1 Introduction
The interactions between charged macroscopic dust particles immersed in a plasma defines the properties of a strongly coupled complex plasma. Often, the complex plasma is subjected to an external electric field (for example, in the plasma sheath, in the positive column etc.) and becomes anisotropic. The anisotropic interaction between the solid particles in a plasma is exhibited in the formation of inhomogeneous dust structures such as string-like structures (Ivlev et al. Reference Ivlev, Morfill, Thomas, Räth, Joyce, Huber, Kompaneets, Fortov, Lipaev and Molotkov2008, Reference Ivlev, Thomas, Räth, Joyce and Morfill2011; Mitic et al. Reference Mitic, Klumov, Khrapak and Morfill2013). The interaction caused by ion flow perturbations is non-reciprocal and can be responsible for heating and instabilities in the dusty plasma (see Melzer Reference Melzer2001; Ivlev et al. Reference Ivlev, Bartnick, Heinen, Du, Nosenko and Löwen2015). When a negatively charged particle is placed in a flow of positive ions, an excess of positive charge is formed downstream of the particle due to the ion focusing (see Melandsø & Goree Reference Melandsø and Goree1995; Lampe et al. Reference Lampe, Joyce, Gaunguli and Gavrishchaka2000; Lapenta Reference Lapenta2000).
In the collisionless case, the anisotropic potential decays at large distances as 
$1/r^{3}$, where 
$r$ is the distance from the particle (see Montgomery, Joyce & Sugihara Reference Montgomery, Joyce and Sugihara1968; Kompaneets, Morfill & Ivlev Reference Kompaneets, Morfill and Ivlev2016). Collisions of the ions with neutrals change the asymptote and give a slower decaying component 
${\sim}1/r^{2}$ at distances larger than an ion mean free path length (Stenflo, Yu & Shukla Reference Stenflo, Yu and Shukla1973; Kompaneets et al. Reference Kompaneets, Konopka, Ivlev, Tsytovich and Morfill2007, Reference Kompaneets, Morfill and Ivlev2016). In a weakly ionised collision-dominated flowing plasma containing electrons and positive ions as mobile charges, the long-range potential around an absorbing charged particle consists of a dipole-like component and an isotropic Coulomb-like one caused by absorption, which is usually dominant (see Chaudhuri, Khrapak & Morfill Reference Chaudhuri, Khrapak and Morfill2007; Filippov et al. Reference Filippov, Zagorodny, Pal’, Starostin and Momot2007; Chaudhuri, Khrapak & Morfill Reference Chaudhuri, Khrapak and Morfill2010). The Coulomb-like potential is a field of the ambipolar diffusion on the dust grain. It is isotropic in a uniform plasma with constant ion mobility (the case of a field-dependent mobility has been considered by Zobnin (Reference Zobnin2018)).
The situation drastically changes in the presence of negative ions in the plasma. The anisotropic part of the potential around the negatively charged absorbing particle decreases with distance as 
$1/r$ in the presence of the negative ion flow, as will be derived below.
2 Theory
Let us consider a small negatively charged absorbing particle in a weakly ionised quasi-neutral collisional plasma containing electrons, single charged positive ions and single charged negative ions, under the action of a uniform external electric field 
$\boldsymbol{E}$. We assume that both kinds of ions have the same temperature 
$T_{i}$ (drifting velocities are sub-thermal), while the temperature of the electrons 
$T_{e}$ may be differ. Assuming that the drift–diffusion approach is valid at the distances of interest and neglecting ionisation and space recombination processes we can write the flow continuity equations for unperturbed densities of the electrons, negative and positive ions as
$$\begin{eqnarray}\unicode[STIX]{x1D735}(z_{\unicode[STIX]{x1D6FC}}n_{\unicode[STIX]{x1D6FC}}\boldsymbol{E}-(k_{B}T_{\unicode[STIX]{x1D6FC}}/e)\unicode[STIX]{x1D735}n_{\unicode[STIX]{x1D6FC}})=0,\end{eqnarray}$$ where the index 
$\unicode[STIX]{x1D6FC}$ denotes ‘e’, ‘-’ or ‘+’ for the electrons, negative ions and positive ions, respectively, 
$z_{\unicode[STIX]{x1D6FC}}$ is the charge number, 
$z_{\text{e}}=z_{-}=-1$ for the electrons and the negative ions and 
$z_{+}=1$ for the positive ions, 
$k_{B}$ is the Boltzmann constant, 
$e$ is the elementary charge. The total electric field can be split into the uniform external field and an additional potential distribution 
$\unicode[STIX]{x1D711}(r)$ staying finite at infinite distances.
The flow continuity equations in the presence of a small absorbing particle are
$$\begin{eqnarray}\unicode[STIX]{x1D735}(z_{\unicode[STIX]{x1D6FC}}(n_{\unicode[STIX]{x1D6FC}}+\hat{n}_{\unicode[STIX]{x1D6FC}})(\boldsymbol{E}-\unicode[STIX]{x1D735}\unicode[STIX]{x1D711}))-(k_{B}T_{\unicode[STIX]{x1D6FC}}/e)\unicode[STIX]{x0394}(n_{\unicode[STIX]{x1D6FC}}+\hat{n}_{\unicode[STIX]{x1D6FC}})=-(F_{\unicode[STIX]{x1D6FC}}/b_{\unicode[STIX]{x1D6FC}})\unicode[STIX]{x1D6FF}(\boldsymbol{r}),\end{eqnarray}$$ where hats over 
$n_{\unicode[STIX]{x1D6FC}}$ denote small perturbations of the corresponding number densities, 
$F_{e}=F_{+}$ are the fluxes of electrons and positive ions on the particle (flux of the negative ions is absent, 
$F_{-}=0$), 
$b_{\unicode[STIX]{x1D6FC}}$ is the mobility of the corresponding species and 
$\unicode[STIX]{x1D6FF}(\boldsymbol{r})$ is the three-dimensional delta function, 
$\boldsymbol{r}$ is the radius vector from the particle centre. The point-like sink ascribed to the delta function has been used early in a number of papers (Chaudhuri et al. Reference Chaudhuri, Khrapak and Morfill2007; Filippov et al. Reference Filippov, Zagorodny, Pal’, Starostin and Momot2007; Khrapak, Klumov & Mofill Reference Khrapak, Klumov and Mofill2008) for the description of the electric field around a small absorbing particle. Neglecting the terms 
$\unicode[STIX]{x1D735}(\hat{n}_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D735}\unicode[STIX]{x1D711})$ and taking into account (2.1) we derive the linearised equations
$$\begin{eqnarray}z_{\unicode[STIX]{x1D6FC}}((\unicode[STIX]{x1D735}\hat{n}_{\unicode[STIX]{x1D6FC}}\boldsymbol{\cdot }\boldsymbol{E})-n_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x0394}\unicode[STIX]{x1D711})-(k_{B}T_{\unicode[STIX]{x1D6FC}}/e)\unicode[STIX]{x0394}\hat{n}_{\unicode[STIX]{x1D6FC}}=-(F_{\unicode[STIX]{x1D6FC}}/b_{\unicode[STIX]{x1D6FC}})\unicode[STIX]{x1D6FF}(\boldsymbol{r}).\end{eqnarray}$$Poisson’s equation is
$$\begin{eqnarray}-\unicode[STIX]{x1D700}_{0}\unicode[STIX]{x0394}\unicode[STIX]{x1D711}=e(\hat{n}_{+}-\hat{n}_{-}-\hat{n}_{\text{e}}-z_{d}\unicode[STIX]{x1D6FF}(\boldsymbol{r})),\end{eqnarray}$$ where 
$\unicode[STIX]{x1D700}_{0}$ is the electric constant and 
$z_{d}$ is the charge number of the particle. After the Fourier transformation of (2.3) and (2.4) in a standard way we have
$$\begin{eqnarray}\unicode[STIX]{x1D711}(\boldsymbol{k})=\frac{-ez_{d}-eJ(\unicode[STIX]{x1D712}_{+}-\unicode[STIX]{x1D712}_{\text{e}}n_{+}b_{+}/(b_{\text{e}}n_{\text{e}}))}{(2\unicode[STIX]{x03C0})^{3/2}\unicode[STIX]{x1D700}_{0}\boldsymbol{k}^{2}(1+\unicode[STIX]{x1D712}_{+}+\unicode[STIX]{x1D712}_{-}+\unicode[STIX]{x1D712}_{\text{e}})},\end{eqnarray}$$ where 
$\unicode[STIX]{x1D712}_{+}=k_{D+}^{2}/(\boldsymbol{k}^{2}+\text{i}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{k}_{E}))$, 
$\unicode[STIX]{x1D712}_{-}=k_{D-}^{2}/(\boldsymbol{k}^{2}-\text{i}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{k}_{E}))$, 
$\unicode[STIX]{x1D712}_{\text{e}}=k_{De}^{2}/(\boldsymbol{k}^{2}-\text{i}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{k}_{E})/\unicode[STIX]{x1D70F})$ are the susceptibilities, 
$J=F_{+}\unicode[STIX]{x1D700}_{0}/(en_{+}b_{+})$ is the dimensionless sink, 
$k_{D+}$, 
$k_{D-}$ and 
$k_{De}$ are the inverse Debye radii for the positive ions, negative ions and electrons, respectively, 
$\boldsymbol{k}_{E}$ is the normalised external electric field or the gas-sound Mach number divided by the ion mean free pass length, 
$\boldsymbol{k}_{E}=e\boldsymbol{E}/(k_{B}T)=\boldsymbol{u}\unicode[STIX]{x1D708}/(2\mathtt{v}_{t}^{2})$, where 
$u$ is the drift velocity vector of the positive ions, 
$\unicode[STIX]{x1D708}$ is the ion–neutral collision frequency, 
$\mathtt{v}_{t}$ is the thermal velocity and 
$\unicode[STIX]{x1D70F}$ is the ratio of electron and ion temperatures. Dependence of the electric potentials on the spatial coordinates is given by the inverse Fourier transformation
$$\begin{eqnarray}\unicode[STIX]{x1D711}(\boldsymbol{r})=\frac{1}{(2\unicode[STIX]{x03C0})^{3/2}}\int \unicode[STIX]{x1D711}(\boldsymbol{k})\exp (\text{i}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{r}))\,\text{d}^{3}\boldsymbol{k}.\end{eqnarray}$$ The term 
$\unicode[STIX]{x1D712}_{\text{e}}n_{+}b_{+}/(b_{\text{e}}n_{\text{e}})$ in (2.5) is small with respect to the term 
$\unicode[STIX]{x1D712}_{+}$ due to the small ratio of the ion and electron mobilities (with the exception of the case of a strongly electronegative plasma with 
$n_{-}/n_{\text{e}}\sim b_{\text{e}}/b_{+}$) and will be omitted below.
3 Large distances asymptote
To investigate the potential profile at large distances, we assume that 
$\unicode[STIX]{x1D712}=\unicode[STIX]{x1D712}_{+}+\unicode[STIX]{x1D712}_{-}+\unicode[STIX]{x1D712}_{\text{e}}\gg 1$, expand (2.5) in the set on the 
$\unicode[STIX]{x1D712}$ negative powers and conserve terms up to 
$\unicode[STIX]{x1D712}^{-1}$. The result is
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}(\boldsymbol{k}) & {\approx} & \displaystyle \frac{-(\boldsymbol{k}^{4}+(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{k}_{E})^{2})(\boldsymbol{k}^{2}-\text{i}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{k}_{E})/\unicode[STIX]{x1D70F})}{(2\unicode[STIX]{x03C0})^{3/2}\unicode[STIX]{x1D700}_{0}k_{D}^{2}\boldsymbol{k}^{4}(\boldsymbol{k}^{2}-\text{i}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{k}_{E})\unicode[STIX]{x1D702})}\nonumber\\ \displaystyle & & \displaystyle \times \left(ez_{d}+\frac{eJk_{D+}^{2}}{(\boldsymbol{k}^{2}+\text{i}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{k}_{E}))}\left(1-\frac{(\boldsymbol{k}^{4}+(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{k}_{E})^{2})(\boldsymbol{k}^{2}-\text{i}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{k}_{E})/\unicode[STIX]{x1D70F})}{k_{D}^{2}\boldsymbol{k}^{2}(\boldsymbol{k}^{2}-\text{i}(\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{k}_{E})\unicode[STIX]{x1D702})}\right)\right),\qquad\end{eqnarray}$$ where 
$k_{D}^{2}=k_{D+}^{2}+k_{D-}^{2}+k_{De}^{2}$ is the square of the inverse Debye length 
$\unicode[STIX]{x1D706}_{D}$, and the parameter 
$\unicode[STIX]{x1D702}$ is defined by the equation 
$\unicode[STIX]{x1D702}=(k_{D+}^{2}(1+1/\unicode[STIX]{x1D70F})-k_{D-}^{2}(1-1/\unicode[STIX]{x1D70F}))/k_{D}^{2}$. The expression (3.1) allows for analytical Fourier transformation, but the resulting equation appears to be too complex for analysis. So, we restrict the analysis via the condition 
$\unicode[STIX]{x1D70F}\rightarrow \infty$. Thus, 
$\unicode[STIX]{x1D702}=(k_{D+}^{2}-k_{D-}^{2})/k_{D}^{2}=(1+2\unicode[STIX]{x1D701})^{-1}$, where 
$\unicode[STIX]{x1D701}=n_{-}/n_{\text{e}}$ is the electronegativity parameter, and the spatial potential distribution can be expressed in the form
$$\begin{eqnarray}\unicode[STIX]{x1D711}(r)=\frac{e}{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D700}_{0}}\left(-\frac{A}{r}+B\frac{k_{E}\cos \unicode[STIX]{x1D703}}{r^{2}k_{D}^{2}}+\exp \left(-\unicode[STIX]{x1D702}k_{E}r\cos ^{2}\frac{\unicode[STIX]{x1D703}}{2}\right)\frac{(C-H(r,\unicode[STIX]{x1D703}))}{r}\right),\end{eqnarray}$$ where 
$k_{E}$ and 
$r$ are the absolute values of 
$\boldsymbol{k}_{E}$ and 
$\boldsymbol{r}$, respectively, 
$\unicode[STIX]{x1D703}$ is the angle between 
$\boldsymbol{E}$ and 
$\boldsymbol{r}$ and 
$A$, 
$B$, 
$C$, 
$D$, 
$E$ and 
$F$ are the parameters
$$\begin{eqnarray}\displaystyle & A=J(1+\unicode[STIX]{x1D701}),\quad B=z_{d}(1+2\unicode[STIX]{x1D701})-J(1+\unicode[STIX]{x1D701})(1+2\unicode[STIX]{x1D701}),\quad C={\displaystyle \frac{2J\unicode[STIX]{x1D701}(1+\unicode[STIX]{x1D701})}{1+2\unicode[STIX]{x1D701}}}, & \displaystyle \nonumber\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle H(r,\unicode[STIX]{x1D703}) & = & \displaystyle \frac{D}{k_{D}^{2}}\left(\frac{k_{E}}{r}\cos \unicode[STIX]{x1D703}+\unicode[STIX]{x1D702}k_{E}^{2}\cos ^{2}\frac{\unicode[STIX]{x1D703}}{2}\right)\nonumber\\ \displaystyle & & \displaystyle -\,8J\unicode[STIX]{x1D701}^{2}\frac{(1+\unicode[STIX]{x1D701})^{2}}{(1+2\unicode[STIX]{x1D701})^{3}}\frac{k_{E}^{2}}{k_{D}^{2}}\left(\sin ^{2}\frac{\unicode[STIX]{x1D703}}{2}-\unicode[STIX]{x1D702}k_{E}r\cos ^{4}\frac{\unicode[STIX]{x1D703}}{2}\right),\nonumber\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle D=4z_{d}\unicode[STIX]{x1D701}\frac{1+\unicode[STIX]{x1D701}}{1+2\unicode[STIX]{x1D701}}-4J\unicode[STIX]{x1D701}\frac{(1+\unicode[STIX]{x1D701})(2+3\unicode[STIX]{x1D701}+2\unicode[STIX]{x1D701}^{2})}{(1+2\unicode[STIX]{x1D701})^{2}}. & \displaystyle \nonumber\end{eqnarray}$$Equation (3.2) is valid when both conditions
$$\begin{eqnarray}r\gg \unicode[STIX]{x1D706}_{D}\quad \text{and}\quad r\gg k_{E}\unicode[STIX]{x1D706}_{D}^{2}/\unicode[STIX]{x1D702}\end{eqnarray}$$are satisfied.
The ion flux on the grain can be easily calculated only for a small enough spherical grain in a collision-dominant plasma, when
$$\begin{eqnarray}l_{i}<a(-e\unicode[STIX]{x1D711}_{s}/k_{B}T)^{1/2}\ll \unicode[STIX]{x1D706}_{D},\end{eqnarray}$$ where 
$a$ is the grain radius, 
$\unicode[STIX]{x1D711}_{s}$ is the surface potential and 
$l_{i}$ is the ion free path length. In this case
$$\begin{eqnarray}J\approx z_{d},\end{eqnarray}$$ which is well known for the two-component plasma (see Su & Lam Reference Su and Lam1963; Khrapak et al. Reference Khrapak, Mofill, Khrapak and D’yachkov2006) and remains true in the electronegative plasma, because (3.5) is derived neglecting space charge in Poisson’s equation. The conditions (3.4) are stronger than necessary for validity of (2.3) because the distances of interest for dust–dust interaction are typically greater than the ion capturing radius and the Debye length. Calculation of the ion flux on the grain in a general case is a rather complicated task even in a two-component plasma. However, equation (3.5) gives the upper limit of the ion flux on the attractive particle, so 
$J\leqslant z_{d}$.
The expression (3.2) contains the components proportional to the parameters 
$A$ and 
$C$, which decrease as 
$1/r$, while other components decrease as 
$1/r^{2}$. The first ones arise from the absorption and are proportional to 
$J$. In the absence of the external field they give a Coulomb-like asymptote
$$\begin{eqnarray}\unicode[STIX]{x1D711}\approx \frac{A-C}{r}=\frac{eJ}{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D700}_{0}r}\frac{k_{D+}^{2}}{k_{D}^{2}}\end{eqnarray}$$ in accordance with Chaudhuri et al. (Reference Chaudhuri, Khrapak and Morfill2007) and Khrapak et al. (Reference Khrapak, Klumov and Mofill2008). In the presence of the external field, the same asymptote remains only in the direction opposite to the field (
$\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}$), while in other directions the asymptote is
$$\begin{eqnarray}\unicode[STIX]{x1D711}\approx \frac{eJ}{4\unicode[STIX]{x03C0}\unicode[STIX]{x1D700}_{0}r}\frac{k_{D+}^{2}}{k_{D}^{2}}(1+2\unicode[STIX]{x1D701}).\end{eqnarray}$$ The anisotropic term decaying as 
$1/r$ appears only in the electronegative plasma and is connected with rarefaction of the negative ion flow behind the negatively charged particle.
Figure 1. Potential distribution profiles for 
$k_{E}\unicode[STIX]{x1D706}_{D}=1$ (the external field is upwards directed), 
$\unicode[STIX]{x1D70F}=100$, 
$J/z_{d}=0.1$ and 1, the negativity parameters 
$\unicode[STIX]{x1D701}=0$, 0.5 and 5, numerically calculated by (2.5), (2.6) (solid lines) and according to (3.2) (dashed lines).
Figure 1 illustrates the potential distributions 
$\unicode[STIX]{x1D711}(r)$ around the charged absorbing particle depending on the electronegativity parameter for the ratios of 
$J/z_{d}=0.1$ and 1.0. The calculations were performed for a fixed normalised external field and a finite temperature ratio 
$\unicode[STIX]{x1D70F}=100$ by the numerical Fourier transformation of (2.5). The additional potential is normalised by the value 
$-ez_{d}k_{D}/(4\unicode[STIX]{x03C0}\unicode[STIX]{x1D700}_{0})$, and the levels 0.3, 0.25, 0.2, 0.15, 0.1, 0.05, 0.02, 
$\pm 0.01$, 
$\pm 0.005$, 0 are shown in the figure by the solid lines. The levels calculated by the approximation (3.2) are shown by the dashed lines. Note, that, according to the conditions (3.3), the approximation (3.2) for 
$\unicode[STIX]{x1D701}=5$ and 
$k_{E}=k_{D}$ is valid only for 
$r\gg 11\unicode[STIX]{x1D706}_{D}$.
4 Conclusions
The obtained results have a direct relation to the dusty gas discharge plasma. Negative ions can appear in chemically active or contaminated plasmas (Amemiya Reference Amemiya1990; Klumov, Ivlev & Morfill Reference Klumov, Ivlev and Morfill2003). The presence of negative ions essentially affects the dust component. It not only decreases the dust particle charge (Klumov et al. Reference Klumov, Ivlev and Morfill2003; Merlino & Kim Reference Merlino and Kim2006), but drastically changes the interaction between grains in the drifting plasma.
The negative ion flow produces the anisotropic interaction even in the hydrodynamic charging mode, when the positive ion flow cannot give significant anisotropy. The anisotropic interaction produced by a negative ion flow fosters string formation. A slow decrease in the potential with distance allows one to expect enhanced capture of the particle in a string with an increasing number of particles. The effect of string formation under the action of the negative ion flow is an analogy to the lane formation in the driven binary complex plasmas, which has been experimentally observed by Sütterlin et al. (Reference Sütterlin, Wysocki, Ivlev, Räth, Thomas, Rubin-Zuzic, Goedheer, Fortov, Lipaev and Molotcov2009) and Du et al. (Reference Du, Sütterlin, Jiang, Räth, Ivlev, Khrapak, Schwabe, Thomas, Fortov and Lipaev2012), because a flow of small dust particles penetrating a cloud of big particles can be treated as the flow of multi-charged negative ions. Of cause, a large charge number limits application of the small perturbation approach used here.
