DefectDefect StructureStructure && MechanicalMechanical BehaviourBehaviour ofof NanomaterialsNanomaterials Defect structure in

 The defect structure in nanomaterials can be altered with respect to their bulk counterparts.

 This in turn can to profound differences in the mechanical behavior of nanomaterials as compared to bulk materials.

Vacancies

 Vacancies are equilibrium thermodynamic defects and at a temperature T (in K) there exists an equilibrium concentration of vacancies (n ) in bulk . v

. nV  the number of vacancies, H nV f . N  number of sites in the lattice (for nV/N << 1) exp NkT. Interaction between vacancies can be ignored  (Hformation (n vacancies) = n . Hformation (1 vacancy)

. Hf  enthalpy of formation of 1 mole of vacancies

T (ºC) n/N 500 1 x 1010 1000 1 x 105 1500 5 x 104 2000 3 x 103

18 Hf = 1 eV/vacancy = 0.16 x 10 J/vacancy

Close to the melting point in FCC Au, Ag, Cu the fraction of vacancies is about 104 (i.e. one in 10,000 lattice sites are vacant)

Vacancies in

. In free-standing nanocrystals of below a critical size, the benefit in configurational entropy does not offset the energy cost of introduce a vacancy. This implies that below the a critical

size (dc), vacancies are not thermodynamically stable.

. Hence, below dc the becomes free of vacancies (assuming that the kinetics permits

so!). This size (dc) is given by [1]:

1/3 3exp(QKT / ) dc  2  4 nv

. Where, nv is the number of per unit volume and Q is the energy for the formation of a vacancy.

. For . Al at 900 K (with Q = 0.66 eV), dc is 6 nm (i.e. Al crystals below 6 nm in size will be free of vacancies- in equilibrium).

. Cu with a higher energy for the formation of a vacancy (of 1.29 eV), has a d of 86 nm at c 900 K.

It is to be noted that bulk statistical thermodynamics, which is based on the assumption of large ensembles, is (strictly speaking) not applicable to small systems like nanocrystals. However, it is seen that the results of statistical thermodynamics can be applied to particles ~100 nm in size with only little errors.

[1] J. Narayan, J. Appl. Phys., 100 (2006) 034309.

MechanicalMechanical BehaviourBehaviour ofof MaterialsMaterials  OverviewOverview &&  NanomaterialsNanomaterials

Density and Elastic properties Plasticity by slip Motion of & dislocations in finite crystals Strengthening mechanisms Grain boundaries in nanocrystals Twinning versus slip Grain size and strength Superplasticity Creep Testing of and Nanomaterials Mechanisms / Methods by which a can Material can FAIL

Elastic deformation Creep Chemical / Physical Fatigue Electro-chemical degradation degradation Slip Microstructural changes Wear

Twinning Erosion Phase transformations Grain growth

Particle coarsening

Failure can be considered as change in desired performance- which could involve changes in properties and/or shape

Density and Elastic properties  Often obtaining full density in a nanostructured bulk material is a challenge. Residual porosity* can affect the density- which is an artifact of specimen preparation.

 The coordination number and packing close to grain boundaries (& triple lines (TL),

quadruple junctions (QJ)/corner junctions) is expected to be lower than that in the bulk of a

/grain (SC/G). This implies that on decreasing grain size the density of the

sample will decrease (albeit marginally).  However, this effect (reduction in density with grain size) is expected to become noticeable when grain size is reduced to the nanoscale regime.

 As a first approximation we can assume that grain boundaries (& triple lines etc.) have a similar character (w.r.t to density) in a nanostructured material, as compared to a micron

grain sized material.

Typical value of t (= dGB) is taken to be about 1nm (for metals about 3 layers of atoms)

*Often 3-5% porosity is considered to be fully dense  The fraction of GB (& TL, QJ) depends on the grain morphology. For simplicity we assume cubic grain morphology to make calculations here. (in cubic grains TL are better referred to as TJ).  The fraction of TL and QJ become important (for t ~ 1nm) when grain size is ~ 10nm (for the cubic morphology assumed) 1 d 3 d 3 f  f 1 0.8 G 3 GB 3 f Grain ()dt ()dt t = 1nm 0.6 f GB f TL 3 2 6t 6td f QJ f  0.4 QJ 3 fTL  3

8(dt ) 4(dt ) Volume fraction 0.2

0 0 1020304050 Grain Size (nm) Effect of these defects on the density of the material (ignoring porosity)

1

y 0.98  Relative Density = polycrystal 0.96 Effects dominate G 0.94 below ~20 nm grain Relative Densit Relative 0.92 size Entity Relative density 0.9 GB 0.95 0 1020304050 Grain Size (nm) TL 0.90

QJ 0.81

Assumed relative density

 [(1 fffGB  TL  QJ). G ]  [ f GB GB  f TL TL  f QJ QJ ]

()ffGGBTLQJ f f 1 Elastic properties

 For now we assume an isotropic material (i.e. properties do not change with position or direction). Note that we have already seen that density varies with position.

 An isotropic material can be described by two independent elastic moduli (e.g. E and ).  Often ‘Modulus’ of some nanostructures are reported as below (it should be noted that

Moduli are bulk macroscopic properties and their definition is extended to be applicable to

these structures).

Young’s Modulus Entity (GPa) Entity Young’s Modulus (GPa) Steel ~200 SW-CNT 1002 (Yu et al. 2000) ~1050 MW-CNT 11-63* (Yu et al. 2000) W ~400 ZnO nanowires 140-200 (Chen et al. 2006) Al ~70 Silica Nanowires 20-100 (Silva et al 2006)

*Range of values is due to diameter differences or number of walls Moduli of composites and nanomaterials

 The modulus of a composite lies between that of the two components. The upper bound and lower bound are given by isostrain and isostress conditions respectively.

.  VEVEE Under iso-strain conditions [m = f = c] mmffc . I.e. ~ resistances in series configuration Voigt averaging

. Under iso-stress conditions [m = f = c] 1 V f Vm  . I.e. ~ resistances in parallel configuration E E E c f m . Usually not found in practice Reuss averaging

in Ef tra sos

→ I s es

c r st For a given fiber fraction f, the modulii of E Iso various conceivable composites lie between an E upper bound given by isostrain condition m and a lower bound given by isostress condition

f A B Volume fraction → of MW-CNT and alumina: Young’s modulus as high as 570 GPa (actually in the range of 200-570 GPa depending on the nanotube geometry and quality and the porosity in alumina). Y ~ 350 GPa alumina

Nanocomposite of MW-CNT and alumina

Modulus of nano-polycrystal

 There is noticeable change in the modulus only when the grain size is below about 20 nm (assuming cube morphology of grains as before and assuming the modulus of the GB is 0.7

that of the Grain).  Presence of porosity can further cause a reduction in the modulus (which is a function of the processing route).

 Early results showed that there is reduction in modulus below even 200nm. These are perhaps because of porosity in the samples and not a characteristic of a fully dense sample.

1

0.9 E Upper bound polycrystal Lower bound Relative Modulus = 0.8 Effects EG dominate below

Relative Modulus Relative Cube morphology 0.7 ~20 nm grain size Assuming EGB = 0.7E G 0.6 0 1020304050 Grain Size (nm) Supermodulus effect

 In early observations of elastic properties it was noticed that there is large (>100%) enhancement of the elastic moduli in multilayers.

 This phenomenon was termed the supermodulus effect.

 More work in this area have attributed this effect to artifacts or anomalies

(it seems now that only about 10% enhancement in the elastic moduli may be real).

 Further work is needed in this area.

Though plasticity by slip is the most important mechanism of plastic deformation, there are other mechanisms as well:

Plastic Deformation in Crystalline Materials

Slip Twinning Phase Transformation Creep Mechanisms ( motion) sliding Vacancy diffusion + Other Mechanisms Dislocation climb

Note: Plastic deformation in amorphous materials occur by other mechanisms including flow (~viscous fluid) and shear banding Plasticity by slip

 The primary mode of plastic (permanent) deformation is by slip.  The simplest test performed to assess the mechanical behaviour of a material is the uniaxial tension test.

Variables in plastic deformation  , , , T

Usually expressed as (for plastic)

n K → strength coefficient Low T  K n → strain / work hardening coefficient

,T ◘ Cu and brass (n ~ 0.5) can be given large

plastic strain more easily as compared to steels with

n ~ 0.15

m High T  C   ,T

C → a constant m → index of strain rate sensitivity

◘ If m = 0  stress is independent of strain rate (stress-strain

curve would be same for all strain rates)

◘ m ~ 0.2 for common metals

◘ If m  (0.4, 0.9) the material may exhibit superplastic behaviour

◘ m = 1 → material behaves like a viscous liquid (Newtonian flow)

Further aspects regarding strain rate sensitivity

 In some materials (due to structural condition or high temperature) necking is prevented by strain rate hardening.

1/mm 1/ m m P m P 1  C   C        ,T A CA From the definition of true strain

m 11dL dA   L dt A dt 1/mm 1/ 11dA P  Adt C A

1/m dA P 1  . If m < 1→ smaller the cross-sectional area, the (1mm ) /  dt C A  more rapidly the area is reduced.

. If m = 1 → material behaves like a Newtonian viscous liquid → dA/dt is independent of A.

Importance of ‘m’ at high temperatures

1/m dA P 1  (1mm ) /  dt C A  Weakening of a crystal by the presence of dislocations  To cause plastic deformation by shear (all of plastic deformation by slip require shear stresses at the microscopic scale*) one can visualize a plane of atoms

sliding past another (fig below**)  This requires stresses of the order of GPa  But typically crystals yield at stresses ~MPa   This implies that ‘something’ must be weakening them drastically #  It was postulated in 1930s and confirmed by TEM observations in 1950s, that the agent responsible for this weakening are dislocations

* Even if one does a pure uniaxial tension test with the tension axis along the z- axis, except for the horizontal and the vertical planes all other planes ‘feel’ shear stresses on them # By Taylor, Orowan and Polyani

 The shear modulus of metals is in the range 20 – 150 GPa

G  The theoretical shear stress will be in  m ~ the range 3 – 30 GPa 2

 Actual shear stress is 0.5 – 10 MPa (experimentally determined)

 I.e. (Shear stress)theoretical > (~)100  (Shear stress)experimental !!!!

DISLOCATIONS

Dislocations severely weaken the crystal

. Whiskers of metals (single crystal free of dislocations, Radius ~ 106m) can approach theoretical shear strengths

. Whiskers of Sn can have a yield strength in shear ~102 G (103 times bulk Sn)

Motion of dislocations and plasticity by slip

 Plastic deformation by slip occurs by the motion of dislocation and their leaving the crystal.

 Dislocations may move under an externally applied stress  At the local level shear stresses on the slip plane can only drive dislocations

 The minimum stress required to move a dislocation is called the Peierls-Nabarro

(PN) stress or the Peierls stress or the Lattice Friction stress (i.e the externally applied stress may even be purely tensile but on the slip plane shear stresses must act in order to move the dislocation)  Dislocations may also move under the influence of other internal stress fields (e.g. those from other dislocations, precipitates, those generated by phase transformations etc.)  In any case the Peierls stress must be exceeded for the dislocation to move  The value of the Peierls stress is different for the edge and the screw dislocations

 The first step of plastic deformation can be considered as the step created when

the dislocation moves and leaves the crystal

→ “One small step for the dislocation, but a giant leap for plasticity”

 When the dislocation leaves the crystal a step of height ‘b’ is created → with it all the stress and energy stored in the crystal due to the dislocation is relieved

More about the motion of dislocations in the chapter on plasticity Edge Dislocation Glide Motion of an edge dislocation leading to the formation of a step (of ‘b’)

Shear stress

Surface step (atomic dimensions)

Dislocations in finite crystals

 Dislocations are attracted to free-surfaces (and interfaces with softer materials) and may move because of this attraction → this force is called the Image Force

 Gb2 F  image )1(4 d Image Forces in Nanocrystals

 In nanocrystals more than one free surfaces will be in proximity to the dislocation → leading to multiple images. [1]

Glide

Image Force (Fimage) (Edge Dislocation, Finite Domain)

22 Gb 11 Gb 2 x Superposition of F  image  22 4(1) dLd(1)  L 4 x two images

"Image forces on edge dislocations: a revisit of the fundamental concept with special regard to nanocrystals" [1] Prasenjit Khanikar, Arun Kumar and Anandh Subramaniam, Philosophical Magazine, 91, p.730, 2011. Dislocations in nanocrystals

 The energy of a dislocation (in bulk crystal) is given by:

2 Gb  . Edl – Energy per unit length of dislocation line E 2ln 0 edge dislocation  . b – Modulus of the Burgers vector 4(1) b . 0 - size of the control volume ~ 70b

 In nanocrystals the energy of a dislocation will be lesser than that in a bulk crystal:  Due to domain deformations  Due to smaller amount of material available to be strained

1.4 Theory Simulation: more constrained domain Simulation:less constrained domain 1.2

(J/m) 1.0 9 

0.8 Energy × 10 Energy

0.6

0.4 100 140 180 Diameter of the domain (in terms of b)  In one or more free surfaces may be in proximity to the dislocation.  As the dislocation is positioned closer and closer to a free surface, at a certain stage the

image force may exceed the Peierls force (  b) PN  the dislocation can spontaneously leave the crystal without the application of any

external stresses [1].

 In nanocrystals the proximity of multiple free surfaces may lead to all dislocations leaving the crystal when: FF (for all dislocations) image PN  Hence, nanocrystals can become completely dislocation free.

 For single crystals of Al and Ni this size is the order of a few tens of nanometers.

 Thus the strength of such a nanocrystal may approach the theoretical strength of the

crystal.

 2w  . G → shear modulus of the crystal    b  . w → width of the dislocation !!!  PN  eG . b → |b|

"Critical Size for Edge Dislocation Free Free-Standing Nanocrystals by Finite Element Method" [1] Prasenjit Khanikar, Anandh Subramaniam, ,Journal of Nano Research, 10, p.93, 2010. Edge dislocation free

Material Properties Material Critical size (nm) Lattice parameter, Slip system, b, G,  Al (FCC) 4.04 Å, <110>{111}, 2.86 Å, 26.18 GPa, 0.348 36 nm Ni (FCC) 3.52 Å, <110>{111}, 2.49 Å, 92.00 GPa, 0.286 20 nm Nb (BCC) 3.30 Å, <111>{110}, 2.86 Å, 36.23 GPa, 0.38 8 nm Si (DC) 5.42 Å, <110>{111}, 3.83 Å, 67.72 GPa, 0.219 < 4 nm

 Work done on Ti and Pd thin films show that with grain size in the range of ~6-9 nm the grains are dislocation free [1]. The study also showed that many grains had twins and stacking faults.  Additionally, in nanocrystals the partials may leave the crystal leaving a stacking fault.  It has also been observed that surface regions of polycrystals can become dislocation free.

1 1 1 [110] → [121] + [211] 2 (111) 6 (111) 6 (111)

Shockley Partials

[1] Interface Structures in Nanocrystalline Materials, Scripta mater. 44 (2001) 1169–1174

 Nanocrystals of certain geometry may bend because of the mere presence of dislocations. One such example is shown in the figure below.

10b

 Thin plates will bend in the presence of edge dislocations and thin cylinders will twist in the presence of screw dislocations. (Eshelby had studied the mechanics of these type of problems in the 1950s) [1]  Dislocations can become mechanically stable in such plates [1,2].

[1] J.D. Eshelby, J. Appl. Phy. 24 (1953) p.176.

[2] "Stable Edge Dislocations in Finite Crystals", Arun Kumar and Anandh Subramaniam, Philosophical Magazine, in press. “Zero Stiffness”! 12 Domain which can bend Constrained domain 11 11.33 7.4625 Referred to primary axis

) (J/m) 10

10 (a) − (a) 10 Zoomed in view Reversible Plastic Deformation × due to Elasticity! [1] 9 (b) Referred to secondary axis Energy ( Energy 11.32 7.462 8 10b 0510 (b) Zero Stiffness Nearly zero 7 stiffness 0 10203040 "Materials Analogue of Zero Stiffness Structures" Distance from centre (in terms of b) Arun Kumar and Anandh Subramaniam, Philosophical Magazine Letters, 91, p.272, 2011. Strengthening mechanisms

 As we noted that crystals are severely weakened by the presence of dislocations.  However, the strength of a crystalline material can be increased by various methods

→ any impediment to the motion of dislocations, increases the strength of the material.

 Strengthening mechanisms include:  solution strengthening → solute atoms increase the resistance to dislocation motion

 Precipitation Hardening and Dispersoid strengthening

 Forest dislocation hardening

 Grain boundary hardening

Strengthening mechanisms

Precipitate & Forest dislocation Grain boundary Dispersoid Twinning versus slip

 Twinning readily occurs in low stacking fault energy materials like Cu and Brass. In high stacking fault energy materials like Al twinning is difficult.

. Twinning can occur either during annealing or during deformation. . Twinning is an important mechanism of plastic deformation and can give to

materials wherein slip may be limited. In BCC metals (e.g. Fe) at low temperatures,

twinning may become the dominant mechanism of plastic deformation. . TWIP (twinning induced plasticity) steels have been developed keeping this in view.

 Process parameters (temperature, strain rate) and material parameters (stacking fault energy, grain size) determine the mechanism operative for plastic deformation (between

slip and twinning). . For a given strain rate, at higher temperatures slip would be favourable as compared to twinning (as slip is thermally activated, while twinning stress is essentially constant with

temperature). . On increasing the strain rate the stress required for both slip and twinning increase and the transition temperature (from twinning to slip) is shifted to higher temperatures.

 To observe significant effects, strain rate variation is over 10 orders of magnitude and the grain sizes should be varied from about 10 to 100s of microns.

 It is seen that for a given strain rate, as we reduce the grain size slip become preferred even at lower temperatures (i.e. slip occupies more region of the temperature-strain rate

space). Superplasticity

 The phenomenon of extensive plastic deformation without necking is termed as structural superplasticity. Superplastic deformation is tension can be >300% (up to even 2000%).

 Typically superplastic deformation occurs when: (i) T > 0.5T m (ii) grain size is < 10 m

(iii) grains are equiaxed (which usually remain so after deformation)

(iv) grain boundaries are glissile (with a large fraction of high angle grain boundaries).

 Presence of second phase (of similar strength to the matrix- reduces cavitation during

deformation) which can inhibit grain growth at elevated temperatures helps (e.g. Al-

33%Cu, Zn-22% Al)).

 Many superplastic alloys have compositions are close to eutectic or eutectoid points.  Superplastic flow is diffusion controlled (can be grain boundary or lattice diffusion

controlled).

  A plot of stress versus strain rate is often sigmoidal and shows three regions:

(i) Region-I - low stress, low strain rate regime (  <105 /s)  m  (0.2,0.33) Sensitive to the purity of the sample. Lower ductility and grain boundary diffusion.

(ii) Region-II- intermediate stress & strain rate regime [    (10–5, 10–2) ]  m  (0.4,0.67) Extended region covering several orders of magnitude in strain rate. Region of maximum

ductility. Strain rate insensitive to grain size and insensitive to purity. Often referred to as the superplastic region.

Mechanism predominantly grain boundary sliding accommodated by dislocation

activity (Activation energy (Q) corresponding to grain boundary diffusion (Qgb)). (iii) Region-III- high stress & strain rate regime (  > 102 /s)  m > 0.33  Creep rates sensitive to grain size.

Mechanism intragranular dislocation process (interacting with grain boundaries).

low ‘m’Note: region I and III in

 C m   ,T Creep  In some sense creep and superplasticity are related phenomena: in creep we can think of damage accumulation leading to failure of sample; while in superplasticity extended plastic

deformation may be achieved (i.e. damage accumulation leading to failure is delayed).  Creep is permanent deformation of a material under constant load (or constant stress) as a

function of time. (Usually at ‘high temperatures’ → lead creeps at RT).

 Normally, increased plastic deformation takes place with increasing load (or stress)

 In ‘creep’ plastic strain increases at constant load (or stress)  Usually appreciable only at T > 0.4 Tm  High temperature phenomenon.  Mechanisms of creep in crystalline materials is different from that in amorphous materials. Amorphous materials can creep by ‘flow’.

 At temperatures where creep is appreciable various other material processes may also active (e.g. recrystallization, precipitate coarsening, oxidation etc.- as considered before).

 Creep experiments are done either at constant load or constant stress.

Phenomenology Harper-Dorn creep Power Law creep Creep can be classified based on Mechanism Stages of creep

I . Creep rate decreases with time . Effect of work hardening more than recovery

. Stage of minimum creep rate → constant II . Work hardening and recovery balanced

. Absent (/delayed very much) in constant stress tests III . Necking of specimen start . specimen failure processes set in

Constant load creep curve → → ) ) I II   → Strain ( )  Strain (

Increasing T Strain ( Increasing stress increases 0  III increases Effect of temperature 0   Effect of stress 0 0 0 0 → Initial instantaneous strain t → t → t → Creep Mechanisms of crystalline materials

Cross-slip Harper-Dorn creep Dislocation related Climb Glide

Coble creep Creep Grain boundary diffusion controlled

Diffusional Nabarro-Herring creep Lattice diffusion controlled

Dislocation core diffusion creep

Diffusion rate through core of edge dislocation more Interface-reaction controlled diffusional flow

Grain boundary sliding

Accompanying mechanisms: creep with dynamic recrystallization Cross-slip

 In the low temperature of creep → screw dislocations can cross-slip (by thermal activation) and can give rise to plastic strain [as f(t)].

Dislocation climb

 Edge dislocations piled up against an obstacle can climb to another slip plane and cause plastic deformation [as f(t), in response to stress].

 Rate controlling step is the diffusion of vacancies.

Grain boundary sliding

 At low temperatures the grain boundaries are ‘stronger’ than the crystal interior and impede the motion of dislocations

 At higher temperature (GB being a high energy region) becomes weaker than the crystal interior

 Above the equicohesive temperature grain boundaries are weaker than grain and slide past one another to cause plastic deformation

Diffusional creep Nabarro-Herring creep → high T → lattice diffusion Coble creep → low T → Due to GB diffusion

 In response to the applied stress vacancies preferentially move from surfaces/interfaces (GB) of specimen transverse to the stress axis to surfaces/interfaces parallel to the stress axis → causing elongation.  This process like is controlled by the diffusion of vacancies → but diffusional creep does not require dislocations to operate.

 

Flow of vacancies Creep mechanism map Testing of Nanostructures and Nanomaterials

The testing procedure for determining the mechanical behaviour of nanocrystalline materials is expected to differ from that used for bulk materials. This is due to: (i) small size of the (nanopillars, , thin films, etc.) or (ii) due to the small quantity of material usually available for testing (as in the case of nano-layers obtained by molecular beam epitaxy). If the nanostructure is a part of a larger hybrid (e.g. nanocrystal of Pb dispersed in an Al matrix), then it becomes even more challenging to determine the properties of an individual nanostructure (a Pb nanocrystal in the example considered). Processing route (e.g. powder consolidation, mechanical milling, high pressure torsion, etc.) often produces variations in the material produced (porocity, dislocation density etc.), thus making it difficult to compare properties across sizes and processing routes (this comparison is necessary to generalize the behaviour at the nanoscale). E.g. Electrodeposited Ni showed a ductility of ~100%; while Ni specimen obtained by powder compaction had a ductility of ~3% (both samples had identical grain size of about 10 nm) [1].

[1] H. Gleiter, Acta Mater. 48 (2000) 1. To summarize, the issues associated with the evaluation of mechanical properties of nanostructures and nanomaterials are:

 Small size of the entity

 Separation of the property of the entity from the support/substrate/matrix

 Small quantity of sample (hence, in literature hardness data is more often found than tensile data)

 Artifacts introduced by the processing route

 Uncertainty in grain size measurement (which can reach 50%) How to synthesize bulk samples which can be tested easily?

Variations in sample 'quality' can lead to altered properties which are not inherent to just the reduced grain size. Here we address the question: "how to produce bulk nanostructured materials which can yield good samples for conventional testing?".

Two popular techniques for the synthesis of bulk nanostructured materials (which makes it easy for making test specimens) can be synthesized by:

(i) consolidation of powders or by

(ii) severe plastic deformation (SPD) techniques.

Consolidation of powders can lead to a high level of porosity in the final sample and SPD techniques are gaining in importance. In SPD techniques it is interesting to note that plastic deformation is leading to a decrease in grain size. Some examples of SPD techniques are (with examples of products are):

(i) Equi-channel Angular Pressing (ECAP)

(ii) High- pressure Torsion (HPT)  nanocrysrtalline Si with grain size ~20 nm

(iii) Accumulative roll bonding,

(iv) Multipass Coin-forging (MCF),

(v) Repetitive corrugation and straightening (RCS) etc. Some common features of these SPD techniques are (though there might be some variations based on technique in question):

 Can produce bulk samples that are practically free of contamination and porosity.

 Both strength and ductility increase with increasing strain.

 In pure materials like Cu (FCC), Fe (BCC), Ti (HCP) the grain size can be reduced to about 100 nm. The range of grain sizes is between 5-100 nm.

 During SPD, at very small grain sizes, grain boundary sliding and grain rotation may occur.

 High elastic strain and nano-distortions to the lattice.

 In the end product (depending on the amount of deformation) a range of grain sizes following a 'log-normal' type distribution may exist. The smallest grains (with d < 50 nm), have no dislocations in them, the intermediate sized grains have a large dislocation density and even larger grains may be divided into sub-grain boundaries.

 These techniques pose serious challenges when processing brittle materials like and some intermetallics. Grain boundaries in nanocrystals

 Grain boundaries and interfaces can comprise of about 50% of the volume fraction in a nanostructured material with grains size of about 5nm.

 Grain boundaries in nanostructured materials seem to be similar in structure to their bulk counterparts (except in specific examples).

 Both sharp and disordered grain boundaries (along with disordered triple junctions) were observed in Ti and Pd nanostructured thin films [1]. The region of disorder at the grain

boundaries (with disorder) was about 0.5 nm. The most natural model to explain grain

boundaries and triple lines is the disclination model.  In a model by Suryanarayana [2], assuming a grain boundary thickness of 1 nm; when

grain size is about 30 nm the volume fraction of grain boundaries is about 10% and that of

triple lines (junctions) is ~1%.

[1] S. Ranganathan, R. Divakar and V.S. Raghunathan, Scripta mater. 44 (2001) 1169–1174. [2] C. Suryanarayana, D. Mukhopadhyay, S.N. Patankar, F.H. Froes, J. Mater. Res. 7 (1992) 2114

Intergranular Glassy Films (IGF) → nanostructures in their own right

 GB are regions where order of one grain changes to order of another grain.  GB can also have some degree of disorder.

 Special grain boundary structures are observed in specific materials like Si N , Al O , 3 4 2 3 SrTiO etc. which have a thin layer of glassy material of constant thickness of about 1-2 3 nm [1]. These are called intergranular glassy films (or IGF for short). These IGFs are characterized by a nearly constant thickness which is basically independent of the

orientation of the bounding grains, but is dependent on the composition of the . The IGF is resistant to and is thought to represent an equilibrium configuration. The presence of the IGF, along with its structure, plays an important role in

determining the properties of the ceramic (as a whole).

0.66 nm Grain-1 High-resolution micrograph from a Lu-Mg

1.4 nm doped Si3N4 sample showing the presence of IGF an Intergranular Glassy Film (IGF).

Grain-2

[1] Anandh Subramaniam, Christoph T. Koch, Rowland M. Cannon, Manfred Rühle, and Engineering A 422 (2006) 3–18.

What kind of mechanical behaviour should we expect in nanomaterials?

If the scaling laws are assumed to be valid at small lengthscales then what can we expect in nanomaterials:

1/2 1. Strength at low temperatures to increase at small sizes due to Hall-Petch relation (y  d ). At very small grain sizes we also expect this relation to break down (as this relation would imply an yield stress higher than the theoretical shear strength of the material).

2. Low strength at high temperatures, wherein creep mechanisms become operative

. 3 ( Coble creep  d ) due to shorter diffusion paths.

3. Superplasticity (usually observed at high temperatures and low strain rates in micron grain sized materials) at lower temperatures or for a given temperature higher strain rates for superplasticity.

4. Higher creep rate by grain boundary diffusion mechanism at lower temperatures. (GB diffusion

creep is usually active at ~0.5Tm and low stresses).

5. With decreasing grain size, the cross-over from slip to twinning is postponed to larger strains and twinning may be suppressed for very small grain sizes. Deformation in Nanomaterials

 Deformation of nanostructured materials seem to occur predominantly by dislocations at interfaces (and not bulk dislocations). These dislocations have Burgers vector smaller than

lattice dislocations. In some in-situ TEM experiments mobile dislocations were not

observed when grain size is less than about 30 nm.

 In brittle materials like ceramics, at small grain sizes grain boundary sliding may be the predominant mechanism of plastic deformation.

 In some cases the nanocrystalline sample showed poorer ductility than their microcrystalline counterparts.

 Paucity of dislocations within the grain and difficulty in generating dislocations. For a mechanism like double ended Frank-Read source to operate to increase the

dislocation density, two pinning points are required. As discussed before the maximum

shear stress required (max) for operation of the source is given by:  max ~/Gb L

Where, L is the distance between the pinning points which can take a maximum value of the grain diameter. In nanocrystals this value can exceed the theoretical shear strength of the crystal! Grain size and strength

 In polycrystalline materials (e.g. polycrystalline Cu, Al or alloys) the dependence of yield strength ( ) on grain size (d) is given by the Hall-Petch relation. y  The relation states that as the grain size decreases the strength of the crystal increases.

 The reason behind this is that the grain boundary is an impediment to the motion of the dislocations (dislocation is a crystallographic defect and its ceases to exist where the

crystallite ends). As the slip plane in one grain is typically not coplanar with the neighbouring grain; higher stress is required for a dislocation to initiate slip in an adjacent grain.  The usual model to explain the grain size dependence on strength is the ‘dislocation pile- up (at grain boundaries) model”.

Hall-Petch Relation 2 . y → Yield stress [N/m ] 2 . i → Stress to move a dislocation in single crystal [N/m ] k 3/2  . k → Locking parameter [N/m ] iy d (measure of the relative hardening contribution of grain boundaries) . d → Grain diameter [m]  Hardness also follows a relationship akin to that of yield stress.

 Nanocrystalline Cu with grain size (d) = 6nm is FIVE times harder that microcrystalline Cu (with d = 50 m).

 The yield stress of nanophase Pd (d = 7 nm) is FIVE times greater than that in the corresponding bulk (d = 100 m).

 This effect understood in terms of the difficulty in creating dislocations and from the existence of barriers to dislocation motion.

 The usual model to explain the origin of the d1/2 dependence is the dislocation pile up model (at grain boundaries). (However, it must be noted that this model has not been universally validated).

 As grain boundaries are obstacles to the motion of dislocations they pile up at the grain boundary and the stress caused by the pile up initiates slip in the next grain.

 This relation is not expected to hold at very small grain sizes:  if this form of the relation continues the yield strength will exceed the theoretical

strength of the crystal.

 at very small grain sizes there is not ‘enough length’ to support a pile up (i.e. at small

sizes only a single dislocation can be accommodated in a grain).

 Hence, at very small sizes the Hall-Petch relation breaks down. This typically occurs at grain sizes of about 10nm.

 As the grain size is reduced from micron sized grains to about 50 nm grain size, normal Hall-Petch behaviour is observed (i.e. d–1/2 dependence of yield stress on grain size).

When decreasing grain size to values less than about 25 nm three types of behaviours may

be observed (data from multiple experiments performed in various materials like Cu, Ni,

Ti, Fe etc., is schematically illustrated in Figure).

 (i) Altered Hall-Petch relation (increasing hardness with decreasing grain size- with an altered hardening function)

(ii) No dependence of hardness on grain size

(iii) Softening with grain size- 'Inverse Hall-Petch effect' (usually found less than 10 nm

grain size)

 Often the data may be so scattered that it may not be possible arrive at precise trendlines.

Other factors to be taken into account . Grain size distribution . Grain orientation distribution

. Grain shape

 In nanostructured materials the dislocations needed for deformation maybe absent or sessile and new ones are prevented from forming.

 Materials with poor ductility (like intermetallics or ceramics), the results of possible increase in ductility by grain-size reduction are contradictory.

(results on some systems and investigators point in one direction while other investigations do not corroborate the results). Nano-laminates k iy   Nanolaminates are hybrids with layer spacing of few nanometers. d  A nanolaminate made of two soft metals like Ni and Cu can have a strength of the order of

few GPa (given that the theoretical shear strength (TSS) is of the order of G/2, the

strength of nano-laminates is very close to the TSS).

 TiN/NbN multilayers can have a hardness value of 50 GPa [3] (the hardness of single TiN or NbN films is around 20 GPa).

 A plot of log of the bilayer period verses tensile strength shows an approximately linear

plot with a slope of ~(0.5) → Hall-Petch like behaviour.

Layer period in laminate

[1] S. Menzies and D.P. Anderson, J. Electrochem. Soc. 137 (1990) 440. [2] D.M. Tench and J.T. White, Metall. Trans. A, 15 (1975) 2039. [3] M. Shinn, L. Hultman, and S. A. Barnett, J.Mater.Res. 7, 901 (1992).

 Hardening in multi-layers and superlattices have been explained using a number of concepts:

(i) barrier strengthening, (ii) coherency stress strengthening, (iii) misfit dislocation density strengthening (iv) dislocation image force (due to discontinuity in the elastic modulus at the interfaces)

(v) dislocation pileup and bowing processes occur (at larger bilayer periods).

In the example cited before: Ni/Cu superlattices [1]  Decreasing bilayer period up 20nm to increase in yield stress in tension test.  Further decrease in bilayer period leads to decrease in strength.

 The peak strength corresponds to one dislocation per layer.

[1] P. M. Anderson, T. Foecke, and P. M. Hazzledine, MRS Bull. 24, 27 (1999).

Nanocomposites: particulate reinforcements

 Precipitation hardening is well known where nanometer (1-100nm) sized precipitates increase the hardness and strength of materials. (Strength of Al → 100 MPa

Strength of Duralumin (Al + 4% Cu + other alloying elements) → 500 MPa).

 When nanosized Mo (5–20 vol.%) is dispersed in micrometer grain sized Al2O3 → there is a considerable improvement in the hardness, fracture strength, and toughness.

 Nanosized SiC dispersions (200 nm) in a matrix of Al2O3, leads to → increase in fracture strength by a factor of 3, enhanced high-temperature mechanical properties

The decrease in hardness with increasing temperature is significantly less in nano- composite Al O /SiC than in monolithic Al O . 2 3 2 3

Creep in Nanomaterials

 In nanocrystalline Pd (~40 nm) and Cu (~20 nm), there seemed to be no increase in creep rate as compared to micron grain sized materials (in some temperature regimes even a

lower creep rate was observed for Pd). This is in direct contradiction with the expectation that nanocrystalline materials will experience a higher creep rate.  Studies on Cu (10-25 nm GS), Pd (35-55 nm GS) (TEM showed porocity in sample) [1] . creep in the low T regime (0.24-0.33 T ) → low creep rate, low grain growth m . creep in the medium T regime (0.33-0.48 T ) → creep rate decreasing even after long m testing time, grain growth (25 nm → 100s of nm )

. Cu creep rates of nc sample was comparable to micron GS sample

. Pd nc sample exhibited lower creep rates

[1] P.G. Sanders, M. Rittner, E. Kiedaisch, J.R. Weertman, H.Kung, Y.C. Lu, Nanostruct. Mater. 9 (1997) 433. [2] D.L. Wang, Q.P. Kong, J.P. Shui, Scr. Metall. Mater. 31 (1994) 47.  In some cases the creep rate increased with a decrease in grain size in the nanoscale regime of grain sizes (e.g. in Ni-P nanocrystalline material the creep rate of ~30 nm grain

sized material was higher than that of 250 nm material [2]).

 In cases where high creep rate expected for nanocrystalline materials (e.g. Pd, Cu) was not observed, the reason attributed are:

(i) presence of low angle grain boundaries and twin boundaries (which are not prone to sliding and have low diffusivity for vacancies), (ii) reduced dislocation activity in nanocrystalline samples.  Creep of nc-Ni at RT (GS: 6, 20, 40 nm) [1]:  Smaller grain size (6nm) showed faster creep rate.

 Behaviour consistent with Grain boundary sliding controlled by grain boundary

diffusion mechanism.

 High stresses and larger GS (20, 50 nm)→ dislocation creep.

[1] N. Wang, Z. Wang, K.T. Aust, U. Erb, Mater. Sci. Eng., A 237 (1997) 150. Superplasticity in Nanomaterials

 In most cases the superplasticity has not fulfilled the initial ‘expectations’.  In many cases superplasticity is only observed in nanocrystalline samples, where it is

already observed in their microcrystalline counterparts.

 Superplasticity was observed in nanocrystalline Ni (20 nm grain size) at 0.36Tm (more than 450C lower than that for the bulk material) [1].

 Nanocrystalline Ni3Al (grain size 50 nm) also became superplastic about 450C below its microcrystalline counterparts. –3 Ni3Al had a ductility of 350% at 650C (strain rate of 10 /s). –1  1420-Al showed superplasticity at a high strain rate of 10 /s. High amount work hardening and higher flow stress for superplastic deformation as compared to micron

grain sized material is observed in these cases.

 Superplasticity was observed in ~40 nm grain size Zn-Al alloy at 373 K, tested at a strain rate of 10–4 /s [2]. Microcrystalline samples showed no superplasticity!

Ni3Al (cP4, Pm-3m)

[1] S. X. McFadden, R. S. Mishra, R. Z. Valiev, A. P. Zhilyaev and A. K. Mukherjee, Nature 398 (1999) 684. [2] R.S. Mishra, R.Z. Valiev, A.K. Mukherjee, Nanostruct. Mater. 9 (1997) 4732.

 Superplasticity at low temperature (or equivalently Superplasticity at high strain rates (> 10–2 /s) at a given temperature in the superplastic regime) is caused by:

 increased diffusion, grain boundary sliding and dislocation activity.

 Grain growth is a serious issue during superplasticity experiments. In the case of nc-Ni it

was seen that the grain size could increase to micron sizes, from the starting grain size of

the order of 20 nm. In other materials the grain growth could be less. Grain growth is expected to be less is two phase mixtures (2nd phase as a precipitate preferred) and nd intermetallic compounds. In two phase mixtures the 2 phase has a pinning effect on the

grain boundaries; while in intermetallics (like Ni3Al) order (with respect to the sublattices) has to be maintained during grain growth, which restrains the process.  In cases where grain boundary sliding is the predominant mechanism for superplasticity (e.g. in some Mg alloys), it is seen that non-equilibrium grain boundaries give lower

elongation as compared to equilibrium grain boundaries (due to the long range stress

fields associated with non-equilibrium grain boundaries, which is expected to hamper grain boundary sliding).

 In Ni3Al the high flow stresses and extensive strain hardening during superplastic deformation has been attributed to depletion of dislocations and high stresses required for

the nucleation of new ones [1].

[1] R.S. Mishra, R.Z. Valiev, S.X. McFadden, A.K. Mukherjee, Mater. Sci. Eng., A 252 (1998) 174. Strain rate sensitivity (m) in nanocrystalline materials  The behaviour of strain rate sensitivity is drastically different between FCC and BCC nanomaterials.

 In FCC materials ‘m’ decreases with grain size (when grain size varies over orders of magnitude) and in BCC materials ‘m’ increases with grain size.

 It should be noted that the trendlines are arrived at by putting together results obtained on

many materials obtained by different processing routes.

 In FCC materials forest dislocations and grain boundary (impediment) seems to play an important role (though the exact reasons for the results are not yet clear).

 In BCC materials mobility of screw dislocations controls the plasticity. Kink pair

formation and propagation plays an important role in this process.

Schematic trendline

Schematic trendline  Deformation of nanostructured materials seem to occur predominantly by dislocations at interfaces (and not bulk dislocations). These dislocations have Burgers vector smaller than

lattice dislocations. In some in-situ TEM experiments mobile dislocations were not observed when grain size is less than about 30 nm.  In brittle materials like ceramics, at small grain sizes grain boundary sliding may be the predominant mechanism of plastic deformation.

Work that needs to be done and issues still to be understood regarding high temperature behaviour of nanocrystalline materials [1]:

 Creep (i) Understand rate controlling mechanisms at various temperatures and strain rates;

(ii) Creep tests need to be conducted over a wide range of temperatures and stress exponent and activation energy need to be determined; (iii) grain size sensitivity needs to be determined by synthesizing materials with a wide range of grain sizes; (iv) large strain steady state creep experiments need to be performed; (v) pure materials should be used (to reduce complications of multiphase materials and impurity content) but grain growth should be minimized; (vi) role of triple junction in creep needs to be ascertained.  Superplasticity (i) Obtained porosity free, surface scratch free samples;

(ii) role of grain boundary sliding in deformation needs to be determined; (iii) role of grain size on flow stress and mechanisms operative need to be determined.

[1] F.A. Mohamed, Y. Li : Materials Science and Engineering A298 (2001) 1–15

Case study: interesting example of the deformation of nanocrystal

 Extrusion of single nanocrystals have been carried in graphitic cages [1].  10nm gold and platinum crystals were confined in spherical graphitic shells

→ experiencing a pressure of about 20 GPa (at 300C)!!  Given that the ideal shear strength of Au is only of 1GPa these are very high pressures  this is the ultra-strong regime!

 The crystal had perfect atomic structure (with grain boundaries and stacking faults

occasionally present).

 Holes where punctured in the shell by electron beam → nanocrystal was extruded (due to the high pressures inside the shell.  The question is that- how is deformation proceeding in the absence of dislocations.

 The mechanism of deformation is postulated to be by nucleation and propagation of

dislocations (/partials).

[1] Plastic deformation of single nanometer-sized crystals, L. Sun et. al, Phys. Rev. Lett., 101, 156101 (2008). Case study: Plastic deformation of single crystals

 Bulk Si (called Si-I) with a diamond cubic structure is a brittle material at room temperature. Si transforms to tetragonal metallic form under pressure (called Si-II structure which is akin to -Sn, at ~12 GPa). The load displacement curve for bulk silicon (in indentation experiments) shows a push-out (PO) while unloading. This PO (or kink- back effect) occurs during unloading and is a signature of phase transformation of Si-II to other allotropic forms of Si (BCC Si-III, rhombohedral Si-XII etc.), which are metastable [1]. Additionally, the load-displacement curve shows an elbow which is a signature of amorphization of Si.  On the other hand, Si nanospheres deformed under compression (size < 57 nm) showed a

pop-in (PI) during loading, while no PO was observed during unloading [2]. This has been termed 'deconfinement effect'. PI can be attributed to the onset of plasticity in a dislocation free crystal.

Loading and unloading behaviour of Si: (a) nanoindentation bulk of form, (b) compression of nanospheres. Typically loadmN isand in displacement in nm.

[1] Domnich, V., Gogotsi, Y. & Dub, S., Appl. Phys. Lett. 76, 22142216 (2000). [2] Dariusz Chrobak, Natalia Tymiak, AaronBeaber, OzanUgurlu, William W. Gerberich, Roman Nowak, NATURE NANOTECHNOLOGY | VOL 6 | 2011, P.480 End Mechanism of deformation

 As the grain size is reduced grain boundary sliding starts to play an important role in plastic deformation. This is more so at high temperatures.

 Nature of bonding plays an important role in deciding this cross-over.

[1] R. W. Siegel and G. E. Fougere, Nanostruct.Mater. 6, 205 (1995).