Parinov I.A. Microstructure and Properties of High-Temperature Superconductors

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198 4 Carbon Problem

non-mechanical energy ﬂow. The material deformation depends on all other

processes due to material expansion, which is caused by carbon dissolution,

carbonate formation and temperature increase. Microscopic models of the

HTSC intergranular cracking during carbon segregation, depicting slow, fast

and steady-state crack growth, screened by linear dislocation array [812] will

be considered in Sect. 4.4. In this section, the governing equations for carbon

embrittlement and fracture of HTSC are derived, taking into account thermal

stresses in the framework of the thermodynamic theory of irreversible pro-

cesses [189], based on the thermal diﬀusion of carbon (Soret’s eﬀect). In this

case, the delayed fracture of carbonate is modeled by using the de-cohesion

model, taking into account de-cohesion energy changed during time due to

the time-dependent carbonate precipitation process [809, 810].

4.3.1 Mathematical Model

for Carbonate Precipitation and Fracture

The model is developed for a superconductor, forming carbonates. The pres-

ence of the carbonates is described by the carbonate volume fraction. Energy

ﬂow and diﬀusion of mass are coupled processes. A temperature gradient

leads to ﬂow of matter, and therefore to a concentration gradient. Conversely,

a diﬀusion process gives rise to a small temperature diﬀerence. A detailed

discussion for thermodynamic treatment of energy ﬂow/diﬀusion is based on

Onsager’s principle of microscopic reversibility [189]. In the following, the the-

ory is applied to the process of carbonate diﬀusion and energy ﬂow, occurring

in YBCO interaction with CO

, forming carbonates and cuprates under stress

and temperature gradient.

According to the empirical law of Fourier, heat ﬂux is linearly related to

the temperature gradient, which is the thermodynamic force, driving heat

ﬂow. In an isothermal system, the ﬂux of a diﬀusing substance is proportional

to the gradient of its chemical potential, that is, chemical potential gradient

is the thermodynamic force, driving diﬀusion under isothermal conditions.

When the above-pointed processes operate simultaneously, coupling is taken

into account by assuming that the non-mechanical energy and carbon ﬂuxes

are linearly related to both thermodynamic forces:

= L

+ L

= L

+ L

= L

, (4.9)

where J

and J

are the components of the non-mechanical energy ﬂux and

the carbon ﬂux, respectively. Note that the energy ﬂux includes conducted

heat, described by Fourier’s law, and the energy transported by the diﬀusing

carbon; X

and X

are the thermodynamic forces, driving non-mechanical

energy ﬂow and carbon diﬀusion, respectively; L

, L

and L

are

phenomenological coeﬃcients. Third relation (4.9) is valid due to Onsager’s

reciprocity relation.

4.3 Carbon Embrittlement and Fracture of YBCO Superconductor 199

The thermodynamic forces are related to the gradients of the absolute

temperature T and chemical potential of carbon in the carbonate μ

= −

∂T

∂x

= −T

∂

∂x





. (4.10)

When carbon and superconductor form carbonate, the carbon ﬂux satisﬁes

the following relation [974]:

= −



∂μ

∂x

∂T

∂x



, (4.11)

where R is the gas constant; C

, D

and Q

are the concentration, diﬀusion

coeﬃcient and heat of transport of carbon in the carbonate, respectively. The

concentration of carbon as well as other components or phases is given in

moles per unit volume. Relation (4.11) is a special case of (4.9) and (4.10),

and therefore satisﬁes for a HTSC under stress. Because of the absence of

carbon diﬀusion in the cuprate, the total carbon ﬂux in cuprate/carbonate

composite is given as

= fJ

, (4.12)

where f is the volume fraction of the carbonate in the material. Mass con-

servation requires that the rate of total carbon concentration, C

,insidea

volume V , is equal to the rate of carbon ﬂowing through the boundary S:



dV +



dS =0. (4.13)

Relation (4.13) is valid for an arbitrary volume. Then, one may derive the

respective diﬀerential equation by using divergence theorem:

= −

∂J

∂x

. (4.14)

Note that the total carbon concentration, C

, is only related to the

concentration of carbon in the carbonate, C

, because of the absence of carbon

in cuprate:

= fC

. (4.15)

This is equal to the carbon terminal solid solubility, C

,whenf =0.

Value of C

is deﬁned with respect to the volume occupied by the carbonate,

that is, fV. Note that in the carbon diﬀusion model, the eﬀect of carbon

trapping in the solid by dislocations and voids is neglected, since the bulk of

the material is assumed to behave elastically. However,the eﬀect of traps is

more important at low temperatures, where the lattice solubility (i.e., ability

to substitute atoms and formation of atom interstices) is relatively small [974].

Corresponding model of carbon trapping by dislocations and crack-like defects

can be developed similar to the hydrogen precipitation in metals [426].

200 4 Carbon Problem

Then, the governing equation for non-mechanical energy ﬂow is deter-

mined. With this aim, the energy ﬂux is found, that is coeﬃcients L

and

are calculated. Comparing (4.9)–(4.11), we have

= L

+ μ

) . (4.16)

The remaining coeﬃcient L

can be determined by taking into account

Fourier’s law for heat conduction:

= −k

∂T

∂x

, (4.17)

where k is the thermal conductivity of the superconductor. Note that (4.17)

is valid when there is no carbon diﬀusion. Then, by stating J

=0,wehave

from (4.11)

∂μ

∂x

= −

∂T

∂x

. (4.18)

Substituting (4.10), (4.17) and (4.18) in (4.9), one may derive L

= kT +

+ μ

)

. (4.19)

Thus, all the coeﬃcients of the phenomenological (4.9) have been deter-

mined. Consequently, one may derive the expression for energy ﬂux in a car-

bonate, substituting (4.16) and (4.19) into (4.9) and taking into account the

relation (4.11) for carbon ﬂux:

=(Q

+ μ

− k

∂T

∂x

. (4.20)

The ﬁrst term of the right-hand side is the energy ﬂux, which is produced

by the diﬀusion of carbon atoms, and the second term is the conducted heat.

According to (4.20), the heat of carbon transport is the heat ﬂux per unit ﬂux

of carbon in the absence of temperature gradient.

Due to the absence of carbon diﬀusion in the cuprate, the energy ﬂow in

the cuprate is only thanks to heat conduction. It is assumed that the thermal

conductivity of the carbonate equals the thermal conductivity of the cuprate;

then by using (4.17), the next relation provides the total energy ﬂux in the

cuprate/carbonate composite:

= fJ

− (1 − f )k

∂T

∂x

, (4.21)

which, when combined with (4.12) and (4.20), leads to

=(Q

+ μ

− k

∂T

∂x

. (4.22)

4.3 Carbon Embrittlement and Fracture of YBCO Superconductor 201

In other to completely describe the process of energy ﬂow, the conservation

of energy should be enforced. It requires that the internal energy rate equals

the energy input rate due to the external stress power and gradient of the

non-mechanical energy ﬂow [661]:

= σ

dε

−

∂J

∂x

, (4.23)

where ρ, u, σ

and ε

are the mass density of material, the internal energy

per unit mass, the stress tensor and the strain tensor, respectively. The minus

leaves the body. According to the discussion of continuum thermodynamics,

based on a caloric equation of state, the rate of internal energy is related

with the speciﬁc entropy, stress–strain state and total carbon concentration

rates:

= ρT

+ σ

dε

+ μ

. (4.24)

Equations (4.14) and (4.22–4.24) lead to the following diﬀerential equation:

ρT

∂

∂x



∂T

∂x



− Q

∂J

∂x

− J

∂μ

∂x

. (4.25)

Entropy rate is also related to the rates of temperature, carbonate vol-

ume fraction and total carbon concentration. This relation is derived taking

into account the dependence of entropy on all thermodynamic variables (i.e.,

temperature, stress as well as carbonate, cuprate and carbon concentrations)

[1111]:

ρT

= ρc

car

+ Q

, (4.26)

where c

is the speciﬁc heat of the superconductor at constant pressure; ΔH

car

is the enthalpy, associated with the formation of a mole of carbonate and V

car

is the carbonate molal volume. In the determination of (4.26), the following

conditions are taken into account: (i) the total number of HTSC moles in

the cuprate and carbonate remains constant; (ii) the partial derivative of

entropy with respect to temperature is related to the speciﬁc heat of the

cuprate/carbonate composite under constant stress, c

= T (∂s/∂T)

(in the

present analysis, c

= c

is assumed); (iii) by neglecting thermo-elastic cou-

pling, the partial derivative of entropy with respect to stress is taken equal

to zero

; (iv) the change of entropy due to carbonate formation is equal to

car

/T ; (v) the change of entropy due to the addition of a mole of carbon

in the carbonate is equal to Q

/T .

Note that, in ceramics, thermo-elastic coupling eﬀects are quite small [74].

202 4 Carbon Problem

Substitution of (4.26) into (4.25), taking into account (4.14), gives the

governing equation for the ﬂow of non-mechanical energy:

ρc

car

∂

∂x



∂T

∂x



− J

∂μ

∂x

. (4.27)

Therefore, the variation of the heat content in the cuprate/carbonate com-

posite depends on conducted heat, heat generated during carbon diﬀusion and

heat released during carbonate formation.

According to the above mathematical formulation for carbon diﬀusion and

energy ﬂow, the knowledge of carbon chemical potential and terminal solid

solubility is necessary. Both quantities depend on applied stress and are de-

rived below.

The chemical potentials of mobile and immobile components in stressed

solids have been derived [620]. The chemical potential of a component B is

given as

= μ

∂w

∂N

− W

, (4.28)

where μ

is the chemical potential of component B under stress-free condi-

tions, for the same concentration as that under stress; w is the strain energy of

the solid and N

is the number of B moles. Therefore, the second term in the

right-hand side of (4.28), ∂w/∂N

, represents the strain energy of the solid

per mole of component B; in the determination of ∂w/∂N

, temperature and

stress are held to be constant. Finally, W

is the work performed by applied

stresses, σ

, per mole of addition of component B. For immobile components,

since the addition or removal of the component takes place at an external

surface or an interface, chemical potential is considered as a surface property.

However, this is not the case for mobile components.

Equation (4.28) is applied to barium carbonate under stress. For material

particle of volume, V , under uniform stress, we have

∂w

∂N

∂

∂N

⎛

⎝



Vσ

dε

⎞

⎠

∂

∂N

⎛

⎝



Vσ

ijkl

dσ

⎞

⎠

⎛

⎝



∂V

∂N

ijkl

∂M

rskl

∂χ

∂N

Vσ

⎞

⎠

dσ

(4.29)

where M

ijkl

is the elastic compliance tensor of the superconductor and χ

is the mole fraction of carbon in the carbonate. The work performed by the

applied stresses per mole of carbon addition in carbonate is given as

= Vσ

∂ε

∂N

= Vσ

. (4.30)

4.3 Carbon Embrittlement and Fracture of YBCO Superconductor 203

Substituting (4.29) and (4.30) in (4.28), the ﬁnal expression is found for

the chemical potential of carbon, being in carbonate under stress:

= μ

+ V



ijkl

−



. (4.31)

In determination of (4.31), it has been taken into account that the deriva-

tive of volume with respect to carbon moles equals the partial molal volume

of carbon

. Moreover, it has been assumed in the ﬁrst approximation that

there is no eﬀect of carbon on the elastic modules of the material (this as-

sumption is discussed in detail in Sect. 4.3.2).Consequently, the derivative of

the elastic compliance with respect to the mole fraction of carbon is equal to

zero. Note that the ﬁrst term in parenthesis in (4.31) is of the order of σ

/E,

where E is Young’s modulus of the superconductor. The second term is of the

order of σ and, therefore, is signiﬁcantly larger than the ﬁrst term. If the ﬁrst

term in parenthesis of (4.31) is neglected, a relation is derived, which is more

oftenusedintheliterature.

The obtained relations for the chemical potential of mobile and immobile

components are used for derivation of carbon terminal solubility in a super-

conductor under stress. According to (4.28), carbonate chemical potential in

a stressed material is given by

car

= μ

car

∂w

∂N

car

− W

car

;

∂w

∂N

car

= w

acc

+ w

int

+ w

;

acc

= −



car

dV ; w

int

= −



car

dV ;



car

dV, W

car

= σ

car

. (4.32)

In determination of relations (4.32) it was taken into account that carbon-

ate formation is accompanied by a deformation, ε

, which is mainly a volume

expansion. Under no external loading, the above deformation leads to the de-

velopment of stresses, σ

, in the carbonate, deﬁning strain energy of material

per mole of precipitating carbonate

acc

. Under externally applied stress, σ

the interaction energy,

int

, as well as the strain energy of the applied ﬁeld,

, should also be taken into account [258]. In the last equation of relations

(4.32), σ

is the normal stress at the location of cuprate/carbonate interface,

where the chemical potential is considered.

The chemical potential of the stressed Ba–O component of barium car-

bonate is deﬁned as

Ba−O

= μ

Ba−O

∂w

∂N

Ba−O

− W

Ba−O

;

∂w

∂N

Ba−O



car

−V

)

dV ;

Ba−O

= σ

car

− V

) . (4.33)

204 4 Carbon Problem

The carbonate is assumed to be in equilibrium with carbon and cuprate

either under stress or under stress-free conditions. Then, taking into account

the chemical formula of barium carbonate BaCO

,wehave:

car

= μ

Ba−O

+ μ

); μ

car

= μ

Ba−O

+ μ

) , (4.34)

where C

and C

are the values of carbon terminal solid solubility un-

der applied stress and stress-free conditions, respectively. By substitution of

relations (4.32) and (4.33) in (4.34), one may derive

) − μ

)=w

acc

+ w

int



dV − σ

. (4.35)

It has been implied that carbonate and Ba–O equilibrium concentra-

tions do not change signiﬁcantly with stress. Because of material continu-

ity, the molal volume of the carbonate equals that of the cuprate at the

cuprate/carbonate interface; therefore

≈ 0 and the last two terms in

the right side of (4.35) can be assumed to be zero. Moreover, in ideal or dilute

solutions (Raoult’s law), the stress-free carbon chemical potential satisﬁes the

following well-known relation:

= μ

+ RT ln(C

V ) , (4.36)

where μ

is carbon chemical potential in the “standard” (i.e., reference) state

and

V is the molal volume of composite. Then, by invoking (4.15) and (4.36)

and substituting (4.31) in (4.35), one may derive the terminal solid solubility

of carbon in the composite under stress:

= C

exp



acc

+ w

int



exp





−

ijkl





. (4.37)

The above derivation is based on similar arguments of [620] for the for-

mation of cementite in ferrite. When the compliance term is neglected, (4.37)

is similar to the relation for the hydrogen terminal solid solubility [872]. Us-

ing (4.37) or its simpliﬁed version implies that chemical equilibrium occurs

under local thermal stress conditions.

Then, it is assumed that all material phases are elastic, and the elastic

properties of carbonate and cuprate are identical in the ﬁrst approximation

and do not depend on carbon concentration. The material strain is coupled

with carbon diﬀusion and energy ﬂow due to the strains, caused by carbon

dissolution, carbonate formation and thermal expansion. Similar to the pro-

cesses of hydride formation in metals [1111] and taking into account carbon

absence in cuprate, the following equation may be obtained:

dσ

= M

−1

ijkl



dε

−

dε

−

dε



; M

−1

ijkl

= λδ

+ μ(δ

+ δ

);

dε

(fθ

car

);

dε

= αδ

, (4.38)

4.3 Carbon Embrittlement and Fracture of YBCO Superconductor 205

where λ and μ are Lame constants of the HTSC; θ

car

= ε

is carbonate

expansion, occurring during its precipitation, and α is the thermal expansion

factor of the composite, assumed to be equal to that of the carbonate.

Simulation of fracture by considering cohesive tractions [51, 228] assumes

that ahead of a crack tip there is a fracture process zone, where the material

deteriorates in a ductile (void growth and coalescence) and/or brittle (carbon-

ate cleavage) mode. According to de-cohesion model, de-cohesion layer with

the thickness as fracture process zone is taken oﬀ the material, along the crack

path. Along the boundaries, created by the cut, the cohesive traction is ap-

plied. All information on the damage is contained in the distribution of the co-

hesive traction, which depends on boundary displacements of the de-cohesion

layer. The shape of the traction–displacement function depends on the failure

process. In the case of tensile separation, the most important features are the

maximum cohesive traction, σ

max

, and the energy of de-cohesion, φ



dδ

, (4.39)

where σ

is the normal cohesive traction and δ

is the respective normal dis-

placement, which equals the sum of the displacements on both sides of the

de-cohesion layer. Moreover, δ

is the normal displacement, which corresponds

to complete fracture and consequently to zero normal cohesive traction. De-

tails of the model for crack growth under plane strain conditions have been

presented in [1108, 1109]. The model has been used for the solution of several

fracture problems [763, 1089, 1112].

Here, cohesive traction is assumed to vary, according to the next relation:

⎧

⎪

⎨

⎪

⎩

≤ δ

max

≤ δ

max

− E

−δ

≤ δ

0 δ

≤ δ

, (4.40)

where δ

is a constant length of the order of carbonate thickness; E

and E

are

the de-cohesion modules, which are assumed to be constant; δ

is the normal

displacement at initiation of damage, at which maximum cohesive traction is

reached. Unloading starts when normal displacement exceeds δ

.Notethatδ

depends on σ

max

and φ

, according to the following relation:



max



−



−1

max

. (4.41)

As shown in [884], the energy of de-cohesion, related to a cohesive zone

ahead of a crack tip, in elastic material, is equal to the critical value, J

,of

J-integral, when fracture is imminent:

= J

1 − ν

, (4.42)

206 4 Carbon Problem

where K

is the critical value of stress intensity factor under plane strain

conditions; E and ν are, respectively, Young’s modulus and Poisson’s ratio of

the material, which ahead of the crack is a composite made of brittle carbonate

and relatively tough cuprate. Therefore, the fracture toughness of the material,

expressed by the energy of de-cohesion, depends on carbonate volume fraction,

f, along the crack plane. In accordance with a mixture rule, we have

= fφ

car

+(1− f )φ

cup

, (4.43)

where φ

car

and φ

cup

are the de-cohesion energies of the material, which relate

with the critical values of the stress intensity factor for carbonate, K

car

,and

cuprate, K

cup

, by using a relationship similar to (4.42). Note that the car-

bonate is surrounded by cuprate, and its delayed cracking is assumed. The

experimental values of the threshold stress intensity factor include the energy

required for the generation of the new surface due to crack growth, as well

as any plastic dissipation in the cuprate matrix, which surrounds the crack

tip carbonate. It is necessary to take these into account by using K

car

.The

cuprate can sustain diﬀerent maximum cohesive tractions from that of the

carbonate. Therefore, the maximum cohesive traction of the composite ma-

terial along the crack plane depends on the carbonate volume fraction. Here,

the maximum cohesive traction is given as

max



fσ

car

+(1− f )σ

cup

, (4.44)

which is derived by assuming that the part of de-cohesion energy during load-

ing satisﬁes a relation similar to (4.43). In (4.44), σ

car

(σ

cup

)isthemaximum

cohesive traction, sustained by the material during crack growth in the pres-

ence of the carbonate (cuprate) along the crack plane over a distance from

the crack tip x>>δ

. According to theoretical studies for the crack tip ﬁeld

in elastic plastic materials [225, 460, 885], the maximum hoop stress, along

the crack plane, is nearly equal to three times the yield stress of the material.

Assuming that σ

car

is equal to the fracture strength of carbonate, relations

(4.43) and (4.44) provide the average properties of the de-cohesion layer over

its thickness. Due to carbon diﬀusion and carbonate formation, carbonate vol-

ume fraction changes locally with time, causing a corresponding change of the

de-cohesion properties. Maximum cohesive traction is reached when δ

corre-

sponds to a normal traction, satisfying relation (4.44). As time increases, the

de-cohesion energy changes, according to ( 4.43), but unloading starts when

relation (4.41) is satisﬁed. It is assumed that the de-cohesion energy does not

change during unloading. The part of de-cohesion energy during unloading is

minimized by choosing the largest value of E

for which quasi-static unloading

exists [1112].

4.3 Carbon Embrittlement and Fracture of YBCO Superconductor 207

4.3.2 Discussion of Results

The developed mathematical model of the carbon embrittlement and fracture

of YBCO takes into account the coupling of the operating physical processes,

namely (i) carbon diﬀusion, (ii) carbonate precipitation, (iii) non-mechanical

energy ﬂow and (iv) cuprate/carbonate composite deformation. Material dam-

age and crack growth are simulated by using the de-cohesion model. Gov-

erning equations are obtained for superconductor/carbon system, in which

brittle carbonate may precipitate and be accommodated elastically, form-

ing cuprate/carbonate composite. It is assumed that carbon does not aﬀect

the elastic modules of the composite. The elastic and thermal properties of

the carbonate and composite are taken to be identical in the development

of the governing equations. The model can be extended to single or multi-

phase alloys without any changes, if the additional elements have a very small

concentration and do not aﬀect operating processes.

The consideration of elastic behavior in the bulk of the body, causing

elastic carbonate accommodation, leads to re-dissolution of the crack tip car-

bonates, after their fracture, and the reduction of the hydrostatic stresses. In

the case of greater plasticity of superconductor, on carbonate precipitation,

cuprate matrix should be yielded similar to the hydride behavior in metal

matrix [1111]. As a consequence, the crack tip carbonates are more stable

and may re-dissolve only partially after fracture. An extension of the present

model, which would take into account elastic-plastic carbonate accommoda-

tion is useful for simulation of processes beyond crack growth initiation as

well as for consideration of all parameters, which play a role in fracture resis-

tance of material. Then, the model of carbon embrittlement and sub-critical

fracture of YBCO may be added to earlier developed simulation of tough-

ening mechanisms, acting in HTSC [800, 804, 812, 814, 821, 822]. Another

direction for further development of the present model is the consideration

of diﬀerent mechanical and thermal properties of cuprate and carbonate. In

this case, the approaches, which are used in composite materials [1175], could

be adopted for developing the governing equations. Along this direction is

also the consideration of the carbon eﬀect on the elastic modules of carbon-

ate, deﬁning carbon chemical potential in carbonate [620], and therefore on

carbonate terminal solid solubility.

Generally, similar to precipitation of hydrogen in metal and intermetallic

compounds [616], the carbon eﬀect on the elastic constants of superconductor

can be characterized by a factor:

r =

− C

)/C

YBCO

, (4.45)

where C

and C

are the elastic moduli of material with and without carbon,

respectively; C

YBCO

is the concentration ratio of carbon to superconduc-

tor. However, the factor r should be regarded only as a rough approximation