed extensively in the original paper,³⁶ the CG bonded interactions, including bonds, angles, and dihedrals, are derived by matching the all-atomistic (AA) bonded probability distributions using the Inverse Boltzmann method (IBM).^37,38 Notably, the non-bonded interactions are captured by the Lennard-Jones (LJ) interactions

where σ is the effective van der Waals radius of the CG model at which $Unonbond$ is zero and ε is the depth of the potential well in energy unit. The cutoff distance $rcut$ is set to be 15 Å. The LJ parameters are tuned to match the density and $Tg$ of the AA bulk systems.

For the 2-bead CG model, there are six different parameters for $Unonbond$⁠: $σAA$ and $εAA$ for backbone and backbone (⁠$AA$⁠) interactions, $σBB$ and $εBB$ for side-group and side-group (⁠$BB$⁠) interactions, and $σAB$ and $εAB$ for backbone and side-group (⁠$AB$⁠) interactions. The cross-interaction terms $σAB$ and $εAB$ are taken as the arithmetic (⁠$σAB=12(σAA+σBB)$⁠) and geometric averages (⁠$εAB=εAAεBB$⁠) of the AA and BB terms, respectively. To generalize our model for the investigation of how the side-group size influences the glass transition of polymer systems, we systematically vary the side-group size by having $σBB$ take the values 3.42 Å, 4.42 Å (original CG PMMA value), 5.00 Å, and 5.42 Å as illustrated in Figure 1(a). We keep $σAA$ as a constant value of 5.5 Å. These polymer systems are denoted as s1, s2, s3, and s4, respectively. Functional forms of the potentials utilized by the CG models can be found in Table S1 in the supplementary material.

All CG-MD simulations in this work are carried out using the LAMMPS package.³⁹ To simulate a bulk system, periodic boundary conditions are applied in all three dimensions. Each system consists of $N=20 000$ CG atoms with a chain length of $n=200$⁠. Using an integration time step of $Δt=4 fs,$ an energy minimization is performed using the conjugate gradient algorithm,⁴⁰ followed by annealing cycles with the NPT ensemble by cycling the temperature from 210 K to 750 K over a period of 4 ns until the density and energy of the systems have been fully equilibrated. Supported thin film systems are simulated with a thickness of ∼18 nm in the z dimension and 9 nm in the x and y dimensions (Fig. 1(b)). Periodic boundary conditions are applied to the x and y dimensions, while the z dimension is kept non-periodic in order to simulate a free surface on the top of the film. An implicit energetic wall is placed at the bottom of the film to represent the substrate. This substrate interacts with the polymer via a LJ 12-6 potential of the form

where $z$ denotes the distance between the substrate and the polymer CG atoms, $εsp$ denotes the adhesive interaction strength, $σsub=4.5$ Å denotes the polymer-substrate distance at which the potential crosses zero, and $zcut=15$ Å is the cutoff interaction distance between the film and substrate. In order to observe a clear substrate effect, we choose a value of $εsp=3$ kcal/mol, which is comparable with the nonbonded cohesive interaction strength ε, leading to an adhesion energy of ∼100 mJ/m² that is achievable experimentally for atomically smooth surfaces. All the simulations of films are performed under the canonical ensemble (NVT). It should be noted that although the NVT ensemble is adopted for thin film simulations, the presence of free surfaces effectively leads to an isothermal-isobaric ensemble (NPT).

Structural relaxation dynamics of both the bulk and film systems is quantified by calculating the self-part of the intermediate scattering function

where N is the total number of beads, $q=|𝒒|$ is a wave number taken from the initial peak of the static structure factor $S(q)$⁠, $𝒓j(t)$ is the position of the jth particle at time $t$⁠, and $⟨…⟩$ is the ensemble average. Although our previous study has identified the q value for calculating $Fs(q,t)$ of the CG model of PMMA,⁴¹ this value will not necessarily be the same if we change the chemical structure of the CG model by changing $σBB$⁠. Accordingly, we calculate the static structure factor $S(q)$ for bulk systems of different $σBB$ and identify q based on the location of the first peak of $S(q)$⁠. $Fs(q,t)$ is then evaluated with the identified q value and then fitted using the Kohlrausch-Williams-Watts (KWW) stretched exponential function of the form: $Fs(q,t)=Cexp[−(tτKWW)β]$⁠, where $C$ and $τKWW$ are the fitting parameters and β is the “stretching exponent.” Consistent with many prior simulation works, we define the segmental relaxation time $τα$ as the time at which $Fs(q,t)=0.2$⁠.^20,41,42 It should be noted that $Fs(q,t)$ is different from that of a full intermediate scattering function, which yields slightly different values of $τα$⁠. However, it has been shown in many prior studies that the trend of glass formation by using the two functions is generally very consistent. The temperature dependent $τα$ results are fitted using the Vogel-Fulcher-Tammann (VFT) function^43–45 

where $τ0$⁠, $T0$⁠, and D are treated as the fitting parameters associated with glass formation: $T0$ is the end-of-glass formation temperature (also known as the Vogel-Fulcher temperature) and D describes the strength of temperature dependence of $τα$ and is inversely related to fragility. To quantify glass transition, we use the commonly employed “computational $Tg$” as defined by the temperature at which $τα$ reaches 1 ns, when the systems have begun to fall out of equilibrium.^24,41,42,46

Prior to studying the confinement behaviors of thin film systems, we first investigate how the local structural and glass formation properties of bulk systems change with increasing the “bulkiness” of the side-group (i.e., increasing $σBB$⁠). To characterize the local molecular structure in the bulk state, we calculate the static structure factor $S(q)$⁠, which provides information on the mean correlations in the positions of segments in the polymer. Fig. 2(a) shows the results of $S(q)$ for different $σBB$ at $T=350$ K. It can be observed that the location of the first major peak of $S(q)$ decreases with increasing $σBB$⁠, indicating a less dense packing as the side-group size becomes larger. This can also be observed from the radial distribution function $g(r)$ as shown in Fig. 2(b), where the first peak in $g(r)$ is located at a greater radial distance r for larger side-group size. These results are reasonable, as decreasing $σBB$ should reduce the excluded volume created by the side-group, making other chains pack more densely. The wave number q at the location of the first peak of $S(q)$⁠, which is listed in Table I, is then used for calculating the segmental relaxation via $Fs(q,t)$ for different CG systems.

Using the segmental relaxation data, we characterize the bulk relaxation, $Tg$⁠, and fragility (defined by $K=1/D$⁠)⁴⁷ of our model systems. As shown in the $τα$ data in Fig. 3(a), CG polymer systems with larger side-groups require longer times to relax at a given temperature. This is manifested by the systems with larger side-groups experiencing lower D (Fig. 3(b)) and larger $Tg$ (Fig. 3(c)). The values of K and $Tg$ are also summarized in Table I. Our finding on the side-group size effect on the glass formation is qualitatively consistent with the GET predictions for the relative flexible backbone and stiff side-group (F-S) class of polymers.⁴⁷ It should be noted that a similar side-group size effect on glass formation is observed by applying the same q as that in s1 for all four systems (Fig. S1 in the supplementary material).

Our findings indicate that the side-group size has a similar effect to increasing the polymer’s stiffness (i.e., either the backbone^25,26,32 or the side-chain stiffness^11,26,30) with regards to increase in fragility. However, the mechanism of the side-group size effect on molecular packing and fragility may not be the same as that of the stiffness effect. Prior studies have shown that the observation of fragility based on stiffness is nonmonotonically dependent on the ratio of backbone and side-group stiffnesses,^23,26 suggesting a fundamental difference between side-group size and stiffness effects on glass formation properties. Moreover, previous studies on antiplasticizers⁴⁸ and diluents²⁴ have both suggested that the inclusion of small molecules or spheres assists in polymer packing behavior and lower bulk fragilities, which support our findings that side-group bulkiness is an important factor in glass formation behavior.

Next, we proceed to examine the confinement behaviors of the supported thin film systems using the CG models. Fig. 4(a) shows the representative $τα$ vs. T results, along with the VFT fits (solid and dashed curves) for all the different systems in the bulk and thin film states. It can be observed that all the systems exhibit faster relaxation in the film compared to bulk. As the side-group size increases, the shift in relaxation dynamics between bulk and film becomes more pronounced. As shown in Fig. 4(b), the confinement effects are observed in all the film systems as indicated by a negative value of $ΔTg(=Tgfilm−Tgbulk)$⁠, which implies that the free-surface effect plays a dominant role in the film confinement behaviors. From our result, it is evident that these confinement effects on film $Tg$ are apparently enhanced for polymers with larger side-group size and higher fragility. Our findings are consistent with recent experimental observations by Torkelson et al., who have shown a one-to-one correlation between fragility and the strength of confinement effect in silicon supported thin films without substantial interactions with the substrate,^12,50 as well as computational results by Xie et al.,³⁰ who have investigated the effects of side-chain stiffnesses on $ΔTg$⁠. However, they do not agree with the results of Shavit et al.³² and Torres et al.,³¹ who investigated polymers with stiffer backbones and found no such correlation between fragility and $ΔTg$⁠. One possible reason for the discrepancy between these findings may be the chain ordering and alignment effect induced by the rigid backbone of the polymers near the free surface.³² Therefore, it is believed that there are other factors that may cause the contrast observations on $Tg$-confinement behaviors and their correlation with polymer fragility.

So far, our result indicates that a larger side-group can reduce packing efficiency, which contributes to an altered confinement effect due to a free surface. However, how the strength of the confinement effect in an attractively supported polymer film depends on the side-group size and polymer fragility remains unclear based on our current results and available experimental data. We can, however, hypothesize that the bulkiness of the side-group and its impact on fragility should play a role in determining the $Tg$ perturbations near the substrate-film interface.

Local relaxation dynamics and spatial $Tg$ distribution within the films should provide deeper insights into the confinement effects by allowing us to partition the relative roles of the free surface and the substrate in influencing the glass transition of the films. As such, we quantify $Tg$ locally in the film by quantifying relaxation as a function of monomer position z in the film, as measured from the substrate-film interface to the free surface. The thickness of the layer to quantify $τα$ in the local region is 2 nm. Fig. 5 shows the results of the representative local $τα$ vs. temperature for the different film systems. For all the systems, we can observe that the temperature dependent $τα(z)$ decreases as the film position shifts from the region near the substrate interface (⁠$z=1$ nm) to the bulk-like center region (⁠$z=9$ nm) and the free surface region (⁠$z=17$ nm), which indicates local $Tg$ variation within the films. Another important observation is that the shift in local $τα$ as marked by the VFT curves is qualitatively different as the side-group size increases. The shift of the local $τα$ from the substrate interface region to the middle region is much larger as the side-group size increases from s1 to s4. However, as the position moves from the middle to the free surface region, the trend seems to reverse—the shift of the local $τα$ is less for the system with smaller side-group size. These results imply that the strength of the confinement effects is distinct for films with different side-group sizes.

Fig. 6 shows the result of the local $Tg$ normalized by bulk $Tg$ in the supported film for different systems. There exists a clear local $Tg$ gradient with the magnitude decreasing from the substrate-film interface to the free surface. Although the overall film $Tg$ is lower than that of the bulk, the local $Tg$ of a polymer in the region near the substrate (z < ∼5 nm) is enhanced compared to the bulk. This can be attributed to the attractive interactions between the polymer and the substrate, which slows down chain relaxation and dynamics near the substrate. In the interior region (∼5 nm < z < ∼10 nm), the $Tg$ becomes close to the bulk values, indicating a bulk-like response within the film. Towards the free surface (z > ∼10 nm), the $Tg$ becomes appreciably lower than the bulk value due to the enhanced mobility near the surface as also observed for $ΔTg$⁠. The observation of the local $Tg$ variation within the supported films is in line with the general picture of a commonly employed tri-layer model:^5,50 a substrate-interface layer, an interior bulk-like layer, and a free-surface layer, which are useful to describe the confinement behaviors of the supported films as illustrated in the inset of Fig. 6.

Comparing the local $Tg$ results for different film systems, we see a more obvious correlation between $σBB$ and confinement effect on the local $Tg$ in the films. As the side-group becomes larger, the magnitude of local $Tg$ reduction near the free surface is greater, which indicates a stronger free-surface effect and is consistent with prior studies and the fragility hypothesis. Interestingly, however, we see a reduced $Tg$-confinement effect near the substrate for a larger side-group size. This result suggests that there exists a correlation between the fragility and the strength of confinement effect induced by the substrate, which contrasts what is observed near the free surface. This effect may be partially attributed to the fact that polymer chain packing on and near the surface is greatly hindered for larger side-groups, which lowers the effective adhesion energy between the surface and the polymer film.⁵¹ 

The result shown in Fig. 7 seems to support this, as it shows a decrease in adhesion energy with increasing the side-group size. Lower adhesion energy will lead to a lower substrate-induced appreciation of local $Tg$⁠. This finding is also consistent with experimental observations. For methacrylate polymers, strong interfacial interaction between silica substrates and films is achieved via hydrogen bonding formed by the polar groups (e.g., ester and hydroxyl groups) in polymer chains. For larger side-group sizes, the packing efficiency and the density of polymer chains near the interface are lower. Larger side-groups can reduce the hydrogen-bond density near the interface and lead to lower adhesion energy, which is similar to our observation from simulations. The side-group size effect on adhesion energy is also evidenced in a previous work by Priestley et al.¹⁵ They observed that the enhancement in local $Tg$ near the silica substrate is greater for PMMA with a smaller side-group than poly(ethyl methacrylate) (PEMA) with a relatively larger side-group, which could be attributed to the difference in local molecular packing and adhesion energy. Overall, this analysis also reveals why $ΔTg$ becomes more substantial with increasing $σBB$⁠, as observed in Fig. 4(b), as it shows that the drop in adhesion energy and the rising influence of the free surface tilt the balance of the competition in favor of the free-surface effect on $Tg$⁠.

We further probe the polymer packing efficiency near the substrate by evaluating the Debye-Waller factor (DWF) in different regions of the film. Recent studies have suggested a fundamental role of the DWF in predicting the glass formation dynamics and stiffness of polymers in the bulk and confined states.^34,52–54 The DWF is a dynamic measurement of the segmental “free volume” explored by the chain segments on the order of picosecond time scale.⁵⁵ This property can also be experimentally measured via incoherent neutron scattering (INS) experiments.^53,56 The DWF also provides information on local molecular stiffness associated with the caging effect of neighboring atoms.^54,57–59 To gain insights into the confinement effect on local dynamics and packing, we analyze the local DWF $⟨u2⟩$ across the supported films. In our simulations, we define $⟨u2⟩$ as the value of the local mean-squared displacement (MSD) $g0$ at $t=4$ ps, which is calculated as

where $ri(t)$ is the position of the ith CG bead at time t, δ is the Dirac delta function, and $⟨…⟩$ denotes the ensemble average. The CG atoms are sorted into each bin with a thickness of 0.25 nm centered at film position z based on their initial z coordinates at $t=0$⁠.

Fig. 8 shows the results of the local $⟨u2⟩$ normalized by their respective values at bulk $Tg$ for different film systems. Similar to the local $Tg$ results, the local $⟨u2⟩$ shows spatial variation as the position moves from the substrate-film interface towards the free surface, although the length scale of local $⟨u2⟩$ variation is smaller than that of local $Tg$⁠. Generally, the local $⟨u2⟩$ is suppressed near the interface and enhanced near the free surface. However, as the side-group size increases (i.e., s1 to s4), the magnitude of local $⟨u2⟩$ suppression near the substrate becomes less, while the magnitude of their enhancement near the free surfaces becomes greater. For film system s4 (i.e., the largest side-group size), there is in fact no decrease in local $⟨u2⟩$ near the substrate. This is likely caused by the increased bulkiness of the polymer, which reduces the packing efficiency of the polymer near the substrate. With bulky side-groups, the packing hindrance near the attractive interface should give rise to less mobility restriction as induced by the substrate, which would lower the $Tg$ enhancement induced by the substrate. The local mobility results corroborate our adhesion energy findings in Fig. 7, and provide a clear picture of how side-group size influences the substrate and free surface effects.

The contribution of our work is two-fold. First, we demonstrate that side-group size plays an important role in governing the glass formation properties. As the side-group size increases, both $Tg$ and fragility increase in the bulk state, which is consistent with theoretical predictions. Second, we show that there exists a correlation between the side-group size and confinement effects, which governs the strength of $Tg$-confinement in supported films. An increase in side-group size causes a stronger free-surface effect as manifested by the greater $Tg$ reduction near the free surface. However, near the substrate, the enhancement of local $Tg$ is less for the polymer with a larger side-group and a greater fragility, indicating a weaker confinement effect induced by the substrate. Through a DWF $⟨u2⟩$ analysis, we find that the strength of the confinement effects on $Tg$ is strongly correlated with local $⟨u2⟩$⁠. The suppression of the local $⟨u2⟩$ near the substrate is less with greater side-group size, which suggests that the diminishing substrate effect for the bulkier side-group is due to the inefficient chain packing and lower adhesion near the attractive interface, leading to less mobility restriction and diminished enhancement in local $Tg$⁠. Our findings help draw a molecular-level picture of $Tg$-confinement effects in supported thin films, and ascertain that polymer side-group size and fragility are important parameters that govern glass formation in both bulk and nanoconfined supported thin film states.

See supplementary material for the full CG potentials of the polymer models and an additional analysis of relaxation dynamics.

W.X., J.S., D.D.H., and S.K. acknowledge support by the National Institute of Standards and Technology (NIST) through the Center for Hierarchical Materials Design (CHiMaD) and from the Department of Civil and Environmental Engineering and Mechanical Engineering at Northwestern University. W.X. gratefully acknowledges the support from the NIST-CHiMaD Fellowship. A supercomputing grant from Quest HPC System at Northwestern University is acknowledged. We thank our collaborator Dr. Jack F. Douglas for fruitful discussions.