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Cite this: Nanoscale, 2017, 9, 12283

Received 7th May 2017,Accepted 27th June 2017

DOI: 10.1039/c7nr03225k

rsc.li/nanoscale

Liquid like nucleation in free-standing nanoscalefilms†

Pooja Rani,a Arun Kumar,a B. Vishwanadh,b Kawsar Ali,b A. Arya,b R. Tewarib andAnandh Subramaniam *a

The concept of a critical nucleus size (r*) is of pivotal importance

in phase transformations involving nucleation and growth. The

current investigation pertains to crystallization in nanoscale thin

films and study of the same using high resolution lattice fringe

imaging (HRLFI) and finite element simulations. Using the CuZrAl

bulk metallic glass system as a model system for this study, we

demonstrate a liquid like nucleation behaviour in nanoscale free-

standing films upon heating. The r* for the formation of the

Cu10Zr7 phase in thin films (of decreasing thickness) approaches

that of the r* for the formation of the crystal from a liquid

(i.e. r*thin film ! r*liquid). Working in the nucleation dominant regime,

we introduce the concept of ‘depth sensitive lattice fringe

imaging’. The thickness of the film is determined by electron

energy loss spectroscopy and the strain energy of the system is

computed using finite element computations.

Diffusional phase transformations occur by nucleation andgrowth.1,2 Embryos with sizes above r* (the critical size fornucleation) grow, while smaller ones tend to revert to theparent phase (‘dissolve’). From a statistical thermodynamicsperspective the nucleation event occurs by a ‘random fluctu-ation’, which is the ‘uphill’ Gibbs free energy (G) and thegrowth occurs downhill in ‘G’, at constant temperature andpressure. From a classical view point, the nucleation of acrystal from a liquid is driven by the reduction in the volumefree energy (ΔGV), which is opposed by an increase in the inter-facial free energy (γ). In the case of the nucleation of a solidphase in a solid matrix (e.g. a crystal from a solid phase like aglass), the nucleation event has to overcome an additionalhurdle in the form of the strain energy, arising from thevolume (and/or shape) mismatch between the phases. The net

Gibbs free energy change for the formation of a crystal from asolid (ΔGhomo) via homogeneous nucleation is given by:

ΔGhomo ¼ �ðΔGV � VÞ þ ðγ � AÞ þ VΔGS ð1Þwhere ΔGV is the Gibbs volume free energy difference betweenthe parent and product phases, ‘V’ is the volume of theproduct phase, ‘A’ is the interfacial area, and ΔGS is the strainenergy associated with the transformation. For a sphericalnucleus of radius ‘r’, eqn (1) can be written as:

ΔGhomo ¼ � 43πr3ðΔGV � ΔGSÞ þ ð4πr2Þ�γ ð2Þ

The critical size of the nucleus is determined by finding theextremum of the function ΔGhomo(r) and in the case of liquidto crystal transformation (i.e. in the absence of the strainenergy term) the critical size of the nucleus is: r* = 2γ/ΔGV.

For the formation of a crystalline nucleus in an amorphousmatrix, the crystal has a lower volume and hence is in a stateof tensile stress. The strain energy depends on the elasticconstants of the parent and product phases (in addition to themisfit strain fm) and can be calculated using:3

EStrain ¼ 8πr3fm2G3

ð1þ νÞð1� νÞ ð3Þ

where ‘fm’ is the linear mismatch between the nucleus and theglass matrix (the linear mismatch is one third of the volumemismatch), ‘G’ and ‘ν’ are the shear modulus and Poisson’sratio, respectively, of the product phase, which is assumed tobe isotropic. It is to be noted that the elastic properties of onlythe product phase appears in eqn (2).4 In the presence ofstrain energy (e.g. in the glass to crystal transformation) thevalue of r* increases and is given by:

r* ¼ 2γðΔGV � ΔGSÞ ð4Þ

where ΔGS ¼ EStrainV

is the misfit strain energy per unitvolume.

In the literature, interesting effects related to the crystalliza-tion of glasses have been studied, which include the formation

†Electronic supplementary information (ESI) available: (i) details of the theore-tical/computational and experimental methodology and (ii) the results (relatedto XRD, DSC & TEM investigations). See DOI: 10.1039/c7nr03225k

aMaterials Science and Engineering, Indian Institute of Technology Kanpur, Kanpur-

208016, India. E-mail: anandh@iitk.ac.inbMaterials Science Division, Bhabha Atomic Research Centre, Trombay, Mumbai-

400085, India

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of a nanoscale liquid like layer during the deformation ofBMGs.5 Furthermore, nanocrystallization in the vicinity of theshear band (with crystallite sizes of about 5 nm) has beenreported, which is localized to a thickness of about 10 nm.6

Sohn et al.7 have studied the effect of the diameters of glassnanorods on the crystallization temperature and have shown anon-monotonic behaviour. Jiang and Ward8 have summarizedimportant effects arising from crystallization of organic com-pounds in nanoscale confinement, which includes suppres-sion of crystallization in very small pores. The effect ofirradiation on the nucleation and growth aspects relatedto crystallization of glasses has also been investigated.9 Inparallel literature, investigators have studied nanoscale effectson phase transformations in general. Chen et al.10 have shownthat in nanoscale samples, the transformation can be accom-plished by a single nucleation event. Zhou et al.11 have studiedheterogeneous nucleation of Al in confined nanoslits usingmolecular dynamics simulations. The finite size effects on thetransformation rates12,13 and coherent to semi-coherent tran-sitions have also been investigated.14

A study of the literature shows the lack of a systematicstudy of the effect of nanoscale domains on the critical sizefor nucleation. Given this scenario, the present work aims atthe following tasks via computations and experiments: (i)determining the critical size for nucleation in bulk samples(r*bulk), (ii) studying the variation of r* with the thickness of thesample (i.e. compute r*thin filmðtÞ & determine the thickness ofthe film below which “nanoscale effects” gain prominence)and (iii) ascertaining the liquid like nucleation behaviouroccurring in nanoscale thin films. Experiments are performedusing Cu–Zr–Al bulk metallic glass (BMG) as a model system;wherein, the size of crystallites formed upon devitrification isdetermined by high resolution lattice fringe imaging (HRLFI).This system has served as a model one in other contexts aswell.15 The BMG alloy serves as a model system for the study ofthe phenomena due to the following reasons: (i) homogeneousnucleation is expected to be dominant in the absence of grainboundaries, dislocations and other crystallographic defects;(ii) it is convenient to determine the size of the nuclei viaHRLFI as the amorphous matrix does not give lattice fringes;(iii) by keeping the temperature of the transformation low, wecan achieve the ‘nucleation dominant regime’ (i.e. practicallysuppressing growth); (iv) the interfacial energy is expected tobe reasonably isotropic (and the error introduced by anapproximate treatment of the glass matrix as a liquid for thecomputation of γ is small) and (v) the diffusion is sluggish atroom temperature and hence any transformation while per-forming experiments (like TEM) can be ignored.

From eqn (3) we see that to compute the value of r* the fol-lowing parameters are required: (i) ΔGV, (ii) γ and (ii) Estrain. Inthe approximation that the material is isotropic, to determineEstrain, we further require the values of the: (i) Young’smodulus (Y), (ii) Poisson’s ratio (ν) and (iii) misfit ( fm). Thevalues of the moduli for both the glass and crystal arerequired. For the glass to crystal transformation, the value ofΔGV is calculated using the method prescribed by Zhou and

Napolitano,16 the value of γ is determined using a formuladerived using the negentropic model17,18 and the strain energyis computed using a finite element model.19 It is worth notingat this juncture that other computational techniques like mole-cular dynamics and density functional theory have also beenused successfully for the study of the nucleation behaviour insolids20–22 in addition to experimental techniques likeHRLFI.23 A simplifying assumption for the calculation of ΔGV

(glass → crystal) often used is to compute the value for liquidto crystal transformation.24 For the formation of the Cu10Zr7crystal (Cmca, oC68, a = 12.68 Å, b = 9.31 Å, c = 9.35 Å (ref. 25))from an amorphous matrix (composition of (Cu64Zr36)96Al4),the value of ΔGV can be determined using the method of Zhouand Napolitano:16 (−16 133 − 1.905T ). Another approach forthe computation of ΔGV is to use the Turnbull approximation:ΔGV = ΔHf(Tm − T )/Tm, where Tm is the melting temperature.It is noteworthy that in the current case both these approachesgive a very similar value of ΔGV. The values of the parametersare:26 ΔHf = 1.23 × 104 J mol−1, Tm = 1185 K and the computedvalue from the Turnbull equation: ΔGV (Turnbull) = 7.39 × 103

J mol−1.The crystal–glass interfacial free energy (γ) is a difficult

quantity to compute and often it is observed that the value ofthe crystal-liquid interfacial energy is a good enough approxi-mation and can be computed using:18

γ ¼ αΔHf

ðNAVm2Þ1=3

ð5Þ

where α is the Turnbull coefficient which depends on thecrystal structure, ΔHf the enthalpy of fusion, NA Avagadro’snumber and Vm the molar volume of the crystal. The com-puted value of γ = 0.03 J m−2.

In the case of nucleation in finite domains, the strainenergy of the system is lower, due to a smaller volume of thestrained material. In the proximity of free surfaces, the strainenergy of the system is further lowered due to relaxationarising from surface deformation.14 In nanoscale thin filmsthese effects gain prominence and the analytical determi-nation of the strain energy becomes practically intractable.Hence, to determine the strain energy of the nuclei as a func-tion of the thickness we employ finite element simulations.A brief outline of the finite element model is described hereand the reader may refer to the work of Kumar et al.4 forfurther details. The schematic of the finite element (FEM)model is shown in Fig. 1a. The (Cu64Zr36)96Al4 BMG system isused in the current investigations, wherein Cu10Zr7 crystals areformed upon nucleation.27 The stress state and strain energyfor various sizes (values of ‘r’) of the crystal positioned at adistance of ‘h’ from the horizontal free-surface in a domain(infinite along the lateral dimension) of thickness ‘t’ is simulatedusing a 2D-axisymmetric numerical model. The eigenstrainscorresponding to the mismatch (‘fm’) are imposed as thermalstrains in region-C of the FEM model. The displacementboundary conditions used for the simulations are also shownin Fig. 1a.

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The data used in the computations are: Yglass = 96.4 GPa,νglass = 0.355, Ycrystal = 131.0 GPa & νcrystal = 0.326. The elasticmoduli for the glass (Y & ν) have been determined by applyingthe ‘rule of mixtures’ and are calculated as: M�1 ¼ P

fiM�1

i ,where M & Mi are the elastic constants for the glass and theconstituent elements, respectively, and fi denotes the atomicpercentage of the constituent elements.28 The elastic constantsof the crystal have been determined by Voigt, Reuss and Hill(VRH) averaging the single crystal data (C11 = 190 GPa, C12 =88 GPa, C13 = 102 GPa, C22 = 185 GPa, C23 = 105 GPa, C33 =167 GPa, C44 = 63 GPa, C55 = 63 GPa and C66 = 47 GPa).29 Thedensity of the glass is determined by the Archimedes method(ρglass = 7.17 g cm−3), while the density of the crystal (Cu10Zr7)is calculated using the volume of the unit cell and atomicmasses (ρcrystal = 7.64 g cm−3).

One special situation arises due to surface nucleation. Inthis case two additional issues have to be taken into account:(i) the amorphous-vacuum interface is replaced by the crystal-vacuum interface and this additional difference in free energyhas to be included in the computation of critical nucleussize (r*surface) and (ii) the shape of the nucleus is expected to bealtered. Assuming a hemi-spherical nucleus, eqn (2) can bemodified for surface nucleation as:

ΔGsurface ¼ � 23πr3ðΔGV � ΔGSÞ þ ð2πr2Þ�γ þ ðπr2Þ�ðγcrystal

� γamorphousÞ ð6Þ

where γamorphous and γcrystal are the surface energies of theamorphous structure and crystal, respectively. The nature ofthe crystallographic surface will determine the value of thesurface energy of the crystal and akin to the bulk nucleation amyriad variety of shapes can be envisaged. These and otherissues involved (ESI†), make the analysis of surface nucleationa very involved process and a detailed study of the sameprovides scope for future work. In the current investigation weconsider an illustrative model, wherein a hemi-spherical

Cu10Zr7 nucleus is considered with a (100) surface (inset toFig. 1a). The values of the surface energies involved (γamorphous

& γcrystal) are computed using molecular dynamics simulation(ESI†) and the values obtained are: γamorphous = 1.335 J m−2

and γcrystal100 = 1.322 J m−2. An interesting feature of surface

nucleation is that, the relative magnitudes of γamorphous andγcrystal will determine, whether the nucleation process will beaided or impeded by the surface energy term. Needless topoint out, r* will be smaller for lower energy crystal surfaces.

The stress state of the system as computed using the modelin Fig. 1a, in the presence of a crystal of radius 1 nm (withh = 2 nm) in a film of thickness of 8 nm, is shown in Fig. 1b.The strain energy of the system as a function of the thicknessof the film, as computed from the model is shown in Fig. 2a.In addition, Fig. 2b includes the strain energy as a function ofthe height (h) of the crystallite within the film (Estrain as a func-tion of ‘h’). As expected Estrain decreases with ‘h’. The shadedregion in (a) corresponds to the film thicknesses, where thenanoscale effect becomes really predominant. From curve (b)it is observed that significant changes in strain energyonly occur for crystallite positions in close proximity to thefree surfaces (<1 nm).

Having determined the strain energy, we are in a positionto compute the Gibbs free energy change during the nuclea-tion event as a function of the radius of the crystal. Fig. 2cshows the plot of ΔGhomo(r) for three cases of nucleation: (i) ina liquid, (ii) at the centre of a film of thickness of 4 nm and(iii) in bulk domains. The peak value of ‘r’ in the plot corres-ponds to r*. As expected, the value of r* increases(r*liquid , r*thin film , r*bulk) with an increased cost of strainenergy. In the present work, the symbol r* without anadditional qualifier, implies the nucleation event at the centreof the sample and r* for a nucleation event at a height ‘h’ isdenoted by r*(h).

Fig. 1 (a) The finite element model used for the computation of strainenergy in the presence of a spherical crystal of radius ‘r’ in a free stand-ing thin film of thickness ‘t’. The boundary conditions used are alsomarked in the figure; along with the region (C), where eigenstrains areimposed to simulate the stress state in the presence of a crystal. Insetshows the model used for surface nucleation. (b) Stress state (plot ofσyy contours) in the presence of a crystal of radius 1 nm, located at adistance of 2 nm from the surface (h = 2 nm) in an 8 nm thick film.

Fig. 2 (a) Plot of the variation of strain energy of the system (Estrain) as afunction of the thickness (t ). (b) Estrain as a function of the height (h) ofthe crystal within the domain. (c) (Inset) Variation in ΔGhomo versusradius of the crystal (r) for nucleation in: (i) bulk, (ii) a liquid & (iii) at thecentre of a domain with t = 4 nm. The maxima in the curves correspondto r*. The shaded region in (a) corresponds to the film thicknesses,where the nanoscale effect becomes really predominant.

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The experimental details are briefly outlined here and theESI† contains further details. The (Cu64Zr36)96Al4 BMG is syn-thesized by arc melting followed by suction casting into acopper mold. Fig. 3a shows the SAD pattern showing the for-mation of an amorphous structure. Two kinds of samples wereannealed to cause nucleation: (i) bulk samples (200 °C for 4 h)and (ii) thin samples (200 °C, 10 min). The choice of the temp-erature is to be in the nucleation dominant regime and tominimize any growth of the crystallites. This is ascertained bytwo methods (ESI†): (i) plotting nucleation and growth ratecurves and (ii) determining the value of the Avrami exponent.It should be noted that the JMAK model used is strictly appli-cable only in the macroscale, wherein ‘random homogeneousnucleation and isotropic growth’ conditions exist. Furtheraspects are discussed in the ESI.†

The choice of the temperature and time is to have only afew nuclei in the field of view in a HRTEM. HRLFI is used forthe determination of the crystallite size30 and electron energyloss spectroscopy (EELS) to determine the sample thickness.31

Fig. 3b shows LFI from a typical crystallite in an amorphousmatrix and Fig. 3c shows the EELS spectrum from a region ofdiameter 10 nm. A total of 16 regions of varying thicknesseswere studied using TEM and the sizes of 30 crystallites weredetermined using LFI.

The variation of r* with the thickness of the sample isshown in Fig. 4. The figure includes the following results:(i) r*bulk (both computed values and experimental values), (ii)r*liquid, (iii) r

*thin film, (iv) r

*surface and (v) experimental data (shown

as a scatter of ‘points’). The r*bulk has been determined byaveraging eight data points and has a value of 1.41 ± 0.02 nm.The data plotted with a circular legend correspond to thelargest crystallite observed within a region having a thicknessas marked along the x-axis. On the other hand the ‘starshaped’ legends correspond to the smallest crystallitesobserved. It is to be noted that multiple crystallites are notobserved in all regions studied and if only one crystallite isobserved, it is marked with a circular legend. The figure alsoincludes the variation of r*(h) for a domain with t = 8 nm. The

following points can be noted from the figure. (i) A goodmatch seen between the computed and experimental values ofr*bulk. (ii) There is a decrease in the value of r* for thin filmswith thicknesses less than about 9 nm and the computedtrendline of r*thin film is in close correspondence to thatobserved experimentally for the nuclei marked with filledcircles. (iii) As h → t/2 (i.e. the nucleus ‘approaches’ the centreof the domain), the r* value increases and (iv) the value ofr*surface (illustrated in the figure for an example of a hemi-spherical crystal with a (100) surface) is similar to r*liquid and‘star shaped’ legends lie close to these values. Based on theseobservations, we can conclude that a nucleation behaviourapproaches that from a liquid (i.e. r* ! r*liquid) under two cir-cumstances: (i) in thin films, wherein the nucleation barrier isreduced and (ii) for nuclei close to the surface. In both thesescenarios the decrease in r* is due to the reduced strain energypenalty. Additionally, for nucleation on the surface, the behav-iour akin to that from the liquid (i.e. r*surface � r*liquid) can occur.This arises due to the surface energy term (the amorphous-vacuum interface is replaced by the crystal-vacuum interfaceupon nucleation), in addition to the strain energy term. Animportant point to be noted from Fig. 4 is that ‘bulk like’ be-haviour is retrieved for films of thicknesses of about 12 nm.

Given that the heat treatment was carried out in the nuclea-tion dominated regime, we can conclude that larger nuclei arelocated closer to the centre (h → t/2) and the smaller ones arecloser to the surface (a decreasing ‘h’ implies a decreasing r*).This implies that we can perform ‘depth (z) sensitive latticefringe imaging’; i.e. determine the position (h) of the nucleifrom lattice fringe images (in conjunction with the compu-

Fig. 3 (a) Selected area diffraction (SAD) pattern showing the formationof an amorphous structure upon suction casting the (Cu64Zr36)96Al4alloy. (b) HRLFI from a crystallite (2r = 2.86 nm) embedded in an amor-phous matrix. (c) A typical EELS spectrum acquired from the region ofdiameter 10 nm, which is used in the computation of the thickness ofthe specimen.

Fig. 4 (i) The computed variation of r* (r*thin film (computed)) with thethickness of the sample (curved line). (ii) The horizontal dashed linescorrespond to r*bulk (computed), r*bulk (experimental) and r*liquid. (iii) Theexperimental data for thin films is shown as a scatter of points. (iv) Thechange in the value of r* with ‘h’ (computed r*(h)) for a film of thicknessof 8 nm (shown for discrete values of h∈[4.0 nm–0.9 nm], enclosed in avertical box). The inset to the figure shows a schematic rendition of thevariation. (v) The horizontal line (solid) corresponds to r*surface (computed)for a (100) oriented hemi-spherical surface nucleus. The experimentaldata points presented in a horizontal box are expected to be surfacenuclei. The errors associated with these measurements are described inrespective sections, including the ESI.†

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tations as above). Exempli gratia, if the crystallite size is 1 nmin a film of thickness of 8 nm, then it must be at a depth of1.2 nm from the surface (the encircled point within the greenvertical box shown in Fig. 4). Similarly, for other values of ‘t’we can calculate ‘h’. It is noteworthy that lattice fringe imagesusually do not contain depth information related to the fea-tures observed.

Most of the material properties used in the present work(e.g. E, ν, γ, etc.) are best suited for the ‘macroscale’. It isexpected the values of these parameters are size dependent,especially at the nanoscale. To date, sufficiently accurate com-putations of the size dependence of these parameters arelacking and hence we have used the best possible approxi-mation. In the future we hope that better models can lead toaccurate values for these material properties, but, we expectthat the qualitative picture presented will be preserved.

In summary, we have demonstrated that the nucleation be-haviour in free-standing nanoscale solid films approaches thatof nucleation from the liquid state (i.e. r*thin film tends to r*liquid)for films of thicknesses less than about 12 nm. Further weargue that the nucleation behaviour on solid surfaces canoccur akin to that from a liquid (i.e. r*surface � r*liquid). We havealso introduced the concept of ‘depth sensitive lattice fringeimaging’ in the context of nucleation. Although this phenom-enon has been elucidated for crystallization in nanoscale thinfilms, the effects observed are expected to be similar for thenucleation step in any solid to solid phase transformationoccurring in any sample having a nanoscale geometry (likerods, spheres, wedges, etc.) as this effect arises due to areduced strain energy penalty for the nucleation.

Acknowledgements

Prof. Kallol Mondal and Anshul Gupta are thanked for theirinvaluable inputs. We thank Mr Ambresh (Advanced ImagingCentre, I.I.T. Kanpur) for his kind help with TEM.

Notes and references

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