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2 Soil Sorptive Potential: Its Determination and3 Predicting Soil Water Density4 Chao Zhang, A.M.ASCE1; and Ning Lu, F.ASCE2

5 Abstract: The soil sorptive potential (SSP) has been recently theorized as the physical source for matric potential and local pore-water6 pressure. It consists of four distinct physicochemical potentials with electromagnetic nature: van der Waals, electrical, cation and surface7 hydration, and osmosis. A general framework is developed to link the SSP, soil water density (SWD), specific surface area, and soil water8 retention curve (SWRC), providing an experimental way to determine the SSP and SWD functions. The experimentally determined SSP and9 SWD functions for different clays accord well with the theoretical SSP and measured SWD data, validating the framework. The parameters

10 governing the SSP, i.e., Hamaker constant and structural parameters, are identified through inverse modeling of the water sorption isotherms11 (SWRC) of different clays, falling within the ranges reported in the literature and thus further confirming the validity of the SSP concept12 and its determination framework. The variability analysis of different clays, i.e., Georgia kaolinite, Wyoming montmorillonite, Denver clay-13 stone, and Denver bentonite, indicates that the SSP can vary to 6 orders of magnitude, resulting in the same orders of magnitude change in14 compressive pore-water pressure and abnormal 1.3 g=cm3 in soil water density. DOI: 10.1061/(ASCE)GT.1943-5606.0002188. © 201915 American Society of Civil Engineers.

16 Author keywords: Soil sorptive potential; Matric potential; Pore-water pressure; Soil water density; Unsaturated soils; Soil water retention;17 Soil physics.

18 Introduction

19 The widely1 used2 concept of water potential or soil suction does20 not possess a spatially varying nature at the intermolecular scale,21 leading to some inabilities in describing some important soil phe-22 nomena and properties. For example, even though the pore-water23 pressure has been inferred as spatially varying in soil at the inter-24 molecular scale (e.g., Bolt and Miller 1958; Mitchell 1960; Iwata25 et al. 1995), it is commonly defined as a constant at a much larger26 representative elementary volume (REV) scale for matric potential27 [3 Fig. 1(a)]. Pore-water pressure, together with temperature, is a28 basic state variable defining the thermodynamic state of soil water.29 As illustrated in Fig. 1(a), defining the pore-water pressure at30 the matric potential REV level further leads to other inabilities,31 e.g., why and how soil water density varies with water content32 (e.g., Martin 1960; Zhang and Lu 2018a, b), and how supercooling33 and depressed cavitation phenomena of adsorbed water can be34 quantified. These inabilities highlight that matric potential is only35 suitable to describe soil behavior at scales greater than the matric36 potential REV and thus is not sufficiently fundamental to describe37 local physical properties of soil water.38 To overcome the aforementioned inabilities, the soil sorptive39 potential (SSP) was coined by Lu and Zhang (2019) as the free40 energy change in a unit volume of soil water generated by soil solid

41matter. In soil, the SSP includes the electromagnetic fields pro-42duced by van der Waals, electrical double layer, surface hydroxyl,43and exchangeable cation, providing a physical mechanism for re-44taining water beside capillarity. The strength of SSP generally de-45cays rapidly with the distance to the source within a few to tens46of water molecular layers but can provide a dominating water-47retention mechanism in clayey soil with large specific surface area48(SSA). This decay nature of the SSP concept allows it to well cap-49ture spatially varying soil properties of pore-water pressure, soil50water density, and phase transitions (Lu and Zhang 2019; Zhang51and Lu 2019).52The objectives of this article are threefold: (1) to develop a53general framework to determine the SSP curve from the experimen-54tally measured soil water retention curve (SWRC); (2) to provide a55scaling law linking the pore-scale SSP to soil properties of soil56water density (SWD) and specific surface area; and (3) to determine57the SWD curve concurrently with the SSP. These objectives are58accomplished as follows. First, two inherent physical characteris-59tics of the SSP are reasoned, yielding the physical basis for deter-60mining the SSP from the SWRC. Thereafter, the relations between61soil sorptive potential, matric potential, and soil water density62are identified, providing the quantitative basis for determining the63SSP and SWD curves from SWRC. An iterative algorithm is pro-64posed to delineate the SWD and SSP curves. Subsequently, the65proposed method is validated using experimentally measured sorp-66tion isotherms and SWD curves, followed by results and implica-67tions of the determined SSP and SWD curves for a variety of clayey68soils.

69Physical Characteristics of Soil Sorptive Potential

70The SSP exhibits two profound physical characteristics: the inher-71ent properties of the soil matrix and the nonlinear decay function of72local distance to particle or cation surface, providing the physical73basis for the proposed framework.

1Professor, Ministry of Education Key Laboratory of BuildingSafety and Energy Efficiency, College of Civil Engineering, Hunan Univ.,Changsha 410082, China. Email: chao_zhang@hnu.edu.cn

2Professor, Dept. of Civil and Environmental Engineering, ColoradoSchool of Mines, Golden, CO 80401 (corresponding author). ORCID:https://orcid.org/0000-0003-1753-129X. Email: ninglu@mines.edu

Note. This manuscript was submitted on October 5, 2018; approved onAugust 9, 2019No Epub Date. Discussion period open until 0, 0; se-parate discussions must be submitted for individual papers. This paperis part of the Journal of Geotechnical and Geoenvironmental Engineer-ing, © ASCE, ISSN 1090-0241.

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74 Inherent Properties of Soil Matrix

75 Physically, SSP originates from the electromagnetic fields associ-76 ated with van der Waals, negative particle charge, and exchangeable77 cation charge. The magnitudes of these electromagnetic fields78 are rooted in the physical properties of soil matrix, e.g., Hamaker79 constant, surface potential, and cation exchange capacity. The pore-80 water properties, e.g., ion concentration and dielectric constant,81 constrain SSP as boundary conditions (Lu and Zhang 2019). There-82 fore, for given pore-water properties, SSP can be treated as an in-83 herent property of the soil matrix, and its magnitude is dominated84 by the spatial distance from soil particle or exchangeable cation85 surface and the physical properties of soil minerals.

86 Nonlinear Decay with Local Distance

87 SSP has been rigorously derived from thermodynamics by Lu and88 Zhang (2019). The components of SSP are unfailingly nonlinear89 decay functions of the local distance to particle surface. Idealizing90 clayey soil as platy particles, the local distance can be further sim-91 plified as the minimum distance to adjacent particle surface4 (x),92 shown in Fig. 1(a). Therefore, SSP is a highly nonlinear function93 of the minimum distance to particle surface and independent of94 total water content and pore shape and structure. The rapidly decay95 nature of SSP suggests that SSP vanishes at a certain distance from96 the particle surface.

97 Framework for Determining Soil Sorptive Potential98 and Soil Water Density

99 With the aid of the aforementioned two physical characteristics100 of SSP, a physically based framework to concurrently determine101 the SSP and SWD curves from which SWRC is developed. Two102 physical links, namely between SSP and SWRC and between SSP103 and SWD, are established, leading to an interrelation among SSP,104 SWRC, SWD, and SSA. This interrelation is utilized in an iterative105 algorithm to use SWRC and SSA data to yield SSP and SWD.

106 Link between Soil Sorptive Potential and107 Soil Water Retention Curves

108 The physical link between the SSP and SWRC can be established109 via the equivalence between SSP and adsorptive part of matric110 potential at the water–air interface, as described subsequently.

111 Soil Water Retention Sequence112 Soil water retention (SWR) describes the capacity of the soil in113 attracting water molecules. It is well recognized that the physical114 mechanism underlying SWR varies with the water content range115 (McQueen and Miller 1974; Lu and Likos 2004; Lu 2016). The116 SWR sequence is unique due to the distinct free energy levels117 of each physical mechanism (Zhang and Lu 2018b). Generally,118 water molecules are first attracted to the soil matrix by intense ad-119 sorptive energies due to the relatively low hydration free energy120 (ΔF) (Zhang and Lu 2018b), e.g., cation and surface hydration121 energy (ΔF < −12.5 kJ=mol) and van der Waals energy (ΔF <122 −1.0 kJ=mol). When the adsorptive water film fully coats the par-123 ticle surface, adsorptive water films get connected as a meniscus at124 the contact area of particles, and thereafter capillarity starts to take125 over the dominant role in retaining water molecules.126 This SWR sequence of switching between adsorption and capil-127 larity has been mathematically incorporated in the generalized128 SWRC model proposed by Lu (2016). The total water content (wt)129 is expressed as follows (Lu 2016):

wtðψmÞ ¼ waðψmÞ þ wcðψmÞ ð1Þ130where wa = adsorptive water content; wc = capillary water content;131and ψm = matric potential. Both adsorptive and capillary water con-132tents can be further expressed as a function of matric potential (Lu1332016)

waðψmÞ ¼ wa;max

�1 −

�exp

�ψm − ψmin

ψm

��m�

ð2Þ

wcðψmÞ ¼1

2

�1 − erf

� ffiffiffi2

p ψm − ψcav

ψcav

��

× ½wsat − waðψmÞ�½1þ ðαψmÞn�1=n−1 ð3Þ134where wa;max = maximum adsorptive water content; ψmin = mini-135mum or most negative matric potential; m = adsorptive strength;136ψcav= mean cavitation matric potential; wsat = saturated gravimetric137water content; n is related to the capillary pore-size distribution;138and 1=α is related to the air-entry matric potential.

139Equivalence between Sorptive Potential and140Matric Potential at Water–Air Interface141In soil, the SSP can convert into either matric potential or water142pressure, depending on the prevailing ambient conditions. At the143water–air interface, the thermodynamic equilibrium links matric144potential (Ψm) to either sorptive potential (Ψsorp) or pressure poten-145tial (Ψspre) 5

ψm½wðxÞ� ¼ ψsorpðxÞ þ ψpreðxÞ ð4Þ146where wðxÞ = total water content encompassed by the water–air147interface with a local distance of x to the particle surface, shown148in Fig. 1(a). Eq. (4) indicates that wðxÞ and ΨpreðxÞ are two un-149knowns involved in linking the SSP to SWRC.

(a)

(b)

F1:1Fig. 1. Scaling from distance to the particle surface (local pressure orF1:2sorptive potential REV) to gravimetric water content (matric potentialF1:3REV): (a) conceptual illustration of matric potential REV and sorptiveF1:4potential REV; and (b) scaled water content for different soils.

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150 The extent to which the SSP is converted to pressure potential151 ΨpreðxÞ is dictated by the thermodynamic equilibrium at the water–152 air interface. As reasoned in the SWR sequence, with increasing153 water content, the dominating water potential switches from ad-154 sorption to capillarity at the wetting or hydrating front of the soil,155 i.e., water–air interface. In the tightly adsorptive regime, the water–156 air interface is governed by adsorption, and thereby the retained157 water behaves as adsorptive water films, exhibiting an almost plane158 water–air interface, suggesting zero curvature and thereby zero-159 pressure potential (i.e., water pressure equal to ambient air pres-160 sure), illustrated in Fig. 2(a). That is, the SSP is fully converted161 or equal to matric potential at the water–air interface.162 In addition, the SWRC is fully dominated by the adsorptive163 part [Eq. (2)] in the tightly adsorptive regime, Therefore, Eqs. (2)164 and (4) can be rearranged to calculate the SSP

ψsorpðxÞ ¼ ψm;ads½wðxÞ� ¼ψmin

1 − 1m ln

h1 − wðxÞ

wamax

i ð5Þ

165 6 where Ψm;ads = adsorptive part of SWR, shown in Eq. (2).166 The rapidly decay nature of SSP suggests that beyond the tightly167 adsorptive regime, the water–air interface is controlled by the pres-168 sure potential rather than the SSP. Therefore, it can be approximated169 that the applicability of Eq. (5) extends to the entire water content170 range, providing a seamless physical equation. Eq. (5) indicates that171 the SSP can be determined from SWRC with the information of172 wðxÞ. Given the inherence of the SSP in soil matrix, this SSP

173determined from water–air interface is transferable to the SSP at174any total water content or local distance.

175Scaling from Local Distance to Soil Water Content

176The formulation of wðxÞ is provided in this subsection. The adsorp-177tive water film can be interpreted as a water layer with a uniform178thickness coating particle surfaces. The uniform thickness implies a179unique local distance (x) to the particle surface, suggesting a unique180value of SSP over the entire water–air interface. Because of this181uniqueness of SSP, the local distance (x) can be scaled to the gravi-182metric water content (w) of adsorptive film water.183Accordingly, a scaling law similar to that of Tuller and Or184(2005) can be formulated

wðxÞ ¼ ðx − x0Þ × SSA × ρavew ðwÞ ð6Þ185where w = gravimetric water content 7; ρavew ðwÞ = average density of186soil water retained in soil at a water content of w, i.e., SWD curve;187and x0 ¼ 0.14 nm represents the dimension of half water molecule188(e.g., Mitchell and Soga 2005), marking the minimum physically189possible distance to the particle surface. As a first theory, both the190stern layer and intercrystalline hydration are not explicitly consid-191ered here. Substitution of Eq. (6) in Eq. (5) leads to

ψsorpðxÞ ¼ψmin

1 − 1m ln

h1 − ðx−x0Þ×SSA×ρavew ðwÞ

wamax

i ð7Þ

(a) (b)

(c)

F2:1 Fig. 2. Soil water retention regimes and corresponding spatial distribution of soil sorptive potential, water pressure, and matric potential: (a) tightlyF2:2 adsorptive regime; (b) capillary regime (shaded area indicates a local water pressure lower than the ambient air pressure); and (c) scaled matricF2:3 potential REV and corresponding position in soil water retention curve.

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192 Eq. (7) suggests that the SSA and SWD are required to deter-193 mine the SSP from SWRC. The SSA is a common variable attain-194 able from conventional geotechnical tests (e.g., Khorshidi et al.195 2017), whereas SWD is intrinsically related to the SSP and can196 be determined from SWRC, as discussed in the next section.

197 Link between Soil Sorptive Potential and198 Soil Water Density

199 The link between SSP and SWD is established by considering how200 SSP induces water pressure and thereby changes SWD.

201 Local Pore-Water Pressure202 Following the local thermodynamic equilibrium principle, the local203 water pressure can be expressed as a function of matric potential,204 SSP, and ambient air pressure (Lu and Zhang 2019; Zhang and Lu205 2019)

uwðx;wÞ ¼ ψmðwÞ − ψsorpðxÞ þ ua ð8Þ206 where uwðxÞ = local water pressure at a distance of x to the particle207 surface; and ua = ambient air pressure.208 Given the spatially varying nature of SSP, the local water pres-209 sure depends on not only the soil water content but also the distance210 to the particle surface, as indicated in Eq. (8). Fig. 2 conceptualizes211 the local pressure profiles in different SWR regimes. The SSP ex-212 hibits a minimum adjacent to the particle surface, i.e., most neg-213 ative. According to Eq. (8), this local difference between sorptive214 and matric potentials results in a local water pressure much higher215 than the ambient air pressure. In Fig. 2(a), the local water pressure216 starts from a maximum adjacent to the particle surface and gradu-217 ally decays to the ambient air pressure as it approaches the water–218 air interface. In the capillary regime, the capillary pressure defined219 at the water–air interface can potentially drag the local water pres-220 sure below the ambient air pressure, shown in Fig. 2(b).221 For the extreme case of SWR, the lowest matric potential oc-222 curs when water molecules are hydrated in the first hydration shell223 around the exchangeable cations or surface hydroxyls (Lu and224 Khorshidi 2015; Zhang et al. 2017; Zhang and Lu 2018b). This225 monolayer coverage on the exchangeable cations or surface hydrox-226 yls is already the water–air interface, suggesting a zero local pore–227 water pressure according to Eq. (8). That is, the pore-water pressure228 concept breaks at the monolayer coverage, but instead the intermo-229 lecular forces between surface hydroxyls or cations and water mol-230 ecules should be interpreted as the interfacial pressure between soil231 matrix and water molecule. Herein, this interfacial pressure is incor-232 porated in the proposed framework by assuming it is equal to the233 negative local sorptive potential.

234 Soil Water Density235 The local pore-water pressure will alter the intermolecular distance236 among water molecules and thereby change SWD. In addition to this237 pressure effect, the structural change induced by surface or cation238 hydration serves as another physical mechanism changing SWD,239 as identified by Zhang and Lu (2018b). As a first theory, this paper240 only focuses on the pressure-effect in the proposed framework.241 The water density’s dependency on local pore pressure can be242 quantified by Tait equation. The local density of water is expressed243 as a function of the local pore-water pressure (Hayward 1967)

ρlocw ½uwðx;wÞ�

¼ ρw0

�1 − uwðx;wÞ − u0

k−10 þmv½uwðx;wÞ − u0� þ nv½uwðx;wÞ − u0�2�−1

ð9Þ

244where uw = local water pressure (bar 8); k0 = isothermal compress-245ibility at T ¼ 298.15 K, i.e., 4.50 × 10−5 bar−1; u0 = atmosphere246pressure, i.e., 1.01 bar; ρw0 = free water density under the atmos-247phere pressure, i.e., 0.997 g=cm3; and mv ¼ 3.32 and nv ¼248−4.54 × 10−5 are fitting parameters calibrated by Cho et al. (2002).249Fig. 3 illustrates the Tait equation expressing the water density250as a function of pressure. In general, the water density increases251with increasing water pressure. With the maximum in the order252of gigapascals, the local water pressure can potentially increase the253water density up to 1.3 g=cm3.254The average SWD and can be calculated from the local SWD

ρavew ðwÞ ¼ 1

x

Zx

x0

ρlocw ðx;wÞdx ð10Þ

255where wi = water content of interest; and xi = corresponding local256distance. Eqs. (8)–(10) provide a tool to predict SWD curves from257the SSP and SWRC.

258Iterative Algorithm

259In the preceding subsections, the SWR sequence and rapidly de-260caying nature of the SSP leads to the fact that the SSP and adsorptive261part of SWR are equivalent at the water–air interface [Eq. (5)]. Then,262the scaling law [Eq. (6)] and inherence of the SSP in the soil matrix263facilitate the extraction of the SSP from SWRC with the information264of SWD and SSA [Eq. (7)]. In return, the extracted SSP can be used265to calculate the local pore-water pressure [Eq. (8)] and thereby SWD266curves [Eqs. (9) and (10)]. These processes are coupled due to the267undetermined SWD curves. Precisely, SSP, SWD, SWRC, and SSA268are interrelated via Eqs. (7)–(10). Knowing two of them, the other269two can be iteratively determined. Here, an iterative algorithm is270developed to decouple the interrelation and offer a tool to determine271the SSP and SWD with given SWRC and SSA.272The proposed iterative algorithm is illustrated in Fig. 4 and im-273plemented in MATLAB 9. Only SWRC and SSA of soil are required274as the input information. Initially, the minimum local distance275of 0.01þ 0.14 nm is considered. Then, a guess value of average276soil water density is set to 0.997 g=cm3, and thereby local soil sorp-277tive potential and water pressure profile can be calculated using278Eqs. (7) and (8). Subsequently, the local soil water density can279be calculated using Eq. (9), resulting in a new average soil water280density using Eq. (10). If the difference between the new and old281average soil water density values is below 0.001 g=cm3, this iter-282ative loop can be considered converged, and the process can move

F3:1Fig. 3. Change of water density with pressure by Tait equation.

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283 to the next local distance value of i × 0.01þ 0.14 nm. If not, the284 new average soil water density is used for the next loop until the285 iterative criteria are satisfied. Once the scaled water content reaches286 the saturated water content, the iteration is terminated, and the287 SWD and SSP curves are determined.

288 Experimental Data Set

289 Four representative clayey soils, i.e., Georgia kaolinite (kaolinite),290 Wyoming montmorillonite, Denver bentonite (montmorillonite),291 and Denver claystone (illite), are selected to illustrate the proposed292 framework. Table 1 summarizes the geotechnical properties of the293 selected clayey soils. The SSA of the selected soils varies from 31294 to 668 m2=g (Khorshidi et al. 2017), and the cation exchange295 capacity (CEC) varies from 0.09 to 1.02 meq=g (Akin 2014).296 Fig. 1(b) illustrates the scaling of local distance (x) to gravimet-297 ric water content (w) for the selected soils. In general, the scaling298 curve is slightly nonlinear due to SWD variation (e.g., Martin 1960;299 Zhang and Lu 2018a, b). In addition, the scaled water content dem-300 onstrates a strong dependence on the SSA of soil. For clayey soils301 with large SSA, a nanoscale distance to particle surface can retain a302 considerable amount of water. Specifically, a monomolecular water303 layer, x ¼ 0.28 nm, indicates a scaled water content of 0.07 and304 0.10 g=g for Wyoming montmorillonite and Denver bentonite, re-305 spectively. The ends of scaling curves mark the saturated water

306content. Therefore, all the water retained in Wyoming montmoril-307lonite and Denver bentonite remains within a distance of 1.5 nm to308the particle surface, illustrated in Fig. 1(b). That is, most water in309clayey soils with large SSA falls within the influence zone of SSP,310and thereby the SSP substantially alters the soil water properties.311The SWR experimental data (Likos and Lu 2003) of the selected312soils are illustrated in Fig. 5. The physical parameters in the gen-313eralized SWRCmodel are determined by fitting to the experimental314data and are summarized in Table 2. Fig. 5 shows the fitted gen-315eralized SWRCs of the selected soils. At the low-water-potential316or high-soil-suction range, the SWR is fully contributed by the ad-317sorptive SWR. For Wyoming montmorillonite, the dominating role318of adsorptive SWR can extend to a water content of 0.32 g=g, sug-319gesting the prevalent influence of SSP on soil water properties.320The generalized SWRCmodel can explicitly separate adsorptive321and capillary water contents, as stated in Eq. (1). Fig. 6 shows the322separated water contents for two of selected soils. The adsorptive323and capillary water contents behave as stepwise functions of total324water content, implying that the interplay between adsorption and325capillarity is not significant. Therefore, the transition regime from326adsorption to capillarity only persists for a negligible water content327range, substantiating the assumed applicability of Eq. (5) to the328entire water content range.

329Experimental Validation

330To examine the proposed framework, the determined SSP curves are331validated against the theoretically calculated ones using inversely332obtained parameters, and the predicted SWD is compared with some333existing experimental data.

334Determined Soil Sorptive Potential

335The dry-end SWRCs reported by Lu and Khorshidi (2015) were336selected to validate the proposed framework in determining soil337sorptive potential curves. In Fig. 7, the scatter plots show the soil338sorptive potential determined from the experimental data using the339proposed framework, and the solid lines illustrate the theoretically340calculated soil sorptive potential curves [formulations provided by341Lu and Zhang (2019)] using the inversely obtained parameters.342Within a distance of 2 nm to the particle surface, the osmotic and343electrical components contribute negligibly to the overall soil sorp-344tive potential (Lu and Zhang 2019). As such and given the lack of345experimental data of the Hamaker constant and surface potential for346Georgia kaolinite and Wyoming montmorillonite, the parameters347controlling osmotic and electrical components, e.g., surface poten-348tial and ion concentration, are fixed during the inverse calcula-349tion. Specifically, the surface potentials (Vedl0) are fixed at 42.7350and 21.2 mV, respectively, for Georgia kaolinite and Wyoming351montmorillonite (Novich 1984). The ion concentration is assumed352to be small and equal to 0.001 mol=m3 for both soil water. Georgia353kaolinite and Wyoming montmorillonite are Kþ- and Naþ-rich clay

Given SWRC and SSAi=1

xi =i 0.01nm+ x0

Calculate pw,j with wave,

SWRC, and SSA usingEqs. (7,8)

Calculate w,newave and

with pw,j using Eqs. (9,10)

wave =0.997 g/cm3

Yes

No

i=i+

1

wav

e =w

,new

ave (

wi)

| |wave

w,newave

0.001 g/cm3

wi wsatYes

Output SSP and SWD

No

F4:1 Fig. 4. Iterative algorithm to concurrently determine SSP and SWDF4:2 functions from SWRC and SSA.

Table 1. Geotechnical properties10 of the selected soils

T1:1 Soil Clay (%) LL (%) PL (%) PI (%)USCS

classification SSAa (m2=g) CECb (meq=g) Source

T1:2 Georgia kaolinite 35 44 26 18 CL 31 0.09 GeorgiaT1:3 Wyoming montmorillonite 100 485 353 132 CH 505 0.71 WyomingT1:4 Denver bentonite 90 118 45 73 CH 668 1.02 ColoradoT1:5 Denver claystone 55 44 23 21 CL 145 0.32 Colorado

aFrom Khorshidi et al. (2017).bFrom Akin (2014).

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(a) (b)

(c) (d)

F5:1 Fig. 5. Soil water retention curve for four clay soils: (a) Georgia kaolinite; (b) Wyoming montmorillonite; (c) Denver bentonite (montmorillonite);F5:2 and (d) Denver claystone (illite). (Experimental data from Likos and Lu 2003; SWRC model from Lu 2016.)

Table 2. Parameter values for Lu (2016) generalized SWRC models of the selected soils

T2:1 Soil Ψmin (−kPa) m Ψcav (−kPa) wsat (g=g) wa;max (g=g) n 1=α (−kPa)T2:2 Georgia kaolinite 1,200,000 0.032 4,460 0.500 0.038 1.296 30.3T2:3 Wyoming montmorillonite 1,200,000 0.013 1,463 0.881 0.316 1.272 200.0T2:4 Denver bentonite 1,200,000 0.087 19,664 0.701 0.171 1.495 76.9T2:5 Denver claystone 1,200,000 0.085 17,116 0.377 0.051 1.542 55.6

Source: Data from Lu (2016).

(a) (b)

F6:1 Fig. 6. Separating soil water into adsorptive water and capillary water using SWRC model from Lu (200611 ): (a) Georgia kaolinite; and (b) WyomingF6:2 montmorillonite.

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354 soils, respectively, and therefore are treated as homoionic in the355 theoretical calculation. Accordingly, the hydration component is356 calculated using the cation hydration component with the CEC357 and SSA in Table 1.358 In Fig. 7, the SSP curves show excellent matches between the359 determined from experimental data and theoretically calculated.360 The coefficient of determination (R2) is higher than 0.95 for both361 soils, suggesting the accuracy of the proposed framework. The co-362 efficient of determination of Wyoming montmorillonite is slightly363 lower than that of Georgia kaolinite. The possible reason is that the364 stepwise hydration pattern in Wyoming montmorillonite caused by365 the intercrystallite swelling is not explicitly considered in the cur-366 rent soil sorptive potential theory. Hamaker constants are inversely367 calculated as 3.0 × 10−20 J for Georgia kaolinite; and 2.2 × 10−20 J368 for Wyoming montmorillonite, showing good agreement with369 3.1 × 10−20 J for kaolinite and 2.2 × 10−20 J for montmorillonite370 reported by Novich (1984). In addition, the structural parameters371 are inversely identified as S0 ¼ 0.08 and λc ¼ 0.55 nm for Georgia372 kaolinite; S0 ¼ 0.02 and λc ¼ 0.11 nm for Wyoming montmoril-373 lonite, falling within the ranges reasoned by Lu and Zhang (2019).

374 Determined Soil Water Density

375 Fig. 8 compares the predicted soil water density and experimental376 data (Bahramian et al. 2017; Zhang and Lu 2018a) for Wyoming377 montmorillonite. In general, the predicted soil water density curves378 show qualitative agreement with existing experimental data. The

379predicted SWD is lower than in the experimental data, especially380at the low-water-content range. The possible reason is that the381structural changes in water molecules due to cation hydration382are not included in the proposed framework. At the low-water-383content range, the soil–water interaction is very likely dominated384by the hydration on the surface hydroxyls or exchangeable cations,385suggesting that the structural changes in water molecules may con-386tribute significantly to the density abnormality (Zhang and Lu3872018b). Considering the discrepancy among different experimental388methods (e.g., Anderson and Low 1958; Bahramian et al. 2017;389Mackenzie 1958; Mooney et al. 1952; Norrish 1954; De Wit and390Arens 1950; Zhang and Lu 2018a, b), the patterns of the predicted391soil water density curve follow the data reasonably well. In addi-392tion, the soil water density serves for upscaling from the statistical393distance to water content, as shown in Eq. (10). Hence, the poten-394tially underestimated soil water density may overestimate the stat-395istical distance but will not significantly influence the magnitude396of SSP.

397Variability of Sorptive Potential and Soil Water398Density

399Soil Sorptive Potential Curves

400SSP curves are determined from SWRC using the proposed frame-401work and illustrated in Fig. 9(a) for the selected soils. The SSP

(a) (b)

F7:1 Fig. 7. Comparisons of soil sorptive potential curves between extracted from experimental data and theoretically predicted using inversely obtainedF7:2 parameters: (a) Georgia kaolinite; and (b) Wyoming montmorillonite.

(a) (b)

F8:1 Fig. 8. Comparisons of experimental data and predicted water density for (a) Georgia kaolinite; and (b) Wyoming montmorillonite.

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402 curves are illustrated as a function of scaled water content, and the403 corresponding local distance can be obtained using the scaling law404 in Eq. (6). The SSP unfailingly rapidly decreases with increasing405 scaled water content or local distance to the particle surface. The406 SSP can be as low as −1.2 GPa and vary in 6 orders with the407 local distance to particle surface. The SSP for Wyoming montmo-408 rillonite and Denver bentonite remain significant at the water con-409 tent of w ¼ 0.15 g=g due to the relatively large SSA. In contrast,410 the SSP decreases to be negligible at the water content of w ¼411 0.05 g=g for Georgia kaolinite and Denver claystone.412 At the dry end, the local distance cannot be directly interpreted413 as real film thickness; rather, it is a statistical average. Here, the414 water molecules are not simultaneously accumulated at the entire415 particle surface, but instead, the hydrated water molecules are con-416 centrated on some local exchangeable cations or surface hydroxyls.417 Therefore, at the dry end, the local distance can be understood as418 the distance normalized by the entire surface area.419 In addition, the SSP curves are only extracted for the tightly ad-420 sorptive regime for simplicity. However, in physical sense, the SSP421 at the water–air interface should persist until the full saturation of422 soil. In the capillary regime, there are actually two types of water–air423 interfaces. In small pores, the adsorptive film forms curved inter-424 face, i.e., meniscus, where the SSP almost vanishes. In large pores,425 the adsorptive film still cannot form a continuous meniscus, i.e., not426 triggering capillary condensation, but instead, some thick adsorptive427 films persist and are dominated by the SSP. However, the water con-428 tent change associated with the thickening of these adsorptive films429 is negligible compared with that due to the capillary condensation.430 Therefore, it is reasonable to neglect the SSP beyond the tightly431 adsorptive regime in practice.432 Despite the insignificance in practice, the SSP at the water–air433 interface of thick films bears physical significance as the source to434 produce SWR capacity, i.e., capillarity. Precisely, the SSP at the435 water–air interface of thick films can be considered as the solid–436 vapor interfacial tension and is intrinsically related to contact437 angles, as depicted in the Derjaguin-Frumkin equation (Derjaguin438 et al. 1987).

439 Soil Water Density Curves

440 Fig. 9(b) illustrates the determined SWD curves for the selected441 soils obtained using the proposed framework. For all soils, the

442SWD starts from a maximum density of 1.27 g=cm3 at the lowest443water content. At a high water content, the SWD abnormality is444insignificant for kaolinite but is still as high as 1.03 g=cm3 for445Wyoming montmorillonite until full saturation of soil. For the se-446lected soils, the average SWD values are always higher than the447free water density in the entire water content range, suggesting that448capillarity plays an insignificant role in changing the SWD (Zhang449and Lu 2018b).450The predicted SWD curves suggest that the density abnormality451should be well considered in defining fundamental soil properties452of clayey soils. For example, the volumetric water content is fre-453quently adopted in geotechnical practice, and the SWD is involved454in defining the volumetric water content. However, the predicted455SWD curves imply that the SWD is coupled with soil water con-456tent, suggesting that the volumetric water content is not a unique457parameter reflecting the real amount of water in soil.

458Pore-Water Pressure Profiles

459It has been speculated that the local pore-water pressure may460change with the soil water content (e.g., Mitchell and Soga 2005).461The SSP theory theoretically and quantitatively justifies this specu-462lation. Fig. 10 illustrates the predicted local pore-water pressure463profiles for Wyoming montmorillonite with three different water464contents of 0.8wa;max, 1.8wa;max, and wsat (2.8wa;max). In Fig. 10,465at a given distance to the particle surface, the predicted pore-water466pressure can be significantly different depending on the soil water467content.468In addition, the pore-water pressure significantly varies with469local distance to the particle surface, shown in Fig. 10. At the water470content of w ¼ 0.8wa;max, the soil water content falls within the471tightly adsorptive regime (solid line in Fig. 10). The predicted472pore-water pressure starts from a significant positive value and rap-473idly decays to the ambient pressure (0.101 MPa) at the water–474air interface, showing consistency with the conceptual model in475Fig. 2(a).476The soil water content 1.8wa;max corresponds to the capillary re-477gime in the SWRC (dotted line in Fig. 10). With increasing distance478to the particle surface, the predicted pore-water pressure rapidly de-479creases to −0.96 MPa below the ambient pressure (0.101 MPa). A480sharp decrease in pore-water pressure occurs around x ¼ 0.75 nm,481which is induced by the rapid decrease in SSP at the wet end,

(a) (b)

F9:1 Fig. 9. Predicted values of (a) soil sorptive potential as a function of distance to the particle surface x but converted to water content by the scalingF9:2 equation w ¼ ðx − x0Þ × SSA × ρavew ðwÞ for different clay soils; and (b) average soil water density curves for different clay soils.

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482 as shown in Fig. 9(a). Beyond the distance of 0.75 nm to the par-483 ticle surface, the pore-water pressure remains −0.96 MPa until484 the water–air interface. This trend is consistent with the concep-485 tual model in Fig. 2(b), and the zone with a constant pressure486 (−0.96 MPa) below ambient pressure is produced by the capillary487 pressure. The capillary pressure is predicted as low as −0.96 MPa,488 falling within the range between binodal cavitation, i.e., upper489 bound (0.003 MPa) and spinodal cavitation, i.e., lower bound490 (−163.966 MPa) (e.g., Azouzi et al. 2012).491 At the saturated water content wsat, the predicted pore-water492 pressure reaches the maximum for all the spatial position in soil493 (dashed line in Fig. 10). As depicted in Eq. (7), the magnitude494 of local pore-water pressure is proportional to matric potential.495 Therefore, when matric potential reaches the maximum (0 MPa)496 at the full saturation of soil, the local pore-water pressure reaches497 the maximum, too.498 The results in Fig. 10 quantitatively illustrate that the local pore-499 water pressure will change with soil water content. This change is500 not monotonic; it can be either an increase or a decrease depending501 on the SWR regimes. Pressure is a fundamental state variable de-502 fining the thermodynamic states of soil water. Consequently, the503 other physical properties of soil water, e.g., soil water density,504 supercooling, and depressed cavitation, depend on not only spatial505 coordinate but also the soil water content reflecting the global506 thermodynamic condition.

507 Practical Implications of Sorptive Potential and508 Soil Water Density

509 The soil water density is a basic physical parameter involved in510 defining or calculating many fundamental variables in geotechnical511 engineering, such as matric potential, volumetric water content, and512 specific surface area (Zhang and Lu 2018b). However, in practice,513 soil water density is frequently treated as identical with the free514 water density (≈0.997 g=cm3). It has been demonstrated that this515 treatment can lead to the overestimation of matric potential by 68%,516 volumetric water content by 40%, and specific surface area by 42%517 (Zhang and Lu 2018b). However, there is still a lack of a physically518 based model to estimate the soil water density from macroscale ex-519 perimental data. The proposed framework facilitates the estimation520 of soil water density from the sorption isotherm of which the

521measurement is commercially available to engineers (e.g., Likos522et al. 2011). Thus, it can be regarded as a step forward toward523the implementation of accurate soil water density in geotechnical524engineering practice.525The proposed framework is able to predict the local water pres-526sure of a soil at any given water content or matric potential. The527local water pressure with temperature are the two thermodynamic528variables defining the phase transition behavior of water. As such,529the proposed framework paves the way to physically predict the530freezing of soil water, which is of critical importance in geotech-531nical engineering in cold regions. In addition, the local water pres-532sure should be the source for swelling pressure and effective stress533variation of unsaturated soils at the high suction range, which are534key elements in geotechnical engineering concerning expansive or535unsaturated soils.

536Summary and Conclusions

537The practical question of how to determine the soil sorptive potential538has been addressed. Specifically, a general framework has been pro-539posed to determine the soil sorptive potential and soil water density540curves from the measured soil water retention curves and specific541surface area. The physical bases underpinning the proposed frame-542work are the two physical characteristics of the SSP, i.e., the inherent543property of soil matrix and nonlinear decay with local distance.544The proposed framework consists of two physical links, namely545between SSP and SWRC and between SSP and SWD, and an iter-546ative algorithm. The SSP is linked to the SWRC with the aid of the547identified equivalence between SSP and adsorptive part of SWRC548at the water–air interface and a scaling law between local distance549and gravimetric water content. The SWD is linked to the SSP by550the fact that the SWD is partly caused by the SSP-induced local551pore-water pressure. As such, the SSP, SWRC, SWD, and SSP are552recognized as interrelated. By decoupling this interrelation, an iter-553ative algorithm is developed to determine SSP and SWD from554SWRC and SSA.555The SSP and SWD determined from the proposed framework556were validated against existing experimental data. The theoretical557equations of SSP showed excellent fit with the experimentally de-558duced SSP curves, and the inversely obtained physical properties of559soil showed good agreement with the values reported in the liter-560ature, both validating the proposed framework for determining SSP.561In addition, the predicted SWD curve showed comparable variation562patterns with existing experimental data, further validating the pro-563posed framework for determining SWD.564The proposed framework makes it possible to experimentally565quantify SSP. The measurement of SSP can be accomplished by566uniquely linking it to some well-established experimental tech-567niques of measuring dry-end SWRC or sorption isotherms. As such,568the determination of SSP is accessible to engineers and researchers,569paving the way to apply the SSP concept in engineering practice and570thus opening a new window to better quantify the fundamental571physical properties of soil water.

572References

573Akin, I. D. 2014. Clay surface properties by water vapor sorption methods.574Madison, WI: Univ. of Wisconsin-Madison.575Anderson, D. M., and P. F. Low. 1958. “The density of water adsorbed by576lithium-, sodium-, and potassium-bentonite.” Soil Sci. Soc. Am. J. 22 (2):57799. https://doi.org/10.2136/sssaj1958.03615995002200020002x.578Azouzi, M. E. M., C. Ramboz, J.-F. Lenain, and F. Caupin. 2012. “A579coherent picture of water at extreme negative pressure.” Nat. Phys.5809 (1): 38–41. https://doi.org/10.1038/nphys2475. 12

F10:1 Fig. 10. Predicted local pore-water pressure as a function of distance toF10:2 particle surface for Wyoming montmorillonite under different soilF10:3 water contents.

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581 Bahramian, Y., A. Bahramian, and A. Javadi. 2017. “Confined fluids in clay582 interlayers: A simple method for density and abnormal pore pressure583 interpretation.” Colloids Surf. A 521 (May): 260–271. https://doi.org/10584 .1016/j.colsurfa.2016.08.021.585 Bolt, G. H., and R. D. Miller. 1958. “Calculation of total and component586 potentials of water in soil.” Trans. Am. Geophys. Union 39 (5): 917–587 928. https://doi.org/10.1029/TR039i005p00917.588 Cho, C. H., J. Urquidi, S. Singh, S. C. Park, and G. W. Robinson. 2002.589 “Pressure effect on the density of water.” J. Phys. Chem. A 106 (33):590 7557–7561. https://doi.org/10.1021/jp0136260.591 DeWit, C. T., and P. L. Arens. 1950. “Moisture content and density of some592 clay minerals and some remarks on the hydration pattern of clay.”593 In Proc., 4th Int. Congress of Soil Science Transactions, 59–62.594 Groningen, Netherlands: Hoitesema Brothers.595 Derjaguin, B., N. V. Churaev, and V. M. Muller. 1987. Surface forces.596 New York: Plenum.597 Hayward, A. T. J. 1967. “Compressibility equations for liquids: A compar-598 ative study.” Br. J. Appl. Phys. 18 (7): 965–977. https://doi.org/10.1088599 /0508-3443/18/7/312.600 Iwata, S., T. Tabuchi, and B. P. Warkentin. 1995. Soil-water interactions:601 Mechanisms and applications. Abingdon, UK: Taylor & Francis.602 Khorshidi, M., N. Lu, I. D. Akin, and W. J. Likos. 2017. “Intrinsic relation-603 ship between specific surface area and soil water retention.” J. Geotech.604 Geoenviron. Eng. 143 (1): 04016078. https://doi.org/10.1061/(ASCE)605 GT.1943-5606.0001572.606 Likos, W., and N. Lu. 2003. “Automated humidity system for measuring607 total suction characteristics of clay.” Geotech. Test. J. 26 (2): 10456.608 https://doi.org/10.1520/GTJ11321J.609 Likos, W. J., N. Lu, andW.Wenszel. 2011. “Performance of a dynamic dew610 point method for moisture isotherms of clays.” Geotech. Test. J. 34 (4):611 373–382. https://doi.org/10.1520/GTJ102901.612 Lu, N. 2016. “Generalized soil water retention equation for adsorption and613 capillarity.” J. Geotech. Geoenviron. Eng. 142 (10): 04016051. https://614 doi.org/10.1061/(ASCE)GT.1943-5606.0001524.615 Lu, N., and M. Khorshidi. 2015. “Mechanisms for soil-water retention and616 hysteresis at high suction range.” J. Geotech. Geoenviron. Eng. 141 (8):617 04015032. https://doi.org/10.1061/(ASCE)GT.1943-5606.0001325.618 Lu, N., and W. J. Likos. 2004. Unsaturated soil mechanics. New York:619 Wiley.

620Lu, N., and C. Zhang. 2019. “Soil sorptive potential: Concept, theory, and621verification.” J. Geotech. Geoenviron. Eng. 145 (4): 04019006. https://622doi.org/10.1061/(ASCE)GT.1943-5606.0002025.623Mackenzie, R. C. 1958. “Density of water sorbed on montmorillonite.”624Nature 181 (4605): 334–334. https://doi.org/10.1038/181334a0.625Martin, R. T. 1960. “Adsorbed water on clay: A review.” Clays Clay Miner.6269 (1): 28–70. https://doi.org/10.1346/CCMN.1960.0090104.627McQueen, I. S., and R. F. Miller. 1974. “Approximating soil moisture628characteristics from limited data: Empirical evidence and tentative629model.” Water Resour. Res. 10 (3): 521–527. https://doi.org/10.1029630/WR010i003p00521.631Mitchell, J. K. 1960. “Components of pore water pressure and their engi-632neering significance.” Clays Clay Miner. 9 (1): 162–184. https://doi.org633/10.1346/CCMN.1960.0090109.634Mitchell, J. K., and K. Soga. 2005. Fundamentals of soil behavior.635New York: Wiley.636Mooney, R. W., A. G. Keenan, and L. A. Wood. 1952. “Adsorption of water637vapor by montmorillonite. II. Effect of exchangeable ions and lattice638swelling as measured by X-ray diffraction.” J. Am. Chem. Soc. 74 (6):6391371–1374. https://doi.org/10.1021/ja01126a002.640Norrish, K. 1954. “The swelling of montmorillonite.” Discuss. Faraday641Soc. 18: 120 https://doi.org/10.1039/df9541800120. 13642Novich, B. E. 1984. “Colloid stability of clays using photon correlation643spectroscopy.” Clays Clay Miner. 32 (5): 400–406. https://doi.org/10644.1346/CCMN.1984.0320508.645Tuller, M., and D. Or. 2005. “Water films and scaling of soil characteristic646curves at low water contents.” Water Resour. Res. 41 (9): 1–6. https://647doi.org/10.1029/2005WR004142.648Zhang, C., Y. Dong, and Z. Liu. 2017. “Lowest matric potential in quartz:649Metadynamics evidence.” Geophys. Res. Lett. 44 (4): 1706–1713.650https://doi.org/10.1002/2016GL071928.651Zhang, C., and N. Lu. 2018a. “Measuring soil-water density by helium652pycnometer.” J. Geotech. Geoenviron. Eng. 144 (9): 02818002. https://653doi.org/10.1061/(ASCE)GT.1943-5606.0001929.654Zhang, C., and N. Lu. 2018b. “What is the range of soil water density?655Critical reviews with a unified model.” Rev. Geophys. 56 (3): 532–656562. https://doi.org/10.1029/2018RG000597.657Zhang, C., and N. Lu. 2019. “Unitary definition of matric suction.” J. Geo-658tech. Geoenviron. Eng. 145 (2): 02818004. https://doi.org/10.1061659/(ASCE)GT.1943-5606.0002004.

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Queries1. [ASCE Open Access: Authors may choose to publish their papers through ASCE Open Access, making the paper freely available

to all readers via the ASCE Library website. ASCE Open Access papers will be published under the Creative Commons-Attribution Only (CC-BY) License. The fee for this service is $1750, and must be paid prior to publication. If you indicateYes, you will receive a follow-up message with payment instructions. If you indicate No, your paper will be published inthe typical subscribed-access section of the Journal.]

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5. Please clarify Ψ_spre or Ψ_pre preceding and in Eq. (4) and throughout.

6. In Eq. (5) and elsewhere, do you mean wa_,max [as per Eq. (2)] rather than wa_max? Please check and ensure consistency ofnotation throughout.

7. SSA was previously defined so it was removed from the explanation of variables following Eq. (6).

8. As per ASCE style, SI units must be used. Please convert all bar measurements to Pa throughout.

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