Dynamics of Premelted Liquid Films Grae Worster Institute of - - PowerPoint PPT Presentation

Dynamics of Premelted Liquid Films

Grae Worster

Institute of Theoretical Geophysics Department of Applied Mathematics and Theoretical Physics University of Cambridge

Collaborators

Greg Dash Larry Wilen Alan Rempel Stephen Peppin Robert Style Mark Hallworth John Wettlaufer

Some Effects of Frost

Something there is that doesn’t like a wall, That sends the frozen ground swell under it And spills the upper boulders in the sun … Robert Frost

Something there is that doesn’t like a wall, That sends the frozen ground swell under it And spills the upper boulders in the sun … Robert Frost

Some Effects of Frost

Something there is that doesn’t like a wall, That sends the frozen ground swell under it And spills the upper boulders in the sun … Robert Frost

Some Effects of Frost

The Forces Responsible…

for frost heave are the same long-range intermolecular forces that underlie surface tension… and also cause most solids close to their melting points to be molten at their surfaces.

Photograph: John Bush

si substrate ice sw substrate ice water wi d bulk interfacial total si sw+wi Free energy Thickness of premelted film, d d

eq

Thermodynamics of Interfacial Premelting

L Tm T Tm = ps pl = pT

phase equilibrium force balance

For van-der-Waals forces pT = A 6d3 d Tm T Tm

• 1/3

Dynamics of Interfacial Premelting

substrate water ice pT ps ps pl

Marangoni versus Thermomolecular Flow

Film thickness determined dynamically Film thickness determined thermodynamically

Ice Water inflow Glass slide Flexible membrane Inset Axis WARM COLD WARM Ice Water Premelted liquid Flow x d(x) h(x,t)

Lubrication Theory

(Wettlaufer & Worster 1995,9

Experiments

(Wilen & Dash 1995

Flow of Premelted Liquid

Lubrication theory gives volumetric flow rate in the premelted film to be Q = d3 12 μ p x = 3Tm 12 μGx p x where the pressure driving the flow is

Elastic wall stress Thermo-molecular pressure

p = hxx

• L

Tm Tm T

( )

Conservation of mass gives h t + Q x = 0 h t + D x 1 x 3h x3

• +
• = 0

Lubrication Theory

where and D = 1 12 3Tm G

• μ

= LG Tm

• 400

400 1 2

Height (μm) RRo ( μm)

64.5 h

ice water

RRo ( μm)

• 400

400 1 2

160 h

ice water

Similarity Solution and Comparison with Experiments

† h = Dt ( )3/5 f ( ) † = x Dt ( )1/5 with †

• f
• f +1
• 1

52 f + 3 5f = 0 †

• f =
• f = 0 = 0

( ), f 0

( )

Multiple Ice Lenses cold warm cold warm V

Taber (1930) Peppin 2005

1mm

particle ice premelted water V d() T = G

Film thickness determined by interfacial pre-melting and curvature Net force on particle is cf Archimedes

Thermodynamic Buoyancy

Where ms is the mass of ice displaced by the particle.

Rempel, Wettlaufer & W PRL 2001

F = pT

S

• n dS = ms

L Tm T msG L Tm T Tm = A 6d3 + sl n

Freezing of soil - formation of ice lenses

warm cold T=0˚C Ice lens water

Single Ice Lens - Complete Particle Rejection V

1mm

• Kaolinite. 60% by weight. Particle size approximately 1 μm.

cold warm

Single Ice Lenses in Nature – Needle Ice

0˚C water Ice lens Contact Pressure Tl Tf

Dynamics of the Lenses and Frozen Fringe

New ice lens

Net vertical inter-particle force is Pp = P

• L

Tm T 1

( )

( ) d z 1 z

( )

( )

z

• + μV

1

( )

2

k

( )

zh z

• d

Overburden Thermodynamic buoyancy Viscous drag

u = k

( )pw

Calculations of ice-lens dynamics

V = 1

( )

zl

• dx po
• 1

( )

2

k

( )

zh zl

• dx
• 1

Calculations of ice-lens dynamics

Calculations of ice-lens dynamics

Calculations of ice-lens dynamics

Calculations of ice-lens dynamics

Calculations of ice-lens dynamics

0.5 1.0 1.5 1.4 1.2 1.0 0.8 0.6 0.4 0.2

Periodic ice lenses No segregated ice Steady lens without fringe

• r periodic lenses

Steady lens with fringe

• r periodic lenses

Steady lens with fringe or no segregated ice Steady lens no fringe

V p0

Modes of Behaviour

Rempel, Wettlaufer & W. JFM 2004

Freezing of a Colloidal Suspension

C t = z D(C) C z

• Ice

Ti = Ti C

( )

Freezing of a Colloidal Suspension

† C t = z D(C) C z

• Slow freezing rate

Fast freezing rate

Different Types of Behaviour

Summary and Conclusions

Long-range intermolecular forces can cause most solids to premelt at their surfaces or at interfaces with other materials Temperature gradients give rise to gradients in thermo-molecular pressure: surface transport; thermodynamic buoyancy Competition between thermodynamic buoyancy and viscous fluid flow determines heaving rates and lens initiation Interplay between morphological instability of lens front, nucleation beyond compaction layer and thermodynamic buoyancy within compaction layer may determine a wide range of different behaviours