Lab on a Chip
PAPER
Ci e his: Lab Chip,2014,14,351
Recei ed 8 h July 2013,
Accep ed 30 h Sep embe 2013
DOI: 10.1039/c3lc50809a
www. sc.o g/loc
Se pen ine channels: mic o- heome e s o luid
elaxa ion imes
Josephine Zilz,
a
Ch is o Schä e ,
b
Ch is ian Wagne ,
b
Robe J. Poole,
c
Manuel A. Al es
d
and Anke Lindne *
a
We p opose a no el de ice capable o measu ing elaxa ion imes o iscoelas ic luids as small as 1 ms.
In con as o mos heome e s, which by hei e y na u e a e conce ned wi h p oducing iscome ic o
nea ly- iscome ic lows, he e we make use o an elas ic ins abili y ha occu s in he low o iscoelas ic
luids wi h cu ed s eamlines. To calib a e he heome e we combine simple scaling a gumen s wi h
elaxa ion imes ob ained om i s no mal-s ess di e ence da a measu ed in a classical shea heome-
e . As an addi ional check we also compa e hese elaxa ion imes o hose ob ained om Zimm heo y
and good ag eemen is obse ed. Once calib a ed, we show how he se pen ine heome e can be used
o access smalle polyme concen a ions and lowe sol en iscosi ies whe e classical measu emen s
become di icul o impossible o use due o ine ial and/o esolu ion limi a ions. In he absence o cali-
b a ion, he se pen ine channel can s ill be a e y use ul compa a i e o index de ice.
1. In oduc ion
Polyme solu ions o long and lexible polyme s a e known o
exhibi s iking non-New onian p ope ies e en a e y small
concen a ions.
1
Fo example, in u bulen pipe o channel
low he ic ion ac o (o d ag) may be signi ican ly educed
by adding a polyme a concen a ions as low as a ew ppm
2
(pa s pe million in weigh ). Such luids a e also used in
enhanced oil eco e y applica ions.
3
Measu ing hei heo-
logical ea u es is a challenging ask and classical heome y
is o en a i s limi s when de e mining o example he elaxa-
ion imes o such dilu e polyme solu ions.
4
He e we de elop a mic o luidic heome e wi h a complex
low geome y o o e come hese di icul ies. Al hough a
numbe o mic o luidic heome e s ha e been p oposed,
mos o hese de ices a e es ic ed o measu emen s o shea
iscosi y,
5–11
al hough de ices which a emp o es ima e
ex ensional iscosi y
12–16
and dynamic p ope ies
17
ha e also
been p oposed. In con as o hese p e ious mic o luidic
de ices, in he cu en s udy we make use o an elas ic ins a-
bili y,
18–26
which occu s in lows o iscoelas ic luids wi h
cu ed s eamlines e en in he absence o ine ia.
20,21
The
h eshold o ins abili y depends on he cu a u e o he low
and he luid elas ici y,
27
desc ibed by he Weissenbe g num-
be . Typically, iscoelas ic e ec s will be obse ed when he
p oduc o a luid elaxa ion ime (λ) and a cha ac e is ic
shea a e eaches o de one. Thus o luids wi h λon he
o de o milliseconds, shea a es on he o de o 10
3
s
−1
a e
equi ed o access such iscoelas ic e ec s. The use o a
mic o luidic de ice enables high shea a es o be ob ained
and hus s ong iscoelas ic e ec s (co esponding o la ge
Weissenbe g numbe s) o be obse ed while keeping ine ial
e ec s, and hence he Reynolds numbe , small.
We ha e ecen ly in es iga ed he low in a se pen ine
mic o-channel o elucida e he scaling o he ins abili y
h eshold wi h he low cu a u e using a dilu e polyme solu-
ion.
28
We ha e shown ha he ins abili y is e y sensi i e o
e en small no mal-s ess di e ences and can hus be used o
de ec hei occu ence. We can now combine ou p ecise
knowledge ega ding he dependence o he ins abili y onse
on he low cu a u e wi h a p ecise knowledge o he heo-
logical p ope ies o a calib a ing luid o quan i a i ely mea-
su e elaxa ion imes. To do so, we ini ially calib a e he
se pen ine heome e using classical shea heome y in he
ange o pa ame e s accessible by his echnology. The se pen-
ine heome e can hen be used wi h luids o smalle
concen a ions o lowe sol en iscosi ies, whe e classical
heome y echniques become di icul ei he due o ine ial
ins abili ies o ins umen esolu ion issues. E en when a p e-
cise calib a ion is no possible, he se pen ine channel can be
used as a compa a i e heome e o compa e he heological
a
PMMH, ESPCI, UPMC, Uni . Pa is-Dide o , CNRS UMR 7636 10, ue Vauquelin,
F-75231 Pa is Cedex 05, F ance. E-mail: anke.lindne @espci.
b
Expe imen alphysik, Saa land Uni e si y, D-66123, Saa b ücken, Ge many
c
School o Enginee ing, Uni e si y o Li e pool, B ownlow Hill, Li e pool,
L69 3GH, UK
d
Depa men o de Engenha ia Quimica, Faculdade de Engenha ia da Uni e sidade
do Po o, Rua D . Robe o F ias, 4200-465 Po o, Po ugal
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p ope ies o wo gi en luids. Finally we p opose me hods o
ully in eg a e he se pen ine channel in o a mic o luidic
lab-on-a-chip de ice capable o measu ing bo h shea iscosi y
and luid elaxa ion ime.
2. Scaling o he onse o elas ic
ins abili y in a se pen ine channel
Pakdel and McKinley
21,27
p oposed a simple dimensionless
c i e ion ha mus be exceeded o he onse o pu ely-elas ic
ins abili y, combining he cu a u e o he low and he ensile
s ess (τ
11
) ac ing along he s eamlines, in he ollowing o m:



11 2

uM
R




c i ð1Þ
wi h ,uand  ep esen ing he local s eamline adius o
cu a u e, eloci y magni ude and shea a e, espec i ely. τ
11
ep esen s he local s eamwise no mal-s ess and η he local
shea s ess, wi h ηbeing he shea iscosi y. The a io τ
11
/η
hus ep esen s a local Weissenbe g numbe (Wi), compa ing
no mal s esses o shea s esses and λu/compa es a ypical
dis ance o e which a polyme elaxes o he adius o cu a-
u e (o can be iewed as a local Debo ah numbe ).
We ha e ecen ly elucida ed he geome ical scaling o
he onse o elas ic- low ins abili y in a se pen ine channel by
adap ing he Pakdel–McKinley c i e ion o he speci ic low
geome y.
28
The se pen ine channel is composed o a se ies
o ci cula hal -loops o al e na ing cu a u e o cons an
wid h (W), heigh (H) and inne adius (R), as shown sche-
ma ically in Fig. 1a. Fo easons o simplici y in ou analysis,
o he shea -domina ed low in he se pen ine channel we
used he uppe -con ec ed Maxwell (UCM) model, neglec ing
he sol en iscosi y (η
s
) con ibu ion. The o al iscosi y (η)
is hus simply equal o he polyme iscosi y (η
p
)(i.e. η=η
p
)
and he no mal-s ess is app oxima ed as τ
11
=2η
p
λ
2
. In his
case he a io τ
11
/ηbecomes equal o 2λ, co esponding o
wice he Weissenbe g numbe . A simple analysis based on
he Pakdel–McKinley c i e ion (eqn (1)) showed ha he
c i ical Weissenbe g numbe a ins abili y onse (Wi
c
) can be
w i en as a squa e oo dependence on he no malized
adius (R/W) wi h a small o -se a small adii. Fo a channel
wi h squa e c oss sec ion, ou nume ical esul s
28
(see Fig. 2)
a e bes desc ibed in he ollowing o m:
WicCR
W
1ð2Þ
No e ha he nume ical alue o he o se a small adii
ound om he nume ical esul s di e s sligh ly om he
heo e ical p edic ion gi en in e . 28 as he low asymme y
occu ing a s ong cu a u e is no cap u ed by ou heo-
e ical model, as has al eady been poin ed ou in e . 28. The
p edic ed scaling is in excellen ag eemen wi h expe imen al
obse a ions, as shown in Fig. 2. He e we wan o go u he
and no only ob ain he scaling o he ins abili y onse wi h
he low geome y, bu each a quan i a i e p edic ion o he
ins abili y h eshold. To do so, one i s has o ake he sol en
iscosi y con ibu ion in o accoun , which canno be neglec ed
o he dilu e polyme solu ions we use. When using an app o-
p ia e model, o example he Old oyd-B model,
1
o desc ibe
he polyme heology so ha he o al iscosi y (η)iscom-
p ised o bo h a polyme con ibu ion (η
p
) and a sol en con-
ibu ion (η
s
), i.e. η=η
p
+η
s
, he scaling o he ins abili y
onse has o be co ec ed
21,29
using τ
11
/η=2(η
p
/η)λ.Bydoing
so and hen ew i ing a modi ied o m o eqn (2) o ob ain he
c i ical shea a e, one ob ains:
c
p





CR
W
1ð3Þ
whe e C/


pcan be iden i ied as he slope om a plo
o he c i ical shea a e (
c
) s. 1RW
.A hisjunc u ei
Fig. 1 a) Schema ic o he mic o luidic se pen ine channel. b)
Snapsho s om he expe imen s showing he ins abili y onse .
Solu ions o PEO a e injec ed in o a mic ochannel ia wo inle s; only
one s eam con ains luo escen dye and is isible on he snapsho s.
Le hand side: s able low below ins abili y onse , igh hand side:
uns able low abo e ins abili y onse .
Fig. 2 Geome ic scaling o he ins abili y onse . The g een iangles
a e nume ical esul s and he ed ci cles a e esul s om expe imen s.
The do ed line is a i o eqn (2). Da a om Zilz e al.
28
using a
solu ion o 125 ppm o PEO 2Mio.
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is also use ul o de ine a pa ame e a=λ/C.Tobeable o
make a quan i a i e p edic ion o he elaxa ion ime (λ) om
measu emen s o he c i ical shea a e (
c
) one hus needs a
calib a ion expe imen o de e mine Cand he a io o he
polyme o he sol en iscosi y. We no e also ha he se pen-
ine heome e s ic ly only allows o a quan i a i e measu e-
men o he polyme elaxa ion ime, as long as he heology
o he solu ion is such ha he a io be ween he no mal
s esses and he shea s esses is p opo ional o Wi.
3. Expe imen al
3.1. The polyme solu ions
A solu ion o he lexible polyme polye hylene oxide (PEO),
supplied by Sigma Ald ich, wi h a nominal molecula weigh
(M
w
)o 2×10
6
g mol
−1
and wo di e en ba ches o PEO wi h
nominal molecula weigh s o 4 ×10
6
g mol
−1
, a concen a-
ions anging om 125 ppm o 500 ppm (w/w), we e used in
wa e –glyce ol mix u es. In he ollowing, he di e en poly-
me s will be e e ed o as 2Mio, 4Mio-1 and 4Mio-2, espec-
i ely. The o e lap concen a ion o 2Mio is c*≃860 ppm
30
and o he 4Mio solu ions, es ima ed using he equa ions
p o ided by Rodd e al.,
30
is c*≃550 ppm and he e o e he
solu ions a e dilu e in all cases (c/c*<1). The sol en iscosi y
(η
s
) a ied om η
s
= 1 mPa s o pu e wa e o 10.7 mPa s, a
20 °C, o a ying concen a ions o glyce ol. All he polyme
solu ions and concen a ions used a e summa ized in Table 1.
3.2. The se pen ine channel
The se pen ine channels used in his wo k consis o a se ies
o 8 hal -loops o wid h (W) = 100 μm, heigh (H)=80μm
and a ying inne adius (R). The numbe o loops and he
geome y o he inle s o he se pen ine channels used in his
s udy a e ep esen ed in Fig. 1a and ha e been desc ibed in
de ail in Zilz e al.
28
Expe imen s pe o med wi h a ying
numbe s o loops (N) con i med ha he esul s p esen ed
he e a e independen o he exac numbe o loops, p o ided
ha 2 <N<15. The channels a e made om PDMS, bu due
o he small iscosi y o he polyme solu ions used, he
applied p essu e emained su icien ly small o a oid de o -
ma ion o he channels. The solu ions a e supplied o he
mic o-channel ia wo inle s, one s eam o which is
luo escen ly dyed. The ligh g ey a ea isible in he snap-
sho s o Fig. 1b shows he loca ion o he luo escen ly
labeled luid. I s wid h a ia ion along he s eamwise di ec-
ion shows he sligh asymme y o he low ield along he
low pa h due o local low accele a ion in he cu ed geo-
me y. No e ha he channel wid h is cons an o e he whole
leng h o he channel. Fig. 1b shows a s able low si ua ion
in he le panel and an uns able low si ua ion in he igh
panel. In his way he s abili y o he low can easily be isually
assessed and will always be moni o ed a he las loop. The
ime-dependen low is easily iden i iable in he eal- ime low
isualiza ion. The low a e (Q) was a ied om 1 o 50 μlmin
−1
,
and was imposed ia a sy inge pump (PHD 2000, Ha a d
appa a us). The Reynolds numbe (Re) is de ined as Re =
ρUW/η,wi hρ ep esen ing he densi y o he luid and U=
Q/WH ep esen ing he a e age eloci y in he channel. The
maximum Re, co esponding o he highes low a e and
lowes iscosi y solu ion, ne e exceeded 5. The low is isu-
alized using an in e ed mic oscope (Axio Obse e , Zeiss)
coupled o a CCD came a (PixeLink). S a ing wi h he lowes
low a e, Qis hen g adually inc eased. A e each change in
Qa su icien ly long ime is allowed o achie e s eady-s a e
low condi ions (on he o de o 10 minu es pe s ep). The
onse o luc ua ions in he low de ines he onse o he
ime-dependen elas ic ins abili y, and he c i ical low a e
(Q
c
) is de e mined. F om he c i ical low a e, we ob ain he
c i ical a e age shea a e (
c
), de ined as 
c
=Q
c
/(W
2
H).
As an example, Fig. 3 illus a es he esul s ob ained o
PEO 2Mio a a concen a ion o 400 ppm. Simila expe i-
men s (no shown) ha e been pe o med a di e en concen-
a ions and o di e en molecula weigh s, as gi en in
Table 1. In pa allel o each expe imen , he sol en iscosi y
and he a io be ween he polyme and sol en iscosi y (η
p
/η
s
)
was de e mined using an Ubbelohde capilla y iscome e . In
Table 1 Polyme solu ions used in he se pen ine channel and i
pa ame e s o λ=Aη
s0.9
Polyme solu ions used in he se pen ine channel
M
w
Concen a ion η
p
/η
s
2Mio 125–500 ppm 7%–39%
4Mio-1 400 ppm 34%
4Mio-2 400 ppm 34%
Fi pa ame e s o λ=Aη
s0.9
om classical heome y
M
w
Concen a ion Ams/(mPa s)
0.9
2Mio 400 ppm 0.25 ± 0.02
4Mio-1 400 ppm 0.59 ± 0.02
4Mio-2 400 ppm 0.95 ± 0.04
Fig. 3 C i ical shea a e (
c
) as a unc ion o he no malized adius
(1 + R/W) o solu ions o PEO 2Mio a a concen a ion o 400 ppm o
di e en sol en iscosi ies (η
s
). The do ed lines ep esen i s o he
da a using eqn (3). Each expe imen was epea ed a leas wo imes
using esh polyme solu ions and he a e age alue is shown oge he
wi h he e o ba s.
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his way i was possible o co ec o small changes in em-
pe a u e ha occu ed in he labo a o y ( ypically be ween
20 °C and 23 °C).
3.3. The classical o a ional heome e
A comme cial o a ional shea heome e (MARS II, The mo
Scien i ic) in combina ion wi h a cone-and-pla e geome y
(diame e (D) = 60 mm, cone angles (α)=2°and 1°) in shea
a e con olled mode was used o measu e he iscosi y (η)
and he i s no mal-s ess di e ence (N
1
=τ
11
–τ
22
) simul a-
neously, ollowing he me hodology laid ou by Zell e al.
31
In
models o dilu e polyme solu ions, τ
22
is negligible
32
in he
s eady simple shea low expe imen s and hus N
1
is iden ical
o he s eamwise no mal-s ess (τ
11
). The shea a e was
inc eased in a s ep-wise p o ocol om 1 s
−1
up o a maximum
o 3000 s
−1
, wi h 15 s o equilib a ion ime a each shea a e.
The empe a u e was kep cons an a T= 20 ± 0.5 °Cby
using a Haake Phoenix II e ige a ed ci cula o . The no mal-
s ess da a o he polyme solu ions we e co ec ed by aking
in o accoun ine ial con ibu ions ha can be easily ob ained
om N
1
measu emen s o he New onian sol en s. The con-
s an sol en iscosi ies (η
s
()≡η
s
) we e also measu ed. The
Ψ
1
da a o he polyme solu ions we e ob ained om qua-
d a ic i s o he co ec ed da a o N
1
acco ding o N
1
()=
Ψ
1

2
wi hin adequa e anges o shea a e. An example o he
no mal-s ess da a (N
1
) oge he wi h he esul s o he shea
iscosi y (η) is shown in Fig. 4a o PEO 2Mio a a concen a-
ion o 400 ppm. A quad a ic i o he no mal-s ess da a is
also indica ed. No e ha o he ep esen a ion o he da a, an
a e age o e se e al uns (a leas 3) has been plo ed. The
elaxa ion ime (λ) o he polyme was de e mined by aking
  



11
22
ps
() [() ()]
 ð4Þ
When calcula ing he elaxa ion ime we e e o he poly-
me iscosi y a a ixed shea a e o = 100 s
−1
and neglec
he sligh shea - hinning beha io o he polyme solu ions.
The associa ed unce ain ies, δη,δη
s
and δΨ
1
( om he s a is-
ics o mul iple independen measu emen s), can be
in e p e ed in e ms o an es ima e o he unce ain y o λ,
i.e.
 
 
22
1
2
1
222
22








pp
swi h mean
alues deno ed by an o e ba . The a ia ions o λ=±δλ
wi h sol en iscosi y a e shown in Fig. 4b and can be
desc ibed as λ=Aη
s0.9
, which has been ound as he bes i
o all h ee cu es p esen ed in Fig. 4b. The i pa ame e s
(A) a e shown in Table 1.
Fo a polyme chain in a good sol en , acco ding o
Zimm’s heo y,
32
he longes elaxa ion ime o a dilu e solu-
ion can be es ima ed using

Zimm
ws
AB
FM
NkT
[] ð5Þ
whe e N
A
is he A ogad o cons an , k
B
is he Bol zmann con-
s an (and he p oduc N
A
k
B
is equal o he uni e sal gas
cons an , R), Tis he absolu e empe a u e and [η] is he
in insic iscosi y. Ti aa madja e al.
33
ha e shown expe i-
men ally ha his can be exp essed as [η] = 0.072M
w0.65
o
he PEO solu ions s udied he e (gi ing [η] in he usual uni s
o ml g
−1
). The p e ac o (F) is gi en by Rodd e al.
34
o be
0.463. Rodd e al.
30
measu ed he in insic iscosi y o di e -
en polyme –wa e –glyce ol mix u es and ha e shown ha i
dec eases due o a dec ease in sol en quali y. As a conse-
quence, he dependence o λon η
s
becomes less han linea .
In Fig. 5a we show how hese es ima es o he Zimm elaxa-
ion ime compa e o hose de e mined om he i s
no mal-s ess di e ence measu ed in he cone-and-pla e o a-
ional heome e , illus a ing ha he ag eemen is e y good.
Gi en he polydispe si y inhe en in such comme cial poly-
me s, and ba ch- o-ba ch a ia ions which he da a in Fig. 4b
highligh , he almos quan i a i e ag eemen be ween he wo
es ima es o λis s iking (especially gi en he a ious con-
s an s used in he de e mina ion o λ
Zimm
om eqn (5)). Such
ag eemen p o ides con idence in he obus ness o ou es i-
ma es o he elaxa ion ime and hence in he calib a ion o
he se pen ine heome e .
Fig. 4 a) Fi s no mal-s ess di e ence (N
1
) as a unc ion o he shea
a e om classical heome y o PEO 2Mio a 400 ppm. A quad a ic i
o he no mal-s ess da a is also indica ed. Inse : shea iscosi y (η)asa
unc ion o shea a e. No e ha o he ep esen a ion o he da a, an
a e age o e se e al uns (a leas 3) has been plo ed. b) λ s. η
s
om
classical shea heome y a a polyme concen a ion o 400 ppm. The
i s co espond o λ=Aη
0.9
s
.
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4. Calib a ion o he se pen ine
heome e
To calib a e he se pen ine heome e he alue o C(eqn (3))
has o be de e mined. To do so we compa e esul s o a=λ/C
om he se pen ine channel o elaxa ion imes om classical
heome y. To ob ain be e accu acy, we no only compa e
hese alues o a single polyme solu ion, bu also use solu-
ions o PEO 2Mio wi h di e en sol en iscosi ies and di -
e en concen a ions.
Fi s ly we de e mine he alues o a=λ/C om he slope
ob ained om he i s o he c i ical shea a e e sus he
adius o cu a u e (Fig. 3) and he a io o he polyme o
he sol en iscosi y o PEO 2Mio o wo di e en concen-
a ions, 125 ppm and 400 ppm. These esul s a e hen com-
pa ed o he esul s o λ om he classical shea heome e
(see Fig. 4b) o PEO 2Mio a a concen a ion o 400 ppm.
No e ha i was no possible o ob ain eliable measu emen s
o he smalle concen a ion o 125 ppm on he classical
heome e . Fig. 5b shows λ oge he wi h aas a unc ion o
he sol en iscosi y (η
s
).
A numbe o hings should be ema ked om Fig. 5b.
Fi s , all h ee da a se s show a compa able dependence o λ
on he sol en iscosi y, which we will con inue o desc ibe
as λ∼η
s0.9
. Second, he esul s ob ained om he se pen ine
heome e o he wo di e en concen a ions a e in good
ag eemen . This shows ha ou co ec ion o he sol en
iscosi y is adequa e and is a i s alida ion ha he p o-
posed heome e wo ks e y well. We adjus ed a=λ/C=
Bη
s0.9
and ob ained B= 4.99 ± 0.23 ms/(mPa s)
0.9
as he bes
i o bo h cu es oge he . Finally by compa ing A=0.25±
0.02 ms/(mPa s)
0.9
(see Table 1) om he classical heome y
o he alue o B om he se pen ine heome e we ob ain
C=0.05.
5. Using he se pen ine channel
We now discuss possible applica ions o he se pen ine
heome e .
5.1. A quan i a i e heome e
5.1.1. Ex ension o lowe concen a ions. Fi s we measu e
he elaxa ion imes o di e en concen a ions (c) o a gi en
polyme . As we a e wo king in he dilu e egime (c<c*), o
a gi en molecula weigh we expec o ob ain iden ical
elaxa ion imes. The esul s o PEO 2Mio a concen a ions
o c= 125 ppm, 250 ppm and 500 ppm o a sol en iscosi y
o η
s
= 4.9 mPa s a e p esen ed in Fig. 6a. These esul s we e
ob ained om an independen se ies o measu emen s and
each expe imen has been epea ed h ee imes using esh
polyme solu ions. The esul s a e compa ed o he alue o
he elaxa ion ime a c= 400 ppm om he calib a ion cu e
(highligh ed by a ci cle in Fig. 6a). No e ha we did no
include he alue o c= 125 ppm om he calib a ion cu e
as we did no pe o m expe imen s wi h he co esponding
sol en iscosi y o his concen a ion.
No wi hs anding he a he la ge unce ain y o he
smalles concen a ion, hese esul s a e e y p omising; a
alue o λ≈1.2 ms is ound independen o he polyme
concen a ion. No e ha i was no possible o measu e he
elaxa ion ime o he low concen a ion, c= 125 ppm, using
he o a ional heome e , indica ing he supe io sensi i i y
o he se pen ine channel, which is able o measu e e y
small elaxa ion imes down o e y small concen a ions.
This is in ag eemen wi h he indings om he calib a ion
cu e ha as long as he co ec ion o he a io be ween he
sol en and he polyme iscosi y is made co ec ly, iden ical
esul s a e ob ained o di e en concen a ions in he dilu e
egime. We ema k ha addi ional minia u iza ion o he
Fig. 5 a) Relaxa ion ime om classical shea heome y (eqn (4)) compa ed o he Zimm elaxa ion imes
30,34
(eqn (5)). b) Relaxa ion ime om
classical shea heome y (eqn (4)) e sus sol en iscosi y (η
s
) o PEO 2Mio a a concen a ion o 400 ppm (le axis). a=λ/C(eqn (3)) om he
se pen ine heome e e sus sol en iscosi y (η
s
) o PEO 2Mio a concen a ions o 125 ppm and 400 ppm ( igh axis). The e o o he da a om
he se pen ine heome e co esponds o he e o o he i o 
c
(eqn (3)), and o he da a om he classical heome e he e o s a e es ima ed
by an e o p opaga ion acco ding o he unce ain ies o Ψ
1
and η
p
.
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se pen ine channel enhances he elas ic e ec s, hus inc eas-
ing u he he sensi i i y o his heome ic de ice. A ecen
e iew
16
discusses he challenges o measu ing iscoelas ic
p ope ies o dilu e polyme solu ions, highligh ing he ele-
ance o using mic o luidic de ices, as in ou wo k, o p obe
he elas ic p ope ies o such polyme solu ions.
5.1.2. Changing he molecula weigh . Secondly we
measu ed he elaxa ion ime o solu ions o PEO o
di e en molecula weigh : 2Mio and wo di e en ba ches o
4Mio, deno ed 4Mio-1 and 4Mio-2. F esh solu ions o all
h ee polyme s we e p epa ed a 125 ppm and a sol en is-
cosi y o η
s
= 5.2 mPa s. The elaxa ion imes ob ained we e
compa ed o he elaxa ion imes om classical shea
heome y measu ed a 400 ppm. As we a e in he dilu e
egime, no dependence o λwi h he polyme concen a ion
is expec ed, and we ha e explici ly shown ha his is ue o
he solu ion o PEO 2Mio in Fig. 6a. The alues used om
he classical o a ional heome y a e calcula ed using he i
pa ame e s om Table 1 o ob ain he elaxa ion imes a he
sol en iscosi y o η
s
= 5.2 mPa s.
The esul s om he se pen ine channel a e plo ed in
Fig. 6b agains he esul s om he classical heome y and
sa is ac o y ag eemen be ween hese wo independen ech-
niques is ob ained. No e ha he ac ha he e o ba s a e
smalle han he di e ences be ween hese wo measu e-
men s in some cases is e y likely due o he ac ha inde-
penden ly p epa ed polyme solu ions ha e been used in
each measu emen on he wo di e en de ices.
In addi ion, as was ac ually al eady appa en om he
classical shea heology da a in Fig. 4b, we no e ha he da a
o he wo ba ches o 4Mio PEO highligh la ge ba ch- o-
ba ch a ia ions ha can occu be ween nominally-iden ical
samples. The e is a ac o o wo di e ence in elaxa ion
ime be ween bo h samples. Such di e ences a e likely a
consequence o di e ing deg ees o polydispe si y in he
wo ba ches.
5.2. In eg a ion in o a mic o luidic lab-on-a-chip de ice
As he o egoing makes clea , o de e mine a quan i a i e
alue o he elaxa ion ime using he se pen ine heome e ,
an independen measu emen o he polyme con ibu ion o
he o al iscosi y (and indeed a measu emen o he sol en
iscosi y i i is unknown) is equi ed such ha he alue o

pcan be de e mined o use in eqn (3). In he cu en
s udy, hese alues we e ob ained om sepa a e measu e-
men s using an Ubbelohde capilla y iscome e . Ideally one
would like o be able o de e mine bo h he iscosi y a io
and he c i ical shea a e om a single mic o luidic lab-on-
a-chip de ice. To do so one could ei he use p essu e d op
measu emen s in a s aigh sec ion ups eam o he se pen-
ine channel, as in Pipe e al.
12
o example, o , al e na i ely,
use he Y-junc ion app oach o Guillo e al.
6
o Nghe e al.
9
As he easies me hod o obse e he pu ely-elas ic ins abili y
in he se pen ine channel is ia an op ical isualiza ion ech-
nique, in eg a ion in o a sys em based on he la e app oach
is p obably o be p e e ed, he eby a oiding he equi emen
o inco po a ing p essu e senso s in o he de ice. To a oid
issues o possible polyme deg ada ion due o he ins abili y,
in he p essu e-d op case he iscome e sec ion o he de ice
should be inco po a ed ups eam o he se pen ine channel
and, in he Y-junc ion case, whe e a e e ence luid is equi ed,
in a sepa a e pa allel mic o-channel on he same chip.
Fig. 6 a) Relaxa ion ime (λ) ob ained om he se pen ine heome e o PEO 2Mio in a sol en wi h iscosi y η
s
= 4.9 mPa s a a ying
concen a ions, anging om 125 ppm o 500 ppm. The da a poin s ep esen he a e age o e h ee se s o expe imen s and he co esponding
e o ba s. The elaxa ion ime a 400 ppm co esponds o he alue om he calib a ion cu e. b) Relaxa ion imes om he se pen ine heome e
e sus elaxa ion imes om classical shea heome y. The esul s om he se pen ine heome e co espond o a concen a ion o 125 ppm o
he h ee di e en polyme s and a sol en iscosi y o η
s
= 5.2 mPa s. The e o ba s a e es ima ed om he e o o he i o 
c
. The esul s om
he classical heome e o 400 ppm solu ions a e calcula ed om he i s o he elaxa ion ime wi h he sol en iscosi y, see Table 1. In his
case, he e o is es ima ed om he e o o he i on A.
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5.3. Using he se pen ine heome e as a compa a o o index
de ice wi h an applica ion o polyme deg ada ion
I one is no conce ned wi h he absolu e alue o a luid's
elaxa ion ime pe se, bu a he wi h indexing di e en
luids acco ding o hei deg ee o elas ici y, hen he se pen-
ine channel ep esen s an ex emely e icien de ice. In his
scena io i is simply su icien o es he di e en luids in a
single channel o known cu a u e and de e mine he c i ical
shea a e in each case. Following eqn (3), his di ec ly leads
o 
c,1
/
c,2
=λ
2
/λ
1
. O cou se, s ic ly speaking, his equali y
only holds o luids whe e he a io η/η
p
emains cons an .
In he absence o quan i a i e in o ma ion ega ding he con-
ibu ion o polyme iscosi y, a p agma ic app oach, i he
c i ical o e lap concen a ion is known, is o use he scaling
32
c*[η] ~ 1, which gi es η
p
~η
s
c/c*,o η
/
η
p
=1+c*/c.
Al e na i ely he c i ical shea a e can be used o mul i-
ple epea expe imen s o he same luid o es o deg ada-
ion. O en one needs o check i polyme deg ada ion has
occu ed due o ei he pho o-induced, mechanical, he mal,
chemical o biological causes.
9,35,36
Simple shea iscosi y
measu emen s a e o en ai ly insensi i e o such e ec s as
deg aded polyme s o en s ill con ibu e o he o e all is-
cosi y o he solu ion, which ends o be domina ed by he
sol en iscosi y in any case o dilu e polyme solu ions
(and is he e o e a small e ec ). In con as , he polyme
elaxa ion ime is a much mo e sensi i e ha binge o deg a-
da ion. In Fig. 7 we show he e ec o pho o-induced deg a-
da ion on a i gin polyme solu ion, i.e. unshea ed, s o ed
a oom empe a u e in a clea bo le exposed o sunligh
o e a pe iod o se e al days. Each day a new measu emen
was made and, a e se en days, a p ecu si e sligh inc ease
in he c i ical shea a e was obse ed, which was ollowed
by des uc ion o he sample ia he o ma ion o bio ilms.
Finally, ou expe ience wi h he se pen ine heome e sug-
ges s i s sensi i i y makes i an ideal ins umen o
moni o ing possible deg ada ion e ec s, ega dless o he
p ecise deg ading mechanism.
6. Conclusions
By unde s anding he scaling beha iou o he onse o a
pu ely-elas ic low ins abili y in a mic o luidic se pen ine
channel, we ha e p oposed a mic o luidic heome ic de ice
ha is capable o measu ing luid elaxa ion imes down o
1 ms. In con as o mos o he heome e s, which aim o
p oduce iscome ic lows o enable he ex ac ion o heolog-
ical p ope ies, he de ice makes use o an inhe en ins abil-
i y wi hin he low o es ima e he luid elaxa ion ime.
Al hough using in e acial ins abili ies has p e iously been
en a i ely p oposed
22,26
as a means o es ima ing ma e ial
p ope ies, as has he onse o iscoelas ic o ices,
37
he cu -
en me hod, which only equi es a single luid, is he i s o
show ha elaxa ion imes can be success ully measu ed
using such an app oach. Also, e y ecen ly, Kose e al.
38
ha e p oposed using c eep- eco e y es s in a mic o luidic
de ice o es ima e polyme elaxa ion imes. Howe e , hei
de ice equi es he use o a high-speed came a and is
es ic ed o elaxa ion imes a leas an o de o magni ude
g ea e han hose measu ed he e. The se pen ine heome e
can ei he (a) be used as a compa a o o indexing de ice, in
which case luids can be anked acco ding o hei elas ici y,
o changes can be moni o ed, such as hose due o deg ada-
ion o (b) be used as a ue heome e once calib a ion es s
using classical cone-and-pla e heome y, o example, ha e
been pe o med. In his la e case, he mic o luidic se pen-
ine de ice can access lowe molecula weigh ma e ials, sol-
en iscosi ies o concen a ions han is possible using s a e-
o - he-a comme cial heome e s.
Acknowledgemen s
Some o his wo k was unde aken whils R. J. P. was a isi -
ing "Chai e Michelin"a he ESPCI Pa is Tech in June 2013
and his suppo is he eby g a e ully acknowledged. M. A. A.
acknowledges unding om he Eu opean Resea ch Council
(ERC), unde he Eu opean Commission "Ideas"Speci ic
P og amme o he Se en h F amewo k P og amme (g an
ag eemen no. 307499).
Re e ences
1 R. Bi d, R. A ms ong and O. Hassage , Dynamics o
Polyme ic Liquids, 1987.
2 P. S. Vi k, AIChE J., 1975, 21, 625–656.
3 D. Jones and K. Wal e s, Rheol. Ac a, 1989, 28, 482–498.
4 A. Lindne , J. Ve man and D. Bonn, Phys. A, 2003, 319,
125–133.
5 P. Guillo , P. Panizza, J.-B. Salmon, M. Joanico , A. Colin,
C.-H. B uneau and T. Colin, Langmui , 2006, 22, 6438–6445.
6 P. Guillo , T. Moulin, R. Kö i z, M. Gui a del, A. Dodge,
M. Joanico , A. Colin, C.-H. B uneau and T. Colin,
Mic o luid. Nano luid., 2008, 5, 619–630.
Fig. 7 C i ical shea a e measu ed o e ime using he se pen ine
heome e wi h wo di e en adii o a PEO sample exposed o
sunligh . The e o ba s co espond o he unce ain y in measu ing
he c i ical low a e.
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7 C. J. Pipe, T. S. Majmuda and G. H. McKinley, Rheol. Ac a,
2008, 47, 621–642.
8 J. Che alie and F. Ayela, Re . Sci. Ins um., 2008, 076102.
9 P. Nghe, P. Tabeling and A. Ajda i, J. Non-New onian Fluid
Mech., 2010, 165, 313–322.
10 E. Li ak-Dahl, J. Lee and M. A. Bu ns, Lab Chip, 2013, 13,
297–301.
11 H. Be he , J. Jund , J. Du i aul , B. Me cie and
D. Angelescu, Lab Chip, 2011, 11, 215–23.
12 C. J. Pipe and G. H. McKinley, Mech. Res. Commun.,
2009, 36, 110–120.
13 M. S. N. Oli ei a, L. E. Rodd, G. H. McKinley and
M. A. Al es, Mic o luid. Nano luid., 2008, 5, 809–826.
14 W. C. Nelson, H. P. Ka ehpou and C.-J. C. Kim, Lab Chip,
2011, 11, 2424–2431.
15 S. J. Hawa d, M. S. N. Oli ei a, M. A. Al es and
G. H. McKinley, Phys. Re . Le ., 2012, 109, 128301.
16 F. J. Galindo-Rosales, M. A. Al es and M. S. N. Oli ei a,
Mic o luid. Nano luid., 2013, 14,1–19.
17 G. F. Ch is ophe , J. M. Yoo, N. Dagalakis, S. D. Hudson and
K. B. Migle , Lab Chip, 2010, 10, 2749–2757.
18 R. G. La son, E. S. G. Shaq eh and S. J. Mulle , J. Fluid
Mech., 1990, 218, 573.
19 A. G oisman and V. S einbe g, New J. Phys., 2004, 6,29–29.
20 E. S. G. Shaq eh, Annu. Re . Fluid Mech., 1996, 28, 129–85.
21 G. McKinley, P. Pakdel and A. Oz ekin, J. Non-New onian
Fluid Mech., 1996, 67,19–47.
22 B. Khomami and K. Su, J. Non-New onian Fluid Mech.,
2000, 91,59–84.
23 A. G oisman and S. Quake, Phys. Re . Le ., 2004, 92,
094501.
24 R. J. Poole, M. A. Al es and P. J. Oli ei a, Phys. Re . Le .,
2007, 99, 164503.
25 A. N. Mo ozo and W. an Saa loos, Phys. Rep., 2007, 447,
112–143.
26 O. Bonhomme, A. Mo ozo , J. Leng and A. Colin, Phys. Re .
E: S a ., Nonlinea , So Ma e Phys., 2011, 83, 065301.
27 P. Pakdel and G. McKinley, Phys. Re . Le ., 1996, 77,
2459–2462.
28 J. Zilz, R. J. Poole, M. A. Al es, D. Ba olo, B. Le aché and
A. Lindne , J. Fluid Mech., 2012, 712, 203–218.
29 R. J. Poole, A. Lindne and M. A. Al es, J. Non-New onian
Fluid Mech., 2013, 201,10–16.
30 L. Rodd, J. Coope -Whi e, D. Boge and G. McKinley, J. Non-
New onian Fluid Mech., 2007, 143, 170–191.
31 A. Zell, S. Gie , S. Ra a and C. Wagne , J. Non-New onian
Fluid Mech., 2010, 165(19), 1–24.
32 R. G. La son, The S uc u e and Rheology o Complex
Fluids, 1999.
33 V. Ti aa madja, G. McKinley and J. Coope -Whi e, Phys.
Fluids, 2006, 18, 043101.
34 L. Rodd, T. Sco , D. Boge , J. Coope -Whi e and
G. McKinley, J. Non-New onian Fluid Mech., 2005, 129,1–22.
35 S. Vanapalli, M. Islam and M. Solomon, Phys. Fluids,
2005, 17, 095108.
36 S. A. Vanapalli, S. L. Ceccio and M. J. Solomon, P oc. Na l.
Acad. Sci. U. S. A., 2006, 103, 16660–16665.
37 L. Campo-Deano, F. J. Galindo-Rosales, F. T. Pinho,
M. A. Al es and M. S. N. Oli ei a, J. Non-New onian Fluid
Mech., 2011, 166, 1286–1296.
38 A. E. Kose , L. Pan, N. C. Keim and P. E. A a ia, Lab Chip,
2013, 13, 1850–1853.
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CEFT
Cen o de Es udos de Fenómenos de T anspo e
T anspo Phenomena Resea ch Cen e