A two-step mechanism for crystal nucleation without supersaturation

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[journal], [year], [vol], 00–00 | 1

This journal is © The Royal Society of Chemistry [year]

A two-step mechanism for crystal nucleation without supersaturation

Tamás Kovács a and Hugo K. Christenson*a

DOI: 10.1039/b000000x [DO NOT ALTER/DELETE THIS TEXT] 5

There is currently considerable interest in two-step models of crystal

nucleation, which have been implicated in a number of systems including

proteins, colloids and small organic molecules. Classical nucleation theory

(CNT) postulates the formation of an ordered crystalline nucleus directly

from dilute vapour or solution. By contrast, the new models explain how 10

crystallisation via a more concentrated but still fluid (disordered) phase can

lead to a significant enhancement of nucleation rates. In this article, we

extend recent work showing that crystal deposition from vapour can also be

greatly accelerated by the operation of a two-step mechanism. The process

relies on a very acute, annular wedge, in which restricted amounts of liquid 15

condense below the bulk melting point Tm. Crystals then nucleate in the

liquid condensates at sufficient temperature depressions ∆T (typically ≥ 30

K) below Tm, followed by rapid growth of these crystals from the saturated

vapour. By using a range of model substances (neo-pentanol, norbornane,

hexamethylcyclotrisiloxane, hexachloroethane, menthol, cyclooctane and 20

pinacol) we show that this is a viable mechanism for substances with

reasonably high absolute vapour pressures (> ca. 1 mm Hg). The lack of

appreciable crystal deposition with substances of significantly lower vapour

pressures (< ca. 0.01 mm Hg) is most likely due to geometrical restrictions

impeding diffusion in our experimental set-up. The results confirm the 25

feasibility of a mechanism for atmospheric ice nucleation that has been

suggested in the literature. Furthermore, there are thermodynamic analogies

with the crystallisation of biominerals via amorphous or fluid-like

precursor phases and protein nucleation in surface topographical features.

1 Introduction 30

Nucleation of solid from vapour is of great importance in cloud formation and

atmospheric precipitation, both on earth and in other planetary atmospheres. It also

plays an essential role in industrial technologies that use chemical or physical

vapour deposition, and an understanding of this process is essential for its control.

The formation of crystals from vapour typically starts with heterogeneous 35

nucleation on a surface such as an impurity particle or the container wall. Although

the free energy barrier is much reduced compared to the case of homogeneous

nucleation, supersaturations of 25-50 % are often required, e.g. for the nucleation of

ice on solid aerosol particles1. Classical nucleation theory (CNT) relates the free

energy barrier towards nucleation to the surface free energy cost of forming the 40

nucleus of a new phase. Although originally devised for the case of liquid

condensing from vapour, it has routinely been applied to the nucleation of crystals

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2 | [journal], [year], [vol], 00–00


both from solution and from vapour. Despite the added complications due to the

anisotropy of the crystalline state, CNT has had some success in qualitatively

accounting for experimental observations.

However, it has become clear that there are many instances where the simple

picture provided by CNT does not agree with experiment or the results of computer 5

simulations2. The new phase does not form simply as a result of a random

fluctuation that brings together a sufficient number of molecules to create a nucleus.

Rather, the nucleation occurs in two stages; first a denser aggregate of molecules

forms and then within this aggregate the actual nucleation takes place. Further

growth of this crystal nucleus then reduces its free energy. In many systems the 10

formation of a denser, metastable fluid state from a dilute solution is kinetically

favoured. The crystalline phase may then nucleate from this metastable state, and the

phase transition to a thermodynamically stable phase thus proceeds via a two-step

mechanism.

The idea of two-step nucleation has been successfully applied to solutions like 15

proteins2-4, small organic molecules2, 4, 5 and colloids6. According to simulations of

protein crystallization7, 8, the first step is the formation of the solute cluster which is

then followed by its reorganisation into a more ordered structure. As the

rearrangement time increases significantly with molecular complexity it was

suggested that the rate-determining step is the reorganisation. 20

Crystallisation from vapour requires the surmounting of a particularly high free

energy barrier due to the large free energy of a crystalline surface in vapour.

However, in the one-step mechanism of CNT changes in the surface topography that

increase the surface-nucleus area can facilitate solid nucleation.9, 10 On the other

hand, as liquids usually have lower surface energies than solids undercooled liquid 25

often deposits from vapour below the bulk melting point Tm. In wedge-shaped

(grooves or conical pits) condensation of liquid, both above and below Tm,. may take

place without any free energy barrier whatsoever, provided that the contact angle of

the liquid on the solid is less than half the wedge or cone angle.10

Condensation of liquid may occur even from undersaturated vapour in fine pores 30

and is termed capillary condensation. It takes place because the liquid phase is

always stabilised relative to the vapour phase in a sufficiently narrow pore as long as

the liquid has a contact angle θ below 90° on the pore walls. This is in practice not a

very restrictive condition and most substances that are liquid under ambient

conditions have low contact angles. The important exceptions are water on 35

hydrophobic surfaces and metals like mercury. A θ < 90° implies a liquid-vapour

interface that is concave towards the vapour phase, and the vapour pressure over this

is lower than over a flat surface. Consequently, the pore-held liquid is in equilibrium

with undersaturated vapour, which is quantitatively described by the Kelvin

equation, 40

)/ln( s

lvM

ppRT

Vr

γ= (1)

where T is the temperature, ps is the saturation vapour pressure, VM is the molar

volume of the condensing substance, γlv the surface tension of the liquid and r the 45

total radius of curvature of the interface – negative for a concave interface. When p

= ps the liquid-vapour interface has zero curvature (r = ∞) and we have the case of

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bulk liquid. Capillary condensation leads to the absorption of vapours by porous

media and hence plays an important role in the processes of drying and moisture

retention in soils, construction materials3, 11 and other porous bodies. It leads to

clogging of finely divided grains and powders at high humidities.

Capillary condensation is a first-order phase-transition and is in general subject to 5

hysteresis like condensation of bulk liquid and crystallisation. In a wedge-shaped, or

conical pit, however, a capillary condensate can grow continuously and reversibly

from the inside of the vertex – as discussed above in the context of the free energy

barrier towards nucleation in a surface cavity.

Below Tm the stable bulk phase is usually a crystalline solid. Just like in the case 10

of liquid and vapour the proximity of two surfaces favours the liquid and is for small

enough pores sufficient to stabilise the liquid relative to the crystal. The liquid

condensate grows until gain in interfacial energy by keeping the condensate liquid is

balanced by the unfavourable entropy of melting. One of us has shown that the

equilibrium radius of curvature r of the liquid-vapour interface of the supercooled 15

condensate is given by12

+

∆+∆−∆∆

=

s

m

m

pmfus

mlvM

p

pRTT

T

T

T

TCTHT

TTVr

lnln1

)(γ (2)

Here ∆T = Tm-T (the undercooling), ∆Hfus is the heat of fusion, ∆Cp is the difference 20

in the molar heat capacity of the liquid and solid, γlv is the (temperature-dependent)

surface tension of the liquid. At saturation (p = ps ) this expression reduces to

∆+∆−∆∆

=

m

pmfus

mlvM

T

T

T

TCTHT

TTVr

ln1

)(γ (3)

25

If the temperature dependence of γlv and ∆Hfus is neglected the expression becomes

very simple;

fus

mlvM

HT

TVr

∆∆=

γ (4)

30

So below Tm the radius of curvature of the condensate is inversely proportional to

the undercooling, instead of being inversely proportional to the ln[p/ps] as above Tm

(the Kelvin equation). In both cases r is equal to VMγlv divided by the difference in

free energy between the condensed liquid and the bulk liquid, i.e. RT ln[p/ps] above

Tm and ∆T∆Hfus/T = ∆T∆S below Tm. 35

The above relationships for the curvature of the liquid-vapour interface both

above and below Tm have been experimentally verified in a series of experiments

with the surface force apparatus (SFA).12-14 This instrument was originally designed

to measure forces between two molecularly smooth mica surfaces in air15 and

liquids.16-18 Multiple-beam interferometry, which is used to measure the surface 40

separation in the SFA also permits the refractive index of the medium between the


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mica surfaces to be determined.19, 20 This makes the instrument ideal for the study of

phase changes in pores, such as capillary condensation or freezing-point depression.

The two mica surfaces are in a crossed-cylinder configuration, which is equivalent to

a sphere-on-a-flat, so that when the two surfaces are in contact an annular, wedge-

like pore is created around the contact point. In this pore capillary condensation 5

may be studied under hysteresis free conditions, but if the surfaces are separated the

local environment approximates a slit-pore, which allows hysteresis effects to be

investigated. Both types of pore coexist with a bulk reservoir in a sealed and

temperature-controlled chamber.

One of us has considered previously the conditions under which the size-limited 10

liquid capillary condensates formed below Tm might freeze.12, 13 A simple argument

based on a comparison of surface and interfacial free energies and reasonable

assumptions about the wetting behaviour of the liquid and the crystalline solid has

suggested that a crystalline condensate would be thermodynamically stable in the

outer part of the wedge as long as γlv > 2γsl, the interfacial tension between the 15

liquid and the crystalline solid. A kinetic argument based on homogeneous

nucleation according to CNT would change this to γlv > 3γsl. These conditions

should be easily met by most liquids, except possibly the second one for water13.

Experimentally, however, no freezing could be observed for cyclohexane down to 13

K below Tm12, or for water down to 9 K below Tm

13. This is hardly surprising as 20

undercoolings of 30-40 K are frequently required for homogeneous nucleation in

pure liquids, and the mica surface is unlikely to significantly promote the nucleation

of water or cyclohexane.

Very recently21 we have shown that capillary condensates will indeed freeze for

large enough ∆T, and that this is followed by the deposition of large amounts of 25

solid, since there is no limit on the growth of the stable bulk phase, unlike the case

with the liquid condensates. In our preliminary study21 deposition of solid from

vapour was observed obtained at large enough undercoolings (∆T > 18 K for neo-

pentanol, ∆T > 33 K for HMCTS and ∆T > 37 K for norbornane) and it was

concluded that deposition of crystals from vapour via liquid condensates is possible 30

when the absolute vapour pressure is high enough (> 7 mm Hg) – one of the reasons

why the above substances were chosen for study.

In an earlier study one of us has shown that capillary condensates of long-chain n-

alkanes confined between mica surfaces will freeze quite easily if the mica surfaces

are separated and the condensate forms a liquid bridge22, 23. This can be related to 35

the surface freezing of these alkanes at the liquid-vapour interface, which means that

this interface very readily nucleates the bulk crystal. For the same reason these

long-chain alkanes will not supercool to any significant extent and their nucleation

behaviour is anomalous.

We here present further investigations of crystal nucleation from vapour via 40

capillary condensates using four additional substances (cyclooctane, menthol,

pinacol, and hexachloroethane), some with significantly lower absolute vapour

pressures (down to ca 10-3 mm Hg at the lowest experimental temperatures). These

experiments confirm the generality of the phenomenon but suggest that diffusion-

related effects play a key role in limiting this type of nucleation if the geometry is 45

too restricted.


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2 Experimental

The experiments were carried out with a simplified Surface Force Apparatus (SFA).

A detailed description of this instrument has been given24-26 and we here restrict

ourselves to a brief summary.

For each experiment 2-5 µm thick mica sheets are cleaved from a 0.1-0.5 mm 5

thick mica block (Paramount Corp., New York) and cut into approximately 1 cm2

squares with a hot platinum wire.27 The pieces are then coated with 50 nm Ag

(99.99%, Advent) by thermal evaporation at p = 10-6 mbar. The silvered mica sheets

are glued onto cylindrically polished silica discs (radius of curvature R = 2 cm)

using an epoxy resin (Epikote 1004), with the lower disk attached to a rigid support. 10

White light from a 150 W 21 V halogen bulb is directed onto the opposing, back-

silvered mica surfaces using an optical fibre and the transmitted light is resolved

into discrete wavelengths (FECO – fringes of equal chromatic order19) with a

monochromator (McPherson, model No. 2035). The interference fringes are

recorded with a CCD camera (Perkin-Elmer Pixcellent) and stored digitally. The 15

apparatus is housed in a thermostated enclosure that allows the temperature to be

controlled and measured to within ±0.1 0C over the range of –10 to +50 0C, with a

platinum thermometer placed 2 cm from the surfaces. The separation of the surfaces

is controlled with a piezoelectric device to within ±0.2 nm and measured with

multiple-beam interferometry. 20

Fig. 1 Left: Schematic cross-section of mica surfaces shown in the equivalent sphere-on-a-

flat configuration, with a liquid capillary condensate around the flattened contact region.

Typically, R = 2 cm, a0 ~ 25 µm, r2 ~ +13-50 µm, h/2 ≈ r1 ~ -(5 nm –1 µm). Right: Close-up

of the surfaces in contact, with the lower surface on a rigid support and the upper on a

piezoelectric actuator. 25

At the beginning of each experiment the surfaces are brought into contact in a N2

atmosphere at 22 0C in order to establish the zero of separation, or mica-mica

contact. Empirical corrections are made to account for the thermal expansion of

mica (more properly changes in the optical path length – typically corresponding to 30

10-5 K-1), by determining mica-mica contact as a function of temperature. The

substance to be studied is then introduced, with at least five times the amount

required to saturate the chamber added. In order to achieve faster equilibration the

temperature is usually set to 30 0C and 2–4 days are allowed for equilibration.

Analytical purity substances were used (cyclooctane: 99%, , neo-pentanol: 99%, 35

Sigma-Aldrich, norbornane: 98%, Sigma-Aldrich, hexamethylcyclotrisiloxane,

henceforth abbreviated HMCTS: 98%, Arcos Organics, menthol: 99%, Sigma-

Aldrich, hexachloroethane: 98%, Alfa Aesar, pinacol: 99%, Arcos Organics). All


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experiments are carried out in the presence of drying agent (molecular sieves, 1.6

mm, Sigma-Aldrich).

A typical experiment involves bringing the surfaces together slowly until they are

pulled together by the formation of a capillary condensate at separations of about 15

nm, although this was not accurately measured. The growth of any annular 5

condensate around the flattened contact zone is then followed, while the contact

separation and the refractive index of the film between the two surfaces is measured.

The condensate size h is determined from the discontinuities in the interference

fringes due to refractive index changes at the condensate vapour interface (Fig. 2).

The radius of curvature of the condensate-vapour interface r is given by12, 28 10

htr =+ 32 (5)

where t is the adsorbed film thickness of the substance on the surfaces at large

separations. It has been shown that an average adsorbed film thickness for many 15

substances below Tm is about 1 nm,12, 29 and we used this value, except where

otherwise noted.

Fig. 2 Interference fringes of mica surfaces in neo-pentanol vapour. The fringes are doublets 20

due to the birefringence of mica, and the wavelength increases towards the left. a) After 2 h in

contact in neo-pentanol vapour at ∆T = 33 K with solid annulus of h ≈ 200 nm showing

discontinuous shifts in the wavelength at the condensate-vapour interface b) after large jump

apart, showing annular, solid residue on the surfaces, c) after 2 h in contact in neo-pentanol

vapour at at ∆T = 28 K with discontinuities barely visible (only on the right-hand fringe) due 25

to the small size of the liquid annulus, d) after separation from contact with very small liquid

bridge in the centre.


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After varying times (from a few s to 3 h) in contact the surfaces are separated and

the outwards jump measured. The surfaces are then left well apart to let deposited

material evaporate, and a number of repeat cycles carried out. When measurements

are concluded at one temperature overnight equilibration is allowed at each new

temperature. 5

The phase state of the condensate is established by monitoring the behaviour of

the condensate on attempting to separate the surfaces. With a liquid condensate the

contact diameter decreases as the load on the surfaces is decreased and as the

surfaces finally jump apart the annular condensate becomes a bridge joining the

surfaces. The adhesion or pull-off force F, when normalised by R, is related to the 10

surface tension of the liquid γLV by F/R = 4π γLV,24 i.e. which translates to typical

outward jumps of 40-60 nm and normalised forces of 0.25-0.4 Nm-1. On further

separation the bridge snaps and droplets spread on each of the two surfaces. By

contrast, a solid condensate does not flow and the contact diameter cannot decrease

as the surfaces are effectively bonded together by the solid. Instead, noticeable 15

deformations are evident on the fringes, at separations beyond the location of the

condensate. The force that has to be used to pull the surfaces apart is an order of

magnitude larger than with the liquid condensates.

Substance Tm

(°C)

psat

(mm Hg) ∆∆∆∆Hfus

(kJ kg-1) ∆∆∆∆Hsubl

(kJ kg-1) γγγγ

(mNm-1)

crystal

structure

Cyclooctane 14.530 3.0–5.431

15–25 0C 21.531 523.132 31.014

simple cubic33

(plastic)

Neo-pentanol 52.530

9.9–63.034

25–56 0C

47.522 635.135 14.836

(53 °C) fcc37 (plastic)

Hexamethyl-

cyclotrisiloxane 64.530

4.2–8.938

24–34 0C 69.638 248.139 16.440 Trigonal41

Norbornane 87.530 27.7–47.042

25–350C 45.242 415.943 31.344 hcp45 (plastic)

(-)-Menthol 43.030

2×10-3–0.0746

0–26 0C 76.122 613.047 28.348 Trigonal 49

Racemic menthol 34.0 50 3×10-3-0.0646

0–26 0C 6646 50446

Pinacol 43.330 0.37 at 25

0C51 124.552 682.052, 53 28.154 Monoclinic 55

Hexachloro-

ethane 186.830

0.32–1.4956

at 25–45 0C 41.2 57 248.758

37.5 at

200C 59

Orthorhombic

(< 45 0C)60

20

In order to study the effect of supersaturating the vapour phase a specially designed

side plate to be bolted to the main chamber was constructed. The crystals of the

substance under study were placed in a stainless steel vessel on the inside of this

plate, about 6 cm from the surfaces. The vessel was heated by external thermistors

that allowed temperature control to ± 0.1 0C. The experiments were carried out in 25

the same way as in the case of saturation but with the source crystals at a positive

temperature difference with respect to the surfaces. The vapour pressure at the

surfaces p was calculated using the Clausius-Clapeyron equation,

−

∆=

TTR

H

p

p sub 11ln

00

(6) 30

where (p/p0) is the ratio of the vapour pressure at the surface to that at the source

crystal, T is the temperature at the surface and T0 the temperature of the vapour


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source. The supersaturation (S) is then S = 1-(p/ps). ∆Hsub is the sublimation

enthalpy, which is 81 kJ/mol for pinacol and 96 kJ/mol for menthol (see Table).

Temperature differential of 2-6 K between the bulk phase and the surfaces were used

and this corresponds to supersaturations of 0.3 – 1.2.

3 Results 5

In what follows observations with the different substances will be described

sequentially. Table 1 lists some relevant physical properties of the substances used.

3.1 Cyclooctane

The cyclooctane condensates remained liquid down –7.5 °C , or ∆T = 22 K, and in

view of results with other substances it is likely that the temperature depression was 10

insufficient to give nucleation of solid. However, the vapour pressure at the lowest

T used is an order of magnitude less than which gave solid deposition with

HMCTS.21 The condensate size h (290 nm) determined at 18 °C, above the Tm of

14.3 °C, was used to calculate a minimum relative vapour pressure p/p0 = 0.993

using the Kelvin equation. Given the slow rate of growth of these condensates it is 15

difficult to ascertain whether or not the equilibrium condensate size has been

reached, as found in previous studies of cyclooctane below Tm.14 The condensate

sizes at six different temperatures below T are plotted in Figure 3. Eq. 3 accounts

accurately for the r of the condensate for ∆T < 15 K, but the reason for the deviation

for larger ∆T is uncertain. 20

Fig. 3 Inverse radii of curvature 1/r (= 2/[h-3t], see text) of the condensate-vapour interface

for liquid menthol and cyclooctane condensates as a function of temperature ∆T below the

bulk melting point. Note the different scales for menthol (right) and cyclooctane (left)

25


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3.2 Racemic menthol

The condensate sizes measured at five different temperatures below Tm with racemic

menthol are also shown in Fig. 3. The condensates remained liquid down to 10 °C

(∆T = 25 K) as shown by their flow and deformation properties, and the adhesion

forces F/R ≈ 0.4 Nm-1. However, the fringes showed that the surfaces became more 5

deformed during separation than at higher temperatures. This might indicate

differences in the state of the adsorbed films on the surfaces - perhaps they became

more solid-like, therefore altering somewhat the adhesion-deformation properties of

the surfaces.. At 1 °C ((∆T = 33 K) no liquid condensate or bridge could be

identified at any time, although F/R increased to ca. 0.8 Nm-1. Separation–approach 10

cycle indicated only the presence of traces of material trapped between the surfaces

at contact. The radius of curvature of the condensate-vapour interface decreased

from 13 to 3.5 nm as ∆T increased from 8 to 27 (Figure 3). The r values at small ∆T

are close to the theoretical predictions but at larger ∆T values they are smaller than

expected (larger 1/r), possibly due to a smaller adsorbed film thickness at the lower 15

temperatures, which would influence r according to eq. 5. Indeed, decreasing the

estimated adsorbed film thickness per surface gives better agreement for the lower

temperatures (Figure 2).

3.3 (1R, 2S, 5R)–menthol

Optically pure (1R, 2S, 5R)-(-)-menthol (L-menthol), which has a higher Tm (42 0C) 20

and a lower psat than the racemic compound (Table) was investigated over the range

17 < ∆T < 46 K. No condensates could be observed from the interference fringes,

but the vapour pressure at ∆T = 46 K is only about 1 × 10-3 mm Hg. When the

apparatus was opened at the end of the experiment ca. 1 mm long, needle-like

menthol crystals were found to have deposited at random places on stainless steel 25

surfaces, suggesting that the atmosphere in the chamber was indeed saturated. No

crystals of menthol (or any other substance) were ever observed to deposit on the

flat mica surfaces, away from the annular wedge. The fact that crystal nucleation

from saturated vapour occurs on the inside of the chamber walls is not surprising

given that the vapour will there be locally supersaturated during cooling cycles. 30

3.4 Neo-pentanol

Experiments were carried out for 8 K< ∆T < 36 K, and have been described previously.21

For 8 K < ∆T < 18 K the condensates were always liquid (Fig. 4), and for ∆T > 34 K the

nucleation and susequent deposition of crystalline material was so rapid that any liquid

condensate cannot be detected. Over an intermediate temperature range (18 < ∆T < 34 K) 35

the liquid condensates could be induced to solidify by mechanical perturbation of the

surfaces. The most effective way was found to be the initial application of an additional

load on the surfaces, thereby increasing the contact diameter, followed by a reduction in

load. The consequent reduction in contact diameter was then immediately followed by a

dramatic increase in the rate of deposition from vapour, and the subsequent properties of 40

the condensate showed clearly that it was solid. The normalised adhesion with liquid

condensates was typically 0.3 Nm-1, and 2.5 Nm-1 with the solid condensates. After the

experiment small crystals of neo-pentanol were found to have deposited on the inside of

the chamber top, showing that the neo-pentanol vapour was saturated.

45


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Fig. 4 The inverse of the radius of curvature, 1/ r (= [2/h-3t]) of the condensate-vapour interface

for condensates of neo-pentanol HMCTS, norbornane and hexachloroethane (inset) as a function of

the temperature ∆T below the bulk melting point. Note the dramatic decrease in 1/r (increase in r)

for the solid deposits compared to the liquid condensates.

5

3.5 Norbornane

As reported21 rapid deposition of solid occurred with norbornane (bicyclo[2.2.1]heptane)

as soon as contact was achieved at all ∆T > 33 K (the maximum attainable temperature).

The adhesion between the surfaces was large, with outward jumps of 750 ± 70 nm, which

corresponds F/R ≈ 4-5 Nm-1. 10

These solid condensates grew at an increasing rate with temperature (Figure 5),

although clearly not just in proportion to the absolute vapour pressure. Even after

40 min there was no sign of a levelling out of the curves. Note that the absolute

vapour pressure of norbornane is higher than that of any of the other substances

studied. The condensate volumes, V were calculated from h using22 15

2

RhV π≈ (7)

Observation of the surfaces from above through a microscope showed that the

norbornane condensates were often overall hexagonal in shape, albeit with a locally 20

rough outline. This could be due to epitaxy directed by the pseudohexagonal symmetry

of the mica basal plane, and/or to close-packed structure of the norbornane, which is

h.c.p. below a transition temperature of 33 °C, and f.c.c. above.45 On attempting to

separate the surfaces the solid norbornane condensate broke into fragments which then

evaporated, confirming that the vapour phase was not supersaturated. The plastic crystal 25

phase that norbornane forms is quite deformable, hence the rather rounded shapes of the


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[journal], [year], [vol], 00–00 | 11


fragments. Note that due to surface tension effects a liquid bridge of this size always

snaps into two distinct droplets and never fragments.

Fig. 5 Condensate growth rates in norbornane vapour. The left hand scale refers the condensate

at 5.8 0C (∆T = 81.7 K), while the right-hand scale to the condensate at at 34.7 0C (∆T = 52.8 K).

5

Fig. 6 Microscope image from above of a solid norbornane deposit in the annular wedge

between crossed mica cylinders formed in saturated vapour at ∆T = 62 K, showing growth after

initial contact (a,b), rupture of crystal on separation (c) and subsequent evaporation of the

norbornane fragments (d-f). The brighter centre region is mica-mica contact, and the norbornane 10

vapour interface is an irregular hexagon.


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3.6 Hexamethylcyclotrisiloxane

As reported previously21, both liquid and solid condensates were observed with

hexamethylcylotrisiloxane. At undercoolings ∆T < 33 K very small liquid condensates

formed around the contact zone, while for ∆T ≥ 34 K solid nucleated in the condensates.

The size h and interfacial radius of curvature r of the liquid condensates could only 5

determined by estimating the size of liquid bridge directly after separation and then

applying eq. 7 on the assumption of unchanged volume (hence the large error bars in Fig.

4). The solid HMCTS deposits gave a very large adhesion between the surfaces with

outward jumps of 1.1–1.9 µm measured, corresponding to normalised adhesion forces of

7-10 Nm-1. 10

3.7 Pinacol

Pinacol (2,3-dimethyl-2,3-butanediol, Tm = 43 °C) was investigated over the ∆T

range of 10–44 K at four different temperatures. As with menthol no condensate

could be observed from the fringes, even after 3 h in contact, although the shift in

contact from the value in nitrogen and the shapes of the fringes during separation 15

pointed to the presence of some adsorbed material on the nanometre scale. The

vapour pressure of pinacol at -1 °C (∆T = 44 K) is estimated from eq. 6 and the

value at 25 °C (Table 1) to be only 10-2 mm Hg, but as with menthol crystals were

found to have deposited on the inside of the chamber at the end of the experiment.

3.8 Hexachloroethane 20

Hexachloroethane has the highest melting point (186 0C) of the substances studied so

these experiments could only be carried out for ∆T > 143 K (23–43 °C). Over this entire

range reasonably large condensates formed immediately after contact was achieved and

their size (inverse interfacial radius of curvature) after 3 h in contact is shown in Fig.4.

Their solid nature was demonstrated by the lack of flow of the condensates on decreasing 25

the load and the large jump out distances of 220–340 nm (F/R ≈ 1.3 – 2. 0 Nm-1. The

condensate size increased with temperature, as found with norbornane.

3.9 Condensation from supersaturated vapour

Slight supersaturations of about 0.3 were induced by applying a 2 °C temperature

difference between the bulk crystal phase (the vapour source) and the surfaces. At these 30

supersaturations deposition of solid was observed with pinacol at ∆T = 38 K, but only

after perturbing the surfaces by first increasing the load on them and then unloading

them, as described earlier with neo-pentanol. The condensate grew to h ≈ 130 nm after 30

min and then remained unchanged for up to 3 h). No solid deposition was observed with

menthol, even at supersaturations of up to 1.2 at ∆T = 42 K (6 K temperature 35

differential).

4 Discussion

The results show beyond doubt that nucleation of crystals in capillary condensates

provides a means of facilitating crystal deposition from vapour. No crystals were

ever observed on the flat mica surfaces outside the annular condensate. There is a 40

very definite correlation between the absolute vapour pressure p and the amount of

crystalline solid that deposits. By far the fastest deposition occurs with norbornane,

which has the highest p, followed by neo-pentanol and then HMCTS, in order of

decreasing p. The precise value of ∆T necessary for nucleation is lower at 18 K for


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[journal], [year], [vol], 00–00 | 13


neo-pentanol than for the other substances, where it is 33 K or larger, although no

lower bound could be determined for norbornane and hexachloroethane as solid

deposits at all experimentally accessible temperatures. The substances with the

lowest vapour pressure, +(-)-menthol and pinacol did not give rise to deposition of

anything more than nanometre-thick films, even at sustantial undercoolings ∆T. At 5

around 0 °C, where nucleation might reasonably be expected to occur, their vapour

pressure is less than 10-2 mm Hg, or three orders of magnitude less than that of

HMCTS when it nucleates in liquid condensates at 30 °C (∆T = 34). The lack of

solid deposition with cyclooctane and racemic menthol is most likely due to a large

enough ∆T not being attainable, as liquid condensates were in evidence except 10

possibly at the lowest temperature with racemic menthol. It is noteworthy that we

have oserved that both menthol and cyclooctane supercool very easily in the bulk.

Detailed investigation of the nucleation process in neo-pentanol, norbornane and

HMCTS has indicated the existence of three temperature regimes.21 For

undercoolings of ∆T ≤ 18 °C (neo-pentanol) or ∆T ≤ 33 °C (HMCTS) a liquid 15

condensate with limited size initially forms around the contact zone. No liquid

condensates were found with norbornane as the minimum undercooling achievable

with our system was still 37 K. At the largest undercoolings the behaviour was very

different, and with all of these substances the condensate grew rapidly immediately

after contact, and definitive evidence for the presence of a liquid condensate could 20

not be obtained. This is the behaviour that we have now found for C2Cl6 as well –

the large ∆T prevents the detection of any liquid condensate. In the intermediate

regime nucleation of solid appeared to be promoted by mechanical perturbation of

the condensates, by loading and unloading the surfaces or separating these from

contact. 25

The temperatures at which nucleation occurs in the liquid capillary condensates

is, with the exception of neo-pentanol, similar to typical homogeneous nucleation

temperatures in bulk liquids. Naturally, most experiments have been carried out

with water, and they point to typical nucleation rates of 1011 to 1013 m-3s-1 at ∆T =

35 K and 1013 to 1015 m-3s-1 at ∆T = 38 K61. The volume V of the annular capillary 30

condensates given by eq. 7 is approximately 10-17 m3 for ∆T = 35-38 K, which

translates to nucleation times of the order of µs – ms. It is hence possible that

classical homogeneous nucleation of crystals in the liquid capillary condensates

explains our results. Heterogeneous nucleation at the mica surface appears much

less likely, although it could perhaps account for the higher nucleation temperatures 35

(smaller undercooling) observed with neo-pentanol.

According to CNT the radius r* of a critical nucleus for nucleation from the melt

is given by

fus

mslM

v

slM

HT

TV

G

Vr

∆∆−=

∆−=

γγ 22* (8) 40

This value of r* differs from the r given by eq. 4 by a factor of 2, and the

replacement of γlv with γsl. This is related to the discussion of metastable liquid

condensates in the introduction, and it appears that our experiments vindicate the

thermodynamic argument that solid capillary condensates are stable in most cases, 45

and may under favourable conditions lead to deposition of large amounts of

crystalline material from vapour.


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The temperature of the experimental chamber is controlled and measured to an

accuracy of ± 0.1 K at best. We cannot guarantee that temperature gradients of this

magnitude are not present, and this leads to possible uncertainties in p/p0. Using the

Clausius-Clapeyron (eq. 6) with typical values of the enthalpy of sublimation of 40-

80 kJ mol-1 an error in T of 0.1 K translates into an error in p/ps of 0.006 to 0.013. A 5

liquid capillary condensate above Tm would grow infinitely large at saturation, but

with typical values of VM (150 cm3) and γlv (30 mJm-2) would be limited to a radius

of curvature r (≈ h/2) of 140-300 nm at 300 K. Since thermodynamically,

undersaturation should prevent the deposition of bulk solid as well, this may well be

part of the explanation for why the size and growth rate of the solid deposits varies. 10

Such an uncertainty cannot, of course explain why nucleation and growth of

crystals was not observed with some substances. Strictly speaking our technique

does not allow us to detect a nucleation event as such - it is rather the accelerated

deposition that follows the nucleation that we observe. We can also note the

increased adhesion that the solid gives rise to, but if only a very small amount of 15

solid is deposited this will not be readily evident. Clearly, if there is insufficient

material in the vapour phase growth will be severely curtailed, whatever the

mechanism behind the nucleation. More importantly, perhaps, the capillary

condensates in the annular wedge are in a very inaccessible location, with vapour

molecules having to diffuse long distances between two walls that are closer 20

together than the mean free path of the vapour molecules. In such a case the

diffusing molecules collide with the wall more frequently than with one another,

leading to a substantial slowing down of diffusion. The Knudsen number Kn is a

relevant measure of this type of restricted diffusion. It is defined as the ratio of the

molecular mean free path λ to the representative physical length scale L62, 25

Lp

Tk

LKn

tot

B

22πσ

λ== (10)

where σ is the molecular diameter and ptot is the total pressure (here ca. 101 kPa). If Kn is

much greater than one Knudsen diffusion is important. In our case it is reasonable to 30

take L as twice the condensate radius, since this represents the final width of the space

through which the molecules have to diffuse before the condensate proper is reached. L is

then of the order of 5 nm, and with σ = 0.5 nm we obtain Kn ≈ 10, meaning that Knudsen

diffusion is important and material transport into the wedge pore is limited. We can

estimate from the annular wedge geometry that the last 10-20 µm of diffusion before the 35

condensate is reached occurs through a slit-pore of less than twice the width of L

assumed above. This approximates a slit-pore with an aspect ratio of about 103, and the

diffusion must be impeded in the extreme. It is very likely that this accounts for why no

capillary condensates appeared to form with (1R, 2S, 5R)–menthol, pinacol, and the

racemic menthol at the lowest temperatures. 40

With a supersaturation of the vapour phase by 30% it was necessary to perturb the

surfaces to initiate the deposition of solid pinacol (h = 140 nm after 30 min) but with

(1R, 2S, 5R)–menthol even 120 % supersaturation gave no discernible condensation or

deposition, even after attempts were made to perturb the surfaces. The pronounced

difference caused by a relatively small increase in the vapour-phase concentration of 45

pinacol is perhaps surprising, as is the complete lack of an effect with the menthol. Over eighty years ago Volmer pointed out in a discussion of heterogeneous

nucleation63 that “embryos” (i.e. nuclei) could be retained in conical cavities on


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surfaces at temperatures above a phase transition and subsequently seed renewed

growth as the phase boundary was recrossed. Similar ideas were later discussed in

the context of atmospheric nucleation and it was proposed that capillary

condensation in conical surface pits could lead to ice deposition on solid aerosol

particles64, although no direct experimental evidence appears to have been 5

published. We believe that we have now shown that this may be a viable mechanism

of nucleation on insoluble aerosol particles, although further studies will have to be

carried out with better models systems. The vapour pressure over ice at temperatures

relevant to atmospheric nucleation at high altitudes varies from about 0.1 mm Hg at

–40 °C to 0.01 mm Hg at –60 °C,65 so the restricted diffusion in our present 10

experimental set-up would be problematic.

The obvious shortcoming of our experiments is that the annular wedge is too

acute and too narrow. In practice such very narrow pits or grooves in a surface

would be rare, so our model is not a completely realistic one. However, it is likely

that the possibility of nucleation via capillary condensation could only be greater 15

with a more open cavity. A larger wedge angle would not reduce significantly the

possibility of capillary condensation without any energy barrier, but make transport

of material for growth of the solid phase much easier. Future experiments should

concentrate on providing a better model system with more obtuse but more

accessible surface cavities. It should then be easier to establish definitively whether 20

nucleation in capillary condensates is a general phenomenon that explains the

enhanced nucleation often seen with rough surfaces.66

Crystal deposition from vapour via liquid condensates has many similarities to

recently discussed models of two-step nucleation from solution. In contrast to CNT

these models propose the initial formation of a denser, liquid-like phase (or clusters) 25

which then crystallises. This was originally found in simulations that showed that a

metastable critical point in colloidal systems with short-range attractive forces could

greatly enhance nucleation rates.7 This metastable critical point and the associated

denser fluid phase are remnants from systems with longer-range attractive forces

where a thermodynamically stable liquid-like phase does exist. The analogy to this 30

denser, liquid-like phase is obvious in our case. Liquid, which is unstable in bulk

below Tm is stabilised by the wedge and leads to nucleation of solid from vapour

without the requirement for any supersaturation. The nucleation rate is thus

significantly enhanced. Recent simulations have suggested that the rate of crystal

nucleation from vapour can increase by proceeding via liquid droplets67, 68 – an 35

analogous mechanism in homogeneous nucleation.

The formation of biominerals like calcite,69 aragonite, hydroxyapatite and even

calcium sulphate70 is known to proceed via an amorphous phase, i.e. a much denser

but in many ways still liquid-like phase. It would be interesting to investigate

whether or not suitable surface cavities with affinity for these precursor phases 40

might act to enhance nucleation rates of such crystalline biominerals from solution.

Simulations have suggested that protein nucleation is facilitated by “capillary

condensation” from solution in surface pits.71

5 Conclusions 45

Nucleation of crystals occurs readily in liquid capillary condensates formed from

saturated vapour below the bulk melting point of the substance, provided that the


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undercooling is large enough and that the condensates are of a reasonable size. The

crystal nucleation is followed by rapid growth of these crystals by deposition from

vapour if the absolute vapour pressure is high enough (above approximately 1-4 mm

Hg). In our specific experimental set-up it is likely that restricted diffusion is

hindering crystal nucleation and growth when the absolute vapour pressure of the 5

substance is too low (of the order of 1 mm Hg or less).

5 Acknowledgement

The Leverhulme Trust is thanked for supporting this project.

a Address, University of Leeds, School of Physics and Astronomy, Leeds, United Kingdom. Fax:+44 10

113 3433879 ; Tel: +44 113 3433900 E-mail: [email protected]

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A two-step mechanism for crystal nucleation without supersaturation

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