1
Concise Mechanis ic Model o Phase Change
Ma e ial Solidi ica ion Kine ics
Louis Desg osseillie s1, Ne ea U anga2, Daniel Ca bonell3, Ignacio Gu uchaga1
1 SPF Ins i u ü Sola echnik, Os schweize Fachhochschule, Rappe swil, Swi ze land
2 Fundación Teknike , Eiba , Spain
3 DCa bo Ene gy Consul ing, San Llo enç Sa all, Spain
E-mail: [email p o ec ed]
Abs ac
A concise powe law eac ion kine ic exp ession o e e sible c ys alliza ion phase change was
p esen ed o use in global modelling o c ys alliza ion and nuclea ion kine ics o solid-liquid
phase change ma e ials (PCMs). This was de eloped as a closed o m, wholly mechanis ic
exp ession o phase change kine ics essen ial o p edic coupled hea -mass la en hea e olu ion
in PCMs. This cons i u ed a signi ican depa u e om he semi-empi ical and
phenomenological o mula ions ha ha e so a domina ed PCM sciences. No ably, he
p esen ed o mula ion exp esses mass supe sa u a ion as he undamen al d i ing o ce o
phase change as opposed o elying on supe cooling deg ee o empe a u e a e. In con as o
p ac ice in he indus ial c ys alliza ion p ocess indus y, i was pos ula ed ha c ys al size
popula ion balances could be neglec ed in PCMs used o he mal s o age due o a e aging
e ec s o c ys al size popula ion balances since he a es o la en hea e olu ion and hea
anspo a e exclusi ely p io i ized a he han yield o a desi ed c ys al size.
A a ou able o mula ion o he global powe law c ys alliza ion kine ic model was ob ained
h ough e alua ion agains he 0D ansien e e ence case de i ed om he cooling cu e o
manually seeded c ys alliza ion o 11 K supe cooled 38.1 %mass NaCHO2 aqueous solu ion,
yielding NaCHO2•3H2O(s), unde de elopmen as a cold s o age PCM. This o mula ion was
based on sequen ial eac ions o nuclea ion and c ys alliza ion, he la e linea ly dependen on
he mass ac ion o nuclei. Values o appa en ac i a ion ene gies o nuclea ion and c ys al
g ow h and hei espec i e p e-exponen ial a e cons an s we e ob ained by leas -squa es
i ing o cooling cu e da a.
Keywo ds: mix u e PCMs, solid-liquid phase change, phase change ma e ials, powe law kine ic model, nuclea ion a e,
c ys alliza ion a e, non-iso he mal kine ics
This p ep in esea ch pape has no been pee e iewed. Elec onic copy a ailable a : h ps://ss n.com/abs ac =5378887
P ep in no pee e iewed
2
1. In oduc ion
A e sa ile and simple, hough mechanis ic, mac oscopic ea men o solid-liquid phase change kine ics would o e dis inc
ad an ages in he dynamic analysis and simula ion o phase change ma e ials (PCMs) and he pe o mance o he mal s o age
sys ems. Such a o mula ion would eadily ind applica ion in simula ion and analysis o coupled hea -mass la en hea
p ocesses o he mal s o age, such as non-equilib ium calo ime ic analysis o PCMs, s udy o hea ing/cooling a e dependen
phase change hys e esis, solidi ica ion om supe cooled liquids, and dynamic he mal s o age sys em pe o mance simula ion.
In ecen yea s, semi-mechanis ic [1] and phenomenological [2] app oaches ha e been in oduced o model supe cooling
and/o phase change hys e esis in s a ic PCM (i.e., no con ec ion anspo ), each one cons i u ing a depa u e om classical
mo ing bounda y p oblems o idealized phase change go e ned only by hea balances. Ba z e al. [2] ha e p esen ed a
con enien phenomenological amewo k o p edic phase ac ion om pa h dependen scaling ela ionships o he appa en ,
hys e esis en halpy p o ile o a PCM. While shown compu a ionally e icien and appa en ly mo e ep esen a i e o
expe imen al da a o incomple e phase change cycling han we e classical models o idealized phase change he modynamics,
phenomenological app oaches canno undamen ally elucida e knowledge o he inhe en phase change p ocesses. Also, as his
was modelled using en halpy- empe a u e p o iles ob ained h ough dynamic calo ime ic expe imen s, such as hose used o
epo en halpy- empe a u e in PCM supplie da ashee s, he da a could be alid only o hea ing/cooling a e in ensi ies
(W.kg-1) nea hose o he calo ime ic measu emen condi ions. In con as , Gün he e al. [1] de eloped a semi-mechanis ic
app oach ha , while emaining classically mechanis ic in i s ea men o he modynamic p ope ies, hea ans e , and
p opaga ion o seconda y nuclea ion, elied on explici empi ical unc ions ob ained om iso he mal expe imen s o single
c ys al g ow h ex apola ed on he bulk o go e n he a e o phase change. In consequence, hei me hod p oduced esul s o
which he phase change solidi ica ion kine ics we e ixed o p esc ibed p o iles. Also, hei model equi ed explici de ini ion
o a nuclea ion empe a u e de i ed om small scale expe imen s, hus no ep esen ing nuclea ion as a pa h dependen dynamic
p ocess. While hei model could be used o ep oduce obse ed phase change beha iou unde iden ical condi ions o hose o
hei sou ce da a o in o ming he model equa ions, i would be howe e incapable o p edic ing beha iou o he same PCM
unde subs an ially di e en expe imen al condi ions, namely in he p edic ion o nuclea ion om supe cooling/supe sa u a ion
o he liquid phase. Fu he mo e, bo h me hods [2] and [1] ha e been de eloped o cong uen ly mel ing PCMs only (i.e., solid
phase and liquid phase ha e iden ical composi ions), hus neglec ing phase change o incong uen PCM mix u es, namely
aqueous solu ions o sal hyd a es.
Bo h [1,2] ha e op ed o simpli y modelling by adop ing exp essions using exclusi ely empe a u e (e.g., supe cooling
deg ee) and empe a u e a e as he sole d i ing o ces o phase change. Howe e , especially conce ning solidi ica ion
p ocesses, empe a u e has only an indi ec e ec on he d i ing o ce o phase change, whe eas i is in ac a mass
supe sa u a ion ha is he esponsible d i ing o ce, long since ecognized in he ields o indus ial c ys alliza ion and
me allu gy [3–6]. Wi h empe a u e, howe e , one can compu e he mass supe sa u a ion by de e mining he equilib ium phase
ac ion ha is ma e ial-sys em speci ic and can be ob ained om ei he a hea balance (cong uen PCMs) o liquidus cu e
da a om mix u e phase diag ams (incong uen PCMs). Many mo e analogies o phase change can be d awn be ween di e en
ma e ial sys ems on he basis o mass supe sa u a ion han a e possible using supe cooling deg ee alone.
While bo h classical nuclea ion heo y (CNT) [4,7] and law o mass ac ion models o bulk c ys alliza ion [3,4,6] employ
mass supe sa u a ion (whe he di ec ly o indi ec ly) as he unde pinning d i ing o ce o nuclea ion and c ys al g ow h, hei
o mula ions a e oo eso e ic and cumbe some o p ac ical use in PCM science and mac oscopic simula ion o PCM he mal
s o age sys ems unde going solid/liquid phase change. CNT is oo o en conce ned wi h he dynamics o ei he only p ima y
nuclea ion o o single c ys al g ow h and gene ally equi es conside able ma e ial p ope y da a gene ally una ailable in he
domain o PCMs. Lane in oduced undamen als o CNT o PCM science in his seminal wo k [8], bu his was no u he
in eg a ed o adap ed o mac oscopic analysis o phase change p ocesses o he mal s o age. Law o mass ac ion models (i.e.,
powe law eac ion kine ics) o bulk c ys alliza ion, on he o he hand, use much mo e concise o mula ions wi h eliance on
ewe physical pa ame e s, bu a e ocused on compu a ion o popula ion balance modelling o c ys alli es a he han jus
global mass balances o phase ac ion yields. This is o cou se mo i a ed by he economic alue o con olling yield o a
desi ed c ys al size and pu i y in indus ial p oduc ion. The JMAK equa ion (Johnson-Mehl-A ami-Kolmogo o ), o he wise
called A ami equa ion [4,5], hough amously used o i s ex eme simplici y o exac solu ion accoun ing o impingemen o
g owing c ys als, ep esen s only iso he mal c ys alliza ion p ocesses, hus unsui able o cyclical and/o non-iso he mal phase
change p ocess modelling and simula ion.
In he ield o PCM science and enginee ing o he mal s o age, he p edic ion and con ol o c ys al size and popula ion
balances a e no gene ally o any g ea conce n, whe eas dynamic p edic ion o bulk phase change yield and i s e ec on he
o e all he mal ene gy balance and hea ans e cons i u e he p ima y objec i es. Fu he mo e, in PCM science and
enginee ing, kine ic exp essions o phase change mus also be able o dynamically achie e a ious s a es o 2-phase equilib ium.
This p ep in esea ch pape has no been pee e iewed. Elec onic copy a ailable a : h ps://ss n.com/abs ac =5378887
P ep in no pee e iewed
3
In his pape , a gene alized powe law o mula ion o mac oscopic, e e sible solidi ica ion phase change kine ics is
p esen ed and u he demons a ed o he case o non-iso he mal cooling, he e ogeneously igge ed nuclea ion in a sal
hyd a e PCM-wa e solu ion and subsequen equilib ium-s age solidi ica ion. This speci ic ocus was mo i a ed by he au ho s’
s udy o T-his o y cooling cu es o sal hyd a e-wa e solu ions manually nuclea ed om he supe cooled s a e ha we e
pe o med unde he auspices o he EU Ho izon unded BEST-S o age p ojec (www.bes -s o age.eu). The ma e ial sys em in
ques ion has been in es iga ed o use in a cold s o age Phase Change Slu y he mal s o age sys em.
The powe law phase change kine ic amewo k has been examined wi h espec o single s ep and wo sequen ial s ep
o mula ions and explo ed o he e ec s o appa en eac ion o de and appa en ac i a ion ene gy. These we e s udied using
as e e ence case T-his o y cooling cu e da a measu ed o NaCHO2-wa e solu ion, manually seeded in he supe cooled s a e.
2. Me hod
This wo k ocused exclusi ely on powe law kine ic exp essions wi h A henius empe a u e dependence and supe sa u a ion
d i ing o ce o he o m:
d
𝑤
s
d
𝑡
=
sign
(
𝜎
)
𝐾
exp
(
―
𝐸
a
𝑅(𝑇
+
273.15)
)
|
𝜎
𝑛
|
, (1)
𝜎
=
𝑤
l
―
𝑤
∗
l
𝑤
∗
l
, and (2)
𝑤
∗
l
=
𝑓
(
𝑇
)
, (3)
whe e w is he mass ac ion, σ he supe sa u a ion quo ien [3,4], K he A henius p e-exponen ial a e cons an , Ea he appa en
ac i a ion ene gy o he ne p ocess, R he ideal gas cons an , T he empe a u e in °C, and n he eac ion a e o de . Subsc ip s
s and l and supe sc ip * indica e he bulk solid phase, solu e in he liquid phase, and chemical equilib ium p ope y,
espec i ely. The appa en eac ion o de n is de ined 1 ≤ n ≤ 3 o c ys al g ow h and seconda y nuclea ion – whe e c ys al
g ow h ≤ seconda y nuclea ion [3] – and as high as n ≈ 20 o p ima y nuclea ion [4]. Las ly, (T) ep esen s a polynomial i
o he liquidus cu e o he solidi ying species in he mix u e’s equilib ium phase diag am. The compiled lis o symbols can
be ound a he end o his documen .
This o mula ion was ounded on ou key axioms:
1. A henius empe a u e dependence o he eac ion a e cons an ,
2. Liquid phase supe sa u a ion d i ing o ce solely esponsible o solidi ica ion p ocesses,
3. Single exp ession o e e sible solidi ica ion p ocess, he e o e able o achie e he equilib ium s a e, and
4. Solidi ica ion a e is gene ally independen o he size o c ys als.
Al hough appea ing decep i ely simple, he second poin was essen ial o a oid concluding in e o ha solidi ica ion
p ocesses exhibi an i-A henius beha iou (i.e., nega i e ac i a ion ene gy), as was ema ked by polyme c ys alliza ion
scien is s in he ea ly 1950s and la e acknowledged o be alse [9]. The e o e, he obse a ion in mos solidi ica ion om
liquids whe e solidi ica ion a es inc ease a empe a u es below hei he modynamic equilib ium ansi ion empe a u es is
owed o he inc ease in mass supe sa u a ion a lowe empe a u es (excep o mix u es wi h liquidus cu es con e ging o he
solidus a low empe a u es). I is impo an o no e also wo impo ance easons o selec ing he liquid phase supe sa u a ion
quo ien as p omo ed by Mye son [3] as d i ing o ce a he han he quo ien o he solid phase con e sion de ici used by
o he s (e.g., Foube e al. and Mazzan i e al. [4]): i) he liquid phase supe sa u a ion ep esen s he excess eagen s o
c ys alliza ion, mo e ap o law o mass ac ion kine ics han he eac ion p oduc de ici , and ii) nonze o alue o he
denomina o wl* a T*, especially impo an o modelling o single componen , cong uen PCM solidi ica ion.
The hi d poin was exploi ed o simpli y he eac ion a e exp ession o a oid sepa a ely exp essing he e e se eac ion a e
in o de o achie e he equilib ium s a e, wl*. In con as o Foube e al.’s o mula ion speci ying sepa a e o wa d and e e se
a e exp essions [4], he cu en o mula ion can accomplish eaching equilib ium in a concise o mula ion due o always
p ese ing he sign o σ in Eq.(1) no ma e he eac ion a e o de ‘n’. Fo his s udy, howe e , he phase change kine ics powe
law model was used only o model mono onic solidi ica ion, so i emained o be seen how he model would pe o m when
ep esen ing ne dissolu ion o he solu e, especially a empe a u es exceeding he sa u a ed liquid empe a u e o he a e age
composi ion o he mix u e PCM.
While i em 4 con adic s p ac ice in indus ial c ys alliza ion, i is none heless an app op ia e simpli ica ion o PCM he mal
s o age sys ems. In he de elopmen o PCM he mal s o age sys ems, he e is no e o aken o ensu e con ol o c ys al size
popula ions, hus lea ing he con ol o he hea ing and cooling a es ee o espond o only he ope a ional equi emen s
(ex e nal hea anspo bounda y condi ions) o hea abso p ion and hea elease. In he case o solid-on-coil PCM hea
exchange, a polyc ys alline bulk laye is o med. The e o e, hese ypical condi ions make i plausible o assume ha he e
would exis a any ime a b oad and andom dis ibu ion o c ys al sizes and nuclei. The e o e, i was assumed ha an a e aging
This p ep in esea ch pape has no been pee e iewed. Elec onic copy a ailable a : h ps://ss n.com/abs ac =5378887
P ep in no pee e iewed
4
e ec domina es he solidi ica ion a e o PCMs, in ha he bi h o new c ys als, he g ow h o ma u e c ys als, impingemen ,
and he ac u e o la ge c ys als all con ibu e simul aneously o bulk solidi ica ion wi h nei he domina ing. O cou se, his
would no always hold ue (e.g., e y low, cons an cooling a es gene a ing ew nuclei and la ge indi idual c ys als), bu was
expec ed o be none heless an expedien and use ul concep wi hin he he mal ene gy s o age ield.
Assump ions on he mos likely an icipa ed eac ion mechanisms (single s ep s. wo sequen ial s eps) and eac ion o de s
ele an o PCMs ha e been examined in he ollowing sec ions o he o mula ion o he comple e sys em o equa ions
ep esen ing 0D ansien solidi ica ion o solu e om a mix u e PCM.
2.1. Single Reac ion S ep
The o mula ion emained as i was in Eqs.(1)-(3). This o mula ion ep esen ed a sys em whe e he nuclea ion a e is
su icien ly s ong ha bulk solidi ica ion does no exhibi a dis inc induc ion pe iod. This could be po en ially sui able o
ep esen PCM ma e ials exhibi ing minimal supe cooling and hose doped wi h nuclea ion addi i es.
In his mode, he eac ion o de was cons ained o 1 ≤ n ≤ 3, ep esen ing as kine ics a low alues o he supe sa u a ion
quo ien , σ. Fas kine ics we e hough o be necessa y o ep oduce equilib ium solidi ica ion beha iou using only a single
eac ion s ep model, whe eas mul iple eac ion s ep models could combine con ibu ions o e ms each ep esen ing as o slow
kine ics.
Independen o his mode succeeding o ep esen he e e ence case, his mode would se e mainly o s udy sepa a ely he
p ope ies o eac ion o de s 1-3 and p io i ize which o be assigned in he s udy o he wo sequen ial eac ions o mula ion as
well as de ine an icipa ed anges o hei eac ion pa ame e s Ea and K.
2.2. Two Sequen ial Reac ions
The de ining ea u e o his mode was he sequen ial na u e o nuclea ion and c ys al g ow h a es whe e he c ys al g ow h
a e was di ec ly p opo ional o he mass ac ion o nuclei, nuclea ion se ing as he o e all a e limi ing s ep. This o mula ion
was an icipa ed o p oduce an ini ial induc ion pe iod in which, a e seeding, he nuclea ion a e would domina e, and up o a
c i ical alue o nuclei mass ac ion, c ys al g ow h would domina e he o e all solidi ica ion kine ics.
The go e ning equa ions o his mode we e:
𝑤
s
=
𝑤
n
+
𝑤
c
, (4)
d
𝑤
n
d
𝑡
=
sign
(
𝜎
)
𝐾
n
exp
(
―
𝐸
a
,n
𝑅
(
𝑇
+
273.15)
)
|
𝜎
𝑛
|
, and (5)
d
𝑤
c
d
𝑡
=
𝑤
n
sign
(
𝜎
)
𝐾
c
exp
(
―
𝐸
a
,c
𝑅
(
𝑇
+
273.15)
)
|
𝜎
𝑚
|
,
(6)
whe e subsc ip s n and c deno e nuclea ion and c ys al g ow h, and n, and m ep esen he espec i e powe s o nuclea ion
eac ion o de , and c ys al g ow h eac ion o de . As has been al eady indica ed in Sec ion 2, he appa en eac ion o de o
seconda y nuclea ion is de ined wi hin he 1 ≤ n ≤ 3 limi s and he one o he c ys al g ow h is cons ained o m ≤ n.
Fo unseeded (p ima y) nuclea ion, one could subs i u e Eq.(5) wi h a o mula ion o sequen ial/pa allel eac ions o p ima y
(deno ed by subsc ip “1”) and seconda nuclea ion (“2”),
d
𝑤
n
d
𝑡
=
sign
(
𝜎
)
[
𝐾
n,1
exp
(
―
𝐸
a
,n,1
𝑅
(
𝑇
+
273.15)
)
|
𝜎
𝑛
1
|
+
𝑤
s
𝐾
n,2
exp
(
―
𝐸
a
,n,2
𝑅
(
𝑇
+
273.15)
)
|
𝜎
𝑛
2
|
]
,
(7)
wi h he seconda y nuclea ion’s dependence on a p io popula ion o c ys alli es, ws and n1 > n2 [3]. This, howe e , was no
pu sued o demons a ion in his s udy as he e e ence case o alida ion was one o manually seeded nuclea ion (seconda y
nuclea ion).
2.3. Ma e ials and Re e ence Expe imen
The e e ence case o demons a ion and pa ial alida ion o he ea u es o he eac ion mechanisms p esen ed in Sec ions
2.1. and 2.2. was ha o a cus om p epa ed T-his o y cooling cu e expe imen pe o med using solu ions o NaCHO2 in wa e .
The T-his o y appa a us and one sample ube assembly a e shown in Figu e 1. The appa a us consis ed o a p ecision
hea ing/chilling ba h (±0.01 K (manu ac u e s a emen ), Julabo DYNEO DD1000F) illed wi h 10 %mass e hanol in wa e
ope a ed a 60 % pump ou pu (27 L/min a 100 %). Th ee sample ubes we e suspended om a cus om lid ix u e in o he ba h:
each consis ed o 10 mm OD/8 mm ID/178 mm long bo osilica e glass sample ube su ounded by polye hylene oam insula ion
o 33 mm OD/14 mm ID and so poly inyl chlo ide slee e added o e he glass ubes o imp o e he i ing. Insula ing he
sample ubes ollowed e inemen s o he T-his o y echnique a ZAE Baye n [10] and helped ensu e ha each sample could
This p ep in esea ch pape has no been pee e iewed. Elec onic copy a ailable a : h ps://ss n.com/abs ac =5378887
P ep in no pee e iewed
5
accu a ely ely on only one single empe a u e p obe despi e he s ongly con ec i e en i onmen o he chilled ba h. Lumped
mass app oxima ion o he indi idual samples was suppo ed by Bi = 0.04 (Bio numbe ) de e mined expe imen ally using
wa e as he sample and assump ions o bo h o a ionally-symme ic (dT/dθ ≈ 0) and axially-independen hea anspo
(dT/dz ≈ 0) due o e y low cooling a e, high iscosi y (~20 mPa.s ), and poo aspec a io o sample olume supp essing
in e nal na u al con ec ion.
Each sample ube was also equipped wi h a op-moun ed 3-po connec o (b in Figu e 1), allowing o nuclea ion agen o
be added manually wi hou dis u bing he T-his o y expe imen . Nuclea ion agen was in oduced ia he ho izon al po o
manually igge solidi ica ion om he supe cooled liquid s a e (closed o e wi h ape when no in use), and a 2 mm OD T- ype
s ainless s eel shea hed he mocouple p obe was inse ed and posi ioned app oxima ely along he ube’s cen eline ia he op
po o he connec o . The heigh s o he sample ube he mocouples we e also adjus ed o he senso ip o lie a app ox. he
mid-dep h o he sample PCM olume (≈ 5 cm om he bo om o he ube). The he mocouples we e calib a ed o ±0.17 °C
unce ain y (95 % con idence) be ween 0 - 17 °C and logged a 0.1 Hz using a Keysigh DAQ970A mul ichannel empe a u e
ansmi e /logge .
The sample ubes we e each illed wi h app ox. 5 mL o solu ion p epa ed om 37.6±0.1 %mass NaCHO2 anh. (o en d ied,
97 %mass pu i y eagen g ade om Sigma, CAS 141-53-7, Table 1) and balance o e e se osmosis wa e , deemed a pseudo-
bina y mix u e o 38.1±0.1 %mass (95 % con idence) NaCHO2 in wa e when omi ing he mass o impu i ies. Fo such a low
ac ion o impu i ies, i was belie ed ha he e should be negligible e ec o highe o de mix u es on he phase equilib ium
ela ionship o wa e -NaCHO2, especially conside ing ha negligible di e en ial hea o solu ion has been epo ed o he
analogous sys em o wa e -NaC2H3O2 [11]. This analogy was jus i iable by he compa able aqueous dissocia ion cons an s o
each compound’s conjuga e acid [12].
The ube assemblies wi h loaded samples we e ini ially condi ioned a oom empe a u e (~16 °C) o achie e he mal
equilib ium, hen inse ed simul aneously in o he chilled ba h a 0±0.01 °C. Nuclea ion agen NaC2H3O2 anh. (99 % pu i y
ACS g ade om Scha lau, CAS 127-09-3, Table 1) was manually added o he sample ubes once each sample eached < 1 °C
(supe cooled > 10 K), a which poin nuclea ion was induced ( eco ded as 5 290 s o sample 1). The samples hen eached
hei espec i e equilib ium condi ions (deemed > 5 460 s o sample 1), ollowed by a slow cooling p ocess app oaching
equilib ium-s age solidi ica ion o he emainde o he expe imen . See sample he mog ams in Figu e 2 o mo e de ail. The
nuclea ion and solid-liquid equilib ium domains whe e solidi ica ion was de ec ed by he e ec o la en hea elease on he
sample empe a u e se ed as he e e ence case o solu ions o he wo kine ic model o mula ions.
Table 1: Ma e ial p o enance
Species
CAS
Supplie
Pu i y
NaCHO2
141-53-7
Sigma
97 %mass
NaC2H3O2
127-09-3
Scha lau
99 %mass
Re e se osmosis wa e
-
Onsi e
-
As seen in he wa e -NaCHO2 equilib ium phase diag am in Figu e 3 (p oduced om digi ized da a [13] and IUPAC-NIST
abula ed da a [14]), nuclea ion o 38.1 %mass NaCHO2 in wa e below 10.6 °C yields he ihyd a e solid compound.
Se ing as he measu ed da a o leas -squa es i ing he models p oposed in Sec ions 2.1. and 2.2., he co esponding
equilib ium mass ac ions o solid ihyd a e, ws*, assumed o ha e been yielded du ing he solid-liquid equilib ium cooling
phase in Figu e 2, we e in e ed using he polynomial i o he ihyd a e compound liquidus (solid-liquid) cu e in Figu e 3
and he Le e Rule (see Eq.(9)):
𝑤
∗
l
=
1.101
×
10
―
4
𝑇
2
+
6.024
×
10
―
3
𝑇
+
3.045
×
10
―
1
±
0.001
, and (8)
𝑤
∗
s
=
𝑤
a e
―
𝑤
∗
l
𝑤
TH
―
𝑤
∗
l
, (9)
whe e T is in °C, wa e co esponds o he bina y mix u e’s a e age composi ion (38.1±0.1 %mass NaCHO2), wTH is he
s oichiome ic mass ac ion o NaCHO2 in he ihyd a e compound (55.720±0.0005 %mass), and o which Eq.(8) ob ained
R2 = 1.000 hough using only he 4 da a poin s ea u ed by IUPAC-NIST Solubili ies Da abase. Howe e , in liquidus cu e
de e mina ion, 4 poin s should su ice o p o ide he wo e mini (pe i ec ic and eu ec ic) and wo poin s in be ween o es ablish
cu a u e and s eepness o slope o a quad a ic i .
Needed o calcula e he mass supe sa u a ion quo ien , σ, in Eq.(2) a e:
𝑤
w
=
1
―
𝑤
a e
𝑤
TH
=
0.316
±
0.002
, and (10)
𝑤
l
=
𝑤
TH
(
1
―
𝑤
w
1
―
𝑤
s
)
, (11)
This p ep in esea ch pape has no been pee e iewed. Elec onic copy a ailable a : h ps://ss n.com/abs ac =5378887
P ep in no pee e iewed
6
whe e ww co esponds o he mix u e’s mass ac ion o wa e in excess o he s oichiome ic a io needed o o m he ihyd a e
compound.
The o dina y di e en ial equa ion (ODE) p oposed in Sec ion 2.1. was sol ed as an ini ial alue p oblem o ws and he
sys em o ODEs in Sec ion 2.2. sol ed o [wn, wc], bo h using he ode15s s i sol e in Ma lab R2023a o educe compu a ion
e o in he equilib ium solidi ica ion egime whe e solu ion s i ness was an icipa ed. The ela i e and absolu e ole ances o
ode15s we e speci ied as 10-7 and 10-10, espec i ely. Ini ial alues o solid ac ions we e each null and he ime in e al o
he solu ion was speci ied as he beginning o he nuclea ion domain o he end o he solid-liquid equilib ium domain shown
in Figu e 2 ([5 290, 10 540] s o sample 1).
The emaining kine ic pa ame e s, Kn, Kc, Ea,n, and Ea,c we e de e mined by leas -squa es i ing o he espec i e ODE
solu ions o expe imen ally de i ed ws*,exp (Eq.(9) calcula ed o he sample 1 cooling cu e a ≥ 5 460 s in Figu e 2). No e
ha ws* in Eq.(9) was alid only o equilib ium solidi ica ion and could no be used o es ima e solid ac ion in he nuclea ion
domain (5 280 ≤ ≤ 5 460 s in Figu e 2) ma ked by as solidi ica ion kine ics. Pa ame e con e gence was achie ed using
mincon cons ained op imiza ion sol e in Ma lab, se o he de aul in e io -poin algo i hm. In each case, in ege alues o n
and m 1 we e speci ied and a ied o s udy he p ope ies o he models.
The gene al o m o he leas -squa es op imiza ion objec i e unc ion and cons ain s was:
min
(
∑
(
𝑤
∗
s,exp
―
𝑤
s
(
[
𝐾
i
]
,
[
𝐸
a
,
i
]
,𝑇,𝜎
)
)
2
)
s. .
{
𝐾
i
>
0
𝐸
a,i
>
0
, (12)
whe e [] indica es a ay se s o each K and Ea, index i = [n,c] (n = nuclea ion and c = c ys al g ow h) and ws is he ODE
imese ies solu ion o o al solid ac ion. The ODE imese ies solu ion, ws, was e imed in Ma lab ( e ime unc ion o
imese ies da a ype) using piecewise cubic spline in e pola ion o ob ain co esponding alues a each ime in e al o ws*,exp
(0.1 Hz). Also, in sol ing ws, linea in e pola ion was used o es ima e T om he sample 1 cooling cu e da a in Figu e 2 when
he imes ep used by ode15s sol e did no coincide wi h he cooling cu e’s 0.1 Hz da a cap u e.
3. Resul s
To simpli y compa ison o he wo 0D kine ic solidi ica ion phase change model o mual ions, he expe imen ally de i ed
equilib ium solid yield, ws*,exp, was calcula ed using Eq.(9) o only he sample 1 equilib ium domain cooling cu e in Figu e
2 ( ≤ 5 460 s) as he o he wo empe a u e cu es we e closely o e lapped wi h ha o sample 1 and so would se e no u he
pu pose. The e o e, all o he model de i ed solu ions, ws, we e gene a ed o a single da a se o di ec compa ison o one-
ano he .
The ini ial ime used o he onse o nuclea ion and c ys alliza ion in he compu a ion o he kine ic models was in e ed
om he sample 1 cooling cu e in Figu e 2-b a which ime he empe a u e p o ile i s showed signs o mono onic sel -
hea ing om la en hea elease due o nuclea ion/c ys alliza ion, deemed = 5 290 s. The end s a e o he nuclea ion domain,
deemed = 5 460 s, was asce ained by he obse a ion o he cessa ion o his mono onic sel -hea ing phase.
The compa ison o con e ged solu ions ob ained using he wo di e en phase change kine ic model o mula ions p esen ed
in his s udy was only he i s phase o compa ison, as he e e ence case could no ye se e as a p ecise alida ion case o he
models, in pa icula since he dynamic e olu ion o he solid ac ion du ing he nuclea ion phase could no be de e mined
di ec ly om he expe imen al esul s. This will ins ead be epo ed in he ollowing s udy o he de ailed, coupled hea -mass
pa ial di e en ial equa ion (PDE) solu ions o all h ee T-his o y cooling cu es in Figu e 2, as only he empe a u e cu e
could se e o accu a ely e eal which o he phase change kine ics solu ions mos ai h ully ep esen ed he dynamic, non-
iso he mal e olu ion o bo h seeded nuclea ion and c ys alliza ion. The esul s o his s udy would he e o e gene a e he sho -
lis o kine ic pa ame e s judged mos p omising o his inal s age o alida ion.
3.1. Single Reac ion S ep
Table 2 shows he lis o con e ged solu ions o n = 1 ound om he sea ch o local minima a ound ini ial guesses o Ea
in he ange [10, 104] J.mol-1 and o K in he ange [10-3, 101] kg.kg-1.s-1. Ini ial guesses supplied o mincon we e ixed a each
alue o Ea = [101, 102, 103, 104] J.mol-1 wi h he ini ial guess o de o magni ude a ied o K un il one o mo e local minima
we e ound o deemed in easible. In cases whe e local minima we e ound, ini ial guesses Ea and K we e epea ed close o hose
minima o u he p obe he neighbou ing solu ion space.
1 A i s , n and m we e also ob ained by leas -squa es op imiza ion, bu hese solu ions did no di e signi ican ly om he
ini ial guesses o n and m as he e we e many local minima possible o
K
n,
K
c,
E
a,n, and
E
a,c o each
n
and
m
.
This p ep in esea ch pape has no been pee e iewed. Elec onic copy a ailable a : h ps://ss n.com/abs ac =5378887
P ep in no pee e iewed
7
No e ha he con e ged solu ions o n = 1 in Table 2 we e ob ained only a e adding o Eq.(12) he addi ional minimiza ion
cons ain
max
(
𝑤
s
(
𝑡
≤
5 460 𝑠
)
)
≤
𝑤
∗
s,exp
(
5 460 𝑠
)
(13)
o supp ess unaway c ys alliza ion in he nuclea ion domain as shown in Figu e 4. While undamen ally un ealis ic, his
oscilla ing unaway c ys alliza ion beha iou se ed a leas o demons a e he inhe en ea u e o he p oposed gene al o m
o he powe law kine ic model o mula ion o encompass bo h he o wa d and e e se eac ions, p oducing he e o e a e u n
o ce o he ini ally obse ed unaway eac ion and la e eaching he co esponding solid-liquid equilib ium s a es.
Table 2: Con e ged kine ic pa ame e s and coe icien o de e mina ion o i o single eac ion model wi h n = 1.
#
Ea (J.mol-1)
K (kg.kg-1.s-1)
R2
1a
9.98×103
4.50×10-1
0.966
1b
9.98×103
4.57×10-1
0.968
1c
1.01×103
1.00×10-2
0.981
1d
1.01×103
1.56×10-2
0.994
1e
9.15×102
1.38×10-2
0.992
1
9.15×102
1.35×10-2
0.992
1g
2.74×102
8.10×10-3
0.982
1h
2.74×102
1.10×10-2
0.993
1i
2.00×102
9.20×10-3
0.989
1j
6.07×101
9.30×10-3
0.992
Whe he om he expe imen al body o knowledge on c ys alliza ion om liquids, CNT o JMAK equa ion iso he mal
c ys alliza ion kine ics, he e was no accep ed basis o his ype o seeded c ys alliza ion oscilla ing unaway kine ic beha iou .
While he e we e no means a ailable o di ec ly measu e he solid mass ac ion du ing he nuclea ion s ep, he gene al
consensus in he li e a u e on bo h iso he mal and con inuously cooled non-iso he mal c ys alliza ion kine ics p oposes ha
nuclea ion solidi ica ion in a iably ollows a sigmoidal p og ession, he e o e s ic ly mono onic [4] and he e o e non-
oscilla ing. The co esponding lis s o con e gence esul s o n = [2,3] a e no shown as hei local minima ollowed simila
pa e ns o solu ions ob ained o n = 1, hus only he solid ac ion p o iles o hei deemed global minima solu ions a e shown
he ein.
The con e ged solu ions o n = 1 in Table 2 o med solu ion amilies g ouped s ongly a ound ini ial guesses o Ea in he
ange [60, 104] J.mol-1, meaning ha o each Ea ini ial guess, mincon i e a ed K in p e e ence o Ea o achie e con e gence,
indica ing a much s onge sensi i i y o K han o Ea. Fo ini ial guesses o Ea ≥ 105 J.mol-1, mincon ailed o i e a e bo h K
o Ea any u he om he ini ial guesses. Howe e , he e emained s ill o iden i y he globally op imum and physically ele an
con e ged solu ion o n = 1.
A b ie su ey o ac i a ion ene gies o nuclea ion and c ys alliza ion p ocesses o ce amics and me als/me alloids sugges ed
ha scaled om 800 – 1 100 K o ≈ 300 K (pe i ec ic empe a u e o NaCHO2•3H2O = 18 °C [14]) using he law o equi alen
s a es (equal en opies o ansi ion), ac i a ion ene gies o bo h nuclea ion and c ys al g ow h would be es ima ed in he ange
≤ 105 J.mol-1 [15–17]. The e o e, along wi h he obse ed con e gence di icul ies o he ini ial guess 105 J.mol-1, hese
ein o ced a easonable uppe bound limi o Ea < 105 J.mol-1 o bo h nuclea ion and c ys al g ow h.
The su ey o he n = 1 solu ion space shown in Table 2 e ealed ha he appa en global op imum was si ua ed in icini y
o solu ion #1d. I became clea ha hose wi h Ea < 103 J.mol-1 (solu ions #1g-1j) we e un ealis ic since a such low alues, he
exponen ial e m would become mos ly in a iable in he sample 1 cooling cu e ange o 0-18 °C, as demons a ed by he
con e ged alues o K ha ing emained nominally equal o 10-2 kg.kg-1.s-1. A compa ison o he A henius exponen ial e m a
Ea = [102, 103, 104] J.mol-1 o 0-18 °C made i clea ha bo h Ea = [102, 104] J.mol-1 ep esen ed ex eme cases o in a iabili y
(ei he nea 0 o nea 1). Fo he case Ea = 103 J.mol-1, he exponen ial e m was e alua ed in he ange 0.6-0.7 o 0-18 °C and
i s slope was 10x g ea e han o ei he Ea = [102, 104] J.mol-1, bo h nominally 10-4 K-1. The e o e, solu ions wi h
Ea ≈ 103 J.mol-1 ep esen ed cases wi h mos ma ked empe a u e dependence o he powe law a e exp ession.
I became clea also om alues o R2 in Table 2 ha hose solu ions wi h Ea ≈ 104 J.mol-1 (solu ions #1a-1b) we e
un a ou able o ai h ully ep esen he appa en equilib ium c ys alliza ion beha iou o NaCHO2•3H2O. The e o e, solu ion
#1d became he deemed globally op imal solu ion in Table 2 as i had also supe io R2 o solu ions #1c and #1e-1 .
Figu e 5 shows he appa en globally con e ged solu ions o he cases o n = [1, 2, 3], whe e he solu ions o n > 1 we e
ob ained in he same manne as was done o n = 1 (solu ion 1d). In he case o n = 2, Ea = 9.81×102 J.mol-1 and
K = 0.134 kg.kg-1.s-1; o n = 3, Ea = 9.98×103 J.mol-1 and K = 39.9 kg.kg-1.s-1. All solu ions we e nume ically i ed o he
This p ep in esea ch pape has no been pee e iewed. Elec onic copy a ailable a : h ps://ss n.com/abs ac =5378887
P ep in no pee e iewed
8
expe imen ally de i ed nominal ws*,exp imese ies da a, ese ing a sensi i i y s udy o he uppe and lowe bounds o ws*,exp o
only he mos p omising kine ic model solu ions.
As seen in Figu e 5 (b), he unaway c ys alliza ion kine ics in he nuclea ion domain we e success ully aba ed by he
op imiza ion condi ion in Eq.(13) o all ins ances o n = [1, 2, 3]. Howe e , ha is whe e he simila i ies ceased be ween
solu ions o di e en eac ion o de . The i s o de appa en op imal solu ion (n = 1) achie ed he mos sui able kine ics o
equilib ium solidi ica ion, ws*,exp (i.e., a low alues o σ), albei a he expense o poo con e gence o ws*,exp a he ansi ion
om he nuclea ion domain o he equilib ium domain in p oximi y o = 5 460 s.
Bo h he second and hi d o de op imal solu ions, while poo ly con e ged o e all o ws*,exp, exhibi ed mo e apid ise in
modelled ws in he nuclea ion domain han did he n = 1 solu ion. These we e he esul s o as e kine ics han o n = 1 when
σ > 0.1, due in bo h cases o o de s o magni ude la ge alues o he A henius exponen ial e m compensa ing o he
educ ions in he powe law σ e m. Wha he wo highe o de solu ions also e ealed was hei inhe en p ope y o sha ply
dampen kine ics below a h eshold alue o σ: o he con e ged n = 2 solu ion his was σ < 0.024, and σ < 0.044 o n = 3.
While ce ainly no imp o ing model i o expe imen ally de i ed ws*,exp in Figu e 5, his ea u e could p o e aluable in
isola ing nuclea ion kine ics when only abo e a σ h eshold alue in wo sequen ial eac ions modelling, allowing he e o e
c ys al g ow h kine ics o domina e below his h eshold. This ype o beha iou would be consis en wi h obse ed no ms in
indus ial c ys alliza ion om solu ion [3].
The e o e, while he i s o de (n = 1) single eac ion model showcased as kine ics a low alues o σ o sui ably ep esen
equilib ium c ys alliza ion kine ics, i could no undamen ally alone cap u e bo h nuclea ion-domina ed kine ics and a
ansi ion o c ys al g ow h dominance. The second and hi d o de models ailed o exhibi equilib ium c ys alliza ion kine ics
bu showed p omise as a ool o egula e he ansi ion om nuclea ion dominance o c ys al g ow h a a h eshold alue o σ in
a mixed o de , wo eac ion model. Fu he mo e, as seen in Figu e 5-b, nei he i s no second and hi d o de s single eac ion
model solu ions could exhibi sigmoidal p o iles no nuclei induc ion pe iods ha a e ypical o c ys alliza ion p ocesses om
liquids [4,5].
Failu e o he single eac ion model o solely cap u e all aspec s o nuclea ion and c ys al g ow h we e an icipa ed bu se ed
none heless as a simple case s udy o he indi idual me i s o eac ion o de s n = [1, 2, 3] o in o m selec ion o eac ion o de s
o he wo sequen ial eac ions models. As such, a sensi i i y s udy o he n = 1 single eac ion model kine ic pa ame e s was
no pe o med.
Fo he emaining wo k on wo sequen ial eac ions modelling, i became clea ha m = 1 would be mos sui able o ep esen
c ys al g ow h in he equilib ium solidi ica ion domain ia Eq.(6) whe e low alues o σ would be encoun e ed. Bo h n = 2 o 3
we e o be explo ed o he seconda y nuclea ion kine ics in Eq.(5) due o hei abili ies o e ec i ely dampen nuclea ion
kine ics upon eaching a h eshold alue o σ.
3.2. Two Sequen ial Reac ions
In he sea ch o local op ima ob ained by leas -squa es i ing o he wo sequen ial eac ions kine ic phase change model,
Ma lab’s pa e nsea ch cons ained op imize was used o u he sea ch he solu ion space nea local minima iden i ied using
mincon in he same manne as was done o inding single s ep eac ion model local minima. This was done o p obe he nea by
space mo e ho oughly using a non-g adien , g id-based op imiza ion algo i hm. The solu ions ob ained a e shown in Table 3.
Solu ions we e ob ained wi h ini ial guesses o ypes Ea,n = Ea,c, Ea,n > Ea,c, and Ea,n < Ea,c o explo e he implica ions o highe
o lowe nuclea ion ac i a ions ene gies o bo h Ea,n and Ea,c in he ange [102, 103, 104] J.mol-1. Values whe e Ea,n > Ea,c we e
hough aluable o explo e as hey could po en ially p o e consis en wi h obse a ions o some PCMs whe e semi-o de ed
molecula clus e s we e ound o pe sis in he bulk liquid phase a T > Tm (mel ing empe a u e) and we e belie ed o se e as
seed nuclei in he cooling phase [18,19]. Highe alues o ne Ea,n o e e sible nuclei o ma ion would heo e ically esul in
nuclei pe sis ing in he bulk liquid phase a T > Tm bu also ha new nuclei a e mo e di icul o o m a T < Tm, esul ing in
deepe supe cooling/supe sa u a ion when he ini ial nuclei popula ion is su icien ly low due o liquid phase supe hea ing p e-
ea men o supp ess bulk c ys alliza ion. The la e ea u e is ep esen ed in he o mula ion and esul s p esen ed he ein o
he wo sequen ial eac ions model, while he o me emains o be demons a ed in simila kine ic modelling o
mel ing/dissolu ion p ocesses.
In Table 3, wo ypes o solu ions we e ob ained (sample p o iles shown in Figu e 6 and Figu e 7): 1) e y close i ing (mos
wi h R2 = 0.995-0.997) wi h kine ic beha iou in wo s ages ma ked by as nuclea ion e mina ing in a pla eau be o e he end
o he nuclea ion domain, hen nea -equilib ium c ys alliza ion o he emainde (no ed as “ wo-s age” in Table 3); and 2) close
i ing (R2 = 0.930-0.990) smoo h kine ic p o iles wi hou an in e media e pla eaux (no ed as “smoo h”). O hese wo solu ion
ypes, he smoo h p o iles we e he mos compelling. While bo h solu ion ypes exhibi ed ini ial nuclea ion induc ion phases
consis en wi h c ys alliza ion om solu ion (owed o wn in Eq.(6)), only he smoo h p o iles exhibi ed ully sigmoidal
This p ep in esea ch pape has no been pee e iewed. Elec onic copy a ailable a : h ps://ss n.com/abs ac =5378887
P ep in no pee e iewed
9
p og ession by also g adually app oaching he equilib ium solidi ica ion domain which was belie ed o be e con o m o he
e e ence empe a u e T-his o y empe a u e p o ile in he nuclea ion domain (Figu e 2) han wo-s age p o iles due o he
slowe and mo e g adual c ys alliza ion a e (hence a e o la en hea elease) o he smoo h p o iles. Howe e , such ques ions
emained inconclusi e in his s udy o lack o sui able da a o di ec alida ion o he model solu ions in he nuclea ion domain.
Compa isons o he cooling cu es in Figu e 2 o he esul ing simula ed hea -mass coupled model empe a u e p o iles ob ained
using he wo-s age and smoo h con e ged kine ic solu ions in Table 3 would be u he examined in a ela ed s udy on he T-
his o y cooling cu es o seeded, supe sa u a ed 38.1 %mass NaCHO2-wa e solu ions.
Table 3: Two sequen ial eac ion model leas -squa es solu ions o n = [2,3] and m = 1. G ey shaded ows indica ed he appa en global
op ima. Low sco ing o solu ions due o e y high wn was based on he con en ional no ion ha nuclei, while numbe ing many, due o
hei small size ep esen gene ally e y li le mass ac ion o e all (assumed < 0.01).
#
n
m
Ea,n
(J.mol-1)
Ea,c
(J.mol-1)
Kn
(kg.kg-1.s-1)
Kc
(kg.kg-1.s-1)
max(wn)
R2
Commen
2a
6.14×10-4
6.32×102
8.00×10-4
1.30×101
0.001
0.9967
Two-s age, e y low Ea,n and
low Ea,c
2b
5.79×101
1.01×103
8.00×10-4
1.33×101
0.0017
0.9948
Smoo h, e y low Ea,n
2c
7.04×101
1.00×103
5.00×10-4
1.22×101
0.0013
0.9787
Smoo h, e y low Ea,n
2d
1.01×102
1.00×103
1.00×10-4
1.01×102
0.0003
0.9966
Two-s age, e y low Ea,n
2e
7.13×102
1.00×104
7.90×10-3
1.00×102
0.012
0.9964
Two-s age, e y low Ea,n
2
9.15×102
9.96×103
4.51×10-2
7.24
0.07
0.9695
Smoo h, e y high ws,n
2g
9.96×102
1.01×103
7.61×10-2
1.14×10-1
0.096
0.9805
Two-s age, e y high ws,n
2h
1.00×103
9.99×102
5.20×10-3
1.02
0.011
0.9195
Smoo h, lowe i han 2m
2i
1.01×103
1.00×104
8.10×10-3
9.92×101
0.01
0.9955
Two-s age
2j
3.78×103
1.00×104
4.30×10-3
7.47×102
0.0018
0.9969
Two-s age
2k
9.98×103
1.12×103
5.13×10-2
2.00×101
0.0015
0.9967
Two-s age
2l
9.99×103
9.99×103
3.60×10-1
1.16×102
0.01
0.9957
Two-s age
2m
1.00×104
1.34×103
1.95×10-2
1.90×101
0.0008
0.9402
Smoo h, bes i n = 2
2n
2
1
9.83×104
9.64×104
1.00×1017
1.00×1017
0.07
0.9327
Smoo h, Kn & Kc e y high,
e y high ws,n
3a
1.58×10-2
6.60×102
3.00×10-2
1.60
0.013
0.9955
Two-s age, e y low Ea,n
3b
1.37×102
1.03×103
2.68×10-2
1.59
0.01
0.99
Smoo h, low Ea,n
3c
2.39×102
9.98×103
3.06×10-2
1.00×102
0.01
0.9958
Two-s age
3d
1.00×103
1.00×104
1.51×10-2
9.92×101
0.005
0.9337
Smoo h, lowe i han 3g
3e
9.98×103
1.33×103
9.80×10-2
3.28×101
0.0006
0.9861
Smoo h, simila o 3g
3
9.98×103
8.98×102
9.00×10-2
4.90×101
0.0005
0.9962
Two-s age
3g
9.99×103
9.63×102
9.33×10-2
3.28×101
0.0006
0.9897
Smoo h, bes i n = 3
3h
3
1
1.00×105
9.98×102
5.00×1015
1.00×102
0.0002
0.9933
Smoo h, Kn oo high
Un il such ime, ou solu ions shown in Table 3 eme ged as he mos p omising (see ows shaded in g ey): solu ions #2j
and #3 o wo-s age p o iles; #2m and #3g o smoo h p o iles. Among he wo smoo h solu ions, he hi d o de solu ion
(#3g) exhibi ed a no able imp o emen in i o he e e ence da a (R2), while he wo-s age solu ion #2j showed he bes i
o e all ( ollowed e y closely by #3 ). Figu e 6 shows he cha ac e is ic di e ences be ween wo-s age and smoo h solu ions,
using he wo bes i solu ions o each kind – #2j and #3g – as ep esen a i e examples and compa ing hem o he single
eac ion s ep bes i solu ion, #1d. Figu e 7 compa es he wo smoo h solu ions o di e ing eac ion o de s, #2m and #3g whe e
he di e ences in R2 in he same solu ion amily we e la ges . O he han hei wn p o iles di e ing by max(wn), he di e ences
be ween bo h he c ys al g ow h ac ion, wc, and he o al solid ac ions, ws, o #2j and #3 we e insigni ican .
In Figu e 6-a can be seen he wo-s age p o ile pla eau o #2j and he smoo hness o he #3g p o ile in compa ison. Figu e
6-b u he shows ha al hough #3g has ul ima ely an a enua ed o e all c ys alliza ion a e in compa ison o #2j, bo h p o iles
ollowed simila ly apid g ow hs a e he ini ial s ages o nuclea ion induc ion, he la e clea ly absen in he #1d p o ile. The
high ini ial alues o σ we e su ely esponsible o his despi e hei di e ences in o al nuclea ion yield, wn. Howe e , he
c ys al g ow h a e o #2j emained highe a lowe alues o σ nea ing he end o he nuclea ion domain, while ha o #3g
became slowed in compa ison. In he equilib ium-solidi ica ion domain in Figu e 6-a, bo h p o iles a e seen o con e ge o
> 5 800 s (including #1d). The e o e, hei indi idual dis inc ions we e s onges in he ini ial phase only, p incipally in he
nuclea ion domain, he e o e whe e condi ions we e conduci e o as eac ion a es.
This p ep in esea ch pape has no been pee e iewed. Elec onic copy a ailable a : h ps://ss n.com/abs ac =5378887
P ep in no pee e iewed
