Experimental and simulation-based investigation of the influence of freezing and annealing on the microstructure in lyophilisates
Abstract
This work explores the impact of freezing and annealing conditions on the microstructure of lyophilisates, using a combination of experimental techniques, such as freeze-drying microscopy and differential scanning calorimetry, and a theoretical phase-field model. The research initially focused on how temperature ramps in freeze-dryers influence the recalescence and crystallisation phases, where the temperature in samples rises during freezing due to an exothermic phase transition. Temperature profiling within vials established a correlation between nucleation temperature and the duration of the recalescence and crystallisation phases. Findings indicated that lower nucleation temperatures reduce the period of elevated temperatures, which is crucial during the initial freezing stage. Moreover, this study examined the microstructural effects of recalescence and crystallisation using a freeze-drying microscope. In cases without a temperature increase, a fine network of crystalline and amorphous phases was observed, complicating phase differentiation. Conversely, samples subjected to controlled temperature increases mimicking in-vial conditions during recalescence and crystallisation showed distinct, grain-like ice crystals, which are comparable to the pores observed in lyophilisates. To complement experimental observations, a two-dimensional phase-field model was developed, later expanded to three dimensions, to simulate and predict morphological changes within the lyophilisate during annealing. The simulations not only provided insights into the dynamics of ice crystal growth and the distribution changes during annealing but also examined additional parameter such as the surface area of the matrix phase, as well as the conjunction areas between neighbouring ice crystals. In conclusion, this work highlights the complex interplay between processing conditions and microstructural outcomes in lyophilisates.
Full text
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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5
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].
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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
7
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
Gene al backg ound
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).
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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
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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
23
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)
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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
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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.
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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.
Gene al backg ound
29
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)
+𝑄𝜂𝜂
,
(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 conjuncon 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.