Crystal plasticity as a mean field depinning transition: results from a phase field crystal model

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by  Georgios Tsekenis

Crystal plasticity as a mean field depinning transition: results from a phase field
crystal model
Georgios Tsekenis, Thomas F. Fehm, Jonathan T. Uhl, Nigel Goldenfeld and Karin A. Dahmen
Department of Physics, University of Illinois at Urbana-Champaign,
Loomis Laboratory of Physics, 1110 West Green Street, Urbana, Illinois, 61801-3080.
Until now, most of our knowledge about the universality class of crystal plasticity has come
from simulations using discrete dislocation dynamics. These are force-controlled, typically at zero
temperature, and deal with the creation and annihilation of dislocations phenomenologically. In
thiswork,wegobeyondtheselimitationsbyusingphasefieldcrystalsimulationsintwodimensions
atfinitetemperaturetoextracttheavalanchestatisticsofasimulatedcrystalunderconstantshear
5 velocity. Inadditiontotheavalanchesizeandenergydistributionsweextracttheavalancheduration
1 distributions and power spectra. All exponents and scaling functions extracted here for the statics
0 and dynamics of crystal plasticity, belong to the mean field elastic depinning universality class,
2 confirming earlier findings based on discrete dislocation dynamics.

n
a PACSnumbers: 62.20.F-,61.72.Bb,89.75.Da,64.60.De
J

9
Crystalplasticityisinsomesenseasolidstateanalogue several avalanche measures and show that they are dis-

] of fluid turbulence, with deformation at micro scales be- tributed according to power-laws over several orders of
i
c ing both intermittent and spatially inhomogeneous [1– magnituderevealinglongrangespatialandtemporalcor-
s 11]. These phenomena have been captured realistically relations. The set of critical exponents we calculate (the
-
l byanumberofapproaches,includingdiscretedislocation duration distribution power law exponent for the first
r
t dynamics (DDD) models [1, 11–17], continuum models time) fully supports the mean field depinning picture for
m
[12], phase field models [18, 19] and phase field crystal crystalplasticity. Thisisinstrongagreementwithearlier
. (PFC) models [20]. The most prominent open question 2D DDD simulations [11]. It also agrees with 3 dimen-
t
a thatremainsunansweredistheonethatunifiestheaccu- sional simulations of dislocation dynamics [15], with an-
m
mulated literature and solidifies the hard-earned knowl- alytics[10,26]andexperiments[9,10,21]. Ourworkad-
- edge: What is the universality class of crystal plasticity? dressesthefundamentalquestionoftheuniversalityclass
d
n A wealth of critical exponent and scaling function of crystal plasticity and is relevant to the deformation of
o information from DDD simulations (in two [11, 12] nano-crystals [21, 27, 28] and micro-crystals [1, 9, 10] as
c and three dimensions [15]) and experiments on slowly the need for miniaturization of devices expands both in
[
compressed nanocrystals and microcrystals [9, 10, 21] breath and depth.

1 strongly suggest that crystal plasticity belongs to the The Phase Field Crystal Model, Sheared: The phase
v
mean field depinning universality class. Nevertheless the field crystal (PFC) model [23, 31, 32] describes how the
0
issue is by no means settled; for example recent DDD local density of atoms changes with time while main-
6
1 simulations in 2D obtained non-mean field results [22]. taining the symmetries and periodicity of the lattice. In
2 Our main aim in this paper is to uncover the univer- addition, the elastic interactions of the atoms are also
0 sal behavior of deforming crystalline matter as it em- captured by the PFC allowing for the elasticity of the
.
1 anates from microscopic origins and percolates through crystal to be expressed. These characteristic properties
0 all scales. To this end we study crystalline plasticity of the phase field crystal model are distinctly different
5 with a phase field crystal model [20, 23–25]. The ele- fromthephasefieldmodel. Inatypicalphasefieldmodel
1
mentary entity in our simulations is a phase field repre- the phase field describes the dynamics of interfaces that
:
v senting local atomic density, which is appropriately con- separate dissimilar regions without keeping track of the
Xi strained to behave as an atomic crystal, and indeed can microscopic information inside those regions. In [18, 19]
be related to density functional theory[24]. The phase Koslowski et al. developed a phase field model to simu-
r
a field crystal exhibits elastic, reversible deformation at late dislocations as interfaces (separating crystal regions
small external loads. It deforms plastically at exceed- with different accumulated slip). In studying plasticity,
ingly large external loads. Large deformations imprint however, it is important to capture the microscopic de-
permanent, irreversible change in the lattice structure. tails such as the dislocations which disrupt the periodic-
Atsufficientshearingtheperiodicityisbrokenandtopo- ity of the perfect lattice. At the same time, it is impor-
logical defects emerge in the system. Dislocations travel tant to capture the macroscopic behavior as well such as
and interact with each other through the lattice forming the collective motion of the dislocation ensemble. The
intricatestructures. Onecanobserveplasticdeformation phase field crystal model is particularly successful in do-
beingmediatedthroughintermittentdislocationmotion, ing that in an elegant way [20].
i.e. through discrete slip-avalanches. Here we extract The total free energy in the phase field crystal (PFC)
2
quantity exponent DDD sims our PFC sims MFT other sims experiments
D (S)∼S−κ κ 1.5 [11] 1.5 (Fig. 1) 3 1.4[12],1.6[13],1.5[15]* 1.5-1.6[9],1.5[21, 27]
S 2
S ∼(cid:0)1− τ (cid:1)−σ1 1 2 [11] 2 2[12],2[15]* 2[12, 21]
max τc σ
D(V )∼V−κA κ 2[29] 1.8[18, 19] 2[2], 2.0±0.1[5]
max max A
” ” 1.5-2[7],1.2-2.2[6]
Dtt(taval)∼∼(cid:0)t1av−al−τ1−(cid:1)−κσ−ννzz1 κt =1νz+ κσ−νz1 2 (Fig. 1) 21
aval,max τc
DE(E)∼E−1+2−κ−σνσzνz κE =1+ 2−κ−σν1z 1.3 [11] 1.3 (Fig. 1) 43 1.8±.2[1]+ 1.6±0.05[1],1.5±.1[4]
E ∼(cid:0)1− τ (cid:1)−2−σσνz 2−σνz 3 [11] 3
max τc σ
(cid:104)S(cid:105)∼tavalσ1νz σ1νz 2 [11] 2 (Fig. 2) 2 1.5[14]+
(cid:104)t (cid:105)∼Sσνz σνz 0.5 [11] 1
aval 2
V(t)shape ∼tavalσ1νz−1 σ1νz ∼1.9 [11] 2 1.5[14]+
PS,PSint(ω)∼ω−σ1νz σ1νz 2 [11] 2 (Fig. 2) 2 1.5[14]+
(cid:104)Sm(cid:105)∼L1+mν−σκ−σ ν 1.0±0.2 [11] 1
(cid:104)V(cid:105)∼(cid:0)τ −1(cid:1)β β 1.1±0.1 [11] 1 1.8[30]+
τc
TABLE I: Table of exponents. Our results from 2D PFC simulations are shown in the fourth column while our results from
2D DDD simulations are shown in the third column. Mean field interface depinning values are in the fifth column. Results
from a full 3D DDD simulation are indicated with an asterisk (*). Results from a 2D DDD simulation with creation and
annihilation in the steady state are indicated with a plus sign (+). Symbol definitions: D(x) stands for the distribution of x,
x isthemaximumofthedistributionofx,PS standsforpowerspectrum,(cid:104)x(cid:105)standsforaverageofx,S isthesizeofaslip
max
avalanche, t is its duration, E its energy. V(t)=(cid:80)N |v (t)| is the collective dislocation speed. τ stands for shear stress,
aval i=1 i
τ is its critical value. Small greek letters are used for critical exponents throughout.
c
model [23, 31] oftemperature(cid:15)∼k T. Thisfreeenergyisgovernedby
B
conservative,relaxational,diffusivedynamicsthatcanbe
F{ρ}=(cid:90) (cid:20)ρ(∇2+1)2ρ+ rρ2+ ρ4(cid:21)ddx (1) derived from density-functional theory [32].
2 2 4 The PFC solid is a perfect triangular lattice in equi-
librium, but its excitations are phonons and topological
isafunctionalofρ((cid:126)x,t)thelocaldensityofthephasefield
defects such as dislocations. When the phase field crys-
(at point (cid:126)x=(x,y) in space and time t). The first term
tal is sheared, it will respond by generating dislocations,
in Eq.(1) penalizes departures of ρ((cid:126)x,t) from periodic-
justasarealcrystal. Theabilitytocreateandannihilate
ity, thusdescribingacrystalstructureasadensitywave.
dislocationsnaturallyandeasilyisoneoftheadvantages
The last two terms impose a double well potential (to
of the phase field crystal method, compared to DDD.
lowest order) similar to the Landau ansatz. The reduced
Weappliedashearstrainratealongthexdirectionat
temperature r is given by (T −T )/T and controls the
c c the y =0,L boundaries by adding the convection term
phase behavior. The material is liquid for temperatures y
v(y)∂ρ/∂x,totheevolutionequationEq.(2). Thebound-
higherthanacriticaltemperatureT whileitcrystallizes
for temperatures below T . Thus, fcor r > 0 one finds a aryshearvelocityprofile, v(y)=±v0e±(y0−y)/Ly (y0 =0
c for+, y =L for−)isdesignedtobemainlycontrolled
liquid, constant ρ, phase (because the potential is single 0 y
by the velocity at the boundary v since its penetration
well)whileforr <0thestablestateisatriangularlattice 0
length λ (cid:28) L does not affect the results strongly. The
or a striped phase (due to the double well potential). y
simulations take place in a square box of sides L ,L in
The PFC density ρ((cid:126)x,t) evolves according to the mas- x y
the x,y direction. The boundary conditions are periodic
sive phase field crystal equation [20, 33]:
in x and fixed at y = 0,L i.e. we design the simula-
y
∂2ρ ∂ρ δF ∂ρ tion cell such that the crystal wraps around in x=0,Lx
+(β) =(α)2∇2 +v(y) +η, (2) and terminates at y =0,L (without wrapping around).
∂t2 ∂t δρ ∂x y
That way we can easily apply a fixed shear rate at the
where α controls the range and β the time scale of y = 0,Ly boundaries and allow the dislocations to flow
phonon excitations of the crystal [33]. Thermal fluctu- unbounded through the x = 0,Lx boundaries effectively
ations are represented by the stochastic noise η which simulatingalargerthermodynamicsystemthanthemere
is assumed to be Gaussian with second moment given dimensions of our basic simulation cell.
bythefluctuation-dissipationtheorem(cid:104)η((cid:126)x,t)η((cid:126)x(cid:48),t(cid:48))(cid:105)= ThePFCmodelhastheaddedvalueoverDDDsimula-
−(cid:15)∇2δ((cid:126)x−(cid:126)x(cid:48))δ(t−t(cid:48)). Thenoiseamplitudesetsthescale tions that it incorporates nonlinear elasticity [34] as well
3
as dislocation creation and annihilation seamlessly with- example the collective dislocation velocity is defined as
outrequiringadditionalphenomenologicalrulestomodel V(cid:48)(t)=(cid:80)N(t)b(cid:126)v (b isthedislocation’sburgersvector)
i=1 i i i
these number-changing operations. The PFC methodol- and represents the strain rate of the crystal [10, 12, 30].
ogyisthusuniquelycapableofaddressingsuchquestions Note that all dislocations, single dislocations or disclina-
as how strain heterogeneity drives dislocation number tionpairsthatmoveathighspeedsandgrainboundaries
fluctuations,whichinturncoupletoplasticityavalanches that move slowly, participate equally in the calculation
[35]. SincethePFCmodelisinessenceanatomicsimula- of V(t) (see figure in Supplementary Material).
tion,thehighlynonlinearinteractionatsmalldislocation Aperfectunshearedtriangularcrystalisdevoidofdis-
distancesiscapturednaturallythroughthecrystallattice locations and therefore every atom has n = 6 nearest
i
that mediates it. The same holds true for the particular neighbors(identifiedwithDelaunaytriangulation). Con-
effects of creation and annihilation of dislocations. versely there exists one dislocation for every atom with
The PFC model handles applied shear velocity na- n = 5 or n = 7 neighbors since vacancies are not al-
i i
tively. DDD models incorporate applied external stress lowed.
naturally. ThusthePFCissuitabletoinvestigatetheslip Wecapturethespeedofthedislocations,V(t),through
avalanchesabovethecriticalpoint(flowstress)inthede- the speed of these defect atoms with n (cid:54)= 6 neighbors,
i
pinned state while the DDD below the critical point, in V˜(t)=(cid:80)Ndefect(t)|(cid:126)u |, where N (t) is the number of
i=1 i defect
thepinnedstate. Inthatsensetheyperformcomplemen- defect atoms and (cid:126)u is the velocity of defect atom i.
i
tarily to each other. In order to partition the signal into individual slip
PFC simulations at Finite Shearing Rate and Temper- avalanchesweapplyathreshold, V , toitforeachtem-
thr
ature: Westudycrystalplasticityasitproceedsintermit- peratureandshearingratewesimulate. Thebeginningof
tently through slip avalanches using the sheared phase theavalancheissignifiedataninstantwhenthecollective
field crystal (PFC) model [20]. We obtain the main scal- dislocation speed V(t) intersects upward the threshold
ing behavior of the distribution of a variety of avalanche while its end is when V(t) crosses the threshold down-
measures and for several different temperatures ((cid:15)) and ward immediately after. We extract the probability dis-
shearing rates (v0). We find remarkable agreement be- tribution of the avalanche duration
tween simulations and analytical mean field theory pre-
dictions of exponents [10, 26, 36]. Our results strongly t =t −t , (4)
aval finish start
support the critical point picture of plasticity, and sug-
gest new experiments. size(alsocalledactivityfluctuationsintheflowingstate)
At every time step we obtain the phase field density S =(cid:82)tfinishV(t)dt,andenergyE =(cid:82)tfinishV2(t)dt,where
ρ in a 2D square simulation cell (i.e. L = L = L) t atsntadrtt are the starting andtsteanrtding time of the
x y start finish
throughEq.(2). LargevaluesofρindicatesPFC‘atoms” event respectively. In order to see the fluctuations that
while low field signifies interatomic space, cooperatively correspond to slip avalanches we applied a threshold
arrangedintoatightcrystal(triangularin2D).Byapply- equal to the average of the signal, V(t), in each realiza-
ing shear along the fixed boundaries y = 0,L the trans- tion of total time t , V = 1 (cid:82)ttotalV(t)dt,
lational symmetry breaks and dislocations are created in total threshold ttotal 0
We also calculate the power spectrum of V(t):
an attempt to relieve the high stress accumulated near
tinhtoesreabctouwnitdhareiaecsh. Ootnhceertthoefdoirsmlocpaatiirosnasnadremcoreraetceodm,tphleexy PS(ω)=(cid:12)(cid:12)(cid:12)(cid:90) ttotalV(t)e−iωtdt(cid:12)(cid:12)(cid:12)2. (5)
(cid:12) (cid:12)
structuressuchaslow-anglegrainboundaries;individual 0
dislocations, dislocation pairs and grain boundaries can Thepowerspectrumrevealsthefrequencycontentofthe
beseeninthesnapshotofthePFCsimulationsinSupple- timeseriesofthecollectivedislocationspeedanditneeds
mentary Material. Dislocations may also glide through- no thresholding. It is equivalent to extracting the time-
out the entire crystal, allowing for slips to self-organize time correlations of the collective dislocation behavior
into slip avalanches. We quantify the avalanche activity and near a critical point it is expected to have no char-
by extracting the total speed of the dislocations, acteristic scale, i.e. be scale-free [37, 38].
For each shearing rate and temperature, we run 48
N(t)
(cid:88) differentrealizationseachwithdifferentseedfortheran-
V(t)= |(cid:126)v |, (3)
i domnumbergeneratorforthenoiseη toobtainsufficient
i=1
statistics. This results in tens of thousands of avalanche
where N(t) is the number of dislocations in the sys- events for each shearing rate v and temperature (cid:15). In
0
tem at time t and (cid:126)v is the speed of the i-th disloca- Fig. 1 we show the event size, duration and energy dis-
i
tion. This measure is similar to the acoustic emission tributions while in Fig. 2 the power spectra and average
signalinWeisset al.’ssinglecrystaliceexperiments(e.g. size versus duration for different shearing rates at the
[1]). Other variants of the avalanche activity measure same temperature. We find that the distributions follow
in the literature can track the avalanches as well. For a power law for small event sizes and cut off at larger
4
sizes, with the maximum avalanche size not exhibiting 104
v0=0.510,ε=1.6
a strong dependence on shear rate over the simulated v0=0.765,ε=1.6
v0=1.020,ε=1.6
range of the parameter v . We suspect that the reason 103
0
that shear rate does not affect the distribution much is
that for our systems the system size sets the cutoff of 102
the avalanche size distribution. For much larger systems 104
the mean field theory predicts that an increase in shear S()ω 101
P 103 1/σ νz=2.0
rate will reduce the cutoff of the avalanche size distribu-
>
tion. This can only be seen in systems that are so large 100 <S102 1/σ νz=2.0
that the system size is much larger than the correlation
length of the avalanches given by the finite shear rate. 10-1 101
100 101 102
InpreviousworkwithPFC[20]adifferentthresholdwas 10-2 taval
applied to the signal so as to extract avalanches. It re- 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100
sulted in a rate-dependent avalanche distribution cutoff. ω
Although both thresholds reveal the same power-law in
FIG. 2: (color online) The power spectrum PS(ω) of
theavalanchedistributionswebelievethethresholdused the collective dislocation speed (main) and the average slip
here (equal to the average of the dislocation activity for avalanche size (cid:104)S(cid:105) versus duration t (inset) for different
aval
each run) is a more natural way to quantify the fluctua- shearing rates v at the same temperature (cid:15) = 1.6. The
0
tions around the mean dislocation activity. powerspectrumscaleswiththeinversesquarefrequencygiv-
ing 1/σνz ≈ 2. The average size scales with the square of
101 the duration, 1/σνz ≈ 2, for avalanches that are sufficiently
100 κ=1.5 small not to touch the sample boundaries. (The boundaries
ofthepower-lawscalingregimeofthepowerspectrumPS(ω)
10-1 κΕ=1.3 areinverselyproportionaltotheboundariesofthepower-law
(E)E10-2 κt=2.0 scalingregimeofthedurationdistributionDt(taval)). Results
S), D 10-3 agree with MFT predictions (Table I).
(S10-4
D
D(t), taval1100--65 ulations [39]. The slip events distribute themselves ac-
10-7 cording to power laws D (t ) ∼ t−2 , D (S) ∼ S−1.5,
1100--98 vvv000===001...570162050,,,εεε===111...666 DthEat(Ear)e∼inEa−g1r.e3e,mPeSnt(ωw)itt∼haωtvha−el2,MweaitanhvaFlcrieitldiScainlteexrpfaocneedntes-
10-1 100 101 102 103 104 105 106
pinning transition universality class (Table I).
t , S, E
aval
Our extended results from the PFC model agree with
FIG. 1: (color online) Probability distributions Dt(taval) the majority of the robust experimental and computa-
(left), D (S) (middle) and D (E) (right) of avalanche du-
S E tional results in the literature (for an extended sum-
ration t , size S and energy E respectively for different
aval mary see Table I). Friedman et al. [21] analyzes the slip
shearingratesv atthesametemperature(cid:15)=1.6. Theprob-
0 statistics of compressed crystalline nano-pillars in dif-
abilitydistributionoftheslipavalanchesizesfollowsapower
law with exponent κ ≈ 1.5, of durations with κ ≈ 2 and of ferent stress bins as the flow stress is approached and
t
energies with κ ≈1.3. These results are in agreement with get κ = 1.5 (also σ = 2). Dimiduk et al. measure
E
MFT (Table I). κ = 1.5−1.6 from compression experiments on micro-
pillars at slowly increasing stress [9]. Similarly, in 2D
Scaling Behavior of the avalanches: The distributions DDD and continuum models with quasi-static stress in-
of the avalanche size, duration and energy, the power crease in the pinned regime, Zaiser et al. calculate
spectra and average size versus duration shown at same κ=1.4 [12, 40]. Slip-event energy amplitudes are power
shearing rate, v = 0.765, and different temperature pa- law distributed with exponent κ = 1.8 [1] in simula-
0 E
rameter values (cid:15) are shown in figures in the Supplemen- tions, and κ = 1.6 in experiments [1]. Richeton et al.
E
tary Material. In Figs. 1 and 2 we presented the dis- [4] report κ = 1.5 for the energy distribution of acous-
E
tributions of the avalanche size, duration, energy, power ticemissiondeformationexperiments. Ofcoursein[11]a
spectraandaveragesizeversusdurationatthesametem- largenumberofscalingexponentswasextractedfrom2D
perature parameter values (cid:15)=1.6 and different shearing DDD simulations and as shown in Table I they corrobo-
rates, v . Each curve is characterized by a power law for ratethatcrystallineplasticitybelongstotheuniversality
0
several decades and a cutoff at large values (smaller val- class of mean field depinning for the discussed static and
ues for the power spectra) which does not change with dynamic properties of the avalanche statistics. This re-
shear rate. Mean field theory predicts the dependence sult is also consistent with analytic results [10, 26] and
on the shear rate, that should be visible in larger sim- simulations in 3 dimensions [15].
5
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