ISSN 01476874, Moscow University Soil Science Bulletin, 2014, Vol. 69, No. 2, pp. 74–77. © Allerton Press, Inc., 2014. Original Russian Text © I.I. Sudnitsyn, 2014, published in Vestnik Moskovskogo Universiteta. Pochvovedenie, 2014, No. 2, pp. 29–33.

SOIL PHYSICS

The Hydration Energy of Elementary Soil Particles of Different Sizes I. I. Sudnitsyn Department of Soil Science, Moscow State University, Moscow, 119991 Russia email: [email protected] Received May 12, 2013

Abstract—The mathematical analysis of the data on the sorption of vapor moisture by the granulometric fractions of a loamy soil has for the first time indicated that an inverse linear relationship occurs: (1) Between the soilmoisture content and the logarithm of the moisture total potential (pressure) (probably as a result of the hydration of exchangeable cations that form a diffuse layer near the surface of the solid phase); (2) Between the soilmoisture content and the logarithm of the mean diameter of elementary soil particles (prob ably, as a result of differences in their specific surface determined by various mineralogical compositions); and (3) Between the hydration energy of elementary soil particles and the logarithm of their diameter. Keywords: soilmoisture pressure, soilmoisture content, principal hydrophysical characteristic, capillary method, moisture availability to plants. DOI: 10.3103/S0147687414020070

the value EW, it is necessary to integrate the expression (δE/δW)WdW (i.e., PWdW) within the range from W = 0 to W:

INTRODUCTION To carry out soil constructions, soils and subsoils different in granulometric composition are used [10, 12]. The optimization of the water and heating condi tions can be achieved only if the predictive calcula tions are sufficiently accurate, which makes it neces sary to know the hydrophysical and thermophysical characteristics of different fractions of the granulom etric composition of soils and subsoils that are employed to develop soil constructions [2]. One of these characteristics is the hydration energy of ele mentary soil particles of different sizes, which is usu ally calculated via the heat of wetting [8]. At the same time, this does not help to asses the results of rapid soil moistening with liquid water, while under natural con ditions the soil is often gradually moistened by vapor water (it is also necessary to mention that the initial and ultimate soilmoisture contents can vary signifi cantly). This is the reason that the standard method based on the heat of wetting cannot be applied to find ing the soilhydration energy. The working out of another method is the subject of this article.

EW =

∫P

W dW.

(1)

The calculation of this integral entails considerable technical difficulties, however, due to the following issues: (1) Within the range of the moisture content from W = 0 to the maximum hygroscopic moisture content (MH), the P(W) relationship is substantially nonlin ear; therefore, the highly accurate calculation of this relationship necessitates a considerable number (at least eight) of experimental points, which is labor and time consuming; (2) If the integrand is not analytic and represented by a table, its calculation necessitates very complex (even with the use of highperformance computing) numerical integration methods. A sufficiently compact mathematical (analytical) expression that describes the P(W) relationship could reduce the complexity of these calculations and improve the accuracy of the results. In this way, the number of experimental points that are necessary to determine the P(W) relationship could be set at two and the integration would amount to quite a simple calculation of an elementary mathematical function. Many analytical expressions to describe the P(W) relationship in the range of moisture contents from W = 0 to MH [8–12, 20–22] have been suggested, but all of them show satisfactory results only for some part of this range. In 1948, K. Terzagi [24], one of the

MATERIALS AND METHODS The soil hydration energy (E) can be determined by analyzing the relationship of the moisture total poten tial, or pressure, (P) from the soilmoisture content (W) [17]. P, which is determined by some W (i.e., PW), indicates the value by which the kinetic energy of water molecules is reduced as a result of their interac tion with the soil’s solid phase (PW = δE/δW). To find 74

THE HYDRATION ENERGY OF ELEMENTARY SOIL PARTICLES

founders of soil physics, while studying the compres sion of watersaturated clays, discovered an elemen tary (inverselinear) relationship between the loga rithm of the mechanical pressure that soil is subject to (P) and the soilmoisture content (W): ln P W = A – BW, (2) where A and B are empirical parameters. This expression implies that there is an exponential relationship between the soilmoisture pressure (or potential) and the soilmoisture content. In 1966, the author of this article discovered a sim ilar relationship for the soils within the range of the hygroscopic moisture content (P –5 ± –200 atm) [18]. This was later confirmed by other researchers (e.g., [2]). In 2009, it was shown that this relationship holds good for many soils within a considerably wider P range: from –30 atm (which corresponds to the maxi mum hygroscopic soilmoisture content) to –2600 atm (which corresponds to a relative vapor pressure of 0.135, or a relative air humidity of 13.5%) [18, 19]. In 2012, the analysis of the results of the research by J.G. Falconer and S. Mattson [22] revealed that this relationship is true for soil suspensions within the range of high P values varying from –0.9 to 0 atm [17]. In all cases ln|PW| and W are quite closely corre lated (the correlation coefficient does not exceed – 0.98 with a significance level lower than 0.05). Such a close correlation between soil parameters can be observed very rarely and is evidence of a strict physical and chemical law. In 1966, the author of this article suggested an explanation for this relationship through the hydration of exchangeable and absorbed cations. Previously, other scientists had also noted a close con nection between moisture sorbed from the air and the hydration energy of cations that saturate exchangable absorbing soil structures (e.g., [8]); however, this was no more than empirical research. The author of this article managed to theoretically derive this relation ship on the basis of fundamental physical laws discov ered by G. Helmholtz, C. Coulomb, J. Maxwell, L. Boltzmann, and M. Gouy [4, 6, 7, 13–15, 23, 25]. An exponential relationship (2) can be represented as: A – BW

(3) PW = e , where e is the base of the natural logarithm. The inte gration of function (3) for a moisture content range from W = 0 to W allows the hydration energy value (EW) to be determined: A

E W = ( e /B ) ( 1/e A

BW 0

– 1/e

BW

)

(4)

BW

= ( e /B ) ( 1 – 1/e ), for a moisture content range from W1 to W2; this takes the form: A

E W2 – E W1 = ( e /B ) ( 1/e

BW 1

– 1/e

BW 2

).

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To determine what effect the size of soil granulomet ric fractions has on the hydration energy, it is necessary to have sufficiently accurate data on the P(W) relationships of these fractions. This data is provided in the fundamen tal monograph by A.A. Rode “Basic Concepts of Soil Moisture Studies” [8]. Table 7a (p. 79) contains infor mation on the moisture quantity adsorbed by different granulometric fractions at different relative vapor pressure rates. The data was checked: the moisture potential (J/g of water) and natural logarithms of their moduli were deduced from the equilibrium relative vapor pressure; adsorbed moisture was the basis to determine the soilmoisture content. Using this data, the hydration energy of different granulometric frac tions was derived from equation (4). RESULTS AND DISCUSSION The moisture content depends to a great extent on the particle size (Table 1). In this way, with the highest relative pressure (p/p0) (0.942, corresponding to the MH calculated with the use of the Mitscherlich method and to a moisture potential of 8.1 J/g of water), the moisture content of particles with a diam eter < 2 μm and > 20 μm achieved 20.6 and 2.05%, respectively. With the lowest p/p0 (0.034 at a moisture potential rate of –459 J/g of water), it was equal to 2.82 and 0.25%, respectively. There is an exponential relationship between the moisture values and moisture potential moduli (Equa tion 3). Taking the log of this relationship gave an inverse linear relationship between the logarithms of moisture potential moduli and moisture content (Equation 2); the correlation between both was assessed. It turned out to be very close: –0.99 with a significance level of less than 0.05 [3]. As for the parameters of equation ln|PW| = A – BW(2), the A parameter cannot be characterized as being dependent on the particle size. It does not exceed the range from 6.62 to 6.75. Moreover, its value is associ ated with the energy of a dry soil adsorbing the first water molecules and varies from –760 to –850 J/g of water (the average value comprises 776 J/g of water, which was equivalent to –7760 atm or 185 cal/g of water). To verify the results, the A parameter was deter mined with the use of another (independent) method to imply the data that were provided in the monograph by A.A. Rode [8]. Table 4 shows that the radius of dry ions of calcium is equivalent to 0.106 nm, while the radius of hydrated ones comprises 0.96 nm. Their vol umes are 0.0054 and 3.7 nm3, respectively. In this way, the volume of water bound to one ion equals 3.7 nm3. According to the information given by S. Hendricks and M. Jefferson ([8], p. 142), the density of bound water is equivalent to 0.88 g/cm3. Therefore, the vol ume of one water molecule equals 0.0325 nm3. Hence, one ion of hydrated calcium binds 114 water mole Vol. 69

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SUDNITSYN

Table 1. The moisture content (as a percentage of the dry soil mass) of a soil and its granulometric fractions (µm) at differ ent rates of the relative vapor pressure (p/p0), soilmoisture total potential (P, J/g of water), and the natural logarithms of the moduli P(ln|P| and pF p/p0 0.942 0.868 0.748 0.582 0.383 0.177 0.069 0.034 Fraction content, %

–P

ln|P|

pF

<2

2–6

6–20

>20

Soil

8.1 19.3 39.3 73.3 130 234 363 459

2.08 2.96 3.67 4.29 4.92 5.45 5.90 6.12

4.91 5.27 5.60 5.87 6.11 6.37 6.56 6.66

20.60 16.82 13.60 10.57 8.00 5.45 3.48 2.82 12.19

13.52 11.32 9.33 7.40 5.50 3.85 2.48 1.78 12.65

9.80 8.50 6.98 5.59 4.16 2.83 1.80 1.25 13.04

2.05 1.63 1.34 1.02 0.82 0.54 0.35 0.25 62.12

7.61 6.24 5.03 3.96 2.91 1.97 1.21 0.88

Table 2. Parameters of the ln(|PW |) = ln(|P0 |) – BW relationship for soils and their granulometric fractions Object

ln(|P0 |)

B

r

Soil Fraction >20 µm Fraction 6–20 µm Fraction 2–6 µm Fraction <2 µm

6.62 6.68 6.75 6.74 6.68

59.2 225.0 45.6 33.8 22.3

–0.99 –0.99 –0.99 –0.99 –0.99

d 40.0 13.0 4.0 1.3

log(|d|)

E × 10–3

1.60 1.11 0.16 0.11

12.9 3.7 17.1 23.3 34.1

W is the soilmoisture content (g of water/g of soil); PW is the soilmoisture total potential (J/g of water) with W known; P0 = P with W = 0; r is the correlation coefficient of these relationships (with a significance level <0.05); E is the hydration energy (J/g of soil); d is the average diameter (µm) of elementary soil particles.

cules, which means that one gramion of hydrated cal cium binds 114 grammolecules of water, or 2050 g of water. Since the hydration energy of calcium ions equals 1570 kJ/gramion [5, p. 515], the moisture total potential is equal to 1570/2050 = 768 J/g of water and the total moisture pressure is equal to –7680 atm. This means that the two potentials of the first moisture por tions adsorbed by dry soil that were derived via inde pendent methods have a 8 J/g water difference, which is just 1% of their average value. This proves the high accuracy of the hygroscopic equilibrium method. The high correlation coefficient between the mois ture total potential and moisture content (Table 2) makes it possible to recommend a simplified hygro scopic method for the assessment of the principal soil hydrophysical characteristics that allows two values of relative air humidity to be used instead of eight. In contrast, the B parameter is considerably depen dent on the sizes of the elementary soil particles (Table 2): the greater the size is the more the value of this param eter grows. In this way, if the average diameter of an elementary soil particles becomes 31 times larger (from 3.1 to 40 μm) the B parameter increases by 10 times. At the same time, an inverse linear relation ship between 1/B values and the logarithms of the

average diameters (logd, μm) of elementary soil parti cles within the range of 40 > d > 1.3 μm was observed: 1/B = 0.048 – 0.026logd. (5) The correlation coefficient of this relationship equals –99 at a significance level less than 0.05 [3]. A nonlinear relationship between 1/B and d exists, which is probably due to the mineralogical composi tions of particles of different sizes. Actually, according to A.A. Rode [8] and A.D. Voronin [1], small granulo metric fractions mainly consist of clay minerals of the montmorillonite and illite types. The platelet shape of their crystals increases the specific surface. Moreover, hydrated exchangeable cations are located in the dif fuse layer not only on the surface of clay mineral crys tals, but in the interpacket spaces, as well. This allows the montmorillonite specific surface to equal 500 m2/g. Larger granulometric fractions are mainly composed of quartz and feldspar, which have no interpacket space, so only the exchangable cations located in the diffuse layer on the external surface of crystals are hydrated. This is the reason that their efficient specific surface does not exceed 100–200 m2/g. Transfer of the B value from equation (5) to equa tion (1) gives the following: W = (0.048 – 0.026logd)(A – lnPW).

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THE HYDRATION ENERGY OF ELEMENTARY SOIL PARTICLES

This relationship makes it possible to determine the moisture content of granulometric fractions within the hygroscopic range. The information on the A and B parameters (Table 2) provides an opportunity to calculate the hydration energy (E, J/g of soil) for the entire hygroscopic moisture range using equation (4). The fraction <20 μm is charac terized by the maximum hydration energy (34.1 J/g of soil); the fraction >20 μm is characterized by the min imum one (3.7 J/g of soil). An inverse linear relation ship was discovered between E and the logarithm of the particle diameter (d, μm): E = 22.5 × 103(1.42 – logd). The correlation coefficient of this relationship equals –0.93 with a significance level that is less than 0.05 [3]. It is necessary to know this relationship to carry out predictive calculations of water conditions of soil constructions [10, 12]. CONCLUSIONS Above all, within the range of the hygroscopic soil moisture content, this was the first discovery of an inverse linear relationship between the logarithms of the soilmoisture total potential (or pF values) and the moisture contents of different loamy soil granulomet ric fractions, which is determined by the hydration of exchangeable cations in the diffuse layer that is situ ated near the surface of the solid phase. This was also the first discovery of an inverse linear relationship between the moisture content of granulo metric fractions and the logarithms of the mean diam eter of soil particles, which is determined by different mineralogical compositions of particles of different sizes. Finally, for the first time an inverse linear relation ship between the hydration energy of an elementary soil particle and the logarithm of its diameter was observed. REFERENCES 1. Voronin, A.D., Strukturnofunktsional’naya gidrofizika pochv (StructuralFunctional Hydrophysics for Soils), Moscow, 1984. 2. Globus, A.M., Eksperimental’naya gidrofizika pochv (Experimental Hydrophysics for Soils), Leningrad, 1969. 3. Dmitriev, E.A., Matematicheskaya statistika v poch vovedenii (Mathematical Statistics in Soil Science), Moscow, 1995. 4. Zhukov, I.I., Kolloidnaya khimiya (Colloid Chemistry), Leningrad, 1949, vol. 1. 5. Karyakin, N.I., Bystrov, K.N., and Kireev, P.S., Kratkii spravochnik po fizike (Short Handbook on Physics), Moscow, 1962. 6. Kireev, V.A., Kurs fizicheskoi khimii (Course of Physical Chemistry), Moscow, 1955. MOSCOW UNIVERSITY SOIL SCIENCE BULLETIN

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7. Levich, V.N., Kurs teoreticheskoi fiziki (Course of The oretical Physics), Moscow, 1962, vol. 1. 8. Rode, A.A., Osnovy ucheniya o pochvennoi vlage (Foun dations of Soil Water Theory), Leningrad, 1965, vol. 1. 9. Smagin, A.V., Theory and methods of evaluating the physical status of soils, Eur. Soil Sci., 2003, vol. 36, no. 3, p. 301. 10. Smagin, A.V., Teoriya i praktika konstruirovaniya pochv (Soils Constructing: Theory and Practice), Moscow, 2012. 11. Smagin, A.V., Manucharov, A.S., Sadovnikova, N.B., et al., The effect of exchangeable cations on the ther modynamic state of water in clay minerals, Eur. Soil Sci., 2004, vol. 37, no. 5, p. 473. 12. Smagin, A.V., Shoba, S.A., and Makarov, O.A., Ekolog icheskaya otsenka pochvennykh resursov i tekhnologii ikh vosproizvodstva (Ecological Estimation of Soils Resource and Ways for Their Restoring), Moscow, 2008. 13. Sudnitsyn, I.I., Soil water content and water supply of plants in the Southern Crimea, Eur. Soil Sci., 2008, vol. 41, no. 1, p. 70. 14. Sudnitsyn, I.I., The role of exchangeable cations in the decrease soil moisture energy (pressure) (dedicated to the 110th Birthday of A.A. Rode), Eur. Soil. Sci., 2006, vol. 39, no. 5, p. 492. 15. Sudnitsyn, I.I., Dvizhenie pochvennoi vlagi i vodopotre blenie rastenii (Soil Water Motion and Plants Water Consumption), Moscow, 1979. 16. Sudnitsyn, I.I., Novye metody otsenki vodnofiz icheskikh svoistv pochv i vlagoobespechennosti lesa (New Ways to Estimate Soils WaterPhysical Properties and Forest Water Supply), Moscow, 1966. 17. Sudnitsyn, I.I., Smagin, A.V., and Shvarov, A.P., The theory of MaxwellBoltzmannHelmholtzGouy about the double electric layer in disperse systems and its application to soil science (on the 100th anniversary of the paper published by Gouy), Eur. Soil Sci., 2012, vol. 45, no. 4, p. 452. 18. Sudnitsyn, I.I., Shvarov, A.P., and Koreneva, E.A., The relationship between soils humidity and total pressure of soil water, Gruntoznavstvo (Pochvoved.), 2009, vol. 10, no. 12(14). 19. Sudnitsyn, I.I., Shvarov, A.P., and Koreneva, E.A., Integral energy of soils hydration, Estestv. Tekhn. Nauki, 2011, no. 1. 20. Teorii i metody fiziki pochv (Theory and Methods of Soils Physics), Shein, E.V. and Karpachevskii, L.O., Eds., Moscow, 2007. 21. Shein, E.V., Kurs fiziki pochv (Course of Soils Physics), Moscow, 2005. 22. Falconer, J.G. and Mattson, S., The laws of soil colloi dal behavior: XIII. Osmotic imbibition, Soil Sci., 1933, vol. 36, no. 4. 23. Gouy, M., Sur la constitution de la charge electrique a la surface d’un electrolyte, J. Phys. Ser. 4, 1910, vol. 9, pp. 457–468. 24. Terzaghi, K. and Peck, R., Soil Mechanics in Engineer ing Practice, New York, London, 1948. 25. Verwey, E.J.W. and Overbeek, J.Th.G., Theory of the Stability of Lyophobic Colloids, Amsterdam, 1948.

Translated by V. Levina Vol. 69

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