Glass-Transition and Side-Chain Dynamics in Thin Films: Explaining. Dissimilar Free Surface Effects for Polystyrene and Poly(methyl methacrylate)

Transcription

1 Supporting Information for Glass-Transition and Side-Chain Dynamics in Thin Films: Explaining Dissimilar Free Surface Effects for Polystyrene and Poly(methyl methacrylate) David D. Hsu, Wenjie Xia, Jake Song, and Sinan Keten* * Corresponding Author: Dept. of Civil & Environmental Engineering and Dept. of Mechanical Engineering, Room A33, Northwestern University, 45 Sheridan Road, Evanston, IL Tel: , s-keten@northwestern.edu Simulation Methodology PS and PMMA coarse-grained CG models were generated using the thermomechanically consistent coarse-graining method (TCCG) with CG bead center locations shown in Figure A and resulting CG representation shown in Figure B in the main manuscript. Both models capture bulk material properties and chemical structural distributions in notable agreement with experiments and all-atomistic (AA) molecular dynamics (MD) simulations. These properties include bond distributions, molecular weight dependent glass transition temperature (T g ) as described by the Flory-Fox relationship, elastic modulus, characteristic ratio, and monomeric diffusion coefficients. The functional forms of the CG bonded and non-bonded potentials and their parameter values for the PS and PMMA models are listed in the Supplemental Tables S-S3. The detailed methodology used to develop CG parameters for these two systems is covered by our previous studies., Here we evaluate the ability for the aforementioned models developed from bulk atomistic simulations to demonstrate chemically specific glass transition behavior of free-standing thin films.

2 Table S. Functional form of force field and optimized potential parameters for PS. A random distribution of meso and racemo B-A-A-B dihedral potentials are applied along the CG chain to reproduce atactic stereoisometry as described in our previous study. Interaction Potential Form Parameters A-A Bond Length () k = 8.69 kcal/mol Å UbondAA l k l l l =.568 Å A-B Bond Length () k = 4.6 kcal/mol Å UbondAB l k l l l =.87 Å A-A-A Angle a =.77e- a = 5.588e- 3 UangleAAA ( ) kb T ln a exp a exp a3 exp b =.76 b = 9.7 b b b 3 θ = 77. θ = 48.8 A-A-B Angle a =.45e- a =.767e- 3 UangleAAB ( ) kb T ln a exp a exp a3 exp b = 6.76 b = b b b 3 θ = 4.5 θ = 93. A-A-A-A Dihedral Angle B-A-A-B Dihedral Angle Non-bonded U U ( ) cos( ) dihedralaaaa A A (atactic) =.5 (kcal/mol) 5 k dihedralbaab ( ) Ak cos ( ) k 6 Unonbond 4 SLJ ( r) r r meso A = 4.36 (kcal/mol) A = -.74 (kcal/mol) A 3 =.5337 (kcal/mol) A 4 = (kcal/mol) A 5 = -.46 (kcal/mol) (See Table 3) a 3 = e- b 3 = 8.78 θ 3 = 48.5 a 3 = 3.99e- b 3 =. θ 3 = 34.7 racemo A = 3.76 (kcal/mol) A =.363 (kcal/mol) A 3 =.8 (kcal/mol) A 4 = (kcal/mol) A 5 = -.88 (kcal/mol)

3 Table S. Functional form of force field and optimized potential parameters for PMMA as described in our previous study. Interaction Potential Form Parameters A-A Bond Length k = 5. kcal/mol Å UbondAA() l k l l, l =.735 Å A-B Bond Length k = kcal/mol Å UbondAB () l k l l, l = Å a =.94e-, a = 4.367e-3, UangleAAA ( ) kb T ln aexp aexp b = 9.493, b = 6., A-A-A Angle b b θ =. θ = A-A-B Angle U k k k A-A-A-A Dihedral Angle B-A-A-B Dihedral Angle Non-bonded angleaab ( ) U U 5 k dihedralaaaa ( ) Ak cos ( ) k 5 k dihedralbaab ( ) Ak cos ( ) k 6 Unonbond 4 SLJ ( r) r r k = 9.88 kcal/mol rad, k 3 = -5. kcal/mol rad 3, k 4 = kcal/mol rad 4, θ =.69 rads A = 4.38 (kcal/mol), A =.8739 (kcal/mol), A 3 = (kcal/mol), A 4 = (kcal/mol), A 5 =.93 (kcal/mol) A = 4.59 (kcal/mol), A = (kcal/mol), A 3 = -.69 (kcal/mol), A 4 =.565 (kcal/mol), A 5 =.956 (kcal/mol) (See Table 3) Table S3. Bead masses and non-bonded -6 LJ potential parameters for PS and PMMA, as described in our previous CG studies., Polymer System ma (g/mol) mb (g/mol) εaa (kcal/m ol) σaa (Angs) εbb (kcal/mol) σbb (Angs) εab (kcal/mol) σab (Angs) Polystyrene Poly(methyl methacrylate)

4 To compare PS and PMMA T g -confinement differences, we first generate bulk systems for PS and PMMA with periodic boundary conditions applied in all directions. The bulk systems contain 5 chains of repeat units per chain at the average literature end-to-end distance (6.7 nm and 6.5 nm for PS and PMMA, respectively) using a random-walk algorithm. The total energy is minimized using the conjugate gradient algorithm, 3 and subsequent annealing cycles are performed with the Nose-Hoover 4 NPT ensemble by cycling the temperature from to 75 K over a period of 4 ns until the energy and density of the system has converged. Free-standing thin film systems are generated by maintaining a 9 x 9 nm cross section that is periodic in the x and y directions and adjusting the number of chains to achieve nm to nm variable thickness conditions in the non-periodic z dimension. Vacuum above and below the polymer layer in the simulation box creates upper and lower film free-surfaces and allows the film thickness to deform freely. After energy minimization, the same annealing cycles as bulk simulations are applied to the film in the Nose-Hoover NVT ensemble until energy and thickness convergence is achieved. We use a timestep of 4 fs for all CG MD simulations and all results presented herein are an average of at least three independent samples per condition. All CG MD simulations are carried out using the LAMMPS molecular simulation package. 5 To evaluate the glass transition temperature and segmental structural relaxation of the polymer, the relaxed systems are heated to 5 K and then cooled in K increments at a step-wise rate of ~7 K/ns as described in our previous study. 6 The self-part of the intermediate scattering function, a measure of the dynamic interparticle correlation, is used to calculate the segmental relaxation time at each temperature and follows the equation: 7 N Fs ( q, t) exp[ iq ( rj ( t) rj ())] N j ()

5 where N is the total number of beads, q is the wave vector taken from the initial peak in the static structure factor and has a magnitude q of 5.9 nm -, rj denotes the time dependent j th particle position, and... represents the ensemble average. We note that usage of a different wave vector q was not found to qualitatively change the results. The resulting correlation function can be fitted using the Kohlrausch-Williams-Watts (KWW) stretched exponential function: KWW F ( q, t) C exp[ ( t / ) ] s KWW () where C and τ KWW are fitting parameters, and β KWW is the exponential stretch factor. The α- relaxation time τ α is estimated as the time where F s (q, t) decays to. and the temperature dependence of τ α can be fitted with the Vogel-Fulcher-Tammann (VFT) equation: 8 DT exp[ ] T T (3) where τ, T and D are fitting parameters and D also provides an estimate of the fragility. Finally, the computational T g is commonly calculated as the temperature at which the relaxation time reaches ns, 9 although we expect that modifying the relaxation time convention to another value near the same order of magnitude will not qualitatively affect our conclusions. This procedure has been extensively used in computational studies to estimate the T g for bulk and nanoscale thin film conditions and here, correlates reasonably well with experimental T g measurements. 7- The average standard deviation in T g measurements is less than ~3 K. We have also used additional methods to validate the T g estimations for the PS and PMMA models including utilizing the mean squared displacement (MSD) method described by Tsige and Taylor, as well as a specific volume methodology. 3 All of these methods give comparable estimates of the T g for our CG models. This procedure is also adopted to generate local relaxation time measurements along the z axis normal

6 to the surface. Specifically, we first generate the thin film with a larger cross sectional area (5 x 5 nm ) to allow for greater sampling. We then partition the film in nm thick layers along the z axis, and measure the relaxation time for each layer at T = T bulk g. We note that in order to compare the correlation lengths in Figure C, we had to use a thicker, 4 nm film for the PS curve, as the half-width of the 8 nm film is below the relaxation convergence length, whereas the PMMA 8 nm film does converge to the bulk relaxation time in the interior. Vibrational Analysis Additional Details The vibrational density of states (VDOS) Φ(ω) has been previously applied to glassy polymer simulations to help understand mechanisms governing changes in bulk and free-standing films such as fragility and local chain mobility differences. 4 For our study, Φ(ω) of CG systems can be calculated through the Fourier transform of the velocity autocorrelation function (VACF), and is expressed as: it ( ) dt e ( t) (4) where ω is the oscillation frequency, and ψ(t) is the VACF which is defined as: () t v() v(t) v() v() (5) where v is the time dependent velocity vector, and... represents the ensemble average. 5 The sampling frequency is commensurate with the simulation timestep. For a given temperature in the vibrational spectrum, the relative intensity of the VDOS peak as a function of frequency provides an indication of the dominant vibrational modes. In Figure 3, we calculate the VDOS near the bulk T g using the VACF shown in Figure S.

7 Figure S. Velocity autocorrelation function ψ(t) of the PS and PMMA side-chains near bulk T g shows larger correlation fluctuations for PMMA than for PS. The magnitude of VACF fluctuations is correlated with the amplitude of vibrations in Cartesian space. The major peak occurring at 9 THz for PMMA in Figure 3 corresponds to the backbone-side-chain bond stretching vibrational mode. An initial broad peak for PS occurring from to 8 THz corresponds to the various angle and dihedral bending vibrations. Likewise, in the PS VDOS, angle and torsion bending occurs at low frequencies ( 5 THz) and a minor peak near 6 THz indicates the backbone-side-chain stretching mode. The frequency of the backbone-side-chain peaks may be corroborated with the approximate frequency of a two-mass harmonic spring system using the CG bead masses and spring constant, where k/ m, and m is the reduced mass, m m m / m m. This yields 8. THz for PMMA and 5.4 THz for PS, which is in reasonable agreement with the peak frequencies. The relative size and intensity of the peak corresponding to the backbone-side-chain stretching mode for PS and PMMA spectra relates to the amplitude of bead fluctuations in Cartesian space for the dominant mode at a given frequency. This can be justified through the connection between the integral of the VACF and Cartesian motion, which, at long timescales quantitatively describes the diffusion coefficient D by the Green-Kubo formula,

8 D dt() t (6) where k T / m is the thermal speed and B () t is the VACF. We note that calculating the VDOS above or below the T g does not qualitatively change the reported results. The VDOS is also calculated in the all atom system for the CG defined side-chain bead force centers near the bulk T g, using the same methodology as described above (Figure 3 inset). Figure S. VDOS Φ(ω) of side-chains in arbitrary units for different mass ratio conditions are measured near bulk T g. The second peak in each spectrum is associated with the backbone-side-chain bond fluctuations. The magnitude of the peak is correlated with the amplitude of side-chain vibrations in Cartesian space, which demonstrates an increase with increasing mass ratio. (Color version can be found online) The VDOS is also calculated as a function of the PS model mass ratio (Figure S). As the mass ratio is increased from. to, the peak intensity associated with the backbone to side-chain bond stretching normal mode monotonically increases, shown in Figure 4B. Concurrently, the frequency of the backbone-side-chain stretching mode goes through a minimum at m A /m B =. In addition, the frequency for each mass ratio and its inverse are equal, which follows from the twomass spring frequency expression above.

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