Ci a ion: Ma qués-Fe nández, J.L.;
Sal ado , M.; Ma ínez-Ga cía, J.C.;
Fe nández-Miaja, P.; Ga cía-A ibas,
A.; Ri as, M. New Pe spec i e on
Plana Induc i e Senso s:
Radio-F equency Re ac ome y o
Highly Sensi i e Quan i ica ion o
Magne ic Nanopa icles. Senso s 2023,
23, 2372. h ps://doi.o g/10.3390/
s23052372
Academic Edi o : Càndid Reig
Recei ed: 22 Decembe 2022
Re ised: 13 Feb ua y 2023
Accep ed: 16 Feb ua y 2023
Published: 21 Feb ua y 2023
Copy igh : © 2023 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
senso s
A icle
New Pe spec i e on Plana Induc i e Senso s: Radio-F equency
Re ac ome y o Highly Sensi i e Quan i ica ion o
Magne ic Nanopa icles
José Luis Ma qués-Fe nández 1, Ma ía Sal ado 1, José Ca los Ma ínez-Ga cía 1, Pablo Fe nández-Miaja 2,
Al edo Ga cía-A ibas 3and Mon se a Ri as 1,*
1Depa men o Physics & IUTA, Uni e si y o O iedo, Campus de Viesques, 33203 Gijón, Spain
2Depa men o Elec ical Enginee ing, Uni e si y o O iedo, Campus de Viesques, 33203 Gijón, Spain
3Depa men o Elec ici y and Elec onics, Uni e si y o he Basque Coun y, 48940 Leioa, Spain
*Co espondence: [email p o ec ed]
Abs ac :
We demons a e how esonan plana coils may be used as senso s o de ec and quan i y
magne ic nanopa icles eliably. A coil’s esonan equency depends on he adjacen ma e ials’
magne ic pe meabili y and elec ic pe mi i i y. A small numbe o nanopa icles dispe sed on a
suppo ing ma ix on op o a plana coil ci cui may hus be quan i ied. Such nanopa icle de ec ion
has applica ion de ec ion o c ea e new de ices o assess biomedicine, ood quali y assu ance, and
en i onmen al con ol challenges. We de eloped a ma hema ical model o he induc i e senso
esponse a adio equencies o ob ain he nanopa icles’ mass om he sel - esonance equency o
he coil. In he model, he calib a ion pa ame e s only depend on he e ac ion index o he ma e ial
a ound he coil, no on he sepa a e magne ic pe meabili y and elec ic pe mi i i y. The model
compa es a ou ably wi h h ee-dimensional elec omagne ic simula ions and independen expe i-
men al measu emen s. The senso can be scaled and au oma ed in po able de ices o measu e small
quan i ies o nanopa icles a a low cos . The esonan senso combined wi h he ma hema ical model
is a signi ican imp o emen o e simple induc i e senso s, which ope a e a smalle equencies
and do no ha e he equi ed sensi i i y, and oscilla o -based induc i e senso s, which ocus on jus
magne ic pe meabili y.
Keywo ds:
sel - esonan equency; induc i e senso ; coil; nanopa icles; magne ic nanopa icles;
magne ic la e al low immunoassays; impedance; e ac ion index; magne ic pe meabili y; elec ic
pe mi i i y
1. In oduc ion
New senso s a e needed o de ec and quan i y nanopa icles (NPs) o inc easing new
applica ions. One example is he de ec ion o biomolecules ( oxins, disease bioma ke s,
o o he s), which a e selec i ely a ached o he pa icles h ough an immunological e-
ac ion [
1
]. The e a e se e al app oaches o i , such as immobilising he pa icle-labelled
molecule on o he senso su ace, using mic o luidics o ake he sample o he senso
o o pass i o e i s sensi i e pa , o pape -based mic o luidics [
2
,
3
]. The la e is well
known o being he basis o he COVID-19 apid diagnos ic es s [
4
]. Such es s ha e wo
main limi a ions ha hinde hei wide applica ion; one is he lack o sensi i i y (many
alse-nega i e esul s), and he o he is hei un eliable quan i ica ion. To imp o e i , some
au ho s ely on magne ic nanopa icles (MNPs) as labels [
5
–
8
], which can be de ec ed by
de ices sensi i e o he inges o hei magne ic ield, as in [
9
]. New needs in li e sciences,
such as heal hca e, ood sa e y, and en i onmen al con ol, equi e u he imp o emen in
he sensi i i y and quan i ying capabili ies o MNP-de ec ing senso s due o he ex emely
small numbe o pa icles immobilised a he es line and hei iny size.
Senso s 2023,23, 2372. h ps://doi.o g/10.3390/s23052372 h ps://www.mdpi.com/jou nal/senso s
Senso s 2023,23, 2372 2 o 12
P e ious wo ks [
10
–
12
] highligh ed he abili y o de ec and quan i y small numbe s
o MNPs by moni o ing he impedance changes in a plana coil a a ixed equency in he
o de o ens o MHz. The senso is sensi i e o magne ic pe meabili y a ia ions a ound
he plana coil [
13
] as a consequence o Fa aday’s elec omagne ic induc ion law and can be
obse ed as changes in i s impedance. The la ge he equency, he la ge he impedance
change, as long as we a e below he coil’s esonance equency. In gene al, eliably ob ain-
ing small impedance changes in he MHz ange equi es expensi e equipmen (such as
impedance o ec o -ne wo k analyse s,) so lowe -cos and po able solu ions would be
mo e accessible and be e combined wi h he inexpensi e LFIAs.
Some epo ed minia u ised de ices ocus on cos educ ion using oscilla o -based
induc i e de ec o s, and hei epo ed sensi i i y is 3 Hz
/µ
g Fe
3
O
4
[
9
,
14
]. Such sensi i i y
is no enough o he equi emen s o many new po en ial applica ions, such as oxin
de ec ion in oods o ea ly diagnosis.
Based on he cen al idea o [
14
], in his wo k, we imp o ed MNP de ec ion sensi i i y
while keeping cos s down using he senso coils’ sel - esonan equency (SRF). The main
oadblock o SRF-based de ec o s is he complexi y o he ma hema ical ela ion be ween
he SRF and MNPs’ mass. We sol ed his bo leneck as desc ibed in Sec ion 2. A usable,
easy- o-implemen , and p ecise ma hema ical model o co ela e he coils’ SRF and he
nanopa icles’ mass ( olume o numbe , depending on he a iable used o he calib a ion)
is indispensable o make his measu emen echnique iable. We de eloped such a a ge ed
model and es ed i agains simula ions and expe imen al esul s o e i y i s alidi y
and explo ed i s limi a ions. E en when he cen al pu pose was o de elop and es he
ma hema ical model, ou analysis e ealed a signi ican sensi i i y and signal- o-noise a io
inc eases.
2. Ma hema ical Fundamen als
The me hod we de eloped o quan i y nanopa icles (NPs) in ol es a plana coil and
moni o ing i s SRF o di e en NP masses (whe he magne ic o no ) placed on op o
i . Gene ally, he senso s ha de ec me al pa icles use single- o mul iple- u n plana
coils [
15
]. Due o i s elec omagne ic- ield uni o mi y, we used a wo- u n coil o his wo k.
The plana na u e o he coil allows he placemen o la samples (as he LFIAs) on op
o he senso o i s measu emen . Figu e 1illus a es a bi-dimensional p ojec ion o he
coil shape.
Figu e 1. Rep esen a ion o he sensing coil used in his wo k and sample placemen .
The main goal o his sec ion is o deduce a ma hema ical exp ession ha ela es he
plana coil’s SRF and he NP mass on op o i . Using some hypo heses, we modelled his
ela ionship in a simple and easy- o-use manne . The i s simpli ying hypo hesis is o
conside only he i s SRF mode o ou sensing coil, which allows us o model he coil’s
impedance as an equi alen ci cui . Using equi alen ci cui s o de ine he beha iou o
ansmission lines is a widesp ead and accep ed p ocedu e [
16
–
19
]. Simple induc o s, such
as hose in Figu e 1a e gene ally modelled using an induc o –capaci i e ne wo k, as in
Figu e 2.
Senso s 2023,23, 2372 3 o 12
Figu e 2. Equi alen ci cui used o desc ibe he beha iou o he induc o .
The model has been used be o e o app oxima e he co e ma e ial p ope ies o he
coil [
20
], including complex magne ic pe meabili y and elec ic pe mi i i y as ollows: way.
ZL=ωLi(µ0
e −iµ00
e )(1)
ZC=1
ωCi(ε0e −iε00
e )(2)
ZEQ =1
ZL+ZR+1
ZC−1
(3)
whe e
ZL
s ands o he induc i e componen ,
ZC
is he capaci i e one, and
ZR=R
is
he esis ance o he equi alen ci cui ;
ω
is he ope a ing angula equency,
L
is he
coil sel -induc ance, and
C
i s capaci ance;
(µ0
e −iµ00
e )
and
(ε0e −iε00
e )
a e, espec i ely,
he e ec i e complex ini ial magne ic pe meabili y and e ec i e complex ini ial elec ic
pe mi i i y. He e, he e m “e ec i e” and he subindex “e ” e e o he e ec o all he
ma e ials su ounding he coil. Equa ion (3) has some equency-dependen pa ame e s,
such as he induc i e and capaci i e eac ances, ma e ial p ope ies ( o example, he mag-
ne ic pe meabili y is equency-dependen , as es ablished wi h he Neél o B ownian model,
depending on he combina ion o ma e ial composi ion and size, magne ic- ield in ensi y,
and i s equency ange), skin e ec s (which, being a consequence o Fa aday’s induc ion,
na ows he cu en pa h wi h inc easing equencies), and o he non-linea i ies. Rega dless,
in he p oximi ies o he SRF, NPs p oduce only mino a ia ions in he equency, so, in a
i s app oxima ion, hese a iables can be aken as cons an s. The equi alen impedance
model allows us o calcula e he SFR as he i s equency alue o which he impedance’s
imagina y componen anishes. This app oach has been p e iously used o cha ac e ise
choke co es [21].
SRF =qLC ε0e µ0
e
3−C2ZR2ε0e
2µ0
e
2+LC ε0e µ00
e
2µ0
e −C ZRε0e µ00
e
2πCL ε0e µ0
e
2+2πLC ε0e µ00
e
2(4)
Gi en ha , in gene al, he numbe o NPs o quan i y is meag e, we expec ha
µ00
e ≈
0. In any case, he nume ical simula ions p o e his assump ion o be accep able.
Fo ou coil geome y,
C2Z2
R
is much smalle han
LC
so ha he exp ession o he SRF can
be simpli ied o
SRF =qLC ε0e µ0
e
3
2πLC ε0e µ0
e
2=1
2πqLCµ0
e ε0e
=1
2π√LCn0
e
(5)
whe e
√LC
is a cons an ( aking in o accoun he desc ibed assump ion) cha ac e is ic o
he coil, and
n0
e
is he eal pa o he e ec i e e ac ion index
n0
e =qµ0
e ε0e
, which
conside s he con ibu ion o all he ma e ials a ound he coil.
Senso s 2023,23, 2372 4 o 12
De e mining he e ec i e e ac ion index o a composi e medium is a complex p ob-
lem explo ed in de ail in he e ec i e medium heo y (EMT) [
22
]. In his con ex , he olume
dis ibu ion o ma e ials, hei p ope ies, and he elec omagne ic ield’s spa ial dis ibu-
ion de e mines he e ec i e alues o elec ical pe mi i i y and magne ic pe meabili y.
Gi en he high complexi y, we need some simpli ica ion o p o ide a compac exp es-
sion. Fo his pu pose, we selec ed Lich enecke ’s model [
23
,
24
], which p o ides enough
accu acy o model he e ec i e e ac ion index nea ou senso wi h he equi ed simplici y.
Ins ead o ma hema ically modelling he sample on op o ou senso ealis ically as a
p ism, we modelled i as a uni o m mass dis ibu ion su ounding he coil ack, as shown
in Figu e 3. In he igu e, he NPs a e ep esen ed as b own do s, and he ma ix ma e ial as
blue do s. On he le , we can see he NPs mingled wi h he ma ix in he shape o a p ism,
whe eas he same amoun s o bo h ma e ials a e sp ead a ound he coil in he igh igu e,
as used in ma hema ics.
Figu e 3.
(
Le
) schema ic ep esen a ion o he sample (NPs and hei embedding ma ix) on op
o he plana coil (he e, he su ounding elec onics and ma e ials a e no depic ed) as used in
expe imen al measu emen s and simula ions; (
igh
) scheme o he uni o m pa icle dis ibu ion
su ounding he coil as used in he ma hema ical model, wi h he same NPs and ma ix amoun s.
In his way, using Lich enecke ’s loga i hmic mixing o mula, one can model he
e ec i e magne ic pe meabili y and he e ec i e elec ic pe mi i i y wi h wo o mally
iden ical exp essions,
ln(µe ) = ln(µs) + (1− )ln(µen )(6)
ln(εe ) = ln(εs) + (1− )ln(εen )(7)
which enables u he simpli ica ion, as will be shown in he nex sec ion. He e,
is he
olume ac ion occupied by he sample conce ning he olume whe e he magni ude o he
elec omagne ic ield is la ge han a gi en h eshold;
µe
and
εe
a e he e ec i e pe me-
abili y and pe mi i i y, espec i ely, de e mined om he alues
µs
and
εs
co esponding
o he sample, and
µen
and
εen
o he en i onmen . In u n,
µs
and
εs
a e ela ed o he
mass o nano ags and hei suppo .
Using he loga i hmic Lich enecke ’s mixing o mula again esul s in
ln(µs) = mNPs
mmax ln(µNPs)+1−mNPs
mmax ln(µm x)(8)
ln(εs) = mNPs
mmax ln(εNPs)+1−mNPs
mmax ln(εm x)(9)
As he NPs and he suppo a e incomp essible,
mNPs/mmax
is he olume ic ela ion,
exp essed as he a io o NP mass in he sample and a e e ence mass co esponding o he
same olume illed by NPs, wi h no oids;
µNPs
,
εNPs
,
µm x
, and
εm x
a e he alues o he
magne ic pe meabili y and elec ic pe mi i i y o he NPs and he ma ix, espec i ely.
Depending on he measu emen p ocedu e, he e may be no ma ix, bu NPs a e
deposi ed on a sample holde o di ec ly on he sensing elemen . The combined en i on-
men e ac ion index accoun s o he sample suppo and ma ix,
nm x =
1. Using his
Senso s 2023,23, 2372 5 o 12
simpli ica ion and combining Equa ions (6)–(9), we ob ain he ollowing e ec i e e ac ion
index:
ne =nen nNPs
nen (mNPs
mmax )(10)
Subs i u ing
ne
in Equa ion (5) leads o a inal exp ession ha co ela es he SRF o
he mass o NPs in he sample as ollows:
SRF =1
2πn0
en √LCn0
NPs
n0
en
mNPs
mmax
(11)
Using na u al loga i hms, exp ession (11) leads o
ln(SRF) = mNPs A1−A0(12)
whe e
A1=−
mmax lnn0
NPs
n0
en (13)
A0=ln(2π√L C n0
en )(14)
One needs o ema k ha
A1
is a unc ion o he ela i e ela ion be ween he NPs
(
n0
NPs
) and hei en i onmen (
n0
en )
, while
A0
depends on he cha ac e is ics o he coil
(
LC
) and he en i onmen . In his way, we can calib a e any senso by de e mining
A1
and
A0
h ough a linea i o he law in (12) using se e al NP masses and he co esponding
expe imen al SRFs. This calib a ion allows us o quan i y unknown NP masses in he same
de ice. As ou senso sensi i i y is de e mined by he a io be ween he e ec i e e ac i e
indexes o he sample and he en i onmen , we de ined his me hod as “ e ac ome y
sensing a adio- equency”.
3. Simula ion and Expe imen al P ocedu es
The ma hema ical model ob ained om he a ge ed hypo hesis needs e i ica ion. We
sough o co obo a e he wo king p inciple o adio- equency e ac ome y, he alidi y
o he hypo hesis, and he eal use o he model. The wo king p inciple and co obo a ion
o he hypo hesis a e be e es ed using simula ion in which we ha e absolu e con ol o e
he de ining measu ing pa ame e s, and i is possible o es si ua ions ha a e no easible
wi h expe imen al echniques. Fo measu emen s, we de eloped an a o dable es ing ig
o measu e di e en ypes o NPs.
3.1. Simula ion Se up
Fo he simula ion o he sensing p inciple, we used a comme cial ini e elemen anal-
ysis ool, Ansys HFSS. Based on he geome y o he sensing se up and he elec omagne ic
p ope ies o he ma e ials, we ex ac ed he impedance o he plana coil. The impedance
was compu ed o a wide ange o equencies o ind he i s SRF poin , which is he
key alue. The coil had wo u ns o 150
µ
m ack wid h and sepa a ion and a leng h
o 12
mm
. The sample was a cuboid o 5
mm ×
1
mm ×
0.25
mm
. The geome y o he
plana ci cui and sample a e schema ically shown in Figu e 3(le ). We used a lumped
po model o exci e he plana induc o . The coil’s ma e ial was coppe , whose p ope ies
we e included in he simula ion. The en i onmen was in oduced wi h a bi a y magne ic
pe meabili y and elec ic pe mi i i y, which could be adjus ed, and he sample wi h i s
elec omagne ic p ope ies. The bounda y o he simula ion pe ec ly abso bed he inciden
adia ion- emo ing e lec ions. All simula ions we e pe o med wi h 1000 equency poin s
in a linea ange be ween 0 and 3 GHz.
Senso s 2023,23, 2372 6 o 12
3.2. Expe imen al Se up
We pe o med he expe imen al measu emen s using a NanoVNA V2 calib a ed a
he end o he connec ion cables and de i ed he equi alen impedance o he coil om he
sca e ing pa ame e s. We made a linea in e pola ion be ween he i s couple o posi i e
and nega i e imagina y impedance alues om he coil’s impedance o ex ac he SRF. This
impedance was measu ed wi h 1000 equency poin s in a linea om 100
kHZ
o 3
GHz
.
The design o he senso p oduced o es ing ma ched he geome y o he simula ed one,
as shown in Figu e 4, in which one can also see an adap e connec ing i o NanoVNA V2.
Figu e 4.
Adap e and plana coil on he p in ed ci cui boa d used o he expe imen al measu emen s.
3.3. Nanopa icles o Tes ing
We used wo ma e ials o es he beha iou obse ed in he simula ions: supe pa am-
agne ic magne i e and gold colloidal suspensions. We chose Fe
3
O
4
due o i s high magne ic
pe meabili y and elec ic pe mi i i y. The suspensions we e chemically syn hesised by
co-p ecipi a ing Fe(II/III) sal s. This ou e is one o he mos s aigh o wa d syn hesis me h-
ods o ob ain la ge amoun s o magne ic ma e ial. B ie ly, a 100 mL solu ion o 27% (w/ )
o Fe(III) was p epa ed in a beake . Consecu i ely, 12.94 g o Fe(II) we e dissol ed in 45 mL
o he o al olume in a g adua ed cylinde . A ew d ops o 37% HCl we e added o bo h
solu ions o ensu e he pe ec dissolu ion o he sal s and a oid he possible oxida ion
o i on ca ions. Then, bo h solu ions we e mixed and s i ed igo ously. Fo he NPs o
p ecipi a e, a basic solu ion was added. The d as ic change in pH allows he nuclea ion
and g ow h o he NPs. To ha e some con ol o hese wo p ocesses, he p ecipi a ing
agen mus be added slowly. The e o e, a solu ion o 75 mL o 25% ammonia was slowly
pou ed. Finally, he NPs ob ained we e magne ically decan ed o elimina e he esidues o
he eac ion h ee imes. Then, he NPs we e esuspended in dis illed wa e .
The a e age NP size was 12
nm
. The pe meabili y and pe mi i i y alues o hese
MNPs we e no measu ed. S ill, simila ones appea ed in he li e a u e [
25
,
26
] p esen ing
alues o he ela i e pe meabili y
µ0
MNPs
a RF be ween 1.5 and 2 and ela i e elec ic
pe mi i i y
ε0MNPs
be ween 6 and 8. The gold NPs we e a comme cial colloid o sphe ical
pa icles wi h a diame e o 5.6
nm
. Thei elec ic pe mi i i y was no measu ed, bu some
s udies [
27
] epo
ε0GNPs
alues be ween 1.5 and 3 in simila pa icles. The NPs we e
Senso s 2023,23, 2372 7 o 12
deposi ed on blo ing pape using a cus om p in e ha d ops a con olled low o solu ion
along a line pa e n. P in ing was pe o med wi h a low o 3.03
µL/s
a a speed o 33
mm/s
.
All he blo ing pape pieces had he same dimensions 80
mm ×
25
mm
, and hei shape
was de e mined o p o ide a uni o m p essu e dis ibu ion on he senso . The shape o he
lines o deposi ed NPs was a ec angle o 25
mm ×
2
mm
. Figu e 5shows he pic u es o
some o he es ed samples.
Figu e 5.
(
Le
) some magne i e NP samples; (
igh
) some gold NPs samples. In all cases, NPs we e
deposi ed on blo ing pape and placed on he sensing coil in he expe imen al se up.
The blo ing pape had o be la ened agains he sensing su ace o elimina e ipples,
hus helping o keep he NP deposi ion a he same dis ance in e e y measu emen . We
achie ed his by applying p essu e on o he blo ing pape . The clamping o ce, which
comp esses he pape , impac s he en i onmen ’s e ac ion index. Keeping i cons an
be ween measu emen s is c i ical o imp o ing epea abili y. Fo his pu pose, we ab ica ed
a cus om sample holde o ix he samples i mly on op o he sensing coil. The sample
holde had a plunge h ough which p essu e was applied on bo h sides o a Te lon
pla e on op o he sensing a ea. The symme ic p essu e esul ed in a uni o m ension
dis ibu ion on he pla e and, hus, a epea able and p edic able con ac . To ensu e his
ension dis ibu ion, we used a 4-ba mechanism ha kep he symme ic plunge pa allel
o he sensing ci cui boa d. The p essu e jig is shown in Figu e 6.
Figu e 6. P essu e jig in combina ion wi h NanoVNA V2.
4. Resul s and Discussion
4.1. Simula ion Resul s
Fi s , we es ed he sensi i i y o he simula ed se up wi h di e en ideal NPs o e i y
he impo ance o he elec ic pe mi i i y and i s impac on he d i o he coil’s SRF.
Speci ically, he ollowing pa ame e alues we e used:
µNPs =
1,
εNPs =
10;
µNPs =
10,
εNPs =1; and µNPs =εNPs =10.
Figu e 7shows he dependence o he
ln(SRF)
on he a io
mNP/mmax
o he NPs in
all h ee cases.
Senso s 2023,23, 2372 8 o 12
Figu e 7. Dependence o he ln(SRF)on he a io mNP/mmax o he simula ed NPs.
Wi h hese simula ion pa ame e s, 0% mass yielded
ln(SRF) =
20.95. When only he
elec ic pe mi i i y ( ed ci cles in Figu e 7) o he magne ic pe meabili y (g een iangles)
we e inc eased om hei minimum uni y alue, he
ln(SRF)
mono onically dec eased in
he same way, as can be seen wi h he supe posi ion o he wo cu es un il 50% o mass
a io. This esul p o es ha , as long as he e ac ion index was he same, ega dless o
whe he i was
µNPs
o
εNPs
ha would change, he
ln(SRF)
e ol ed in he same way as he
NP mass. When he e ac ion index inc eased, as in he cu e wi h blue squa es (
Figu e 7
),
he dec ease in he
ln(SRF)
wi h he mass was accen ed. This simula ion beha iou is
consis en wi h ou ma hema ical model as long as he a ia ion in
ln(SRF)
was less
han 0.04, which co esponds o an SRF change o app oxima ely 50
MHz
(he e, a mass
a io o 50%). This maximum a ia ion in he SRF depends on he speci ic sensing coil,
he en i onmen , and NP p ope ies. Unde his limi a ion, he signal’s esponse is linea
e sus he mass, and he slope ollows Equa ion (13). These esul s p o e ha induc i e
senso s, wo king in hei SRF poin , a e sensi i e o bo h magne ic pe meabili y and
elec ic pe mi i i y and, o small a ia ions in he SRF (such as hose p oduced wi h small
amoun s o NPs), a e well desc ibed using he ma hema ical model p esen ed in his a icle.
Consequen ly, he me hod can be used o ake ad an age o he induc i e and capaci i e
e ec s in he coil o de ec NPs wi h inc eased sensi i i y.
One o his wo k’s mos signi ican simpli ying assump ions is he dismissal o he
imagina y pa s o he magne ic pe meabili y and he elec ic pe mi i i y. To check he
impac o his simpli ica ion, we simula ed he sys em in he p esence o di e en NPs,
speci ically wi h loss angen s o
an(δe) = an(δm) =
0,
an(δe) = an(δm) =
0.05,
and an(δe) = an(δm) = 0.1, whe e δm=µ00/µ0and δe=ε00/ε0.
Figu e 8shows ha he e is no e ec o
µ00
and
ε00
on he senso esponse o a sample
wi h
µ0
NP =ε0
NP =
10. This allows us o conclude ha neglec ing he imagina y pa s o
µ
and εis an accep able assump ion.
Figu e 8. Dependence o he ln(SRF)on he a io mNP/mmax o di e en lossy NPs.
The ollowing simula ion ocuses on he en i onmen ’s e ec . Manu ac u ing an
ac ual senso in ol es using ma e ials whose e ac ion index signi ican ly in luences
i s beha iou . We pe o med simula ions wi h he expec ed alues o he en i onmen
Senso s 2023,23, 2372 9 o 12
pe meabili ies and pe mi i i ies in h ee cases: when he coil is in acuum,
µen =εen =
1;
o he coil on op o a ib eglass ma e ial, o which we assumed
µen =
1 and
εen =
3;
and i he senso is su ounded by ib eglass, we used
µen =
1 and
εen =
4. In all cases,
we assumed ha no e o- o e i-magne ic ma e ial, apa om he sample, was nea
he coil.
Fo he simula ion wi h he same NP pa ame e s as be o e,
µNPs =εNPs =
10, we
ob ained he esul s shown in Figu e 9.
Figu e 9.
Dependence o he
ln(SRF)
on he a io
mNP/mmax
o se e al en i onmen e ac ion
indexes. The solid lines a e hei i ness o ou model (12).
Using Equa ion (12), we i ed he simula ed da a in all h ee cases, hus ob aining he
pa ame e s shown in Table 1.
Table 1.
Fi ing o Equa ion (12) o he simula ion esul s o se e al e ac ion indexes in he
en i onmen .
A1A0R2
nen =1−1.7313 ×10−1−20.9629 0.9776
nen =1.732 −8.7557 ×10−2−20.4109 0.9981
nen =2−7.6000 ×10−2−20.2684 0.9940
In his simula ion, he NPs we e he same, so he di e ences in
A1
and
A0
alues
came exclusi ely om he a ia ion in he en i onmen ’s e ac ion index. Based on he
R2
alues, i could be in e ed ha he model’s alidi y was comp omised when he di e ence
be ween he en i onmen and he NP e ac ion indexes signi ican ly inc eased. This
p oblem esul ed om ying o accommoda e la ge a ia ions in he SRF agains ou
model’s p emises. When
nen
inc eases
A1
, he line slope’s absolu e alue dec eases, hus
dec easing he sensi i i y o he measu emen .
4.2. Expe imen al Resul s
Figu e 10 shows he esul s o measu ing gold ( ed ci cles) and magne i e (g een
iangles) NPs wi h ou senso o alida e he ma hema ical model (12). The co esponding
linea eg ession pa ame e s a e shown in Table 2.
We can obse e some di e ences be ween he expe imen al (Table 2) and simula ed
(Table 1)
A0
alues. This is due o he change in he LC p oduc (see Equa ion (14))
associa ed wi h he coil geome y (mo e p ecisely o he adap e , which can be seen in
Figu e 4). Rega dless, hese esul s p o e ha , despi e he low concen a ion o NPs,
he magne i e NPs exhibi i e imes he signal o he gold ones, wi h he same mass. This is
e idenced by he sensi i i y alues
A1
and easily explained by he highe e ac ion index
o MNPs. This di e ence also explains he low
R2
alue o he gold. This esul alida es
ou analysis o he senso based on changes in he SRF and con i ms he possibili y o easily
quan i ying he MNP mass in a es sample.
