Strain Rate Dependent Behavior of Nanocrystalline Gold Films
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Proceedings of the XIth International Congress and Exposition
June 2-5, 2008 Orlando, Florida USA
©2008 Society for Experimental Mechanics Inc.
Strain Rate Dependent Behavior of Nanocrystalline Gold Films
Liwei Wang, PhD student, Mechanical Engineering, 275 Wilmore Engineering Labs,
Auburn University, Auburn, Al-36849
B. C. Prorok, Assistant Professor, Mechanical Engineering, 283 Wilmore Engineering Labs,
Auburn University, Auburn, Al-36849
ABSTRACT: We This paper reports on the influence of strain rate on the onset of mechanical softening of
nanocrystalline gold at room temperature. Micro-tensile testing was performed with applied strain rates on the
order of 10−4 s−1 to 10−6 s−1. Our results defined a threshold strain rate, whereby plastic deformation at larger
rates was dominated by dislocation processes and at smaller rates by one or more other deformation
mechanisms. Furthermore, the data suggested that the critical grain size for inverse Hall-Petch behavior was
strain rate sensitive.
INTRODUCTION: The Nanostructured metals hold the promise of becoming the next generation of engineered
materials that possess both high strength and high ductility [1]. The underlying characteristic of these materials is
their nanoscale grain structure, which can impart a variety of extraordinary material properties [2]. In terms of
mechanical properties, they achieve their high strength via their grain size, which places limits to the amount of
dislocation formation and motion available to accommodate plastic deformation, a correlation described by the
well known Hall-Petch relation [3-5]. In this regard, if grain size drops too far into the nanoscale regime the
material undergoes a mechanical softening process, whereby further decreases in grain size result in more
softening, known as the inverse Hall-Petch effect [6-8]. Even though a plethora of very good experimental work
and model development has been accomplished and published describing this transition, there is still need to
examine how these effects manifest under loading conditions in real-world application. One of the more important
loading conditions that has received recent attention is low applied strain rates that have been shown to induce
considerable diffusion-based deformation at room temperature in nanostructured gold and other metals [9-14]. A
recent model on the strain hardening behavior of nanostructured metals addressing these low strain rates, called
the Phase-Mixture model, was proposed by Kim and Estrin [15]. In this work, the authors generated a deformation
mechanism map for nanocrystalline copper based upon the competition of dislocation-based processes and grain
boundary diffusion processes as a function of grain size and applied strain rate. An important prediction of the
model was that the critical grain size for the onset of inverse Hall-Petch behavior was sensitive to the applied
strain rate. Garnering a better fundamental understanding of this effect and verifying its existence is important and
critical in the design of components fashioned from these nanostructured materials.
To this end, this work was designed to garner experimental evidence of strain rate dependent effects in
nanostructured gold films. Our work was performed on nanocrystalline gold films that were microfabricated into
tensile specimens and deformed by the membrane deflection experiment (MDE) [16-18]. Our results defined a
threshold strain rate, whereby plastic deformation at larger rates was dominated by dislocation processes and at
smaller rates by one or more other deformation mechanisms. Furthermore, the data suggested that the critical
grain size for inverse behavior was sensitive to strain rate, in a manner similar to predictions of the Phase-Mixture
model. These results offer important experimental insight of how nanostructured materials behave at very low
strain rates.
EXPERIMENTAL DETAILS: The membrane deflection experiment was employed to subject freestanding,
nanocrystalline gold films to pure tension, the operation principle and complete details are described elsewhere
[16-18]. It should be noted that each specimen tested was considered as an individual, thus thickness and other
dimensions and parameters were measured for each specimen to obtain the most accurate data reduction
procedures. Films of different thickness were grown by e-beam evaporation, namely 0.25, 0.50 and 1.00 µm, to
achieve a variety of film microstructures. The specimen microstructure and dimensions were determined with a
JEOL JSM-7000F field-emission scanning electron microscope (SEM). In terms of dimensions, the specimens
tested in this study possessed a membrane half length of ~300 µm and gauge width of ~10 µm. The average in-plane grain size of all films was determined by employing ASTM Standard E112-96E2 [19] to the SEM
micrographs in Fig. 1. In this analysis 10 regions of each film were examined at two different magnifications,
whereby 5 at each were randomly selected to provide a reasonable statistical representation. The 0.25 µm and
0.50 µm thick films shared similar grain sizes on the order of approximately 40 nm and 50 nm respectively while
the structure of the 1.00 µm thick films possessed an average grain size of approximately 100 nm. It should also
be mentioned that the 0.25 µm film possessed a distribution of grain sizes that may indicate the potential for
multiple deformation mechanisms operating at the same time [20, 21]. The cross-sectional microstructure was
columnar for all films and each possessed a strong (111) texture, see Fig. 2. As a final point, the columnar grain
structure has both an in-plane and transverse grain size, which may complicate microstructural-based analyses.
Fig. 1: Scanning electron micrographs illustrating Fig. 2: An SEM micrograph of the cross-
the microstructure of the nanocrystalline gold sectional microstructure and inverse pole figure
films (a) 0.25, (b) 0.50 and (c) 1.00 µm thick. illustrating the film texture for the 0.50 µm film.
In order to subject the freestanding films to a variety of strain rates, the vertical displacement rate in the MDE
tests was altered in a systematic manner at rates of 10 nm/s, 50 nm/s, 200 nm/s, 500 nm/s and 1000 nm/s. Due
to the particular loading geometry of the MDE technique, the strain rate in the plane of the membrane is not
constant and is related to the vertical displacement rate, see Espinosa et al. [18]. In order to assign a relative in-
plane strain rate to the specimens subjected to different loading rates we chose to report the strain rate achieved
at the specimen’s yield stress.
RESULTS and DISCUSSIONS: Table I Table 1 lists the values of yield strength and ultimate strain obtained from
the MDE results at each displacement rate for each film thickness. Each reported value represents the average
of 5 specimens with nearly identical dimensions, microstructure and testing conditions. Also included is the
standard deviation in each group of 5 specimens. Values are within the range reported by other researchers;
however, it should be mentioned that the mechanical properties of gold in particular are highly sensitive to minute
impurity levels [22-24] and results may vary from lab to lab. Two very distinct trends are seen in the data: (1)
yield strength was found to decrease appreciably as the displacement/strain rate decreased, more so for the
thinner films; and (2) the yield strength was found to significantly increase as film thickness decreased,
particularly at the larger displacement/strain rates. These trends can be attributed to both the film microstructure
and change in dominant plastic deformation mechanism.Table 1: Yield strength and Hall-Petch Constants of E-Beam evaporated gold films subjected to various
applied displacement rates.
Displacement
Rate 500 200 50 10
1000 nm/s
Thickness nm/s nm/s nm/s nm/s
(Grain Size)
Yield 0.25 µm (40 nm) 390 ± 14 358 ± 11 308 ± 4 275 ± 7 230 ± 14
Strength 0.50 µm (50 nm) 355 ± 7 330 ± 7 313 ± 7 275 ± 7 233 ± 15
(MPa) 1.00 µm (100 nm) 277 ± 25 250 ± 10 235± 9 237 ± 15 218 ± 15
Ultimate 0.25 µm (40 nm) 2.4 ± 0.5 2.8 ± 0.2 3.1 ± 0.2 4.2 ± 0.5 6.0 ± 0.6
Strain 0.50 µm (50 nm) 2.3 ± 0.2 2.4 ± 0.2 2.9 ± 0.1 5.7 ± 0.4 7.9 ± 0.8
(%) 1.00 µm (100 nm) 1.4 ± 0.3 1.5 ± 0.3 1.5 ± 0.4 1.6 ± 0.6 2.0 ± 0.7
-1
Ė(s ) Ave. strain rate at σy 3.0 x 10-4 1.5 x 10-4 5.5 x 10-5 1.5 x 10-5 5.0 x 10-6
KH-P (MPa nm1/2) 1932.8 1872.3 1375.7 704.1 190.1
2
R 0.9994 0.9991 0.8888 0.9222 0.8908
Given that the grain size for all films was within the nanoscale size regime, they likely possessed
increased grain boundary volume normally attributed to this size regime [25], which has been shown to force a
change in the dominant plastic flow mechanism at small enough grain size [26-28]. This effect would be
intensified in the thinner films where grain size is significantly smaller than in the 1.00 µm film. Further evidence
to support this supposition includes the convergence of yield strength values, regardless of microstructure, at the
lowest applied strain rate and the appreciable increases in ultimate strain supported by the films at the lower rates,
approaching 8% in the 0.50 µm thick film. In this regard, it is clear that the 0.50 µm film exhibits more ductility
than the 0.25 µm film when considering that the film thickness is doubled between the 0.25 and 0.50 µm films and
yet the average grain size only increased from 40 to 50 nm. One can then state that the 0.50 µm film possessed
a larger portion of grain boundary volume per total volume, which may point to the influence of grain boundary-
based mechanisms.
In terms of film thickness, the yield strength was found to significantly increase as film thickness
decreased, particularly at the larger displacement/strain rates. This is not surprising as the grain size was
relatively correlated to film thickness and the films are exhibiting Hall-Petch boundary hardening. The
dependence of yield strength on grain size can be described according to the Hall-Petch relationship [3-5]:
σ y = σ 0 + K H − P d −1 / 2 , (1)
where σ0 is the intrinsic stress caused by lattice friction, KH-P is the Hall-Petch constant and σy and d are the yield
strength and grain size respectively. Fig. 3 is a Hall-Petch plot illustrating the yield strength vs. d -1/2 relationship
for the 5 applied displacement rates. The average strain rate at each displacement rate is indicated on the plot
and listed in Table 1. In examining the slope of each curve (solid line), or the so-called Hall-Petch constant KH-P,
one can see a gradual decrease as displacement/strain rate decreased. Physically, the Hall-Petch constant
quantifies the strength of the microstructural barriers in resisting plastic flow. In this case, the strength of the
microstructural barriers is not changing with strain rate. The actual change in slope reflects the influence of other
deformation mechanisms, which become more prominent and dominate the onset of plastic flow as rate
decreases.
One can generate a plot of the Hall-Petch slope versus strain rate, which enables a direct observation of
the degree to which the contribution of dislocation-based processes influences plastic flow, see Fig. 4. A clear
trend is seen whereby at small displacement rates there is little contribution from boundary hardening, while at
larger rates their affect increases and appears to saturate at a value of approximately 2 x 103 MPa·nm1/2 near a
strain rate of approximately 1.5 x 10-4 s-1 (displacement rate of 500 nm/s). This suggests that at room
temperature, boundary hardening processes dominate plastic flow in thin gold films when displacement rate is on
the order of this value or greater. In contrast, below 1.5 x 10-4 s-1, other deformation mechanisms activate and
begin to influence and control the plastic flow rate in an increasing manner as displacement rate decreases.
These results may, in part, shed some light on the wide variability of yield strengths and Hall-Petch relationships
of nanocrystalline gold reported by many researchers [9-14].In order to investigate which deformation mechanisms are responsible for the observed strain rate
behavior, analyses were performed using models from known mechanism. The values of the stress exponent, n,
for power-law creep were found to be 8.7, 9.3 and 15.9 for a film thickness of 0.25, 0.50 and 1.00 µm respectively.
Both of the thinner films were within the range of 3 to 10 for metals experiencing this climb-glide controlled creep
[29, 30]. Alternatively, the creep rates for grain boundary diffusion and sliding can be estimated using the in-plane
grain size in the Coble [31] and Ashby [32] models and compared to the experimental strain rates. These
estimates showed that the Coble creep rate was at least one to two orders of magnitude lower than the
experiments and the Ashby rate was also significantly lower for all but the thinnest film. This analysis suggests
that climb and glide controlled creep may be the likely rate controlling mechanisms at the lower strain rates.
However, our applied strain rate was not constant and there was a noticeable scatter in the experimental strength
data, the later may in part be related to the grain size distribution of the 0.25 µm film [20, 21]. Therefore, other
mechanisms, such as the nucleation of partial dislocations, grain boundary sliding, twinning, etc., may be
participating as well.
Displacement Rate (nm/s)
100 nm 50 nm 40 nm 200 400 600 800 1000
2500
400
1000 nm/s
500 nm/s
Hall-Petch Slope (MPa·nm1/2)
200 nm/s 2000
360 50 nm/s
Yield Strength (MPa)
10 nm/s
320 1500
280
1000
240
500
200
0 0
0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
d-1/2 (nm-1/2) Strain Rate x 10-4 (s-1)
Fig. 3: A Hall-Petch plot illustrating the relationship Fig. 4: A plot of the Hall-Petch slope versus
of grain boundary hardening on plastic flow as a strain rate illustrating the competition between
function of applied strain rate (the solid line is linear dislocation-based and grain boundary diffusion
regression). deformation mechanisms.
In light of the apparent transition in rate-controlling deformation mechanism beginning below a strain rate of 1.5 x
10-4 s-1, one can reinterpret the Hall-Petch plot. The three lowest strain rates can then be re-interpolated by
considering the trends in the data rather than performing linear regression, see dashed lines in Fig. 3. The re-
assessment indicated possible evidence that inverse Hall-Petch behavior may be occurring at the lower strain
rates in a grain size range of 40 to 50 nm. It is interesting to note that these observations are comparable with
predictions of the Phase Mixture model for nanocrystalline metals proposed by Kim and Estrin [15], which is
based upon grain boundary diffusion mechanisms. However, it is unclear at this time exactly which mechanism(s)
is controlling the flow rate.
CONCLUSIONS: In summary, the applied strain rate was found to have significant influence on the plasticity of
nanocrystalline gold films at room temperature. When subjected to a strain rate on the order of 1.5 x 10-4 s-1 or
greater, Hall-Petch boundary hardening controlled the plastic flow rate while at lower strain rates flow rate
transitioned to control by one or more other mechanisms. The results also indicated the possibility that inverse
behavior was occurring and that the critical grain size denoting this transition appeared to increase as strain rate
decreased.Acknowledgement: This work was sponsored by the National Science Foundation, Directorate for Engineering,
Civil & Mechanical Systems Division under award number CMS-0528265. The authors would like also like to
thank to Conventor Ware, Inc. for providing software to complete the simulation work.
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