Enhanced mobility of dislocation network nodes and its effect on dislocation multiplication and strain hardening

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Enhanced mobility of dislocation network nodes and its eﬀect on

dislocation multiplication and strain hardening

Nicolas Bertina,∗

, Wei Caib, Sylvie Aubrya, Athanasios Arsenlisa, Vasily V. Bulatova

aLawrence Livermore National Laboratory, Livermore, CA, USA

bDepartment of Mechanical Engineering, Stanford University, Stanford, CA, USA

Abstract

Understanding plastic deformation of crystals in terms of the fundamental physics of dislocations has re-

mained a grand challenge in materials science for decades. To overcome this, the Discrete Dislocation

Dynamics (DDD) method has been developed, but its lack of atomistic resolution leaves open the possibility

that certain key mechanisms may be overlooked. By comparing large-scale Molecular Dynamics (MD) with

DDD simulations performed under identical conditions we uncover signiﬁcant discrepancies in the predicted

strength and microstructure evolution in BCC crytals under high-strain rate conditions. These are traced

to unexpected behaviors of dislocation network nodes forming at dislocation intersections, that can move

in ways not previously anticipated as revealed by MD. Once these newfound freedoms of nodal motion

are incorporated, DDD simulations begin to closely match plastic evolution observed in MD. This addi-

tional mechanism of motion whereby non-screw dislocations can change their glide plane profoundly aﬀects

fundamental processes of dislocation multiplication, recovery and storage that deﬁne strength of metals.

Keywords: Dislocation multiplication, dislocation network nodes, dislocation mobility, dislocation

cross-slip, Molecular Dynamics, Discrete Dislocation Dynamics

1. Introduction

Over nearly 90 years since their initial discovery, lattice dislocations have been ﬁrmly established as the

main agents of crystal plasticity and yet quantitative prediction of macroscopic crystal plasticity directly

from dislocation behavior remains a challenge [1]. Computational method of Discrete Dislocation Dynamics

(DDD) is widely viewed as a promising approach in which motion of each individual dislocation as well as

interactions among dislocation lines are explicitly accounted for and thus directly deﬁne the overall crystal

plasticity response [2–7]. Given the drastic reduction in the degrees of freedom – from all atoms to just

dislocation lines – the DDD method has the promise to reach length and time scales relevant for crystal

plasticity that are inaccessible to fully atomistic simulations. As a method bridging between microscopic

dislocation theory and macroscopic crystal plasticity, DDD still faces serious computational limitations.

However, algorithmic advances coupled with steadily growing computing capacity are bringing about DDD

simulations on ever-increasing length- and time-scales [8].

Here we focus on a challenge altogether diﬀerent from the method’s computational eﬃciency, namely

the limited or unknown physical ﬁdelity of mechanisms of dislocation behavior presently included in DDD

models. Being a mesoscopic approach, a DDD simulation only includes mechanisms that the developer

knew about and chose to account for in the model formulation. But what if one or several mechanisms of

dislocation behavior are not known a priori and thus never included in the DDD model?

Historically, dislocation theory has been concerned with the behavior of individual dislocations, includ-

ing mechanisms of dislocation mobility, dislocation core structure, cross-slip, climb, etc. Until ﬁrst TEM

∗Corresponding author

Email address: bertin1@llnl.gov (Nicolas Bertin)

Preprint submitted to Acta Materialia April 3, 2024

arXiv:2210.14343v3 [cond-mat.mtrl-sci] 2 Apr 2024

observations of dislocations in the late 1950’s, understanding of dislocation behaviors was based to a great

extent on physical intuition and deductive reasoning. Yet, despite subsequent impressive developments in

imaging techniques, including in situ transmission electron microscopy (TEM), experiments do not resolve

dislocations in details suﬃcient to conﬁrm some previously hypothesized mechanisms or to discover un-

known atomistic mechanisms of dislocation motion. Since the 1960’s, atomistic simulations of individual

dislocations and, subsequently, small groups of dislocations have been increasingly used as means of inquiry

into dislocation behaviors augmenting experiment. Presently, direct MD simulations performed at the limits

of super-computing are reaching previously unachievable scales of simultaneous motion and interactions of

thousands of dislocation lines statistically representative of macroscopic crystal plasticity at deformation

rates of the order 105s−1and higher [9–11].

Equally important as their scales is that such direct MD simulations are fully atomistically resolved so

that every feature in a simulated stress-strain curve can be unambiguously connected to the underlying

dynamic events in the life of dislocations. In tandem with the recently developed accurate and eﬃcient

methods for dislocation extraction and indexing (DXA) [12,13], direct MD simulations now serve as a

powerful in silico computational microscope. Unlike the more traditional atomistic simulations invariably

probing behaviors of single dislocations or small groups of dislocations in conﬁgurations presumed relevant

for crystal plasticity, in massive MD simulations one observes how dislocations collectively and naturally

respond en masse to applied straining. Unbiased by human intuition, such simulations can reveal previously

unanticipated mechanisms of dislocation behavior.

Direct MD simulations of crystal plasticity are especially informative when contrasted against DDD

simulations performed under identical conditions, a practice we will refer to as cross-scale (X-scale) matching.

In this paper we present one example where X-scale matching bears fruit by exposing glaring discrepancies

between MD and DDD predictions that are traced to a distinct type of dislocation network nodes and their

modes of evolution that have not been previously considered. Colloquially referred to as sticky hereafter,

such nodes are immobile and restrict further motion of dislocation lines in DDD simulation. In contrast, in

MD simulation the same sticky nodes can dissociate into mobile nodes thus preventing formation of dense

dislocation tangles and excessive strain hardening. Discovery and kinematics of sticky nodes via topological

rearrangement are the main focus of this paper. We further show that, once the physics missing in DDD is

added, its predictions fall close in line with corresponding MD simulations precisely where the two previously

disagreed.

2. Computational methods

We employ the X-scale matching approach whereby simulations of metal plasticity are performed using

MD and DDD simulations by subjecting model single crystals of BCC tantalum (Ta) to the same loading

conditions on the same length and time scales where both methods overlap. The mesoscale approach of

DDD has gained widespread recognition as a successful method for materials simulations, yet how well it

reﬂects the underlying atomistic dynamics in strained crystals remains largely unknown [8]. In the context

of this study, we regard MD simulations as the ground truth for which we wish the DDD model to be a

faithful proxy. Here we demonstrate how direct one-to-one comparisons of dislocation trajectories initiated

from the same conﬁgurations are used to assess the diﬀerences between the MD and DDD predictions and

help us identify previously overlooked or missing physical mechanisms. Once uncovered, these mechanisms

can be included as new rules in the DDD model to enable better agreement with MD predictions.

2.1. MD simulations

MD simulations were performed in LAMMPS [14] using a previously reported interatomic potential for

Ta [15]. Large-scale MD simulations were performed in a crystal volume containing ∼33 million atoms

which was determined in our previous work [9,16,17] to be suﬃcient for statistically representative simula-

tions of single crystal plasticity under compression at a rate of 2 ×108/s. Twelve hexagon-shaped prismatic

dislocation loops of vacancy type were seeded at random locations in an initially perfect BCC crystal [9].

After initial annealing, the crystals were subjected to uniaxial compression along the [001] crystallographic

axis at a strain rate of 2 ×108/s. Uniaxial stress conditions and constant 300K temperature were main-

tained using the langevin thermostat and nph barostat. Dislocation networks were extracted along the MD

trajectories using the DXA algorithm [12,13] at time intervals ranging from 0.1 to 1 ps between snapshots.

Elemental dislocation networks containing sticky nodes with degree three (3-nodes) were created to

verify the new mechanisms of 3-node motion in §4.3. Full 3D periodic boundary conditions (3D-PBC) are

convenient for our purposes as they enable relatively straightforward and accurate control of applied stress

and eliminate unwanted boundary eﬀects while preserving translational invariance. The price one pays for

using 3D-PBC is that a minimum of two dislocation networks have to be inserted for the net Burgers vector

of the entire ensemble to remain zero. Upon insertion in a ∼500k atoms cell, CNA and DXA analyses as

implemented in Ovito [18] were used to ensure that no defects other than the two dislocation networks were

introduced in the simulation volume.

2.2. DDD simulations

DDD simulations were performed using the ParaDiS code developed and maintained at LLNL [7]. In the

DDD model, dislocation lines are discretized into a set of segments interconnected through dislocation nodes

[19]. The dislocation system is then evolved in time by calculating nodal velocities using a mobility function

which describes the local response of the dislocation nodes to the driving force. Material parameters for

the DDD model were computed using the same interatomic potential model of Ta as in the corresponding

MD simulation: lattice constant a0= 0.33032 nm, shear modulus µ= 55 GPa and Poisson’s ratio ν= 0.34.

We employed linear mobility functions both for the edge and the screw dislocations. However, the drag

coeﬃcient for the screws was more than 100 times greater than that of the edges, at 5.0×10−2Pa·s and

3.8×10−4Pa·s respectively. The screw mobility function was of a simple pencil type [7,20]. In all cases the

drag coeﬃcient for climb motion was set to a high value of 104Pa·s so that climb was eﬀectively suppressed.

Central to the DDD method is the kinematics of the dislocation nodes by which motion of the dislocation

network is prescribed. In the following section, kinematics of network nodes as presently implemented in

our DDD code is reviewed.

3. Kinematics of conservative motion of network 3-nodes

Our work concerns dislocation junction nodes that form when two dislocation lines – parents – collide and

merge with each other resulting in zipping a third common line, a product or junction dislocation. Forming

at the junction’s ends is a network 3-node in which two parent dislocations and the product dislocation

connect together (Fig. 1). Lying at the intersection of the parent glide (habit) planes the associated network

3-nodes have been assumed to act as strong pinning points hindering further motion of all three dislocations

resulting in an increase in the ﬂow stress, the phenomenon referred to as strain hardening. Furthermore,

junction 3-nodes have been predicted to enhance dislocation multiplication [21–24] thus further adding to

strain hardening.

Before describing our ﬁndings, it is useful to ﬁrst review the kinematics of nodal motion as presently

described in the literature (e.g. [6,20]) and implemented in our ParaDiS DDD model and code [7]. Based on

extensive literature search and communications with DDD practitioners, it is our understanding that most

of existing DDD models presently in use rely on similar if not identical rules. Consider the schematic in

Fig. 1where dislocation lines numbered i= 1, 2 and 3 merge in a 3-node with their three Burgers vectors bi

and unit vectors lideﬁning their tangent line orientations numbered accordingly. A dislocation with Burgers

vector band line tangent vector lcan move conservatively, i.e. not requiring any diﬀusional mass transport,

in its geometric glide plane deﬁned by the plane normal n= (b×l). Except for conditions not considered

here, conservative motion or glide is much easier than any non-conservative motion such as climb that takes

dislocations out of their glide planes. Expressed algebraically, for a dislocation to move conservatively its

velocity vector vshould satisfy the condition n·v= 0. For a 3-node to be able to glide conservatively with

velocity v, the same condition should be simultaneously satisﬁed for all three lines merging at the node

ni·v= 0; i= 1,2,3.(1)

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EnhancedmobilityofdislocationnetworknodesanditseffectondislocationmultiplicationandstrainhardeningNicolasBertina,∗,WeiCaib,SylvieAubrya,AthanasiosArsenlisa,VasilyV.BulatovaaLawrenceLivermoreNationalLaboratory,Livermore,CA,USAbDepartmentofMechanicalEngineering,StanfordUniversity,Stanford,CA,USAAbstra...

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Enhanced mobility of dislocation network nodes and its effect on dislocation multiplication and strain hardening

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