Formation of brine channels in sea iceK. Morawetz1,2,3,B. Kutschan1, S. Thoms4

1Munster University of Applied Sciences, Stegerwaldstrasse 39, 48565 Steinfurt, Germany2 International Institute of Physics (IIP), Av. Odilon Gomes de Lima 1722, 59078-400 Natal, Brazil

3 Max-Planck-Institute for the Physics of Complex Systems, 01187 Dresden, Germany4Alfred Wegener Institut fur Polar- und Meeresforschung, Am Handelshafen 12, 27570 Bremerhaven, Germany

Different ice textures as habitat for micro-organisms

Columnar Col.-granular granular plate

Weissenberger, Ber. Polarforsch. 111 (1992)

Diatoms in brine pockets bacteria Dieckmann

• carbon consump-tion of organisms inbrine channels:18% in southernocean

Minimal microscopic model

1. Molecular structure of water and iceby ”tetrahedricity’

u =1

15 < l2 >

∑

i,j

(li − lj)2

li lengths of six edges by four nearest neighbors

ice: ideal tetrahedron MT = 0water: random structure MT = 1

2. Salinity vfree energy density

Dice

2(∇u)2+

a12u2−

a23u3+

a34u4

+h

2u2v+

Dsalt

2v2

a1... freezing parameter (phase transition)a3... structure parameter (nonlinear)a2... first-order phase transitionh... ice-water coupling (reaction rate)Dice, Dsalt diffusion coefficients of ice and salt

Time evolution of Cahn-Hilliard-type

• conservation of salt mass, demand balance equation ∂v/∂t = −∇~j

• current generalized force ~j ∼ ~F by potential ~F = −∇P• potential in turn by free energy density P = δf/δv

dimensionless time τ = Dsalta22t/h

2, space ξ = a2x/h, order parameterψ = h2u/Dsalta2, and salinity ρ = h3v/Dsalta

22

∂ψ

∂τ= −α′

1ψ + ψ2 − α3ψ3 − ψρ +D

∂2ψ

∂ξ2,

∂ρ

∂τ=

1

2

∂2ψ2

∂ξ2+∂2ρ

∂ξ2.

time-dependent Ginzburg-Landau equations couple 2 order parameters by3constants:

1. freezing parameter α′1 = a1h

2/a22Dsalt

2. structure parameter α3 =a3Dsalth2

3. diffusivity D = DiceDsalt

with α1,′ α3, D > 0

Thermodynamics

uniform stationary free energy density f (Ψ0, ρ0) =α12 ψ

20 −

13ψ

30 +

α34 ψ

40

• Freezing-point depression since α1(T ) = α′1(T ) + ρ0 (higher temperature

than α′1)

• at T 0c = 233.15K vanishes α1(T ) = α1(T − T 0

c )

• freezing point depression ∆T = −ρ0α1

→ latent heat of water-ice phase transition ∆H = 6kJ/mol and a dis-sociation ratio of x = (nNa+ + nCl−)/nH2O = 1/50, Clausius-Clapeyron:∆T = −xRT 2/∆H = −2K

→ specific heat c = −T 0c∂2f(ψ+0 (T ))

∂T 2 = 881

T 0c

α33(Tc−T0c )

2

energy scale difference of latent heat of water freezing KE =L(0◦C)− L(−40◦C) = 98J/g, our theory: cspec = KEc = 2.14J/gK

→ α3 = 0.9

Supercooling/heating only by α1, α3

• α1 > 1/4α3, minimum at 0: disordered state• α1 ≤ 1

4α3= α1(T1) second minimum, coexis-

tence curve, borderline represents super-heating• α1(Tc) = 2

9α3< α1(T ) <

14α3

ordered phase

metastable, (ice formation possible)• α1 ≤ α1(Tc) ordered phase stable, jump at Tclatent heat, first-order phase transition• super-cooling T 0

c , freezing-point Tc, super-heating

T1 =98Tc −

18T

0c

• homogeneous nucleation at super-heating >5◦C, fresh water (Tc = 273.15K and T 0

c =233.15K) we have upper limit of super-heatingT1 = 4.96◦C in agreement with experimentK. Baumann et al. Z. Phys. B Cond. Matt.56 (1984) 315

0 0,5 1 1,5ψ0

-0,1

-0,05

0

0,05

f(ψ

0)

T=T1

α1=1/4α3T=T

c α1=2/9α3

T=T0

c,s α1=0

230 240 250 260 270 280T [K]

0

0.2

0.4

0.6

0.8

1

ψ0+

(T)

Tc

0 Tc

T1

α1=1/4a3

α1=2/9a3

→ α3 = 0.9, α1 = 0.2

Linear stability analysis

linear stability analysis ρ = ρ0eλ(κ)τ+iκξ around the disordered and ordered

phaseλ(κ) > 0 allows fluctuation with wave-vector κ to grow exponentially intimetime-oscillating structures would appear only if Imλ(κ) 6= 0, never in ourmodel

0 0.1 0.2 0.3α1

0

0.5

1

1.5

2

2.5

α 3

uniform uniform liquid-

spatial structures

ice phase water phasemelting

phase diagram where spatial struc-tures can occur (checked)

• structure parameters α1 determines brinechannel formation:• small α1 means low temperatures or lowsalinities and consequently freezing process• uniform ice phase for sufficiently large α3

and a precipitate of saltat higher α1 higher temperatures or highersalinities inducing a melting with uniformliquid water phase and dissolved salt• spatial structures can only appear in theinstability region (maximal point α1 = 1/9at α3 = 2)

-60 -40 -20 0 20 40 60 ξ

-0,8-0,6-0,4-0,2

00,20,40,60,8

11,21,41,6

ψ, ρ

ψ ρ

-60 -40 -20 0 20 40 60 ξ

-0,5

0

0,5

1

1,5

ψ, ρ

ψ ρ

-60 -40 -20 0 20 40 60 ξ

-0,5

0

0,5

1

1,5

ψ, ρ

ψ ρ

Time evolution

1D (left) and 2D (right)vs. spatial coordinatesτ = 10, 150, 500 (from above)with α3 = 0.9, α1 = 0.2,

D = DsaltDice

≈ 0.5

by Dsalt at T0c = −1.9oC

[S. Maus 2007]

and Dice by reorientation rateH2O [A. Bogdan 1997] andcorrelation length [D. Eisenbergand W. Kauzmann 2002]

Comparison with experiment: 1. morphology

(b)

(c)

(d)

(a) Imaging brine porespace with X-ray com-puted tomographyD. J. Pringle et al. J.Geophys. Res. Oceans114 (2009) C12017upper images: alongbrine layers, bottom:across(b) Scanning electronmicroscopy image of castof brine channels Weis-senberger(c) long-time Turing(d) phase field structurewith salinity conserva-tion

Comparison with experiment: 2. percolation threshold

histogram of connected clusters

above: along brine layers and below: acrosswith temperatures from −18,−8,−4◦C

α′1 = 0.1, and D = 0.5 and α3 = 0.9 (left) compared to α3 = 1 (right)

Comparison with experiment: 3. structure size

• fastest-growing wave-vector κc(D,α1, α3) sets structure by 2π/κc• critical domain size as function of freezing point depression

λc =2πkc

= 2πκc

ha2= 2π

κc

√

Dsaltρ0a1|∆T |

1. Pure sea ice:• our choice α3 = 0.9, α1 = 0.2 (super-cooling ∆Tsup = 6.3K) andD = Dice/Dsalt = 0.5: dimensionless pattern size of 13.81• rate of re-orientations of H2O-molecules determines a1 = 1250K−1s−1

• we obtain critical domain size λc = 0.8µm in agreement with seaice platelet spacing λmax ≈ 1µm from morphological stability analysis(Weissenberger, Prible, Golden)

2. Size for natural conditions• by upper limit of the instability region α3 = 1.99 and freezing parameterα1 = 0.111482, realistic description of seawater at 0.032K super-coolingand a lower limit of the super-cooling region of fresh water at −18.78◦C

• we obtain dimensionless structure size 2π/κc = 4975.25 and critical do-

main size λc = 198µm in agreement with observed values 3 − 1000µm

average 200µm

Summary

• model for brine channel formation with two coupled order parameters,tetrahedricity and salinity preserving mass conservation

• linear stability analysis provides phase diagram (two parameters)

• region of instabilities determined exclusively by freezing parameter andspecific heat or structure parameter not by diffusivity (Touring model)

• parameters describe thermodynamical properties of water like super-heating, super-cooling, freezing temperature and latent specific heatsimultaneously

• time-dependent evolution solved and brine channel texture in agreementwith the experimental values

• physical justification of parameters by other properties of water leadsto a better description of brine channel texture by mass conservation:internal consistency of the model

• Outlook: New dynamical mechanism of antifreeze proteins (new project)

supported by grant MO 621/22