4. Effects of sand electrification on plants

Numerous studies have shown that the electric fields, as a physical stimulus, have a wide range of effects on the physiological processes of plant; for example, an appropriate strength of the electric field can affect cell proliferation, enzyme activity, biofilm permeability, and DNA synthesis, which will have an impact on plant growth and even improve the plant tolerance [21]. Some experiment also reported that the high-level electric field accelerated the drying of the hydrous plants [21]. In general, plants used to adapt a couple of abiotic effects like drought, ice, and sandstorm causing spectacular physical damage in plant tissues, especially the psammophyte. However, the psammophyte is the product of evolution, but others are more common, which would not adapt the damage from electric field, for example, the crop on the desertification land. In my knowledge, no any references concerned about the impact of windblown sand electric field on plant's physiological processes. In this chapter I want to discuss the effect of environmental electric field on the plant sap flow, the potential root water, and the cell membrane permeability.

#### 4.1. Effect on plant sap flow and root absorption

3. Wind-blown sand electric field

88 Community and Global Ecology of Deserts

Figure 2. The effect of wind-blown sand electric field on plant.

Figure 1. The effect of environmental electric field on the sand's electrification.

On the research of wind-blown sand electric field, Rudge firstly found that the atmospheric electric potential is obviously enhanced in the dusty weather and the direction is reversed [49]. Gill found the strong electric field and the spark phenomenon in the dust storm [50]. Freier found that the electric field in strong dust storm can be up to 60 kV/m [51]. Schmidt et al. measured the electric field in sand flow, and they found the electric field up to 160 kV/m at a distance of 5 cm from the ground [52]. These researchers just measured the vertical electric field. Jackson and Farrell found that the horizontal electric field of dust devil can be up to 120 kv/m [53]. Bo and Zheng found that the horizontal electric field is much larger than the vertical electric field [27]. Zhang et al. measured the electric field range from 0 to 30 m, in the sand

For the part between the crown and the root, which also named as xylem duct system, Parlange et al. proposed a theoretical model to simulate the sap flow of plant, and he suggest that the Poiseuille equation can be used to describe the sap flow in the stalk [55]. The driving force of the sap flow is the water potential differences between the plant root and the soil water. We set it as Δp. The water flux in a duct of the stem can be calculated through the Poiseuille equation:

$$Q = \frac{\pi r^4}{8\eta L} (\rho g \Delta h + \Delta p) \tag{8}$$

here, Q is the water flux, r is the duct radius, L is the duct length, Δh is the height difference of the duct, η is the fluid viscosity coefficient, and rgΔh represents the gravity of water in the duct.

It is clear that Eq. (8) does not consider the effect of environmental electric field. The plants in desertification land are located in a strong wind-blown sand electric field. The duct of the stem can be equivalent to tubules, the water in the duct can be simplified to a tiny conductor rod,

Figure 3. The plant in wind-sand electricity field.

and it must be affected by the electrostatic field force. So, we need to add one term in Eq. (8) to describe the effect from electric field (Figure 3).

If we set the vertical height of stem as 2L, the electric field is Eh, σ<sup>h</sup> is the polarization charge density, and the electric field force can be calculated through Eq. (9):

$$F\_E = \int\_{-L}^{L} \sigma\_\hbar E\_\hbar dh \tag{9}$$

v ¼ Q= πr

we need to find a related model to describe it.

so,

qXð Þ<sup>z</sup> (mm�<sup>3</sup>


the van den Honert's constant flow equation:

here, a can be thought as the radius of stem duct. After some mathematics calculation, we can obtain

(A) Soil water potential is linear function. We supposed the soil water potential is

The water potential at root cap ψXð Þ L is

d2 ψXð Þz

Eq. (18) can be used as the control equation of root suction model.

condition and climatic conditions and the plant with deep roots.

<sup>2</sup> <sup>¼</sup> <sup>r</sup>

the root absorption is RR, and the following relation is established:

here, z is the distance to the root tip; for the root tip, z ¼ 0.

2 ð Þ <sup>8</sup>η<sup>L</sup> �<sup>1</sup>

Based on Bernoulli's theory, the change of fluid velocity can induce the change of pressure. Those meanings of the electric field also can influence the root absorption process. Therefore,

Supposedly, the resistance of radial direction and the one of axial direction for root absorption are RR RX, and they all keep constant. In addition, supposedly the water potential inside and outside the root is ψsð Þz ,ψXð Þz , the water flux from soil to root is qRð Þz , the radial resistance for

The water potential gradient along the root equals to the axial flux of root water in unit length:

) is the volume flux of water from soil to root, which can be calculated through

dψXð Þz

dqXð Þz

dz<sup>2</sup> <sup>¼</sup> <sup>2</sup>πaRX RR

The water potential function ψ<sup>s</sup> must vary with the research region. For example, the linear function can be used for the region where plant with shallow roots and well irrigation, and the exponential function can be used to describe the region with large changes in geological

In here μ is an experimental constant, and we set it as 0.5. z is the distance from the soil surface.

½ � rgΔh þ Δp þ signð Þ E ∗FE (14)

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Sand Electrification Possibly Affects the Plant Physiology in Desertification Land

RRqRð Þ¼ z ψsð Þ� z ψXð Þz (15)

dz ¼ �RXqXð Þ<sup>z</sup> (16)

dz <sup>¼</sup> <sup>2</sup>πaqRð Þ<sup>z</sup> (17)

ψsð Þ¼ z μz þ ψ<sup>0</sup> (19)

<sup>ψ</sup>Xð Þ� <sup>z</sup> <sup>ψ</sup>sð Þ<sup>z</sup> (18)

Δh ¼ 2L. To simplify, we set that the environment electric field is equivalent to the field in the stem duct, and we directly use the experimental results of wind-blown sand electric field reported by Schmidt et al. [52]:

$$E\_{\rm h} = E\_0 h^{-0.6} \tag{10}$$

here, E<sup>0</sup> is a constant and h is the height from the ground. The charge on the water rod is [56]:

$$\sigma\_h = \frac{E\_h h}{\ln\left(4^{\left(L^2 - h^2\right)} / \_{d^2}\right) - 2} \tag{11}$$

Now, we can obtain the electric field force on the water rod:

$$F\_E = \int\_{-L}^{L} \frac{E\_0^2 h^{-0.2}}{\ln\left(4^{\left(L^2 - h^2\right)}/\_{d^2}\right) - 2} dh\tag{12}$$

Then, we can modify Eq. (8) to contain the effect of electric field:

$$Q = \frac{\pi \mathbf{r}^4}{8\eta L} [\rho \mathbf{g} \Delta h + \Delta p + \text{sign}(E) \ast F\_E] \tag{13}$$

The corresponding flow velocity is

$$\sigma = \mathbb{Q}/(\pi r^2) = r^2 (8\eta\mathcal{L})^{-1} [\rho\text{g}\Delta\!\!h + \Delta\!p + \text{sign}(E) \ast F\_E] \tag{14}$$

Based on Bernoulli's theory, the change of fluid velocity can induce the change of pressure. Those meanings of the electric field also can influence the root absorption process. Therefore, we need to find a related model to describe it.

Supposedly, the resistance of radial direction and the one of axial direction for root absorption are RR RX, and they all keep constant. In addition, supposedly the water potential inside and outside the root is ψsð Þz ,ψXð Þz , the water flux from soil to root is qRð Þz , the radial resistance for the root absorption is RR, and the following relation is established:

$$R\_R \eta\_R(z) = \psi\_s(z) - \psi\_X(z) \tag{15}$$

here, z is the distance to the root tip; for the root tip, z ¼ 0.

and it must be affected by the electrostatic field force. So, we need to add one term in Eq. (8) to

If we set the vertical height of stem as 2L, the electric field is Eh, σ<sup>h</sup> is the polarization charge

Δh ¼ 2L. To simplify, we set that the environment electric field is equivalent to the field in the stem duct, and we directly use the experimental results of wind-blown sand electric field

here, E<sup>0</sup> is a constant and h is the height from the ground. The charge on the water rod is [56]:

E0 2 <sup>h</sup>-0:<sup>2</sup>

ln <sup>4</sup> <sup>L</sup>2�h<sup>2</sup> ð Þ=a<sup>2</sup> � � � <sup>2</sup>

<sup>σ</sup><sup>h</sup> <sup>¼</sup> Ehh ln <sup>4</sup> <sup>L</sup>2�h<sup>2</sup> ð Þ=a<sup>2</sup>

σhEhdh (9)

<sup>E</sup><sup>h</sup> <sup>¼</sup> <sup>E</sup>0h�0:<sup>6</sup> (10)

� Þ � <sup>2</sup> (11)

<sup>8</sup>η<sup>L</sup> ½ � <sup>r</sup>gΔ<sup>h</sup> <sup>þ</sup> <sup>Δ</sup><sup>p</sup> <sup>þ</sup> signð Þ <sup>E</sup> <sup>∗</sup>FE (13)

dh (12)

FE ¼ ðL -L

describe the effect from electric field (Figure 3).

Figure 3. The plant in wind-sand electricity field.

90 Community and Global Ecology of Deserts

reported by Schmidt et al. [52]:

The corresponding flow velocity is

density, and the electric field force can be calculated through Eq. (9):

Now, we can obtain the electric field force on the water rod:

FE ¼ ðL -L

Then, we can modify Eq. (8) to contain the effect of electric field:

<sup>Q</sup> <sup>¼</sup> <sup>π</sup>r<sup>4</sup>

The water potential gradient along the root equals to the axial flux of root water in unit length: so,

$$\frac{d\psi\_X(z)}{dz} = -R\_X q\_X(z) \tag{16}$$

qXð Þ<sup>z</sup> (mm�<sup>3</sup> -s�<sup>1</sup> ) is the volume flux of water from soil to root, which can be calculated through the van den Honert's constant flow equation:

$$\frac{dq\_X(z)}{dz} = 2\pi a q\_R(z) \tag{17}$$

here, a can be thought as the radius of stem duct.

After some mathematics calculation, we can obtain

$$\frac{d^2\psi\_X(z)}{dz^2} = \frac{2\pi a R\_X}{R\_{\mathbb{R}}} \left[\psi\_X(z) - \psi\_s(z)\right] \tag{18}$$

Eq. (18) can be used as the control equation of root suction model.

The water potential function ψ<sup>s</sup> must vary with the research region. For example, the linear function can be used for the region where plant with shallow roots and well irrigation, and the exponential function can be used to describe the region with large changes in geological condition and climatic conditions and the plant with deep roots.

(A) Soil water potential is linear function.

We supposed the soil water potential is

$$
\psi\_s(z) = \mu z + \psi\_0 \tag{19}
$$

In here μ is an experimental constant, and we set it as 0.5. z is the distance from the soil surface. The water potential at root cap ψXð Þ L is

$$\left. \psi\_X(z) \right|\_{z=L} = \psi\_X(L) \tag{20}$$

Figure 5 showed the effect of electric field on the speed of stem flow, and we can see that the stem flow increases exponentially with the electric field increasing. From these two results, we can see that the effect of wind-blown sand on the plant is obvious than that of the clean wind.

Sand Electrification Possibly Affects the Plant Physiology in Desertification Land

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For the dielectric response of the cell media under the electric field, the weak conductor-coated spherical model proposed by Prodan et al. can be used directly [57]. We used it to simplify

4.2. Effect on the permeability of the cell membrane

expression derived by Di Biasio A. [57]:

Figure 4. The effect of stem parameters on the stem flow.

Figure 5. The effect of environmental electric field on the speed of stem flow.

At the influence of environmental electric field, the sap flow is accelerated. Considering the Bernoulli equation, the parameter ψXð Þ L can be predicted as follows:

$$
\psi\_X(L) = k/v^2 \tag{21}
$$

In here v is the velocity of the stem flow at z ¼ L, which can be calculated by Eq. (14). k is an experimental constant, and we set it as 0.5.

In addition, considering the cross section of root tip is too small, we set.

$$q\_X(0) = 0\tag{22}$$

Now, we can obtain that the water potential in root is

$$\psi\_X(z) = \left[\psi\_X(L) - \left(\mu z + \psi\_0\right)\right] \frac{\cosh(az)}{\cosh(aL)} + \frac{\mu}{a} \frac{\sinh(aL - az)}{\cosh(aL)} + \left(\mu z + \psi\_0\right) \tag{23}$$

(B) Soil water potential is exponential function.

We supposed the soil water potential is as follow:

$$
\psi\_s(\mathbf{z}) = \psi\_s(\mathbf{0}) \mathbf{e}^{-\mu z} \tag{24}
$$

μ is a constant. Then, we can obtain the water potential function in root:

$$q\_X(z) = \mathbf{A}e^{\alpha z} + \mathbf{B}e^{-\alpha z} - \frac{2\pi a\mu}{\left(\mu^2 - \alpha^2\right)R\_R}\Psi\_s(0)e^{-\mu z}$$

$$\frac{dq\_X(z)}{dz} = \mathbf{A}\alpha e^{\alpha z} - B\alpha e^{-\alpha z} + \frac{2\pi a\mu^2}{\left(\mu^2 - \alpha^2\right)R\_R}\Psi\_s(0)e^{-\mu z}$$

$$\psi\_x(z) = \psi\_x(0)e^{-\mu z} - \frac{R\_R}{2\pi a}\left[\mathbf{A}\alpha e^{\alpha z} - B\alpha e^{-\alpha z} + \frac{2\pi a\mu^2}{\left(\mu^2 - \alpha^2\right)R\_R}\Psi\_s(0)e^{-\mu z}\right] \tag{25}$$

$$\mathbf{G} = \frac{R\_R}{2\pi a}\left(\alpha e^{-\alpha L} + e^{\mu L}\right) \quad M = \frac{\mu\psi\_s(0)}{\mu^2 - \alpha^2}\left(\mu e^{-\mu L} + e^{\mu L}\right)$$

$$A = \frac{2\pi a\mu}{\left(\mu^2 - \alpha^2\right)R\_R}\psi\_S(0) - B \quad B = \frac{1}{G}\left[\psi\_s(\mathbf{L}) - \psi\_s(0)e^{-\mu L} + M\right]$$

Now, we want to make some discussion on them.

Firstly, we showed the effect of electric field on the stem flow speed, which is shown in Figure 4. From it we can see that the velocity of stem flow obviously increased when we consider the effect of the environmental electric field, and it also increases with the stem radius increasing, but it decreases with the stem length increasing.

Figure 5 showed the effect of electric field on the speed of stem flow, and we can see that the stem flow increases exponentially with the electric field increasing. From these two results, we can see that the effect of wind-blown sand on the plant is obvious than that of the clean wind.

#### 4.2. Effect on the permeability of the cell membrane

ψXð Þz � �

Bernoulli equation, the parameter ψXð Þ L can be predicted as follows:

In addition, considering the cross section of root tip is too small, we set.

� � � � coshð Þ αz

μ is a constant. Then, we can obtain the water potential function in root:

dz <sup>¼</sup> <sup>A</sup>αe<sup>α</sup><sup>z</sup> � <sup>B</sup>αe�α<sup>z</sup> <sup>þ</sup>

2πa

qXð Þ¼ <sup>z</sup> Ae<sup>α</sup><sup>z</sup> <sup>þ</sup> <sup>B</sup>e�α<sup>z</sup> � <sup>2</sup>πa<sup>μ</sup>

experimental constant, and we set it as 0.5.

92 Community and Global Ecology of Deserts

Now, we can obtain that the water potential in root is

ψXð Þ¼ z ψXð Þ� L μz þ ψ<sup>0</sup>

(B) Soil water potential is exponential function.

We supposed the soil water potential is as follow:

dqXð Þz

<sup>ψ</sup>xð Þ¼ <sup>z</sup> <sup>ψ</sup>xð Þ<sup>0</sup> <sup>e</sup>�μ<sup>z</sup> � RR

<sup>G</sup> <sup>¼</sup> RR 2πa αe �α<sup>L</sup> <sup>þ</sup> <sup>e</sup>

<sup>A</sup> <sup>¼</sup> <sup>2</sup>πa<sup>μ</sup> <sup>μ</sup><sup>2</sup> � <sup>α</sup><sup>2</sup> � �RR

Now, we want to make some discussion on them.

increasing, but it decreases with the stem length increasing.

At the influence of environmental electric field, the sap flow is accelerated. Considering the

In here v is the velocity of the stem flow at z ¼ L, which can be calculated by Eq. (14). k is an

coshð Þ <sup>α</sup><sup>L</sup> <sup>þ</sup> <sup>μ</sup>

α

<sup>μ</sup><sup>2</sup> � <sup>α</sup><sup>2</sup> � �RR

<sup>A</sup>αe<sup>α</sup><sup>z</sup> � <sup>B</sup>αe�α<sup>z</sup>

Firstly, we showed the effect of electric field on the stem flow speed, which is shown in Figure 4. From it we can see that the velocity of stem flow obviously increased when we consider the effect of the environmental electric field, and it also increases with the stem radius

<sup>α</sup><sup>L</sup> � � <sup>M</sup> <sup>¼</sup> μψsð Þ<sup>0</sup>

<sup>ψ</sup>Sð Þ� <sup>0</sup> B B <sup>¼</sup> <sup>1</sup>

2πaμ<sup>2</sup> <sup>μ</sup><sup>2</sup> � <sup>α</sup><sup>2</sup> � �RR

þ

<sup>μ</sup><sup>2</sup> � <sup>α</sup><sup>2</sup> <sup>μ</sup><sup>e</sup>

<sup>z</sup>¼<sup>L</sup> <sup>¼</sup> <sup>ψ</sup>Xð Þ <sup>L</sup> (20)

<sup>ψ</sup>Xð Þ¼ <sup>L</sup> <sup>k</sup>=v<sup>2</sup> (21)

qXð Þ¼ 0 0 (22)

coshð Þ <sup>α</sup><sup>L</sup> <sup>þ</sup> <sup>μ</sup><sup>z</sup> <sup>þ</sup> <sup>ψ</sup><sup>0</sup>

<sup>ψ</sup>sð Þ¼ <sup>z</sup> <sup>ψ</sup>sð Þ<sup>0</sup> <sup>e</sup>�μ<sup>z</sup> (24)

Ψsð Þ0 e �μz

2πaμ<sup>2</sup> <sup>μ</sup><sup>2</sup> � <sup>α</sup><sup>2</sup> � �RR

" #

Ψsð Þ0 e �μz

�μ<sup>L</sup> <sup>þ</sup> <sup>e</sup> <sup>α</sup><sup>L</sup> � �

<sup>G</sup> <sup>ψ</sup>sð Þ� <sup>L</sup> <sup>ψ</sup>sð Þ<sup>0</sup> <sup>e</sup>�μ<sup>L</sup> <sup>þ</sup> <sup>M</sup> � �

Ψsð Þ0 e �μz

� � (23)

(25)

sinhð Þ αL � αz

For the dielectric response of the cell media under the electric field, the weak conductor-coated spherical model proposed by Prodan et al. can be used directly [57]. We used it to simplify expression derived by Di Biasio A. [57]:

Figure 4. The effect of stem parameters on the stem flow.

Figure 5. The effect of environmental electric field on the speed of stem flow.

$$\psi(r,\theta) = \left\{-rE\_0 + \Im R\_1^2 E\_0 \left[p\_1 \nu(r, R\_1) + p\_2 \nu(r, R\_2)\right]\right\} \cos\theta \tag{26}$$

Φ ¼ 3 p<sup>1</sup>

is the Faraday constant, and <sup>∂</sup><sup>E</sup>

Eq. (28) can be changed:

The thickness of cell membrane is Δx:

Do some mathematic operation:

Then

And then

So, we obtain the ion flux:

R2 <sup>3</sup> � <sup>R</sup><sup>1</sup> 3 � �

and outside the cell. For the j � th ion, the net flux is

the second term is stem from the potential gradient.

Then, the last equation can be changed as follows:

þ p<sup>2</sup>

Jj <sup>¼</sup> ujRT γj

R2 <sup>3</sup> � <sup>R</sup><sup>2</sup> 1 3R<sup>2</sup>

The charged ion can be partially transported across the cell membrane and exchanged inside

here, uj is the ion mobility, R is the thermodynamic constant, T is the temperature, γ<sup>j</sup> is the active coefficient in a solution, cj is the ion concentration, zj is the number of valence electron, F

In general, the first term repressed the ion flux originate from the concentration gradient, and

If we supposed that the active coefficient inside and outside the cell keeps a constant, then

∂cj

<sup>∂</sup><sup>x</sup> � ujcjzjF <sup>∂</sup><sup>ϕ</sup>

<sup>Δ</sup><sup>x</sup> <sup>¼</sup> <sup>Φ</sup> Δx

dx ¼ � dcj

dx ¼ � <sup>ð</sup><sup>c</sup><sup>i</sup> j c0 j

> c0 <sup>j</sup> <sup>þ</sup> JjΔ<sup>x</sup> ujzjFΦ

ci <sup>j</sup> <sup>þ</sup> JjΔ<sup>x</sup> ujzjFΦ

zjFΦ RT � � <sup>c</sup>

i <sup>j</sup> þ

JjΔx ujzjFΦ � �

cj <sup>þ</sup> JjΔ<sup>x</sup> ujzjFΦ

> dcj cj <sup>þ</sup> JjΔ<sup>x</sup> ujzjFΦ

<sup>∂</sup><sup>x</sup> is the potential gradient.

Jj ¼ ujRT

∂E <sup>∂</sup><sup>x</sup> <sup>¼</sup> Δϕ

zjFΦ RTΔx

zjFΦ RTΔx

> zjFΦ RT <sup>¼</sup> ln

JjΔx ujzjF<sup>Φ</sup> <sup>¼</sup> exp

ð<sup>Δ</sup><sup>x</sup> 0

c 0 <sup>j</sup> þ <sup>∂</sup><sup>x</sup> � ujcjzjF <sup>∂</sup><sup>E</sup>

∂γj cj

� � � � <sup>E</sup>0cos<sup>θ</sup> <sup>þ</sup> <sup>E</sup>0ð Þ <sup>R</sup><sup>1</sup> � <sup>R</sup><sup>2</sup> cos<sup>θ</sup> (27)

Sand Electrification Possibly Affects the Plant Physiology in Desertification Land

<sup>∂</sup><sup>x</sup> (28)

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<sup>∂</sup><sup>x</sup> (29)

In here <sup>p</sup><sup>1</sup> <sup>¼</sup> <sup>C</sup>�<sup>B</sup> AC�<sup>B</sup> , p<sup>2</sup> <sup>¼</sup> <sup>A</sup>�<sup>1</sup> AC�<sup>B</sup> , and <sup>D</sup><sup>0</sup> <sup>k</sup> <sup>¼</sup> <sup>2</sup>Dk <sup>þ</sup> <sup>i</sup>ωR<sup>2</sup> <sup>k</sup> , k ¼ 1, 2. This expression is based on the spherical coordinates, r is radial component, and θ is azimuth angle (Figure 6):

$$A = \frac{1 + \frac{D\_1^\flat}{2R\_{1\uparrow\uparrow}} \left(\varepsilon\_1^\* + 2\varepsilon\_0^\*\right)}{1 + \frac{D\_1^\flat}{2R\_{1\uparrow\uparrow}} \left(\varepsilon\_1^\* - \varepsilon\_0^\*\right)}\\B = \frac{R\_2}{R\_1} \frac{1 - 2\frac{D\_1^\flat}{2R\_{1\uparrow\uparrow}} \left(\varepsilon\_1^\* - \varepsilon\_0^\*\right)}{1 + \frac{D\_1^\flat}{2R\_{1\uparrow\uparrow}} \left(\varepsilon\_1^\* - \varepsilon\_0^\*\right)}\\C = \frac{R\_1^2}{R\_2^2} \frac{1 + \frac{D\_2^\flat}{2R\_{1\uparrow\uparrow}} \left(\varepsilon\_2^\* + 2\varepsilon\_1^\*\right)}{1 + \frac{D\_2^\flat}{2R\_{1\uparrow\uparrow}} \left(\varepsilon\_2^\* - \varepsilon\_1^\*\right)}$$

$$\nu(r, R\_1) = \frac{R\_1}{3r^2} \quad \nu(r, R\_2) = \frac{R\_2}{3r^2} \qquad (r > R\_1)$$

$$\nu(r, R\_1) = \frac{r}{3R\_1^{-2}} \qquad \nu(r, R\_2) = \frac{R\_2}{3r^2} \quad (R\_1 > r > R\_2)$$

$$\nu(r, R\_1) = \frac{r}{3R\_1^{-2}} \qquad \nu(r, R\_2) = \frac{r}{3R\_2^{-2}} \quad (r < R\_2)$$

The electric potential difference between inside and outside of the cell is

$$\begin{aligned} \Phi &= \psi(R\_1, \theta) - \psi(R\_2, \theta) \\\\ &= 3R\_1^2 \left[ p\_1 \left( \frac{R\_2}{3R\_1^2} - \frac{1}{3R\_1} \right) - p\_2 \left( \frac{1}{3R\_2} - \frac{R\_2}{3R\_1^2} \right) \right] E\_0 \cos\theta + E\_0 (R\_1 - R\_2) \cos\theta \end{aligned}$$

After further simplification, we obtain

Figure 6. The coated spherical cell model and its physical parameters.

$$\Phi = 3\left[p\_1\left(\frac{R\_2}{3} - \frac{R\_1}{3}\right) + p\_2\left(\frac{R\_2}{3} - \frac{R\_1^2}{3R\_2}\right)\right]E\_0\cos\theta + E\_0(R\_1 - R\_2)\cos\theta\tag{27}$$

The charged ion can be partially transported across the cell membrane and exchanged inside and outside the cell. For the j � th ion, the net flux is

$$J\_{\dot{j}} = \frac{u\_{\dot{j}} \mathcal{R} T}{\gamma\_{\dot{j}}} \frac{\partial \gamma\_{\dot{j}} c\_{\dot{j}}}{\partial \mathbf{x}} - u\_{\dot{j}} c\_{\dot{j}} z\_{\dot{j}} F \frac{\partial E}{\partial \mathbf{x}} \tag{28}$$

here, uj is the ion mobility, R is the thermodynamic constant, T is the temperature, γ<sup>j</sup> is the active coefficient in a solution, cj is the ion concentration, zj is the number of valence electron, F is the Faraday constant, and <sup>∂</sup><sup>E</sup> <sup>∂</sup><sup>x</sup> is the potential gradient.

In general, the first term repressed the ion flux originate from the concentration gradient, and the second term is stem from the potential gradient.

If we supposed that the active coefficient inside and outside the cell keeps a constant, then Eq. (28) can be changed:

$$J\_j = \mu\_j RT \frac{\partial c\_j}{\partial \mathbf{x}} - \mu\_j c\_j \mathbf{z}\_j F \frac{\partial \phi}{\partial \mathbf{x}} \tag{29}$$

The thickness of cell membrane is Δx:

$$\frac{\partial E}{\partial \mathbf{x}} = \frac{\Delta \phi}{\Delta \mathbf{x}} = \frac{\Phi}{\Delta \mathbf{x}}$$

Then, the last equation can be changed as follows:

$$\frac{z\_{\vec{j}}F\Phi}{RT\Delta x}d\mathbf{x} = -\frac{dc\_{\vec{j}}}{c\_{\vec{j}} + \frac{l\_{\vec{j}}\Delta x}{u\_{\vec{j}}F\Phi}}$$

Do some mathematic operation:

$$\int\_{0}^{\Delta x} \frac{z\_{j}F\Phi}{RT\Delta x} d\mathbf{x} = -\int\_{c\_{j}^{0}}^{c\_{j}^{i}} \frac{dc\_{j}}{c\_{j} + \frac{f\_{j}\Delta x}{u\_{j}z\_{j}F\Phi}}$$

Then

<sup>ψ</sup>ð Þ¼ � <sup>r</sup>; <sup>θ</sup> rE<sup>0</sup> <sup>þ</sup> <sup>3</sup>R<sup>2</sup>

AC�<sup>B</sup> , and <sup>D</sup><sup>0</sup>

νð Þ¼ r;R<sup>1</sup>

νð Þ¼ r;R<sup>1</sup>

νð Þ¼ r;R<sup>1</sup>

Φ ¼ ψð Þ� R1, θ ψð Þ R2, θ

R2 3R<sup>2</sup> 1 � <sup>1</sup> 3R<sup>1</sup>

Figure 6. The coated spherical cell model and its physical parameters.

!

<sup>¼</sup> <sup>3</sup>R<sup>2</sup> <sup>1</sup> p<sup>1</sup>

After further simplification, we obtain

In here <sup>p</sup><sup>1</sup> <sup>¼</sup> <sup>C</sup>�<sup>B</sup>

AC�<sup>B</sup> , p<sup>2</sup> <sup>¼</sup> <sup>A</sup>�<sup>1</sup>

1 2R1γ<sup>1</sup> ε∗ <sup>1</sup> <sup>þ</sup> <sup>2</sup>ε<sup>∗</sup> 0 � �

<sup>A</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>D</sup>'

94 Community and Global Ecology of Deserts

<sup>1</sup> <sup>þ</sup> <sup>D</sup>' 1 2R1γ<sup>1</sup> ε∗ <sup>1</sup> � <sup>ε</sup><sup>∗</sup> 0 � � <sup>1</sup>E<sup>0</sup> p1νð Þþ r;R<sup>1</sup> p2νð Þ r;R<sup>2</sup>

<sup>k</sup> <sup>¼</sup> <sup>2</sup>Dk <sup>þ</sup> <sup>i</sup>ωR<sup>2</sup>

1-2 <sup>D</sup>' 1 2R1γ<sup>1</sup> ε∗ <sup>1</sup> � <sup>ε</sup><sup>∗</sup> 0 � �

<sup>1</sup> <sup>þ</sup> <sup>D</sup>' 1 2R1γ<sup>1</sup> ε∗ <sup>1</sup> � <sup>ε</sup><sup>∗</sup> 0 � �

<sup>2</sup> νð Þ¼ r;R<sup>2</sup>

<sup>2</sup> νð Þ¼ r;R<sup>2</sup>

1 3R<sup>2</sup> � <sup>R</sup><sup>2</sup> 3R<sup>2</sup> 1

R2

R2

r 3R<sup>2</sup>

<sup>3</sup>r<sup>2</sup> ð Þ <sup>r</sup> <sup>&</sup>gt; <sup>R</sup><sup>1</sup>

<sup>3</sup>r<sup>2</sup> <sup>ν</sup>ð Þ¼ <sup>r</sup>; <sup>R</sup><sup>2</sup>

spherical coordinates, r is radial component, and θ is azimuth angle (Figure 6):

<sup>B</sup> <sup>¼</sup> <sup>R</sup><sup>2</sup> R1

R1

r 3R<sup>1</sup>

r 3R<sup>1</sup>

The electric potential difference between inside and outside of the cell is

� p<sup>2</sup>

" # !

� � � � cosθ (26)

<sup>C</sup> <sup>¼</sup> <sup>R</sup><sup>2</sup> 1 R2 2

<sup>3</sup>r<sup>2</sup> ð Þ <sup>R</sup><sup>1</sup> <sup>&</sup>gt; <sup>r</sup> <sup>&</sup>gt; <sup>R</sup><sup>2</sup>

<sup>2</sup> ð Þ r < R<sup>2</sup>

<sup>k</sup> , k ¼ 1, 2. This expression is based on the

<sup>1</sup> <sup>þ</sup> <sup>D</sup>' 2 2R2γ<sup>2</sup> ε∗ <sup>2</sup> <sup>þ</sup> <sup>2</sup>ε<sup>∗</sup> 1 � �

<sup>1</sup> <sup>þ</sup> <sup>D</sup>' 2 2R2γ<sup>2</sup> ε∗ <sup>2</sup> � <sup>ε</sup><sup>∗</sup> 1 � �

E0cosθ þ E0ð Þ R<sup>1</sup> � R<sup>2</sup> cosθ

$$\frac{z\_{j}F\Phi}{RT} = \ln \frac{c\_{j}^{0} + \frac{f\_{j}\Delta x}{u\_{j}z\_{j}F\Phi}}{c\_{j}^{i} + \frac{f\_{j}\Delta x}{u\_{j}z\_{j}F\Phi}}$$

And then

$$c\_j^0 + \frac{J\_j \Delta x}{\mu\_j z\_j F \Phi} = \exp\left(\frac{z\_j F \Phi}{RT}\right) \left[c\_j^i + \frac{J\_j \Delta x}{\mu\_j z\_j F \Phi}\right].$$

So, we obtain the ion flux:

$$J\_{j} = \frac{\mu\_{j} z\_{j} F \Phi}{\Delta x} \frac{1}{\left[ \exp\left(\frac{z\_{j} F \Phi}{RT}\right) - 1 \right]} \left[ c\_{j}^{0} + c\_{j}^{i} \exp\left(\frac{z\_{j} F \Phi}{RT}\right) \right] \tag{30}$$

5. Conclusions and perspective

Acknowledgements

from the reviewers and the editor.

frequency of electric field on the plant physiology process.

Figure 8. Ion flux changed with the azimuth angle (θ) of cell in spherical coordinate.

This chapter discussed the electrification of sand flow and proposed a simple physical model to reveal the mechanism of it. In addition, we also further discussed the effect of wind-blown sand electric field on the physiological process of plants. The simulation results showed that the electric field strongly enhanced the sap flow and root-water-uptake rate, and the permeability of the cell membrane also changed and then influences the growth of plants. However, these results are derived from the theoretical model, so we hope someone can carry out a series of relevant experimental studies to verify them. In addition, there is a lack of detailed experiment and discussion on the frequency of wind-blown sand electric field and on the effect of

Sand Electrification Possibly Affects the Plant Physiology in Desertification Land

http://dx.doi.org/10.5772/intechopen.74976

97

We acknowledge support from the National Natural Science Foundation of China (Grant Nos. 11562017 and 11302111), and the Major Innovation Projects for Building First-Class Universities in China's Western Region (Grant No. ZKZD2017006). We also appreciate the comments

Now, we will discuss the effect of environmental electric field on the ion flux in and out of the cell. The cell radius R<sup>1</sup> ¼ 1μm, the thickness of cell membrane d ¼ 7nm, its dielectric constant, and conductivity are ε<sup>1</sup> ¼ 150, ε<sup>2</sup> ¼ 50, ε<sup>0</sup> ¼ 80, σ<sup>1</sup> ¼ 0:15, σ<sup>2</sup> ¼ 0:2, and σ<sup>0</sup> ¼ 0:1; then the complex permittivity ε<sup>∗</sup> <sup>i</sup> <sup>¼</sup> <sup>ε</sup><sup>i</sup> <sup>þ</sup> <sup>σ</sup>i=ð Þ <sup>i</sup>εm<sup>ω</sup> , <sup>i</sup> <sup>¼</sup> <sup>1</sup>, 2, and <sup>ε</sup><sup>m</sup> <sup>¼</sup> <sup>8</sup>:<sup>857</sup> � <sup>10</sup>�12; the frequency of incident electric field <sup>ω</sup> <sup>¼</sup> 103 Hz, and the temperature T ¼ 300K. In Figure 7 we just showed the results, while θ ¼ 0. From it we can see that with the increasing of electric field, the negative ion flux increased, but the positive ion flux decreases, and the number of valence electron is larger; the influence is more obvious.

Figure 8 showed that the ion flux changed with the azimuth angle; from it we can see that with the increasing of azimuth angle, the net flux of positively charged ion became increased, but the one for negative valence ions decreases. In addition, with the increase of the number of valence electrons, the net flux of negative ions in the upper part of the cell is increasing, but the net flux of positive ions is constantly disappearing, which is opposite for the lower part of the cell. This is due to the opposite polarization charge in the upper and lower parts of the cell.

Figure 7. Effect of electric field on ion flux while it is with different polarity charges.

Sand Electrification Possibly Affects the Plant Physiology in Desertification Land http://dx.doi.org/10.5772/intechopen.74976 97

Figure 8. Ion flux changed with the azimuth angle (θ) of cell in spherical coordinate.

## 5. Conclusions and perspective

Jj <sup>¼</sup> ujzjF<sup>Φ</sup> Δx

complex permittivity ε<sup>∗</sup>

lower parts of the cell.

incident electric field <sup>ω</sup> <sup>¼</sup> 103

96 Community and Global Ecology of Deserts

electron is larger; the influence is more obvious.

Figure 7. Effect of electric field on ion flux while it is with different polarity charges.

1 exp zjF<sup>Φ</sup> RT � �

� 1 h i <sup>c</sup>

Now, we will discuss the effect of environmental electric field on the ion flux in and out of the cell. The cell radius R<sup>1</sup> ¼ 1μm, the thickness of cell membrane d ¼ 7nm, its dielectric constant, and conductivity are ε<sup>1</sup> ¼ 150, ε<sup>2</sup> ¼ 50, ε<sup>0</sup> ¼ 80, σ<sup>1</sup> ¼ 0:15, σ<sup>2</sup> ¼ 0:2, and σ<sup>0</sup> ¼ 0:1; then the

the results, while θ ¼ 0. From it we can see that with the increasing of electric field, the negative ion flux increased, but the positive ion flux decreases, and the number of valence

Figure 8 showed that the ion flux changed with the azimuth angle; from it we can see that with the increasing of azimuth angle, the net flux of positively charged ion became increased, but the one for negative valence ions decreases. In addition, with the increase of the number of valence electrons, the net flux of negative ions in the upper part of the cell is increasing, but the net flux of positive ions is constantly disappearing, which is opposite for the lower part of the cell. This is due to the opposite polarization charge in the upper and

0 <sup>j</sup> þ c i j exp zjFΦ RT

(30)

� � � �

<sup>i</sup> <sup>¼</sup> <sup>ε</sup><sup>i</sup> <sup>þ</sup> <sup>σ</sup>i=ð Þ <sup>i</sup>εm<sup>ω</sup> , <sup>i</sup> <sup>¼</sup> <sup>1</sup>, 2, and <sup>ε</sup><sup>m</sup> <sup>¼</sup> <sup>8</sup>:<sup>857</sup> � <sup>10</sup>�12; the frequency of

Hz, and the temperature T ¼ 300K. In Figure 7 we just showed

This chapter discussed the electrification of sand flow and proposed a simple physical model to reveal the mechanism of it. In addition, we also further discussed the effect of wind-blown sand electric field on the physiological process of plants. The simulation results showed that the electric field strongly enhanced the sap flow and root-water-uptake rate, and the permeability of the cell membrane also changed and then influences the growth of plants. However, these results are derived from the theoretical model, so we hope someone can carry out a series of relevant experimental studies to verify them. In addition, there is a lack of detailed experiment and discussion on the frequency of wind-blown sand electric field and on the effect of frequency of electric field on the plant physiology process.

#### Acknowledgements

We acknowledge support from the National Natural Science Foundation of China (Grant Nos. 11562017 and 11302111), and the Major Innovation Projects for Building First-Class Universities in China's Western Region (Grant No. ZKZD2017006). We also appreciate the comments from the reviewers and the editor.
