Coarsegrained model quantitatively reproduces experimental observations of wrinkles on patterned substrates
We are primarily interested in universal phenomena, rather than graphene-specific behavior. We therefore use a linearized model that captures the essential physics of a sheet interacting with a substrate, in which the sheet is a triangular lattice of springs that provide a stretching cost, with pairs of plaquettes joined by hinges that have a harmonic energy favouring a flat sheet, which provides a bending cost. Both stretching and bending energies must be accounted for in order to model accurate wrinkle profiles17, in particular, without bending cost, sharp tent-like structures form rather than smooth wrinkles. Similar models have previously proved successful for studying thin sheets8,28. The sheet is pushed towards to the substrate except at short distances where it is repelled. To reduce spurious edge effects and allow for effects of friction between the sheet and substrate, particles near the sheet edge are coupled to their original lateral positions with harmonic springs, to constrain their lateral (but not vertical) motion.
We verify our model through comparison with experimental studies of graphene sheets deposited on substrates decorated with nanoparticles of diameter ~10 nm, as published recently17. We simulate 1 × 1 μm2 regions of monolayer and 5-layer graphene, with particles of radius 8 nm placed at positions determined by digitizing the atomic force micrograph of a system with nanoparticle density of 90 μm−217. As seen in Fig. 1, the agreement between monolayer simulation and experiment is good, with both systems displaying similar networks of long narrow wrinkles connecting particles. We note that agreement is only expected in the centre of the region simulated, because the experimental image is a sample of a larger region and wrinkles near the edges are affected by particles not shown in the limited field of view.
Importantly, our linearized simulations exhibit avoiding pairs of wrinkles, an example of which is highlighted in Fig. 1(d,e). The pairs are stable and their appearance is robust against changes of simulation details such as substrate interactions. In addition, like Yamamoto et al., we find increasing the thickness of the graphene sheet hinders its conformation to the substrate (compare our Fig. 1 with Fig. 6 of Ref. 17). This is especially apparent in regions where close-spaced particles are positioned on the vertices of a polygon, as in the region indicated in Fig. 1(a,b). As highlighted in Fig. 1(f,g), the sheet in this region “pops up” when its thickness (and hence ratio of bending to stretching costs) increases. In the rest of this paper, we study these two characteristic behaviours in greater detail.
Beyond visual comparisons, we compare simulation and experiment statistically, by measuring the distribution of the number of wrinkles emanating from each particle for the graphene monolayers, as plotted in Fig. 1(h). The agreement is good, although the simulations have a slightly larger mean number of wrinkles per particle (2.48 for simulation, vs 2.15 for experiment). We attribute this to lower noise levels in simulation data allowing more wrinkles to be detected.
Features reminiscent of avoiding pairs of wrinkles can be observed also on macroscopic lengthscales, as demonstrated in Fig. 1(i,j), which shows a geomembrane used to line a landfill. That these features are not graphene-specific further supports our choice of coarse grained, rather than atomistic, model.
Coalescence of growing wrinkles depends on their initial separation
We first study the growth and interactions of wrinkles, with a focus on avoiding pairs. In order to characterize these pairs, we require a controlled method for creating and growing them. Inspired by studies of interacting cracks grown from a pair of notches21, we study propagating and interacting wrinkles using simulations in which an initially flat sheet of dimensions 600 × 600 nm2 is subject to upward pulling in “wrinkle nuclei” regions. The tips of the nuclei are separated by distances X and Y, as defined in Fig. 2(a), which are our control parameters. Because the nuclei are long and narrow and the sheet has high stretching cost, the upward driving primarily causes the wrinkle to propagate forward, with minimal lateral broadening, as shown in Fig. 2(a). We run each simulation until a stable configuration is attained, which occurs when the applied upward forces are balanced by stresses in the sheet.
Whereas an isolated wrinkle propagates directly forward, parallel to its nucleus, nearby wrinkles interact and either join or form an avoiding pair. To uncover the role of interactions, we fix driving conditions and make a systematic study of the effect of wrinkle nucleus separation, which we characterize by aspect ratio X/Y and distance . As shown in Fig. 2(b), when X/Y > 1, the joining condition is simply a distance threshold of ~30 nm. However, for X/Y < 1, wrinkle merging becomes more favourable and the minimum distance required for the formation of an avoiding pair increases rapidly.
Wrinkle morphology is controlled by stress fields
In addition to the conditions for avoiding pair formation, we also characterize how the morphology of such pairs depends on wrinkle nuclei separation. Typical wrinkle shapes are shown in Fig. 2(c), for fixed X/Y = 1. For small R, wrinkles are directed toward each other, but for larger R they are initially repelled before being attracted. To quantitatively describe the wrinkle shapes, we focus on two characteristics which are indicated in Fig. 2(a): wrinkle extension, which is the distance from base to tip, and aspect ratio μ/ν. For fixed X/Y, ξ depends on R and is maximized at a distance ~230 nm, as shown in Fig. 2(d). As shown in Fig. 2(e), the maximum extension decreases with increasing X/Y, and avoiding pairs of wrinkles formed by nearly head-on nuclei are substantially longer than wrinkles formed from nuclei that are separated in the X direction. Remarkably, for nucleus separations , X/Y has little effect on the aspect ratio μ/ν of wrinkles, which decreases approximately linearly with R.
To understand how wrinkle interactions affect their shape, we characterize the stresses induced by wrinkles, using the trace of the stress tensor Tr(σ) = σxx + σyy + σzz. Unlike, e.g, tensile fracture, where hoop stress σθθ determines crack propagation29, the symmetry of wrinkle propagation makes the isotropic stress the relevant quantity. We first consider a single wrinkle. As shown in Fig. 3, stresses emanating from the tip of a wrinkle are compressive (Tr(σ) < 0). The wrinkle grows via buckling of the sheet, and deformation occurs primarily as bending rather than stretching which is highly unfavourable for graphene.
As indicated in Fig. 3(b), the stress distribution displays an angular dependence and compressive stresses are largest along a line parallel to the wrinkle. The stress decays rapidly, faster than a power law. We have also calculated the stresses generated by an idealized wrinkle in an infinite sheet under the assumption of zero in-plane deformation by imposing a wrinkle height profile and numerically calculating the stresses using the Föppl-von Kármán equations30, with Youngs modulus E = 2.4 TPa (Ref. 31, 32). As shown in Fig. 3(c,d), the idealized wrinkle also generates a stress field with an angular dependence. Far from the wrinkle tip, the stresses decay as a power law with exponent 1 parallel and 2 perpendicular to the wrinkle axis.
When two wrinkles are grown in proximity, their behaviour is determined by the superposition of their stress fields. A video showing the evolution of stresses for a growing pair of wrinkles is provided as Supplementary Information. As shown in Fig. 4, wrinkles propagate along the steepest stress gradient in front of them. Pairs of wrinkles propagate in a spiral around the centre of the line joining their tips. As seen in Fig. 4(a,b), wrinkle tips initially propagate outwards slightly, and the high-stress central region expands. As the tips pass each other, their motion changes to an inwards propagation, as seen in Fig. 4(c), and the high-stress region contracts. The wrinkles continue to propagate around the centre of the line joining their tips, and come to a stop in the configuration seen in Fig. 4(d). This configuration is stabilized by the balance between applied forces and stresses in the sheet. The deformations required to merge two wrinkles are highly unfavourable and avoiding pairs are hard to remove. As seen in Fig. 2(d), when wrinkle nuclei are close, wrinkles are short because they are unable to grow far before blocking each other. On the other hand, for large nuclei distances, stresses due to interactions are reduced and wrinkle propagation is decreased accordingly. The existence of an optimal tip distance for wrinkle growth is due to the competition between these factors.
Friction suppresses wrinkle formation
We also test how wrinkle propagation is affected by friction. We take two complementary approaches to modelling frictional effects, which give remarkably similar results, despite one approach being much simpler than the other. In agreement with intuition, we find that friction hampers the propagation of wrinkles.
The first approach is to introduce quenched disorder on the substrate to induce pinning and friction. Substrate characteristics are given in Table S1. Figure 5(a–c) shows how this disorder affects propagation of wrinkles. In the limit of low friction, wrinkles propagate a long distance without the wrinkle nuclei growing substantially. As friction is increased, wrinkle growth is reduced and the response to the upward pulling of the nuclei is dominated by detachment of the nuclei.
We have also tested the effective friction used in our simulations presented in this paper, by varying the strength of harmonic coupling of the edges of the sheet with their original lateral positions, for a sheet in a Lennard-Jones potential without random pinning. The stronger the coupling, the larger the effective friction, as the cost of sliding of the sheet increases. In the limit of very stiff coupling, for the sheet to buckle out of plane it must stretch, a mechanism which is otherwise avoided due to large energetic costs. As seen in Fig. 5(d–f), increasing the coupling is qualitatively the same as increasing substrate pinning. The coupling k = 0.0001 eV/Å2, shown in Fig. 5(e), corresponds to the value used in the other simulations reported in this paper.
Wrinkles across length scales: simulations of multilayers
An advantage of coarse grained simulations is that sheet thickness enters as a change in energetics, without a change in the number of particles in the system, so that multilayer sheets can readily be simulated. Bending costs grow faster than stretching costs with sheet thickness: bending energy scales as n3 and stretching as n, where n is the number of layers. Therefore, the minimum feature size of a sheet's morphology increases with its thickness. We examine the consequences of this through simulations of multilayer graphene sheets, which serve as a bridge between micro and macro length scales. We consider two scenarios: interacting growing wrinkles and deposition on patterned substrates.
Figure 6 illustrates how wrinkle configurations change as sheet thickness is increased, up to 10 layers. We remark on how the wrinkles in the 10-layer sheet are reminiscent of those seen in geomembranes (see Fig. 1(i,j)). In general, wrinkles become shorter and wider with increasing sheet thickness, and in thick sheets wrinkle nuclei broaden considerably during wrinkle growth. Stresses in the sheet are reduced as its thickness increases, as shown in Fig. 6(d–f). Similar results are also seen in simulations of deposition of mono- and multilayers on substrates decorated with four particles (Fig. S5).
In general (if not always), films are about some theme or topic, and fictional films involve characters and types of characters. There are several complications in researching this topic, in part because sometimes there's a distinction between how the Library describes films vs how it describes books about those films ... and sometimes there doesn't seem to be. Please read the explanations below carefully.
On this page:
Research into what films are about often overlaps with other topics in Film Studies. The following pages may also be relevant:
Finding Films on Particular Topics, Themes or Characters
Quick and Dirty Indexes of Topics in Films
These three sources are very useful, but they are of course limited by their publication dates and the number of films they cover.
American Film Scripts Online
AFSO identifies screenplays' subject(s) and genre, and lets you search by these criteria. Note that AFSO only includes American films. See Screenplays/Filmscripts for more information.
Encyclopedia of film themes, settings and series. Richard B. Armstrong and Mary Willems Armstrong. Jefferson, N.C.; London: McFarland, 2001.
SML Reference, PN1997.8 +A75 2001
This encyclopedia is especially useful for obtaining examples of films with themes like amnesia, deafness, genies, ice skating, mirrors, photographers, sewers, tattoos, and windmills.
The Ultimate Movie Thesaurus. Christopher Case. New York: Henry Holt and Co., 1996.
SML Reference, PN1998 C323X 1996
Consists of two major sections. The first is a list of films by title, providing a one-sentence synopsis, its genre(s), and its topics or themes. The second section lists the genres and topics, and the films that provide examples.
Using Orbis to Find Films on Particular Topics, Themes or Characters
|From the Crawford Collection, Yale University Library Manuscripts and Archives|
Films on a specific character, character type or group are typically in the form [Character]--Drama or [Character] films. The "--Drama" subheading is the proper one, but the reason there may be two forms is that often these are considered a genre (although sometimes the subject heading is applied by mistake). See also Genres, Styles & Categories.
- In many cases you will need to search both forms.
- "Drama" covers fiction movies, plays and TV shows, including fictionalized accounts of real people or events. To some extent you can limit to films by using the strategies described on the Films & Videos page.
- These headings may be used for both fiction and non-fiction (documentary) films. For instance, there may be a non-fiction film about the rise of the vampire movie genre.
Films on a theme or event usually have a subject heading taking the form [Topic]--Drama, but there are also related genres (e.g., War films).
Films that are part of a series (or implied series) may receive [Series name] films. A series is usually treated as though it's a genre.
NOTE: Catalogers are not required to apply any headings of this type to films. You may need to supplement searches for films on a particular topic by looking at filmographies and similar sources.
Works About a Topic, Theme or Character as Represented in Films
Critical works, including documentaries, about a topic as seen in motion pictures. (Sometimes these headings are also applied to fiction films. This is usually an error, but it's appropriate for a film like Hollywood Shuffle, which is in fact about African-American actors in Hollywood.)
- Note that nearly all of these have the form [Topic] in motion pictures, but that's not true for war movies and topics treated as genres. See also Genres, Styles, Categories and Series. The phrase "in motion pictures" often covers films that aren't specifically about some topic etc., but happens to include it.
- The lists below are far from exhaustive, but they should indicate the many possibilities.
Selected Reference Resources
General Sources on Film Themes and Topics
Bibliographies concerning the movies, race, ethnicity, and gender (University of California--Berkeley)
Includes bibliographies on film pertrayals of African-American, Arabs, Asians, Latinos/Chicanos, Jews, Indigenous peoples, Italian-Americans, Gay, lesbian, and transgendered People, and Immigrants and outsiders.
Sources for Specific Film Themes and Topics
Be sure to look at the Bibliographies & Filmographies page for more resources on how particular social groups are portrayed!
The American Indian in film. Hilger, Michael. Metuchen, N.J.: Scarecrow Press, 1986.
SML, Reference PN1995.9 I48 H54 1986
The bent lens: a world guide to gay & lesbian film. Edited by Claire Jackson and Peter Tapp. St Kilda, Vic.: Australian Catalogue Company, 1997.
SML Reference, PN1995.9 H55 B38 1997
Blacks in black and white: a source book on Black films. Sampson, Henry T. Edition: 2nd ed. Metuchen, N.J.: Scarecrow Press, c1995.
SML, Reference PN1995.9 N4 S2 1995
The Columbia companion to American history on film: how the movies have portrayed the American past. Ed. Peter C. Rollins. New York: Columbia University Press, 2003.
SML Reference, PN1995.9 U64 C65 2003
The encyclopedia of ethnic groups in Hollywood. James Robert Parish. Facts on File, Inc. : New York, NY, 2003.
SML Reference, PN1995.9 M56 E63 2003
Encyclopedic dictionary of women in early American films, 1895-1930. Denise Lowe. New York: Haworth Press, 2005.
SML Reference, PN1998.2 .L686X 2005
African Americans in motion pictures.
Steven Spielberg Jewish Film Archive
The Spielberg Archive is the world's largest collection of Jewish/Israeli documentary material. This includes extensive holdings covering Israel, both before and after the establishment of the State, the Holocaust and Jewish communities abroad throughout the century. The archive is digitizing films and making them accessible online.