High-Throughput Characterization of Pattern Formation in ... › files › 2017 › 09 › 851825.pdfcation.6–9Many of the techniques used to prepare libraries for inorganic materials

High-Throughput Characterization of Pattern Formation inSymmetric Diblock Copolymer Films

ARCHIE P. SMITH, JACK F. DOUGLAS, J. CARSON MEREDITH,* ERIC J. AMIS, ALAMGIR KARIM

Polymers Division, National Institute of Standards and Technology, Mail Stop 8542, Gaithersburg, Maryland 20899

Received 17 November 2000; revised 10 May 2001; accepted 21 May 2001Published online xx Month 2001

ABSTRACT: Surface-pattern formation in thin block copolymer films was investigatedby utilizing a high-throughput methodology to validate the combinatorial measurementapproach and to demonstrate the value of the combinatorial method for scientificinvestigation. We constructed measurement libraries from images of subregions ofblock copolymer films having gradients in film thickness and a range of molecular mass,M. A single gradient film covers a wide range of film morphologies and containsinformation equivalent to a large number of measurements of films having a fixedthickness, h. Notably, the scale of the surface patterns is generally much larger thanthe molecular dimensions so that the interpretation of the patterns is more subtle thanordering in bulk block copolymer materials, and there is no predictive theory of thistype of surface-pattern formation. We observed a succession of surface patterns thatrepeat across the film with increasing h [extended smooth regions, regions containingcircular islands, labyrinthine (“spinodal”) patterns, holes, and smooth regions again].The extended smooth regions and the labyrinthine patterns appear to be novel featuresrevealed by our combinatorial study, and these patterns occurred as bands of h thatwere quantized by integral multiples of the bulk lamellar period, Lo. The magnitude ofthe height gradient influenced the width of the bands, and the smooth regions occupiedan increasing fraction of the film-surface area with an increasing film gradient. Theaverage size of the spinodal patterns, l, was found to scale as l ; Lo

22.5 or l ; M21.65

and reached a limiting size at long annealing times. The hole and island features hada size comparable to l, and their size likewise decreased with increasing M. The smoothregions were attributed to an increase in the surface-chain density in the outer brush-like block copolymer layer with increasing h, and the scaling of l with M was inter-preted in terms of the increasing surface elasticity with M. © 2001 John Wiley & Sons, Inc.J Polym Sci Part B: Polym Phys 39: 2141–2158, 2001Keywords: combinatorial measurements; block copolymer; thin film; surface pattern

INTRODUCTION

Combinatorial analysis, in conjunction with high-throughput multivariate measurements, is a newand rapidly developing approach to materials sci-

ence research. This methodology is made possibleby the development of the computational re-sources needed to store and analyze the largeamounts of information that such studies natu-rally generate. This method involves the creationof large “material libraries” and high-throughputscreening techniques that allow the efficient ex-ploration of the multiparameter space governingcomplex physical phenomena. This approach canbe utilized to rapidly identify regions of parame-ter space where particularly interesting phenom-

* Present address: School of Chemical Engineering, Geor-gia Institute of Technology, Atlanta, GA 30332

Correspondence to: A. Karim (E-mail: [email protected])Journal of Polymer Science: Part B: Polymer Physics, Vol. 39, 2141–2158 (2001)© 2001 John Wiley & Sons, Inc.

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ena occur, and the method is also helpful in es-tablishing basic parameter trends that assist ingaining an understanding of the origin of the ob-served phenomena. This information is not onlyuseful in the practical design of new materials1,2

but can contribute greatly in the development andtesting of theoretical models.

Combinatorial methodologies have revolution-ized the pharmaceutical industry because of theircapacity to bring new drugs to the market in atimely, cost-effective manner.3,4 The applicationof combinatorial principles to materials sciencecharacterization likewise promises to influencethe future direction of research and development.Application of the combinatorial process is al-ready occurring in the discovery of new catalystsand in a wide variety of inorganic materials in-vestigations utilizing semiconducting, phospho-ric, and superconducting materials.5

The characterization of the physical propertiesof polymeric materials is one area where combi-natorial methods are just beginning to find appli-cation.6–9 Many of the techniques used to preparelibraries for inorganic materials are inapplicableto organic and polymeric materials; thus, newstrategies must be developed that are suitable tothese materials. As an initial application of com-binatorial methods to organic materials, a tech-nique has been developed to create polymer thinfilms with continuous gradients in film thicknessand composition.7–9 In the present article, theutility of this technique is demonstrated by inves-tigating the well-studied phenomenon of surface-pattern formation in block copolymer films. Inaddition to reproducing former results, our inves-tigation has uncovered important new aspects ofblock copolymer film-pattern formation thatshould be helpful in the development of a theoryof this type of surface-pattern formation.

Symmetric diblock copolymers are composed ofpolymer components of nearly equal molecularmass joined at a covalent chemical junction. Pre-vious studies10–33 have shown that in orderedthin films of these materials, the polymer-surfaceinteractions induce the formation of lamellae par-allel to the substrate surface that have a thick-ness nearly equal to the bulk equilibrium lamel-lae spacing, Lo. When one block segregates toboth the substrate and air interfaces, smoothfilms are formed with a total thickness hs 5 mLo,where m is an integer. Conversely, smooth filmsof thickness hs 5 (m 1 1

2) Lo are observed whenone block segregates to the substrate and theother to the air interface. In the case where film

thickness h deviates from hs, the excess materialforms an incomplete surface lamella in the formof islands or holes with height Lo with the natureand percentage of surface coverage of these fea-tures dependent on the deviation of h from hs.This dependence of the surface morphology on his exploited to demonstrate the efficacy of combi-natorial methods applied to polymer character-ization by creating thickness gradient thin filmsof symmetric polystyrene-b-poly(methyl methac-rylate) (PS-b-PMMA) copolymers of different mo-lecular masses M. Of primary interest in thepresent study is the location and nature of tran-sitions in the surface-pattern morphology, the ef-fect of the gradient slope on the morphology, andfactors that govern the lateral size of the surfacepatterns.

LIBRARY CREATION ANDCHARACTERIZATION

To create thin-film thickness gradient libraries, aspecially designed automated solution flow coaterhas been developed. This flow coater, shown inFigure 1(a), is based on a computer-controlledlinear motion stage, which moves at variablespeeds to create the film-thickness gradient. Thesubstrates utilized for these gradients are Si wa-fers (10 cm diameter, Polishing Corp. of Ameri-ca34) that have been cleaned with the standard“Piranha-etch” technique to form a SiOx/SiOHsurface layer.35 After cleaning, the Si wafer wassecured to the robotic stage, and a 30-mm-wideknife edge was placed ' 300 mm above the sub-strate surface at an approximate 5° angle withrespect to the substrate. Approximately 50 mL ofpolymer solution (mass fraction 1–10%) were sub-sequently placed under the knife edge, and therobotic stage was driven with constant accelera-tion spreading the solution. Solvent evaporationoccurs in a matter of seconds, producing a thinfilm with thickness related to the stage velocity(i.e., higher velocity produces thicker films). Typ-ical thin films are 25–35 mm in length and havethickness gradients of 40–50 nm, although thesevalues can be varied. Control of the initial filmthickness and gradient slope is achieved by mod-ifying the solution concentration, solution vol-ume, solvent, knife-edge height, and stage accel-eration. Films as thin as 2 nm and as thick as10 mm with gradient ranges as small as 10 nmand as large as 5 mm have been produced withthis methodology.

2142 SMITH ET AL.

The film thickness of the gradient library sam-ples was measured by a Filmetrics F20 ultravio-let–visible (UV–vis) interferometer with a0.5-mm spot diameter and a standard uncertaintyof 61 nm for a 500 nm film [shown in Fig. 1(b)].This interferometer is connected to an automatedtranslation stage where measurements are ac-quired at intervals across the sample (usuallyhaving 2–3 mm spacing). These thickness mea-surements are used to create a two-dimensionallibrary of the film thickness, which can be used tocreate a thickness contour map of the sample.Figure 2(a) shows a thickness map for a filmprepared from a solution with a mass fraction

Figure 1. Diagrams of the basic elements of high-throughput polymer film measurements: (a) automatedflow coater used to make films having a gradient inthickness and (b) UV–vis interferometry, optical mi-croscopy, and AFM used to characterize the h-gradientfilms.

Figure 2. (a) Plot of the film thickness for a gradientlibrary. The thickness measurements were performedby spot interferometry and acquired on a 2.5-mm grid.(b) Composite optical micrograph of a gradient filmlibrary showing typical film features. Thicker regionsare represented by darker colors. Dark diamonds areartifacts of the CCD digitization.

SYMMETRIC DIBLOCK COPOLYMER FILMS 2143

of 4% polymer and a stage acceleration of0.175 mm/s2. In this example, thickness measure-ments were acquired with 2.5-mm spacing acrossa library approximately 30 mm in length with athickness varying between 35 and 75 nm. Flowbehavior near the free boundary of the knife edgeproduces a slight thickening of the film (' 5 nm)within ' 2 mm of each edge, and this region isexcluded from consideration. The inner region,however, has a nearly uniform thickness gradientthat can be utilized for quantitative experimentalinvestigations. For example, the relative thick-ness variation in the center of the gradient (or-thogonal to the thickness gradient direction)shown in Figure 2(a) is less than 4% at any posi-tion along the gradient. The viscoelastic nature ofthe polymer film ensures that the gradients re-main stable for small spatial scales and long time-scales.36

Optical microscopy [OM, depicted in Fig. 1(b)]was also utilized to characterize the gradient li-braries. A Kodak Megaplus camera (1024 3 1024,8-bit pixels) was connected to a Nikon Optiphot 2microscope operating in the reflection mode. Themicroscope’s standard sample stage was replacedwith a motorized two-dimensional translationstage possessing a 5-cm range of motion and a0.5-mm resolution. A computer synchronouslycontrols the translation stages and acquires digi-tal images from the CCD while scanning acrossthe sample. These micrographs can be assembledtogether to form a composite image of the entirelibrary, and an example of this is found in Figure2(b). In this figure, low-magnification optical mi-crographs (253) are acquired from a film castwith a mass fraction of 2% polymer and assem-bled to show the film structure of the entire li-brary. The contrast is provided by the opticalinterference within the thin film, and the darkershades represent thicker areas for films of thisthickness range. The gradient library creationmethodology described in this work allows forcombinatorial samples possessing up to 100 dis-

tinct state points (Dh 5 0.5 nm) per library. Thislarge amount of data greatly expands the poten-tial for scientific investigation and practical char-acterization of polymer-film properties.

SYMMETRIC DIBLOCK COPOLYMERMORPHOLOGY

To demonstrate the advantages of using the com-binatorial libraries previously described forstudying block copolymers, thin-film gradient li-braries of the symmetric diblock copolymer poly-(styrene-b-methyl methacrylate) (PS-b-PMMA,Polymer Source Inc.) were created. The molecularcharacteristics37 of the PS-b-PMMA copolymers(as provided by the supplier) are given in Table I.Unreacted homopolymer contamination is esti-mated to be less than 4% based on gel permeationchromatographic and NMR data provided by thesupplier. Solutions with mass fractions of 2–5% intoluene were used to create gradient film librarieson Si wafers with SiOx/SiOH surface layers. Up tofour libraries having different initial h, gradientslopes, and molecular masses were placed on asingle 100 mm Si wafer to extend the h range andeliminate experimental processing variability. In-terferometric measurements, as previously de-scribed, were performed on the as-cast films, andthe libraries subsequently annealed at 170 °C forup to 96 h under vacuum. After annealing, themorphology of the resultant films was character-ized by OM and atomic force microscopy (AFM) ona Digital Nanoscope Dimension 3100.

Previous work12–14,18 has determined thatwhen thin films of PS-b-PMMA are placed on Siwith a SiOx/SiOH surface layer and annealed, thePMMA block preferentially segregates to the sub-strate, whereas the PS block segregates to the airinterface. This behavior results in smooth filmswith hs 5 (m 1 1

2) Lo for this copolymer andincomplete surface lamellae with various surfacepatterns when the film deviates from this thick-

Table I. Molecular Characteristics and Lamella Thickness (Lo) for the Block Copolymers in This Study (ErrorsGiven Are the Standard Uncertainty)

LabelMn S Block37

(g/mol)Mn MMA Block

(g/mol) Mw/Mn

Lo Calculated14

(nm)Lo Measured

(nm)

26k 12,800 12,900 1.05 17.1 17.8 6 0.851k 25,300 25,900 1.06 26.8 30.2 6 0.8104k 50,000 54,000 1.04 42.4 42.3 6 1.4

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ness. These observations were acquired fromstudying a number of individual samples, eachhaving a discrete h such that the sample setspanned the h range of interest. By utilizing thethickness gradient libraries used here, the com-plete, continuous addition of successive lamellaeon a single sample can be observed directly, asshown in Figure 3. True-color OM micrographsobtained from a continuous 26k PS-b-PMMA gra-dient film annealed for 6 h at 170 °C are assem-bled together to form a montage showing the ad-dition of four lamellae. This micrograph is ob-tained near the gradient film edge to compressthe lateral scale, and h varies nonlinearly from' (45 to 117) nm over a distance of ' 1.1 mm.Labels denote h at the approximate locations of hsfor m 5 2, 3, 4, 5, and 6. The island and holemorphologies observed in this micrograph areconsistent with previous observations from sam-ples having a fixed h.10,15,16,25,26 Identification ofthe labyrinthine morphology separating the is-land and hole patterns and observation of thewidth of the bands where the films remain smoothare apparently new results that that have notbeen reported previously.

A more detailed picture of the morphology evo-lution with increasing h is presented in Figure 4.This figure shows OM micrographs obtained froma continuous 51k PS-b-PMMA gradient library

annealed for 30 h at 170 °C, placed together toform a composite image of the film-surface mor-phology. These micrographs were obtained from asection of a gradient ' 2.5 mm in length havingan initial h range of ' 70–100 nm (or equiva-lently 2.5–3.5Lo for this molecular mass). Con-trast is provided by the thin-film optical interfer-ence, and the darker gray corresponds to the3.5Lo layer thickness. As h increases above 70 nm,discrete islands initially form [shown in theenlargement in Fig. 4(a)] and adopt a circularappearance as a result of surface-energy minimi-zation, as previously observed.16,17 Increasing hfurther causes the islands to grow in size andconsolidate until a labyrinthine pattern of islandsand holes forms [Fig. 4(b)] near certain criticalfilm thicknesses, hc. This morphology resemblesspinodal decomposition in polymer blends38 andspinodal dewetting39 patterns in other polymericsystems so that we term these “spinodal pat-terns.” A “phase-separation” model of block copoly-mer pattern formation has been proposed40–42 withthe variation of film height relative to a criticalvalue playing the role of an order parameter, andthis model is considered subsequently. With a fur-ther increase in h, the morphology undergoes a“phase inversion” where the islands become contin-uous and form the next lamella (h 5 3.5Lo) withlarge irregular shaped holes down to the previous

Figure 3. True color optical micrograph of a continuous 26k PS-b-PMMA gradientfilm. Film was annealed for 6 h at 170 °C, and the image shows the addition of foursuccessive lamellae to the block copolymer film with increasing thickness. Labelsindicate the approximate position of hs 5 (m 1 1

2) Lo, where m 5 2–6. The lower

section is a continuation of the upper section.

SYMMETRIC DIBLOCK COPOLYMER FILMS 2145

lamella “surface” (h 5 2.5Lo). The size and numberof holes decrease as h increases further [Fig. 4(c)]until the holes eventually disappear, leaving asmooth film when h 5 3.5Lo ('100 nm for this M).The morphologies observed in this figure are nextinvestigated at higher resolution using AFM.

Figure 5 depicts AFM micrographs obtainedfrom 51k PS-b-PMMA gradient films annealed for6 h at 170 °C, where the brighter colors corre-spond to higher topography. The micrographswere acquired from two gradient libraries with hbetween 35 and 70 nm and 65 and 110 nm, re-spectively. This figure illustrates the morpholog-ical variation with h observed in Figure 4 but at a

higher magnification. The h values given in thefigure correspond to h ' 1.5–3.5Lo and are de-termined from interferometer measurements onthe unannealed sample with an estimated error of62 nm. These micrographs show a smooth sur-face morphology (h 5 42 nm), followed by theformation of circular islands (h 5 46 nm, 51 nm)that become larger and irregular in shape (h5 54 nm). The islands eventually grow largeenough to join together to form the next continu-ous lamella with large, irregular holes down tothe previous lamella (h 5 61 nm, 63 nm). Theseholes become smaller and more circular (h 5 68 nm)followed by an increase in size accompanied by

Figure 4. Composite optical micrograph of a continuous 51k PS-b-PMMA film an-nealed for 30 h at 170 °C. The unannealed film thickness range is roughly 70–100 nm(or 2.5–3.5Lo) with the new lamella appearing dark relative to the underlying layer.The enlargements show (a) islands of the new lamella on the surface of the existinglamella, (b) a labyrinthine island/hole pattern (“spinodal pattern”), and (c) holes in thenew lamella down to the underlying lamella. The bright white areas correspond to filmdewetting.

2146 SMITH ET AL.

a decrease in number (70 nm) as h is furtherincreased. The holes eventually disappear to re-form a smooth surface (h 5 72 nm). This se-quence of surface-pattern evolution is then re-peated as h increases from 72 to 104 nm. In ad-dition to illustrating the detailed nature of thetransition from island to spinodal to hole-patternformation, these micrographs show that the mor-phological pattern repeats for each additional la-mella added to the film. Although this sequence ofpattern formation is only shown for two lamellae,it is repeatedly observed in films with h up to6.5Lo. AFM is also used to measure Lo from mul-tiple points across the films, and these values aregiven in Table I along with the calculated stan-dard deviation. The measured Lo values agreewell with those calculated from the previouslydetermined14 empirical relation Lo } M0.66 (Ta-ble I), corresponding to the expected strong seg-regation scaling.43 This agreement, as well as thesimilarity in morphology between the combinato-rial samples and the uniform h samples observed

previously, provide an important consistencycheck for the continuous gradient methods uti-lized for the present experiment.

We next focus attention on aspects of the sur-face morphology observed in the gradient samplesthat have not been previously identified in uni-form h samples. The first feature of interest is thelabyrinthine island/hole morphology (“spinodalpattern”), shown in Figures 4(b) and 5 (83 nmmicrograph), and the critical film thickness, hc,where this morphology occurs. This morphologytype is observed for all annealing times andshould not be confused with transient patternsobserved at short times in a thickness regime thatlater evolves into islands or holes.17 Similar lab-yrinthine patterns have also been observed inblock copolymer films on patterned sub-strates.32,33 As seen in Figure 4, the area wherethe spinodal pattern exists is quite narrow rela-tive to the entire lamellar width, corresponding toa small film-thickness regime relative to Lo. Uti-lizing OM and AFM, the location of the spinodal

Figure 5. AFM micrographs showing the morphology variation with h. Bright areascorrespond to the higher topography of the 51k PS-b-PMMA samples annealed for 6 hat 170 °C. Labels indicate the unannealed film thickness with an estimated standarduncertainty of 62 nm. These micrographs show that successive lamellae form accordingto the same pattern-formation series (islands, spinodal patterns, holes, etc.).

SYMMETRIC DIBLOCK COPOLYMER FILMS 2147

pattern was determined to be at 0.41 6 0.03Loabove the smooth film thickness hs, or equiva-lently hc 5 [m 1 (0.91 6 0.03)]Lo. The value ofhc is invariant with respect to M within experi-mental uncertainty and stable under the anneal-ing conditions applied in the present work. Ofsignificance is the fact that the hc value is notexactly 0.5Lo above hs, such that the hole mor-phologies occupy a larger fraction of the lamellarmorphology relative to the island morphologies.This finding indicates that the holes are a moreenergetically favorable surface morphology thanthe islands, as previously suggested.17,26

The next feature of interest is the smooth re-gions between the island and hole-surface pat-terns shown in Figures 3 and 4. These regions arecentered about hs (within experimental error)and are defined by abrupt transitions separatingthe areas where islands and holes exist. Thesmooth regions account for a large percentage ofthe gradient film area and thus correspond to arange of h deviating significantly from hs, andthis fraction becomes increasingly large as the hgradient is increased. These smooth regions arehypothesized to arise from deformations in theouter block copolymer layer and were investi-gated experimentally. The hypothesis of chain ex-pansion and compression of the surface block co-polymer layer relative to Lo implies an h varia-tion across the smooth region that should bedetectable. The continuous gradient method uti-lized here allows the measurement of this h vari-ation in the smooth “transition” region illustratedin Figure 6. This figure shows a true-color OM

micrograph of a 104k PS-b-PMMA gradient filmwith the film thickness indicated. The sample wasannealed for 22 h at 170 °C, and the micrographwas obtained near the edge of the sample to com-press the lateral dimension of the smooth region.The orange features on the left of Figure 6 areholes, whereas the green/yellow features on theright are islands. The island and holes lack defi-nition in this micrograph because their size isnear the resolution limit of optical microscopy.The color of the smooth region between the holesand islands varies from purple to blue/green be-cause of interference changes caused by increas-ing h. This color change indicates an h variationacross the smooth region of ' 25 nm centeredabout 105 nm (hs for m 5 2) for this sample.

To further quantify the film-thickness changeacross these smooth regions, gradients were an-nealed for 30 h at 170 °C to obtain approximate“steady-state” (or at least very slowly varying)film patterns. Positions of the interface betweenthe holes and smooth areas as well as between thesmooth region and islands were recorded. Theunannealed h of these positions was then utilizedto determine the change in film thickness (Dh)across the entire smooth region. Because hs isfound to be near the smooth region center, Dh/ 2 isthe height deviation at the edges of the smoothfilm region relative to the smooth region center,hs. The values of Dh/ 2 obtained for each M inves-tigated are listed in Table II, along with thechange in this quantity relative to Lo, dh 5 (Dh/2)/Lo. The data in Table II show that dh is nearlyinvariant within standard uncertainty for the

Figure 6. True color optical micrograph across smooth band. The image is of a 104kPS-b-PMMA gradient film. The outer block copolymer lamella becomes increasinglystretched as the chain density increases with increasing h and the layer expands as aresult of increased interchain interactions.

2148 SMITH ET AL.

films studied, regardless of the block copolymerM. Furthermore, no statistically significant vari-ation in dh was found for h , hs and h . hs orwith an increase in the total number of lamellae,up to films having a thickness h ' 6.5Lo. This hindependence of dh directly supports our hypoth-esis that the chain deformation is confined to theouter block copolymer layer because dh wouldotherwise depend on the total number of lamellae.

We next examined the effect of the gradientslope, « [ Dh/Dw, where w is the width of theregion under consideration, on the morphology ofthe system. Although the gradients achievablewith the gradient flow coater are limited, we caninvestigate higher gradients by using a combina-tion of the cast gradients in the central regions ofthe film and the steeper, nonlinear gradientstructures located at the film edges. The influenceof « on the block copolymer surface morphology isdemonstrated by comparing Figures 3, 4, and 7.Figure 4 demonstrates that the island and holemorphologies are significantly wider than thesmooth surface areas for « ' 10 nm/mm. Thismorphology trend is also observed for slopes aslow as ' 1 nm/mm (not shown here) with a cor-responding decrease in the fraction of the smoothregions relative to the regions where the hole andisland morphologies are observed. Figure 3 showsthe morphology in a film where « ' 80 nm/mmand demonstrates the increase in the width of thesmooth region relative to the island and hole pat-terns with increasing «. Micrographs of films witheven higher « are shown in Figure 7, illustratingthe morphological effect of changing « in the high-gradient regime. These micrographs were ob-tained from 26k PS-b-PMMA films annealed for30 h at 170 °C. Figures 7(a,b) are optical mi-crographs of films with « of approximately (a)150 nm/mm (uniform across the sample) and(b) 500 nm/mm (left side) to 750 nm/mm (rightside). Figure 7(c) is an AFM micrograph of a film

with the steepest gradient, « ' 1800 nm/mm witha line scan of the micrograph indicating the film hchange. Notably, the step-like height variation isonly apparent in films having steep gradients. Forcompleteness, we note that « ' 1 nm/mm in Fig-ure 5 and « ' 180 nm/mm in Figure 6. We nextconsider the crossover between the low- and high-gradient film morphologies.

The micrographs shown in Figures 3, 4, and 7demonstrate that the smooth regions centeredabout hs become an increasingly large part of thetotal lamella morphology as « increases, until thelamellae eventually adopt a step-like morphology.To further investigate this behavior, the width ofthe smooth region, ws, and the width of the entirelamella, wL, as shown in Figure 7, were measuredas a function of « from samples annealed for 30 hat 170 °C. Figure 8 depicts a plot of the ratiowS/wL as a function of « for all three molecularmasses investigated. Inspection of this plot re-veals that this ratio goes to its maximum value ofunity (corresponding to adoption of the step-likemorphology) with increasing «, independent of M.Once a ratio of unity is attained, a further in-crease of « does not alter the step morphology, but

Table II. Measured Parameters for the SmoothRegions in the Block Copolymer Morphology as aFunction of M (Errors Given Are the StandardUncertainty)

Mn

(kg/mol) Dh/2 (nm) dh 5 (Dh/2)/Lo «c (nm/mm)

26k 2.1 6 0.7 0.12 6 0.03 720 6 5051k 4.2 6 0.5 0.14 6 0.01 2400 6 300104k 5.8 6 0.8 0.14 6 0.01 3300 6 400

Figure 7. Influence of « on surface-pattern formationof 26k PS-b-PMMA samples: (a) optical micrograph ofan area with « ' 150 nm/mm showing that the smootharea width (wS) is a larger fraction of the entire lamellamorphology width (wL) than in Figures 3 and 4, (b)optical micrograph of regions with « ' 500 nm/mm(left side) and « ' 750 nm/mm (right side) showinghow the morphology becomes steplike with increas-ing slope, and (c) AFM micrograph of a defect with« ' 1800 nm/mm with a line scan (denoted as the whiteline) demonstrating the steplike nature of the lamellaeabove the critical slope values.

SYMMETRIC DIBLOCK COPOLYMER FILMS 2149

the width of the step decreases. The value of «where the ratio wS/wL initially plateaus to unity,«c, increases with increasing M. The measured «cvalues (see Table II) correspond to the pointwhere islands or holes can no longer form as aresult of the steep gradient, and the step-like filmmorphology is then adopted. This behavior is dis-cussed in more detail subsequently.

The next aspect of the block copolymer filmsinvestigated is the effect of the copolymer molec-ular mass on the surface-pattern formation andevolution. Changes in M have previously beenfound13 to affect the equilibrium lamellar thick-ness Lo (as described previously) and the kineticsof lamellar formation and evolution. In addition,we observed an M dependence on the lateral (in-plane) size scale of the spinodal surface patterns,as demonstrated in Figure 9. AFM micrographsobtained from PS-b-PMMA gradient films withM’s of 26k (Figs. 9(a,d)], 51k [Figs. 9(b,e)], and104k [Figs. 9(c,f)] annealed at 170 °C for 6 h [Figs.9(a–c)] and 30 h [Figs. 9(d–f)] are shown. Theseimages were obtained at constant magnificationnear h ' hc where the spinodal patterns occur,and a line scan obtained from the area designatedby the white line is plotted below each micro-graph. The effects of M on the inplane surface-

pattern dimensions are visible for the samplesannealed for only 6 h [shown in Figs. 9(a–c)]. Thelateral pattern scale in the 26k, 51k, and 104kfilms become increasingly small with increasingM. Further annealing of these materials for 30 hproduces an increase in the size of the lateralpattern scale, but the significantly smaller mor-phology found for the high M samples is retained.

To quantify the M dependence of the surface-pattern sizes and evolution demonstrated in Fig-ure 9, two-dimensional fast-Fourier transforms ofmicrographs obtained from samples annealed upto 96 h were performed. The circular average ofthe transform data yielded a peak in the intensitydata at wavevector q*. A characteristic size, l,defined as l [ (q*)21, was also calculated. Figure10 shows a plot of this calculated l as a functionof annealing time for films of all three molecularmasses annealed up to 96 h. The plot shows thatthe differences in l because of M observed inFigure 9 are retained for long annealing times.Therefore, the lateral scale of the surface patternsalways decreases strongly with increasing M forthe times investigated. In addition, Figure 10 il-lustrates that the lateral size of the morphologyceases to evolve and becomes nearly stationary

Figure 8. Plot of the ratio wS/wL versus « for the 26k(F, solid line), 51k (■, dashed line), and 104k (}, dottedline) M samples showing how the relative width of thesmooth area increases with increasing gradient slope.The slope at which wS/wL first equals 1 increases withincreasing M. Lines serve as a guide to the eye, anderror bars are the standard uncertainty.

Figure 9. Molecular mass dependence of the blockcopolymer spinodal pattern size. AFM images: (a,d)26k, (b,e) 51k, and (c,f) 104k of symmetric PS-b-PMMAcopolymer films annealed for (a–c) 6 h and (d–f) 30 h at170 °C. The 26k (a,d) sample has the largest surfacefeatures, whereas the higher M samples have smallersurface features. Line scans from each micrograph (de-noted by white lines) demonstrate how the patternshoulders become more rounded, reducing the rough-ness of the sample.

2150 SMITH ET AL.

for all M values and times used. A similar cessa-tion of growth of block copolymer surface featureshas been previously reported26 where the size ofholes and islands remained almost invariant forabout 2 decades in annealing time. This cessationof growth of the surface pattern or “pinning” isindicative of viscoelastic restoring force in theblock copolymers, and this will be explored fur-ther subsequently. A future article will furtherexplore the kinetics of pattern evolution includingthe temperature dependence of the “pinned”value of l and the characteristic time t governingthe approach to this steady state.44

Another means of characterizing the effect ofM on the surface-pattern morphology is by exam-ining the roughness of the surface. The line pro-files given in Figure 9 demonstrate the effect of Mon the roughness. For the 6-h annealed data, the

low M sample [Fig. 9(a)] has relatively sharp cor-ners in the line scan, whereas the corners for thehigher M samples [Figs. 9(b,c)] are more rounded.With further annealing (30 h), the 54k sample[Fig. 9(e)] becomes sharper whereas the 104ksample [Fig. 9(f)] retains the rounding of the fea-tures. To quantify this, the root-mean-squareroughness, r, [defined as r 5 =(¥ Zi

2/N), whereZi is the difference in height from the mean valuefor each pixel, and N is the number of pixels] wascalculated. The r values calculated from samplesannealed for 6, 30, and 96 h are given in Table III.By normalizing r by Lo, the relative roughness ofthe different M samples can be compared, andthese values are also given in Table III (r/Lo

5 0.5 corresponds to a bicontinuous step functionprofile with two heights, h 5 0 and h 5 Lo). Thisratio shows that the high M sample has a lessrough surface and that the corners of the patternsare rounded relative to the lower M samples.Thus, increasing the molecular mass at this mod-est annealing time decreases r on the scale of Lo,again indicative of viscoelastic effects in the blockcopolymers. Previous measurements13 haveshown that the roughness of high M filmschanges slowly in time; thus, studies at very longtimes may be needed to confirm these roughnesseffects.

To further investigate the dependence of thesurface-pattern morphology on M, a plot of l as afunction of Lo for representative films annealed 6and 30 h is represented in Figure 11. We note thelarge scale of the surface patterns (l 5 0.5–8 mm)in comparison with Lo and even the entire filmthickness h.45 The l versus M data are nearlylinear in a log–log plot and yield the scaling rela-tion l (mm) } Lo

22.5 for all annealing times.46

Using relation previously discussed, l } M0.66,l has been found to scale with M according to therelation l (mm) } M21.65.47 The hole and islandpatterns that form for h Þ hc also remain stable(or at least evolve very slowly), and the average

Figure 10. Plot of the average feature size, l, versusannealing time, t, for the 26k (F, solid line), 51k (■,dashed line), and 104k (}, dotted line) M samplesshowing how the pattern size becomes nearly station-ary for long annealing times. The sample was annealedat 170 °C. The lines serve as guides for the eye, anderror bars correspond to the standard uncertainty.

Table III. Measured RMS Roughness Values for the AFM Micrographs in Figure 9 (Errors Given Are theStandard Uncertainty)

Mn (kg/mol) r (6 h) (nm)r (30 h)

(nm)r (96 h)

(nm) r/Lo (6 h) r/Lo (30 h) r/Lo (96 h)

26k 8.3 6 0.2 8.3 6 0.2 8.3 6 0.2 0.47 6 0.03 0.47 6 0.03 0.47 6 0.0351k 13.1 6 0.2 14.1 6 0.2 13.9 6 0.2 0.44 6 0.02 0.47 6 0.02 0.46 6 0.02104k 11.3 6 0.2 15.5 6 0.2 14.2 6 0.2 0.27 6 0.02 0.37 6 0.02 0.34 6 0.02

SYMMETRIC DIBLOCK COPOLYMER FILMS 2151

size of these morphological features likewise di-minishes strongly with increasing M.

Utilizing this dependence of l on M, we nowreturn to the observation in Figure 8 that higher Mfilms could sustain surface pattern formation witha higher «. Specifically, the critical gradient, «c,where the width of the smooth region wS becomescomparable to wL, increases strongly with increas-ing M. This phenomenon is understandable giventhat the existence of a surface pattern requires thatthe difference wL 2 wS must be larger than theminimal surface-pattern size. The sizes of the is-lands, holes, and spinodal patterns all have a com-parable order of magnitude in our measurements(for fixed M) so that l calculated previously can beapproximately identified with this minimal patternsize. Thus «c can be expected to vary in some in-verse relation to l. Figure 12 shows a plot of «cversus l, and the limited data suggest a nearlylinear relation, «c ' 2470l 1 3600 where the slopeis notably negative.46 Thus, the parameters l and «care indeed strongly correlated. Alternatively, a plot(not shown) of «c versus M indicates the approxi-mate scaling «c ; 2aM21.5 1 3500, where a is apositive constant (' 1.2 3 1010). This relation sug-gests a tendency of «c to saturate at a constant valuefor high M.

DISCUSSION

The aforementioned observations raise questionsabout the nature of surface-pattern formation inblock copolymer films that have not been previ-ously addressed. At this point, we summarize ourtentative views about the physical origin of thesurface features observed in block copolymer thinfilms. These arguments provide a working modelfor understanding our data and guiding furthermeasurements.

Brush Interpretation of Smooth Region

Previous studies48 on the formation of polymerbrush layers grafted in a poor solvent and dried inair have found a progression from island to laby-rinthine to hole-surface patterns and finallysmooth polymer films with an increase in thesurface density, s, of the grafted polymer chains.The outer layer of block copolymer films exposedto air can likewise be considered a brushlike layerwith thickness, hl, where the extended smoothregions about hs of the block copolymer films cor-respond to h ranges where s is relatively high.For films with h , hs, s should be smaller thanthat found in a “complete” lamella (h 5 hs). In-terchain interactions must be relatively weak inthis regime, and the chain conformations within

Figure 11. Size of labyrinthine pattern size, l, versusbulk lamellar thickness, Lo, with standard uncertain-ties displayed. Data obtained for samples annealed for6 h (F, solid line) and 30 h (■, dashed line) (Fig. 9) showa decrease in the lateral pattern size with M. Solidlines are power-law fits of the data yielding l (mm)' 7200Lo

22.5.

Figure 12. Plot of «c versus l for the molecular masspolymers used here with the standard uncertaintiesgiven. The value of « where the island and hole mor-phology disappear is linearly dependent on the averagefeature size of the spinodal pattern.

2152 SMITH ET AL.

the “chain-deficient” block copolymer regime isexpected to correspond to weak segregation scal-ing (hl ; M1/ 2). In contrast, an increase of habove hs should lead to an increase in s and afurther stretching of the outer chain layer into anextended “brushlike” form. We then expect thelayer height of the outer block copolymer for h. hs to have a strong segregation scaling (hl; aM0.66), but with a prefactor a different14 fromthe scaling relation for the bulk lamellar thick-ness Lo ; aeM

0.66.The stretching of the outer block copolymer

layer between hs and the swollen edge of thesmooth band equals (a 2 ae) Lo, and this stretch-ing in comparison to the equilibrium layer thick-ness should be constant (a 2 ae)/ae in a fullydeveloped scaling limit (hl ; aM0.66), where a isindependent of M.49 Previously, it was found thatthe height change relative to Lo measured fromthe center of the smooth region, dh 5 (Dh/ 2)/Lo,is nearly constant (dh ' 0.12–0.14) in our mea-surements, in accord with these scaling argu-ments. The existence of chain stretching for hl. Lo is supported by previous results from Rus-sell et al.20 who directly observed significantswelling in the outer block copolymer layer basedon a (single) neutron reflection measurement on afilm of constant thickness where h exceeded hs by13% Lo. The stretching of the outer layer was alsoobserved by de Jeu et al.50 but on a smaller scale(' 2% Lo). The continuous change in the color ofthe smooth film region OM micrographs shown inFigure 6 provides evidence for a gradient in filmthickness in these smooth film regions. This hgradient is attributed to a continuous transitionfrom compressed to stretched chains within theouter block copolymer layer. We next focused onthe topographic patterns observed for the remain-der of the block copolymer thickness ranges.

Phase-Separation Model of Block CopolymerSurface-Pattern Formation

A number of researchers have suggested that theordering of block copolymer films can be describedby two-dimensional “phase separation”16,24,40–42

based on a phenomenological similarity of thesurface-pattern growth in the block copolymerfilms to phase separation in fluid mixtures. In thismodel, the film height relative to a critical valuedefines an order parameter for the phase transi-tion. The counterpart order parameter in fluid-phase separation is the composition relative tothe critical composition.51 Although this picture

remains highly conjectural, it does have implica-tions relevant to our measurements. This phase-separation model implies the formation of “spi-nodal decomposition” patterns for h near somecritical height and formation of holes and islandsstructures for h greater or less than this criticalheight.40 The block copolymer holes and islandscorrespond to droplet formation in off-criticalfluid mixtures. This predicted behavior is remark-ably consistent with our observations where hc isidentified with the critical height in the phase-transition model. In this type of picture, the ob-served deviation of hc from (m 1 1) Lo for thiscopolymer seems to correspond to an asymmetricphase boundary (critical relative compositionÞ 0.5). The phase-separation model of block co-polymer pattern formation further suggests thatthe surface patterns in block copolymer filmsshould morphologically resemble phase separa-tion of “ultrathin” polymer blend films (wherephase separation occurs quasi-two-dimensionallywithin the plane of the film). In studies of theseblends, a progression from islands to spinodal tohole patterns with varying compositions is indeedobserved,48 and the morphology appears similarin form to the block copolymer surface patterns.We take this observation as encouraging evidencein qualitative support of the phase-separationmodel of surface-pattern formation in block copol-ymer films.

Surface Elasticity and Pattern Pinning

Our measurements on the block copolymer pat-tern formation have shown that the size of thepatterns “pins” at long timescales (i.e., patternsstop growing). The effect has also been observedin phase-separating blend films,52,53 although theorigin of the pinning may be quite different inthese liquid-like systems. Surface elasticity hasbeen suggested to play a large role in determiningthe scale and form of surface patterns in phaseseparating fluids when long-range interactionsare present.54,55 In addition, this type of model(phase separation plus surface elasticity) hasbeen invoked to rationalize pattern formation inthe lipid bilayer films arising in cell mem-branes.54,55 The phase-separation model of blockcopolymer pattern formation discussed previouslyalong with a generalization to account for filmelasticity then has potential for guiding under-standing of the observed M dependence of thespinodal pattern scale, l.

SYMMETRIC DIBLOCK COPOLYMER FILMS 2153

To explore this interpretation of the surface-pattern formation, preliminary calculations ofphase separation with surface elasticity were per-formed based on a Cahn–Hillard model of phaseseparation and the Helfrich model of surface elas-ticity.56 These calculations indicate that the pres-ence of a finite-surface elastic constant, k, can“pin” the scale of the phase separation over ap-preciable timescales. The surface elasticity of thefilm inhibits long wavelength phase separationmuch like the forming of the chemical junction inthe block copolymer leading to an elastic restoringforce acting against macroscopic phase separa-tion. These calculations also indicate that if k isnot sufficiently large, the observed pinning istransient in nature, and growth will continue atlong times. Because a similar “pinning” in theblock copolymer surface patterns is observed, sur-face elasticity is then implicated as a potentiallyimportant parameter in the pattern formation ofblock copolymer films.

Inclusion of surface elasticity as a significantparameter in block copolymer pattern formationdirectly leads to specific expected behavior thatcan be studied. First, an increase in the surfaceelasticity is anticipated to decrease the surfaceroughness because of the larger energetic cost ofchain displacements away from the average layerheight. This is observed as rounding of the high Mplateau features in Figure 9 and the decreasingvalue of r/Lo in Table III.57 As the surface elas-ticity increases with increasing M, it is more dif-ficult to form the sharp plateau boundaries be-cause of the increased bending modulus. In addi-tion, a larger bending modulus should result in alarger areal compressibility modulus, which in asimple mechanical model of a membrane is pro-portional to the bending modulus.58,59 This in-creased areal compressibility modulus leads us toexpect that the formation of shallow surface pat-terns over large areas of the film surface requir-ing large-scale inplane displacements should like-wise decrease with increasing surface elasticity.The increased elasticity acts as a restoring forceto inhibit molecular migration over the lengthscales necessary to form large-scale patterns.This behavior is indeed observed in Figures 9, 10,and 11 for the block copolymer patterns. (Large-scale labyrinthine patterns are also observed inthin films of (neutral) gels exposed to a poor sol-vent, and the inplane size of the patterns de-creases with increasing crosslink density similarto the outer block copolymer layer observedhere.60) To further understand the M dependence

of the block copolymer surface patterns, we nextinvestigate the M dependence of k and the depen-dence of the surface pattern scale l on k.

The surface rigidity, k, of block copolymer lay-ers has been theoretically predicted61 to increasestrongly with M. We propose that the decrease inl with an increase in M arises from the increasingenergetic cost of the surface deformation as aresult of the increasing elasticity that accompa-nies surface-pattern formation. Specifically, thepreceding observations and discussions suggestan inverse relation between l and k, and we as-sume an inverse power relation, l ; k2b, where 0, b , 1. This relation is consistent with theobservations that the patterns become large inlow M films where the surface elasticity should besmaller. For strong segregation block copolymerslayers61 and dense low molecular mass surfactantlayers,62 k is predicted to scale as k ; M23. Byapplying this relation to the current block copol-ymer study, the measurements of the previoussection where l ; M21.65 yield the preliminaryindication that b is near 1

2, that is, b ' 0.55.47

Direct measurements of k for the block copoly-mers used in the present work would be helpful inbetter establishing the relation between l and k,however.62 The increase of «c with increasing Min Figure 8 is also interpreted to arise from theincreasing elasticity of the block copolymer layerwith increasing M.57 Films having a higher M aremore elastic allowing for a more gradual bendingof the free boundary on the scale of the surfacepattern l, which in turn allows the film to supportlarger h gradients.

The importance of surface elasticity in blockcopolymer pattern formation can be further dem-onstrated by additional observations. A change ofthe block copolymer monomer structure from therelatively rigid (PS-b-PMMA) to the more flexiblePS-b-poly-(n-butyl methacrylate) has been ob-served42 to give a large increase of the block co-polymer pattern scale. These data are interpretedas an increase in pattern scale as a result of areduced surface rigidity. Temperature measure-ments are also informative about the surface-pat-tern scale and origin. In a companion study,44 adramatic increase of the block copolymer patternsize is observed upon approaching the estimatedfilm-ordering transition temperature. This obser-vation is consistent with our hypothesis that sur-face elasticity (bending and compressional) con-trols pattern size because fluctuation effects nearthe ordering temperature should diminish k andthus increase the size of the surface patterns.63,64

2154 SMITH ET AL.

Compositional fluctuations should also have theadditional effect of slowing down the rate atwhich the surface patterns form near the surface-ordering transition (termed “phase-coherencetransition” by Mansky et al.42), and this effect hasalso been observed.63,64 These observationsstrongly suggest that surface elasticity and com-positional fluctuations associated with the sur-face-ordering transition have a large influence onsurface-pattern formation in thin block copolymerfilms. These effects should be investigated morethoroughly, both experimentally and theoreti-cally.

Future Considerations

As a final point, recent work65 on the modeling ofpattern formation in block copolymer films pre-dicts the formation of spiral patterns and otherfeatures not seen in our measurements. However,this modeling assumes the presence of strong hy-drodynamic interactions in idealized two-dimen-sional block copolymer films. The importance ofhydrodynamic interactions in ordered films withsurface elasticity seems doubtful, but this modelhas potential interest in the disordered regimewhere the block copolymer film is more fluidlikein its viscoelastic properties. It would then beinteresting to make films in the disordered re-gime to check if qualitatively different block co-polymer patterns are then observed. In the disor-dered state, the surface viscosity66,67 should playa large role in the rate at which the surface pat-terns grow. Recent measurements,68 of the sur-face viscosity, hs, of homopolymer films indicate ascaling relation hs ; M1.3, suggesting that therewould be a strong reduction in the rate of growthof the patterns with increasing M. Block copoly-mer pattern formation in the disordered regimewould be an interesting topic for a future inves-tigation by combinatorial methods.

CONCLUSIONS

High-throughput methods are used to examinethe properties of block copolymer thin films. Ourmethod involves creating controlled, continuousgradients in film thickness using a specially de-signed automated flow coater and performinghigh-throughput characterization on these poly-mer films as a function of thickness, thicknessgradient, and molecular weight. Automated anal-yses of the data generated by these libraries pro-

vides greatly enhanced experimental efficiency,providing samples with a large number of inde-pendent state points. The film gradient libraryinformation is validated through comparison toprevious observations on surface-pattern forma-tion in symmetric diblock copolymer films. Thisillustrative example shows the efficiency of thecombinatorial method in exploring a complexphysical phenomenon.

A single thickness gradient library reproducesthe entire range of surface pattern formation, andnew phenomena are identified in the process (for-mation of smooth films for extended thicknessranges and labyrinthine surface-pattern forma-tion separating the familiar island and hole pat-terns). We attribute the thickness ranges wherethe film remains smooth to an increase of thesurface chain density of the brushlike outer blockcopolymer layer with an increase in the filmthickness. In this view, the smooth block copoly-mer films observed correspond to the last stage ofbrush formation where the brush layer is contin-uous, but increasingly filled as the film thicknessincreases. The scaling relation for the stretchedchains in the overfilled layers (h $ hs) is believedto still obey the strong segregation scaling rela-tion (hl ; aM2/3) but with an altered limitingprefactor a at the edge of the band where themaximum extension is presumed to occur. Thisargument indicates that the extent of layerstretching across a band should be a constantfraction of Lo, in accord with our measurements.Specifically, the height change across the smoothregions, normalized by Lo, was found to be nearlyindependent of molecular mass and the magni-tude of the surface gradient (provided the gradi-ent was small enough to geometrically accommo-date surface patterns). Once the gradient slopebecame sufficiently large, these smooth regionsincreasingly dominated the morphology untilabove a critical slope a step-like morphology wasadopted. This critical slope is found to vary lin-early with pattern dimension.

The size and evolution of the block copolymersurface patterns at long times were also investi-gated. The average sizes of these patterns arefound to become nearly stationary at long anneal-ing times with the larger M samples have a sig-nificantly smaller size. The average size of thespinodal block copolymer pattern was found toscale as l ; Lo

22.5 (or equivalently l ; M21.65).This effect is attributed to a strong increase in thesurface elasticity of the outer block copolymerlayer with M and the increased energetic cost of

SYMMETRIC DIBLOCK COPOLYMER FILMS 2155

large-scale surface deformation accompanyingsurface pattern formation. These observationsprovide a confirmation of the value of combinato-rial methodology to rapidly explore complex phe-nomena dependent on many relevant variables,validating the technique against previous mea-surements and showing the power of the method-ology to identify new and important phenomenaeven in mature fields of polymer science.

The authors thank Dr. M. VanLandingham for use ofand assistance with the AFM. The authors also ac-knowledge the helpful discussions with Drs. P. Greenand I. Szleifer about block copolymer film morphologyand surface elasticity, respectively.

APPENDIX: INTERPRETATION OF CRITICALLAYER THICKNESS, HC

The formation of island, hole, and complex clusterpatterns is a ubiquitous phenomenon in the for-mation of self-assembled monolayers,69–71 thegrowth of grafted polymer layers,48 and the sur-face structure of growing or melting crystals.72

Although the growth of the outer block copolymerlayer is phenomonologically similar to the forma-tion of physi-sorbed grafted polymer layers, wecan also draw an analogy to the growth of crystalswhere hole and island features form near thecrystal surface even near equilibrium and wherethe interaction between the ordered layers withinthe crystal and the crystal surface is important.72

We may consider ordered block copolymer films asvarieties of smectic phases similar to those ob-served in liquid crystals. The viewpoint of sur-face-pattern formations as a variety of “surfaceroughening” in block copolymer crystals has someinteresting implications. The equilibrium fea-tures of the surface-pattern formation should bevery dependent on T, and we can expect the sur-face-pattern properties to be described by a phasetransition in some cases.72 The type of surface-phase transition (second or infinite order) or evenwhether a thermodynamic phase transition existsat all depends on the energetic asymmetry offorming island and hole structures in the outercrystal layer.72 The nature of the order–disordersurface “roughening transition” is somewhat spe-cific to the molecular structure of the crystallizingspecies.

As previously mentioned, the observed devia-tion of hc from (m 1 1) Lo for this copolymerseems to correspond to an asymmetric phase

boundary (critical relative composition Þ 0.5).This asymmetry arises from a breaking of theexchange symmetry between the fluid particles influid mixtures and the breaking of the “particlehole” symmetry in the spin model of crystalroughening.72 In the case of crystals, the symme-try breaking is associated with an energetic in-equivalence between the formation of island andhole-surface patterns having the same shape andsize.72 As previously mentioned, this symmetrybreaking has important ramifications for the na-ture of the surface-roughening transition. An en-ergetic asymmetry between the formation of is-lands and holes has been predicted to act like anapplied field. Thus, the surface-roughening tran-sition near the ordering transition becomes“rounded” (no phase transition exits), leading to atendency of the crystal surface to graduallychange from smooth to rough over a large temper-ature range.72

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