pattern formation mechanisms in sphere-forming diblock ......two-dimensional thin ﬁlm systems is...

PAPERS IN PHYSICS, VOL. 10, ART. 100001 (2018)

Received: 22 August 2017, Accepted: 12 December 2017Edited by: R. DickmanReviewed by: A. Peters, Dept. Chemical Engineering, Louisiana Tech Univ., Ruston, USA.Licence: Creative Commons Attribution 4.0DOI: http://dx.doi.org/10.4279/PIP.100001

www.papersinphysics.org

ISSN 1852-4249

Pattern formation mechanisms in sphere-forming diblock copolymer thin films

Leopoldo R. Gómez,1 Nicolás A. García,1 Richard A. Register,2 Daniel A. Vega1∗

The order–disorder transition of a sphere-forming block copolymer thin film was numerically stud-ied through a Cahn–Hilliard model. Simulations show that the fundamental mechanisms of patternformation are spinodal decomposition and nucleation and growth. The range of validity of each re-laxation process is controlled by the spinodal and order–disorder temperatures. The initial stagesof spinodal decomposition are well approximated by a linear analysis of the evolution equation ofthe system. In the metastable region, the critical size for nucleation diverges upon approaching theorder–disorder transition, and reduces to the size of a single domain as the spinodal is approached.Grain boundaries and topological defects inhibit the formation of superheated phases above the order–disorder temperature. The numerical results are in good qualitative agreement with experimental dataon sphere-forming diblock copolymer thin films.

I. Introduction

Many practical applications of polymers and other softmatter involve the self-assembly of the system intocomplex multidomain morphologies [1]. The finalproperties and applications of such materials dependon the ability to control the morphology by adjustingmolecular features and macroscopic variables. For ex-ample, by appropriate control over their molecular ar-chitecture, block copolymers can be designed as rigidand transparent thermoplastics, or as soft and flexibleelastomers, depending on the features of their buildingblocks [1].

The most important features of the block copolymerphase behavior are already captured by the simplest A–B diblock architecture [1, 2]. Here, the unfavorable in-teractions between blocks, and the constraints imposed

∗E-mail: [email protected]

1 Instituto de Física del Sur (IFISUR), Consejo Nacional de Inves-tigaciones Científicas y Técnicas (CONICET), Universidad Na-cional del Sur, 8000 Bahía Blanca, Argentina.

2 Department of Chemical and Biological Engineering, PrincetonUniversity, Princeton, New Jersey, 08544, USA.

by the connectivity between their constituents, results ina nanophase separation leading to the formation of peri-odic morphologies. For example, depending on molec-ular size, temperature, and relative volume fraction ofthe two blocks, diblock copolymer melts can developbody-centered-cubic (BCC) arrays of spheres, hexago-nal patterns of cylinders, gyroids, or lamellar structures.For these systems, the periodicity of the self-assembledpattern is mainly controlled by the average molecularweight, and typically is in the range of 10–100 nm.Also, since the magnitude of the interblock repulsiveinteraction generally diminishes with temperature, anorder–disorder transition can be induced thermally atthe order–disorder transition temperature TODT [1, 2].

During recent decades, the properties of self-assembling copolymers have received great attentionbecause of their potential use in nanotechnology [3–10]. Applications of block copolymer systems include,among others, templates for nanoporous materials, so-lar cells, and photonic crystals. Perhaps the most press-ing application for understanding pattern formation intwo-dimensional thin film systems is block copolymerlithography [6–9]. This process uses self-assembledpatterns, such as single layers of cylinders or spheres in

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PAPERS IN PHYSICS, VOL. 10, ART. 100001 (2018) / L. R. Gómez et al.

Figure 1: AFM phase images of the PS-PEP block copolymer thin film supported on a silicon substrate after annealing at T =333 K. Left and right panels show the pattern configuration after 5 minutes and 255 minutes of thermal annealing, respectively.Image size: 1.0 µm x 1.0 µm. The schematic in the central panel shows the block copolymer configuration in the thin film.Here, the minority phase (polystyrene) forms hexagonally-packed arrays of nearly spherical domains (red domains in thescheme). Note the presence of a wetting layer of PS blocks on the silicon substrate.

copolymer thin films, as templates to fabricate devicesat the nanometer length scale. However, the use of thesematerials to obtain lithographic masks requires the pro-duction of structures with long-range order. Since thedegree of ordering is controlled mainly by the densityof topological defects, it is important to determine thephysical mechanisms involved in the nanophase sepa-ration process and the effects of the thermal history onthe pattern morphology.

Experimentally, the time evolution of the densityof topological defects and correlation lengths at T <TODT has been studied through atomic force mi-croscopy (AFM) in block copolymer thin films withdifferent morphologies [11–16]. On the one hand, bycomparing the density of disclinations (orientational de-fects) and the correlation length, it was shown that incylinders (smectic phase) the dominant mechanism ofcoarsening involves mostly the annihilation of com-plex arrays of disclinations [11]. On the other hand,in monolayers of sphere-forming thin films with hexag-onal order it was found that the majority of the defectsare condensed in grain boundaries [16]. The orienta-tional correlation length was found to grow following apower law, but with a higher exponent than the trans-lational correlation length [16, 17]. However, as thetypical exponents observed in the scaling laws of theseexperimental systems are relatively small (∼ 15 for thetranslational correlation length and 14 for the orienta-tional correlation length), they are hard to distinguishfrom logarithmic, glass-like, dynamics [18].

The process of phase separation and the kineticsof ordering in different two-dimensional systems havebeen studied through different phase field models [17–20]. In particular, simulations of hexagonal systemswith a Cahn–Hilliard model were found to be in goodagreement with experimental data for block copolymerthin films. Through simulations, it was also shown thatthe orientational correlation length grows via annihila-tion of dislocations [17]. In addition, simulations havealso shown that triple points, regions where three grainsmeet, control the dynamics of defect annihilation andcan lead to the formation of metastable configurationsof domains that slow down the dynamics, with correla-tion lengths growing logarithmically in time [18].

In block copolymer systems, the degree of order andcontent of defects depend not only on TODT, but alsoon thermal history, depth of quench, and other charac-teristic temperatures, like the glass transition tempera-ture for each block, the spinodal temperature, and thetemperature of crystallization (if any) [1, 2].

In general, in first-order phase transitions, the depthof quench determines whether the system relaxes toequilibrium by means of spinodal decomposition or nu-cleation and growth [21,22]. Spinodal decomposition isthe relaxation process of a system quenched into an un-stable state. In this case the phase transition is a sponta-neous process, beginning with the amplification of fluc-tuations which are small in amplitude but large in ex-tent. Nucleation and growth is the physical mechanismof relaxation emerging when the system is quenched

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into a metastable state. This is an activated process,and a free energy barrier must be overcome in order torelax to the stable phase. In polymeric systems bothmechanisms can be inhibited due to crystallization orvitrification of any of the copolymer blocks [1].

In this paper, we study the disorder–order transi-tion and the ordering kinetics in sphere-forming blockcopolymer thin films as a function of the depth ofquench. The dynamics are studied through a Cahn–Hilliard equation and compared with experiments ondiblock copolymer thin films. We have organized thepaper as follows: Section II presents the experimen-tal system. In section III we present the equations ofevolution of the order parameter for diblock copolymersystems and the classical linear instability analysis ofthe Cahn–Hilliard equation. Section III.i describes thenumerical scheme employed to solve the model equa-tions. Results and concluding remarks are presented insections IV and V, respectively.

II. Experimental System

The sphere-forming diblock copolymer used in thiswork consists of a chain of polystyrene (PS, 3300g/mol) covalently bonded to a chain of poly(ethylene-alt-propylene) (PEP, 23100 g/mol). The copolymerwas synthesized through sequential living anionic poly-merization of styrene and isoprene followed by selec-tive saturation of the isoprene block [23]. Becausethe two polymer species are immiscible, the minoritypolystyrene forms spherical microdomains within themajority poly(ethylene-alt-propylene). The bulk mor-phology of the PS-PEP diblock copolymer consists ofarrays of spherical domains of PS packed with a BCCorder, with TODT = 394 ± 2 K, according to small-angle X-ray scattering experiments. The glass transi-tion temperature for the PS block was estimated to beTPSg ∼ 320 K while the glass transition temperature ofthe PEP block is well below room temperature [24].

The block copolymer was deposited (film thicknessca. 30 nm) on silicon substrates via spin coating froma disordered state in toluene, a good solvent for bothblocks, to produce a quasi-two-dimensional periodichexagonal lattice of polystyrene spheres within a ma-trix of poly(ethylene-alt-propylene). The film thick-ness was measured using ellipsometry (Gaertner Scien-tific LS116S300, λ = 632.8 nm). Order was inducedthrough vacuum annealing above the glass transitiontemperature of both blocks and below the TODT of the

block copolymer. In this system, the PEP block wets theair interface, while PS wets the silicon oxide substrate[25, 26], i.e., there is an asymmetric wetting condition(see schematic in Fig. 1).

Samples were imaged with a Veeco Dimension 3000AFM in tapping mode, using phase contrast imaging.The contrast is provided by the difference in elasticmodulus between the hard polystyrene spheres and thesofter poly(ethylene-alt-propylene) blocks. The repeatspacing for the block copolymer is 25 nm (as measuredon thin films by AFM). The spring constant of the tip(uncoated Si) was∼ 40 N/m and its resonant frequency300 kHz.

During spin coating, most of the toluene evaporatesrapidly (within a few seconds), so the block copoly-mer thin film suffers a relatively quick quench well be-low the glass transition temperature of the PS domains,inhibiting the relaxation of the early nanophase sepa-rated structure towards equilibrium. The deep and quickquench below the spinodal and order–disorder tempera-tures forms a nanodomain structure that contains a highdensity of defects. Figure 1 shows tapping-mode AFMphase images of the PS-PEP system at a very early stageof annealing. Note in this figure the large content ofdefects and that the spherical domains are not well de-fined. On the other hand, after ∼ 4 hours of annealingabove the glass transition temperature of the PS block,the pattern shows a higher degree of order and a well-defined structure of spherical domains. In order to bet-ter quantify the degree of order in this system, here wecalculate the orientational order parameter ξ6. Usingstandard image tools to identify the position of the in-dividual spheres and applying a Delaunay triangulation[13,16] it is possible to determine the inter-sphere bondorientation θ(r), with regard to a reference axis. Then,we can evaluate the local orientational order parameterat a position r: Φ(r) = exp[6i(θ(r1) − θ(r2))], wherer = r1 − r2 [17, 18]. The azimuthally-averaged cor-relation function g6(r) = 〈Φ(r)〉 was then calculatedand the correlation length ξ6 was measured by fittingg6(r) with an exponential exp(−r/ξ6). Figure 2 showsthe azimuthally-averaged correlation function g6(r) forthe two pattern configurations shown in Fig. 1. Duringannealing, the correlation length of the film increasesfrom . 10 nm to 80 nm upon increasing the annealingtime from 5 to 255 minutes.

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III. Model

The phase transition and the dynamics of nanophaseseparation for a diblock copolymer can be describedthrough the Cahn–Hillard model [1]. A convenient or-der parameter for the diblock copolymer can be definedin terms of the local volume fractions for each blockas ψ = φA − φB − (1 − 2f). Here φA and φB arethe local densities for the A and B blocks, respectively,and f is the volume fraction of one block in the copoly-mer. The dynamics can be described by the followingtime-dependent equation for a conserved order parame-ter [27]

∂ψ

∂t= M ∇2 δF{ψ}

δψ. (1)

In this equation, M is a phenomenological mobility co-efficient, and F{ψ} is the free energy functional. In thismodel, all the dynamics are rescaled by the mobilitycoefficient M , which fixes the characteristic timescaleof the diffusive phenomena that drive the dynamics to-wards equilibrium. In block copolymer systems, thediffusion process depends on several parameters, in-cluding the monomeric friction coefficients of the in-dividual blocks, molecular weight distribution, compo-sition, symmetry of the mesophase structure, and de-gree of segregation between blocks [28]. In addition,the diffusion also depends on the molecular architecture

Figure 2: Orientational correlation function g6(r) for the twopatterns shown in Fig. 1. As annealing time increases, thereis an increasing order in the system, as shown by the slowerdecay in g6(r) with r and the increase in the correlation lengthξ6.

and degree of entanglement, that dictate whether the re-laxation occurs by a Rouse, reptation, or arm retractionmechanism [28–30]. Thus, a detailed description of thedynamics involves an enormous complexity and conse-quently, at present, there is no phase field model equa-tion like Eq. (1) able to capture all of the controllingparameters. However, as the experiments here are con-ducted at temperatures well above the glass transitiontemperatures of both blocks, we found that a constantmobility provides a good approximation to the dynamicresponse.

In the mean field approximation, the free energyfunctional for a diblock copolymer can be decomposedinto a sum of short-range and long-range terms [31, 32]

F{ψ} = FS{ψ}+ FL{ψ}. (2)

The short-range contribution FS has the form of a Lan-dau free energy and can be expressed as

FS{ψ} =∫dr{W (ψ) + 1

2D (∇ψ)2}, (3)

where W (ψ) represents the mixing free energy ofthe homogeneous blend of disconnected A and B ho-mopolymers, the term containing the gradient repre-sents the free energy penalty generated by the spatialvariations of ψ (interfacial energy), and D is a pa-rameter related to the Kuhn segment length a0 andthe number fraction of A monomers per chain f as:D = b

2

48f(1−f) [33].The free energy W (ψ) has the form of a non-

symmetrical double well

W (ψ) = −12

[τ −a(1− 2f)2]ψ2 + v3ψ3 +

u

4ψ4. (4)

Here, the parameter τ is related to temperature bymeans of the Flory-Huggins parameter χ through [34,35]

τ = 8 f (1− f) ρ0 χ−2 s(f)

f (1− f)N, (5)

where ρ0 is the monomer density, N = NA + NB isthe total number of monomers in the diblock copoly-mer chain, s(f) is a constant of order one [31], v =ν(1 − 2f) and b = 9[2Na0(1−f)]2 . Here, a, ν, and u arephenomenological constants derived by Leibler usingthe random phase approximation [31, 33].

The Flory-Huggins interaction parameter χmeasuresthe incompatibility between the two monomer unitsand depends on the absolute temperature T as χ =

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Figure 3: Phase separation process after 4.5× 104 time stepsfor a sphere-forming diblock copolymer monolayer quenchedwithin the spinodal region. Top and bottom panels correspondto τr = τ/τs − 1 = 0.34 and τr = 0.025, respectively. Inorder to describe the structure of domains developed by theminority phase, here we employ a contour plot at a fixed valueof the order parameter (ψ = −0.22).

A + B/T , where A and B are phenomenological con-stants [1]. Thus, note that here τ increases as tempera-ture decreases.

The term FL{ψ}, representing the long-range con-tribution that accounts for the connectivity betweenblocks, can be expressed as [32]

FL{ψ} =b

2

∫dr1

∫dr2G(r1 − r2)ψ(r1)ψ(r2),

(6)where G is the solution of∇2G(r) = −δ(r).

Then, using Eqs. (2)–(6), Eq. (1) takes the form

1

M

∂ψ

∂t= ∇2(f̃(ψ)−D∇2ψ)− bψ, (7)

where f̃(ψ) is given by

f̃(ψ) = −[τ − a (1− 2 f)2]ψ + v ψ2 + uψ3. (8)

Equation (7) is actually a coarse-grained phase fieldmodel. In recent years, such models have been usedto study a variety of systems under different condi-tions, including shear and other external fields, curva-ture and patterned substrates, and pattern formation in3D [20,33,36–40]. In this work, the free energy param-eters were selected to capture the desired symmetry andsegregation strength of the block copolymer hexagonalphase [20, 34, 35]. More details about the equilibriumphase diagram and the different spinodals can be foundelsewhere [34, 35, 41–44].

i. Numerical methods

In this work, Eq. (7) was numerically solved under con-finement between impenetrable walls that represent theinterfaces with the air and the silicon wafer. To de-scribe the confinement interactions we consider an ex-ternal field that couples with the two components ofthe block copolymer system (see details in Ref. [45]).The film thickness was fixed at the monolayer value.We employed periodic boundary conditions along the(x,y)-axis. The evolution equation, Eq. (7), was solvedthrough the cell dynamics method, which has been ex-tensively used in non-equilibrium studies of this kind ofsystem [35]. One of the main advantages of this modelis that it is efficient over the time scales involved in thedynamics of defect annihilation, and is thus appropriateto describe the dynamics of coarsening (see Ref. [46]for an extensive and detailed analysis of the cell dynam-ics method and a comparison with the Cahn–Hilliard–Cook model). Another remarkable advantage of thismodel is its matricial nature (continuum discretization),which allows the implementation of a transparent paral-lelization into a GPU code. Here, we solved this modelusing a dual buffering scheme (Ping-Pong technique)and an optimized use of the different GPU memories[47, 48].

We study how the process of pattern formation varieswith the parameter related to the temperature τ , whilekeeping the rest of the parameters fixed at the followingvalues: M = 1.0,D = 0.1, a = 1.5, v = 2.3, u = 0.38and b = 0.01. The initial homogeneous (disordered)state is simulated by a random noise distribution of theorder parameter.

Figure 3 shows two snapshots of typical pattern con-figurations obtained through simulations of monolay-ers, at different conditions of annealing when an ini-tially disordered system is quenched into the spinodalregion (τs < τ < τODT).

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To obtain a better comparison with the experimen-tal AFM phase images shown in Fig. 1, rather thanconsidering a contour plot to describe the individualPS spheres, in Fig. 4 we integrate the order parameterdescribing the density fluctuations along the directionperpendicular to the substrate. In this way, the inter-faces are smoothed out, facilitating the comparison withAFM. Note the qualitative agreement between experi-mental data (Fig. 1) and the numerical results shown inFig. 4 for systems at different annealing conditions.

ii. Linear Instability Analysis

Due to the strong degree of confinement in the mono-layer, in general the formation of ordered structures andpatterns can be well approximated as a 2D process. Inthis section we consider temperatures in the vicinity ofthe order–disorder transition (T . TODT) and study thepattern evolution of a modulated structure in the planez = 0, whose order parameter profile can be describedby the following sum of Fourier modes

ψ(r, t) =∑

k

ψk exp (ik · r + λ t), (9)

where ψk is the Fourier coefficient at t = 0 (we con-sider that k is restricted to the thin film’s middle planeand neglect the spatial variation of ψ along the directionperpendicular to the thin film surfaces).

The stability of the solution ψ = 0 (high temperaturephase) for a system quenched into an unstable state canbe studied by linearizing Eq. (7) around ψ = 0 [49,50].Substituting Eq. (9) into the linearized Eq. (7), it canbe easily shown that the amplification factor λ satisfies

λ(k) = −Dk4 + [τ − a (1− 2 f)2] k2 − b, (10)

where k = ‖k‖. The spinodal/nucleation and growthtransition line can be calculated as the lowest value ofτ for which an extended mode can grow, i.e., λ > 0 forsome k

τs = 2√D b+ a (1− 2 f)2. (11)

Thus, for τ < τs, no extended mode can be amplifiedand the state ψ = 0 is a stable or metastable configura-tion. If τ > τs, some modes grow exponentially withtime. These growing modes are constrained accordingto

k21 ≤ k2 ≤ k22, (12)

Figure 4: Two-dimensional pattern configurations for the 3Dsimulation data shown in Fig. 3. Observe the similaritiesbetween these patterns and those in the experimental systemshown in Fig. 1.

where

k21 =Γ−√

Γ2 − 4D b2D

,

k22 =Γ +√

Γ2 − 4D b2D

,

(13)

with Γ = τ −a (1−2 f)2. Moreover, the most unstable

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mode (where λ is maximal) is

ks =

√Γ

2D. (14)

If τ → τs, this has the form ks → (b/D)1/4, and is theonly mode which is unstable.

Note that, in general, the spinodal temperature doesnot coincide with the order–disorder transition temper-ature, TODT. If a homogeneous system is quenched be-tween these temperatures, it can remain in a metastablestate, which can relax only by nucleation and growth.For block copolymers the two temperatures coincideonly for symmetric lamellar morphologies, f = 12 [1].

Figure 5 shows the range of unstable modes as givenby Eq. (12). Note that the distribution of unstablemodes is not symmetrically distributed around ks, thewave vector at the spinodal. This skewness becomesprogressively larger as the depth of quench increases.Consequently, for systems quenched in the neighbor-hood of the spinodal, one can expect a very stronglength scale selectivity and a strong free energy penaltyfor elastic distortions which shift the system from theoptimum ks. On the other hand, deep quenches stabi-lize the presence of different elastic distortions and thus,more disordered patterns can be expected. In addition,due to the skewness of the distribution, phase-separatedsystems generated at deep quenches are characterizedby an average wave vector k > ks.

Figure 5: Range of unstable modes as a function of the re-duced temperature. Inset: k2/k1 as a function of the reducedtemperature. Note that the unstable modes are not evenly dis-tributed around ks.

It is known that the linear instability analysis de-scribes only the initial stage (short times) of the spin-odal decomposition mechanism. As time proceeds, thedynamics become highly nonlinear and higher-orderwave numbers emerge [51–53]. We will discuss theseother stages of spinodal decomposition in the next sec-tion.

IV. Results

i. Spinodal Decomposition

Spinodal decomposition is the process of relaxation ofthermodynamically unstable states. Early studies onspinodal decomposition date back to the 1960s, withthe pioneering works of Cahn and Hilliard [53], Hillert[54], and Lifshiftz and Slyozov [55]. Such studies weremainly focused on the mechanism of phase formationand macroscopic phase separation in solid binary alloysand fluid binary mixtures.

At present, it is well known that spinodal decom-position in binary mixtures has three different regimes[56]. Initially, some modulations present in the homo-geneous phase grow exponentially with time, follow-ing the early linear evolution dynamics. With time, thenonlinear coupling between these growing modes slowstheir growth. In this second stage, the pattern has well-defined interfaces delimiting domains of the differentstable phases. At longer times, the average domain sizeincreases in order to reduce the total interfacial area.The asymptotic long-time state consists of two macro-scopic domains, one for each phase.

In block copolymers, most of the studies of pat-tern formation and ordering investigate the later stageof spinodal decomposition (the coarsening stage), fo-cusing on the kinetic exponents of evolution, andcomparison with other related systems, like the self-assembled structures observed in Rayleigh-Benard con-vection [17, 18, 46, 56–60]. Here, we consider the earlystages of spinodal decomposition in sphere-formingblock copolymer systems under confinement in mono-layers. We have observed two leading factors that con-trol the degree of order during annealing below the spin-odal. For τ > τs the system is nanophase-separated butdisordered and the kinetics of ordering are completelyinhibited, while for τ & τs the strong length scale se-lectivity observed in Fig. 5 leads to well-defined hexag-onal patterns with a lower density of defects and fasterordering kinetics. The differences in mode selectivityas a function of τ can be visualized in Fig. 3, where it

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can be appreciated that deep quenches lead to patternswith poorly defined symmetry.

In order to explore the dynamics and to facilitatecomparison with the experimental results, here we havecomputed the circularly averaged structure factor forboth experiments and simulations. The circularly av-eraged structure factor is defined as

Sk = 〈ψ̃(k) ψ̃∗(k)〉, (15)

where ψ̃(k) is the Fourier transform of the order param-eter.

For the experimental data, we have calculated Skthrough the phase fields obtained by imaging the blockcopolymer with AFM. Although the density maps ob-tained by AFM are correlated with the local elasticityof the system, rather than with density fluctuations, theyadequately describe the main pattern features: symme-try, defects, and degrees of local and long-range order.

Figure 6 shows Sk for systems quenched and an-nealed at different conditions, as obtained from the sim-ulations. Here, if the high-temperature phase is deeplyquenched into the spinodal region (τr = τ/τs − 1 =0.2), the length-scale selectivity imposed by the ra-dius of gyration of the molecule leads to a nanophase-separated system with poorly defined symmetry. Theinsets in Fig. 6 show the 2D scattering function for thissystem under two different conditions. The broad halo

Figure 6: Circularly averaged scattering function Sk fromsimulations, at different depths of quench and annealingtimes. The insets show the 2D patterns of Sk. The higher-order peaks, located at

√3kmax,

√4kmax and

√7kmax, are

the signature of hexagonal order.

of intensity is consistent with a poorly defined sym-metry and liquid-like order. In this case, there is notonly one mode which can grow, but rather a continuousrange of modes delimited by an annulus determined byk1 ≤ k ≤ k2 (see Fig. 5). As the temperature is lowand a wide spectrum of modes is stable, the kineticsof coarsening are frozen. Observe in Fig. 6 that evenafter 3 × 105 time steps, Sk remains unaffected whenτr = 0.2. Similar results were found by Yokojimaand Shiwa, and by Sagui and Desai, in a related sys-tem [61–63]. By including thermal fluctuations it hasbeen shown that the system can move towards equilib-rium via defect annihilation and grain growth [17, 64].

When the same system is subjected to a shallowquench, τr = 3 × 10−4, Sk shows the typical fea-tures of hexagonal patterns, with a sharper main peak atkmax and well-defined higher-order peaks at

√3kmax,√

4kmax and√

7kmax. The rings of nearly uniform in-tensity are also a signature of isotropy. However, in thiscase the isotropy emerges as a consequence of the poly-crystalline structure and not a liquid-like order, as inthe case of systems deeply quenched below the spin-odal temperature TS.

The dominant features of the scattering functionduring spinodal decomposition are in good qualitativeagreement with the experimental data shown in Fig. 7.In the experimental system, the early patterns are char-

Figure 7: Circularly averaged scattering function Sk for theexperimental data at two different annealing times. Annealingtemperature: T=333 K. Note that as annealing time increases,the main peak sharpens, shifts towards lower values of k, anddevelops a second-order peak at

√3kmax, characteristic of a

hexagonal pattern.

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acterized by a strong length-scale selectivity, but short-range order (see also Fig. 1). As the annealing time in-creases, there is a shift of the main peak in Sk towardslower values of k. In addition, the main peak sharpens,and a weak higher-order peak located at

√3kmax can be

clearly identified. These features are in good agreementwith the results of Figs. 1 and 2, which show increasingorder with annealing time.

Although there is qualitative agreement between ex-periments and simulations, it must be emphasized thatthe model employed here cannot capture the detaileddynamics of the system as a function of temperature.In block copolymers, the chain mobility depends on aneffective monomeric friction coefficient that is stronglydependent on temperature, the glass transition and/orcrystallization temperatures of the individual blocks,the degree of segregation between blocks, the symme-try of the self-assembled structure, and the degree ofentanglement. Although some of these properties canbe qualitatively captured through an effective mobilitycoefficient [M in Eq. (1)], at present it is not quantita-tively clear how these factors affect the dynamics.

ii. Nucleation and growth

When the high-temperature phase is quenched to tem-peratures slightly above the spinodal (quenches withτ . τs), all the modes decrease exponentially in timeand the order parameter goes to zero across the wholesystem. That is, the initially randomly distributed or-der parameter ψ(r), characterized by 〈ψ(r)〉 = 0 and〈ψ(r)ψ(r’)〉 6= 0, dies off. However, in the same rangeof temperatures, an initial crystalline hexagonal patternis completely stable, indicating a region of metastabil-ity τs > τ > τODT, where τODT denotes the limit ofmetastability.

In addition, within this metastable region, a poly-crystalline structure can improve its degree of ordervia the annihilation of defects, with a similar mecha-nism to the one observed for quenches only slightly be-low the spinodal. Figure 6 also shows Sk for a systemquenched into this metastable region (τs > τ > τODT,τr = −5 × 10−2 < 0). Note that at the same time ofannealing (3×105 timesteps), as compared with the sys-tem quenched below the spinodal (τr = 3×10−4 > 0),this system shows an improved order, as the main peaksharpens and higher-order peaks become better defined.

The most distinctive feature of the metastable regionis that only the nuclei or grains above a critical size cangrow to develop the equilibrium phase, while smaller

Figure 8: Critical nucleus sizeRc vs. the reduced temperatureτ/τODT− 1. Here a is the lattice constant. Note that Rc ∼ afor τ ∼ τs.

nuclei collapse due to the surface free energy. Accord-ing to the classical picture of nucleation and growth in2D, the variation in the free energy ∆F due to the for-mation of a nucleus of radius R can be expressed as[21, 22]

∆F = 2σπR− π|∆F0|R2, (16)

where σ represents the line tension and ∆F0 ∝(τODT − τ) is the difference in the local free ener-gies of the high-temperature (disordered) and the low-temperature (ordered) phases, which drives the transi-tion. Due to the competition between these surface andvolume contributions, only those nuclei whose size ex-ceeds a critical valueRc can propagate to form the equi-librium phase.

In order to obtain the dependence of the critical sizefor nucleation Rc on the depth of quench, we explorethe stability of crystalline nuclei of different sizes.

To study the stability of a crystal seed, we changeτ back and forth around the critical value to obtaina defect-free crystal patch. The partial melting andrecrystallization involved in this process removes de-fects and long-wavelength elastic distortions and yieldsa crystal seed that is then used as an initial conditionin the numerical solution of Eq. (1). Figure 8 showsthe critical size for nucleation Rc as a function of τ .Around the order–disorder temperature τODT we foundthat the critical size for grain growth Rc diverges asRc ∼ 1/(τ/τODT−1), in agreement with classical the-

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Figure 9: Time evolution of a crystalline domain as a functionof temperature for a system quenched to τODT < τ < τs.Here the left panel corresponds to a hexagonal grain of initialsize R0/a = 4.6, where a is the lattice constant. The rightpanels correspond to a grain that collapses [top, τ < τc(R)]or propagates [bottom, τ > τc(R)].

ories of nucleation and growth [22]. Note also that asthe spinodal temperature is approached, the critical sizefor nucleation becomes on the order of the lattice con-stant a. This result indicates that the process of spinodaldecomposition could be inhibited, as the nucleation andgrowth process precedes spinodal decomposition [22].

Figure 9 shows typical snapshots of the evolution ofnuclei as a function of temperature for quenches withinthe metastable region (τODT < τ < τs). For aninitial crystalline nucleus of a given size R, and withRc ∝ (τODT − τ)−1, if the temperature is lowered toa value where R is larger than Rc, grain growth occurs.Otherwise, the crystal collapses since line tension over-comes the bulk contribution.

iii. Melting

A perfect crystalline structure may get trapped in a su-perheated state, such that the system may remain or-dered when annealing above τODT. However, we ob-served that for polycrystalline structures, topologicaldefects trigger the phase transition, inhibiting super-heating and disordering the system when τ & τODT.Figure 10 shows a polycrystalline structure obtainedduring annealing at τr = 1.64 × 10−2. This figureshows the local orientation of the different hexagonalgrains and also the defect structure. In 2D crystals withhexagonal symmetry, the elementary defects, nameddisclinations, are the domains which have a number ofneighbors different from six (a five-coordinated domainis a positive disclination and a seven-coordinated do-main is a negative disclination). In general, these de-

Figure 10: Orientational map θ(r) (left panel) and defectstructure (right panel) of the hexagonal lattice determined bya Delaunay triangulation. Spheres with seven neighbors areindicated with a green dot, those with five neighbors in red.Dislocations are formed by a pair of 5-7 disclinations sepa-rated by one lattice constant, and are indicated by a connect-ing yellow line segment (see also the inset with a dislocation).A comparison between the defect structure and local orienta-tion indicates that grain boundaries are decorated with lineararrays of dislocations. The bottom of the figure for θ(r) showsthe color scale used to indicate the local orientation. Here thepatterns correspond to τr = 1.64 × 10−2 and 9 × 104 timesteps.

fects are too energetic to be found in isolation: theycouple in pairs to form dipoles, also known as dislo-cations (see Fig. 10). Note in Fig. 10 that disloca-tions are piled up in linear arrangements, decorating thegrain boundaries. Figure 11 shows two snapshots of thesystem during annealing at temperatures above TODT.Note in this figure the strong correlation between theliquid-like phase (ψ = 0) and the position of the ini-tial grain boundaries. In addition, it can also be ob-served that melting does not start uniformly at all grainboundaries, but depends on the orientational mismatchbetween neighboring crystals.

The average distance between the dislocations lo-cated along a grain boundary depends on the orien-tation mismatch between neighboring grains, ∆θ, asdds ∝ a/∆θ−1, where a is the lattice constant. Notethat the largest possible mismatch in orientation be-tween crystals in a hexagonal pattern is 30 Degrees.Compared with small-angle grain boundaries (SAGB),large-angle grain boundaries (LAGB, those for whichthe orientational mismatch is in the range 10–30 De-grees), are more energetic and contain a larger num-ber of dislocations per unit length. Consequently, thephase transition is initially triggered at LAGB, wherethe crystal is more disordered. We also observe that the

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kinetics of disordering are affected by the initial defectstructure. Note in Fig. 12 that a polycrystal with smallcrystal size, and thus a higher content of dislocations,leads to a larger fraction of the liquid phase at the sameannealing time.

Figure 11: Upon increasing the temperature above TODT, thecrystals of Fig. 10 melt first at LAGB (left panel). As timeproceeds, the liquid phase (ψ = 0) propagates through thesystem. τ = 0.938τs. Left panel: t = 1.5 × 103 time steps.Right panel: t = 5× 104 time steps.

Figure 12: Melting of the crystal phase for a system contain-ing a relatively large initial content of topological defects(leftpanel). Although the temperature and annealing time are thesame as in the left panel of Fig. 11, due to the larger initialcontent of defects, there is a larger fraction of liquid phasehere. τ = 0.938τs, t = 1.5× 103 time steps.

V. Conclusion

In this work, we studied the different mechanismsleading to the self-assembly of sphere-forming blockcopolymer thin films. Using a Cahn–Hilliard free en-ergy model valid for diblock copolymers, we followedthe evolution of the system with numerical simulationsfor different temperatures, and presented a unified viewof the phase transition process. The results are con-sistent with a first-order transition. Below a spinodal

temperature TS, the initially homogeneous disorderedstate relaxes towards equilibrium by spinodal decom-position, by the spontaneous growth of characteristicmodes. Above this temperature, the system can onlyself-assemble into an ordered phase by nucleation andgrowth (TS < T < TODT). Above the order–disordertemperature TODT a well-ordered system can in princi-ple remain superheated without disordering. However,dislocations and grain boundaries trigger the melting ofthe structure, such that defective states cannot be super-heated above TODT.

For systems quenched into the metastable region, wefound that the critical size for nucleation and growth di-verges as the disorder–order temperature is approached,following the law: Rc ∼ 1/(τ/τODT−1), in agreementwith classical theories of nucleation and growth.

Acknowledgments

We gratefully acknowledge financial support from theNational Science Foundation MRSEC Program throughthe Princeton Center for Complex Materials (DMR-1420541), Universidad Nacional del Sur and the Na-tional Research Council of Argentina (CONICET). ThePS-PEP block copolymer was hydrogenated by JohnSebastian.

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pattern formation mechanisms in sphere-forming diblock ......two-dimensional thin ﬁlm systems is...

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