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Data on interdiffusion coefficients in quaternary and higher order systems have been almost absent in the literature due to the scarcity in proper experimental techniques. In the present work, interdiffusion coefficients are reported at 1000 °C in FeNiCoCr and FeNiCoCrMn systems using square root diffusivity analysis of the diffusion couples prepared based on body diagonal diffusion couple approach. Strong diffusional interactions have been manifested in terms of the large values of the cross interdiffusion coefficients compared with the respective main interdiffusion coefficients. D ~ NiCo Cr and D ~ CoNi Cr were both found to be positive and, 54.5-66.7% and 30-50% of the respective main coefficients in the case of quaternary Fe-Ni-Co-Cr system. D ~ FeNi Cr and D ~ FeCo Cr were found to be positive and, 74.5-85.2% and 50.9-59% of the main coefficient D ~ FeFe Cr whereas D ~ NiMn Cr was found to be negative and 57.5-63.6% of the main coefficient D ~ NiNi Cr in magnitude in the case of quinary Fe-Ni-Co-Cr-Mn system. Errors associated with the interdiffusivities determined using the square root diffusivity approach are higher in comparison to the Kirkaldy’s method for quaternary system. However, the trend is reversed for the quinary system. Successful application of the square root diffusivity technique to the body-diagonal diffusion couples prepared in these systems indicate that the interdiffusion coefficients are constant over a large composition range in the two studied systems.

Binary interdiffusion data as functions of composition in the Mg-Al system are essential in modeling kinetics of phase transformations in multicomponent Mg and Al alloys. Interdiffusion and phase growth kinetics were studied in the binary Mg-Al system using multiphase diffusion couples assembled between pure Mg and pure Al at 380, 400 and 420 °C. Two phases, Al 3 Mg 2 (β) and Mg 17 Al 12 (γ) were formed between Al and Mg at the three temperatures studied. Both β and γ phases were observed to follow parabolic growth with time, which suggests that the growth of the two phases is controlled by bulk diffusion mechanisms. The activation energies for the growth of β and γ phases in the temperature range of 380-420 °C were found to be 37.3 ± 4.1 and 187.7 ± 1.9 kJ/mol, respectively. The interdiffusion coefficients were evaluated as functions of compositions in various phases at the three temperatures studied, which were further utilized for evaluating the activation energies and frequency factors for interdiffusion in each phase. The activation energy for interdiffusion in FCC-Al is found to increase with increasing Mg-content whereas the activation energies for interdiffusion in HCP-Mg and γ phases do not vary significantly with composition.

In non-ideal solutions, partial molar volumes change with composition, which means a diffusion process is always accompanied by change in volume of the system. To account for this change in volume while solving diffusion equation, it is necessary to know the velocity of the local center of volume ( U V ). An expression is derived for U V , using a treatment that is applicable to a multicomponent system. Simulations of multicomponent diffusion profiles with composition dependent partial molar volumes have been absent in the literature so far. The expression derived in this work is also used to generate diffusion profiles in a hypothetical ternary diffusion couple. Significant difference is observed between the concentration profiles obtained with and without the assumption of constant molar volume. Exact calculation of U V also enables the estimation of the expansion or contraction accompanied by diffusion, which in turn would help in assessing diffusion induced stresses and dimensional changes.

Quaternary isotherm of Fe–Ni–Co–Cu at 950 °C is determined by multiphase diffusion couple experiments, focused on establishing the (BCC + FCC) two-phase regions. The present findings, combined with our previous study, discover the presence of five separate two-phase fields in this isotherm namely (BCC + Cu-rich FCC), (BCC + Cu-lean FCC richer in Fe), (BCC + Cu-lean FCC richer in Co), (Cu-rich FCC + Fe-rich FCC), and (Cu-rich FCC + Co-rich FCC). This also indicates the existence of two three-phase fields. Based on diffusion couples exhibiting planar interfaces between BCC and FCC phases, three tie lines in (BCC + Cu-rich FCC) two-phase field were also determined. It is observed that Fe–Co–Cu ternary isotherm at 950 °C has a wider BCC region (up to 8 wt% Cu) than the commonly accepted phase diagram. A qualitative representation of the entire quaternary isotherm is proposed in the form of multiple iso-Ni concentration sections. Graphical Abstract

For the first time in the literature, experimental determination of entire sets of exact interdiffusion coefficients in quaternary and quinary alloy systems is reported. Using the method of body-diagonal diffusion couple, a set of nine quaternary interdiffusion coefficients were evaluated in Fe–Ni–Co–Cr and a set of sixteen quinary interdiffusion coefficients were determined in a Fe–Ni–Co–Cr–Mn system, both at approximately equimolar compositions. Regions of uphill interdiffusion and zero flux planes were observed for nickel and cobalt in quinary couples, indicating the existence of strong diffusional interactions in Fe–Ni–Co–Cr–Mn alloys. The strong diffusional interactions were also manifested in the large magnitudes of cross coefficients in both the systems. The existence of strong diffusional interactions in high-entropy alloys (HEAs) as observed through experimentally determined interdiffusion coefficients in this study establishes beyond doubt the fact that cross interdiffusion coefficients cannot be ignored in HEAs.

Interdiffusion study was carried out in β-(Ni,Pt)Al system at 1100 °C by solid–solid diffusion couple technique. Using Kirkaldy’s approach, ternary interdiffusion coefficients were determined at 27 different compositions covering broad range of the β phase region. Zero flux planes (ZFP) and regions of uphill diffusion were manifested in several diffusion couples indicating presence of strong diffusional interactions in the system. Among the three components, Pt and Al were found to be the slowest and the fastest diffusing elements, respectively. For a given Al content, the main interdiffusion coefficients D~AlAlNi and D~PtPtNi increased with increase in Pt content but they decreased with increase in Al concentration at a given Pt-content. The cross interdiffusion coefficient D~NiAlPt is negative and about one order magnitude higher than the main coefficient D~NiNiPt which implies that the interdiffusion flux of Ni is enhanced up the Al concentration gradient. Based on the ternary interdiffusion coefficients, it is inferred that Ni and Al have stronger interaction than Pt and Al. The effect of Pt in reducing spinel oxide formation in TGO layer is explained based on the diffusional interactions of Al and Ni as determined in this study.

Interdiffusion data have gained utmost significance for predictive alloy design and process development with ever-increasing utilization of integrated computational materials engineering (ICME), Artificial Intelligence (AI) and Machine Learning (ML) in automobile industry. Titanium-Aluminium based alloys have great potential as lightweight alternative for steels. Hence, a review of interdiffusion studies reported in the β phase field of Ti-Al based ternary systems is presented. Availability of a variety of diffusivity terms at times create confusion in the users’ minds. Hence, the review starts with a brief discussion on some basic concepts in diffusion. Significance of diffusional interactions is specifically emphasized in this review as observed in the several Ti-based systems reviewed here. New analytical expressions are presented for evolution of concentration profiles in multicomponent systems for two types of boundary conditions, one for periodic boundary conditions typically observed in homogenization process and another for diffusion couples with finite boundary conditions. The newly derived expressions are then used for simulating concentration profiles in two hypothetical ternary systems, mimicking Ti-Al-Fe and Ti-Al-Nb. The simulation results clearly bring out the fact that diffusional interactions in these ternary systems can be effectively utilized to selectively enhance or deplete certain regions of a particular component by appropriately setting up the relative concentration gradients of the diffusing species.

Interdiffusion was studied in FCC FeNiCoCrMn high entropy alloy (HEA) system with the help of two quinary diffusion couples annealed at 1000 °C for 100 hours. The terminal alloys of the two couples were selected based upon the knowledge of binary thermodynamic interactions so as to have enhancement or reduction of interdiffusion of particular components. Interdiffusion fluxes of nickel and manganese, which have highest negative binary enthalpy of mixing, were observed to be enhanced up the gradients of each other and reduced down the gradients of each other. Regions of uphill interdiffusion observed for chromium and iron and presence of a zero flux plane observed for iron in one of the diffusion couples indicate the existence of strong diffusional interactions in this HEA. Quinary interdiffusion coefficients were also calculated at various compositions of the FeNiCoCrMn system based upon Manning’s model, utilizing the knowledge of tracer diffusivities of constituent elements and thermodynamic factors. The calculated cross interdiffusion coefficients were shown to be consistent with the diffusional interactions observed in the two diffusion couples. Nickel and Manganese, which are slowest and fastest diffusing species in the FeNiCoCrMn HEA and, which also possess highly negative binary enthalpy of mixing were observed to play particularly significant role in determining the diffusional interactions in this HEA system. Validity of the interdiffusion coefficients evaluated by Manning’s approach was established by regenerating the concentration profiles of the experimental diffusion couples based on transfer matrix method (TMM).

It has been generally accepted and often mentioned in the text books that gradient in chemical potential of a species is the fundamental driving force for its diffusion. However, a general derivation of the interrelation between the diffusion flux of a component and chemical potential gradients in a non-ideal solution is lacking. Although there have been various studies in the literature reporting such interrelations for a binary system, they all assume constant molar volume. In a non-ideal system though molar volume changes with composition. Hence, in the present work, kinetic theory is used to derive a relation between diffusion flux and the chemical potential gradients for a multicomponent system with composition dependent molar volume. It is shown that the velocity of the marker as measured in a diffusion couple experiment should consist of the drift velocity (UN) due to change in molar volume accompanied by diffusion as well as the Kirkendall velocity caused by vacancy equilibration process. For the assumption of volume change occurring only in the direction of diffusion, the Kirkendall velocity is same as the marker velocity measured in a diffusion couple. However, if the lattice is allowed to relax in all directions, the contribution of UN to the marker velocity can be significant. This is shown to be as high as 20% for Cu in a Cu-Ni diffusion couple.

There are seven approaches of determination of interdiffusion coefficients with respect to volume-fixed frame given in the literature. These methods are by Boltzmann–Matano, Sauer–Freise, den Broeder, Wagner, Balluffi, Guy et al. and Danielewski et al. It is important to verify the applicability of these approaches under real conditions such as when molar volume Vm is a function of composition. This is one of the primary objectives of the present study and to achieve this goal, both qualitative and quantitative analyses of the said methods are carried out. These seven methods are examined critically for possible errors and implicit assumptions, if any, in their derivation. Subsequently, quantitative estimation of errors in these approaches is done through MATLAB simulation. Five hypothetical cases of variation of Vm with composition are treated including constant, ideal and non-ideal dependence on composition. Concentration profiles are generated for each of these cases in a hypothetical binary diffusion couple by employing a code written in MATLAB. Using the concentration profiles, errors are evaluated in the diffusion coefficients determined based on the seven approaches. Based on the error analysis, methods which give the least error in interdiffusion coefficients are finally proposed. Effect of assuming constant Vm in alloy systems which show dependence of Vm on composition is also investigated.