Improved Determination of Q Quality Factor and Resonance Frequency in Sensors Based on the Magnetoelastic Resonance Through the Fitting to Analytical Expressions

ma e ials
A icle
Imp o ed De e mina ion o Q Quali y Fac o and
Resonance F equency in Senso s Based on he
Magne oelas ic Resonance Th ough he Fi ing o
Analy ical Exp essions
Bea iz Sisniega 1,* , Jon Gu ié ez 1,2 , Vi ginia Mu o 3and Al edo Ga cía-A ibas 1,2
1Depa amen o de Elec icidad y Elec ónica, Uni e sidad del País Vasco/Euskal He iko
Unibe si a ea (UPV/EHU), Ba io Sa iena s/n, 48940 Leioa, Spain; [email p o ec ed] (J.G.);
al edo.ga [email p o ec ed] (A.G.-A.)
2BCMa e ials, Basque Cen e o Ma e ials, Applica ions and Nanos uc u es, UPV/EHU Science Pa k,
48940 Leioa, Spain
3
Depa amen o de Ma em
á
ica Aplicada y Es ad
í
s ica e In es igaci
ó
n Ope a i a, Uni e sidad del Pa
í
s Vasco
UPV/EHU, P.O. Box 644, 48080 Bilbao, Spain; [email p o ec ed]
*Co espondence: [email p o ec ed]
Recei ed: 17 Sep embe 2020; Accep ed: 19 Oc obe 2020; Published: 22 Oc obe 2020


Abs ac :
The esonance quali y ac o
Q
is a key pa ame e ha desc ibes he pe o mance o
magne oelas ic senso s. I s alue can be easily quan i ied om he wid h and he peak posi ion o he
esonance cu e bu , when he esonance signals a e small, o ins ance when a lo o damping is
p esen (low quali y ac o ), his and o he simple me hods o de e mine his pa ame e a e highly
inaccu a e. In hese cases, nume ical i ings o he esonance cu es allow o accu a ely ob ain he
alue o he quali y ac o . We p esen a s udy o he use o di e en exp essions o nume ically i he
esonance cu es o a magne oelas ic senso ha is designed o moni o he p ecipi a ion eac ion
o calcium oxala e. The s udy compa es he pe o mance o bo h i ings and he equi alence o
he pa ame e s ob ained in each o hem. Th ough hese nume ical i ings, he e olu ion o he
di e en pa ame e s ha de ine he esonance cu e o hese senso s is s udied, and hei accu acy in
de e mining he quali y ac o is compa ed.
Keywo ds: magne oelas ic esonance; quali y ac o ; esonance cu e i
1. In oduc ion
Magne oelas ic esonance senso s a e usually made o amo phous, ibbon-shaped e omagne ic
alloys [
1
]. Magne os ic ion in hese ma e ials p o ides a s ong coupling o hei magne ic and
mechanical p ope ies, so ha when an ex e nal al e na ing magne ic ield is applied o a magne oelas ic
ibbon, elas ic wa es a e induced in i . Con e sely, he magne ic s a e o hese ma e ials is highly
dependen on he applica ion o ex e nal mechanical o ces and loads, and he e o e hese mechanical
a ia ions gene a e changes in he magne ic lux ha can be emo ely de ec ed by a de ec ion coil
placed on he ibbon. Ma ching he physical dimensions and elas ic p ope ies o he ma e ial used in
he senso , he phenomenon o magne oelas ic esonance occu s a speci ic equencies ha can be
exp essed as:
n=n
2LsE
ρ(1)
whe e
L
is he leng h o he ibbon,
E
is he Young’s modulus o he ma e ial,
ρ
i s densi y, and
n=
1
co esponds o he undamen al esonan equency. This esonan beha io is highly sensi i e o
Ma e ials 2020,13, 4708; doi:10.3390/ma13214708 www.mdpi.com/jou nal/ma e ials
Ma e ials 2020,13, 4708 2 o 15
di e en ex e nal pa ame e s, which has gene a ed a g ea in e es in he use o hese ma e ials as
a main pa o di e en sensing sys ems [
2
]. Magne oelas ic senso s ha e been used o measu e
se e al en i onmen al pa ame e s such as p essu e, humidi y o empe a u e [
3
,
4
], liquid iscosi y
and densi y [
5
,
6
], chemical agen s and pH [
7
,
8
], and biological agen s [
9
–
11
]. Speci ically, and wi h
espec o i s use as a mass senso , when he magne oelas ic ibbon is loaded wi h a uni o m and small
amoun o mass (
δm
), a shi in i s esonance equency occu s, ha can be e alua ed om Equa ion (1)
o be, o i s o de app oxima ion:
δ =−1
2
δm
m0
0(2)
whe e 0is he esonance equency o he ba e ma e ial and m0 he ini ial mass o he senso .
The pe o mance o his ype o mass senso is mainly de e mined by wo pa ame e s: he
sensi i i y o he senso (
S
) and he quali y ac o (
Q
) [
12
]. The sensi i i y indica es he minimum
de ec able equency change when he mass o he senso inc eases:
S=−δ
δm(3)
The quali y ac o
Q
is a dimensionless pa ame e ha desc ibes how sha p he esonance cu e is,
and dec eases when he esona o is damped. Damping in luences he shape o he magne oelas ic
esonance cu e and he esonance equency i sel , conside ed as he equency o he maximum o
he cu e. A high
Q
alue indica es a lowe a e o ene gy loss, a sha p magne oelas ic esonance peak
(Figu e 1) and he e o e, good esolu ion when de e mining he esonance equency and equency
shi s. Fo de eloping high pe o mance senso s, he highes Sand Qa e desi able.
Ma e ials 2020, 13, x FOR PEER REVIEW 2 o 15
o di e en ex e nal pa ame e s, which has gene a ed a g ea in e es in he use o hese ma e ials as
a main pa o di e en sensing sys ems [2]. Magne oelas ic senso s ha e been used o measu e
se e al en i onmen al pa ame e s such as p essu e, humidi y o empe a u e [3,4], liquid iscosi y
and densi y [5,6], chemical agen s and pH [7,8], and biological agen s [9–11]. Speci ically, and wi h
espec o i s use as a mass senso , when he magne oelas ic ibbon is loaded wi h a uni o m and
small amoun o mass (𝛿𝑚), a shi in i s esonance equency occu s, ha can be e alua ed om
Equa ion (1) o be, o i s o de app oxima ion:
𝛿𝑓=−12𝛿𝑚
𝑚
𝑓
 (2)
whe e 𝑓 is he esonance equency o he ba e ma e ial and 𝑚 he ini ial mass o he senso .
The pe o mance o his ype o mass senso is mainly de e mined by wo pa ame e s: he
sensi i i y o he senso (𝑆) and he quali y ac o (𝑄) [12]. The sensi i i y indica es he minimum
de ec able equency change when he mass o he senso inc eases:
𝑆=−𝛿𝑓
𝛿𝑚 (3)
The quali y ac o 𝑄 is a dimensionless pa ame e ha desc ibes how sha p he esonance cu e is,
and dec eases when he esona o is damped. Damping in luences he shape o he magne oelas ic
esonance cu e and he esonance equency i sel , conside ed as he equency o he maximum o
he cu e. A high 𝑄 alue indica es a lowe a e o ene gy loss, a sha p magne oelas ic esonance
peak (Figu e 1) and he e o e, good esolu ion when de e mining he esonance equency and
equency shi s. Fo de eloping high pe o mance senso s, he highes 𝑆 and 𝑄 a e desi able.
Figu e 1. Two magne oelas ic esonance cu es om he same esona o wi h di e en 𝑄 alues,
caused by di e en amoun o mass loading. The peak co esponding o he highe 𝑄 (blue) is
sha pe and na owe , he esonance cu e wi h a lowe 𝑄 is wide ( ed).
The quali y ac o can be es ima ed om he esonance cu e as he expe imen ally measu ed
esonance equency 𝑓 (a which he ampli ude is maximum, 𝐴), di ided by he wid h o he
signal ∆𝑓 (measu ed as he ull wid h a hal maximum, co esponding o ampli udes 𝐴/√2) [13]:
𝑄=
𝑓

∆
𝑓
(4)
Bu he accu acy o his es ima ion me hod is e y low (i can lead o e o s up o 20% in he
de e mina ion o 𝑄) [14], so inding al e na i e and obus me hods o de e mine his pa ame e wi h
g ea e accu acy is o g ea impo ance o he s udy o he pe o mance o hese senso s, especially
in cases whe e he cu es ha e conside able noise o damping, as i is he case when he
Figu e 1.
Two magne oelas ic esonance cu es om he same esona o wi h di e en
Q
alues, caused
by di e en amoun o mass loading. The peak co esponding o he highe
Q
(blue) is sha pe and
na owe , he esonance cu e wi h a lowe Qis wide ( ed).
The quali y ac o can be es ima ed om he esonance cu e as he expe imen ally measu ed
esonance equency
max
(a which he ampli ude is maximum,
Amax
), di ided by he wid h o he
signal
∆
(measu ed as he ull wid h a hal maximum, co esponding o ampli udes
Amax/√2
) [
13
]:
Q= max
∆ (4)
Bu he accu acy o his es ima ion me hod is e y low (i can lead o e o s up o 20% in he
de e mina ion o
Q
) [
14
], so inding al e na i e and obus me hods o de e mine his pa ame e wi h
Ma e ials 2020,13, 4708 3 o 15
g ea e accu acy is o g ea impo ance o he s udy o he pe o mance o hese senso s, especially in
cases whe e he cu es ha e conside able noise o damping, as i is he case when he magne oelas ic
senso ope a es imme sed in a luid. Some s udies ha e been made on he di e en me hods o
de e mining his quali y ac o
Q
[
13
,
15
], and Lopes e al. [
16
] epo ed an ex ensi e s udy o he
de e mina ion o he quali y ac o o a magne oelas ic senso using di e en s a egies. In his las
wo k, a nume ical i ing o he expe imen al da a o an exp ession o he magne ic suscep ibili y o he
sample a ound he esonance is ca ied ou and used as a sui able echnique o calcula e he alue o
Q
.
In he p esen wo k, he nume ical i ings o he esonance cu es ob ained wi h a magne oelas ic
senso ha e been ca ied ou using wo di e en exp essions: on he one hand, he exp ession
used by Lopes e al. [
16
], ha appea s as Equa ion (6) below in his wo k and, on he o he hand,
a phenomenological app oach ha desc ibes he equency esponse o hese magne oelas ic senso s
and ha has al eady been used o i hese esonance cu es [
17
,
18
], appea ing as Equa ion (9) wi hin
his wo k. The pe o mance o each o he i ing exp essions o de e mine he quali y ac o
Q
has
been s udied, and hei compa ison and equi alence ca ied ou .
The esonance cu es used in he i analysis we will show in he ollowing, co espond o hose
ob ained in he moni o ing o he p ecipi a ion eac ion o calcium oxala e sal c ys als (ino ganic sal s
esul ing om a ious me abolic ac i i ies in humans) using a magne oelas ic senso . This moni o ing
p ocess was i s desc ibed by Bou opoulos e al. [
19
], and he measu emen s ob ained in his eac ion
sensing and used in he p esen wo k a e explained in de ail in a p e ious publica ion [
20
]. Th ough he
nume ical i ing o hese cu es, we ha e s udied he e olu ion o he di e en pa ame e s ha de ine
he esonance cu e as he p ecipi a ion eac ion p og ess, which p o ides a be e unde s anding o
he e ec ha he changes o mass in he senso and damping ha e on he esonance signal ob ained.
2. Ma e ials and Me hods
2.1. Magne oelas ic Ma e ial o he Senso
The ma e ial used as senso pla o m is a magne oelas ic ibbon o composi ion
Fe73C 5Si10B12
p o ided by Vacuumschmelze GmbH & Co. KG, Hanau Ge many. This ma e ial has an excellen
co osion esis ance beha io due o he ac ha i con ains a small amoun o ch omium (5%
a
.),
which o ms a supe icial passi a ion laye in he ma e ial. Magne ic, magne oelas ic and co osion
esis ance p ope ies o his ma e ial we e s udied by Sagas i e al. [
21
], and a e summa ized in Table 1.
The s ip was lase cu wi h dimensions 20 mm ×2 mm ×25 µm (leng h o wid h a io R=10).
Table 1.
Magne ic, magne oelas ic, and co osion beha iou pa ame e s o he
Fe73C 5Si10B12
sample [
21
].
Ms
is he spon aneous magne iza ion,
λs
he sa u a ion magne os ic ion,
∆E
he change in he Young’s
modulus wi h he applied magne ic ield (o
∆E
e ec ),
k
he magne oelas ic coupling coe icien and
Eco he co osion po en ial.
Composi ion µ0Ms(T)λs(ppm) ∆E(%)kEco
(mV)
Co osion Ra e
(µm/Yea )
Fe73C 5Si10B12 1.12 14 17 0.41 47 0.035
2.2. P ecipi a ion Reac ion Measu emen s
The measu emen s used in his wo k o make he nume ical i ings co espond o he moni o ing
o he p ecipi a ion eac ion o calcium oxala e c ys als (CaC2O4):
CaCl2(aq)+H2C2O4(aq)→CaC2O4(s)+2HCl(aq)(5)
The o ma ion o he sal c ys als du ing he p ecipi a ion p ocess was acked by de ec ing changes
in he senso esonance equency
max
, as ully desc ibed by Sisniega e al. [
20
]. The magne oelas ic
senso desc ibed in he p e ious subsec ion was placed in a ial wi h a mix u e o equal pa s o he
Ma e ials 2020,13, 4708 4 o 15
eagen s (
CaCl2
and
H2C2O4
) a he same concen a ion ( o h ee di e en concen a ions: 30, 50,
and 100 mM) and he changes in he esonance equency o he senso (de e mined by Equa ion
(2)) we e moni o ed along ime, as he c ys als p ecipi a e and se le on o he senso (inc easing i s
mass). The expe imen al se up used o egis e he esonance cu es consis s o h ee coaxial solenoids:
one o apply a cons an bias ield; a second one o p oduce he al e na ing ield o magne os ic i ely
exci e he sample; and he hi d one, consis ing o a compensa ed pick-up coil, o de ec he induced
magne iza ion oscilla ions on he senso . A spec um analyze (HP 3589A, Hewle -Packa d, Palo Al o,
CA, USA) wo king in swep mode is used o p oduce he exci a ion and o ecei e he signal induced
in he pick-up coil. A e eco ding he esonance cu es, he equency o he maximum
max
and i s
ampli ude a e measu ed using he buil -in analysis p ocedu es o he analyze and ansmi ed o a
con ol compu e . The sweep speed was se looking o a comp omise be ween good cu e esolu ion
and he necessa y speed o be able o eco d he whole esonance-an i esonance cu e as enough o
ollow he p ecipi a ion p ocess. Wi h his con igu a ion, he expe imen al unce ain y in he equency
de e mina ion is 100 Hz, and he measu emen noise can be up o 1 kHz when he esonance cu es
a e wide and wi h low ampli ude.
Figu e 2p esen s an example o he esul s ob ained in he p ecipi a ion moni o ing measu emen s.
The cu es in Figu e 2a show he dec ease o he senso esonance equency (
max
) as he p ecipi a ion
akes place, o he h ee solu ions wi h di e en concen a ions. Figu e 2b shows, o one pa icula
solu ion (30 mM), he e olu ion o he comple e esonance cu e as he eac ion p og esses (cu es
aken a di e en eac ion imes). I can be seen ha , in addi ion o he dec ease in he esonance
equency, he e is also a educ ion o he quali y ac o
Q
ha cha ac e izes he esonance (o inc ease
in damping), and a dec ease in he ampli ude o he senso signal. The esonance cu es displayed in
Figu e 2b we e used in he p esen wo k o ca y ou he nume ical i ings, oge he wi h analogous
cu es co esponding o he solu ions o concen a ion 50 and 100 mM. Successi e expe imen s in he
same condi ions yield simila esul s. He e we p esen he analysis in a single un o each concen a ion.
Nine esonance cu es co esponding o di e en eac ion imes we e i ed o each concen a ion.
Ma e ials 2020, 13, x FOR PEER REVIEW 4 o 15
he senso (inc easing i s mass). The expe imen al se up used o egis e he esonance cu es consis s
o h ee coaxial solenoids: one o apply a cons an bias ield; a second one o p oduce he al e na ing
ield o magne os ic i ely exci e he sample; and he hi d one, consis ing o a compensa ed pick-up
coil, o de ec he induced magne iza ion oscilla ions on he senso . A spec um analyze (HP 3589A,
Hewle -Packa d, Palo Al o, CA, USA) wo king in swep mode is used o p oduce he exci a ion and
o ecei e he signal induced in he pick-up coil. A e eco ding he esonance cu es, he equency
o he maximum 𝑓 and i s ampli ude a e measu ed using he buil -in analysis p ocedu es o he
analyze and ansmi ed o a con ol compu e . The sweep speed was se looking o a comp omise
be ween good cu e esolu ion and he necessa y speed o be able o eco d he whole esonance-
an i esonance cu e as enough o ollow he p ecipi a ion p ocess. Wi h his con igu a ion, he
expe imen al unce ain y in he equency de e mina ion is 100 Hz, and he measu emen noise can
be up o 1 kHz when he esonance cu es a e wide and wi h low ampli ude.
Figu e 2 p esen s an example o he esul s ob ained in he p ecipi a ion moni o ing
measu emen s. The cu es in Figu e 2a show he dec ease o he senso esonance equency (𝑓)
as he p ecipi a ion akes place, o he h ee solu ions wi h di e en concen a ions. Figu e 2b shows,
o one pa icula solu ion (30 mM), he e olu ion o he comple e esonance cu e as he eac ion
p og esses (cu es aken a di e en eac ion imes). I can be seen ha , in addi ion o he dec ease in
he esonance equency, he e is also a educ ion o he quali y ac o 𝑄 ha cha ac e izes he
esonance (o inc ease in damping), and a dec ease in he ampli ude o he senso signal. The
esonance cu es displayed in Figu e 2b we e used in he p esen wo k o ca y ou he nume ical
i ings, oge he wi h analogous cu es co esponding o he solu ions o concen a ion 50 and 100
mM. Successi e expe imen s in he same condi ions yield simila esul s. He e we p esen he
analysis in a single un o each concen a ion. Nine esonance cu es co esponding o di e en
eac ion imes we e i ed o each concen a ion.
(a)
(b)
Figu e 2. (a) Tempo al e olu ion o he magne oelas ic esonance equency o he senso du ing he
eac ion o p ecipi a ion o di e en concen a ions (30 mM, 50 mM and 100 mM) and a con ol cu e
(senso in a ial wi h dis illed wa e ). The alue o he esonance equency, 𝑓, is ob ained as he
equency a he maximum ampli ude o measu ed esonance cu es (Figu e 2b); (b) Measu ed
magne oelas ic esonance cu e signals o he senso a di e en imes du ing he p ecipi a ion
p ocess o he concen a ion 30 mM. Da a aken om Re . [20].
2.3. Nume ial Fi ing o he Senso Resonance Cu es
The quali y ac o 𝑄 o he esonance can be quan i ied om he wid h and he peak posi ion o
he esonance cu e. Howe e , when he esonance signals a e hea ily damped his and o he simple
me hods a e unable o de e mine his pa ame e o a e highly inaccu a e. We p opose o use a
equency esponse i ing me hod o de e mine he quali y ac o and o he pa ame e s ha
Figu e 2.
(
a
) Tempo al e olu ion o he magne oelas ic esonance equency o he senso du ing
he eac ion o p ecipi a ion o di e en concen a ions (30 mM, 50 mM and 100 mM) and a con ol
cu e (senso in a ial wi h dis illed wa e ). The alue o he esonance equency,
max
, is ob ained as
he equency a he maximum ampli ude o measu ed esonance cu es (Figu e 2b); (
b
) Measu ed
magne oelas ic esonance cu e signals o he senso a di e en imes du ing he p ecipi a ion p ocess
o he concen a ion 30 mM. Da a aken om Re . [20].
Ma e ials 2020,13, 4708 5 o 15
2.3. Nume ial Fi ing o he Senso Resonance Cu es
The quali y ac o
Q
o he esonance can be quan i ied om he wid h and he peak posi ion
o he esonance cu e. Howe e , when he esonance signals a e hea ily damped his and o he
simple me hods a e unable o de e mine his pa ame e o a e highly inaccu a e. We p opose o
use a equency esponse i ing me hod o de e mine he quali y ac o and o he pa ame e s ha
cha ac e ize he esonance cu es. In 1978, Sa age e al. [
22
] de i ed an exp ession o he suscep ibili y
o he magne oelas ic esona o a ound he magne oelas ic esonance gi en by
χ(ω)=χ0
1−8k2
π2X
n
1
n2×1
1−ω2
n
ω2+jQ−1ωn
ω
(6)
whe e
k
is he magne oelas ic coupling coe icien (
k=sπ2
8 1−
a2!
,
and
a
being he esonance
and an i- esonance equencies, espec i ely),
ωn=
2
π n
is he esonance equency o he n h ha monic
o he exci ed undamen al mode (
n=
1,
1=
),
Q−1
is a phenomenological damping coe icien ,
χ0
is he magne ic suscep ibili y measu ed a a equency a below he esonance and j
=√−1
. Due o
he expe imen al p ocedu e used o measu e he esonance cu es [
20
], he ol age induced in he
pick-up coil (which is displayed in Figu es 1and 2b) is p opo ional o he magne ic suscep ibili y.
The pa ame e s o be i ed in Equa ion (6) a e:
ω
,
ωa
,
χ0
, and
Q
. The nume ical i ing o he
expe imen al da a o he modulus o magni ude o he Equa ion (6) o i s i s esonan mode (
n=
1)
was pe o med using Ma hema ica
©
so wa e. The pa ame e s
ω
,
ωa
, and
χ0
, we e allowed o a y
a ound hei expe imen al co esponding alues (ex ac ed om he expe imen al da a), un il he
alue o
Q
ha minimizes he no m be ween he i and he expe imen al da a was ound. The no m
(o esidual) is de ined as:
R=1
NX
i=1,N χexp,i−χ i ,i
χmax !2
(7)
whe e
χmax =maxχmax,exp,χmax, i 
,
N
is he numbe o expe imen al poin s (
N=
401, in ou
measu emen s), and 0 ≤ R ≤ 1, meaning alues close o 0 ha he i ing is good.
Addi ionally, a phenomenological exp ession using he o malism o linea sys ems was de eloped
o desc ibe he equency esponse o a magne oelas ic ma e ial [
17
]. The magne oelas ic esonance
is cha ac e ized by maximum ampli ude a he esonance equency
, and a null minimum a he
an i- esonance equency
a
. The analy ical exp ession o he ans e ence unc ion (
G
) ha desc ibes
a sys em wi h such equency esponse is composed o a couple o complex conjuga ed poles in he
denomina o (desc ibing he esonance) and a couple o complex conjuga ed ze os in he nume a o
(desc ibing he an i- esonance):
G(s)=ω2
ω2
a·s2+2δaωas+ω2
a
s2+2δ ω s+ω2
(8)
whe e
ω =
2
π
is he esonance equency,
ωa=
2
π a
he an i- esonance equency, and
δ
and
δa
a e
damping pa ame e s. The ans e unc ion is exp essed as a Laplace ope a o wi h
s=jω
, whe e
j=√−1 and ωis he equency.
The ampli ude o he equency esponse o he sys em, when submi ed o ha monic exci a ion,
based on he equency esponse desc ibed on he ans e ence unc ion is:
V(ω)=A·ω2
ω2
a
ω2−2jδaωaω−ω2
a
ω2−2jδ ω ω−ω2

+aω+b(9)

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whe e
A
accoun s o he ampli ude and
a
and
b
o he backg ound signal o he expe imen (assumed
o be linea ). The pa ame e s o be i ed in he exp ession a e:
ω
,
ωa
,
δ
,
δa
,
A
,
a
and
b
. No e ha
in his exp ession, he quali y ac o
Q
does no appea explici ly. Ins ead, he damping pa ame e s
δ
and
δa
ca y he ele an in o ma ion. The nume ical i ing o Equa ion (9) o he expe imen al
da a was ca ied ou using MATLAB
®
so wa e and a nonlinea leas -squa es i ing. The pa ame e s
ω
,
ωa
, and A o be i ed a e ini ially aken as hei alues ob ained om he expe imen al cu e.
All pa ame e s a e i ed un il hose ha bes i he expe imen al cu e a e ound, in his case hose
ha co espond o a local minimum o a unc ion ha is a sum o squa ed esiduals (being he esidual
he di e ence be ween he expe imen al alue o he dependen a iable and he alue p edic ed by
he i ing model).
3. Resul s and Discussion
3.1. Nume ical Fi ing o he Resonance Cu es and Resonance F equency
Nume ical i ings we e pe o med o he esonance cu es measu ed a di e en imes in he
p ecipi a ion eac ion, o solu ion wi h di e en concen a ions (30, 50 and 100 mM). Figu e 3shows
an example o he nume ical i ings made wi h Equa ion (6), while Figu e 4shows he same i ing bu
using Equa ion (9). I can be seen ha bo h models i he expe imen al da a easonably well, esul ing
in a low alue o he co esponding esiduals.
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3. Resul s and Discussion
3.1. Nume ical Fi ing o he Resonance Cu es and Resonance F equency
Nume ical i ings we e pe o med o he esonance cu es measu ed a di e en imes in he
p ecipi a ion eac ion, o solu ion wi h di e en concen a ions (30, 50 and 100 mM). Figu e 3 shows
an example o he nume ical i ings made wi h Equa ion (6), while Figu e 4 shows he same i ing
bu using Equa ion (9). I can be seen ha bo h models i he expe imen al da a easonably well,
esul ing in a low alue o he co esponding esiduals.
(a)
(b)
Figu e 3. Fi o Equa ion (6). (a) Measu ed magne oelas ic esonance cu es o he senso a di e en
imes du ing he p ecipi a ion p ocess o eac an s o concen a ion 30 mM, and co esponding
nume ical i ings (in dashed lines) o Equa ion (6); (b) Measu ed magne oelas ic esonance cu e,
i ing o Equa ion (6) (in dashed line), and co esponding e o (in ed) o he senso a ime = 125 s
o eac an s o concen a ion 30 mM.
(a)
(b)
Figu e 4. Fi o Equa ion (9). (a) Measu ed magne oelas ic esonance cu es o he senso a di e en
imes du ing he p ecipi a ion p ocess o eac an s o concen a ion 30 mM, and co esponding
nume ical i ings (in dashed lines) o Equa ion (9); (b) Measu ed magne oelas ic esonance cu e,
i ing o Equa ion (9) (in dashed line), and co esponding e o (in ed) o he senso a ime = 125 s
o eac an s o concen a ion 30 mM.
Figu e 3.
Fi o Equa ion (6). (
a
) Measu ed magne oelas ic esonance cu es o he senso a di e en
imes du ing he p ecipi a ion p ocess o eac an s o concen a ion 30 mM, and co esponding
nume ical i ings (in dashed lines) o Equa ion (6); (
b
) Measu ed magne oelas ic esonance cu e,
i ing o Equa ion (6) (in dashed line), and co esponding e o (in ed) o he senso a ime =125 s
o eac an s o concen a ion 30 mM.
Howe e , when he deposi ed mass is high (a high eac ion imes o highe concen a ion o
eagen s), he i ing o Equa ion (9) beha es sligh ly be e (wi h lowe e o be ween he expe imen al
da a and he alues ob ained h ough he model) han he one using he Equa ion (6).
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3. Resul s and Discussion
3.1. Nume ical Fi ing o he Resonance Cu es and Resonance F equency
Nume ical i ings we e pe o med o he esonance cu es measu ed a di e en imes in he
p ecipi a ion eac ion, o solu ion wi h di e en concen a ions (30, 50 and 100 mM). Figu e 3 shows
an example o he nume ical i ings made wi h Equa ion (6), while Figu e 4 shows he same i ing
bu using Equa ion (9). I can be seen ha bo h models i he expe imen al da a easonably well,
esul ing in a low alue o he co esponding esiduals.
(a)
(b)
Figu e 3. Fi o Equa ion (6). (a) Measu ed magne oelas ic esonance cu es o he senso a di e en
imes du ing he p ecipi a ion p ocess o eac an s o concen a ion 30 mM, and co esponding
nume ical i ings (in dashed lines) o Equa ion (6); (b) Measu ed magne oelas ic esonance cu e,
i ing o Equa ion (6) (in dashed line), and co esponding e o (in ed) o he senso a ime = 125 s
o eac an s o concen a ion 30 mM.
(a)
(b)
Figu e 4. Fi o Equa ion (9). (a) Measu ed magne oelas ic esonance cu es o he senso a di e en
imes du ing he p ecipi a ion p ocess o eac an s o concen a ion 30 mM, and co esponding
nume ical i ings (in dashed lines) o Equa ion (9); (b) Measu ed magne oelas ic esonance cu e,
i ing o Equa ion (9) (in dashed line), and co esponding e o (in ed) o he senso a ime = 125 s
o eac an s o concen a ion 30 mM.
Figu e 4.
Fi o Equa ion (9). (
a
) Measu ed magne oelas ic esonance cu es o he senso a di e en
imes du ing he p ecipi a ion p ocess o eac an s o concen a ion 30 mM, and co esponding
nume ical i ings (in dashed lines) o Equa ion (9); (
b
) Measu ed magne oelas ic esonance cu e,
i ing o Equa ion (9) (in dashed line), and co esponding e o (in ed) o he senso a ime =125 s
o eac an s o concen a ion 30 mM.
I we analyze he e olu ion o he pa ame e s ob ained by he i ing p ocedu e, we can see ha
he wo damping pa ame e s in Equa ion (9) (
δ
and
δa
) inc ease p og essi ely as he mass deposi ed
in he senso inc eases (Figu e 5). As he eac ion ad ances, he c ys als a e o med in he solu ion
o he ial and a e deposi ed on he su ace o he senso , inc easing i s mass, and his inc eases he
damping su e ed by his senso . In he same manne , as he concen a ion o he eagen s inc eases
(30, 50, 100 mM), he mass o calcium oxala e c ys als is g ea e and, he e o e, so is he damping.
The nume ical i ings show his di ec ela ionship be ween he deposi ed mass and he damping in
he esonance cu es ob ained. Simila ly, i can be seen ha he
Q
pa ame e ob ained by i ing he
da a o Equa ion (6) dec eases as he p ecipi a ion eac ion p og esses and he mass is deposi ed on he
senso (Figu e 6).
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Howe e , when he deposi ed mass is high (a high eac ion imes o highe concen a ion o
eagen s), he i ing o Equa ion (9) beha es sligh ly be e (wi h lowe e o be ween he
expe imen al da a and he alues ob ained h ough he model) han he one using he Equa ion (6).
I we analyze he e olu ion o he pa ame e s ob ained by he i ing p ocedu e, we can see ha
he wo damping pa ame e s in Equa ion (9) ( 𝛿 and 𝛿) inc ease p og essi ely as he mass
deposi ed in he senso inc eases (Figu e 5). As he eac ion ad ances, he c ys als a e o med in he
solu ion o he ial and a e deposi ed on he su ace o he senso , inc easing i s mass, and his
inc eases he damping su e ed by his senso . In he same manne , as he concen a ion o he
eagen s inc eases (30, 50, 100 mM), he mass o calcium oxala e c ys als is g ea e and, he e o e, so
is he damping. The nume ical i ings show his di ec ela ionship be ween he deposi ed mass and
he damping in he esonance cu es ob ained. Simila ly, i can be seen ha he 𝑄 pa ame e
ob ained by i ing he da a o Equa ion (6) dec eases as he p ecipi a ion eac ion p og esses and he
mass is deposi ed on he senso (Figu e 6).
(a)
(b)
Figu e 5. E olu ion o he damping pa ame e s ob ained h ough he nume ical i ing o Equa ion
(9) o he esonance cu es du ing he p ecipi a ion ime o di e en concen a ion o eagen s (30,
50, and 100 mM). (a) Damping pa ame e 𝛿; (b) Damping pa ame e 𝛿.
Figu e 6. E olu ion o he quali y ac o 𝑄 o he esona o du ing he p ecipi a ion p ocess o
di e en eagen s concen a ions (30, 50, and 100 mM) ob ained h ough he nume ical i ings o
Equa ion (6) o he esonance cu es.
Figu e 5.
E olu ion o he damping pa ame e s ob ained h ough he nume ical i ing o Equa ion (9)
o he esonance cu es du ing he p ecipi a ion ime o di e en concen a ion o eagen s (30, 50, and
100 mM). (a) Damping pa ame e δ ; (b) Damping pa ame e δa.
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Howe e , when he deposi ed mass is high (a high eac ion imes o highe concen a ion o
eagen s), he i ing o Equa ion (9) beha es sligh ly be e (wi h lowe e o be ween he
expe imen al da a and he alues ob ained h ough he model) han he one using he Equa ion (6).
I we analyze he e olu ion o he pa ame e s ob ained by he i ing p ocedu e, we can see ha
he wo damping pa ame e s in Equa ion (9) ( 𝛿 and 𝛿) inc ease p og essi ely as he mass
deposi ed in he senso inc eases (Figu e 5). As he eac ion ad ances, he c ys als a e o med in he
solu ion o he ial and a e deposi ed on he su ace o he senso , inc easing i s mass, and his
inc eases he damping su e ed by his senso . In he same manne , as he concen a ion o he
eagen s inc eases (30, 50, 100 mM), he mass o calcium oxala e c ys als is g ea e and, he e o e, so
is he damping. The nume ical i ings show his di ec ela ionship be ween he deposi ed mass and
he damping in he esonance cu es ob ained. Simila ly, i can be seen ha he 𝑄 pa ame e
ob ained by i ing he da a o Equa ion (6) dec eases as he p ecipi a ion eac ion p og esses and he
mass is deposi ed on he senso (Figu e 6).
(a)
(b)
Figu e 5. E olu ion o he damping pa ame e s ob ained h ough he nume ical i ing o Equa ion
(9) o he esonance cu es du ing he p ecipi a ion ime o di e en concen a ion o eagen s (30,
50, and 100 mM). (a) Damping pa ame e 𝛿; (b) Damping pa ame e 𝛿.
Figu e 6. E olu ion o he quali y ac o 𝑄 o he esona o du ing he p ecipi a ion p ocess o
di e en eagen s concen a ions (30, 50, and 100 mM) ob ained h ough he nume ical i ings o
Equa ion (6) o he esonance cu es.
Figu e 6.
E olu ion o he quali y ac o
Q
o he esona o du ing he p ecipi a ion p ocess o di e en
eagen s concen a ions (30, 50, and 100 mM) ob ained h ough he nume ical i ings o Equa ion (6) o
he esonance cu es.
Figu e 6shows he in e se ela ion be ween he quali y ac o
Q
and he damping pa ame e s
(
δ
and
δa
). All hese pa ame e s show a clea and ma ked endency as he p ecipi a ion eac ion occu s.
The e olu ion o esonance pa ame e s, p e e en ially he esonance equency, can be used o
moni o he eac ion by he changes caused by he mass deposi ion on he senso signal. Nume ical
i ings imp o e he in o ma ion ob ained h ough he changes obse ed in he measu ed esonance
equency, since he esul comes om he whole esonance cu e, no only om he poin o maximum
ampli ude and he wid h a hal maximum. In addi ion, he i ing p ocedu e is use ul o educe
he noise, especially when he signal is low o highly damped, as e idenced in Figu e 7a. This plo
displays he equency o he maxima ( ha is, he esonance equency
max
) o all he esonance cu es
measu ed du ing he p ecipi a ion p ocess o he solu ion wi h concen a ion 50 mM. The da a ob ained
di ec ly om he measu emen s p esen a conside able ipple wi h an ampli ude close o 1 kHz, which
se e ely limi s he esolu ion o he senso . I is mainly caused by he limi ed esolu ion wi h which
equency di e ences a e disc imina ed by he measu ing sys em. The e olu ion o he equency o
he maxima ob ained om he i ing o he esonance cu es displays a smoo h mono onous endency,
which acili a es he calib a ion and use o he senso .
One impo an consequence o he i ing p ocedu e is ha he alues o he
pa ame e ob ained
in he i ings do no co espond o
max
, he posi ion o he maxima o he esonance cu es. Figu e 7b
compa es he i ed alues om bo h i ing exp essions wi h he expe imen al esonance equency
( max, aken as he maximum o he esonance cu e).
This ells us ha wha we usually ake as he esonance equency, ha is he equency o he
maximum o he esonance cu e, is no eally he physical esonance equency o he senso ,
which
can be ob ained om he i ing o he analy ical exp essions. The equency o he maximum o he
measu ed cu es,
max
, is sys ema ically lowe han he esonance equency,
, since i is a ec ed
by he damping o he cu e. The same applies o he equency o an i- esonance (minimum o he
cu e), which, when a ec ed by damping due o he mass deposi ion, inc eases wi h espec o he
alue,
a
, which is ob ained om he i s. Figu e 8a illus a es his disc epancy, showing he equency
esponse o he sys em, o ideal esonance and an i- esonance sepa a ely, and o he ull cu e (sum o
esonance and an i- esonance) a ec ed by damping. In Figu e 8b, he di e ence be ween he equency
o he maximum o he esonance
max
and he esonance equency
is shown as a unc ion o he
damping, and a linea ela ion be ween hem is obse ed.
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Figu e 6 shows he in e se ela ion be ween he quali y ac o 𝑄 and he damping pa ame e s
(𝛿 and 𝛿). All hese pa ame e s show a clea and ma ked endency as he p ecipi a ion eac ion
occu s.
The e olu ion o esonance pa ame e s, p e e en ially he esonance equency, can be used o
moni o he eac ion by he changes caused by he mass deposi ion on he senso signal. Nume ical
i ings imp o e he in o ma ion ob ained h ough he changes obse ed in he measu ed esonance
equency, since he esul comes om he whole esonance cu e, no only om he poin o
maximum ampli ude and he wid h a hal maximum. In addi ion, he i ing p ocedu e is use ul o
educe he noise, especially when he signal is low o highly damped, as e idenced in Figu e 7a. This
plo displays he equency o he maxima ( ha is, he esonance equency 𝑓) o all he esonance
cu es measu ed du ing he p ecipi a ion p ocess o he solu ion wi h concen a ion 50 mM. The da a
ob ained di ec ly om he measu emen s p esen a conside able ipple wi h an ampli ude close o 1
kHz, which se e ely limi s he esolu ion o he senso . I is mainly caused by he limi ed esolu ion
wi h which equency di e ences a e disc imina ed by he measu ing sys em. The e olu ion o he
equency o he maxima ob ained om he i ing o he esonance cu es displays a smoo h
mono onous endency, which acili a es he calib a ion and use o he senso .
One impo an consequence o he i ing p ocedu e is ha he alues o he 𝑓 pa ame e
ob ained in he i ings do no co espond o 𝑓, he posi ion o he maxima o he esonance cu es.
Figu e 7b compa es he i ed alues om bo h i ing exp essions wi h he expe imen al esonance
equency (𝑓, aken as he maximum o he esonance cu e).
(a) (b)
Figu e 7. (a) E olu ion o he equency co esponding o he maximum o he esonance cu es
esul ing om he i ing o Equa ion (9) o he esonance cu es measu ed du ing he p ecipi a ion,
when he concen a ion o he eac an s is 50 mM, compa ed o expe imen al da a (𝑓𝑚𝑎𝑥); (b)
E olu ion o he 𝑓 ob ained in he nume ical i ing o Equa ions (6) and (9) o he measu ed
esonance cu es, compa ed o expe imen al da a (𝑓𝑚𝑎𝑥).
This ells us ha wha we usually ake as he esonance equency, ha is he equency o he
maximum o he esonance cu e, is no eally he physical esonance equency o he senso , 𝑓
which can be ob ained om he i ing o he analy ical exp essions. The equency o he maximum
o he measu ed cu es, 𝑓, is sys ema ically lowe han he esonance equency, 𝑓, since i is
a ec ed by he damping o he cu e. The same applies o he equency o an i- esonance (minimum
o he cu e), which, when a ec ed by damping due o he mass deposi ion, inc eases wi h espec
o he alue, 𝑓, which is ob ained om he i s. Figu e 8a illus a es his disc epancy, showing he
equency esponse o he sys em, o ideal esonance and an i- esonance sepa a ely, and o he ull
cu e (sum o esonance and an i- esonance) a ec ed by damping. In Figu e 8b, he di e ence
Figu e 7.
(
a
) E olu ion o he equency co esponding o he maximum o he esonance cu es
esul ing om he i ing o Equa ion (9) o he esonance cu es measu ed du ing he p ecipi a ion,
when he concen a ion o he eac an s is 50 mM, compa ed o expe imen al da a (
max
); (
b
) E olu ion
o he
ob ained in he nume ical i ing o Equa ions (6) and (9) o he measu ed esonance cu es,
compa ed o expe imen al da a ( max ).
Ma e ials 2020, 13, x FOR PEER REVIEW 9 o 15
be ween he equency o he maximum o he esonance 𝑓 and he esonance equency 𝑓 is
shown as a unc ion o he damping, and a linea ela ion be ween hem is obse ed.
(a)
(b)
Figu e 8. (a) F equency esponse o he sys em o esonance and an i- esonance beha io , and he
o al esponse a ec ed by he damping. Co esponding esonance and an i- esonance equencies a e
indica ed; (b) E olu ion o he di e ence be ween he expe imen al esonance equency 𝑓

and
he esonance equency ob ained h ough he i ing o Equa ion (9) as he damping pa ame e 𝛿

inc eases, o he case o concen a ion 50 mM. The shi in he esonance equency becomes g ea e
as he damping inc eases.
The cu es in Figu e 8a we e ob ained wi h he magni ude o he equency esponse
co esponding o he ans e unc ions ep esen ing sys ems o : a pu e esonance (𝐺), a pu e an i-
esonance (𝐺), and he combina ion o bo h (𝐺), which ep esen s he obse ed esonance-
an i esonance beha iou . 𝐺(𝑠)= 𝜔
𝑠+2𝛿𝜔𝑠+𝜔 (10)
𝐺(𝑠)=𝑠+2𝛿𝜔𝑠+𝜔
𝜔 (11)
𝐺(𝑠)=𝐺∙𝐺=𝜔
𝜔𝑠+2𝛿𝜔𝑠+𝜔
𝑠+2𝛿𝜔𝑠+𝜔 (12)
The equency alues ob ained o he esonance (maximum alue) and an i- esonance
(minimum alue) o he espec i e cu es in Figu e 8a, coincide wi h hose ob ained in he i ings
(𝑓 𝐹𝑖𝑡 and 𝑓 𝐹𝑖𝑡 in Figu e 8a). In he combined cu e, howe e , he maximum and minimum a e
shi ed wi h espec o hese alues, esul ing in he alues o he esonance and an i- esonance
equencies (𝑓 𝐷𝑎𝑚𝑝𝑒𝑑 and 𝑓 𝐷𝑎𝑚𝑝𝑒𝑑 in Figu e 8a), which ma ch he expe imen al da a.
The mass inc ease on he esona o p oduces bo h a dec ease o he maximum o he esonance
equency (𝑓) and a dec ease o he quali y ac o . Figu e 9 illus a es he co ela ion be ween bo h
pa ame e s. Howe e , he obse ed educ ion o 𝑓 combines he e ec o mass loading (Equa ion
(2)) and he e ec desc ibed in Figu e 8a. The analysis o he esonance cu es h ough he i ing
p ocedu e allows he de e mina ion o 𝑓, which is, in p inciple, he one ha is conside ed in
Equa ion (2).
Figu e 8.
(
a
) F equency esponse o he sys em o esonance and an i- esonance beha io , and he
o al esponse a ec ed by he damping. Co esponding esonance and an i- esonance equencies
a e indica ed; (
b
) E olu ion o he di e ence be ween he expe imen al esonance equency
max
and
he esonance equency ob ained h ough he i ing o Equa ion (9) as he damping pa ame e
δ
inc eases, o he case o concen a ion 50 mM. The shi in he esonance equency becomes g ea e as
he damping inc eases.
The cu es in Figu e 8a we e ob ained wi h he magni ude o he equency esponse co esponding
o he ans e unc ions ep esen ing sys ems o : a pu e esonance (
G1
), a pu e an i- esonance (
G2
),
and he combina ion o bo h (
G3
), which ep esen s he obse ed esonance-an i esonance beha iou .
G1(s)=ω2
s2+2δ ω s+ω2
(10)