sw—soil and water: sorption of phosphorus by soil, part 1: principles, equations and models

Biosystems Engineering (2002) 82 (1), 1–24doi:10.1006/bioe.2002.0054, available online at http://www.idealibrary.com onSW}Soil and Water

15

REVIEW PAPER

Sorption of Phosphorus by Soil, Part 1: Principles, Equations and Models

M. B. McGechan; D. R. Lewis

Environment Division, SAC, West Mains Road, Edinburgh EH9 3JG, UK; e-mail of corresponding author: [email protected]

(Received 26 January 2001; accepted in revised form 1 February 2002)

The very extensive literature on phosphorus (P) sorption studies is reviewed with the intention of selectingequations and parameter values for use in a soil P dynamics model. Processes considered are fast reversiblesorption of P onto surface sites, and various slower processes including reactions which deposit P at depthbelow surfaces of iron or aluminium oxide minerals in soil or precipitate calcium phosphate. Sorption isconsidered to take place both onto static soil components and onto mobile sediments or colloids. Phosphorustransport in sediments moving in surface runoff flows has been extensively studied, but problems of colloidfacilitated through soil P pollution flows have only recently received attention. There is almost no publishedinformation about sorption onto such colloids. Equations are considered for processes, column experimentsand P dynamics systems studies. Isotherm equations representing fast reversible sorption have been identified.Instantaneous equilibrium can be assumed for most applications of the fast sorption equations, with theexception of surface erosion studies. Details of some very complex mechanistic models of the slow reactionand deposition processes are presented and discussed. Some simpler equations for these processes fromexisting field-scale P dynamics models are also presented. It is concluded that, at least in the short term, themechanistic approach is too complex for incorporation into a systems model of the whole range of Pprocesses, and that further development should represent time-dependent processes by adaptation of thesimple equations. # 2002 Silsoe Research Institute. Published by Elsevier Science Ltd. All rights reserved

1. Introduction

Plant nutrients from chemical fertilizer or animalmanure are positive assets if retained in the soil foruptake by plants, but become environmental pollutantsif leached into watercourses or groundwater (McGechan& Wu, 1998; McGechan & Lewis, 1998, 2000; Lewis &McGechan, 1998). There has been much attention in thepast devoted to nitrogen as a nutrient and pollutant, dueto its high solubility and leachability through fielddrains and to groundwater (McGechan et al., 1997; Wuet al., 1998), and its high potential for conversion toharmful volatile or gaseous emissions (McGechan &Wu, 1998). Attention is currently moving more towardsthe less soluble and non-volatile contaminating nutrientphosphorus (P). The study of P as a plant nutrient andsoil component is complicated by its long residence timein the ground due to sorption, and its tendency to movewith soil water in colloidal or particulate form. Unlike asolute such as nitrate, there can be restrictions onmovements of colloids in soil pores. This paper

37-5110/02/$35.00 1

addresses sorption of P by soil components, and somefeatures of P sorption which make the process complexcompared to sorption of other reactive chemicals.Weather-driven simulation modelling has become an

important component of studies of soil nutrients, bothfor crop growth and for their losses by leaching tobecome environmental pollutants. Such modellingstudies using the soil nitrogen dynamics model SOILN(Johnsson et al., 1987) and other models have beendescribed for nitrogen as a nutrient and pollutant by Wuand McGechan (1998a, 1998b) and Wu et al. (1998).The authors, Lewis and McGechan, have also reviewedthe field-scale models ‘GLEAMS’, ‘ANIMO’ and‘CENTURY’, which are being used to study P (as wellas nitrogen) cycling processes. Their review covers thewhole range of transformation and transport processesapplicable to P in the soil, including a short descriptionof sorption processes. However, due to the complexityof sorption processes for P and the extensive publishedliterature on it, this topic is the subject of a moredetailed review in the current paper, including ‘models’

# 2002 Silsoe Research Institute. Published by

Elsevier Science Ltd. All rights reserved

M. B. MCGECHAN; D. R. LEWIS2

Notation

A coefficient in Eqn (20) (Hansen et al.,1999)

Aa coefficient in Eqn (22) (Barrow, 1974b)Alox oxalate-extractable aluminium content,

mmol kg�1 or mg kg�1

[Alox+Feox] oxalate-extractable aluminium plus ironcontent, mmol kg�1

Ar coefficient in the Arrhenius equation,Eqn (22) (Barrow & Shaw, 1975a)

As specific surface area of soil particles inEqn (58) (Sharpley & Ahuja, 1983),cm2 g�1

a0–a3 fitted coefficients in Eqn (40)ab proportion of P converted to an ineffec-tive form [in Eqn (21), Barrow 1974a]

ad coefficient in diffusivity function in Eqn(58) (Sharpley & Ahuja, 1983)

ar coefficient in Eqn (60) (Ahuja et al.,1982)

as coefficient in Eqn (34) (Sharpley, 1982)aw slope constant in Eqn (57) (Sharpley &Ahuja, 1982)

B coefficient in Eqn (20) (Hansen et al.,1999)

B1, B2, B3 coefficients in Eqn (23) (Barrow & Shaw,1975a)

Bsat base saturation by ammonium acetatemethod in GLEAMS, %

b1 exponent in the Freundlich sorptionisotherm equation, Eqn (2)

b2 exponent in time-dependent term of theFreundlich sorption isotherm equation,Eqn (5), and of desorption equation,Eqn (52)

b3 third exponent in Eqn (52) (Barrow,1979)

bk coefficient in Eqn (7) (Kuo, 1988)br coefficient in Eqn (60) (Ahuja et al.,1982)

C concentration of P in solution, mg l�1

%CC average concentration of P in runoffwater [Eqn (60), Ahuja et al., 1982]

Ce concentration of P at the interfacebetween oxide and phosphate [Eqn(41)], mg l�1

Cd,0 Limiting concentration of P in solu-tion for no desorption (Barrow, 1979),mg l�1

Ci, Ci�1 concentrations of P in solution at times tiand ti�1 [Eqn (37), Barrow, 1983a]

C0 concentration of P in solution at timezero, mg l�1

C0,0 coefficient representing limiting concen-tration of P in solution for zero deso-rption time (Barrow, 1979)

Cd,0 composite coefficient in desorption equa-tions, Eqns (50) and (52) (Barrow, 1979)

Cro instantaneous concentration of desorbedP in runoff water, Eqn (59) (Sharpleyet al., 1981a)

Ct concentration of P in solution at time tin Eqn (23) (Barrow & Shaw, 1975a),mg l�1

CCaCO3 calcium carbonate concentration inGLEAMS

CL clay content of soil, %cr coefficient in Eqn (60) (Ahuja et al.,1982)

D diffusion coefficientDSSP degree of saturation with P, %

E energy of activation of chemical compo-nent [in Eqns (7) and (14), Kuo, 1988 andEqn (28), Barrow 1974a]

Er kinetic energy of rainfall per unit areaper unit time in Eqn (60) (Ahuja et al.,1982)

eT rate constant adjustment factor for soiltemperature in EPIC and GLEAMS

es power coefficient in Eqn (60) (Ahujaet al., 1982)

ey rate constant adjustment factor for soilwater content in EPIC and GLEAMS

F the FaradayFeox oxalate-extractable iron content,


f fraction of sorption sites that take partin the fast sorption process (Chen et al.,1996)

fb parameter in Eqns (37) and (38) (Bar-row, 1983a)

fl fraction of fertilizer P labile after 6months incubation period in EPIC

fs diffusion impedance factor (Nye &Staunton, 1994)

I integral with respect to time of differ-ences in P concentration in solution andat the interface between oxide andphosphate within a particle, Eqn (40)

Ir rainfall intensity, cm h�1

i index for summation in generalizedisotherm equation, Eqn (15) (Goldberg& Sposito, 1984)

is coefficient in Eqn (34) (Sharpley, 1982)j index for summation in Eqns (84) and (87)

K coefficient in Eqn (23) (Barrow & Shaw,1975a)

K2,j, K3, j coefficients in ANIMO, Eqn (84)(Schoumans, 1995)

Ks constant in Eqn (56) (Sharpley et al.,1981b)

SORPTION OF PHOSPHORUS 3

k1, k2, k3, k4 coefficients in multiple-component sorp-tion and desorption equations

kb the Langmuir coefficient in Eqn (18)(Barrow, 1983a)

kE1, kE2 coefficients in the Elovich sorption iso-therm equation, Eqn (16)

kF coefficient in the Freundlich sorptionisotherm equation, Eqn (2)

ki coefficient in generalized sorption iso-therm equation, Eqn (15)

kL coefficient in the Langmuir sorptionisotherm equation, Eqn (6), lmg�1 [P]

kL1, kL2 coefficients in two-component Langmuirsorption isotherm equation, Eqn (11),lmg�1 [P]

kT1, kT2 coefficients in the Temkin sorption iso-therm equation, Eqn (1)

kZ2, kZ3 coefficients in kinetic form of Langmuirsorption isotherm equation, as presentedby Van der Zee et al. (1989a), Eqns (9)and (10)

L length of slope in Eqn (60) (Ahuja et al.,1982)

M mass of soil in interaction zone [Eqns(59) and (60)], kg

m proportion of added P in solution attime zero in Eqn. (25) (Barrow & Shaw,1975a)

md Power coefficient in diffusivityfunction in Eqn (58) (Sharpley &Ahuja, 1983)

mr Power coefficient in Eqn (60) (Ahujaet al., 1982)

m1, m2 coefficients in electrostatic potentialequations (Barrow, 1983a, 1983b)

n number of components in generalizedisotherm equation, Eqn (15) (Goldberg& Sposito, 1984)

nb power coefficient in Eqns (21) and (29)(Barrow & Shaw, 1975b)

nr power coefficient in Eqn (60) (Ahujaet al., 1982)

OM organic matter, %Pact active P pool in EPIC and GLEAMS

Pacto previous value of Pact in Eqn (86)Pd quantity of P desorbed, g [P] kg

�1 [soil]Pfe fertilizer P added, g [P] kg

�1 [soil]Pilf labile P after fertilization and incubationin EPIC and GLEAMS

Pili initial labile P (prior to fertilization)Plab labile P pool in EPIC and GLEAMSPom minimum quantity of desorbable P in

Eqn (57) (Sharpley & Ahuja, 1982)Pox oxalate-extractable P content,


Psp proportion of added P which remainslabile after incubation in EPIC andGLEAMS

Pspcl clay content related proportion of addedP which remains labile after incubationin ICECREAM

Pspph soil pH related proportion of added Pwhich remains labile after incubation inICECREAM

Pstab stable deposited P poolP0 initial quantity of desorbable P in Eqn(56) (Sharpley et al., 1981a, 1981b)

pH pH in GLEAMSQ quantity of P sorbed on surface sorptionsites, including that sorbed by fast time-dependent processes, mg [P] kg�1 [soil]

Qe ‘equilibrium value’ of quantity of P sorbedon surface sorption sites, mg [P]kg�1 [soil][Eqn (43), Raats et al., 1982]

Qmax maximum P sorption capacity for sur-face sorption sites, including that sorbedby fast time-dependent processes, mg[P] kg�1 [soil]

Qmax,1,Qmax,2

maximum P sorption capacity for sur-face sorption sites in two-componentLangmuir sorption isotherm equation,Eqn (11), mg [P] kg�1 [soil]

Q0 surface-sorbed P in soil prior to the startof a soil P test, Eqn (3)

Qr runoff rate per unit area in Eqn (60)(Ahuja et al., 1982)

q coefficient in Eqns (31) and (32) (Bar-row, 1983b)

R universal gas constantRas flow rate for slow P adsorption in EPIC

and GLEAMS, kg ha�1 day�1

Rla flow rate for rapid P adsorption in EPICand GLEAMS, kg ha�1 day�1

S quantity of P deposited below sorptionsurfaces by slow reaction, or quantitysorbed by time-dependent processeswhich include slow deposition, g[P] kg�1 [soil]

S1, S2, S3 three components of S (for j=1, 2 and 3)given by Eqn (87) (Schoumans, 1995)

Sl angle of slope in Eqn (60) (Ahuja et al.,1982), %

Smax maximum quantity of P deposited belowsorption surfaces by slow reaction, ormaximum quantity sorbed by time-de-pendent processes which includes slowdeposition, g [P] kg�1 [soil]

Stot quantity of P sorbed on surface sorptionsites plus that deposited below sorptionsurfaces by slow reaction, g [P] kg�1

[soil]s coefficient in Eqn (34) (Sharpley, 1982)T absolute temperature, K

T1, T2 temperatures in Eqn (32) (Barrow 1983c)Tc soil temperature, 8Ct time, day


t1, t2 time periods in Eqn (32) (Barrow &Shaw, 1983b)

td desorption time, h (Barrow, 1979)tL lag time which elapses before the slowdeposition process becomes establishedin Eqn (20) (Hansen et al., 1999)

t0 start of time period, dayV constant relating quantity of desorbableP to effective concentration in Eqn (58)(Sharpley & Ahuja, 1983)

Vr total rain volume in the rainfall event perunit area in Eqn (60) (Ahuja et al., 1982)

v water velocity in convection/dispersionequation

W water:soil ratio in Eqn (56) (Sharpleyet al., 1981a, 1981b)

Wk interaction energy in Eqn (14) (Kuo,1988)

z depth in soil in convection/dispersionequation

zi valency, including sign (�2 for HPO42�)

zk number of nearest neighbour surround-ing a central phosphate species in Eqn(14) (Kuo, 1988)

a proportion of phosphate present asHPO4

2� (Barrow, 1983a)as power constant in Eqn (56) (Sharpley

et al., 1981a, 1981b)b rate constant in kinematic sorptionisotherm equations, and also in Eqn(45) (Raats et al., 1982)

bas rate constant for slow adsorption inEPIC, day�1

bb rate constant in Eqn (21) (Barrow 1974a)be rate variable [Eqns (43) and (45), Raats

et al., 1982]bs power constant in Eqn (56) (Sharpley

et al., 1981a, 1981b)g activity coefficient of ions in solution

ja electrostatic potential in Eqns (18), (19)and (40) (Barrow, 1983a)

jao initial value of electrostatic potential inEqns (19) and (40) (Barrow, 1983a)

y volumetric water content of soil, fractiony0�03 volumetric soil water content at a ten-

sion of 30 kPayd proportion of sites occupied (Barrow,1983a)

yfc volumetric soil water content atfield capacity in GLEAMS model (as-sumed to be at a tension of 33 kPa inNorth America) (Knisel, 1993)

yk replaces Q/Qmax in Eqn (7) (Kuo, 1988)yw volumetric soil water content at thewilting point (assumed to be at1500 kPa) in GLEAMS model (Knisel,1993)

rd dry bulk density of soil, gm�3

s standard deviation of electrostaticpotential (Barrow, 1983a)

of sorption processes applied to laboratory experiments.The availability and usefulness of data for estimatingparameters of equations which may be incorporated intofield-scale models is later assessed in a follow-on Part 2of this paper (McGechan, 2002).

2. General principles of phosphorus sorption

Sorption is the process by which reactive chemicalsbecome attached to surfaces, sometimes of otherwiserelatively harmless solids. Small particles have a largespecific surface area, so tend physically to have a highsorption capacity (but this is further modified by theirchemistry). In addition to those on immobile particles inthe soil matrix, there are competing sorption sites onotherwise non-polluting sediments and colloids whichmove relatively freely in water flows along the surface orthrough the soil. Small soil particles readily becomedetached to become mobile sediments or colloids, andland-spread manures and wastes contain additionalcolloidal material. However, attention in the extensiveliterature has been directed mainly to sorption onto

immobile soil components, with a few references tosorption onto sediments moving in surface runoffflows.In common with other reactive chemicals, the extent

to which P is adsorbed relative to that in solution ishighly non-linear, as energy levels vary between differentbinding sites on the solid surfaces, high-energy sitesbecoming occupied before low-energy sites. This non-linearity is commonly represented mathematically by anumber of alternative equations (‘isotherms’), withlogarithmic or other transformations to make linearapproximations.For P (which may differ from some other reactive

chemicals), the sorption process is complicated for onemain reason. Apparent sorption of P can be thought ofas being a combination of several processes, including afast (almost instantaneous) reversible true sorptionprocess on soil particle surfaces, plus various slowertime-dependent processes, some of which lead todeposition of P at a depth below the surface of particles.These slower processes can be further subdivided intorelatively faster and very slow components, and aredescribed by various authors as ‘slow adsorption’, ‘the


slow reaction’, ‘deposition’, ‘fixation’, ‘precipitation’ or‘solid-state diffusion’; opinions differ about the extent towhich such processes are reversible. A description of theslow deposition process, and the role of soil mineralssuch as iron (Fe), aluminium (Al) and calcium (Ca)compounds (including the somewhat different precipita-tion process in calcareous soils compared to that onmetal oxides in acid soils), is presented by Hemwall(1957). The distinction between fast and slow sorptionprocesses, and that between sorption on the surface andat depth below surfaces, are in both cases indistinct;Addiscott and Thomas (2000) have suggested that theseprocesses could better be regarded as belonging to acontinuum. The multiple sorption processes complicatewhat happens when desorption is induced by dilution ofthe soil solution, since the extent to which slowdeposition has progressed influences the quantity ofsorbed material available for fast desorption from thesurface (rapid) sorption sites. This complication isillustrated by the observations of Barrow (1979) andothers that curves representing the reverse process ofdesorption do not retrace the paths of the sorptioncurves. In particular, they observed that dilution of thedissolved P occurs after long incubation periods withhigh P concentrations (or alternatively after shorterincubation periods at raised temperatures), a muchlower quantity of P was desorbed than the quantityoriginally adsorbed. Also, where dilution occurred afteronly short incubation periods, desorption more nearlyfollowed the path of adsorption, but this was followedby a period of re-adsorption.Soil P can be considered as being contained in a

number of ‘pools’, including (amongst others) dissolvedinorganic P, inorganic P sorbed onto surface sites,inorganic P sorbed or deposited by various slow time-dependent processes and various organic P pools(including unbound precipitates). The quantity of P ineach pool at a given time depends on the history of Pfertilizer application, including the lapsed time since themost recent applications. The term ‘labile P’ iscommonly used to represent mobile P which is available(or rapidly becomes available by reactions with fastkinetics) as a nutrient for plant growth, includingsoluble P and that which is sorbed onto surfacesites, but not that which has been deposited by theslow reaction (which is not readily available). Also,the ‘buffering capacity’ is commonly used as anindication of the quantity of P sorbed on surfacesites which will rapidly desorb when dilution occurs,and again P deposited by the slow reaction isexcluded from this. The multiple sorption processesare further complicated if competing sorption sites in thestatic soil matrix and on mobile particles are bothconsidered.

The concept of saturation is mentioned by manyauthors (e.g. Beauchemin & Simard, 1999), togetherwith the degree of saturation with P (DSSP). However,the quantity of P deposited below the surface by theslow reaction in soil saturated with P is not clearlydefined. A ‘change point’ soil P content (not related tosaturation but generally much lower than the saturationP content) has been discussed by some researchers,including Heckrath et al. (1995). If the P content isabove the change point there is a tendency for P to beleached to field drains at a much higher level than if thecontent is below the change point.

3. Factors influencing extent of sorption

The physical and chemical mechanisms of P sorptionin soil are described in a number of textbooks, e.g.Wild(1988). The extent to which a soil adsorbs P (bufferingcapacity or sorption capacity) differs widely betweendifferent soils. It tends to be high in soils with a highproportion of small-size particles (and hence a highspecific surface area) such as clay. Manure or slurryadded to the soil have large contents of both P andcolloidal material on which P is sorbed (as discussed byDe Willigen et al., 1982), and such colloids provideadditional sorption sites when distributed by ploughing.The effect of manure applications on the sorptioncapacity of soil has been studied by Eghball et al.(1996). Chemical considerations mean that clay soilscontaining high proportions of Fe or Al oxide mineralshave particularly high buffering capacities, as discussedby Bowden et al. (1977). Another important environ-mental factor is pH (e.g. Barrow, 1984), which has amajor influence on ionic mechanisms of sorption (asdiscussed in Section 4.2.2). This results in a distinctsorption behaviour for alkaline calcareous soils whichdiffers from that of more acid soils.

4. Equations and models representing sorption processes

4.1. Isotherm equations

There are a number of standard isotherm equationswhich are commonly used to fit experimental data foradsorption of P and other reactive solutes, taking intoaccount the non-linearity of these relationships. Theseinclude the Temkin equation, the Freundlich equationthe Langmuir equation, the two-surface Langmuirequation and the Elovich equation. They are describedin various textbooks and papers, including Barrow(1978), Travis and Etnier (1981), Mead (1981), Chienand Clayton (1980) and Kinniburgh (1986). In their


simplest form, these equations are defined to assumeinstantaneous approach to equilibrium, but they canalso be modified to represent a time-dependent(‘kinematic’) approach to equilibrium.

4.1.1. Temkin equationThe Temkin equation is described by Barrow (1978)

and Mead (1981) as follows:

Q ¼ kT1 lnðkT2CÞ ð1Þ

where Q is the quantity of P sorbed in g [P] kg�1 [soil], Cis the concentration of P in solution in mg l�1 and kT1

and kT2 are coefficients.

4.1.2. Freundlich equationThe equilibrium Freundlich equation has the general

form

Q ¼ kFCb1 ð2Þ

where non-linearity is introduced by the exponent b1and kF is a coefficient [other symbols defined as for Eqn(1)]. The form of the Freundlich equation (Fig. 1) is suchthat good fits can be obtained to sorption data fornearly all soils. However, the equation has the dis-advantage that it does not define a maximum (satura-tion) value. Barrow (1978) discusses the need to considerP already present in the soil when fitting Eqn (2) toexperimental sorption data, suggesting an extra term inthe isotherm:

Q þ Qo ¼ kFCb1 ð3Þ

where Qo is the surface-sorbed P in the soil prior to thestart of the test andQ is the P sorbed during the test. Chenet al. (1999) discuss units and conversion transformationsfor kF , so values of kF can be compared between differentsorbents where the value of the exponent b1 varies.

Fig. 1. Langmuir and Freundlich isotherms for some English soil sisotherms, from Holford et al. (1974) and from Holford and Mat

Incorporating the kinetic component into the Freun-dlich equation requires a solution to the following first-order differential equation:

@Q

@t¼ bðkFCb1 � QÞ ð4Þ

where b in day�1 is the kinetic rate constant for thereaction. An alternative form of the Freundlich equationincorporating time dependence is presented by Barrow(1983a):

Stot ¼ kFCb1 tb2 ð5Þ

where Stot is the sum of instantaneous and time-dependent sorption and b2 is a second exponent fortime t.

4.1.3. Langmuir equationThe equilibrium Langmuir equation is usually written

in the form

Q ¼ QmaxkLC

1þ kLC

� �ð6Þ

where Qmax in g [P] kg�1 [soil] corresponds to the

maximum sorption capacity (saturation) and kL is acoefficient.An alternative form of the Langmuir equation has

been presented by Kuo (1988) as follows:

ykð1� ykÞ ¼ bkCðE=RT Þ ð7Þ

In this form compared to the standard form, yk hasreplaced Q=Qmax and bkE=RT has replaced kL, where Eis the energy derived from the chemical component, T isthe absolute temperature and R is the universal gasconstant.The kinetic version of the Langmuir equation can be

represented by the following first-order differential

eries (for parameter values see Table 1). , double Langmuirtingley (1995); , Freundlich isotherm, from Barrow (1978)


equation:

Q ¼ QmaxkLC � ð1=bÞð@Q=@tÞ

1þ kLC

� �ð8Þ

An alternative form of equation to represent the kineticsof approach to equilibrium for a Langmuir isotherm ispresented by Van der Zee et al. (1989a):

dQ

dt¼ kZ2CðQmax � QÞ � kZ3Q ð9Þ

where

kL ¼ kZ2=kZ3 ð10Þ

4.1.4. Two-surface Langmuir equationHolford et al. (1974) and Holford and Mattingly

(1975, 1976) present a ‘two-surface’ modification to theLangmuir equation, for the situation where the simpleLangmuir equation tends to give poor fits to sorptiondata. Equation (6) is extended to include two terms eachwith different coefficient values:

Q ¼ Qmax;1kL1C

1þ kL1C

� �þ Qmax;2

kL2C

1þ kL2C

� �ð11Þ

In this case, there are two sorption maxima Qmax,1 andQmax,2 so saturation can be defined as Qmax,1+Qmax,2.This represents sorption by a solid or soil whichcontains two distinct populations of sorption sites, oneof high bonding strength, the other with much lowerbonding strength (typically around one hundredth of thestrength, i.e. kL1 � 100� kL2, but with a populationthree times that of the high bonding strength sites, i.e.Qmax;2 � 3� Qmax;1). Holford et al. (1997) postulatethat no leaching should occur before a quantity of P hasbeen applied equivalent to the high-strength sorptioncapacity Qmax,1 (although this is not the case in practiceas leaching always occurs to some degree). The high-strength sorption capacity may correspond roughly tothe ‘change point’ as discussed by Heckrath et al. (1995)(Section 2), although this is more loosely defined basedon a simple P extraction procedure as used for advisorypurposes. The shape of the two-surface Langmuirisotherm can be similar to the Freundlich isotherm(Fig. 1), but (unlike with Freundlich) it still has definedsorption maxima.Holford and Mattingly (1976) define buffering capa-

city as being the change in the quantity of adsorbed Pper unit change in concentration of dissolved P, whichfollows from Eqn (11) as being

dQ

dC¼ Qmax;1

kL1C

ð1þ kL1CÞ2

� �þ Qmax;2

kL2C

ð1þ kL2CÞ2

� �ð12Þ

Also, the maximum value of the buffering capacityoccurs at very low concentrations, i.e.

dQ=dC� �

c!0¼ Qmax;1kL1 þ Qmax;2kL2 ð13Þ

Selim et al. (1976) describe a kinetic version of a two-siteequation, with first-order approach to equilibrium, butwithout any non-linearity.

4.1.5. Modified Langmuir equationKuo (1988) has described a ‘modified Langmuir

equation’ as follows:

yk=ð1� ykÞ ¼ bkC exp ðE � zkWkykÞ=RT� �

ð14Þ

where Wk is the interaction energy from which the netcontribution to the total energy of sorption can increaseor decrease depending on the number of the nearest-neighbour zk surrounding a central phosphate speciesand on the fraction of the sites occupied, and the othersymbols are as for Eqn (7).

4.1.6. Generalized isotherm equationGoldberg and Sposito (1984) present a generalized

form of equilibrium isotherm:

Q ¼Xn

i¼1

Qmax;ikiCbi

1þ kiCbi

� �ð15Þ

They demonstrate that this generalized form converts tothe standard equation forms, n=1 and b1=1 for theLangmuir equation, n=2 and b1=b2=1 for the two-surface Langmuir equation and n=1, 05b151 andki51 for the Freundlich equation.

4.1.7. Elovich equationThe Elovich equation describes P reaction kinetics,

although it can be thought of as taking the placeof a kinematic sorption isotherm. It is generallyexpressed as

Q ¼ ð1=kE2ÞlnðkE1kE2Þ þ ð1=kE2Þlnðt þ t0Þ ð16Þ

where kE1 and kE2 are constants and t0 is the start of thetime period in days. Chien and Clayton (1980) present asimplified form of the Elovich equation, by assumingkE1kE2t41:

Q ¼ ð1=kE2Þlnð1þ kE1kE2tÞ ð17Þ

4.2. The fast sorption process

4.2.1. Application of isotherm equations to representfast sorption processThe term ‘model’ is used widely in literature on P

sorption to describe fitting of experimental data toequilibrium isotherm equations, or the kinetics ofapproach to such an equilibrium. Barrow (1978) andother authors have discussed the application of thealternative standard isotherm equations to the adsorp-tion of P in soils.


There have been a number of attempts to fit Langmuirisotherms, as discussed by Van der Zee et al. (1989a). Insome cases, this concerns fitting data on sorption of P tospecific metal oxide minerals (e.g. McLaughlin et al.,1977; Bowden et al., 1980), where there is a clear valueof Qmax, so good fits can be obtained with thisequation form. However, other researchers such asVan der Zee and Gjaltema (1992) have used thisequation for sorption by soil. Barrow (1978)compared the standard isotherm equations for fittingto P sorption to soils data, concluding that Freundlichwas superior to Langmuir but even this was satisfactoryfor limited concentration ranges only. Shayan andDavey (1978) and Sibbesen (1981) both found that fitscould be improved by modifications to the Freundlichequation.

4.2.2. Ionic mechanismsParfitt et al. (1975) discuss the mechanism of

phosphate fixation by Fe oxides in terms of molecularstructures and the electrical charges on the ions. Bowdenet al. (1977, 1980) and Barrow et al. (1980, 1981a,1981b) have worked with an equation representing theelectrochemical energy levels of the process of binding tosurface sites for particular clay minerals. Sposito (1980)showed how the Freundlich isotherm could be generatedfrom a distribution of these binding parameters (eachrepresented by the Langmuir equation) for a series ofminerals. Based on this approach, Barrow (1983a)developed a mechanistic model consisting of thefollowing equations:

yd ¼kbagC expð�ziFja=RTÞ1þ kbagC expð�ziFja=RT Þ

ð18Þ

ja ¼ ja0 � m1yd ð19Þ

where yd is the proportion of sites occupied, ja is theelectrostatic potential in the plane of adsorption, andthe initial value of the potential ja0 is considered tobe a normal distribution (of 30 elements with widths/3) with mean ja0 and standard deviation s (leadingto yd being the weighted mean of the 30 elements).Also, a is the proportion of phosphate presentas HPO4

2�, g is the activity coefficient of those ionsin solution, zi is the valency, with sign (�2 for HPO4

2�),T is the absolute temperature, F is the Faraday, R isthe universal gas constant and kb is a coefficient(similar to kL in the Langmuir isotherm equation).Posner and Bowden (1980) discuss a model withthree Langmuir isotherm terms of form similar to0Eqn (18).

4.2.3. Kinetics of fast sorptionFast sorption has been described by Van der Zee and

Van Riemsdijk (1991) in Eqn (9) as a dynamic process

with a forward and reverse reaction term. Onceequilibrium has been reached dQ/dt=0, rearrangementof Eqn (9) gives the Langmuir isotherm. In practice,sorption of P onto surface sites happens so fast that formost purposes it can be considered to be instantaneous.For instance, the ANIMO weather-driven systemssimulation model (Groenendijk & Kroes, 1999) assumesa Langmuir isotherm based on Eqn (9) from Van derZee and Van Riemsdijk (1991) reaching instantaneousequilibrium.A few authors have presented equations for the

kinetics of reaching equilibrium for the fast sorptionprocess, as the faster component of time-dependentsorption (as distinct from the slower components whichrepresent slow deposition as discussed in Section 4.3).Staunton and Nye (1989a) postulate that any delay inreaching equilibrium for fast sorption arises not becauseof slow exchange at the solid/liquid interface butbecause of the time taken for dissolved P to move intocontact with sorption sites within soil aggregates aslimited by diffusion transport. Staunton and Nye(1989b) present three approaches to modelling thisdiffusion process, one of which is selected and testedagainst P sorption data for aggregated soils by Nye andStaunton (1994). Parameter values (including a‘diffusion impedance factor’ fs) are chosen for theselected model, which has diffusion processes repre-sented according to cylindrical rather than sphericalaggregate geometry, and reasonable fits are found todata for a 10 day contact time. Over a longer period(57 day) fits are improved if representation of the slowdeposition reaction is included in addition to fastsorption. Hansen et al. (1999) present an empiricalequation for the kinetics of phosphate sorption inmacropores of aggregated subsoils, which they foundto be a better fit to their experimental data than kineticversions of any of the standard isotherm equations:

lnC ¼A; 05t4tL

A � B lnt þ 1tL þ 1

� ; t > tL

8><>: ð20Þ

where tL is the ‘lag-time’ which elapses before the slowdeposition process becomes established (although forsome subsoils a simplified form of the equation with tLset to zero was found to be adequate), and A and B arecoefficients.

4.3. Slow reaction processes

Earlier studies assumed P sorption to be a simple butnon-linear process which could be represented by one ofthe standard equilibrium isotherm equations (as pre-sented in Section 4.1), with parameters estimated by a


range of standard techniques as discussed in Part 2(McGechan, 2002). The most common technique forestimating isotherm parameters involved measurementsafter a fixed period of contact between P and theabsorbant, usually 24 h. However, parameters wereobserved to vary with variation in the contact timeand temperature. This had the effect of giving areduction in dissolved P (and of P in a form readilyavailable as a plant nutrient) with increase in time sinceinitial contact (or since fertilizer application to soil). Linet al. (1983a, 1983b) represented this simply by adding atime-dependent term to the isotherm equations [as Eqn(4), (5) or (8)]. However, this approach was generallyfound not to adequately describe the process ofdesorption which occurs when the soil solution isdiluted.

4.3.1. Empirical equations for slow reaction from BarrowBarrow (1974a) described the slow reaction effect by

an empirical rate equation

dab

dt¼ bb 1� abð Þnb ð21Þ

where ab is the proportion of added P fertilizer Pfe

converted to an ineffective form and nb is an exponent.The effect of temperature T in K on the rate constant bb

was represented by a form of the Arrhenius equation:

bb ¼ Ar exp �E=RT� �

ð22Þ

where E is analogous to the energy of activation, R isthe gas constant and Ar is a coefficient. Barrow andShaw (1975a) then combined these equations with theFreundlich isotherm [Eqn (2)] to give a general empiricaltime- and temperature-dependent equation for theconcentration of P in solution Ct at time t:

lnðCtÞ ¼ K þ B1 lnðPaÞ � B2 lnðtÞ þ B3=T ð23Þ

where

K ¼ ð1=b1Þlnðm=k1Þ � ðb2=b1ÞlnðA=b2Þ ð24Þ

m ¼ C0=Pa ð25Þ

B1 ¼ 1=b1 ð26Þ

B2 ¼ b2=b1 ð27Þ

B3 ¼ b2E=ðRb1Þ ð28Þ

b2 ¼ 1=ðnb � 1Þ ð29Þ

and B1, B2 and B3 are constants arising from logarithmictransformations of equations with the power coefficientsb1 and b2; C0 is the concentration of P in solution at timezero and Eqn (29) is a simplification of the integral ofEqn (21) between the limits 0 and ab. Barrow and Shaw(1975a) also present a version of Eqn (23) for constant

temperature, with a different value for the constant term(equal to K+B3/T). The constant temperature version isin effect a logarithmic transformation and rearrange-ment of Eqn (5) from Barrow (1983a) where

K þ B3=T ¼ �ð1=b1ÞlnðkF Þ ð30Þ

As presented here, the subscripts for b1 and b2 have beenreversed compared to those used by Barrow and Shaw(1975a), in order to correspond to those used by Barrow(1983a). The time course following fertilizer spreading ofsurface-sorbed P and P which has undergone the slowreaction, as given by Eqn (5) or (23) with coefficients forone soil at 258C from Barrow and Shaw (1975a), isillustrated in Fig. 2. Barrow (1983b) later presents aslightly different form of Eqn (30) without the logarith-mic transformation and including another coefficient q:

Ct ¼ ðPa=kF Þ1=b1ft expð�qb2=b1TÞg�b2=b1 ð31Þ

He used this to compare the results of incubationexperiments at different temperatures, equating timeperiod t1 at temperature T1 to corresponding period t2 atanother temperature T2, since it follows from Eqn (31)that

t1

t2¼expð�qb1=b2T2Þexpð�qb1=b2T1Þ

ð32Þ

4.3.2. Empirical equations based on the work of SharpleySharpley (1982) measured the decline in ‘water

extractable P’ Psp during a period following fertilizerspreading. This decline represents in effect the quantityof P removed from the surface (water-extractable)sorbed pool by the slow reaction. He fitted the followingequations to his data:

Psp ¼ Pili þ sPfe ð33Þ

s ¼ is þ as log t ð34Þ

as ¼ �0�232þ 0�00301CL ð35Þ

is ¼ 0�722� 0�00679CL ð36Þ

where Pili is the water-extractable P before spreading,Pfe is the P added, CL is the soil clay content in % and s,is and as are coefficients. The time course of Psp

following fertilizer spreading as given by Eqns (33)and (34) is illustrated in Fig. 2. Sharpley et al. (1984)determined parameter values for Eqn (33) for a numberof soils at a fixed incubation time of 6 months, as acomponent of a field-scale model [Eqn (77), see Section4.7].

4.3.3. Extension of Barrow’s ionic mechanism equationBarrow (1979) found that the empirical Eqns (23)–

(29) could not adequately explain his and others’ obser-vations that desorption does not retrace the path of

Fig. 2. Time course variation in phosphorus pools indicated by Barrow (1983a) and by EPIC and ANIMO models, at twotemperatures: (a) 158C; EPIC, labile; EPIC, active; EPIC, slow; ANIMO, surf; ANIMO, S1 þ S2;ANIMO, S3. (b) 258C EPIC, labile; EPIC, active; EPIC, slow; Sharpley, labile; Barrow, surface;Barrow, deposited. Results are shown at both temperatures for EPIC as this is the only model to include temperature dependence;

S1;S2 and S3 are three components of S (for j ¼ 123) given by Eqn (87) in ANIMO


previously occurring sorption but tends to occur a lesserextent than adsorption. After initially developingempirical equations to represent desorption [Barrow(1979), as discussed in Section 4.5], Barrow (1983a) laterdeveloped a mechanistic model that assumed thatreversible fast sorption to sites on the surface of theadsorbant material is accompanied by a parallel slowreaction by which P is deposited at a depth below thesurface. The mechanism for this slow process is assumedto be solid-state diffusion, represented by modificationof a standard diffusion equation (which includes thediffusion coefficient D, based on Fick’s Law) presentedby Crank (1964):

S ¼ ð2=ffiffiffip

pÞ½C0

ffiffiffiffiffiffiffiffiffiffiffiffiffiðDfbtÞ

p

þXi

0

ðCi � Ci�1ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðDfbðt � tiÞ

pÞ ð37Þ

where the parameter fb is given by

fb ¼ 1=ð1� yd Þ � 1 for small values of yd ð38Þ

and where S is the quantity of P deposited by the slowreaction, C0 is the initial concentration of P in solution,Ci is the concentration of P in solution at time ti andCi�1 is the concentration of P at the previous timestep.

The concept of electrical charge described by Eqn (18) isretained (Section 4.2.2), including a distribution ofLangmuir isotherm equations together with an exten-sion of Eqn (19):

ja ¼ ja0 � m1yd � m2S=Smax ð39Þ

where Smax is the maximum quantity of P deposited bythe slow reaction, and m1 and m2 are coefficients. Oneimplication of Eqn (39) is that P transferred from surfacesites to a depth below the surface partially (but notentirely) frees up surface sites for further fast sorption.

4.3.4. Unreacted shrinking core modelVan Riemsdijk et al. (1984a) also describe a model of

the slow reaction of P with soil based on diffusionaccording to Fick’s Law. However, they assume areaction with the bulk of metal oxides with diffusiontowards the reaction zone being rate limiting, and anemphasis on the sizes and geometry of particles, ratherthan diffusion according to penetration and electricalcharge theories assumed by Barrow (1983a). This is laterdescribed by Van der Zee et al. (1989b) and Van der Zeeand Van Riemsdijk (1991) as being an example of an‘unreacted shrinking core (USC) model’, as mentionedin chemical engineering literature (Wen, 1968). Van


Riemsdijk et al. (1984a) developed the USC model byconsidering a sphere of metal oxide which is graduallyconverted (starting at the surface of the sphere) to metalphosphate. At each point in time, there is a distinctinterface between the inner sphere of unconverted metaloxide and the outer shell of metal phosphate. Solid-statediffusion limits the rate of transfer of new P through theever-increasing thickness of the metal phosphate shell.Complex equations relating diffusion to the square rootof solute concentration, similar to Eqn (37), arepresented for different geometric shapes (a cylinderand a platelet as well as a sphere). A set of equations forthe spherical case of this shrinking core model ispresented by Van der Zee et al. (1989a). However, VanRiemsdijk et al. (1984b) considered such idealisticequations to be of little practical use for real soils withvarious shapes and sizes of particles. For application ofthe model to a real soil with varying quantities ofdifferent metal oxides, and a range of sizes of particleswith different and largely irregular shapes, they showedthat the diffusion equations are equivalent to

S ¼ a0 þ a1 lnðIÞ þ a2ðlnðIÞÞ2 þ a3ðlnðIÞÞ

3 ð40Þ

I ¼Z t

0

ðC � CeÞ dt ð41Þ

where C is the variable P concentration in solution, Ce isthe concentration at the interface between oxide andphosphate (a fitted parameter), I is the integral withrespect to time of the concentration differences and a0–a3 are fitted coefficients of a third-order polynomial for aparticular soil.Van der Zee and Van Riemsdijk (1986) and Van der

Zee et al. (1989a) reduce Eqn (40) to a second-orderpolynomial, while Van der Zee and Van Riemsdijk(1988) further reduce it to a first-order polynomial anddrop the constant term. Freese et al. (1995) replace thepolynomial in Eqn (40) by an expression similar to theLangmuir isotherm equation to give the followingequation for total sorption:

Q þ S ¼ SmaxðkLIÞb1

1þ ðkLIÞb1ð42Þ

where I is given by Eqn (41).

4.3.5. Other equations for time-dependent sorptionand slow reaction processesThe slow reaction between P and soil is discussed in

other papers. Munns and Fox (1976) call it ‘slowreaction’ or ‘slow fixation’, describing the kinetics by asimple first-order reaction. Raats et al. (1982) use theterms ‘fixation’ or ‘chemisorption’ (regarding it astotally irreversible), representing the process by a first-order reaction equation with a rate constant modified by

the extent of fast sorption, as follows:

@S

@t¼ beðQ � QeÞ if Q > Qe ð43Þ

¼ 0 if Q5Qe ð44Þ

where (Q�Qe) is the excess ‘mobile solute’ (i.e. solubleplus surface-sorbed P) above the equilibrium value Qe sothat be=b when Q=0, and be is the rate variabledetermined from the rate constant b by

be ¼ 1� S=Smax� �

b ð45Þ

Enfield et al. (1976) tested a model consisting of a fastreversible sorption process (according to either aLangmuir or Freundlich isotherm) plus a solid-statediffusion process, showing better fits to data (represent-ing high P loadings associated with wastewater treat-ment by soil) than either a simple first-order kineticequation or a kinematic Freundlich isotherm equation.Enfield et al. (1981b) developed this model further usingvarious summation expressions to generate an integralof the form given in Eqn (41). Chien and Clayton (1980)found that their modification to the Elovich equation[Eqn (17)] gave better fits to data on sorption overvarious time periods than either an equilibrium isothermequation or a simple first-order kinetic equation.Aharoni et al. (1991) concluded that a diffusion-basedmodel is valid over a longer time period than a modelbased on first-order kinetics or the Elovich equation.Similarly, Polyzopoulos et al. (1986) had found that theElovich equation has limited applicability, particularlyduring initial fast sorption and also much later when thesorption rate drops to a very low value. In contrast,Agbenin and Tiessen (1995) concluded that the Elovichequation was most satisfactory out of a number ofmodels including one based on diffusion.The slow rate transfer of P between different ‘P

minerals’ pools accounts for reductions over time inboth soil solution P and P available for plant growth, asobserved by Barrow (1974a, 1974b) and Barrow andShaw (1975a, 1975b). This description can also explaintime-dependent sorption/desorption isotherms pre-sented by Sawhney (1977).

4.4. Degree of saturation with phosphorus

The concept of degree of saturation of soil with P hasrecently become widely used as an indication of thesusceptibility of a site to exporting polluting losses of Pto the environment. Beauchemin and Simard (1999)have reviewed papers describing ‘P sorption indices’determined by various test procedures, as a measure ofsoil P saturation degree. These include a single-point ‘Psorption index’ described by Bache and Williams (1971),estimated as Q/logC in Eqn (1) following the addition of


1�5 g [P] kg�1 [soil], which they considered to be a usefulreference index to characterize the sorption properties ofsoils. Use of the sorption index concept is heavilydependent on a satisfactory definition of what is meantby soil saturated with P, i.e. the maximum or limitingsorption capacity of the soil. A common definition ofthe limiting sorption capacity is Qmax in the Langmuirisotherm, Eqn (6), but this represents the maximum forfast, reversible sorption and opinions differ aboutwhether any or all of the pools into which P is adsorbedor deposited by a time-dependent process should also besaturated. Van der Zee and Van Riemsdijk (1988)related the sorption coefficient a1 in Eqn (40) (with first-order term only) to the oxalate-extractable Fe and Alcontents [Feox+Alox] of the soil. Borggaard et al. (1990)suggest that the Fe component should be subdividedbetween crystalline and non-crystalline forms, withdifferent coefficients for one aluminium and two ironcomponents. Freese et al. (1992) after fitting a largebody of experimental data (including that for soilsheavily contaminated by animal slurry and industrialwastes) concluded that separating Feox into twocomponents was unnecessary, but the quantity of Ppreviously adsorbed (Pox, also determined by oxalateextraction) had to be considered. Also, the limits of bothfast adsorption and the slow reaction were related to[Feox+Alox]. Overall, there appear to be two distinctapproaches to defining saturation in terms of the variouspools, each with an associated procedure for determin-ing the degree of saturation.The first approach, described by Mozaffari and Sims

(1994), Sharpley (1995) and Maguire (1996), is to definesaturation (for the fast reversible surface-sorbed P poolonly) as being equal to Qmax in the Langmuir isotherm[Eqn (6)], determined by shaking soil samples withvarious quantities of P for 24 h as described in Part 2(McGechan, 2002). With this approach, the degree ofsaturation is determined by a soil P test such as Mehlich-3 (Mehlich, 1984, see Part 2; McGechan, 2002), whichindicates the quantity of labile or plant-available P heldin the reversible surface-sorbed pool. Beauchemin andSimard (1999) note that there is some confusion aboutwhether the quantity of P already absorbed is taken intoaccount when determining the Langmuir sorptionisotherm and Qmax as the basis for estimating the degreeof saturation.The second approach is to determine a saturation

index (sometimes referred to as the Dutch P saturationindex) using ammonium oxalate extraction methods,also described in Part 2 (McGechan, 2002). Thisprocedure extracts all the P associated with Al and Feoxides and hydroxides in the soil. Saturation is alsodetermined by using ammonium oxalate, to determinethe quantities of Al and Fe which can be extracted using

this reagent. Schoumans (1995) describes the chemicalprocesses by which P becomes associated with Al and Fe(hydr)oxides, showing that the theoretical maximumquantity of Pox which can be sorbed in all the pools (fastreversible sorption to the surface sites plus diffusion-limited time-dependent sorption or deposition belowsurfaces) is equal to [Alox+Feox], where Pox, Alox andFeox are expressed on a molar basis. This leads to onedefinition of the Dutch saturation index as

DSSP ¼ Pox=½Alox þ Feox ð46Þ

Schoumans (1995) and Schoumans and Groenendijk(2000) found experimentally that in practice the max-imum sorbed P determined by oxalate extraction isequal to roughly 0�5[Alox+Feox], consisting of a max-imum for fast reversible surface sorption (Qmax in theLangmuir isotherm) of roughly 1/6[Alox+Feox], and themaximum for time-dependent sorption of roughly1/3[Alox+Feox]. This leads to the more commonly useddefinition of the Dutch saturation index as

DSSP ¼ Pox=f0�5½Alox þ Feox g ð47Þ

4.5. Desorption process

4.5.1. Quantity of phosphorus desorbedBarrow (1979) studied the process of desorption

which occurs following the dilution of soil water aftera period of incubation of P with soil. At that time, hedeveloped a set of empirical equations to represent thedesorption process (and how it differs from sorption) forone particular soil, using a linearizing procedure similarto the standard Freundlich isotherm equation. For avarying quantity of added P (Pfe g [P] kg

�1 [soil]) but afixed incubation period and temperature (22 day at258C), the quantity of P desorbed Pd after dilution tovarious degrees was given by an equation which includesa term of the Freundlich form

Pd ¼ k2 � k1Cb1 ð48Þ

where coefficient k1=415, index b1=0�4 and coefficientk2 is given by

k2 ¼ 0�227Pfe þ 71�6 ð49Þ

For a fixed quantity of added P (1�5 g [P] kg�1 [soil]) butvarying incubation periods and temperatures, Eqn (48)was modified to

Pd ¼ k1ðCb1d;0 � Cb1Þ ð50Þ

where

k1 ¼ k3tb2d ð51Þ

and

Cb1d;0 ¼ Cb1

0;0ð1þ k4tdÞ�b3 ð52Þ


For the desorption time td (and b1 retaining the value0�4), values of the coefficients k1, k3, k4 and C0,0 plus theindices b2 and b3 were fitted for each set of incubationperiod and temperature, to give the composite coeffi-cient Cd,0. These equations illustrated his observationsthat: firstly, dilution of the dissolved P after longincubation periods with high P concentrations (oralternatively after shorter incubation periods at raisedtemperatures), a much lower quantity of P is desorbedthan the quantity originally adsorbed, and secondly,where dilution occurred after only short incubationperiods, desorption more nearly followed the path ofadsorption, but this was followed by a period of re-adsorption. Later, Barrow (1983a) found that hismechanistic models of the parallel fast sorption andslow reaction processes [Eqns (18), (19) and (37)–(39)]gave an equally good fit to the experimental data fordesorption, along with a more satisfactory mechanisticexplanation of the processes. By assuming the parallelslow reaction to be irreversible (or reversible at a veryslow rate), desorption as the reverse of the fast truesorption process could be assumed to be reversiblefollowing the same isotherm equation as for sorption,but allowing for the portion of P no longer available fordesorption due to progress of the slow precipitation ordiffusion reaction.Raven and Hossner (1993) also studied the process of

desorption which occurs on dilution of soil water afterincubation (for 31 day at 248C) of P with soil. Theyfitted the following relationship to their data:

Pd ¼ k1C�0�1 þ k2 lnðC þ 1Þ þ k3 ð53Þ

They also found that Eqn (48) (from Barrow, 1979) gaveequally good fits, and both Eqns (48) and (53) gavemuch better fits than the following equation fromBrewster et al. (1975):

Pd ¼ k2 lnðCÞ þ k3 ð54Þ

Hooda et al. (2000) fitted desorption data to a linearisedFreundlich model:

log Pd ¼ log k1 þ 1=ðb1 logCÞ ð55Þ

4.5.2. Rate of phosphorus desorptionSharpley et al. (1981b) studied the kinetics of P

desorption over short time periods (as the reverse of thefast sorption process), as an indication of the release ofP from agricultural soils in surface runoff arising duringheavy rainfall. In a review of previous similar studies,they note that Amer et al. (1955), Li et al. (1972), Kuoand Lotse (1972, 1973), Griffin and Jurinak (1973) andEvans and Jurinak (1976) all found poor fits of data to afirst-order kinetic reaction equation. Most used a higherorder kinetic equation, while Kuo and Lotse (1973) and

Barrow (1979) favoured an adaptation of the Freundlichisotherm equation. Sharpley et al. (1981b) also adaptedthe Freundlich isotherm, finding that their experimentaldesorption data (obtained by shaking soil samples withvarious soil-to-water ratios for periods up to 3 h) couldbe described by the equation

Pd ¼ KsP0tasWbs ð56Þ

where Pd is the quantity of P desorbed in time t at awater-to-soil ratio of W ; Ks, as and bs are constants fora particular soil; and P0 is described as the initialquantity of ‘desorbable P’ after incubation for 3 day at258C. Sharpley and Ahuja (1982) investigated the effectsof varying the time period, temperature and soil watercontent during incubation prior to desorption, on theparameters in Eqn (56). They found that the fittedvalues of Ks, as and bs remained almost unchanged for aparticular soil, but there was a very large effect of thesefactors on P0. The effect of time and temperature couldbe described by Eqn (23) from Barrow and Shaw(1975a), while the effect of soil water content y (fraction)could be described by the following equation:

P0 ¼ Pom þ awð25� 100yÞ ð57Þ

where Pom is the minimum quantity of desorbable P (aty=0�25 for the particular soil tested) and aw is the slopeconstant. Sharpley and Ahuja (1983) later describe a‘diffusion interpretation’ of desorption. They suggestthat (for the fast sorption process from surface sites)reaction times are limited by diffusion of desorbed Pthrough static films of water surrounding particles andwithin aggregates. They present a differential equationfor desorption consisting of multiplicative area, diffu-sivity and concentration gradient terms, as follows:

dPd

dt¼ Asad

P0 � Pd

P0

� md

:P0 � Pd

V�

Pd

W

� �ð58Þ

where dPd/dt is the rate of desorption, As is the specificsurface area in cm2 g�1 of the soil particles and P0 andW are defined as for Eqn (56). The expression ad �½ðP0 � Pd Þ=P0 md with parameters ad and md is thevariable diffusivity function in cm s�1, and (P0�Pd)/V isthe effective concentration of desorbable P at any time t.The constant V , which relates the quantity of desorbableP P0�Pd to the effective concentration, is interpreted asthe volume of the diffusion layer on the surface or insideparticles which contains the P in the soil. Presentation ofEqn (58) is followed by a number of steps ofmathematical manipulation to produce Eqn (56).Sharpley et al. (1981a) developed Eqn (56) into

further equations representing the release of P in runoffevents following heavy rainfall, which they test experi-mentally (see McGechan, 2002, Part 2). They noted thatthe instantaneous concentration of desorbed P in runoff


water (and also in infiltration water) Cro would be equalto the rate of P desorption dPd/dt [given by differentiat-ing Eqn (56) with respect to time] divided by the rainfallrate Ir:

Cro ¼ ½KsP0Mtas�1Wbs =Ir ð59Þ

where M is the mass of soil in the interaction zone.Ahuja et al. (1982) present a series of equations based onEqns (56) and (57) (not given here), representing runoffdown a slope of length L, leading ultimately to anequation for the average concentration of P in runoff %CCin relation to rainfall, runoff and soil characteristics:

%CC ¼KsP0 arSl þ brð ÞEr þ crLmrSes

l Qnrr

� �1�bs

Ir þ 0�5LQrð Þbs=Iasr V1�as

r ð60Þ

where Ir is the rainfall intensity in cmh�1, Sl is the angle

of slope in %, L is the length of slope, Er is the kineticenergy of rainfall per unit area per unit time, Qr is therunoff rate per unit area, Vr is the total rain volume (inthe whole rainfall event) per unit area and ar, br, cr, es,mr and nr are constants for a given soil.Lookman et al. (1995) describe an equation for

desorption kinetics with two first-order terms fordesorption from two pools Q (fast, P sorbed ontosurface sites) and S (slow, P which has undergone theslow reaction)

Pd ¼ Q 1� exp �k1tð Þf g þ S 1� exp �k2tð Þf g ð61Þ

Q þ S ¼ Pox ð62Þ

where Pox is the oxalate-extractable P content of the soil.The fast pool Q is described as P sorbed onto surfacesites, but the time constant (around 0�5 day�1, muchslower than that measured by Sharpley et al., 1981b)suggests that this might represent the faster componentof the diffusion-limited reaction described by otherauthors.Kirk (1999) and Geelhoed et al. (1999) describe

simulation models of another process, by which someplants exude organic anions or acids from their roots togain access to P in the deposited or precipitated poolwhich is not normally available to plants. Thisprocess appears to be confined to a few particular plantspecies (such as white lupin and rape), and also toconditions of low soil P fertility, so will not beconsidered further here.

4.6. Models for analysis of column experiment data

Models of the sorption processes described in Sections4.1–4.5 have been developed into simulation modelsrepresenting the sorption kinetics and transport of P foruse in analysis of results of laboratory column experi-

ments. For such simulations, equations representing thesorption processes have to be combined with represen-tation of the transport processes by the convection/dispersion equation:

@C

@t¼ D

@2C

@z2� v

@C

@zð63Þ

where z is the depth below the soil surface and v is thewater velocity.Raats et al. (1982) in the first of a series of three

papers (each with a different first author), link theconvection/dispersion equation to their sorption equa-tions [Eqns (43)–(45)], then explore analytically solu-tions of limiting cases based on specific initial andboundary conditions as well as fixation capacities. In thesecond paper in the series, De Willigen et al. (1982)develop the equations into a computer simulation modelfor a column 1m long divided into 20 equal layers.Simulations start with an application of P of 140 kg ha�1

(corresponding to a heavy dose of pig slurry), and theeffects of a range of scenarios regarding precipitation,evaporation, initial levels of sorbed and fixed P andsorption equation parameters (isotherms for fast sorp-tion and fixation rate for the slow reaction) are tested.Results indicated that the fixed P pool created by theslow reaction would go on rising for 30 yr or morebefore becoming saturated. In the third paper in theseries, Gerritse et al. (1982) compared simulation resultswith experimental data for scenarios with a range ofhigh P application rates.Van der Zee et al. (1989a) fit the second-order version

of Eqn (40) to data from column experiments. Van derZee and Gjaltema (1992), in the first of two papers onsimulating P transport in column experiments, add aterm for total sorption (fast reversible sorption plusfixation by the slow reaction) to the convection/dispersion equation:

r@Stot

@tþ y

@C

@t¼ yD

@2C

@z2� yv

@C

@zð64Þ

where y is the volumetric water content, r is the dry bulkdensity of the soil and

Stot ¼ Q þ S ð65Þ

They also discuss extensively alternative assumptionsabout boundary conditions and whether or not there is aneed for daily interaction between the sorption andtransport processes. In the second paper, Van der Zeeet al. (1992) test assumptions and options are tested bysimulation (but not by comparison with experiments).Enfield and Shew (1975), Enfield et al. (1976, 1981a)

and Stuanes and Enfield (1984) also use Eqn (64) (withor without the dispersion term), with various versions ofthe equations for rapid sorption and the slow reaction,to represent P movement in laboratory soil samples (in


tests carried out with a view to develop of a system forwaste water treatment by soil). Similar sets of equationswere used in column experiments relating to applicationof wastes to soil by Shah et al. (1975) and Mansell et al.(1977, 1985), and in other applications of columnexperiments by Murali and Aylmore (1981), Aylmoreand Murali (1981), Lin et al. (1983a, 1983b), Cho (1991)and Akinremi and Cho (1991). Beauchemin et al. (1996)use Eqn (53) (from Raven & Hossner, 1993) as a bulkmodel of a column experiment, to fit data on phosphateconcentration in water passing out of the bottom of thecolumn during desorption. For analysis of columnexperiments in which sorbed and solution P concentra-tions were measured at various depths after surfaceapplications of phosphate solution, Chen et al. (1996)assume two identical kinetic Freundlich equation forms(with different coefficient values) to represent the twoparallel fast and slow sorption processes:

@Q

@t¼

yr

k1Cn � k2Q ð66Þ

@S

@t¼

yr

k3Cn � k4S ð67Þ

where the total sorption Stot is given by Eqn (64), and

Q ¼ fStot ð68Þ

S ¼ ð1� f ÞStot ð69Þ

where a fraction f of the sorption sites which take partin the fast sorption process.

4.7. Weather-driven soil phosphorus dynamics models

Representation of sorption processes has also beenincorporated into sub-models which are components ofmore wide-ranging weather-driven simulation models ofsoil P dynamics. Such models include ‘pools’ represent-ing categories of P, as well as processes of transforma-tion and transfer between pools. The relevant pools forP sorption as discussed in this paper are described asdissolved inorganic P, surface-sorbed inorganic P(resulting from the fast, reversible sorption process)and various categories of inorganic P which has under-gone time-dependent sorption (including that depositedat a depth below sorption surfaces due to the slowreaction). The models also include pools for organic Pfor which sorption processes are of less importance.

4.7.1. EPIC and related modelsFor one such model, Erosion/Productivity Impact

Calculator ‘EPIC’ (Jones et al., 1984a, 1984b; Sharpleyet al., 1984), separate equations are included for whatare called rapid adsorption and slow adsorption of

inorganic P. However, no soluble P is considered in themodel, so the processes described are not sorption out ofsolution in the manner in which the term sorption isused in other studies discussed in this paper. In fact, therapid process represents a flow from the labile pool Plab

(consisting of soluble plus surface-sorbed inorganic P) toan ‘active’ inorganic pool Pact. The active pool resemblesa component of P which has undergone time-dependentsorption or deposition (as described in other studies), towhich the flow is relatively fast and also reversible. Slowadsorption represents an irreversible flow from theactive pool to a stable deposited pool Psta at a muchslower rate. The flow rate for rapid adsorption Rla isgiven by

Rla ¼ 0�1eyeT Plab � PactPsp

1� Psp

� � �ð70Þ

The initial value of Pact is given by

Pact ¼ Plab 1� Psp

� �=Psp ð71Þ

The proportion of added P which remains labile afterincubation Psp is given by

Psp ¼ Pilf � Pili

� �=Pfe ð72Þ

where Pili is the initial labile P (prior to fertilization), Pfe

is the fertilizer P added and Pilf is the labile P afterfertilization and incubation. The temperature andmoisture content adjustment factors eT and ey (both inthe range 0–1) are given by

eT ¼ expð0�115Tc � 2�88Þ ð73Þ

ey ¼ y=y0�03 ð74Þ

where Tc is the soil temperature in 8C, y is thevolumetric soil water content and y0�03 is the volumetricsoil water content at a tension of 30 kPa. The flow ratefor slow adsorption Ras is

Ras ¼ basð4Pact � PstaÞ ð75Þ

where the rate constant bas takes a value of0�00076 day�1 for calcareous soils, and for non-calcar-eous soils is given by the equation

bas ¼ expð�1�77Psp � 7�05Þ ð76Þ

The CREAMS (Knissel, 1988) and GLEAMS (Knis-sel, 1993) weather-driven systems simulation models aredevelopments of the EPIC model and use similarequations. For the proportion of added P remainingafter incubation, Psp, alternative equations are providedfor different categories of soils. For non-calcareous soilsthey also differ between the original EPIC paper(Sharpley et al., 1984), the EPIC Model Documentation(Sharpley & Williams, 1990) and the GLEAMS ModelDocumentation (Knissel, 1993) which quotes Sharpley


and Williams (1990). They are presented here as inGLEAMS, as follows:

No

Psp ¼

0�58� 0�0061CCaCO3 calcareous soils

0�0054Bsat þ 0�116pH � 0�73 slightly weathered non-calcareous soils

0�46� 0�0916 ln CLð Þ highly weathered non-calcareous soils

8><>: ð77Þ

with 0�054Psp40�75 where CCaCO3 is the calciumcarbonate concentration, Bsat is the base saturation bythe ammonium acetate method in %, pH is the soil pHand CL is the clay content in %.Compared to the original EPIC (Jones et al., 1984a),

the soil moisture content adjustment factor ey inGLEAMS (Knissel, 1993) is modified to

ey

y� yw

yfc � yw; y4yfc

0; y > yfc

8><>: ð78Þ

where yw is the volumetric soil water content at thewilting point of 1500 kPa and yfc is the volumetric soilwater content at field capacity (assumed to be at atension of 33 kPa, a larger tension value than commonlyassumed for field capacity in Europe).The ICECREAM model (Simes et al., 1998) is a

development of GLEAMS for Finnish soils. It retainsthe same equation forms, but the coefficients in Eqn (77)for Psp required different values, reflecting higher Pretention by Finnish soils compared to that for NorthAmerican soils:

Psp ¼Pspph ¼ 0�0054Bsat þ 0�116pH � 0�73

if ððpH� 0�73ÞCL42 ð79Þ

Psp ¼ Pspcl ¼ 0�46� 0�0916 lnðCLÞ if OM410% ð80Þ

Psp ¼ ð0�0025CL þ 0�65ÞPspcl þ ð0�35� 0�0025CLÞPspph

for other soils ð81Þ

where organic matter is abbreviated as OM.

4.7.2. ANIMOAnother weather-driven simulation model of soil P

dynamics is ANIMO (formerly known as the

Table

Parameter values in Eqn (85) for slow-deposition proc

Sorption class, j

K2,j, day�1

1 1�17552 0�03343 0�0014382

te: rd , dry bulk density, kgm�3; ½Alox þ Feox , aluminium and

‘Agricultural Nitrogen Model’ before the P routines wereadded) from the Netherlands (Kroes & Rijtema, 1998;

Groenendijk & Kroes, 1999). For fast, reversible sorption,ANIMO works with an equation based on the equili-brium Langmuir isotherm, Eqn (6). Parameter values inm3kg�1 [P], estimated by Schoumans (1995), are

kL ¼ 1129 ð82Þ

Qmax ¼ 5�167� 10�6r½Alox þ Feox ð83Þ

where [Alox+Feox] is the Al plus Fe content of the soil inmmol kg�1.For the time-dependent processes including the slow

diffusion-limited reaction, the authors of ANIMOconsidered using the shrinking core model (as describedby Van Riemsdijk et al., 1984a, see Section 4.3.3),but found it to be impractical as a component of a field-scale model. Instead, they estimated a reaction rate asthe sum of three widely differing rates for three distinctsorption (or deposition) sub-pools (designated j=1–3),according to the following equation:

@S

@t¼

P3j¼1

K2;jðK3;jCb1 � SÞ; K3;jCb1 > S ð84Þ

Coefficients K2,j and K3,j listed in Table 1, fitted bySchoumans (1995) to data reported by Schoumans et al.(1986), were found to be suitable for a wide range ofsandy soils in the Netherlands for sorption. Therelationships represented by these equations werealso checked against those for column experimentspresented by Van der Zee et al. [1989a, with thesecond-order version of Eqn (40) fitted to columnexperiment data] and Van der Zee and Gjaltema(1992). As the equations were simply empirical fits todata, the authors do not state which processes arerepresented; however, the fastest component (indexj=1) may represent the kinetic component of

1

ess in ANIMO model (Groenendijk & Kroes, 1999)

Coefficients Exponent b1

K3,j, kg m�3 ðkg m�3Þ�b1

11�87� 10�6 rd ½Alox þ Feox 0�53574�667� 10�6 rd ½Alox þ Feox 0�19959�711� 10�6 rd ½Alox þ Feox 0�2604

iron content, mmol kg�1.


fast sorption as described in Section 4.2.3, while theslower components (index j=2, 3) may represent theslow deposition reaction. An equation of similar form toEqn (84) is suggested for desorption, but the samecoefficient values as for sorption have to be assumed asdifferent values have not been determined experimen-tally.

4.7.3. Comparison of kinetic transfer ratesbetween modelsA comparison of time series solutions of Eqns (70),

(75) and (84) representing the pools simulated by EPIC(at two alternative temperatures) and ANIMO is shownin Fig. 2, alongside results from equations presented byBarrow (1983a) and Sharpley (1982). The comparison at158C (the temperature of the experiments on whichANIMO is based) suggests that the ‘labile’ pool in EPICdeclines in a manner roughly similar to the surface-sorbed P pool in ANIMO. At 258C, the decline in thelabile pool in EPIC is similar to that for Psp (water-extractable P) in the equations from Sharpley (1982) andsurface-sorbed P in the work of Barrow (1983a). Also,the ‘active’ pool in EPIC corresponds roughly to thesum of the first two (index j=1, 2) time-dependentsorption subpools in ANIMO. Similarly at 158C, the‘stable’ pool in EPIC corresponds roughly to thethird (index j=3) time-dependent sorption subpoolin ANIMO. These time series were obtained bysolving the first-order decay processes represented byEqns (70), (75) and (84), as follows. For the decline inthe labile pool (due to transfer to the active pool) inEPIC:

Plab ¼ Pili þ PspPfe þ Pfe 1� Psp

� �exp �0�1eyeTtð Þ ð85Þ

for the active pool in EPIC:

Pact ¼Pacto 4� 3 exp �bast� ��

� PspPfe

� Pfe 1� Psp

� �exp �0�1eyeTtð Þ ð86Þ

where Pacto is the initial value of Pact; for ANIMO (aspresented by Schoumans, 1995):

S ¼X3j¼1

K3;jCb1 1� exp �K2;j t

� �� ð87Þ

4.7.4. Other soil phosphorus dynamics modelsTwo other soil models with P dynamics routines are

‘CENTURY’ and ‘ecosys’. CENTURY (Parton et al.,1987; Metherell et al., 1993) differs from the othermodels in that it operates with a monthly timestep as itis designed to indicate long-term trends, but there is arecent version ‘DAYCENT’ operating on a daily time-step which has been reviewed by the authors, McGechanand Lewis (2001). It operates with a number of soil Ppools, dynamic flow rates between pools and fast

sorption of inorganic P following the Langmuirisotherm. The ecosys model (Grant & Heaney, 1997)has a very complex treatment of individual reactions(including those representing sorption) between P and arange of Al, Fe, Ca and other soil mineral compounds.Equilibrium associations are listed for over 50 sets of ionpairs. The model was initially tested to represent Ptransformations and transport in laboratory columnexperiments, then linked to a P uptake routine(comprising a further 19 equations) to represent Puptake by root systems (Grant & Robertson, 1997).

4.7.5. Further developments of sorption equationsin phosphorus dynamics modelsThe two existing weather-driven soil P dynamics

models, GLEAMS (with the associated models EPIC,CREAMS and ICECREAM) and ANIMO, use rela-tively simplistic equations to represent the parallel fastsorption, slower sorption and very slow depositionprocesses. Some of the sources on which these equationsare based are either old or represent preliminaryinvestigations. Also, equation parameters may applyonly to specific soils in the countries where measure-ments were made. Meanwhile, there has been extensiveresearch into the mechanisms of sorption processes(including the slow, diffusion-limited depositionreaction), leading to more mechanistic and generallyapplicable equations. A logical next step would be toincorporate such mechanistic equations into field-scalemodels, and to determine values of coefficients andparameters applicable to a wide range of soils.

5. Sorption onto mobile particulate and colloidal material

5.1. Sorption onto sediments in surface runoff

Phosphorus sorption as described up to now in thispaper, and as described in nearly all published papers onthe subject, generally refers to sorption onto static soilcomponents. However, some desorption studies fromNorth America refer to sediment material that hasbecome detached from the main body of soil and movein surface runoff water (overland flow) leading to Ppollution of receiving water bodies. Models of theprocess have been described in Section 4.5.2, andexperimental data are discussed in Part 2 (McGechan,2002).

5.2. Sorption onto colloidal particles moving through thesoil

The concept that, in addition to surface flow Ptransport, there is also a substantial flow of P to water


bodies through soil movement of P-laden particulatematerial, has only recently been proposed. Particulatematerial passing through the soil (mainly by macroporeflow) tends to be very finely divided colloids (derivedfrom animal manures, soil organic matter or the clayfraction of soil minerals) with a very high specificsurface area and extensive sorption surfaces. There are afew recent studies from Europe reporting substantialthrough-soil losses of P. Stamm et al. (1998) observedraised concentrations of P during large rainfall/drainageevents in grassland soils. This suggests mobilization ofcolloid material, since if this did not occur a lowerconcentration due to dilution of the soil solution byincoming rain would be expected. Similar raised Pconcentrations during rainfall events have been ob-served in through-soil leaching from grassland byHawkins and Scholefield (1996) and by Haygarth et al.(1998). These studies also support the hypothesis thatcolloid-facilitated transport plays an important role inthrough-soil leaching of P to receiving waters. Addiscottet al. (2000) observed high P losses on arable croppedland, which they attribute to colloid transport; theysuggest that colloidal clay particles are initially mobi-lized in surface runoff flows until they reach amacropore through-the-soil pathway leading to a moledrain channel. All these results tend to refute a widelyheld misconception that where all incident rainwaterinfiltrates through the soil surface (so overland flow isabsent) nearly all phosphorus present is sorbed ontostatic soil components and removed from leaching flows.This mobile solid represents a third phase for thecontaminant (Fig. 3), in addition to that in solution andthat sorbed onto the static soil matrix, as described byMcCarthy and Zachara (1989).Modelling of colloid-facilitated pollutant transport

processes is dependent on an accurate model descriptionof the dynamics of the carrier colloids. It is commonlyassumed that colloids can be generated by detachmentof soil particles from the static body of soil. Jarvis et al.(1999) reviewed literature on this subject, and concludedthat detachment occurs mainly by rainfall impact atthe soil surface rather than due to scouring bywater passing through sub-surface soil pores. Animal

Fig. 3. Simplified schematic representation

manures, particularly slurry (liquid manure), containlarge quantities of both colloid-sized particles and P, soland application of manures provides an additionalimportant source of P-laden colloids. The only knownmodel of colloid-facilitated transport of contaminantsthrough soil is an adaptation of MACRO (Jarvis, 1994)as described by Jarvis et al. (1999). This model focusseson macropore flow, in view of its importance fortransport of colloids. It also includes representation ofprocesses by which colloids become trapped by strainingand filtration. Separate filtration equations are definedfor micropores (where most trapping of colloids occurs)and for macropores. While developed mainly torepresent transport of sorbed pesticides (Villholth et al.,2000), McGechan et al (2002) have recently applied thisadaptation of MACRO to describe this process asapplied to P transport. With this model, it is necessaryto specify sorption parameters for both sorption ontostatic soil material and onto the mobile colloid particles(according to the Freundlich equation for MACRO).The limited available information about parametervalues for sorption of P onto colloidal material isdiscussed in Part 2 (McGechan, 2002).

5.3. Sorption onto organic material

While there is evidence supporting the concept oftransport of inorganic P in association with organiccolloids (mainly in manure or slurry but perhaps alsothose derived from plant residues/litter or soil organicmatter), the mechanism by which P is sorbed on suchmaterial is less clear. There is a clear ionic mechanismfor P sorption onto metal oxides (which are oppositelycharged), but this will not apply for sorption ontoorganic material since phosphate anions will not beattracted to organic colloids which also tend to have anegative charge. ‘Cation bridging’ involving othersubstances may have a part to play. Gerke andHermann (1992) studied this bridging process in theadsorption of orthophosphate onto humic-FE-com-plexes, observing a large increase in the extent ofsorption in relation to the quantity of iron present.

of main inorganic soil phosphorus pools


6. Conclusions

6.1. General comments regarding phosphorus sorptionrelationships

The processes of sorption of P in soil are verycomplex, and the very extensive literature on the subjectcan present a very confusing picture regardingthe mechanisms of sorption. Nevertheless, there aresome broad trends and some general comments canbe made.In some of the earlier works, attempts at fitting

simple, single mathematical equations (isotherms) toexperimental sorption data were confused by the timedependence factor. In early studies on desorption,empirical relationships were fitted to measured datawith no explanation of why desorption did not retracethe path of sorption. Later work clarified that thereappear to be at least two distinct processes, a fastreversible sorption onto solid surfaces, plus a slowalmost irreversible process consisting of diffusionthrough the sorbing layer followed by precipitation ordeposition below the sorption surfaces. The sorbingsurfaces consist mainly of iron and aluminium oxides (inthe clay components) in acid soils, or calcium carbonatein calcareous soils for which the process differs some-what from that in acid soils. Simple Langmuir isothermscan represent sorption onto individual minerals, but forreal soils either the two-surface Langmuir or theFreundlich isotherm equation (which can be consideredto be in effect an integration of several Langmuirisotherms) has sometimes been found to be moresatisfactory. Equilibrium sorption represented by suchan isotherm equation is reached within about 1 day offertilizer or manure application, but following this there isa gradual reduction (over several months) of dissolved Pin the soil solution (available for plant growth) due tovarious ongoing slow time-dependent diffusion or pre-cipitation processes. The distinction between almostinstantaneous sorption on surface sites and the slowprocess of deposition below surfaces is not clear-cut.Some researchers have described an intermediatetime-dependent process, faster than slow deposition,where there is some time dependence due to diffusionthrough the soil water to sites within the soil matrix;others suggest that all the processes should be regarded asa continuum.

6.2. Applications of sorption relationships in simulationmodels and other studies

Weather-driven simulation models of soil P dyna-mics require mathematical equations to describe P

transformation processes such as reversible sorptionon particle surfaces and the slow deposition processbelow surface sites. Identifying such relationships formodels on the basis of literature sources was the reasonfor carrying out this review.

6.2.1. Sorption phases and pollutant transport modelsA simplified representation of the phases and compo-

nents of sorption by P in soil is shown in Fig. 3, makingthe distinction between P sorbed onto static componentsof the soil matrix and that sorbed onto mobile sedimentsor colloids. This also shows the over-simplistic repre-sentation with distinct compartments for instantaneoussurface sorption and slow deposition (when in fact theremight also be some intermediate faster time-dependentpools or else representation as a continuum). The mobilesorbed components consist of both P associated withsediments transported in surface runoff and that sorbedonto finer colloidal material transported through the soilby macropore flow. The ideal model will consider allthese components and all possible pollutant transportroutes. Currently, EPIC ignores the solute phasecompletely, so cannot indicate leaching of soluble Pvia field drains or to deep groundwater. In contrast,soluble P leaching to field drains is the main loss routeconsidered in ANIMO. Transport in mobile sedimentsin surface runoff flows is the main loss processconsidered in EPIC. No models currently considerlosses of P attached to mobile colloidal material movingthrough the soil by macropore flow, but a modificationto the MACRO model to represent this is currentlybeing undertaken.

6.2.2. Selection of timestep in models in relationto sorptionFor many purposes, models can operate on a daily

timestep, assuming instantaneous equilibrium for fastreversible sorption on surface sites represented either bythe Freundlich, single Langmuir or double Langmuirisotherm equations. Surface runoff, and the associatedprocesses of sediment erosion and pollutant transport of Pas well as components of surface spread manure or slurry,tend to happen during short-lasting high-intensity rainfallevents where fast-acting processes are important. Unlikewhere other transport processes are represented by themodels, it may not be adequate to assume instantaneousequilibrium for fast, reversible sorption on surface sites.Also, model simulations need to be carried out overa timestep of minutes rather than days. The equationsand approach adopted by Sharpley are appropriatein this situation, including representation of timedependence particularly for desorption of P from surfacesites.


6.2.3. Required complexity for representation ofsorption in modelsFor fast reversible sorption on surface sites, repre-

sentation by isotherm equations (either Freundlich,single Langmuir or double Langmuir) is satisfactoryfor most modelling processes. The more complexmechanistic model of Barrow is in effect an extensionof the Langmuir equation. The main debate aboutcomplexity concerns the slow, time-dependent sorptionor deposition processes. For these slower processes, theequations in mechanistic models may be unnecessarilycomplex for their representation alongside other trans-formations, as well as having parameters which areunknown for many soils. More practical is to assume asimple time-dependent relationship such as Eqn (5) or(84) to represent transfer from labile (readily availablefor plant uptake) pools to the less-available depositedpool. To this should be added the constraint of amaximum total P content for all time-dependent pools,based on oxalate-extracted Al plus Fe. This was theapproach adopted in the Dutch ANIMO model, inpreference to the equations from the USC model whichwas considered to be too complex.

Acknowledgements

Funds to carry out this work were provided by the Sco-ttish Executive Environment and Rural Affairs Department.

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sw—soil and water: sorption of phosphorus by soil, part 1: principles, equations and models

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