Glass-F eeze Analysis P o ocol: CPA + Cons ain Ra e Compa ison
Deb a S. Ga an ∗
DPΦIni ia i e, USA
Ch is ian E. P ecke †
Independen Physicis , Po ugal
No embe 2025
Abs ac
The CPA + Cons ain (CPA + C) o mula ion o he Dynamic P esen Theo y (DP
Φ
) was e alua ed ac oss
h ee canonical iscosi y da ase s: he o ho- e phenyl (OTP) measu emen s o Laughlin and Uhlmann (1972),
he independen OTP da ase o Plazek (1994), and h ee glyce ol–wa e mix u es epo ed by Kuma e al.
(1994). Ac oss all e alua ed da ase s, he CPA + C model ep oduced he Vogel–Fulche –Tammann (VFT)
model wi h s a is ical pa i y (
𝑅2>0.99
) and sligh ly ou pe o med i in he Plazek OTP da ase . Unlike
he empi ical s uc u e o he VFT equa ion, he CPA + C o mula ion de i es he same cu a u e om a
physically mo i a ed cons ain -modula ed a e o Con inuous P esen Ac ualiza ion (CPA). These indings
indica e ha he esidual noise obse ed in simple models ep esen s an ac ual physical signal: he inc ease
in con igu a ional cons ain as i app oaches a CPA Lock-In (o cohe ence s all), which CPA + C e ec i ely
isola es. Combined wi h mechanis ic in e p e abili y, his nume ical pa i y es ablishes CPA + C as a physically
g ounded al e na i e o pu ely empi ical iscosi y ela ions, po en ially acili a ing he a ional design o
ma e ials wi h cus omized low p ope ies.
1 INTRODUCTION
The CPA + Cons ain (CPA + C) o mula ion in e p e s iscosi y as a measu able consequence o cons ain -
induced esis ance o con igu a ion change. As con igu a ional cons ain inc eases, he sys em’s capaci y
o econ igu e slows, a p ocess modeled wi hin Dynamic P esen Theo y (DP
Φ
) [
1
] as Con inuous P esen
Ac ualiza ion (CPA), an i e e sible, ene gy-limi ed a e o s a e change go e ned by local ene gy densi y and
s uc u al load.
Wi hin his amewo k, iscosi y e lec s esis ance o ac ualiza ion, a educ ion in accessible mic ocon igu a-
ions as s uc u al cons ain accumula es. The CPA + C model o malizes his by linking iscosi y g ow h o a
physically mo i a ed, cons ain -modula ed ac ualiza ion a e.
This s udy e alua es whe he he CPA + C o mula ion can quan i a i ely ep oduce he Vogel–Fulche –Tammann
(VFT) equa ion [
2
,
3
,
4
], one o he mos success ul empi ical ela ions in ma e ials science, ac oss h ee
chemically dis inc iscosi y da ase s: he o ho- e phenyl (OTP) measu emen s o Laughlin and Uhlmann [
5
],
he independen OTP da ase o Plazek [6], and glyce ol–wa e mix u es cha ac e ized by Kuma e al. [7].
2 METHODS
Viscosi y- empe a u e da a o o ho- e phenyl (OTP, 240–385 K) we e i ed using ou models: he A henius
equa ion, he Vogel-Fulche -Tammann (VFT) equa ion, a Simple Exponen ial, and CPA + Cons ain (CPA + C).
∗Co esponding au ho : [email p o ec ed], h ps://o cid.o g/0009-0004-5593-713X.
†[email p o ec ed], h ps://o cid.o g/0000-0003-0828-6835.
1
The analysis e alua ed h ee chemically dis inc sys ems: he canonical o ho- e phenyl (OTP) measu emen s o
Laughlin and Uhlmann (1972) [
5
], he independen OTP da ase o Plazek (1994) [
6
], and h ee glyce ol-wa e
mix u es epo ed by Kuma e al. [
7
]. Iden ical GUI expo s we e u ilized o all pa ame e ex ac ion, esidual
analysis, and consis ency checks
Nonlinea leas -squa es op imiza ion (Le enbe g-Ma qua d ) was used o minimize he oo -mean-squa e
(RMS) e o in
log10(𝜂)
. Model selec ion was based on AIC and BIC sco es, wi h 95% con idence in e als
calcula ed om he pa ame e co a iance ma ix. The alue
𝑘
was de ined as he numbe o ee- i ed pa ame e s;
ixed hype pa ame e s and duplica e ep esen a ions (e.g., 𝜂0 s. log10 𝜂0) we e no coun ed.
Model i ing and e i ica ion we e pe o med on a cus om Glass GUI V1.1 (2025) de eloped by C. E.
P ecke , which implemen ed he CPA + C o mula ion alongside e e ence models.
Pos -p ocessing by D. S. Ga an included heo e ical syn hesis, in eg a ion o analy ical inpu s, collabo a ion
managemen , and manusc ip d a ing. Fo mal de ini ions o he ounda ional p inciples, Con inuous P esen
Ac ualiza ion (CPA) and Cons ain Load (
𝐶
), a e p o ided in Dynamic P esen Theo y I (DP
Φ
) [
1
]. This s udy
ope a ionalizes hese p inciples as he CPA + C iscosi y model.
P io o he acquisi ion o esul s, he e alua ion c i e ia and analysis plan in he p e egis a ion p o ocol [
8
]
ou lined a quali a i e expec a ion ha he CPA + C model would ep oduce he cu a u e o he VFT equa ion.
The nea -exac quan i a i e pa i y ac oss chemically dis inc sys ems was no assumed be o ehand.
3 RESULTS
The CPA + Cons ain (CPA + C) o mula ion ep oduced he i o he Vogel–Fulche –Tammann (VFT) equa ion
o o ho- e phenyl iscosi y (240–385 K) wi h s a is ical pa i y (
𝑅2=0.9967
, RMSE
≈
0.235), success ully
cap u ing he supe -A henius cu a u e ac oss 14 o de s o magni ude.
The same pa i y was main ained o all h ee glyce ol–wa e mix u es epo ed by Kuma e al. [
7
], each
yielding 𝑅2≈0.9981 unde iden ical pa ame e ex ac ion p ocedu es.
In he independen OTP da ase o Plazek e al. [
6
], he CPA + C model sligh ly ou pe o med he VFT
e e ence (AIC =
−33.81
s.
−33.42
;
𝑅2=0.9932
) wi h esiduals exhibi ing no sys ema ic cu a u e ac oss he
empe a u e domain.
Toge he , hese esul s demons a e ha he CPA + C model ep oduces VFT-le el accu acy using a compa able
numbe o i ed pa ame e s while also o e ing a physically in e p e able mechanism based on cons ain -limi ed
ac ualiza ion.
Table 1: Fi me ics o iscosi y models applied o o ho- e phenyl (OTP) da a (240–385 K).
Model RMSE MAE Bias MAD 𝑅2AIC
VFT 0.2346 0.1989 −1.8×10−11 0.1770 0.9967 -95.50
CPA + Cons ain 0.2346 0.1989 −1.7×10−70.1770 0.9967 -91.50
Simple Exponen ial 1.6053 1.2747 0.0357 0.8978 0.8447 39.13
A henius 2.2890 1.9479 -0.2821 0.5397 0.6843 61.97
2
Figu e 1: Viscosi y o o ho- e phenyl e sus empe a u e. The CPA + Cons ain model achie es s a is ical pa i y wi h he
empi ical VFT equa ion, success ully cap u ing he supe -A henius cu a u e o e 14 o de s o magni ude.
4 DISCUSSION
The CPA + Cons ain (CPA + C) o mula ion ep oduces VFT-le el cu a u e wi h negligible s a is ical de ia ion
while p o iding a physically mo i a ed mechanism o iscosi y g ow h; as con igu a ional cons ain inc eases,
he sys em’s abili y o econ igu e slows, app oaching a CPA Lock-In (o cohe ence s all) h eshold.
In his iew, he VFT
𝑇0
singula i y eme ges as a non-physical a i ac : an ex apola ion e o om i ing an
empi ical cu e beyond i s alid ange. The CPA + C model eplaces his di e gence wi h a physically g ounded
al e na i e: he Cons ain Load (
𝐶
), which modula es he baseline CPA Ra e as he sys em app oaches a eal,
ini e CPA Lock-In empe a u e (𝑇𝑔).
Residual pa e ns in simple ela ions (e.g., he A henius o unmodi ied exponen ial o ms) a e ein e p e ed—
no as noise—bu as s uc u ed de ia ions: signa u es o cons ain eedback on he ac ualiza ion a e. The
CPA + C o mula ion isola es and quan i ies his s uc u e, p o iding a di ec physical in e p e a ion o he
supe -A henius egime.
Cons ain and Ac ualiza ion. In he CPA + C amewo k, cons ain e e s o he e ec i e limi a ion on
accessible mic ocon igu a ions as a sys em becomes mo e s uc u ally loaded. Physically, cons ain e lec s he
inc easing di icul y o ea angemen due o geome ic us a ion, molecula c owding, o bond-ne wo k igidi y.
In o ma ionally, i co esponds o a na owing o iable nex -s a e ansi ions unde local ene gy and opology
3
condi ions. As cons ain accumula es, he con igu a ion-change a e slows, culmina ing in a CPA Lock-In nea
𝑇𝑔. Con inuous P esen Ac ualiza ion (CPA) models his p ocess as a p esen -only, cons ain -modula ed low.
Ac oss all es ed sys ems, CPA + C ep oduced he VFT ela ion while educing empi ical o e head and
in oducing a physically in e p e able mechanism. This c oss-da ase ag eemen sugges s ha he longs anding
success o he VFT o m may a ise om deepe cons ain -d i en dynamics ha CPA + C explici ly cap u es.
By oo ing iscosi y scaling in cons ain -modula ed ac ualiza ion dynamics, he CPA + C o mula ion
enables p incipled ex apola ion and may suppo he a ional design o ma e ials wi h ailo ed low p ope ies,
pa icula ly in egimes whe e con en ional models ex apola e poo ly.
Acknowledgmen s
The p epa a ion o his manusc ip bene i ed om mode n la ge-language-model ools, which assis ed wi h
d a ing, edi ing, and e i ica ion.
Funding
No ex e nal unding was ecei ed o his wo k.
Da a and Code A ailabili y
The ull da ase s, esidual expo s, and pa ame e iles a e included in his eposi o y. The analysis ool (Glass
GUI V1.1) was a chi ed by C.E. P ecke and is a ailable upon eques .
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