Defect Structure & Mechanical Behaviour of Nanomaterials

1. Defect Structure&Mechanical Behaviour of Nanomaterials

2. Defect structure in nanomaterials • The defect structure in nanomaterials can be altered with respect to their bulk counterparts. • This in turn can lead to profound differences in the mechanical behavior of nanomaterials as compared to bulk materials.

3. Vacancies • Vacancies are equilibrium thermodynamic defects and at a temperature T (in K) there exists an equilibrium concentration of vacancies (nv) in bulk crystals. • nV the number of vacancies, • N number of sites in the lattice (for nV/N << 1) • Interaction between vacancies can be ignored (Hformation (n vacancies) = n . Hformation (1 vacancy) • Hf  enthalpy of formation of a vacancy • Close to the melting point in FCC metals Au, Ag, Cu the fraction of vacancies is about 104(i.e. one in 10,000 lattice sites are vacant)

4. Vacancies in nanocrystals • In free-standing nanocrystals 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 crystal becomes free of vacancies (assuming that the kinetics permits so!). This size (dc) is given by [1]: • Where, nv is the number of atoms 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. Alcrystals 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 dc of 86 nm at 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.

6. Mechanical Behaviour of Materials Overview &  Nanomaterials Density and Elastic properties Plasticity by slip Motion of dislocations & dislocations in finite crystals Strengthening mechanisms Grain boundaries in nanocrystals Twinning versus slip Grain size and strength Superplasticity Nanocomposites Creep Testing of Nanostructures and Nanomaterials

7. Mechanisms / Methods by which a Material can FAIL Elasticdeformation Chemical /Electro-chemicaldegradation Creep Physicaldegradation Fatigue Fracture Slip Microstructuralchanges Wear Twinning Erosion Phase transformations Grain growth Particlecoarsening Failure can be considered as deterioration in desired performance- which could involve changes in properties and/or shape

8. 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 single crystal/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

9. 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)

10. Effect of these defects on the density of the material (ignoring porosity) Effects dominate below ~20 nm grain size Assumed relative density

11. 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). *Range of values is due to diameter differences or number of walls

12. 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. • Under iso-strain conditions [m = f = c] • I.e. ~ resistances in series configuration Voigt averaging • Under iso-stress conditions [m = f = c] • I.e. ~ resistances in parallel configuration • Usually not found in practice Reuss averaging Ef Isostrain Ec→ Isostress For a given fiber fraction f, the modulii of various conceivable composites lie between an upper bound given by isostrain conditionand a lower bound given by isostress condition Em f A B Volume fraction →

13. Nanocomposite 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). Yalumina ~ 350 GPa Nanocomposite of MW-CNT and alumina

14. 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. Effects dominate below ~20 nm grain size Cube morphology

15. 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.

16. Though plasticity by slip is the most important mechanism of plastic deformation, there are other mechanisms as well: Plastic Deformation in Crystalline Materials Creep Mechanisms Slip(Dislocation motion) Twinning Phase Transformation Grain boundary sliding Vacancy diffusion + Other Mechanisms Dislocation climb Note: Plastic deformation in amorphous materials occur by other mechanisms including flow (~viscous fluid) and shear banding

17. 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.

18. Variables in plastic deformation Usually expressed as (for plastic) K → strength coefficientn → strain / work hardening coefficient ◘ Cu and brass (n ~ 0.5) can be given large plastic strain more easily as compared to steels with n ~ 0.15 Low T High T C → a constantm → 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)

19. Further aspects regarding strain rate sensitivity • In some materials (due to structural condition or high temperature) necking is prevented by strain rate hardening. From the definition of true strain • If m < 1→ smaller the cross-sectional area, the more rapidly the area is reduced. • If m = 1→ material behaves like a Newtonian viscous liquid → dA/dt is independent of A. 

20. Importance of ‘m’

21. 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

22. The shear modulus of metals is in the range 20 – 150 GPa • The theoretical shear stress will be in the range 3 – 30 GPa • 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)

23. 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, dislocations are driven only by shear stresses (on the slip plane). • 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

24. Edge Dislocation Glide Motion of an edge dislocation leading to the formation of a step (of ‘b’) Shear stress Surface step(atomic dimensions)

25. 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

26. 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) Superposition of two images [1]

27. Dislocations in nanocrystals • The energy of a dislocation (in bulk crystal) is given by: • Edl – Energy per unit length of dislocation line • b – Modulus of the Burgers vector • g0 - size of the control volume ~ 70b • In nanocrystals the energy of a dislocation will be less than that in a bulk crystal: Due to domain deformations Due to smaller amount of material available to be strained

28. In nanocrystal 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 (PN  b) 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: (for all dislocations) • 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. • G → shear modulus of the crystal • w → width of the dislocation !!! • b → |b| [1]

29. Edge dislocation free • 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. → + Shockley Partials [1] Interface Structures in Nanocrystalline Materials, Scripta mater. 44 (2001) 1169–1174

30. 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].

31. “Zero Stiffness”! Referred to primary axis (a) (a) Zoomed in view Reversible Plastic Deformation due to Elasticity! [1] (b) Referred to secondary axis (b) Nearly zero stiffness Zero Stiffness

32. 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: Solid solution strengthening → solute atoms increase the resistance to dislocation motion Precipitation Hardening and Dispersoid strengthening Forest dislocation hardening Grain boundary hardening Strengthening mechanisms Precipitate & Dispersoid Solid solution Forest dislocation Grain boundary

33. 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 ductility 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.

34. 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). (At a point like ‘X’, small grain size material will plastically deform, while large grain size material will twin). • In the micron size regime, twinning stress decreases with grain size. I.e. large grain size material will twin more readily. With a decreasing grain size, the twinning stress increases more rapidly than the full dislocation slip stress. X

35. Next slide inserted on ref’s comments

36. Before we leave this topic let us take up few additional points about twinning • Twinning/de-twinning is the main mechanism of deformation in shape memory alloys (though twinning typically does not contribute to large deformations). • Twinning can lead to large orientations changes, which is of help in formation of deformation textures in HCP metals.

38. Superplasticity • The phenomenon of extensive plastic deformation without necking is termed as structural superplasticity. Superplastic deformation in tension can be >300% (up to even 2000%). • Typically superplastic deformation occurs when: (i) T > 0.5Tm(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 a 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).

39. A plot of stress versus strain rate is often sigmoidal and shows three regions: • 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. • 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)). • Region-III- high stress & strain rate regime ( > 102 /s)  m > 0.33Creep rates sensitive to grain size. Mechanism intragranular dislocation process (interacting with grain boundaries). Note: low ‘m’ in region I and III

40. 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. Harper-Dorn creep Phenomenology Power Law creep Creep can be classified based on Mechanism

41. Stages of creep I • Creep rate decreases with time • Effect of work hardening more than recovery • Stage of minimum creep rate → constant • Work hardening and recovery balanced II • Absent (/delayed very much) in constant stress tests • Necking of specimen start • specimen failure processes set in III

42. Creep Mechanisms of crystalline materials Cross-slip Harper-Dorn creep Climb Dislocation related Glide Coble creep Creep Grain boundary diffusion controlled Nabarro-Herring creep Diffusional 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

43. 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

44.   Flow of vacancies Nabarro-Herring creep → high T → lattice diffusion Diffusional creep 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 dislocation creep is controlled by the diffusion of vacancies → but diffusional creep does not require dislocations to operate.

45. Creep mechanism map

46. Testing of Nanostructures and Nanomaterials

50. 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