Dynamics of Premelted Liquid Films Grae Worster Institute of Theoretical Geophysics Department of Applied Mathematics and Theoretical Physics University of Cambridge
Collaborators John Wettlaufer Greg Dash Stephen Peppin Larry Wilen Robert Style Alan Rempel Mark Hallworth
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 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
The Forces Responsible… for frost heave are the same long-range intermolecular forces that underlie surface tension… Photograph: John Bush and also cause most solids close to their melting points to be molten at their surfaces.
Thermodynamics of Interfacial Premelting Free substrate � si energy ice � si total bulk substrate � sw water d interfacial � wi � sw+ � wi d ice eq Thickness of premelted film, d
Dynamics of Interfacial Premelting substrate p s water p T p l ice p s � L T m � T = p s � p l = p T T m force phase balance equilibrium � 1/3 � � A � d � T m � T p T = � � For van-der-Waals forces 6 � d 3 T m � �
Marangoni versus Thermomolecular Flow Film thickness determined Film thickness determined dynamically thermodynamically
Flow of Premelted Liquid Experiments (Wilen & Dash 1995 Axis Flexible membrane Inset Water Ice inflow WARM COLD WARM Glass slide Lubrication Theory (Wettlaufer & Worster 1995,9 Premelted liquid Flow d(x) h(x,t) Water Ice x
Lubrication Theory Lubrication theory gives volumetric flow rate in the premelted film to be = � � 3 T m � p d 3 � p Q = � 12 μ Gx � x 12 � x μ where the pressure driving the flow is � L ( ) T m � T p � � h xx = � T m Elastic Thermo-molecular wall stress pressure Conservation of mass gives � t + � Q � 3 h � h � � h 1 � � � � � t + D � � x = 0 � = 0 � + � � � � x x � x 3 � � � � � 3 T m � 1 � = � LG where D = and 12 G � T m μ
Similarity Solution and Comparison with Experiments x ) 3/5 f � ( ( ) with h = � Dt � = ) 1/5 ( Dt f + 1 � � � � 1 f + 3 5 � 2 � f � � � � � 5 � f = 0 � † † ( ) f = � f = 0 � = 0 � � ( ) , f � 0 � � � † 2 2 64.5 h 160 h † Height ( μ m) 1 1 ice water ice water 0 0 -400 0 400 -400 0 400 R � Ro ( μ m) R � Ro ( μ m)
Multiple Ice Lenses Taber (1930) cold warm V 1mm cold warm Peppin 2005
Rempel, Wettlaufer & W Thermodynamic Buoyancy PRL 2001 Film thickness determined by interfacial pre-melting and curvature � L T m � T A = 6 � d 3 + � sl �� n T m V premelted water d( � ) � T = G particle ice L � F = p T n dS = m s � T � m s G Net force on particle is T m S Where m s is the mass of ice displaced by the particle. cf Archimedes
Freezing of soil - formation of ice lenses cold Ice lens water T =0 ˚ C warm
Single Ice Lens - Complete Particle Rejection warm V 1mm cold Kaolinite. 60% by weight. Particle size approximately 1 μ m.
Single Ice Lenses in Nature – Needle Ice
Dynamics of the Lenses and Frozen Fringe Contact Pressure Ice lens T l New ice lens u = � k � ( ) � p w T f water 0 ˚ C Net vertical inter-particle force is � z � z 2 ( 1 � � ) o � � L � � � ( ) d � � z 1 � � z ( ) � P p = P � T 1 � � � ( ) ( ) + μ V d � T m k � ( ) � � � � 0 z h Overburden Thermodynamic buoyancy Viscous drag
Calculations of ice-lens dynamics � 1 � � � � z l z l 2 ( 1 � � ) � � � � � � V = ( 1 � � ) dx � p o dx k � ( ) � � � � � � � � 0 z h
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
Modes of Behaviour Rempel, Wettlaufer & W. JFM 2004 1.4 No segregated ice Periodic ice lenses 1.2 1.0 Steady lens without fringe V or periodic lenses 0.8 Steady lens with fringe 0.6 or periodic lenses Steady lens 0.4 with fringe or no segregated ice 0.2 Steady lens no fringe 0 0.5 1.0 1.5 p 0
Freezing of a Colloidal Suspension � � � C � z D ( C ) � C � t = � � � � � � z T i = T i C ( ) Ice
Freezing of a Colloidal Suspension � � � C � z D ( C ) � C � t = � � � � � � 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 fl uid fl ow 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
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