Science of Sintering, 44 (2012) 147-160 ________________________________________________________________________

_____________________________

*) Corresponding author: aray@barc.gov.in

DOI:10.2298/SOS1202147R

UDK 622.785:661.884 Construction of Master Sintering Curve of ThO2 Pellets using Optimization Technique Aditi Ray1*), Joydipta Banerjee2, T. R. G. Kutty2, Arun Kumar2, Srikumar Banerjee3 1 Theoretical Physics Division, Bhabha Atomic Research Centre, Mumbai, India, 400094 2 Radio Metallurgy Division, Bhabha Atomic Research Centre, Mumbai, India, 400094 3 Department of Atomic Energy, Anushakti Bhavan, Mumbai 400 001, India Abstract:

Sintering kinetics and densification behavior of pure ThO2 have been studied using high temperature dilatometer experiments at constant rate of heating. Sintering activation energy has been determined by Wang and Raj method. Master sintering curve (MSC) for densification is a functional sintering model that describes densification under arbitrary time-temperature excursion of a particular material during sintering. MSC for pure ThO2 has been constructed by fitting experimental relative density versus work of sintering data with modified sigmoid function. Five independent parameters of the fitting function are determined by Nelder-Mead optimization technique with the objective of minimizing fitting error in terms of mean residual square. A FORTRAN program has been developed for efficient construction of best converged master curve. It is shown that activation energy of pure ThO2 found by MSC approach is consistent with those obtained by other methods. Keywords: Activation Energy, Master Sintering Curve, Work of Sintering, Optimization, Mean Residual Square

1. Introduction

Thorium utilization in Indian nuclear power program to produce fissile 233U has of late become important because of its limited uranium reserves and vast thorium deposits [1]. The three stage nuclear power program was conceptualized accordingly to utilize its thorium reserves in the Advanced Heavy Water Reactor (AHWR) [2]. Fuel property wise, thoria (ThO2) has a number of advantages over urania (UO2), like higher melting point, better thermal transport and radiation stability, more resistance to chemical interactions, low vapor pressure etc. The recent revival of interest in thorium-based fuel cycles is motivated by its potential to address concerns related to proliferation and waste disposal.

ThO2 pellets are usually fabricated by conventional powder metallurgy technique. Large-scale production of these pellets are carried out by the process of cold compaction followed by high temperature sintering in reducing atmosphere so as to attain pellet density close to its theoretical value. Pure ThO2 is not easily sinterable due to several reasons. Normally it is doped with various other ceramic materials.

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Sintering is a complex process involving micro-structural evolution by densification through several different transport mechanisms. The factors that influence the sintering are temperature, annealing time, green density and bulk composition. Therefore, there is a need for optimizing the sintering process parameters. The importance of sintering study is to predict densification behavior under any arbitrary thermal histories for a given processing method. In materials with high melting points, like, nuclear ceramics that require high sintering temperatures, it is beneficial to design sintering cycles that minimizes the energy consumption while attaining a certain target density and uniform microstructure. Minimizing sintering time reduces grain growth giving better sintered strength and other physical properties [3]. Even though there is continuing interest and considerable effort in modeling the sintering process, synthesis of nuclear fuel materials, multi-layered functionally graded materials have left many newer problems in accelerating fabrication of sophisticated ceramic systems. To determine the underlying physical mechanisms of sintering for a given material system, it is necessary to know the kinetic mechanisms by which the densification and grain growth occur during sintering. This can be realized by determining the activation energy for the corresponding consolidation process.

MSC is a unique representation wherein densification data is expressed in terms of a suitable master variable that combines time and temperatures for any arbitrary heating schedules. It was first proposed by Su and Johnson [4, 5] under the assumptions that i) single sintering mechanism, either grain boundary diffusion (GBD) or volume diffusion (VD), is dominant and ii) densification during sintering is thermally activated. MSC is expressed mathematically by a functional relationship between relative density and natural logarithm of master variable, referred to as ‘work of sintering’, and has the potential to characterize sintering kinetics regardless of heating profiles. The work of sintering represents time-temperature path followed by the material during sintering process and is characterized by apparent activation energy (Q) for densification. Thus, once mapped for density versus work of sintering, MSC enables process optimization.

MSC models for both grain-growth and densification are shown to be very useful for better interpretation of sintering data [4, 5, 6, 7]. Most of the current research in sintering studies is directed towards building MSC for densification. The common approach for developing MSC is based on fitting the time-temperature evolution of relative density data with a unique nonlinear curve. Su and Johnson [4] had used polynomial function for this purpose. However, polynomial function has the disadvantage of unphysical oscillation for higher order terms.

In 2002, Teng et al [8] had first proposed the use of sigmoid function for fitting sintered density data and applied it to construct MSC for a number of ceramics. Successively, it has been shown that sigmoid function best describes the relationship between relative density and work of sintering integral [9, 10]. However, usual form of sigmoid function has the limitation of generating accurate MSC for materials that are not easily sinterable, like undoped ThO2 powders.

Besides fitting of density–temperature trajectory, there are other attempts for MSC construction. For example, Kiani et al [11] has developed a scheme based on finite element shape function to represent densification data.

Development of a computer program to facilitate efficient construction of MSC is very useful. There are number of reports of development of self-sufficient computer programs for this purpose. However, the predictive accuracy with which MSC of different materials is constructed so far, are not completely satisfactory. Thus there is a scope for further improvement in existing method of constructing MSC.

In the current paper, we have investigated the influence of temperature on densification growth and sintering kinetics of pure ThO2. This is realized by carrying out constant heating rate experiments in a high temperature dilatometer. We have also evaluated the sintering activation energy from usual method of Arrhenius plot of logarithm of strain rate versus

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reciprocal of temperature. Since the available values of activation energy of pure ThO2 are very limited, and there is large scattering of the available values, it is prudent to estimate the same from a completely different approach. We have employed MSC concept for this purpose.

To that end, we report numerical construction of MSC for pure ThO2 pellet from sintered density data collected at four different heating rates (HRs). In order to realize best converged MSC, we have developed a new numerical algorithm based on Nelder-Mead optimization scheme [12] in multi-dimension. Modified sigmoid function with five independent parameters is chosen for fitting the relative density vs. work of sintering data collected in constant heating rate experiments. The fitting error in terms of mean residual square (MRS) defined as the spread in measured density from that predicted by fitting function, is chosen as the variable to be optimized (minimized). For best fit of the master curve with experimental density profiles, MRS found by optimization technique will be least for a particular value of activation energy. This value is accepted as apparent activation energy for the material under consideration. Steps of the numerical scheme and flowchart of the FORTRAN code developed are presented in this paper. It is shown that the experimental relative density data exactly superimpose on to the master curve indicating that the code provides a versatile tool for designing optimal sintering cycles.

MSC theory has been applied to many materials with high melting points, nanocrystalline and microcrystalline ceramics, etc., but so far it has not been used for analyzing sintering behavior of nuclear materials. Before this work, the only report of development of MSC for nuclear materials is by Kutty et al, where sintering of ThO2 doped with either CaO [13] or U3O8 [14] have been studied. The present paper is the first report of utilizing optimization technique to construct very accurate MSC of any material and provides a detailed study of sintering of pure ThO2 utilizing MSC concept.

Further, the apparent activation energy for densification of pure ThO2 estimated using MSC principle is shown to be consistent with those obtained by other deterministic methods. Our calculation brings out the important fact that detecting global minimum of MRS is necessary for accurate estimation of activation energy.

The organization of the paper is as follows. In section II we briefly describe experimental details. The temperature dependence of pellet shrinkage and relative density profiles, and determination of Q from Arrhenius plot are also presented in this section. The theory of MSC is presented in Section III. Section IV provides the details of numerical model, based on Nelder-Mead optimization scheme, for construction of MSC. Flowchart of the FORTRAN code developed is given in Section V. Numerically constructed MSC of pure ThO2 and analysis of MSC results are presented in section VI. Finally important conclusions are summarized in section VII. 2. Experimental details: Fabrication of pellets:

Green pellets of ThO2 for this study are prepared by conventional powder metallurgy route. The measured apparent density of the starting ThO2 powders is 0.7 gm/cm3 with specific surface area of 3.7 m2/gm. The total impurity content is less than 1000 ppm and O/M ratio is found to be 2. Green density of the compact is about 67% of theoretical density (TD). As-received ThO2 powders are milled for 8 hours in a planetary ball mill followed by pre-compaction at 150 MPa, granulation and final compaction at 300 MPa to form the green pellet [15]. To facilitate compaction and to impart handling strength to the green pellets, 1 wt.% zinc behenate was added as lubricant/binder during the last 1 h of the mixing/milling procedure. Pellets thus formed are about 7 mm in diameter and around 8 mm in length.

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Shrinkage measurements:

The shrinkage behavior of ThO2 green pellets are studied using a high temperature vertical dilatometer (make: Setaram Instrumentation, France; model: Setsys Evolution 24). Details of the experimental setup and procedures are same as ThO2 -4 % UO2 powder as described in Ref [16]. A calibrated thermocouple (W-5/26% Re) is placed just above the sample to record sample temperature. Heating rate used for the above studies are 2, 5, 10 and 15 K/min. Samples are heated up to 1923 K and cooled to room temperature at a rate of 20 K/min.

Fig. 1. SEM picture for sintered ThO2 pellet. Average grain size is 7.5 µm.

The experimental data from dilatometer are obtained in terms of shrinkage against time

and temperature. Experiments are carried out in reducing (Ar–8%H2) atmospheres at a dynamic gas flow rate of 20 cm3/min. Impurity contents of the cover gas used in this study includes N2 (~10 volume ppm) and O2, moisture, hydrocarbon, CO, CO2 etc each with less than 5 volume ppm. The microstructure of sintered ThO2 pellet, shown in Fig. 1, is evaluated by scanning electron microscope (SEM). Average grain size of the matrix is 7.5 µm.

Density-temperature-time data:

Pellet shrinkages as a function of temperature collected at four heating rates are converted to corresponding to relative density. In Fig. 2 we have shown temperature variation of relative density for ThO2 pellet. Blue, green, cyan and violet curves correspond to 2, 5, 10

and 15 K/min, respectively. In all the cases curves have familiar truncated sigmoid shape and are shifted to higher temperatures with increasing heating rates. It can be noticed that sintered densities obtained at any temperature shows a systematic dependence on rate of heating. At any given temperature, maximum attainable sintered density reduces with increase in rate of heating. Moreover, it can be noticed that temperature for onset to densification increases with increase in heating rate. The maximum sintered density observed for un-doped ThO2 powder is less than 90% of TD.

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Fig. 2. Density vs. temperature profile of ThO2 for constant heating rate experiments. Blue,

green, cyan and violet curves represent the cases of 2 K/min, 5 K/min, 10 K/min and 15 K/min, respectively.

Determination of Q by Wang and Raj method:

Following method of Wang and Raj [17], the slope dρ/dT is obtained from density (ρ) vs. temperature (T) plots of Fig. 2 for four HRs. In order to draw the Arrhenius plot of logarithm of strain rate (or equivalent) vs. reciprocal of temperature, for same density levels, we have calculated the quantity ln[T(dρ/dT) (dT/dt)], where dT/dt is the HR, and plotted against 104/T as shown in Fig. 3. The data for each of the density cases follow almost a linear curve. Q is determined from the slope (Q/R) of least square linear fit to these data at each density. Q estimated by this method at ρ=70%, 72%, 75%, 80% are found to be 485, 447, 454 and 464 kJ/mol, respectively, leading to an average value of 462.5 ± 14 kJ/mol.

Fig. 3. The plot of ln[T (dρ/dT) (dT/dt)] vs. 104/T for determination of activation energy using

Wang and Raj method.

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Alternatively, following the method of Young and Cutler [18], Q is estimated from the slope of linearized curve of ln[T 5/3d(ΔL/L0)/dT] vs. 1/T, for a particular heating rate assuming GBD, where ΔL is the change in length of the pellet and L0 is its initial length. Q estimated at HRs 2, 5 and 10 K/min are found to be 452, 423 and 458 kJ/mol, respectively. Thus the average value of activation energy, considering three HRs is 444.3 ± 15 kJ/mol.

3. MSC theory:

Hansen et al. had [19] proposed a generalized model for sintering induced shrinkage rate quantifying sintering as a continuous process from beginning to end. The model relates the linear shrinkage rate of a compact at any given instant to the grain boundary and volume diffusion coefficients, surface tension and aspects of instantaneous microstructure of the compact. This led to the foundation of combined stage sintering model. MSC, as formulated and constructed by Su and Johnson [4, 5], derived from combined stage sintering model of Hansen [19], is based on the following relationship between density (ρ), time (t) and temperature (T),

⎟⎟⎠

⎞⎜⎜⎝

⎛+

Ω= 3

vv4

bb

))G(ρ()(ρΓD

))G(ρ()(ρΓδD

kTdt3ρdρ γ

, (1)

where γ is the surface energy, Ω the atomic volume, δ the width of grain boundary and k the Boltzmann constant. Γb and Γv are the lumped geometric scaling parameters for GBD and VD respectively [19] and depend only on density. The mean grain diameter G(ρ) is a function of density only. Db and Dv are the coefficients for GBD and VD and expressed as

⎟⎠⎞

⎜⎝⎛−=

RTQexpD(T)D b

b0b (2)

and

⎟⎠⎞

⎜⎝⎛−=

RTQexpD(T)D v

v0v . (3)

In Eq. (2) and Eq. (3), Qb and Qv are the activation energy descriptive of densifying mechanism, Db0, Dv0 are constant coefficients of diffusion mechanism and R universal gas constant. Eq. (1) can be rewritten in a more general form as given below,

⎟⎠⎞

⎜⎝⎛−

ΓΩ=

RTQexp

))kT(G(ρ)ρ(D

dt3ρdρ 0

nγ

, (4)

where D0=Dv0 and n=3 for VD and D0=δDb0 and n=4 for GBD. By separating all micro-structural and materials properties from temperature, Eq. (4) can be rearranged and integrated in the following way,

∫∫ =⎟⎠⎞

⎜⎝⎛−

ρ

ρ

n

0

t

0 0

dρΓ(ρ)3ρ

))(G(γΩD

kdtRTQexp

T1 ρ

. (5)

Now we introduce the master variable, ))T(tΘ(t, , referred to as ‘work of sintering’, describing the time-temperature excursion as

dtRTQexp

T1T(t))Θ(t,

t

0∫ ⎟

⎠⎞

⎜⎝⎛−= , (6)

and grain size-density dependent master variable, Φ(ρ) as

∫=ρ

ρ

n

0 0

dρΓ(ρ)3ρ

))(G(γΩD

kΦ(ρ) ρ. (7)

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Hence, Eq. (5) reduces to ( )T(t)t,ΘΦ(ρ) = . (8)

This empirical relationship between ρ and Φ(ρ) or ρ and Θ(t, T(t)) is the basis of master

sintering curve. MSC is unique for a given powder and green-body density and is independent of sintering path. Thus combined stage sintering model gives rise to two types of MSC. Master sintering for grain growth (G-MSC) is the grain size-density trajectory determined by integrating Eq. (7), as described in Ref [20].

Master sintering for densification (ρ-MSC) is obtained empirically by integration of Eq. (6). The current paper is devoted to development of MSC for densification and finding the corresponding activation energy for ThO2 pellet. It can be noticed from Eq. (6) that work of sintering depends on time-temperature pathway and contains unique value of apparent activation energy. While the dominant densification mechanism is volume or grain boundary diffusion, most materials densify through a mixture of diffusion mechanisms, each with changing roles during heating and as the microstructure changes. Because of these mixed events and their implicit dependence on temperature and grain size, the ‘apparent activation energy’ used in Eq. (6) deviates from the activation energies Qb and Qv of Eqs (2) and (3).

For any time-temperature pathway, it is possible to calculate the work of sintering Θ(t, T(t)) for any point where the density is measured. For construction of MSC, it is customary to convert the instantaneous density data for certain temperature into a plot of same density corresponding to work of sintering. In doing so one uses pre-assumed value of apparent activation energy, Q. If the correct value of Q is chosen then ρ vs. Θ(t, T(t)) curves obtained for different HRs will exactly superimpose on each other. The single curve that carries all the information and characteristics of sintering and interpolates the experimental sintering data for different HRs is referred to as master sintering curve.

There are two different approaches for developing MSC of any powder. In one approach, the apparent activation energy, Q is determined by some standard method. Then, using this Q, work of sintering is evaluated and ρ-ln[Θ(t, T(t))] trajectories are drawn with experimental data. The coincidence of curves of different cases is analyzed. If the superposition of curves is not satisfactory then value of Q is changed and process is repeated till it reaches the level of satisfaction. A curve is then fitted with ρ-ln[Θ(t, T(t))] trajectory where all data points are expected to collapse. The fitted curve is the required MSC. This method has been adopted for construction of MSC of ThO2 powder [13, 14] and nanocrystalline and microcrystalline ZnO [21] and many other materials.

In the second approach (as followed in this paper), MSC is constructed by varying Q method wherein one starts the iteration from a low value of Q and proceeds by increasing its value. For every value of Q, the coincidence of different HRs curves with a pre-decided fitting function is analyzed till the error reaches minimum. The value of Q for which exact superposition of curves occurs (and corresponds to minimum fitting error in Q-space), is accepted as activation energy.

Different numerical algorithms are employed for determining the unknown parameters of the fitting function. Levenberg-Marquardt method [22] of nonlinear curve-fitting is normally used to derive the desired curve [8, 23]. Generalized Newton-Rhapson method is utilized by Blaine et al [9] for obtaining the two constants of the sigmoid function. However, the accuracy of fitted curve in published literature is not always impressive. We have employed an altogether different approach that relies on optimization of fitting error and determination of optimized values of the parameter of fitting function. 4. Numerical scheme for MSC construction:

In this section we describe the methodology adopted for construction of numerical

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MSC. The sequential steps as incorporated in FORTRAN program are described below. 1. Conversion of shrinkage data to relative density: Pellet shrinkage, 0ΔL/L as a function of temperature, collected for different constant HR experiments is converted to corresponding relative density using their governing equation,

3

00 ΔL/L1

1ρρ ⎟⎟⎠

⎞⎜⎜⎝

⎛+

= , (9)

where L0 is initial length of the pellet in axial direction and ∆L is its increment. ρ and ρ0 are relative densities, measured in percentage of theoretical density (TD), of sintered and green pellet respectively. Some guess value (very low) of apparent activation energy Q is chosen. 1. Numerical evaluation of work of sintering, Θ(t,T(t)): For constant rate of heating,

dtdTc = , Eq. (6) can be rewritten as

dTRTQexp

T1

c1Θ(T)

T

T0

∫ ⎟⎠⎞

⎜⎝⎛−= , (10)

where T0 is the temperature below which no sintering takes place. Note that, there is no explicit time dependence in expression of work of sintering now; it depends only on final temperature and assumed value of Q.

As a first step towards improvement over earlier works, we have employed variable step-size Simpson’s rule for numerical evaluation of work of sintering. Simpson method is preferred since it uses parabolic arcs to approximate the function to be integrated. Moreover, this method gives rise to more accurate results since local truncation error is related to fifth order derivative of the function. Earlier studies in this regard make use of trapezoidal rule which is less accurate due to its straight line approximation and truncation error being related to second derivative of the function. 2. Next, different sets of HR data for ρ-T(t) of Fig. 2 are converted into ρ-ln(Θ(T)) profile. It is generally observed that ρ-ln(Θ(T)) curves resemble sigmoid shape. If correct value of Q is chosen then the separation between different ρ-ln(Θ(T)) curves reduces and all data points can be interpolated by a single curve. 3. Choosing the fitting function: We assume the following form, first used by Teng et al [8], of the modified sigmoid function for representing ρ-ln(Θ) curve

( ) CSig

BA))ln(Θexp1

DDln(Θρ

⎥⎦

⎤⎢⎣

⎡⎥⎦⎤

⎢⎣⎡ −−+

′+= , (11)

where A, B, C, D, D' are parameters to be determined by fitting experimental data of ρ-ln(Θ) for all HRs studied. We compare Eq. (11) with frequently used sigmoid function given by [9, 10, 24],

[ ]))ln()(ln(nexp1ρ~1ρ~ρ~

0

00 Θ−Θ−+

−+= , (12)

where ρ~ and 0ρ~ are the fractional relative density of sintered and green pellet. Note that ‘A’ of Eq. (11) is related to ln(Θ0) of Eq. (12), i.e., midway of sintering cycle. It can be shown that, under some condition ‘B’ and ‘C’ together becomes equivalent to ‘n’. In the situation when C = 1 then B reduces to 1/n. Finally D and D' are analogous to 0ρ~ and )ρ~1( 0− . Eq. (12) assumes that final density is close to TD. Five parameters, A, B, C, D and D' determine the exact form of S-curve that gives smooth MSC and is shown to be better fit to data.

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4. Determination of function parameters: The five unknown parameters of ( ))ln(ρSig Θ are determined by minimization of MRS, σ(Q), for a particular value of Q. For this purpose we have invoked Nelder–Mead scheme [22] of optimization in multi-dimensions. For the current problem of moderate computational requirement, this method is extremely efficient and robust. 5. In multi-dimensional optimization of five independent variables of the function σ(Q), we feed our numerical algorithm with some guess values for each of the parameters as input to start the iteration. The algorithm then proceeds downhill through a 5-dimensional topography, until it encounters a minimum. 6. Fitting error: In order to check the fitting error, we define MRS, i.e., sum of the squares of relative errors divided by number of data points as

2N

1i iSig

iiSig

))(ln(ρ))(ln(ρ))(ln(ρ

N1σ(Q) ∑

= ⎥⎥⎦

⎤

⎢⎢⎣

⎡

ΘΘ−Θ

= i . (13)

In the above, ))(ln(Θρ ii denotes the experimentally measured value of percentage relative density at ln(Θi), i.e., logarithm of work of sintering corresponding to temperature Ti. In the same way, ))(ln(Θρ iSig is the value of sintered density obtained from sigmoid function of Eq. (11) at same Ti and hence same Θi. The index ‘i’ runs over all the experimental data points collected for ‘m’ different constant heating rates, each containing Mj readings. N is the total number of data points including all HRs, i.e,

∑=

=m

1jjMN . (14)

The advantage of using such discretization is that the number of data may vary in each run. For a given set of experimental data σ(Q) depends on the value of Q. 7. For a particular value of Q, optimized values of the parameter, i.e., coordinates of simplex point and the corresponding error, σ(Q) are noted down for the last iteration. On successful execution of optimization routine, last iteration returns the lowest value of function σ(Q) and optimized set of parameter values. 8. The value of Q is increased to Q+δQ and all the steps from 3 to 9 are repeated so as to obtain the new value of σ(Q). We demonstrate that increment for Q should be small enough for increasing accuracy of MSC. 9. If σ(Q) obtained for new Q is lower than the preceding value then steps 3 to 10 are repeated. 10. The program is terminated when σ(Q) is found to be increasing for three/four consecutive values of Q. 11. Thus an array of σ(Q) values for increasing Q is generated until the global minimum of σ(Q)-Q curve is reached. The value of Q that leads to minimum in σ(Q) is accepted as the apparent activation energy of sintering process. 12. The MSC is now obtained by evaluating the function ))(ln(Θρ iSig from Eq. (11) by substituting the optimized values of parameter set corresponding to minimum σ(Q). 13. To demonstrate the convergence, ))(ln(Θρ ii vs. )ln(Θi curves for different HRs are plotted with master curve, i. e, ))(ln(Θρ iSig vs )ln(Θi overlaid on it. Inputs required for optimization routine: i) Number of independent parameters of the function, which is 5 for this function; ii) guess values of the parameters for starting iteration; iii) terminating limit for variance of the function, generally very small (~10-6); iv) size and shape of initial simplex; v) number of iterations for convergence check, generally very large.

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5. Flowchart of the FORTRAN program:

For the sake of completeness and to make the steps of numerical algorithm more transparent, in the following we provide flowchart of the FORTRAN code developed.

6. MSC of ThO2 - Results and discussion:

With the numerical scheme and FORTRAN code described in sections IV and V respectively, next we proceed with the construction of MSC for pure ThO2. Using sintered density data of Fig. 2 with Q = 200 kJ/mol and some guess values for coefficients (A- D') of the function, we initiated the iteration and noted the value of σ for that Q. This has been continued until the global minimum in σ(Q) is reached. It is found that the least value of σ(Q) is arrived for Q = 422 kJ/mol. Hence the apparent activation energy for densification of undoped ThO2 in Ar - H2 reducing atmosphere is 422 kJ/mol. For each of the calculation with different Q, terminating limit for variance of the function is maintained at very low value

Read exp. data, ΔL/L0 vs T for different HRs

Calculate ρ% vs T from shrn. data

Guess a value of Q

Supply required inputs

Define function to be minimized, σ(Q)

Calculate Θ(T) and ln(Θ) for every T

Store ρ vs. ln(Θ)

Call opt. subroutine

Store σ(Q) for last iteration

Increase Q i.e, Qi =Qi +δQ

If for any ith step σi(Qi) < σi-1(Qi-1)

If for any jth step, σj+2(Qj+2) > σj+1(Qj+1) > σj(Qj)

Store σj(Qj), opt. parameter values for Qj

Accept Qj as opt. value, end of opt. routine

Find ( ))ln(ρSig Θ using opt. parameters to get MSC

Define fitting function, ( ))ln(ρSig Θ

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(~10-6), hence σ(Q) is very low for any Q. Every calculation with different Q starts with same initial guess of five parameters.

Fig. 4. Relative density plotted as a function of logarithm of work of sintering for four heating rates. The red line in all the cases illustrates the master sinter curve obtained numerically by fitting four HR data. Blue, green, cyan and violet symbols represent the cases of 2 K/min, 5

K/min, 10 K/min and 15 K/min, respectively. Fig. 4 displays the growth of densification with logarithm of work of sintering for four

different HRs separately, with MSC constructed by using Eq. (11) and activation energy, Q=422 kJ/mol, overlaid on each of them. Thus Fig. 4 compares the experimental relative density data for HRs: 2 K/min, 5 K/min, 10 K/min and 15 K/min, in blue, green, cyan and violet symbols respectively, with best converged master curve (red line). Excellent agreement of the experimental data for all HRs with MSC can be observed for entire sintering cycle. This provides an indication of the quality of Q used to construct MSC. However, there appears to be some scattering and dispersion of the sintering data in the density range of 70% -75% for lower heating rates (2 K/min and 5 K/min). However, the scattering is reasonably small and appears to be random.

Activation energy, Q obtained from MSC route, that is, 422 kJ/mol, is lower than that estimated by both Wang and Raj method (462.5 kJ/mol) and Young and Cutler method (444 kJ/mol). Q predicted by MSC approach should be lying between activation energy for surface diffusion (SD) at lower end and grain boundary diffusion (GBD) or volume diffusion (VD) at higher end. The reported value of activation energy for VD in ThO2 is 600 kJ/mol [25]. This sets the upper limit of apparent activation energy. It is known that activation energy for surface diffusion that plays significant role in densification at low temperatures is generally lower than lattice or grain boundary diffusion. Thus inclusion of low temperature density data (ρ < 70%) in MSC has led to lower value of apparent activation energy obtained by MSC method. Discrepancy of this order is expected since MSC imposes the condition of exact superposition of measured data for entire sintering cycle from onset of sintering to saturation, which is the main essence of combined stage sintering model. On contrary deterministic methods such as Wang and Raj as well as Young and Cutler method are applicable for initial

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stage sintering only. Moreover, good agreement of Q for ThO2, obtained by MSC route with those of other

deterministic methods validates the numerical algorithm and provides confidence for applying the same for other compositions, where experimental results are limited. Good convergence of constructed MSC reveals that in spite of different diffusion mechanisms dominating at different density range of the entire sintering cycle, it is still possible to predict an effective value of activation energy responsible for the combined stage sintering model of Hansen.

Finally, for graphical representation of determination of Q, in Fig. 5 we have shown the variation of fitting error, MRS with increasing value of activation energy. In order to cover a wide range of Q, MRS values are plotted in logarithm scale. The minimum of MRS, i.e., σ(Qopt)=0.286*10-5, has been found for Qopt=422 kJ/mol. Fluctuations in (σ(Q)-Q) plot suggests that finer steps of Q intervals is very essential for arriving at global minimum. Note the appearance of three prominent local minima between 300 kJ/mol and 400 kJ/mol, where σ(Q) is order of magnitude higher than the global minimum.

Fig. 5. Mean residual square, σ(Q) vs. activation energy, Q calculated by MSC program. The

optimum activation energy 422 kJ/mol leads to global minimum of MRS. Finally, the optimized set of parameter values of modified sigmoid function,

))(ln(ρSig Θ for Q=422 kJ/mol are: A = -29.3444, B = 2.89055, C= 0.93405, D = 66.78431 and D'= 25.32310. It can be noticed that maximum attainable sintered density is D+D' =92.12%. The exponent C is very close to 1 for this composition. The ratio (C/B), which is analogous to the exponent of power law variation of densification ratio with work of sintering, is 0.323. Further, D and D+D' are analogous to green density and maximum sintered density predicted by master curve for the material under consideration. 7. Conclusions

In the present paper we have investigated the densification growth of pure ThO2 using high temperature dilatometer and determined its activation energy. We have introduced optimization based numerical scheme for fitting relative density vs. integral of thermal history data with modified sigmoid curve. The algorithm has been incorporated into a FORTRAN

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program for construction of very accurate master sintering curve for pure ThO2. The apparent value of the activation energy of ThO2 estimated by this code is in agreement with those determined by other methods. Important conclusions emerging from our study are listed below. 1. The sintering activation energy determined by Arrhenius plot according to Wang and Raj method is 462.5 kJ/mol, whereas it is 444 kJ/mol in Young and Cutler method. 2. MSC code yields apparent activation energy of 422 kJ/mol for ThO2 which is lower than other two methods. 3. Excellent coincidence of experimental densification profiles with master curve affirms the possibility to predict densification behavior using MSC theory. 4. Nelder-Mead optimization scheme appears to be very useful tool for nonlinear curve fitting and determination of optimized parameters of the fitting function in this case. Acknowledgement One of us (AR) is grateful to Dr. S. V. G. Menon, Retired Distinguished Scientist of Theoretical Physics Division, Bhabha Atomic Research Centre, for providing important suggestions and conducting useful discussions during the course of this study. References

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Садржај: Кинетика синтеровања и процес денсификације чистог ThO2 праћен је високо-температурским дилатометром при константној брзини загревања. Енергија активације синтеровања одређена је методом Ванга и Раја. Мастер синтеринг крива (МСК) даје функционални модел синтеровања који описује процес денсификације под произвољним режимом време-температура за појединачни материјал. MСК чистог ThO2 је конструисана фитовањем експерименталних резултата релативне густине у зависности од података синтеровања модификованих сигмоидном функцијом. Пет независних параметара фитоване функције одређени су Нелдер-Мед оптимизационом техником са циљем минимизације грешки фитовања. Програм FORTRAN је развијен ради ефикасније конструкције мастер криве. Показано је да енергија активације синтеровања чистог ThO2 израчуната из МСК одговара вредностима енергије активације израчунате другим методама. Кључне речи: Енергија активације, Мастер синтеринг крива, Синтеровање, Оптимизација