Table Of ContentNucleation of a new phase on a surface that is changing irreversibly with time
∗
Richard P. Sear
Department of Physics, University of Surrey
Guildford, Surrey GU2 7XH, United Kingdom
Nucleation of a new phase almost always starts at a surface. This surface is almost always
assumed not to change with time. However, surfaces can roughen, partially dissolve and change
4 chemically with time. Each of these irreversible changes will change the nucleation rate at the
1 surface, resulting in a time-dependent nucleation rate. Here we use a simple model to show that
0 partialsurface dissolution can qualitatively changethenucleation process, in away thatis testable
2 inexperiment. Thechanging surface meansthat thenucleation rate isincreasing with time. There
n is an initial period during which nonucleation occurs, followed byrelatively rapid nucleation.
a
J
2 I. INTRODUCTION
] Nucleationofanewphase,suchasacrystaloraliquid,
t
f almostalwaysoccurswith the nucleus at a surface [1–3].
o
Therateofnucleationisknowntobeextremelysensitive
s
. tomicroscopicdetailsofthesurface. Forexample,micro-
t
a scopic changes in the surface geometry [3–7] can change FIG.1. (Coloronline)Snapshotofnucleationofanewphase
m rates by orders of magnitude. It is also well known that (red)inasolution (white)incontactwithaslowly dissolving
- surfaces change irreversibly over time. Mineral surfaces surface (black). The snapshot is of a simulation in a box of
d 100 (x axis) by 400 (y axis) lattice sites, but it is cropped
in the atmosphere are subject to weathering [8], smooth
n along the vertical axis. The dissolution rate rD = 2×10−6,
metal surfaces corrode and become pitted [9], and poly-
o 2h/kT = 0.12, and the snapshot is taken with the nucleus a
c mersurfacesdegrade[9]. Despitethe factthatforminga little over thenucleation barrier.
[ nanoscale pit is known to hugely increase the nucleation
rate [4, 6, 7], no attempts have been made to model and
1
v understand the effect on nucleation, of pitting and other inexperimentbypreparingasetoftensormoreidentical
5 dynamic changes in a surface. We model this here, us- droplets and then plotting the fraction of them in which
0 ing a simple generic model of nucleation of a new phase crystals have nucleated, as a function of time. The sig-
5 on a slowly dissolving surface. In this simple model, we natureofarateincreasingwithtime,isaninitialplateau
0 find that dissolution, by roughening the surface, greatly with P(t) ≃ 1 due to a small initial nucleation rate, fol-
.
1 increases the nucleation rate, i.e., the nucleation rate, lowedby P(t) dropping faster andfaster, andso curving
0 rN(t), becomes an increasing function of time. downwards,as the nucleation rate increases. This signa-
4 The classical picture of nucleation of a new crystal is turehasbeeninobservedinthecrystallisationofthe ex-
1 thatthenucleusisalargeandrarethermalfluctuationof plosiveRDXbyKimet al.[17]. Thusourmodelprovides
:
v the nucleatingphase. Therateis lowdue totherareness apossiblequalitativeexplanationforthisobservation. It
i of the fluctuation; it is necessary to wait a long time for isonlyqualitativeasourmodelisasimplelatticemodel,
X
such a rare fluctuation to occur. In this classical pic- not an accurate model of RDX in solution, and the ex-
r
a ture of nucleation, the fluctuation occurs in a system perimentsdonotcharacterisethesurfacethatnucleation
that is at a metastable local equilibrium and so is not is occurring on. Further experiments to characterise the
changing with time [1–3]. Nucleation is then a stochas- source of the increasing nucleation rate, and simulation
tic process, with a time-independent nucleation rate, rN of more detailed models will be needed to develop our
(that depends exponentially on the free energy cost of understanding of time-dependent nucleation rates.
the fluctuation). For a stochastic process with a time- In the next section we will describe our simple model,
independentrate,theprobabilitythatnucleationhasnot and the simulation algorithm we will use. We will then
occurred at a time t, is an exponential function of time: present and discuss results, before comparing these re-
P(t) = exp(−rNt). However, if the surface is changing sults with a simple analytic theory. We discuss the gen-
with time, so will the free energy of forming a nucleus erality of our observed P(t) in nucleation and note that
and hence the nucleation rate, and P(t) is no longer an a similar form is seen for cancer. The final section is a
exponential function of time. conclusion.
The property P(t) is easily measurable in experiment,
and this has been done for a number of crystallisingsys-
tems [10–17]. For crystallisation, P(t) can be estimated
II. MODEL
Our simple model for nucleation on a slowly dissolv-
∗ [email protected] ing surface is a modified two-dimensionalIsing model or
2
1
2
0.8
1.5
ρ 1 0.6
0.5 P
0.4
0
0 5×105 1×106 2×106
t 0.2
FIG. 2. (Color online) Plot of the surface roughness, ρ, for 0
a single surface, as a function of time t in units of cycles. 0 2×106 4×106 6×106
Therunisuntilnucleationoccurs. ThedissolutionraterD = t
2×10−6.
FIG. 3. (Color online) Plot of the cumulative probability
thatnucleationhasnotoccurred,P(t),asafunctionoftime,
lattice gas (the two models are equivalent [18]), with t, in units of cycles. The black (right) and green (mid-
Glauber Monte Carlo dynamics [19]. See Fig. 1 for a dle) curves are with dissolution, at rates rD = 10−6 and
simulation snapshot. We will use lattice-gas terminol- rD = 2 × 10−6, respectively. In both cases, the super-
saturation 2h/kT = 0.12. The purple (left) curve is at a
ogy here. Each lattice site is either empty (left white
higher supersaturation 2h/kT = 0.16, and with no dissolu-
in Fig. 1), filled with a particle (red) or part of the
tion (rD =0). The magenta dotted curveis a fit of an expo-
surface and so filled by a surface particle (black). We
nential function to the P(t) without dissolution. The brown
will study nucleation of the high-particle-density (liquid
andturquoisedottedcurvesarefitsofthefunctionofEq.(4)
so mostly red) phase from the dilute phase (vapour so
to the black and green P(t)’s respectively. In all cases the
mostly white), as we did in earlier work on nucleationin simulation P(t)’sare obtained from 250 nucleation runs.
pores [4]. Nucleation is via a rare barrier-crossing ther-
mal fluctuation, that occurs on an irreversibly changing
surface. To obtain statistics of the nucleation times we simply
Neighbouring particles interact via an energy ǫ, and runmanysimulations,eachtimestartingwithaperfectly
a particle interacts with a neighbouring surface particle smoothsurface. Nucleationisdefinedashavingoccurred
with an energy ǫ/2. This results in the nucleus of the when the fraction of particles is greater than 10%. This
liquid phase having a contact angle θ = 90◦ with the fraction is defined as being the ratio of the number of
surface. Weworkatatemperaturesuchthatǫ/kT =3.0. particlestothetotalnumberofparticlesandemptysites.
The chemicalpotentialof the particlesµ/kT =2h/kT− The equilibrium fraction of sites occupied by particles
2ǫ/kT,where2h/kT isthesupersaturationinunitsofkT in the dilute phase is ≃ 0.5%, so this threshold is only
(h isthe magneticfieldinspinlanguage). Allourresults crossed once nucleation has occurred.
areforsimulationsofsystemsof100(x)by400(y)lattice
sites. Ourunitoftimeisonecycle: oneattemptedMonte
Carlo move per site. III. COMPUTER SIMULATION RESULTS
Eachsurfacestartsoutasperfectlyflat,andparallelto
the x axis, but dissolves at a rate r along y during the Results for the probability that nucleation has not
D
simulation. Dissolution and nucleation is shown in Sup- occurred, P(t), are plotted in Fig. 3. In Fig. 3 we
plementary Movie 1 [20]. Our model for dissolution is plot P(t) for systems with and without surface disso-
simple. In our Monte Carlo simulations we select lattice lution. The purple curve is for no dissolution, and at
sites at random. If a site is occupied by a surface parti- a higher supersaturation. We see that this P(t) is well
cle,thenifall4ofitsneighboursarealsosurfaceparticles fit by a simple exponential function. The fitted rate is
we do nothing. If between 1 and 3 of its neighbours are 1.22×10−6. This agrees well with the nucleation rate
surface particles, we flip it irreversibly to an empty site of 1.35×10−6±0.22×10−6 for this surface calculated
with probability r . If none of its 4 neighbours are sur- usingtheForwardFluxSampling(FFS)methodofAllen
D
face particleswe flip itirreversiblyto anempty site with et al. [4, 21, 22]. This is just what we expect for a time-
probability 1. This is a simple model of irreversible dis- independentnucleationrate. Notethatthecurvewithout
solution that creates rough surfaces. We measure height dissolution in Fig. 3 cannot be directly compared with
variations in the surface by ρ, the standard deviation of the curves for the surfaces with dissolution, because the
the surface height. The height is defined as the y coor- supersaturations are different.
dinate of the highest lattice site occupied by a surface However, with slow dissolution (black and green
particle. The roughness as a function of time is plotted curves) the functional form is very different. In particu-
for a single run in Fig. 2. lar,whereastheexponentialfunctionissteepestatt=0,
3
with dissolution the P(t) is almost horizontalthere. Ini- ation rate to increase (by orders of magnitude) until it
tially, there is a waiting period during which almost no is fast enough to observe nucleation. Although this is
nucleation occurs and so P(t) is almost horizontal, then not an example of self-organised criticality (SOC) [28],
significantnucleationoccursandP(t)drops. Thewaiting it does have features in common with SOC systems. In
time isincreasedifthedissolutionrateisdecreased. The both casesthe dynamics drive the system until a thresh-
black curve is for a r half that of the green curve, and oldisexceeded,atwhichpointasuddeneventoccurs(an
D
we see that the initial plateau at short times is longer. avalanche in the sand pile model of self-organised criti-
The form of P(t) with dissolution is also very differ- cality, and nucleation here). Future work could perhaps
entfromthestretchedexponentialP(t)thatresultsfrom take ideas from the study of the approach to the point
quenched(i.e.,time-independent)disorder[23,24]. Thus whereanavalancheoccursinSOC,andapplythemhere.
the shape of P(t) allows us to distinguish between sys-
tems with a single time-independent rate (simple expo-
nential), systems with quenched disorder (stretched ex- IV. ANALYTICAL MODEL: DERIVATION,
ponential), and systems with a time-dependent nucle- RESULTS AND COMPARISON WITH
ation rate due to dissolution (initially almost horizontal COMPUTER SIMULATION RESULTS
P(t)). See Refs. 10 and 11 for examples of experimen-
tal systems with simple exponential P(t), and Ref. 25 We will now develop a simple general model of nucle-
for an experimental system showing stretched exponen- ation with a time-dependent rate. We do this in order
tial behaviour. I am aware of two experimental studies, to generalise our findings for our lattice model, and to
which show P(t)’s with a plateau at short times. Both make experimentally testable predictions. We can con-
areoncrystallisationfromsolution,ofglycineinthecase struct a simple model for the increasing rate as follows.
of Badruddoza et al. [26], and of the explosive RDX in The probability that nucleation has not occurred, P(t),
the case of the work of Kim et al. [17]. In both sets of satisfies the differential equation
results, the effective nucleation rate is clearly increasing
with time, and so they are both consistent with nucle- dP(t)
=−P(t)r (t) (1)
ation occurring on a surface that is changing with time dt N
in such a way as to increase the nucleation rate. De-
where r (t) is the nucleation rate of the system at time
termining if a time-dependent surface is responsible for N
t. It has dimensions of one over time. Our observations
the form of the experimental P(t), would require find-
requirearatethatincreasesbyordersofmagnitude,and
ing the surface that nucleationis occurringon. This was
so we assume that r (t) is an exponentially increasing
notdoneintheseexperimentsbutcouldbeattemptedin N
function of time:
futurework. AsestimatingP(t)isstraightforwardinex-
periment[12–15],P(t)isprobablythebestwaytoobtain
evidence for a time-dependent nucleation rate. rN(t)=r0exp[λt] (2)
Inthesnapshotshowingnucleation,Fig.1,andinSup-
wherer0 isthe rateatt=0,andλisthe rateofincrease
plementary Movie 1 [20], we see that nucleation occurs
ofthenucleationrate. Thisexponentialtimedependence
with the nucleus in a concave part, or pit, in the sur-
willgiveustherequiredlargeincreaseinrate,andwould
face. This is as expected [2–7]. The free energy barrier
followifthefree-energybarrierisdecreasinglinearlyasa
to nucleation comes from the cost of creating the inter-
functionoftime. Theroughnessisnotincreasinglinearly
face aroundthe growingnucleus. The shorter the length
withtime,seeFig.2,sothisisunlikelytobeexactlytrue,
of interface that needs to be created, the lower the free-
but it does provide a simple model. Note that with this
energy barrier. At any surface there is a pre-existing
assumptionofanexponentiallyincreasingrateourmodel
interface. This pre-existing part of the interface does
becomes the Gompertz model [29], a model widely used
not contribute to the barrier, which is why the barrier
forpredictinglifetimes,forexamplethelifetimesofliving
is lower at interfaces. At any concave part of a surface,
organisms.
the length of this pre-existing interface is larger than at
PuttingEq.(2)intoEq.(1),andsolvingweobtain[29]
a flat or convex surface, and so nucleation is faster there
[1–3]. P(t)=exp[(r0/λ)(1−exp[λt])] (3)
At a supersaturation 2h/kT = 0.12, the radius of the
∗ P(t)≃exp[−exp[λt+ln(r0/λ)]] when r0/λ≪1(4)
criticalnucleus r =γ/(2h)=8.75. This used Onsager’s
exact expression [27], for the surface tension γ, which We are interested in systems where the initial rate r0 is
∗
gives γ = 1.05kT at ǫ/kT = 3. This critical radius r is much lower than λ, as then the rate increases for some
larger but of the same order of magnitude as the rough- time before nucleation occurs. So, in the second line
ness of the surface seenin Fig.2. A roughnessofaround we took the r0/λ ≪ 1 limit and simplified the equa-
2,oraboutonequarterofthecriticalradiusisenoughto tion. It is worth noting that Eq. (4) has the functional
dramatically increase the nucleation rate here. form of the Gumbel distribution of extreme-value statis-
The key finding of our computer simulations is that tics [30], which can also arise in nucleation problems via
roughening of the surface with time causes the nucle- other mechanisms [31].
4
Fits of Eq. (4) to the simulation data are shown as increasing rate of a stochastic process. In other systems,
the brown and turquoise dotted curves in Fig. 3. They otherprocessesmaycausethesurfaceandhencetherate
provide reasonably good fits to the data. The best fit to change with time. Another example of such a process
valuesof(λ,ln(r0/λ))are(1.10×10−6,−3.75)and(1.77× may be a chemical reaction that modified the surface
10−6, −3.65), for the fits to data with dissolution rates such as to reduce the contact angle of the nucleus at the
r = 10−6 and 2×10−6 respectively. So, the best-fit surface. Another might be nucleation on some growing
D
values of r0/λ satisfy our assumption that r0/λ≪1. aggregateinsolution,wherethenucleationrateincreases
We can obtain an independent estimate of the value as the size of the aggregate grows [23]. Nucleation of
of λ by approximating it as the rate of change of the lysozyme crystals is known to be affected by aggregates
free energy barrier to nucleation, ∆F∗, with time: λ ≈ [33]. All members of this class should have qualitatively
(∂(∆F∗/kT)/∂ρ)×(∂ρ/∂t), which is expressedin terms similar P(t)’s.
of the rate of change of the roughness. In Fig. 2 we see Finally, we note that our surfaces are an example of a
that the rate of increase of ρ is not a constant but is al- system where multiple random steps (not just one) are
waysoforder10−6 forr =2×10−6,i.e., (∂ρ/∂t)≈r . required before the event of interest occurs. A number
D D
We can estimate (∂(∆F∗/kT)/∂ρ) by assuming that of erosion steps is needed, followed by a nucleation step.
∆F∗ approximately halves when ρ becomes of order r∗. Systems where multiple steps are required may generi-
At the supersaturation 2h/kT = 0.12, the initial nucle- callyresultincumulativeprobabilitiesP(t)thataresim-
ation rate on a flat surface is 1.76×10−8±0.74×10−8 ilar to those in Fig. 3. An example in a very different
(fromFFSsimulations). Takingthebarrierforaflatsur- context,butwithaP(t)ofasimilarform,isthatoflung
face ∆F∗/kT ≈ ln(r0) ≈ 18. As r∗ ≈ 8, we then have cancer. The probability that a smoker does not have
(∂(∆F∗/kT)/∂ρ) ≈ (18/2)/8 ≈ 1. Thus, our estimate lung cancer has a similar form to our P(t)’s in Fig. 3;
for λ at r = 10−6 is λ ≈ 10−6, which is close to the see Refs. 34 and 35, where they plot the probability of
D
best fit value. dying of cancer, equivalent to 1−P(t). Smoking greatly
Equation (4) gives a standard deviation of nucleation increases the probability of getting lung cancer, but for
times σ = (π/61/2)/λ (obtaining this expression re- typical smokers who start when young, this risk remains
quires extending the integration over −∞ < t < ∞ low until their 40s [34, 35], after which it rises rapidly.
but for our parameter values this is a very good ap-
proximation). Thus experimental measurements of the
spreadofnucleationtimesdirectlymeasurethetimescale VI. CONCLUSION
for the increase in nucleation rate. The ratio of the
standard deviation to the mean, ht i, is given by
N Nucleation occurs on surfaces, and in practice all
σ/htNi = (π/61/2)/(ln(λ/r0) − γ), for γ ≃ 0.577 the surfaces change over time, it is simply a question of
Euler-Mascheroni constant. In other words, for large
whether this change is faster, slower, or comparable to
λ/r0,σ/htNiscalesas1/ln(λ/r0),i.e.,decreasesbutonly the timescale of the experiment. Almost all prior work
asthelogoftheratio. Thusthepredictionisthatforsys-
studying nucleation has implicitly assumed that it is
tems with low initial rates, r0, the ratio σ/htNi should muchslower. Here we introduced a simple model for nu-
be significantly less than one, but due to the logarith-
cleation on a slowly dissolving surface, and showed that
mic dependence, values will presumably almost always
it predicts a characteristic P(t) (Fig. 3). This should
bearound0.1orabove. Verysmallvaluesarenotachiev-
be straightforward to observe in experiment. It is also
able.
indicativeofamechanismdeterminingthetimeuntilnu-
There is experimental data in which σ/ht i < 1, for
N cleation,thatisfundamentallydifferenttothatpredicted
example the work of Fasano and Khan [32] on the crys-
by classical nucleation theory. Here, the time until nu-
tallisationfromsolutionofcalciumoxalate. Asmallvalue
cleation is the time taken for dissolution to increase the
ofthisratioischaracteristicofasystemwithanucleation
nucleation rate to the point that it is fast enough to oc-
rateincreasingwithtime. Inthecalciumoxalatesystem,
curbeforefurtherchangeatthesurface. Thisisofcourse
thesurfacethecalciumoxalatecrystalsarenucleatingon
qualitatively different from the mechanism in classical
may be changing with time in such a way as to increase
nucleation theory, where the nucleation time is the time
the rate of nucleation at this surface.
for a rare thermal fluctuation to occur.
A P(t) consistent with a nucleation rate increasing
with time has been in observed by Kim et al. [17]. Our
V. GENERALITY OF OUR RESULTS AND modelprovidesapossiblequalitativeexplanationforthis
COMPARISON WITH THE INITIATION OF observation. Toldy et al. [16] also see a similar P(t), in
CANCER
that case for glycine crystallising in solution, but there
thecrystallisingdropletsmaynotbeindependentofeach
Our Eq. (4), by definition, will apply to any process other. Theysometimesobservethatonceonedroplethas
in which the nucleation rate varies exponentially with crystallised, then neighbouring droplets may crystallise.
time. Our model of a slowly dissolving surface is just Thismeansdropletsarenotindependent,whichcangive
one member of a class of systems, defined by having an an effective nucleation rate for the ensemble of droplets
5
that increases with time, even when the rate in an iso- willbeneededtobetterunderstandhowcrystalsnucleate
lateddropletmaynotbeincreasingwithtime. Thiseffect on a time-dependent rough surface.
complicates understanding the nucleation behaviour.
Further experimental studies will be needed to under-
standwhatsortsof crystallisingsystems havenucleation
rates that vary with time, and to understand what de-
ACKNOWLEDGMENTS
terminesthe keyparameter: the rateatwhichthe nucle-
ationrateincreaseswithtime,λ. Ifwearetopredictand
controlsystemswithtime-dependentnucleationrateswe ItisapleasuretoacknowledgediscussionswithSathish
will need to determine what are the physical processes Akella, Seth Fraden, James Mithen, and Patrick War-
that control this time dependence. Finally, computer ren. I acknowledge financial support from EPSRC
simulations of crystallisationin simple off-lattice models (EP/J006106/1).
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