Finite size effects on liquid-solid phase coexistence and the estimation of crystal nucleation barriers

0.73 MB·English

by  Antonia Statt

Finite size effects on liquid-solid phase coexistence and the estimation of crystal
nucleation barriers
Antonia Statt1,2, Peter Virnau1 and Kurt Binder1
1Institut fu¨r Physik, Johannes Gutenberg-Universita¨t Mainz,Staudinger Weg 9, 55128 Mainz, Germany
2 Graduate School of Excellence Materials Science in Mainz, Staudinger Weg 9, 55128 Mainz, Germany
(Dated: January 8, 2015)
AfluidinequilibriuminafinitevolumeV withparticlenumberN atadensityρ=N/V exceeding
the onset density ρ of freezing may exhibit phase coexistence between a crystalline nucleus and
f
surrounding fluid. Using a method suitable for the estimation of the chemical potential of dense
fluidsweobtaintheexcessfreeenergyduetothesurfaceofthecrystallinenucleus. Thereisneither
5
1 a need to precisely locate the interface nor to compute the (anisotropic) interfacial tension. As a
0 testcase,asoftversionoftheAsakura-Oosawamodelforcolloidpolymer-mixturesistreated. While
2 ouranalysisisappropriateforcrystalnucleiofarbitraryshape,wefindthenucleationbarriertobe
compatible with a spherical shape, and consistent with classical nucleation theory.
n
a
J
Nucleation of crystals from fluid phases and their
7
subsequent growth is one of the most important phase

transformations in nature [1–3]; applications range from
]
h ice crystal formation in the atmosphere, to metallurgy,
c nanomaterials, protein crystallization, etc. Despite its
e
overwhelming importance, crystal nucleation still is only
m
poorly understood.
-
t For the nucleation of a liquid drop from supersatu-
a
rated vapor, clearly the average nucleus shape is spheri-
t
s cal. Onlythecurvaturedependenceoftheinterfacialten-
.
t sion [4–9] presents a stumbling block for the prediction
a
m of nucleation barriers. Unlike interfaces between fluid
phases, the crystal-fluid interface tension γ((cid:126)n) depends
-
d ontheorientationoftheinterfacenormal(cid:126)nrelativetothe
n crystallatticeaxes[10–12]. Forisotropicγ thenucleusis
o a sphere of radius R (volume V = 4πR3/3) and its sur- FIG. 1. Schematic plot of the chemical potential µ versus
c face excess free energy is F = 4πR2γ = A γV2/3, density ρ for a system undergoing a liquid-solid transition in
[ surf iso
with A = (36π)1/3. For crystals the term A γ is a finite box volume Vbox with periodic boundary conditions.
1 replacedisoby a complicated expression, iso (A plot of pressure p vs. ρ would qualitatively look just the
v same). Due to interfacial effects, non-negligible in finite sys-
2 tems, the isotherm deviates from p=pcoex in the two-phase
(cid:90)
5 F (V)= γ((cid:126)n)d(cid:126)sV2/3 A γ¯V2/3 . (1) coexistence region, ρf < ρ < ρm. The features in the curve
5 surf ≡ W (kinks in reality are rounded due to fluctuations) are due to
1 AW transitions between the different states shown in the figure
0 via snapshots of the simulated generalized Asakura-Oosawa
. model. Only the part where the solid phase is the minority
1 HereA isthesurfaceareaofaunitvolumewhoseshape
W phase is discussed. For further explanations cf. text.
0 is derivable from γ((cid:126)n) via the Wulff construction [10–
5
12] , and the average interface tension γ¯ is defined as
:1 γ¯ =A−W1(cid:82) γ((cid:126)n)d(cid:126)s. critical nucleus volume V∗ and barrier ∆F∗,
v
In the classical nucleation theory [1–3], the formation
i (cid:20) (cid:21)3
X freeenergyofanucleusiswrittenintermsofvolumeand V∗ = 2AWγ¯ , ∆F∗ = 1A γ¯V∗2/3 = 1(p p )V∗
W c l
r surface terms as 3(pc pl) 3 2 −
a − (3)
Even if V∗ is large enough so that correction terms to
∆F = (p p )V +F (V). (2)
− c− l surf Eq. 2 can be neglected, the application of Eq. 3 is diffi-
cult due to lack of knowledge on A and γ¯. This lack
W
Here p is the pressure in the crystal nucleus and p in of knowledge has hampered the comparison of observed
c l
the(metastable)liquidphasesurroundingit. Inthether- nucleationrates[13–16](andthebarriersextractedfrom
modynamic limit, the configuration with one nucleus on them)andsimulations[17–20]where∆F∗ wasestimated
top of the free energy barrier in the metastable phase is directlybybiasedsamplingmethods. Thesecomparisons
a saddle point in configuration space. The condition for were made for suspensions of (hard sphere-like) colloidal
(unstable) equilibrium, ∂(∆F)/∂V = 0, then yields the particles; the large size of the colloids has the advantage
2
20 namic limit (where the chemical potential µ=µcoex and
18 1.0 the pressure p = pcoex for all densities from the onset
] densityoffreezingρ totheonsetdensityofmeltingρ )
16 0.5 T f m
1124 -00..05 rk()[B cFaigu.s1edcobryrefispnoitnedsinizge.toTthhues,htohmeopgaernteooufsthfleuiidsotfohrerfimnitine
U volumeV exceedsρ andcontinuesuptothe“droplet
p˜10 -1.0 box f
evaporationcondensationtransition”[26]atρ ,wherefor
8 1 1.051.101.15 1
thefirsttimeacrystallinedropletinthesystembecomes
6 r
stable. Note that this transition is a sharp phenomenon
4
soft EffAO only when V (and then ρ ρ , consistent
box 1 f
2 EffAO with the lever rul→e [2∞7]). At a second s→pecial density ρ
2
0
the “droplet” changes its shape from compact to cylin-
0.2 0.3 0.4 0.5 0.6 0.7
drical (stabilized by the periodic boundary conditions).
η
Ataboutρ=ρ aslabconfiguration,separatedfromthe
3
fluidbytwoplanarinterfaces,appears(Fig.1). Inthisre-
FIG.2. Normalizedpressurep˜=pσ3/k T plottedvs. pack-
c B gion µ=µ and p=p holds true also in the finite
ing fraction η ≡ ρπσ3/6 of the colloids, for the effective AO coex coex
c system, if the linear dimensions in the directions parallel
model (henceforth denoted as Eff AO, asterisks) and its soft
to the planar interfaces are chosen such that the crystal
version (soft Eff AO, squares). Curves are guide to the eye
only. These data were obtained from simulations of homoge- (at density ρm) is commensurate without any distortion.
neous liquid and solid (fcc) phases, while the pressure where The analogous behavior for vapor to liquid transitions is
two-phase coexistence occurs was found from the “interface well studied [9, 28–30]. Here we show that the descend-
velocity method” [37], namely p˜= 8.44±0.04 (soft Eff AO) ing part of the p(ρ) and µ(ρ) isotherms can be used to
and p˜=8.06±0.06 (Eff AO). The coexistence packing frac- extractinformationonF ,V∗ and∆F∗ fortheliquid-
surf
tions are ηf = 0.495 (1) and ηm = 0.636(1) for the soft Eff solid transition as well.
AO case. The insert compares the potentials of the Eff AO
In the snapshots the particles in the fluid region are
(which is singular at r=σ =1) and soft Eff AO models.
c
shown in blue, in the crystal in red color, using the av-
eraged Steinhardt local bond order parameters [31, 32]
to distinguish the character of the phases (see Ref.[32]
to allow direct microscopic observations of crystal-liquid
for definitions and implementation details). Particles in
interfaces [21] and nucleation events [22, 23]. Since ki-
theinterfacialregion,forwhichthisclassificationyielded
neticprocessesforcolloidsaremanyordersofmagnitude
ambiguous results, are shown in green color. The face-
slower than for small molecules, colloids are model sys-
centeredcubic(fcc)packingofthecrystalisclearlyseen,
tems for the study of the liquid-solid transition [24, 25],
andthecrosssectionthroughthe“droplet”alsosuggests
and well suited to separate nucleation from the subse-
that the shape may non spherical.
quent crystal growth.
However, to elucidate the persisting discrepancies be- The model of our simulations qualitatively describes
tween simulations and experiments one needs to know colloid polymer mixtures [33–36]. In the Asakura-
moreaboutthetheoreticalnucleationbarriers: Howlarge Oosawa (AO) model [33], colloids are described by hard
mustV∗besothatEq.3isagoodapproximation? What spheres of diameter σc, polymers as soft spheres (which
is the physical origin of corrections to ∆F∗ Eq. 3 and may overlap each other without energy cost) of diameter
their magnitude? Is it legitimate to assume{a sph}erical σp. Of course, the mutual overlap of colloids and poly-
shape of the nucleus, in spite of its crystalline structure? mers is also strictly forbidden. Polymers create the (en-
And so on. Understanding the general conditions under tropic)depletionattractionbetweencolloids[33];varying
which the classical description Eqs. 2, 3 holds will be thesizeratioq =σp/σc andthepolymerdensityonecan
usefultounderstandliquid-solid{transition}sincondensed tune the phase diagram [34–36] and interfacial proper-
matter in general. ties [37, 38]. A useful feature of this model occurs for
Inthepresentletter,weaddresstheseissues,andshow q < q∗ = 0.154 [35, 39]: then one can integrate out the
how both V∗ and ∆F∗ can be obtained, considering the polymer degrees of freedom exactly, and one is left with
equilibrium of the system at fixed finite particle number an effective pairwise potential, which is attractive in the
N in a finite simulation box Vbox. For a suitable range range σc < r < σc+σp (and zero for r > σc+σp), but
ofdensityρ=N/Vbox,theequilibriumbetweenthecrys- infinitely repulsive for r < σc. The strength of the po-
talline nucleus and surrounding fluid is perfectly stable. tential of this “effective” AO model is controlled by the
We explain how both V∗ and pc pl can be estimated fugacity zp of the polymers [39] (Fig. 2, insert).
directlyandaccurately. Usingthen−∆F∗ =(p p )V∗/2 However, it is computationally more convenient to re-
c l
−
Eq. 3 , the need of dealing with γ(n¯) and use of Eq. 1 place the Eff AO model by a similar but continuous po-
{ }
is bypassed. So we do not need to assume anything on tential, the soft Eff AO model [39] (Fig. 2, insert). For
the shape of the nucleus. this model the pressure (in the fluid phase) is straight-
Thus,thecentralideaofthepresentworkistoexplore forwardly obtained in the simulation from the Virial ex-
the deviations from phase coexistence in the thermody- pression [39, 40], while for the Eff AO model due to the
3
discontinuity at r = σc this is very cumbersome [38]. 2.5
4
Fig. 2 shows that the variation of p with η is very simi-
lar for both potentials. Since real colloids never are de- 2.0 2
scribed by hard spheres precisely [41], nor are polymers µ0
precisely modeled by ideal soft spheres [42], a quantita- -2
1.5
tively accurate modeling of real systems anyway cannot -4
η
be attempted. The soft Eff AO model is proposed here
1.0 0.511.522.5
as a coarse-grained qualitative model of colloid-polymer
z
mixtures which is practically useful in a simulation con-
0.5
text.
Using the Virial expression the pressure p of the liq-
l
0.0
uidintheregionsurroundingthecrystalnucleusinFig.1
0 5 10 15 20 25 30
(farawayfromtheinterfacialregion)canbereadilymea- (a)
z
sured,butobtainingp insidethenucleusforsmallnuclei
c
is not reliably possible. It is necessary to base the anal- 7
Widom
ysis of the two-phase equilibrium in Vbox on the chem- 6 5 5 20
ical potential µ, because µ is strictly constant in equi- 5 7×7×30
librium also in a spatially inhomogeneous situation. But 4 7×7×40
the standard particle insertion method [40, 43] does not 10 ×10×40
3
workathighpackingfractionsηc nearηm. Thus,wehave µ2 10××10××50
extendedanapproach[44]tosamplethechemicalpoten-
1
tial of a dense fluid by studying a system where walls
0
are present; using a soft wall that reduces the density
-1
suitablysuchthatthereparticleinsertionworks(Fig.3).
Of course, it is important to choose L large enough so -2
z
that outside of the range of z, for which the walls affect
0.36 0.39 0.42 0.45 0.48 0.51 0.54
thedensityprofile,actuallyaconstantdensityisreached. (b) η
Fig.3demonstratesthatinthiswaythechemicalpoten-
tialcanbeobtainedaccuratelyevenforη >ηf. Thepres- FIG.3. (a)Illustrationofthemethodtocomputethechemi-
surep(computedintheregionwhereη(z)=ηbulk =con- calpotentialofaverydensefluid,usingaL×L×Lzslabgeom-
stant) agrees with the corresponding bulk data of Fig. 2. etry,withasoftwallatz=0andahardwallatz=L =30
z
Now we exploit the fact that µ is constant through- (lengths being measured in units of σc, L=7, and 4 choices
out the system also when a crystalline nucleus is present ofN areused,N =750,950,1100and1250,respectively). In-
(Fig.1): thechemicalpotentialinthefluidµf(pl)equals sertshowsµinunitsofkBT asafunctionofz,forthe4choices
shown, over the regions of z where particle insertion works.
that of the crystal nucleus µ (p ). From µ (p ) =
c c c coex
(b) Chemical potential µ (in units of k T) plotted vs. η, for
µ (p )=µ we readily find, using the expansions B
l coex coex differentchoicesofLandL ,asindicated,toshowthatfinite
z
π 1 size effects are negligible. The data labeled by “Widom” at
µ (p ) µ + (p p ), (4)
c c coex c coex notsolargeη areobtainedbythestandardparticleinsertion
≈ 6η −
m
method for homogeneous bulk systems. Arrows on abscissa
π 1
µl(pl) µcoex+ (pl pcoex), (5) and ordinate indicate ηf and µc(pcoex)/kBT, respectively.
≈ 6η −
f
that (p p )η = (p p )η . Since we have
c coex f l coex m
− −
recorded both functions µ (η) and p (η), we also know corresponding packing fractions differ from their coex-
l l
µ (p ) and hence can verify that the data indeed fall istence values. Initializing the simulation by putting a
l l
in the regime where the linear expansion, Eq. 5, holds. crystalofabouttherightvolumeV∗ andabouttheright
Finding µ (p ) via thermodynamic integration (using choice for η (p ) in the box, after a long period of equi-
c c c c
µ (p )=µ (p ) as starting point), we have verified libration we measure both p and η (p ) in the fluid re-
c coex l coex l l l
that Eq. (4) also introduces only negligible errors. gion (far away from the crystal) and verify (from the
Thetwo-phaseequilibriumofacrystallinedropletsur- dataofthebulksimulation,Fig.3b)thatequilibriumhas
rounded by fluid has been studied for three system sizes, been reached. Since we know also the chemical potential
keepingthenumberofcolloidsinthesimulationboxfixed (µ (p ) = µ (p ) is constant), we can obtain p and also
c c l l c
(at N = 6000, 8000, and 10 000, respectively) and vary- η (p ) and hence Eq. 6 determines V∗ unambiguously.
c c
ing Vbox and hence ρ=N/Vbox. In thermal equilibrium, Fig. 4 shows the data for ∆p = pl pcoex versus η.
we then have a finite-size variant of the lever rule Actually when we use the chemical pot−ential µ (p ) from
l l
ηV =η (p )(V V∗)+η (p )V∗ (6) Fig. 3b and obtain ∆p from Eq. 5, the data are precisely
box l l box c c
− reproduced,whichjustisaconsistencycheck. Fromsim-
While for V we would have p = p = p ulations determining γ for interfaces parallel to 111,110
box l c coex
→ ∞
and η (p ) = η , in the finite system p ,p and the and 100 planes [45] it is found that γ((cid:126)n) depends only
c coex m l c
4
very weakly on (cid:126)n. For comparison with classical nucle- 2.1
ation theory, we neglect the dependence on (cid:126)n and take 2.0 1.07
γ γ111 γ((cid:126)n) = 1.013 [45]. Assuming a spherical 1.9 p
sha≈peV∗ =≈4πR∗3/3wefind∆p=(2γ/R∗)/(η /η 1). µ
m f 1.8
Using the observed values of V∗ one then obtain−s a
1.7
prediction for the curves ∆p(η). We find that these
p1.6
∆
predicted curves fall slightly below the actual observed
1.5
data. Theycanbebroughtingoodagreementiftheyare
1.4
rescaled by a constant factor of c=1.07. This small en-
1.3
hancementcanbeduetotheratioA/A orerrorsinthe
iso
1.2
estimation of γ((cid:126)n). Unexpectedly, we hence find that for
1.1
ourmodelofcolloidpolymermixturestheassumptionof
0.526 0.527 0.528 0.529 0.53
asphericalnucleusshapeworksratherwell, butitwould (a)
η
not be needed to predict the nucleation barrier. Using
Eq. 2, knowledge of p p and V∗ suffices to predict 220
c l
∆F∗. One can expect, h−owever, that significant deriva- 200
tions from spherical nucleus shape will appear for large 180
ηr in our model, where the fluid is a vapor-like phase, 160
p
and γ((cid:126)n) will depend more strongly on (cid:126)n. Gratifyingly, 140
∗
Fig. 4b shows that the three choices for N superimpose F
∆120 6000
to a common curve, so in the shown regime finite size
8000
100
effects essentially are negligible. 10000
80
In summary, we have shown that for the liquid-solid-
transition a description of nucleation barriers in terms 60
of the classical nucleation theory holds, at variance with 40
studies of nucleation with hard sphere-like colloids [13–
40 50 60 70 80 90 100 110 120 130
20, 46]. However, we feel the latter studies are inconclu- (b) V 2/3
∗
sive, do to their use of too large η (0.53 < η < 0.57)
l l
where the slowing down due to the kinetic prefactor of FIG. 4. (a) Pressure difference ∆p=p −p between the
l coex
the nucleation rate matters [47]. While the range of pressure p in a fluid surrounding a crystal nucleus of finite
l
∆F∗ in Fig. 4b corresponds to ηl/ηf 1 0.06 the size and the coexistence pressure, plotted versus the average
− ≤
range of the experiments in Fig. 4b would correspond packing fraction η in the simulation box, for particle number
to 5<∆F∗ <10 only. N = 6000, 8000 and 10000 (symbols, from bottom to top).
Analyzingfinitesizeeffectsonphasecoexistence,both Curves show the formula ∆p = (2γ/R∗)c/(ηm/ηf −1) with
V∗, p , p and the chemical potential for this stable c = 1.07, extracting R∗ from the assumption of a spherical
l c nucleus (V∗ = 4πR∗3/3) and taking γ ≈ γ˜ ≈ 1.013 [45].
two-phase coexistence in a finite simulation box can be 111
(b)∆F∗ computedfromp,p andV∗ (usingEqs.2,5and6)
reliably estimated. The numerical results also clearly l c
plotted vs. V∗2/3, straight line is Eq.(3) with A = A and
show that in the regime where ∆F∗ 80 the relation iso
∆F∗ V∗2/3 holds precisely, as vis≥ible from the fit γ˜=γ111. The broken line is a fit illustrating ∆F∗ ∝V∗2/3.
∝
in Fig.4b; thus we have verified that classical theory of
homogeneous nucleation for crystals is accurate, in this
regime of barriers, provided one takes into account that
the nucleus shape is in general nonspherical. However,
sincethetwostraightlinesinFig.4balmostcoincide,the
spherical approximation is shown here to be almost per-
fect. Since crystal faces in contact with a dense fluid are
frequentlyatomicallyrough,thesphericalapproximation
is expected to be quite good generally, in particular for
somewhat smaller nuclei, for which the nucleation rates
also would be larger.
Acknowledgments: Thisresearchwassupportedbythe
Deutsche Forschungsgemeinschaft (grant No. VI237/4-
3). We thank the H¨ochstleistungsrechenzentrum
Stuttgart (HLRS) and the Zentrum fu¨r Datenverar-
beitung Mainz for generous grands of computing time
at the HERMIT and MOGON supercomputers.
5
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