INTERACTIONS BETWEEN DISLOCATIONS AND

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Collaborative Proposal:
The Role of Grain Boundaries in Non-Nano Polycrystal Deformation
Robert H. Wagoner, The Ohio State University (OSU), PI
Brent L. Adams, Brigham Young University (OSU), PI
Elizabeth A. Holm, Sandia National Laboratories (SNL), Principal Advisor
(These are shown on a separate page to show the page limit for the Project Summary.)
PROJECT SUMMARY
(next page)
1
Grain size is a critical aspect determining the strength of polycrystalline metals. Grain size
reduction alone can increase the strength of steels by a factor of 3 for “normal” (non-nano) grain
sizes (>1 m). In spite of its identification more than 50 years ago (Hall-Petch Effect [1,2]) and
its undisputed practical importance, its origins are poorly understood and quantitative predictions
have been elusive.
Intellectual Merit - PI’s Adams and Wagoner have been developing ways to simulate and
measure the interactions of large numbers of dislocations (1014/m2) with grain boundaries.
Recently, through an NSF EAGER award, two breakthroughs were made:
1) A multi-scale simulation scheme was demonstrated. It predicted quantitatively, for the
first time, the Hall-Petch effect.
2) New EBSD techniques were developed, thus enabling the resolution of dislocation
densities and characters for typical densities in annealed polycrystals (i.e. about two
orders of magnitude lower than were previously resolvable).
Both of these rely on knowing the grain boundary obstacle strength, τobs. The proposed work
capitalizes on these advances to measure and verify τobs, thus solving a long-standing challenge in
metallurgical science and enabling a new phase of polycrystalline material design.
A “Critical Experiment” will be carried out by the three collaborating groups. It involves the insitu loading and deformation of an Fe bicrystal, simulation using atomistic and the multi-scale
methods, and the high-resolution measurement of dislocation tensor distributions within the
specimen. Each of these aspects requires new, novel techniques to be developed. The result will
be a verified value of τobs and its variation with boundary character and strain.
Broader Impacts – New capabilities in computational metal plasticity and its characterization
will enable material design, a benefit to society. The methods developed will bridge longstanding length-scale gaps between single-dislocation behavior and continuum plasticity without
introducing arbitrary unknowns and fitting parameters. Benefits will include design of higher
strength / higher ductility structural materials, improved processing routes, and new multi-scale
measurement techniques.
EBSD, already widely used, will be extended to recover detailed tensor dislocation content at
previously unimaginable resolutions (HR-EBSD), a contribution to enhanced infrastructure. The
multi-scale model predicts polycrystal properties and interpretation of collective dislocation
characterization corresponding to measurements from HR-EBSD. For the first time, the full
lattice distortion can be predicted and measured for any array of dislocations and with any
boundary conditions. This capability allows critical probing of long-held assumptions of the links
between single-dislocation / atomistic methods and large-scale constitutive models.
Each investigator is committed to wide dissemination of results: publicationin peer-reviewed
journals, graduate theses/dissertations, and reporting at national/international meetings. For
example, output from two recently-completed NSF grants to Wagoner and Adams included 53
publications 10 invited/plenary presentations, and 4 book chapters.
The collaboration of two universities and a government laboratory will provide exciting learning
opportunities for the three funded Ph.D. students, for undergraduate students to be funded
separately by REU proposals, and for the senior researchers. Periodic meetings and summer
internships will deepen the interactions. All of the investigators are committed to broadened
participation by under-represented groups. 10 of the 44 graduate students advised by Wagoner to
completion of their degrees are women. Adams currently mentors 3 undergraduate women in
materials design.
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SPECIAL NOTES TO REVIEWERS
Two unusual aspects of this proposal need clarification The first relates to the structure of the
proposal, which is dictated by the unusual history of the prior work. The second clarifies the key,
unfunded role of the Sandia National Laboratory personnel.
History of Collaboration. A significant part of this proposal focuses on two recent research
advances made as part of a current NSF EAGER award. These enabling advances have not all
appeared in print yet, so significant space is devoted to introducing them, and links are provided
to manuscripts for more detail. The “Past Results” section of this proposal is thus longer and
more detailed than usual. It reflects many years of work although the EAGER funding has been
in place for only one year, as explained by the brief history below.
The PIs, during a discussion at a chance professional meeting in 2004, identified the following
“grand challenge:” understand and predict, without unrealistic models or arbitrary length scales,
the Hall-Petch Effect. An unfunded collaboration was begun that continues today. New,
promising approaches were developed but two major problems were uncovered that appeared
unsolvable until very recently:
1) A new idea for a multi-scale simulation technique treating the interactions of many
dislocations with grain boundaries showed good numerical characteristics and practical
computation times, but did not properly link strain with the motion of dislocation
populations. It predicted Hall-Petch slopes 20-50 times smaller than those measured.
2) The most accurate EBSD techniques developed to resolve dislocation tensor content were
only suitable for high dislocation densities, typically greater than 1014/m2, such as are
found only near grain boundaries or in highly-strain-hardened states. Thus, the critical
grain-scale spatial distributions of dislocation density throughout grain interiors predicted
by the multi-scale could not be compared with sufficient precision.
A series of 4 “Hall-Petch” proposals were declined by NSF between 2004 to 2008. The reviews
grew progressively more positive as the unfunded research progressed. The last proposal
received reviews of Excellent and Very Good, but was declined for budgetary reasons. Part of
that proposal was revived by new Program Director Ardell as an EAGER award, thus making
possible the advances outlined in the Past Results section.
The main criticisms of the Hall-Petch proposals related to two issues:
1) Unlikely to succeed; simulation method untried and unproven, and
2) Not based on nano-scale experiments and theory
In answer to the first of these, the results in the Prior Results section (and in links provided)
demonstrate the validity of the simulation and experimental approach. In answer to the second
criticism, atomistic simulation of a key parameter, the obstacle stress, is a key focus in the current
project, however, the target microstructural scale (“non-nano”) remains as before. Non-nanograined materials have overwhelming importance for structural applications, and it is clear that
distinct mechanisms operate in nano and non-nano grains..
Role of Sandia National Laboratory (SNL). The PI’s have insufficient experience in atomistic
simulation to carry out an essential part of the proposed project. Fortunately, Elizabeth A. Holm,
Stephen Foiles, and Christopher R. Weinberger of the Computational Materials Science and
Engineering group at SNL volunteered to assist the project in the following immediate ways:
1) Train and advise a graduate student at OSU on atomistic methods
2) Provide summer internships to accelerate and improve the training
3) Host and/or attend annual project review meetings
3
4) Provide access to state-of-the-art computation facilities at SNL
Furthermore, Dr. Holm will apply for internal SNL funding to allow direct performance of
simulations and developments by Holm, Foiles, and Weinberger. The success of that application
cannot be guaranteed, but funding from NSF would make it more probable.
TECHNICAL BACKGROUND
Hall-Petch Models. The widely accepted and experimentally verified Hall-Petch equation [1, 2]
relates the strength of polycrystalline metals,  y , to the grain size, D, as follows:
 y   0  ky D1/2
(1)
where σ0 and ky are material constants usually referred to as the friction stress and the Hall-Petch
slope, respectively. For FCC materials, ky values are generally less than 0.3 MN/m3/2 while those
for BCC materials are higher by a factor of 2 to 3, the origin of the large difference being unclear.
The sole quantitatively predictive model is the original one of Hall and Petch. Dislocations on a
slip plane pile-up until a obstacle stress is exceeded at the head of the pile-up, the length of which
is taken to be half of the grain diameter, giving a Hall-Petch slope as follows:
  b obs 
ky  M 

 k 
1/2
(2)
where M is the Taylor factor, µ is the shear modulus, b is the Burgers vector, and k=1 for screw
dislocations and k=1-ν for edge dislocations.
While an attractive, predictive model, pile-ups are seldom seen in deformed polycrystals,
particularly BCC ones, where the effect of grain size is most pronounced. When seen (at very
small strains and in very thin specimens of FCC materials), pile-ups do not extend across a
sizable fraction of a grain but instead are local to a grain boundary [6-8]. Furthermore, the value
of k y computed for iron using Eq. 2 (with one of the few measured values of  obs ) is
approximately an order of magnitude smaller than measured.
Alternative models of the Hall-Petch effect are phenomenologically fit to the macroscopically
observed behavior. The composite model, for example, envisions a strong grain boundary
“phase” of constant width [11-13]. The width is fit to macroscopic behavior. More recent straingradient models introduce an arbitrary material length scale to adjust dislocation density [14-19]
or introduce a similar term directly into a continuum constitutive description [20-27]. The
physical origin of the new length scale is murky or unmeasurable, and its relationship to grain
boundary character is not addressed.
Progress has been made for mechanisms for nano-scaled grains, including dislocation dynamic
simulations [28-33] and nanoindentation near a grain boundary [34-36] but the nano-scale
mechanisms are generally recognized as distinct from “normal”grain sizes, and yield different
slopes.
Grain Boundary Obstacle Stress, obs. The only known experimental determinations of obs
utilize TEM observations of pile-up spacings in stainless steel. Shen, Wagoner, and Clark [37]
found that a simple geometric criterion [37] reproduced observed emitted slip systems across a
boundary from a pile-up:
N  L1  Li  g1  gi
(3)

 

4
where L1 and Li are the intersection lines between grain boundary and slip planes and g1 and gi are
the slip directions of incoming and emitted dislocations, respectively. The transmissivity, N,
ranges from 0 to 1 representing maximum and minimum obstacle stress, respectively.
Eq. 3 was recently used to prepose a quantitative relationship for the obstacle stress for any grain
boundary as follows [REF]:
 obs  (1  N ) *
(4)
where * is the maximum grain boundary strength. Shen et al. [35] calculated a lower-bound
value * equal to 5 times the macroscopic yield stress for 304 stainless steel.
Atomistic simulation to obtain obs for specific interfaces or grain boundaries for FCC metals have
appeared [38-41] but their accuracy for BCC metals, for which interatomic potentials are less
satisfactory, has not been tested. This aspect is a key part of the proposed work.
RESULTS FROM PRIOR NSF SUPPORT
DMR-0936349: EAGER/ Interactions between dislocations and grain boundaries in BCC
metals: Hall-Petch effect, R. H. Wagoner, B. L. Adams, Sep.09- Aug. 11, $155,462 OSU,
$139,841 BYU.
This project was intended to predict the strengthening effect of grain boundaries (Hall-Petch
effect) in metallic polycrystals in a physically consistent way, and without invoking arbitrary
length scales. In the first year of the two-year project (and after many unfunded years of prior
developments) two major advances have been made:
1. A multi-scale method to treat the interactions of large numbers of dislocations with grain
boundaries predictively has been developed, implemented, and tested successfully.
2. High-resolution EBSD methods (HR-EBSD) based on cross-correlation techniques have
proven capable of resolving the tensor character of dislocation densities as low as
1012/m2, two orders of magnitude lower than is possible for traditional Hough-based
methods.
These advances, summarized in this section, lend optimism to the overall goal of predicting the
mechanical properties of a non-nano-scale polycrystal based on the grain / grain-boundary
structure.
The two advances are summarized in the remainder of this section of the proposal.
Multi-scale Model. Two simulation scales are sequentially iterated in the model. At the first
scale, a standard finite-element model of a polycrystal (up to 100 grains currently) [3-7] uses a
new, dislocation-based constitutive equation for single crystals [8]. The single-crystal
constitutive model represents measured single-crystal behavior. (In contrast, typical singlecrystal constitutive models are inferred from fitting textured polycrystals using simulations that
ignore grain size, shape, and character.) The new constitutive equations represent real singlecrystal stress-strain response by a factor of two better overall than conventional PAN (Peirce,
Asaro, Needleman [9]) models back-fit from texture simulations. An example comparison is
shown in Fig. 1.
5
120
100
Cu single crystal
Fit to multiple slip
100
y= 81 MPa
std.dev.=22.6 MPa
80
Taylor Model (SCCE-D)
[001] Measured
80
[123] Prediction (PAN)
[001] Fitted
60
y (MPa)
Eng. Stress (MPa)
Taylor Model (SCCE-T)
Sample 1
PAN model
40
[123] prediction
(Two-scale model)
40
y= 80 MPa
std. dev.=18.6 MPa
60
y= 83 MPa
std.dev.=19.8 MPa
Two-scale model
20
20
y= 72 MPa
std. dev.=4.6 MPa
Measured
y= 86 MPa
[123] Measured
0
0.00
0.02
0.04
0.06
Eng. Strain
(a)
0.08
0.10
0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Eng. Strain
(b)
Figure 1: Comparison of predicted stress-strain curves with the measurement for (a) copper single
crystal [8], and (b) Fe multi-crystal. [10] (Fix reference number?)
At the second scale, dislocation densities demanded by the first scale simulation are lumped into
superdislocations on each slip system at the center of each finite element. The dislocation
population is distributed at each time step in order to be consistent with the plastic strain
throughout the polycrystal, and the dislocation configuration is used to compute the associated
inter-dislocation back stress. Finally, the second scale simulation enforces local slip transmission
criteria at grain boundaries, i.e. so slip across a boundary is blocked until the total stress acting on
a single dislocation near the boundary exceeds a critical obstacle stress (characteristic of the slip
systems involved, the boundary orientation and grain-to-grain misorientation).
Only three arbitrary parameters are used in the multi-scale model, all related to the strain
hardening of single crystals. Comparisons of preliminary tests of the new method and
experiments revealed the following:
1) It is practical, taking only 5% more CPU time compared with a standard elastic-plastic FE
model of a polycrystal.
2) It reproduces solutions consistent with results for analytical solution of simple single
pileups.
3) It predicts Hall-Petch slopes for iron of 1.2 ± 0.3 MN/m3/2 and 1.5 ± 0.3 MN/m3/2, in
agreement with the measured value of 0.9 ± 0.1 MN/m3/2.
4) It predicts the tensile response of an Fe multicrystal (9 to 39 grains) 2-4 times more
accurately than by standard CP-FEM or Taylor-type texture models.
5) It predicts the magnitude of lattice curvature near grain boundaries of deformed Fe-3% Si
multicrystals within the experimental scatter, Fig. 2. (The techniques for measuring
lower lattice curvature or dislocation content were not available at the time this work was
done. See below for the promising new developments in that area.)
5) It predicts the dislocation content and character at a scale represented by the finite
elements. Examples of predictions for an idealized single crystal within a crystal are
shown in Fig. 3.
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G1
G2
B
A
G4
G3
(b) Initial mesh
(a) Initial grain orientations (OIM)
rad/μm
rad/um
0.004
0.002
0
(d) Lattice Curvature (Predicted)
(c) Lattice Curvature (Measured)
(a)
(b)
Figure 2: Deformed Fe-3% Si specimen images after 8% strain, (a) measured lattice curvature (b)
predicted lattice curvature using the multi-scale mode
 (m2 )
 (m2 )
 (m2 )
e=0.01
e=0.05
e=0.10
(b)
 (m2 )
 (m2 )
 21 1  1 11
1 12  111
(c)
Figure 3: First predicted dislocation densities for cylindrical grain within a grain tensile specimen:. (a)
evolution of total dislocation density distribution with deformation, and (b) dislocation densities for
two slip systems at the same deformation state.
High-Resolution-EBSD. A key experimental challenge to verifying any dislocation-based
model of metal deformation lies in the resolution and characterization of large populations of
dislocations, not only in scalar form but in tensor form. (For small numbers of isolated
dislocations, as for example in nano structures, TEM and other methods can resolve all
aspects of individual dislocations.)
Standard EBSD using Hough-based indexing of EBSD patterns is limited to 0.5 in angular
resolution of lattice orientation, corresponding to greater than 1014 m-2 of dislocation density.
New HR-EBSD methods, based upon cross-correlation techniques [11, 12] can now resolve
lattice orientations and dislocation densities of ~ 0.005, and 1012 m-2, respectively. [13-16].
These double-order-of-magnitude improvements allow mapping of incredible detail of
dislocation densities, as shown in Figure 4, where even cell formation and other such
complex dislocation arrangements are imaged. While polishing techniques for Ni (as shown
7
in Fig. 4) are well established, the ones for softer steels have been developed at BYU and are
still being improved.
Figure 4. GND dislocation density estimates for a wedge indented Ni sample, at a step size of
100nm[17].
An equally important advance promises to allow recovery of the complete field of the elastic
displacement gradient tensor, e(x), and the complete elastic incompatibility, curle(x), which
is Nye’s geometrically-necessary dislocation (GND) tensor, (x) [18-22]. HR-EBSD
methods enable direct, partial recoveries of these tensor fields. Cross correlation is insensitive
to the spherical components of e; but given the near-surface nature of EBSD, if the local
elastic properties are known, the traction free surface condition can be exploited to recover
the missing spherical component [11, 15]. The stress equilibrium relation (div  = 0) can be
exploited to recover estimates of the additional components of the dislocation tensor, as long
as the elastic properties of the local material are known. Only 5 components of the 9
component dislocation tensor can be measured; however the form of the solution to the
equilibrium relation suggests that additional information can be recovered by solving a large
system of linear equations.
This approach has now been verified by recovering the four missing, and previously unretrievable components of a simulated dislocation distribution; these recoveries were for a W
grain, selected for its isotropic elastic simplicity, Fig. 5. The ability to recover the principal
features of the simulated dislocation field is evident.
Figure 5. Demonstration of principle of recovery of the complete dislocation tensor
for a single grain of Tungsten, of 6.26 m2.
8
Human Resources Developed. Hojun Lim received his Ph. D. in Spring 2010; is now Postdoctoral researcher. .Ji Hoon Kim was post-doc, now is Research Scientist, Korea Institute of
Materials Science.
Publications
1) Lim, H., Lee, M. G., Kim, J. H., Adams, B. L., Wagoner, R. H., (2010). "Simulation of
Polycrystal Deformation with Grain and Grain Boundary Effects", Int. J. Plast.
(submitted). LINK (to be added):
2) Lee, M. G., Lim, H., Adams, B. L., Hirth, J.P., Wagoner, R. H., (2010). "A dislocation
density-based single crystal constitutive equation." International Journal of Plasticity 26:
925-938. LINK (to be added):
3) Lim, H., Lee, M. G., Kim, J. H., Hirth, J. P., Adams, B. L., Wagoner, R. H., (2010).
"Prediction of Polycrystal Deformation with a Novel Multi-scale Approach", 1st Conf.
Adv. Interaction & Multiscale Mech. (AIMM'10). Eds. C.-K., Choi, Y. -B. Yang,
Techno-Press, pp.1164-1192
4) BRENT – CAN YOU CONTRIBUTE SOMETHING HERE, SUBMITTED OR IN
PREPARATION? [YOUR REFERENCE 5]
A CRITICAL EXPERIMENT
This section outlines, in a condensed and simplified way, the core of the proposed work: a
“Critical Experiment” aimed at measuring the stress to push1 a dislocation through a grain
boundary, obs .
A value for obs has seldom been measured and is poorly understood. Its determination is an
important scientific advance in its own right; it is the key to understanding the mechanical
behavior of non-nano-scale polycrystals (i.e. virtually all structural alloys). It is also needed to
enable effective material design, the eventual technical and societal benefit of the work proposed
here. The goal is not only to measure a single value of τobs, but more importantly its dependence
on grain boundary character, grain misorientation, and orientation of slip systems on both sides of
the grain boundary. Only through such knowledge can the varying properties depending on grain
boundary character (not only texture) be understood. The open question of whether τobs depends
on the magnitude of slip transmitted will also be answered.
For the moment, the developments required to carry out the “experiment” will be ignored. These
will be addressed subsequently. In its simplest presentation, the experiment consists of the
physical in-situ deformation and simultaneous EBSD characterization of a bcc bicrystal, and at
least two kinds of corresponding simulation, as portrayed schematically in Fig. 6. The
simulations will be conducted at two widely removed length scales: meso and atomistic. The
experiment will be repeated for various combinations of active slip systems and grain/ground
boundary orientations.
The critical importance of obs for understanding macroscopic grain boundary behavior was
revealed in the Past Results section. Accurate prediction of the Hall-Petch slope and the
More formally, obs is the local stress required to propagate plastic slip across a boundary obstacle. The
detailed mechanisms are not of immediate concern at the meso scale, except as they affect the effective
obstacle strength. Thus, obs may be determined by the nucleation of a new dislocation on the outgoing side
of the boundary, or other mechanisms involving cross-slip, absorption and emission events. The proposed
atomistic scale simulations will probe these detailed mechanisms automatically as they simulate the grain
boundary resistance to slip transmission.
1
9
dislocation distribution within the grains required one main critical parameter, obs . The other
parameters inherent in the multi-scale simulation are either well-known and standard (elastic
constants, slip system orientations, burgers vectors, for example), or are readily determined (3
strain hardening parameters, obtainable from single crystal tensile tests), or have little effect on
the model predictions (for example, the proportion of edge and screw dislocation character). In
preparation of this proposal, simulations were performed to probe the sensitivity of the Hall-Petch
slope by changing each parameter by a factor of ½ and 2. The change of Hall-Petch slope for the
various parameters is as follows: obs: +/-80%, o (initial dislocation density): +/-15%, all others:
maximum of 3%.
Figure 6: Conceptual views of a bicrystal “critical experiment” including EBSD
measurement of tensor dislocation density distributions and two two kinds of interpretative
simulations: multi-scale model of the overall configuration and atomistic simulation of a
small region of the physical boundary.
The values of obs for various grain boundaries were obtained using Eq. 4 and a value of the
maximum obstacle stress, * , 375 MPa, inferred from TEM measurements of dislocation pile-up
spacings in thin films. There are many possible sources of error in this procedure, but there have
been few alternative schemes for determining obs or * (as reviewed in the Background Section).
It is unclear, for example, whether the obstacle strength varies with slip passage, with the crystal
structure, or with the magnitude of slip transmitted.
The Basic Physical Experiment. The Critical Experiment, in its simplest form, Fig. 7, consists
of the following steps :
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1) Prepare a columnar bi-crystal tensile specimen oriented for single slip in a planar
direction. Materials: originally very coarse grained Fe or Fe-3%Si for which singlecrystal constitutive behavior has already been measured and equations have been verified.
The specimen size will depend on the grain size achievable, but will be suitable for insitu deformation in an EBSD-equipped SEM. (The procedures for obtaining large grains
and tensile specimens are in place. They were used to obtain the Past Results.)
2) Specimens will be polished using techniques for Fe developed at BYU. (Already
developed.)
3) Specimens will be OIM scanned to obtain initial crystallographic orientations and
dislocation densities near the grain boundary. New HR-EBSD techniques will be
employed.
4) Fiducial marks will be applied to each sample to determine the local extension.
5) A pre-set extension will be applied using special micro-tensile stage developed for in-situ
SEM2. Load and extension will be recorded, and the shape of the specimen and the grain
boundary.
5) Full components of dislocation tensors will be measured and recovered using HR-EBSD
and special analysis. The dislocation content will be obtained at locations throughout the
specimen with a focus on locations approaching the grain boundary along the traces of
the active slip planes. The step size will approximate the element size for two-scale
modeling.
6) After EBSD in-situ testing is complete, the specimen will be sent to OSU for nanoindentation experiments similar to ones reported by Aifantis et al. [REF] These will
provide a third route to the measurement of obs.
Figure 7: Schematic of a tensile specimen for the “Critical “Experiment.”
Analysis. In parallel with the physical EBSD bicrystal tensile experiment, meso-scale and
atomistic simulations will be conducted using identical orientations of grains, grain boundary, and
extension direction.
The simplest analysis will compare the measured distribution of scalar dislocation density
(function of distance from the grain boundary, (x)), the measured stress-strain response, and the
2
The tensile testing apparatus was designed by Luke Brewer and Brad Boyce at SNL, especially for use
with EBSD capabilities. The design has been altered at BYU to accommodate the short (12mm) working
distance on the available FEI-SFEG-based EBSD system. The modified system is currently under
construction.
11
measured shape changes of grain boundaries and specimens with corresponding quantities from
meso-scale simulations based on obs computed atomistically. If everything agrees within
experimental and simulation scatter, the computed obs will have been confirmed. This would be
the first such measurement and confirmation of a grain boundary obstacle stress.
Assuming that the agreement is not perfect, a second layer of analysis will be initiated to recover
the effective obs from the scalar dislocation distribution, (x), ignoring for the moment the value
of obs simulated atomistically. This will be accomplished using an overall reverse analysis, i.e.
computing (x) for several choices of obs, finding the best fit value, and then comparing details
of the simulations and measurements to verify the consistency of the solution. The task will then
be to determine why obs from meso-scale measurements and simulation do not agree with
atomistic ones.
Assuming that the nature of the discrepancy has not yet been revealed, the next step will be to
look at detailed dislocation content in tensor form, and its gradient from one element or
measurement to another. Discrepancies in these quantities may reveal detailed local mechanisms
not considered in the meso-scale model, such as cross-slip, complex interactions among the longrange fields of among various types of slip systems, slip-affecting boundary conditions at free
surfaces (currently treated as traction free only). These will guide future developments of the
meso-scale model. Similarly, such analysis can also be used to guide the atomistic simulations
and to probe the accuracy of the BCC potentials used.
One particular phenemonen to look for in the experiments and measurement is the non-Schmid
nature of slip in BCC crystals [REF]. This should appear automatically in the atomistic
simulations if the potential is sufficiently accurate, and will need to be added in a continuum
fashion [REF – Anand, others?] to the meso-scale simulation as guided by the atomistic results
and EBSD experimental results.
Extensions of the Basic Experiment. Two basic extensions are planned at the outset. Others
will be guided by the results and the experimental difficulties that are encountered.
The first planned extension involves stopping the deformation of the bicrystal at various
extensions, then performing the measurements as shown for the basic experiment. The number of
EBSD scans is limited by the time in the SEM chamber with voltage on, on the order of a few
hours until the signal is degraded. Nonetheless, even obtaining three sets of (x), will allow
determination of whether obs evolves with the magnitude of local slip passed through a boundary.
Such a determination would be a first. If such scans cannot be performed in their entirety because
of time limitations, coarser scans of smaller areas would be attempted to get the required variation
with strain.
The second planned extension involves testing a series of bicrystals with markedly different
crystal misorientations and boundary orientation. It is hoped that 6-10 such configurations can be
tested in order to verify or modify the simple rule for computing obs from a known value of the
maximum ground boundary resistance () and geometric orientation information, Eqs. 3 and 4.
Having a confirmed rule for general grain boundary character would also be the first result of its
kind, although inferences for TEM of thin films at very small strain have tended to confirm
relationships such as Eqs. 3 and 4.
The remainder the extensions are conceived of, but will be guided by the results and by the
practical limitations of cost, time, and equipment. A high priority is to use the basic method to
12
examine dislocation distributions (measured and simulated) around grain boundary triple
junctions. If the results make sense, then the next step would be to test and analyze multicrystals.
These configurations are shown schematically in Fig. 9.
Figure 8 : Extended tests following the critical experiment.
The last kind of extension envisaged is the testing of other materials, perhaps jumping to multicrystal configurations if results are good for bicrystals and multicrystals of Fe or Fe-Si. Testing
other BCC materials, such as Ta or Ni, would test whether the relationships for obsand carry
over. Ta is of particular interest because the SNL partners are working on it in a related project,
and verified potentials are being developed. Ni would be a good target because it can readily be
polished to a very fine surface finish, using procedures already developed [REF]. Finally, and
particularly if difficulty understanding the results for BCC materials persists, it would helpful to
test and simulate an FCC material for which the atomic potentials are known as well as the
polishing procedures. Examples would include Cu and 304 stainless steel.
In the remaining technical sections of this proposal, brief introductions to the methods and
developments needed to carry out the Critical Experiment are outlined.
EBSD TECHNIQUES AND CHALLENGES (BYU).
The EBSD work (primarily conducted at BYU) has three key challenges and solutions:
1. Recover the complete intra-grain dislocation density fields using HR-EBSD for
direct components and by applying stress equilibrium relation (div  = 0) for the
remaining components. These measurements are for comparison with, verification
of, and improvement of the OSU multi-scale model.
2. Fine-scale HR-EBSD recoveries of the inter-grain elastic incompatibility (also
known as the surface dislocation tensor). These will be compared with
corresponding incompatibilities obtained by atomic-scale simulations.
3. If contributions 1 and 2 are successful, pursue methods for accelerating the HREBSD methods in order to facilitate characterizations at the multi-crystal and
poly-crystal levels.
The technical basis for each of these challenges/solutions is discussed briefly in this section.
Each of these contributions of HR-EBSD to the critical experiment requires extension of current
HR-EBSD methods. Key elements of the work include:
1. Completion and experimental verification of the methodology for recovering
estimates of the complete GND tensor using Green’s function methods, as
mentioned in the Prior Results section.
2. Develop methods for determining the elastic strain fields in loaded and unloaded
samples. Using reference EBSD images taken far from the bi-crystal interface, in
the unloaded condition, the elastic strain field will be recovered as close to the
interface as possible. These recoveries will be used to make direct estimates of
τobs.
13
3. Development of elastic incompatibility recoveries as proximate as possible to
grain boundaries, and methodology for comparison with atomic-scale simulations.
4. Acceleration of HR-EBSD methods for characterization of multi-crystals and
poly-crystals
A brief description of these activities follows.
Methodology for recovering complete GND tensor. This approach has been outlined briefly in
the Prior Results section. Finite-difference solutions to the stress equilibrium equations with
known or assumed boundary conditions provide additional constraints on GND fields. To verify
consistency with measurements, Ni samples will first be prepared and tested because removal of
polishing effects is much easier. The results will then be extended large-grain Fe, and Fe-Si steel.
A Ni sample will be serially sectioned to measure directly the missing derivatives parallel to the
sample normal direction. Relaxations corresponding to the new surface creation will occur, but
this alteration does not significantly affect all components of the dislocation tensor; thus it will be
possible to conduct a partial experimental validation of the new methodology.
Fine Scale Recovery of Elastic Fields. Determination of local elastic strain requires knowing the
PC very precisely [8] along with a reference strain-free EBSD pattern. The lattice orientation of
the strain-free EBSD pattern must be within 20 of the field point, otherwise corrections for crosscorrelation must be made [9]. These techniques will be extended to the plastic regime by
comparing the loaded and unloaded states. The presence and character of internally-equilibrated
elastic strain fields will be compared between OSU simulations and BYU experimental
characterizations. BYU has recently presented techniques related to full elastic characterization
[5]. [Brent: Add link here or elsewhere.]
Use of a reference strain-free EBSD pattern has been shown to allow measurement the tensor
dislocation content of a grain boundary, although this is at the limit of HR-EBSD resolution. For
boundaries subjected to large plastic deformation (slip transmission), the strength of
incompatibility will increase, and HR-EBSD should be able to resolve the interaction of
dislocations with particular boundaries by considering the changes in the surface dislocation
tensor. Equally novel and interesting, measured incompatibilities can be compared with those
predicted from equilibrated atomic positions obtained by molecular simulations of selected
boundaries. Thus, the HR-EBSD results, which characterizes structure at ~ 100nm, will be
compared with the multi-scale simulations of OSU at a length scale of ~ 1,000nm, and with
atomistic simulations at ~ 10nm.
Acceleration of HR-EBSD methods. Adaptive EBSD techniques [16,17] will be devloped to
allow faster partial scans, i.e. focusing on finer length scales near grain boundaries and triple
junctions. A coarse step size ~ d/10 (where d is the grain size) will used to scan the
microstructure, where differences in adjacent lattice orientation identify the presence of one or
more grain boundaries. The scan is subsequently and sequentially refined in the local
neighborhood. A typical final step size might be ~ d/1000, leading to accelerations of ~100 in
typical applications. Adaptive methods will become important in the proposed work as larger
numbers of interfaces and grains are considered within the simulations and the corresponding
experimental campaigns. Similarly, multiple scans will be required for a single sample stopped at
several extensions for characterization.
ATOMISTIC SIMULATION OF τOBS. (OSU, SNL).
14
The initial atomistic simulation work will be done by PI Wagoner and an OSU graduate student
co-advised by the SNL group (Holm, Foiles, Weinberger). Later stages will depend on internal
SNL funding to allow direct participation in large-scale simulations. Under both arrangements,
SNL will make available large-scale computing facilities in addition to facilities at the Ohio
Supercomputer Center (see supplementary information). SNL has a project in place to simulate
the mechanical response of bcc Ta from the atomic to the continuum scales. SNL will supply
computational methods and experimental data that will enable the measurement of the obstacle
stress for Ta grain boundaries (and indirectly for Fe grain boundaries). The results will benefit the
projects at SNL with improved constitutive and continuum models for Ta deformation.
Technical Developments. The direct interaction of a dislocation and grain boundary occurs at
distances less than 100 nm, in reference volumes making it suitable for atomistic simulation. In
concept, the appropriate bicrystal is built, a dislocation is inserted, and the bicrystal stressed until
slip passes through the boundary. However, each step requires special knowledge and techniques
for which SNL has the needed expertise. SNL is the developer of the widely used LAMMPS code
for molecular dynamics [1] and the inventor of the embedded atom method (EAM). It therefore
has as unique capabilities in atomistic simulation.
Dislocation motion in bcc materials such is highly sensitive to the Peierls stress, which is not
accurately reproduced by EAM potentials. Current research programs at SNL are developing
atomistic simulation methods for dislocations in Ta based on bond order potentials; this work will
provide a platform for simulations required in this program. These potential will be directly
applicable to Ta, but variations will likely be suitable for Fe. Fe represents an additional
challenge because of ferromagnetic effects. These will be ignored in the original simulations.
Building an atomic bicrystal with an equilibrated grain boundary structure is required [3]. SNL
has developed an automated grain boundary construction method that provides well-minimized
bicrystal structures with arbitrary grain boundary crystallography [4]. Ts technique will be used to
build the required bicrystals.
The magnitude of the applied stress needed to move a dislocation through a grain boundary can
affect the structure of the dislocation and the grain boundary, and the stress is affected by the
strain rate applied in the simulation. The solution is to decrease strain rate as much as possible by
running very long simulations. SNL recently performed the longest MD simulation of
polycrystalline grain growth to date [5], and can apply such simulations to this program. The
large number of large-scale simulations required necessitates access to high performance
computing resources, such as those that will be provided by SNL.
An alternate method that might prevent the distortions caused by large applied stresses is the
“synthetic driving force” MD method, recently developed at SNL [6-8]. This method has shown
to simulate grain boundary motion identically to physical driving forces, but without the artifacts
caused by the physical mechanism [8]. A similar development applied to dislocations might
allow simulation of realistic dislocation motion while still avoiding long CPU times. SNL will
investigate both low-rate conventional MD and synthetic driving force MD to determine the more
appropriate method. Both the physical insights gained and the computational technique developed
will be important outcomes from this program.
Complexities are expected in the details of the simulated results. Dislocations may be absorbed
by the grain boundary and disappear from the system, thus changing the boundary structure for
the next slip event. After enough absorption events, the grain boundary structure may be
modified sufficiently to then transmit a dislocation or to undergo a structural transformation [9].
15
These sorts of changes will produce a changing τobs with regard to slip magnitude. Unrealistically
high stresses might be required because thermally activated or assisted mechanisms are typically
not amenable the MD time frame for complex problems such as this one. Nonetheless, MD
simulations should predict the relative transmission strength of boundaries despite the influence
of high stresses, and perhaps a small number of detailed simulations can be used to estimate τ* ,
such as appears in Eq. 4, thus effectively scaling the 0 K simulations for a variety of
configurations.
Investigating, understanding, and incorporating these effects into a model that can be applied on
the continuum scale is the ultimate goal – and challenge – of the atomic scale simulation effort.
Research Tasks and Approximate Schedule
 Develop a synthetic driving force MD method for dislocation motion and compare with
conventional stress-driven MD. (Y01Q02)
 Define atomistic simulation platform, including accurate and appropriate interatomic
potential for defects in Ta. (Y01Q04)
 Build a catalog of atomic scale Ta bicrystals containing dislocations. (Y02Q02)
 Measure obs for individual dislocations approaching bicrystal grain boundaries. (Y02Q04)
 Investigate the phase space of dislocation/boundary interactions, including multiple
dislocation effects. (Y03Q02)
 Develop a model for obs that can be implemented in continuum deformation simulations.
(Y03Q04)
MULTI-SCALE MODELING AND MATERIAL PREPARATION (OSU)
The OSU multi-scale model has been introduced in the Prior Results section along with
various verifications to-date. More detailed information about its formulation and the
dislocation-density-based single crystal constitutive equations which are utilized is
available the respective links provided. No further detail will be provided here because
of space limitations.
Simulations performed using the multi-scale model (described in the XX section) showed
that obs is the only parameter likely to be of critical importance in the accuracy of its
predictions, thus many of the details of the formulation may not be of immediate
significance. However, the original dislocation density (related to the yield stress of
single crystals) has secondary importance, about 1/5 the sensitivity as for obs.
The form of single-crystal constitutive models used so far was confirmed by single-crystal
experiments for Fe and Cu appearing in the literature, with parameters set by limited experiments
with small multi-crystal specimens of Fe and Fe-3%Si. Large columnar grains are attainable for
these metals using strain annealing. One of the OSU tasks will focus on preparing single-crystal
specimens as well as the bicrystals and mulit-crystals required for the critical experiment and
extension. With single-crystal specimens of known orientation, and utilizing techniques similar
to the critical experiment, the constitutive equations and initial dislocation densities will be
verified.
Extensions to other metals such as Ni and Ta will in any case require single-crystal
measurements. If large single crystals can be obtained by thermo-mechanical treatments, similar
16
procedures will be followed. If attaining very large grains proves difficult, special micro tests
being developed at OSU [1, 2], Figure 9.
Figure 9: Hardening curve (left) of Rene 80 from a 6 m-diameter tensile specimen (right). [2]
Research Tasks and Approximate Schedule
 Heat treatment of Fe to obtain large grain sizes/ fabricate single- and bi-crystal specimens
oriented for various directions (Y01Q02)
 Formulate the multi-scale model in a form that can be parallelized to make larger-scale
simulations (~100-1000 3-D grains) possible. (Y01Q04)
 Conduct tensile tests of single crystals, fit to two scale model to obtain best fit parameters
(Y01Q04)
 Perform pileup analysis of bicrystals to obtain obs (Y02Q02)
 Simulate lattice curvature for more complicated grain structure, e.g. triple junctions
(Y03Q04)
 Simulate Hall-Petch slope for 1000 3-D grains (Y04Q02)
BROADER IMPACT (BYU, OSU)
Benefits to Society – Fundamentally new capabilities in computational metal plasticity and its
characterization will enable material design and improve applications such as metal forming. The
methods developed bridge length-scale gaps between single-dislocation behavior, computational
methods, and continuum plasticity for finite-strain applications without introducing arbitrary
unknowns and fitting parameters. The methods will enable a range of new developments that
cannot readily be envisaged at the moment, but will include higher strength / higher ductility
structural materials, improved processing routes to obtain optimal grain structures, and new
measurement techniques that rely on multi-scale simulation for interpretation.
Enhanced Infrastructure – Orientation Imaging Microscopy, invented by PI Adams, is today
widely used to investigate the structure and properties of a range of materials. The proposed
collaborations in developing HROIM, and 3D EBSD techniques and software will allow direct
measurement of deformation tensors and GND content. For metal applications, this ability
provides a quantitative method to characterize the local dislocation density content that is
otherwise unavailable and badly needed.
17
The multi-scale model developed at OSU allows both the prediction of polycrystal properties but
also the interpretation of collective dislocation characterization such as is available with EBSD.
For the first time, the full lattice distortion can be predicted for any array of dislocations and with
any boundary conditions. This capability allows critical probing of long-held assumptions of the
links between single-dislocation / atomistic methods and large-scale constitutive models.
Dissemination of Results – Each investigator is committed to wide publication of results in peerreviewed journals, graduate theses/dissertations, and undergraduate project reports. This
commitment is readily demonstrated by the most-recently completed NSF grants to Wagoner and
Adams. For DMR 0139045 (Wagoner), 23 publications, 2 book chapters, 6 invited/international
presentations, and numerous domestic presentations were made. For DMR 0079996 (Adams), 30
publications, 2 book chapters, 4 invited and plenary addresses and various domestic presentations
were made.
Teaching, Training, Learning – The collaboration of two universities and a government
laboratory will provide exciting broadening opportunities for the three funded Ph.D. students, for
undergraduate students to be funded separately by REU proposals, and for the senior researchers.
Two meetings are planned per year to maintain close contact, in addition to weekly conference
calls and e-mail exchanges. The summer internship opportunities at SNL will also be used to
deepen and broaden the educational opportunities for the students.
Educational modules will be developed presenting basic concepts of how microstructure
influences properties, including the interpretation and simulation of OIM data. Wagoner will
incorporate the modules into MSE 661: Ferrous Metallurg. Funds to Wagoner from a 5-year
curriculum development grant from AISI and the AIST Foundation will leverage these
developments. Adams is leading a group of 4 faculty from ME and Math developing a 2-year
undergraduate capstone sequence. The modules will be incorporated into this sequence as well as
project assignments for ME 500 Design and Materials Applications, and ME 503 Plasticity and
Fracture..
Broadened Participation – All of the investigators are committed to advising graduate students
from under-represented groups and mentoring undergraduate students through the transition to
graduate education. 10 of the 44 graduate students advised by Wagoner to completion of their
degrees are women who now work at a variety of places, including Honda of America, Idaho
National Laboratory, British Petroleum R&D, and Daimler-Chrysler. Adams currently mentors 3
undergraduate women in materials design.
18
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OSU REFERENCES (micro-pillar)
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Samples, unpublished research.
22
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