A Stochastic Model for Nucleation in the Boundary Layer during Solvent Freeze-Concentration

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Communication

A stochastic model for nucleation in the boundarylayer during solvent freeze-concentration

Geoffrey G. Poon, and Baron PetersCryst. Growth Des., Just Accepted Manuscript • Publication Date (Web): 27 Sep 2013

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layer thickness, and a constant solvent crystallization growth velocity. Whether heterogeneous

on the ice surface, or homogeneous in the boundary layer, the models suggest that nucleation is

dramatically accelerated by the growing ice. For methane hydrates, which form at conditions

similar to that of ice, induction times for hydrate nucleation can be reduced by as much as 10105

times because of the moving supersaturation zone.

TEXT

Freeze-concentration refers to phenomena in which a crystallizing solvent excludes and

thereby concentrates a solute. Freeze-concentration is important in atmospheric chemistry,1 zone

refining,2-4

fruit juice concentration,5 protein crystallization,

6 amorphous calcium carbonate

nucleation,7 and perhaps in the formation of Liesegang precipitation rings.

8 In many of these

processes, solvent crystallization (often ice formation in aqueous solutions) induces the

nucleation of a dissolved solute. This communication discusses the interplay between

spatiotemporal solute concentration profiles and nucleation kinetics which depend strongly on

the solute concentration.9

Pohl10

and Tiller et al.3, 4

showed that a localized boundary layer of high solute concentration

forms ahead of the growing ice front. This feature makes freeze-concentration an intriguing way

to control the conditions and location where nucleation occurs. For example, when the bulk

solution is saturated (not supersaturated), then solute nucleation can only occur at or very near

the ice surface at the freezing temperature. By contrast, nucleation in many other environments

is a messy affair. For example, it is difficult to be certain whether nucleation is homogeneous

(spontaneous organization of solutes in solution) or heterogeneous (catalyzed by some dust

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particle or defect in the glassware). Such difficulties in the mechanistic interpretation impede the

development of accurate nucleation theories. Freeze-concentration allows nucleation sites away

from the ice growth front to be excluded from consideration.

Previous investigators have noted that nucleation of gas bubbles in the moving ice front leaves

a record of nucleation events as bubbles trapped in ice.11-13

As the ice grows, solute accumulates

in a thin boundary layer ahead of the ice front until a nucleus forms. The nucleus grows creating

a bubble near the advancing ice. Eventually, the bubble depletes the local supersaturation and is

engulfed by the ice. The cycle continues as the ice continues to grow and solute accumulates

near the front again. Although nucleation is a stochastic process, previous investigators have

exclusively used deterministic models to understand the spacing and size of entrapped bubbles.11,

12

This communication presents a dimensionless stochastic model for nucleation within the

concentrated solute boundary layer formed by the moving front. We make several simplifying

assumptions. First, we only consider the case where solute is completely rejected from the ice.

Complete rejection is realistic when nucleation occurs at solute concentrations much lower than

those required to force solute inclusions into the ice. However, our general framework is still

valid when there is some finite solute incorporation in the ice if the concentration profile is

adjusted accordingly. Note that Smith et al. found the concentration profile when the solute has

a constant partition coefficient at the interface.4 Second, we assume vertical ice growth in a

perfectly quiescent environment with no convection. Note that even without forced convection,

buoyancy driven flows can result from density differences between the concentrated boundary

layer and the bulk solution.14

Third, we ignore freezing point depression and the possible

consequences of cellular/dendritic growth. Therefore, the analysis here is quantitatively accurate

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only for nucleation of sparingly soluble solutes.15

Fourth, we assume that diffusion is slower

than conduction, i.e. that the thermal boundary layer is thicker than the concentration boundary

layer. This fourth assumption is typically true for liquid solutions.16

Combining the third and

fourth assumptions implies that the entire concentration boundary layer is isothermal and at the

solvent freezing temperature.

Figure 1. A diagram of the semi-infinite freeze-concentration process in our analysis. We

compute the local concentration ��, �� in the solution ahead of the ice growth front when the

bulk solution is maintained at a constant concentration �� and the ice grows at a constant rate �.

If the planar ice-solution interface moves at a constant growth velocity �, if the solute

diffusion coefficient is , and if � is position relative to the moving ice-water interface (Figure

1), then the local concentration ��, �� satisfies � �⁄ � � � ⁄ � � � �⁄ . This equation

can be non-dimensionalized to yield

∆� �

∆� �

∆� �1�

where Δ��, �� ≡ �� ⁄ is dimensionless concentration enrichment, � ≡ �� ⁄ is

dimensionless time, � ≡ �� ⁄ is dimensionless position, and �� ≡ lim�→� ��, �� is the bulk

concentration. The complete rejection of solute by the growing ice corresponds to a no flux

boundary condition at z = 0, i.e. �Δ � Δ �⁄ �� 1. Calculations in this paper are for a bulk

solution that is perfectly saturated with solute, i.e. lim�→� Δ � 0 and initial condition Δ��, 0� �0. Fedorchenko et al.

17 obtained the concentration profile for � 0 and � ! 0 as

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Δ��, �� "�# exp '�� 4� ) � �1 � � � ��2 +,�erfc 0� � �2√� 2

� 12 erfc 0� � �2√� 2

�2�

Local supersaturation S��, ��, the solute concentration relative to the solubility limit, is easily

calculated from Δ��, ��. S��, �� 456 71 � Δ��, ��8 �3�

Figure 2 shows how the concentration profile develops with time after the onset of ice growth.

The growth rate of ice and the solute diffusivity determine the extent of supersaturation in the

boundary layer and the boundary layer thickness. For all but extremely short times, the

concentration boundary layer is approximately �⁄ thick. In this model, the interfacial

concentration continually increases without bounds until nucleation occurs.

Figure 2. The interfacial supersaturation :�;, <� (dashed curve) and its large < limit = � <

(dotted line) at large <, Several supersaturation profiles :�>, <� (solid curves) for corresponding

interfacial supersaturations are also shown.

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The dimensionless induction time for the first nucleation event �? is a stochastic random

variable that depends on the rate of nucleation @, which increases rapidly with supersaturation.9,

18, 19 Classical nucleation theory (CNT) predicts that @ is approximately

@ � Aexp 0� Bln D2 �4�

where A is a prefactor, B ln D⁄ � ΔE‡/HIJ is the dimensionless free energy barrier, and

D � �/�456 is the supersaturation.9 Although CNT predictions of A and B are notoriously

inaccurate, nucleation kinetics often do exhibit a linear relationship between ln @ vs. 1/ ln D.9, 19,

20 Equation (4) has also been used in the analysis of metastable zone width (MZW) data.

21-23

Stochastic models of MZW data are based on the Poisson survival probability, i.e. the probability

that nucleation has not yet occurred after waiting for a time �.21, 24 For homogeneous nucleation

in an observation volume K and with a time-variant supersaturation D��, the survival probability

is L�� exp7�M @��KN�6� 8 where @�� is the time-variant homogeneous nucleation rate.

21, 22, 25

Note that L�� exp7�M @��KN�6� 8 assumes a spatially uniform supersaturation, so it is not

valid for nucleation during solvent freeze-concentration.

The appropriate stochastic model for nucleation near or at the ice growth front depends on

whether homogeneous nucleation (HON) is occurring in the concentrated solution ahead of the

front or whether heterogeneous nucleation (HEN) is occurring at the ice-water interface. The

L�� for HON and HEN are described by Equation (6) and (9) respectively. The B parameter for

HEN depends on properties of the ice-water interface as well as properties of the solute in the

water. Additionally, the prefactor A for HEN has units nuclei/area/time instead of

nuclei/volume/time as in HON.

For HON, the stochastic survival probability analysis should consider the probability of

nucleation anywhere in the solution volume ahead of the ice growth front. However, the

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homogenous nucleation rate @OPQ vanishes at points far from the ice growth front when

�� 456⁄ � 1, and this is the special case that we model. The probability that nucleation has not

happened in the semi-infinite volume ahead of the moving front up to time t is

LOPQ�� exp '�RS N�′S N�@OPQ��, �′��

6�

) �5� where @OPQ��, �� AOPQ exp7�BOPQ/ ln D��, ��8 is the local HON rate from Equation (4) and

R is the area of the planar interface. We non-dimensionalize equation (5) by introducing the

Damkohler number ROPQ ≡ RAOPQ �V⁄ and our previously defined dimensionless variables.

LOPQ�� exp '�ROPQS N�′S N�exp 0� BOPQln D��, �′�2��

W�

) �6� LOPQ�� resembles an inverse sigmoidal curve. LOPQ�� 1 for small times and then rather

suddenly drops to 0 as D�0, �� crosses through the metastable zone. The probability distribution

for the moment of first nucleation Y�� NL N�⁄ is used to find the average dimensionless

induction time for the first HON event to occur ⟨�OPQ⟩.

⟨�OPQ⟩ � S N��YOPQ��

� S N�LOPQ��

�7� The final expression in Equation (7) is obtained using integration by parts. Using Equations

(2), (3), (6), and (7), we can predict the dependence of ⟨�OPQ⟩ on BOPQ and ROPQ and how

stochastic or deterministic nucleation times are. The relative 90% confidence interval width

(90% RCIWHON) for HON in the boundary layer is

90%_`abOPQ � ��c � �dc� �e?f⁄ �8� where LOPQ��dc� � 0.95, LOPQ��c� � 0.05, and �e?f � ��c � �dc� 2⁄ . When 90% RCIWHON

is small, the distribution of first nucleation times is narrow and nucleation can be treated as a

deterministic event. Conversely, a large 90% RCIWHON implies that the stochastic nature of

nucleation cannot be ignored. Numerical results for ⟨�OPQ⟩ ≡ ⟨� �OPQ ⁄ ⟩ and 90% RCIWHON

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are shown in Figure 3. Figure 3 reveals that nucleation is slower and more stochastic when

ROPQ is small, i.e. ice growth rate is fast. As expected, induction times increase exponentially

with BOPQ, i.e. barrier height.

Figure 3. (a) logk�⟨�OPQ⟩ as a function of logk� ROPQ and logk� BOPQ. Contours of constant

logk�⟨�OPQ⟩ are spaced evenly by 0.5. (b) Contours of constant 90% RCIWHON (spaced evenly

by 0.1) are plotted against logk� ROPQ and logk� BOPQ. The numerical accuracy in the yellow-

black-striped region is questionable because of the precision of our numerical integration in

Equation (7).

A similar survival probability analysis for HEN on the ice-water interface can provide average

induction times and relative confidence interval widths for HEN, ⟨�OlQ⟩ and 90% RCIWHEN.

The rate of HEN at the interface depends only on the time-dependent interfacial supersaturation

D�0, ��. A new Damkohler number for HEN, ROlQ ≡ RA � ⁄ , can be introduced to

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nondimensionalize the arguments of the survival probability. The probability of no

heterogeneous nucleation up to a dimensionless time � is

LOlQ�� exp '�ROlQS N�mexp 0� BOlQln D�0, �′�2W�

) �9�

As for HON, Equation (8) and (9) can be used to find ⟨�OlQ⟩ and 90% RCIWHEN respectively

if the HON subscripts are changed to HEN. Figure 4 shows ⟨�OlQ⟩ and 90% RCIWHEN as a

function of BOlQ and ROlQ. If both HON and HEN can occur and if both 90% RCIWs are

small, the nucleation mechanism with the lowest average induction time will tend to occur first.

Like HON, ⟨�OlQ⟩ and 90% RCIWHEN increase with decreasing ROlQ, and ⟨�OlQ⟩ has a strong

exponential dependence on BOlQ.

Figure 4. (a) logk�⟨�OlQ⟩ as a function of logk� ROlQ and logk� BOlQ. Contours of constant

logk�⟨�OlQ⟩ are spaced evenly by 0.5. (b) Contours of constant 90% RCIWHEN (spaced evenly

by 0.1) are plotted against logk� ROlQ and logk� BOlQ.

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The quasi-stationary rate laws in our stochastic models of nucleation require that the local

supersaturation is approximately constant over the length scale of the nucleus. Specifically, the

critical nucleus radius n‡ at the moment and location of nucleation should be much smaller than

the characteristic length of the boundary layer, i.e. n‡ ≪ �⁄ . According to the classical

nucleation theory (CNT), the critical size for a spherical nucleus in HON is

n‡ � 2p�q HIJ ln D⁄ �10�

where v0 is the volume per solute molecule in the nucleus, and γ is the interfacial free energy9.

The nucleation barrier is

BOPQln D �ΔE‡HIJ �

16#3p� ln D 0

p�qHIJ2V

�11�

The left equality is the definition of the parameter B, and the right equality is from CNT.

Equations (10) and (11) imply a relation that can be used to eliminate the difficult to ascertain

parameter n‡ in favor of BOPQ. Specifically, n‡ ln D � �3Bp� 2#⁄ �k/V for HON with spherical

nuclei.9 This gives us the requirement for validity for the spatially uniform approximation.

0ℓ�� 2 03BOPQ2# 2k/V ≪ lnD��⟨�?⟩� �12�

where ℓ� � p�k/V is approximately the solute molecular diameter. For a spherical cap nucleus

in HEN, the cap height s � n‡�1 � cos u� where u is the contact angle must be much smaller

than �⁄ .26

Since BOlQ � v�u�BOPQ, Equation (13) is modified to

0ℓ�� 2 �1 � cos u� 0 3BOlQ2#v�u�2k/V ≪ ln D��⟨�?⟩� �13�

where v�u� � �2 � 3 cos u � cosV u� 4⁄ . For a solute diffusivity of 10-9

m2/s and an ice growth

rate of 10-6

m/s, the boundary layer thickness �⁄ is 1 mm which is much greater than a typical

nuclei size. This indicates that the spatially uniform approximation is valid in many cases.

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Additionally, our model is only valid within the quasi-stationary regime, when the cluster size

distribution relaxes to the new stationary distribution quickly relative to the rate at which

supersaturation changes. Kashchiev showed that this requirement is satisfied when

|N ln D N�⁄ | x 16 y#z‡��{5|��},k where z‡�� is the critical nucleus size for a static system

at the supersaturation at time � and �{5|�� is the lag time for the redistribution of nuclei size

after the change in supersaturation at time �.9, 19, 27 According to CNT, the requirement for

validity of the quasi-static approximation is

~N ln,� D��N� ~ x #p�AOPQ6BOPQ � �14�

for HON and

~N ln,� D��N� ~ x #p�AOlQv �u�6BOlQ � �15�

for HEN. Both requirements must be satisfied for all times prior to the first nucleation event.

To illustrate the impact of freeze concentration, consider the nucleation of natural gas hydrates.

Methane hydrate nucleation from an aqueous solution of methane is fast in simulations at

extremely high supersaturation.28

However, at more realistic supersaturations, Knott et al. found

that homogeneous hydrate nucleation from an aqueous solution is impossibly slow.29

HEN on

the surface of ice remains a possibility, but Shepherd et al. found that clathrate cages did not

form within the quasi-liquid layer that forms at the ice-methane interface during their simulation.

In addition, they found no enhancement in methane solubility within the quasi-liquid layer,

suggesting that static ice surfaces do not catalyze hydrate nucleation.30

Yet nucleation does

occur on the surface of ice in experiments.31

Here, we take values of AOPQ � 5.8 � 10kdcm,V�,k and BOPQ � 930 from the calculations

of Knott et al.29

Even with a bulk supersaturation of D � 5.8, the induction time in a 1 cm3

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volume is over 10k�V years. In contrast, for an observation area of 1 cm2, no bulk

supersaturation, an ice growth rate of 1 µm s-1

, and a methane diffusivity of 1.46×10-9

m2

s-1

(diffusivity at 273 K), the average induction time is drastically reduced to 22 hours (Table 1)!

The calculation illustrates how freeze-concentration in the boundary layer can provide a

sufficiently high driving force to induce hydrate nucleation. Molinero and coworkers find, in

preliminary tests of the new model, that freeze-out in methane-water solutions can generate

methane bubbles instead.32

Understanding how pressure and other factors influence selectivity

between alternative nucleation processes is therefore an interesting direction for future work.

Table 1. Average induction times for methane hydrate nucleation with and without freeze-

concentration

Freeze-conc. Quiescent

initial supersaturation, D�� 0� 1 5.8

observation area or volume, R or K 1 cm2

1 cm3

ice growth velocity, � 1 µm/s --

methane diffusivity, 1.46 × 10-9

m2/s --

average induction time, ⟨�⟩ 58 hours 10103

years

Average induction time is drastically reduced when freeze-concentration occurs. In both cases,

we take values of AOPQ � 5.8 � 10kdcm,V�,k and BOPQ � 930 from the calculations of Knott

et al.29

In conclusion, a boundary layer of giant solute supersaturation develops ahead of the moving

crystallization front when a freezing solvent (often ice) excludes the solute. We have developed

stochastic models of boundary layer nucleation during freeze-concentration by using quasi-

steady nucleation kinetics and a time-dependent one-dimensional model of the concentration

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profile in the boundary layer. We provide numerical calculations of average induction times and

90% relative confidence interval widths as a function of two dimensionless variables. Whether

nucleation is heterogeneous on the ice surface or homogeneous in the solution, nucleation is

dramatically accelerated by freeze-concentration. For methane hydrates, which can form

concurrently with ice, induction times for homogeneous nucleation could be reduced by as much

as 10105

times because of freeze-concentration. The stochastic model of nucleation in the freeze-

concentration boundary layer is widely applicable and requires only two dimensionless

parameters. These can be obtained or estimated to understand a variety of freeze-concentration

induced solute nucleation processes as listed in the introduction. However, future work should

also improve upon some oversimplifications in our model. For example, our stochastic treatment

should be combined with more comprehensive models that account for surface charging33

, fluid

mechanics in the process of engulfment11, 34

, spontaneous convection driven by buoyancy in the

boundary layer35

, and cellular/dendritic growth.36

Such a comprehensive model might enable

quantitative comparisons to interferometry measurements of concentration profiles and bubble

formation during freeze-concentration.36

FIGURES

Figure 1. A diagram of the semi-infinite freeze-concentration process in our analysis. We

compute the local concentration ��, �� in the solution ahead of the ice growth front when the

bulk solution is maintained at a constant concentration �� and the ice grows at a constant rate �.

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Figure 2. The interfacial supersaturation :�;, <� (dashed curve) and its large < limit = � <

(dotted line) at large <, Several supersaturation profiles :�>, <� (solid curves) for corresponding

interfacial supersaturations are also shown.

Figure 3. (a) logk�⟨�OPQ⟩ as a function of logk� ROPQ and logk� BOPQ. Contours of constant

logk�⟨�OPQ⟩ are spaced evenly by 0.5. (b) Contours of constant 90% RCIWHON (spaced evenly

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by 0.1) are plotted against logk� ROPQ and logk� BOPQ. The numerical accuracy in the yellow-

black-striped region is questionable because of the precision of our numerical integration in

Equation (7).

Figure 4. (a) logk�⟨�OlQ⟩ as a function of logk� ROlQ and logk� BOlQ. Contours of constant

logk�⟨�OlQ⟩ are spaced evenly by 0.5. (b) Contours of constant 90% RCIWHEN (spaced evenly

by 0.1) are plotted against logk� ROlQ and logk� BOlQ.

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TABLES

Table 1. Average induction times for methane hydrate nucleation with and without freeze-

concentration.

Freeze-conc. Quiescent

initial supersaturation, D�� 0� 1 5.8

observation area or volume, R or K 1 cm2

1 cm3

ice growth velocity, � 1 µm/s --

methane diffusivity, 1.46 × 10-9

m2/s --

average induction time, ⟨�⟩ 58 hours 10103

years

Average induction time is drastically reduced when freeze-concentration occurs. In both cases,

we take values of AOPQ � 5.8 � 10kdcm,V�,k and BOPQ � 930 from the calculations of Knott

et al.29

AUTHOR INFORMATION

Corresponding Author

Email: [email protected]

Author Contributions

The manuscript was written through contributions of all authors.

Funding Sources

We gratefully acknowledge support from the National Science Foundation through CE-1125235.

ACKNOWLEDGMENT

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We gratefully acknowledge support from the National Science Foundation through CE-1125235.

We thank Valeria Molinero for helpful discussions about methane hydrate and methane bubble

nucleation.

SUPPORTING INFORMATION AVAILABLE

Wolfram Mathematica codes for computing average induction times for HON and HEN. This

information is available free of charge via the Internet at http://pubs.acs.org.

ABBREVIATIONS

MZW, metastable zone width; HON, homogeneous nucleation; HEN, heterogeneous nucleation;

RCIW, relative 90% confidence interval width.

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SYNOPSIS

We develop stochastic models that predict the induction time for the first nucleation event in the

boundary layer ahead of the moving ice front during one-dimensional freeze-concentration.

Freeze-concentration greatly accelerates both homogeneous and heterogeneous nucleation and is

a potential ice-assisted mechanism for gas hydrate nucleation.

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For Table of Contents Use Only

A stochastic model for nucleation in the boundary layer during solvent freeze-concentration

Geoffrey G. Poon and Baron Peters

SYNOPSIS

We develop stochastic models that predict the induction time for the first nucleation event in the

boundary layer ahead of the moving ice front during one-dimensional freeze-concentration.

Freeze-concentration greatly accelerates both homogeneous and heterogeneous nucleation and is

a potential ice-assisted mechanism for gas hydrate nucleation.

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Documents

A Stochastic Model for Nucleation in the Boundary Layer during Solvent Freeze-Concentration