Polymers and rheology: A tale of give and take

Polyme 271 (2023) 125811
A ailable online 1 Ma ch 2023
0032-3861/© 2023 The Au ho s. Published by Else ie L d. This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/).
Polyme s and heology: A ale o gi e and ake
Lei e Sang oniz, Me cedes Fe n´
andez, An xon San ama ia
*
POLYMAT and Depa men o Polyme s and Ad anced Ma e ials: Physics, Chemis y and Technology, Facul y o Chemis y, Uni e si y o he Basque Coun y UPV/EHU,
Paseo Manuel de La diz´
abal, 3, 20018, Donos ia-San Sebas i´
an, Spain
ABSTRACT
Polyme s and heology we e bo h bo n in he wen ies o he las cen u y, ui o he dis up i e and c ea i e scien i ic a mosphe e o his decade. The de elopmen o
polyme science and echnology in he las almos 100 yea s has been colossal, whe eas heology has been able o c ea e i s own ou e as a b anch o Physics. In his
e iew pape we desc ibe he in e ac ions be ween bo h scien i ic a eas, demons a ing ha many o he c ucial aspec s o he p og ess o one o hem owes o
con ibu ion o he o he and ice e sa. This in e ela ionship is shown h ough a his o ical su ey o he p incipal miles ones, s a ing om he co ela ion o he
molecula weigh wi h he in insic iscosi y. A ponde ed analysis o he con ibu ions o heology o polyme science and echnology, lead us o asse ha he ole o
he o me is a he ounded on 50 yea s old disco e ies.
1. The umul uous beginnings o polyme s and he quie bi h o
heology
The idea o long-chain molecules in ol ing only chemical bonds,
in oduced by He mann S audinge in 1924 and nowadays a unda-
men al concep in biology, chemis y and physics, was dis ega ded and
e en a acked by he chemis s o he ime. The eac ion o he scien i ic
communi y was logical in iew o he appa en soundness o he no ion
o colloids as agg ega es o small molecules, which was able o explain
he high iscosi y solu ion, gela ion and o he physical ea u es o gums,
cellulose and o he subs ances cu en ly de ined as polyme s. The
o ensi e agains he new dis up i e idea was in many occasions un-
pleasan and agg essi e o S audinge , who su e ed pe sonally om his
lack o conside a ion; his was ecognized by he scien i ic communi y
when he Ge man scien is was awa ded wi h he Nobel P ize o
Chemis y in 1953 [1,2].
Some examples o wha we now conside un ounded c i icisms a e
epo ed in he li e a u e. Fo ins ance, as ema ked by Mo a e z [3], he
e y in luen ial o ganic chemis Paul Ka e dismissed S audinge ’s idea
conside ing i idiculous ha s a ch would consis o hund eds o
glucose uni s joined by glucoside bonds, because “i is imp obable ha a
plan in con e ing suga o a ese e subs ance om which i migh
soon ha e o be eco e ed would pe o m such complex wo k”. Mo -
awe z e e s also o an unpleasan episode in he a ewell lec u e gi en
by S audinge a he ETH Zu ich in 1925, in which one o he speake s
compa ed him and his concep o long-chain molecules, wi h a a ele
in A ica ha had seen a 400 m long zeb a. In his lis o dis espec ul
commen s, i is also known [4] he iendly le e sen o S audinge by
he Nobel P ize lau ea e in 1927 Wieland, wi h his cen al message:
“Dea colleague, abandon you idea o long molecules, o ganic mole-
cules wi h molecula weigh s exceeding 5000 do no exis . Pu i y you
p oduc s such as ubbe , hey will c ys allize and u n o be low mo-
lecula weigh compounds”.
As poin ed ou by Mulhaup [1,2], a i m opposi ion o he concep o
polyme s came om he pionee ing scien is s in he ield o incipien
c ys allog aphy, who belie ed ha i was impossible o long-chain
molecules o i in he c ys allog aphic uni cell. Cu en ly i is known
ha only chain segmen s a e p esen in he uni cell. Pa adoxically,
c ys alliza ion o polyme s emains a p incipal subjec o s udy and
discussion in polyme s. Acco ding o he epo o T.P. Lodge on he
occasion o he 50 h anni e sa y o he e y in luen ial jou nal Mac o-
molecules [5], he heo y o polyme c ys alliza ion is posi ioned in i s
place in he lis o cu en challenges in polyme s.
Compa ed o he he oic e o s o S audinge o in oduce he
concep o polyme chains in 1920s, he bi h o heology in 1928 was
humble and peace ul. Ma cus Reine emembe ed [6] he summe o
1928 when he a i ed om Pales ine o Eas on Pennsyl ania in i ed by
Eugene Bingham who said o him: “He e you, a ci il enginee , and I, a
chemis , a e wo king oge he a join p oblems. Wi h he de elopmen
o colloid chemis y, such a si ua ion will be mo e and mo e common.
We he e o e mus es ablish a b anch o physics whe e such p oblems
will be deal wi h”. Reine ’s esponse was: “This b anch o physics
al eady exis s; i is called mechanics o con inuous media o mechanics
o con inua”. Bingham eplied: “No, his will no do. Such a designa ion
will igh en away he chemis s”. Then, he consul ed a p o esso o
classical languages a La aye e College and a i ed a he designa ion o
* Co esponding au ho .
E-mail add ess: [email p o ec ed] (A. San ama ia).
Con en s lis s a ailable a ScienceDi ec
Polyme
jou nal homepage: www.else ie .com/loca e/polyme
h ps://doi.o g/10.1016/j.polyme .2023.125811
Recei ed 6 Oc obe 2022; Recei ed in e ised o m 18 Feb ua y 2023; Accep ed 22 Feb ua y 2023
Polyme 271 (2023) 125811
2
he e m Rheology, in concomi ance wi h he sen ence o He acli us
“Pan a ei” (E e y hing lows).
I can be said ha , humbly and elegan ly, heology was bo n wi h he
in en ion o pleasing and a ac ing he chemis s (a leas o no dis u b
hem). The de ini ion o heology coined by Bingham was “Fundamen al
and p ac ical knowledge conce ning he de o ma ion o low o ma e ”.
Acco ding o Do aiwamy [7], he ounda ions o he Socie y o Rheology
we e es ablished in Ap il 1929, a a mee ing in Columbus (Ohio)
a ended by He schel, Os wald and o he scien is s, in addi ion o
Bingham and Reine . The Socie y o Rheology was o icially ounded on
Decembe 19, 1929 and became one o he i e ounding membe s o he
Ame ican Ins i u e o Physics. Bu , heology was no ee o c i icisms
and yea s la e T uesdell [8] emembe ed ha a nega i e de ini ion o
heology was in oduced a he ime h ough he sen ence: “Rheology
conce ns he luids ha luid-dynamicis s igno e”. Besides he ac ha
e lec s a ce ain igno ance, since heology also conce ns solids, i s
malicious in en ion ailed comple ely in iew o he ex ao dina y pe -
spec i es open o heologis s wi h he de elopmen o ma e ials o
peculia physical ea u es, such as polyme s. The de elopmen o
heology o become an independen science is soundly explained in he
book o Tanne and Wal e s “Rheology: An his o ical pe spec i e” [9],
wi h p ecise de ails o he o igins o heological socie ies, cong esses and
jou nals.
In his pape we show he c ucial ole played by heology in polyme
science and echnology and we ecall ha he p incipal ad ances in
heology, as non New onian low beha io and iscoelas ici y, owe o
he s udy o polyme s. This mu ual de o ion and s imulus be ween
polyme s and heology, has been emphasized, among o he s, by Kausch-
Blecken [10].
2. In insic iscosi y-molecula weigh ela ionships: A g ea
s ep o polyme s
The high iscosi ies o polyme solu ions, e en a dilu e concen a-
ions, d ew S audinge ’s a en ion who used his ci cums ance o
consolida e he concep o polyme s, and o ace hose who main ained
he hypo hesis o colloidal agg ega es. Based on iscosi y esul s o
polys y ene/benzene solu ions measu ed in an Os wald capilla y
iscome e , S audinge [11] obse ed a linea co ela ion be ween he
molecula weigh and he in insic iscosi y [
η
] =K
m
M, and so p oposed
an expe imen al me hod o deduce he molecula weigh o polyme s.
The molecula weigh conside ed by S audinge is now de ined as he
iscous a e age molecula weigh M
, and lies be ween he numbe
a e age molecula weigh , M
n
, and he weigh a e age molecula
weigh , M
w
. The in insic iscosi y is ob ained om he ela i e is-
cosi ies o polyme s solu ions o di e en concen a ions:
[
η
] ≡ lim
c→0(
η
−
η
s
c
η
s)(1)
η
being solu ion iscosi y,
η
s
he sol en iscosi y and c he concen-
a ion [12].
S audinge ’s expe imen s we e ollowed by o he au ho s, including,
o ins ance, he 1975 Nobel P ize lau ea e P.J. Flo y [13] who used he
ollowing co ela ion, es ablished by Goldbe g, Hohens ein and Ma k o
polys y ene [14]:
log MV=log[
η
]+4.013
0.74 (2)
M
being he iscosi y a e age molecula weigh .
La e , linea equa ions o he ela ionship be ween in insic is-
cosi y and molecula weigh we e subs i u ed by a be e app oach, he
powe law co ela ion o Ma k-Houwink equa ion [15,16]:
[
η
]=kMa
V(3)
K and a pa ame e s depend on he polyme /sol en couple and
empe a u e and hei espec i e alues a e a ailable in li e a u e o
mos polyme s. M
is he iscosi y a e age molecula weigh .
Molecula heo ies on dynamics o polyme s, de eloped a he end o
he o ies, ha e explained he co ela ions ound be ween in insic
iscosi y and molecula weigh [17]. The s udy o he kine ics o
monome ic uni s on he assump ion ha monome uni s do no in e ac
wi h each o he and do no dis o low, lead Debye [18] o demons a e
S audinge ’s law. This was co ec ed by Debye and Bueche [19] and by
Ki kwood and Riseman [20] who in oduced he idea o he shielding
e ec o he pe iphe al monome s o e in e io monome s, using,
espec i ely, di e en models o he in e nal s uc u e o he polyme
molecule. In e es ingly, hese molecula models gi e a physical meaning
o he a exponen o he empi ical Ma k Houwink equa ion. The hy-
d odynamic shielding de e mines he alue o a (be ween 0.5 and 1):
when p o ec ion is comple e, a =1 (S audinge law), whe eas when he
sol en eely pene a es he polyme , i d ains all he monome s inde-
penden ly o hei posi ion, hen a =0.5. These models a e alid o
polyme coils, since in he case o od-like polyme s, which gi e ise o
liquid c ys al polyme s, he exponen is a >1 [21]. The de elopmen o
models sus ained on iscosi y esul s o dilu e solu ions opened new
ou es o he s udy o he con o ma ion o polyme chains, as e lec ed
in Flo y’s book “P inciples o Polyme s Chemis y” [22].
Al hough he implica ions o he heology o polyme solu ions in he
ield o he Molecula Biology a e no in he scope o his e iew, i is
compulso y o ecall he wo k o Zimm e al. [23–25] on he iscoelas ic
beha io o ch omosomal-sized DNA. Thei heological analysis p o ed
ha he DNA o an euka yo ic cell consis s o jus one, ex emely big
mac omolecule (M
w
≈4 ×10
9
g/mol and con ou leng h ≈2 cm).
In ecen decades o he echniques, like ligh sca e ing and, in
pa icula , size exclusion ch oma og aphy (SEC), which allows de e -
mining M
n
, M
w
and highe a e ages o he molecula weigh dis ibu-
ion, ha e gained g ound. No wi hs anding, heological e alua ion o
he in insic iscosi y emains cu en ly a cheap and e y widely used
ool o ha e a i s app oach o he molecula weigh o he analyzed
polyme . The a he edious p ocedu e o ob aining he in insic is-
cosi y by means o Os wald o Ubbelohde [26–28] iscome e s is
cu en ly acili a ed by he use o au oma ed capilla y and o a ional
iscome e s.
3. The non-New onian low and i s implica ion in polyme
p ocessing: ga he ing scien is s and enginee s
In he opening session o a Gene al Discussion on colloids o ganized
by he Fa aday Socie y in 1913, Os wald [29] opined abou he iscosi y
o hese subs ances. He men ioned 10 ac o s a ec ing hei low p op-
e ies. Flow a e o he associa ed shea a e was no among hese
ac o s.
The no ion o he iscosi y o a liquid being dependen on he applied
shea a e was unknown un il he end o he wen ies and i s disco e y is
he mos impo an inding o he beginnings o heology as a newly
de ined b anch o physics. The analysis o colloidal dispe sions ca ied
ou by Os wald in 1925 [30] using he capilla y iscome e designed by
himsel , e ealed a dependence o he iscosi y on low eloci y. The
de ia ion om he so-called New onian low beha iou , acco ding o
which he iscosi y is independen o he applied shea a e, was clea ly
e idenced in he pape published by Reine in No embe 1929, on he
iscosi y o ubbe /benzene solu ions [31]. The esul s showed a
dec easing iscosi y as he shea a e o shea s ess is inc eased. Fig. 1
displays he summa y o he iscosi y da a ob ained by Reine using a
capilla y iscome e in which he applied p essu e and he low a e a e,
espec i ely, di ec ly p opo ional o he shea s ess and he shea a e.
Os wald’s analysis led o es ablish a powe law be ween he s ess
and a e in a shea low
σ
21 =K˙γn, which is ex ended o a powe law o
he iscosi y,
η
=K˙γn−1, whe e K and n a e cha ac e is ic pa ame e s o
he s udied sys em. Acco ding o he Os wald’s powe law, when n <1
he iscosi y dec eases as he shea a e inc eases; a beha io which has
L. Sang oniz e al.
Polyme 271 (2023) 125811
3
been e med as pseudoplas ic o shea hinning. On he con a y, o n >
1 he iscosi y augmen s wi h he shea a e, ollowing a beha iou
de ined as dila an o shea hickening. Fo n =1 he iscosi y emains
cons an a any applied shea a e, as was assumed by New on, Hagen,
Poiseuille, Coue e and all scien is s p eceding he ad en o heology.
The powe law equa ion is e y help ul o ex end he dynamics o
New onian liquids o he dynamics o polyme liquids, as i is demon-
s a ed in he essen ial book o Bi d, A ms ong and Hassage [32],
which con ains a g ea numbe o low cases.
The p og essi e di usion o he no ion o non-New onian low, and
in pa icula he sp ead o he idea o shea hinning liquids, had an
eno mous ele ance in polyme echnology. I was possible o compa e
he p ocessing ease and ene gy consump ion o di e en polyme s by
measu ing hei cha ac e is ics pa ame e (k and n) a any empe a u e
in home-made pis on d i en capilla y ex usion heome e s which we e
de eloped in he 1940s [33]. This was he i s s ep o he sound and
ui ul ela ionship be ween heology and polyme s, as can be seen in
he li e a u e abou polyme p ocessing [34–37]. F om he 11,320 o al
esul s ound un il July 2022 in he ISI Web o Knowledge [38] o he
jou nal Polyme Enginee ing and Science edi ed by he e y in luen ial
Socie y o Plas ics Enginee s (which was ounded in 1942 and accoun s o
mo e han 20,000 a ilia es cu en ly), 4220 e e o he e m p ocessing
and include, a a la ge o sho ex en , heological da a.
The in e es o polyme heologis s o s udy he iscosi y cu es (i.e.
iscosi y e sus shea a e) o polyme mel s, has led o combine he
esul s o o a ional (pla e-pla e and cone-pla e) and capilla y ex usion
heome e s. This allows co e ing a e y wide in e al o shea a es o
ob ain a inge p in by shi ing he iscosi y cu es a di e en em-
pe a u es o p ocu e a mas e cu e o each polyme , aking ad ance o
he ime- empe a u e supe posi ion (TTS) me hod [39]. Fig. 2 shows he
iscosi y esul s o a low densi y polye hylene ob ained a di e en
empe a u es [32,40]; simila esul s compa ing he ea u es o di e en
polyme s can be ound abundan ly in he li e a u e. In o de o show he
ele ance o his kind o heological esul s o he p ocessing o poly-
me s, in his igu e we ha e included he co esponding shea a es
anges in ol ed in he mos common p ocesses like ex usion, injec ion
e c. The shea a es ange equi ed o he mo e mode n 3D p in ing
p ocedu e [41] a e also included.
The a o emen ioned Os wald’s law is only alid o i he da a in he
shea hinning egion ha is o say abo e he c i ical shea a e which
ma ks he end o he New onian low. Cu en ly he mos used model o
i he iscosi y da a o polyme s is he Ca eau equa ion [32]:
η
=
η
∞+(
η
0−
η
∞)
[1+(
α
˙γ)a](1−n)/a(4)
whe e
η
0
is he so-called New onian iscosi y,
η
∞
he iscosi y pla eau a
high shea a es,
α
a cha ac e is ic ime whose in e se is he c i ical
shea a e o he onse o he non-New onian beha io , and n is he low
index. Gene ally, he con ibu ion o
η
∞
is neglec ed.
The ac ual use o compu e aided ools o polyme p ocessing, such
as he simula ion so wa e Mold low o injec ion and comp ession
molding, equi es he ine i able loading o iscosi y da a. In doing so,
he e ec o shea a e, empe a u e and p essu e on iscosi y can be
exp essed h ough he equa ion:
η
(˙γ,T,P)=
η
0exp(Ea
RT)exp (βP)
[1+(
η
0exp(Ea
RT)exp(βP)˙γ/
τ
)a](1−n
a)(5)
Fig. 1. O iginal able aken om he pape o Reine [31]. The mos signi ican da a a e hose o he i s and he second columns, which co espond, o he applied
p essu e and he low a e, espec i ely. The esul s indica e ha he e is no a linea ela ionship be ween hem, e lec ing a non-New onian beha iou .
Fig. 2. Viscosi y o a low densi y polye hylene a se e al empe a u es
measu ed wi h a capilla y iscosime e (high shea a es, abo e 5 10
−2
s
−1
) and
a Weissenbe g Rheogoniome e (low shea a es). In he igu e he shea a es
co esponding o se e al p ocessing echniques a e shown [32,40]. The co e-
sponding empe a u es a e included in he igu e. The lines a e d awn o guide
he eye.
L. Sang oniz e al.
Polyme 271 (2023) 125811
4
In his equa ion he e is an A henius like dependence o he iscosi y
on bo h, empe a u e and p essu e: E
a
is he ac i a ion ene gy o low
and β he iscosi y-p essu e coe icien . The pa ame e
τ
is he c i ical
s ess le el a he ansi ion o shea hinning (
α
=
η
0
/
τ
). Ins ead o he
exponen ial dependence o he iscosi y on empe a u e, he well known
William-Landel-Fe y equa ion [42,43] can be also used.
log aT=−C1g(T−Tg)
C2g+T−Tg
(6)
The physical meaning o hese pa ame e s is linked o he heo y o
“ ee olume” [43] which is e y ele an o he physical chemis y o
polyme s. The ela i e iso- ee olume s a e
g
=V
/V =0.025, whe e V
is he ee olume and V he o al olume, is he same o all polyme s a
he co esponding glass ansi ion empe a u e T
g
. Fo a ypically lex-
ible polyme like polye hylene he T
g
is −120 ◦C, ha co esponds o a
ela i ely low he mal ene gy o each
g
=0.025. Mo e ene gy is
necessa y, o ins ance, o each his alue in he case o a mo e igid
chain, like polys y ene, which gi es ise o a highe glass ansi ion
empe a u e, T
g
=100 ◦C [12]. I has been obse ed [43] ha he highe
he glass ansi ion empe a u e, he highe is he ac i a ion ene gy o
low.
In o de o ace indus ial polyme p ocessing, he coo dina ed wo k
o polyme chemis s who c ea e polyme s, and heologis s is c ucial and
cons i u es one o he mos in e es ing ad ances o bo h, polyme s and
heology. Cu en ly i is known how he basic ea u es o polyme s, such
as chemical s uc u e o monome , a e age molecula weigh , poly-
dispe si y o he molecula weigh dis ibu ion and e en ual p esence o
sho and long b anches, a ec he pa ame e s o equa ion (5). As an
example o he e ec o some polyme pa ame e s on iscosi y, in Fig. 3
[44] he combined e ec o he molecula weigh , M
w
, and he molec-
ula weigh dis ibu ion b oadness, exp essed in e ms o he poly-
dispe si y index M
w
/M
n,
is obse ed o wo polys y ene samples: he
e ec o M
w
is no iced on
η
0
, while he in luence o he polydispe si y is
e lec ed in he c i ical shea a e o he onse o non-New onian
beha iou . Fo he bene i o polyme scien is s and enginee s he is-
cosi y dependency on shea a e, closely linked o he p ocessing con-
di ions, can be con olled moni o ing he polyme iza ion condi ions.
The pa adigma ic ela ionship polyme iza ion- heology-p ocessing has
been consolida ed wi h he e up ion o new polyme iza ion me hods,
like me allocene ca alys polyme iza ion, a om ans e adical poly-
me iza ion (ATRP) [45], e e sible addi ion− agmen a ion chain
ans e (RAFT) [46], ing opening polyme iza ion (ROP) [47], ing
opening me a hesis polyme iza ion (ROMP), con olled anionic/ca-
ionic polyme iza ion [48] and he p og essi e ad ance o polyme s
ob ained om biological sou ces ha has led o new polyme s and
complex opologies.
4. New onian iscosi y and polyme chain en anglemen s: he
imagina i e ube and ep a ion model
The a o emen ioned molecula heo ies ha explain he ela ion
be ween molecula weigh and in insic iscosi y a e based on he in-
di iduali y o mac omolecules, dis ega ding in e ac ions be ween hem.
This is an accep able easoning, since ob aining [
η
] implies measu ing
he iscosi y o e y dilu ed polyme solu ions. Howe e , in he case o
concen a ed solu ions and polyme mel s his indi idual esponse o
polyme chains becomes ques ionable. Using he concep o empo a y
c osslinks o en anglemen s be ween chains [49–51], in oduced o
explain he elas ic beha iou o polyme mel s, G aessley [52] p oposed
a simple and in ui i e model o he shea a e dependence o he is-
cosi y. Acco ding o he model, du ing low, polyme chains disen angle
and en angle in a dynamics go e ned by he applied shea a e. A su -
icien ly low shea a es, ha is o say below he c i ical shea a e, ˙γc, o
he incep ion o shea hinning, he chains ha e enough ime o disen-
angle and en angle again, because he e is a low ela i e eloci y be-
ween hem. The e o e, he densi y o en anglemen s emains cons an ,
gi ing ise o he New onian iscosi y. Howe e , a shea a es abo e ˙γc,
he en anglemen s densi y begins o dec ease, because he con ac ime
be ween mac omolecules is oo sho o b ing abou e ec i e in-
e ac ions. As shea a e is inc eased he en anglemen s densi y de-
c eases p opo ionally, which is e lec ed in he iscosi y educ ion
obse ed in Fig. 3.
Pa adoxically, he disco e y o he non-New onian iscosi y o
polyme s led o inc ease he scien i ic in e es on he New onian o shea
independen iscosi y,
η
0
, o polyme mel s. The co ela ion o he
New onian iscosi y wi h he leng h o he polyme chain aised scien-
i ic expec a ions, opening he ou e o he models o physical polyme
chain in e ac ions.
Expe imen al e idences o he co ela ion be ween New onian is-
cosi y and molecula weigh we e epo ed mo e han 50 yea s ago,
leading o esul s simila o hose displayed in Fig. 4 aken om he
pape o Be y and Fox [53]. I is wo h no ing ha in a g ea majo i y o
he epo ed New onian iscosi y-molecula weigh co ela ions he
da a e e o he weigh a e age molecula weigh M
w
. In e es ingly
enough, i has been obse ed o e he yea s ha he double loga i hmic
η
0
-M
w
plo s o lexible polyme s show he same end, which is exp essed
by he equa ions:
η
0=kMw( o Mw<Mc)(7)
And
η
0=KM3.4
w( o Mw>Mc)(8)
whe e k and K a e cons an s ha e lec he e ec o o he ac o s han
molecula weigh , like empe a u e and he molecula s uc u e o he
monome , and M
c
is a c i ical molecula weigh ha depends on each
polyme . I is assumed ha o M
w
<M
c
he chains a e no su icien ly
long o en angle. Ac ually, he Rouse model [54] was he i s molecula
heo y p edic ing
η
0
=k M
w
on he basis o no in e ac ion be ween
polyme chains.
No iceable M
c
di e ences a e ound o di e en polyme s; o
ins ance, be ween polye hylene, M
c
=3500 Da, and polys y ene, M
c
=
31,000 Da [44], e lec ing a g ea e easiness o en angle o he o me . In
1986 G aessely and Edwa ds [55] ound a co ela ion be ween M
c
and
he mic os uc u e o he monome , in e ms o he bond a e age leng h,
Fig. 3. Mas e cu e o iscosi y as a unc ion o shea a e o wo polys y enes
[44], PS
260
M
w
=260,000 g/mol, M
w
/M
n
≈2.4, and PS
160
M
w
=160,000
g/mol, M
w
/M
n
<1.1. The espec i e e ec s o molecula weigh and molecula
weigh dis ibu ion b oadness (polydispe si y) a e obse ed (see ex ). The lines
a e d awn o guide he eye.
L. Sang oniz e al.
Polyme 271 (2023) 125811
5
he monome molecula weigh and he cha ac e is ic a io C
∞
which
s ands o he igidi y o he chain.
Acco ding o mo e exhaus i e analysis o he ela ion be ween
η
0
and
M
w
abo e M
c
, ca ied ou wi h a g ea numbe o polyme s, he alue 3.4
o he exponen should be conside ed as an app oxima ion, since ac u-
ally alues be ween 3 and 4 ha e been ound. In any case, he uni e -
sali y o he scaling law
η
0
=k M
w
a
, wi h 3 <a <4, is ou o ques ion,
since i is ollowed by all in es iga ed polyme s, including biopolyme s
[56–58].
To highligh he ele ance o his esul we conside i om he
polyme cha ac e iza ion pe spec i e. The New onian iscosi y becomes
he physical pa ame e mos suscep ible o molecula weigh changes,
hus i is used as a con ol pa ame e in many p ocesses like he mo-
mechanical deg ada ion du ing low, pos -polyme iza ion p ocesses,
physical ageing and o he s [59]. Fo ins ance, conside ing a alue o he
powe law o a =3.4, a molecula weigh change o only 5 % is e lec ed
in a
η
0
change o nea 20 %.
The expe imen ally obse ed unique beha iou o polyme mel s
a ac ed he in e es o ele an physicis s. Ad ancing on he concep o
physical chain in e ac ions, Edwa ds [60,61] in oduced he ube model
o en angled s a e o monodispe se polyme s (M =M
n
=M
w
) on he
ollowing hypo hesis: he mo ion o a polyme chain is es ic ed by
su ounding chains, so i is pic u ed o be con ined o a ube-like egion.
The Nobel P ize o Physics (1991) Pie e Giles de Gennes en isaged ha
polyme linea chains a e cons ained o ep a e in he ube [62] and
de ined a ep a ion o disengagemen ime,
τ
d
, as he ime i akes he
chain o di use ou o he ube, which is p opo ional o N
3
being N he
polyme iza ion deg ee. This leads o he scaling law
η
0
=k M
3
, which is
weake han he expe imen ally ound
η
0
=k M
w
a
wi h 3 <a <4, being a
=3.4 he mos o en obse ed. This di e ence be ween expe imen al
and ep a ion model exponen was conside ed o be due o ube leng h
luc ua ions [63]. O he hypo hesis o his disc epancy a e gi en, o
ins ance, in he book o Doi and Edwa ds [64].
Ini ially he ube- ep a ion model was en isaged o linea polyme
chains, bu in he las decades i has been adap ed o mo e complex
a chi ec u es [65–69], like long chain b anched, s a polyme s, cyclic
polyme s and physical polyme gels [70], inc easing he scien i ic in-
e es o his model o iscosi y and di usion ela ed phenomena.
Acco ding o he analysis ul illed by T.P. Lodge among he pape s
published in Mac omolecules since i s beginning [5], he heo y o
polyme chain ep a ion is among he six mos ele an achie emen s in
polyme science in he las 50 yea s, e ealing he anscendence o his
heological con ibu ion o polyme s. Ac ually, he ube and ep a ion
model includes he s udy o he di usion o polyme chains, which is
ca ied ou expe imen ally by mic oscopy and neu on sca e ing. Fo
ins ance, Nobel P ize winne o Physics 1997 S e en Chu p o ided
expe imen al e idences o he key assump ions o ube and ep a ion
model, using luo escence mic oscopy o analyze he dynamics o luo-
escen ly labelled molecules o DNA [71]. As an in e es ing co olla y o
he con luence o polyme s and heology, we ema k ha he men ioned
Nobel lau ea e, Chu, also p o ed expe imen ally ( h ough elonga ional
lows) [72] he chain coil-s e ch ansi ion, which was p edic ed by
ano he Nobel lau ea e, Pie e Giles de Gennes [73].
5. Viscoelas ici y: polyme s, he pa adigm o iscoelas ic
ma e ials
In non-polyme ic liquids and solids espec i e de ia ions om
iscous and elas ic beha iou a e negligeable, because hey a e
composed by small molecules which a e only able o espond in a simple
mode in a ield o o ce. On he con a y, polyme s a e iscoelas ic
ma e ials, owing o he capaci y o he chains o o e mechanical e-
sponses a di e en leng h scales, om sho - ange o long- ange, when
a o ce is applied. When a mechanical o ce is applied o a polyme , hen,
local, segmen al and comple e mo ions o he chain can be induced,
depending on he magni ude o he o ce, he empe a u e and he
applica ion ime. This is schema ically exp essed in Fig. 5, aken om
he a o emen ioned undamen al book o Fe y [43].
Fig. 4. Co ec ed iscosi y, log
η
ξ
, conside ing he cons an ic ion ac o ξ as a
unc ion o log X
w
=log [((s
2
)
0
/M)Z
w
/
υ
2
] o se e al linea polyme s, being X
w
a pa ame e ha cha ac e izes polyme coil dimension. The lines show a slope
o 1.0 and 3.4. The cu es ha e been shi ed in he o dina e scale [53].
Fig. 5. Mo ions induced along ime unde s ess a e depic ed o poly-
isobu ylene in he scheme [43]: segmen al, local and comple e chain le el ( om
igh o le ).
L. Sang oniz e al.

Polyme 271 (2023) 125811
6
In addi ion o he dynamics o an indi idual polyme chain en isaged
in his igu e, en anglemen s among chains, which gi e ise o empo-
a y elas ic ne wo ks, c ea e a unique and inhe en ly iscoelas ic
amewo k.
In oducing he concep o iscoelas ici y, Maxwell desc ibed gases
as iscoelas ic luids wi h iscosi y and elas ici y and p oposed he i s
equa ion o iscoelas ici y [32,74], which can be adap ed as:
σ
21 +
η
0
G
∂σ
21
∂
=
η
0
˙γ21 (9)
whe e
σ
21
is he shea s ess, ˙γ21 is he s ain a e,
η
0
is he New onian
iscosi y and G he Hookean elas ic modulus.
A simple ma hema ical analysis o his equa ion shows ha i is able
o ep esen he wo ex emes: he pu ely iscous esponse in he s eady
s a e low whe e d
σ
/d =0, o he pu ely elas ic beha iou in a sudden
change o s ess o which
σ
=0.
The e m
η
0
/G is ac ually de ined as he elaxa ion ime λ and can be
ob ained expe imen ally by applying he Maxwell equa ion o a s ess
elaxa ion a e a sudden s ain implemen a ion. The elaxa ion ime λ
=
η
0
/G is used o e alua e he Debo ah numbe , De, in oduced by
Reine in he beginnings o he heology, as was explained in 1964 [75]:
De = ime o elaxa ion/ ime o obse a ion. This is a undamen al
concep o heology which si ua es solids and liquids unde he same
physical concep , since he g ea e is De, he mo e solid (elas ic) is he
ma e ial, whe eas as De is smalle he ma e ial is mo e luid. Ideally, o
De =∞ he ma e ial is pu ely elas ic and o De =0 is me ely iscous.
Ce ainly, he idea o polyme ic chains gi ing ise o a single elax-
a ion ime is in appa en con adic ion wi h he polyme dynamics
exp essed in Fig. 5, and he Maxwell model should be ex ended wi h a
ce ain numbe o elaxa ion imes o i expe imen al esul s o poly-
me s. This leads o an in eg al o m o he gene alized linea iscoelas ic
model [32]:
σ
( )= ∫
−∞
G( − ′)dγ( ′)
d ′d ′(10)
whe e G ( − ’) is he elaxa ion modulus which allows de e mining a
con inuous spec um o elaxa ion imes H(λ).
The app oach o e ed in p eceding pa ag aphs is pa icula ly help ul
o in oduce he concep o iscoelas ici y o polyme chemis s and en-
ginee s. Bu we ha e o ema k ha Coleman and Noll [76] showed hei
c i icism on he lack o physical igo o he sp ings and dashpo model.
Ins ead, hey asse ha he in eg al o m o he iscoelas ici y (Eq. (10))
can be deduced om he Bol zmann [77,78] supe posi ion p inciple,
acco ding o which he s ess a any ime can be desc ibed as a unc ion
o he his o y o he a e o change o s ain.
The a o emen ioned excellen ea ise on polyme dynamics w i en
by Bi d, A ms ong and Hassage [32], as well as he book o Tanne on
enginee ing heology [79] b ing many examples o he applica ion o
his equa ion o di e en iscoelas ic expe imen s.
Fo a his o ical su ey o iscoelas ic me hods in polyme s we u n o
he ollowing wi y commen o Plazek [80]: “Whe eas Leade man was
conside ed o be he King o C eep du ing he 40′and 50′, A hu V.
Tobolsky was he King o S ess om he 40′ h ough he 60′and John D.
Fe y was he King o Dynamic Mechanical P ope ies om he 50’
h ough he 70”. A summa y o he physical pa ame e s and iscoelas ic
unc ions ha can be de e mined in linea iscoelas ic measu emen s
using shea and ex ensional de o ma ion is shown in Table 1, aken om
he pape o Dealy on heological nomencla u e [81].
All he men ioned me hods ha e indeed con ibu ed o he de el-
opmen o he s udy o he iscoelas ici y o polyme s, bu we a e
compelled o ecognize ha dynamic o oscilla o y es s leading o
de e mine s o age and loss moduli, G′and G′′, cons i u e he mos
ele an and en iching heological p ocedu e in polyme science.
Conside ing he a ie y o e en ual mechanical esponses o
polyme s, and he e ec o ime and empe a u e in e ms o he
dimensional scales o hese esponses, he combined e ec o bo h pa-
ame e s was conside ed, b inging abou one o he miles ones o
polyme s iscoelas ici y: The ime- empe a u e supe posi ion (TTS)
me hod. The issue a s ake was whe he in polyme iscoelas ici y,
empe a u e change is equi alen o a shi o he loga i hmic ime scale.
Pionee ing wo ks on he e ec o empe a u e we e ca ied ou by
Leade man [82], analyzing c eep esul s o plas icized poly inyl chlo-
ide, and by Tobolsky and And ews [83] s udying elaxa ion and c eep
esul s o ubbe gum. F om hese esul s a ou e was opened o achie e
a educed o iscoelas ic mas e cu e by shi ing o supe pose da a
ob ained a di e en empe a u es along a loga i hmic ime o equency
axis. This p ocedu e was explained and popula ized in he essen ial book
o Fe y “Viscoelas ic P ope ies o Polyme s” [43]. An example o he
applica ion o TTS me hod o ob ain a mas e cu e which allows
ex ending hugely he ime scale o s ess elaxa ion da a is shown in
Fig. 6, hanks o he use o a shi ac o which is shown in he inle .
The applica ion o TTS me hod is no ob ious o all kind o poly-
me s. Ac ually, i is only alid o “ he mo heologically simple” poly-
me s, as was coined by Schwa zl and S a e man [39]. TTS ails in
complex and mul iphasic polyme ic sys ems, like immiscible polyme
Table 1
Nomencla u e o linea iscoelas ici y om Socie y o Rheology [81].
Quan i y Symbol S.I. uni s
Simple shea
Shea s ain γ ─
Shea modulus (modulus o igidi y) G Pa
Shea elaxa ion modulus G( ) Pa
Shea compliance J Pa
−1
Shea c eep compliance J( ) Pa
−1
Equilib ium shea compliance J
e
Pa
−1
S eady-s a e shea compliance J
s
0
Pa
−1
Complex iscosi y
η
*(
ω
) Pa s
Dynamic iscosi y
η
′(
ω
) Pa s
Ou -o -phase componen o
η
*(
ω
)
η
’’(
ω
) Pa s
Complex shea modulus G*(
ω
) Pa
Shea s o age modulus G’(
ω
) Pa
Shea loss modulus G’’(
ω
) Pa
Complex shea compliance J*(
ω
) Pa
−1
Shea s o age compliance J’(
ω
) Pa
−1
Shea loss compliance J’’(
ω
) Pa
−1
Tensile ex ension
S ain (T ue s ain)
ε
─
Young’s modulus E Pa
Tensile elaxa ion modulus E( ) Pa
Tensile compliance D Pa
−1
Tensile c eep compliance D( ) Pa
−1
Fig. 6. In he le , s ess elaxa ion modulus o an unc osslinked poly-
isobu ylene (PIB) sample measu ed a 11 di e en empe a u es om −80 o
50
◦C is shown. On he igh he mas e cu e ob ained by shi ing s ess
elaxa ion cu es ho izon ally along he ime axis a a e e ence empe a u e o
25 ◦C is shown [84]. The shi ac o , a
T
a ies wi h empe a u e as shown in
he inse a uppe igh . S ess and ime uni s a e depic ed as de ined in
he e e ence.
L. Sang oniz e al.
Polyme 271 (2023) 125811
7
blends and phase sepa a ed block copolyme s and mas e cu es canno
be accomplished.
Simila , o he p ocedu e shown in Fig. 6, mas e cu es can be ob-
ained wi h c eep and dynamic o oscilla o y iscoelas ic da a. When a
p ope ime span is eached h ee cha ac e is ic zones, which e lec he
uniqueness o polyme s as iscoelas ic ma e ials, a e de ined: glassy
s a e, ubbe y s a e and e minal o low egion. Howe e , in he case o
c osslinked polyme s, like cu ed elas ome s, only wo egions a e
de ec ed, since he e minal o low egion is supp essed due o he
impossibili y o polyme chains o di use. Ins ead o a con inuous
dec ease o G ( ) as ime inc eases (Fig. 6), an equilib ium modulus, G
e
,
is obse ed. The equilib ium modulus is p opo ional o he molecula
weigh M
x
be ween wo chemical c oss-links:
Ge=
ρ
RT
Mx
(11)
whe e
ρ
is he densi y, R is he gas cons an and T he empe a u e.
This equa ion, which deno es he en opic cha ac e o he elas ici y
o c osslinked polyme s, is de i ed om he ne wo k heo y o he
ubbe elas ici y [85]. The p ac ical ele ance o his simple equa ion in
e e y day polyme echnology is huge, in pa icula o he mose s,
which include ele an indus ial ields like ubbe s and adhesi es.
The h ee egions depic ed in Fig. 6 a e in e p e ed on he basis o he
a ious leng h scale dynamics o polyme chains, wi hin he amewo k
o he gene al concep s o he physics o polyme s. The in e media e
pla eau, which co esponds o he so-called ubbe y zone, is assumed o
a ise om polyme chain en anglemen s [43,86]. Consis en ly, no any
o he ma e ial bu polyme s show his pla eau zone. The alue o he
s ess elaxa ion modulus a he in e media e pla eau zone is called he
en anglemen modulus, G
N
0
, and is concomi an o he equilib ium
modulus, G
e
, o c osslinked polyme s. This pa allelism be ween
c oss-links leading o a pe manen ne wo k and en anglemen s which
lead o empo a y ne wo ks, has b ough abou a ela ionship be ween
G
N
0
and he molecula weigh be ween en anglemen s, M
e
:
G0
N=
ρ
RT
Me
(12)
The liaison be ween he en anglemen s in ol ed in iscosi y esul s
and he iscoelas ic pa e n o Fig. 6, is unde s ood assuming ha o
la ge imes he e minal o low zone is eached, as en anglemen s
slippage occu s. The e o e, wo cha ac e is ic molecula weigh s o en-
anglemen s a e de ined in polyme heology: The a o emen ioned
c i ical molecula weigh , M
c
, o he dependence o he New onian
iscosi y on molecula weigh and he molecula weigh be ween en-
anglemen s, M
e
. As i is asse ed in he book o Vinog ado and Malkin
“Rheology o Polyme s” [86], independen de e mina ions o hese
molecula weigh s ha e shown ha M
c
≈M
e
. The physico-chemical
conside a ions made abou he e ec o he mic os uc u e o he
monome on he c i ical molecula weigh , M
c
, which ma ks he limi
be ween he linea and he powe law dependence o
η
0
on M
w
, a e also
alid o he molecula weigh , M
e
, ob ained om he pla eau modulus
G
N
0
. A lis o G
N
0
alues wi h hei co esponding M
e
alues o di e en
polyme species is gi en, o ins ance, in he ea ise o G aessley [44].
Besides i s uniqueness and undamen al physical ele ance, G
N
0
,
which de e mines he densi y o en anglemen s G
N
0
=
ρ
N
A
/M
e
(whe e
ρ
and N
A
a e espec i ely he densi y and he A ogad o’s numbe ), is also
unc ionally ele an in he mechanical p ope ies o polyme s, as is
ema ked by Hans-Henning Kaus ch [10] in his e iew commemo a ing
80 yea o polyme s. Fo ins ance, his au ho ex ols he esul s o Wu
[87] demons a ing he close co ela ion be ween he en anglemen s
densi y and c azing, p ecu so o ac u e, o polyme solids. Also, he
en anglemen modulus is linked o he di usion o mac omolecules in
polyme -polyme blends in e aces [88,89] and o ackiness o imme-
dia e adhesion o adhesi es [90,91]. Chain in e di usion is also c ucial
o each a good welding in laye by laye addi i e manu ac u ing p ocess
[92].
In he glassy s a e, which co esponds o e y sho imes o low
empe a u es (see le side o Fig. 6), he mo ion o he chains is e y
local co esponding ypically o a ew monome s, as is depic ed in Fig. 5.
Inc easing ime o empe a u e, a ansi ion om he glassy s a e o he
ubbe y s a e is obse ed. This ansi ion is de ined as he glass ansi-
ion empe a u e, T
g
, when he isoch onal elaxa ion modulus is plo ed
as a unc ion o empe a u e gi ing ise o G(T) which is equi alen o G
( ). The anscendence o he glass ansi ion empe a u e, which can be
also de e mined by dila ome y and calo ime y, is eno mous in polyme
science and echnology. Many physico-chemical s udies o polyme s a e
cen e ed on he glass ansi ion empe a u e, as can be no iced in any o
he exis ing gene al books o polyme s. Re e ing again o he pape o T.
P. Lodge on 50 yea s o Mac omolecules [5], we ema k ha he heo y o
he glass ansi ion appea s as he second mos ele an challenge in
polyme science, only a e he heo y o polyme c ys alliza ion. The
heo y o he ee olume, which sus ains he concep o T
g
as an iso- ee
olume s a e, has been soundly de eloped in polyme science in
consonance wi h heological esul s. On he o he hand, om he
pe spec i e o polyme enginee ing, i is wo h ecalling ha he glass
ansi ion empe a u e ma ks o a g ea ex en he p ocessing condi ions
o amo phous polyme s. Whe eas in he case o semi-c ys alline poly-
me s, ex usion, injec ion and o he p ocessing me hods should be,
ob iously, ca ied a empe a u es abo e he mel ing empe a u e, T
m
,
o amo phous polyme s he ecommended empe a u e is T
g
+100 ◦C
[36]. An in e es ing link be ween he glass ansi ion empe a u e and
he heology o polyme p ocessing is ha , in gene al, he highe is T
g
he bigge is he ac i a ion ene gy o low, E
a
[43], which s ands o he
e ec o empe a u e on iscosi y depic ed in Equa ion (5). The impli-
ca ions o hese esul s in he ene gy consump ion du ing ab ica ion o
plas ic pa s, a subjec o inc easing in e es , a e e iden .
6. Dynamic iscoelas ic measu emen s: A powe ul ool o
cha ac e ize polyme s
The applica ion o al e na ing s esses o s ains o undamen al
esea ch in ma e ials was ini ia ed in he middle o he pas cen u y. In
pa icula , he use o dynamic measu emen s as a ool o in es iga ion
in me als was e iewed by Zene in 1948 in his book “Elas ici y and
Anelas ici y o Me als” [93]. In he same yea , Nolle [94] desc ibed
se e al me hods o he de e mina ion o he dynamic iscoelas ic
p ope ies o ubbe solids and ubbe solu ions unde e y small sinu-
soidal s ains. In 1952 dynamic iscoelas ic esul s o polyisobu ylene
solids and liquids we e epo ed by Ma ko i z e al. [95]. The wo ks o
Fe y [43], in eg a ed al eady in he i s edi ion (1960) o his a o e-
men ioned book, ep esen a undamen al s ep o consolida e he
esea ch on dynamic iscoelas ici y o polyme s. Also, he emendous
de elopmen o elec onics and compu ing science in he las decades o
XX h cen u y, acili a ing da a acquisi ion, has s ongly con ibu ed o
he sp ead o heology, like in any o he b anch o science. We ha e o
ecall, o ins ance, ha in con as wi h he powe ul calcula ion sys-
ems employed nowadays by con empo a y heome e s, nomog aphs
we e used by Nolle in 1948 o calcula e he eal and imagina y pa s o
Young o ensile modulus, E′(s o age) and E′′ ( iscous) o ubbe s unde
a ia ion o equencies om 0.1 o 1 cycles/s and empe a u es om
−60 o 100 ◦C. The de elopmen o expe imen al me hods o ob ain he
dynamic iscoelas ic unc ions o polyme s has led o he ollowing
p e e en ial s ain modes: Bending, ba o sion and simple ex ension, o
ob ain he ensile s o age (elas ic) modulus, E′, and he ensile loss
( iscous) modulus, E′′, o polyme solids. To sion in coaxial cylinde s,
cone-pla e and pla e-pla e, o de e mine he shea s o age (elas ic)
modulus, G′, and he shea loss ( iscous) modulus, G′′. Oscilla o y
comp ession expe imen s o ob ain he comp ession s o age (elas ic)
modulus, K′, and he comp ession loss ( iscous) modulus, K′′, a e much
less employed, al hough hey a e o g ea in e es in he ield o physical
gels, o ins ance hyd ogels o medical pu poses [96,97].
L. Sang oniz e al.
Polyme 271 (2023) 125811
8
As is explained in Fe y’s book [43], he simples way o in oduce
he physical basis o dynamic o oscilla o y iscoelas ic unc ions is o
conside a sinusoidal shea s ain γ:
γ=γ0sin
ω
(13)
whe e γ
0
is he s ain ampli ude and
ω
is he angula equency o
oscilla ion.
P o ided ha he iscoelas ic beha io is linea , i is ound ha he
shea s ess
σ
also a ies sinusoidally, al hough ou o phase wi h s ain:
σ
=
σ
0sin (
ω
+δ)(14)
whe e δ is he phase angle be ween he s ess and he s ain.
Using he gene alized in eg al equa ion o he linea iscoelas ic
model (Eq. (10)) he s o age o elas ic shea modulus and he loss o
iscous shea modulus a e espec i ely de ined:
G’=
σ
0
γ0
cos δ (15)
G’’ =
σ
0
γ0
sin δ (16)
F om hese unc ions he loss angen is de e mined:
an δ =G’’
G’(17)
To ensu e ha he esul s a e ob ained in he linea egime, i.e. same
espec i e moduli independen ly o he applied s ess o s ain ampli-
ude, low ampli udes a e eques ed. This has led o coin he e m Small
Ampli ude Oscilla o y Shea (SAOS) measu emen s, s anding o linea
dynamic es s, whe eas he name LAOS (La ge Ampli ude Oscilla o y
Shea ) is used o non-linea measu emen s.
P obably he mos ou s anding esul s o he dynamic iscoelas ic
beha io o polyme s a e hose which display he espec i e mas e
cu es o s o age and loss moduli and an δ as a unc ion o equency, in
measu emen s ca ied ou a di e en empe a u es. An example is
shown in Fig. 7 [98]. The e iden esemblance o he G′(
ω
) unc ion wi h
he elaxa ion modulus, G( ), conside ing he in e se p opo ion be-
ween equency and ime, has led o many esea che s o use his ype o
plo s o de ine he h ee iscoelas ic zones. Compa ed o he in o ma ion
ha can be eached om G( ) plo s, simul aneous plo s o G′(
ω
) and G′′
(
ω
) allow unde s anding he iscoelas ic cha ac e o each iscoelas ic
zone in a mo e in ui i e way. The chain dynamics in he e minal zone
(low equencies o la ge imes/ empe a u es) is cha ac e ized by he
mo ion o he polyme chain as a whole (Fig. 5), which implies mo e
ene gy dissipa ion han s o age. This is e lec ed in Fig. 7 by G′′ >G′,
which also leads o loss angen alues o an δ >1. The ubbe y zone,
obse ed a in e media e imes in G( ) measu emen s and cha ac e ized
by he en anglemen s modulus, G
N
0
, is no iced in Fig. 7 by an in e me-
dia e equency in e al a which he elas ic modulus is p ac ically in-
dependen o equency and G′>G′′, wi h a consequen an δ minimum.
This deno es he elas ic cha ac e o he en anglemen s ne wo k, whose
modulus is de e mined ypically by he cons an alue o G′a he in-
e media e equency in e al o he alue o G′a he G′′ minimum.
O he p ocedu es o ob ain G
N
0
om dynamic iscoelas ic measu emen s
ha e been p oposed in li e a u e [99], showing ha SAOS measu e-
men s allow de e mining G
N
0
in an easie and mo e accu a e way han
s ess elaxa ion expe imen s.
The ease o use and he ela i e low p ice, as compa ed o o he
polyme cha ac e iza ion echniques, has led o a g ea popula i y o
SAOS es s o in es iga e he e minal and ubbe y zones o polyme
solu ions and mel s. The applica ion o he gene al linea iscoelas ic
model [32] o e s he possibili y o linking low equency G′and G′′
esul s o s eady s a e low pa ame e s, like he New onian iscosi y,
η
0
,
and he s eady s a e o eco e able compliance, J
e
0
, which can be ypi-
cally ob ained om c eep and eco e y es s and ep esen s he elas ici y
o a polyme liquid du ing low.
η
0=lim
ω
→0G’’ /
ω
(18)
J0
e=lim
ω
→0G’/G’’2(19)
These esul s imply ha acco ding o he linea iscoelas ic model G′
should be p opo ional o
ω
2
and G′′ p opo ional o
ω
as equency is
educed.
Since he New onian iscosi y depends on he molecula weigh
ollowing a powe law equa ion (see abo e) and J
e
0
inc eases wi h he
molecula weigh o monodispe se polyme s [44], as well as wi h he
b oadness o he molecula weigh dis ibu ion and long chain b anch-
ing [100–102], he e minal zone becomes e y sensi i e o any mo-
lecula change in polyme chains a chi ec u e. Taking ad ance o his
ea u e, app oaches on he co ela ion be ween SAOS esul s in he
e minal zone and molecula weigh dis ibu ion ha e been made in
li e a u e [100,103] co ela ing each ime om he spec um o elax-
a ion imes H(λ) o a ce ain molecula weigh . Nume ical me hods o
de e mine elaxa ion spec a om s o age and loss moduli a e a ailable
in he classical book o Tschoegl “The Phenomenological Theo y o
Linea Viscoelas ic Beha io ” [104] and in he mo e ecen book o Cho
“Viscoelas ici y o Polyme s. Theo y and Nume ical Algo i hms” [105],
among o he s. Fo ins ance, he ollowing equa ion, p oposed by
Tschogel, can be used o ob ain he elaxa ion spec a [43]:
H(
τ
)= dG′
dln
ω
+1
2
d2G′
d(lnw)21
ω
=
2
τ
√
(20)
Howe e , he p ocedu e o ob aining elaxa ion spec a om he
app op ia e esul s (c eep da a b ing abou disc e e o con inuous
e a da ion spec a, whe eas s ess elaxa ion and oscilla ion da a gi e
elaxa ion spec a) equi es ad anced ma hema ical me hods o a oid
expe imen al e o s, as hose de eloped by Hone kamp and Weese [106,
107] included in he so wa e cu en ly a ailable in some comme cial
heome e s.
Ac ually, besides polyme a chi ec u e, any ac o which a ec s
mobili y o he chain as a whole, like mic ophase sepa a ion in a poly-
me blend o a block copolyme , al e s signi ican ly bo h, he elas ic and
loss moduli, in he e minal zone, educing conside ably he espec i e
dependences G’∝
ω
2 and G’’∝
ω
. An example o his is gi en in Fig. 8
aken om he pape o Ba es [108] which shows he heological
Fig. 7. Mas e cu e o G
′and G′′ o polys y ene a a e e ence empe a u e
equal o 150 ◦C. The moduli we e ac ually measu ed in he equency ange
10
−3
o 10
2
1/s a di e en empe a u es and hen supe posi ion me hod was
applied [98].
L. Sang oniz e al.
Polyme 271 (2023) 125811
9
beha io in he o de ed s a e (mic ophase sel -assembly) and in he
diso de ed s a e abo e a ce ain c i ical empe a u e.
On he o he hand, in he con ex o wha we can call p ac ical
ou ine heology, he analysis o he e minal zone o indus ial polyme
p oduc s is used o quali y con ol; ypically, a G′and G′′ e sus e-
quency oo p in o each sample is ob ained h ough SAOS es s and
compa ed o he model esul s o he sample aken as a e e ence.
The mos ex ended dynamic iscoelas ic es s in polyme science a e
he ones pe o med inc easing he empe a u e om he glassy s a e, a a
cons an s ain ampli ude and equency (isoch onal). The mos usual
measu emen s a e pe o med in bending mode a a equency o 1 Hz o
de e mine E′, E′′ and an δ as a unc ion o empe a u e. The pu pose o
he so-called Dynamic Mechanical Analysis o DMA es s is o de e mine
he glass ansi ion empe a u e o glass ansi ion empe a u es o any
polyme sys em. The e is, he e o e, a i al y be ween his heological
echnique and di e en ial scanning calo ime y, DSC, which is o en
used o ob ain T
g
. Indeed, he ad an age o he la e echnique compa ed
o DMA lies mo e on i s g ea capaci y o he analysis o he c ys alline
phase. Bu , a oiding any passiona e a o i ism o heology, i should be
ecognized ha DMA is mo e accu a e o de e mining glass ansi ions
in complex polyme sys ems, such as semic ys alline and c osslinked
polyme s, polyme blends, andom and block copolyme s and polyme
composi es, among o he s.
The sensibili y o dynamic iscoelas ic measu emen s o de ec he
ansi ion om he glass o he ubbe y s a e, is obse ed in Fig. 7 which
shows ypical G′and G′′ esul s as a unc ion o equency o a homo-
polyme : In he ansi ion zone a maximum in G′′, which gi es ise o a
an δ maximum (no shown), is no iced. When ins ead o equency
scans, empe a u e scans a e pe o med, like in DMA, he esul s a e he
opposi e in e ms o x axis, since on he basis o ime- empe a u e
equi alence empe a u e is equi alen o he in e se o equency.
Then, he glass ansi ion empe a u e, T
g
, is de e mined as he em-
pe a u e a which he maximum on E′′ o an δ akes place.
In he i s edi ion o he Fe y’s book (1960), as well as in he e y
use ul book o beginne heologis s o Mu ayama “Dynamic Mechanical
Analysis o Polyme ic Ma e ials” [109] men ions o pape s on T
g
de e mined by DMA, published al eady in he six ies o he pas cen u y,
a e included [110,111]. So a , he numbe o pape s e e ing o his
kind o measu emen s, unde he name o “Dynamic Mechanical Anal-
ysis” (DMA) o “Dynamic Mechanical The mal Analysis” (DMTA), is
subs an ial. And, in iew o he la ge numbe o equipmen pu chased by
indus ial polyme companies, a sizable quan i y o empe a u e scans o
γ =γ
0
sin
ω
a e ca ied by polyme enginee s e e y day. I is wo h
men ioning he ex ao dina y in o ma ion gi en by Bell and Mu ayama
50 yea s ago [111] in hei pape abou DMA esul s o nylons and
polye hylene e eph hala e. As an example, Fig. 9 aken om Mu -
ayama’s wo k, shows he e ec o he deg ee o c ys alliza ion on he
glass ansi ion empe a u e (de ined as
α
ansi ion) and sub-glass
ansi ion empe a u es o a Nylon 6.
The obse ed capaci y o his heological echnique o de ec local
mo ions, like side-mo ions and c anksha o a ion, besides segmen al
mo ions which s and o T
g
, is absolu ely ema kable. In polyme science
DMA esul s a e o en combined wi h o he esul s ob ained by dielec ic
and NMR me hods o s udy he e ec o mic os uc u e on dynamics o
polyme chains.
Conside ing he physical basis o he dynamic iscoelas ic expe i-
men s, es s in ol ing s ain ampli ude scans a e also ele an . Fi s ,
because hey ma k he way o de e mine he linea iscoelas ic egion, a
which he iscoelas ic unc ions a e independen o he applied s ain o
s ess ampli ude. Also, s udies o he e ec o s ain ampli ude ou side
he linea egion a e pa icula ly in e es ing o polyme sys ems which
con ain physical in e ac ions, like in e ac ions among c ys alli es o
seconda y bonds, as is he case o polyme s p one o in e chain
hyd ogen bonds. This is he case o he mo e e sible polyme gels whose
ansi ion om weak solids o liquids depends on empe a u e and he
applied s ess. I is gene ally assumed ha he anishing o he gel
ne wo k akes place when G′′ o e comes G′in an iso he mal and
isoch onal s ain ampli ude scan. Howe e , he p ominen pape o
Win e and Chambon [112] (1879 ci a ions un il July 2022), which
p oposes a sound iscoelas ic c i e ion o he gel poin o c osslinked
polyme s, discloses he e ec o equency on G′′ >G′ ule.
Ad anced elec onics and compu a ional echniques applied o cu -
en heome e s has led o a p og essi e popula iza ion o La ge
Ampli ude Shea Oscilla o y, LAOS, measu emen s in polyme s
heology. In he las en yea s (2013–2022) 230 pape s on LAOS ha e
been published. Bu , despi e he inc easing p opo ion o LAOS pape s
wi h espec o SAOS pape s, he new ou comes ob ained by hese
Fig. 8. Reduced elas ic and loss moduli o 1,4-polybu adiene-1,2-polybu a-
diene diblock copolyme . The loss moduli da a has been shi ed 1 o de o
magni ude in he e ical axis [108]. A a empe a u e o 111
◦C and abo e he
sample is in a diso de ed s a e and a beha io ypical o homopolyme s is
obse ed. Howe e , a 87 ◦C and below he esponse co esponds o ha o a
mic ophase sepa a ed copolyme . Moduli uni s a e depic ed as de ined in
he e e ence.
Fig. 9. an δ s empe a u e o Nylon 6 ob ained employing di e en p o-
cedu es a 100 Hz [109]. The e ec o he deg ee o c ys alliza ion is no ed in
he
α
ansi ion, close o 100 ◦C: he highes peak co esponds o he less
c ys alline sample. T ansi ions a lowe empe a u es, called γ (lowes em-
pe a u e) and β, co espond o local mo ions o he chains.
L. Sang oniz e al.
Polyme 271 (2023) 125811
16
[40] J. Meissne , Kuns s o e 61 (1971) 576–586.
[41] I. Cala el, R.H. Agui esa obe, M.I. Pe˜
nas, A. San ama ia, M. Tie no, J.I. Conde,
B. Pascual, Sea ching o heological condi ions o FFF 3D p in ing wi h PVC
based lexible compounds, Ma e ials 13 (2020) 178, h ps://doi.o g/10.3390/
ma13010178.
[42] M.L. Williams, R.F. Landel, J.D. Fe y, The empe a u e dependence o elaxa ion
mechanisms in amo phous polyme s and o he glass- o ming liquids, J. Am.
Chem. Soc. 77 (1955) 3701–3707, h ps://doi.o g/10.1021/ja01619a008.
[43] J.D. Fe y, Viscoelas ic P ope ies o Polyme s, i s ed., J. Wiley & Sons, New
Yo k, 1960 second ed. 1970, hi d ed. 1980.
[44] W.W. G aessley, Polyme ic Liquid and Ne wo ks: Dynamics and Rheology,
Ga land Science Taylo and F ancis G oup, New Yo k, 2008.
[45] K. Ma yjaszewski, J. Xia, A om ans e adical polyme iza ion, Chem. Re . 101
(2001) 2921–2990, h ps://doi.o g/10.1021/c 940534g.
[46] M.R. Hill, R.N. Ca mean, B.S. Sume lin, Expanding he scope o RAFT
polyme iza ion: ecen ad ances and new ho izons, Mac omolecules 48 (2015)
5459–5469, h ps://doi.o g/10.1021/acs.mac omol.5b00342.
[47] Z. Li, M. Tang, S. Liang, M. Zhang, G.M. Biesold, Y. He, S.-M. Hao, W. Choi, Y. Liu,
J. Peng, Z. Lin, Bo leb ush polyme s: om con olled syn hesis, sel -assembly,
p ope ies o applica ions, P og , Polym. Sci. 116 (2021), 101387, h ps://doi.
o g/10.1016/j.p ogpolymsci.2021.101387.
[48] S. I o, R. Goseki, T. Ishizone, A. Hi ao, Syn hesis o well-con olled g a polyme s
by li ing anionic polyme iza ion owa ds exac g a polyme s, Polym. Chem. 5
(2014) 5523–5534, h ps://doi.o g/10.1039/C4PY00584H.
[49] W.F. Busse, The physical s uc u e o elas ic colloids, J. Phys. Chem. 36 (1932)
2862–2879, h ps://doi.o g/10.1021/j150342a002-.
[50] L.R.G. T eloa , Elas ic eco e y and plas ic low in aw ubbe , Rubbe Chem.
Technol. 13 (1940) 795–806, h ps://doi.o g/10.5254/1.3546559.
[51] M.S. G een, A.V. Tobolsky, A new app oach o he heo y o elaxing polyme ic
media, J. Chem. Phys. 14 (1946) 80–92, h ps://doi.o g/10.1063/1.1724109.
[52] W.W. G aessley, Molecula en anglemen heo y o low beha io in amo phous
polyme s, J. Chem. Phys. 43 (1965) 2696–2703, h ps://doi.o g/10.1063/
1.1697198.
[53] G.C. Be y, T.G. Fox, The iscosi y o polyme s and hei concen a ed solu ions,
Ad . Polym. Sci. 5 (1968) 261–357, h ps://doi.o g/10.1007/BFb0050985.
[54] P.E. Rouse, A heo y o he linea iscoelas ic p ope ies o dilu e solu ions o
coiling polyme s, J. Chem. Phys. 21 (1953) 1272–1280, h ps://doi.o g/10.1063/
1.1699180.
[55] W.W. G aessley, S.F. Edwa ds, En anglemen in e ac ions in polyme s and he
chain con ou concen a ion, Polyme 22 (1981) 1329–1334, h ps://doi.o g/
10.1016/0032-3861(81)90231-7.
[56] M.P. G os eno , J.N. S ani o h, The e ec o molecula weigh on he heological
and ensile p ope ies o poly(e-cap olac one), In e n. J. Pham. 135 (1996)
103–109, h ps://doi.o g/10.1016/0378-5173(95)04404-3.
[57] M. Ga in, L. Tighze , I. V oman, S. Ma inko ic, B. Es ine, The in luence o mola
mass on heological and dilu e solu ion p ope ies o poly(bu ylene succina e),
J. Appl. Polym. Sci. 131 (2014), 40887, h ps://doi.o g/10.1002/app.40887.
[58] J.J. Coope -Whi e, M.E. Mackay, Rheological p ope ies o poly(lac ides). E ec
o molecula weigh and empe a u e on he iscoelas ici y o poly(l-lac ic acid),
J. Polym. Sci. B Polym. Phys. 37 (1999) 1803–1814, h ps://doi.o g/10.1002/
(SICI)1099-0488(19990801)37:15<1803::AID-POLB5>3.0.CO;2-M.
[59] X. Colin, J. Ve du, Polyme deg ada ion du ing p ocessing C, R. Chimie 9 (2006)
1380–1395, h ps://doi.o g/10.1016/j.c ci.2006.06.004.
[60] S.F. Edwa ds, The s a is ical mechanics o polyme ized ma e ial, P oc. Phys. Soc.
92 (1967) 9–16, h ps://doi.o g/10.1088/0370-1328/92/1/303.
[61] S.F. Edwa ds, S a is ical mechanics wi h opological cons ain s: I, P oc. Phys.
Soc. 91 (1967) 513–519, h ps://doi.o g/10.1088/0370-1328/91/3/301.
[62] P.G. De Gennes, Rep a ion o a polyme chain in he p esence o ixed obs acles,
J. Chem. Phys. 55 (1971) 572–579, h ps://doi.o g/10.1063/1.1675789.
[63] M. Doi, Molecula dynamics and heological p ope ies o concen a ed solu ions
o odlike polyme s in iso opic and liquid c ys alline phases, J. Polym. Sci.,
Polym. Phys. Ed. 19 (1981) 229–243, h ps://doi.o g/10.1002/
pol.1981.180190205.
[64] M. Doi, S.F. Edwa ds, The Theo y o Polyme Dynamics, Ox o d Science
Publica ions, Ox o d, 1986.
[65] E. Na imissa, M.H. Wagne , Re iew on ube model based cons i u i e equa ions
o polydispe se linea and long-chain b anched polyme mel s, J. Rheol. 63
(2019) 361–375, h ps://doi.o g/10.1122/1.5064642.
[66] V.A.H. Bouda a, D.J. Read, J. Ramí ez, Rep a e heology so wa e: oolki o he
analysis o heo ies and expe imen s, J. Rheol. 64 (2020) 709–722, h ps://doi.
o g/10.1122/8.0000002.
[67] M. Abadi, M.F. Se ag, S. Habuchi, En angled polyme dynamics beyond ep a ion,
Na . Commun. 9 (2018) 5098, h ps://doi.o g/10.1038/s41467-018-07546-7.
[68] S. Habuchi, N. Sa oh, T. Yamamo o, Y. Tezuka, M. Vacha, Mul imode di usion o
ing polyme molecules e ealed by a single-molecule s udy, Angew. Chem. In .
Ed. 49 (2010) 1418–1421, h ps://doi.o g/10.1002/anie.200904394.
[69] M. Kapnis os, M. Lang, D. Vlassopoulos, W. Pyckhou -Hin zen, D. Rich e , D. Cho,
T. Chang, M. Rubins ein, Unexpec ed powe -law s ess elaxa ion o en angled
ing polyme s, Na . Ma e . 7 (2008) 997–1002, h ps://doi.o g/10.1038/
nma 2292.
[70] L. Leible , M. Rubins ein, R.H. Colby, Dynamics o e e sible ne wo ks,
Mac omolecules 24 (1991) 4701–4707, h ps://doi.o g/10.1021/ma00016a034.
[71] T.T. Pe kins, D.E. Smi h, S. Chu, Di ec obse a ion o ube-like mo ion o a single
polyme chain, Science 264 (1994) 819–822, h ps://doi.o g/10.1126/
science.8171335.
[72] H.P. Babcock, R.E. Teixei a, J.S. Hu , E.S. Shaq eh, S. Chu, Visualiza ion o
molecula luc ua ions nea he c i ical poin o he coil-s e ch ansi ion in
polyme elonga ion, Mac omolecules 36 (2003) 4544, h ps://doi.o g/10.1021/
ma034073p.
[73] P.G. de Gennes, Coil-s e ch ansi ion o dilu e polyme solu ion in ul ahigh
elonga ional low, J. Chem. Phys. 60 (1974) 5030, h ps://doi.o g/10.1063/
1.1681018.
[74] J.C. Maxwell, On he dynamical heo y o gases, Phil. T ans. Roy. Soc. A157
(1867) 49–88, h ps://doi.o g/10.1098/ s l.1867.0004.
[75] M. Reine , The Debo ah numbe , Phys. Today 17 (1964) 62, h ps://doi.o g/
10.1063/1.3051374.
[76] B.D. Coleman, W. Noll, Founda ions o linea iscoelas ici y, Re . Mod. Phys. 33
(1961) 239–249, h ps://doi.o g/10.1103/Re ModPhys.33.239.
[77] L. Bol zmann, Zu heo ie de elas ischen nachwi kung, Si zbe . Akad. Wiss.
Wien, Ma h.-Na u w. 70 (1874) 275–306.
[78] L. Bol zmann, Zu Theo ie de elas ischen Nachwi kung, Pogg. Ann. Phys. Chem.
7 (1876) 624–654.
[79] R.I. Tanne , Enginee ing Rheology, Ox o d Science Publica ions, Ox o d, 1985.
[80] D.J. Plazek, 995 Bingham Medal Add ess: oh, he mo heological simplici y,
whe e o e a hou? J. Rheol. 40 (1996) 987–1014, h ps://doi.o g/10.1122/
1.550776.
[81] J.M. Dealy, O icial nomencla u e o ma e ial unc ions desc ibing he esponse
o a iscoelas ic luid o a ious shea ing and ex ensional de o ma ions, J. Rheol.
39 (1995) 253–265, h ps://doi.o g/10.1122/1.4765670.
[82] H. Leade man, C eep and c eep eco e y in plas icized poly inyl chlo ide, Ind.
Eng. Chem. 35 (1943) 374–378, h ps://doi.o g/10.1021/ie50399a024.
[83] A.V. Tobolsky, R.S. And ews, Sys ems mani es ing supe posed elas ic and iscous
beha io , J. Chem. Phys. 13 (1945) 3–27, h ps://doi.o g/10.1063/1.1723966.
[84] L.E. Nielsen, Mechanical P ope ies o Polyme s, Van Nos and Reinhold, New
Yo k, 1962.
[85] L.R.G. T eloa , The Physics o Rubbe Elas ici y, hi d ed., Ox o d Uni e si y
P ess, Ox o d, 1975.
[86] V. Vinog ado , A. Ya, Malkin, Rheology o Polyme s, Mi Publishe s Moscow,
Sp inge -Ve lag, Be lin, 1980.
[87] S. Wu, Chain s uc u e, phase mo phology, and oughness ela ionships in
polyme s and blends, Polym. Eng. Sci. 30 (1990) 753–761, h ps://doi.o g/
10.1002/pen.760301302.
[88] R.P. Wool, B.-L. Yuan, O.J. McGa el, Welding o polyme in e aces, Polym. Eng.
Sci. 29 (1989) 1340–1367, h ps://doi.o g/10.1002/pen.760291906.
[89] A. Ka a an os, R.J. Compos o, K.I. Winey, M. K ¨
oge , N. Cla ke, Modeling o
en angled polyme di usion in mel s and nanocomposi es: a e iew, Polyme s 11
(2019) 1–29, h ps://doi.o g/10.3390/polym11050876.
[90] A. Zosel, The e ec o bond o ma ion on he ack o polyme s, J. Adhes. Sci.
Technol. 11 (1997) 1447–1457, h ps://doi.o g/10.1163/156856197X00237.
[91] C. C e on, M. Cicco i, F ac u e and adhesion o so ma e ials: a e iew, Rep.
P og. Phys. 79 (2016), 046601, h ps://doi.o g/10.1088/0034-4885/79/4/
046601.
[92] J.E. Seppala, S. Hoon Han, K.E. Hillga ne , C.S. Da is, K.B. Migle , Weld
o ma ion du ing ma e ial ex usion addi i e manu ac u ing, So Ma e 13
(2017) 6761–6769, h ps://doi.o g/10.1039/C7SM00950J.
[93] C. Zene , Elas ici y and Anelas ici y o Me als, Uni e si y o Chicago P ess,
Chicago, Illinois, 1948.
[94] A.W. Nolle, Me hods o measu ing dynamic mechanical p ope ies o ubbe -like
ma e ials, J. Appl. Phys. 19 (1948) 753–774, h ps://doi.o g/10.1063/
1.1698201.
[95] H. Ma ko i z, P.M. Ya o sky, R.C. Ha pe , L.J. Zapas, T.W. DeWi , Ins umen
o measu ing dynamic iscosi ies and igidi ies, Re . Sci. Ins um. 23 (1952)
430–437, h ps://doi.o g/10.1063/1.1746338.
[96] A. Tejo-O e o, F. Fenollosa-A ´
es, I. Achae andio, S. Rey-Vinolas, I. Buj-Co al,
M.´
A. Ma eos-Timoneda, E. Engel, So - issue-mimicking using hyd ogels o he
de elopmen o phan oms, Gels 8 (2022) 1–14, h ps://doi.o g/10.3390/
gels8010040.
[97] H.K. Lau, S. Ra an, H. Fu, C.G. Ga cia, D.M. Ba be , K.L. Kiick, A.J. C osby,
Mic omechanical p ope ies o mic os uc u ed elas ome ic hyd ogels,
Mac omol. Biosci. 20 (2020) 1–12, h ps://doi.o g/10.1002/mabi.201900360.
[98] S.H. Wasse man, W.W. G aessley, E ec s o polydispe si y on linea
iscoelas ici y in en angled polyme mel s, J. Rheol. 36 (1992) 543, h ps://doi.
o g/10.1122/1.550363.
[99] C. Liu, J. He, E. Van Ruymbeke, R. Keunings, C. Bailly, E alua ion o di e en
me hods o he de e mina ion o he pla eau modulus and he en anglemen
molecula weigh , Polyme 47 (2006) 4461–4479, h ps://doi.o g/10.1016/j.
polyme .2006.04.054.
[100] J.F. Vega, M. Aguila , J. Pe´
on, D. Pas o , J. Ma ínez-Salaza , E ec o long chain
b anching on linea - iscoelas ic mel p ope ies o polyole ins, E-Polyme s 2
(2002) 1–35, h ps://doi.o g/10.1515/epoly.2002.2.1.624.
[101] C. Gab iel, H. Müns ed , In luence o long-chain b anches in polye hylenes on
linea iscoelas ic low p ope ies in shea , Rheol. Ac a 41 (2002) 232–244,
h ps://doi.o g/10.1007/s00397-001-0219-6.
[102] H. Müns ed , Rheological measu emen s and s uc u al analysis o polyme ic
ma e ials, Polyme s 13 (2021) 1123, h ps://doi.o g/10.3390/polym13071123.
[103] E. Van Ruymbeke, R. Keunings, C. Bailly, De e mina ion o he molecula weigh
dis ibu ion o en angled linea polyme s om linea iscoelas ici y da a,
J. Nonnew on. Fluid Mech. 15 (2002) 153–175, h ps://doi.o g/10.1016/S0377-
0257(02)00080-0.
[104] N.W. Tschoegl, The Phenomenological Theo y o Linea Viscoelas ic Beha io ,
Sp inge -Ve lag, Be lin, Heidelbe g, 1989.
L. Sang oniz e al.

Polyme 271 (2023) 125811
17
[105] K.S. Cho, Viscoelas ici y o Polyme s. Theo y and Nume ical Algo i hms, Sp inge
Science, D od ech , 2016.
[106] J. Hone kamp, J. Weese, A nonlinea egula iza ion me hod o he calcula ion o
elaxa ion spec a, Rheol. Ac a 32 (1993) 65–73, h ps://doi.o g/10.1007/
BF00396678.
[107] C. Els e , J. Hone kamp, J. Weese, Using egula iza ion me hods o he
de e mina ion o elaxa ion and e a da ion spec a o polyme ic liquids, Rheol.
Ac a 31 (1992) 161–174, h ps://doi.o g/10.1007/BF00373238.
[108] F.S. Ba es, Block copolyme s nea he mic ophase sepa a ion ansi ion. 2. Linea
dynamic mechanical p ope ies, Mac omolecules 17 (1984) 2607–2613, h ps://
doi.o g/10.1021/ma00142a025.
[109] T. Mu ayama, Dynamic Mechanical Analysis o Polyme ic Ma e ial, Else ie
Scien i ic, Ams e dam, 1978.
[110] M. Ma suo, T. Ueno, H. Ho ino, S. Chujyo, H. Asai, Fine s uc u es and physical
p ope ies o s y ene-bu adiene block copolyme s, Polyme 9 (1968) 425–436,
h ps://doi.o g/10.1016/0032-3861(68)90052-9.
[111] J.P. Bell, T. Mu ayama, Rela ions be ween dynamic mechanical p ope ies and
mel ing beha io o nylon 66 and polye hylene e eph hala e, J. Polym. Sci. 2
Polym. Phys. 7 (1969), h ps://doi.o g/10.1002/pol.1969.160070607.
[112] H.H. Win e , F. Chambon, Analysis o linea iscoelas ici y o a c osslinking
polyme a he gel poin , J. Rheol. 30 (1986) 367–382, h ps://doi.o g/10.1122/
1.549853.
[113] K. Hyun, M. Wilhelm, C.O. Klein, K.S. Cho, J.G. Nam, K.H. Ahn, S.J. Lee, R.
H. Ewold , G.H. McKinley, A e iew o nonlinea oscilla o y shea es s: analysis
and applica ion o la ge ampli ude oscilla o y shea (Laos), P og. Polym. Sci. 36
(2011) 1697–1753, h ps://doi.o g/10.1016/j.p ogpolymsci.2011.02.002.
[114] K.S. Cho, K. Hyun, K.H. Ahn, S.J. Lee, A geome ical in e p e a ion o la ge
ampli ude oscilla o y shea esponse, J. Rheol. 49 (2005) 747–758, h ps://doi.
o g/10.1122/1.1895801.
[115] R.H. Ewold , A.E. Hosoi, G.H. McKinley, New measu es o cha ac e izing
nonlinea iscoelas ici y in la ge ampli ude oscilla o y shea , J. Rheol. 52 (2008)
1427–1458, h ps://doi.o g/10.1122/1.2970095.
[116] P.E. Boukany, S.Q. Wang, Use o pa icle- acking elocime y and low
bi e ingence o s udy nonlinea low beha io o en angled wo mlike micella
solu ion: om wall slip, bulk disen anglemen o chain scission, Mac omolecules
41 (2008) 1455–1464, h ps://doi.o g/10.1021/ma702527s.
[117] P.A. Smi h, G. Pe ekidis, S.U. Egelhaa , W.C.K. Poon, Yielding and c ys alliza ion
o colloidal gels unde oscilla o y shea , Phys. Re . E - S a . Nonlinea So Ma e
Phys. 76 (2007) 1–11, h ps://doi.o g/10.1103/PhysRe E.76.041402.
[118] M.C. Roge s, K. Chen, L. And zejewski, S. Na ayanan, S. Ramak ishnan, R.
L. Leheny, J.L. Ha den, Echoes in x- ay speckles ack nanome e -scale plas ic
e en s in colloidal gels unde shea , Phys. Re . E - S a . Nonlinea So Ma e
Phys. 90 (2014) 1–7, h ps://doi.o g/10.1103/PhysRe E.90.062310.
[119] S.A. Roge s, In sea ch o physical meaning: de ining ansien pa ame e s o
nonlinea iscoelas ici y, Rheol. Ac a 56 (2017) 501–525, h ps://doi.o g/
10.1007/s00397-017-1008-1.
[120] J. Ching-Wei Lee, J.D. Pa k, S.A. Roge s, S udying la ge ampli ude oscilla o y
shea esponse o so ma e ials, J. Vis. Exp. 146 (2019), e58707, h ps://doi.o g/
10.3791/58707.
[121] H.Y. Song, L. Faus , J. Son, M. Kim, S.J. Pa k, S. Kyun Ahn, M. Wilhelm, K. Hyun,
Small and medium ampli ude oscilla o y shea heology o model b anched
polys y ene (PS) mel s, Polyme s 12 (2020) 365, h ps://doi.o g/10.3390/
polym12020365.
[122] E. Aliabadian, M. Kamka , Z. Chen, U. Sunda a aj, P e en ion o ne wo k
des uc ion o pa ially hyd olyzed polyac ylamide (HPAM): e ec s o sal ,
empe a u e, and umed silica nanopa icles, Phys. Fluids 31 (2019), 013104,
h ps://doi.o g/10.1063/1.5080100.
[123] H.T. Lim, K.H. Ahn, J.S. Hong, K. Hyun, Nonlinea iscoelas ici y o polyme
nanocomposi es unde la ge ampli ude oscilla o y shea low, J. Rheol. 57 (2013)
767–789, h ps://doi.o g/10.1122/1.4795748.
[124] R. Salehiyan, H.Y. Song, K. Hyun, Nonlinea beha io o PP/PS blends wi h and
wi hou clay unde la ge ampli ude oscilla o y shea (Laos) low, Ko ea Aus .
Rheol. J. 27 (2015) 95–103, h ps://doi.o g/10.1007/s13367-015-0010-3.
[125] L. Sang oniz, J.K. Palacios, M. Fe n´
andez, J.I. Eguiazabal, A. San ama ia, A.
J. Mülle , Linea and non-linea heological beha io o polyp opylene/
polyamide blends modi ied wi h a compa ibilize agen and nanosilica and i s
ela ionship wi h he mo phology, Eu . Polym. J. 83 (2016) 10–21, h ps://doi.
o g/10.1016/j.eu polymj.2016.07.026.
[126] L. Sang oniz, T. Ganche a, B.D. Fa is, A.J. Mülle , A. San ama ia, Rheology o
complex biobased qua e na y blends: poly(lac ic acid) [poly(e hylene oxide)]/
poly(e he -b-amide)/poly(amide 11), J. Rheol. 65 (2021) 437–451, h ps://doi.
o g/10.1122/8.0000202.
[127] S.A. Roge s, B.M. E win, D. Vlassopoulos, M. Cloi e, A sequence o physical
p ocesses de e mined and quan i ied in Laos: applica ion o a yield s ess luid,
J. Rheol. 55 (2011) 435–458, h ps://doi.o g/10.1122/1.3544591.
[128] G. Huang, H. Zhang, Y. Liu, H. Chang, H. Zhang, H. Song, D. Xu, T. Shi, S ain
ha dening beha io o poly( inyl alcohol)/Bo a e hyd ogels, Mac omolecules 50
(2017) 2124–2135, h ps://doi.o g/10.1021/acs.mac omol.6b02393.
[129] K. Hyun, J.G. Nam, M. Wilhellm, K.H. Ahn, S.J. Lee, La ge ampli ude oscilla o y
shea beha io o PEO-PPO-PEO iblock copolyme solu ions, Rheol. Ac a 45
(2006) 239–249, h ps://doi.o g/10.1007/s00397-005-0014-x.
[130] C. Gau hie , J.-F. Chailan, J. Chaucha d, U ilisa ion de l’analyse isco´
elas ique
dynamique `
a l’´
e ude de la c is allisa ion iso he me du poly( ´
e ´
eph ala e
d’e hyl`
ene) amo phe. Applica ion `
a des composi es unidi ec ionnels a ec ib es
de e e, Mak omol. Chem. 193 (1992) 1001–1009, h ps://doi.o g/10.1002/
macp.1992.021930416.
[131] C. Ca o , J. Guille , K. Bou aha , Rheological beha io o a semi-c ys alline
polyme du ing iso he mal c ys alliza ion, Rheol. Ac a 32 (1993) 566–574,
h ps://doi.o g/10.1007/BF00369073.
[132] N.V. Pogodina, H.H. Win e , S. S ini as, S ain e ec s on physical gela ion o
c ys allizing iso ac ic polyp opylene, J. Polym. Sci. B Polym. Phys. 37 (1999)
3512–3519, h ps://doi.o g/10.1002/(SICI)1099-0488(19991215)37:24<3512::
AID-POLB12>3.0.CO;2-%23.
[133] A.P. Ko ula, A equency-dependen e ec i e medium model o he heology o
c ys allizing polyme s, J. Rheol. 64 (2020) 505–514, h ps://doi.o g/10.1122/
1.5132407.
[134] A.P. Ko ula, K.B. Migle , Pe cola ion implica ions in he heology o polyme
c ys alliza ion, Polym. C ys alliza ion 4 (2021), e10162, h ps://doi.o g/
10.1002/pc 2.10162.
[135] J.G. Old oyd, On he o mula ion o heological equa ions o s a e, P oc. Royal
Soc. London, Se A Ma h. Phys. Sci. 200 (1950) 523–541, h ps://doi.o g/
10.1098/ spa.1950.0035.
[136] K. Weissenbe g, A con inuum heo y o heological phenomena, Na u e 159
(1947) 310–311, h ps://doi.o g/10.1038/159310a0.
[137] D.V. Boge , K. Wal e s, Rheological Phenomena in Focus, Else ie , Ams e dam,
1993.
[138] M. Reine , De o ma ion, s ain and low, in: An Elemen a y In oduc ion o
Rheology, H.K. Lewis and Company, London, 1960.
[139] A.S. Lodge, Elas ic Liquids, Academic P ess, 1964.
[140] A.C. Me ing on, Flow o Visco-elas ic ma e ials in capilla ies, Na u e 152 (1943)
663, h ps://doi.o g/10.1038/152663a0.
[141] C. Allain, M. Cloi e, P. Pe o , Expe imen al in es iga ion and scaling law
analysis o die swell in semi-dilu e polyme solu ions, J. Non-New onian Fluid
Mech. 73 (1997) 51, h ps://doi.o g/10.1016/S0377-0257(97)00051-7.
[142] R.S. Spence , R.E. Dillon, The iscous low o mol en polys y ene, J. Colloid Sci. 3
(1948) 163–180, h ps://doi.o g/10.1016/0095-8522(48)90066-X.
[143] R.A. Mendelson, F.L. Finge , E.B. Bagley, Die swell and eco e able shea s ain in
polye hylene ex usion, J. Polym. Sci. C, Polym. Symp. 35 (1971) 177–188,
h ps://doi.o g/10.1002/polc.5070350114.
[144] W.W. G aessley, S.D. Glasscock, R.L. C awley, Die swell in mol en polyme s,
T ans. Soc. Rheol. 14 (1970) 519–544, h ps://doi.o g/10.1122/1.549177.
[145] R.I. Tanne , A heo y o die-swell, J. Polym. Sci. A-2 Polym. Phys. 8 (1970)
2067–2078, h ps://doi.o g/10.1002/pol.1970.160081203.
[146] R.I. Tanne , Some me hods o es ima ing he no mal s ess unc ions in
iscome ic lows, T ans. Soc. Rheol. 14 (1970) 483–507, h ps://doi.o g/
10.1122/1.549175.
[147] J.D. Fe y, M.L. Williams, D.M. S e n, Slow elaxa ion mechanism in concen a ed
polyme s solu ions, J. Phys. Chem. 58 (1954) 987–992, h ps://doi.o g/10.1021/
j150521a012.
[148] R.S. Spence , R.E. Dillon, The iscous low o mol en polys y ene. II, J. Colloid
Sci. 4 (1949) 241–255, h ps://doi.o g/10.1016/0095-8522(49)90007-0.
[149] M. Reine , De o ma ion and Flow, an Elemen a y In oduc ion o Theo e ical
Rheology, H. K. Lewis and Company, L d., London, 1949.
[150] J.P. To della, F ac u e in he ex usion o amo phous polyme s h ough
capilla ies, J. Appl. Phys. 27 (1956) 454–458, h ps://doi.o g/10.1063/
1.1722401.
[151] J.P. To della, Chap e 2, in: F.R. Ei ich (Ed.), Rheology Theo y and Applica ions,
Academic P ess, 1969. Ed .
[152] M. Denn, Ex usion ins abili ies and wall slip, Annu. Re . Fluid Mech. 33 (2001)
265–287, h ps://doi.o g/10.1146/annu e . luid.33.1.265.
[153] J.J. Benbow, R.V. Cha ley, P. Lamb, Uns able low o mol en polyme s, Na u e
192 (1961) 223–224, h ps://doi.o g/10.1038/192223a0.
[154] E.B. Bagley, H.P. Sch eibe , E ec o die en y geome y on polyme mel ac u e
and ex uda e dis o ion, T ans. Soc. Rheol. 5 (1961) 341–353, h ps://doi.o g/
10.1122/1.548904.
[155] S.-Q. Wang, P.A. D da, Y.-W. Inn, Explo ing molecula o igins o sha kskin,
pa ial slip, and slope change in low cu es o linea low densi y polye hylene,
J. Rheol. 40 (1996) 875–898, h ps://doi.o g/10.1122/1.550766.
[156] S.-Q. Wang, Nonlinea Polyme Rheology: Mac oscopic Phenomenology and
Molecula Founda ion, John Wiley & Sons, Inc., 2018.
[157] K.B. Migle , Y. Son, F. Qiao, K. Flynn, Ex ensional de o ma ion, cohesi e ailu e,
and bounda y condi ions du ing sha kskin mel ac u e, J. Rheol. 46 (2002) 383,
h ps://doi.o g/10.1122/1.1445186.
[158] S.G. Ha ziki iakos, K.B. Migle (Eds.), Polyme P ocessing Ins abili ies: Con ol
and Unde s anding, CRC P ess, 2004, h ps://doi.o g/10.1201/9781420030686.
[159] J.F. Agassan , D.R. A da, C. Combeaud, A. Me en, H. Müns ed , M.R. Mackley,
L. Robe , B. Ve gnes, Polyme p ocessing ex usion ins abili ies and me hods o
hei elimina ion o minimisa ion, In . Polym. P ocess. 21 (2006) 239–255,
h ps://doi.o g/10.3139/217.0084.
[160] D. Kane , E. Takacs, J. Vlachopoulos, Rheological e alua ion and obse a ions o
ex usion ins abili ies o biodeg adable polyes e s, In . Polym. P ocess. 22 (2007)
395–401, h ps://doi.o g/10.3139/217.2053.
[161] N. O hman, B. Jaz awi, P. Meh khoda andi, S.G. Ha ziki iakos, Wall slip and mel
ac u e o poly(lac ides), Rheol. Ac a 51 (2012) 357–369, h ps://doi.o g/
10.1007/s00397-011-0613-7.
[162] B. Jaz awi, N. No oozi, M. Ansa i, S.G. Ha ziki iakos, P ocessing aids o
biodeg adable polyme s, J. Appl. Polym. Sci. 128 (2013) 3592–3600, h ps://doi.
o g/10.1002/app.38562.
[163] S.G. Ha ziki iakos, J.M. Dealy, Role o slip and ac u e in he oscilla ing low o
HDPE in a capilla y, J. Rheol. 36 (1992) 845–884, h ps://doi.o g/10.1122/
1.550320.
L. Sang oniz e al.
Polyme 271 (2023) 125811
18
[164] J. Vlachopoulos, M. Alam, C i ical s ess and eco e able shea o polyme mel
ac u e, Polym. Eng. Sci. 12 (1972) 184–192, h ps://doi.o g/10.1002/
pen.760120305.
[165] J.-M. Piau, N. El Kissi, F. Toussain , A. Mezghani, Dis o ions o polyme mel
ex uda es and hei elimina ion using slippe y su aces, Rheol. Ac a 34 (1995)
40–57, h ps://doi.o g/10.1007/BF00396053.
[166] M. Aguila , M.T. Exp´
osi o, J.F. Vega, A. Mu˜
noz-Escalona, J. Ma ínez-Salaza ,
Elimina ion o ex uda e dis o ions in me allocene-ca alyzed polye hylene,
Mac omolecules 37 (2004) 681–683, h ps://doi.o g/10.1021/ma035173y.
[167] N. Bha dwaj, S.C. Kundu, Elec ospinning: a ascina ing ibe ab ica ion
echnique, Bio echnol. Ad . 28 (2010) 325–347, h ps://doi.o g/10.1016/j.
bio echad .2010.01.004.
[168] P. Gup a, C. Elkins, T.E. Long, G.L. Wilkes, Elec ospinning o linea
homopolyme s o poly(me hyl me hac yla e): explo ing ela ionships be ween
ibe o ma ion, iscosi y, molecula weigh and concen a ion in a good sol en ,
Polyme 46 (2005) 4799–4810, h ps://doi.o g/10.1016/j.polyme .2005.04.021.
[169] C.J.S. Pe ie, Elonga ional Flows Aspec s o he Beha iou o Model Elas ic
Viscous Fluids, Pi man, 1979.
[170] C.J.S. Pe ie, One hund ed yea s o ex ensional low, J. Non-New onian Fluid
Mech. 137 (2006) 1–14, h ps://doi.o g/10.1016/j.jnn m.2006.01.010.
[171] F.T. T ou on, On he coe icien o iscous ac ion and i s ela ion o ha o
iscosi y, P oc. Roy. Soc. A77 (1906) 426–440, h ps://doi.o g/10.1098/
spa.1906.0038.
[172] J. Meissne , De elopmen o a uni e sal ex ensional heome e o he uniaxial
ex ension o polyme mel s, T ans. Soc. Rheol. 16 (1972) 405, h ps://doi.o g/
10.1122/1.549258.
[173] J. Meissne , T. Raible, S.E. S ephenson, Ro a y clamp in uniaxial and biaxial
ex ensional heome y o polyme mel s, J. Rheol. 25 (1981) 1–28, h ps://doi.
o g/10.1122/1.549612.
[174] J. Meissne , J. Hos e le , A new elonga ional heome e o polyme mel s and
o he highly iscoelas ic liquids, Rheol. Ac a 33 (1994) 1–21, h ps://doi.o g/
10.1007/BF00453459.
[175] C.D. Han, Rheology and P ocessing o Polyme ic Ma e ials, Ox o d Uni e si y
P ess, USA, 2007.
[176] C. Gab iel, H. Müns ed , S ain ha dening o a ious polyole ins in uniaxial
elonga ional low, J. Rheol. 47 (2003) 619–630, h ps://doi.o g/10.1122/
1.1567752.
[177] T. Le che (Ed.), Plas ic Was e and Recycling, Else ie , Ams e dam, Ne he lands,
2020.
[178] H. Müns ed , Ex ensional heology and p ocessing o polyme ic ma e ials, In .
Polym. P ocess. 33 (2018) 594–618, h ps://doi.o g/10.3139/217.3532.
[179] G.J. Field, P. Micic, S.N. Bha acha ya, Mel s eng h and ilm bubble ins abili y o
LLDPE/LDPE blends, Polym. In . 48 (1999) 461–466, h ps://doi.o g/10.1002/
(SICI)1097-0126(199906)48:6<461::AID-PI169>3.0.CO;2-7.
[180] W. Minoshima, J.L. Whi e, Ins abili y phenomena in ubula ilm, and mel
spinning o heologically cha ac e ized high densi y, low densi y and linea low
densi y polye hylenes, J. Non-New onian Fluid Mech. 19 (1986) 275–302,
h ps://doi.o g/10.1016/0377-0257(86)80053-2.
[181] H.M. Laun, H. Schuch, T ansien elonga ional iscosi ies and d awabili y o
polyme mel s, J. Rheol. 33 (1989) 119–175, h ps://doi.o g/10.1122/1.550058.
[182] V. Rauschenbe ge , H.M. Laun, A ecu si e model o Rheo ens es s, J. Rheol. 41
(1997) 719–737, h ps://doi.o g/10.1122/1.550880.
[183] M.H. Wagne , A. Be nna , V. Schulze, The heology o he heo ens es , J. Rheol.
42 (1998) 917–928, h ps://doi.o g/10.1122/1.550907.
[184] M.H. Wagne , H. Bas ian, A. Be nna , S. Ku zbeck, C.K. Chai, De e mina ion o
elonga ional iscosi y o polyme mel s by RME and Rheo ens expe imen s,
Rheol. Ac a 41 (2002) 316–325, h ps://doi.o g/10.1007/s00397-002-0228-0.
[185] A.K. Dou as, Analysis o he heo ens expe imen wi h iscoelas ic cons i u i e
equa ions o p obing ex ensional heology o polyme mel s, J. Rheol. 50 (2006)
749–769, h ps://doi.o g/10.1122/1.2243338.
[186] M.L. Sen mana , Minia u e uni e sal es ing pla o m: om ex ensional mel
heology o solid-s a e de o ma ion beha io , Rheol. Ac a 43 (2004) 657–669,
h ps://doi.o g/10.1007/s00397-004-0405-4.
[187] M.L. Sen mana , B.N. Wang, G.H. McKinley, Measu ing he ansien ex ensional
heology o polye hylene mel s using he SER uni e sal es ing pla o m, J. Rheol.
49 (2005) 585–606, h ps://doi.o g/10.1122/1.1896956.
[188] F.J. S adle , A. Nishioka, J. S ange, K. Koyama, H. Müns ed , Compa ison o he
elonga ional beha io o a ious polyole ins in uniaxial and equibiaxial lows,
Rheol. Ac a 46 (2007) 1003–1012, h ps://doi.o g/10.1007/s00397-007-0190-y.
[189] Q. Huang, M. Mangnus, N.J. Al a ez, R. Koopmans, O. Hassage , A new look a
ex ensional heology o low-densi y polye hylene, Rheol. Ac a 55 (2016)
343–350, h ps://doi.o g/10.1007/s00397-016-0921-z.
[190] B. Li, W. Yu, X. Cao, Q. Chen, Ho izon al ex ensional heome y (HER) o low
iscosi y polyme mel s, J. Rheol. 64 (2020) 177–190, h ps://doi.o g/10.1122/
1.5134532.
[191] H. Müns ed , F.R. Schwa zl, De o ma ion and Flow o Polyme ic Ma e ials,
Sp inge , Swi ze land, 2014, h ps://doi.o g/10.1007/978-3-642-55409-4.
[192] J.M. Dealy, R.G. La son, S uc u e and Rheology o Mol en Polyme s, Hanse
Publishe s, Munich, 2006, h ps://doi.o g/10.3139/9783446412811. m.
[193] C.W. Macosko, Rheology P inciples, Measu emen s, and Applica ions, Wiley-
VHC, New Yo k, 1994.
[194] M. Kemp , D. Ahi wal, M. Cziep, M. Wilhelm, Syn hesis and linea and nonlinea
mel heology o well-de ined comb a chi ec u es o PS and PpMS wi h a low and
con olled deg ee o long-chain b anching, Mac omolecules 46 (2013)
4978–4994, h ps://doi.o g/10.1021/ma302033g.
[195] A.D. Go sis, B.L.F. Zee enho en, A.H. Hog , The e ec o long chain b anching on
he p ocessabili y o polyp opylene in he mo o ming, Polym. Eng. Sci. 44 (2004)
973–982, h ps://doi.o g/10.1002/pen.20089.
[196] R. Al-I y, K. Lamnawa , A. Maazouz, Biopolyme blends based on poly (lac ic
acid): shea and elonga ion heology/s uc u e/blowing p ocess ela ionships,
Polyme s 7 (2015) 939–962, h ps://doi.o g/10.3390/polym7050939.
[197] M.C. Mis e a, L. Bo a, R. A igo, F. Le o, G. Malucelli, F.P. La Man ia,
Bionanocomposi e blown ilms: insigh s on he heological and mechanical
beha io , Polyme s 13 (2021) 1167, h ps://doi.o g/10.3390/polym13071167.
[198] P. Spi ael, C.W. Macosko, S ain ha dening in polyp opylenes and i s ole in
ex usion oaming, Polym. Eng. Sci. 44 (2004) 2090–2100, h ps://doi.o g/
10.1002/pen.20214.
[199] K. Taki, K. Taba a, S.I. Kiha a, M. Ohshima, Bubble coalescence in oaming
p ocess o polyme s, Polym. Eng. Sci. 46 (2006) 680–690, h ps://doi.o g/
10.1002/pen.20521.
[200] J. S ange, H. Müns ed , Rheological p ope ies and oaming beha io o
polyp opylenes wi h di e en molecula s uc u es, J. Rheol. 50 (2006) 907–923,
h ps://doi.o g/10.1122/1.2351880.
[201] N. Weinga , D. Raps, M. Lu, L. Endne , V. Al s ¨
ad , Compa ison o he oamabili y
o linea and long-chain b anched polyp opylene— he legend o s ain-ha dening
as a equi emen o good oamabili y, Polyme s 12 (2020) 725, h ps://doi.o g/
10.3390/polym12030725.
[202] F.N. Cogswell, Con e ging low and s e ching low: a compila ion, J. Non-
New onian Fluid Mech. 4 (1978) 23–38, h ps://doi.o g/10.1016/0377-0257(78)
85004-6.
[203] D.H. C a e , J.A. Cuculo, E. Boud eaux, Flow-induced c ys alliza ion o polyme
mel s in con e gen low egimes and imposed empe a u e g adien o o m
ul a-o ien ed ibe s uc u es, Polym. Eng. Sci. 20 (1980) 324–329, h ps://doi.
o g/10.1002/pen.760200503.
L. Sang oniz e al.