Changes in the shape of microdroplet aggregates exposed to an alternating magnetic field with fre-quency from 0.01 to 1 Hz are investigated both experimentally and using a computer simulation. Micro-droplet aggregates are spherical droplets of a highly concentrated magnetic fluid with a radius from one to 
several tens of microns dispersed in a liquid with a low concentration of magnetic particles. For values  of
interfacial tension  σ > 2·10– 6
N/m small oscillations in a weak alternating field could not be obtained, but 
periodic forced oscillations were obtained at strong elongations. Oscillations of elongated microdroplet ag-gregates are of a periodical nonsinusoidal type with higher harmonics and subharmonics in the spectrum.
When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/3611

Drozdova, V.I.

Romanenko, M.G.

Shagrova, G.V.

Electronic Sumy State University Institutional Repository

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ Vol. 6 No 3, 03013(3pp) (2014) Том 6 № 3, 03013(3cc) (2014)   The article was reported at the International Conference «The Advanced Technology, Equipment and Analytical Systems for Materials», Kursk, 13-14 May, 2014 2077-6772/2014/6(3)03013(3) 03013-1  2014 Sumy State University The Effect of Interface Tension on Forced Oscillations of Elongated Microdroplet  Aggregates in Magnetic Fluids  V.I. Drozdova, G.V. Shagrova, M.G. Romanenko  North-Caucasian Federal University, 2, Kulakova Pr., 355009 Stavropol, Russia   (Received 19 May 2014; revised manuscript received 03 July 2014; published online 15 July 2014)  Changes in the shape of microdroplet aggregates exposed to an alternating magnetic field with fre-quency from 0.01 to 1 Hz are investigated both experimentally and using a computer simulation. Micro-droplet aggregates are spherical droplets of a highly concentrated magnetic fluid with a radius from one to several tens of microns dispersed in a liquid with a low concentration of magnetic particles. For values of interfacial tension  > 2·10 – 6 N/m small oscillations in a weak alternating field could not be obtained, but periodic forced oscillations were obtained at strong elongations. Oscillations of elongated microdroplet ag-gregates are of a periodical nonsinusoidal type with higher harmonics and subharmonics in the spectrum.   Keywords: Magnetic fluid, Forced oscillations, Interfacial tension, Microdroplet aggregate.   PACS numbers: 75.50.Mm, 79.60.Jv   1. INTRODUCTION  Application and research of liquid magnetizable nanodisperse systems known as magnetic fluid (MF), started over 50 years ago. MF are the stabilized colloids of magnetic particles of size ~ 10 nanometers in differ-ent non-magnetic host liquids. The combination of flu-idity with the possibility to be magnetized in weak magnetic fields is still attracting the attention of engi-neers and researchers. Complex heterogeneous systems that are based on the MF, in particular, microemul-sions and MF containing microdroplets are promising new materials for defectoscopy. The properties of boundary between heterogeneous magnetic phases are unique in having both low values of interfacial tension values  ~ 10 – 7 H/m and high values (for liquid artificial media) of permeability units. In the absence of external field, microdroplet aggregates have a spherical shape; in a weak field, they can be strongly deformed assuming the shape of needles oriented along the field. The static de-formation of suspended MF ” droplets and microdroplet aggregates is well studied experimentally and theoretical-ly [1, 2]. Deformation of microdrops under weak magnetic field corresponded closely to the model that uses assump-tions about the linear nature of the magnetization and ellipsoidal shape of the aggregates. When using energy approach equilibrium forms of microdrops can be obtained from the Lagrange equations                m Ed T T E EdT,   where T – total energy, E – surface energy, Em – mag-netic energy, E  – viscous dissipation rate,  – the ra-tio of the major and minor axes of the ellipsoid. From the equation (1) it follows that deformation of aggre-gates in magnetic field has a hysteresis nature. When the magnetic field strength is lower than crit-ical value H1 the microdrop takes the shape of prolate ellipsoid of revolution oriented along the field with the axes ratio a / b  2. At H  H1 a jumpwise elongation of a microdrop occurs and it acquires the form of a body of revolution with tapered ends and the axis ratio a / b  10. When the external magnetic field strength decreases, the transition from highly elongated aggre-gates to the aggregates with ratio a / b  2 take place in the critical field H2  H1. In an ac magnetic field H0sin(t) microdrops can make as small oscillations close to sinusoidal and nonlinear large-amplitude oscil-lations [3]. At the same time, the influence of the alter-nating field on the form of microdrop and large-amplitude oscillation has been insufficiently studied.  2. DESCRIPTION OF THE OBJECT AND METHODS OF INVESTIGATION  We have studied both experimentally and using a computer simulation not previously discussed effect of the interfacial tension on the nature of the oscillations of elongated aggregates. The changes in the shape of microdrops were studied by the optical method with registration of light transmitted through sample. Using a microscope, the boundary between a microdrop and surrounding fluid was projected on a cathode of photo-multiplier through a slit diaphragm placed in parallel to the oscillation axis. The experimental set-up is de-scribed in [3].  The experimental samples were magnetic fluids of the «magnetite in kerosene» type containing micro-droplets. The interfacial tension values  differ by one order of magnitude and varies between 2·10 – 7 H/m and 3,7·10 – 6 H/m. For exciting forced oscillation, the root-mean-square value of ac magnetic field Н(t)  H0sin(t) exceeds the critical value H1. At the same time, the value H0 was experimentally chosen so that the aggre-gates are not merged into a larger ones under the in-fluence of the magnetic field, and the disperse composi-tion of the fluid during oscillation is not changed.   3. EXPERIMENT  In a weak alternating sinusoidal field Н(t)  H0sin(t) at frequency f   / 2 when 0.01 Hz  f  0,1 Hz for a period microdrops have time to take strongly elongated shape and return to equilib-rium (spherical) shape during period. Oscillations of elongated microdroplet aggregates are of a periodical  V.I. DROZDOVA, G.V. SHAGROVA, M.G. ROMANENKO J. NANO- ELECTRON. PHYS. 6, 03013 (2014)   03013-2   a b  Fig. 1 – Photographs of the oscillograms: f  0,2 Hz – a; f  0,3 Hz – b  nonsinusoidal type with period equal to period of the driving force which has sinusoidal character with fre-quency 2 f (proportional to HH).  Figure 1 shows photograph of the oscillograms, in which two curves are presented. The curves 1 corre-spond to magnetic field Н(t)  H0sin(t), the curves 2 corresponds to changes in the light flux. As the frequency increases, and the ratio Tel/Т0 in-creases reaching Tel/Т0  1. Т0 is period of oscillation and Tel is the residence time of droplets in an elongated state. Figure 2 shows the dependence of Tel/Т0 on the ac field frequency when   3,7·10 – 7 N/м, Н0  373 А/м.    Fig. 2 – The ratio Tel / Т0 versus frequency  To analyze the character of the microdrops oscillation, periodic nonsinusoidal curves that corresponded to the light flux were expanded into Fourier series. For har-monic components of the discrete spectrum, we calcu-lated coefficients ki  Ai/A, represent the ratios of root-mean-square values of harmonics to the effective value of the entire curve, which was determined without the constant component. The frequency of fundamental harmonic (k  1) of oscillations of microdroplets coin-cides with the frequency of driving force (2f). The sub-garmonic (ks) has frequency f which coincides with the frequency of external magnetic field. It was shown in [3] that contribution of subgarmonic to the effective value of the light flux signal increases with the fre-quency, and the fraction of fundamental harmonic de-creases. The distinctive property of samples which have   2·10 – 6 N/m is that it is impossible to obtain small-amplitude oscillation due to strong damping. But with strong elongations for these samples the forced oscilla-tion were obtained in an ac field which efficient value is greater than H1. Elongation of microdroplet aggregates during the forced oscillation is composed of a fixed part, which does not coincide with the equilibrium shape, and a variable part, which is caused by oscillations sharpened end. At   3,7·10 – 6 N/m and f  0,5 Hz we obtained k1  0,75 and when f  1 Hz we obtained k1  0,67. A further increase in frequency is accompa-nied by the fact that the fraction of the fundamental harmonic decreases and the fraction of the subgarmon-ic increases and character of frequency dependence of k1 and ks for samples which have   2·10 – 6 N/m corre-sponds to that obtained previously at lower values of the interfacial tension. The surprising thing is that we obtained forced os-cillation of microdroplets which have   2·10 – 6 N/m. To explain this fact, we performed numerical calcula-tions.   4. COMPUTER SIMULATION  To compare the effects of viscosity, surface and magnetic forces on microdroplets oscillation at high elongations the computer modeling were performed. For this purpose, equation (1) is represented as:   221 2 3 42( ) ( ) ( )( ( )) ( ( )) ( ( )) ( ) ( ( )) ( )d t d t d tf t f t f t t f t tdt dtdt                     where  21 2( ) 2( ( )) ;( ) 2 ( ) 1tf tt t    2/32 22160 ( )( ( )) ;2 ( ) 12tf ttR             2 2 23 22 2 23 145 ( ) ( ) 4 ( ) 1 2( ( )) arcsin 1 ;( ) ( )2 ( ) 1 ( ( ) 1) ( ) 142t t tf tt tt t tR                                 THE EFFECT OF INTERFACE TENSION ON FORCED OSCILLATIONS… J. NANO- ELECTRON. PHYS. 6, 03013 (2014)   03013-3        22 2 2042 8/3 2/3145 ( ) 1( ( ))2 ( ) ( ) 2 ( )2i e H t tf tR t t t                             3/22 2 2 2 2 123/22 2 2 2ln 2 ( ) 2 ( ) ( ) 1 1 2 1 ( ) 2 ( ) 1 ( ) 1 2 ( ).( ) / 1 ln 2 ( ) 2 ( ) ( ) 1 1 2 1 ( ) 2 ( ) 1i et t t t t t tt t t t t t                                        a b   c d  Fig. 3 – Functions of time: ratio  (t) –  a, viscous forces f2 ((t)) – b, surface forces  f3 ((t)) – c, magnetic forces f4((t))  – d  Numerical calculations performed by the Runge-Kutta methods, allow us to compare the contribution of inertia f1((t)), viscous f2((t)), surface f3((t)) and mag-netic f4((t)) forces in the process of oscillation. In the calculations the following parameters were used: inter-facial tension between microdroplet aggregate and the surrounding fluid   3,7·10 – 6 N/m, viscosity   0,068 Pa·s; permeability unit i  60; permeability of surrounding fluid e  2; density of microdroplets   1900 kg/m3 density of the surrounding fluid ρ1  1360 kg/m3, H0  440 A/m, f  1 Hz. The results are presented in Figure 3. 5. CONCLUSION  From the calculation results, we can conclude that for samples with   210 – 6 N/m the effect of viscous dissipation forces less pronounced than the surface and magnetic forces. Near the sharpened end a dominant influence on the deformation and oscillation can exert magnetic forces.  Therefore, oscillations of strongly elongated aggre-gates at H  H1 in contrast to small oscillations at H  H1 are not suppressed by surface forces.   REFERENCES  1. J.C. Bacri, D. Salin, J. Physique-Lett. 43, L649 (1982). 2. E.Ya. Blum, M.M. Maiorov, A.O. Tsebers, Magnitnye zhidkos-ti (Magnetic Fluids) (Zinatne: Riga: 1989) [in Russian]. 3. V.I. Drozdova, V.V. Kushnarev, G.V. Shagrova, Colloid J. 68, 142 (2006).  

The Effect of Interface Tension on Forced Oscillations of Elongated Microdroplet Aggregates in Magnetic Fluids

SocioEconomic Challenges Journal (Sumy State University)

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ 
Vol. 6 No 3, 03013(3pp) (2014) Том 6 № 3, 03013(3cc) (2014) 
 
 
The article was reported at the International Conference «The Advanced Technology, Equipment and Analytical Systems for Materials», Kursk, 13-14 May, 2014 
2077-6772/2014/6(3)03013(3) 03013-1  2014 Sumy State University 
The Effect of Interface Tension on Forced Oscillations of Elongated Microdroplet  
Aggregates in Magnetic Fluids 
 
V.I. Drozdova, G.V. Shagrova, M.G. Romanenko 
 
North-Caucasian Federal University, 2, Kulakova Pr., 355009 Stavropol, Russia  
 
(Received 19 May 2014; revised manuscript received 03 July 2014; published online 15 July 2014) 
 
Changes in the shape of microdroplet aggregates exposed to an alternating magnetic field with fre-
quency from 0.01 to 1 Hz are investigated both experimentally and using a computer simulation. Micro-
droplet aggregates are spherical droplets of a highly concentrated magnetic fluid with a radius from one to 
several tens of microns dispersed in a liquid with a low concentration of magnetic particles. For values of 
interfacial tension  > 2·10 – 6 N/m small oscillations in a weak alternating field could not be obtained, but 
periodic forced oscillations were obtained at strong elongations. Oscillations of elongated microdroplet ag-
gregates are of a periodical nonsinusoidal type with higher harmonics and subharmonics in the spectrum.  
 
Keywords: Magnetic fluid, Forced oscillations, Interfacial tension, Microdroplet aggregate. 
 
 PACS numbers: 75.50.Mm, 79.60.Jv 
 
 
1. INTRODUCTION 
 
Application and research of liquid magnetizable 
nanodisperse systems known as magnetic fluid (MF), 
started over 50 years ago. MF are the stabilized colloids 
of magnetic particles of size ~ 10 nanometers in differ-
ent non-magnetic host liquids. The combination of flu-
idity with the possibility to be magnetized in weak 
magnetic fields is still attracting the attention of engi-
neers and researchers. Complex heterogeneous systems 
that are based on the MF, in particular, microemul-
sions and MF containing microdroplets are promising 
new materials for defectoscopy. The properties of 
boundary between heterogeneous magnetic phases are 
unique in having both low values of interfacial tension 
values  ~ 10 – 7 H/m and high values (for liquid artificial 
media) of permeability units. In the absence of external 
field, microdroplet aggregates have a spherical shape; in a 
weak field, they can be strongly deformed assuming the 
shape of needles oriented along the field. The static de-
formation of suspended MF ” droplets and microdroplet 
aggregates is well studied experimentally and theoretical-
ly [1, 2]. Deformation of microdrops under weak magnetic 
field corresponded closely to the model that uses assump-
tions about the linear nature of the magnetization and 
ellipsoidal shape of the aggregates. When using energy 
approach equilibrium forms of microdrops can be obtained 
from the Lagrange equations 
 
 

   
   
    
   
m Ed T T E E
dT
,  
 
where T – total energy, E – surface energy, Em – mag-
netic energy, E  – viscous dissipation rate,  – the ra-
tio of the major and minor axes of the ellipsoid. From 
the equation (1) it follows that deformation of aggre-
gates in magnetic field has a hysteresis nature. 
When the magnetic field strength is lower than crit-
ical value H1 the microdrop takes the shape of prolate 
ellipsoid of revolution oriented along the field with the 
axes ratio a / b  2. At H  H1 a jumpwise elongation of 
a microdrop occurs and it acquires the form of a body of 
revolution with tapered ends and the axis ratio 
a / b  10. When the external magnetic field strength 
decreases, the transition from highly elongated aggre-
gates to the aggregates with ratio a / b  2 take place in 
the critical field H2  H1. In an ac magnetic field 
H0sin(t) microdrops can make as small oscillations 
close to sinusoidal and nonlinear large-amplitude oscil-
lations [3]. At the same time, the influence of the alter-
nating field on the form of microdrop and large-
amplitude oscillation has been insufficiently studied. 
 
2. DESCRIPTION OF THE OBJECT AND 
METHODS OF INVESTIGATION 
 
We have studied both experimentally and using a 
computer simulation not previously discussed effect of 
the interfacial tension on the nature of the oscillations 
of elongated aggregates. The changes in the shape of 
microdrops were studied by the optical method with 
registration of light transmitted through sample. Using 
a microscope, the boundary between a microdrop and 
surrounding fluid was projected on a cathode of photo-
multiplier through a slit diaphragm placed in parallel 
to the oscillation axis. The experimental set-up is de-
scribed in [3].  
The experimental samples were magnetic fluids of 
the «magnetite in kerosene» type containing micro-
droplets. The interfacial tension values  differ by one 
order of magnitude and varies between 2·10 – 7 H/m and 
3,7·10 – 6 H/m. For exciting forced oscillation, the root-
mean-square value of ac magnetic field Н(t)  H0sin(t) 
exceeds the critical value H1. At the same time, the 
value H0 was experimentally chosen so that the aggre-
gates are not merged into a larger ones under the in-
fluence of the magnetic field, and the disperse composi-
tion of the fluid during oscillation is not changed.  
 
3. EXPERIMENT 
 
In a weak alternating sinusoidal field 
Н(t)  H0sin(t) at frequency f   / 2 when 
0.01 Hz  f  0,1 Hz for a period microdrops have time 
to take strongly elongated shape and return to equilib-
rium (spherical) shape during period. Oscillations of 
elongated microdroplet aggregates are of a periodical 
 V.I. DROZDOVA, G.V. SHAGROVA, M.G. ROMANENKO J. NANO- ELECTRON. PHYS. 6, 03013 (2014) 
 
 
03013-2 
  
a b 
 
Fig. 1 – Photographs of the oscillograms: f  0,2 Hz – a; f  0,3 Hz – b 
 
nonsinusoidal type with period equal to period of the 
driving force which has sinusoidal character with fre-
quency 2 f (proportional to HH).  
Figure 1 shows photograph of the oscillograms, in 
which two curves are presented. The curves 1 corre-
spond to magnetic field Н(t)  H0sin(t), the curves 2 
corresponds to changes in the light flux. 
As the frequency increases, and the ratio Tel/Т0 in-
creases reaching Tel/Т0  1. Т0 is period of oscillation 
and Tel is the residence time of droplets in an elongated 
state. Figure 2 shows the dependence of Tel/Т0 on the ac 
field frequency when   3,7·10 – 7 N/м, Н0  373 А/м. 
 
 
 
Fig. 2 – The ratio Tel / Т0 versus frequency 
 
To analyze the character of the microdrops oscillation, 
periodic nonsinusoidal curves that corresponded to the 
light flux were expanded into Fourier series. For har-
monic components of the discrete spectrum, we calcu-
lated coefficients ki  Ai/A, represent the ratios of root-
mean-square values of harmonics to the effective value 
of the entire curve, which was determined without the 
constant component. The frequency of fundamental 
harmonic (k  1) of oscillations of microdroplets coin-
cides with the frequency of driving force (2f). The sub-
garmonic (ks) has frequency f which coincides with the 
frequency of external magnetic field. It was shown in 
[3] that contribution of subgarmonic to the effective 
value of the light flux signal increases with the fre-
quency, and the fraction of fundamental harmonic de-
creases. 
The distinctive property of samples which have 
  2·10 – 6 N/m is that it is impossible to obtain small-
amplitude oscillation due to strong damping. But with 
strong elongations for these samples the forced oscilla-
tion were obtained in an ac field which efficient value is 
greater than H1. Elongation of microdroplet aggregates 
during the forced oscillation is composed of a fixed part, 
which does not coincide with the equilibrium shape, 
and a variable part, which is caused by oscillations 
sharpened end. At   3,7·10 – 6 N/m and f  0,5 Hz we 
obtained k1  0,75 and when f  1 Hz we obtained 
k1  0,67. A further increase in frequency is accompa-
nied by the fact that the fraction of the fundamental 
harmonic decreases and the fraction of the subgarmon-
ic increases and character of frequency dependence of 
k1 and ks for samples which have   2·10 – 6 N/m corre-
sponds to that obtained previously at lower values of 
the interfacial tension. 
The surprising thing is that we obtained forced os-
cillation of microdroplets which have   2·10 – 6 N/m. 
To explain this fact, we performed numerical calcula-
tions.  
 
4. COMPUTER SIMULATION 
 
To compare the effects of viscosity, surface and 
magnetic forces on microdroplets oscillation at high 
elongations the computer modeling were performed. 
For this purpose, equation (1) is represented as: 
 
 
22
1 2 3 42
( ) ( ) ( )
( ( )) ( ( )) ( ( )) ( ) ( ( )) ( )
d t d t d t
f t f t f t t f t t
dt dtdt
  
     
 
        
 
  
 
 where 
 
2
1 2
( ) 2
( ( )) ;
( ) 2 ( ) 1
t
f t
t t


 


   
2/3
2 2
21
60 ( )
( ( )) ;
2 ( ) 1
2
t
f t
tR
 

 

 
    
 
  
 
 
 
2 2 2
3 22 2 2
3 1
45 ( ) ( ) 4 ( ) 1 2
( ( )) arcsin 1 ;
( ) ( )2 ( ) 1 ( ( ) 1) ( ) 14
2
t t t
f t
t tt t tR
   

    
          
            
 
  
 THE EFFECT OF INTERFACE TENSION ON FORCED OSCILLATIONS… J. NANO- ELECTRON. PHYS. 6, 03013 (2014) 
 
 
03013-3 
 
     
 
22 2 2
0
4
2 8/3 2/31
45 ( ) 1
( ( ))
2 ( ) ( ) 2 ( )
2
i e H t t
f t
R t t t
   


    
 
 
 
   
 
  
 
      
      
3/2
2 2 2 2 2 1
2
3/2
2 2 2 2
ln 2 ( ) 2 ( ) ( ) 1 1 2 1 ( ) 2 ( ) 1 ( ) 1 2 ( )
.
( ) / 1 ln 2 ( ) 2 ( ) ( ) 1 1 2 1 ( ) 2 ( ) 1i e
t t t t t t t
t t t t t t
      
       

 

         
  
         
 
  
 
 
 
a b 
 
 
c d 
 
Fig. 3 – Functions of time: ratio  (t) –  a, viscous forces f2 ((t)) – b, surface forces  f3 ((t)) – c, magnetic forces f4((t))  – d 
 
Numerical calculations performed by the Runge-
Kutta methods, allow us to compare the contribution of 
inertia f1((t)), viscous f2((t)), surface f3((t)) and mag-
netic f4((t)) forces in the process of oscillation. In the 
calculations the following parameters were used: inter-
facial tension between microdroplet aggregate and the 
surrounding fluid   3,7·10 – 6 N/m, viscosity 
  0,068 Pa·s; permeability unit i  60; permeability 
of surrounding fluid e  2; density of microdroplets 
  1900 kg/m3 density of the surrounding fluid 
ρ1  1360 kg/m3, H0  440 A/m, f  1 Hz. The results are 
presented in Figure 3. 
5. CONCLUSION 
 
From the calculation results, we can conclude that 
for samples with   210 – 6 N/m the effect of viscous 
dissipation forces less pronounced than the surface and 
magnetic forces. Near the sharpened end a dominant 
influence on the deformation and oscillation can exert 
magnetic forces.  
Therefore, oscillations of strongly elongated aggre-
gates at H  H1 in contrast to small oscillations at 
H  H1 are not suppressed by surface forces. 
 
 
REFERENCES 
 
1. J.C. Bacri, D. Salin, J. Physique-Lett. 43, L649 (1982). 
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https://essuir.sumdu.edu.ua/bitstream/123456789/36118/1/The%20Effect%20of%20Interface%20Tension%20on%20Forced%20Oscillations%20of%20Elongated%20Microdroplet.pdf

The Effect of Interface Tension on Forced Oscillations of Elongated Microdroplet Aggregates in Magnetic Fluids

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