A physical model of frozen ground considered as a complex macrosystem

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1 Permafrost, Phillips, Springman & Arenson (eds) 2003 Swets & Zeitlinger, Lisse, ISBN A physical model of frozen ground considered as a complex macrosystem A.D. Frolov Scientific Council on Earth Cryology of the Russian Academy of Sciences, Moscow, Russia ABSTRACT: The physical model of frozen ground represented as a complex macrosystem with corresponding hierarchy, which takes into account the constitution, structure and main physical properties of this complicated medium, is the subject of this paper. A systems approach is the most appropriate for such model development for the present state of knowledge on frozen formations. The component composition and specific spatial cryogenic coagulant-crystalline structure, as well as the physical subsystems of frozen soil, which condition its physical properties, are considered. Examples of comparison of this model with experimental data are presented and discussed for better comprehension and explanation of the electrical, mechanical property changes and probable interrelations between them for various sandy-clayey frozen soils. The proposed model is a useful tool for planning experiments and interpreting their results. 1 INTRODUCTION Frozen soils are complex polycrystalline, multicomponent, polyphase and polydisperse physical-chemical systems, which usually are in a state of unstable quasiequilibrium. It is essential to have a sufficiently adequate physical model of frozen ground, based on information available at present, in order to comprehend fundamental and specific features of this medium. The development of a model elucidating the formation and changes of physical properties of frozen ground would be of great significance for the solution of scientific and engineering problems. A systems approach seems the most appropriate for such model development, with a frozen soil being considered as a complex macrosystem. In this context, the following notions are important: a system is a multitude of correlated elements with a definite structural integrity, which ensures its particular response to environmental impact. Integrity is one of the principal distinctive features of a complex system, because the physical properties of it as a whole are not an additive sum of corresponding properties of constituents, but depend on interactions between them. In general, any real heterogeneous complex system lies between two extremes: absolute additivity and absolute integrity (Ostreikovsky 1997). Structure reflects the most essential interrelations between elements and subsystems that ensure the system functions and determines its principal properties. Communicativity means that the system forms a specific relationship with an environment, namely the system is not isolated. It is extremely difficult, however, to formalize a description of frozen soils, which are very complicated formations. Current analytical models of granular and capillary-porous media or additive mixtures etc. are inadequate for frozen soils (Frolov 1976, 1998, Lunardini 1991). For the present state of knowledge, a mathematical model of such a system would be a considerable simplification. Basing a qualitative conceptual physical model of frozen ground on a system approach would help to improve comprehension. Such a model is proposed and described below, with new details as well as some examples of its usefulness in explaining experimental data. 2 FROZEN SOIL COMPLEX MACROSYSTEM Frozen ground is a complex macrosystem distinguished not simply by the coexistence of ice and unfrozen water but by a specific cryogenic spatial structure, which accounts for the response to external actions. Within certain limits, it is an adaptive, self-restoring (relaxing) system, which may be represented by hierarchies with relatively weak bonds (Ostreikovsky 1997). As the basis for a model, two correlative hierarchy levels are suggested as follows: 1. In the context of the medium s constitution: A. Component composition of the frozen soil and volume relationships between its components. B. Spatial Cryogenic Coagulant-crystalline Structure (SCCS), which appears and develops in the process of freezing and a constitutional ice formation. C. Principal elements (matrices of soil). 2. In the context of the macrosystem s response to external impact: A. Physical subsystems. B. Mechanisms of the responses on external excitations. C. Physical properties. Each of the points mentioned above are discussed briefly below. 259

2 2.1 Component composition The component composition of frozen soil consists of: mineral solids (soil skeleton) silicate grains and grain aggregates; pore and segregated constitutional ice (usually impure); the unfrozen liquid phase, (actually, pore solution, but frequently named unfrozen water ); and the gaseous phase. To describe component composition one may assume that V m V i V u V g 1, where: V m, V i, V u and V g are volumes of mineral solids, ice, unfrozen liquid and gas, respectively, in a unit volume of frozen soil. It should be noted, that the unit chosen must be big enough to ensure representativeness of all characteristics of the given soil. It is necessary to distinguish between the two important cases of the component composition changes in freezing soil, when its volumetric moisture content W v increases: a) V m const. 0, constant porosity, partial initial saturation, b) V g 0, complete initial moisture saturation, variable porosity. In the first case, the frozen soil is considered to be a four-component medium where, as W v increases until it equals the volume of pores, the pore ice-cement content increases and V g decreases finally reaching zero. In the second case, initially a three-component medium exists, where V m and V u are reduced when W v and therefore V i increase. Finally both V m and V u drop to minimum when the medium is in its ultimate state, corresponding to polycrystalline ice. All these components are briefly discussed below. Mineral solids of frozen sandy-clayey soils consist of grains of both primary silicate minerals and secondary ones phyllosilicates. The latter are distinguished by a greater specific surface, by complicated non-uniform grain morphology and enhanced capacity for sorption, ion exchange, swelling, etc. The soil skeleton may also contain poorly soluble salts, clathrates, organic and organic-mineral substances. The characteristics of the skeleton are polydispersity and heteroporosity. These features should be taken into account when considering unfrozen liquid phase formation, especially in clayey frozen soils. Constitutional ice in frozen soil is highly diversified in size, configuration, texture, impurities and structures of polycrystalline aggregates. Two kinds of ice are defined: ice-cement and segregated ice. Their spatial distribution controls the cryotexture of the soil. During freezing of clayey water-saturated soil, ice rich clusters, including non-aggregated clay particles, frequently occur. However, even so- called pure monomineral ice is actually heterogeneous matter containing liquid phase and impurities in the inter-grain boundary zones of crystallites and between them. The unfrozen liquid phase in frozen ground differs noticeably from common water, which is very important to bear in mind when examining the physical properties. The differences may be summarized briefly as follows: 1. liquid matter in soils is always a solution; saline sandy-clayey frozen soils differ from non-saline ones only in certain initial critical (limit) concentration of the pore solution, depending on its ionic composition and on the soil skeleton (Frolov 1998), 2. unfrozen pore solution resides near (or between) different surfaces of solid grains (mineral and ice); the grain surfaces are irregular in micro-morphology, distribution of unneutralized electric charges, existence of quasi-liquid layers etc., 3. as a result, the liquid phase in soil is always inhomogeneous, i.e. it is marked by allotropy. In essence, it presents discrete (semi-discrete) domains of different liquid phases with diverse structure of hydrogen bonds, with differences in internal energy and entropy, and therefore in eutectic and other parameters of the solution states within the soil, 4. these phases are in equilibrium with impure ice at temperatures below 0 C, and cannot be considered as metastable states due to external influence of grain surfaces; they have a somewhat reduced (when compared to free water) chemical potential, 5. the presence of impure ice noticeably modifies the energetic state of the liquid phase in discrete domains as well as the electrochemical activity of the medium, and consequently it changes the electric polarizability and dielectric permittivity of frozen soil with respect to a thawed soil. This leads to better understanding of soil freezing over an extended interval of negative temperatures. The gaseous phase composition in frozen soils affects their properties only in the case when pores are not completely filled with ice and unfrozen liquid phase. Besides air and water vapor, there are gaseous hydrogen compounds (CH 4, H 2 S, P 2 H 4, etc.) in noticeable quantities in soils enriched in organic matter. However, even detailed knowledge of the component composition would not enable reliable estimates of the frozen-soil physical properties to be obtained, because of interaction between the components and resulting violation of the additive principle. 2.2 Spatial Cryogenic Coagulant-crystalline Structure (SCCS) In conformity with the physical-chemical mechanics (e.g. Rebinder 1979) heterogeneous, disperse, liquidcontaining media form stable spatial structures by way of two principal types of contacts: coagulation (through the liquid interlayer) and phase or crystallizationcondensation, namely due to sintering or mutual 260

3 coalescence of solid phases. Combined contacts may exist, varying both in configuration and size of contacting areas and in thickness and composition of liquid and quasi-liquid interlayers. Cohesion forces at coagulant contacts are about 10 3 or more times less then that ( 10 3 ) at phase contacts. Applying this approach to various frozen soils and ices implies that the specific SCCS appears during the formation and offers the key distinguishing feature of these geological objects (Frolov 1976, 1998). The principal constituents of the SCCS are: ice matrix ice grains (crystals and crystallites); mineral matrix grains of various minerals including poorly dissolving salts; matrix of defects inter-grain zones including unfrozen liquid phase, gases, impurities, defective boundary layers of solid phases. The latter matrix is the most variable and informative element of the SCCS state and exerts principal control over the most sensitive electrical and mechanical characteristics of the frozen soil as a macrosystem. A solid salt matrix may forms at temperatures below eutectic values in saline frozen soils. The presence of the liquid phase accumulating within inter-grain zones accounts for the viscousplasticity, and the zones often act as traps for expanding microfissures, which in turn influence the frozen soil strength (especially the tensile strength). These intergrain zones in frozen soil are undoubtedly nonuniform. Electrically uncompensated active centers are arranged in strips or in spots on grain surfaces, which accounts for insular adsorption of molecules or ions from liquid or gaseous phases. The discrete distribution is traced as a distorted structure within adsorbed phases over some distances (up to 10 3 nm and more) from the adsorbent surfaces. The concept of homogeneous films of adsorbed water still persisting since twentieth years of last century (Langmuir s adsorption theory) is highly simplified and is not adequate for moist soils. All matrices of the SCCS are integral elements of a new (cryogenic) macrosystem. Spatial patterns, quantitative relationships and correlations between the SCCS matrices depend on initial conditions of the soil freezing, as well as on its further evolution in changing environments. Four principal stages of SCCS formation and development may be recognized, from the appearance of the odd individual ice crystals to practically complete freezing of the liquid phase (Frolov 1976, 1998). Specific features of cryogenesis and properties of the macrosystem allow a unified concept of the physical-chemical basis to be produced to describe the laws and mechanisms relating to the physical properties and the variability typical for each stage. The specific character of inter-grain zones and impure constitutional ice produces a noticeable effect on structural cohesive forces, which in turn control frozen soil deformability and strength. Bonds within the ice and clay aggregates, as well as between the grains of the medium, vary considerably in strength, which results in their successive breakage (beginning with the weaker ones) in the mechanical stress field. Such a mechanism can produce a partial relaxation of stress and, consequently, damping (stabilization) of the microcracking. The macrorupture of the medium will begin when an average load per single bond is equal to an average strength value for unbroken bonds. 2.3 Physical subsystems and physical properties In studies of frozen ground as a complex macrosystem, it is common practice to select a certain kind of physical field acting upon the object and to vary its intensity, duration and regime of application, frequency and temperature ranges etc. In the process, physicalfiltration is performed and the result reflects processes taking place primarily within a certain subsystem of the complex macrosystem. Part of the information is inevitably lost (as in any data processing), but the other part yields much finer details. It is reasonable to designate as separate physical subsystems, those degrees of freedom of internal movements (i.e. physicalchemical transformations), which are characteristic of processes controlling the specific properties under consideration. Therefore, a complex macrosystem should be considered as an assemblage of physical subsystems combined into SCCS and related to its principal elements (Fig. 1). It is essential that a subsystem after excitation should evolve towards quasi-equilibrium (relaxation) at greater rate than the macrosystem as a whole. In that case, certain external action on the macrosystem may result in selective excitation of individual subsystems. This opens up new possibilities in the investigations of their relaxation mechanism, hence the peculiarities of corresponding physical properties, which are the parameters characterize the medium s response to external physical fields (Frolov 1998). In the context of the matter s structure and response to various force fields, a number of physical (nuclear, electronic, ionic) subsystems may be distinguished (Fig. 1), each of which may include several groupings of mechanisms and properties. In subsystems that are bound under external field action the corresponding particles (i.e. electrons or ions) would oscillate and characterize alternative molecular microdipoles. In ones that are quasi-free, the particles can move progressively over a considerable distance and would control transfer phenomena (electrical and heat conduction). Also possible are semi-bound ionic and electronic subsystems, where electrically charged particles may move in translatory 261

4 Macrosystem -FROZEN SOIL Component Composition Spatial Cryogenic Coagulant-crystalline Struture PRINCIPAL ELEMENTS mineral matrix matrix of defctes ice matrix PHYSICAL SUBSYSTEMS nuclear -spin nuclear -dipolar electron -spin electron -dipolar natural induced electron -hole free semi-bound bound ionic free semi-bound bound PROPERTIES MECHANISMS Nuclear - Electron - magnetic magnetic relaxation relaxation proton mobility magnetization Mechanical relaxation heat conductivity heat capacity elasticity viscosity Strength D.C conduction creep compression tensile shift electronic ionic mode, though only within a confined volume which results in temporary surplus charge accumulation and formation of polarized discrete portions of the medium (macro-dipoles). When comparing the different physical properties of a complex macrosystem, it is important that they should have a common causal factor, i.e. their variations should be controlled by response of the same subsystems or ones reflecting similar changes in the medium (e.g. changes of the phase composition in frozen soils). Only in these cases it is possible to establish reliable correlations between different properties; otherwise the correlations are either purely formal, or applicable only in particular cases. Studies of relaxation processes in heterogeneous media aimed at excitation of separate subsystems are therefore exceedingly important. Sometimes they offer a unique route for recognizing the kinetics of complex physical-chemical transformations and obtaining information on the role of individual mechanisms and their interrelationships. 3 EXPERIMENTAL DATA EXPLANATION USING THE MODEL Dielectric relaxation permittivity loss factor polarization Figure 1. Hierarchy of frozen soil considered as a complex macrosystem. An indicator of the integrity of frozen ground as a macro system is that its properties are inconsistent Figure 2. Relative dielectric permittivity of the frozen sand vs. frequency and temperature; initial W v ~ 20%. with the additive principle. It is especially apparent in frozen soils with respect to the real part of the relative complex dielectric permittivity. Thus, according to experimental data (e.g. Frolov 1976, 1998, Gurov 1983), at frequencies of Hz, with the volumetric moisture content varying from 1 2 to 40% and temperatures up to 40 C (for clays) the value, even of frozen non-saline sandy-clayey soils, may vary by a factor of tens, hundreds or even thousands (Olhoeft 1978, Maeno et al. 1992). But the value for water and ice does not exceed 10 2, and 10 for silicates. Hence, equations for additive mixtures cannot explain the experimental results. The abnormally high values of obtained for frozen sandy-clayey soils as well as their frequency and temperature dependence (Fig. 2), could be explained by help of the model of the unfrozen liquid phase in discrete domains within a matrix of defects. Application of an external electromagnetic field results in the migration of ions within domains, accumulation of divided charge, and rise of electric macrodipoles, with the values of electric dipole moments depending essentially on the mobility and concentration of ions as well as on the duration of the action of the electromagnetic field, hence on its period (or frequency). Existence of such macrodipoles was confirmed by the fact that it allowed to explain of the non-linear electro acoustic effect in frozen soil observed at doubled frequency compared to that of the applied electric field (Frolov 1998). The contribution of the macro-dipole polarization mechanism is reduced (Fig. 2) with an increase of the frequency of the electromagnetic field or a decrease of the frozen soil temperature. In the first case, this is due to decrease of the time for ion migration and charge division within liquid phase domains; in the second case, it is due to decrease in domain sizes 262

5 because of the liquid phase freezing out and tending to lessen the mobility of ions within it. Thus at a high frequency range MHz, 4 5 was obtained (Delaney & Arcone 1982) for frozen aeolian sand and this value does not practically depend either on total moisture content (which was varied over two-fold), or on temperature in the range from 2 to 22 C. This means no macro-dipole effect has been displayed. However, according to this work the temperature dependence is quite clear for frozen loess: the value decreases from (at 2 C) to 7 6 (at 18.8 and 21.8 C), and only the latter values approximately correspond to equations for statistical mixtures. Similar results have obtained for clayey soils by Olyphant (1985). At temperatures higher 15 C in frozen clayey soils there is a noticeable contribution from the macrodipole mechanism of polarization up to frequencies 1Ghz. This is due to considerable unfrozen liquid-phase content ( 5 6% by volume up to 15 C) distributed in a large number of small-size domains forming in inter-grain zones of these soils, as well as to sufficiently high mobility of ions within these domains. Another confirmation of the dominant role of the matrix of defects is the existence of boundary temperatures, at which the dynamic elastic moduli of the initially water-saturated frozen soil were found to be independent of the volumetric ice content over a wide range of values (Frolov 1976). Figure 3 shows this phenomenon and the modulus values obtained for Glukhovetsky kaolin at the boundary temperatures: for the shear modulus G at t 14 C, Young s modulus E at t 10 C, and bulk modulus K at t 6 C. The values of moduli (E and K GPa, G GPa) correspond to those of the polycrystalline ice at the temperatures indicated, beginning from a volumetric ice content of about 35%. At temperatures above the boundary values, the intergrain zones in frozen kaolin are less elastic (higher liquid phase content) than those of polycrystalline ice. So, an increase in ice volumetric content in the soil results in an increase of the elastic moduli of the medium due to the decrease in the proportion of the mineral solids (consequently of the liquid phase) and coagulation (weak) contacts between the grains. At temperatures below these boundary values, the rigidity of the inter-grain zones in the frozen kaolin exceeds this in polycrystalline ice; therefore the elastic moduli decreases with an increase of the ice proportion (by volume). Similar results were obtained (Vinson et al. 1983) during studies of the dynamic Young s moduli of natural samples of frozen loess differing in water content. At a boundary temperature of about 2.5 C, E GPa, which closely corresponds to that of polycrystalline freshwater ice. The boundary temperatures under K,E,G, GP a t,ºc discussion must depend on the composition and dispersiveness of the SCCS mineral matrix. Thus, for coarse frozen soils (sand, gravel), such temperatures are probably within the range from 0 to 0.5 C (this remains to be tested), while for frozen heavy swelling clays they may be much lower than those obtained for kaolin. The model also allows the established empirical relationship (Frolov 1976, 1998) between the dynamic Young s modulus of saturated frozen sandy-clayey soils with massive cryotexture and the degree of their mineral matrix dispersiveness to be understood easily: E (t) k(t)log d E (t), where d is the average particle size in m, E is the modulus in GPa for clay (d 1 m) and the parameter k(t) increases from 9 to 11.5 GPa with a temperature increase from 40 to 2 C. The E value changes from 14 to 3 GPa over this temperature range. During the process of freezing various mineral matrices from clays to sands, increasingly rigid inter-grain zones are formed and these are marked by decreasing content of the liquid phase and increasingly homogeneous states; this accounts for the noticeable growth of the moduli of elasticity. It should be noted that composition and state of the matrix of defects and the ice matrix have a pronounced influence on the frozen soil strength, on the thermal deformations and on the creep. Concerning strength, it is clear from the model that there must be a reasonably close correlation between parameters of strength and elasticity. Such a correlation for compressive strength and compression stress wave velocity for sandy-clayey soils have obtained by various investigators. In line with the prediction (Frolov 1976), the tensile strength is more sensitive to changes in state of the matrix of defects, which determine the K E G W,% 100 Figure 3. Boundary temperatures for the values of dynamic elastic moduli in frozen kaolin during saturation by ice (defined by increasing W v ). 263

6 a f /a th λf / λth a af /a th λf / λ 1.5 th transition from quasi-plastic to brittle modes of failure of the medium. Therefore, the peaks of tensile strength should occur as the frozen soil temperature falls. Such maxima were confirmed in laboratory experiments carried out by Shusherina, Zaitzev and Rogov in undisturbed samples of frozen soils and underground ice (see Frolov 1998). Finally, in saline frozen soils the matrix of defects is of even greater importance for determining the thermal, electrical and mechanical properties of the media. At a certain initial concentration C i of the pore solution, the influence of the mineral matrix may even be suppressed or attenuated considerably (Frolov 1998). According to Figure 4, the salinity effect on the thermal properties of sandy soils consist of a marked reduction of thermal conductivity ( ) and diffusivity (a) in its frozen state (Frolov & Komarov 1993), which is contrary to response in non-saline soils. Note, that in states th and ath for the soils saturated with solutions, when their concentration grows, decreased only 5 7%, while in frozen state f and af became 15 25% less than in thawed state. This effect, showing a decrease of the thermal parameters, is due to increasing accumulated unfrozen pore solution and results in an abrupt rise of concentration during the freezing of saline soils. 4 CONCLUDING REMARKS The data discussed above lead to the conclusion that the physical model considered describes frozen soil behavior quite adequately. The model provides a consistent explanation and better understanding of the specific features and experimentally established laws of changes in mechanical and electric properties of frozen soils during the processes of freezing and thawing. The expedience of joint studies and analysis of the processes of mechanical and electrical relaxation is demonstrated by the model. It also offers a reliable C g/i C g/i Figure 4. Thermal conductivity (dotted line) and thermal diffusivity (solid line) vs initial concentration of pore NaCl solution for: (a) sand, (b) sandy loam. Indices correspond to: f frozen state, t 12 to 15) C ; th thawed state, t 5 C. b correlation between the mechanical and electrical characteristics of frozen soils. Such a correlation would provide the basis for developing non-destructive acoustic and electromagnetic (including remote sensing) methods to estimate and control the strength and deformation characteristics of the frozen soils in situ. It seems to be possible to develop the knowledge of the way in which the physical subsystems of the SCCS would be influenced, in order to determine some appropriate properties for frozen soil or to predetermine the regime for changing them under certain conditions. ACKNOWLEDGEMENT This work was supported in part by the Russian Fund for Basic Research (grant ) for which I would like to express my gratitude. REFERENCES Delaney, A.J. & Arcone, S.A Laboratory measurements of soil electric properties between 0.01 and 5 GHz. CRREL Rep Frolov, A.D Electric and elastic properties of cryogenic grounds. Moscow: Nedra Press. Frolov, A.D Electric and elastic properties of frozen earth materials. Pushchino: ONTI PSC RAS Press. Frolov, A.D. & Komarov, I.A Characteristics of the thermophysical and electric property changes of saline frozen soils. In V.J. Lunardini and S.L. Bowen (ed.), Proc. of Fourth Intern. Symp. on Thermal Engineering and Science for Cold Regions: Hanover, N.H.: USA. CRREL Special Rep Gurov, V.V Technique and some experimental results of frozen soil dielectric properties studies. Geocryological studies 21: Moscow: Moscow University Press. Lunardini, V.J Heat transfer with freezing and thawing. Amsterdam: Elsevier. Maeno, N., Araki, T., Moore, J.C. & Fukuda, M Dielectric response of water and ice in frozen soils. In Maeno, N. and T. Hondoh (ed.), Proc. Intern. Symp. on the Physics and Chemistry of Ice: Sapporo: Hokkaido University Press. Olhoeft, G.R Electric properties of natural clay permafrost. Canad. J. Earth Sci. 14(1): Olyphant, J.I A model for dielectric constants of frozen soils. In Freezing and thawing of soil-water system: New York: Amer. Soc. Civil. Eng. Ostreikovsky, V.A Theory of systems. Moscow: Vysshaya Shkola Press. Rebinder, P.A Surface phenomena in disperse systems. Physical-chemical mechanics. Moscow: Nauka Press. Vinson, T.S., Wilson, C.R. & Bolander, P Dynamic properties of naturally frozen silt. In Proceedings of the 4th International Conference on Permafrost: Washington: National Academy Press. 264