Thermodynamics-based soil freezing characteristic curve model considering unequal-diameter particle contact

Abstract

The invention relates to a thermodynamics-based soil freezing characteristic curve model considering unequal-diameter particle contact, and the model comprises the following steps: (1) carrying out soil unequal-diameter particle contact calculation to obtain a contact mode of a soil sample; and (2) carrying out water freezing calculation on the soil to obtain freezing points of capillary water and adsorbed water of the soil. And (3) calculating freezing characteristic curves of the soil sample at different temperatures through the soil-water characteristic curve model. On the basis of thermodynamics, according to the similarity of a soil-water characteristic curve and a freezing characteristic curve, the determination effect of a complex particle contact mode of soil on the conversion relation of the soil-water characteristic curve and the freezing characteristic curve is considered, and the capillary effect and the adsorption effect of the soil on water are considered from the angle of the action relation of unequal-diameter particle contact influence specific interface energy; a soil-water characteristic curve model is introduced to deduce and establish a freezing characteristic curve theoretical model. According to the method, multiple influence factors such as particle size, interface energy and void ratio are fully considered, the influence factors are clear, the physical significance is definite, and the method can be used for calculating and predicting the relation between the temperature and the water content of the frozen soil. In addition, the method can also provide theoretical support for analyzing permeability, water and salt migration characteristics, soil shear strength and volume deformation of frozen soil, and provide theoretical reserve for perfecting an engineering numerical calculation model.

Description

Technical Field

[0001] The invention relates to the technical field of geotechnical engineering in cold regions, and in particular to a thermodynamic theoretical model for estimating a freezing characteristic curve according to soil contact modes. Background Art

[0002] The freezing characteristic curve (SFCC) is an important hydraulic characteristic that characterizes the relationship between the temperature and unfrozen water content of frozen soil. It is crucial to estimate the characteristics of rock and soil in cold regions and is the basis for analyzing soil water migration, solute migration, and water retention in cold regions. Whether in engineering or agriculture, the freezing characteristic curve has a wide range of applications. Direct measurement to obtain SFCC is very time-consuming and labor-intensive, while obtaining SFCC through indirect methods is a more economical and effective method. Existing indirect methods are often divided into empirical fitting models and physical concept models. However, the physical meaning of the empirical fitting model proposed based on experimental data is insufficient. Although the physical concept model has sufficient physical meaning, the model accuracy is often not ideal. The reason can be summarized as the model's assumptions about the stacking mode of soil particles are too ideal, and it often only considers capillary water, while ignoring the adsorbed water content when the water content is lower. Therefore, it is not enough to accurately describe the relationship between the temperature and unfrozen water content of real soil.

[0003] In summary, with the urgent need for research development and engineering construction, there is an urgent need to propose a theoretical gap in the freezing characteristic curve model with sufficient physical significance and considering the contact of particles of different diameters based on thermodynamics. In-depth research on the relationship between unfrozen soil water and temperature will help to further establish a high-precision freezing characteristic curve model and lay a more complete theoretical basis for numerical calculations. Summary of the invention

[0004] The technical problem to be solved by the present invention is to provide a freezing characteristic curve prediction method taking into account factors such as soil contact mode, porosity, soil type, capillary action and adsorption.

[0005] In order to solve the above problems, the thermodynamic soil freezing characteristic curve model considering the contact of particles of different diameters described in the present invention comprises the following steps:

[0006] (1) Calculate the contact between soil particles of different diameters and obtain the contact mode of the soil sample.

[0007] (2) Calculate the freezing point of soil moisture and obtain the freezing point of soil capillary water and adsorbed water.

[0008] (3) The freezing characteristic curve of the soil sample at different temperatures is calculated through the soil-water characteristic curve model.

[0009] The theoretical basis for linking SWCC and SFCC is based on the similarity of the mechanisms of SWCC and SFCC during drying and freezing. The flow to the freezing front in frozen soil is similar to the flow to the evaporation front in dry soil. During soil drying, the difference between air pressure and pore water pressure at the curved air-water interface can be expressed in the form of the Young-Laplace equation:

[0010]

[0011] In the formula, p a is the air pressure (M·L -1 ·t -2 ), p w is the pore water pressure (M·L -1 ·t -2 ), σ aw is the specific energy of the air-water interface (M·t -2 ), r aw is the average radius of curvature of the air-water interface (L). Similarly, in airless frozen soil, the corresponding expression for the ice-water interface can be obtained:

[0012]

[0013] Where r wi is the radius of curvature of the ice-water interface (L), σ iw is the ice-water interface specific energy (M·t -2 ).

[0014] Due to the different surface energies of the phase interfaces, different phases of water can be at different pressures but still be in thermodynamic equilibrium. A comparison of the two can establish a relationship:

[0015] p a -p w =n r (p i -p w ) (3)

[0016] where n r =σ aw / σ iw For soils with rigid pore structures and solid-solid direct contact (SS) between particles, such as sand and silt, capillary force plays a dominant role. r = 2.2; however, for soils with liquid contact (SLS) between particles (such as clay), the dominant force is adsorption rather than capillary force, so n r =1(Koopmans&Miller,1966).

[0017] Depending on the contact method of different types of soil, different values of n are selected. However, it is not practical to arbitrarily determine the value of n as a binary value and apply it to all soils. Most soils in nature are neither completely SLS nor completely SS. Natural soil is composed of sand grains, silt grains, and clay grains. For sand grains and silt grains, the contact between them is SS. For clay grains, due to the existence of a relatively thick adsorbed water film controlled by its high surface energy, the contact with any other particles is of the SLS type. As the clay content increases, the contact method between soil particles will also change, transitioning from SS contact to SSLS contact and finally to SLS contact. This means that for SSLS soils with different particle compositions, it is necessary to correctly measure the proportion of SS contact and SLS contact in the particle contact method in order to accurately determine the value of n. r However, it is not practical to arbitrarily determine the value of n r as a binary value and apply it to all soils. Most soils in nature are neither completely SLS nor completely SS. Natural soil is composed of sand grains, silt grains, and clay grains. For sand grains and silt grains, the contact between them is SS. For clay grains, due to the existence of a relatively thick adsorbed water film controlled by its high surface energy, the contact with any other particles is of the SLS type. As the clay content increases, the contact method between soil particles will also change, transitioning from SS contact to SSLS contact and finally to SLS contact. This means that for SSLS soils with different particle compositions, it is necessary to correctly measure the proportion of SS contact and SLS contact in the particle contact method in order to accurately determine the value of n. r value.

[0018] To reasonably correct the value of the soil specific interfacial energy, this paper established a method to measure the soil particle contact method by describing the contact of unequal-diameter particles. The definition of the coordination number is the total number of ligand particles directly connected to it in soil particles. The porosity determines to a certain extent the density of the particle spacing, that is, it determines the coordination number. According to the random contact experiment of unequal-diameter spheres, the relationship between the average coordination number and the porosity can be obtained (Trong & Mamoru, 2023):

[0019]

[0020] where N t is the average coordination number, N tmax is the average maximum coordination number, b and k are material coefficients. For the sphere packing system, b = 1.23 and k = 1.4.

[0021] In the contact of two unequal-diameter particles, assume that the small spherical particle with an average radius of r is arranged around the large spherical particle with an average radius of R in the most closely packed manner (r < R). Assume that the average maximum coordination number of soil particles can be calculated by the average value of the upper and lower limits of the maximum coordination number of unequal-diameter particle contact.

[0022] The solid angle of the whole large sphere is 4π (ignoring the unit). The solid angle profile of the spherical cap surface area corresponding to each small particle in close contact around the center of the large particle is φ. The ratio of the two is the upper limit of the maximum coordination number. Because there will be gaps between small particles, a curvilinear triangular area will be left between every 3 small particles. The solid angle blocked by all small particles cannot cover the entire spherical surface. Therefore, if these small curvilinear triangular areas are not considered, the obtained contact number is the upper limit value. According to the geometric relationship, the solid angle θ corresponding to the spherical cap is:

[0023]

[0024] Where h = R-RcosΦ is the height of the spherical cap; Φ is the cross-sectional angle corresponding to each small particle in close contact with the center of the large particle.

[0025] The final number of contacts is the upper limit value N up for:

[0026]

[0027] Then consider those curved triangle areas, calculate their areas on the sphere, and get the corresponding solid angle to calculate the lower limit of the contact number. Establish a rectangular coordinate system, assume that the three small balls are balls A, B, and C, and the coordinate origin is located at the center of the large ball, denoted as O. The three small balls are located in the xOy plane, ball A is located in the positive direction of the x-axis, balls B and C are located in the negative direction of the x-axis, and are symmetrical about the xOz plane. The center O′ of the equilateral triangle formed by the centers of the three small balls is located on the negative z-axis, with coordinates (0,0,-h). The coordinates of the centers of the three small balls are Where h is the height of O-ABC, according to the geometric relationship

[0028] The area of ​​the triangle with equal curved sides cut by the triangular pyramid O-ABC is A1, and the surface area of ​​the spherical crown cut by the cone OA is A0. A1 is based on the spherical triangle area formula, A1 = (3β-π)R 2 ; β is the dihedral angle formed by any two side faces of the regular triangular pyramid O-ABC, and the three spherical angles are equal in size.

[0029] Taking the angle between plane OAB and plane OBC as an example, the cosine value cosβ is calculated as:

[0030]

[0031] The area S of the curved triangle is:

[0032]

[0033] The corresponding solid angle is:

[0034]

[0035] Get the lower limit of the number of contacts N down for:

[0036]

[0037] Average the upper and lower limits to get N tmax for:

[0038]

[0039] In the contact mode of particles, as the content of fine particles increases, SS contact changes to SSLS contact and even SLS contact, and the contact mode changes with the type of adjacent particles. The coordination number in the initial unit changes accordingly. The initial contact unit is defined as equal-diameter particles, and the initial coordination number in the unit is:

[0040]

[0041] Among them, 12 is the coordination number of equal-diameter particles under the loosest simple cubic packing, 6 is the coordination number of equal-diameter particles under the tightest tetrahedral packing; D is the SC and TH stacking ratio divided according to the density of particle packing, which is used to calculate the relative density of the initial unit coordination number.

[0042] With the addition of particles of different diameters, the coordination number changes, which affects the contact mode. According to the ratio of the number of coarse and fine particles and the change rate of the coordination number relative to the initial unit coordination number, the growth rate of the average coordination number relative to the initial unit coordination number of the same particles is obtained, which represents the change of the contact mode with the change of particle content. r The influence strength of the value is obtained, and the contact type influence coefficient is:

[0043]

[0044] Where N a is the number of coarse particles, which is determined by the mass fraction of coarse particles ω a , soil sample mass m s The average radius r of the coarse particles a calculate, N b is the number of fine particles, which is determined by the mass fraction of fine particles ω b , soil sample mass m s The average radius r of fine particles b calculate,

[0045] When there are no fine particles in the soil, the contact mode is only SS contact. As the fine particle content increases, the contact mode changes to SSLS contact, and the SS contact will inevitably disappear completely, becoming SLS contact. Therefore, the interface energy value is compared and corrected:

[0046]

[0047] According to the Gibbs-Duheim equation, the chemical potential is:

[0048] dμ=-SdT+Vdp (15)

[0049] The chemical potential of liquid water at any pressure p and temperature T can be calculated as follows:

[0050]

[0051] in is the chemical potential of pure water (J·mol -1 ), depends on the temperature T w (K) and water pressure p w (kPa); R s is the universal gas constant (R s =8.31432 J·mol -1 ·K -1 );a w is the water activity, which describes the non-ideality of salt solutions and depends on the type and concentration of solutes. Ignoring salt concentration, a w =1.

[0052] Likewise, the chemical potential of ice is:

[0053]

[0054] If the two phases are in chemical equilibrium, their chemical potentials are equal. For the atmospheric pressure p0 and the freezing point (273.15K, denoted as T0). Therefore, at p0 and T0, liquid water (μ w ,J.mol -1 ) and ice (μ i ,J.mol -1 ) are the same (in all derivations, the subscripts w and i represent liquid water and ice, respectively). At the reference states p0 and T0, the chemical potentials of water and ice are equal:

[0055] μ w (p w , T w )=μ i (p i , T i ) (18)

[0056] Combining equations (16), (17), and (18), considering thermal equilibrium, the temperature of ice and water is the same as the ambient temperature t. The control equation for pressure change and temperature is obtained as:

[0057]

[0058] The freezing process of water in soil is similar to the drying process. During the freezing process, capillary water turns into ice first, while adsorbed water freezes last. Therefore, based on the similarity between SWCC and SFCC, the freezing mechanisms of adsorbed water and capillary water are analyzed by analogy. The ice pressure in soil water is assumed to be air pressure based on the analogy of freezing and drying processes. For capillary water, Ψ m =p a -p w Substituting formula (3) into:

[0059]

[0060] The change in volume of ice-water phase transition is ignored, V i / V w = 1. Particle contact does not affect the pressure exerted by adsorbed water on the soil, so the water content of adsorbed water at the corresponding points of SWCC and SFCC is the same, and the corresponding potential energy difference is the same. So there is p a -p w =p i -p w Therefore n r =1, and substituting into formula (20) we get:

[0061]

[0062] Thus, the freezing control equations of adsorbed water and capillary water considering the changes of pore water pressure and ice-water interface are obtained. Equations (20) and (21) can describe the freezing points at different pore water pressure positions in the soil-water system during the freezing process. In other words, under given temperature and solute concentration, the corresponding pore water pressure can be solved, so as to combine it with SWCC. The SWCC model is selected from (Lu, 2016), initialized according to the literature values ​​and the least squares algorithm is used to obtain the fitting results of the parameters, which are in the following form:

[0063]

[0064] θ z =θ c +θ a (24) where θ z is the total water content, θ a is the adsorbed water content, θ c is the capillary water content, θ amax is the amount of adsorbed water, θ s is the saturated water content, ψ is the matrix potential, ψ max is the maximum matrix potential, ψ cav is the cavitation matrix potential, m is the adsorption strength parameter, n is the pore size distribution parameter, and a is the intake matrix potential parameter.

[0065] The present invention is based on thermodynamics, according to the similarity between the soil-water characteristic curve and the freezing characteristic curve, and considers the decisive role of the complex soil particle contact mode on the conversion relationship between the two. Starting from the perspective of the relationship between the contact of unequal diameter particles affecting the specific interfacial energy, and considering the capillary action and adsorption of soil on water, the soil-water characteristic curve model is introduced to derive and establish a theoretical model of the freezing characteristic curve. The present invention fully considers many influencing factors such as particle size, particle size distribution, soil composition, porosity, etc. The influencing factors are clear and have clear physical meanings, which can be used to calculate and predict the relationship between the temperature and water content of frozen soil. In addition, the present invention can also provide theoretical support for analyzing the permeability of frozen soil, water-salt migration characteristics, soil shear strength and volume deformation, and provide theoretical reserves for improving engineering numerical calculation models.

[0066] Compared with the prior art, the present invention has the following advantages:

[0067] 1. The present invention is based on thermodynamics, according to the similarity between the soil-water characteristic curve and the freezing characteristic curve, considering the decisive role of the complex soil particle contact mode on the conversion relationship between the two, starting from the perspective of the relationship between the contact of unequal diameter particles affecting the specific interface energy, and considering the capillary action and adsorption of soil on water, the soil-water characteristic curve model is introduced to derive and establish a theoretical model of the freezing characteristic curve. The present invention finally obtains a freezing characteristic curve model whose input parameters are porosity, void ratio, particle size distribution, soil composition, and soil-water characteristic curve.

[0068] 2. The present invention can be used to analyze and calculate the relationship between the temperature and moisture content of frozen soil, taking into full account many influencing factors such as porosity, porosity ratio, particle size distribution, soil composition, soil-water characteristic curve, etc. The influencing factors are clear and their physical meanings are clear. BRIEF DESCRIPTION OF THE DRAWINGS

[0069] The specific implementation modes of the present invention are described in detail below with reference to the accompanying drawings.

[0070] Figure 1 Schematic diagram of soil contact mode

[0071] Figure 2 Schematic diagram of the contact mode of particles of different diameters in plane and three-dimensional space

[0072] Figure 3 The calculated frozen characteristic curve

[0073] Specific implementation instructions

[0074] Based on thermodynamics, according to the similarity between the soil-water characteristic curve and the freezing characteristic curve, the soil freezing characteristic curve model is estimated by considering the complex soil particle contact mode. The model method includes the following steps:

[0075] (1) Calculate the contact between soil particles of different diameters and obtain the contact mode of the soil sample.

[0076] (2) Calculate the freezing point of soil moisture and obtain the freezing point of soil capillary water and adsorbed water.

[0077] (3) The freezing characteristic curve of the soil sample at different temperatures is calculated through the soil-water characteristic curve model.

[0078] Figure 1 This is a schematic diagram of soil contact mode. The soil contact is regarded as the arbitrary contact mode between two unequal diameter particles of coarse and fine particles. The surface energy of coarse and fine particles determines the different contact modes between particles. Figure 2 The plane and three-dimensional schematic diagrams of the contact mode of particles of unequal diameters can generally reflect the contact mode of the soil through the content of coarse and fine particles, porosity ratio and coordination number.

[0079] The theoretical basis for linking SWCC and SFCC is based on the similarity of the mechanisms of SWCC and SFCC during drying and freezing. The flow to the freezing front in frozen soil is similar to the flow to the evaporation front in dry soil. During soil drying, the difference between air pressure and pore water pressure at the curved air-water interface can be expressed in the form of the Young-Laplace equation:

[0080]

[0081] In the formula, p a is the air pressure (M·L -1 ·t -2 ), p w is the pore water pressure (M·L -1 ·t -2 ), σ aw is the specific energy of the air-water interface (M·t -2 ), r aw is the average radius of curvature of the air-water interface (L). Similarly, in airless frozen soil, the corresponding expression for the ice-water interface can be obtained:

[0082]

[0083] Where r wi is the radius of curvature of the ice-water interface (L), σ iw is the ice-water interface specific energy (M·t -2 ).

[0084] Due to the different surface energies of the phase interfaces, different phases of water can be at different pressures but still be in thermodynamic equilibrium. A comparison of the two can establish a relationship:

[0085] p a -pw =n r (p i -p w ) (3)

[0086] where n r =σ aw / σ iw For soils with rigid pore structures and solid-solid direct contact (SS) between particles, such as sand and silt, capillary force plays a dominant role. r = 2.2; however, for soils with liquid contact (SLS) between particles (such as clay), the dominant force is adsorption rather than capillary force, so n r =1(Koopmans&Miller,1966).

[0087] According to the contact mode of different types of soil, the selected n r However, arbitrarily setting n r It is not practical to apply a binary value to all soils. Most soils in nature are neither completely SLS nor completely SS. Natural soils are composed of sand, silt, and clay. For sand and silt, the contact between them is SS. For clay, due to the existence of a thicker adsorbed water film controlled by its higher surface energy, the contact with any other particles is SLS. With the increase of clay content, the contact mode between soil particles will also change, from SS contact to SSLS contact, and finally to SLS contact. This means that for SSLS soils with different particle compositions, it is necessary to correctly measure the ratio of SS contact and SLS contact in the contact mode of the particles in order to accurately determine n. r value.

[0088] In order to reasonably correct the value of soil specific interfacial energy, this paper establishes a method to measure the contact mode of soil particles by describing the contact between particles of unequal diameters. The coordination number is defined as the total number of ligand particles directly connected to the soil particle. The porosity determines the density of the particle spacing to a certain extent, that is, the coordination number. According to the random contact experiment of spheres of unequal diameters, the relationship between the average coordination number and porosity can be obtained (Trong & Mamoru, 2023):

[0089]

[0090] Where N t is the average coordination number, N tmax is the average maximum coordination number, b and k are material coefficients, and for the sphere stacking system, b=1.23, k=1.4.

[0091] When two particles of unequal diameters are in contact, it is assumed that small spherical particles with an average radius of r are arranged around a large spherical particle with an average radius of R in the most closely packed manner (r < R). It is assumed that the average maximum coordination number of soil particles can be calculated from the average of the upper and lower limits of the maximum coordination number of particles with unequal diameters in contact.

[0092] The solid angle of the entire large sphere is 4π (ignoring units). The solid angle profile of the spherical cap surface area corresponding to each small particle in close contact around the center of the large particle is φ. The ratio of the two is the upper limit of the maximum coordination number. Since there will be gaps between small particles, a curved triangular area will be left between every three small particles. The solid angle blocked by all small particles cannot cover the entire spherical surface. Therefore, if these small curved triangular areas are not considered, the obtained contact number is the upper limit value. According to geometric relationships, the solid angle θ corresponding to the spherical cap is:

[0093]

[0094] where h = R - RcosΦ is the height of the spherical cap; Φ is the profile angle corresponding to each small particle in close contact around the center of the large particle,

[0095] The finally obtained contact number is the upper limit value N up which is:

[0096]

[0097] Then, considering those curved triangular areas, calculate their areas on the spherical surface, so as to obtain the corresponding solid angles to calculate the lower limit value of the contact number. Establish a rectangular coordinate system. Let the three small particles be sphere A, B, and C respectively, and the origin of coordinates is located at the center of the large sphere, denoted as O. The three small particles are below the xOy plane. Sphere A is on the positive x-axis, and spheres B and C are on the negative x-axis and are symmetric about the xOz plane. The center O′ of the equilateral triangle formed by the centers of the three small particles is on the negative z-axis, and the coordinates are (0, 0, -h). Taking the coordinates of the centers of the three small particles as where h is the height of O - ABC. According to geometric relationships

[0098] The area A1 of the curved triangular area intercepted by the triangular pyramid large sphere O - ABC, and the surface area of the spherical cap intercepted by the cone OA is A0. Among them, A1 is calculated according to the spherical triangle area formula, A1 = (3β - π)R 2 ; β is the dihedral angle formed by any two side faces of the regular triangular pyramid O - ABC, and the three spherical angles are of equal size.

[0099] Taking the angle between the plane OAB and the plane OBC as an example, the cosine value cosβ is calculated as:

[0100]

[0101]

[0102] The area S of the curved triangle is:

[0103]

[0104] The corresponding solid angle is:

[0105]

[0106] Get the lower limit of the number of contacts N down for:

[0107]

[0108] Average the upper and lower limits to get N tmax for:

[0109]

[0110] In the contact mode of particles, as the content of fine particles increases, SS contact changes to SSLS contact and even SLS contact, and the contact mode changes with the type of adjacent particles. The coordination number in the initial unit changes accordingly. The initial contact unit is defined as equal-diameter particles, and the initial coordination number in the unit is:

[0111]

[0112] Among them, 12 is the coordination number of equal-diameter particles under the loosest simple cubic packing, 6 is the coordination number of equal-diameter particles under the tightest tetrahedral packing; D is the SC and TH stacking ratio divided according to the density of particle packing, which is used to calculate the relative density of the initial unit coordination number.

[0113] With the addition of particles of different diameters, the coordination number changes, which affects the contact mode. According to the ratio of the number of coarse and fine particles and the change rate of the coordination number relative to the initial unit coordination number, the growth rate of the average coordination number relative to the initial unit coordination number of the same particles is obtained, which represents the change of the contact mode with the change of particle content. r The influence strength of the value is obtained, and the contact type influence coefficient is:

[0114]

[0115] Where N a is the number of coarse particles, which is determined by the mass fraction of coarse particles ω a , soil sample mass m s The average radius r of the coarse particles a calculate, N b is the number of fine particles, which is determined by the mass fraction of fine particles ω b , soil sample mass m s The average radius r of fine particles b calculate,

[0116] When there are no fine particles in the soil, the contact mode is only SS contact. As the fine particle content increases, the contact mode changes to SSLS contact, and the SS contact will inevitably disappear completely, becoming SLS contact. Therefore, the interface energy value is compared and corrected:

[0117]

[0118] According to the Gibbs-Duheim equation, the chemical potential is:

[0119] dμ=-SdT+Vdp (15)

[0120] The chemical potential of liquid water at any pressure p and temperature T can be calculated as follows:

[0121]

[0122] in is the chemical potential of pure water (J·mol-1), which depends on the temperature T w (K) and water pressure p w (kPa); R s is the universal gas constant (R s =8.31432 J·mol -1 ·K -1 );a w is the water activity, which describes the non-ideality of salt solutions and depends on the type and concentration of solutes. Ignoring salt concentration, a w =1.

[0123] Likewise, the chemical potential of ice is:

[0124]

[0125] If the two phases are in chemical equilibrium, their chemical potentials are equal. For the atmospheric pressure p0 and the freezing point (273.15K, denoted as T0). Therefore, at p0 and T0, liquid water (μ w ,J.mol -1 ) and ice (μ i ,J.mol -1 ) are the same (in all derivations, the subscripts w and i represent liquid water and ice, respectively). At the reference states p0 and T0, the chemical potentials of water and ice are equal:

[0126] μ w (p w , T w )=μ i (p i , T i ) (18)

[0127] Combining equations (16), (17), and (18), considering thermal equilibrium, the temperature of ice and water is the same as the ambient temperature T. The control equation for pressure change and temperature is obtained as:

[0128]

[0129] The freezing process of water in soil is similar to the drying process. During the freezing process, capillary water turns into ice first, while adsorbed water freezes last. Therefore, based on the similarity between SWCC and SFCC, the freezing mechanisms of adsorbed water and capillary water are analyzed by analogy. The ice pressure in soil water is assumed to be air pressure based on the analogy of freezing and drying processes. For capillary water, Ψ m =p a -p w Substituting formula (3) into:

[0130]

[0131] The change in volume of ice-water phase transition is ignored, V i / V w = 1. Particle contact does not affect the pressure exerted by adsorbed water on the soil, so the water content of adsorbed water at the corresponding points of SWCC and SFCC is the same, and the corresponding potential energy difference is the same. So there is p a -p w =p i -p w Therefore n r =1, and substituting into formula (20) we get:

[0132]

[0133] Thus, the freezing control equations of adsorbed water and capillary water considering the changes of pore water pressure and ice-water interface are obtained. Equations (20) and (21) can describe the freezing points at different pore water pressure positions in the soil-water system during the freezing process. In other words, under given temperature and solute concentration, the corresponding pore water pressure can be solved, so as to combine it with SWCC. The SWCC model is selected from (Lu, 2016), initialized according to the literature values ​​and the least squares algorithm is used to obtain the fitting results of the parameters, which are in the following form:

[0134]

[0135] θ z =θ c+θ a (twenty four)

[0136] Where θ z is the total water content, θ a is the adsorbed water content, θ c is the capillary water content, θ amax is the amount of adsorbed water, θ s is the saturated water content, ψ is the matrix potential, ψ max is the maximum matrix potential, ψ cav is the cavitation matrix potential, m is the adsorption strength parameter, n is the pore size distribution parameter, and a is the intake matrix potential parameter.

[0137] Model results verification

[0138] The data used in this paper to verify the soil SFCC comes from experimental studies by multiple groups of scholars (Ma et al., 2017; Tenget al., 2020; Ren & Vanapalli, 2019). Due to the small number of data points for particle size distribution, the soil samples cannot obtain a unified soil sample classification standard. Therefore, the sand, silt, and clay content of the soil samples and the soil type are classified and named according to the standards adopted by the source. The detailed physical parameters of the soil samples are shown in Table 1. In summary, the data selected in this study include sand, silt, clay, poor clay, and silty clay, with a wide range of working conditions and strong representativeness. The detailed physical parameters of the soil samples are summarized in Table 1 below:

[0139]

[0140]

[0141] The detailed fitting parameters used for the soil samples are summarized in Table 2 below:

[0142]

[0143] The data in the above table include a variety of soil types and are highly representative. Through the prediction method of the present invention, SFCC can be obtained by inputting sample parameters. As a theoretical model, the calculation results of the present invention show high reliability for samples of different soil types.

[0144] In summary, the calculation method of the present invention is not only widely applicable and has clear physical meaning, but also has high model accuracy, and can meet the prediction requirements of most soil freezing characteristic curves. Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention is described in detail with reference to the preferred embodiments, ordinary technicians in this field should understand that the technical solution of the present invention can be modified or replaced by equivalents without departing from the purpose and scope of the technical solution of the present invention, which should be included in the scope of the claims of the present invention.

Claims

1. A soil freezing characteristic curve model based on thermodynamics considering the contact of particles of unequal diameters, characterized in that: Including the following steps: (1) Conduct the calculation of the contact of soil particles with different diameters to obtain the contact mode of the soil sample. (2) Conduct the calculation of the freezing of soil moisture to obtain the freezing points of capillary water and adsorbed water in the soil. (3) Through the soil-water characteristic curve model, calculate the freezing characteristic curve of the soil sample at different temperatures.

2. The soil freezing characteristic curve model considering the contact of particles with different diameters based on thermodynamics according to claim 1, wherein: The theoretical basis for connecting the SWCC and SFCC is mainly based on the similarity of the mechanisms of the SWCC and SFCC during the drying and freezing processes. The process of water flowing towards the freezing front in frozen soil is similar to the process of water flowing towards the evaporation front in dry soil. During the drying process of the soil, the air pressure and pore water pressure difference at the curved air-water interface can be expressed in the form of the Young-Laplace equation: In the formula, p a is the air pressure (M·L -1 ·t -2 ), p w is the pore water pressure (M·L -1 ·t -2 ), σ aw is the specific energy of the air-water interface (M·t -2 ), r aw is the average radius of curvature of the air-water interface (L). Similarly, in airless frozen soil, the corresponding expression for the ice-water interface can be obtained: Where r wi is the radius of curvature of the ice-water interface (L), σ iw is the ice-water interface specific energy (M·t -2 ). Due to the different surface energies of the phase interfaces, different phases of water can be at different pressures but still in thermodynamic equilibrium. The ratio of the two can establish a relationship: p a -p w =n r (p i -p w ) (3) Where n r =σ aw / σ iw For soils with rigid pore structures and solid-solid direct contact (SS) between particles, such as sand and silt, capillary force plays a dominant role. r = 2.2; however, for soils with liquid contact (SLS) between particles (such as clay), the dominant force is adsorption rather than capillary force, so n r =1(Koopmans&Miller,1966). According to the contact mode of different types of soil, the n r However, arbitrarily setting n r It is not practical to apply a binary value to all soils. Most soils in nature are neither completely SLS nor completely SS. Natural soils are composed of sand, silt, and clay. For sand and silt, the contact between them is SS. For clay, due to the existence of a thicker adsorbed water film controlled by its higher surface energy, the contact with any other particles is SLS. With the increase of clay content, the contact mode between soil particles will also change, from SS contact to SSLS contact, and finally to SLS contact. This means that for SSLS soils with different particle compositions, it is necessary to correctly measure the ratio of SS contact and SLS contact in the contact mode of the particles in order to accurately determine n. r value. In order to reasonably correct the value of the specific interfacial energy of the soil, this paper established a method to measure the contact mode of soil particles by describing the contact of particles with different diameters. The coordination number is defined as the total number of ligand particles directly connected to it in the soil particles. The porosity determines to a certain extent the density of the particle spacing, that is, it determines the coordination number. According to the random contact experiment of spheres with different diameters, the relationship between the average coordination number and the porosity can be obtained (Trong & Mamoru, 2023): Where N t is the average coordination number, N tmax is the average maximum coordination number, b and k are material coefficients, and for the sphere stacking system, b = 1.23, k = 1.

4. Under the contact of two particles with different diameters, it is assumed that small spherical particles with an average radius of r are arranged around a large spherical particle with an average radius of R in the closest packing manner (r < R). It is assumed that the average maximum coordination number of soil particles can be calculated by the average value of the upper and lower limits of the maximum coordination number of the contact of particles with different diameters. The solid angle of the whole large sphere is 4π (ignoring the unit). The solid angle profile of the spherical cap surface area corresponding to each small particle in close contact around the center of the large particle is φ. The ratio of the two is the upper limit of the maximum coordination number. Because there will be gaps between small balls, a curved triangular area will be left between every three small balls. The solid angle blocked by all small balls cannot cover the entire spherical surface. Therefore, if these small curved triangular areas are not considered, the obtained contact number is the upper limit value. According to the geometric relationship, the solid angle θ corresponding to the spherical cap is: Where h = R-RcosΦ is the height of the spherical cap; Φ is the cross-sectional angle corresponding to each small particle in close contact with the center of the large particle. The final number of contacts is the upper limit value N up for: Then consider those curved triangle areas, calculate their areas on the sphere, and get the corresponding solid angle to calculate the lower limit of the contact number. Establish a rectangular coordinate system, assume that the three small balls are balls A, B, and C, and the coordinate origin is located at the center of the large ball, denoted as O. The three small balls are located in the xOy plane, ball A is located in the positive direction of the x-axis, balls B and C are located in the negative direction of the x-axis, and are symmetrical about the xOz plane. The center O′ of the equilateral triangle formed by the centers of the three small balls is located on the negative z-axis, with coordinates (0,0,-h). The coordinates of the centers of the three small balls are in, h is the height of O-ABC, according to the geometric relationship The area of ​​the triangle with equal curved sides cut by the triangular pyramid O-ABC is A1, and the surface area of ​​the spherical crown cut by the cone OA is A0. A1 is based on the spherical triangle area formula, A1 = (3β-π)R 2 ; β is the dihedral angle formed by any two side faces of the regular triangular pyramid O-ABC, and the three spherical angles are equal in size. Taking the angle between the plane OAB and the plane OBC as an example, calculate the cosine value cosβ as: The area S of the curved triangle is: The corresponding solid angle is: Get the lower limit of the number of contacts N down for: Average the upper and lower limits to get N tmax for: In the contact mode of particles, with the increase of the fine particle content, the SS contact changes to the SSLS contact and even the SLS contact, and the contact mode changes with the type of adjacent particles. The coordination number in the initial unit changes accordingly. The initial contact unit is defined as particles with the same diameter, and the initial coordination number in the unit is: Among them, 12 is the coordination number of equal-diameter particles under the loosest simple cubic packing, 6 is the coordination number of equal-diameter particles under the tightest tetrahedral packing; D is the SC and TH stacking ratio divided according to the density of particle packing, which is used to calculate the relative density of the initial unit coordination number. With the addition of particles with different diameters, the coordination number changes, affecting the contact mode. According to the ratio of the number of coarse and fine particles and the change rate of the coordination number relative to the initial unit coordination number, the growth rate of the average coordination number relative to the initial unit coordination number of the same particles is obtained, which represents the effect of the contact mode changing with the change of particle content on n. r The influence strength of the value is obtained, and the contact type influence coefficient is: Where N a is the number of coarse particles, which is determined by the mass fraction of coarse particles ω a , soil sample mass m s The average radius r of the coarse particles a calculate, N b is the number of fine particles, which is determined by the mass fraction of fine particles ω b , soil sample mass m s The average radius r of fine particles b calculate, When there are no fine particles in the soil, the contact mode is only SS contact. As the fine particle content increases, the contact mode changes to SSLS contact, and the SS contact will inevitably disappear completely, becoming SLS contact. Therefore, the interface energy value is compared and corrected: According to the Gibbs-Duheim equation, the chemical potential is: dμ=-SdT+Vdp (15) The chemical potential of liquid water at any pressure p and temperature R can be calculated as follows: in is the chemical potential of pure water (J·mol -1 ), depends on the temperature T w (K) and water pressure p w (kPa); R s is the universal gas constant (R s =8.31432 J·mol -1 ·K -1 );a w is the water activity, which describes the non-ideality of salt solutions and depends on the type and concentration of solutes. Ignoring salt concentration, a w =1. Likewise, the chemical potential of ice is: If the two phases are in chemical equilibrium, their chemical potentials are equal. For the atmospheric pressure p0 and the freezing point (273.15K, denoted as T0). Therefore, at p0 and T0, liquid water (μ w ,J.mol -1 ) and ice (μ i ,J.mol -1 ) are the same (in all derivations, the subscripts w and i represent liquid water and ice, respectively). At the reference states p0 and T0, the chemical potentials of water and ice are equal: m w (p w ,T w )=μ i (p i ,T i ) (18) Combining equations (16), (17), and (18), considering thermal equilibrium, the temperature of ice and water is the same as the ambient temperature T. The control equation for pressure change and temperature is obtained as: The freezing process of water in soil is similar to the drying process. During the freezing process, capillary water turns into ice first, while adsorbed water freezes last. Therefore, based on the similarity between SWCC and SFCC, the freezing mechanisms of adsorbed water and capillary water are analyzed by analogy. The ice pressure in soil water is assumed to be air pressure based on the analogy of freezing and drying processes. For capillary water, Ψ m =p a -p w Substituting formula (3) into: The change in volume of ice-water phase transition is ignored, V i / V w = 1. Particle contact does not affect the pressure exerted by adsorbed water on the soil, so the water content of adsorbed water at the corresponding points of SWCC and SFCC is the same, and the corresponding potential energy difference is the same. So there is p a -p w =p i -p w Therefore n r =1, and substituting into formula (20) we get: Thus, the freezing control equations of adsorbed water and capillary water considering the changes of pore water pressure and ice-water interface are obtained. Equations (20) and (21) can describe the freezing points at different pore water pressure positions in the soil-water system during the freezing process. In other words, under given temperature and solute concentration, the corresponding pore water pressure can be solved, so as to combine it with SWCC. The SWCC model is selected from (Lu, 2016), initialized according to the literature values ​​and the least squares algorithm is used to obtain the fitting results of the parameters, which are in the following form: i z =θ c +θ a (24) Where θ z is the total water content, θ a is the adsorbed water content, θ c is the capillary water content, θ amax is the amount of adsorbed water, θ s is the saturated water content, ψ is the matrix potential, ψ max is the maximum matrix potential, ψ cav is the cavitation matrix potential, m is the adsorption strength parameter, n is the pore size distribution parameter, and a is the intake matrix potential parameter.

3. The thermodynamic soil freezing characteristic curve model considering the contact of particles of different diameters according to claim 2 is characterized by: The derived particle contact relationship can describe the contact mode of the real soil, obtain the corresponding transformation relationship between the soil-water characteristic curve and the freezing characteristic curve, solve the moisture content related to temperature, and obtain the model of the freezing characteristic curve.

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