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in recent years , advanced production techniques have made it possible to create metals , alloys and ceramics with grain sizes down to 5 nm . for metals ,
this represents a reduction of the grain size by approximately four orders of magnitude compared to most conventionally produced metals .
as can be expected , such a dramatic change in the microstructure leads to significant changes in the mechanical properties of the metals .
for example , the hardness of typical nanocrystalline metals is far higher than what is seen in their coarse - grained counterparts ( * ? ? ? * ; * ? ? ? * and references therein ) .
nanocrystalline metals are an attractive group of metals to model , as the small grain size provides a `` cut - off '' of the typical length scales , where structures appear during deformation
. in coarse - grained materials structures appear on vastly different length scales , making it very difficult to model the properties of these materials .
the models must include processes that occur on length scales from the sub - nanometer scale of the atomic processes in dislocation cores , to the micro- or even millimeter scale of grain and subgrain structures @xcite .
atomic - scale simulations of systems of these sizes are beyond the reach of even the most powerful of todays supercomputers .
one is thus forced to split the system into sub - problems at different length - scales , and only treat sub - problems at the atomic scale with atomistic models .
on coarser length scales other modeling paradigms must be used , such as dislocation dynamics and continuum plasticity calculations .
dividing the problem into sub - problems at different length scales often results in a better understanding of the problem , as it draws attention to the structures which are relevant at a given length scale ( for example , when studying the formation of dislocation structures it is clearly more relevant to focus on dislocations as the fundamental concept rather than on individual atoms ) .
on the other hand , one becomes dependent on this understanding , when creating the coarse - grained models , as many assumptions about the relevant phenomena at different length - scales will by necessity be built into the multi - scale model .
the low grain size of nanocrystalline metals `` compresses '' this range of length scales to a range , where the whole deformation problem can be modeled at the atomic scale , as many grains in the polycrystalline material can be handled in an atomic - scale simulation .
this makes it possible to perform unbiased simulations of the deformation process , where no _ a priori _ assumptions are made about the deformation mechanisms .    in recent papers
, we have presented simulations of the plastic deformation of nanocrystalline cu and pd @xcite .
other authors have presented simulations of the structure and elastic properties of nanocrystalline metals and semiconductors @xcite , and of the plastic deformation of ni under constant stress loading @xcite . in this paper
we review our simulations of the deformation mechanisms in nanocrystalline metals , and present simulations of the effects of impurities in the grain boundaries .
we have attempted to generate three - dimensional systems with a microstructure similar to that observed experimentally in samples generated by inert gas condensation ( igc ) .
the grains appear to be randomly oriented , approximately equiaxed , and dislocation free .
the grain size distribution is close to log - normal ( * ? ? ?
* ; * ? ? ?
* and references therein ) .
we try to create systems that match this description .
the grains are produced using a three - dimensional voronoi tesselation : random grain centers are chosen , and space is divided into regions in such a way that each region consists of the points in space closer to a given grain center than to any other grain center .
each region is then filled with a randomly oriented fcc lattice .
an example of a two - dimensional voronoi tessellation is shown in figure [ fig : voronoi ] .
the generated samples are annealed for 50ps at 300k to relax the grain boundary structure .
we found that the annealing time and temperature are uncritical , but that the properties of the system are different if _ no _ annealing is done .
the interactions between the atoms are modeled using the effective medium theory ( emt ) @xcite .
emt is a many - body potential providing a realistic description of the metallic bonding , in particular in face - centered cubic ( fcc ) metals and alloys of fcc metals .
computationally , emt is not much more demanding than pair potentials , but provide a significantly more realistic description of the metallic bonding .
the systems were deformed using a molecular dynamics ( md ) procedure .
a conventional md simulation was performed , but at each timestep the atomic coordinates in the pulling direction ( the @xmath0 coordinates ) were rescaled in order to deform the system gradually . at the same time , the box dimensions in the transverse directions are allowed to shrink to keep the transversal components of the stress ( @xmath1 and @xmath2 ) close to zero @xcite .
the change in system dimensions is slow at the timescale of the simulation , the relative elongation is @xmath3 each timestep . with a timestep of 5 fs , this nevertheless results in a very high strain rate ( @xmath4 ) .
the results presented here are not very sensitive to the strain rate , although some dependence is seen ( typical stresses increase by 20% when the strain rate is changed from @xmath5 to @xmath6 ) @xcite .
+    figure [ fig : evolution ] shows the deformation of a typical sample with an average grain size of 5.21 nm .
the atoms have been color coded according to the local crystal structure @xcite .
white atoms are in local fcc order , and thus situated inside the grains .
light grey atoms are in local hexagonal close - packed ( hcp ) order , these atoms are at stacking faults .
atoms in all other local environments are colored dark grey .
these are typically atoms at grain boundaries and in dislocation cores .
some dislocation activity is seen in the system , as witnessed by the generation of stacking faults .
the dislocation activity is not sufficient to account for the observed plastic deformation .
a detailed analysis of the deformation shows that the main deformation mode is sliding in the grain boundaries @xcite .    as the volume fraction of atoms in the grain boundary increases with decreasing grain size
, one would expect that increasing the grain size increases the strength of the material as long as the grain boundaries remain the carriers of the deformation .
this is indeed what we see in the simulations .
figure [ fig : stress ] shows the stress - strain curves obtained from simulations of systems with various average grain sizes . a `` reverse hall - petch effect '' , i.e. a softening of the material with decreasing grain size ,
is observed .
the simulations of the deformation of nanocrystalline metals show a reverse hall - petch effect in cu and pd for the grain sizes studied .
this softening of the material , when the grain size is reduced is caused by plastic deformation in the grain boundaries .
there appears to be two different deformation mechanisms active at different grain sizes . in metals with the very small grain sizes studied here , the dominating deformation mechanism is sliding in the grain boundaries through a large number of essentially uncorrelated events , where a few atoms move with respect to each other at each event @xcite .    at much coarser grain sizes ,
dislocations are known to be the dominating carriers of deformation .
in that regime , a hardening of the material is seen , when the grain size is reduced , as the grain boundaries act as barriers to the dislocation motion .
experimentally , this behavior is seen to continue far down into the nanocrystalline range .
as the grain size is reduced , dislocation - mediated deformation becomes more and more difficult . on the other hand , the volume fraction of the grain boundaries increases , favoring a deformation mechanism where the grain boundaries carry the deformation .
furthermore , as the grain size is approaching the grain boundary thickness , it becomes geometrically easier for slip to occur on more than one grain boundary , without large stress concentrations where the grain boundaries meet @xcite .
the emerging picture is one where two deformation mechanisms compete .
one is active at very small grain sizes , another at larger grain sizes .
this leads to a maximum in the yield stress and hardness of nanocrystalline metals at intermediate grain sizes , see figure [ fig : maximum ] .
there have been many reports in the literature of a `` reverse hall - petch effect '' at sufficiently small grain sizes . however , the hardness measurements are very sensitive to sample defects , and in particular to sample porosity @xcite . in high - quality cu samples , the hall - petch effect is seen to continue at least down to grain sizes around 15 nm @xcite
.    there does not appear to be any _ unequivocal _ experimental evidence for a reverse hall - petch relationship in porosity free nanocrystalline metals .
there are , however , indications of a break - down of the ordinary hall - petch relation at grain sizes below 15 nm in high quality copper samples produced by inert gas condensation @xcite .
the ordinary hall - petch effect appears to cease , although a reverse hall - petch relation is not seen .
it is still difficult to make direct comparisons between simulations and experiment , as there is little overlap in grain size .
as the experimental techniques improve , there is hope that more experimental data will be gathered on high - quality samples with grain sizes below 1015 nm .
it should then become clear if the hall - petch effect does indeed break down at these grain sizes . at the same time , computer simulations of larger systems might lead to observations of the cross - over region between the reverse and the ordinary hall - petch effect . that cross - over region is beyond the reach of the largest simulations presented here .
the grain boundaries remain the main carriers of the deformation even when the grain size is increased to 13.2 nm , but be _ do _ observe a slight increase in dislocation activity as the grain size is increased .
as the main deformation mechanism is sliding in the grain boundaries , it could be expected that altering the structure and composition of the grain boundaries might have an effect on the properties of the material .
one such modification is the addition of impurity atoms in the grain boundaries .
we have chosen to study the effects of silver impurities .
a reason for choosing silver was that silver and copper are immiscible , low concentrations of silver impurities in nanocrystalline copper can therefore be expected to segregate to the grain boundaries . we have not studied the segregation process itself , as it is beyond the scope of this study .
segregation is a slow , diffusional process that can not be studied directly with md simulations due to the timescales involved .
other approaches ( such as monte carlo or kinetic monte carlo simulations ) may be more appropriate .
instead of simulating the segregation process , we have generated systems that model nanocrystalline copper after such a segregation has occurred .
it is done by replacing 25% of the atoms in the grain boundaries with silver atoms , see figure [ fig : initag ] .
the system is then annealed and deformed in the same way as the pure systems .
+   +    figure [ fig : compare ] shows how the stress - strain curves have changed , when silver is introduced in the grain boundaries .
the general trend seems to be a slight softening of the material , although the effect is very weak and not seen in all systems .
figure [ fig : flowstress ] shows the flow stress levels for the different simulations , i.e.  the stress level at the horizontal part of the stress - strain curve ( for simplicity the flow stress was defined as the average stress for @xmath7 ) .
we again see the tendency for the silver - containing systems to be softer .    in the simulations of impurities in the grain boundaries , the mechanical behavior is close to what is seen in the pure systems .
as the major part of the deformation happens in the grain boundaries , one could expect that adding impurities in the grain boundaries could have a relatively large effect on the mechanical properties .
the simulations presented here show that this is not the case , when silver is used as impurities in copper .
we see a tendency towards a weakening of the material , but the effect is barely detectable .
the atomic bonding in copper and silver are of a similar nature , and the size of the atoms are not very different .
this may account for the absence of a stronger effect of alloying in the grain boundaries .
atomic - scale simulations have been used to study the deformation mechanisms in nanocrystalline copper with and without impurities in the grain boundaries . in both cases , we find that the main deformation mode is sliding in the grain boundaries through a large number of apparently uncorrelated events , each involving only a few atoms .
some dislocation activity is seen in the grains , the dislocations are probably necessary to allow the grains to deform a little , as they glide past each other .
we observe a _ reverse hall - petch effect _ ,
i.e.  a hardening of the material as the grain size is increased : as the amount of grain boundary atoms is decreased , the deformation becomes harder . at the same time
the dislocation activity is seen to increase a little with grain size .
this effect is seen in the entire range of grain sizes that we have studied ( 3 to 13 nm ) , but at some point we expect that dislocation motion will begin to dominate the behavior of the system .
when that happens , the yield strength should begin to _ decrease _ with increasing grain size .
adding silver to the grain boundaries has only a weak effect on the properties of the material .
the strength of the material is seen to decrease marginally , when the silver is introduced .
we expect that other elements , which chemically behave in a way that is more different from copper will have a larger effect , but such simulations remain to be made .
this work was financed by the danish technical research council through grant no .
parallel computer time was financed by the danish research councils through grant no .
center for atomic - scale materials physics is sponsored by the danish national research council .
r.  w. siegel and g.  e. fougere , in _ nanophase materials : synthesis  properties  applications _ , edited by g.  c. hadjipanayis and r.  w. siegel , vol .
260 of _ nato - asi series e : applied sciences _ , p.  233
( kluwer , dordrecht , 1994 ) .
j.  v. carstensen , t.  leffers , t.  lorentzen , o.  b. pedersen , b.  f. srensen , and g.  winther , eds . , _
modelling of structure and mechanics of materials from microscale to product _ , proceedings of the 19th ris international symposium on materials science , ris national laboratory , roskilde ( 1998 ) .
nanocrystalline metals , i.e.  metals with grain sizes from 5 to 50 nm , display technologically interesting properties , such as dramatically increased hardness , increasing with decreasing grain size . due to the small grain size , direct atomic - scale simulations of plastic deformation of these materials 
 are possible , as such a polycrystalline system can be modeled with the computational resources available today .    
 we present molecular dynamics simulations of nanocrystalline copper with grain sizes up to 13 nm . 
 two different deformation mechanisms are active , one is deformation through the motion of dislocations , the other is sliding in the grain boundaries . at the grain sizes studied here the latter dominates , leading to a softening as the grain size is reduced . 
 this implies that there is an `` optimal '' grain size , where the hardness is maximal .    since the grain boundaries participate actively in the deformation , it is interesting to study the effects of introducing impurity atoms in the grain boundaries . 
 we study how silver atoms in the grain boundaries influence the mechanical properties of nanocrystalline copper .