Sma Ma e ials and S uc u es
Sma Ma e . S uc . 32 (2023) 065016 (11pp) h ps://doi.o g/10.1088/1361-665X/acd0e5
T ansien dynamics o he ield
induced o ce in he iso opic
magne o heological elas ome
M Kubík1,2, D Bo in1,∗and S Odenbach1
1Ins i u e o Mecha onic Enginee ing, Chai o Magne o luiddynamics, Measu ing and Au oma ion
Technology, Technische Uni e si ä D esden, 01062 D esden, Ge many
2Facul y o Mechanical Enginee ing, B no Uni e si y o Technology, B no, Czech Republic
E-mail: dmi y[email p o ec ed]
Recei ed 15 Feb ua y 2023, e ised 24 Ma ch 2023
Accep ed o publica ion 27 Ap il 2023
Published 11 May 2023
Abs ac
The ansi ion dynamics in silicon ubbe based iso opic magne o heological (MR) elas ome s
in e ms o he no mal o ce induced by an ex e nal homogeneous magne ic ield is
expe imen ally add essed. The p ima y goal was o e alua e dynamic pe o mances o he MR
elas ic iso opic composi e using a anspa en ly p esen ed measu ing sys em wi h known
cha ac e is ics in con as o ew p e ious s udies on he opic. I was ound ha an inc ease in
he magne ic ield leads o an inc ease in he induced o ce and a dec ease in he esponse ime
o he MR elas ome . A he same ime, bo h he use o coa se pa icles as magne ic ille and a
signi ican educ ion in he s i ness o he polyme ma ix educe he esponse ime o he MR
elas ome unde s udy. The analysis ca ied ou akes in o accoun he dynamics o he
elec omagne ic coil and he eddy cu en s induced in he magne ci cui . The sho es esponse
imes ob ained o a ious MR elas ome samples a e in he ange o 27–72 ms o he maximal
used magne ic ield wi h an induc ion o 230 mT. These imes co espond o he as es
p e iously epo ed ones o MR elas ome s and MR elas ome based sys ems. In addi ion, he
ob ained esul s indica e he p esence o di e en mechanisms esponsible o he measu ed
magne ode o ma ional e ec obse ed in MR elas ome s.
Supplemen a y ma e ial o his a icle is a ailable online
Keywo ds: magne o-ac i e ma e ials, MR elas ome s, magne ode o ma ion, ansien ,
esponse ime
(Some igu es may appea in colou only in he online jou nal)
∗Au ho o whom any co espondence should be add essed.
O iginal con en om his wo k may be used unde he e ms
o he C ea i e Commons A ibu ion 4.0 licence. Any u -
he dis ibu ion o his wo k mus main ain a ibu ion o he au ho (s) and he
i le o he wo k, jou nal ci a ion and DOI.
1361-665X/23/065016+11$33.00 P in ed in he UK 1 © 2023 The Au ho (s). Published by IOP Publishing L d
Sma Ma e . S uc . 32 (2023) 065016 M Kubík e al
1. In oduc ion
Magne o heological (MR) elas ome s ep esen an in ensely
explo ed class o con olled sma ma e ials [1]. These com-
posi es a e based on a so elas ic ma ix wi h embedded mag-
ne ic mic opa icles. The applica ion o an ex e nal magne ic
ield makes i possible o e e sibly con ol he a ious p ope -
ies o he MR elas ome . The e a e a g ea numbe o s udies
examining hese composi es om a ious pe spec i es [2]. In
addi ion o magne ic ield-dependen iscoelas ici y, one e y
a ac i e ea u e o MR elas ome is he magne ode o ma ion
e ec [3]. The con ollable change in dimensions o MR elas -
ome allows i o be used as an ac ua o , e.g. o al es e c
[4,5]. I is ob ious ha apa om quan i a i e and quali a i e
assessmen o he changes in geome ic dimensions as well as
a ious mac oscopic physical p ope ies o he MR elas ome
unde he ac ion o a applied magne ic ield, he esponse ime
o he composi e is ex emely impo an o p ac ical applica-
ions. Ne e heless, cu en ly, he in o ma ion abou he an-
sien esponse o MR elas ome on an applied ield is a he
limi ed and ambiguous. Gene ally, ib a ion isola ion de ices
and heome e s a e used o e alua e he esponse ime o MR
elas ome specimens o MR elas ome -based sys ems. In [6]
a esponse ime o he bushing wi h an in eg a ed na u al ub-
be based MR elas ome is measu ed. As demons a ed, he
measu ed o ce ises exponen ially wi h a cha ac e is ic ime
( esponse ime) o abou 9.5 ms when he cu en is applied o
he elec omagne ic coil o he de ice. I is also s a ed ha a
small pa o he o ce g ows and elaxes a longe ime scales.
This cha ac e is ic ob ained in [6] is a speci ic alue o he
de ice conside ed by he au ho s unde speci ic condi ions and
is no di ec ly ela ed solely o he physical pa ame e s o he
MR composi e. In [7] a esponse ime o a mul i-laye ed MR
elas ome base isola o has been in es iga ed. Using a ious
app oaches he au ho s ob ained he as es o ce ise ime o
he s udied sys em o 52 ms. In [8] a MR elas ome based ac u-
a o was analyzed and a esponse ime o 350 ms was ob ained
o he o ce measu emen s. Again, in bo h las men ioned
cases [7,8], a complex sys ems was e alua ed, no solely a spe-
cimen o MR elas ome o a ce ain composi ion. Thus, se e al
independen s udies gi e a ange in he esponse ime o MR
elas ome -based sys ems o se e al o de s o magni ude. O he
esea che s measu ed he ansien esponse o a MR elas ome
using a heome e equipped wi h a magne ic cell. In [9] he
au ho s e alua ed he ansien esponse using dynamic o sion
a ixed exci a ion equency and assume a s ep change o mag-
ne ic ield exci a ion. The pa ame e es ima ed by he au ho s
is he s o age modulus and is p opo ional o he measu ed
o que. I is epo ed ha he MR elas ome has a leas h ee
cha ac e is ic ime cons an s, while he as es one is unde
10 s. A simila me hod o assessing he ansien dynamics in
elas ome s has been used in [10,11]. In [10] he au ho s p o-
posed a comp ehensi e model based on magne ic dipola and
iscoelas ic pa ame e s o an aniso opic MR elas ome . The
epo ed sho es i s -o de ime cons an is longe han 3 s. In
[11] he au ho s ied o imp o e he ansien esponse o MR
elas ome s by using he pa icles syn hesized in a special p o-
cedu e (CIP@FeNi) and epo ed he sho es i s -o de ime
cons an o 0.68 s. I should be no ed, howe e , ha using a
pla e-pla e con igu a ion in he con ex o heome e s udies
o solid ma e ials in ol es a numbe o c i ical issues and he
absolu e alues o he physical quan i ies ob ained in his way
may no be co ec [12–14]. In pa icula , he e is a p oblem o
adhesion o he sample o he measu ing geome y and impe -
ec ans e o s ain om he mo ing pla e o he sample and
hence impe ec ans e o o que om he sample o he pla e.
In an a emp o esol e his issue and a oid global o local
adhesi e ailu e, he esea che s load he specimen wi h a sig-
ni ican no mal s ess (no mal o ce up o se e al ens o N).
This no mal s ess can signi ican ly comp ess he sample and
can be much g ea e han he shea s ess, esul ing in inco ec
measu emen s. Fu he mo e, he shea modulus is no a pa a-
me e di ec ly quan i iable by a heome e . In ac , i is a ma e
o applying a sinusoidal s ain and measu ing he co espond-
ing shea s ess in oscilla o y dynamics. The ou pu modulus
alues a e he esul o p ocessing he dynamic da a by he
co esponding heome e i mwa e and so wa e. An analysis
o he aw da a would be necessa y o a co ec assessmen
when using his me hod. On he whole, he me hod emains
applicable only o small de o ma ions, o which he beha-
io o he ma e ial is linea . The e is no possibili y o ake
a non-linea i y in o accoun , due o a non-uni o m s ain dis-
ibu ion in he pla e-pla e con igu a ion. Thus, obse ed and
epo ed wi hin his con igu a ion non-linea e ec s emain in
doub s. In [15] he ansien esponse o a MR elas ome is sys-
ema ically es ed using a heome e unde comp ession mode.
The au ho s used he Maxwell model o he e alua ion o he
ime cons an s. I is concluded ha an aniso opic MR elas -
ome specimen has a be e ansien esponse han an iso opic
one, and he MR elas ome sample unde small comp essi e
s ain is as e han unde a highe one. The sho es epo -
ed esponse ime o he aniso opic sample is 12 ms. No e-
wo hy is he end o he no mal o ce- ime cu es demon-
s a ing o e shoo a all elec ical cu en s, i.e. magne ic ields,
used by he au ho s. No e idence is gi en ha his is a ea u e
o he ma e ial and no o he used heome e sys em. Consid-
e ing he abo e, he sca e ing o cha ac e is ic imes o MR
elas ome s epo ed in he publica ions o a ious au ho s is
no su p ising.
The objec i e o his s udy is o e alua e he ansi ion
dynamics in MR elas ome s in e ms o he no mal o ce
induced by an ex e nal homogeneous magne ic ield and using
a anspa en ly p esen ed measu ing sys em wi h known cha -
ac e is ics. The no mal o ce e e s o he magne ode o ma-
ion e ec in MR elas ome s. The e ec is ha , unde he
in luence o an ex e nal magne ic ield, he MR elas ome
changes i s o iginal shape [16,17]. Acco ding o known
s udies o highly illed MR elas ome specimens, he mag-
ne ode o ma ion e ec co esponds o hei elonga ion in he
di ec ion o he applied magne ic ield [18–20]. F om he he-
o e ical poin o iew he e a e wo main con ibu ions o he
magne ode o ma ion o he MR elas ome s and hey ope a e a
2
Sma Ma e . S uc . 32 (2023) 065016 M Kubík e al
di e en scales [3,21,22]. The i s one is ela ed o he gen-
e al mac oscopic p ope ies o he composi e, i.e. i s shape and
ne magne iza ion, which de e mine demagne izing ield c e-
a ing he su ace magne ic p essu e jump and some in e nal
o ces in he sample. The second one is mic os uc u e ela ed
and c ucially depends on he local spa ial dis ibu ion o he
magne ic pa icles in he composi e ma ix. Pe o med in [3]
3D modeling has shown ha a low ields he magne ode o m-
a ion is de e mined by pa icles s uc u ing and he mac o-
scopic and mic oscopic con ibu ions wo k agains each o he .
Consequen ly, he de o ma ion o he ma e ial is qui e low. As
pa icle s uc u es eme ge, he p ocesses o o a ion o s uc-
u es in he di ec ion o he ield and o ien a ion along he
ield become dominan . In his case bo h con ibu ions wo k
oge he and he de o ma ion is signi ican . In [23], using a
mic oscopically mo i a ed con inuum app oach, he au ho s
also claim pa icle mo ion in an elas ic ma ix as he main
cause o he magne ode o ma ion e ec in MR elas ome s.
The magne ode o ma ion e ec is impo an in poin o iew
o p ac ical ma e ial applica ions, he e o e, u he mo i a ion
and he need o in es iga e he co esponding dynamic cha ac-
e is ics is ob ious.
This wo k is o ganized as ollows: i s , we p esen he used
es ig (sec ion 2.1) and a de ailed desc ip ion o he me hodo-
logical aspec s o de e mining he ansien ime (sec ion 2.2).
This is ollowed by a summa y p esen a ion o he elas -
ome MR samples used in he measu emen s (sec ion 2.3).
The ob ained esul s a e p esen ed and discussed in sec ion 3,
including he no mal o ce da a and he ansien esponse o
he MR elas ome s. Addi ionally, an issue o he ini ial speci-
men p e ension is add essed. A summa y is gi en in sec ion 4.
2. Ma e ials and me hods
2.1. MR elas ome samples
Th ee ypes o MR elas ome samples we e ab ica ed o he
s udy in his wo k. The wo-componen silicone ubbe Elas -
osil RT623 (Wacke Chemie AG, Ge many) was used as ma -
ix. The liquid polyme was dilu ed wi h silicone oil M1000
(GE Baye Silicones, Ge many) o educe he s i ness o he
ma ix, as he add essed e ec is a ea u e o su icien ly so
composi es [17]. Two ypes o i on powde we e used as mag-
ne ic ille . The p ima y ille was CC g ade ca bonyl i on
powde (BASF, Ge many) con aining sphe ical mic opa icles
wi h an a e age size o ∼3–5 µm. Addi ionally, an i on ille
wi h coa se pa icles (∼80–100 µm) ob ained by ac iona-
ion o a highly polydispe se powde o pa icles wi h i egula
shape on a sie e (Ha e &Boecke , Ge many) was used. Be o e
mixing he ille wi h he ini ially liquid ma ix componen s,
he magne ic pa icles we e ea ed wi h silicone oil o be e
compa ibili y wi h he ma ix. Speci ically, he mic opa icle
powde was mixed wi h a solu ion o silicone oil in hexane,
passed h ough a dispe san , and d ied. The o al ille concen-
a ion o all ab ica ed specimens was 80 w .% which co es-
ponds o abou 35 ol.%. A e mixing, he composi ion was
acuum degassed o emo e ai bubbles, pou ed in o molds
and polyme ized o a leas 2 h a 70 ◦C. As a esul , disc-
like MR elas ome specimens wi h a diame e o 14 mm and
a heigh o 4 mm we e ob ained. In o ma ion on he magne ic
powde con en in he samples as well as on he elas ic modu-
lus o he polyme ma ix is p o ided in able 1. Samples 1 and
2 ha e he same ille , bu sample 2 has a signi ican ly so e
ma ix, e ec i ely being gel-like. The ma ix o sample 3 co -
esponds o sample 1, bu he magne ic ille is a mix u e o
he wo ypes o powde s men ioned abo e.
Thus, sample 1 co esponds o a ypical MR elas ome
based on so silicone ubbe [17], sample 2 can be desc ibed as
a so magne ic gel, and sample 3 has an inc eased ini ial mag-
ne ic suscep ibili y due o he p esence o a signi ican ac ion
o coa se pa icles [24,25].
2.2. Tes ig and me hodology measu emen
The expe imen al es ig is schema ically p esen ed in igu e 1
and is composed o a magne ic ci cui made o low-ca bon
s eel (g ey), an elec omagne ic coil (o ange), a non-magne ic
sample holde (g een) consis ing o a s a ically ixed i an pla e
(lowe pa ) and a sha (uppe pa ) made o he moplas ic
polyes e , and he MR elas ome sample (yellow). The ai gap
be ween he i an pla e and he uppe pa o he magne ic ci -
cui is 9 mm. The diame e o he pa o magne ic ci cui unde
he pla e is 20 mm on op. The sha o he sample holde
is connec ed o a o ce senso KM 1403 K 50 N (Mega on
Elek onik Gmbh, Ge many) ha is ixed o he ame. A sc ew
on he sample holde se s he ini ial sample p e ension. The di -
ec ion o he homogeneous ex e nal magne ic ield Band he
o ce F o be measu ed a e coaxial. As men ioned abo e, he
applica ion o an ex e nal homogeneous magne ic ield o he
MR elas ome causes i o elonga e in he di ec ion o he ield.
In ou es se up, he specimen is ixed be ween wo pa s o he
sample holde and he con olled pa ame e co esponds o he
elonga ion o ce Finduced by he specimen unde he ac ion
o he applied ield B. A minimum ini ial o ce o 0.1 ±0.01 N
is applied o he specimen when i is ixed. The magne ic lux
densi y B, which exci es he sample, is measu ed by a Hall
p obe connec ed o a Gaussme e (Lake Sho e 460, Lake Sho e
C yo onics, Inc., USA). A as cu en con olle is used o
he exci a ion o he elec omagne ic coil o imp o e he an-
sien esponse. A labo a o y powe supply Gen300-17 (TDK
Lambda Co p., Japan) powe ed he cu en con olle . A sig-
nal gene a o HMF2550 (Rohde & Schwa z GmbH & Co KG,
Ge many) gene a es he ol age signal Uas inpu o he cu -
en con olle . A cu en clamp Fluke i30 (Fluke Co p., USA)
moni o s he elec ic cu en I, which exci es he elec omag-
ne ic coil.
All ou signals (U, I, B, and F) a e acqui ed o he DAQ
sys em Dewe-50 (Dewe on GmbH, Aus ia) wi h a sampling
equency o 25 kHz. Each measu emen is composed o six
con ol signal s ep ises wi h a equency o 0.2 Hz, see
igu e 2. Fi e ise amps a e used o he subsequen e al-
ua ion (numbe 2–6). The du a ion o he e alua ed sec ion
is 2.5 s. A ac o limi ing he du a ion o he measu emen
is he g adual hea ing up o he sys em, due o elec ical
3
Sma Ma e . S uc . 32 (2023) 065016 M Kubík e al
Table 1. Composi ion o he s udied MR elas ome samples.
Sample ID
To al ille
concen a ion
BASF CC pa icles
(∼3–5 µm)
La ge pa icles
(∼80–100 µm)
Ra io o he silicon
ubbe o silicon oil
Elas ic modulus
o he ma ix
Sample 1 ∼35 ol.% 100% 0% 1:1 ∼30 kPa
Sample 2 ∼35 ol.% 100% 0% 1:1.5 ∼3 kPa
Sample 3 ∼35 ol.% 35% 65% 1:1 ∼30 kPa
Figu e 1. Expe imen al es ig: sys em block diag am (le ) and schema ic ep esen a ion o he MR elas ome specimen (MRE) unde
expe imen al condi ions ( igh ).
Figu e 2. An example o a measu emen o a 1 A (∼96 mT) exci a ion o sample 2 o demons a e he expe imen al p ocedu e.
cu en applied o he coil. Wi hin he amewo k o he
me hodology used, a hea ing o he sample unde s udy was
a oided.
The s anda d de ia ion σwas de e mined om he selec-
ed measu ed o e alua ed signals ( o ce and ime cons an s).
The s anda d de ia ion σwas ob ained using i e consecu i e
measu emen s (exci a ions) on one sample. I should be no ed
ha he p esen ed da a a e om one MR elas ome specimen
o each ma e ial.
2.3. Me hodology e alua ion o MR elas ome esponse ime
Se e al de ini ions o ansien esponse ( esponse ime, ime
cons an , e c) o dynamic sys ems such as dampe s, clu ches,
o elas ome isola o s exis . The ansien esponse o hese
de ices is usually assumed o be a i s -o de dynamic sys em.
Acco ding o a ailable in o ma ion, MR elas ome s ha e mo e
ime cons an s [6,9]. Howe e , we ocused on he sho es ime
cons an and simpli ied he beha io o he MR elas ome as
a i s -o de sys em. In he case o a s ep change o he inpu ,
he esponse ime τ63 ( ime cons an ) is o be de e mined as
he ime o he inc ease o 63.2% o he s eady-s a e alue,
see igu e 3. Time cons an τ63 ep esen s he ime i would
ake o he esponse o each a s eady s a e while main aining
he ini ial a e o change ( angen o he ansien cha ac e is ic
o =0: g een line). On he o he hand, he esponse ime o
86.5% (2τ63), 90% (2.3τ63), 95% (3τ63) o 98.2% (4τ63) o
he s eady-s a e alue a e also commonly used in he ansien
analysis. In he cu en s udy, we u ilize he c i e ium o 63.2%
(τ63) and 90% (τ90 =2.3τ63).
4
Sma Ma e . S uc . 32 (2023) 065016 M Kubík e al
Figu e 3. Response o a i s -o de sys em on a s ep inpu wi h he ele an ime cons an s.
The expe imen al se up desc ibe abo e con ains se e al
ime cons an s ha a e due o (i) he dynamics o he elec o-
magne ic coil τ1, (ii) he eddy cu en s induced in he mag-
ne ic ci cui τ2, and (iii) he MR elas ome i sel τ. The main
goal o ou expe imen s was o ob ain he ime cons an o he
MR elas ome , i.e. τ. The elec omagne ic coil induc ance c e-
a es a ime delay be ween ol age and elec ic cu en . F om
an elec ical poin o iew, he used es ig can be simpli ied
as induc ance Lconnec ed in se ies wi h a esis ance R. The e-
o e, he con ol cu en canno make a s ep change since he
elec omagne ic induc ance Lneeds ime o s o e and elease
ene gy. The di e en ial equa ion o an elec ic con ol ci cui
acco ding o Ki chho ’s laws can be as ollows:
Ldi( )
d +Ri( ) = U( ),
whe e is ime, and Uis ol age. In he case o ol age con-
ol, he ime de elopmen o he elec ic cu en i( ) a e he
ol age is swi ched on can be exp essed as:
i( ) = U
R(1−e−
τ1).
In [7] is s a ed ha he majo i y o he esponse ime o MR
elas ome isola o de ices is consumed du ing he con e sion
om he con ol ol age command o he elec ic cu en . In
his case [7], i was 73% o he o e all esponse ime o he
de ice. The e o e, we used a as cu en con olle wi h an
o e - ol age me hod o imp o e he dynamics o he elec o-
magne ic coil. Mo e de ailed in o ma ion abou his me hod is
o be ound elsewhe e [26,27]. We used a maximum ol age
o 120 V. In ou case, he longes ime cons an (wo s case)
o he elec omagne ic coil was τ1=4 ms; see igu e 4(a). The
elec ic cu en luc ua ion be ween app oxima ely 50 ms and
80 ms is due o he homemade cu en egula o p inciple unc-
ion. The esponse ime was sligh ly sho e o a lowe inal
alue o he elec ic cu en .
In ou expe imen s, a ime lag be ween he end o he elec-
ic cu en Iand he end o he magne ic lux densi y Bcan be
ecognized, which is caused by he eddy cu en induced in he
magne ic ci cui , see igu e 4(b). Eddy cu en s low pe pen-
dicula o he changing magne ic lux in he magne ic ci cui .
They induce a magne ic ield ha ends o oppose he changes
in he ield acco ding o Lenz’s law which c ea es a ime lag
(τ2). In ou case, i can be assumed he e is a s ep change
(exci a ion) in elec ic cu en . The e o e, he ime cons an o
he magne ic ci cui , which is τ2=105 ms, can be de e mined
(see igu e 4(b)). This alue can be conside ed as independ-
en o he elec ic cu en magni ude and om he ype o MR
elas ome sample. The ocus o he cu en s udy is he MR
elas ome ime cons an τ, which is gi en by he ime delay
be ween he magne ic ield and he no mal o ce, see igu e 5.
The exci a ion o he MR elas ome by he magne ic ield
canno be conside ed as s epwise. The e o e i is necessa y o
ind he ans e unc ion be ween he magne ic ield and he
induced no mal o ce. The ans e unc ion is conside ed as a
i s -o de sys em in he o m:
G(s) = Kp
1+Tp.s,
whe e Kpis p opo ional gain and Tpis ime cons an
(Tp=τ63). The Ma lab sys em iden i ica ion oolbox is used
o he iden i ica ion o he cons an s Kpand Tp. I is neces-
sa y o no e ha he assump ion o a linea dynamic sys em
is a simpli ica ion. The esponse imes lis ed in he esul s a e
de e mined as s ep exci a ion o he ans e unc ion.
3. Resul s and discussion
Fi s we conside he e ec o he applied magne ic ield on
he maximum no mal o ce induced by he in es iga ed MR
elas ome specimens, i.e. he alues co esponding o ∼100%
5

Sma Ma e . S uc . 32 (2023) 065016 M Kubík e al
Figu e 4. The empo al de elopmen o elec ic cu en o a s ep ol age exci a ion (a) and he empo al de elopmen o magne ic lux
densi y (b) o e alua e ime cons an s τ1and τ2and o show a co ec ed simpli ica ion o elec ic cu en s ep exci a ion.
o he s eady s a e (a >1 s). Figu e 6shows he depend-
ence o he measu ed s a iona y no mal o ce Fs a on he mag-
ne ic lux densi y B. I should be no ed ha he alues o he
s anda d de ia ion σa e oo small o be shown in he g aphs.
The Fs a (B) dependence is non-linea o all es ed samples.
The highes o ce exhibi s he MR elas ome sample 3; he
smalles one is ound o sample 2 o he highes ield used.
A he same ime, in a ield up o ∼150 mT, he o ce Fs a
induced by he sample 2 is sligh ly highe han ha ha o
sample 1. The maximum e ec obse ed o sample 3 is o
be a ibu ed o he inc eased ini ial magne ic suscep ibili y
o he la ge ille pa icles compa ed o he suscep ibili y o
he o he ille s. Simila esul s on he g ea e no mal s ess
induced by MR composi es wi h g ea e magne ic suscep ibil-
i y we e p e iously epo ed in [25,28]. The dependence Fs a
(B) ob ained o sample 1 can be i qui e accep ably using a
quad a ic unc ion Fs a ∼B2 o he whole ange o he ield B
used in he expe imen , while o samples 2 and 3 a su icien ly
good co ela ion can be ound only up o ∼170 mT. The quad-
a ic dependence o Fs a (B) is consis en wi h he quad a ic
6
Sma Ma e . S uc . 32 (2023) 065016 M Kubík e al
Figu e 5. An example o he esponse o he magne ic lux densi y B and he o ce F on a s ep exci a ion in elec ic cu en o show a ime
delay ela ed o MR elas ome .
Figu e 6. Dependence o he measu ed s a iona y no mal o ce Fs a on he magne ic lux densi y B (poin s). The alues o he s anda d
de ia ion σa e oo small o be shown in he g aphs, he e o e he maximal s anda d de ia ion σmax ob ained is addi ionally p o ided. Do ed
and solid lines co espond o he ma hema ical i ing o he da a using a Fs a ∼B2 unc ion, wi h he B ange o 0–170 mT aken in o
accoun o samples 2 and 3, while he ull ange o he used B- alues was aken in o accoun o sample 1.
dependence o he magne ic ee ene gy on he applied ield,
o which he componen o he p opo ionali y ac o is he
magne ic suscep ibili y in acco dance o elec odynamics o
con inuous media [29]. I is hus con i med ha he depend-
encies ob ained o a leas a limi ed ange o magne ic ields
can be explained conside ing he ini ial magne ic suscep ibil-
i y, which is expec ed o be equal o samples 1 and 2 and is
highe o sample 3.
F om a ce ain ield (∼170 mT) he dependencies o
samples 2 and 3 do no co espond o he unc ion Fs a ∼B2,
e en hough he magne iza ion cu e o simila ly illed
composi es is s ill linea in his ange, i.e. he magne ic
suscep ibili y is cons an , see e.g. [16,28]. Ob iously, he
physical p ocesses which de e mine he beha io o complexly
illed composi es (sample 3) and e y so gel-like ma ix
composi es (sample 2) should be qui e di e en om hose
o he s anda d sample 1. As o be assumed, he di e en
beha io o di e en samples can be a ibu ed o bo h, mac-
oscopic de o ma ion and mic os uc u al change p ocesses
in ol ing di e en displacemen s and s uc u ing o magne ic
pa icles wi hin he polyme ma ix. F om a mic oscopic poin
o iew, pe haps a dis inc ion should be made be ween he
7
Sma Ma e . S uc . 32 (2023) 065016 M Kubík e al
Figu e 7. Raw o ce esponse F o a magne ic ield exci a ion B o e ime. Resul s a e gi en o elec omagne ic coil exci a ions o 1 A and
2.5 A which co esponds o s eady-s a e magne ic lux densi ies o ∼95 mT and ∼230 mT. Addi ionally he magne ic lux densi y esponse
is gi en.
s uc u ing p ocess and he o a ion o o he mo emen s o
magne ic mic opa icles and pa icles s uc u es wi hin he
ma ix, as ea lie obse ed in some s udies o he mic o-
s uc u e o low concen a ed MR composi es [30,31]. I
is o be assumed ha o a sample 3 wi h la ge pa icles
and he e o e wi h highe in e pa icle dipola o ces, he
mic os uc u al e ec s a e mo e p onounced han o sample
1. In he con ex o sample 2, one can speak o inc eased
mobili y o he pa icles and hei agg ega es inside he ma -
ix. F om a mac oscopic poin o iew, a ious changes in
he shape o he samples can be expec ed, e.g. changes in
he e ec i e con ac a ea o he composi e wi h he senso
sha canno be uled ou . Howe e , he compac magne ic
cell wi h a closed magne ic ci cui un o una ely does no
allow an accu a e isual check o he shape o he sample in
he expe imen . The e o e, wi hou app op ia e mac oscopic
and mic oscopic isual obse a ions, i would be inco ec o
p o ide such speci ic explana ions. The p oposi ion and jus i-
ica ion o a de ailed magne ode o ma ion mechanism emains
beyond he scope o his s udy and should be addi ionally
conside ed.
Figu e 7shows he aw o ce esponse F o magne ic ield
exci a ions (B) exempla y gi en o elec omagne ic coil exci -
a ions o 1 A and 2.5 A o e ime o all h ee MR elas ome
samples unde s udy. F om he g aphs, he di e ences in he
ansien dynamics obse ed o he di e en samples a e e id-
en . No able is he change in he esponse o samples 1 and 2
o he highe ield (I=2.5 A, B∼230 mT) compa ed o
hei esponse o he lowe ield (I=1 A, B∼95 mT). This
change demons a es an e ec o he so gel-like ma ix. The
so e composi e (sample 2) eac s as e o he ex e nal ield
and o he low ield he induced o ce is sligh ly highe han
ha obse ed o sample 1. When he ex e nal magne ic ield
eaches a ce ain h eshold alue (a ound 170 mT), he o ce
induced by sample 1 becomes g ea e han he o ce induced
by sample 2. The dynamics o he magne ic ield g ow h mus
be as well aken in o accoun he e. In he ini ial pa o he
exci a ion, i.e. up o he in e sec ion o he ansien esponse
o samples 1 and 2, he ield Bis s ill signi ican ly lowe han i s
s eady s a e alue. Tha is, in ac , in his pa we s ill obse e
he esponse o he MR composi e samples o he low ield.
The appea ance o an in e sec ion poin in he esponse o
samples 1 and 2, i.e. some h eshold alue o he magne ic
ield, is con i med o o he exci a ion cu en s (see igu e S1
in supplemen a y).
Le us now conside he e ec o he magne ic ield on
he esponse ime de e mined using he me hodology p esen-
ed abo e (see sec ion 2.2). Figu e 8shows he depend-
encies o he esponse ime τ63 (a) and τ90 (b) on he
magne ic ield B. The s anda d de ia ion is e y low com-
pa ed o he absolu e alues o he ansien ime and
he e o e is gi en in sepa a e g aphs in he supplemen a y
( igu e S2).
As he alue o he magne ic lux densi y Binc eases, he
esponse ime dec eases o all samples. Sample 1 exhibi s
a signi ican ly longe esponse ime han he o he samples
(τ63 =72 ms; τ90 =165 ms a I=2.5 A, B∼230 mT).
Sample 3 achie ed he sho es esponse ime (τ63 =27 ms;
τ90 =62 ms a I=2.5 A, B∼230 mT). Tha is, bo h, he
use o coa se pa icles in he magne ic ille and a signi ic-
an educ ion in he s i ness o he polyme ma ix educe he
esponse ime o he MR elas ome . The esponse ime cu es
o samples 2 and 3 a e o a simila quali a i e and quan i a -
i e na u e, excep o one poin co esponding o he minimum
8
Sma Ma e . S uc . 32 (2023) 065016 M Kubík e al
Figu e 8. Response ime τ63 (a) and τ90 (b) s applied magne ic ield B ob ained o he MR elas ome samples unde s udy.
ield used (B=50 mT). As in he case o he discussion
o no mal o ce induced by he magne ic ield gi en abo e,
a de ailed unde s anding o he esul s ob ained is only pos-
sible wi h accu a e isual obse a ions o changes in bo h, he
mic os uc u e and he mac oscopic s a e o he samples unde
s udy. This was no ealizable wi hin he se up being used.
I would also be p ac ical o compa e he expe imen wi h a
heo e ical p edic ion. Howe e , we a e no awa e o analy ical
heo e ical app oaches di ec ly ela ed o he ansien dynam-
ics o MR elas ome s. As he mic os uc u al changes induced
by he applied magne ic ield a e ela ed o he in e -pa icle
dipola o ces, one can expec as e changes in sample 3
which ha e la ge pa icles. On he o he hand, hese dipola
o ces induce he coun e ac ion elas ic s esses depending on
9