Identification of the Identification of the Mathematical Models of Mathematical Models of

Complex Relaxation Complex Relaxation Processes in SolidsProcesses in Solids

Bakhrushin V.E.Bakhrushin V.E.

University of Humanities University of Humanities “ZISMG”, Zaporozhye, Ukraine“ZISMG”, Zaporozhye, Ukraine

Relaxation processes:- internal friction;- dispersion of modulus;- stress relaxation;- elastic aftereffect.

Parameters:- interstitial concentrations for different states;- interstitial solubility;- local diffusion coefficients;- activation energies for jumps.

Identification tasks

1. To choose the type of mathematical model: ideal Debay peak (model of the standard linear body); the sum of ideal peaks (processes); enhanced Debay peak; the sum of enhanced peaks; the sum of peaks + background.

2. To determine the quantity of relaxation processes

3. To determine the parameters of relaxation processes

n

1 1 1 1 i0 0i

i 1 0i

E E 1 1Q T Q exp Q cosh

RT R T T

0ii 0i

kTE RT ln

hf

Model of spectrum at Snoeck relaxation area:

10Q ,E – background intensity and activation energy;

10i 0iQ ,T – i-th peak height and temperature

– i-th peak activation energy

f – sample vibration frequency

Parameters, which must be determined are: 1

0i 0in,Q ,T or 10i in,Q ,E

0,14139 0,0032450T 12,89967 2,706674f 0,04547 0,04929f E,

An error for 300 – 800 К interval at f = 20 – 60 Hz is not more, then 1 %

From Wert & Marx formula such approximation may be obtained:

m 2

1 1j j

j 1

S Q Q T min

21 1

mj j

1 2j 1 j

Q Q TS min

2

exp z /z

mz ln / z2

Peak enhance may be taken into account by the model of log-normal distribution of relaxation time:

max

max id

20,0853 0,197 0,970

n1 1 1 i

0ii 1 0i

E 1 1Q Q cosh

R T T

In this case:

mln

For value from N.P. Kushnareva & V.S. Petchersky data such approximation may be obtained:

Graphic decomposition

2 peaks without error

-8

-4

0

4

1,4 1,7 2 2,3

arcch(Q-1m/Q-1)

103/T

4 peaks + error

-6

-3

0

3

6

1,2 1,5 1,8 2,1 2,4

arcch(Q-1m/Q-1)

Linear least-squares method:

n

1j i i j i

i 1

x A cosh [B (x c )]

1i 0i j j i 0iA Q , x 1/ T , c 1/ T

ii i

1 kB ln

c hfc

nj

j jj 1 1

nj

j jj 1 2

nj

j jj 1 n

xy x 0;

A

xy x 0;

A

xy x 0,

A

j 1i j i ij

i

xcosh [B (x c )] F

A

n n n n2

1 1j 2 1j 2 j n 1j nj j 1jj 1 j 1 j 1 j 1

n n n n2

1 2 j 1j 2 2 j n 2 j nj j 2 jj 1 j 1 j 1 j 1

1 nj 1j 2 nj

A F A F F A F F y F ;

A F F A F A F F y F ;

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A F F A F

n n n n2

2 j n nj j njj 1 j 1 j 1 j 1

F A F y F

1 11 2 12 n 1n 1

1 21 2 22 n 2n 21

1 n1 2 n2 n nn n

A W A W ... A W Z ;

A W A W ... A W Z ;

. . . . . . . . . . . . . . . . . . . . . . . . . .

A W A W ... A W Z ,

n1 1

ik i kj 1 j i j k

1 1 1 1W cosh B cosh B

x c x c

n1

i i ij 1 j i

1 1Z y cosh B

x c

The advantages of linear least-squares method:- simple realization;- sufficient accuracy (up to 10 % for the main peaks heights)

Method disadvantages:- linearization error; - necessity of peak temperatures preliminary definition;- possibility to obtain an ill-condition system;- supposition of uniformly precise of the data;- supposition of absolute accuracy of temperature measurements;- possibility of obtaining the negative values of peak heights.

The method of gradient descent(linearized least square method)

Main differences:- an expression of ideal peak is linearized by Taylor series expansion in the neighborhood of some point (initial estimate) with abandonment of only linear terms;- the possibility to choose the different types of objective function (cancellation of supposition that the data have the same errors)

[L. Crer et. al, 1969; M.S. Ahmad et. al, 1971; O.N. Razumov et. al., 1974; A.I. Efimov et. al., 1982.]

1 1 1mj j

2j 1 j k

Q Q T Q T0, k 1,2,...q

a

1 1 10 01 0n 1 n 01 0na E,Q ,Q ,...,Q ,E ,...,E ,T ,...,T

q 3n 2

21 1

mj j

1 2j 1 j

Q Q TS min

From

we can obtain:

at general case, q 3n without background,

q 2n for ideal Debay peaks.

After linearization we obtain:

m

0 1k k k mkm

1

a a a M Z ,

1 1m

k 2j 1 j k

Q T Q T1M ,

a a

1 1 1mj j

2j 1 j

Q Q T Q TZ ,

a

where:

derivatives are determined in0ka

Adjusted values:

0k k ka a a , 0 1.

From the definition of ka follows, that it corresponds

with the general formula of gradient search methods:

0k k 1a a gradS .

Gradient methods realize an iteration procedure, in which such stopping conditions may be used:

p p 1k ka a ; p p 1

1 1S S ; 1gradS ; , , 0.

Problems and disadvantages:

- poor convergence at the case of large number of peaks;- possibility of iteration stopping at the critical point, which is not the point of minimum;- possibility of getting into a loop, when the objective functional S is ravine;- absence of realization at standard libraries of the most popular software packages;- М matrix must be positively defined at the every step of iterations

(k 1) (k) 1 (k) (k)H , X X X G X

Quasi-Newton algorithm

1 1 10 01 0n 1 n 01 0na E,Q ,Q ,...,Q ,E ,...,E ,T ,...,T

2 2 2

21 1 2 1 q

2 2 2

22 1 2 2 q

2 2 2

2q 1 q 2 q

S S S...

a a a a a

S S S...

a a a a aH

... ... ... ...

S S S...

a a a a a

1

2

q

S

a

S

aG

...

S

a

ijij

h , i j;c

0, i j.

n1 T

1 k k kk 1

P z ,

v v

1 1 11H C PC ,

zk are eigenvalues and vk are eigenvectors of matrix:

Grinshtadt technique:

1 1P C HC ,

It is necessary to provide the positive definiteness of Hesse matrix or to find an approximation of Н-1

and

F is a Fisher criterion value for the corresponding numbers of degrees of freedom and significance level, 2 - sum of errors squares (relative errors) of experimental points.

Adequacy criteria for spectrum models:

2

SF

2

FS

0,052

SF F 2,0...2,5

number of model parameters must be increased;

2

FS

number of model parameters must be decreased.

Quasi-unimodelity (an absence of physically different minimums) of objective functional, that is all minimums of objective functional correspond to the same physical model of a spectrum.

Deviation from quasi-unimodality may be caused with:- the presence of excess peaks in the model;- absence of some essential peak in the model;- presence at the real spectrum of some collateral peak, which height is close to measurement error.

Absence of model residuals serial correlation (Darbin & Watson criterion):

m 2

j j 1j 2

m2t

j 1

e e

d ,e

1 1j j je Q Q T - model residuals.

d 2- serial correlation is absent;

d 0d 4

- positive serial correlation;- negative serial correlation(there are excess peaks).

0

5

10

500 600 700 800

Q-1·103

T, K

Given data (4 peaks + error)

Initial approach (4 peaks) =0,1

0

4

8

12

500 600 700 800 T, K

Q-1·103

The result of decomposition

S=0,30

F=0,74

-0,25

-0,1

0,05

0,2

500 600 700 800

Q-1·103

T, K

Residuals

d=2,18

0

4

8

12

500 600 700 800 T, K

Q-1·103

Initial approach 1 (3 peaks) =0,1

0

4

8

12

500 600 700 800 T, K

Q-1·103

The result of decomposition 1 (3 peaks)

S=20,52

F=50,31

-2

-1

0

1

500 600 700 800

Q-1·103

T, K

Residuals

d=0,33

0

4

8

12

500 600 700 800 T, K

Q-1·103

Initial approach 2 (3 peaks) =0,1

0

4

8

12

500 600 700 800 T, K

Q-1·103

The result of decomposition 2 (3 peaks)

S=13,58

F=33,28

-2

-1

0

1

500 600 700 800

Q-1·103

T, K

Residuals

d=0,54

0

4

8

12

500 600 700 800

Q-1·103

T, K

Initial approach 1 (5 peaks) =0,1

Initial approach 2 (5 peaks)

0

4

8

12

500 600 700 800

Q-1·103

T, K

=0,1

  Given 3_1 3_2 4 5_1 5_2

T1 570 571,3 578,6 569,4 569,4 569,4

T2 620 632,5   619,8 619,8 619,8

T3 690 698,1 685,5 690,2 690,2 690,2

T4 750   742,4 749,2 749,2 749,2

T5         499,6 718,2

Q1 6 6,3 7,4 5,9 5,9 5,9

Q2 3 3,5   3,1 3,1 3,1

Q3 12 13,0 12,2 12,0 12,0 12,0

Q4 3   3,6 2,9 2,9 2,9

Q5         0,0 0,0

=0,1

0

4

8

12

500 600 700 800 T, K

Q-1·103

=0,3Initial approach 1 (4 peaks)

0

4

8

12

500 600 700 800 T, K

Q-1·103

Initial approach 2 (4 peaks) =0,3

0

4

8

12

500 600 700 800 T, K

Q-1·103

Initial approach 3 (4 peaks) =0,3

0

4

8

12

500 600 700 800 T, K

Q-1·103

The result of decomposition

s=3,01

F=0,76

-0,5

-0,2

0,1

0,4

500 600 700 800

Q-1·103

T, K

d=1,20

Residuals

0

4

8

12

500 600 700 800 T, K

Q-1·103

Initial approach 1 (3 peaks) =0,3

0

4

8

12

500 600 700 800 T, K

Q-1·103

Initial approach 2 (3 peaks) =0,3

0

4

8

12

500 600 700 800 T, K

Q-1·103

Initial approach 3 (3 peaks) =0,3

The result of decomposition

0

4

8

12

500 600 700 800 T, K

Q-1·103s=15,87

F=3,99

-2,5

-1

0,5

500 600 700 800

Q-1·103

T, K

Residuals

d=0,57

0

4

8

12

500 600 700 800

Q-1·103

T, K

Initial approach 1 (5 peaks) =0,3

0

4

8

12

500 600 700 800

Q-1·103

T, K

The result of decomposition

s=2,92

F=0,74

-0,5

-0,2

0,1

0,4

0,7

500 600 700 800

Q-1·103

T, K

Residuals

d=1,21

0

4

8

12

500 600 700 800

Q-1·103

T, K

Initial approach 2 (5 peaks) =0,3

0

4

8

12

500 600 700 800

Q-1·103

T, K

The result of decomposition

s=1,41

F=0,35

-0,5

-0,2

0,1

0,4

0,7

500 600 700 800

Q-1·103

T, K

Residuals

d=2,47

  Given 3 4 5_1 5_2

T1 570 578,7 569,8 571,2 569,7

T2 620   621,8 622,8 621,3

T3 690 686,5 691,2 691,2 690,6

T4 750 749,2 757,1 757,2 751,7

T5       537,5 851,0

Q1 6 7,2 5,9 5,7 5,8

Q2 3   3,0 2,9 2,9

Q3 12 12,4 12,2 12,2 12,0

Q4 3 3,6 2,9 2,9 2,9

Q5       0,3 0,6

=0,3

0

10

20

450 550 650 750

Q-1·103

T, K

0,180,360,420,60

[N], at.%:

Nb – 2 at.% W – N (3 peaks, 1 result)

-3,5

-2,5

-1,5

-0,5

0,5

1,5

2,5

450 550 650 750

Q-1·103

T, K

d: 0,69; 0,521,94; 1,14

Residuals

0

10

20

450 550 650 750

Q-1·103

T, K

0,180,360,420,60

[N], at.%:

Nb – 2 at.% W – N (3 peaks, 2 result)

-3,5

-2,5

-1,5

-0,5

0,5

1,5

2,5

450 550 650 750

Q-1·103

T, K

d: 0,85; 1,331,88; 1,27

Residuals

0

10

20

450 550 650 750

Q-1·103

T, K

[N], at.%: 0,180,360,420,60

Nb – 2 at.% W – N (4 peaks)

-2,2

-1,2

-0,2

0,8

1,8

2,8

450 550 650 750

Q-1·103 d: 1,83; 2,102,84; 1,67

Residuals

0

10

20

450 550 650 750 T, K

Q-1·103 [N], at.%: 0,180,360,420,60

Nb – 2 at.% W – N (5 peaks, 1 result)

Residuals

-2

-1

0

1

2

450 550 650 750

Q-1·103

T, K

d: 1,60; 2,363,09; 1,83

0

10

20

450 550 650 750 T, K

Q-1·103 [N], at.%: 0,180,360,420,60

Nb – 2 at.% W – N (5 peaks, 2 result)

-2

-1

0

1

2

450 550 650 750

Q-1·103

T, K

d: 2,54; 2,362,96; 2,25

Residuals

  T1 T2 T3 T4 Т5 S F

3_1 539,1 657,2 685,2     70,1 2,0

3_2   647,1 673,6 748,7   86,5 1,6

4 537,9 650,9 674,7 749,0   43,3 3,3

5_1 528,8 656,9 676,4 748,0 593,0 35,7 4,0

5_2 535,5 641,4 674,4 745,5 665,5 34,1 4,2

  Q1 Q2 Q3 Q4 Q5

3_1 2,2 18,2 13,4    

3_2   9,3 21,3 2,7  

4 2,1 10,8 19,5 2,6  

5_1 1,5 13,6 15,9 2,6 2,1

5_2 2,0 7,0 17,9 2,6 5,8

  Nb – 12 at.% W Nb – 6 at.% W

  4 peaks

5 peaks 4 peaks1 set 2 set

E1, kJ/mol   102,2 86  

E2, kJ/mol 109,5 111,5 110,1 109,8

E3, kJ/mol 116,3 116,9 116,5 116,5

E4, kJ/mol 128,9 129,6 129,1 128,3

E5, kJ/mol 1456 145,9 145,9 145,9

0

0,3

0,6

0,9

500 600 700 800

Nb – 2 at.% Hf – 0,32 at.% N (3 peaks):E = 1,47; 1,61; 1,76 kJ/mol;d = 0,90; F = 1,73.

0

0,3

0,6

0,9

500 600 700 800

Nb – 2 at.% Hf – 0,32 at.% N (4 peaks):E = 1,29; 1,48; 1,62; 1,79 kJ/mol;d = 1,26; F = 2,64.

0

0,3

0,6

0,9

500 600 700 800

Nb – 2 at.% Hf – 0,32 at.% N (5 peaks, 1 result):E = 1,44; 1,54; 1,63; 1,77; 1,91 kJ/mol;d = 1,06; F = 3,21.

0

0,3

0,6

0,9

500 600 700 800

Nb – 2 at.% Hf – 0,32 at.% N (5 peaks, 2 result):E = 1,26; 1,46; 1,57; 1,64; 1,80 kJ/mol;d = 1,55; F = 3,58.

n

N ii 1

0ii 2 2 2

i

i

0ii

M(T) M M (T);

MM (T) ;

1 4 f

E 1 1exp

R T T.

2 f

The temperature dependence of dynamic elastic modulus in a case of n processes, which satisfy the model of standard linear body, may be determined from a system:

MN – non-relaxed modulus.

Model parameters, which must be identified, are:

0i 0iM ,T .

m 2

1 exp j jj 1

S M T M T min,

We are to solve such minimization problem:

exp jM Twhere are experimental data for modulus at Tj.

(*)

Functional (*) has a great number of minimums, so the result of minimization strongly depends on initial assumption.

Adequate model may be obtained by using as T0i initial values the results of relaxation spectrum decomposition and setting initial values as

0iM

10i N 0iM 2M Q .

T0i values after minimization are very close with the initial ones, and values change essentially. But there is a correlation (r = 0,90 – 0,97) between partial Snoek peaks heights and results for :

0iM

0i1

0i H

M2,00 0,15.

Q M

0iM

Nb – 12 at.% W – N

[N], at.%: - 0,11; ■ – 0,16; ▲ – 0,22; - 0,31