Entanglement-Stabilized Nanoporous Polymer Films Made by Mechanical Deformation

We present a new simulation-guided process to create nanoporous materials, which does not require specific chemical treatment and solely relies on mechanical deformation of pure highly entangled homopolymer films. Starting from fully equilibrated freestanding thick polymer melt films, we apply a simple “biaxial expansion” deformation. Upon expansion holes form, which are prevented from growing and coalescing beyond a characteristic size due to the entanglement structure of the melt. We investigate the local morphology, the void formation upon expansion, and their stabilization. The dependence of the average void (pore) size and void fraction (porosity) on the total strain and subsequent relaxation is investigated. Furthermore, the stabilization of the porous structure of the thin expanded films through cooling below the glass transition temperature Tg is discussed.


SIII. FREE-STANDING FILM SUBJECT TO BIAXIAL EXPANSION
The expansion protocol is as follows: Starting from a fully equilibrated free-standing films, the film is first instantaneously stretched by a factor of 1.02 along the x, and then along the y− direction with periodic boundary condition along the two lateral dimensions.The strain rate along each direction is defined by After each instantaneous stretch, the film is relaxed for dt = 0.02/ ε = (0.02τR /C) with the strain rate ε = C/τ R (τ R being the Rouse time) we set.Two strain rates C = 77 (slow) and 32000 (fast) are chosen such that subchains of chain length 8N e and 0.4N e (N e ≈ 28 being the entanglement length), respectively, are relaxed after each instantaneous stretch.This deformation step is so small that it does not induce any instabilities in the simulations and mimics a quasi continuous deformation.Moreover, to stabilize the free surfaces, we keep the lateral dimensions of expanded film after the strain is increased by a factor of (1.02) 3 × (1.02) 3 , and let the film relax and adjust its film thickness until the pressure in the direction perpendicular to the interfaces, P zz ≈ 0.0ϵ/σ 3 .We repeat this same procedure up to a strain of 4 × 4 ≈ (1.02) m × (1.02) m with m = 69, where the resulting expanded film is in the thin film regime, i.e., the film thickness h < R (0) g (e.g. Figure S4a).Here R  0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2 . Change in lateral dimensions (Lx(t) × Ly(t))/L 2 w plotted versus the relaxation time t on a semi-log scale for films subject to slow (a) and fast (b) expansion.The effective strain rate C/τR is determined by the slope, see Eq. ( S1).
Eq. ( S1), the strain rate should also be determined by the fractional change in lateral dimensions, (L , with respect to the relaxation time t as shown in Figure S3.Indeed, the slope is given by C/τ R for films subject to slow expansion.For fast expansion, the initial effective strain rate is much slower (C ≈ 2800 for λ < 1.5) than the average effective strain rate (C ≈ 14710), i.e., the required relaxation time for P zz (t) (the retraction force) recovering to zero is longer than the average relaxation time, see Figure S4b.
Considering the pairwise potential U ij , the three-body bond bending potential U ijk , and the total effective volume of film, V film = hL x L y , here the components of pressure tensor P αβ = −σ αβ (σ αβ being the components of stress tensor defined via the virial theorem) is given by where m i and v α i are the mass and the αth component of the velocity vector of the ith monomer, respectively for i = 1, 2, . . ., n c N , r ij =| ⃗ r j −⃗ r i | is the distance between the ith monomer and the jth monomer, and is the αth component of the force vector acting on the ith monomer by the jth monomer.
Linear dimensions of a thick free-standing film subject to expansion at a fast effective strain rate ετ R = C = 14710, L x,y (t) and h(t), and the resulting three diagonal terms of pressures tensor P αα (t) with α = x, y, and z during the expansion process are shown in Figure S4.Results of the rescaled internal mean-square distances right after S5, normalized by the affine deformation scaling parameter C R , as indicated.Here n is the chemical distance between two bonds along the path of the same chain.Snapshot configurations illustrating the formation of a typical pore is shown in Figure S10.

SIV. ESTIMATES OF POROSITY AND PORE SIZE DISTRIBUTION
The porosity in expanded film is defined by with where as mentioned in the second section (main text), see Figures 1 and 6.Practically, we estimate ϕ of our systems by simply performing a Monte Carlo (MC) integration of V void .The radius R of any pore, assuming a spherical shape of diameter D pore = 2R in the void space is determined by first randomly selecting a point ⃗ r p in the void space, and then find the largest radius R of hard sphere located at ⃗ r c containing ⃗ r p in the void space satisfying the following conditions: [7, 8] where is thus given by the negative derivative of the cumulative histogram H(D pore ) that counts the probability of finding a point in the void space with a pore size equal and smaller than D pore .We combine the grid-based method [9] with the grid spacing l c = 0.8σ and the Hoshen-Kopelman method [10] in MC simulations to look for unoccupied grid sites in the void space and find the maximal pore radius R according to the criterion given in Eq. ( S5).The maximum pore size D   To investigate the variation in lateral dimensions as the restoring force approaches zero, i.e., P xx,yy ≈ 0.0ϵ/σ 3 , we relax expanded films at T = 0.4ϵ/k B < T g starting from the final configurations of expanded films subject to cooling by performing molecular dynamics simulations in the NPT ensemble.The percentage change in the lateral dimensions λ p (t)/λ×100 % versus the relaxation time t is shown in Figure S17, where λ p = L x,y (t)/L w with L w ≈ 526σ.

Figure S1 .Figure S2 .
Figure S1.A schematic drawing of MSD (usually called g1(t)) of monomers.The crossover points between two different scaling regimes correspond to the characteristic microscopic relaxation time τ0, the entanglement time τe = τ0N 2 e (Ne being the entanglement length), the Rouse time τR = τ0N 2 , and the reptation (disentanglement) time τ d = τ0N 2 (N/Ne) 1/4 .Between τe and τR, monomers are restricted to move only along an imaginary tube, which is geiven by the averaged contour of the very chain.The tube diameter dT ∝ N 1/2 e and its contour length LT = dT (N/Ne).Rg(N ) and l b denote the radius of gyration of chains and bond length between monomers, respectively.

Figure S4 .Figure S5 .
FigureS4.Time series of two lateral dimensions Lx(t), Ly(t), and thickness h(t) of film (a), and three diagonal terms of pressure tensor P αβ (t) (b) for films subject to fast expansion (C = 14710) at T = 1.0ϵ/kB.The configurations of expanded films of two selected lateral dimensions Lx(t) = Ly(t) ≈ 399σ and 526σ, i.e. the strain of λ = Lx,y/Lw ≈ 3.0 and 4.0, in the thin film regime (h < R (0) g ) at Pzz(t) ≈ 0.0ϵ/σ 3 are indicated by arrows for later study.

10σ RelaxationFigure S10 .
Figure S10.Snapshot configurations of an expanded film at fixed strain of λ ≈ 3.5 before and right after relaxation where the film thickness h freely adjusted from 11.0σ to 11.3σ.The formation process of a marked pore having a spherical-like cross section is illustrated by the changes of local structures from a different perspective.
is the distance between monomer i and the test particle for i = 1, 2, . . ., N tot (= n c N ) and Θ(d) is the Heaviside step function.Here the effective film thickness h = z density distribution ρ(z) in the direction perpendicular to the interfaces located at z is the distance between monomer i and the center of sphere, ⃗ r c .The distribution P (D pore ) = − dH(Dpore) dDpore .{all D pore values in one Monte Carlo block}.Typical snapshot configurations of a free-standing porous film and the corresponding unoccupied grid sites are shown in Figure S11.Choosing l c = 0.8σ, pores are well represented by clusters of free grid sites as shown in Figure S11b.However, cutting expanded films into slices, we see that not all pores are permeable, see Figures S8, S9, S14, and S15.Pores can either form in the surface or in the interior of films.

Figure S11 .
Figure S11.Typical snapshot of a free-standing porous film of lateral dimensions Lx = Ly ≈ 526σ, and film thickness h ≈ 9.7σ (a), and the corresponding clusters (pores) of unoccupied grid sites (b).Detailed structure in the interior of film, see Figure S15.