EXPERIMENTAL AND SIMULATION-BASED INVESTIGATION
OF THE INFLUENCE OF FREEZING AND ANNEALING ON THE
MICROSTRUCTURE IN LYOPHILISATES
Doc o al Thesis
Submi ed in pa ial ul ilmen o he equi emen s o
he deg ee o doc o o na u al sciences
o he Facul y o Ma hema ics and Na u al Sciences
a Kiel Uni e si y, Ge many
by
Tig an Le ono ic Kha a yan
Kiel, 2024
P in ed wi h he pe mission o he Facul y o Ma hema ics and Na u al Sciences o Kiel Uni e si y.
This wo k was ca ied ou om May 2019 un il Ap il 2023 a he Depa men o Pha maceu ical
De elopmen o Daiichi Sankyo Eu ope GmbH, P a enho en an de Ilm, Ge many. The hesis was
p epa ed unde supe ision o P o . D . Regina Sche ließ. In his hesis, DeepL W i e has been used o
iden i y and co ec language mis akes (g amma and spelling).
1s Re e ee: P o . D . Regina Sche ließ
2nd Re e ee: D . No a A. U bane z
Submission o he PhD applica ion: 21.05.2024
Da e o examina ion: 06.09.2024
Dean: P o . D . F ank Kempken
– Dedica ed o my amily –

Resea ch ac i i ies
Publica ions
The disse a ion includes ex and igu es om he publica ions o he au ho s lis ed below.
I. Quan i a i e Analysis o Glassy S a e Relaxa ion and Os wald Ripening du ing
Annealing Using F eeze-D ying Mic oscopy by Tig an Kha a yan, S ikan h R. Gopi eddy,
To u Ogawa, Ta suhi o Kodama, No ihi o Nishimo o, Sayaka Osada, Regina Sche ließ
and No a A. U bane z in Pha maceu ics (Volume 14, Issue 6, June 2022, 1176)
II. Impac o pos - eeze annealing on sh inkage o suc ose and ehalose lyophilisa es by
Tig an Kha a yan, Shunya Igawa, S ikan h R. Gopi eddy, To u Ogawa, Ta suhi o
Kodama, Regina Sche ließ and No a A. U bane z in In e na ional Jou nal o
Pha maceu ics (Volume 641, June 2023, 123051)
Table o con en s
Table o con en s
In oduc ion ..................................................................................................................... 1
Objec i es......................................................................................................................... 2
1 Gene al backg ound ....................................................................................................... 3
1.1 F eeze-d ying in he pha maceu ical ield ........................................................................ 3
1.2 F eeze-d ye uni .............................................................................................................. 4
1.3 Pha maceu ical o mula ions in eeze-d ying ................................................................. 5
1.4 P ocess s eps in a eeze-d ying cycle .............................................................................. 7
1.4.1 F eezing ...................................................................................................................... 9
1.4.1.1 He e ogeneous and homogeneous nuclea ion ................................................. 10
1.4.1.2 F eeze-concen a ion o he ma ix phase........................................................ 12
1.4.2 Annealing du ing he eezing phase ....................................................................... 15
1.4.3 P ima y d ying .......................................................................................................... 17
1.4.4 Seconda y d ying ..................................................................................................... 20
1.5 Mic os uc u e o ma ion in an aqueous solu ion du ing eezing................................ 21
1.6 Coa sening o he ice c ys al phase du ing annealing .................................................... 23
1.7 Mo phological assessmen o he mic os uc u e in a lyophilisa e ............................... 25
1.8 Nume ical simula ions and phase- ield me hods in he pha maceu ical ield .............. 27
2 Ma e ials and me hods ................................................................................................ 31
2.1 Ma e ials ......................................................................................................................... 31
2.1.1 D-(+)-suc ose ............................................................................................................ 31
2.1.2 D-(+)- ehalose ......................................................................................................... 32
2.2 Me hods .......................................................................................................................... 32
2.2.1 Lyophilisa ion ........................................................................................................... 32
2.2.2 F eeze-d ying mic oscopy ........................................................................................ 33
2.2.3 Pola ised ligh mic oscopy ....................................................................................... 35
2.2.4 Di e en ial scanning calo ime y ............................................................................ 36
2.2.5 Phase- ield compu a ion and isualisa ion .............................................................. 36
2.2.6 Image analysis .......................................................................................................... 37
3 Resul s and discussion .................................................................................................. 38
3.1 Mic os uc u e o ma ion du ing eezing ..................................................................... 38
Table o con en s
3.1.1 Recalescence and c ys allisa ion in samples du ing eezing .................................. 38
3.1.2 Cha ac e isa ion o he he mal p o ile du ing eezing.......................................... 41
3.1.3 Heigh -dependen ecalescence and c ys allisa ion in lyophilisa ion ials ............. 48
3.1.4 Assessmen o he mic os uc u e in a ozen solu ion ........................................... 52
3.1.5 Impac o he mal his o y du ing eezing on he mic os uc u e .......................... 62
3.1.6 Gene al discussion – impo ance o ecalescence and c ys allisa ion .................... 66
3.2 Mic os uc u e e olu ion du ing annealing ................................................................... 67
3.2.1 Me hodological amewo k o in es iga e he mic os uc u e ............................... 68
3.2.2 Implemen a ion and calib a ion o he phase- ield model ..................................... 72
3.2.2.1 Fo mulism o he mul iphase- ield model ........................................................ 72
3.2.2.2 Expe imen al de e mina ion o ele an calib a ion pa ame e s..................... 74
3.2.2.2.1 Mass ac ions du ing annealing ............................................................... 75
3.2.2.2.2 Volume ac ions du ing annealing/maximum eeze-concen a ion ...... 78
3.2.2.2.3 Rec ys allisa ion a es du ing annealing ................................................... 80
3.2.3 Ini ial condi ions o wo-dimensional and h ee-dimensional simula ions ........... 86
3.2.4 Calib a ion o inpu pa ame e s o he phase- ield simula ions ............................ 95
3.2.4.1 Adjus men o he a ea and olume ac ions o he pa icula e phase .......... 95
3.2.4.2 Coa sening pa ame e s o he wo-dimensional phase- ield model ............. 101
3.2.5 Ad ancemen o he phase- ield simula ion o h ee-dimensional ...................... 107
3.2.6 Final adjus men o pa icle olume ac ions ....................................................... 110
3.2.7 Mo phological p ope ies in he h ee-dimensional phase- ield model ............... 111
3.2.7.1 Desc ip ion o mo phological p ope ies in a po ous mic os uc u e ............ 112
3.2.7.2 Ice c ys al size dis ibu ion .............................................................................. 116
3.2.7.3 Conjunc ion a ea o ice c ys als ...................................................................... 120
3.2.7.4 Su ace a ea o he ma ix phase .................................................................... 126
3.2.8 Gene al discussion – po e size de e mina ion in lyophilisa es ............................. 129
4 Fu u e applica ions ..................................................................................................... 134
4.1 Reo ganisa ion kine ics du ing ecalescence and c ys allisa ion ................................. 134
4.2 CFD simula ions h ough he po ous mic os uc u e ................................................... 135
5 Appendices ................................................................................................................. 137
Appendix A – Valida ion and s abili y analysis o he phase- ield simula ion .................. 137
i. Gibbs-Thomson (capilla i y) e ec on composi ion .................................................... 137
ii. Ene gy ba ie coe icien on in e acial ee ene gy .................................................. 138
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The solu ion in ended o d ying is placed in con aine s such as glass ials, which a e a ailable
in a ious sizes and ma e ials. Fo ins ance, ambe glass ials a e used o p oduc s sensi i e
o UV adia ion [17]. Be o e d ying, he p oduc mus be ho oughly ozen a empe a u es
ypically anging om -50 °C o -80 °C. This ensu es maximum phase sepa a ion wi hin he
sample due o eezing, a p ocess ha will be de ailed la e . Consequen ly, a cooling sys em
connec ed o he shel es whe e he ials a e placed is essen ial. Some eeze-d ye s lack an
in eg a ed cooling sys em, equi ing p oduc s o be p e- ozen ou side he d ye .
To acili a e sublima ion, he p essu e wi hin he d ying chambe mus be educed o a leas
below he iple poin o he sol en , making a acuum sys em indispensable. The d ying
chambe is connec ed o a condense chambe and a acuum pump h ough al es, wi h he
acuum pump egula ing he p essu e by emo ing non-condensable gases [16]. The
condense , kep a empe a u es lowe han he shel es, enables he ansi ion o wa e apo
in o ice ia deposi ion/ esublima ion. The d i ing o ce behind he eeze-d ying p ocess is he
empe a u e di e ence be ween he p oduc and he condense . This di e ence c ea es
a ied sa u a ion apo p essu es o ice a he sublima ion on in he p oduc and a he ice
on he condense , he eby es ablishing a p essu e g adien ha acili a es he ans e o
wa e molecules om he p oduc o he condense [18]. Addi ionally, hese condense s a e
designed wi h high su ace a eas o enhance hei capaci y o accumula ing ice [2]. The ials
a e hea ed ia he shel es in o de o expedi e he sublima ion p ocess by u he inc easing
he empe a u e di e ence be ween he p oduc and he condense [19].
1.3 Pha maceu ical o mula ions in eeze-d ying
F eeze-d ying is ypically u ilised o Ac i e Pha maceu ical Ing edien s (APIs) whe e gen le
d ying p ocesses and s abilisa ion a e c ucial [3]. This is pa icula ly he case o
biopha maceu ical APIs such as monoclonal an ibodies, he apeu ic p o eins and accines,
which a e p one o loss o unc ionali y, expensi e o p oduce, and equi e me iculous
downs eam p ocessing [5,20]. Fu he mo e, empe a u e-sensi i e small molecule d ugs
such as an ibio ics o an i i als a e also subjec o eeze-d ying in o de o p ese e hei
unc ionali y. Some examples o eeze-d ied pha maceu icals a e lis ed in Table 1.1.
Besides he API, eeze-d ied pha maceu ical p oduc s con ain inac i e excipien s, de ined by
he FDA as “any componen o a d ug p oduc o he han an ac i e ing edien ” [21].

Gene al backg ound
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App oxima ely 67 % o lyophilised small molecule pha maceu icals and nea ly all
bio echnology-de i ed p oduc s con ain excipien s [22,23]. Lyophilisa ion excipien s a e
classi ied acco ding o hei unc ions as s abilise s, bu e s, su ac an s, o bulking agen s [4].
Ce ain subs ances may ul il mul iple oles in a o mula ion. Fo example, disaccha ides such
as suc ose may be used as s abilise s and bulking agen s a he same ime [24].
Table 1.1: Examples o eeze-d ied pha maceu icals wi h FDA app o al [25].
D ug name (FDA
app o al yea ) API Ca ego y/Indica ion Manu ac u e
Enb el® (1998) E ane cep Rheuma oid a h i is,
Pso iasis Amgen Inc.
Enhe u® (2019) T as uzumab
de ux ecan B eas /gas ic cance Daiichi Sankyo
Company, Limi ed
A onex® (1996) In e e on be a-1a Mul iple scle osis Biogen
O encia® (2005) Aba acep Rheuma oid a h i is B is ol-Mye s
Squibb
Bo ox® (1991) Daxibo ulinum oxin A Va ious Alle gan
Dan ium® (1979) Dan olene sodium Muscle elaxan The P oc e &
Gamble Company
Cosmegen® (2009) Dac inomycin An ibio ic oncologic Me ck KGaA
S abilise s se e o p o ec APIs om agg ega ion o s uc u al changes du ing eezing o
d ying, ac ing as c yop o ec an s o lyop o ec an s [4,26–28]. Examples o s abilise s include
disaccha ides such as suc ose and ehalose, along wi h syn he ic polye he s such as
polye hylene glycol and poly inylpy olidone. The mechanisms h ough which s abilise s
ope a e include i i ica ion and wa e eplacemen [29]. Vi i ica ion in ol es he
immobilisa ion o he API in an amo phous ma ix phase, which p e en s in e ac ions be ween
API molecules. Wa e eplacemen desc ibes he subs i u ion o hyd ogen bonds be ween he
API and wa e wi h bonds be ween he s abilise and he p o ein. As he s abilise is no
emo ed du ing d ying, he API is less likely o be o ced ou o i s na i e o m upon deso p ion
o un ozen wa e .
Bu e s can be inco po a ed in o a o mula ion o egula e pH and main ain he solubili y and
s abili y o he API. Fo ins ance, p o eins ypically ha e he lowes solubili y a hei isoelec ic
poin due o he neu al ne cha ge on hei su ace, which leads o in e ac ions be ween
Gene al backg ound
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p o eins a he han be ween p o eins and dipola wa e molecules [30]. Howe e , he
selec ion o a sui able bu e is c ucial, as phospha e bu e s, o example, can lead o
signi ican pH shi s du ing eezing, esul ing in p o ein dena u a ion and p ecipi a ion
[31,32]. Bu e s commonly used in lyophilisa ion, such as ci a es and his idine bu e s, a e
less p one o c ys allisa ion and exhibi a smalle empe a u e-dependen pH ange [4].
Su ac an s, such as polyso ba e 80 and leci hin, a e amphiphilic and e ec i e a lowe ing he
su ace ension o wa e . In pha maceu ical o mula ions, su ac an s bind o he hyd ophobic
egions o p o eins, dec easing agg ega ion. Fu he mo e, su ac an s aise he ene gy
equi ed o p o eins o un old, he eby s abilising hei na i e s uc u e [33,34].
Bulking agen s a e added o o mula ions when he API is p esen in small amoun s pe dose
due o i s high po ency, he e o e inc easing he o al mass o he pha maceu ical p oduc
[35]. F equen ly u ilised bulking agen s o biopha maceu icals include disaccha ides like
suc ose and ehalose, o suga alcohols such as xyli ol and manni ol, which o m ei he
amo phous o c ys alline phases upon eezing [2,36,37]. As bulking agen s may cons i u e a
signi ican po ion o a lyophilisa e, hei beha iou du ing eezing and d ying is impo an o
he o e all p ocess.
1.4 P ocess s eps in a eeze-d ying cycle
A eeze-d ying cycle is di ided in o h ee main p ocess s eps: eezing, p ima y d ying, and
seconda y d ying. Du ing each s age, he pha maceu ical solu ion o lyophilisa e unde goes
physicochemical changes on bo h he mic o- and mac oscale, as illus a ed in Figu e 1.3.
The subp ocesses o a lyophilisa ion cycle a e egula ed by he shel empe a u e and chambe
p essu e. The ini ial eezing s ep in ol es lowe ing he empe a u e o he samples un il he
solu ion solidi ies comple ely, esul ing in he o ma ion o a mic os uc u e composed o
dispe sed indi idual ice c ys als wi hin a con inuous ma ix phase. Solu ions con aining glass-
o ming solu es eeze o e a wide empe a u e ange. Fo example, a solu ion migh s a as
a slush a ound -10 °C and solidi y a ound -30 °C o -40 °C, depending on he solu e [38].
The e o e, o ho oughly eeze such a solu ion, empe a u es well below he iple poin o
he sol en a e necessa y. An op ional annealing subp ocess can be included in he eezing
s ep, whe e he empe a u e is sligh ly aised bu kep below he mel ing poin o he solu ion
o p e en hawing.
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Figu e 1.3: Illus a ion o a eeze-d ying p ocess including an annealing s ep du ing eezing.
This phase is shown in g ea e de ail o highligh he annealing empe a u e (Ta), annealing
ime ( a), and he a es o eezing and cooling.
In p ima y d ying, mos o he wa e , in he o m o ice c ys als, is emo ed om he ials by
educing he chambe p essu e o a leas below he iple poin o he sol en . Since
sublima ion is an endo he mic phase ansi ion, he empe a u e inside he ials dec eases
du ing his s age. To coun e ac his cooling and accele a e sublima ion, he samples a e
hea ed ia he shel es. Du ing his phase, only he ice in di ec con ac wi h he gas phase
unde goes he phase ansi ion, c ea ing a sublima ion on ha p og esses om he su ace
o he ozen solu ion owa ds he bo om o he ial. To comple e he d ying p ocess, i is
necessa y o emo e he esidual wa e om he ma ix phase. This is ypically achie ed by
u he inc easing he shel empe a u e, which causes wa e molecules o di use and deso b
om he ma ix phase, educing he wa e con en in he lyophilisa e o he a ge ange o
usually a ound 1 % (w/w) o 3 % (w/w) o eeze-d ied biological p oduc s [2]. Once he
desi ed wa e con en is eached, he ials a e sealed, o en wi h an ine gas, o p e en
mois u e om he ai om we ing he lyophilisa e.
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1.4.1 F eezing
In he eezing s ep, he shel empe a u e is lowe ed wi h he goal o ensu ing ha mos o
he sol en , such as wa e , c ys allises. This leads o pa ial phase sepa a ion, commonly
e e ed o as eeze-concen a ion, which is c ucial o he subsequen emo al o he sol en
h ough sublima ion in la e s ages o he p ocess. This phase includes se e al subp ocesses,
which a e illus a ed in Figu e 1.4.
Figu e 1.4: Tempe a u e p o ile o sample and shel du ing he eezing s ep: p ecooling (1),
eezing/c ys allisa ion (2), eezing pla eau (3), annealing amp (4), annealing (5),
e eezing (6), e eezing pla eau (7).
A e an op ional p ecooling s ep aimed a equalizing he condi ions ac oss ials in he eeze-
d ye o app oxima ely 5 °C, he shel empe a u e is u he dec eased o ypically be ween
-40 °C and -50 °C o eeze he samples. Du ing his p ocess, a no able inc ease in he sample's
empe a u e occu s a e eaching 10 °C o 15 °C below i s equilib ium eezing/mel ing poin ,
i.e., he empe a u e a which a ozen solu ion wi h he same componen s would comple ely
mel . This empe a u e inc ease signals an exo he mic phase ansi ion, speci ically he
c ys allisa ion o wa e o ice. Jus be o e he empe a u e begins o ise, he solu ion emains
liquid despi e being below i s equilib ium eezing empe a u e, he eby becoming
supe cooled. A supe cooled solu ion is no in equilib ium and is only me a-s able [39]. The
g ea e he deg ee o supe cooling, he mo e he solu ion becomes o e sa u a ed wi h
espec o he sol en , which in u n inc eases he p obabili y o nuclea ion [13]. Once
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nuclea ion begins, exo he mic c ys allisa ion apidly occu s h oughou he en i e sample. This
ise in empe a u e du ing solidi ica ion is known as ecalescence, occu ing when he hea
eleased du ing he ansi ion exceeds he a e a which hea can dissipa e om he ma e ial
[40]. This phenomenon, depic ed in Figu e 1.5, is essen ial o unde s anding he impac o
eezing condi ions on he lyophilisa ion p ocess.
Figu e 1.5: Schema ic empe a u e p o ile o a solu ion du ing eezing: supe cooling (1),
nuclea ion (2), ecalescence (3), eeze-concen a ion (iso he mal c ys allisa ion) (4), end o
solidi ica ion (5). The equilib ium eezing empe a u e (T ) is shown as a g ey dashed line.
Modi ied om [41].
Once he sample empe a u e nea ly eaches i s equilib ium eezing empe a u e, i emains
s eady o pe iod o ime. Meanwhile, wa e con inuously c ys allises, eleasing hea ha mus
be emo ed om he sample by he cooling shel es. A de ailed desc ip ion o he phenomena
con ibu ing o he ypical empe a u e p o ile o he sample du ing eezing will be p esen ed
in he ollowing sec ions. Addi ionally, an in-dep h examina ion o ecalescence and he
subsequen c ys allisa ion will be explo ed in he Resul s sec ion o his wo k. This ocus is
mo i a ed by he o en unde es ima ed impac o he empe a u e inc ease du ing eezing
on he mic os uc u e o he lyophilisa e.
1.4.1.1 He e ogeneous and homogeneous nuclea ion
The onse o c ys allisa ion equi es he o ma ion o ini ial nuclea ion poin s, known as nuclei,
whe e wa e molecules s a o p ecipi a e and o m la ge ice c ys als [42]. Nuclea ion does

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no necessa ily occu a he equilib ium eezing empe a u e bu can happen a much lowe
empe a u es, in luenced by p ocess condi ions and ma e ial p ope ies. Fo ins ance, pu e
wa e , wi h a mel ing poin a 0 °C, can emain un ozen a empe a u es as low as -42 °C [43].
Howe e , he p esence o impu i ies migh ini ia e nuclea ion a empe a u es close o he
equilib ium eezing poin , such as -5 °C [44,45]. These impu i ies could include pa icula e
con aminan s o la ge molecules like p o eins, ac ing as in e aces o acili a e nuclea ion [46].
This phenomenon is explained by classical nuclea ion heo y, which di e en ia es be ween
homogeneous and he e ogeneous nuclea ion, and is illus a ed in Figu e 1.6.
Figu e 1.6: Size-dependen ee ene gy o homogeneous (A) and he e ogeneous (B)
nuclea ion. Modi ied om [47].
In homogeneous nuclea ion, small nuclei o m as a solid phase wi hin he o e sa u a ed
solu ion. These nuclei a e clus e s o wa e molecules a anged and bonded simila ly o ice.
Thei o ma ion, d i en by B ownian mo ion wi hin he bulk olume, is s ochas ic in na u e,
ye he p obabili y o o ma ion inc eases as he empe a u e dec eases [48]. The ene gy
equi ed o o m and sus ain a nucleus depends on i s size, wi h he Gibbs ee ene gy (ΔG)
comp ising su ace ee ene gy and bulk ee ene gy componen s. Su ace ee ene gy
inc eases wi h nucleus size, while bulk ee ene gy has a s abilising e ec , educing he
sys em's ee ene gy as he nucleus g ows [49]. The e o e, he s abili y o a nucleus hinges on
i s su ace a ea o olume a io, o simply i s adius in he case o a sphe ical nucleus. The ee
ene gy peak indica es he c i ical pa icle size ( c) ha mus be su passed o la ge ice c ys als
o g ow om he nucleus. Failing o exceed his size, he sys em's ee ene gy can only
dec ease by educing he su ace a ea o he nuclei, po en ially leading o hei comple e
b eak up. In he e ogeneous nuclea ion, he new solid phase o ms on an exis ing in e ace,
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such as he su ace o an impu i y, which educes he equi ed ene gy o nucleus s abilisa ion
due o a smalle su ace a ea compa ed o homogeneous nuclea ion o he same adius,
he eby making he e ogeneous nuclea ion mo e p e alen [50]. In eeze-d ying, a ious
echniques a e u ilised o in luence he nuclea ion empe a u e du ing he eezing s ep.
These include he ice og echnique, quench eezing, elec o eezing, ul asound-con olled
ice nuclea ion, acuum-induced su ace eezing, high-p essu e shi and dep essu isa ion,
and he addi ion o nuclea ion agen s [44].
1.4.1.2 F eeze-concen a ion o he ma ix phase
Once s able nuclei wi h adii abo e he c i ical size a e o med, wa e molecules om he
supe cooled liquid phase can a ach o hei su aces and s a o o m ice c ys als [51]. In he
condi ions o a eeze-d ye , ice ypically o ms in he Ih phase, also known as no mal
hexagonal c ys alline ice. Wi hin a solu ion, he dense s uc u e o hexagonal ice p e en s
mos solu es om being inco po a ed in o he c ys alline la ice, leading o he o ma ion o
a nea ly pu e ice phase and a solu e-en iched ma ix phase [16]. This e e sible p ocess,
known as eeze-concen a ion, is go e ned solely by empe a u e. The lowe he
empe a u e, he mo e wa e p ecipi a es om he solu ion in o he exis ing ice c ys als, as
illus a ed in Figu e 1.7. Howe e , his p ocess can only con inue un il he ma ix phase
eaches i s maximum eeze-concen a ed s a e, which a ies depending on he solu e in he
aqueous solu ion [52].
Figu e 1.7: F eeze-d ying mic oscopy o a pa ially ozen 10 % (w/w) suc ose solu ion a -4 °C
(le ) and a -10 °C ( igh ).
The eezing beha iou o a ma ix phase can be classi ied as c ys allising o i i ying and he
choice o excipien s in he o mula ion will de e mine which o hese beha iou s is p esen o
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13
whe he a combina ion o bo h is obse ed. C ys alline sys ems a e ypically cha ac e ised by
he o ma ion o a eu ec ic mix u e upon eezing, as illus a ed in he phase diag am o a
wo-componen sys em shown in Figu e 1.8.
Figu e 1.8: Phase diag am o a bina y sys em wi h c ys allising componen s A and B. Modi ied
om [16].
A eu ec ic mix u e has a lowe eezing poin han any o he mixing a io o i s componen s.
This means ha he mix u e ab up ly eezes a he eu ec ic empe a u e (Teu), and he
componen s, A and B, o m a ine s uc u e o indi idual c ys alline phases, illus a ed in
Figu e 1.8 as α and β. Non-eu ec ic mix u es, ei he hypoeu ec ic o hype eu ec ic, begin o
o m a p ecipi a e o he excess componen as he empe a u e dec eases. A hypoeu ec ic
mix u e emains en i ely liquid un il i eaches he equilib ium eezing empe a u e, also
known as he liquidus line. A his poin , componen A begins o p ecipi a e (1→2). This
p ocess con inues along he liquidus line un il he eu ec ic poin (5), whe e a po ion o
componen A o ms a eu ec ic mix u e wi h all o componen B. Upon u he cooling, his
eu ec ic mix u e c ys allises. A hype eu ec ic mix u e ollows a simila pa h (3→4→5). Upon
solidi ica ion, such eu ec ic sys ems may o m mic oscopic a angemen s, whe e one phase is
embedded as lamella o globula s uc u es wi hin he o he phase [53,54].
Many pha maceu ical o mula ions con ain majo componen s ha do no c ys allise upon
eezing bu ins ead emain in a ubbe y phase and unde go glass ansi ion upon su icien
cooling. Du ing eezing o a solu ion wi h a glass- o ming componen , p ecipi a ion o wa e
Gene al backg ound
14
and ice o ma ion ollow he liquidus line once nuclea ion occu s, simila o a c ys allising
sys em. Howe e , ins ead o o ming a eu ec ic mix u e, i.e., a mix u e wi h a single mel ing
poin , mo e wa e c ys allises o e a b oad empe a u e ange. This p ocess con inues un il
he amo phous phase canno be u he concen a ed by empe a u e educ ion. A his poin ,
he sys em is e e ed o as maximally eeze-concen a ed and is cha ac e ised by he onse
empe a u e o ice mel ing Tm’ and he solu e concen a ion in he ma ix phase cg’ [38]. The
phase diag am in Figu e 1.9 illus a es he eezing beha iou o such solu ions.
Figu e 1.9: Phase diag am o an aqueous sys em wi h a glass- o ming solu e. The glass
ansi ion empe a u e o he solu e-en iched phase (Tg) changes du ing eeze-concen a ion
(blue) and deso p ion o un ozen wa e ( ed). Modi ied om [55,56].
In case o glass- o ming disaccha ides, he o ma ion o hyd ogen bonds be ween solu e and
wa e as well as be ween solu e molecules exhibi s compa able ene gy and con igu a ion [57].
A dec ease in empe a u e shi s he chemical po en ial (µ) o hese hyd ogen bonds. Once
he ma ix phase becomes maximally eeze-concen a ed, he µ o hyd ogen bonds be ween
he solu e and wa e alls below he µ o wa e -wa e in e ac ions, hus inhibi ing u he
c ys allisa ion o wa e [38]. Roos (1991) showed ha app oxima ely 20 % (w/w) o esidual
wa e emains in a maximally eeze-concen a ed ma ix o a ozen suc ose solu ion. Fu he
in es iga ion by Sei e e al. (2020) de e mined ha he esidual wa e con en in suc ose
anges be ween 22.6 % and 24.6 % (w/w).
Gene al backg ound
21
eeze-concen a ed ma ix be o e d ying [67,94]. The condi ions and du a ion o seconda y
d ying allow he desi ed esidual mois u e in he lyophilisa e o be achie ed. This is impo an
because APIs a e no necessa ily bes s abilised when he esidual mois u e is as minimal as
possible [95–97].
1.5 Mic os uc u e o ma ion in an aqueous solu ion du ing eezing
The li e a u e on he p ope ies o he po ous mic os uc u e in lyophilisa es gene ally ag ees
ha po e size and numbe a e in luenced by wo key physical phenomena: nuclea ion and
Os wald ipening. The nuclea ion empe a u e is c ucial as i a ec s he numbe o s able
nuclei ha o m and subsequen ly g ow in o ice c ys als. A lowe nuclea ion empe a u e
esul s in a g ea e numbe o s able nuclei [13,16,86,98]. This is u he suppo ed by
molecula simula ions, which indica e ha o homogeneous ice nuclea ion a 15 °C below he
equilib ium eezing empe a u e, he c i ical size o a nucleus is app oxima ely
8000 molecules, co esponding o a c i ical adius o 4 nm. This c i ical size dec eases o
1.7 nm wi h a g ea e deg ee o supe cooling o 35 °C [99].
The olume ic nuclea ion a e B0 desc ibes he numbe o s able nuclei ha o m in a gi en
olume pe uni ime and is de ined by Colucci e al. (2020) as:
𝐵

=
𝑘


𝑇

−
𝑇



, (Eq. 1)
whe e kb and b a e kine ic pa ame e s, T is he equilib ium eezing empe a u e, and Tn is
he nuclea ion empe a u e. The kine ic pa ame e s in his model can be adjus ed o e a ange
o alues un il an accep able i is achie ed be ween he model p edic ions and expe imen al
da a. The expe imen al da a o his pu pose can be de e mined by examining images o he
d ied lyophilisa e wi h a scanning elec on mic oscope (SEM) and quan i ying he numbe o
po es.
Acco ding o Equa ion 1, he c i ical ac o de e mining he numbe o s able nuclei is he
di e ence be ween he equilib ium eezing empe a u e o he mix u e and he ac ual
nuclea ion empe a u e. Howe e , his does no imply ha nuclei o m uni o mly ac oss he
en i e sample olume a he momen o nuclea ion. Ins ead, nuclea ion ypically begins wi h
he o ma ion o a leas one p ima y nucleus, likely h ough he e ogeneous nuclea ion. F om
his ini ial si e, he ice c ys al on ad ances, wi h new nuclei o ming ia seconda y

Gene al backg ound
22
nuclea ion, o en simply e e ed o as c ys allisa ion, h oughou he en i e supe cooled
olume o he sample [86].
The a e o c ys allisa ion, o mo e speci ically, he seconda y nuclea ion on eloci y o he
p opaga ion a e o he ice on , is in luenced by he deg ee o supe cooling wi h g ea e
supe cooling esul ing in a as e p opaga ion a e [2,100]. Empi ical equa ions ha e been
o mula ed o depic he ela ionship be ween he p opaga ion a e o he ice on and he
nuclea ion empe a u e. Fo ins ance, a an a e age nuclea ion empe a u e o -10 °C ± 3 °C
o wa e o injec ion, he p opaga ion a e o he ice on is app oxima ely 5.2 cm/s.
Howe e , his a e signi ican ly inc eases o abou 23.7 cm/s a a lowe nuclea ion
empe a u e o -20 °C [16]. When a solu e is p esen , he p opaga ion a e o he ice on
slows down o he o de o mm/s and a ies depending on he solu e concen a ion in he
solu ion [101,102].
Once p ima y and seconda y nuclei ha e o med, wa e molecules p ecipi a e om he
supe cooled solu ion in o hese s able nuclei, o ming ice c ys als. Al hough his p ocess does
no change he numbe o c ys als, i does inc ease hei size. These ice c ys als ha e
c ys allog aphic o ien a ions ha p e en hem om coalescing when he e is a misma ch
be ween he o ien a ions o neighbou ing c ys als [103]. The olume o wa e ha
p ecipi a es is solely dependen on he ambien empe a u e, and his p ocess con inues un il
he maximum eeze-concen a ed composi ion o he solu e-en iched ma ix phase is
achie ed [59]. Assuming ha no addi ional nuclea ion occu s once he s able p ima y and
seconda y nuclei a e p esen , he size o he esul ing ice c ys als p ima ily depends on he
ini ial numbe o hese nuclei, as illus a ed in Figu e 1.11. This a ionale also unde pins he
me hodology used o de e mine he pa ame e s o Equa ion 1, whe e each indi idual po e is
a ibu ed o a co esponding indi idual ice c ys al and, by ex ension, o a p eceding indi idual
s able nucleus.
In hei s udy, Colucci e al. (2020) measu ed po e sizes in a eeze-d ied 5 % (w/w) suc ose
solu ion using scanning elec on mic oscopy (SEM), inding sizes anging om app oxima ely
25 µm o 50 µm based on axial posi ion wi hin he lyophilisa e. Thomik e al. (2022) epo ed
po e sizes o 23.73 µm ± 11.13 µm in a simila 5 % (w/w) suc ose solu ion, measu ed using
mic o-compu ed omog aphy (µ-CT). Addi ionally, Fang e al. (2020) employed low-p essu e
me cu y in usion po osime y and B unaue -Emme -Telle (BET) analysis o de e mine
Gene al backg ound
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a e age po e sizes anging om 20 µm o 60 µm in hei s udy o a simila solu ion. Al hough
he eezing p o ocols and equipmen a ied ac oss hese s udies, he esul s consis en ly
showed a simila magni ude wi h he minimum po e size a ound 20 µm.
Figu e 1.11: Schema ic depic ion o he in luence o supe cooling on he size o ice c ys als in
a ozen solu ion.
These indings assume ha size and numbe o po es in non-annealed samples a e solely
in luenced by nuclea ion kine ics when no addi ional annealing s ep is included. Howe e , his
explana ion simpli ies he o ma ion o mic os uc u es in aqueous solu ions du ing eeze-
d ying. The phase ansi ion du ing eezing gene a es la en hea o c ys allisa ion, which
aises he sample empe a u e and esul s in a complex he mal his o y h oughou he
eezing s age o a lyophilisa ion cycle. In summa y, i is assumed ha he numbe o s able
nuclei o med du ing eezing dic a es he numbe o indi idual ice c ys als in he ozen
solu ion, and consequen ly, he numbe o po es in he d ied lyophilisa e. This wo k, howe e ,
seeks o cla i y he impac o he la en hea o c ys allisa ion du ing ecalescence and he
subsequen c ys allisa ion phase on he mic os uc u al o ma ion in solu ions in ended o
eeze-d ying.
1.6 Coa sening o he ice c ys al phase du ing annealing
An annealing s ep can be inco po a ed in o he p ocess pos - eezing o coa sen he dispe se
ice phase, as depic ed in Figu e 1.12. Nakagawa e al. (2018) demons a ed ha a 20 % (w/w)
Gene al backg ound
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suc ose solu ion, ini ially wi h an a e age po e size o app oxima ely 60 µm a e 1 hou o
annealing a -5 °C, expanded o abou 150 µm a e 6 hou s o annealing. Simila ly, in he s udy
by Thomik e al. (2022), a 5 % (w/w) suc ose solu ion annealed o 11 hou s a -5 °C showed
an inc ease in po e size om 23.73 µm ± 11.13 µm o 33.04 µm ± 26.95 µm.
Figu e 1.12: Schema ic depic ion o he in luence o annealing on he size o ice c ys als in a
ozen solu ion.
As desc ibed in Chap e 1.4.2, he phenomenon ha occu s du ing annealing is called Os wald
ipening, and i is d i en by he educ ion o he sys em’s ee ene gy. This e ec is
ma hema ically exp essed by he classical Li shi z-Slyozo -Wagne (LSW) heo y o Os wald
ipening, which desc ibes he coa sening beha iou o pa icles (he e: ice c ys als) o d ople s
in a ma ix phase o solu ion. I is o en used in he con ex o ma e ial science, especially o
me al alloys, when cha ac e ising he g ow h o he a e age pa icle size o e ime. In he
wo k o Nie hamme (2008), a de ailed desc ip ion o assump ions and limi a ions ega ding
he LSW heo y a e gi en. The LSW heo y assumes ha he pa icles a e sphe ical and exhibi
ini ially a uni o m size dis ibu ion. Addi ionally, he dis ance be ween pa icles is conside ed
la ge compa ed o he indi idual pa icle size, so ha di usion o molecules om he
pa icula e phase is he dominan mechanism o pa icle g ow h. Fu he mo e, he
concen a ion in he su ounding medium o ma ix phase is assumed o be cons an , i.e., no
concen a ion g adien s a e p esen in he ma ix phase. The ma hema ical exp ession o he
empo al change o pa icle adius due o dissolu ion and edeposi ion is gi en by [105]:
𝑑𝑟
𝑑𝑡
=
𝐷
𝑟
󰇡
∆
−
𝜀
𝑟
󰇢
,
(Eq. 2)
whe e D is a sys em-dependen di usion cons an , Δ is he di e ence be ween concen a ion
a he pa icle su ace and he equilib ium concen a ion in he liquid phase, ε is a e m
Gene al backg ound
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con aining su ace ension, a omic olume o he solu e, solubili y, he ideal gas cons an , and
empe a u e.
Unde he assump ion ha di usion o molecules h ough he liquid phase is he a e limi ing
ac o , he ollowing equa ion desc ibing he change o he mean pa icle size can be
de i ed [106]:
𝑟

(
𝑡
)

=
𝑟


+
𝑘𝑡
,
(Eq. 3)
whe e  is he mean pa icle adius, 0 is he ini ial pa icle size, and k is he iso he mal
ec ys allisa ion a e, gi en by [105]:
𝑘
=
8
𝜎
𝑐

𝑣

𝐷
9
𝑅𝑇
,
(Eq. 4)
whe e σ is he su ace ension, c∞ is he solubili y o he pa icle ma e ial, ν is he mola olume
o he pa icle ma e ial, D is he di usion coe icien o he pa icle ma e ial, R is he ideal gas
cons an , and T is he absolu e empe a u e.
Acco ding o Equa ion 3, he mean pa icle size inc eases wi h ime, and he a e o inc ease
is p opo ional o he cube o he pa icle adius. In an ideal sys em ha ul ils all he
equi emen s o LSW heo y, Equa ion 4 can be used o calcula e he ec ys allisa ion a e,
i.e., he a e a which he pa icula e phase in he sys em coa sens, om i s p inciples. In
such a case, he a e o Os wald ipening can be deduced om he physical and chemical
p ope ies o he sys em.
1.7 Mo phological assessmen o he mic os uc u e in a lyophilisa e
Va ious echniques a e employed in he li e a u e o assess he impac o annealing in a ozen
disaccha ide solu ion o he d ied lyophilisa e. Lyophilisa es sh ink due o deso p ion o he
un ozen wa e con en in he maximum eeze-concen a ed ma ix phase du ing p ima y
and seconda y d ying [91,67]. Al hough sh inkage p ima ily e lec s a educ ion in he bulk
olume o he lyophilisa e, i also implies changes in he in e nal mic os uc u e. In he
simples scena io, a p opo ional educ ion occu s in he indi idual s uc u es, i.e., i he bulk
olume dec eases by app oxima ely 20 %, he indi idual po es a e simila ly educed in hei
olume. In o he scena ios, in e nal s uc u es may collapse o compensa e o he o e all
educ ion in bulk olume. The e o e, i is c ucial o speci y whe he he discussion pe ains o
ice c ys als in he ozen solu ion o po es in he d ied lyophilisa e.
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F eeze-d ying mic oscopy is commonly used o de e mine he collapse empe a u e o a
solu ion du ing he p ima y d ying s ep and was used in his wo k o in es iga e he impac o
eezing and annealing on he ini ial o ma ion and he subsequen e olu ion o he
mic os uc u e in ozen disaccha ide solu ions. Nakagawa e al. (2018) used a di e en
app oach o in es iga e he mic os uc u e, whe e c oss-sec ions o ozen hodamine dye-
s ained solu ions we e cu using a mic o ome, ollowed by measu emen o ice c ys al size
wi h a ligh mic oscope. Al e na i ely, scanning elec on mic oscopy (SEM) can be u ilised o
analyse he mic os uc u e a e d ying [107,108]. Ad ancemen s in his ield include scanning
elec on c yomic oscopy (c yo-SEM), which enables di ec obse a ion o ozen and und ied
samples [109,110]. Addi ionally, X- ay mic o omog aphy was employed by Thomik e al.
(2022) o assess he po e size dis ibu ion in eeze-d ied samples.
Howe e , each o hese me hods p esen s ce ain d awbacks when cha ac e ising and
quan i ying he mic os uc u e ha o ms as a esul o eezing and e ol es du ing annealing.
F eeze-d ying mic oscopy, o ins ance, con ines he sample be ween wo glass pla es a he
han in a la ge bulk olume, which can a ec he coa sening o he ice c ys al phase.
Techniques equi ing he slicing o samples, such as mic o ome use, yield esul s ha depend
on he ac u e o slice plane in he ozen solu ion o lyophilisa e. Since ice c ys als o po es
a e exposed a andom and a ying posi ions, di e en ia ion becomes challenging i only he
na owes o wides pa s o an ice c ys al o po e a e isible, making all measu ed pa ame e s
imp ecise. Simila ly, SEM and c yo-SEM analyses p oduce wo-dimensional da a om a h ee-
dimensional s uc u e due o he opaci y o he sample su ace, limi ing obse a ions o an
a ea a he han a olume. Techniques ha equi e d ied samples also pose p oblems, as he
emo al o wa e may al e he s uc u e wi hin he ma ix phase. Addi ionally, conduc ing an
en i e lyophilisa ion cycle jus o es a single annealing condi ion can be excessi ely ime-
consuming. Mo eo e , he esolu ion o me hods like X- ay mic o omog aphy may no be
su icien o cap u e all ine s uc u es wi hin he lyophilisa e. This limi a ion was illus a ed
by Thomik e al. (2022), who epo ed a ying po osi ies o annealed e sus non-annealed
samples. This inding is con a y o expec a ions since annealing should inc ease he a e age
po e size, ye he o al olume o he ma ix phase, and consequen ly he po osi y, should
emain la gely unchanged.

Gene al backg ound
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Due o he limi a ions men ioned abo e, cu en me hods o de e mining he mic os uc u e
in a ozen solu ion o in a d ied lyophilisa e a e insu icien o conclusi ely assess he impac
o eezing and annealing condi ions on he mic os uc u e. Consequen ly, a di e en
app oach is necessa y. This app oach should no be limi ed o expe imen al me hods alone
bu should also inco po a e compu a ional echniques. These compu a ional me hods can
simula e phenomena ha a e no di ec ly obse able o measu able, p o iding a mo e
comp ehensi e unde s anding o he p ocesses in ol ed.
1.8 Nume ical simula ions and phase- ield me hods in he pha maceu ical ield
Nume ical simula ions a e compu a ional echniques ha a e used o model phenomena o
en i e sys ems as hey e ol e o e a pe iod o ime. By sol ing algo i hms and equa ions on a
compu e , hese simula ions make p edic ions abou he sys em o in e es [111,112]. These
in silico me hods can be used complemen a y o expe imen al indings, i.e., expe imen al da a
can be used o calib a e simula ions, bu p ocesses can also be simula ed when he acquisi ion
o expe imen al da a is no easible. O e all, simula ion echniques can p o ide aluable
insigh s in o complex sys ems and phenomena, bu hei meaning ul use depends on
app op ia e conside a ion o limi a ions and assump ions [113,114].
In gene al, simula ion models can be classi ied in o wo ypes: i s -p inciple and
phenomenological models. Fi s -p inciple models, also known as ab ini io (“ om he
beginning") models, a e buil di ec ly on undamen al physical o chemical laws and p inciples.
This means ha he unde lying equa ions ep esen he mos basic, i educible elemen s o
hese laws. The ad an age o i s -p inciple app oaches is hei u ili y in explo ing a eas wi h
limi ed da a a ailabili y, p o iding insigh s based on heo e ical ounda ions. On he o he
hand, phenomenological models a e empi ical, de eloped based on obse a ions and
expe imen al da a. These models ma hema ically desc ibe phenomena, such as he a e o
change o a pa ame e , and a e calib a ed o eplica e obse ed beha iou as accu a ely as
possible. While ypically simple han i s -p inciple models, phenomenological models ely
mo e hea ily on assump ions, which may a ec he eliabili y o hei p edic ions.
Va ious simula ion echniques exis , each wi h i s ad an ages and disad an ages depending
on he applica ion a ea and he sys em unde in es iga ion. These simula ions can model
physical, chemical, biological, and o he ypes o sys ems, and a e ex ensi ely used ac oss
Gene al backg ound
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scien i ic esea ch, enginee ing design, and nume ous o he ields. Fo ins ance,
compu a ional luid dynamics (CFD) is widely employed o simula e he low beha iou o
luids in h ee-dimensional spaces. CFD simula ions a e g ounded in he undamen al laws o
physics, as hey de i e om he conse a ion laws o mass, momen um, and ene gy, and do
no depend on empi ical ela ionships o assump ions. Applica ions o CFD in he ield o
lyophilisa ion include op imising he design o eeze-d ye s [115] o condense s [116], as well
as in es iga ing he hea ans e wi hin eeze-d ye s [117,118], o he mass lux in ials [119].
Fu he mo e, he scale-up o p ocesses is also suppo ed by CFD analysis [120].
Simula ion me hods o he han h ee-dimensional modeling a e o en employed o desc ibe
pa s o he eeze-d ying p ocess. Fo ins ance, Colucci e al. (2020) de eloped a one-
dimensional popula ion balance model ha p edic s he po e size in a lyophilisa e based on
i s axial posi ion wi hin he ial. Simila ly, Chun e al. (2020) p oposed a ma hema ical model
ha co ela es p ima y d ying imes wi h he size o ice c ys als in ozen solu ions.
Addi ionally, A siccio e al. (2017) u ilised a mechanis ic model o p edic he dis ibu ion o
ice c ys al sizes a e eezing. While de ailed in o ma ion on h ee-dimensional simula ions
o ice c ys als in pha maceu ical solu ions du ing annealing is sca ce, he o ma ion o simila
mic os uc u es, which consis o pa icula e and ma ix phases, and hei coa sening is well-
es ablished in o he ields such as me allu gy. Phase- ield models, especially o alloys, a e
used in me allu gy o be e unde s and mic oscopic e olu ion and changes in he ma e ial,
as e idenced by ecen s udies by Geslin e al. (2015), Radhak ishnan e al. (2018), and Ansa i
e al. (2021).
The phase- ield me hod is a nume ical simula ion echnique used o desc ibe phase
bounda ies and mic os uc u es wi hin ma e ials. Phase- ield models a e a powe ul ool in
ma e ials science, physics, medicine, biology, and ea h sciences o simula ing mo phological
changes in ma e ials. These simula ions use nume ical me hods o ep esen he spa ial and
empo al e olu ion o indi idual phases, including hei in e acial mo ions, and can conside
di e en cha ac e is ics such as chemical composi ions and c ys allog aphic o ien a ions. The
mic os uc u e esul s om he shape and olume o he indi idual phases, which in u n a e
o med by di usi e, mechanical, he mal, elec ochemical o magne ic d i ing o ces [126].
Phase- ield models i s gained p ominence in he 1970s and 1980s when scien is s explo ed
phase ans o ma ions, speci ically ocusing on solidi ica ion and mic os uc u e e olu ion.
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Pionee s like J.W. Cahn and J.E. Hillia d made signi ican con ibu ions, laying he ounda ional
g oundwo k o unde s anding he ma hema ics o phase sepa a ion and ansi ions. In he
1990s, esea che s such as W.J. Rappel and A. Ka ma expanded hese s udies o include
solidi ica ion and dend i ic g ow h. Wi h ecen ad ancemen s in high-pe o mance
compu ing, phase- ield models now enable de ailed simula ions, leading o echnological
p og ess h ough new ma hema ical models, algo i hms, and a b oadening ange o
applica ions.
A de ining cha ac e is ic o phase- ield models is hei abili y o depic dis inc phases wi hin
a sys em using an o de pa ame e . This pa ame e gene ally anges om 0 o 1, wi h 0
ep esen ing one phase, 1 ep esen ing ano he , and alues be ween 0 and 1 indica ing he
in e ace ha sepa a es hese phases. The in e ace demons a es di usi i y, c ea ing a
g adual ansi ion ac oss a ini e a ea. This inhe en di usi i y enables a ealis ic
ep esen a ion o phase changes as hey occu . Addi ionally, phase- ield models s eamline
he depic ion o complex in e acial mo ions, such as me ging, dissolu ion, and b eakup, by
econs uc ing he in e ace h ough a se o con inuous ield a iables. The basic p inciple o
a phase- ield model is depic ed in Figu e 1.13.
In a phase- ield model, he d i ing o ce is cha ac e ised by he ee ene gy unc ional.
Di e en componen s o he ee ene gy can be included, each inco po a ing dis inc e ms
ailo ed o he speci ic aspec s o he ene gy balance ha need ep esen a ion in he
simula ion. In gene al, he ee ene gy unc ional can be de ined as ollows [126]:
𝐹
=


𝑓

+
𝑓

+
𝑓

+
𝑓


𝑑𝑉

,
(Eq. 5)
whe e c is he chemical ee ene gy densi y, g is he g adien ee ene gy densi y, m is he
mechanical ee ene gy densi y, and e is he ex e nal ee ene gy densi y.
The chemical ee ene gy densi y, also known as homogeneous o bulk ee ene gy densi y,
p omo es phase sepa a ion in he absence o an in e ace, while he g adien ene gy densi y
penalises he o ma ion o sha p in e aces by imposing penal ies. The mechanical ee ene gy
densi y includes con ibu ions om elas ic displacemen s caused by s ess and s ain.
Addi ionally, all ex e nal o ces, such as elec os a ic o magne ic in luences, a e inco po a ed
in o he ex e nal ee ene gy densi y. Du ing simula ions, conce ed e o s a e made o
minimise he sys em's ee ene gy, which leads o changes in he phases wi hin he sys em.
Gene al backg ound
30
Addi ionally, phase- ield models also ha e he capaci y o desc ibe mul iple phases
concu en ly as mul i-phase sys ems.
Figu e 1.13: P inciple o he o de pa ame e (η) in phase- ield modeling. The phases a e
cap u ed by alues be ween 0 (black) and 1 (whi e). Modi ied om [127].
The ele ance o phase- ield models in he pha maceu ical en i onmen is inc easingly
e iden , wi h a g owing body o li e a u e add essing he mic oscopic beha iou o subs ances
wi hin his con ex . Van de Sman (2016) enhanced a wo-dimensional phase- ield model wi h
heo ies on he modynamics and di usion kine ics o simula e ice c ys al g ow h in a suga
solu ion du ing eezing. Simila ly, Fan e al. (2018) de eloped a sophis ica ed wo-
dimensional phase- ield model o desc ibe he phenomenon o eeze-concen a ion du ing
eezing, based on i s -p inciple assump ions. Li and Fan (2020) modelled mac oscopic
eezing in a cylind ical essel, while Li e al. (2022) applied phase- ield modeling o explo e
dend i ic mo phologies and ice c ys al g ow h inhibi ion in a suc ose solu ion. The
in es iga ions in his wo k a e based on he model used by Mukhe jee e al. (2009), which was
o iginally used by he au ho s o elucida e he e ec o mis i s ain and in e ace cu a u e
on he g ow h o a single p ecipi a e in a supe sa u a ed ma ix. A de ailed desc ip ion o he
model can be ound in he Resul s sec ion o his wo k.
Ma e ials and me hods
37
analyses we e also ca ied ou and a e included in Appendix A. The simula ion esul s we e
isualised using he open-sou ce da a analysis so wa e Pa aView (Sandia Na ional
Labo a o ies, Ki wa e Inc, Los Alamos Na ional Labo a o y).
2.2.6 Image analysis
To de e mine ec ys allisa ion a es in annealing expe imen s as well as in es iga e he
eezing beha iou o disaccha ide solu ions, indi idual ice c ys als on FDM images we e
ma ked in he open-sou ce scalable ec o g aphics edi o Inkscape (Inkscape Communi y).
The au oma ic coun ing unc ion in Inkscape was hen used o de e mine he numbe o ice
c ys als in each image. This p ocess is shown in Figu e 2.6.
Figu e 2.6: Coun ing o indi idual ice c ys als om an FDM image sec ion o a 10 % (w/w)
ehalose solu ion du ing annealing a -6 °C o 10 min (le ) and 60 min ( igh ). The blue do s
we e manually placed o ma k each indi idual ice c ys al.

Resul s and discussion
38
3 Resul s and discussion
The indings o his hesis a e p esen ed in he ollowing sec ions. As desc ibed in he
Objec i es chap e , he main aspec s can be summa ised as ollows. The i s pa ocuses on
he in es iga ion o he mic os uc u e consis ing o a pa icula e (he e: ice c ys al) and ma ix
phase du ing eezing o an aqueous disaccha ide solu ion. In he second pa , a phase- ield
simula ion was implemen ed and calib a ed wi h expe imen al da a o model he coa sening
beha iou o he mic os uc u e in he ozen solu ion du ing annealing. The inal s ep o his
wo k was o compa e and discuss da a ob ained om he li e a u e wi h da a gene a ed om
he phase- ield simula ions.
3.1 Mic os uc u e o ma ion du ing eezing
In Chap e 1.5, he o ma ion o mic os uc u e du ing he eezing o an aqueous solu ion, as
o en discussed in he li e a u e, was explained. I was also no ed ha samples de elop a
complex he mal his o y du ing he eezing p ocess, namely, he empe a u e o he sample
does no me ely ollow he shel empe a u e (see Chap e 1.4.1). This aises he ques ion o
how much he he mal his o y, de i ed om he measu ed empe a u e p o ile du ing
eezing, in luences he o ma ion o he mic os uc u e. To assess his, he empe a u e
p o iles o samples du ing eezing in a lyophilise we e analysed, ollowed by an in es iga ion
o he impac o he mal his o y on he mic os uc u e in a ozen suc ose solu ion using a
eeze-d ying mic oscope. Gi en ha disaccha ide solu ions anging om 5 % o 20 % (w/w)
a e commonly used in he ield o lyophilisa ion [70,83,71], a 10 % (w/w) suc ose solu ion was
employed o his wo k.
3.1.1 Recalescence and c ys allisa ion in samples du ing eezing
To in es iga e he he mal his o y o samples du ing eezing, ials we e illed wi h a
10 % (w/w) suc ose solu ion and placed in a lyophilise . The mocouples we e posi ioned close
o he glass bo om inside he samples o eco d he empe a u e. A e a sho p ecooling
phase o equilib a e he sample empe a u e o app oxima ely 5 °C, he shel es we e cooled
o -40 °C a a a e o 1.2°C/min. Images o he illed ials we e aken con inuously du ing his
p ocess. The esul s a e shown in Figu e 3.1.
Resul s and discussion
39
Figu e 3.1: Shel and sample empe a u e du ing eezing a 1.2 °C/min. The sample
empe a u e was measu ed nea he bo om o he ial. Pic u es we e aken o samples a
di e en s ages: supe cooling (A), nuclea ion and ecalescence (B), c ys allisa ion (C), and
comple e solidi ica ion (D).
Du ing he cooling o he shel es, he empe a u e inside he samples also dec eased. No ably,
a delay be ween shel and sample empe a u e became appa en due o he impe ec
conduc ion o hea and he placemen o he he mocouple wi h a small dis ance o
app oxima ely 5 mm om he bo om o he ial. The sample empe a u e dec eased o nea ly
-13 °C wi hou eezing, which means ha he solu ion became supe cooled once alling below
i s equilib ium eezing empe a u e (A). Upon he o ma ion o s able nuclei in he
supe cooled solu ion, he ini ia ion o ice c ys al g ow h lead o a sudden inc ease in he
sample empe a u e as a esul o he hea o c ys allisa ion (B). Once he sample eached i s
maximum empe a u e du ing ecalescence (B), he empe a u e emained ele a ed o a
pe iod o ime (B→C). This is due o he hea balance be ween he la en hea gene a ed by
he eezing wa e in he sample and he hea emo ed by he cooling shel o he lyophilise ,
Resul s and discussion
40
as will be explained in he ollowing. The o al amoun o la en hea gene a ed by wa e
du ing eezing is app oxima ely 79.7 cal/g [66]. This means ha abou 79.7 calo ies o hea
a e gene a ed o comple ely eeze one g am o wa e . In he expe imen , howe e , his hea
was no immedia ely emo ed om he samples. Ins ead, he supe cooled sample ook up
only a po ion o ha hea , which in u n aised i s empe a u e close o i s equilib ium
eezing empe a u e (B→C). Depending on he deg ee o supe cooling, app oxima ely
15 cal/g o hea a e nea ly immedia ely aken up by he sample as sensible hea [2]. This
indica es ha only a po ion o he wa e was apidly ozen once nuclea ion occu ed.
Subsequen ly, he empe a u e o he sample was con inuously educed ia he cooling
shel es, which in u n allowed mo e wa e o c ys allise and gene a e mo e la en hea . In
o de o comple ely solidi y he sample, he o ali y o la en hea o ice c ys allisa ion
(79.7 cal/g) had o be emo ed ia he cooling shel es o he eeze-d ye h oughou he
en i e eezing s ep once nuclea ion had occu ed (B→C→D). As soon as he wa e con en
a ailable o eezing was comple ely c ys allised, and no addi ional hea could be gene a ed,
he sample empe a u e dec eased and con e ged almos o he shel empe a u e (D). In
summa y, he “ eezable” wa e in he sample did no eeze di ec ly bu necessi a ed he
emo al o he hea i gene a ed while u he c ys allising. The ime equi ed o achie e his
caused a delay in he sample empe a u e d op ela i e o he shel empe a u e. This delay
esul ed in a ce ain ime he sample emained abo e i s nuclea ion empe a u e (B→C).
This en i e p ocess was also obse ed ia a came a in he eeze-d ye . As a side no e, he
placemen o he samples o he image eco ding did no ollow he scheme p e iously shown
in Chap e 2.2.1. Ins ead, he samples we e posi ioned (wi hou any he mal senso s) di ec ly
in on o he came a in o de o ake pic u es. The supe cooled and anspa en sample (A)
became sligh ly opaque once nuclea ion occu ed (B). O e he cou se o eezing a g adual
inc ease in opaqueness was obse ed om he bo om o he op o he ial (C), un il he
sample became comple ely opaque a e comple e solidi ica ion (D). I can be in e ed ha
he opaqueness o he sample is dependen on he amoun o ice in he sample. Mo e
speci ically, he liquid phase and he c ys alline ice phase ha e di e en e ac i e indices,
which means ha he o al su ace a ea o ice c ys als, whe e e ac ion o ligh occu s, de ines
he opaqueness o he sample. Du ing ecalescence, he olume ac ion o he c ys alline
phase is lowe han in a comple ely solidi ied sample due o he inc eased sample
Resul s and discussion
41
empe a u e. The e o e, he o al su ace a ea o he c ys alline phase is also lowe han in a
comple ely solidi ied sample. Fu he mo e, he g adien o opaqueness du ing eezing (C) can
be explained by he di ec ional emo al o hea om he sample (and he e o e u he
c ys allisa ion) a he bo om o he ial by he cooling shel es. This phenomenon is
pa icula ly o in e es , because i indica es ha he empe a u e condi ions in he sample a e
no uni o m bu a he dependen on he dis ance o he cooling shel .
3.1.2 Cha ac e isa ion o he he mal p o ile du ing eezing
The empe a u e p o ile shown in Figu e 3.1 is ypical o lyophilisa ion samples, and i is o en
used o desc ibe he phenomenon o supe cooling du ing eezing [2,16,44,86]. Se e al
s udies ha e been ca ied ou o in es iga e he he mal e olu ion o samples du ing
lyophilisa ion, ocusing on he in luence o he empe a u e amp o he shel es and he
nuclea ion empe a u e o he samples [86,134–136].
These s udies ypically compa e he de ec ed nuclea ion empe a u e and eezing a e wi h
p ima y d ying imes. The unde lying assump ion is ha he nuclea ion empe a u e
de e mines he numbe o s able nuclei ha will g ow in o ice c ys als and la e o m he po es
in he lyophilisa e (see Chap e 1.5). This seems easonable since he amoun o wa e
a ailable o eezing is ini e, which means ha he highe he numbe o ice c ys als, he
smalle he a e age ice c ys al size mus be. I is also assumed ha his po e size will a ec
he d ying ime o he sample. Speci ically, smalle po e sizes should esul in inc eased
esis ance o apou low h ough he lyophilisa e, hus educing he a e o sublima ion
du ing d ying.
Howe e , hese s udies end o o e look he ac ha du ing lyophilisa ion he samples a e
exposed o ele a ed empe a u es due o bo h ecalescence and he subsequen
c ys allisa ion phase un il he o al hea o c ys allisa ion is emo ed om he sample. Fo he
pu poses o his wo k, he ime du ing which he empe a u e o he samples is inc eased as
a esul o ecalescence and c ys allisa ion is e e ed o as he esidence ime ( ). The
esidence ime can be de ined o di e en empe a u e limi s, i.e., when does he sample
empe a u e exceed o all below a ce ain empe a u e, as shown in Figu e 3.2. In addi ion,
he a ea unde he cu e (AUC) can be de i ed om he empe a u e p o ile and he
co esponding empe a u e limi o he esidence ime. I should be no ed ha nei he he
Resul s and discussion
42
esidence ime no he AUC o he espec i e esidence ime a e physical alues in ended o
any kind o he modynamic quan i ica ion, bu se e in his wo k as a measu e o e alua e he
e ec o eezing on he exposu e o he sample o ele a ed empe a u es.
Figu e 3.2: Tempe a u e p o ile o a 10 % (w/w) suc ose solu ion du ing eezing wi h
nuclea ion empe a u e (Tn) and esidence imes ( ) o empe a u e limi s o -5 °C and -10 °C.
The a ea unde he cu e (AUC) is also shown o empe a u e limi s o -5 °C ( ed) and -10 °C
( ed and blue).
To in es iga e he esidence ime du ing eezing, ials we e illed wi h 3 ml o a 10 % (w/w)
suc ose solu ion and placed in he lyophilise . Two emp y ials we e le be ween each sample
o educe hea ans e be ween samples, as discussed in Chap e 2.2.1. The mocouples and
wi eless empe a u e senso s we e placed app oxima ely 1 mm abo e he bo om o each
sample ial o ensu e compa abili y o indi idual measu emen s, as a he mal g adien is
expec ed in he samples due o cooling om he bo om o he ials. I should be no ed ha
he use o he mocouples/wi eless empe a u e senso s in oduced addi ional su aces whe e
nuclea ion could occu . Howe e , his could no be a oided as he only way o measu e
empe a u e in his wo k was by his in asi e me hod. The samples we e p e-cooled o
app oxima ely 5 °C and hen ozen o -40 °C using wo di e en eezing a es. The i s a e
was 1.2 °C/min, which is close o he uppe limi possible wi h he eeze-d ye in his wo k,
while he second a e o 0.1 °C/min se ed as a e y slow eezing a e. The nuclea ion
empe a u e was i s de e mined o each measu emen and is shown in Figu e 3.3.

Resul s and discussion
43
Figu e 3.3: E ec o eezing a e on nuclea ion empe a u e (Tn) in a 10 % (w/w) suc ose
solu ion. S anda d de ia ions (e o ba s) we e de e mined o he mocouples wi h n = 8 and
wi eless senso s wi h n = 16.
The nuclea ion empe a u e did no show a signi ican dependence on he eezing a e o
bo h he mocouple and wi eless senso measu emen s (p > 0.05). This is consis en wi h he
indings o Sea les e al. (2001), who epo ed no e ec o eezing a e on nuclea ion
empe a u e in he ange o 0.05 °C/min o 1 °C/min. I can be concluded ha he highe
eezing a e was no su icien o lowe he empe a u e as enough o a ec nuclea ion.
Ins ead, he supe cooled samples all became uns able a a ound -11.9 °C ± 2.1 °C and s a ed
o o m he i s s able nuclei.
The esidence ime was hen examined in ela ion o he eezing amp and is shown in
Figu e 3.4. Fo his pu pose, empe a u e limi s be ween -2 °C and -10 °C we e chosen and he
co esponding esidence imes we e ex ac ed om he empe a u e p o iles. The lowe
empe a u e limi o -10 °C was chosen o he cu en s udy because i is close o he a e age
nuclea ion empe a u e o he 10 % (w/w) suc ose solu ion in he eeze-d ye , while he
uppe limi o -2 °C is only sligh ly below he equilib ium eezing empe a u e o he solu ion.
In addi ion, empe a u e in e als o 2 °C we e chosen o mo e accu a ely cap u e he
empe a u e p o ile obse ed du ing ecalescence and c ys allisa ion. I should be no ed ha
his app oach is only an a emp o cha ac e ise he he mal p o ile and does no include any
he modynamic conside a ions ega ding he eezing solu ion.
Resul s and discussion
44
Figu e 3.4: E ec o eezing a e on esidence imes ( ) in a 10 % (w/w) suc ose solu ion.
S anda d de ia ions (e o ba s) we e de e mined o he mocouples wi h n = 8 and wi eless
senso s wi h n = 16.
Signi ican di e ences in esidence imes we e obse ed be ween eezing amps o
0.1 °C/min and 1.2 °C/min o each empe a u e limi (p < 0.05). A he highes eezing a e
o 1.2 °C/min, which is close o he maximum capaci y o he equipmen and he e o e
ep esen s he uppe limi o hea emo al possible in a lyophilise , he esidence ime was
be ween 6.3 min ± 3.7 min and 9.5 min ± 1.1 min, depending on he empe a u e limi . The
lowe eezing a e o 0.1 °C/min esul ed in esidence imes be ween 13.0 min ± 3.7 min and
23.0 min ± 6.1 min, also dependen on he empe a u e limi . I can be concluded ha he
eezing amp can be used o in luence he esidence imes o he samples du ing eezing.
An al e na i e me hod o assessing he e ec o eezing on he exposu e o he sample o
ele a ed empe a u es was by de e mining he AUC o he ecalescence and c ys allisa ion
phase using he same empe a u e limi s o -2 °C and -10 °C. This app oach conside s he non-
linea na u e o he empe a u e p o ile du ing eezing, and migh he e o e cap u e de ails
be e han he esidence ime. The esul s o he e alua ion a e shown in Figu e 3.5.
Signi ican di e ences we e obse ed be ween he eezing amps o 0.1 °C/min and
1.2 °C/min (p < 0.05), excep o a empe a u e limi o -2 °C (p > 0.05). The AUC showed simila
ends o he e alua ion o he esidence ime. Howe e , he AUC exhibi ed a linea inc ease
wi h dec easing empe a u e limi , whe eas he esidence ime showed a la ened p o ile.
Resul s and discussion
45
This di e ence can be a ibu ed o he non-linea empe a u e p o ile du ing ecalescence
and c ys allisa ion
Figu e 3.5: E ec o eezing a e on he a ea unde he cu e (AUC) o he he mal p o ile o
a 10 % (w/w) suc ose solu ion du ing eezing. S anda d de ia ions (e o ba s) we e
de e mined o he mocouples wi h n = 8 and wi eless senso s wi h n = 16.
In he ield o lyophilisa ion, he e is a consensus ha he nuclea ion empe a u e a ec s he
p ima y d ying ime [86,137–140]. Speci ically, highe nuclea ion empe a u es a e associa ed
wi h sho e p ima y d ying imes due o ewe s able nuclei and consequen ly la ge a e age
ice c ys al sizes. This obse a ion aligns wi h classical nuclea ion heo y, which explains he
ela ionship be ween nuclea ion empe a u e and nuclea ion a e (see Chap e 1.5). Howe e ,
p e ious s udies ha e la gely o e looked he ole o he ecalescence and c ys allisa ion
phase, ocusing ins ead on a ibu ing ice c ys al size p io o d ying o nuclea ion kine ics. The
goal o his wo k is no o dispu e classical nuclea ion heo y bu o explo e whe he he
empe a u e inc ease om he la en hea o c ys allisa ion impac s he mic os uc u e and i
i s signi icance has been unde es ima ed.
This i s aises he ques ion whe he nuclea ion empe a u e has an impac on he he mal
p o ile o a sample du ing eezing. To in es iga e his, he esidence imes and AUC om he
eezing expe imen s we e plo ed agains he nuclea ion empe a u es o each sample. The
esul s a e shown in Figu e 3.6. Fo cla i y, only da a pe aining o a empe a u e limi o -6 °C
wi h wi eless senso s a e displayed. Addi ional da a can be ound in Appendix B.
Resul s and discussion
46
Figu e 3.6: Compa ison o esidence ime ( ) and a ea unde he cu e (AUC) o he
ecalescence and subsequen c ys allisa ion phase wi h he nuclea ion empe a u e (Tn)
du ing eezing o a 10 % (w/w) suc ose solu ion measu ed wi h wi eless empe a u e senso s.
The esidence ime and he AUC o he ecalescence bo h exhibi ed a posi i e co ela ion wi h
he nuclea ion empe a u e. Al hough he coe icien o de e mina ion R² is a he low, i has
o be conside ed ha uni o m empe a u e condi ions canno be gua an eed in a eeze-d ye ,
and ha he ials we e dis ibu ed o e nea ly he en i e shel (see Chap e 2.2.1).
Ne e heless, he quali y o he da a is compa able o o he wo ks in his ield [86]. The
obse ed dependence be ween ecalescence and subsequen c ys allisa ion, he e quan i ied
by esidence ime and he AUC, and he s ochas ic nuclea ion empe a u e suppo s he
ollowing hypo hesis abou he o ma ion o he mic os uc u e in a ozen solu ion.
In he ollowing, a hypo hesis will be made ha explains he ela ionship be ween he he mal
p o ile o a sample du ing eezing and he nuclea ion empe a u e. Du ing he eezing s ep,
samples ypically become supe cooled as desc ibed in Chap e 1.4.1. As soon as nuclea ion
occu s, la en hea o c ys allisa ion is gene a ed du ing he ecalescence and c ys allisa ion
phase and mus be emo ed by he cooling shel . Howe e , he amoun o la en hea eleased
immedia ely a e nuclea ion depends on he deg ee o supe cooling. This phenomenon is
illus a ed in Figu e 3.7.
The ini ial up ake o he la en hea o c ys allisa ion by he sample is only possible un il he
sample empe a u e nea ly eaches i s equilib ium eezing empe a u e (T ). Exceeding his
empe a u e would mean ha c ys allisa ion could p oduce enough hea o mel he
c ys alline phase comple ely, which would be absu d om a he modynamic poin o iew.
Resul s and discussion
53
ha e a conca e bo om, which esul s in con ac wi h he cooling shel only a he ou e edge
o he ial, u he educing he con ac a ea.
Fo he subsequen expe imen s, 1 µl o a 10 % (w/w) suc ose solu ion was pipe ed on o a
glass pla e, co e ed wi h a second glass pla e, and placed in o he eeze-d ying s age o he
mic oscope ( e e o Chap e 2.2.2). The empe a u e o he cooling block was apidly educed
o -40 °C a a a e o 50 °C/min o eeze he sample. This s eep eezing a e and he low
olume- o-con ac a ea a io we e aimed a minimising changes in he mic os uc u e ha
could be enabled by a ise in he sample empe a u e due o hea o c ys allisa ion, he eby
add essing he issues o empe a u e g adien s and he e ogenei y ypical o a ial se up.
Expe imen s we e pe o med wi h and wi hou a space be ween he glass pla es o
de e mine he mos sui able sample p epa a ion. Once comple ely ozen, he empe a u e
was aised o -5 °C a a a e o 10 °C/min and main ained o anneal he samples, simula ing
he empe a u e condi ions du ing he ecalescence and he subsequen c ys allisa ion phase.
The esul s o hese expe imen s a e shown in Figu e 3.11.
Figu e 3.11: F eeze-d ying mic oscopy o a 10 % (w/w) suc ose solu ion wi h ( op) and wi hou
(bo om) a space a e annealing a -5 °C o a ious annealing imes ( a). The space
inc eased he sample heigh o app oxima ely 70 μm.
The ozen suc ose solu ion showed a lack o well-de ined s uc u es unde he eeze-d ying
mic oscope when a space was used. E en a e a sho annealing phase o 5 minu es, du ing
which he pa icula e ice phase was expec ed o coa sen due o Os wald ipening, he

Resul s and discussion
54
iden i ica ion o indi idual s uc u es emained specula i e. In con as , omi ing he space
acili a ed he o ma ion o a hin laye , wi h a calcula ed sample heigh o app oxima ely
5 µm. This se up allowed o he iden i ica ion o dis inc ice c ys als wi h g ain-like shapes.
This concep is u he illus a ed in Figu e 3.12.
Figu e 3.12: F eeze-d ying mic oscope image o a 10 % (w/w) suc ose solu ion a e annealing
a -6 °C o 10 min (le ) and schema ic depic ion o he c oss sec ion o he sample ( igh ) wi h
bo de ing ice c ys als (A) o ice c ys als sepa a ed by he ma ix phase (B).
The clea isibili y o indi idual ice c ys als can be a ibu ed o hei simul aneous con ac
wi h bo h he lowe and uppe glass pla es. This a angemen causes he phase bounda y
be ween he ice c ys al and he amo phous ma ix o ex end ac oss he en i e dis ance om
one glass pla e o he o he . In cases whe e ice c ys als a e su icien ly small o allow mul iple
c ys als o i be ween he glass pla es, phase bounda ies would o m pa allel o he pla es,
hinde ing he iden i ica ion o indi idual ice c ys als. No ably, when he ozen sample was no
subjec ed o an addi ional annealing s ep, e en wi hou a space , he isual cha ac e is ics o
he solu ion immedia ely a e eezing did no align di ec ly wi h he p esence o g ain-like
ice c ys als as obse ed a e he annealing p ocess, as shown in mo e de ail in Figu e 3.13.
Ins ead, a ine ex u e was obse ed, and he de e mina ion o i s mic os uc u al p ope ies
was no possible due o he insu icien magni ica ion o he eeze-d ying mic oscope.
The images o non-annealed samples shown in Figu e 3.13 p omp an in es iga ion in o he
a angemen o he c ys alline and amo phous componen s wi hin he ozen sample. Two
hypo heses can be conside ed. The i s sugges s ha , simila o he annealed samples, g ain-
like ice c ys als migh be p esen wi hin a con inuous ma ix phase, bu signi ican ly smalle in
Resul s and discussion
55
size. The o e lap o hese small ice c ys als, in addi ion o hei small size, may hinde hei
clea obse a ion using a eeze-d ying mic oscope. Al e na i ely, i is plausible ha he ice
c ys als in non-annealed samples o m di e en ly, no appea ing as g ain-like s uc u es.
Al e na i ely, i is plausible ha he ice c ys als in non-annealed samples a e p esen in a
di e en o m a he han g ain-like ice c ys als. This phenomenon is in iguing because o he
signi ican size di e ence be ween he s uc u es obse ed in non-annealed samples unde
he eeze-d ying mic oscope and he po es ypically ound in lyophilised p oduc s. As
men ioned in Chap e 1.5, di e en s udies using di e en me hods ha e epo ed a e age
po e diame e s o app oxima ely 20 µm o 60 µm in non-annealed 5 % (w/w) o 20 % (w/w)
suc ose samples [70,83,98,71]. This o de o magni ude is compa able o ha obse ed in
annealed samples om he eeze-d ying mic oscope (see Figu e 3.13). Howe e , his does
no hold ue o non-annealed samples wi h hei ine s uc u es.
Figu e 3.13: F eeze-d ying mic oscopy o a 10 % (w/w) suc ose sample wi hou (le ) and a e
10 minu es o annealing a -6 °C ( igh ). The eezing empe a u e was app oxima ely -17 °C.
The nex s ep in ol ed es ing di e en eezing amps in he eeze-d ying mic oscope o
iden i y a wha empe a u e nuclea ion occu ed and whe he he nuclea ion empe a u e
could be a ec ed by he eezing amp in his expe imen al se up. Nuclea ion empe a u es
we e de e mined by cap u ing images as he sample cooled un il c ys allisa ion was isible.
This ansi ion was obse ed in he came a iew as ex u e-like ine s uc u es began o o m,
as depic ed in Figu e 3.13. Due o came a limi a ions a apid eezing a es, images we e
cap u ed e e y 1 °C, esul ing in a nuclea ion empe a u e da a esolu ion o 1 °C. Figu e 3.14
displays he nuclea ion empe a u es o di e en eezing a es obse ed wi h he eeze-
d ying mic oscope.
Resul s and discussion
56
Figu e 3.14: E ec o eezing a e on nuclea ion empe a u e (Tn) in a 10 % (w/w) suc ose
solu ion unde a eeze-d ying mic oscope. S anda d de ia ions (e o ba s) we e de e mined
om iplica es.
The a e age nuclea ion empe a u e in he eeze-d ying mic oscope was de e mined o be
-16.9 °C ± 1.1 °C. No signi ican di e ences in nuclea ion empe a u es we e obse ed
be ween eezing amps om 0.1 °C/min o 20 °C/min (p > 0.05). In con as o he eezing
amps used in he lyophilise , s eepe amps we e es ed in he eeze-d ying mic oscope o
de e mine whe he he highes possible eezing amp in a lyophilise o 1.2 °C/min was simply
insu icien o a ec he nuclea ion empe a u e. Consequen ly, he eezing amp was ound
o be ine ec i e in in luencing he nuclea ion empe a u e in bo h lyophilise and eeze-
d ying mic oscopy expe imen s.
Howe e , he nuclea ion empe a u e in he lyophilise was ound o be -11.9 °C ± 2.1 °C,
which signi ican ly di e ed om he alues in he eeze-d ying mic oscope (p < 0.05). To
ensu e compa abili y be ween eeze-d ying mic oscopy and lyophilisa ion expe imen s, he
nuclea ion empe a u e in he eeze-d ying mic oscope had o be adjus ed acco dingly.
E o s we e made o induce nuclea ion a app oxima ely -12 °C, including a emp s o hold
he empe a u e o a p olonged ime, ab up ly educing and inc easing he p essu e in he
eeze-d ying s age, and oscilla ing empe a u e wi hin ± 1 °C o he desi ed nuclea ion
empe a u e. The mos ep oducible esul s we e ob ained when nuclea ion was induced by
epea ed gen le aps on he sample holde while main aining he sample empe a u e a
-12 °C. Fu he examina ion o he samples was ca ied ou in he eeze-d ying mic oscope
using pola ised ligh , as shown in Figu e 3.15 o ozen and non-annealed samples. The
Resul s and discussion
57
p inciples o pola isa ion and bi e ingence ha e al eady been desc ibed in Chap e 2.2.3. The
samples we e cooled o -12°C and nuclea ion was induced by apping he sample holde . The
samples we e hen cooled o -40°C a 10°C/min and images we e aken.
Figu e 3.15: F eeze-d ying mic oscopy o a 10 % (w/w) suc ose sample wi h ( igh ) and wi hou
(le ) pola isa ion a -40 °C wi h p eceding nuclea ion a a ound -12 °C.
Resul s and discussion
58
The eeze-d ying mic oscope images show iden ical sec ions wi h and wi hou pola ised ligh .
In addi ion, he ou e edge o he ozen sample was delibe a ely included in he 100x and
200x magni ica ions (bo om igh -hand co ne o each image). Due o he high eezing a e
o 10 °C/min, he samples we e cooled o -40°C wi hin 3 minu es o nuclea ion. No sepa a e
annealing s ep was pe o med o hese samples. The pola ised ligh mic oscopy images also
indica e he p esence o ea he -like s uc u es wi hou a clea pa e n simila o an ice lowe
on a window. Addi ionally, he s uc u es seem o ha e o igina ed and sp ead ou om he
phase bounda y o he solu ion. To in es iga e he ansi ion om ice c ys als in non-annealed
samples o g ain-like ice c ys als, he sample was hea ed om -40 °C o -6 °C a 10 °C/min and
annealed o 120 minu es. The co esponding images a e shown in Figu e 3.16. I egula
s uc u es we e obse ed in he non-annealed sample a he op o Figu e 3.16, while single
g ain-like ice c ys als became isible a e sho annealing imes ( a = 10 min). Each indi idual
ice c ys al appea ed o ha e a single uni o m colou . I is also no iceable ha neighbou ing ice
c ys als o en exhibi ed he same colou a ion. These clus e s o ice c ys als o he same colou
we e o en loca ed whe e i egula s uc u es o a simila colou had p e iously been p esen .
In o de o u he in es iga e he e ec o he ecalescence and c ys allisa ion phase on he
mic os uc u e in a ozen solu ion, he annealing s ep in he eeze-d ying mic oscope was
designed o mimic he condi ions expe ienced by he samples in he lyophilisa ion
expe imen s om Chap e 3.1.2.
Howe e , he empe a u e p o ile o he sample in he lyophilise du ing eezing was no
linea o s epwise, so i would no ha e been easible o implemen such a p o ile in he eeze-
d ying mic oscope. In addi ion, ca e mus be aken o ensu e ha he sample empe a u e
does no exceed he equilib ium eezing empe a u e o he solu ion, as his would esul in
comple e hawing o he sample. Some empe a u e luc ua ions a e o be expec ed in he
eeze-d ying mic oscope and he e o e i would ha e been oo isky o use a se empe a u e
sligh ly below he equilib ium eezing empe a u e o a 10 % (w/w) suc ose solu ion o abou
-0.6 °C [66]. The e o e, in he ollowing expe imen s, an annealing empe a u e o -6 °C was
chosen, which, based on p e ious expe ience, clea ly demons a ed he e ec o annealing on
he mic os uc u e, bu was su icien ly a om he equilib ium eezing empe a u e o
p e en he comple e hawing o he sample. The esidence imes in he lyophilisa ion
expe imen s om Chap e 3.1.2 we e aken in o accoun o he annealing ime o he eeze-

Resul s and discussion
59
d ying mic oscopy expe imen s. Fo a empe a u e limi o -6 °C, he esidence imes o
eezing amps o 1.2 °C/min and 0.1 °C/min we e ounded o o 10 minu es and 20 minu es,
espec i ely.
Figu e 3.16: F eeze-d ying mic oscopy wi h pola ised ligh o a 10 % (w/w) suc ose sample
du ing annealing a -6 °C.
Resul s and discussion
60
The mic os uc u e o a ozen solu ion was assessed by coun ing indi idual ice c ys als a e
mimicking he empe a u e condi ions o he ecalescence and c ys allisa ion phases
expe ienced du ing eezing in a lyophilise . As p e iously discussed in Chap e 1.5, he
numbe o ice c ys als se es as an indica o o he e en ual po e size a e d ying. This is
because he eezable wa e in he solu ion is dis ibu ed among all he ice c ys als o med
du ing eeze-concen a ion, i.e., a la ge numbe o ice c ys als co esponds o smalle
a e age ice c ys al sizes. Fo his e alua ion, samples we e ei he cooled using di e en
eezing amps un il nuclea ion occu ed o nuclea ion was induced a -12 °C. Following
nuclea ion, he samples we e apidly cooled o -40 °C a a a e o 50 °C/min o minimise
s uc u al changes ha migh occu abo e he glass ansi ion empe a u e. The samples we e
hen hea ed o an annealing empe a u e o -6 °C a 10 °C/min and held o 20 minu es.
Images we e aken a e 10 minu es and 20 minu es, co esponding o esidence imes o
eezing amps o 1.2 °C/min and 0.1 °C/min in a eeze-d ye . The numbe o ice c ys als was
quan i ied h ough image analysis, as de ailed in Chap e 2.2.6, ac oss a iewing a ea o
app oxima ely 1.12 × 10³ μm². The esul s a e shown in Figu e 3.17.
Figu e 3.17: E ec o eezing a e and annealing ime ( a) on he numbe o ice c ys als in a
10 % (w/w) suc ose solu ion. S anda d de ia ions (e o ba s) we e de e mined om
iplica es.
Fo all eezing amps, as well as o en o ced nuclea ion a -12 °C, no signi ican di e ence
was obse ed in he numbe o indi idual ice c ys als (p > 0.05), indica ing ha unde uni o m
Resul s and discussion
61
annealing condi ions, he coa sening o he c ys alline phase occu ed independen ly o he
p eceding eezing condi ions. Howe e , a signi ican ly lowe numbe o ice c ys als was
coun ed a e 20 minu es o annealing compa ed o 10 minu es, ac oss all eezing amps and
en o ced nuclea ion condi ions (p < 0.05).
The ela ionship be ween he numbe o ice c ys als and he nuclea ion empe a u e is
illus a ed in Figu e 3.18. I is impo an o no e ha he nuclea ion empe a u e o each
measu emen was de e mined wi h an accu acy o 1 °C, due o limi a ions in cap u ing mo e
images wi h he came a du ing he s eep eezing amps.
Figu e 3.18: E ec o nuclea ion empe a u e (Tn) and annealing ime ( a) on he numbe o
ice c ys als in a 10 % (w/w) suc ose solu ion.
These esul s also show ha o e a ange o nuclea ion empe a u es om -12 °C o -18 °C,
he esul ing numbe o indi idual ice c ys als did no change signi ican ly a e he samples
we e exposed o ei he 10 minu es o 20 minu es o annealing (p > 0.05). Howe e , he e was
a signi ican di e ence be ween he da ase s co esponding o he wo di e en annealing
imes a any gi en nuclea ion empe a u e (p < 0.05), simila o he esul s om Figu e 3.17.
The esul s o he annealing expe imen s in he eeze-d ying mic oscope can be summa ised
as ollows:
 F eezing a small olume o sample while p e en ing he empe a u e om ising due
o ecalescence and c ys allisa ion leads o he o ma ion o ine ea he -like
s uc u es.
Resul s and discussion
62
 I hese s uc u es a e composed o small indi idual ice c ys als, hen hese ice c ys als
a e one o mo e o de s o magni ude smalle han any po e ound in a lyophilisa e.
 When hese ozen samples a e annealed in a eeze-d ying mic oscope unde
condi ions ha mimic ypical esidence imes in a lyophilise , indi idual g ain-like ice
c ys als wi h sizes compa able o po es in a d y lyophilisa e a e o med.
 The quan i y (and he e o e size) o hese ice c ys als depends mainly on he condi ions
du ing he ecalescence and c ys allisa ion phases, and no on he nuclea ion
empe a u e.
I is impo an o no e ha he educ ion in sample hickness can impac he a e o Os wald
ipening o ice c ys als du ing annealing. The spa ial con inemen be ween glass pla es al e s
he shape o ice c ys als by la ening hem, as illus a ed p e iously in Figu e 3.12. Os wald
ipening, desc ibed in Chap e 1.4.2, is a cu a u e-dependen p ocess and may decele a e
when ewe cu ed su aces a e p esen . The e o e, i is c ucial o cla i y ha he objec i e o
hese expe imen s is no o de e mine an exac po e size based on eezing and annealing
condi ions. Ra he , he goal is o assess he quali a i e e ec o ecalescence and he
subsequen c ys allisa ion phase on he mo phological changes wi hin he mic os uc u e o
he ozen solu ion.
3.1.5 Impac o he mal his o y du ing eezing on he mic os uc u e
The ollowing hypo hesis is p oposed o in e p e he expe imen al da a om p e ious
chap e s, wi h addi ional con ex on he eezing o a glass- o ming disaccha ide solu ion.
The g ow h o ice c ys als om a nucleus in a solu ion can be classi ied in o h ee main
ypes: dend i es, i egula dend i es, and sphe uli es, in luenced by ac o s such as eezing
a e, solu e composi ion, and concen a ion [48]. Regula dend i es o m when he e is
su icien ime o ice c ys al g ow h, while as e c ys allisa ion leads o he de elopmen o
i egula dend i es and sphe uli es [2]. Gi en ha he samples expe ienced supe cooling bo h
in he eeze-d ying mic oscope and in he lyophilise , i is likely ha he c ys allisa ion p ocess
occu ed ela i ely quickly a e nuclea ion. This apid o ma ion is e iden in Figu e 3.15,
whe e he s uc u es appea less o ganized han, o ins ance, a snow lake, which exempli ies
dend i ic ice c ys al g ow h. S uc u es like dend i es, i egula dend i es, and sphe uli es a e
Resul s and discussion
69
se up o a eeze-d ying mic oscope. The concep o planes o cu a u e is illus a ed in
Figu e 3.22.
Figu e 3.22: Schema ic ep esen a ion o he mo phology o ice c ys als be ween wo spa ial
bounda ies. I he dis ance be ween he bounda ies is su icien ly la ge, sphe ical ice c ys als
wi h wo planes o cu a u e a e o med (A), whe eas i he dis ance is small, cylind ical ice
c ys als wi h only one plane o cu a u e a e p esen (B).
Ice c ys als in h ee-dimensional space wi hou any bounda ies exhibi wo cu a u e planes,
namely a ho izon al and a e ical plane. Howe e , when he dis ance be ween he bounda y
su aces ha limi he ice c ys als becomes smalle han he diame e o he ice c ys als, hen
he ice c ys als s a o de o m un il hey assume cylind ical shapes ha only ha e a single
cu a u e plane.
The d i ing o ce behind Os wald ipening, he phenomenon ha is o in e es o his wo k,
is based on he Gibbs-Thomson e ec , which e e s o he dependence o apou p essu e o
chemical po en ial on he in e acial ene gy/cu a u e o a su ace [146]. As explained in
Chap e 1.4.2, he sa u a ion apou p essu e de ia es om a pu ely empe a u e-dependen
ma e ial p ope y when he ma e ial is p esen in su icien ly small pa icles such as
mic oscopic ice c ys als. A ma hema ical ela ionship can be es ablished be ween cu a u e
and he de ia ion in sa u a ion apou p essu e, which is desc ibed in Appendix A o phase-
ield models. I a ci cle ( wo-dimensional) and a sphe e ( h ee-dimensional) wi h adius a e
conside ed, he mean cu a u e is 1/ and 2/ , espec i ely. This means ha he de ia ion in
sa u a ion apou p essu e doubles when a ci cle is expanded in o a sphe e, i.e., ano he plane
o cu a u e is added (see Equa ion App. 1). Acco dingly, phenomena such as Os wald ipening
would also p oceed a a di e en a e o a ci cle and a sphe e o he same adius.

Resul s and discussion
70
In eeze-d ying mic oscopy expe imen s, he sample olume is con ined o a hin ilm
be ween wo glass pla es. The dis ance be ween he wo glass pla es can be con olled using
a space . I no space is used, he dis ance be ween he wo glass pla es will adjus acco ding
o he sample olume (see Chap e 3.1.4). When he dis ance be ween he glass pla es is
su icien ly small compa ed o he size o he ice c ys als, he ice c ys als a e o ced in o a
cylind ical shape (see Figu e 3.22). Subsequen ly, he plane o cu a u e in he e ical
di ec ion becomes negligible, educing he sys em o a single plane o cu a u e. In his s a e,
annealing s ill causes Os wald ipening o occu , bu he cu a u e-dependen coa sening o
he ice c ys als is expec ed o be al e ed due o he educ ion o he sys em o a single plane
o cu a u e.
P ecisely his p ope y was u ilised o ca y ou a simula ion-based in es iga ion o he
coa sening, because wo-dimensional simula ions a e ep esen a i e o he eeze-d ying
expe imen s due o he single cu a u e plane, while h ee-dimensional simula ions wi h hei
wo cu a u e planes e lec he ice c ys als in a lyo ial. Phase- ield modelling was chosen as
he simula ion ype o he ollowing easons. Phase- ield models can be easily ex ended om
wo-dimensional o h ee-dimensional, while he o e all ma hema ical basis and simula ion
pa ame e s emain iden ical. In addi ion, phase- ield models inhe en ly inco po a e he
concep s o cu a u e (in e acial ene gy) and sa u a ion apou p essu e
(composi ion/chemical po en ial). They p o ide a s aigh o wa d way o desc ibing he
empo al e olu ion o phases in such a sys em and a e o en used o p edic coa sening
phenomena. The simula ion s a egy is illus a ed in Figu e 3.23 and explained below.
Fi s , samples o eeze-d ying mic oscopy expe imen s we e p epa ed wi h small olumes in
o de o educe he sample heigh as much as possible. The aim was o educe he e ical
plane o cu a u e o such an ex en ha i s con ibu ion o he coa sening o he ice c ys als
became negligible. In Chap e 3.2.2.2.3, annealing expe imen s we e ca ied ou wi h hese
samples in he eeze-d ying mic oscope o de e mine he ec ys allisa ion a es o a single
plane o cu a u e a di e en annealing empe a u es. This expe imen al da a was used o
calib a e a phenomenological wo-dimensional phase- ield model in o de o de e mine
simula ion pa ame e s ha would cap u e he coa sening beha iou as accu a ely as possible
(see Chap e 3.2.4).
Resul s and discussion
71
Figu e 3.23: App oach o he p edic ion o mo phological p ope ies o he mic os uc u e in
a lyophilisa e in his wo k: sample p epa a ion o eeze-d ying mic oscopy (1), calib a ion o
a wo-dimensional phase- ield model (2), de e mina ion o simula ion pa ame e s (3),
p edic ion o mic oscopic mo phological pa ame e s (4).
A h ee-dimensional phase- ield simula ion was hen implemen ed using he simula ion
pa ame e s de e mined om he wo-dimensional simula ion in Chap e 3.2.5. The aim o his
s ep was o e-in oduce he ho izon al plane o cu a u e, hus e e sing he educ ion in he
numbe o planes o cu a u e om he expe imen al se up. This would hen allow
Resul s and discussion
72
conclusions o be d awn o he impac o he annealing p ocess on a ozen disaccha ide
solu ion in a lyo ial. Finally, wi h he h ee-dimensional ex ension, mo phological pa ame e s
we e p edic ed in hei empo al e olu ion, such as he ice c ys al size dis ibu ion, he su ace
a ea o he ma ix phase and he conjunc ion a ea be ween ice c ys als. The de ails o each
mo phological pa ame e a e gi en in he ele an chap e s.
3.2.2 Implemen a ion and calib a ion o he phase- ield model
In he ollowing sec ions, a phase- ield model was implemen ed as a nume ical me hod o
in es iga e he mic os uc u al changes in a ozen solu ion du ing annealing, and
expe imen al da a we e collec ed o calib a e he ele an model pa ame e s.
3.2.2.1 Fo mulism o he mul iphase- ield model
The conside ed phase- ield model is based on a pa icula e phase p (he e: ice c ys als) and a
con inuous ma ix phase m. These phases a e desc ibed in space and ime by he o de
pa ame e η( , ) (c ys alline ice phase = 1, amo phous ma ix phase = 0) and hei composi ion
c( , ) (pu e wa e = 1, pu e solu e = 0). The ansi ion be ween he phases is ep esen ed in
he phase- ield model by a smoo h in e ace wi h alues be ween he co esponding minima
and maxima.
The e olu ion o he mic os uc u e in his phase- ield model is go e ned by he Cahn-Hillia d
di usion equa ion o conse ed ield a iables:
𝜕𝑐
𝜕𝑡
=
∇
𝑀
∇
𝛿𝐹
𝛿𝑐
,
(Eq. 6)
and he Allen-Cahn elaxa ion equa ion o non-conse ed ield a iables:
𝜕𝜂
𝜕𝑡
=
−
𝐿
∇
𝛿𝐹
𝛿𝜂
,
(Eq. 7)
whe e M is he mobili y coe icien , L is he elaxa ion coe icien and F is he ee ene gy o
he sys em [126].
Since Os wald ipening is a di usi e p ocess, pa icle-pa icle in e ac ions such as coalescence
mus be inhibi ed. In na u e, usion o ice c ys als is p e en ed h ough a misma ch o he
c ys allog aphic o ien a ions o neighbou ing c ys als [103].
Resul s and discussion
73
This concep can also be adop ed in a phase- ield model, whe e η( , ) is ex ended o he
mul iphase- ield ηi( , ) and is gi en by:
𝜕
𝜂

𝜕𝑡
=
−
𝐿
∇
𝛿𝐹
𝛿
𝜂

(
𝑟
,
𝑡
)
,
(Eq. 8)
wi h i = 1, 2, …, n and he pa icula e phase is di ided be ween he n elemen s o he phase
ields. The ee ene gy o he sys em F is composed o he bulk ee ene gy and he
con ibu ions om he g adien ee ene gies:
𝐹
(
𝑓
,
𝑐
,
𝜂

)
=


𝑓
(
𝑐
,
𝜂

)
+
𝜅

(
∇
𝑐
)

+
𝜅

(
∇
𝜂

)

𝑑
Ω


,
(Eq. 9)
whe e (c,ηi) is he bulk ee ene gy densi y, and κc and κη a e he g adien ene gy coe icien s
o composi ion and o de pa ame e , espec i ely. The bulk ee ene gy densi y (c,ηi)
p omo es phase sepa a ion whene e g adien and in e acial ene gies a e no p esen and is
gi en by:
𝑓
(
𝑐
,
𝜂

)
=
𝑓

(
𝑐
)

1
−

𝑊
(
𝜂

)



+
𝑓

(
𝑐
)

𝑊
(
𝜂

)


+𝑃𝜂(1−𝜂)




+𝑄𝜂𝜂








,
(Eq. 10)
whe e P is a coe icien o he magni ude o he ene gy ba ie be ween he m- and he p-
phase, Q is a coe icien o he magni ude o he ene gy ba ie be ween he n pa icula e
phases, and m and p a e he ee ene gies o he m- and he p-phase, espec i ely.
The in e pola ion unc ion W(ηi) is used in Eq. (7) and is de ined as [147]:
𝑊
(
𝜂

)
=

0
;
𝜂


(
10
−
15
𝜂

+
6
𝜂


)
;
1
;
o
𝜂

<
0
,
o
0
≤
𝜂

≤
1
,
o
𝜂

>
1
.
(Eq. 11)
The ee ene gies m and p a e gi en by:
𝑓

(
𝑐
)
=
𝐴
(
𝑐
−
𝑐


)

,
(Eq. 12)
𝑓

(
𝑐
)
=
𝐵

𝑐


−
𝑐


,
(Eq. 13)
whe e A and B a e posi i e cons an s, and 𝑐
 and 𝑐
 a e equilib ium composi ions o he m-
and he p-phase, espec i ely.
Resul s and discussion
74
In his wo k, he ec ys allisa ion a e in he model was calib a ed using he simula ion
pa ame e s M and L. Bo h alues emained cons an , i.e., when M was changed, L was
adjus ed o he same alue, e ec i ely only impac ing he ec ys allisa ion a e and no he
mo phological p ope ies o he pa icles. I dissimila alues o M and L had been used in a
simula ion, he a io o di usion o elaxa ion o he pa icles would ha e changed, po en ially
a ec ing he mo phology adop ed by he pa icles du ing coa sening. The gene a ion o he
wo-dimensional and h ee-dimensional ini ial condi ions is discussed la e in Chap e 3.2.3.
The dimensionless simula ion pa ame e s o he phase- ield model a e lis ed in Table 3.1.
Table 3.1: Dimensionless pa ame e s used in his wo k.
Pa ame e Value
κc 1
κη 1
A 1
B 1
P 1
Q 2
A semi-implici Fou ie spec al scheme was implemen ed acco ding o Chen and Shen (1998)
by ea ing linea and second-o de ope a o s implici ly and esidual e ms explici ly. Some
s abili y es s and pa ame e de e mina ion es s a e p o ided in Appendix A.
3.2.2.2 Expe imen al de e mina ion o ele an calib a ion pa ame e s
The phenomenological phase- ield model employed o p edic he e ec o annealing on he
mic os uc u e o disaccha ide solu ions, bo h du ing annealing and in hei maximally eeze-
concen a ed s a e, equi ed calib a ion o ele an simula ion pa ame e s. Ini ially,
expe imen al alues we e ob ained o he olume ac ions o ice c ys als and he amo phous
ma ix phase, which a y wi h sample empe a u e and hus di e based on whe he
annealing is pe o med o i he solu ion eaches i s maximally eeze-concen a ed s a e.
Rec ys allisa ion a es du ing annealing, which a e also empe a u e-dependen , we e hen
de e mined expe imen ally. These expe imen s included bo h suc ose and ehalose solu ions
o also conside a second commonly used excipien in eeze-d ying.

Resul s and discussion
75
3.2.2.2.1 Mass ac ions du ing annealing
As desc ibed in Chap e 1.4.1.2, he mass ac ions o solu e and sol en in he eeze-
concen a ed ma ix phase o a ozen solu ion depend solely on he sample empe a u e.
This is impo an because he empe a u e is inc eased du ing annealing abo e he onse
empe a u e o mel ing Tm’ a which wa e molecules dissol e om he c ys alline phase in o
he ma ix phase. Consequen ly, he a io o solu e and sol en in he amo phous phase
changes. These mass ac ions in he ma ix phase ollow he liquidus line o he solu e-sol en
sys em. Fo example, i a 30 % (w/w) suc ose solu ion becomes comple ely liquid a -1.6 °C, a
eeze-concen a ed ma ix a his exac empe a u e ( ega dless o he ini ial concen a ion
in he solu ion) will also con ain a 30 % (w/w) a io o suc ose o wa e . The e o e, he eezing
poin dep ession o di e en solu e concen a ions can be used o de e mine he
empe a u e-dependen mass ac ions in he eeze-concen a ed ma ix phase [60].
In he ollowing, he mel ing empe a u e Tm o suc ose and ehalose solu ions wi h
concen a ions be ween 30 % (w/w) and 55 % (w/w) was de e mined using di e en ial
scanning calo ime y (DSC) wi h he p o ocol om Chap e 2.2.4. An example hea low is
shown in Figu e 3.24.
Figu e 3.24: Exempla y he mog ams o suc ose solu ions wi h di e en solu e concen a ions
du ing mel ing. The mel ing peak was used o de e mine he mel ing empe a u e (Tm) o he
espec i e solu ion.
Resul s and discussion
76
The glass ansi ion du ing hea ing can be obse ed a app oxima ely -33 °C by he shi in he
hea low om one pla eau o ano he . Acco ding o Chap e 1.4.1.2, he glass ansi ion
empe a u e o a maximally eeze-concen a ed solu ion (Tg’) should heo e ically be
iden ical o all samples, as i is p esumed independen o solu e concen a ion. Howe e ,
obse ed di e ences be ween he solu ions can be a ibu ed o he dissimila he mal
his o ies a ising om a ying solu e concen a ions. Gi en ha nuclea ion empe a u e is
in luenced by solu e concen a ion [46], and conside ing he demons a ed co ela ion
be ween nuclea ion empe a u e and esidence ime due o ecalescence and c ys allisa ion
(as shown in Chap e 3.1.2), i can be in e ed ha he samples unde wen di e en condi ions
du ing eezing.
A e he glass ansi ion, all cu es depic ed in Figu e 3.24 exhibi ed simila beha iou du ing
hea ing, al hough wi h a ia ions in peak posi ions and peak a eas. These di e ences in peak
a ea can be a ibu ed o he a ying amoun s o wa e p esen in each solu ion. A sample
wi h a lowe solu e concen a ion con ains mo e wa e , he eby p o iding mo e ice o
mel ing. Since mel ing is an endo he mic phase ansi ion, i appea s as a posi i e hea low in
he he mog am. As men ioned in Chap e 1.4.2, ca e mus be aken when using e ms such
as mel ing in he con ex o solu ions. Mel ing is used he e o desc ibe he dissolu ion o
p ecipi a ed ice in o he ma ix phase, which con ains suc ose and wa e , and no he dis inc
phase change ha occu s a a pa icula empe a u e when an en i e subs ance changes
phase. Speci ically, he mel ing empe a u e Tm o he solu ion has been de ined in his wo k
as he empe a u e a which he hea lux peaks. The empe a u e o he mel ing peak,
indica ing he empe a u e a which he la ges ac ion o ice in he sample con e s o wa e ,
is obse ed o dec ease wi h inc easing solu e concen a ion. This ela ionship is c i ical o
he calib a ion o he phase- ield model and is de ailed in Figu e 3.25 o bo h solu es.
As he solu e concen a ion inc eased, he mel ing empe a u e o he co esponding suc ose
and ehalose solu ions dec eased. This is consis en wi h he phenomenon o eezing poin
dep ession, whe e he eezing empe a u e o he solu ion is nega i ely co ela ed wi h he
amoun o solu e added o a sol en [149]. Fu he mo e, he eco ded da a a e in ag eemen
wi h alues epo ed in he li e a u e [72,150], wi h no signi ican di e ence be ween he
liquidus lines o suc ose and ehalose solu ions (p > 0.05).
Resul s and discussion
77
Figu e 3.25: Liquidus line o a suc ose-wa e (le ) and ehalose-wa e ( igh ) sys em. The
mel ing empe a u e o 30 % (w/w) o 55 % (w/w) solu ions we e measu ed ia DSC. S anda d
de ia ions (e o ba s) we e calcula ed om iplica es.
A second-deg ee polynomial end line was i ed o he da a and can be ep esen ed as:
𝑦
=
−
0
.
0062
𝑥

+
0
.
2188
𝑥
−
2
.
4362
,
(Eq. 14)
whe e y is he mel ing empe a u e, and x is he solu e concen a ion, wi h R² = 0.9985.
Equa ion 14 ela es he solu e mass ac ion in he ma ix phase o he ambien empe a u e
o suc ose and ehalose solu ions wi h concen a ions anging om 30 % (w/w) o
55 % (w/w). This expe imen al ange co esponds o mel ing empe a u es o -1.6 °C ± 0.0 °C
and -9.3 °C ± 0.1 °C, espec i ely. Thus, Equa ion 14 can be used o de e mine he composi ion
o he ma ix phase a annealing empe a u es wi hin his ange.
Nex , app op ia e annealing empe a u es we e selec ed o pe o m coa sening expe imen s
o he calib a ion o he phase- ield model. Annealing should always be pe o med abo e he
glass ansi ion empe a u e Tg’ o he solu ion, o he wise changes in he mic os uc u e will
be in he o de o mm/yea due o he high iscosi y o he ma e ial [60]. The a e o
coa sening, o ec ys allisa ion a e, o he ice phase is dependen on he annealing
empe a u e, wi h highe empe a u es esul ing in as e coa sening [72]. Howe e , while
highe annealing empe a u es sho en he p ocess ime, i is impo an o a oid se ing he
empe a u e oo close o he mel ing empe a u e o he solu ion as e en small empe a u e
luc ua ions in he equipmen can lead o he comple e hawing o he sample. As discussed
Resul s and discussion
78
in Chap e 3.1.4, he mel ing empe a u e o a 10 % (w/w) suc ose solu ion is a app oxima ely
-0.6 °C, which, based on expe ience, allows annealing empe a u es o a ound -2 °C in he
eeze-d ying mic oscope wi hou isking he hawing o he sample.
Commonly, -5 °C is used in li e a u e o s udy he e ec s o annealing on lyophilisa e
mic os uc u e [70,71], p o iding a subs an ial bu e om he mel ing poin while allowing
sho annealing du a ions o no iceably impac he mic os uc u e. The e o e, annealing
empe a u es o -4 °C and -6 °C we e chosen o u he expe imen s and simula ions,
conside ing he p ocess du a ion and a he same ime main aining ealis ic condi ions
compa able o published s udies. In addi ion, wo di e en annealing empe a u es we e
chosen o demons a e he empe a u e dependency o he p ocess. Since -4 °C and -6 °C we e
wi hin he ange o he expe imen ally de e mined polynomial om Equa ion 14, and suc ose
and ehalose solu ions did no di e signi ican ly in hei eezing poin dep ession, he solu e
o wa e mass ac ions o bo h disaccha ides a -4 °C and -6 °C we e calcula ed o be
0.398 g/g and 0.461 g/g, espec i ely.
3.2.2.2.2 Volume ac ions du ing annealing/maximum eeze-concen a ion
The nex s ep in ol ed con e ing he mass ac ions in o olume ac ions o bo h he ice
and ma ix phases unde annealing condi ions and a maximum eeze-concen a ion. Due o
he una ailabili y o di ec measu emen s o he ma ix phase densi y a -4 °C, -6 °C, and
maximum eeze-concen a ion, li e a u e da a o he densi y o concen a ed suc ose and
ehalose solu ions a 20 °C we e u ilised. Po en ial expansion o con ac ion o he ma ix
phase due o low empe a u es was no conside ed. In he i s -p inciples simula ion wo k by
Fan e al. (2018), he impac o empe a u e on he densi y change o he amo phous phase
was limi ed o he con ac ion o he esidual wa e con en , which is pa icula ly small
compa ed o he solu e mass ac ion a maximum eeze-concen a ion. The e o e, any
de ia ion in he olume ac ion was deemed negligible in his con ex . The ice phase did no
p esen his issue, as densi y da a o ice unde he speci ied eezing condi ions in a lyophilise
we e a ailable (see Table 3.3). Fi s , he wa e mass ac ions in he maximum eeze-
concen a ed ma ix phase o bo h suc ose and ehalose solu ions we e ga he ed om
a ious li e a u e sou ces and a e summa ised in Table 3.2.
Resul s and discussion
85
su p ising, as a di e ence in ec ys allisa ion a e was p e iously ound in a sepa a e
publica ion [68]. Howe e , i should be no ed ha he aim o his p eceding publica ion was
no o de e mine an exac ec ys allisa ion a e, bu a he o assess Os wald ipening and
glassy s a e elaxa ion quali a i ely. Acco dingly, he annealing expe imen was only
pe o med and e alua ed once o bo h solu es, and no s anda d de ia ion could be aken.
Klinmalai e al. (2017) also ound a sligh di e ence be ween he wo solu es du ing
coa sening. Howe e , he quali y o he da a is ques ionable as he s anda d de ia ion a ies
g ea ly depending on he se ies o measu emen s. In addi ion, i can also be obse ed om
he mic oscopy images ha non-cylind ical 3-dimensional c ys als o m in some expe imen s.
This is pa icula ly p oblema ic as he adius o an ice c ys al can no longe be eliably
de e mined and he measu emen now s ongly depends on he laye hickness o he sample.
An inc ease in he annealing empe a u e esul ed in a signi ican inc ease in he
ec ys allisa ion a e om an a e age o suc ose and ehalose o 26.13 µm³/min a -6 °C o
56.80 µm³/min a -4 °C (p < 0.05). The linea ela ionship obse ed be ween he cube o he
ci cle equi alen adius and annealing ime sugges s ha he coa sening beha iou o suc ose
and ehalose solu ions in he eeze-d ying mic oscope ollows he g ow h law o he LSW
heo y. Howe e , his obse a ion only alida es Equa ion 3, as no all he condi ions o he
LSW heo y a e me wi hin he expe imen al se up o he eeze-d ying mic oscope.
The e o e, he slopes de i ed om he linea i s in Figu e 3.28 we e used o di ec ly
de e mine he ec ys allisa ion a e using Equa ion 3 ins ead o he i s -p inciples app oach
om Equa ion 4. Howe e , his is pe ec ly adequa e as he phase- ield simula ion in his wo k
is in ended o be a phenomenological model anyway, i.e., he simula ion should only
quan i a i ely desc ibe he espec i e phenomenon om he expe imen and does no ha e
o be de i ed om physical laws. All i ing pa ame e s a e summa ised in Table 3.4.
The linea i o he cube o he ci cle equi alen adius o ice c ys als du ing he annealing
expe imen se es as he basis o calib a ing he wo-dimensional phase- ield simula ion.
Al hough o he i s we e also de e mined, he use o he cube o he ci cle equi alen adius
is a con enien choice as i is al eady es ablished in he li e a u e and allows o easy
compa ison be ween simula ion and expe imen due o i s linea ela ionship.

Resul s and discussion
86
Table 3.4: Fi ing pa ame e o numbe o pa icles (NoP, powe law i ), ci cle equi alen
adius (CER, loga i hmic i ) and he cube o he ci cle equi alen adius (CER³, linea i ) a
di e en annealing empe a u es (Ta).
Solu ion Ta
[°C]
NoP CER CER³
a
[-]
b
[1/min]
R² a
[µm]
b
[µm] R² k
[µm³/min]
R²
10 %
(w/w)
suc ose
-4 3895 -0.592 0.9958
4.943
-2.38 0.9919
54.74 0.9849
-6 8815 -0.628 0.9997
4.024
-3.27 0.9695
27.74 0.9973
10 %
(w/w)
ehalose
-4 8482 -0.689 0.9940
5.433
-4.90 0.9930
58.86 0.9914
-6 5970 -0.563 0.9994
3.685
-1.37 0.9862
24.52 0.9891
3.2.3 Ini ial condi ions o wo-dimensional and h ee-dimensional simula ions
The speci ica ion o ini ial condi ions is a c ucial s ep o in silico me hods, whe e ele an
pa ame e s mus be de ined as he s a ing poin o he simula ions. Fo he phase- ield model
used in his wo k, he ini ial placemen and size dis ibu ion o he pa icles (=ice c ys als) a e
essen ial o simula e he subsequen coa sening p ocess. This sec ion desc ibes he me hod
used o gene a e he dis ibu ion and posi ioning amewo k o he pa icula e phase o all
wo-dimensional and h ee-dimensional phase- ield simula ions.
Ini ially, a basic s uc u e o he posi ioning o indi idual pa icles was es ablished. Fo wo-
dimensional simula ions, wo simple geome ic a angemen s we e conside ed: a squa e and
a iangula la ice, as depic ed in Figu e 3.29. These con igu a ions ep esen examples o
close-packing ha maximize he numbe o equal-sized pa icles in a gi en dis ibu ion. I is
impo an o no e ha he choice o he basic geome ic s uc u e is no in luenced by physical
conside a ions bu is ins ead chosen solely o es ablish a s a ing poin o c ea ing he ini ial
condi ions. The choice o close-packing s uc u e a ec s he dimensions and pa icle a ea
ac ions. The iangula la ice esul s in a y-coo dina e leng h o 𝑁𝑦= √3/2∙𝑁𝑥, due o
he posi ioning o he pa icle cen es. This s uc u e also achie es a highe a ea ac ion o
he pa icula e phase, as i u ilises he spaces be ween pa icles mo e e icien ly. The a ea
ac ion o a squa e close-packed sys em is app oxima ely 0.79, while o a iangula close-
packed sys em i is abou 0.91 [155].
Resul s and discussion
87
Figu e 3.29: Illus a ion o he close-packing o ci cles in squa e (A) and iangula (B) la ices.
Bo h a angemen s exhibi an equidis an dis ibu ion ega ding he cen e o he pa icles.
To ei e a e, olume and a ea ac ions a e used in e changeably o wo-dimensional
simula ions, as hey a e assumed o be iden ical in he expe imen al se up due o he low laye
hickness and cylind ical pa icle shape. This assump ion would be in alid i he pa icles had
oo much cu a u e in he e ical heigh axis. The olume ac ions o he pa icula e phase
du ing annealing we e calcula ed in Chap e 3.2.3.2 and amoun ed o an a e age o 0.793 and
0.828 o annealing empe a u es o -4 °C and -6 °C, espec i ely. The e o e, he iangula
la ice was chosen o he wo-dimensional simula ions in his wo k, as i is easie o educe
he a ea ac ion o a close packing by simply educing he pa icle size han by inc easing he
a ea ac ion o he squa e la ice. The exac se ing o he a ea ac ion in wo-dimensional
simula ions and he olume ac ion in h ee-dimensional simula ions will be discussed in he
nex chap e .
Fo he ini ial condi ions o he h ee-dimensional simula ions, he basic s uc u e o pa icle
dis ibu ion needed o be es ablished as well. When ex ended in o h ee dimensions, he wo-
dimensional iangula s uc u e ansi ions in o a e ahed al dis ibu ion o pa icles. Two
di e en close-packings based on a e ahed al building block a e possible, as depic ed in
Figu e 3.30. Bo h close-packings in h ee-dimensional ha e simila packing ac ions o
app oxima ely 0.74 [156]. Consequen ly, he choice is he e o e no based on physical
conside a ions o olume ac ions, bu a he on ease o implemen a ion. The hexagonal
close-packing has a mo e epe i i e posi ioning and is he e o e sligh ly easie o implemen .
Analogous o he educ ion o he y-coo dina e in he iangula wo-dimensional case, he z-
Resul s and discussion
88
coo dina e also changes in he hexagonal h ee-dimensional case o 𝑁𝑧=√6/3∙𝑁𝑥. In
summa y, he pa icle posi ioning was chosen o be iangula o wo-dimensional
simula ions and hexagonal o h ee-dimensional simula ions o simpli y he se up and
implemen a ion p ocess.
Figu e 3.30: A angemen o hexagonal and cubic close-packings in di e en laye s. In he
hexagonal sys em, he pa icles in he hi d laye a e posi ioned exac ly he same as in he i s
(blue), while in he cubic sys em he posi ion is sligh ly shi ed (g een).
Nex , he g id size o he simula ion domain and he ini ial numbe o pa icles we e
es ablished based on compu a ional easibili y. The scaling o he o al numbe o g id poin s
in such a sys em is depic ed in Figu e 3.31.
Figu e 3.31: Scaling o he simula ion g id size (Nx · Ny · Nz) wi h he hexagonal posi ioning o
pa icles. The e ahed al basic s uc u e esul s in 𝑁𝑦= √3/2∙𝑁𝑥 and 𝑁𝑧= √6/3∙𝑁𝑥.
Resul s and discussion
89
When de e mining he domain size, i was c ucial o manage he numbe o g id poin s o
p e en un easonably long compu a ion imes. This conside a ion is especially impo an o
h ee-dimensional simula ions, whe e he numbe o g id poin s subs an ially inc eases due
o he addi ion o he z-coo dina e. Consequen ly, he o al size o he h ee-dimensional
domain was used o de ine he simula ion domain. The base g id size was chosen based on
expe ience wi h an uppe limi o app oxima ely 10 million g id poin s (gp). Acco ding o
Figu e 3.31, a base g id size o Nx = 240 gp was chosen, esul ing in a o al o jus unde
9.8 million g id poin s, wi h Ny = 208 gp and Nz = 196 gp.
The nex s age o he p ocess in ol ed he selec ion o he numbe o pa icles o be used o
he s a ing poin o he simula ion. In he case o wo-dimensional simula ions, a o al o
n pa icles we e placed in each coo dina e, esul ing in a o al o n² pa icles. In he h ee-
dimensional se up wi h hexagonal posi ioning, a o al o 4n³ pa icles we e used. As wi h he
g id size, an issue may a ise when he numbe o pa icles becomes oo la ge due o he need
o inc ease he numbe o o de pa ame e s in he mul iphase- ield app oach. To ei e a e,
he o de pa ame e in he mul iphase- ield model is he equi alen o he c ys allog aphic
o ien a ion o ice c ys als. This app oach allows a de ined numbe o pa icles o be compu ed
in a single i e a ion, p o ided ha he pa icles in he same o de pa ame e a e no in close
p oximi y o each o he . I pa icles o he same o de pa ame e come in o con ac wi h each
o he , unin en ional in e ac ions such as usion o neighbou ing pa icles will occu .
The e o e, i is impo an ha he ini ial pa icles a e dis ibu ed o e a su icien numbe o
o de pa ame e s in o de o p e en con ac - o-con ac in e ac ions be ween pa icles wi hin
he same o de pa ame e . The ad an age o placing mul iple pa icles in o he same o de
pa ame e is ha his educes he compu a ional ime by a ac o equal o he a e age numbe
o pa icles pe o de pa ame e . Fo example, placing 10 pa icles pe o de pa ame e
educes he compu a ional ime by app oxima ely 10- old. The e o e, he choice o how many
pa icles a e placed in each o de pa ame e is a balance be ween p e en ing unwan ed
in e ac ions while dec easing compu a ional e o .
Based on expe ience, n was chosen o be 6 o h ee-dimensional simula ions, esul ing in
864 ini ial pa icles. All pa icles we e dis ibu ed o e 86 o de pa ame e s, wi h an a e age
o 10 pa icles pe o de pa ame e . This means ha a maximum o 86 i e a ions had o be
calcula ed o a single ime s ep ins ead o 864. The pa icles we e gene a ed wi h a la ge
Resul s and discussion
90
spacing wi hin he same o de pa ame e o a oid unwan ed in e ac ions a e sligh
coa sening. In he case o wo-dimensional simula ions, n was inc eased o 10 o ha e
100 ini ial pa icles in o de o obse e coa sening o e a la ge numbe o pa icles. Due o
he sho compu a ional imes o wo-dimensional simula ions, 100 o de pa ame e s we e
used, i.e., each pa icle was placed in i s own o de pa ame e .
To ini ia e coa sening in he phase- ield simula ion, i egula i ies in he ini ial pa icle size,
shape, o posi ion had o be in oduced, as will be explained below. Figu e 3.32 illus a es he
p ocess o cons uc ing he ini ial condi ion o he wo- and h ee-dimensional simula ions.
Figu e 3.32: Gene a ion o he ini ial condi ions wi h a iangula base s uc u e o wo-
dimensional simula ions (uppe ow) and a hexagonal base s uc u e o h ee-dimensional
simula ions (lowe ow).
A sys em wi h equidis an pa icles o simila size would no coa sen because he in e acial
ene gy o cu a u e o all pa icle su aces would be iden ical, hus lacking a d i ing o ce o
change. In oducing i egula i ies leads o de ia ions om ideal packing, a ec ing he
expec ed a ea and olume ac ions. These de ia ions we e, howe e , a e o no consequence

Resul s and discussion
91
as he na u al packing ac ions o iangula and hexagonal close-packings di e ed om he
calcula ed ac ions. This means ha a sepa a e adjus men o he a ea and olume ac ions
ega dless o ini ial i egula i ies needed o be pe o med anyway. To in oduce i egula i y,
he pa icle size was i s changed om he denses packing, whe e adjacen pa icles a e in
con ac , o a size dis ibu ion. A sligh andomisa ion o pa icle posi ioning was hen
in oduced, mode a ely a ying he dis ances be ween pa icles while main aining he basic
iangula o hexagonal posi ioning.
The choice o Nx, Ny and Nz is based on he compu a ional e o , as he g id size inc eases
exponen ially as shown in Figu e 3.31. On he o he hand, he dis ances be ween wo g id
poin s dx, dy and dz in he phase ield simula ions can be chosen mo e lexibly as hey do no
a ec he compu a ional e o . Fo he sake o simplici y, only dx will be e e ed o as he
dis ance be ween wo g id poin s, as he same alue is usually used o dx, dy and dz in a
simula ion. Inc easing dx will esul in a la ge physical domain o he simula ion as he ac ual
edge leng h o he simula ion is calcula ed using Nx · dx. Howe e , se ing dx o a high alue
may esul in in e aces being co e ed by an insu icien numbe o g id poin s. This migh lead
o g id e ec s, whe e he angle o an in e ace a ec s i s in e acial ene gy, causing nume ical
p oblems du ing he simula ion. An example o his is shown in Figu e 3.33.
An inc ease in he dis ance be ween g id poin s dx was accompanied by de o ma ion o he
pa icles when hey we e il ed (dx = 1.2) o e en loss o cu a u e when dx became oo la ge
(dx = 1.5). As dx inc eased, so did he physical simula ion domain, bu he in e aces also
became na owe and a a c i ical poin i was no longe possible o accu a ely ep esen
inclina ions a he in e aces. The choice o dx was he e o e a ade-o , as lowe dx equi ed
highe Nx o cap u e he same physical space, while highe dx educed accu acy and e en led
o e o s. A e ca e ul conside a ion, a alue o dx = 0.9 was chosen o all subsequen
simula ions, as his p o ided he la ges physical domain size wi hou no iceable g id e ec s.
Addi ional es s we e ca ied ou o e i y he phase- ield simula ion implemen a ion and
pa ame e s. These es s included e i ying he co ec ep esen a ion o he Gibbs-Thomson
e ec and ensu ing ha he in e acial ene gy was independen o he ene gy ba ie
coe icien . The esul s o he o he es s o ensu e he s abili y o he simula ion a e
p esen ed in Appendix A.
Resul s and discussion
92
Figu e 3.33: G id e ec in a wo-dimensional phase- ield simula ion wi h wo pa icles
ho izon ally aligned (uppe ow) and sligh ly il ed (lowe ow). The il ed simula ions a e
o a ed o isualise he di e ence compa ed o he ho izon ally aligned simula ions.
In he nex s ep, he spa ial disc e isa ion was dimensionalised o gi e physical meaning o he
space in he simula ion domain. A con e sion ac o α was in oduced o con e he dis ance
be ween wo g id poin s in o a physical leng h. Fo example, a con e sion ac o o 1 µm/gp
would imply ha o Nx = 240 gp he x-coo dina e co esponds o 240 µm. The me hod o
de i ing α will be explained in he ollowing.
In his wo k, Nx = 240 gp and Np = 864 we e selec ed due o easibili y and compu a ional
cons ain s. Unde his con igu a ion, he pa icles each an a e age size (de ined he e as he
sphe e equi alen adius) o 14.1 gp ± 1.7 gp when occupying he space in he simula ion
domain wi h olume ac ions o he pa icula e phase o 0.915. This alue ep esen s he
a e age ice c ys al olume ac ion o maximum eeze-concen a ed 10 % (w/w) suc ose and
ehalose solu ions, as p e iously de e mined in Chap e 3.2.2.2.2. In Chap e 1.5, published
da a on po e sizes in non-annealed samples we e p esen ed. These po e sizes we e he
p oduc o con en ional eezing and d ying and we e con e ed o ci cle equi alen adii o
Resul s and discussion
93
10.0 µm o 30.0 µm [83], 12.5 µm o 25.0 µm [98], and 11.9 µm ± 5.6 µm [71]. A wide ange
o po e sizes can be obse ed wi hin he indi idual publica ions, which can be a ibu ed o
a ious ac o s such as a he e ogeneous dis ibu ion o po e sizes, di e ences in po e sizes
depending on he axial posi ion in he ial [98], and measu emen unce ain ies associa ed
wi h he me hods used. Un o una ely, his means ha he e is no uni o m alue o
expe imen ally de e mined po e sizes ha can be used as a s a ing poin o he phase- ield
simula ions. The e o e, a ange o ini ial po e sizes has also been conside ed in his wo k.
Due o he mic oscopic scale o he simula ion domain, simula ing a wide a ie y o ini ial
pa icle sizes wi hin a single simula ion is imp ac ical. The e o e, sepa a e simula ions we e
un o di e en ini ial pa icle sizes a he s a o annealing. A his poin , wo se up op ions
we e conside ed:
 Adap ing he g id size: The i s op ion was o de i e he con e sion ac o α and adjus
Nx (and consequen ly Ny and Nz) o each ini ial pa icle size. Howe e , his app oach
could esul in a simula ion domain ha is ei he excessi ely la ge o so small ha he
in e acial wid h would become limi ing, i.e., mos o he simula ion domain would be
occupied by in e ace alone i Nx becomes oo small as he in e acial wid h is mo e o
less ixed.
 Using di e en con e sion ac o s: The al e na i e was o a y he ini ial condi ions
using di e en con e sion ac o s wi hou al e ing he h ee-dimensional simula ion
domain. This mean main aining he same numbe o ini ial pa icles and g id poin s
ac oss simula ions bu adjus ing hei physical meaning h ough he con e sion ac o .
In his wo k, he second op ion was chosen in o de o a oid po en ial excessi e
compu a ional imes du ing he execu ion o he simula ion. The con e sion ac o was
de ined as:
𝛼
=
𝑟

𝑟

,
(Eq. 20)
whe e exp is he expe imen ally de e mined ci cle equi alen adius, and sim is he sphe e
equi alen adius om h ee-dimensional simula ions.
Th ee di e en ini ial pa icle sizes o 10.0 µm, 12.5 µm, and 15.0 µm we e selec ed, based on
he expe imen al da a, and con e sion ac o s we e calcula ed o align he a e age pa icle
Resul s and discussion
94
size in he h ee-dimensional simula ion domain (14.1 gp ± 1.7 gp) wi h hese physical sizes.
Fo example, he con e sion ac o was calcula ed ha would ansla e he ini ial physical
pa icle size o 10.0 µm o he simula ion pa icle size o 14.1 gp, in his case α = 0.68 µm/gp.
These con e sion ac o s we e hen used o calcula e he base g id size o wo-dimensional
phase- ield simula ions, ensu ing alignmen wi h FDM expe imen al da a. Fo a con e sion
ac o o α = 0.68 µm/gp, he pa icle size a he beginning o he simula ion should be 9.2 gp.
This is based on he expe imen al inding ha pa icles had an a e age adius o 6.2 µm a he
ea lies quan i iable ime poin o 10 minu es a e annealing began. In o de o place
100 pa icles in he simula ion domain, each wi h an a e age adius o 9.2 gp, he base g id
size was de e mined o be Nx = 197.
In summa y, his means ha he con e sion ac o s we e used o de e mine he size o he
wo-dimensional simula ion domains ha could ep esen a i ely desc ibe 100 pa icles om
FDM expe imen s. Howe e , he a ious con e sion ac o s hemsel es we e p e iously
calcula ed by compa ing he es ablished h ee-dimensional phase- ield simula ion and
li e a u e da a on ice c ys al sizes/po e sizes. This s a egy allowed o adap a ions only in he
wo-dimensional simula ions, while he leng h scales o he mo e esou ce-in ensi e h ee-
dimensional simula ions we e simply adjus ed by he co esponding con e sion ac o s. A
summa y o all da a equi ed o calcula e he con e sion ac o s and Nx o wo-dimensional
simula ions is p o ided in Table 3.5.
Table 3.5: De e mina ion o con e sion ac o s (α) and base g id size o wo-dimensional
simula ions (Nx2D) o he expe imen ally de e mined ci cle equi alen adius o pa icles om
eeze-d ying mic oscopy a e 10 minu es o annealing a -6 °C ( exp), he ini ial sphe e
equi alen adius o pa icles in h ee-dimensional ( sim,3D), and he ini ial ci cle equi alen
adius o wo-dimensional simula ions ( sim,2D).
exp
[µm]
sim,3D
[µm]
sim,2D
[gp]
α
[µm/gp]
Nx2D
[gp]
6.2
10.0 9.2 0.68 197
12.5 7.4 0.85 157
15.0 6.2 1.00 134
Resul s and discussion
101
Table 3.6: Pa ame e s o he calib a ion o he wo-dimensional and h ee-dimensional
phase- ield model. The polynomial unc ion o he ini ial supe sa u a ion o he ma ix phase
(𝑐
) was de e mined dependen on he a ea and olume ac ion o he pa icula e phase
(𝜑) and he base g id size o he simula ion domain (Nx).
𝜑

[-]
Dimensions
[-]
Nx
[gp] Polynomial unc ion R²
[-]
𝑐


[-]
0.793
2D
197 y = 14.133x³ - 22.555x² + 12.313x – 1.3377 1 0.338
157 y = 9.9716x³ - 16.137x² + 9.0861x – 0.8157 1 0.336
134 y = 5.4224x³ - 8.8472x² + 5.226x – 0.1402 1 0.313
3D 240 y = 7.5809x3 – 13.627x2 + 8.5475x – 0.8864 1 0.369
0.828
2D
197 y = 12.225x³ - 21.028x² + 12.45x – 1.549 1 0.386
157 y = 8.1264x³ - 14.481x² + 9.0219x – 0.9637 1 0.385
134 y = 5.073x³ - 9.2468x² + 6.0656x – 0.4106 1 0.372
3D 240 y = -1.7858x3 – 0.9729x2 + 3.3107x – 0.2685 1 0.426
3.2.4.2 Coa sening pa ame e s o he wo-dimensional phase- ield model
In he ollowing, he simula ion pa ame e s o he wo-dimensional phase- ield simula ions
we e de e mined which co espond o he expe imen al ec ys allisa ion a es om
Chap e 3.2.2.2.3. The objec i e was o adjus he mobili y coe icien M and elaxa ion
coe icien L simul aneously o ma ch he empo al e olu ion o pa icle sizes be ween
expe imen s and simula ions. An example o coa sening in a wo-dimensional phase- ield
simula ion is depic ed in Figu e 3.38.
A he s a o he simula ion, he a ea ac ion o he pa icula e phase inc eased apidly un il
i pla eaued a sim = 300 d , indica ing ha he phases had eached hei espec i e
equilib ium composi ions. The heigh o he pla eau was adjus ed by he ini ial
supe sa u a ion o he ma ix phase as discussed in Chap e 3.2.4.1. In o de o educe he
o al numbe o pa icles in he sys em, comple e dissolu ion o indi idual pa icles was
equi ed. Howe e , a he s a o he simula ion he supe sa u a ion o he ma ix phase
s abilised he small pa icles un il he equilib ium composi ions we e eached. The e o e, as

Resul s and discussion
102
long as he olume lux in o small pa icles due o supe sa u a ion was g ea e han ha ou
o hem due o Os wald ipening, he numbe o ice c ys als did no change. A e equilib ium
composi ions we e eached, he s anda d de ia ion o he a e age pa icle a ea s eadily
inc eased, indica ing a b oadening o he pa icle size dis ibu ion. Consequen ly, some
pa icles ell below he c i ical pa icle size and dissol ed, leading o an inc ease in local
composi ion wi hin he ma ix phase and sligh supe sa u a ion. In esponse, neighbou ing
pa icles began o g ow o educe he supe sa u a ion.
Figu e 3.38: Tempo al e olu ion o numbe o pa icles (○), a ea ac ion o he pa icula e
phase 𝜑 (Δ), and he a e age pa icle a ea (x) wi h s anda d de ia ion (e o ba s) in a wo-
dimensional phase- ield simula ion wi h M = 1, L = 1, 𝑐
 = 0.539, and Nx = 197 gp.
A e a lag phase o sim = 1000 d , he pa icle numbe s a ed o dec ease wi h a hype bolic
cu e p o ile. Expe imen al da a om eeze-d ying mic oscopy do no show such a lag phase
due o he p esence o smalle pa icles a he ime o obse a ion ( a = 10 min), which
immedia ely all below he c i ical adius and dissol e when he sample empe a u e is
inc eased. This in u n immedia ely educes he numbe o ice c ys als om he s a o he
obse a ion. F om sim = 3000 d , he dec ease in he numbe o pa icles slowed down and
app oached an appa en pla eau. The mean pa icle a ea and i s s anda d de ia ion inc eased
s eadily om sim = 1000 d , indica ing coa sening. In addi ion, based on he Gibbs-Thomson
e ec (see Appendix A), a sligh inc ease in he a ea ac ion o he pa icula e phase was
obse ed. The simula ion esul s a e isualised in Figu e 3.39.
Resul s and discussion
103
Figu e 3.39: Visualisa ion o he composi ion ield in a wo-dimensional phase- ield simula ion
wi h M = 1, L = 1, 𝑐
 = 0.539, and Nx = 197 gp.
The simula ion esul s we e isualised s a ing om sim = 300 d , when he ma ix phase
eached i s equilib ium composi ion. Be ween sim = 300 d and sim = 1000 d , he pa icula e
phase unde wen a edis ibu ion esul ing in a b oade pa icle size dis ibu ion wi h he
p esence o bo h smalle and la ge pa icles. F om sim = 1000 d o sim = 5000 d , he numbe
o pa icles isibly dec eased, esul ing in a coa sening o he pa icula e phase.
The nex s ep was o assign physical meaning o he empo al disc e isa ion o he simula ion.
This means ha he ime be ween wo i e a ions in he simula ion (d ) had o be de ined in
SI uni s. Howe e , d i sel is a pa ame e used o ensu e he s abili y o he simula ion and o
minimise inaccu acies and e o s. I d is se oo high, he simula ion may no be able o
cap u e he empo al e olu ion co ec ly as excessi e change occu s be ween i e a ions. I d
is se oo low, mo e i e a ions will ha e o be calcula ed o simula e he same phenomenon,
inc easing compu a ional ime. Fo his wo k, a d o 0.05 has p o en op imal, as highe alues
Resul s and discussion
104
led o c ashes and lowe alues o p olonged simula ion imes. The empo al scale, i.e., wha
each simula ion i e a ion means in physical ime, was hen adjus ed based on he coa sening
p o ile in Figu e 3.38, as will be explained in he ollowing.
Fo M = 1 and L = 1, he es ima ed lag phase las ed abou 1000 i e a ions and he numbe o
pa icles emained ela i ely cons an a e 4000 i e a ions. In his wo k, M and L we e chosen
while he empo al con e sion ac o τ was de e mined. Technically, he empo al con e sion
ac o could ha e been chosen i s and hen he app op ia e M and L de e mined.
Ma hema ically, he esul s would ha e been iden ical. As he expe imen al da a co e ed
360 minu es o annealing and he ac ual coa sening occu ed wi hin simula ions wi h M = 1
and L = 1 o abou 3000 i e a ions, each simula ion i e a ion was se o ep esen 0.1 minu es
o 6 seconds. The empo al con e sion ac o was calcula ed wi h:
𝜏
=
6
𝑠
𝑑𝑡
,
(Eq. 23)
and esul ed in τ = 120 s o all simula ions.
As p e iously men ioned, he ec ys allisa ion a e in he phase- ield simula ions was
manipula ed by adjus ing he alues o M and L simul aneously. This app oach allowed he
ec ys allisa ion a e o be modi ied while keeping he in e acial ene gy o he sys em
cons an . Al e ing he a io o M and L would ha e a ec ed he con ibu ions o di usion,
dissolu ion and edeposi ion o he o e all coa sening, which is ou side he scope o his wo k.
In o de o in es iga e he ela ionship be ween M, L and he ec ys allisa ion a e (including
he empo al con e sion ac o ), a ious simula ions we e ca ied ou wi h inc emen ally
changing M and L. The esul s a e illus a ed in Figu e 3.40. I should be no ed ha he
simula ion da a was no used un il coa sening s a ed o occu . This means ha he s a ing
poin a ied be ween simula ions as highe alues o M and L would esul in a sho e lag
phase.
Simila o he expe imen al da a, he simula ion esul s displayed a linea inc ease in he cube
o he ci cle equi alen diame e as coa sening commenced. The cu es may appea almos
s epwise. Howe e , his is a misconcep ion. This appea ance can be explained by he ac ha
when pa icles dissol e om one i e a ion o he nex , he o al numbe o pa icles changes,
leading o e a ic luc ua ions in he calcula ed a e age cube o ci cle equi alen adius.
Resul s and discussion
105
Figu e 3.40: Impac o simula ion pa ame e s M and L on he coa sening beha iou o a wo-
dimensional phase- ield simula ion wi h 𝑐
 = 0.539 and Nx = 197 gp.
The slope o he eg ession inc eased wi h highe se ings o M and L, indica ing he success ul
inc ease o he ec ys allisa ion phenomenon dependen on M and L. Following hese
obse a ions, he condi ions speci ied in Table 3.6 we e used o conduc wo-dimensional
phase- ield simula ions. These simula ions we e aimed a de e mining ec ys allisa ion a es
based on he slopes o he cube o he ci cle equi alen adii. The esul s o hese simula ions
a e depic ed in Figu e 3.41.
Figu e 3.41: Impac o M and L on he ec ys allisa ion a e in a wo-dimensional phase- ield
simula ion.
Resul s and discussion
106
The ec ys allisa ion a e exhibi ed a posi i e linea co ela ion wi h he simula ion
pa ame e s M and L, as expec ed. This is because an inc ease in bo h M and L simul aneously
equa es ma hema ically o an inc ease in d . This ela ionship can be simpli ied as ollows In
he simula ions, i does no ma e whe he a p ocess akes, o example, 10 minu es wi h a
da a poin collec ed e e y minu e, o i he p ocess uns wice as as wi h a da a poin
collec ed e e y hal minu e. Ul ima ely, he amoun o wo k done, and he ou pu da a emain
iden ical.
I was also obse ed ha he ec ys allisa ion a e scales as e wi h M and L as he g id size
dec eases. The e ec o M and L on he ec ys allisa ion a e also inc eased wi h he
equilib ium concen a ion o he ma ix phase. Using he eg ession lines om Figu e 3.41,
he simula ion pa ame e s M and L co esponding o he expe imen al ec ys allisa ion a es
we e de e mined and a e shown in Table 3.7.
Table 3.7: De e mina ion o simula ion pa ame e s mobili y coe icien M and elaxa ion
coe icien L dependen on expe imen al ec ys allisa ion a e and base g id size (Nx).
Rec ys allisa ion a e
[µm³/min]
Nx
[gp] Polynomial unc ion R²
[-]
M & L
[-]
56.80
197 y = 7.2702x + 0.5562 0.9992 7.74
157 y = 12.822x + 1.4688 0.9994 4.32
134 y = 27.29x – 1.0529 0.9999 2.12
26.13
197 y = 6.0147x + 1.5804 0.9978 4.08
157 y = 10.318x + 1.9271 0.9988 2.35
134 y = 19.78x + 1.0279 0.9988 1.27
Nex , he simula ion pa ame e s M and L, as de e mined in his chap e , we e applied o
h ee-dimensional phase- ield simula ions. These simula ions aimed o p edic he coa sening
beha iou o ozen 10 % (w/w) suc ose and ehalose solu ions a annealing empe a u es o
-4 °C and -6 °C wi h ini ial pa icle adii o 10 µm, 12.5 µm, and 15 µm.

Resul s and discussion
107
3.2.5 Ad ancemen o he phase- ield simula ion o h ee-dimensional
Fo he implemen a ion o he h ee-dimensional phase- ield simula ion, all ields (e.g.,
composi ion, o de pa ame e s, e c.) we e ex ended o include he z-coo dina e. Spa ial
ope a o s (∇) used o calcula ing g adien s and di e gences o hese ields we e acco dingly
modi ied o accommoda e he addi ional dimension. The pa icle posi ion and size
dis ibu ions, as es ablished in Chap e 3.2.3, se ed as he ini ial condi ions. Volume ac ions
we e also adjus ed based on he ini ial supe sa u a ions o he ma ix phase, as discussed in
Chap e 3.2.4.1. Wi h he con e sion ac o s p e iously de e mined, he ollowing simula ions
a e all dimensionalised in ega d o space and ime, i.e., olumes a e exp essed in μm³ and
ime in minu es. Figu e 3.42 illus a es an example o he esul s om a h ee-dimensional
phase- ield simula ion.
Figu e 3.42: Tempo al e olu ion o numbe o pa icles (○), olume ac ion o pa icula e
phase 𝜑 (Δ), and he a e age pa icle olume (x) wi h s anda d de ia ion (e o ba s) in he
h ee-dimensional phase- ield simula ion wi h M = 2.35, L = 2.35, 𝑐
 = 0.539, and Nx = 240 gp.
Du ing he annealing simula ion, he a e age pa icle size con inuously inc eased, and i did
no each a pla eau e en a e sim = 360 min. Th oughou he coa sening p ocess, he s anda d
de ia ion o he a e age pa icle size also inc eased, indica ing an inc easing deg ee o
he e ogenei y in pa icle sizes as annealing p og essed. Simila o he wo-dimensional
simula ions a lag phase was p esen , whe e he olume ac ion o he pa icula e phase
showed a apid inc ease up o sim = 10 min while he numbe o pa icles emained ela i ely
Resul s and discussion
108
cons an . Du ing his ini ial s age, he olume ac ion o he pa icula e phase inc eased due
o supe sa u a ion in he ma ix phase, causing all pa icles o g ow un il eaching he desi ed
olume ac ions de e mined expe imen ally in Chap e 3.2.2.2.2. A s eady inc ease in he
olume ac ion o he pa icles was obse ed be ween sim = 10 minu es and sim = 120 minu es
as he simula ion p og essed, pa icula ly when he numbe o pa icles dec eased apidly. As
wi h he wo-dimensional simula ions, his phenomenon is based on he Gibbs-Thomson
e ec and is ela ed o he cu a u e o he pa icles, i.e., he smalle a pa icle, he g ea e
i s cu a u e and he g ea e he a ia ion in composi ion wi hin he pa icle. The e o e, he
olume ac ion o he pa icle inc eases wi h coa sening. A desc ip ion o his phenomenon
can be ound in Appendix A.
The lag phase obse ed in he h ee-dimensional phase- ield simula ions was no ably sho e
compa ed o he wo-dimensional simula ions. This di e ence is p ima ily in luenced by wo
ac o s. Fi s , he addi ion o an ex a plane o cu a u e in h ee dimensions accele a es
coa sening, which e ec i ely educes he lag phase. Secondly, he c i ical pa icle size a which
dissolu ion begins di e s be ween wo-dimensional and h ee-dimensional simula ions.
Speci ically, he c i ical adius o dissolu ion was de e mined o be 6 g id poin s in wo-
dimensional simula ions ( ep esen ing he adius o a ci cle) and 7 g id poin s in h ee-
dimensional simula ions ( ep esen ing he adius o a sphe e), as de ailed in Appendix A.
Consequen ly, pa icles in h ee-dimensional simula ions dissol e ea lie and coa sen as e ,
esul ing in a sho ened lag phase.
The esul s we e also isualised and a e shown in Figu e 3.43. The s eady coa sening o he
pa icula e phase was clea ly e iden om he h ee-dimensional isualisa ions, which also
showed an inc ease in pa icle size he e ogenei y. Mo eo e , he isualisa ions unde sco ed
ha he simula ion canno con inue inde ini ely. E en ually, pa icles could g ow la ge enough
o span om one bounda y o he opposi e bounda y. This poses a p oblem as pe iodic
bounda y condi ions we e employed in all simula ions. Unde hese condi ions, he le bo de
o he simula ion domain e ec i ely con inues om he igh bo de , allowing he simula ion
domain o be concep ually s acked inde ini ely om le o igh , and simila ly om bo om o
op and on o back, while p ese ing he con inui y o he in e nal mic os uc u e.
Resul s and discussion
109
Figu e 3.43: Visualisa ion o he composi ion ield du ing coa sening in a h ee-dimensional
phase- ield simula ion wi h M = 2.35, L = 2.35, 𝑐
 = 0.539, and Nx = 240 gp.
This also means ha a pa icle g owing om any bounda y o he opposi e side could
po en ially come in o con ac wi h i sel , c ea ing an un ealis ic scena io i i becomes
su icien ly la ge. This issue is illus a ed in Figu e 3.44. To add ess his, all subsequen
e alua ions we e ca e ully moni o ed, and simula ions we e e mina ed i his issue a ose,
ensu ing he in eg i y and ealism o he simula ion esul s.
Resul s and discussion
110
Figu e 3.44: Illus a ion o he pe iodici y o he simula ion domain and he associa ed
p oblems. The o iginal simula ion domain (A, ed squa e) can be eplica ed and placed a each
bo de due o i s pe iodic bounda ies. Howe e , i he pa icles become oo la ge, he e is a
isk ha a pa icle will come in o con ac wi h i sel (B).
3.2.6 Final adjus men o pa icle olume ac ions
In lyophilisa ion, he sample empe a u e mus be educed again a e annealing o enable
he ma ix phase o each i s maximum eeze-concen a ed s a e. This s ep p omp s wa e
molecules o p ecipi a e in o ice c ys als, al e ing he wa e -solu e a io in he ma ix phase,
and consequen ly a ec ing i s olume ac ion in he lyophilisa e. I is impo an o implemen
his in phase- ield simula ions o accu a ely model he pa icle sizes ha co espond o he
po es in he lyophilisa e.
The adjus men o he olume ac ions in he h ee-dimensional phase- ield simula ions was
achie ed by es a ing he simula ion a e he desi ed annealing s ep, bu wi h an adjus ed
ma ix equilib ium composi ion. Elegan ly, his is also wha happens physically in a eal
solu ion. As he empe a u e dec eases, he solubili y o he wa e in he amo phous phase
changes and so does i s equilib ium concen a ion. Figu e 3.45 shows how he pa icle olume
ac ion inc eases as a esul o supe sa u a ion un il i eaches he a ge olume ac ion o
0.913, as calcula ed in Chap e 3.2.2.2.2.
Resul s and discussion
117
Figu e 3.50: Tempo al e olu ion o he a e age ice c ys al size in he h ee-dimensional phase-
ield simula ion wi h di e en ini ial pa icle sizes ( sim,3D). The s anda d de ia ion was omi ed
o be e cla i y.
I should be no ed ha he s anda d de ia ion became e y high o e he cou se o he
simula ions and has he e o e been omi ed om he esul s o cla i y. This phenomenon was
obse ed p e iously in Figu e 3.42 and can be explained by he ac ha du ing coa sening,
la ge pa icles end o con inue o inc ease in size while he es o he pa icles con inue o
sh ink un il hey each he c i ical pa icle adius and dissol e. The e o e, e y la ge and e y
small pa icles exis a he same ime, which b oadens he pa icle size dis ibu ion and
inc eases he he e ogenei y in pa icle size. No ably, he indi idual da a se s do no ollow a
pe ec cu e, bu end o “jump” be ween annealing imes. This can be explained by he
comple e dissolu ion o pa icles du ing he p ocess. This dissolu ion changes he o al
numbe o pa icles, leading o e a ic shi s in he a e age pa icle size calcula ions.
Nex , he sphe e equi alen adius was calcula ed wi h:
𝑟

=

3
∙
𝑉

4
∙
𝜋

,
(Eq. 25)
whe e Vp is he dimensionalised olume o an indi idual pa icle.

Resul s and discussion
118
I mus be men ioned ha he pa icles in his simula ion we e no pe ec ly sphe ical, and
he e o e he sphe e equi alen adius is jus an app oxima ion o he pa icle size. Howe e ,
ci cle o sphe e equi alen adii a e commonly used in he li e a u e ega ding po e sizes in
lyophilisa es, and acili a e he compa ison wi h o he wo ks. The esul s a e isualised in
Figu e 3.51.
Figu e 3.51: Tempo al e olu ion o he sphe e equi alen adius in he h ee-dimensional
phase- ield simula ion wi h di e en ini ial pa icle sizes ( sim,3D) and annealing empe a u es
(Ta). The s anda d de ia ion was omi ed o be e cla i y.
The con e sion o he ice c ys al olume o he sphe e equi alen adius also shows he
inc ease o e he annealing pe iod and he dependence o he esul ing pa icle adius on he
ini ial pa icle size. The inc ease is no comple ely linea , and he e may be se e al easons o
his. Fi s , changes in he mass ac ion in he ma ix phase due o he Gibbs-Thomson e ec
migh play a ole, as la ge pa icles exhibi lowe composi ional alues. This phenomenon
sligh ly in luences he coa sening beha iou , as he d i ing o ce o Os wald ipening is
dependen upon he composi ional di e ence be ween he pa icles and he ma ix phase.
Addi ionally, he in e ace may also con ibu e o he non-linea inc ease. Gi en ha he
in e ace main ains a ixed hickness, i s impac is mo e p onounced on smalle pa icles han
on la ge ones. Consequen ly, small pa icles migh appea la ge in e alua ions because he
Resul s and discussion
119
in e ace is included in hei measu emen . As p e iously demons a ed in Figu e 3.42, he
s anda d de ia ion o pa icle size inc eased conside ably as annealing p og esses. To u he
illus a e his, he equency o di e en pa icle sizes a a ious annealing imes is depic ed
in Figu e 3.52.
Figu e 3.52: Exempla y ice c ys al size dis ibu ions in he h ee-dimensional phase- ield
simula ion wi h Ta = -4 °C and sim,3D = 10 μm. The equency al e s be ween he plo s.
Resul s and discussion
120
The la ening o he cumula i e equency o pa icle olumes, no malised be ween 0 and 1,
clea ly indica es a shi owa ds la ge pa icle sizes. I is impo an o no e ha he
equencies depic ed in he plo s (y-axis) a e no consis en ac oss di e en plo s. These
esul s a e quali a i ely consis en ac oss all se s o simula ion pa ame e s and con i m ha
coa sening in such sys ems leads o inc eased he e ogenei y in pa icle sizes.
Fu he mo e, i can be obse ed, especially in Figu e 3.51, ha he a e age pa icle adius a
he beginning o he simula ion is be ween 10 μm and 15 μm. This indica es ha he s a egy
in ol ing he con e sion ac o , as de ailed in Chap e 3.2.3, was success ully implemen ed in
he h ee-dimensional simula ions and has yielded meaning ul esul s.
3.2.7.3 Conjunc ion a ea o ice c ys als
The nex s ep in ol ed he de e mina ion he conjunc ion a eas as de ined in Chap e 3.2.7.1
and hei size dis ibu ions, o mo e p ecisely a ea dis ibu ions. As an in e media e s ep, he
olume esul ing om he o e lap o he in e aces o wo adjacen ice c ys als was
de e mined, he ea e e e ed o as he o e lapping olume. The concep o o e lapping
olumes is illus a ed in Figu e 3.53 and explained below.
Figu e 3.53: Concep o o e lapping olume in a wo-dimensional domain. The space occupied
by he in e ace o bo h pa icles (blue and ed) o ms an a ea in wo-dimensional space
(g een) and a olume in h ee-dimensional space.
The o e lapping olume (o a ea) exis s because he in e ace o he pa icles exhibi a ce ain
wid h, which means ha some space in he simula ion domain can be occupied by wo (o
e en mo e) pa icle in e aces simul aneously. No ably, his alue depends on how he
in e acial wid h is e alua ed. I he in e acial wid h is de ined as he leng h be ween wo
Resul s and discussion
121
phases whe e he alues de ia e om he equilib ium composi ions, hen la ge wid hs o 6
o 8 g id poin s can be measu ed. Al e na i e me hods, such as in Appendix A, use he
in lec ion poin o he in e ace and de e mine lowe alues. Ne e heless, he hickness o
he in e ace, gi en ha g adien ene gy coe icien s a e no al e ed, emains cons an .
The o e lapping olumes appea as hin and pa ially cu ed discs, as shown in Figu e 3.54 o
h ee-dimensional simula ions. These shapes de elop because he in e aces be ween
adjacen pa icles de o m du ing pa icle g ow h and coa sening p ocesses. The o al
o e lapping olume be ween all pa icles is gi en by:
𝑉
,
=


𝜂

𝜂











,
(Eq. 26)
whe e Np is he o al numbe o pa icles, and η is he o de pa ame e .
Figu e 3.54: Th ee-dimensional phase- ield simula ion wi h an ini ial pa icle size o
sim,3D = 10 μm a e annealing a Ta = -6 °C o a = 360 min. Ice c ys als a e shown on he le ,
while he o e lapping olumes a e isualised on he igh .
The conjunc ion a ea ha co esponds o an o e lapping olume app oxima es abou hal o
i s su ace a e and can be de e mined wi h:
𝐴

=
1
2
𝐴

,
(Eq. 27)
whe e Ao is he su ace a ea o an o e lapping olume.
Resul s and discussion
122
The concep o conjunc ion a eas is no commonly add essed in he li e a u e conce ning he
mic os uc u e o lyophilisa es. The e o e, in his wo k, se e al pa ame e s we e ini ially
examined o iden i y po en ially ele an aspec s o conjunc ion a eas. Simila ly o how ice
c ys al p ope ies we e assessed, he numbe , a e age a ea, and o al a ea o conjunc ions
we e de e mined. An example o he empo al e olu ion o hese p ope ies is shown in
Figu e 3.55.
Figu e 3.55: Tempo al e olu ion o numbe o conjunc ions (○), a e age conjuncon a ea wi h
s anda d de ia ion (Δ), and o al conjunc ion a ea (x) du ing annealing in he h ee-
dimensional phase- ield simula ion wi h an ini ial pa icle size o sim,3D = 10 μm a e annealing
a Ta = -4 °C.
As annealing p og essed, he numbe o conjunc ions dec eased. This is ela ed o he ac
ha he numbe o ice c ys als also dec eased du ing coa sening, so ha ewe ice c ys als
we e in con ac . Meanwhile, an inc ease in he a e age conjunc ion a ea was obse ed, which
was also accompanied by an inc ease in i s s anda d de ia ion. This indica es ha , simila o
he ice c ys als, he sys em became mo e he e ogeneous in e ms o conjunc ion sizes.
Addi ionally, i was no ed ha he o al conjunc ion a ea dec eased o e he cou se o he
simula ion. This decline is consis en wi h he ypical beha iou o coa sening, whe e su ace
a eas wi hin he sys em a e minimized in a ou o la ge olumes. The ends in he numbe
o conjunc ions o all simula ion condi ions a e depic ed in Figu e 3.56.

Resul s and discussion
123
Figu e 3.56: Compa ison o a ious simula ion pa ame e s on he numbe o conjunc ions in
h ee-dimensional phase- ield simula ions.
The numbe o conjunc ions dec eased mo e apidly a highe annealing empe a u es, a
esul o as e coa sening which educes bo h he numbe o pa icles and hei con ac s.
Fu he mo e, smalle ini ial pa icle sizes esul ed in mo e conjunc ions due o he inc eased
equency o con ac among nume ous smalle pa icles. This does no imply a change in he
numbe o conjunc ions pe pa icle, bu a he indica es ha he o al numbe o
conjunc ions is dependen upon he o e all pa icle coun .
Nex , he a e age conjunc ion a ea o all simula ion condi ions was de e mined wi h:
𝐴
󰆹

=
𝛼

∑
𝐴

,






𝑁

,
(Eq. 28)
whe e α is he con e sion ac o , Ac is he espec i e conjunc ion a ea, and Na is he o al
numbe o conjunc ions.
This alue is pa icula ly in e es ing because i desc ibes how na ow, on a e age, he pa hs
o mass low o apou du ing sublima ion can become. The idea is ha he na owes pa h
ac s as a bo leneck and po en ially limi s he o al mass low. The da a illus a ing his
phenomenon is shown in Figu e 3.57.
Resul s and discussion
124
Figu e 3.57: Compa ison o he impac o a ious simula ion pa ame e s on he a e age
conjunc ion a ea in h ee-dimensional phase- ield simula ions.
The esul s demons a e a clea ela ionship be ween he a e age conjunc ion a ea and he
ini ial pa icle size, which is in ui i e as la ge pa icles a e likely o ha e la ge con ac a eas.
Addi ionally, he e appea s o be a empe a u e-dependen in luence on his ela ionship. This
can be explained by he ac ha mo e apid coa sening a highe empe a u es leads o he
o ma ion o la ge pa icles and, consequen ly, la ge conjunc ion a eas. This empe a u e
e ec is pa icula ly p onounced o pa icles wi h an ini ial adius o 15 μm, as e idenced by
he s eepe inc eases in conjunc ion a ea du ing annealing. No ably, he da a shows e a ic
luc ua ions, which a e due o he educ ion in he numbe o conjunc ions o e ime. These
educ ions a ec he calcula ion o he a e age conjunc ion a ea.
Finally, he o al conjunc ion a ea wi hin he sys em was examined. I is impo an o no e ha
he simula ions accoun ed o di e en o al olumes a e con e ing o physical leng h uni s,
meaning ha he dis ance be ween g id poin s a ies depending on he applied con e sion
ac o . The e o e, i was c ucial o calcula e he speci ic conjunc ion a ea o enable meaning ul
compa isons o he simula ion esul s.
Resul s and discussion
125
Fo his pu pose, he o al olume o a simula ion was calcula ed using:
𝑉

=
𝛼

∙
𝑁𝑥
∙
𝑁𝑦
∙
𝑁𝑧
,
(Eq. 29)
whe e α is he con e sion ac o , and Nx, Ny, and Nz a e he numbe o g id poin s in x, y, and
z di ec ion, espec i ely.
The uni con en ionally used in he scien i ic li e a u e o de e mine speci ic su ace a ea is
m²/g. This app oach is also applicable o simula ions, whe e he physical weigh o he
simula ion domain can be de e mined by calcula ing i s olume using Equa ion 29 and hen
applying he densi y o he ma ix phase. The physical mass o he ma ix phase in a simula ion
is gi en by:
𝑚

,

=
𝑉

∙

1
−
𝜑


∙
𝜌

,
(Eq. 30)
whe e Vsim is he physical olume o he simula ion domain, ϕp is he olume ac ion o he
pa icula e phase, and ρm is he densi y o he ma ix phase.
The alues o he de e mina ion o he physical mass o he ma ix phase a e summa ised in
Table 3.8. Since disaccha ides such as suc ose and ehalose ha e simila densi ies, only he
alues o suc ose we e employed in his analysis. Subsequen ly, he conjunc ion a ea was
no malised o he speci ic conjunc ion a ea, de ined as:
𝑆𝐶𝐴
=
𝛼

∑
𝐴

,






𝑚

,

,
(Eq. 31)
whe e α is he con e sion ac o , Ac is he espec i e conjunc ion a ea, Na is he o al numbe
o conjunc ions, and mm,sim is physical mass o he ma ix phase wi hin he simula ion domain.
The esul s a e depic ed in Figu e 3.58.
Table 3.8: Con e sion alues o he calcula ion o he physical mass o he ma ix phase in he
simula ion domain (mm,sim) om he con e sion ac o (α), he olume o he simula ion
domain (Vsim), he pa icle olume ac ion (φp), and he ma ix densi y (ρm).
α
[µm/gp]
Vsim
[μm³]
φp
[cm³/ cm³]
ρm
[g/cm³]
mm,sim
[g]
0.68 3.08 x 106 0.913 1.404[1] 3.28 x 10-7
0.85 6.01 x 106 0.913 1.404[1] 6.41 x 10-7
1.00 9.78 x 106 0.913 1.404[1] 1.04 x 10-6
1 Lescu e (1995)
Resul s and discussion
126
Figu e 3.58: Compa ison o he impac o a ious simula ion pa ame e s on he speci ic
conjunc ion a ea in h ee-dimensional phase- ield simula ions.
Gene ally, he speci ic conjunc ion a ea dec eased o e he cou se o annealing, which can be
a ionalised by conside ing he undamen al d i e o minimise su ace a eas du ing
coa sening. In e aces be ween phases con ibu e o he en halpy o he sys em and a e
consequen ly educed du ing elaxa ion p ocesses such as Os wald ipening. No ably, he
speci ic conjunc ion a ea dec eased mo e du ing annealing when he ini ial ice c ys als we e
la ge and when highe annealing empe a u es we e used.
3.2.7.4 Su ace a ea o he ma ix phase
In he nex s ep, he su ace a ea o he ma ix phase was de e mined. An exempla y
isualisa ion o he su ace a ea is shown in Figu e 3.59.
As p e iously men ioned, i is impo an o no e ha he simula ion domain co esponds o
di e en physical olumes depending on he se o simula ion pa ame e s. This means ha
analogous o he speci ic conjunc ion a ea, he su ace needed o be no malised as well o
ob ain compa able esul s be ween di e en simula ions. Elegan ly, he ma ix su ace a ea
can be de e mined using alues al eady acqui ed o pa icle su ace a ea and conjunc ion
a ea, as shown in he ollowing.
Resul s and discussion
133
In his hesis di e en ini ial pa icle sizes we e used in he simula ions e lec ing he b oad
spec um o alues ound in he exis ing li e a u e o non-annealed samples. The esul s o
he h ee-dimensional annealing simula ions in he con ex o a ozen disaccha ide solu ion
can be summa ised as ollows:
 Annealing leads o an inc ease in he a e age size o ice c ys als and b oadens hei
size dis ibu ion. The inc ease in ice c ys al size du ing annealing, which la e ansla es
in o he po e sizes in he d ied lyophilisa e, aligns wi h obse a ions ha la ge po es
can enhance he sublima ion a e du ing eeze-d ying.
 The speci ic su ace a ea o he ma ix phase dec eases o e he cou se o annealing.
This phenomenon migh play a ole in he di usi e and deso p ion p ocesses du ing
d ying. Fu he mo e, he educ ion in su ace a ea is bene icial as i could help
minimise p o ein agglome a ion by minimising he a ailable su aces o p o ein
p ecipi a ion, he eby enhancing he s abili y and quali y o he inal p oduc .
 Du ing annealing, he numbe o conjunc ions, which o m he na owes gaps in he
po ous mic os uc u e, dec eases. Howe e , he a e age size o hese conjunc ions
inc eases, indica ing a shi owa ds ewe bu la ge channels wi hin he
mic os uc u e. Quan i ying pa ame e s such as he conjunc ion a ea migh o e
be e explana ions why annealing e ec i ely educes p ima y d ying imes.
 The phenomena desc ibed abo e depend on bo h he annealing empe a u e and he
ini ial ice c ys al size. In gene al, highe annealing empe a u es accele a e hese
p ocesses, while la ge ini ial ice c ys al sizes educe he ime equi ed o each
pla eaus.
The indings om his esea ch o e p ac ical applica ions in he ield o eeze-d ying,
speci ically h ough he use o phase- ield simula ions. By simula ing di e en annealing imes
and empe a u es, i is possible o de e mine when mic os uc u al changes each pla eau
phases. This app oach can help op imise eeze-d ying p ocess condi ions, ensu ing ha he
bene i s o annealing (as men ioned abo e) a e achie ed e icien ly. The goal is o maximise
he posi i e impac s o annealing while minimising he du a ion o he annealing s ep, making
he p ocess mo e ime-e ec i e and cos -e icien . This me hodology allows o a mo e
p ecise con ol o e he lyophilisa ion p ocess, po en ially leading o be e quali y p oduc s
wi h educed p ocessing imes.

Fu u e applica ions
134
4 Fu u e applica ions
In he ollowing chap e some ideas o u u e esea ch a e explo ed, which can be based on
he esul s o his wo k. These include bo h expe imen al and simula ion opics.
4.1 Reo ganisa ion kine ics du ing ecalescence and c ys allisa ion
In Chap e 3.1.4, he expe imen al indings showed ha he ecalescence and c ys allisa ion
phase signi ican ly in luences he mic os uc u e o ma ion in a ozen solu ion. To ei e a e,
du ing ecalescence and c ys allisa ion, he inc eased mobili y wi hin he pa ially ozen
sys em led o a eo ganisa ion o he c ys alline phase, esul ing in he o ma ion o ice c ys als
esembling g ains and app oxima ely ma ching he sizes o he po es wi hin he lyophilisa e.
Quan i ying his eo ganisa ion e en o e en unde s anding i s kine ics migh be help ul o
a ious easons.
F om a scien i ic s andpoin , i could p o ide an al e na i e explana ion o he po e sizes in a
lyophilisa e. This wo k challenges he common unde s anding ha po es in he lyophilisa e
di ec ly co espond o s able nuclei o med du ing eezing. Ins ead, i he ecalescence and
c ys allisa ion phase p edominan ly de e mines ice c ys al size, i is plausible ha mul iple
i egula dend i ic ice c ys als migh coalesce o o m each g ain-like ice c ys al obse ed.
Thus, in e ing he numbe o nuclei solely based on po e size obse a ions may be misleading.
F om he pe spec i e o phase- ield modeling, unde s anding he kine ics o his
eo ganisa ion e en would allow o de ine he s a ing poin o he subsequen annealing
simula ion, since ini ial pa icle sizes would be known. Howe e , he issue a ises on how o
obse e and quan i y such an e en . I a eeze-d ying mic oscope is used, he sample
hickness o a ew mic ome es migh al eady be oo la ge o his e en o occu in a pseudo-
wo-dimensional manne , since he ini ial s uc u es seem o be much ine han he heigh o
he sample. Resol ing his challenge migh equi e u he educ ion in laye hickness o
po en ially ansi ioning o a h ee-dimensional phase- ield simula ion, whe e he simula ion
heigh ma ches he dis ance be ween he wo glass pla es. I such a model is calib a ed and
alida ed, i would enable he p edic ion o po e sizes in a lyophilisa e when no addi ional
annealing s ep is pe o med. Fu he mo e, i would allow o op imise he eezing s ep in
espec o he homogenei y o he lyophilisa e wi hin a ial, o when used wi h empe a u e
da a om he eeze-d ye he homogenei y be ween indi idual ials.
Fu u e applica ions
135
4.2 CFD simula ions h ough he po ous mic os uc u e
This idea could be used complemen a y wi h he simula ion ou pu om his wo k. The h ee-
dimensional phase- ield model gene a es a da a objec ha can be used as a simula ion
domain o CFD simula ions. This in eg a ion p esen s a signi ican ad an age, as ab ica ing
such de ailed and complex mic os uc u es using al e na i e me hods would be subs an ially
mo e challenging. As men ioned in Chap e 3.2.8, he esolu ion o expe imen al me hods
such as μ-CT a e simply insu icien o cap u e he mic os uc u e , especially he hin ma ix
walls.
The implemen a ion o a obus CFD simula ion, ed wi h he mic os uc u e as a simula ion
domain, could be used o make p edic ions abou he pa ial apou p essu e in he po es
immedia ely a he sublima ion on .
The Knudsen-Langmui equa ion desc ibes he mass low (sublima ion a e) as [161]:
𝑚
󰇗
=
𝑎
(
𝑝

−
𝑝


)
∙




 , (Eq. 34)
whe e a is an accommoda ion coe icien , p0 is he sa u a ion apou p essu e o he solid
phase, 𝑝 is he pa ial p essu e in he bulk apou , M is he molecula mass o he apou , R
is he uni e sal gas cons an , and T is he in e ace empe a u e.
F om Equa ion 34 i can be de i ed ha he pu ely physical sublima ion a e, which does no
ake geome ic ea u es o p oduc esis ance in o accoun , depends on h ee ac o s:
1) The di e ence be ween he equilib ium apo p essu e o he solid phase and he
pa ial apo p essu e in he gas phase.
2) The empe a u e in he solid phase a he in e ace.
3) The accommoda ion coe icien .
The Accommoda ion coe icien se es as a undamen al physical pa ame e , desc ibing he
in e ac ion o gas o apou molecules when hey collide wi h he su ace o a solid o liquid.
The empe a u e a he in e ace, on he o he hand, eme ges as a mul i ace ed unc ion
in luenced by se e al ac o s. These ac o s include he hea ex ac ed by he ozen p oduc ,
hea ans e ed om su ounding su aces, hea eleased du ing ice c ys allisa ion, and
changes in en halpy a ising om he o ma ion o new in e aces.
Fu u e applica ions
136
The p essu e e ms in his equa ion indica e ha he sublima ion a e is slowed down when
he pa ial apou p essu e inc eases. This means ha he o al sublima ion a e is adjus ed
o a balance be ween he p essu e educ ion due o mig a ion o wa e apou away om he
in e ace and he pu ely physical sublima ion a e ha is mos ly dependen on he
empe a u e a he in e ace. The p essu e educ ion e m can be assumed o be dependen
on he ambien p essu e, as well as po e geome y, possibly po e sizes o he smalles
conjunc ion a eas (bo leneck p inciple). The e o e, a CFD simula ion could be used o e i y
i he mic os uc u e p edic ed om he h ee-dimensional phase- ield model is accu a e by
compa ing he expe imen al sublima ion a e wi h he balanced sublima ion a e om he
combina ion o CFD simula ion and he po ous mic os uc u e om he phase- ield model.
Appendices
137
5 Appendices
Appendix A – Valida ion and s abili y analysis o he phase- ield simula ion
The ollowing sec ion desc ibes mul iple es s ha we e conduc ed o ensu e ha he
simula ion had been implemen ed co ec ly. On he one hand, ca e is aken o e i y ha
simula ion alues co espond o heo e ical alues (see Gibbs-Thomson e ec ), bu also ha
he simula ion ou pu scales p ope ly wi h he inpu pa ame e s (e.g., in e acial wid h scaling
wi h g adien ene gy coe icien s).
i. Gibbs-Thomson (capilla i y) e ec on composi ion
The composi ion o he phases in a phase- ield model a e dependen on he cu a u e o he
in e ace, i.e., he mo e cu ed an in e ace he highe he composi ion o he co esponding
phase. This phenomenon, namely he Gibbs-Thomson e ec , is also p esen in he eal wo ld
as he cu a u e-dependen inc ease o he sa u a ion apou p essu e in su icien ly small
pa icles [162]. The de ia ion o he composi ion o a pa icle wi h a cu ed in e ace is gi en
by [163]:
∆
𝑐

=
𝜒

∙
𝛾

𝑐


−
𝑐



∙
𝜓

,
(Eq. App. 1)
whe e χp is he mean cu a u e o he in e ace, γ is he in e acial ene gy densi y, and 𝑐
 and
𝑐
 a e equilib ium composi ions o he m- and he p-phase, espec i ely, and ψm is a quan i y
de ined as [163]:
𝜓

=
󰇩
𝜕

𝑓

𝜕
𝑐

󰇪



,
(Eq. App. 2)
whe e 0 is he bulk ee ene gy densi y, and c is he composi ion.
The impac o he Gibbs-Thomson e ec was calcula ed o he wo-dimensional as well as he
h ee-dimensional model. Subsequen ly, simula ions we e pe o med wi h bo h models and
di e en ini ial pa icle sizes. The esul s in Figu e App. 1 show ha simula ion ou pu and
heo e ical calcula ions ag eed, and ha bo h models exhibi ed a c i ical pa icle size. Once
he pa icle adius ell below he c i ical pa icle size, he co esponding pa icle dissol ed
comple ely, as can be in e ed om he misma ch o simula ed and calcula ed alue a 6 g id
Appendices
138
poin s and 7 g id poin s o he wo-dimensional and he h ee-dimensional simula ions,
espec i ely.
Figu e App. 1: Inc ease o pa icle composi ion dependen on pa icle adius o wo-
dimensional ( op) and h ee-dimensional (bo om) wi h c i ical pa icle adius ( c) o dx = 0.9.
ii. Ene gy ba ie coe icien on in e acial ee ene gy
The ene gy ba ie coe icien Q is used in his mul iphase- ield model o inc ease he ee
ene gy a loci in he simula ion domain whe e he in e ace o wo adjacen pa icles o
di e ing o de pa ame e s come in o con ac . Consequen ly, his penalises he o e lapping o
he in e aces o pa icles, and he e o e p omo es phase sepa a ion. The addi ion o such a
coe icien allows o inc ease d (inc emen be ween imes eps), because “mo ing” in e aces

Appendices
139
due o pa icle g ow h a e al eady inhibi ed by hei di ec ional expansion ea ly on when close
o o he in e aces. I Q is se o 1, hen mo e o e lapping o pa icle in e aces is necessa y
o inc ease he bulk ee ene gy, which in u n means ha d has o be low enough o p e en
he o e lapping om pa icles om one imes ep o he nex .
When choosing he ene gy ba ie coe icien , ca e mus be aken ha he in e acial ee
ene gy σ o he sys em is no al e ed, o he wise he simula ion esul s would become
dependen on Q. In o de o iden i y sui able alues o Q, a phase- ield simula ion was se up
whe e in e aces we e in con ac wi h each o he . Speci ically, a single ci cula pa icle was
placed wi h a hin ing o ano he phase sepa a ing he pa icle om he es o he simula ion
domain, as shown in Figu e App. 2, he eby c ea ing o e lapping in e aces.
Figu e App. 2: wo-dimensional simula ion domain (le ) and c oss-sec ion ( igh ) wi h sligh ly
o e lapping in e aces. The pa icle in he middle and he su ounding phase we e bo h
assigned hei own o de pa ame e o inhibi usion.
Nex , he ene gy ba ie coe icien Q was a ied, and he o e all in e acial ee ene gy o he
c oss-sec ion was calcula ed wi h:
𝜎
=

󰇩
𝑓
(
𝑐
,
𝜂

)
+
𝜅


𝑑𝑐
𝑑𝑥


+
𝜅


𝑑𝜂

𝑑𝑥


𝑑
x
󰇪
,
(Eq. App. 3)
whe e (c,ηi) is he bulk ee ene gy densi y, and κc and κη a e he g adien ene gy coe icien s
o composi ion and o de pa ame e , espec i ely. The esul s a e depic ed in Figu e App. 3.
Appendices
140
Figu e App. 3: Impac o he ene gy ba ie coe icien (Q) on he in e acial ee ene gy (σ) in
he mul iphase- ield simula ion used in his wo k.
The in e acial ee ene gy σ emained cons an o alues o he ene gy ba ie coe icien Q
up o 4. I Q was u he inc eased, a small inc ease o σ was obse ed, wi h a subsequen
s eep dec ease e en up o nega i e alues. The esul s o his es indica e ha alues o Q
up o 4 a e accep able wi hou al e ing he in e acial ene gy and he e o e he simula ion
esul s. A e some ial and e o , a alue o 2 was chosen o Q as i allowed o su icien ly
inc ease d o all wo-dimensional and h ee-dimensional simula ions.
iii. G adien ene gy coe icien s on in e acial wid h and in e acial ene gy
The g adien ene gy coe icien s κc and κη o composi ion and o de pa ame e , espec i ely,
bo h in luence he in e acial wid h ω and he in e acial ene gy σ in phase- ield simula ions.
In pa icula , he ollowing exp essions mus be ul illed i he phase- ield simula ion was
implemen ed co ec ly [164]:
𝜔
~

𝜅

𝛾
,
(Eq. App. 4)
𝜎
~

𝛾
∙
𝜅

,
(Eq. App. 5)
whe e γ is a pa ame e ela ed o he heigh o he ene gy ba ie be ween he wo phases,
and κi is he espec i e g adien ene gy coe icien .
Appendices
141
The in e acial ene gy was calcula ed wi h Equa ion App. 3, while he in e acial wid h had o
be de e mined ia he in lec ion poin o he in e ace, as depic ed in Figu e App. 4.
Figu e App. 4: De e mina ion o he in e acial wid h (ω) ia a angen line going h ough he
in lec ion poin o he composi ion p o ile.
The de e mina ion o he uppe and lowe limi s o he in e ace was achie ed h ough he
iden i ica ion o he poin s o in e sec ion be ween he angen line and he ex ema
(maximum/minimum alues) o he composi ion p o ile. Subsequen ly, he in e acial wid h
was quan i ied by measu ing he dis ance be ween hese uppe and lowe limi s. Al hough
al e na i e me hodologies, o example wi h he de i a i e o he composi ion p o ile, a e
a ailable, his echnique is commonly employed o he es ima ion o in e acial wid h due o
i s simple applica ion. I is no ewo hy ha , in ins ances o in e aces exhibi ing mo e wid h,
his me hod may no encompass he en i e in e ace. None heless, he c i ical aspec o his
p ocedu e is o main ain consis ency in he measu emen . Nex , and he in e acial ene gy
and wid h we e de e mined o a ious ene gy g adien coe icien s, as shown in
Figu e App. 5.
Fo his es , ei he κc o κη o bo h we e a ied simul aneously. The esul s show ha he
p opo ionali y exp essions om Equa ion App. 4 and Equa ion App. 5 we e ul illed, as he
in e acial wid h as well as he in e acial ene gy co ela ed linea ly wi h he squa e oo o
he ene gy g adien coe icien s.
Appendices
142
Figu e App. 5: Dependency o in e acial wid h (ω) and in e acial ene gy (σ) on he g adien
ene gy coe icien s wi h γ = 1. The in e acial wid h depends on he e alua ion me hod.