U.P.B. Sci. Bull., Series A, Vol. 70, Iss. 2, 2009
ISSN 1223-7027
ACCOMMODATION AND DIS-ACCOMMODATION
PROCESSES IN PHYSICS AND BIOPHYSICS*
Dan A. IORDACHE1
Lucrarea analizează rezultatele obţinute – prin colaborarea grupurilor de
cercetare (ale catedrei de Fizică a UPB) în Fizica laserilor, respectiv în Fizica
materialelor, în anii 1978-1981, cu privire la procesele de acomodaredezacomodare specifice acţiunii radiaţiilor electromagnetice (în particular, laser)
asupra unor materiale magnetice, în baza constatărilor recente privind clasele de
Universalitate ale fenomenelor fizice şi – în general – procesele de creştereacomodare din Fizică şi Biofizică .
Se constată că descrierea proceselor de acomodare-dezacomodare din
Fizică şi – respectiv – Biofizică necesită: a) utilizarea unor modele de similitudine
ale proceselor de creştere-acomodare, care pot fi puse în corespondenţă directă cu
anumite clase de Universalitate, b) folosirea unor metode numerice adecvate pentru
evaluarea parametrilor corelaţiilor teoretice specifice claselor de Universalitate, c)
verificarea compatibilităţii locale şi – respectiv – globale a relaţiilor teoretice
investigate cu rezultatele experimentale. Este propus un nou model teoretic bazat pe
cuplajul proceselor auto-catalitice, respectiv de stagnare, şi se constată că acest
model poate descrie rezultatele experimentale obţinute în anii 1979-1981.
This work analyzes the obtained results – by means of the cooperation of
the research groups (from the Physics Department of the “Politehnica” University
from Bucharest) on Lasers Physics, and on Physics of Materials, respectively during
the years 1978-1981 – concerning the accommodation - dis-accommodation
processes specific to the action of the laser radiation on some magnetic materials,
starting from the findings about the Universality classes of the physical phenomena
and the growth-accommodation processes met in Physics and Biophysics,
respectively. One finds that the description of the accommodation – disaccommodation requires: a) the use of some similitude models of the growthaccommodation processes, that present a direct correspondence with some certain
Universality classes, b) the use of some suitable numerical methods for the
evaluation of the parameters of the theoretical relations specific to the Universality
classes, c) the check of the local and global compatibility of the studied theoretical
relations with the existing experimental data. A new theoretical model (of the autocatalytic growth and stagnation coupling) is proposed, finding the possibility to
describe the experimental results obtained during the 1979-1981 interval.
1. Introduction
Starting from the basic works concerning the experimental methods of
magnetism [1] - [4], the initial permeability and the basic parameters of the
1
Prof., Physics Department of the University POLITEHNICA Bucharest, Romania, e-mail:
daniordache2003@yahoo.com
*
Paper presented at the Scientific Meeting on 2008, organized by The Department of Physics I,
Faculty of Applied Science, UPB, on the occasion of the 70-th birthday of Prof. C.P. Cristescu
48
Dan A. Iordache
magnetic hysteresis loops of some ferri-magnetic materials were evaluated
following the approach discussed in refs [5] and [6]. The correlation between the
main parameters of the magnetic hysteresis and the corresponding permeability
for the mono-parametric technological series of spinelic ferri-magnetic materials,
was also included in [6]. The results on the photo-magnetic effects (in particular,
accommodation and dis-accommodation processes) induced by the natural light
and coherent (laser) radiation on some industrial Romanian ferri-magnetic
materials were studied in the frame of works [7], [8].
2. Experimental results
References [7] and [8] pointed out that the photo-irradiation of some very
thin2 spinelic soft ferri-magnetic torus shaped samples leads to a certain disaccommodation (worsening of their magnetic performance, i.e. decrease of the
initial permeability  ri and increase of the magnetic coercive strength H c ); see
Table 1 and Figures 1 and 2.
Table 1
Efficiency of the non-coherent and coherent (laser) radiation concerning the photo-magnetic
dis-accommodation effects on Zn0.58Mn0.32Fe2.10O4 ferrite
Radiation
Non-coherent
Coherent Radiation
Radiation
Non-stabilized laser
Stabilized laser
  632 .8 nm
 /  (%)
H c / H c (%)
- 7.92
- 9.73
- 12.80
8.59
10.76
14.67
Fig.1. The time dependence of the photoinduced permeability decrease under the
action of non-coherent and coherent radiation
2
Fig. 2. The photo-induced increase of the magnetic
coercive (field) strength under the action of some
non-coherent and coherent radiation
In order to ensure a good penetration of the light radiation, the width of the studied magnetic
samples was 0.2 mm (while its external and internal diameters were equal to 5 mm and 2.5 mm).
Accommodation and dis-accommodation processes in physics and biophysics
49
3. Classical similitude models of the growth processes
Starting from the differential equation of the growth (accommodation) of
dY
  (t )  Y (t ) ,
an arbitrary physical parameter Y(t):
(1)
dt
where π(t) is the time density of the growth (accommodation) probability, with the
 (t )  1 .
physical dimension:
(2)
T
Introducing the similitude criteria (functions):
Y (t )
 (t )
   (0)  t , y (t ) 
, and: p(t ) 
,
(3)
Y (0)
 (0)
dy
dz
 p ( )  y ( ) , i.e.: p ( ) 
one obtains the similitude growth equation:
, (4)
d
d
z  ln y .
by means of the similitude variable:
(5)
Assuming that z is a function solely of p(τ), one obtains:
p 

z
p

  n  p n .

dz dp
  n p n n 1
(6)
n0
The growth processes can be classified according to the degree N of the
algebraic polynomial p ( p) in the following Universality Classes [9]: a) U0
(corresponding to a constant value of the probability density p(τ)), b) U1, for a
linear p  f ( p) dependence, … c) U3, for a p  f ( p) dependence expressed by
a 3rd degree polynomial, etc.
3.1. The particular case of the auto-catalytic growth processes (U0 model)
Many results concerning the basic features of the growth processes in
Physics [10], Biology [11] and even Cosmology [12], point out the outstanding
importance of the auto-catalytic (exponential) growth type. The auto-catalytic
growth equation can be obtained from equation (6), for:  ( )  const. 
  
 .
τac is the time constant of the auto-catalytic growth: Y ( )  exp 
  ac 
1
 ac
, where
(7)
3.2. The particular case 2: the U1 model
The U1 Universality model was proposed by Gompertz [13] in 1825, in
strong connection with the study of some tables concerning the mortality.
50
Dan A. Iordache
Starting from relation (8), with null coefficients, excepting 1  1 , one
obtains:
(8)
dp d   p .
Integrating the equation (6) with the probability time density (8), one
obtains:
(9)
ln y   exp    C , hence: y  expC  exp(  ) ;
from the condition: y(0) = 1, one finds the similitude expression of the growth
equation, according to Gompertz’s model: y  exp1  exp   .
(10)
The Gompertz’s model is valid for some tumors growth processes [14],
[15], as well as for the descriptions of the population dynamics [16], etc.
3.3. The 3rd particular case: the U2 model
The similitude expression of the U2 (West’s) growths [17], [18a]:
(11)
y  1  b  b  exp(  )1/ b ,
generalizes the equation (10), because it leads to the similitude expression of the
Gompertz’s growth for b  0 . Obviously, the growth rate is given in this case by
dy 1  b 1b 1

y
 y .
the nonlinear expression:
(12)
d
b
b
Figure 3 presents the similitude plots of the growths of U2 type for
different values of the parameter b, the value b = 0 corresponding to the
Gompertzian growths, while the value b  0.25 corresponds to the original
West’s model [17].
Fig. 3. The nondimensional (similitude) plots of the growths corresponding to the
U2 model [17], [18a]
We consider as useful to point out that the U2 model:
(i) describes the growth processes of the living beings, in the range
protozoa – plants – mammals, by means of the similitude equation:

m( )  M b  b  exp(  )
1/ b , where: M  lim m( )  1  b1/ b ;
(13)
Accommodation and dis-accommodation processes in physics and biophysics
51
defining the development remainder (rest) by means of the expression:
b
one finds that:
m
z  1   ,
M 
z  exp(  ) , where:     ln b  b  ln M
(14)
(15)
is the so-called biological time,
(ii) can be used for the tumors growth description, with values of the
parameter p depending on the fractal nature of the biological channels (e.g. in
angio-genesis) [18].
3.4. The 4th particular case: the U3 model [18b], [18e]
The similitude growth rate of the probability time density corresponding to
the U3 model is:


p   p 1  b  p  c  p 2 .
(16)
Because according to relations (4) and (6):
dy
dy
,
p y 
 p 
d
dp
dy
p  dp
dp
one obtains for the U3 model: K  ln y     
. (17)

y
p
1 b  p  c  p2
One finds that the U3 model involves an additional parameter (c) relative
to the previous U2 model, hence this model (U3) presents important advantages
for the detailed description of some particular features of the growth processes
[18e].
4. Accommodation (adaptation) processes. The model of the auto-catalytic –
stagnation processes coupling [19]
4.1. Basic Elements
As it results from figure 3, the growth models of the Gompertz’s type do
not involve usually some inflexion points of the growth (adaptation)
characteristics, as it was found rather frequently both in Physics [10] and Biology
[11], and even (as it seems) in Cosmology [12]. A typical plot of the
accommodation (adaptation) processes is that of the skeletal muscles contractions
under the action of an increased concentration of the Ca2+ myoplasmic ions (see
figure 4 and [20]). Assuming a uniform injection of the Ca2+ myoplasmic ions,
figure 4 can be generalized for the coupling of any arbitrary auto-catalytic and
stagnation stages intervening in the description of the accommodation (growth)
processes of different parameters p(t), as it is shown by figure 5. For this reason,
we studied also the possibilities of description of some growth (adaptation)
52
Dan A. Iordache
phenomena, synthesized (for an arbitrary parameter p) by means of the plot
presented in figure 5.
Fig. 4. The dependence on the myoplasmic
Ca2+ concentration of the force corresponding to
the skeletal muscle contraction [20]
Fig. 5. Typical plot of the coupling
of the auto-catalytic and stagnation
growth processes
The accomplished study pointed out [19] the possibility of descriptions of
some growth (accommodation) processes like those indicated by figures 4 and 5
by means of some (master) differential equations of the type:
dp
dp
dp
dt
r


.
(18)
p
p  p p ms  ac
From the definition of the inflexion point of the growth (accommodation)
d2p

  0 , one obtains: pinf  p ,
plot:
(19)
 dt 2 
1

r

 inf
and:


2
r  1  r
1 

, respectively.
(20)
 rel  p
p ms 


p
For the alnico alloy, we obtained [19]:  rel  52 min and: inf  0.1 min 1 .
p
p max  p inf 
4.2. Classification of the time dependencies corresponding to the above
theoretical model of the growth/accommodation processes [22]
Taking into account that the experimental data referring to the photoirradiation induced magnetic dis-accommodation correspond mainly to the last
(relaxation) part of the accommodation process, the corresponding master
equation can be simplified as:
Accommodation and dis-accommodation processes in physics and biophysics
dp
dp
dt
r

.
p
p  p  ac
Integrating the above-indicated differential equation, one obtains:
t
ln p  r  ln  p  p   ln C 
.
53
(21)
 ac
For t = 0, one finds: C  p(0)   p  p(0)r , hence the time dependence
of the studied parameter p is described by the implicit equation:
 t 
 ,
(22)
p(t )   p  p(t ) r  p(0)   p  p(0) r  exp 

 ac 
 p  p(t )r
 p  p(0)r
 t 
 .
 exp  
(22’)
p(t )
p (0)

 ac 
For: p(0), p(t )  p , the relation (22) leads to the time dependence:
equivalent to:

 t 
 ,
(23)
p(t )  p(0)  exp 
  ac 
i.e. to the equation of the auto-catalytic growth.
 t 
   , hence:  p  p(t )r   and:
For t → ∞, we have: exp 
  ac 
lim p(t )  p  . In such conditions, the relation (22’) reaches its asymptotic
t 
expression:
p(t )  p  p(t )   p  p(0) r

p
t
 exp  
p(0)
  rel

 ,

(22”)
 rel  r   ac ,
where:
(24)
is the relaxation time of the studied parameter p.
The master equation corresponding to the studied growth differential
1
p p  p 
dp
1 1
r 
p 

equation is:
.
(25)
 
 
dt  acg  p p  p 
 acg  p  p(r  1)
It results that if p  p , i.e.: p  p(0) , the slope p is positive, or
conversely. As it concerns the curvature, one finds:
p 
where:
d2p
dt 2

p  p   p 

r  1

 P   p  p   p  p r  1 ,
P
2
 acg
  p  p(r  1)3
(25’)
(25”)
54
Dan A. Iordache
is a positive factor if:
(26)
1  r  p p(t ) .
The relations (25) - (25”) point out the following specific behaviors:
(i) If p  p(0)  r  1 , the plot p(t) has always a positive slope (i.e. it is
monotonically increasing) and an inflexion point for the ordinate:


p inf l. 
p
r 1
(26’)
;
this behavior corresponds to the “complete” growth plot represented by figure 5.
(ii) If p(0)  p  p(0)  r  1 , the plot p(t) presents also a positive
slope (i.e. a monotonic increase), but it has always a negative curvature, and no
inflexion point. Such a shape of the time dependence (fig. 6) corresponds to the
U2 growths (see figure 3), to the magnetic coercive field H c accommodation
after photo-irradiation [8] (fig. 2, as a completely different example), etc.
(iii) Finally, if p  p(0) , then the plot has always a negative slope, a
positive curvature and no inflexion point. Such a time dependence (fig. 7)
corresponds to the decrease of some populations in unfavorable conditions, to the
initial permeability  i accommodation after photo-irradiation [8] (fig. 1, as a
completely different example), etc.


Fig. 6. Particular case of the auto-catalytic/stagnation model of the accommodation processes
corresponding to the photo-irradiation induced increase of the coercive magnetic strength (fig. 2).
Fig. 7. Particular case of the auto-catalytic/
stagnation model of the accommodation
processes corresponding to the photoirradiation induced decrease of the initial
permeability (see fig. 1).
4.3. Special conditions of coincidence of the Universality U2 growth
model and of the auto-catalytic/stagnation model of the growth/accommodation
processes
In order to ensure the coincidence of these models, it is necessary firstly to
have similar qualitative shapes of the time dependence of the growing physical
parameter. Comparing the plots from figures 5, 6 and 7 with that from figure 3,
one finds that only the regime: p(0)  p  p(0)  r  1 ,
(27)
with the time dependence given by relation (22”):


Accommodation and dis-accommodation processes in physics and biophysics
55

p
t 
 ,
 exp  
p(0)
  rel 
could lead to a time dependence similar to the U2 model, expressed by relation
p(t )  p  p(t )   p  p(0) r
y  1  b  b  exp(  )1 / b .
Denoting:   t  rel , it is possible to rewrite relation (24”) as:
(28)
p(t )  p( )  p 1  exp(  )  p(0)  exp(  ) .
Comparing this relation with the above expression (13), one finds that for
their coincidence it is necessary that the West-Delsanto’s parameter b be: b  1 . In
this case, the relation (13) becomes: y  2  exp(  ) .
(13’)
In following is very easy to find that – in order to ensure that relation (28)
leads to relation (13’) – we have to denote: y  p(t ) p(0) and choose the value:
p  2 p(0) .
(13):
5. Comparison of the quantitative descriptions of some experimental results
of some dis-accommodation processes by means of the auto-catalytic –
stagnation (ACS) model and of the U2 model [19]
3.4. Quantitative ACS description of the permeability dis-accommodation
The accomplished study of the relaxation part (for irradiation duration of 6
minutes or more) of the dis-accommodation of the relative initial permeability of
the investigates ferrite (see Fig. 1) pointed out a very good agreement of the
experimental data relative to the auto-catalytic – stagnation model. For the
stabilized laser irradiation, the correlation coefficient corresponding to relation
(22”) was:   0.999947 , for a relaxation time:  rel  4.343 min , a value:
r  0.248 of the ratio of the relaxation and auto-catalytic characteristic times, and
an auto-catalytic growth time:  ac  17.5 min . One finds easily that the condition
(26) is also fulfilled in this case: 1  r  0.752   ri ,   ri (0)  0.874 . The values
of the relative initial permeability after different irradiation duration were
evaluated by means of the classical gradient method [23], starting from the above
reported zero-order approximation values and the implicit equation (22’). The
obtained results were plotted in Figure 8.
3.5. Quantitative ACS description of the coercive field dis-accommodation
The similar study of the dis-accommodation effects after photo-irradiation
of the coercive magnetic field led to: a) a very good correlation of the relaxation
process (22”) for the entire time domain (correlation coefficient:   0.9948 ), b)
the limit value: H c  42.8 A / m , and a value: r  0.01 without physical
56
Dan A. Iordache
meaning of the ratio of characteristic times. It results that for the coercive
magnetic field H c , the photo-irradiation effects resume basically to relaxation.
3.6. U2 model description of the coercive field dis-accommodation
Because according to relation (13), the maximum value of the
corresponding U2 parameter is:
y max  (1  b)1 b ,
(13’)
H c
42.8

 1.132275 , one
H c (0) 37.8
finds that the corresponding U2 parameter is: b  26.75 . In such conditions, the
value ( 42.7 A/m) of the coercive magnetic field after 12 minutes of stabilized
laser irradiation is obtained for a U2 characteristic time [see relations (3a) and
and for the stabilized laser irradiation: y max 
(13)]:  ch   (0)1  4.341 min . Because this value agrees very well with the
relaxation times corresponding to the relative initial permeability (see above) and
with that of the coercive magnetic field (evaluated for the 8…12 irradiation
minutes):  rel , H c  4.365 min , we adopted it for the U2 model evaluation of the
coercive magnetic field dis-accommodation after photo-irradiation.
Both the obtained results by means of the proposed ACS model and those
resulted using the U2 model, are presented in Figure 9.
Fig. 8. Comparison with the experimental
Fig. 9. Comparison of the theoretical results given
results for the relative initial permeability dis- for the coercive magnetic field dis-accommodation
accommodation of the theoretical predictions
by the proposed theoretical ACS model and that
given by the theoretical model of the autocorresponding to the U2 Universality class growth
catalytic – stagnation growth model (ACS)
Accommodation and dis-accommodation processes in physics and biophysics
57
3.7.Interpretation of the obtained results
Taking into account that the relative errors affecting the measurements of
the relative initial permeability  ri (accomplished by means of the Tesla Q-meter
BM 311E) and of the coercive magnetic field H c (done by means of the low
frequency LF-1 device of magnetic measurements) are of the magnitude order of
3…10% and 5…15% [5c], respectively, it results that both studied theoretical
models are compatible with the experimental results. Another choice of the
characteristic time corresponding to the Universality U2 model (different than the
relaxation times of the magnetic permeability and coercive field), could improve
the general agreement of this model predictions with the experimental results
concerning the photo-irradiation induced dis-accommodation of the studied ferrimagnetic materials.
6. Conclusions
The accomplished study achieved: a) to examine some basic existing
theoretical models of the continuous growth/accommodation processes, b) to
propose a new (ACS) model of the coupling of the auto-catalytic and stagnation
processes, and to describe by means of it the experimental findings about the
photo-induced increase of the coercive magnetic strength and decrease of the
initial permeability, respectively, c) to compare (for some photo-irradiation
induced dis-accommodation processes) the predictions of the proposed ACS
model with those of the Universality models.
Acknowledgements
The author thanks very much to Professors Pier Paolo Delsanto
(Politecnico di Torino) and Constantin Cristescu (“Politehnica” University of
Bucharest) for their important suggestions.
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