Supporting Information Methods S1 and S2
Method S1 Details of image processing
Neutron radiographs were referenced to flat field (radiography without sample) and dark current
(signal recorded by the camera when there was no beam). The referenced radiographs, Inorm,
include the neutron attenuation of the soil solid phase (µs), aluminum (µal), normal water (µH2O)
and deuterated water (µD2O). The neutron attenuations of aluminum and dry soil were determined
by scanning a slab filled with dry soil (Idry (x,y)). The attenuation coefficients of H2O (μH2O=3.65
cm-1) and D2O (μD2O=0.61 cm-1) were measured using control samples with known contents of
H2O and D2O. After subtraction of the contribution of the aluminum and dry soil, the remaining
values gave the neutron attenuation of H2O and D2O in the sample as
 I (x , y ) 
 log  norm
 H 2O d H 2O  D 2O d D 2O
 I (x , y ) 
dry


(S1.1)
where dH2O and dD2O are the thickness of H2O and D2O in the sample [cm].
For our convenience, we define
soil 
( H 2O d H 2O   D 2O d D 2O )
d tot
(S1.2)
which gives the neutron attenuation coefficient of the liquid phase across the thickness of the
sample (dtot). In the pixels containing roots, the value of µsoil is composed of attenuation
coefficients of the root and of the soil in front of and behind the root in the beam direction. The
actual contributions of H2O and D2O in the root, µroot, were calculated assuming that the amount
of H2O and D2O in soil in front of and behind of the root were equal to that of the soil at the
Supporting Information Methods S1 and S2
sides of the root (i.e. we assumed a radial symmetry around the roots). The pixel-wise neutron
attenuation coefficient in roots μroot is calculated as
root
I 
 log  root   s (1   )d root  soil d root
 I soil 

d root
(S1.3)
where Iroot is the value of Inorm in the most center pixel in the roots, Isoil is the average values of
Inorm in soil near to the roots, ϕ is the soil porosity, and droot is the root diameter. The attenuation
coefficient of root (μroot) depends on the volumetric content of H2O and D2O in the root. To
calculate the D2O content in roots we assumed that root swelling after D2O injection was
negligible. It follows that
D 2O
d root

root (t )  root (t  0)
d root
 D 2O   H 2O
(S1.4)
𝐷2𝑂
Where (𝑑𝑟𝑜𝑜𝑡
) is the thickness of D2O in roots, and t=0 refers to the radiograph before D2O
𝐷2𝑂
injection. The thickness of D2O in soil (𝑑𝑠𝑜𝑖𝑙
) is given by
D 2O
d soil

 soil   H 2O
d
 D 2O   H 2O soil
liq
(S1.5)
𝑙𝑖𝑞
𝑙𝑖𝑞
where (𝑑𝑠𝑜𝑖𝑙
) is the total liquid content in soil. To calculate 𝑑𝑠𝑜𝑖𝑙
we assumed that the liquid
content (H2O+D2O) inside the compartment quickly reaches a uniform distribution after
𝑙𝑖𝑞
injection. We calculated 𝑑𝑠𝑜𝑖𝑙
from the total volume of H2O and D2O divided by the area of the
compartment. We used the volumetric definition of D2O concentration in the root (Croot) and soil
(C soil) as the thickness of D2O divided by the total liquid thickness in root and soil,
respectively:
Supporting Information Methods S1 and S2
C root 
C soil 
D 2O
d root
liq
d root
(S1.6)
D 2O
d soil
liq
d soil
(S1.7)
𝑙𝑖𝑞
The total liquid thickness in root (𝑑𝑟𝑜𝑜𝑡
) is calculated as H2O thickness in the first radiograph
before
D2 O
was
injected.
Supporting Information Methods S1 and S2
Method S2 Derivation of the model of D2O transport into roots
We assume that when roots are immersed in D2O, the apoplastic free space of the root cortex
rapidly saturates with D2O. It follows that all cortical cells and the root enodermis are
simultaneously immersed in an identical concentration of D2O, which is equal to that of the soil
at the root surface, C0. In this way, the average concentration of D2O across the cortex
corresponds to that of a single cortical cell. To simplify the complex three-dimensional geometry
of the cortical tissue, we assumed that the cortical cells are long cylinders with a surface of
2πrcL - i.e. we assumed that the radius of the cortical cell is much smaller than their length.
Additionally, we assumed that the concentration inside the cell is uniform (well stirred
compartment). The reflection coefficient of membrane to D2O transport is assumed to be zero
(Henzler & Steudle 1995). Under these assumptions, the total flow of D2O into the cortical cells
𝑛
at nighttime, 𝐽𝑟,𝑐
[m3 s-1], is described by a diffusion equation (House, 1974)
J rn,c  2 rc LPD,c (C0  Cc )
(S2.1)
where rc [m] is the radius of the cortical cells, L [m] is the length of the root segment immersed
in D2O, PD,c [m s-1] is the diffusional permeability of the cortical cell, and Cc [-] is the D2O
concentration inside the cortical cell. From mass conservation it follows that
 rc2 L
Cc
 2 rc LPD ,c (C0  Cc )
t
(S2.2)
Under the boundary conditions
Cr  0
C0 (t )  C0
t 0
t 0
(S2.3)
Supporting Information Methods S1 and S2
the solution of Eq. (S2.2) is

Cc  C0 1  exp kc t
n

(S2.4)
where
kcn 
2 PD ,c
(S2.5)
rc
𝑛
Similarly, during the nighttime, the total flow of D2O into the root stele, 𝐽𝑟,𝑠
[m3 s-1], is
described by a diffusion equation as
J rn,s  2 Rs LPD,e (C0  Cs )
(S2.6)
where Rs [m] is the root radius, PD,e [m s-1] is the diffusional permeability of the endodermis, and
Cs [-] is the D2O concentration inside the root stele. Similar to the assumption for the cortical
cell, we assumed uniform concentration inside the stele (well stirred compartment). From the
mass conservation it follows that
 Rs2 L
Cs
 2 Rs LPD ,e (C0  Cs )
t
(S2.7)
Under the boundary conditions of Eq. (S2.3), the solution of Eq. (S2.7) is

Cs  C0 1  exp ks t
where
n

(S2.8)
Supporting Information Methods S1 and S2
ksn 
2 PD ,e
(S2.9)
Rs
The average D2O concentration in the root, Cr [-] is the sum of D2O concentration in the cortex
and stele:
Cr 
( Rr  Rs )Cc  Rs Cs
Rr
(S2.10)
where Rr [m] is the roots radius. By substituting Eq. (S2.4) and Eq. (S2.8) in Eq. (S2.10) it
follows that the increase of Cr at nighttime is
Cr 



n
n
Rr  Rs
R
C0 1  exp  kc t  s C0 1  exp  ks t
Rr
Rr

(S2.11)
During the daytime, transpiration induces a net flow of water into the roots. The convective
transport of D2O into the cortical cells depends on the ratio between the cell-to-cell water flow
and the total flow. We call λ [-] the ratio of the cell-to-cell to total water flow. The net flow of
𝑑
D2O into the cortical roots during daytime, 𝐽𝑟,𝑐
(m3 s-1), is described as
J rd,c  2 rc LPD ,c (C0  Cc )  
2 rc
Ljr (C0  Cc )
4
(S2.12)
where jr [m s-1] is the radial flux of water into the root calculated at the endodermis, and λ [-] is
the ratio between the cell-to-cell water flow and the total water flow across the cortex. The factor
1/4 in the right term of Eq. (S2.12) depends on the fact that the convective flow into each cortical
cell crosses approximately one fourth of the cortical cell perimeter.
From the mass balance it follows that
Supporting Information Methods S1 and S2
 rc2 L
Cc
2 rc
 2 rc LPD ,c (C0  Cc )  
Ljr (C0  Cc )
t
4
(S2.13)
Under the boundary conditions set in Eq. (S2.3), the solution of Eq. (S2.13) is

Cc  C0 1  exp kc t
d

(S2.14)
where
kcd 
2 PD ,c  
jr
2
(S2.15)
rc
𝑑
The net flow of D2O into the root stele during daytime, 𝐽𝑟,𝑠
[m3 s-1], is described by
J rd,s  2 Rs LPD,e (C0  Cs )  2 Rs Ljr C0   Rs2 jxout Cs
(S2.16)
where 𝑗𝑥𝑜𝑢𝑡 Are the axial flux [m s-1] calculated for the stele cross-section. In Eq. (S2.16) it is
assumed that the only the root segment of length L is immersed in D2O, while the other root
segments are immersed in normal water. From the mass conservation for CD2O it follows that
 Rs2 L
Cs
 2 Rs LPD ,e (C0  Cs )  2 Rs Ljr C0   Rs2 jxout Cs
t
(S2.17)
Under the boundary conditions set in Eq. (S2.3), the solution of Eq. (S2.17) is

Cs   C0 1  exp ks t
where
d

(S2.18)
Supporting Information Methods S1 and S2
k 
d
s
2 PD ,e
Rs
jxout

L
2 PD ,e  2 jr
Rs

2 PD ,e jxout

Rs
L
(S1.19)
By substituting Eq. (S2.14) and Eq. (S2.18) in Eq. (S2.10) it follows that the increase of Cr at
daytime is
Cr 



d
d
Rr  Rs
R
C0 1  exp  kc t  s  C0 1  exp  ks t
Rr
Rr

(S1.20)
Reference
House CR. 1974. Water transport in cells and tissues. London, UK: E. Arnold.