Deformation in amorphous-crystalline nanolaminates -- an effective-temperature theory and interaction between defects
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec Deformation in amorphous-crystallinenanolaminates – an effective-temperature theoryand interaction between defects
Charles K C Lieou , Jason R Mayeur and Irene J Beyerlein , Earth and Environmental Sciences and Center for Nonlinear Studies, Los AlamosNational Laboratory, Los Alamos, NM 87545, USAE-mail: [email protected] Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545,USAE-mail: [email protected] Mechanical Engineering and Materials Departments, University of California, SantaBarbara, CA 93106, USAE-mail: [email protected]
November 2016
Abstract.
Experiments and atomic-scale simulations suggest that the transmissionof plasticity carriers in deforming amorphous-crystalline nanolaminates is mediated bythe biphase interface between the amorphous and crystalline layers. In this paper, wepresent a micromechanics model for these biphase nanolaminates that describes defectinteractions through the amorphous-crystalline interface (ACI). The model is basedon an effective-temperature framework to achieve a unified description of the slow,configurational atomic rearrangements in both phases when driven out of equilibrium.We show how the second law of thermodynamics constrains the density of defects andthe rate of configurational rearrangements, and apply this framework to dislocationsin crystalline solids and shear transformation zones (STZs) in amorphous materials.The effective-temperature formulation enables us to interpret the observed movementof dislocations to the ACI and the production of STZs at the interface as a “diffusion”of configurational disorder across the material. We demonstrate favorable agreementwith experimental findings reported in (Kim et al., Adv. Funct. Mater., 2011), anddemonstrate how the ACI acts as a sink of dislocations and a source of STZs.
1. Introduction
Amorphous-crystalline nanolaminates are heterogeneous structures fabricated byalternately stacking nano-thick layers of nanocrystalline materials (commonlynanocrystalline copper) and amorphous materials (often metallic glasses) upon oneanother [1, 2, 3, 4, 5], often by means of magnetron sputtering. Figure 1 shows aCuZr/Cu amorphous-crystalline nanolaminate subject to uniaxial loading. Experiments eformation in amorphous-crystalline nanolaminates Figure 1.
CuZr/Cu amorphous-crystalline nanolaminate sample undergoing tensiletesting. (a, b) Bright-field TEM images for cross-sections of the nanolaminates withamorphous layer thickness h a = (a) 17 nm and (b) 128 nm; the crystalline layerthickness is h c = 16 nm. (c, d) SEM images of the freestanding tensile samples (c)before and (d) after tension. Adapted with permission from [3]. have shown that the addition of metallic glass layers greatly enhances the strengthand ductility of the nanocrystalline material [1], and that this effect is especiallypronounced when the metallic glass thickness is below some threshold [3, 2]. Whilesome microscopy imaging and molecular dynamics (MD) simulations suggest that theexceptional strength and ductility may be accounted for by the inhibitory effect ofthe metallic glass layer on shear band propagation in the crystalline layer [1], otherssuggest that the crystalline layer obstructs shear band propagation in the metallicglass [5]. In either case, the amorphous-crystalline interface (ACI) mediates the plasticinteraction between the two constituent materials. The ACI absorbs plasticity carriers– dislocations in the crystalline material and shear transformation zones (STZs) in theamorphous material – coming in from one side and triggers emission into the other,thereby playing an important role in controlling the deformation of the heterogeneousnanolayered structure.A predictive description that links the physical mechanisms underlying theseobservations in atomic-scale simulation to the macroscopic deformation response wouldbe useful in understanding their behavior and ultimately designing their microstructures.However, the classes of models that treat amorphous materials alone and crystallinesolids alone, in practice, use different formulations or frameworks. The commonly usedtheories of dislocations in crystalline solids are based on the thermodynamics of slipovercoming obstacles (e.g., [6, 7]).Those that describe the development of STZs often directly address the flow ofenergy and entropy in the deforming material or the principle of symmetry. It isunclear whether the basis of defect theories of two dissimilar materials can be chosenindependently and still provide a reliable mechanics model. Adopting a generalizedframework, in which the kinetics of both types of defects can be described, ought tobe the more straightforward path to take. Accordingly, for the present ACI system of eformation in amorphous-crystalline nanolaminates eformation in amorphous-crystalline nanolaminates
2. The effective temperature
This section provides a brief introduction to the concept of effective temperature.In this discussion we will make no reference to the nature of the structural flowdefects (dislocations or STZs), or that of the deforming material, but shall show thatimportant conclusions about the flow defect densities can be drawn in a systematic andstraightforward manner. For a full review of effective-temperature thermodynamics thereader is referred to [18, 19, 20].In a deforming solid, the atomic configurational degrees of freedom – those thatdescribe their interactions and relative positions – are driven out of equilibrium with thefast, kinetic-vibrational degrees of freedom by external forces. The configurational andkinetic-vibrational subsystems are only weakly coupled to one another. Despite this, thetwo subsystems do exchange energy with one another when clusters of atoms rearrange ina nonaffine, irreversible manner, albeit extremely slowly when compared to the atomictime scale of order τ ∼ − s. Thus we focus exclusively on the configurationalsubsystem; denote by U C and S C its energy and entropy. U C is a function of S C andthe density ρ of defects, and perhaps of other order parameters that we omit for thetime being. Conversely, S C is the configurational entropy computed by counting thenumber of atomic configurations, or the number of possible arrangements of defects, atfixed energy U C and defect density ρ . Next, define the effective temperature (here withunits of energy): χ = (cid:18) ∂U C ∂S C (cid:19) ρ . (1)In a deforming solid undergoing atomic rearrangements, defects are driven byexternal forces to explore a large swath of the available configurational phase space;thus, the configurational degrees of freedom must be maximizing the entropy S C duringthis process. The configurational energy U C , meanwhile, is determined by the balancebetween the external work rate and the rate at which energy is dissipated into kinetic-vibrational subsystem. This process minimizes the configurational free energy givenby F C = U C − χS C . (2)If e D is the characteristic energy of a single defect, and ρ is the population densityof defects, then in the noninteracting defects approximation, U C = V e D ρ , where V isthe total volume of the solid. Meanwhile, elementary statistical mechanics shows that S C ∝ − ρ ln ρ + ρ . Minimizing F C , we find that the instantaneous, steady-state defectdensity ρ ss must be given by ρ ss ∝ e − e D /χ . (3)This argument applies directly to dislocations in a crystalline solid, with the caveat thatwhen we refer to the dislocation density per unit area, we replace the volume V by somecross-sectional area A . For STZs there is an extra order parameter m that denotes the eformation in amorphous-crystalline nanolaminates ss = 2 e − e Z /χ , (4)where e Z is a characteristic STZ formation energy. Thus, the evolution of the defectdensities is largely controlled by the temporal evolution of the effective temperature,governed by the input power and the energy dissipation rate.Next, we proceed to derive the evolution equation for the effective temperature χ from the first law of thermodynamics as follows. Let U K and S K denote the energy andentropy, respectively, of the kinetic-vibrational degrees of freedom, and denote by θ thethermal temperature in energy units. Then the total energy equals U K + U C , and theenergy balance equation reads˙ U C + ˙ U K = V σ ˙ ǫ pl = χ ˙ S C + (cid:18) ∂U C ∂ρ (cid:19) S C ˙ ρ + θ ˙ S K . (5)In (5), σ and ˙ ǫ pl denote the stress and plastic strain rate. From here onwards, wespecialize to the case of tensile deformation, so that it is permissible in most cases touse the magnitudes σ and ˙ ǫ pl instead of writing the tensorial product for the inputpower V σ : ˙ ǫ pl in full. (Note that only the plastic work of deformation plays a role; theelastic work of deformation cancels out of this equation following the argument in [19].)Meanwhile, the second law of thermodynamics says that ˙ S C + ˙ S K ≥
0; eliminating S C using (5), we find V σ ˙ ǫ pl − (cid:18) ∂U C ∂ρ (cid:19) S C ˙ ρ + ( χ − θ ) ˙ S K ≥ . (6)Since this holds for all possible motions of the state variables, each independent termmust separately be non-negative. Because V σ ˙ ǫ pl ≥ − (cid:18) ∂U C ∂ρ (cid:19) S C ˙ ρ ≥
0; (7)( χ − θ ) ˙ S K ≥ . (8)In order for each of these inequalities to hold, both multiplicative factors in the inequalitymust switch signs at the same point. Thus the first inequality constrains the steady-statedefect density ρ ss in the same manner as before. (In the case of STZs, for which there isan extra orientational order parameter, an extra constraint for the transition rate arises;see for example [20].) The second inequality says that ˙ S K must be proportional to thetemperature difference χ − θ , which must be non-negative. We write this in the form θ ˙ S K = K ( χ, θ )( χ − θ ) , (9)where K is a non-negative coupling coefficient between the configurational and kinetic-vibrational subsystems. Finally, substituting this into (5), and using χ ˙ S C ≃ V c eff ˙ χ , eformation in amorphous-crystalline nanolaminates c eff is an effective specific heat capacity, we arrive at the evolution equation forthe effective temperature: V c eff ˙ χ = V σ ˙ ǫ pl − K ( χ, θ )( χ − θ ) − (cid:18) ∂U C ∂ρ (cid:19) ˙ ρ. (10)We shall now apply this effective-temperature formulation to STZs in amorphous solidsand dislocations in crystalline solids independently.
3. Effective-temperature theory of STZs in amorphous solids
This section provides a brief review of the STZ theory of plastic deformation inamorphous solids. The interested reader is referred to, for example, [9, 10, 20], fordetails and derivations.In the STZ description, the order parameters of interest are the STZ density Λ andthe orientational bias m . STZs fluctuate into and out of existence due to the thermalmotion of the atoms and the mechanical work input. The former is unimportant if weconfine ourselves to metallic glasses below the glass transition temperature as in [1, 3],while the latter is described by a mechanical noise strength Γ. When subjected toexternal stresses, the atoms in an STZ undergo irreversible arrangements and produceplastic strain. The tensorial relation between the plastic strain rate and the stress is τ ˙ γ pl ij = ǫ C (¯ s )Λ s ij ¯ s ( T (¯ s ) − m ) . (11)Here, τ is the fundamental molecular time scale or the inverse attempt frequency, and ǫ is the ratio of the STZ plastic core volume to the atomic volume. C (¯ s ) and T (¯ s ) aresymmetric and antisymmetric combinations of the forward and backward STZ transitionrates R ( ± ¯ s ): C (¯ s ) ≡
12 ( R (¯ s ) + R ( − ¯ s )) ; T (¯ s ) ≡ R (¯ s ) − R ( − ¯ s ) R (¯ s ) + R ( − ¯ s ) , (12)where ¯ s ≡ q s ij s ij , and s ij is the deviatoric stress tensor. In a similar vein theplastic strain rate tensor and the deviatoric plastic strain rate are related through¯˙ γ pl ≡ q ˙ γ pl ij ˙ γ pl ij . Thus, in the case of tensile loading, if we choose coordinate systemssuch that the only nonzero element of the total stress tensor is σ xx = σ , and that thecorresponding plastic strain rate is ˙ ǫ pl , then the nonzero elements of the deviatoric stresstensor s ij = σ ij − tr( σ ij ) are s xx = 23 σ ; s yy = s zz = − σ, (13)and the deviatoric plastic strain tensor has nonzero elements˙ γ pl xx = ˙ ǫ pl ; ˙ γ pl yy = ˙ γ pl zz = − ˙ ǫ pl / . (14)Thus, σ = √ s and ˙ ǫ pl = (2 / √ γ pl . The plastic work of deformation, or the dissipationrate excluding those attributed to the change of internal state variables, is σ ˙ ǫ pl = 2¯ s ¯˙ γ pl .The rest of the paper is devoted to tensile deformation. For convenience, from nowon we use the experimentally measured tensile stress σ instead of the deviatoric stress eformation in amorphous-crystalline nanolaminates s in the arguments for the STZ transition rate factors C (¯ s ) and T (¯ s ). The tensile stressevolves with time according to linear elasticity; that is,˙ σ = E ( ˙ ǫ − ˙ ǫ pl ) , (15)where E is the Young modulus of the amorphous material, and ˙ ǫ is the applied strainrate. The plastic strain rate ˙ ǫ pl evolves according to τ ˙ ǫ pl = 2 √ ǫ Λ C ( σ ) ( T ( σ ) − m ) , (16)where, below the glass transition temperature [20], as is the case in the experiments ofinterest [1, 3], C ( σ ) = exp (cid:18) − T E T (cid:19) cosh (cid:18) ǫ σa √ χ (cid:19) , (17)and T ( σ ) = tanh (cid:18) ǫ σa √ χ (cid:19) ; m = ( T ( σ ) if σ T ( σ ) ≤ σ σ /σ if σ T ( σ ) > σ . (18)Here, T E is an activation temperature and a is the atomic radius. The stress σ maybe interpreted as a yield stress parameter. It emerges from the proportionality betweenthe mechanical noise strength Γ and the plastic dissipation per STZ as a proportionalityconstant [20]: Γ = √ τ σ ˙ ǫ pl ǫ σ Λ . (19)The STZ density Λ evolves according to the equation˙Λ = Γ τ (2 e − e Z /χ − Λ) . (20)As before, the quantity e Z is the STZ formation energy, and in section 2, we argued thatat steady state Λ ss = 2 e − e Z /χ . Finally, the effective temperature χ evolves according to(10). After some algebraic simplifications, we find c eff ˙ χ = σ ˙ ǫ pl (cid:18) − χχ (cid:19) − e Z a ˙Λ . (21)In (21), c eff is the so-called effective heat capacity. It has the dimensions of inversevolume. The effective temperature evolves to some constant value χ in the steadystate. Strictly speaking, the steady-state value should be a function of the strainrate. At strain rates slower than the internal relaxation rate controlled by the atomicvibration frequency (i.e., when τ ˙ γ pl ≪ χ is sufficient.Note that the equation of motion for Λ, (20), does not contain an overall factorof Λ, which we have assumed to be small. On the other hand, the equation of motionfor χ , (21), contains the small factor of Λ through ˙ ǫ pl . Thus Λ is a fast variable while χ is a slow variable. For most purposes, we can use the steady-state approximationΛ ≈ Λ ss = 2 e − e Z /χ , which we shall do from here onwards. eformation in amorphous-crystalline nanolaminates σ = E ( ˙ ǫ − ˙ ǫ pl ) , (22) c eff ˙ χ = σ ˙ ǫ pl (cid:18) − χχ (cid:19) , (23) τ ˙ ǫ pl = 4 √ ǫ e − e Z /χ C ( σ ) ( T ( σ ) − m ) . (24)
4. Effective-temperature theory of dislocations in crystalline solids
Deformation in the crystalline layer is mediated by dislocations. Like STZs, themotion of dislocations can be analyzed in a statistical-thermodynamic framework. Thedevelopment here closely follows that of [16, 17]. As in the amorphous layer, the tensilestress increases linearly with the elastic strain rate; thus,˙ σ = E ( ˙ ǫ − ˙ ǫ pl ) . (25)Note, however, that both the Young’s modulus E and the plastic strain rate ˙ ǫ pl generallydiffer from those in the amorphous layer.The derivation of the expression for the plastic strain rate starts with the Orowanrelation for the plastic shear rate ˙ γ pl :˙ γ pl = ρbv. (26)In this equation, ρ is the areal density of mobile dislocations, to be distinguished from thevolume density or number density in section 2 above. (Here, we do not study the motionsof individual dislocations; rather, we apply coarse-graining and use a dislocation densitydescription.) The quantity b is the length of the Burgers vector, and v = l/τ P ( σ ) is theaverage speed at which dislocations move in the crystal, expressed in terms of averagespacing l = 1 / √ ρ between dislocations, and the depinning rate 1 /τ P ( σ ). Depinning isa thermally activated process with an assumed stress-dependent barrier of the form U P ( σ ) = k B T P e − s/σ T , (27)where s is the shear stress σ T is the Taylor (depinning) stress σ T = µ T b √ ρ, (28)with µ T being an effective shear modulus on the order of 1 /
30 times the shear modulus µ . As such, the depinning rate is1 τ P ( s ) = 1 τ f P ( s ) , (29)where f P ( s ) = exp (cid:18) − T P T e − s/σ T (cid:19) . (30) eformation in amorphous-crystalline nanolaminates γ pl = √ ¯ ρτ [ f P ( s ) − f P ( − s )] , (31)where ¯ ρ ≡ b ρ is a non-dimensional dislocation density. The second term on the RHSof (31) accounts for reverse transitions; it is typically neglected in practice, and will bedropped in the following.To convert these expressions to a form appropriate for describing tensiledeformation, we first rewrite equations (26) and (31) using the deviatoric stress andplastic strain rate tensors s ij and ˙ γ pl ij , and the stress and strain rate invariants ¯ s and¯˙ γ pl , as in section 3. Thus, the Orowan relation, (26), becomes˙ γ pl ij = ρ s ij ¯ s bv. (32)(This reduces directly to (26) in the case of simple shear, for which the only nonvanishingelements of the stress and strain rate tensors are s xy = s yx = s and ˙ γ pl xy = ˙ γ pl yx = ˙ γ pl / ǫ pl = ρ σ/ σ/ √ v = 1 √ ρbv, (33)so that (31) becomes q ≡ τ ˙ ǫ pl = p ˜ ρf P (¯ σ ) , (34)where now ˜ ρ = ¯ ρ/ f P is now expressed as function of the von Mises effective stress¯ σ = √ s , i.e. f P (¯ σ ) = exp (cid:18) − T P T e − ¯ σ/σ T (cid:19) . (35)The dislocation density evolves according the second law of thermodynamics.Following the analysis in [17], it approaches some steady state ρ ss ( χ ) = (1 /a ) e − e D /χ ,controlled by the effective temperature χ , with e D being the energy per dislocation. Therate at which ρ approaches ρ ss ( χ ) is assumed to be proportional to the rate of plasticwork, and inversely proportional to the dislocation energy per unit length γ D . Thus,˙ ρ = κ ρ σ ˙ ǫ pl γ D (cid:20) − ρρ ss ( χ ) (cid:21) , (36)where κ ρ is a dimensionless conversion factor that determines the fraction of energyinput that is converted into dislocations.Meanwhile, the equation for the effective temperature describes the flow of entropyand, as in the amorphous case, is a statement of the first law of thermodynamics: c eff ˙ χ = σ ˙ ǫ pl (cid:18) − χχ (cid:19) − γ D ˙ ρ. (37) eformation in amorphous-crystalline nanolaminates b = √ a , or ˜ ρ = a ρ , for the dislocationdensity ρ . Next, note that (34) can be solved explicitly for the stress as a function ofthe strain rate and the dislocation density: σσ T = ln (cid:18) T P T (cid:19) − ln (cid:20) ln (cid:18) √ ˜ ρq (cid:19)(cid:21) ≡ ν ( T, ˜ ρ, q ) . (38)Here we have taken advantage of the fact that ¯ σ = σ under unaxial loading conditions.Because the elastic modulus E is much larger than the other stress scales in the problem,we make use of the approximation ˙ ǫ pl ≈ ˙ ǫ , or q ≈ q ≡ τ ˙ ǫ . As such, the only dynamicalequations would concern the effective temperature χ and the normalized dislocationdensity ˜ ρ . Their equations of motion are c eff ˙ χ = σ ˙ ǫ pl (cid:18) − χχ (cid:19) − γ D ˙˜ ρa , (39)˙˜ ρ = κ ρ a σ ˙ ǫ pl γ D (cid:20) − ˜ ρe − e D /χ (cid:21) . (40)The tensile stress is directly given by σ = ¯ µ T p ˜ ρ ν ( T, ˜ ρ, q ) , (41)where ¯ µ T is proportional to the reduced shear modulus µ T , defined above in (28):¯ µ T = √ µ T .We close this section with some comments on the dimensionless conversion factor κ ρ in (40), which determines the fraction of input power that is stored in the form ofdislocations. To understand the physics behind this parameter, we consider the onset ofstrain hardening, when q = q but the dislocation density ˜ ρ is still small and has yet toreach its steady-state value. The stress at the onset of hardening is simply the Taylorstress, so that from (40), we get (cid:18) d ˜ ρdǫ (cid:19) onset ≈ κ ρ a σ T γ D = κ ρ a ¯ µ T γ D p ˜ ρ. (42)This can be substituted into (41) to give (cid:18) dσdǫ (cid:19) onset ≈ (cid:18) dσ T dǫ (cid:19) onset = κ ρ ¯ µ T a γ D . (43)However, if we directly use the full versions of equations (40) and (41) to compute theonset rate, we get an extra factor ν ( T, ˜ ρ, q ) multiplying κ ρ on the right-hand side of(43). Thus we conclude that κ ρ = ˜ κ ρ ν ( T, ˜ ρ, q ) , (44)where ˜ κ ρ is a constant of order unity. Then, after some algebra, the evolution equationfor ˜ ρ becomes ˙˜ ρ = κ √ ˜ ρq ν ( T, ˜ ρ, q ) (cid:18) − ˜ ρe − e D /χ (cid:19) , (45) eformation in amorphous-crystalline nanolaminates κ ≡ ˜ κ ρ a ¯ µ T γ D (46)is of order unity.
5. Coupled amorphous-crystalline layers – interaction between STZs anddislocations
We are now in a position to combine the effective-temperature descriptions of STZs anddislocations from sections 3 and 4, and model the interaction between the dislocations inthe crystalline layers and the STZs in the amorphous layers in simple terms. From nowon, we use the subscripts a and c to denote the quantities relevant to the amorphousand crystalline layers, respectively. Under isostrain conditions in the two constituents,the experimentally measured tensile stress is σ ≡ σ a h a + σ c h c h a + h c , (47)where h a and h c denote the layer thickness of the amorphous and crystalline layers,respectively. The assumption of co-deformation (isostrain) also implies that ˙ ǫ a = ˙ ǫ c ≡ ˙ ǫ ,and in general ˙ ǫ pl a = ˙ ǫ pl c , and σ a does not necessarily equal σ c .Experiments and simulations (e.g. [1]) indicate that the amorphous-crystallineinterface (ACI) acts as a sink of dislocations; an arriving dislocation from the crystallinelayer gets absorbed and triggers an STZ that moves into the amorphous layer. Otherstudies (e.g. [5]) seem to suggest that the stress concentration of an STZ near the ACImay be accommodated locally by the emission of a dislocation or an array of dislocationsthat moves into the f, which is also a plausible scenario. One way to interpret thesedislocation/STZ interactions is through the lens of effective-temperature dynamics andthe flow of entropy. Specifically, if the effective temperature of the amorphous layersomehow increases more slowly than in the crystalline layer during the deformationprocess, it is possible for entropy to flow from the crystalline layer to the amorphouslayer, or for the effective temperature to “diffuse” into the amorphous layer. Thisentropy flow is manifested by the movement of dislocations in the crystalline layer intothe amorphous-crystalline interface to trigger STZs that move into the amorphous layer.The opposite movement may occur if the effective temperature of the amorphous layerincreases more quickly, and stays above that of the crystalline layer. In either case, thediffusion term that describes this process is of the form (cid:18) dχdt (cid:19) diff = D a ˙ ǫ pl ∂ χ∂y , (48)where χ = χ c or χ a , D = D c or D a , where the conduction coefficients D c and D a in the two layers need not be equal, and y is the spatial coordinate in the directionnormal to the interface. A diffusion term of this type, proportional to the divergence ofthe “configurational heat flux”, was invoked elsewhere [13, 14, 11] to model the shear-banding instability. eformation in amorphous-crystalline nanolaminates ǫ as the independentvariable in the dynamical equations, the equations of motion for the coupled amorphous-crystalline nanolaminate are d ˜ σ a dǫ = 1 − q L a Z L a q a ( y ) dy ; (49) d ˜ χ a dǫ = κ a ˜ σ a qq (cid:18) − ˜ χ a ˜ χ (cid:19) + D a a q a q ∂ ˜ χ a ∂y ; (50) d ˜ χ c dǫ = κ c p ˜ ρν ( T, ˜ ρ, q ) (cid:18) − ˜ χ c ˜ χ (cid:19) + D c a ∂ ˜ χ c ∂y ; (51) d ˜ ρdǫ = κ √ ˜ ρν ( T, ˜ ρ, q ) (cid:18) − ˜ ρe − β/ ˜ χ c (cid:19) , (52)where ˜ σ a = σ a /E a is the tensile stress in the amorphous layer of width L a normalizedby the Young modulus, and β = e D /e Z . The effective temperatures have been non-dimensionalized by the STZ formation energy e Z : ˜ χ a ≡ χ A /e Z and ˜ χ c ≡ χ C /e Z . Also, κ a ≡ E a / ( c eff a e Z ) and κ c ≡ ¯ µ T / ( c eff c e Z ). We have also dropped the term proportionalto γ D in the equation for the effective temperature ˜ χ c in the crystalline layer, since theresults are apparently not sensitive to that term [17]. The plastic strain rate in theamorphous layer is q a = τ ˙ ǫ pl a = 4 √ ǫ e − / ˜ χ a C (˜ σ a ) ( T (˜ σ a ) − m ) , (53)where C (˜ σ a ) = exp (cid:18) − T E T (cid:19) cosh (cid:18) ǫ ˜ σ a √ e Z ˜ χ a (cid:19) ; (54) T (˜ σ a ) = tanh (cid:18) ǫ ˜ σ a √ e Z ˜ χ a (cid:19) ; (55) m = ( T (˜ σ a ) if ˜ σ a T (˜ σ a ) ≤ ˜ σ ,˜ σ / ˜ σ a if ˜ σ a T (˜ σ a ) > ˜ σ . (56)˜ e Z is the STZ formation energy scaled by E a a : ˜ e Z ≡ e Z / ( E a a ). Also, the tensile stressin the crystalline layer is directly given by σ c = ¯ µ T p ˜ ρν ( T, ˜ ρ, q ) . (57)
6. Model predictions and comparison with experiments
The equations of motion, (49) through (52), are integrated using an adaptivetime-stepping scheme based on the Crank-Nicolson method, with uniform spatialdiscretization (distance between two adjacent grid points is 0.5 nm). Because ofsymmetry, we confine ourselves to a transverse, one-dimensional domain stretching fromthe middle of an amorphous CuZr layer to the middle of the adjacent crystalline Cu layer,perpendicularly crossing the ACI. This is illustrated in Figure 2. The initial conditionsare ˜ σ a = 10 − (small but nonzero to facilitate numerical solution), ˜ χ a = ˜ χ c = 0 . eformation in amorphous-crystalline nanolaminates Figure 2.
Schematic illustration of the domain numerical solution of the evolutionequations that describe the amorphous-crystalline nanolaminate subject to tensiledeformation. Assuming symmetry about the center plane of each layer, it suffices tosolve the equations on the one-dimensional domain that stretches from the middle ofone amorphous CuZr layer to the middle of the adjacent crystalline Cu layer, depictedby the red line. across the sample, and ˜ ρ = 10 − . The thickness of the crystalline Cu layer is fixed at h c = 16 nm, while the amorphous CuZr layer thickness is varied in order to compare tothe Kim, Jang and Greer experiment [3]. The parameter values are listed in Table 1.Many of these parameter values are documented in the literature (e.g., [17]), with a fewexceptions. For example, with a Young’s modulus of E a = 72 GPa for amorphous CuZrinferred from [3], and an STZ formation energy e Z of the order of 1 eV, the dimensionlessSTZ formation energy roughly equals ˜ e Z ∼ O (1). Then, χ /e D = 0 .
25 according to [17];but because we have chosen ˜ χ = 0 .
04 here, which is roughly consistent with estimatesin, for example, [9], we choose β = e D /e Z = ( χ /e Z ) / ( χ /e D ) = 0 .
16. Next, the grain-size-dependent conversion factor κ that specifies the fraction of energy converted intodislocations was of order O (1) in [17] for grain sizes of order 10 µ m, and is an increasingfunction of decreasing grain size. For nanolaminates κ should be considerably larger,and we have chosen κ = 30. Finally, we choose for the effective temperature diffusioncoefficients D a = 10 and D c = 1 . × ; our choice stipulates that the diffusion ofdisorder in the crystalline layer is much slower than in the amorphous layer.Figure 3 shows the variation of the tensile stress σ with the accumulated strain ǫ ,for various values of the amorphous layer thickness h a . Our choice of the parameters ˜ σ , E a , and ¯ µ T ensures that the heterogeneous material yields at strain ǫ ≈ .
03 and stress σ ≈ h a = 128 and 215 nm break off at strains ǫ ∼ . ǫ pl at the edge of the amorphous layer falls off to zero. Indeed, this behavior,dependent on the initial conditions as well as the choice of parameters – especially D a –is seen in the numerical solutions to the equations of motion. Figure 4 shows the strainrate profile at a thickness h a = 68 nm, below the critical thickness for early material eformation in amorphous-crystalline nanolaminates Figure 3.
Variation of tensile stress σ with strain ǫ , for various values of the amorphousCuZr layer thickness h a . The crystalline layer thickness is h c = 16 nm, and the strainrate is ˙ ǫ = 10 − s − . The open circles and squares are the stress levels captured from[3]. The curves and data points have been offset vertically by 0.5 MPa for each pair ofadjacent values of amorphous layer thickness h a for clarity. Figure 4.
Variation of dimensionless plastic strain rate q across the half-width ofthe amorphous CuZr layer, at various snapshots of total accumulated strain ǫ . Theamorphous layer thickness is h a = 68 nm, so that y = −
34 nm is the center axis ofthe amorphous layer. The applied loading rate is ˙ ǫ = 10 − s − ; with τ = 10 − s, thedimensionless loading rate is q = 10 − . eformation in amorphous-crystalline nanolaminates Table 1.
List of variables and parameter values.Variable Description Value˜ χ Steady-state effective temperature 0.04 E a Young’s modulus of CuZr 72 GPa¯ µ T Effective shear modulus 10 GPa T Thermal temperature 298 K T P Depinning temperature 4 . × K [17] T E STZ activation temperature 600 K [9] κ Conversion factor 30 κ c Conversion factor 11 [17] κ a Conversion factor 80 τ Dimensionless loading rate 10 − s [9, 17] ǫ STZ core volume in units of a e Z Rescaled STZ formation energy 1.0 β Dislocation-STZ energy ratio 0.16˜ σ STZ yield stress parameter 0.02 [9] D a Diffusion constant in amorphous layer 10 D c Diffusion constant in crystalline layer 1 . × a Atomic size 0.167 nm failure. The strain rate near the amorphous-crystalline interface is close to zero, at leastimmediately after the onset of plastic deformation in the amorphous layer, while for h a = 68 nm the strain rate profile quickly becomes more or less uniform. This rapidapproach to uniformity may not be the case for nanolaminates with a thicker amorphouslayer. This point will be discussed in more detail afterwards, in conjunction with theeffective temperature profile shown in figure 6.Figure 5 shows the nondimensionalized dislocation density ˜ ρ across the half-widthof the crystalline Cu layer at various snapshots of the total accumulated strain ǫ or,equivalently, time. The dislocation density increases with increasing strain, as it should,and decreases towards the amorphous-crystalline interface at position y = 0. Thusour choice of parameters suggests the absorption of dislocations by the interface, inconcordance with simulations such as [1]. It is worth noting that the interface is a moreeffective sink of dislocations prior to the yielding of the amorphous layer than after.Finally, Figure 6 shows snapshots of the effective temperature distribution acrossthe half-width from the center of the amorphous CuZr layer to the center of thecrystalline layer. The effective temperature in the amorphous layer ˜ χ a remains constantprior to yield ( ǫ = 0 . y = 0 via equation (50). The direction ofdiffusion of the effective temperature is largely determined by its gradient across theinterface. For the present choice of parameters – specifically with κ c = 10 – it seemsthat the effective temperature in the amorphous layer increases more slowly than in the eformation in amorphous-crystalline nanolaminates Figure 5.
Nondimensionalized dislocation density ˜ ρ across the half-width of thecrystalline Cu layer, at various snapshots of total accumulated strain ǫ . Here κ c = 10;other parameters are listed in Table 1. The position y = 0 is the interface with theamorphous CuZr layer, as indicated by the arrow, while y = 8 nm is the center axis ofthe crystalline layer. As tensile deformation continues the dislocation density increasesin the Cu layer, but decreases towards the amorphous-crystalline interface. Figure 6.
Nondimensionalized effective temperature ˜ χ across the amorphous-crystalline nanolaminate, at various snapshots of total accumulated strain ǫ . Here κ c = 10; other parameters are listed in Table 1. The position y = 0 is the amorphous-crystalline interface. Here the thickness of the amorphous CuZr layer is h a = 68 nm,while that of the crystalline Cu layer is h c = 16 nm. Thus y = −
34 nm is the centeraxis of the amorphous layer, while y = 8 nm is the center axis of the crystalline layer.The diffusion of effective temperature and hence configurational disorder is largelydetermined by the effective temperature gradient across the interface y = 0. eformation in amorphous-crystalline nanolaminates κ a /κ c , it is possible for the effective temperaturein the amorphous layer to increase faster than in the crystalline layer, setting up aneffective temperature gradient across the ACI opposite to the one in the present case.In such a case, the reverse may occur, i.e., the interface would act as a sink of STZsand a source of dislocations, as in [5]. Delineation of the exact mechanism is likelymaterial-dependent and requires further microscopic imaging during laboratory studieson a case-by-case basis.If we compare figures 4 and 6, however, it becomes evident that an increasedSTZ density through a higher effective temperature does not automatically imply anelevated plastic strain rate. This is a purely entropic effect. To understand why thishappens, recall from (53) that the effective temperature ˜ χ a controls the plastic strainrate not just through the STZ density Λ = 2 e − / ˜ χ a , but also through the rate factors C (˜ σ a ) ∝ cosh[ ǫ ˜ σ a / ( √ e Z ˜ χ a )], and T (˜ σ a ) = tanh[ ǫ ˜ σ a / ( √ e Z ˜ χ a )]. The argument ofthese hyperbolic trigonometric functions is a decreasing function of increasing effectivetemperature ˜ χ a in the amorphous layer. While the STZ density is an increasingfunction of ˜ χ a , there is a range of ˜ χ a over which q decreases as a function of increasing˜ χ a . Physically, while the effective temperature near the edge of the amorphous layerincreases as a result of effective heat transfer – or diffusion of disorder – from thecrystalline layer, the STZs produced at the interface do not contribute to plastic strainuntil they move deeper into the amorphous layer. Importantly, if ˜ χ a is large enough,the plastic strain rate goes to zero since ˜ σ a T (˜ σ a ) < ˜ σ such that T (˜ σ a ) − m = 0. Theamorphous material near the interface becomes so disordered that the applied stress canno longer sustain the strain and the material fails.
7. Summary and concluding remarks
In this paper, we presented an effective-temperature framework that statisticallydescribes the motion of and interaction between plasticity carriers (dislocations in thecrystalline layers, and STZs in the amorphous layers) across an ACI in a natural manner.The effective temperature controls the dynamics of defects in a deforming solid, anddescribes the slow, configurational degrees of freedom that correspond to the infrequentatomic rearrangements associated with irreversible plastic deformation. The absorptionof plasticity carriers on one side of the ACI and the subsequent production of plasticitycarriers that move deep into the other side is interpreted as the flow of configurationaldisorder across the interface. Given our choice of parameters, we find the ACI tobe a sink of dislocations in the crystalline Cu layer and a source of STZs that moveinto amorphous CuZr, as observed in experiments and simulations such as [1]. In eformation in amorphous-crystalline nanolaminates
Acknowledgments
CL was partially funded by the Center for Nonlinear Studies at the Los Alamos NationalLaboratory over the course of this work. JRM acknowledges the support of the LosAlamos National Laboratory Directed Research and Development (LDRD) Early CareerAward 20150696ECR. eformation in amorphous-crystalline nanolaminates [1] Wang Y, Li J, Hamza A V and Barbee T W 2007 Proceedings of the National Academy of Sciences
Preprint ) URL [2] Arman B, Brandl C, Luo S N, Germann T C, Misra A andain T 2011
Journal of Applied Physics http://scitation.aip.org/content/aip/journal/jap/110/4/10.1063/1.3627163 [3] Kim J Y, Jang D and Greer J R 2011
Advanced Functional Materials http://dx.doi.org/10.1002/adfm.201101164 [4] Brandl C, Germann T and Misra A 2013 Acta Materialia [5] Zhang J, Liu G and Sun J 2013 Scientific Reports http://dx.doi.org/10.1038/srep02324 [6] Follansbee P and Kocks U 1988 Acta Metallurgica
81 – 93 ISSN 0001-6160 URL [7] Kocks U and Mecking H 2003
Progress in Materials Science
171 – 273 ISSN 0079-6425 URL [8] Falk M L and Langer J S 1998
Phys. Rev. E (6) 7192–7205 URL http://link.aps.org/doi/10.1103/PhysRevE.57.7192 [9] Langer J S 2008 Phys. Rev. E (2) 021502 URL http://link.aps.org/doi/10.1103/PhysRevE.77.021502 [10] Falk M L and Langer J 2011 Annual Review of Condensed Matter Physics Journal of the Mechan-ics and Physics of Solids
269 – 288 ISSN 0022-5096 URL [12] Langer J S 2015
Phys. Rev. E (1) 012318 URL http://link.aps.org/doi/10.1103/PhysRevE.92.012318 [13] Manning M L, Langer J S and Carlson J M 2007 Phys. Rev. E (5) 056106 URL http://link.aps.org/doi/10.1103/PhysRevE.76.056106 [14] Manning M L, Daub E G, Langer J S and Carlson J M 2009 Phys. Rev. E (1) 016110 URL http://link.aps.org/doi/10.1103/PhysRevE.79.016110 [15] Rycroft C H and Bouchbinder E 2012 Phys. Rev. Lett. (19) 194301 URL http://link.aps.org/doi/10.1103/PhysRevLett.109.194301 [16] Langer J, Bouchbinder E and Lookman T 2010
Acta Materialia [17] Langer J S 2015 Phys. Rev. E (3) 032125 URL http://link.aps.org/doi/10.1103/PhysRevE.92.032125 [18] Bouchbinder E and Langer J S 2009 Phys. Rev. E (3) 031131 URL http://link.aps.org/doi/10.1103/PhysRevE.80.031131 [19] Bouchbinder E and Langer J S 2009 Phys. Rev. E (3) 031132 URL http://link.aps.org/doi/10.1103/PhysRevE.80.031132 [20] Bouchbinder E and Langer J S 2009 Phys. Rev. E (3) 031133 URL(3) 031133 URL