Usually several deformation mechanisms interact to accommodate
plastic deformation. Quantifying the contribution of each to the total
strain is necessary to bridge the gaps from observations of microstructures, to
geomechanical descriptions, to extrapolating from laboratory
data to field observations. Here, we describe the experimental and computational
techniques involved in microscale strain mapping (MSSM), which allows
strain produced during high-pressure, high-temperature deformation
experiments to be tracked with high resolution. MSSM relies on the analysis
of the relative displacement of initially regularly spaced markers after
deformation. We present two lithography techniques used to pattern rock
substrates at different scales: photolithography and electron-beam
lithography. Further, we discuss the challenges of applying the MSSM
technique to samples used in high-temperature and high-pressure experiments. We
applied the MSSM technique to a study of strain partitioning during creep of
Carrara marble and grain boundary sliding in San Carlos olivine, synthetic
forsterite, and Solnhofen limestone at a confining pressure,
During plastic deformation of a crystalline material, strain is accommodated by a variety of deformation mechanisms, including atomic diffusion, mechanical twinning, grain boundary sliding, and dislocation glide, climb, and cross slip, which may operate separately or in combination. Deformation mechanism maps are conceptual tools that divide stress, temperature, and strain rate space into different fields with a particular deformation mechanism prevalent in each. Simplified flow laws are established by assuming specific micromechanical models corresponding to the prevalent mechanisms and using experimental data to fit the material constants in the theoretical flow laws (Ashby, 1972; Frost and Ashby, 1982). Such simplified flow laws are a useful first step for the extrapolation of laboratory results to larger-scale geomechanical problems, but it is also possible that the relative activity of the various deformation processes might change significantly when extended to natural conditions of rate, mean pressure, chemical environment, and temperature. Thus, the quantification of the strain due to each process and the determination of the interactions among the processes is crucial for establishing robust microphysical models that can be used to interpret observations of natural microstructures and nanostructures and that can describe natural deformation at larger spatial and temporal scales.
Quantification of the strain accommodated by each mechanism is not an easy task. Ideally, the microstructural evolution of a polycrystalline material would be monitored continuously as a sample is strained. However, high pressures and temperatures are required when deforming rocks through plastic or viscous processes in the laboratory, making microscopic observations of deformation in situ a significant experimental challenge. Fortunately, even stepwise observations of the amount of strain accommodated by individual intracrystalline mechanisms could significantly enhance our understanding of high-temperature creep.
In this paper, we describe a technique for microscale strain measurement (MSSM), which permits strain mapping at micrometer and submicrometer scales, and discuss several observations of strain partitioning in rocks deformed at high pressures and temperatures. To quantify the deformation, the coordinates of the material points (at the microscale) must be identified before and after deformation. Then, the strain tensor can be computed from their relative displacements. We used two lithography techniques, described in Sect. 2, to print a grid of points onto the polished surface of a rock. After deformation, we computed strain following the formulation described in Sect. 3. We applied the MSSM technique to four different high-temperature experiments described in Sect. 4.
The literal meaning of lithography, writing on stones, is very appropriate
for our application. More generally, lithography refers to different
printing techniques that allow a pattern to be transferred onto a substrate.
Microlithograpy and nanolithography are widely used in the semiconductor industry
(Razegui, 2006; Mack, 2006) to pattern micrometric and nanometric circuits onto
silicon wafers. Applying these techniques to nonconductive, heterogeneous
materials is challenging, but we developed two protocols using
photolithography and electron-beam (e-beam) lithography to produce patterns
on the surfaces of Carrara marble, Solnhofen limestone, San Carlos olivine, and
synthetic forsterite. The patterns had features with spatial dimensions on
the order of 1
Figure 1 summarizes the steps needed for the lithographic process. Both photolithography and electron-beam lithography require special equipment, which is usually present in standard clean laboratories used to fabricate solid-state devices. The process described here is commonly referred as “lift-off” in the microfabrication community. It uses a sacrificial layer of polymer, which is a film that can be spread uniformly on the sample, to create a pattern on the sample surface (Razeghi, 2006). Note that the microfabrication processes used in solid-state engineering require extra steps to ensure the functionalities of the devices (conductivity and reflectivity, among others). We present the steps required to deposit micrometric and nanometric patterns on a rock surface with the main objective of printing a particular grid designed to track strain at a granular level. Quintanilla-Terminel (2014) describes in more detail the different exploratory steps that were taken before developing the following protocols.
Lithographic process: (1) rock, (2) photoresist or PMMA, and (3) metal layer (sputtered or evaporated). Each step is described in more detail in the text: (I) sample preparation, (II) coating with a polymer (photoresist for photolithography, PMMA for e-beam lithography), (III) exposure to UV rays for photolithography (IIIa) or an electron beam for e-beam lithography (IIIb), (IV) development of the coating, (V) etching, (VI) metal deposition, (VII) dissolution of the remaining polymer, and (VIII) patterned sample.
Sample preparation: the surface to be patterned needs to be polished to a
roughness of less than 0.5 Coating with a polymer: the surface then needs to be uniformly coated with a
polymer sensitive to the energy used to transfer the pattern. A photoresist
is sensitive to UV light (photolithography) and poly(methyl methacrylate), or
PMMA, is sensitive to an electron beam (e-beam lithography). The coating
method will depend on the sample geometry. Spin coating is the standard
method for flat samples. The sample is held on a rotating device by a
vacuum chuck and the polymer is spread onto the surface to uniform thickness by
spinning the sample at a controlled speed, typically around 3000 rpm. For
more challenging geometries, such as half cylinders or curved surfaces, a
paintbrush or an aerosol spray can be used, but care must be taken to ensure
that the polymer thickness is relatively uniform from sample to sample. Exposure to UV rays for photolithography (IIIa) or an electron beam
for e-beam lithography (IIIb). Although both techniques allow for the transfer
of a custom-designed pattern, the process by which this is achieved is
different. Photolithography necessitates the creation of a shadow mask, which is a
glass mask imprinted with the wanted design analogous to a negative in the
photographic process. UV light is then shined through the mask onto the
sample coated with the photoresist. E-beam lithography does not necessitate
a mask; instead the pattern is directly rastered with an electron beam at a
specific dose (typically set at around 500 to 1000 Development of the coating: the structure of the polymer exposed to the UV
light or to the electron beam changes, and a chemical bath is used to
dissolve this sacrificial layer. The pattern is now exposed on the sample
surface. The development time depends on the exposure time and on the
resolution of the pattern. Some play exists between the exposure time and
the development time; ultimately, the lithographer must determine the
optimum match for the intended application. Etching: the exposed surface is slightly etched before proceeding to the
metal deposition. This step was found to be important in geomaterials to
allow for a good adhesion of the metal layer. The etching step can be dry
(via reactive or nonreactive gas) or wet (typically in a diluted HCl or HF
bath). If the sample cannot be in contact with water, a dry etch would be
preferable. Metal deposition: a physical vapor deposition is used to coat the sample
surface with a thin film of a chosen metal. There is latitude in the choice
of the metal sputtered. Our applications require a metal that can be seen
before and after deformation and that has a minimum interaction with the
rock. Therefore, the parameters that are important in the choice of metal
include adhesion, melting temperature, visibility, and interactions with the
rock. Dissolution of the remaining polymer: the remaining polymer is dissolved,
leaving only the metal deposited onto the sample surface. Patterned sample: the patterned sample is now ready to be deformed.
Patterned Carrara marble using photolithography.
Both photolithography and e-beam lithography can be used for decorating rocks; a combination of both would also be possible. The main factors to take into account are the size of the area on which the pattern will be deposited, the design of the pattern (shape and resolution), and the sample geometry. Photolithography is a good choice when many samples are required, when large areas are to be patterned, and when the sample geometry is more complicated. For applications requiring a pattern with submicron features, e-beam lithography will be more appropriate. Because e-beam lithography does not necessitate a mask, it would be faster to implement for single applications. In the following sections we provide a detailed protocol for each technique. With some adaptation, particularly regarding the exposure, developing steps, and the choice of the metal to be deposited, these protocols might be used on other geomaterials.
Photolithography was carried out at the MIT Microsystems Technology
Laboratory (MTL), where half cylinders of Carrara marble were patterned.
First, each half cylinder was polished with a succession of aluminum oxide
suspensions down to a 0.5 The polished half cylinders were dried in a convection oven at
130 The sample surfaces were manually coated with a light-sensitive
photoresist, Fujifilm OCG 825 g-line, using a paintbrush; the thickness of
the polymer was determined to be The pattern inscribed in a photomask was transferred onto the polymer
coating by exposing the coating to ultraviolet light in a mask aligner for
60 s. The exposure time depends on the substrate, the photoresist, and
photoresist thickness. Our photomask was created using computer-aided design
(CAD) software to create the pattern and a laser to draw the pattern onto a
glass plate covered with chrome and photoresist, followed by a typical
etch-back technique (Mack, 2006). Photomasks are also available from various
commercial sources in the microfabrication industry. The nonexposed polymer was dissolved using a developer, Fujifilm OCG
934, for 30 to 60 s. The samples were then very lightly etched with an acidic solution (a 1 mass % dilute solution of HCl for 1 s). The surface of the samples was sputtered with a double layer composed of
20 nm of chromium and 30 nm of gold using a plasma sputterer. Finally, the protective layer of photoresist was removed with
2-(2-aminoethoxy)ethanol, N-methyl-2-pyrrolidine at 100
Photolithography is a very flexible patterning technique. Patterns can be
deposited on large surfaces with a customized design, little limitation
exists regarding the geometry of the sample, and a wide selection of metals
can be used. For example, the 10
The e-beam lithography process was carried out at the Minnesota Nano Center
(MNC). The forsterite and San Carlos olivine samples were polished using
diamond-lapping film down to a 0.5 A wafer of San Carlos olivine, Solnhofen limestone, or synthetic
forsterite was initially left in a vacuum oven at 130 The sample was first coated with PMMA diluted in 4 vol %
chlorobenzene. The coating was done with a precision spin coater spinning at
3000 rpm for 60 s to produce a thickness of the PMMA film of 350 nm
(evaluated with ellipsometry). The sample was then baked on a hot plate at
180 The patterning was achieved using a Vistec EBPG5000 The first step in the developing process required the dissolution of the
gold layer using a solution of KI, routinely called a gold etcher. The sample
was first submerged in the KI solution for 30 s, then rinsed in deionized
(DI) water for 10 s, and finally rinsed in isopropanol for 10 s. The sample
was then developed. Two developing processes were explored. The typical
developing solution for PMMA is a Two etching processes were explored; a wet etch using HF diluted to
1 vol % applied for 1 s and a chlorine-gas etch using an Oxford plasma
etcher for 1 min. Although the two processes produced similar results, the wet
etch was simpler to implement. A 110 nm thick layer of Cr was deposited in a plasma evaporator. To dissolve the remaining PMMA, the sample was immersed in
2-(2-aminoethoxy)ethanol, N-methyl-2-pyrrolidine at 120
Secondary electron images of patterned San Carlos olivine
Atomic force micrographs of patterned San Carlos olivine
The resulting patterned surfaces of San Carlos olivine and Solnhofen
limestone are illustrated in Fig. 4. Notice the difference in scale
compared to the patterned marble surface in Fig. 2; the images of the
patterned Carrara marble in Fig. 2 are optical light micrographs, whereas
the images of the San Carlos and Solnhofen limestone in Fig. 4 are
scanning secondary-electron micrographs. The high resolution, the sharpness
of lines, and the pattern profile can be better appreciated in images of the
patterns deposited on San Carlos olivine obtained by atomic-force microscopy
(AFM; Fig. 5). It is apparent that electron-beam lithography permits much
higher resolution than photolithography; in theory, nanometer scales can be
achieved (Vieu et al., 2000; Manfrinato et al., 2013). For geomaterials, a
resolution of 20 nm is easily attained. The pattern is rastered directly
onto the sample without the need for a mask, which facilitates tests with
different designs. However, because the rastering process has to be repeated
for each sample, e-beam lithography is much slower and more costly than
photolithography. Furthermore, samples have to fit into the electron-beam
lithography system, limiting the geometry of the sample that can be
patterned. In the Vistec EBPG5000
The regular grid that is printed onto the sample is used to track the strain accommodated by the sample. Indeed, the ultimate goal of using the MSSM technique is to understand which micromechanical processes accommodate strain in a deforming rock and to relate the observed microstructure to the imposed macroscopic deformation conditions. For this purpose, it is necessary to calculate the local strains using microscale gauge lengths within larger regions of interest.
Different techniques allow for the computation of strain at a local scale; most are image-based particle tracking techniques and therefore use a Lagrangian description in which strain is calculated by following a material point before and after deformation (Reddy, 2013; Malvern, 1969). The more widely used strain analysis technique, digital image correlation (DIC), has been successfully applied in a variety of rock mechanics applications. For instance, DIC was used to map the localization of damage in a heterogeneous carbonate (Dautriat et al., 2011), to quantify the role of crystal slip and grain boundary sliding during creep of synthetic halite (Bourcier et al., 2013), and to better understand creep of ice (Chauve et al., 2015; Grennerat et al., 2012). DIC is an excellent tool for extracting displacement and strain measurements at a microscopic scale (Bruck et al., 1987; Sutton et al., 2009) by comparing digital micrographs of the sample at different stages of deformation. The strain is computed using various algorithms that allow the observer to track blocks of pixels and cross correlate them between deformation steps (Bornert et al., 2008). Each block of pixels therefore needs to be unique and recognizable between deformation steps.
High-temperature and high-pressure deformation often changes the reflectivity of the material to a point at which blocks of pixels are no longer recognizable between deformation steps. Because our experimental setup makes it difficult to analyze incrementally strained samples, we require a technique that allows us to identify a material point independently of the amount of deformation the material has experienced. We therefore rely on the analysis of an initially regular grid characterized after deformation (Allais et al., 1994; Ghadbeigi et al., 2012; Karimi, 1984; Moulart et al., 2007; Sharpe, 2008; Wu et al., 2006; Martin et al., 2014). Because our experiments are performed at high temperature and high pressure with a metal jacket surrounding the sample, the grid is introduced in the middle of the sample, following different variations of a split-cylinder assembly (Raleigh, 1965). The initial reference grid is formed by deposition using either photolithography or e-beam lithography. Other deposition methods have been used, such as sputtering through a commercial screen (Xu and Evans, 2010) or introducing a metal mesh between the sample halves (Spiers, 1979). Recently, Hiraga (2015) used a focused ion beam to groove lines on samples composed of synthetic forsterite and diopside in order to track grain displacement and rotation during diffusion creep. In all the techniques, the main challenge is to accurately identify each individual marker or line before and after deformation. Various complications can arise; for instance, the markers can be unstable at high temperature, or the split-cylinder surfaces can weld together at high pressures, making the recovery of the patterned surface challenging or impossible. Nevertheless, if the identification of individual markers after deformation is accomplished, then spatial variations in relative displacement can be used to compute the strain field across the reference surface. Lithography has the advantages that higher resolutions can be achieved, the patterns are more robust and they survive high-temperature, high-pressure deformation, and the specific patterns can be custom designed to account for the grain size of the material or the research questions to be answered. Furthermore, because lithographic techniques allow the researcher to create extremely regular patterns with known dimensions, the strain can be computed by locating the markers only after deformation. It is, however, advisable to image the sample before deformation as well, as this step could provide additional information (particularly if the grain boundaries are visible) and eliminate the potential error introduced by irregularities in the undeformed grid.
Different image processing techniques can be used for locating the markers before and after deformation, such as a convolution (Biery et al., 2003) or a Hough transform algorithm (Quintanilla-Terminel and Evans, 2016). Depending on the application, the user could rely on the regularity of the grid and not identify the markers before deformation. Often, some manual input is required to ensure that the marker is correctly identified after deformation.
The
The deformation gradient tensor,
The deformation gradient tensor
The Hencky strain tensor,
An optical or electron micrograph allows for a 2-D strain inversion. The
strain tensor,
Strain maps for the three components of the 2-D strain tensor for
a sample of Carrara marble deformed at
The resolution of the strain analysis technique depends on two main factors that should be assessed for each application before interpreting the strain results. The main sources of error are related to the location of the markers and the strain induced by the sample preparation. Any error in the marker locations will propagate into the strain calculation. This error is a function of the resolution of the images and the image analysis technique used to locate the markers. The second source of error comes from actual local strains produced by the preparation of the sample. This error has to be estimated with a “zero strain” experiment by measuring the local strain in a sample left under the pressure and temperature conditions for a time length typical of the experiment.
The
Figure 6 illustrates the different
The choice of grid design and patterning technique should reflect the test geometry, physical characteristics of the target rocks, and research goals, including specific hypotheses regarding constitutive behavior. The design choices made for tests on Carrara marble and San Carlos olivine, synthetic forsterite, and Solnhofen limestone can be used for illustration. In the first case, Carrara marble deformation by power-law creep (Renner and Evans, 2002; Renner et al., 2002), we wished to investigate the partitioning of the strain between different deformation mechanisms to determine which were dominant and to identify the internal state variables needed for a more accurate flow law (Evans, 2005). Patterns in this study were made using photolithography. In the second case, we wished to observe microstructures produced during dislocation creep in San Carlos olivine (Hansen et al., 2011; Hirth and Kohlstedt, 2003) and during diffusion creep in synthetic forsterite (Dillman, 2016) and Solnhofen limestone (Schmid et al., 1977). Here, our goals included determining the strain contribution of grain boundary sliding in different creep regimes and testing current constitutive models. For this work, we produced patterns with e-beam lithography.
The average grain size of our Carrara marble samples was 130
Experimental setup for studying creep in Carrara marble, San
Carlos olivine, and fine-grained Solnhofen limestone at high temperatures
and
Strain maps over different regions of interest for a Carrara
marble split cylinder deformed at
Strain maps for
Strain maps for
Secondary electron images of a gridded fine-grained San Carlos
olivine after deformation at
Strain inversion on a recovered grid in a fine-grained sample of San
Carlos olivine deformed at
Atomic force micrograph of a deformed grid in synthetic forsterite
deformed at
Secondary electron micrographs of a patterned Solnhofen limestone
sample
Photolithography is a flexible technique that can be used to mark surfaces with varying geometry, including split cylinders or cylindrical surfaces (Figs. 2a, b, 8a). E-beam lithography has better spatial resolution but limits the surface geometry. As seen in Sect. 2, the sample has to be flat and not more than 1 mm thick. Two different composite cylinder sample configurations were used to introduce a 1 mm thick patterned sample. In the first, a 1 mm thick disk of San Carlos olivine disk was placed between two short olivine cylinders (Fig. 8b). In the second, a 1 mm thick rectangular slab of patterned Solnhofen limestone was inserted between two half cylinders (Fig. 8c).
We used a split-cylinder setup following Raleigh (1965) and Spiers (1979).
Half cylinders of “Lorano Bianco” Carrara marble, a standard material for
deformation experiments (Molli
and Heilbronner, 1999; Heege et. al., 2002), were polished down to a
0.5
The greatest challenge in studying strain distributions during creep in
olivine rocks is the separation of the surfaces after deformation. Because
creep tests must be carried out at
We used electron-beam lithography to pattern monophase polycrystalline
samples prepared from powders of San Carlos olivine, Fo
Recovering the grid after deformation was more challenging because of the
high temperatures and normal loads on the marked surface. We experimented
with different metals and separation techniques. Sputtering a window shape
with a high-melting-temperature metal, such as tungsten, on the facing sample (a
technique similar to the one used for Carrara marble) was unsuccessful, and
the two faces still could not be separated. Inserting a thin (0.01 mm),
ring-shaped, nickel foil between the gridded surface and the facing surface
was still not sufficient, but the gridded surface was successfully recovered
if a full nickel foil was placed between both surfaces. Secondary electron
micrographs of grids with lines 200 nm and 1
Samples of synthetic forsterite were synthesized following
Koizumi et al. (2010). After
gridding, they were deformed at
Three-piece samples of Solnhofen limestone were prepared (Fig. 8c), gridded
using e-beam lithography, and deformed in compression in gas-medium apparatus
(Paterson, 1990) at
Patterning using photolithography and e-beam lithography can provide maps of
strain calculated over spatial scales of 10–0.5
All figures and underlying research data used in this paper
can be accessed at
The authors declare that they have no conflict of interest.
This article is part of the special issue “Analysis of deformation microstructures and mechanisms on all scales”. It is a result of the EGU General Assembly 2016, Vienna, Austria, 17–22 April 2016.
Matej Pec is acknowledged for his help in acquiring the SEM pictures. Amanda Dillman is acknowledged for providing the data related to the forsterite sample and for her help in acquiring the AFM data. The lithography techniques were developed at the Minnesota Nano Center (MNC) of the University of Minnesota and at the MIT Micro Technology Laboratories (MTL). AQT thanks Bryan Cord at the MNC and Kurt Broderick at the MTL for their guidance through the fabrication processes. We benefitted from enriching discussions with Matej Pec, Amanda Dillman, Renée Heilbronner, Holger Stünitz, and William Durham and thoughtful reviews by Lars Hansen and an anonymous reviewer. Support through NSF grants EAR-1520647 (UMN) and 145122 (MIT) is gratefully acknowledged. Edited by: Florian Fusseis Reviewed by: Lars Hansen and one anonymous referee