Supplementary Figure 1. Circular parallel lamellae grain size as a function of annealing time at 250 C. Error bars represent the 2σ uncertainty in

Transcription

1 Supplementay Figue 1. Cicula paallel lamellae gain size as a function of annealing time at 50 C. Eo bas epesent the σ uncetainty in the measued adii based on image pixilation and analysis uncetainty contibutions in measued aeas, and ae geneally within the data points. PS-PMMA film thicknesses ae (A) H 0.8 L0, (B) H 1. L0 and (C) H 1.6 L0. 1

2 Supplementay Figue. Finite diffeence calculation of paallel lamellae aeas fo initial masking featue aspect atios anging fom 0.5 to 4.0. Numbes shown ae fo nanomete units. The vaiable input paametes fo each gaph ae (A) tfd = 0; (B) Δf/γ = 0.045, tfd = 5000; (C) Δf/γ = 0.015, tfd = 5000; (D) Δf/γ = 0, tfd = 000; (E) Δf/γ = 0, tfd = Supplementay Figue 3. Calculated paallel lamellae aeas fo Δf = 0. (A) Calculated aeas vs. design aspect atio fo thee diffeent finite diffeence calculation times (tfd). (B) Aveage of calculated aeas shown in (A) vs. tfd.

3 A Paallel lamellae Film thickness (H) R o Masking featue A masking Supplementay Figue 4. Geomety used fo model deivation. A paallel lamellae gain of adius and height H is consideed to be diectly above a cicula mask featue of adius R0. Supplementay Discussion. The deivation of the phenomenological model descibed in the main text shaes a numbe of chaacteistics with a pevious teatment of seconday gain gowth in polycystalline thin films by Thompson 1. The model is deived based on the geomety shown in Supplementay Figue 4. The volume of a paallel lamellae (V) gain is given by: V HA = Hπ = (1) The time ate of change of the paallel gain (v = d/dt) is negatively popotional to the total ate in which lamella chains tansfe fom the paallel gain to the suounding pependicula lamellae matix. That is, when a substate paallel lamellae tansitions to a substate pependicula lamellae at the bounday, the substate paallel gain shinks. The fowad and evese ates in which lamellae tansfe fom the paallel gain to the 3

4 suounding pependicula lamellae matix ae assumed to be zeoth ode. Then fom basic ate theoy the time ate of change of the paallel gain is given by: v d µ Rate = K 1 exp () dt kt = Whee K is the fequency in which lamellae change fom a pependicula to a paallel oientation, µ is the change in chemical potential of polyme chains fom the paallel to the pependicula oientation, k is Boltzmann s Constant and T is the annealing tempeatue. At conditions of constant tempeatue, pessue, and system volume, Δμ may be elated to the change in fee enegy fom a paallel gain to a pependicula gain of the same adius above the masking featue ( F = F F ) by: FV µ = c (3) V whee Vc is the volume of a polyme chain. The fee enegy tems ae given by: F F π Ai ( γ γ ) π = f H (4) HSQ ( γ γ ) π γ πh = f Hπ Ai HSQ (5) In these equations, f and f ae the fee enegy pe unit volume fo the paallel and pependicula lamellae, espectively, γ Ai is intefacial tension at the ai inteface, γ HSQ is intefacial tension at the HSQ inteface, and γ is the gain bounday intefacial tension. Combining Supplementay Equations 1, 4 and 5, the change in fee enegy pe unit volume is 4

5 F = V = f 0 ( f f ) γ H Int γ ( γ γ ) ( γ γ ) Ai HSQ H Ai HSQ γ (6) which is the diving foce fo ectification. Notably, Δf0 is expected to be negative, while γ Int is positive, and the diection of bounday motion is equal to the sign of the diving foce. The Δf0 paamete possesses a complicated dependence on the film thickness H due to equiements of commensuability between the thickness of the paallel lamellae gains and an integal numbe of polyme layes. Howeve, AFM data we have acquied indicate that the paallel lamellae gains can adjust thei thickness to meet this equiement, and that pefoated lamellae may emege within gain fo H < L0. The exact mechanism by which these phenomena occu and thei effects on the enegetic tems which depend on H ae the subject of futhe study. When the magnitude of Δμ is << kt, the ate equation may be lineaized using a Maclauin seies. Then the bounday velocity may be descibed by: ( H ) γ v f (7) whee Δf encompasses the Δf0 and ΔγInt/H tems. Actual magnitudes of Δf ae difficult to estimate owing to the likely existence of PMMA-wetting layes and non-bulk BCP mophologies 3 above the masking featues in the initial o fully ectified states. Regadless, fo thicke films (inceasing H), Δf deceases in magnitude compaed to the gain bounday enegy contibution to the diving foce, but not necessaily monotonically as Δf0 also depends on H. 5

6 The height diffeence at the edge of masking featues may have an impact on the initial stages of ectification; this likely entails efinements to ou model, but does not invalidate the geneal conclusions outlined in this investigation. Supplementay Methods The model-calculated contous shown in Fig. 4 of the main text wee pefomed using a finite diffeence implementation in ImageJ. Fist, ectangles with slightly ounded cones and dimensions coesponding to analogous masking featues wee geneated paametically. Then, at each finite diffeence time step (ΔtFD), the local adius of cuvatue (ρ) is computed fo each point by the elation: 1 = ρ d x ds d y ds (9) whee s is the ac length of the cuve. Each x and y coodinate is then updated: x = x γ tfd tfd tfd ρ f t FD nˆ x (10) y = y γ t FD t FD t FD ρ f t FD nˆ y (1) Whee nˆ x and nˆ y ae the x and y components of the unit nomal vecto, espectively. The signs ae evesed because the nomal vecto points inwad. Fo analysis, γ was held constant, while Δf was vaied systematically. The esults of Δf vaiation wee qualitatively inspected and quantitatively compaed by measuing the esulting aeas. Calculated esults fo the post-anneal paallel lamellae aeas ae shown in Supplementay Figue. 6

7 In thick films the enegetic contibutions in the model othe than the BCP gain bounday enegy ae expected to appoach zeo, and thus in this limit Δf = 0. The calculations fo this case show that the paallel lamellae aeas shink at a constant ate egadless of the initial aspect atio (ie. the design aspect atio), as shown in Supplementay Figue 3. Supplementay Refeences 1. Thompson, C.V. Seconday gain gowth in thin films of semiconductos: Theoetical aspects. J. Appl. Phys. 58, (1985).. Walton, D.G., Kellogg, G.J., Mayes, A.M., Lambooy, P. & Russell, T.P. A fee enegy model fo confined diblock copolymes. Macomolecules 7, (1994). 3. Liu, G. et al. Nonbulk Complex Stuctues in Thin Films of Symmetic Block Copolymes on Chemically Nanopattened Sufaces. Macomolecules 45, (01). 7