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PHYSICAL REVIEW B 1 MARCH 1999-IVOLUME 59, NUMBER 9

Grain-size limit of polycrystalline materials

N. X. Sun* and K. LuState Key Laboratory of Rapidly Solidified Alloys, Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110

People’s Republic of China~Received 28 September 1998!

Through combining an experimental investigation on the isothermal nanocrystallization thermodynamics ofbulk amorphous selenium and a theoretical analysis, an easy and reliable way to derive the lower grain-sizelimit of polycrystalline materials was proposed. The grain-size limit for the polycrystalline selenium wasdetermined to be about 4.0 nm.@S0163-1829~99!00310-0#

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Solid polycrystalline materials are composed of crystlites that are separated by grain boundaries, and it hasof general curiosity that whether there is a lower boundthe mean grain size, or a grain-size limit for a polycrystallimaterial. Up to now, there are indications that there existhermodynamics-based grain-size limit~GSL! for each poly-crystalline material, below which the amorphous state willenergetically favored.1–4 Obtaining the thermodynamicsbased GSL is very important, as it can cast light onunderstanding of the transition between the polycrystaland amorphous state; in addition, it can provide guidancpractical manipulation of materials at the atomic level. Hoever, efforts to obtain the GSL for a specific material are rand far from being successful.

Theoretical considerations on this issue are rare. Throatomistic computer simulations, Wolfet al.1 found that thereexists the possibility of a reversible, free-energy based trsition between the amorphous and the polycrystalline stwith a critical mean grain size~about 1.4 nm in their modematerial!. Vepreket al.2 made a rough evaluation of the GSof silicon by assuming the excess energy of the nanocrysline silicon to be the elastic energy associated with lattexpansion which leads to a transition from the crystallstate to amorphous state. These GSL’s are thermodynabased, as they are controlled only by the thermodynaparameters of the polycrystalline materials, without regardthe kinetics involved in the particular experimental metho

Experimentally, Vepreket al.2 reported that the lowesobtainable mean grain size for vapor-phase deposited pcrystalline silicon was about 2.0 nm for films with compresive stress and 3.0 nm for stress-free films through x-diffraction ~XRD! analysis. Another way to determine thGSL is through mechanical attrition-induced solid-staamorphization, which usually proceeds via a nanocrystalprecursor state with a minimum grain size. However,minimum grain sizes determined for the same matethrough the ball-milling method often differ significantly, foexample, the minimum grain sizes for the ball-milled silicare 8 and 3 nm, respectively, in two investigations.3,4 Boththe deposition method and ball-milling method for determing the thermodynamics-based GSL’s are flawed due tofollowing facts. First, the experimentally available minimugrain sizes obtained through deposition and mechanical ation are actually the result of the competition betweenkinetic and thermodynamic factors, and will change sign

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cantly with different experimental conditions. Second, whthe mean grain sizes of polycrystalline materials approthe minimum grain size, e.g., 2–3 nm, it is rather difficultdistinguish between the amorphous and nanocrystalstates. Third, extended defect structures, voids, and contnation are nearly inevitable in the materials obtained throuboth the mechanical attrition and deposition methods.these factors will affect the experimentally obtained mimum grain sizes significantly,2–4 and result in error in determining the thermodynamics-based GSL. Other methshould be explored to obtain the thermodynamics-based Gin polycrystalline materials. Here, through combining an eperimental study of the nanocrystallization thermodynamof amorphous selenium and theoretical analysis, a simplereliable way to obtain the GSL of polycrystalline materialspresented.

Nanocrystallization of amorphous solids provides anfective way to obtain nanocrystalline materials with denand clean interfaces, low microstrain, and nearly perfcrystallite structures.5 The nanocrystalline materials synthsized through this method are excellent model materialsdetermining the thermodynamics-based GSL. After a stematic study on the thermodynamics of the crystallizatprocess of amorphous selenium, a simple and reliamethod to obtain the thermodynamics-based GSL’s of pocrystalline materials is presented in this paper.

Bulk amorphous selenium~purity better than 99.999%!was produced by quenching the Se melt that was sealed ievacuated quartz ampoule into a mixture of water andThe obtained amorphous Se was then fully relaxed in a wbath before thermal analysis. Isothermal annealing ofamorphous Se at different annealing temperatures ranfrom 373 to 433 K was monitored and recorded by a diffential scanning calorimeter~DSC-7, Perkin-Elmmer! to ob-tain the different grain-sized nanocrystalline Se and to msure the isothermal crystallization enthalpyDHcryst of theamorphous Se at the same time. Quantitative x-ray diffrtion ~XRD! analyses and transmission electron microsco~TEM! observations were carried out to determine the avage grain sizes of the isothermally crystallized Se sampMeasurements of the heat capacity at constant pressureCp)were carried on the same differential scanning calorimetethe temperature range of 223–500 K on the fully relaxamorphous, nanocrystalline~with a mean grain size of 10

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nm!, and conventional polycrystalline Se. Details of the eperiments are available in Refs. 6,7.

The measured excess heat capacity of the undercoliquid Se with respect to the conventional polycrystalline SDCp

c2 l(T), can be expressed as7 DCpc2 l(T)543.6

20.079T, when 320<T<493 K. Then, the enthalpy difference between the undercooled liquid Se and the conventipolycrystalline Se,DHl 2c, can be calculated through the folowing equation:DH5DH f1*Tf

T DCpc2 l(T)dT, whereDH f

is the enthalpy of fusion andTf is the equilibrium meltingtemperature of Se. It was found out that the calculaDHl 2c is evidently not equal to the measuredDHcryst; andthat the difference between the two enthalpies is actuallyexcess enthalpy of the nanocrystalline Se with respect toconventional polycrystalline Se,DHc2nc, which can be ex-pressed by the following equation:7

DHc2nc5DHcryst2DHl 2c. ~1!

So, the excess enthalpy values of the nanocrystallized pucts with different mean grain sizes can also be calculat

It has been widely accepted that at low temperatures,entropy effects will not be large and the enthalpy can bgood approximation of the Gibbs free energy.8 Based on thisapproximation, a linear extrapolation method to determthe GSL in polycrystalline Se is presented below.

1. Linear extrapolation betweenDH cryst and 1/d

If we assume that the excess enthalpy of the nanocrylized Se concentrates in the grain boundaries, then the exenthalpy of the nanocrystalline Se with respect to the cventional polycrystalline state,DHc2nc, can be expressed afollows:9

DHc2nc52ggVm /d, ~2!

where g is the grain-boundary enthalpy,Vm is the molarvolume,d is the average grain size, andg is the geometricalfactor depending on the shape and size distribution ofgrains. Substituting Eq.~2! into Eq. ~1!, we can get

DHcryst5DHl 2c12ggVm /d. ~3!

If we assume that the enthalpy difference between the uncooled liquid and conventional polycrystalline SDHl 2c(T), keeps constant@although actuallyDHl 2c(T) isnot constant in the temperature range of 373–433 K, sucassumption is acceptable, as can be seen from the followresults#, and that the product of the grain-boundary enthag and the geometry factorg does not vary with the meagrain size, then we can obtain a linear relationship betwDHcryst and 1/d according to Eq.~3!. Figure 1 shows themeasured enthalpy of crystallization of the amorphousDHcryst, as a function of 1/d, the reciprocal of the meangrain size of the crystallized products. It is clear that all tdata points are on an approximate straight line. Also,intercept on the inordinate is about25.5 kJ/mol, corresponding with the average value ofDHl 2c(T), where T is thetemperature range of 373–433 K. When linearly extrapoing the measuredDHcryst values toDHcryst50, we can obtainthe intercept on the abscissa, 1/dlim . The implication ofdlimis clear, when the mean grain size of the crystallized

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reachesdlim , the enthalpy, or approximated Gibbs free eergy of the nanocrystallized product, the nanocrystallinewill be equal to that of the amorphous state. Therefore,obtaineddlim is actually the thermodynamics-based GSLpolycrystalline Se. Here, the obtained GSL isdlim53.9 nm.

The merit of the above approach to obtain the GSL issimplicity, GSL can be obtained just through measuringDHcryst and d. However, this approach has to assume tDHl 2c(T) keeps constant during the temperature ranwhere the enthalpies of crystallization of the amorphous sare measured. This is correct for most elemental metalsusually have lowDCp

c2 l(T),10 but is not often true for mul-ticomponent alloy systems with comparatively highDCp

c2 l(T).11

2. Linear extrapolation betweenDH c2nc and 1/d, animproved approach

According to Eq.~2!, the excess enthalpy of the nancrystallized products,DHc2nc, will scale linearly with thereciprocal of the mean grain size of the nanocrystallizproducts with a zero intercept on the inordinate, if we asume the product ofg andg keeps constant with varyingd.Figure 2 shows the calculated excess enthalpy of the nacrystallized product,DHc2nc, as a function of the reciprocaof the mean grain size of the nanocrystalline Se. As indicain Fig. 2, all the data points are on a straight line withexperimental error; and also, the intercept on the inordinis close to zero, just as what is predicated in Eq.~2!. Whenlinearly extrapolatingDHc2nc to DHc2nc5DHc2a, whereDHc2a is the enthalpy difference between the amorphostate and the conventional polycrystalline Se, the enthalpthe approximate Gibbs free energy of the nanocrystalliproduct will be equal to that of the amorphous state, andcorresponding mean grain size will be the thermodynambased GSL. Here, the enthalpy difference between the amphous and conventional polycrystalline Se is chosenDHc2a53.4 kJ/mol, the nearly constant enthalpy differenbetween the amorphous and conventional polycrystallinewhenT,250 K, whereDCp

c2a(T)50.7 As indicated in Fig.2, DHc2nc reachesDHc2a at dlim54.1 nm. The two GSL’sobtained in the two approaches are very close, in the rang4.060.1 nm, and are comparable with that obtained for S4

FIG. 1. Linear extrapolation of the experimentally measuredthalpy of crystallization for amorphous Se,DHcryst, as a function of1/d, the reciprocal of the mean grain size of the nanocrystallized

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It is notable that in this method for determining GSL, itassumed that the excess energy of the nanocrystallizeduct concentrates in the grain boundaries, and that the crylites have the same lattice structure as those in the cograined counterpart. Generally, a detectable lattice expan

FIG. 2. Linear extrapolation of the excess enthalpy of the nacrystallized Se,DHc2nc, as a function of 1/d, the reciprocal of themean grain size.DHc2a is the excess enthalpy of the amorphouswith respect to the conventional polycrystalline Se whenT,250 K.

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~with volume expansion less than 1%! often exists in thenanocrystalline materials crystallized from the amorphostate,5 and the elastic energy associated with this expansis negligible ~less than 10 J/mol in Se! compared with theother enthalpy values, such asDHcryst andDHc2nc.

Compared with the other methods for determining tthermodynamics-based GSL, this linear extrapolatimethod possesses its unique advantages. First, the matewhich are obtained through annealing the amorphous stahave nearly perfect crystallite structure and can serve ascellent model material for accessing their thermodynamproperties. Second, the GSL is obtained by extrapolatingexperimental data guided by theoretical analysis, the comnation of the experimental results and theoretical analydistinguishes this method from all the other methods fevaluating the GSL’s, which depend on either experimenobservation or theoretical calculation. Third, this methodrather reliable. In the two approaches for accessing GSLthe experimental data are all on a very good line with propintercept on the inordinate, just as what has been predicain theoretical analysis. The good agreement between theperimental results and theoretical predication indicates tthis method is very reliable.

The support from NSFC and the Chinese AcademySciences is gratefully acknowledged.

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*Present address: Department of Materials Science and Enging, Stanford University, CA 94305. Electronic addrenxsun@crism.Stanford.EDU1D. Wolf, J. Wang, S. R. Phillpot, and H. Gleiter, Phys. Lett.

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