Modeling of carbon segregation and accompanying processes during HTSC manufacture


I. A. Parinova[1], L. I. Parinovab, and E. V. Rozhkova


aMechanics and Applied Mathematics Research Institute, Rostov State University,

Rostov-on-Don, 344090, Russia


bDepartment of Mechanics and Mathematics, Rostov State University,

Rostov-on-Don, 344090, Russia



The formation of microstructure defects and weak links, which have considerable influence on the structure-sensitive properties of high-temperature superconductors (HTSC) is discussed, taking into account the carbon segregation, which embrittles intergranular boundaries and constructs the weak links. The carbon segregation processes are associated with slow, fast and steady states of the dislocation-screened crack growth. The solutions obtained can be applied to the finite element formulations and other numerical codes by which the stress-strain states, distributions, kinetics and parameters of intergranular defects during manufacture of the HTSC systems can be predicted.


PACS codes: 82D55


Keywords: HTSC, carbon segregation, dislocation-screened crack





It is obviously, that an optimization of the manufacture techniques and material compositions is the most important to obtain HTSC with improved and more controlled structure-sensitive properties. The techniques used to prepare the HTSC bulks (e.g., cold or hot pressing) are very complex and usually include some intermediate stages, the goal of which is to obtain a highly connected and align grain structure. Nevertheless, the microstructure defects and weak links formation is very difficult to avoid into HTSC as any heterogeneous solids. The defects as rule located at and near interfaces [1], have first degree importance in research of structure transformations. During certain thermal and mechanical loading treatments the secondary phases and segregations form into the composition, rendering, generally, non-simple effect on the final HTSC properties. For example, the melt-processing has been successfully applied to obtain the large-grain superconductive YBa2Cu3O7-x (YBCO) ceramics with decreased content of the intergranular boundaries and improved critical current density (Jc) [2, 3]. However, there are essential problems connected with

preparation of the optimum compositions (in particular, with the size and concentration of the normal Y2BaCuO5 particle dispersion) and thermal regimes. Moreover, for HTSC bulks there is considerable problem of carbon, which segregates to the intergranular boundaries, embrittles them and constructs the weak links. Carbon can be introduced into yttrium and bismuth HTSC ceramics by carbon-containing gases or liquids. By this, Ba and Sr are the most reactive superconductor constituents. During the processing, it is possible to form nanometer-scale carbon inclusions with superconductor grains and enhance flux pinning substantially [4]. However, it is well known, that retained carbon can adversely affect grain boundaries [5, 6], superconducting transition temperatures (Tc) [7], and transport critical current density (Jc) [6, 8-11]. Carbon or carbon dioxide segregated to the grain boundaries can degrade them essentially. The material density change caused by the CO2 gas release from the liquid phase during the Bi2Sr2Ca1Cu2Ox (Bi-2212) formation can be tremendous. A rough calculation has indicated that 200 ppm carbon can cause about 36% porosity in the core, if all carbon forms CO2 at high temperature [12].

It is the aim of this paper to discuss the next problem that could help to improve the processing techniques and HTSC compositions, namely the carbon segregation, to intergranular boundaries. The proposed microscopic theory of carbon-induced intergranular cracking (CIIC) is based on the corresponding models of the slow, fast and steady states of the dislocation-screened crack growth.



The computer models of the microstructure transformations during manufacture of different YBCO and BSCCO compositions have been developed using the finite difference method and Monte-Carlo procedures [13-19]. The more powerful finite element methods and other numerical codes need detailed information on the critical characteristics, microstructural features, material properties, damage formation and growth, etc. It could be proposed, that carbon can segregate not only to grain boundaries, but also to crack surfaces and dislocations, where lattices are distorted. Therefore, two microcracking processes are possible: continuously slow crack growth and discretely rapid crack growth associated with high amounts of acoustic emissions. Then the carbon segregation processes can be studied by using the microscopic models of the equilibrium slow and fast crack propagation, and also a steady state crack growth, which are screened by dislocation field.


2.1. Equilibrium slow and fast crack growth

Consider an intergranular crack with the length, 2a, in a carbonated HTSC. The crack lies along x axis in an elastic-plastic isotropic body with shear modulus, G, Poisson ratio, n, yield strength, sy, and work hardening factor, n. The body is loaded by a remote stress, sa, parallel to the y axis at a constant temperature, T. At the x axis two linear dislocation arrays with the length ry locate at the distance d from the crack tips. This model proposes that an intergranular crack tip maintains an atomistic sharpness and a local equilibrium condition in the presence of screening dislocations. It is assumed that all geometrical parameters of the crack tip (in particular, the size of the arc-shaped crack tips, q, and a crack tip displacement, dc) remain constant during plastic deformation. The condition of the local equilibrium at the crack tip is that the crack must be screened by dislocation field and maintains a dislocation free zone with the size d. The loaded system "crack - dislocation arrays" maintains a local stress, sd, in the dislocation free zone and produces the next stress intensity in the screening dislocation zone, given by Hutchinson, Rice and Rosengren as [20]


syy=sd, a< êx ê< a+d (1a)


syy=bsy(Ka/sy)2n/(n+1)/( êx ê- a) n/(n+1),


a+d<êxê<a+d+ry (1b)


where Ka is the applied stress intensity, and b is the factor depended on the elastic and plastic deformation properties. The carbon segregation process is found by the crack tip profile and the stress field ahead of the crack tip. The chemical potentials of carbon and superconductor can be stated following [21] in various grain boundary and crack surface zones, namely: I - zone not affected by the stress intensity (çxç>a+d+ry), II - zone of screening dislocations (a+d<çxç< a+d+ry), III - dislocation free zone ahead of the crack tip (a < çxç< a+d), IV - arc-shaped crack tip zone (a-q < çxç< a) and V - parallel flat crack surface zone (çxç< a-q). At equilibrium, the chemical potentials of carbon and superconductor must be the same in the all regions, respectively. So, the equilibrium carbon segregation depends on the binding energies and crack tip conditions. The binding energies of carbon at grain boundaries and crack surfaces (HB)b and (HB)s, respectively, are found through the standard chemical potentials of C and HTSC as


(HB)b = (mm0)C - (mb0)C -

- (mm0)HTSC + (mb0)HTSC (2a)

(HB)s = (mm0)C - (ms0)C -

- (mm0)HTSC + (ms0)HTSC (2b)


Here and further the subscripts of different parameters indicate the next: m is the matrix, b is the grain boundary, and s is the crack surface. The basic assumption of the model is that the embrittlement occurs as a reduction of the surface and grain boundary energies due to the carbon segregation. Moreover, it is taken into account, that slow fracture occurs, when the solute is sufficiently rapid to maintains the same chemical potential of solute between the grain boundary and the crack surface. Fast fracture occurs, when the solute concentration remains the same at the grain boundary and the crack surface. Then, from the thermodynamic theory proposed by Seah, Rice and Hirth [22, 23] it can be obtained the ideal works, expended in slow (gs) and fast (gf) fracture as





here the equilibrium carbon concentrations at zones III and V have forms








where F2 = exp[(HB)s/(RT)], Cm is the bulk carbon concentration, Vh is the molar volume of carbon, R is the gas constant, 1/Wi is the carbon coverage at interfaces, and are the critical values of carbon concentration at zone III, required for slow and fast fracture, respectively, g0 is the ideal work of intergranular fracture in the absence of carbon, and is the chemical potential difference between the crack surface and the stressed grain boundary. The equations for constant carbon concentrations also can be found at zones I and IV. By this, CI coincides with CV, in which (HB)s is replaced by (HB)b, and for CIV we have equation



where r is the curvature radius of the arc-shaped crack tip, and gs0 is the crack surface energy in the absence of carbon. At the same time, due to the variable stress distribution (1) the carbon content at zone II is not constant. The relationship between the critical stress intensity required to propagate the crack (for slow, fast or steady state fracture) and to change the ideal work due to the carbon segregation is stated by using the local energy balance condition as [21]




where the superscript c corresponds to certain fracture state, Kd is the local stress intensity factor, connected with the dislocation free zone size ahead of the crack tip (d) and local stress (sd) in this zone by the equation approximately derived from the load balance condition between a crack with a linear stress intensity and that with local stress [21]: pd = (Kd /sd)2. Moreover, a relation between sd, Kd and dc follows from the condition, that the elastic energy release rate is the same as the J integral, i.e.. Then the threshold apparent stress intensity, , is given by the relationships (1b) and (7)



where K0 is the fracture toughness, is the critical crack opening displacement (CCOD) required for various fracture processes (superscript c), and dc0 is the CCOD in the absence of carbon, defined as


Note, that in order to the crack to maintain the dislocation free zone during the growth, besides the inequality (7), it is necessary to satisfy additional condition, namely the total energy balance criterion in the form [21]


where gp is the plastic work due to the generation and motion of screening dislocations, which could be found numerically, e.g. in the case of a linear dislocation array.


2.2. Steady state crack growth

Assume that the carbon diffusion along stressed boundaries and crack surfaces is the mechanism, which controls the intergranular embrittlement and affects the crack growth rate. By this, the bulk diffusion effects on CIIC are neglected. Under the geometrical and loading conditions of above considered equilibrium crack growth problem, the steady state case indicates subcritical intergranular crack growth with constant velocity. Taking into account the grain boundary and crack surface zones (II-V) the fluxes of carbon in these regions, , can be stated as




where Di is the diffusivity of carbon, is the carbon concentration, i is the subscript indicating b or s, and j is the superscript indicating various interface zones, are the corresponding chemical potentials. The differentiation with respect to s is carried out only at zone IV, by this, s is the variable arc length at the correspondent part of the arc-shaped crack tip. The continuity equation of fluxes is




where t is the time. Based on equations. (1), (10) and (11), and also the relationships between the interface energies, , and the amounts of carbon, , in the various zones, derived from the Gibbs theory and dilute solute approximation as , the second-order differential equations, controlling the carbon diffusion in the intergranular cracking regions, can be obtained, analogously to [25]. It is assumed, the steady state crack growth maintains the equilibrium values at the crack center and at the triple point of grain boundaries ahead of the crack. It should be noted the present boundary condition is a first order approximation because the equilibrium content of carbon at the triple grain junction is difficult to attain, especially at sufficiently high velocity of crack. The interface conditions find, that the chemical potentials and fluxes of carbon must be the same at each interface in order to maintain the continuity of the carbon flux. So, it is stated the boundary value problem for solution of which some relationships defined in the equilibrium crack growth to be used, namely: the local equilibrium condition at the crack tip, the geometrical crack tip conditions, and also the crack tip condition derived from the local energy criterion (7). The carbon diffusivity effect is determined by the ideal work of steady state fracture






the superscript indicates the steady state fracture. The boundary value problem can be solved numerically, e.g. by using the Runge-Kutta method. The boundary conditions at the triple points allow to study the effects of grain sizes on the kinetics of CIIC. Then, the model of the steady state crack growth caused by the carbon segregation can be added to previously developed modeling of the toughening mechanisms, acting into HTSC systems [1, 14-18, 26]. At the same time, the size effects can not be estimated in the cases of the equilibrium slow and fast cracks.




The numerical results in the case of equilibrium crack growth obtained for different values of the bulk carbon concentration, Cm, are presented in Table 1 (slow crack) and Table 2 (fast crack). Note that the spreads of values of the key material parameters could be considerable in dependence on the manufacture techniques, compositions, thermal and mechanical treatments, etc. Therefore, an optimum selection should be realized by using the maps of material properties and fracture features [1]. Nevertheless, in the present paper we are limit by concrete set of parameters, only, namely:g0 = 1 J/m2,1/Wb = 1/Ws = 8.1*10-5 mol/m2, (HB)s = 50KJ/mol, , (HB)b = 10 KJ/mol, n = 0.1, sy = 10MPa, Vh = 2*10-6 m3/mol, R = 8.316 J/mol K, n = 0.2, K0 = 1MPa m1/2, G = 50GPa, T = 1110K. By equating the right parts of the equations (1a) and (1b) at the |x| = a+d and Ka = K0, and following consideration of the relations (3a), (4), (5), and (7) for slow crack and (3b), (4), (5), and (7) for fast crack the problem is reduced to numerical solution of transcendental algebraic equations. These equations allow to state the relationships gc and the critical tip conditions () to Cm. Then, the values of are determined from equation (8) using the calculated values of gc and . As it is shown by the numerical results, all auxiliary parameters ()change monotonously with Cm for both slow and fast fracture. In particular, the normalized parameter, gc/g0, decreases with increase of Cm. At the same time, the normalized parameter, , increases together with Cm. These alternative contributions into cause its nonnmonotonous behavior in the dependence on Cm. By this, the strengthening effect on (i.e., when > 1) occurs when the segregation of carbon in the crack regions strongly affects the crack tip condition (i.e. reduction of ) but it does not produce a substantial reduction in gc [see equation (8)]. More evident change all auxiliary parameters in the case of slow crack to compare with the fast crack at the considered range of Cm proposes the more susceptibility of slow growth on CIIC increase. Then, it is apparent, that under the condition of a dislocation-screened crack, the carbon segregation induces slow fracture more readily than fast fracture. The weak change of on Cm in both cases of slow and fast cracks is apparently caused by small range of Cm (while this is real bulk carbon concentration into HTSC systems). The presented numerical example should be specified with more accurate selection of key parameters for certain HTSC.

For the used local energy criterion, which is controlled by CIIC, the dependence of on the ideal work of fracture, the crack tip conditions and the plastic deformation properties [see equation (8)] is somewhat similar to that obtained by Weertman [27]. The difference consists in that the present paper explicitly includes not only the embrittlement effect of carbon but also the crack tip conditions affected by the carbon segregation. It should be noted, the presence of a dislocation-screened crack and the microscopic behavior of plastic deformation associated with CIIC have not yet been experimentally verified in HTSC compositions. However, as it has been shown by Rice and Thomson [28] an intergranular crack remains atomistically sharp when an energy barrier for the nucleation of a dislocation loop at a crack tip is present. This barrier is produced due to a low level of stress intensity at the crack tip in the presence of screening dislocations stated by the dislocation sources (e.g., such as intergranular boundaries, particles, defects, etc.) [29]. Thus, the relative strength of energy barriers for dislocation nucleation produced by the crack tip or/and the other dislocation sources states whether a crack tip maintains an atomistical sharpness or emits dislocation loops, Obviously, the carbon segregation in HTSC causes a complex effect on the dislocation behavior. Then, the occurrence of the dislocation-screened crack tips is possible in the presence of carbon depending on how carbon affects the generation of dislocation at the crack tips and the other dislocation sources. Besides the crack tip conditions, the strength of carbon binding at crack surface and intergranular boundaries also controls the carbon embrittlement of the grain boundaries. It is known, that the value of (HB)s is much higher than that of (HB)b, and they are found by the conditions at interfaces such as solute or carbon coverage, structure and roughness. Moreover, a high degree of lattice incoherence (e.g., at the interphase boundary [29]) can also state a higher susceptibility to carbon embrittlement of HTSC. Obviously, there are also another factors controlling the strength of carbon binding at interfaces. So, the values of carbon binding need an accurate experimental foundation for considered HTSC systems.

The detailed analysis of numerical results for steady state crack outruns of the present paper. Note, only, the parameters corresponding to obtained ones for slow and fast cracks are found in dependence on the crack velocity and the carbon diffusivities at grain boundaries and crack surfaces. In particular, the numerical results obtained for steady state crack indicate that the crack growth rate is higher for smaller grain size, i.e. as the grain size decreases the susceptibility to CIIC increases. In total, the solutions and numerical results obtained in the paper can be used in the finite element formulations and other numerical codes by which the stress-strain states distributions, kinetics and parameters of intergranular defects during HTSC bulk manufacture can be predicted.

Acknowledgements: This work was supported by the Russian Department of Education (Program of Basic Researches in the Natural Science). I.A.P. thank the COBASE Grant Program for financial support.





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Table 1. Numerical results for slow crack.




50 ppm


100 ppm


150 ppm


200 ppm


250 ppm


300 ppm

d s, mm





















g s/g0





















Table 1. Numerical results for fast crack.




50 ppm


100 ppm


150 ppm


200 ppm


250 ppm


300 ppm

d f, mm





















g f/g0





















[1] Corresponding author: Ivan A. Parinov, 200/1, Stachki Ave, Rostov-on-Don 344090, Russia, Tel: 7-8632-434688; Fax: 7-8632-434885; e-mail:;