Hydrodynamic Transitions with Changing Particle Size That Control

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Hydrodynamic Transitions with Changing Particle Size Control Ice Lens Growth Tomotaka Saruya, Alan W Rempel, and Kei Kurita J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/jp505366y • Publication Date (Web): 01 Jul 2014 Downloaded from http://pubs.acs.org on July 8, 2014

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The Journal of Physical Chemistry

Hydrodynamic Transitions with Changing Particle Size Control Ice Lens Growth Tomotaka Saruya,∗,† Alan W. Rempel,‡ and Kei Kurita† Earthquake Research Institute, University of Tokyo, Tokyo, 113-0032, Japan, and Department of Geological Sciences, University of Oregon, Eugene, Oregon 97403, USA E-mail: [email protected]

∗

To whom correspondence should be addressed Earthquake Research Institute, University of Tokyo ‡ Department of Geological Sciences, University of Oregon †

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Abstract Ice lenses are formed during soil freezing by the migration and solidification of premelted water that is adsorbed to ice–particle interfaces and confined to capillary regions. We develop a model of ice lens growth that clearly illustrates how the freezingrate dependence on particle size and soil micro-structure changes in response to changes in the relative importance of permeable flow and thin-film flow in governing the water supply. The growth of an ice lens in fine-grained porous media is primarily constrained by low permeability in the unfrozen region. In contrast, the constraints offered by film flow decrease the lens growth-rate adjacent to larger particles. The trade-off between resistance to permeable flow and film flow causes the growth rate for ice lenses to be maximized for particles of intermediate size. Moreover, since film flow along particle surfaces adjacent to a growing lens is not strongly affected by the microstructure of the pore space, our analysis predicts that lensing in coarse-grained porous media is insensitive to pore microstructure and porosity, but the permeable flow that governs lens formation in fine-grained porous media causes their growth to be much more affected by these details.

keywords: premelted water, freezing rate, frost susceptibility, Peclet number, GibbsThomson effect


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Introduction When mixtures of water and granular materials are frozen, layers of pure ice can be spontaneously segregated from the surrounding granules. Such segregated ice lenses form by the migration and solidification of premelted water that remains unfrozen in capillary regions and adsorbed films on particle surfaces – even below the nominal melting temperature Tm . Liquid remains in equilibrium at these temperatures because of intermolecular forces (including van der Waals and electrostatic forces) and surface energy (the Gibbs-Thomson effect), 1,2 and many aspects of its behavior are well understood. 3 For example, the intermolecular forces that act between the ice and particles across the liquid film are responsible for producing the net thermomolecular force that drives particle motion and causes ice lenses to form. 1,4,5 A simple model of steady-state ice lens growth that traces the interactions between a particle and a growing ice surface was developed by Worster and Wettlaufer. 6 The strength of the net themomolecular force that pushes particles ahead of the ice lens surface increases with the depression in temperature beneath Tm . To balance this particle motion and allow the ice lens to grow, water is drawn from warmer regions. Building on this understanding in a refinement to the “rigid-ice model” of O’Neill and Miller, 7 Rempel et al. 8 showed how the transmission of the thermomolecular force through connected networks of pore-space ice could unload particle contacts and enable ice lens nucleation. More recently, Style et al. 9 argued that soil heterogeneities could allow ice to nucleate in isolated larger pores as a result of “geometrical supercooling”, with subsequent enlargement by crack-like growth causing incipient lenses to extend laterally in the direction normal to heat flow. Whichever mechanism prevails, once a lens nucleates, imbalances between the gravitational force and the thermomolecular force produce the hydrodynamic force and associated fluid pressure gradient needed to drive the flow of unfrozen water to the lens growth surface. 5,8 Particle size has an important influence on the characteristics of ice lenses. The Gibbs– Thomson effect helps to control the amount of premelted water that is present and set the temperature at which ice can first form. 2,10,11 Because the thermomolecular force increases in 3 ACS Paragon Plus Environment


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strength as the temperature drops, lenses near the warmest limit of ice–liquid equilibrium set by the Gibbs–Thomson effect adjacent to smaller particles are capable of supporting larger net combinations of gravitiational and hydrodynamic loads. At the same time, since larger particles produce less flow resistance (have a lower surface area and higher permeability) than an equivalent volume of smaller particles, the hydrodynamic force for a given rate of water supply should tend to increase as particle size is reduced. Indeed, Penner reported that the maximum frost susceptibility, or rate of surface heave due to lens growth, is produced by siltsized particles, while frost susceptibility is decreased with smaller and larger particles. 12 This observation is consistent with the force considerations just described if lens growth is limited by hydrodynamic constraints for small particles and by the reduced thermomolecular force that can be generated within porous media consisting of larger particles. Recently, Saruya et al. observed the growth of ice lenses as a function of particle size using homogeneous glass beads 13 and observed ice lenses growing rapidly in smaller glass beads, with the rate decreasing as the host particle size was increased. However, with the resistance to water supply attributed to permeable flow, calculations showed that for the small overburden forces that these experiments imposed, the drop in thermomolecular force as particle size increased was not sufficient to explain the reduced lens thicknesses that were observed. 13 A promising alternative explanation was recognized by Style and Peppin, who highlighted the potential importance of the resistance to flow in the thin premelted films immediately adjacent to growing ice lenses, 14 as suggested previously for certain limiting cases. 6,15 These kinetic effects had been ignored in many past treatments because the path length for film flow is only on the scale of the particle diameter, which is very short compared with the distance over which permeable flow is supplied in most natural and laboratory settings. Nevertheless, there is increasing recognition that kinetic effects due to liquid film flow can play an important role in constraining ice lens growth, and can even provide a mechanism for probing the behavior of the thin films themselves. 16 The suggestion explored here is that hydrodynamic constraints are the dominant restrictions on the rate of lens growth near the


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ground surface both for large particles because of film flow and for small particles because of permeable flow. We use a simple force balance model 8,13,17 to analyze ice lens behavior with the small overburden loads and transient heat flow conditions that are characteristic of soil freezing at shallow depths. In a step-freezing configuration under conditions that are predicted to fall within the “multiple lensing regime”, 17 the heat flow is initially very rapid and though the soil surface heaves upwards, it does so without lenses growing large enough to be visually detected. 13 Following this first stage of freezing, the heat flow decreases sufficiently that a single macroscopic ice lens is observed to form and grow for the remaining experimental duration. Motivated by these observations and analogous behavior in nature, we focus here on the hydrodynamic controls on the relationship between ice lens growth and particle size. Dimensional analysis highlights the importance of the Peclet number and Stefan number for describing ice lens behavior and the character of the temperature profiles that result. Our results demonstrate how field observations of changes in frost susceptibility with sediment particle size can arise primarily from the changing dominance of permeable versus film flow in constraining the liquid supply, with particle-size induced changes in the strength of the net thermomolecular force occupying a secondary role.

Modeling Film thickness Premelted water is central to enabling ice lenses to form because it causes the (thermomolecular) force between ice and particles to depend on temperature rather than simply balancing the gravitational load, and the difference gives rise to the hydrodynamic gradient needed to supply growth. 2 Moreover, the thickness of the premelted water films around particle surfaces can dominate the resistance to water supply. 18,19 The thickness of the premelted film d can be approximated by a power law and expressed within a general equilibrium relationship



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between undercooling ∆T and curvature K as

ρL

∆T = γK + Λd−ν , Tm

(1)

where ρL is the volumetric latent heat, γ is the ice–liquid surface energy, and Λ is a prefactor that gauges the strength of ice–liquid–particle interactions (e.g. Λ = A/6π in the case of ν = 3, where A is the Hamaker constant that controls the strength of van der Waals interactions). Here, the exponent ν represents the type of dominant interfacial interactions that controls film behavior: non-retarded van der Waals forces produce ν=3, retarded van der Waals forces produce ν=4, and different limits of electrostatic forces (short-range or long-range) produce ν=3/2 and 2. 20–22 Near a planar surface, we can neglect the curvature term above, allowing the thickness to be expressed as d=

( Λ T )1/ν m . ρL ∆T

(2)

By contrast, in pore throats of radius Rp the interfacial curvature can reach K = 2/Rp and d becomes large enough that the above expression reduces to the Gibbs-Thomson equation

∆T ≈ ∆Tf ≡

2γTm . ρLRp

(3)

This represents the depression of the freezing temperature ∆Tf in porous media and it describes how the high surface curvature due to smaller particles and pore throats induce large depressions to the melting temperature in fine-grained porous media. The film thickness against a planar surface at the depressed melting temperature Tm −∆Tf can be used to define λ as d≈

( ΛR )1/ν p

2γ

= λ.


(4)

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However, around spherical particles of radius R, the negative surface curvature can affect the film thickness significantly so that

ρL

∆T 2γ = − + Λd−ν , Tm R

(5)

and taking Rp = αR, the film thickness at ∆T = ∆Tf is [

ΛαR d= 2γ (1 + α)

]1/ν = λK .

(6)

This indicates that curvature effects cause the film thickness against a rounded particle surface to be thinner by a factor of 1/(1 + α)1/ν in comparison with the film thickness against a flat surface.

Ice lens growth The velocity of the freezing front plays an important role in the behavior of ice lenses. When the freezing rate is high, even though ice lenses can be nucleated, they cannot grow large enough to become visible before the supply of unfrozen water is cut off by the nucleation of subsequent lenses. The growth rate of an ice lens depends on the supply of unfrozen water, and comparisons between heat and mass transport constraints can be used to define different regimes of freezing behavior. For example, Rempel et al. 8,17 examined the behavior of ice lenses as a function of the freezing rate and overburden pressure and showed that at high freezing rates with low overburden, multiple lenses are expected, whereas lower freezing rates can support the growth of a single macroscopic lens. This is consistent with experiments in which Saruya et al. 13 found that the progressive decrease in heat flow in a step-wise freezing configuration resulted in two distinct stages of behavior; the first stage is characterized by measurable surface heave with no visible lenses, and the second stage begins with the appearance of a macroscopic ice lens, followed by a gradual decrease in the rate of its growth and the associated heave of the sediment surface (see Fig. 1). The evolution of this behavior 7 ACS Paragon Plus Environment


can be described with a simple one-dimensional treatment, in which lens nucleation is not explicitly treated, but instead assumed to occur infinitesimally close to the Tf isotherm in

Heaving and thickness (mm)

view of the very low overburden forces that are imposed.

Macroscopic lens! Sediment heaving!

0.5" Growth rate μm/s

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Elapsed time (hour)

0.4"

(a)

(b)

0.3" 0.2" 0.1" 0" 0"

5" 10" Elapsed time (hour)

15"

Figure 1: (a) Measured surface heaving (open circles) and lens thickness (filled circles) and (b) estimated growth rate found by differentiating the data shown in (a). In the course of a typical step-freezing experiment, (a) shows that heave of the sediment surface begins before the appearance of the single macroscopic lens. (b) shows that the heave rate of the sediment surface approximates the growth rate of the ice lens, and that the heave rate is nearly constant prior to the appearance of the macroscopic lens and close to the initial ice lens growth rate. Defining zf as the location of the depressed melting point Tf (due to the Gibbs–Thomson effect), the freezing rate dzf /dt during the first stage is much higher than the growth rate Vf of an ice lens with undercooling ∆Tf = Tm − Tf . In this stage the water supply is insufficient for a single macroscopic lens to grow, but heave of the ground surface is still observed to occur – presumably as numerous unseen lenses are nucleated in succession. The heat balance


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in the unfrozen portion of the porous medium satisfies ∂2T ∂T ∂T ≈ κe 2 − Vf Γ , ∂t ∂z ∂z

(7)

where z is oriented upwards in the direction of increasing temperature, κe is the effective thermal diffusivity, the rate of water supply Vf is taken as the lens growth rate at temperature Tf , and Γ = O(1) is the ratio of the volumetric heat capacities of the fluid and the bulk ¯ ≈ ρp Cp (1 − ϕ)+ρCl ϕ, where ϕ is porosity, porous medium. The latter is approximated as ρC Cp and Cl are the specific heats of the particles and liquid water and ρp Cp and ρCl are the ¯ The heat heat capacities of the particles and liquid water respectively, so that Γ = ρCl /ρC. balance condition along the lens boundary is written as

Ke0

∂T Tf − T0 − ρLVf , ≈ Ke− ∂z z=z+ zf

(8)

f

where Ke0 ≈ Klϕ Kp1−ϕ is the water-saturated thermal conductivity and the average thermal conductivity beneath the lens boundary is approximated as Ke− ≈ 0.5Ke0 +0.5Kiϕ Kp1−ϕ , with Kp , Ki , and Kl the thermal conductivities of the particles, ice, and liquid water. Because of latent heat release, the temperature gradient is well-approximated as linear from the fixed boundary at z = 0 where T = T0 up to the moving Tf isotherm at zf . Equation (7) must be ¯ solved in the region above zf where κe ≈ Ke0 /ρC. We have assumed a one-dimensional regime for illustrative purposes because this is a reasonable approximation for common near-surface natural conditions. However, lateral heat loss is often a significant factor in experiments and it is worth a brief digression to outline how it can be approximated in a fairly straightforward manner. For example, ice lenses grown during freezing experiments by Saruya et al. 13 were observed to have a shallow convex-upward curvature. Assuming ice lenses formed along isotherms, this configuration implies a radial temperature gradient that drove lateral heat flow. The experimental chamber of diameter D was surrounded by insulation with conductivity Ks and thickness hs . Heat 9 ACS Paragon Plus Environment


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flow through the chamber can be approximated by averaging the heat balance over axial slices and treating the radial heat flow as proportional to the local temperature difference with the exterior temperature Tex . Defining the radial heat transfer coefficient H = Ks /hs , the heat balance equation (7) is replaced by ∂ 2T 4H ∂T ∂T ≈ κe 2 − V f + ¯ (Tex − T ) . ∂t ∂z ∂z ρCD

(9)

In practice H can be constrained through comparisons between modeled and observed interior temperatures. We consider a configuration in which the lens growth rate is determined by force balance considerations that involve the buoyancy of the overlying sediments, the fluid pressure gradient due to flow from water ponded at the sediment surface at height zT (see Fig.2), and the effects of flow in a thin film of thickness d that separates particles of radii R from the lens boundary at zf so that cf. Eq. 6 in Saruya et al. 13

Vf =

ρL∆Tf /Tm − (zT − zf ) (1 − ϕ) ∆ρg . µ [(zT − zf ) /k0 + µ′ (R2 /d3 )f ]

(10)

In the numerator, the first term is the net force per unit area due to thermomolecular pressure and the second term is the overburden pressure that is produced by the weight of particles above the lens, reduced by the displaced weight of fluid, where ∆ρ is the density differences between the particles and water, g is the acceleration of gravity, and the lens is located at zf . The denominator accounts for the hydrodynamic force associated with the flow of fluid at viscosity µ to supply lens growth: the first part accounts for permeable flow as described by Darcy’s law, with permeability k0 ; the second part arises since fluid still needs to be drawn through the premelted film around particle surfaces against the ice lens (see Fig.2). Flow through the liquid thin film is sensitive to film thickness and the pressure drop is inversely dependent on film thickness cubed, as described by lubrication theory. Because the pressure drop associated with film flow is proportional to the fluid flux needed to supply the lens over 10 ACS Paragon Plus Environment

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an area spanned by the particle surface, the second term in the denominator is proportional to R2 /d3 . The factor of µ′ accounts for the possibility that the film viscosity may differ from the bulk value µ, and f is a geometrical parameter of order unity. 15 warmer ZT

permeable flow

film flow

Tm

Z Zf

Tf Tl

Zl

ice lens

T0

Z0

colder

Figure 2: Schematic diagram of partially frozen porous media and growing ice lens. Defining the pressure scale Pf = ρL∆Tf /Tm that measures the net thermomolecular force along the lens boundary, the growth rate can be expressed as Pf k0 Vf ≈ µzT 0

1− zT −zf zT 0

zT −zf w zT 0

+ζ

( R )3/ν .

(11)

0

R

The permeability of the water-saturated particle pack can be modeled using the Kozeny– Carmen relation so that k0 ≈ R2 ϕ3 /[180(1 − ϕ)2 ]. The initial location of the particle– water interface is zT 0 . The exponent ν is used to describe power-law variations in d with undercooling so that d = λK (R/R0 )1/ν for a reference particle size R0 that is defined so that d = λK when ∆Tf = 2γTm /(ρLαR0 ). The overburden term is proportional to the 11 ACS Paragon Plus Environment


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constant w = ∆ρ (1 − ϕ) gzT 0 /Pf . The film-flow term on the denominator is scaled by ζ = k0 µ′ f /(zT 0 R) (R/λκ )3 , where µ′ ≡ 1 when the viscosity in the film is equal to the bulk fluid viscosity. It is useful at this stage to scale distances by zT 0 , time by zT2 0 /κe , and temperature by ∆Tf so that the dimensionless form of equation (7) can be written as ∂ 2 T˜ ∂ T˜ ∂ T˜ ≈ − P eΓ , ∂ z˜2 ∂ z˜ ∂ t˜

(12)

where the Peclet number represents the importance of fluid heat transport relative to conduction so that Pf k0 V f zT 0 = Pe = κe µκe

1− zT −zf zT 0

zT −zf w zT 0

+ζ

( R )3/ν . 0

R

The interface condition from equation (8) becomes Ke− (Tf − T0 ) /∆Tf ∂ T˜ ≈ − SP e , ∂ z˜ Ke0 zf /zT 0

(13)

¯ where the Stefan number is S = ρL/(ρC∆T f ), which reflects the relative importance of latent to sensible heat. A far-field upper boundary condition can be specified as the temperature of the liquid reservoir at zT . As the freezing rate slows during step-wise freezing, the lens growth regime transitions to a second, macroscopic stage when the freezing rate matches Vf . This second stage of growth is governed by the same heat balance constraints as the first stage, but since the freezing rate continues to drop, the growth rate of the macroscopic ice lens must decrease over time. This is accomplished by increasing the lens temperature Tl above Tm − ∆Tf , so that the growth rate is P f k0 Vl ≈ µzT 0

Tm −Tl ∆Tf zT −zf zT 0

+ζ

− [

zT −zf w zT 0

R0 (Tm −Tl ) R∆Tf

]3/ν .

(14)

The Peclet number introduced above P e = Vl zT 0 /κe gauges the importance of fluid flow


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to the temperature profile above the lens boundary, and in combination with the Stefan number, P e also helps determine the degree to which the temperature gradient deviates from that set by the heat flux through the frozen sediments below. In the scaled expressions for Vl and Vf , the approximate magnitude of the growth rate is the lesser of Pf k0 /(µzT 0 ) and Pf k0 /(µzT 0 ζ). In the former case (i.e. when Darcy flow with permeability k0 dominates the hydrodynamic pressure drop) we have that ( Pe ≈ O

Pf k0 µκe

(

) ≈O

2γRϕ3 180µκe α (1 − ϕ)2

) ,

(15)

whereas in the latter case (i.e. when film flow is more important than permeable flow) ( Pe ≈ O

Pf k0 µκe ζ

)

( ≈O

2γzT 0 λ3K µµ′ f κe αR3

) .

(16)

When P e ≪ 1 the temperature gradient above the lens is nearly linear, and this is the basis for a simple approximate treatment of lens growth that can work reasonably well. When the hydrodynamic pressure drop is dominated by Darcy flow, the Peclet number is proportional to R, which highlights the permeable restriction to lens growth around smaller particles. Conversely, when film flow dominates the drop of hydrodynamic pressure, P e is inversely proportional to R3 , which reflects the restriction to film flow around larger particles. Since both permeable and film flow are required for ice lens growth, particles of intermediate size are associated with the highest rates of heave. The Stefan number tends to be large in the parameter regime of interest, but whether latent heat release exerts an important control on the temperature gradient at the lens boundary depends on the product SP e, which can be approximated by ( SP e ≈ O

ρ2 L2 RR0 ϕ3 180µKe0 Tm (1 − ϕ)2

)


,

(17)


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when permeable flow dominates, and ( SP e ≈ O

ρ2 L2 zT 0 λ3K R0 µµ′ f Ke0 Tm R3

) (18)

when film flow dominates. SP e represents the difference of heat flux between the unfrozen and frozen regions that is associated with the latent heat of fusion, so that SP e = Qf rozen − Qunf rozen . Lens growth stops when the numerator in Eq. (14) vanishes so SP e tends to zero and the heat flux on either side of the lens boundary is the same.

Results and Discussion Regimes of ice lens growth In a step-freezing configuration that approximates typical near-surface freezing conditions, ice lens behavior can be subdivided into two regimes that are defined in relation to the velocity of the freezing front. Ice lenses are nucleated and can grow to macroscopic scales only after the freezing rate slows to become comparable to Vf . 13 However, prior to this “second stage” where the macroscopic-lens grows to millimeter scales, there is clear evidence that periodic-lenses are also formed in a “first stage” of growth, during which the freezing rate is much more rapid than Vf . The total frost heave at the ground surface reflects not only the thickness of macroscopic lenses, but also the integration of the numerous thin, short-lived lenses from the first stage. Figure 1(a) shows the time evolution of macroscopic-lens and total sediment heave obtained during a step-freezing experiment (modified from Fig.8 of Saruya et al. 13 ). The final thickness of the macroscopic lens after 15 hours cooling is ∼ 4.3 mm, while the total sediment heave is ∼ 8 mm. This difference is associated with periodic lenses in the first stage that were too small to be visually detected. We average over their nucleation, growth, and termination and approximate their role in perturbing the temperature profile using a constant heave rate Vf in numerical calculations. Macroscopic lens growth begins


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only 4 hours after the beginning of the experiment, approximately 2 hours after the sediment surface begins to heave. Temporal variations in the sediment heaving rate and macroscopiclens growth rate are shown in Fig.1 (b). When the freezing regime transitions from first to second stage, i.e., at the beginning of macroscopic-lens nucleation, the heaving rate of the sediment surface and growth rate of the macroscopic-lens approximately coincide.

Particle size dependence The growth of an ice lens depends on the water supply to the lens surface, which is controlled by the permeable flow through the unfrozen region and lubrication flow in the thin liquid film adjacent to the lens surface, as represented by the Peclet number defined by equations (15) and (16). Permeable flow distances from the water reservoir at zT to the lens surface zf are much larger than the particle diameter, so the permeability has been considered most important to the water supply in most previous models. The permeability of watersaturated porous media depends on the particle radii and the porosity so that the reduction in permeability for smaller particles restricts ice lens growth in very fine-grained porous media. In coarse-grained porous media, the permeability is high, but the path length for film flow scales with the particle diameter, so lens growth amongst larger particles is constrained by film flow. The growth rate and total thickness of an ice lens are determined by the balance between these two restrictions. It should be noted that the particle size also affects the thermomolecular force by restricting the maximum temperature at which pore ice can form and contribute to lens nucleation. However, in near-surface regions where the overburden is low, the restrictions on water supply are often expected to exert the largest controls on lens growth and total heave. When permeable flow dominates the hydrostatic pressure drop and water flow, the Peclet number is proportional to the particle radii R, assuming that the pore throat radius Rp ∼ αR. In the opposite limit, when film flow is more important than permeable flow, P e is proportional to R−3 . This reflects the drastic restriction to film flow around larger particles, which leads to lower frost susceptibility in coarse-grained sediments. 15 ACS Paragon Plus Environment


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Figure 3 shows the Peclet number of permeable flow and film flow described in eqn. 15 and 16 when non-retarded van der Waals forces are dominant (ν=3). Increased porosity induces larger values of P e because of the higher permeability described by the CarmanKozeny equation. By contrast, film flow does not depend on the porosity since it involves only transfers in mass along particle surfaces. 108 106 104 102 100 10-2

Figure 3: Peclet numer of permeable flow and film flow a function of particle sizes. Figure 4 shows S Pe values for permeable flow and film flow with a range of labeled porosities when ∆T = ∆Tf . Here we assume that ice lenses are nucleated near ∆Tf because of the low overburdens near the sediment surface. As ice lenses grow, the heat flux to the lens surface decreases and eventually tends to zero, so that ∆T has to increase and Pe decreases so that S Pe drops as well. However, most of the time during growth S Pe is much bigger than unity and this implies that there is a significant jump in the temperature gradient at the boundary of a growing lens. The growth rate of the macroscopic ice lens is at its highest at the beginning of the second stage, when ∆T = ∆Tf . As the lens temperature increases to approach the melting point, the growth rate becomes smaller (e.g. see Figure 1). The dependence of the maximum growth rate on particle size is shown in Figure 5 for various assumed porosities, using the parameters summarized in Table 1. A decrease in particle size induces further constraints 16 ACS Paragon Plus Environment

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109 107 105 103 101

Figure 4: SPe values of permeable flow and film flow as a function of particle sizes. on permeable flow, so that the maximum growth rate becomes lower. Such a relationship was observed in a sequence of step-wise freezing experiments 13 in which the lens thickness following 15 hours of cooling was thickest at 5 µm and decreased with increasing particle size. Table 1: Parameters that controls the ice lens behaviors. Parameter ρi L Tm γsl A α κe zT ∼ zT 0 ∆ρ g µ f Kl Kp

Value 920 [kg m−3 ] 3.3 × 105 [J kg−1 ] 273 [K] 29 × 10−3 [J m−2 ] 10−19 [J] 0.7 2.3 × 10−9 [m2 s−1 ] 0.05 [m] 1650 [kg m−3 ] 9.8 [m s−2 ] 1.8 × 10−3 [Pa s] 1 0.57 [W m−1 K−1 ] 1.1 [W m−1 K−1 ]

Figure 5 also shows that the particle size with maximum growth rate is expected to vary with porosity. When granular materials are densely-packed, porosity is determined by the 17 ACS Paragon Plus Environment


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packing conditions (e.g., random closed packing, thinner regular packing). The porosity difference is not particularly large in the case of homogeneous materials such as mono-sized glass beads, but natural soils often have vesiculated structures that can alter the packing and porosity. Local structure and porosity might be responsible for permeability variations that play an important role in the water supply. Even in homogeneous materials, mixtures of fine-grained particles and liquid can achieve a colloidal suspension state. In such a case, the porosity is increased due to double-layer forces between particles. 23 We note that ice lens formation and periodic banding in colloidal dispersions have been observed in experiments 24,25 and investigated theoretically. 24 Although we assume that the porosity is spatially and temporally constant in the illustrations provided here, it is important to recognize that particle ejection from the lens surface can increase the particle number density nearby and decrease the local permeability. This further emphasizes the importance of the balance between different restrictions on water flow that are highlighted by Figs. 3 and 5. The growth rates of ice lenses are determined by the balance between the restrictions of permeable flow and film flow. Here we assume mono-dispersed particles in the estimation of permeability using the Carman-Kozeny equation. However, in natural soils, particles have size distributions and vesiculated structures that can produce more heterogeneous and complex permeability structures. In addition, small-scale structures including vesiculated particle surfaces can induce high-surface curvatures that promote enhanced saturation levels of unfrozen water in and around particles. The importance of such complications for modifying freezing morphology are well-recognized within the field of frozen ground engineering. 26 When assessing and predicting the susceptibility for frost heave and ice lens formation in a particular natural soil, empirical parameters that better represent the material characteristics might be required. The essential qualitative behavior emphasized here is nevertheless expected to be robust, with permeable flow acting as a dominant control on lens growth for smaller particles, and film flow gaining enhanced importance for larger particles. Further experiments and modeling efforts focussed on lensing in fine-grained particles, natural soils, and the effects of pore


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micro-structure are ongoing.

Figure 5: Maximum growth rate of ice lenses at each porosities against particle sizes.

Conclusion Ice lens growth can be subdivided into two stages that are marked by changes in freezing rate. When the freezing rate is much higher than the lens growth rate, the initiation and termination of growth follow upon each other in quick succession. Evidence for the growth of periodic lenses in this “first stage” is provided by observations of surface heave even without the appearance of macroscopic lenses. At later times, a macroscopic lens can begin to grow in a “second stage” that begins when the freezing rate slows sufficiently to allow a stable water supply. The total frost heave is observed as the accumulation of ground deformation following both the first and second stages of lens growth. In addition to the freezing rate, the particle size in porous media exerts a dominant control on the ice lens behavior. Water supply that is required for the growth of the ice lens must involve both permeable flow through the unfrozen region and film flow around the particle surfaces adjacent to the lens boundary. While the Peclet number Pe that represents conditions in which the water supply is primarily limited by permeable flow is proportional to the particle radii R, when film flow dominates the hydrodynamic pressure drop Pe is proportional to R−3 . Since these pathways 19 ACS Paragon Plus Environment


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are both required to supply lens growth, the net flow is governed by the smaller Pe. These restrictions lead to a maximum in the rate of ice lens growth for particles of intermediate size. Moreover, since the film flow that governs lens formation in more coarse grained sediments does not depend strongly on porosity or other microstructural characteristics of the particle packing, it is only when lenses form in smaller particles such as colloidal suspensions that lensing is sensitive to these variables.

Acknowledgement The authors thank the two anonymous reviewers and the support of a grant from the Global COE Program, From the Earth to Earths, from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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(6) Worster, M. G.; Wettlaufer, J. S. Fluid Dynamics at Interfaces; edited by W. Shyy and R. Narayanan; Cambridge University Press: Cambridge, U.K., 1999; pp 339–351. (7) O’Neill, K.; Miller, R. D. Exploration of a Rigid Ice Model of Frost Heave. Water Resour. Res. 1985, 21, 281–296. (8) Rempel, A. W.; Wettlaufer, J. S.; Worster, M. G. Premelting Dynamics in a Continuum Model of Frost Heave. J. Fluid Mech. 2004, 498, 227–244. (9) Style, R. W.; Peppin, S. S. L.; Cocks, A. C. F.; Wettlaufer, J. S. Ice-lens Formation and Geometrical Supercooling in Soils and Other Colloidal Materials. Phys. Rev. E 2011, 84, 0414021-04140212. (10) Cahn, J. W.; Dash, J. G.; Fu, H. Y. Theory of Ice Premelting in Monosized Powders. J. Cryst. Growth 1992, 123, 101–108. (11) Ishizaki, T.; Maruyama, M.; Furukawa, Y.; Dash, J. G. Premelting of Ice in Porous Silica Glass. J. Cryst. Growth 1996, 163, 455–460. (12) Penner, E. Particle Size as a Basis for Predicting Frost Action in Soils. Soils Found. 1968, 8, 21–29. (13) Saruya, T.; Kurita, K.; Rempel, A. W. Experimental Constraints on the Kinetics of Ice Lens Initiation and Growth. Phys. Rev. E 2013, 87, 0324041-0324049. (14) Style, R. W.; Peppin, S. S. L. The Kinetics of Ice-lens Growth in Porous Media. J. Fluid Mech. 2012, 692, 482–498. (15) Rempel, A. W. A Theory for Ice-till Interactions and Sediment Entrainment beneath Glaciers. J. Geophys. Res. 2008, 113, F01013. (16) Saruya, T.; Kurita, K.; Rempel, A. W. Indirect Measurement of Interfacial Melting from Macroscopic Ice Observations. Phys. Rev. E 2014, 89, 0604011-0604015. 21 ACS Paragon Plus Environment


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(17) Rempel, A. W. Formation of Ice Lenses and Frost Heave. J. Geophys. Res. 2007, 112, F02S211-F02S2117. (18) Wilen, L. A.; Dash, J. G. Frost Heave Dynamics at a Single-crystal Interface. Phys. Rev. Lett. 1995, 74, 5076–5079. (19) Wettlaufer, J. S.; Worster, M. G. Dynamics of Premelted Films: Frost Heave in a Capillary. Phys. Rev. E 1995, 51, 4679–4689. (20) Wettlaufer, J. S.; Worster, M. G.; Wilen, L. A.; Dash, J. G. A Theory of Premelting Dynamics for all Power Law Forces. Phys. Rev. Lett. 1996, 76, 3602–3605. (21) Wettlaufer, J. S.; Worster, M. G.; Wilen, L. A. Premelting Dynamics: Geometry and Interactions. J. Phys. Chem. B 1997, 101, 6137–6141. (22) Hansen-Goos, H.; Wettlaufer, J. S. Theory of Ice Premelting in Porous Media. Phys. Rev. E 2010, 81, 0316041-03160413. (23) Roller, P. S. The Bulking Properties of Microscopic Particles. Ind. Eng. Chem. Res. 1930, 22, 1206–1208. (24) Peppin, S. S. L.; Wettlaufer, J. S.; Worster, M. G. Experimental Verification of Morphological Instability in Freezing Aqueous Colloidal Suspensions. Phys. Rev. Lett. 2008, 100, 2383011-2383014. (25) Anderson, A. M.; Worster, M. G. Periodic Ice Banding in Freezing Colloidal Dispersions. Langmuir 2012, 28, 16512–16523. (26) Andersland, O. B.; Ladanyi, B. An Introduction to Frozen Ground Engineering; Chapman and Hall: New York, 2004 .


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TOC Graphic millimeter ice lens

micrometer ice premelted film substrate

nanometer


Hydrodynamic Transitions with Changing Particle Size That Control

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