Goudriaan, J.

Keulen, H., van

Seligman, N.G.

English

Wageningen Yield 

Reprint Neth. J. agric. Sci. 23 (1975): 131-138 
Availability of anions in the growth medium to roots of an 
actively growing plant 
H. van Keulen, N. G. Seligman\ J. Goudriaan 
Department of Theoretical Production Ecology, Agricultural University, Wage-
ningen, the Netherlands 
Accepted: 17 January 1975 
Summary 
Analysis of anion transport to the root of an actively growing plant with a normally 
dense root system showed that virtually the whole of the anion store in the rooting 
zone is available to the plant within a few days at the most. Transport by diffusion 
only is enough to account for most of the depletion, but mass flow will speed up 
the process. The effcet of mass flow will be considerable in soils with a high dis-
persion coefficient (loess), but very small in soils with a low dispersion coefficient 
(clay and sand). 
A rule is proposed to determine whether a given root density is sufficient to 
supply the nitrogen and water needs of the plant by diffusion only. 
Introduction 
Transport of ions in the growth medium to roots of plants is so well studied a 
subject that there would appear to be very little to add (Nye, 1969; Tinker, 1969). 
The processes of ion diffusion and mass transport are indeed well understood in 
basic physical terms. The. ecological significance of these processes to an actively 
~~owing plant has received far less attention. In particular no quantitative treatment 
pf the relative importance of diffusion and mass flow in such a situation comes to 
inind. The difference between these processes is fundamental to an understanding 
of ion availability to roots because when transport is by mass flow of water, avail-
ability of ions is mainly dependent on concentration; whereas when transport is by 
diffusion, availability is dependent mainly on root density. 
In the treatment offered here, anions only will be considered in order to avoid 
the complicating processes of exchange between soil solution and the solid phase 
of the soil. It is relevant to nitrate ions which are the main source of available 
nitrogen to the plant under many conditions. As nitrogen is quantitatively the 
dominant 'mineral' element in normal plant tissue, an understanding of the pro-
1 Present address: Agricultural Research Organisation, Volcani Center, Bet Dagan, Israel. 
131 
H. VAN KEULEN, N. G. SELIGMAN AND J. GOUDRIAAN 
cesses that determine its availability is important when the dynamic relations be-
tween nitrate availability and plant growth are being studied. 
A simple calculation makes it clear that transport by mass flow alone is often 
inadequate to explain nitrate availability to plants. A crop canopy growing at near 
optimum rate can increase in dry weight at the rate of 200 kg ha-1 day-1 • If the 
nitrogen content of the plant is 2.5 % and the transpiration rate is 5 mm day-t, 
transport by mass flow alone would mean that the nitrogen concentration of the 
soil solution must be at least 100 mg/kg. As the nitrogen concentration of the soil 
water is much lower in most situations, it would appear that diffusion must at least 
make up the difference. The purpose of this paper is to determine from first prin-
ciples the potential contribution of diffusion and mass flow to anion availability as 
a function of root density and anion concentration. 
Theoretical background 
The root can be depicted as a cylinder with radius R, surrounded by a growth 
medium, bounded by half the distance between two roots. A treatment of ion 
transport by diffusion and mass flow into such a cylindrical system has been given 
by de Wit & van Keulen (1972). The basic equations are taken from Crank (1956) 
and were programmed for numerical solution in CSMP (see Appendix). 
Assuming that there are no gradients in the vertical or tangential directions the 
equation for the diffusion flux per unit area through the wall of a cylinder, sur-
rounding the root, is: 
oC F = -D -, where D = 8D0L 
or 
F diffusion flux per unit area 
C concentration of solute 
D 0 diffusion coefficient in water 
e = water-filled pore volume 
L labyrinth factor 
r distance from centre of cylinder 
(mmol cm-2 day-1) 
(mmol cm-3) 
(cm2 day-1) 
(cm3 cm---3) 
(dimensionless) 
(em) 
To obtain the flux per unit length of root F must be multiplied by 2nr: 
oC Fl=-2nrD-
or 
where F1 = diffusion flux per unit length (mmol cm-1 day-1). 
When mass flow also is considered, there are two additional terms in the 
equation for F: 
oC F=-(D+sJvl)- + vC 
or 
v = rate of water flow in the radial direction (em day-1) 
s = dispersion length (em). 
132 Neth. J. agric. Sci. 23 (19751 
AVAILABILITY OF ANIONS TO ROOTS OF AN ACTIVELY GROWING PLANT 
The terms s lvl describes the dispersion. It must be added to the true diffusion 
coefficient (Frissel et al., 1970) to give the apparent diffusion coefficient. The term 
vC represents the transport by mass flow. While transport by mass flow is de-
pendent on the concentration of anions around the root (or sink), transport by 
diffusion is dependent on the concentration gradient. 
The conservation equation for the radial system, when vertical and tangential 
gradients are absent, is 
8 oc =_oF_ 1 F 
ot or r 
Substitution of the expression for F gives the Fokker-Planck equation, which be-
comes rather lengthy in the case with mass flow. When there is only diffusion it 
becomes: 
oC = D'(o2C + ~ oC) 
()t or2 r or 
where D' = D.L. Note that (9 has cancelled out. 
The initial and boundary conditions are: 
t=O r>R , C=C0 
t;:::o r=R , C=O 
The inner cylinder with radius R, or root, is thus regarded as a zero sink. 
The analytical solution of this partial differential equation, with the above bounda-
ry and initial conditions, is given by Crank (1956, p. 82) and Carslaw & Jaeger 
(1959, pp. 335-336) for the case where C=C0 is constant at r=CXJ. This solution 
can be used to check the numerical solution (de Wit & van Keulen, 1972). 
Although an analytical solution for the combined diffusive and convective flow 
is available (Olsen & Kemper, 1968), it is very complicated and a numerical 
solution is more practical. The partial differential equation for this case is therefore 
not written out here. 
The listing of a program to solve the radial flux equation numerically is given 
by de Wit & van Keulen (1972, Chap. 3). This makes it possible to calculate the 
depletion of anions in a finite zone around the roots. The zone is defined by giving 
rmax a value, dividing the zone into concentric compartments and calculating the 
radial flux across each of the compartments and into the root. By changing the 
value of rmax the effect of distance between roots (or root density) on the radial 
flux under different flow conditions can be determined. 
The radial flux is then integrated over time and the amount of anion removed 
expressed as a fraction of the original amount of anion in the defined root zone 
(see Appendix). 
Problem specifications 
Representation of the root zone as cylindrical means that the corners of the square 
columns into which a soil volume would be subdivided are neglected. To take the 
Neth. J. agric. Sci. 23 (1975) 133 
H. VAN KEULEN, N. G. SELIGMAN AND J. GOUDRIAAN 
corners into account, rmax should be increased. A fair approximation should be 
rmax V (4/:n) or 1.13 rmax. The rmax values given below, therefore refer more accu-
rately to the given root distances/1.13. 
The radial flux equation was solved for the following conditions: 
R 0.03 em 
rmax 0.5 em (1 em between roots) 
rmax 1.0 em (2 em between roots) 
D0 1.0 
s 7.0 em (for loess soil) 
and s 0.8 em (for clay and sand) 
(these values are from Frissel et al., 1970). 
w=O (no mass flow, diffusion only) 
and w= 0.025 (2.5 mm transpiration day-\ roots 1 em apart, 10 em long) 
and w = 0.050 (5.0 mm transpiration day-\ roots 1 em apart, 10 em long) 
where w is water uptake per unit root length ( cm3 day-1 cm-1), related to v ac-
cording to w = -2:nrv. 
The values for w are based on the assumptions that roots are evenly distributed 
1 em apart and their effective depth is 10 em. w is then determined as the amount 
fraction of 
salt removed 
1.0 
.9 
.8 
.7 
.6 
.5 
.4 
3 
. 2 
.1 
w 
CD 0 . 
® 0.025 
0 0.050 
8 0.025 
® 0. 
® 0.025 
s rmax 
2. 
7. 2. 
7. 2. 
0.8 2 . 
1. 
7. 1. 
time 
Fig. 1. Amount of anion removed from soil solution with different root densities, flow rates 
and dispersion factors. 
134 Neth. J. agric. Sci. 23 (1975) 
AVAILABILITY OF ANIONS TO ROOTS OF AN ACTIVELY GROWING PLANT 
of water withdrawn from each cm3 of soil in the given root depth. When transpi-
ration is 2.5 mm day-t, then w= 0.25 cm3 water (10 cm3 soil)-1day-t, or 0.025 cm3 
(em length of root )-1day-1• 
If effective rooting is increased, w is decreased proportionately to maintain the 
same transpiration rates. When roots are 2 em apart, the transpiration rate for a 
given value of w is ).;4: of the above rates. 
Initial values of anion concentration were taken as 0.005 mmol cm-3 water which 
is equivalent to 70 mg N/kg. It should however be noted that for the purposes of 
this study, the fraction of anion removed to a zero sink in the root was calculated 
and this result would have been unaffected by initial concentration. The absolute 
amount transported to the zero sink would then be directly proportional to con-
centration. 
Results 
Some results are given in Fig. 1. Depletion is slowest when root density is low 
(1 root per 4 cm2) and there is no mass flow. However, even here more than 90% 
of the anions are depleted within five days (curve 1). A slight mass flow, equivalent 
to a transpiration rate of 0.6 mm day-1 reduces the time to arrive at a similar 
depletion level to just over two days in loess soil with a dispersion coefficient of 
7 (curve 2), but only to four days in sand and clay soils with dispersion coefficients 
of around 0.8 (curve 4). Doubling the transpiration rate in a loess soil, adds very 
little to the rate of depletion (curve 3). 
Increasing root density to 1 root cm-2, however, dramatically increases transport 
by diffusion to the extent that a similar depletion level is attained in less than one 
day (curve 5). In a loess soil with a transpiration rate of 2.5 mm day-1 similar 
depletion levels are attained in half the time (curve 6). 
Thus anion transport can be increased considerably by mass flow especially 
when the dispersion coefficient of soils is high. For most soils with much lower 
dispersion coefficients than loess, transport must be mainly by diffusion, especially 
when the anion concentration in the soil solution is low. 
In order to gauge the ecological significance of these results, the time scale of 
these processes should be noted. Depletion of 90 % of the anions with a sparse 
root system within four days means that for actively growing plants virtually the 
whole anion store is available if they need it and this can be transported to the 
root by diffusion only. With a denser root system, virtually the whole anion store 
is potentially~ available in one day,~ again even· by diffpsi0rr alone. This is ·particu-
larly important in understanding nitrate uptake by growing plants and explains 
how nitrogen fertilizer can be so efficiently utilized even when the actual increase 
in concentration of the nitrate in the soil solution that is being taken up for transpi-
ration is small. 
When nitrogen is limiting growth, virtually all the nitrate will be taken up soon 
after it is released (by mineralization for instance). As a result the soil under a 
growing sward will have a very low nitrate concentration, as is so often found 
Neth. J. agric. Sci. 23 (1975) 135 
H. VAN KEULEN, N. G. SELIGMAN AND J. GOUDRIAAN 
(Nye, 1969). However, this is not necessarily due to the supply being so low as 
much as it is due to the transport to (and into) the root being so rapid. 
Discussion 
Theoretical considerations of transport of anions to a root of an actively growing 
plant indicate that the anion store can be made readily available to the plant by 
diffusion only. The rate of depletion of a given anion by diffusion is dependent on 
the root density, the diffusion coefficient of the anion and the period during which 
the anion store must become available. It is not necessarily dependent on the 
moisture content of the soil because as soil moisture drops the anion concentration 
increases while diffusion continues even at low soil moisture levels (see 'Theoretical 
background'). A given density may be inadequate, if depletion must occur within a 
given period but it may be adequate if the permissable depletion period is longer; 
or if an anion with a larger diffusion coefficient is considered. The relationships 
between these factors can be represented as follows: 
RDN=D0.t/X2 
where RDN = a dimensionless root density number 
X = distance between roots/1.13 (see 'Problem specifica-
tions') (em) 
diffusion coefficient of the anion ( cm2 day-1) 
= time during which the anion store must become avail-
able to supply the needs of the plant (day). 
The value of RDN for which root density is adequate for depletion by diffusion 
can be determined by substituting the values obtained in this study. Both sets, 
X=1, D0 =1, t=1 and X=2, D0 =1., t=4 give an RDN value of 1. Thus, when 
RDN > 1 the root system is sufficiently dense to deplete the anion ~tore by dif-
fusion within the period t. When RDN < 1, the root density is too low to deplete 
the store in period t by diffusion only. Mass transport will become more important 
in such a case. 
It may be necessary for the root system to be dense enough to deplete the anion 
store in one day when supply is low but continuous. This would occur when nitrate 
is being released by micro-organisms during the growth period of a plant. 
Micro-organisms that compete with the plant roots for the available anions wil1 
directly affect the availability of anions to the plant. As they are located between 
the roots where anion concentration is higer, it is reasonable to suppose that they 
will have first call on the available anions. Depending on their activity and the 
aVallability of anions, they can make the anion unavailable to the plant. In order 
to compete with micro-organisms for a small amount of anion, the root density 
must equal the density of the micro-organisms. Thus in the presence of strong 
competition from micro-organisms, the root density will be too low to supply the 
needs of the plant even when RDN = 1. 
The number, RDN, can also be used to determine the adequacy of a given root 
density for water withdrawal from the soil. As the diffusion coefficient of water 
(diffusivity) depends on the moisture content of the soil, the adequacy of the root 
136 Neth. J. agric. Sci. 23 (1975) 
AVAILABILITY OF ANIONS TO ROOTS OF AN ACTIVELY GROWING PLANT 
system will depend largely on the soil moisture content. For a moist soil, D0 is 
much larger than 1 and the value drops to 1 only as the soil water potential ap-
proaches 4 to 5 atmospheres (Childs, 1969). Thus for most situations where the 
soil moisture is somewhat above the wilting point a root density that is adequate 
for supplying anions is also adequate for supplying water. 
The availability to the plant of anions, and in drier conditions of water too, is 
thus dependent mainly on root density. As the root system responds to nutrient and 
sometimes even to water shortage by better growth (Brouwer & de Wit, 1969) it 
will often increase its density when the need arises. It must therefore be concluded 
that in most cases of a normal, well developed root system, virtually the whole 
anion store in the rooting zone will be almost immediately available for plant 
growth. It would seem that a similar situation holds for water as long as soil mois-
ture is not too low. 
Acknowledgment 
The authors are indebted to Prof. C. T. de Wit for the stimulating discussions and 
the provocative comments that initiated the ideas presented here. 
Appendix: Programming the numerical solution for the flow equation 
The core of the program given by de Wit & van Keulen (1972, Chap. 3) consists 
of the following statements: 
CONCn 
FLOW1 
FLOWi 
APDIFi 
SALTUP 
FSR 
where: 
n 
CONCi 
ICONCi 
FLOWi 
VOLWi 
APDIFi 
VOLFLW 
AREAi 
LAB 
INTGRL(ICONCi, (FLOWi-FLOWi+ 1)/VOLWi) where i 1, 
... n-1 
INTGRL(ICONCn, FLOWn/VOLWn) 
= -CONC1(APDIF1 + VOLFLW) 
APDIFi(CONCi_ 1-CONCi)-VOLFLW(CONCi + CONCi_ 1)/2 
where i=2, ... ,n 
AREAi(LAB.WC.DIF + DISP.RFLOWi)/DISTi, where i = 1, ... ,n 
INTGRL(O.,-FLOW1) 
SALTUP/TIAS 
number of concentric compartments into which the zone around 
the root is devided 
concentration of anion (mmol cm--a) in compartment 
= initial concentration of anion in compartment (mmol cm-3) 
= flow of anion over a boundary (mmol day-1) 
volume of water in compartment ( cm3) 
= apparent diffusion coefficient for anion flow ( cm3 day-1) 
= volume of water flow ( cm3 day-1) 
area of cylinder compartment ( cm2) 
labyrinth factor (dimensionless) 
Neth. J. agric. Sci. 23 (1975) 137 
---~-- -~~--~---~ ~-~ ~--
wei 
DIP 
DISP 
RFLOWi 
DISTi 
SALTUP 
FSR 
TIAS 
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H. VAN KEULEN, N. G. SELIGMAN AND J. GOUDRIAAN 
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diffusion coefficient of anion ( cm2 day-1) 
dispersion factor (em) 
VOLFLW I AREAi = flow rate of water ( cm3 cm-2 day-1) 
diffusion distance between the centres of compartment i and i-1 
amount of anion removed to the zero sink (mmol) 
fraction of anion removed from original amount 
total initial amount of anion in the zone around the roots (mmol) 
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138 Neth. J. agric. Sci. 23 (1975) 


An introduction to the physical basis of soil water phenomena.

Chromatographic transport through soils.

Conduction of heat in solids.

Movement of nutrients to plant roots.

Numerical values for the temperature in radial heat flow.

Simulation of transport processes in soils.

The mathematics of diffusion.

The soil model and its application to plant nutrition. In:

The transport of ions in the soil around plant roots. In:

Availability of anions in the growth medium to roots of an activily growing plant

Name not available

Availability of anions in the growth medium to roots of an activily growing plant

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