Carbon Diffusion in Concentrated Fe-C Glasses
CCarbon Diffusion in Concentrated Fe-C Glasses
Siavash Soltani , J¨org Rottler , , Chad W. Sinclair Department of Materials Engineering, The University of British Columbia,Vancouver, BC, Canada V6T 1Z4, Department of Physics and Astronomy, The University of British Columbia,Vancouver, BC, Canada V6T 1Z1 Stewart Blusson Quantum Matter Institute, The University of British Columbia,Vancouver, BC, Canada V6T 1Z4E-mail:
Abstract.
By combining atomistic simulations with a detailed analysis of individualatomic hops, we show that the diffusion of carbon in a binary Fe-C glass exhibitsstrong (anti-)correlations and is largely determined by the local environment. Higherlocal carbon concentrations lead to slower atomic mobility. Our results help explainthe increasing stability of Fe-C (and other similar metal-metalloid glasses) againstcrystallization with increasing carbon concentration.
1. Introduction
Unlike most crystalline materials, the far from equilibrium nature of metallic glassesinduces a significant time dependence to structural and functional properties. Diffusioncontrolled de-vitrification, phase separation and structural aging all impact the abilityto fabricate materials with technologically desirable properties that are stable underoperating conditions. The fact that all technologically important metallic glassesare multicomponent alloys increases the complexity as one must consider the atomicmobility of all species and their interaction, under non-dilute conditions. In the case ofde-vitrification, solute repartitioning to or away from the growing crystalline phase canchange the local composition, atomic mobility and potentially the interfacial structure.The rejection of metalloids like B and C from the growing crystalline phase in Fe-B andFe-C glasses has been suggested to lead to a kinetic slowing of de-vitrification due to astrong dependence of diffusivity on the solute metalloid content in the glass [1].The effect of solute ‘size’ (e.g. metallic diameter) on atomic mobility in metallicglasses has been a point of discussion for some time. Technologically the importance ofthis is driven by the key role metalloids (like C and B) play in all iron-based bulk metallicglasses [2]. Metalloids not only improve glass forming ability [3, 4, 5], but also increasethe crystallization temperature [3, 6, 1]. Generally, though with several discrepancies, itis found from experiments that the activation enthalpy for diffusion is order of magnitudesimilar to that found for diffusion in crystals and increases with atomic size. The a r X i v : . [ c ond - m a t . m t r l - s c i ] S e p arbon Diffusion in Concentrated Fe-C Glasses x B − x glasses [7, 6], it was suggested that B atoms occupy ‘interstitial-like’ sites in low concentration and ‘substitutional-like’ sites in high concentrations. Asthe substitutional-like character of the solute increases, more cooperative movementis required for diffusion of Fe and B, resulting in lower diffusivity and higher stabilityagainst crystallization with increasing solute concentration [6]. This distinction betweendiffusion controlled by single atomic motion for ‘small’ solute atoms (e.g. [8]) andcollective motion for ‘large’ solute atoms (e.g. [9, 10, 11]) has now been more widelyreported, though much more evidence exists for diffusion in systems with ‘large’ solutesthan ‘small’.More recently, understanding the underlying correlation between composition andshort to medium range ordering in metallic glasses has been facilitated by simulation(see e.g. [12, 13, 14]). Rather than a simple random packing of hard spheres, metallicglass alloys, and in particular metal-metalloid glasses, show a complex local structurewhich depends on composition and atom type. While atomistic simulations have shownthat pure or dilute glasses are dominated by icosahedral short ranged order, additionof metalloid elements reduces the degree of icosahedral ordering, and the ordering ofsolvent atoms around the metalloid element takes very specific forms [14, 13]. In thecase of Fe based metallic glasses, the surroundings of the metalloid become associatedwith structures characteristic of the compositionally nearest intermetallic phase, e.g.tricapped trigonal prisms in Fe-B and Fe-C glasses. With increasing metalloid contentthe competition between the formation of metalloid centered structures and theicosehedral ordering preferred by the Fe leads to a form of ‘geometrical frustration’[14]. The change in the local potential energy landscape associated with this change instructure impacts the atomic mobility and thus the bulk diffusivity.In this work, we study the influence of carbon concentration on the carbontracer diffusivity in model Fe-C glasses with molecular dynamics simulations. Goingbeyond a conventional analysis of mean-square displacements, we resolve individual Ctrajectories in terms of atomic jumps, which provides access to statistical distributionsof jump lengths and times. The choice of the Fe-C system is motivated by our recentexperimental work, which has shown strong effects of local carbon concentration on thecrystallization kinetics in binary Fe-C glasses [1, 15]. Previously, the observed stasis inisothermal crystallization kinetics was attributed to thermodynamic effects [15]. Here,we provide evidence that the slowing of diffusion due to the accumulation of carbon atthe crystal/glass interface may be dominant.
2. Computational Methods
Molecular dynamics simulations were performed using LAMMPS [16] and the 2NN-MEAM Fe-C interatomic potential developed by Liyanage et al. [17]. This potential, arbon Diffusion in Concentrated Fe-C Glasses et al. [18] andUddin et al. [19], was explicitly fit to the properties of Fe C, including the behaviour ofliquids containing up to 25 at.%C. All simulations reported here were performed underisothermal-isobaric (NPT) conditions employing a Nos´e-Hoover thermotat/barostat. Atimestep of 2 fs was used throughout.Glasses with three different carbon concentrations were generated starting froma periodic simulation box with dimensions of 51 ˚A ×
67 ˚A ×
32 ˚A. In all three cases,the starting structure was stoichiometric cementite, Fe C (25 at.%C), containing 8400Fe atoms. To obtain boxes containing 8 at.% and 16 at.% carbon, carbon atoms wererandomly removed from the stoichiometric phase and a conjugate gradient minimizationof the energy was performed to create the three starting materials.The resulting models were melted by heating to T= 2000 K and held for 5 ns.Following this, constant rate quenches were performed to below 100 K to form a glass(figure 1). The slowest quench rate allowing all three compositions to be quenchedwithout crystallization was found to be 600 K/ns.Figure 1: Volume/atom as a function of temperature during quenching of the threealloys. The vertical grey line shows the temperature range 900 - 1100 K containing theglass transition temperature for all three compositions.The highest temperature at which data could be collected without crystallizationof the most unstable (8 at.%C) glass was 700 K. All results shown here are thereforelimited to 700 K. Each run was heated to 700 K and held for 5 ns to equilibrate. Thiswas followed by a 200 ns hold during which the positions of all atoms were recorded every δt = 0.1 ns. The mean square displacement of carbon and iron atoms were calculated,treating the atoms as tracers, according to a time averaging scheme, arbon Diffusion in Concentrated Fe-C Glasses (cid:68) (∆ r ( i ) ( t )) (cid:69) = (cid:104) (cid:104) r ( i ) ( t n ) − r ( i ) (0) (cid:105) (cid:105) (1)= 1 N (cid:80) Ni =1 (cid:80) n obs − nj =1 (cid:104) r ( i ) (( n + j ) δt ) − r ( i ) ( jδt ) (cid:105) n obs − n We denote by r ( i ) ( t n ) the position of the i th atom at time t n = nδt . For a given atomictrajectory, n obs is the total number of observation times (i.e. 200 ns/0.1 ns for all caseshere) and N is the total number of observed (carbon or iron) atoms in the system. A‘hop detection algorithm’ adopted from the work of Smessaert and Rottler [20] was thenapplied to the same trajectories to resolve individual carbon atom ‘hops’ or ’jumps’, fromwhich statistics on residence times, jump distances and jump correlations were studied.Ovito [21] has been used for visualization as well as for the structural characterizationof the glasses described below.
3. Results and Discussion
The three quenched glasses were found to exhibit many similarities. First, all threewere found to have glass transition temperatures between T g ≈ − K (figure1). No clear evidence of a composition dependence of T g could be observed. The shortto medium range atomic order of the glasses, reflected in the Fe-Fe, Fe-C and C-Cpartial radial distribution functions (figure 2), also shows only minor differences. In allthree cases, the Fe-Fe nearest neighbour distance is 2 . . ∼ r = 4 ˚A, centered on each carbon atom. A radius of r = 4 ˚Awas selected as it is located between the first and second nearest neighbour positions inthe C-C partial radial distribution function (figure 2) and thus identify carbon atomswithin the nearest-neighbour coordination shell.If randomly distributed in space, the statistical distribution of carbon neighbourswithin the sampling volume should be Poissonian, P ( k ) = ¯ k k exp (cid:16) − ¯ k (cid:17) k ! (2) arbon Diffusion in Concentrated Fe-C Glasses k is the number of carbon atoms within an observation volume and ¯ k is theaverage expected observation number (average carbon concentration multiplied by thespherical observation volume). As shown in figure 3, the agreement between equation (2)and the collected data is excellent, with only the sample containing 25 at.%C showingdeviations at the high number of neighbours. This likely reflects the repulsive nature ofcarbon atoms at close proximity.The carbon concentration dependence of the topological short-range order can beinvestigated in more detail by considering the coordination polyhedra, computed byVoronoi spatial tessellation, centered on Fe and C atoms (figure 4). Unsurprisingly,the most common environment for Fe centred polyhedra was found to be icosahedral( (cid:104) , , , (cid:105) ). For the sample containing 8 at.%C, the next most prevalent polyhedrabelong to the icosahedral-like (cid:104) , , , x (cid:105) and (cid:104) , , , x (cid:105) classes (where x = 0 , , , , (cid:104) , , , (cid:105) , figure 4(b). This represents a tri-capped trigonal prism arbon Diffusion in Concentrated Fe-C Glasses r = 4 ˚A ( k ). Solid lines show equation (2) where the values of ¯ k for eachglass were calculated from the bulk compositions as, ¯ k (8% C ) = 2 .
6, ¯ k (16% C ) = 5 . k (25% C ) = 8 . C) structure. The prevalence of this ordering has been previouslynoted in simulations of Fe-C liquids [22] and in the case of B centered environmentsin Fe-B glasses [14]. The prevalence of the (cid:104) , , , (cid:105) environment drops rapidly withincreasing carbon content of the glass, this also being a strong indicator of the increasinggeometrical frustration experienced in the glass.The geometrical frustration noted above, can be seen in a more local fashion ifrather than correlating the polyhedra type with bulk composition we correlate it withthe local composition. In figure 4 (c) we show the prevelance of the two most commoncarbon-centered polyhedra ( (cid:104) , , , (cid:105) and (cid:104) , , , (cid:105) ) as a function of the number ofcarbon atoms within a radial distance of r = 4 ˚A. The analysis performed for the threedifferent glasses shows that, universally, the fraction of (cid:104) , , , (cid:105) centered polyhedradecreases with local carbon fraction. Interestingly, this suggests that the change inthe local topological arrangement around a carbon atom is a unique function of thenumber of nearest neighbour carbon atoms rather than bulk composition. The changein fraction of (cid:104) , , , (cid:105) polyhedra as a function of the bulk carbon content (figure 4(b)) is therefore a reflection of the proportion of number of carbon-carbon neighbours(cf. figure 3) rather than a reflection of some larger scale change in the structure of theglass. During the 700 K isothermal-isobaric holds, the atomic trajectories for both Fe and Catoms were used to compute mean square displacements (eq. (1)). The results are shownin figure 5. In the case of the Fe atoms, the mean square displacement remains small ( < ) and sub-linear with respect to time over times of up to 200 ns. No dependence arbon Diffusion in Concentrated Fe-C Glasses . . have been filtered out. Panel (c) shows thefrequency of < , , , > and < , , , > centered by C atoms as a function of numberof carbon neighbors, k , within radial distance of r = 4 ˚A.of the mean square displacement on the carbon content was observed in this case.By contrast, the mean square displacement of carbon atoms exhibits a strongdependence of the carbon concentration of the alloy. In all three cases, a significantregime of sub-diffusive ( t ∼ t α , α <
1) behaviour is observed with linear, diffusivescaling, being approached asymptotically at the longest observation times (see the insetto figure 5). Taking a simple linear fit to the mean square displacement over the last100 ns of the trajectory indicates that the diffusivity of the 16 at.%C sample is 60%that of the 8 at.%C sample, while the 25 at.%C is 35% that of the 8 at.% sample. Inaccord with the discussion in the introduction, the mean square displacements observedhere are within the same order of magnitude as those expected for carbon diffusion incrystalline BCC α -Fe. The concentration dependence of the mean square displacementis, however, smaller for the glass compared to the composition dependence of C diffusionin BCC α -Fe at the same temperature. In the case of diffusion in the crystal, longrange elastic repulsion between carbon atoms is predicted to significantly increase thebarrier for diffusion as the carbon concentration is raised. Experimentally, however, theobservation of this composition dependence is not encountered owing to the very limitedsolubility of carbon in α -Fe. arbon Diffusion in Concentrated Fe-C Glasses >
100 ns.
Although figure 5 reveals the anticipated decrease in carbon mobility with increasingcarbon concentration, it does not provide any insight into the origins of this dependence.To delve further into the reasons for these trends, we have identified individualdiffusive carbon jumps. From this it is possible to reconstruct each carbon atom’strajectory with knowledge of distributions of jump rates, jump distances and correlationsbetween jumps. Such analysis has proven to be useful in elucidating various relaxationphenomena, for instance structural aging in strong [23] and fragile [24] glass formers.To capture the diffusive hops of C atoms during the isothermal annealing, thepositions of C atoms were saved every N obs = 100 time steps, and a hop detectionalgorithm was implemented following ref. [20]. For each C atom, starting from t = 0, atime window containing N hist = 100 frames of the trajectory is input to the algorithm.The algorithm divides this evaluation time window into two sections A [0 , N hist /
2) and B [ N hist / , N hist ), each containing N hist / p hop according to [20], p hop ( t ) = (cid:113) (cid:104) ( r A − r B ) (cid:105) . (cid:104) ( r B − r A ) (cid:105) (3)which is a measure of average distance between the mean position r A in section A,and all trajectory points in section B, r B , and vice versa. The oldest time frame is thenremoved and the next time frame of the trajectory is added to the end of the evaluation arbon Diffusion in Concentrated Fe-C Glasses N hist frames.The procedure is repeated until the final time frame of the trajectory. The power ofthis algorithm is that p hop changes rapidly when a hop occurs, as shown in figure 6 (b).A threshold p th is then defined and a hop is identified when p hop > p th . The time of thejump is then recorded where p hop is at its maximum and the jump vector, l = r B − r A can be obtained using the mean positions in section A and B. It should be mentionedthat N hist sets the resolution of the algorithm. A large value results in low resolution,whereas a small value causes small fluctuations in displacement to produce large peaksin the profile of p hop .Figure 6: a) Partial trajectory of a carbon atom, the x,y and z coordinates of its positionbeing shown. The evaluation time window and sections A and B are also highlighted(not to scale). The evaluation time window sweeps the entire trajectory. b) p hop profileof the trajectory shown in (a). The dashed lines show the threshold value p th . c)Trajectory of an atom which makes several jumps to an adjacent cage and then quicklyjumps back. The hop algorithm does not detect all backward jumps, as indicated witharrows in (d).Frequently, atoms were observed to jump from one cage to another only to returnquickly to their starting point. An example of such behaviour is shown in figure 6 (c)and (d). Since the size of the evaluation time window, N hist , sets the resolution ofthe algorithm, a hop with smaller residence time than sections A or B ( N hist / p hop < p th . Reducing N hist can allow one to capture these short lived jumps,though at significant computational cost. To ensure all hops are recorded one can alsoreduce p th . However, as one does this the distinction between a cage breaking jumpand position fluctuations within a cage breaks down. To overcome these problems, avalue of p th = 1 . was fixed which easily captures hops with long sojourn time (figure6(b)). This value can however lead to missing some quick backward jumps as shownin figure 6(d). Therefore the position of each atom obtained from the hop algorithm arbon Diffusion in Concentrated Fe-C Glasses i ) having N i ( t ) discretejumps, j , each having a jump vector l ( i ) j ( t ) is computed as ∆ r ( i ) ( t ) = (cid:80) N i ( t ) j l ( i ) j . Withthis assumption, the mean square displacement is (cid:68) (∆ r ( i ) ( t )) (cid:69) = (cid:42) N i ( t ) (cid:88) j l ( i ) j N i ( t ) (cid:88) k l ( i ) k (cid:43) (4)= (cid:42) N i ( t ) (cid:88) j ( l ( i ) j ) (cid:43) + 2 (cid:42) N i ( t ) (cid:88) j N i ( t ) (cid:88) k>j l ( i ) j l ( i ) k (cid:43) where the angle brackets (cid:104)(cid:105) ≡ (cid:104)(cid:105) i denote an average over all atoms as before. Figure7a shows the mean square displacement reconstructed from the detected atomic hops(open circles) alongside the data obtained directly from MD simulations (same data asshown in figure 5), while figure 7b shows the distribution of displacements (Van Hovedistribution) observed at the end of the annealing time, t =200 ns, obtained from boththe atom positions at the beginning and end of the MD run and from the hop algorithm.The good agreement between the data obtained in these two ways provides confidencethat the hop detection algorithm is able to adequately track the displacements of atoms.From the hop detection algorithm it is possible to obtain separate statistics on theindividual contributions to the mean square displacement arising from atomic jumps.Figure 8(a) shows the distribution of the magnitudes of the jump lengths observed overthe 200 ns anneal. It shows that the jump lengths for all three glasses fall within anarrow distribution from 1 to 2 ˚A with an average of 1 .
58 ˚A. The distribution of thecomponents of the jump vector along the coordinate system of the simulation box areshown for the case of the sample containing 8 at.%C in the inset. This distributionshows a plateau followed by an exponential decay, similar to that observed in otherglasses [25].Figure 8(b) shows the probability distribution function for the residence times,the distribution of times for the first observed jump shown in the inset, for the threeglasses. The distributions for all three cases collapse to the same (truncated) powerlaw distribution (exponent between -1.3 and -1.4), commonly observed for a variety ofglasses [20, 25, 26]. While the probability distributions are nearly identical for the threeglasses, the number of observed jumps per atom in the 200 ns simulations was not. Forinstance, the glass containing 8 at%C produced nearly twice as many jumps/atom as arbon Diffusion in Concentrated Fe-C Glasses (a) (b) Figure 7: (a) Mean square displacement obtained directly from MD trajectory (solidline) compared to the MSD obtained from the hop detection algorithm. Open symbolsshow the results obtained when all detected hops are used (equation (4)) while closedsymbols show results when strongly correlated jumps are removed from the trajectory(equation (5)). For the results from uncorrelated jumps only the symbols represent thecase where a threshold of cos ( θ ) > − . t = 200 ns. Solid circles are obtained directly from theatom positions at time = 0 and time = 200 ns. The open symbols are the displacementsobtained by summing the displacements obtained from the hop detection algorithm.The lines are best fit Gaussian distributions. Blue corresponds to 8 at.%C, orange to16 at.%C and green to 25 at.%C.the glass containing 25 at.%C.Many of these jumps are, however, often highly anticorrelated as already noted withrespect to the partial trajectory shown in figure 6. A jump in direction l is followedimmediately by a reverse jump in direction − l . In order to characterize jumps for eachparticle as correlated or uncorrelated, N i ( t ) = N i,u ( t ) + N i,c ( t ), the following procedurewas used. For a given particle, two consecutive jumps were compared. If the anglebetween the two consecutive jump vectors ( θ i,i +1 ) was greater than a critical angle thenthe two jumps were tagged as correlated. The choice of this critical angle is arbitrary,though as shown below, the results are not drastically changed by its choice. Fromthe resulting list of uncorrelated jumps the atomic trajectory was reconstructed anda new set of jump vectors ( l ( i ) j,u ) calculated ensuring that the final position of eachatom matched that obtained from the full set of jumps. If one considers only theseuncorrelated hops, the cross terms in equation (4) vanish and we can approximate the arbon Diffusion in Concentrated Fe-C Glasses (cid:96) = | l | ) measured from the hop detectionalgorithm. The average jump length is (cid:104) (cid:96) (cid:105) = 1 .
58 ˚A. Blue corresponds to 8at.%C,orange to 16at.%C and green to 25 at.%C. The line is intended as a guide to the eye.The inset shows the distribution of spatial components of the jump vectors for the samplecontaining 8 at.%C sample. (b) Distribution residence times, τ . The inset shows thedistribution of first jump times ( τ ). Blue corresponds to 8 at.%C, orange to 16 at.%Cand green to 25 at.%C.mean-square displacement as (cid:104) (∆ r ( i ) ( t )) (cid:105) = (cid:104) N i,u ( t ) (cid:88) j ( l ( i ) j,u ) (cid:105) (5)where the mean square jump length in this equation is obtained from the recalculatedjumps as described above. The solid symbols in figure 7a show this approximation andtrack the open symbols from equation (4) very well, which supports the decomposition.Going one step further, we assume that on average all atoms make the same numberof uncorrelated jumps (a strong assumption given the broad power law distribution ofjump times) and that all of these jumps have the about the same squared jump length.The resulting expression, (cid:104) (∆ r ( i ) ( t )) (cid:105) = (cid:104) N i,u ( t ) (cid:105)(cid:104) ( l ( i ) j,u ) (cid:105) . (6)is evaluated in figure 9. The approximation still captures the overall trend withconcentration, but systematically over predicts the mean-square displacement. It isnotable that (cid:104) ( l ( i ) j,u ) (cid:105) ≈ . (cid:104) ( l ( i ) j ) (cid:105) ≈ . (cid:68) ∆ r ( i ) ( t ) (cid:69) = (cid:42) N i ( t ) (cid:88) j (cid:16) l ( i ) j (cid:17) (cid:43) N i ( t ) (cid:88) n C auto ( n ) (7) arbon Diffusion in Concentrated Fe-C Glasses θ i,i +1 ) > − .
8. As this threshold is arbitrary, the resultsbounded by cos ( θ i,i +1 ) > − . θ i,i +1 ) > − . C auto ( n ) = (cid:68)(cid:80) N i ( t ) − nj l ( i ) j l ( i ) j + n (cid:69)(cid:28)(cid:80) N i ( t ) j (cid:16) l ( i ) j (cid:17) (cid:29) (8)The first term on the right hand side of equation (7) is the uncorrelated mean squaredisplacement; the mean square displacement that would be predicted if the observedjumps were selected in random order to construct the trajectory. The second termcontains all effects from correlations, C auto ( n ) being the jump auto-correlation functionfor jumps separated by n jumps. The ratio of the true MSD to the purely uncorrelatedMSD is usually called the correlation factor f ( t ) = (cid:104) (∆ r ( i ) ( t )) (cid:105) / (cid:104) N i ( t ) (cid:88) j ( l ( i ) j ) (cid:105) (9)Figure 10 shows the magnitude of the autocorrelation plotted as a function ofthe number of jumps separating two given jumps. Here we see a significant differencebetween the three glasses. The range of the correlation increases from 8 at.%C to arbon Diffusion in Concentrated Fe-C Glasses f ( t ) evaluated according to equation (9).Here the lines are only intended as a guide to the eye. Blue corresponds to 8 at.%C,orange to 16 at.%C and green to 25 at.%C.25 at%C. For all three glasses the long correlation lengths reveal long periods of highlycorrelated jumps.It is also valuable to examine how f ( t ) evolves with time. Figure 10b shows that forthe glasses containing 8 at.%C and 16 at.%C the value of the correlation factor saturateswithin the first ∼
50 ns of the simulation, while f ( t ) for the glass containing 25 at.%Cstill has still not reached a constant value by the end of the simulation.The picture that emerges is that carbon atoms make multiple coordinated jumpsbetween two or more nearby ‘basins’. Many such jumps tend to be made before thecarbon atom will escape from this ‘super-basin’. The highly correlated jumps madewithin one such ‘super-basin’ then do not contribute to diffusive behaviour, this beingreflected by the magnitude of the correlation factor. Using the results in figure 10 andequation (7) we find correlation factors for the three glasses of f = 0 . f = 0 . f = 0 .
02 after 200 ns.With values for the correlation factor in hand, we can propose yet anotherapproximation of equation (4), which consists again in assuming that on average allatoms jump about the same number of times and all these jumps have the about thesame squared jump length. This yields (cid:104) (∆ r ( i ) ( t )) (cid:105) = (cid:104) N i ( t ) (cid:105)(cid:104) ( l ( i ) j ) (cid:105) f ( t ) , (10)the evaluation of which is shown in figure 9b, where f ( t ) has been taken from the datashown in fig. 10b. This approximation captures the time and concentration dependenceof the mean square displacement very well. The only case not matching the direct arbon Diffusion in Concentrated Fe-C Glasses f had not yet converged by the end of the run (fig. 10).The results so far indicate that the variation of the MSD with carbon concentrationis carried by the number of jumps and the correlation factors. Our simulations allowto further resolve a dependence of the average number of jumps in terms of the localenvironment. As noted in the introduction, there has been an increasing recognition ofthe importance of the local structure of the glass on atomic mobility. Following the samelogic as that used in the construction of figure 4c, we start by looking at the dependenceof atomic motion on the local chemistry.Figures 11a and b show that both total and uncorrelated number of hops peratom decrease strongly as the number of carbon atoms in their immediate environmentincreases. Given that the total number of hops is much larger than the number ofuncorrelated hops, the total number of hops is nearly identical to the number ofcorrelated hops detected. In both cases the trend appears to be independent of thebulk carbon concentration of the glass as a single straight line approximates the resultsobtained from the three glasses. This would suggest that the composition dependence ofthe MSD (figure 5) is not due to intrinsic differences in the structure of the glasses tiedto the bulk composition, but rather to the local number of carbon neighbours (figure3). A linearly decreasing dependence of average number of (uncorrelated) hops with k would naturally arise in a system where site-blocking is prevalent. In the Fe-C system,the strong repulsion between carbon atoms at short separations [27, 12] means thattrajectories bringing a carbon atom close to another would be highly unfavourable.Under these circumstances, the probability of a hop trajectory being ’unsuccessful’ dueto its proximity to another carbon atom will be proportional to the number of carbonatoms within the immediate vicinity of the hop.This explanation would justify the dependence of the number of observeduncorrelated jumps on the local, and global, carbon composition of the glass. It doesnot, however, explain the carbon concentration dependence of the correlation factor, f .The results in figures 11a and b give another way to estimate f . Equating equation (6)and equation (10), accepting that these are only approximations to the true MSD (cf.figures 9a and b) we obtain, f ( t ) ≈ (cid:104) N i ( t ) (cid:105)(cid:104) ( l ( i ) j ) (cid:105) / (cid:104) N i,u ( t ) (cid:105)(cid:104) ( l ( i ) j,u ) (cid:105) (11)Using this and taking the ratio of the data in figures 11a and b, provides an estimateof f , evaluated at 200 ns, as a function of k (figure 11c).First, it is notable that back extrapolating to k = 0 indicates a very low ( ∼ arbon Diffusion in Concentrated Fe-C Glasses k ) within a neighbour distance of 4 ˚Aat the time of the jump of a given atom. (b) Same as (a) but after removing highlycorrelated jumps. The insets in (a) and (b) show the total average number of jumpsobserved in 200 ns for each glass. (c) the ratio of the data in (a) and (b), this beingrelated to the correlation factor f , also separated into bins based on the number ofneighbour carbon atoms at the time of the observed jump. The inset in this case showsthe ratio in the case of the total average number of jumps per atom (open symbols). Thevalues of the correlation factors for the three glasses determined based on equation (9)at 200 ns are shown as filled symbols in the same inset. Lines are shown only as a guideto the eye. Blue corresponds to data from the 8 at.%C glass, orange to the 16at.%Cglass and green to 25 at.%C glass. In all cases the data are shown for a threshold ofcos ( θ ) > − . arbon Diffusion in Concentrated Fe-C Glasses θ i,i +1 > ◦ ). The average location of the endpoints of these correlated jumps is shown by the large orange points in figure 11.Figure 12: An example of a carbon atom observed every 2 ps for 2 ns surrounded byneighbouring Fe atoms. The large, dark blue spheres show the average location of theFe atoms, while the diffuse cloud of small blue points shows the actual Fe positionsobserved every 2 ps. Similarly, the diffuse cloud of orange points indicates the positionof the carbon atom observed every 2 ps. A large number of long ( > . .
46 ˚A. The atom positions havebeen positioned such that the horizontal direction is the direction along which the jumpdistance is maximum.The dependence of the estimated correlation factor on local carbon concentration(figure 11c) shows the same, bulk composition independent, decay with k as shown infigure 11a and b. Site blocking, cited above as a possible explanation for the composition arbon Diffusion in Concentrated Fe-C Glasses k , figure 4. Recallingfigure 4, it was shown that the presence of the dominant tri-capped trigonal prismssurrounding carbon atoms (Voronoi indices (cid:104) , , , (cid:105) ) also decreased linearly with k ,again independent of the bulk composition of the glass. Sheng et al. [12] examined therelationship between ‘void space’ and structural ordering in simulated Ni-P glasses. Itwas found in these studies that increasing structural order leads to large voids beingreduced in size or annihilated. On top of this, changes in the structural ordering impacton medium range ordering; the way in which the structural polyhedra fill space. In thecase of Fe-B glasses it was shown that the addition of B led to decreasing icosahedralordering of Fe due to the need to preserve space filling [14]. Such structural changes tothe glass would be expected to impact on the amount of connected void space availablefor correlated jumps, but the impact of such changes on the average number of jumpsis difficult to discern.
4. Summary
By means of molecular dynamics simulations and statistical analysis of individualatomic hops, we have shown a strong composition dependence of the atomic mobilityof carbon in an Fe-C glass, this being qualitatively in agreement with experimentalobservations. The origins of this composition dependence were traced to two sources;the decreasing number of productive (uncorrelated) atomic jumps and an increasingdegree of correlation with atomic hops. The composition dependence was also shownto be related to the local composition rather than the bulk composition of the glass;carbon atoms having a large number of neighbouring carbon atoms were much lesslikely to make atomic hops than those with no neighbouring carbon atoms. It has beensuggested that the decrease in total number of productive (uncorrelated) jumps couldresult from site blocking, carbon atoms having strong repulsion at close proximity. Thenumber of correlated jumps, on the other hand, has been hypothesized to be morestrongly tied to changes in the underlying structure of the glass.To return to the central question posed at the beginning of this work, the diffusion ofcarbon in a model Fe-C glass has been shown here to be strongly carbon concentrationdependent, the mobility of the solute atoms decreasing with increasing bulk carbonconcentration. This would appear to be coherent with experimental observations whichshow a continuous slowing of α -Fe growth during the devitrifaction of a binary Fe-C glass. Our results would suggest that the slowing growth is attributable to theaccumulation of carbon, rejected from the crystalline α -Fe, at the glass/crystal interface.As carbon accumulates at the interface, mobility of carbon atoms also reduces leadingeventually to an apparent halt to the crystallization. This interpretation would beconsistent with the experimental observation of a sudden jump in growth rate attendingan increase in annealing temperature. This strong coupling between composition arbon Diffusion in Concentrated Fe-C Glasses
5. Acknowledgements
The authors acknowledge funding for this work from the Natural Science andEngineering Research Council of Canada. JR thanks the Alexander von HumboldtFoundation for financial support.
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