2 Assumptions and velocity distribution function

Consider a spherical particle moving at velocity $\boldsymbol{V}_{p}$ in a linear shear flow, where $\boldsymbol{V}_{p}$ is in the same direction as the velocity of the shear flow. The local mass velocity of the gas mixture is $\boldsymbol{V}_{0}$ , and $\boldsymbol{v}_{i}$ is the velocity of the gas molecule relative to $\boldsymbol{V}_{0}$ , i.e. $\boldsymbol{v}_{i}$ is the peculiar velocity and the subscript $i$ denotes the gas species $i$ . The particle velocity relative to the local mass velocity of the gas is $\boldsymbol{V}$ ( $\boldsymbol{V}=\boldsymbol{V}_{p}-\boldsymbol{V}_{0}$ ), and its direction is perpendicular to the velocity gradient of the shear flow $\boldsymbol{G}$ . A reference frame is introduced with origin located at the mass centre of the particle, as shown in figure 1. The relative velocity $\boldsymbol{V}$ is in the positive $z$ -direction and the velocity gradient $\boldsymbol{G}$ is in the negative $y$ -direction.

Figure 1.

(a) Collision model between a gas molecule and a small particle in a linear shear flow. (b) Relationship among various vectors.

Figure 2.

Gas molecules travel in a cylindrical region.

Based on the gas kinetic theory (Chapman & Cowling Reference Chapman and Cowling1970; Bird Reference Bird1994; Li & Wang Reference Li and Wang2003a ,Reference Li and Wang b ), the force acting on the particle can be obtained by finding the total momentum transfer from the gas molecules to the particle. The gas molecules travelling in a cylindrical shell with the same impact parameter $l$ and the same relative velocity $\boldsymbol{g}_{i}$ ( $\boldsymbol{g}_{i}=\boldsymbol{v}_{i}-\boldsymbol{V}$ ) will undergo collisions with the particle during a short time $\,\text{d}t$ . Then, the length of this cylinder is $L=\boldsymbol{g}_{i}\,\text{d}t$ (see figure 2), and the number of gas molecules in the small sector of this cylindrical shell is $f_{i}g_{i}l\,\text{d}l\,\text{d}\unicode[STIX]{x1D709}\,\text{d}t$ , where $f_{i}$ is the velocity distribution of the gas molecules. The total momentum transfer of the gas molecules in the whole real space and velocity space can be obtained by integrating the momentum transfer of the gas molecules over $l$ , $\unicode[STIX]{x1D709}$ and $\boldsymbol{v}_{i}$ . As a result, the force experienced by the particle upon collisions with gas molecules is given by

(2.1) $$\begin{eqnarray}\boldsymbol{F}_{i}=m_{ip}\int _{\boldsymbol{v}_{i}}g_{i}\boldsymbol{g}_{i}\,f_{i}Q(g_{i})\,\text{d}\boldsymbol{v}_{i},\quad i=1,2,\end{eqnarray}$$

where $m_{ip}=m_{i}m_{p}/(m_{i}+m_{p})$ is the reduced mass, with $m_{p}$ and $m_{i}$ being the particle mass and gas molecular mass of species $i$ , respectively. In (2.1), $Q(g_{i})$ represents the collision cross-section, which depends on the scattering scenario of the gas molecules upon collisions with the particle. Specular and diffuse scattering are the two limiting models of collision (Hirschfelder et al. Reference Hirschfelder, Curtiss, Bird and Mayer1954; Chapman & Cowling Reference Chapman and Cowling1970). For specular scattering,

(2.2) $$\begin{eqnarray}Q_{s}(g_{i})=2\unicode[STIX]{x03C0}\int _{0}^{\infty }(1-\cos \unicode[STIX]{x1D712})l\,\text{d}l,\end{eqnarray}$$

with $\unicode[STIX]{x1D712}$ being the scattering angle given by

(2.3) $$\begin{eqnarray}\unicode[STIX]{x1D712}=\unicode[STIX]{x03C0}-2l\int _{r_{m}}^{\infty }\frac{\text{d}r}{r^{2}\sqrt{1-{\displaystyle \frac{l^{2}}{r^{2}}}-{\displaystyle \frac{U(r)}{m_{ip}g_{i}^{2}/2}}}},\end{eqnarray}$$

where $r$ is the separation between the particle and gas molecule, $r_{m}$ is the distance of closest encounter and $U(r)$ is the gas–particle interaction potential. The collision cross-section for diffuse scattering is defined as

(2.4) $$\begin{eqnarray}Q_{d}(g_{i})=2\unicode[STIX]{x03C0}\left[\int _{0}^{l_{0}}\left(1+\frac{1}{g_{i}}\sqrt{\frac{\unicode[STIX]{x03C0}k_{B}T}{2m_{ip}}}\sin \frac{\unicode[STIX]{x1D712}}{2}\right)+\int _{l_{0}}^{\infty }(1-\cos \unicode[STIX]{x1D712})\right]l\,\text{d}l,\end{eqnarray}$$

where $k_{B}$ is the Boltzmann constant, $T$ is the temperature $l_{0}$ is the critical impact parameter. Diffuse scattering can only occur if the impact parameter is less than $l_{0}$ and the gas molecule collides with the particle physically; if the impact parameter is greater than $l_{0}$ , then the gas molecule just grazes the particle and the scattering is considered to be specular.

As mentioned above, the velocity distribution of the gas molecules is required to calculate the force acting on the particle. In the free-molecule regime, whereby the mean free path of the gas is much larger than the radius of the spherical particle, the interactions between the incident gas molecules and those reflected by the particle can be ignored. Hence, the velocity distribution of gas molecules can be obtained by neglecting the presence of the particle (Liu & Bogy Reference Liu and Bogy2008). For a binary gas mixture, the velocity distribution function of the two gas species is more complicated than that of pure gases. According to the Chapman–Enskog theory (Chapman & Cowling Reference Chapman and Cowling1970), in the presence of a velocity gradient, the second-order approximation to the velocity distribution function is $f_{i}=f_{i}^{(0)}+f_{i}^{(1)}$ , where the first term $f_{i}^{(0)}$ is the molecular velocity distribution function in uniform state, and the second term $f_{i}^{(1)}$ accounts for the flow velocity gradient. Thus, the integral in (2.1) consists of two terms, the first of which is the drag force and the second of which is the lift force. The present paper focuses on the lift force, which involves $f_{i}^{(1)}$ (Chapman & Cowling Reference Chapman and Cowling1970; Bird Reference Bird1994)

(2.5) $$\begin{eqnarray}f_{i}^{(1)}=-2f_{i}^{(0)}\unicode[STIX]{x1D63D}_{i}\boldsymbol{ : }\unicode[STIX]{x1D735}\boldsymbol{V}_{0},\quad i=1,2,\end{eqnarray}$$

where $f_{i}^{(0)}$ is the velocity distribution function in uniform state given by

(2.6) $$\begin{eqnarray}f_{i}^{(0)}=n_{i}\left(\frac{m_{i}}{2\unicode[STIX]{x03C0}k_{B}T}\right)^{3/2}\exp \!\left(\!-\frac{m_{i}v_{i}^{2}}{2k_{B}T}\right),\quad i=1,2,\end{eqnarray}$$

where $n_{i}$ is the number density of gas $i$ , and $\unicode[STIX]{x1D63D}_{i}$ is a symmetrical and non-divergent tensor, which is obtained as a series of Sonine polynomials. Under the first-order approximation,

(2.7) $$\begin{eqnarray}\unicode[STIX]{x1D63D}_{i}=b_{i}\frac{m_{i}}{2k_{B}T}\mathop{\circ }^{\boldsymbol{v}_{i}\boldsymbol{v}_{i}},\quad i=1,2,\end{eqnarray}$$

where $b_{i}$ is a function of gas composition, temperature, gas molecular mass and the intermolecular potential among gas molecules. The superscript ‘ $\circ$ ’ above $\boldsymbol{v}_{i}\boldsymbol{v}_{i}$ denotes that the tensor is non-divergent. For the flow in figure 1, $f_{i}^{(1)}$ can be written as

(2.8) $$\begin{eqnarray}f_{i}^{(1)}=-f_{i}^{(0)}b_{i}\frac{m_{i}}{k_{B}T}v_{y,i}v_{z,i}G,\quad i=1,2,\end{eqnarray}$$

where $v_{y,i}$ and $v_{z,i}$ are the peculiar velocity components in $y$ - and $z$ -directions, respectively.

3 Lift force in a binary gas mixture

3.1 Lift force formula on a spherical nanoparticle by gas $i$

By substituting (2.8) into (2.1), the lift force $F_{L,i}$ can be written as

(3.1) $$\begin{eqnarray}\boldsymbol{F}_{L,i}=-\frac{m_{ip}m_{i}b_{i}G}{k_{B}T}\int _{\boldsymbol{v}_{i}}g_{i}\boldsymbol{g}_{i}\,f_{i}^{(0)}v_{y,i}v_{z,i}Q(g_{i})\,\text{d}\boldsymbol{v}_{i}.\end{eqnarray}$$

As shown in figure 1(b), $\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x1D703}$ are the colatitude and azimuthal angles of $\boldsymbol{g}_{i}$ in the frame of reference, respectively. The relative velocity vector $\boldsymbol{g}_{i}$ is therefore given by

(3.2) $$\begin{eqnarray}\boldsymbol{g}_{i}=g_{i}\sin \unicode[STIX]{x1D719}\cos \unicode[STIX]{x1D703}\,\boldsymbol{i}+g_{i}\sin \unicode[STIX]{x1D719}\sin \unicode[STIX]{x1D703}\,\boldsymbol{j}+g_{i}\cos \unicode[STIX]{x1D719}\,\boldsymbol{k}.\end{eqnarray}$$

Since the lift force is collinear with the velocity gradient, only the $y$ -component of $\boldsymbol{g}_{i}$ needs to be considered, and it is easy to check that the first and third terms in (3.2) vanish upon integration over the angles in spherical coordinates. This simplifies (3.1) to

(3.3) $$\begin{eqnarray}\boldsymbol{F}_{L,i}=\frac{m_{ip}m_{i}b_{i}G}{k_{B}T}\int _{\boldsymbol{v}_{i}}g_{i}^{3}\sin ^{2}\unicode[STIX]{x1D719}\sin ^{2}\unicode[STIX]{x1D703}\,(g_{i}\cos \unicode[STIX]{x1D719}+V)\,f_{i}^{(0)}Q(g_{i})\,\text{d}\boldsymbol{v}_{i}.\end{eqnarray}$$

Substituting (2.6) into (3.3), we obtain the lift force (a detailed derivation is given in appendix A)

(3.4) $$\begin{eqnarray}\boldsymbol{F}_{L,i}={\textstyle \frac{8}{15}}m_{ip}\sqrt{2\unicode[STIX]{x03C0}k_{B}T/m_{i}}\,\boldsymbol{G}VR^{2}n_{i}b_{i}[5\unicode[STIX]{x1D6FA}_{ip}^{(1,1)\ast }-6\unicode[STIX]{x1D6FA}_{ip}^{(1,2)\ast }],\end{eqnarray}$$

where $\unicode[STIX]{x1D6FA}_{ip}^{(k,q)\ast }$ is the reduced collision integral given by

(3.5) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{ip}^{(k,q)\ast }=\int _{0}^{\infty }\frac{\text{e}^{-{\unicode[STIX]{x1D6FE}_{i}}^{2}}{\unicode[STIX]{x1D6FE}_{i}}^{2q+3}Q^{k}(g_{i})\,\text{d}\unicode[STIX]{x1D6FE}_{i}}{[(q+k)!/2]\{1-[1+(-1)^{k}]/(2+2k)\}\unicode[STIX]{x03C0}R^{2}},\end{eqnarray}$$

and $\unicode[STIX]{x1D6FE}_{i}=g_{i}\sqrt{m_{i}/(2k_{B}T)}$ . More details can be found in appendix A.

3.2 Total lift force on a spherical nanoparticle in binary gas mixtures

Since the mass of the particle is usually much larger than that of a gas molecule, i.e. $m_{i}/m_{ip}\approx 1$ , then the total lift force can be expressed as

(3.6) $$\begin{eqnarray}\displaystyle \boldsymbol{F}_{L}=\boldsymbol{F}_{L1}+\boldsymbol{F}_{L2} & = & \displaystyle \frac{8\boldsymbol{G}VR^{2}\sqrt{2\unicode[STIX]{x03C0}k_{B}T}}{15} [n_{1}b_{1}\sqrt{m_{1}}(5\unicode[STIX]{x1D6FA}_{1p}^{(1,1)\ast }-6\unicode[STIX]{x1D6FA}_{1p}^{(1,2)\ast })\nonumber\\ \displaystyle & & \displaystyle +\,n_{2}b_{2}\sqrt{m_{2}}(5\unicode[STIX]{x1D6FA}_{2p}^{(1,1)\ast }-6\unicode[STIX]{x1D6FA}_{2p}^{(1,2)\ast })\!]\!.\end{eqnarray}$$

In (3.6), the coefficients $b_{1}$ and $b_{2}$ are given by (Chapman & Cowling Reference Chapman and Cowling1970)

(3.7a,b ) $$\begin{eqnarray}b_{1}=\frac{\unicode[STIX]{x1D6FD}_{1}b_{22}-\unicode[STIX]{x1D6FD}_{2}b_{12}}{{\mathcal{B}}},\quad b_{2}=\frac{\unicode[STIX]{x1D6FD}_{2}b_{11}-\unicode[STIX]{x1D6FD}_{1}b_{12}}{{\mathcal{B}}},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FD}_{1}=5n_{1}/(2n^{2})$ , $\unicode[STIX]{x1D6FD}_{2}=5n_{2}/(2n^{2})$ and ${\mathcal{B}}$ is the symmetric determinant,

(3.8) $$\begin{eqnarray}{\mathcal{B}}=\left|\begin{array}{@{}cc@{}}b_{11} & b_{12}\\ b_{12} & b_{22}\end{array}\right|.\end{eqnarray}$$

For binary gas mixtures, the expressions for the elements of determinant ${\mathcal{B}}$ are

(3.9) $$\begin{eqnarray}\displaystyle & b_{11}=\sqrt{{\displaystyle \frac{k_{B}T}{2\unicode[STIX]{x03C0}m_{0}}}}\left\{{\displaystyle \frac{4n_{1}^{2}}{n^{2}}}\sqrt{{\displaystyle \frac{2}{M_{1}}}}\unicode[STIX]{x1D6FA}_{11}^{(2,2)}+{\displaystyle \frac{16n_{1}n_{2}}{3n^{2}}}M_{2}\sqrt{{\displaystyle \frac{1}{M_{1}M_{2}}}}\left[5M_{1}\unicode[STIX]{x1D6FA}_{12}^{(1,1)}+{\displaystyle \frac{3}{2}}M_{2}\unicode[STIX]{x1D6FA}_{12}^{(2,2)}\right]\right\}, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(3.10) $$\begin{eqnarray}\displaystyle & b_{22}=\sqrt{{\displaystyle \frac{k_{B}T}{2\unicode[STIX]{x03C0}m_{0}}}}\left\{{\displaystyle \frac{4n_{2}^{2}}{n^{2}}}\sqrt{{\displaystyle \frac{2}{M_{2}}}}\unicode[STIX]{x1D6FA}_{22}^{(2,2)}+{\displaystyle \frac{16n_{1}n_{2}}{3n^{2}}}M_{1}\sqrt{{\displaystyle \frac{1}{M_{1}M_{2}}}}\left[5M_{2}\unicode[STIX]{x1D6FA}_{12}^{(1,1)}+{\displaystyle \frac{3}{2}}M_{1}\unicode[STIX]{x1D6FA}_{12}^{(2,2)}\right]\right\}, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(3.11) $$\begin{eqnarray}\displaystyle & b_{12}=\sqrt{{\displaystyle \frac{k_{B}T}{2\unicode[STIX]{x03C0}m_{0}}}}\left\{\sqrt{{\displaystyle \frac{1}{M_{1}M_{2}}}}\left(-{\displaystyle \frac{16n_{1}n_{2}}{3n^{2}}}\right)M_{1}M_{2}\left[5\unicode[STIX]{x1D6FA}_{12}^{(1,1)}-{\displaystyle \frac{3}{2}}\unicode[STIX]{x1D6FA}_{12}^{(2,2)}\right]\right\}, & \displaystyle\end{eqnarray}$$

where $M_{1}=m_{1}/m_{0}$ , $M_{2}=m_{2}/m_{0}$ and $m_{0}=m_{1}+m_{2}$ . Here, $\unicode[STIX]{x1D6FA}_{ij}^{(k,q)}$ is defined as (Hirschfelder et al. Reference Hirschfelder, Curtiss, Bird and Mayer1954)

(3.12) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{ij}^{(k,q)}=\int _{0}^{\infty }\text{e}^{-\unicode[STIX]{x1D6FE}_{ij}^{2}}\unicode[STIX]{x1D6FE}_{ij}^{2q+3}Q^{k}(g_{ij})\,\text{d}\unicode[STIX]{x1D6FE}_{ij},\end{eqnarray}$$

which depends on the intermolecular interactions between gas molecules. It should be noted that the mass in the dimensionless quantity $\unicode[STIX]{x1D6FE}_{ij}$ is the reduced mass between two gas molecules.

For simplicity, hereafter, (3.9)–(3.11) will be represented by

(3.13a-c ) $$\begin{eqnarray}b_{11}=\sqrt{\frac{k_{B}T}{2\unicode[STIX]{x03C0}m_{0}}}\tilde{b}_{11},\quad b_{22}=\sqrt{\frac{k_{B}T}{2\unicode[STIX]{x03C0}m_{0}}}\tilde{b}_{22},\quad b_{12}=\sqrt{\frac{k_{B}T}{2\unicode[STIX]{x03C0}m_{0}}}\tilde{b}_{12},\end{eqnarray}$$

where $\tilde{b}_{11}$ , $\tilde{b}_{22}$ and $\tilde{b}_{12}$ denote the terms in the curly braces in (3.9), (3.10) and (3.11), respectively. Then (3.7) can be rewritten as

(3.14a ) $$\begin{eqnarray}\displaystyle b_{1} & = & \displaystyle \frac{\unicode[STIX]{x1D6FD}_{1}b_{22}-\unicode[STIX]{x1D6FD}_{2}b_{12}}{{\mathcal{B}}}=\frac{\unicode[STIX]{x1D6FD}_{1}\tilde{b}_{22}-\unicode[STIX]{x1D6FD}_{2}\tilde{b}_{12}}{\sqrt{{\displaystyle \frac{k_{B}T}{2\unicode[STIX]{x03C0}m_{0}}}}(\tilde{b}_{11}\tilde{b}_{22}-{\tilde{b}_{12}}^{2})}=\sqrt{\frac{2\unicode[STIX]{x03C0}m_{0}}{k_{B}T}}\tilde{b}_{1},\end{eqnarray}$$

(3.14b ) $$\begin{eqnarray}\displaystyle b_{2} & = & \displaystyle \frac{\unicode[STIX]{x1D6FD}_{2}b_{11}-\unicode[STIX]{x1D6FD}_{1}b_{12}}{{\mathcal{B}}}=\frac{\unicode[STIX]{x1D6FD}_{2}\tilde{b}_{11}-\unicode[STIX]{x1D6FD}_{1}\tilde{b}_{12}}{\sqrt{{\displaystyle \frac{k_{B}T}{2\unicode[STIX]{x03C0}m_{0}}}}(\tilde{b}_{11}\tilde{b}_{22}-{\tilde{b}_{12}}^{2})}=\sqrt{\frac{2\unicode[STIX]{x03C0}m_{0}}{k_{B}T}}\tilde{b}_{2}.\end{eqnarray}$$

Putting (3.14) into (3.6), the total lift force reads

(3.15) $$\begin{eqnarray}\displaystyle \boldsymbol{F}_{L}=\frac{16\unicode[STIX]{x03C0}m_{0}\boldsymbol{G}VR^{2}}{15}[n_{1}\tilde{b}_{1}\sqrt{M_{1}}(5\unicode[STIX]{x1D6FA}_{1p}^{(1,1)\ast }-6\unicode[STIX]{x1D6FA}_{1p}^{(1,2)\ast })+n_{2}\tilde{b}_{2}\sqrt{M_{2}}(5\unicode[STIX]{x1D6FA}_{2p}^{(1,1)\ast }-6\unicode[STIX]{x1D6FA}_{2p}^{(1,2)\ast })]. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

It should be noted that $M_{i}$ and $\unicode[STIX]{x1D6FA}_{ip}^{(k,q)\ast }$ in (3.15) are dimensionless, whereas $n_{i}\tilde{b}_{i}$ has the same unit as $1/R^{2}$ . Letting $\unicode[STIX]{x1D6F1}_{i}^{\ast }=R^{2}n_{i}\tilde{b}_{i}\sqrt{M_{i}}(5\unicode[STIX]{x1D6FA}_{ip}^{(1,1)\ast }-6\unicode[STIX]{x1D6FA}_{ip}^{(1,2)\ast })$ , the lift force can be expressed as

(3.16) $$\begin{eqnarray}\boldsymbol{F}_{L}=\frac{16\unicode[STIX]{x03C0}m_{0}\boldsymbol{G}V}{15}(\unicode[STIX]{x1D6F1}_{1}^{\ast }+\unicode[STIX]{x1D6F1}_{2}^{\ast }),\end{eqnarray}$$

where $\unicode[STIX]{x1D6F1}_{1}^{\ast }$ and $\unicode[STIX]{x1D6F1}_{2}^{\ast }$ are dimensionless quantities, which contain the interactions between gas species and the particle. It is worth mentioning that $\boldsymbol{F}_{L1}$ and $\boldsymbol{F}_{L2}$ in (3.6) are coupled with each other via the coefficients $b_{1}$ and $b_{2}$ , which depend on the interactions between the two gas species. Thus, the total lift force $\boldsymbol{F}_{L}$ is not simply the linear summation of the lift force exerted by each gas component.

3.3 Lift force in a pure gas

It is expected that (3.6) can be simplified to (1.2) for pure gases if the two species are identical. Letting $\unicode[STIX]{x1D6FA}_{1p}^{(1,1)\ast }=\unicode[STIX]{x1D6FA}_{2p}^{(1,1)\ast }=\unicode[STIX]{x1D6FA}_{gp}^{(1,1)\ast }$ , $\unicode[STIX]{x1D6FA}_{1p}^{(1,2)\ast }=\unicode[STIX]{x1D6FA}_{2p}^{(1,2)\ast }=\unicode[STIX]{x1D6FA}_{gp}^{(1,2)\ast }$ and $m_{1}=m_{2}=m_{g}$ , equation (3.6) becomes

(3.17) $$\begin{eqnarray}\boldsymbol{F}_{L}=\frac{8\boldsymbol{G}VR^{2}\sqrt{2\unicode[STIX]{x03C0}k_{B}Tm_{g}}}{15}(5\unicode[STIX]{x1D6FA}_{gp}^{(1,1)\ast }-6\unicode[STIX]{x1D6FA}_{gp}^{(1,2)\ast })(n_{1}b_{1}+n_{2}b_{2}),\end{eqnarray}$$

and (3.9)–(3.11) can be rewritten as

(3.18) $$\begin{eqnarray}\displaystyle & \displaystyle b_{11}=\sqrt{\frac{k_{B}T}{\unicode[STIX]{x03C0}m_{g}}}\left\{\frac{4n_{1}^{2}}{n^{2}}\unicode[STIX]{x1D6FA}_{g}^{(2,2)}+\frac{8n_{1}n_{2}}{3n^{2}}\left[\frac{5}{2}\unicode[STIX]{x1D6FA}_{g}^{(1,1)}+\frac{3}{4}\unicode[STIX]{x1D6FA}_{g}^{(2,2)}\right]\right\}, & \displaystyle\end{eqnarray}$$

(3.19) $$\begin{eqnarray}\displaystyle & \displaystyle b_{22}=\sqrt{\frac{k_{B}T}{\unicode[STIX]{x03C0}m_{g}}}\left\{\frac{4n_{2}^{2}}{n^{2}}\unicode[STIX]{x1D6FA}_{g}^{(2,2)}+\frac{8n_{1}n_{2}}{3n^{2}}\left[\frac{5}{2}\unicode[STIX]{x1D6FA}_{g}^{(1,1)}+\frac{3}{4}\unicode[STIX]{x1D6FA}_{g}^{(2,2)}\right]\right\}, & \displaystyle\end{eqnarray}$$

(3.20) $$\begin{eqnarray}\displaystyle & \displaystyle b_{12}=\sqrt{\frac{k_{B}T}{\unicode[STIX]{x03C0}m_{g}}}\left(\frac{8n_{1}n_{2}}{3n^{2}}\right)\left[\frac{3}{4}\unicode[STIX]{x1D6FA}_{g}^{(2,2)}-\frac{5}{2}\unicode[STIX]{x1D6FA}_{g}^{(1,1)}\right]. & \displaystyle\end{eqnarray}$$

By using (3.7) and (3.8),

(3.21) $$\begin{eqnarray}n_{1}b_{1}+n_{2}b_{2}=\frac{5}{2}\left[\frac{n_{1}^{2}b_{22}-2n_{1}n_{2}b_{12}+n_{2}^{2}b_{11}}{n^{2}(b_{11}b_{22}-b_{_{12}}^{2})}\right].\end{eqnarray}$$

Then, by substituting (3.18)–(3.20) into (3.21), we obtain

(3.22) $$\begin{eqnarray}n_{1}b_{1}+n_{2}b_{2}=\frac{5}{8\sqrt{{\displaystyle \frac{k_{B}T}{\unicode[STIX]{x03C0}m_{g}}}}\unicode[STIX]{x1D6FA}_{g}^{(2,2)}}.\end{eqnarray}$$

As a result, (3.17) is simplified to

(3.23) $$\begin{eqnarray}\boldsymbol{F}_{L}=\frac{\sqrt{2}\unicode[STIX]{x03C0}m_{g}}{3\unicode[STIX]{x1D6FA}_{g}^{(2,2)}}\boldsymbol{G}R^{2}V[5\unicode[STIX]{x1D6FA}_{1p}^{(1,1)\ast }-6\unicode[STIX]{x1D6FA}_{1p}^{(1,2)\ast }],\end{eqnarray}$$

which is the same as (1.2) by using $\unicode[STIX]{x1D70C}=m_{g}n$ , $\unicode[STIX]{x1D6FA}_{g}^{(2,2)\ast }=\unicode[STIX]{x1D6FA}_{g}^{(2,2)}/(2\unicode[STIX]{x03C0}\unicode[STIX]{x1D70E}_{g}^{2})$ and $\unicode[STIX]{x1D706}=1/(\sqrt{2}\unicode[STIX]{x03C0}\unicode[STIX]{x1D70E}_{g}^{2}n)$ , where $\unicode[STIX]{x1D70E}_{g}$ is the collision diameter of the gas molecules.

As another test of consistency, we consider the case in which the concentration of one gas component tends to zero. For instance, in the limit of $n_{2}\rightarrow 0$ and $n_{1}\rightarrow n$ , $b_{1}n_{1}=5\sqrt{\unicode[STIX]{x03C0}m_{1}}/(8\unicode[STIX]{x1D6FA}_{11}^{(2,2)}\sqrt{k_{B}T})$ , $b_{2}n_{2}=0$ and (3.6) converges to that in a pure gas.

For large particles, the gas–particle interactions can be neglected and the rigid-body collision model is reasonable. In this case, (3.23) can be easily reduced to (1.1) by assuming rigid-body collisions. However, as the particle size decreases to the nanoscale, the influence of the van der Waals interactions between the gas molecules and particle on lift force becomes notable. This influence is manifested in the term  $\unicode[STIX]{x1D6FA}_{ip}^{(k,q)\ast }$ .

The lift force given by (1.2) could be positive $(5\unicode[STIX]{x1D6FA}_{gp}^{(1,1)\ast }>6\unicode[STIX]{x1D6FA}_{gp}^{(1,2)\ast })$ or negative $(5\unicode[STIX]{x1D6FA}_{gp}^{(1,1)\ast }<6\unicode[STIX]{x1D6FA}_{gp}^{(1,2)\ast })$ , depending on the gas–particle interactions. In most cases, the direction of the lift force on nanoparticles in pure gases (1.2) is negative, i.e. opposite to the velocity gradient, which is consistent with (1.1). However, the positive lift force takes place in a certain temperature range for a given gas–particle interaction. The physical reason for the direction change is that the gas molecules on the low-velocity side transfer more momentum to the nanoparticle due to its larger scattering angles upon non-rigid-body collisions. As for binary gas mixtures, the total lift force is a nonlinear combination of individual lift forces contributed from species 1 and 2. A direction change of the lift force is expected if the lift forces of gas species 1 and 2 are in opposite directions, as will be demonstrated in the following section.

4 Lift force on a silver spherical nanoparticle in gas mixtures

As an example, a suspended silver nanoparticle in a Ne–Ar binary gas mixture is considered. A widely used model to describe the van der Waals interactions is the Lennard-Jones (LJ) 12–6 potential,

(4.1) $$\begin{eqnarray}U_{12-6}(r)=4\unicode[STIX]{x1D700}\left[\left(\frac{\unicode[STIX]{x1D70E}}{r}\right)^{12}-\left(\frac{\unicode[STIX]{x1D70E}}{r}\right)^{6}\right],\end{eqnarray}$$

where $\unicode[STIX]{x1D700}$ and $\unicode[STIX]{x1D70E}$ represent the binding energy and collision diameter, respectively. The LJ potential parameters for Ne ( $\unicode[STIX]{x1D70E}_{1}=2.82$  Å and $\unicode[STIX]{x1D700}_{1}/k_{B}=32.8~\text{K}$ ), Ar ( $\unicode[STIX]{x1D70E}_{2}=3.47$  Å and $\unicode[STIX]{x1D700}_{2}/k_{B}=114.0~\text{K}$ ) and Ag ( $\unicode[STIX]{x1D70E}_{p}=2.57$  Å and $\unicode[STIX]{x1D700}_{p}/k_{B}=4075.0~\text{K}$ ) are adopted from the literature (Svehla Reference Svehla1962; Hippler, Troe & Wendelken Reference Hippler, Troe and Wendelken1983; Agrawal, Rice & Thompson Reference Agrawal, Rice and Thompson2002). The potential parameters for the interactions between different species 1 and 2 are obtained using the conventional Lorentz–Berthelot rules, i.e. $\unicode[STIX]{x1D70E}_{12}=(\unicode[STIX]{x1D70E}_{1}+\unicode[STIX]{x1D70E}_{2})/2$ and $\unicode[STIX]{x1D700}_{12}=\sqrt{\unicode[STIX]{x1D700}_{1}\unicode[STIX]{x1D700}_{2}}$ .

Figure 3 shows the dimensionless quantity $\unicode[STIX]{x1D6F1}_{i}^{\ast }$ of a silver nanoparticle in a rarefied Ne–Ar gas mixture as a function of reduced Ne concentration $n_{1}/n$ . It is seen that $\boldsymbol{F}_{L}$ converges to $\boldsymbol{F}_{L,1}$ (open circle) or $\boldsymbol{F}_{L,2}$ (open triangle) as $n_{1}\rightarrow n$ or $n_{2}\rightarrow n$ , respectively. It is also found that $\unicode[STIX]{x1D6F1}_{i}^{\ast }$ varies nonlinearly with Ne concentration. In (3.6), it seems that the total lift force is simply a linear superposition of the lift forces given by the gas components. However, both $b_{1}$ and $b_{2}$ depend on the composition ratio and the interactions between the two gas species, by which the dependence of the total lift force on the concentration is nonlinear.

Figure 3.

Variation of $\unicode[STIX]{x1D6F1}_{i}^{\ast }$ as a function of reduced Ne concentration $n_{1}/n$ at $T=100~\text{K}$ (a), $T=300~\text{K}$ (b), $T=500~\text{K}$ (c) and $T=700~\text{K}$ (d). Open symbols corresponding to the results in pure gases.

As discussed by Luo et al. (Reference Luo, Wang, Xia and Li2016), if the LJ 12–6 potential function is employed for the gas–particle interaction, a positive lift force that directs towards the high-velocity side exists in a certain reduced temperature range, $0.417<T^{\ast }=k_{B}T/\unicode[STIX]{x1D700}<0.951$ (Wang & Li Reference Wang and Li2012; Khrapak Reference Khrapak2014; Luo et al. Reference Luo, Wang, Xia and Li2016). Therefore, the lift forces acting on the nanoparticle in pure neon gas ( $F_{L1}$ ) and pure argon gas ( $F_{L2}$ ) become positive when the temperature is roughly in the ranges of 154–347 K and 286–648 K, respectively. If the temperature is extremely high or low, as shown in figure 3(a) for $T=100~\text{K}$ and figure 3(d) for $T=700~\text{K}$ , since the lift forces in both gas species are negative, the total lift forces are definitely negative. At moderate temperatures, positive $F_{L1}$ (Ne) and/or $F_{L2}$ (Ar) can take place, as shown in figures 3(b) and 3(c). As previously mentioned, the total lift force consists of two contributions from species 1 and 2. If the lift forces of gas species 1 and 2 are in opposite directions, it is expected that the direction of the total lift force changes as the gas concentration varies. For example, in the case of $T=500~\text{K}$ (see figure 3 c), $F_{L1}$ (Ne) is negative and $F_{L2}$ (Ar) is positive; as a result, the direction of $F_{L}$ changes from positive to negative as the Ne concentration varies. This means that in a certain temperature range, there exists a critical value of $n_{1}/n$ , where the total lift force changes its direction. However, the complex nonlinear dependence of the lift force on the gas concentrations makes it difficult to predict this critical value.

Figure 4 shows the dimensionless quantity $\unicode[STIX]{x1D6F1}_{i}^{\ast }$ of a silver nanoparticle in other gas mixtures (He–Ar and He–Kr gas mixtures). The collision diameter $\unicode[STIX]{x1D70E}$ and binding energy $\unicode[STIX]{x1D700}/k_{B}$ for He and Kr are 2.55 Å, 10.22 K and 3.66 Å, 178.9 K, respectively (Svehla Reference Svehla1962). As can be seen from (3.16), the total molecular mass of the two species $m_{0}$ appears in the coefficient of $(\unicode[STIX]{x1D6F1}_{1}^{\ast }+\unicode[STIX]{x1D6F1}_{2}^{\ast })$ , and $\unicode[STIX]{x1D6F1}_{1}^{\ast }$ depends on $1/m_{0}$ as $n_{1}/n$ converges to 1. Since the total molecular mass of He and Ar is different from that of He and Kr, the values of $\unicode[STIX]{x1D6F1}_{1}^{\ast }$ in figures 4(a) and 4(b) are not equal to each other as $n_{1}/n$ converges to 1 (pure He gas). It has been checked that the values of $\unicode[STIX]{x1D6F1}_{1}^{\ast }$ in He–Ar and He–Kr gas mixtures are equal to each other if $\unicode[STIX]{x1D6F1}_{1}^{\ast }$ is multiplied by $m_{0}$ . By comparing figures 3(b) with 4, it is seen that the dependence of the total lift force on the concentration is still nonlinear and the degree of the nonlinearity between $\unicode[STIX]{x1D6F1}_{i}^{\ast }$ and $n_{1}/n$ is different for different gas mixtures. The nonlinearity in the case of figure 4(b) is the most distinct one because the differences between the potential parameters and the molecular masses of He and Kr are the largest. Similarly, for the case in figure 3(b), the nonlinearity is weak because the potential parameters and the molecular mass of Ne are close to those of Ar. This is consistent with the discussion in § 3.3: if the potential parameters and molecular mass of species 1 are identical to those of species 2, the formulae reduce to those in pure gases and the nonlinearity disappears.

Figure 4.

Variation of $\unicode[STIX]{x1D6F1}_{i}^{\ast }$ as a function of reduced He concentration $n_{1}/n$ in He–Ar gas mixtures at $T=300~\text{K}$ (a) and in He–Kr gas mixtures at $T=300~\text{K}$  (b).

It should be noted that the LJ 12–6 potential might not be an adequate model to describe the interactions between gas molecules and a particle. A rigorous description of the gas–particle interaction is unavailable. As a coarse approach, the potential for gas–particle interactions can be determined as a sum of the LJ interactions of gas molecules with all atoms in the particle. If the distribution of atoms in the particle is assumed to be continuous, integration can be carried out in place of summation (Rouquerol et al. Reference Rouquerol, Rouquerol, Llewellyn, Maurin and Sing2013). In this way, the interaction between gas molecules and a single solid lattice plane is given by (Everett & Powl Reference Everett and Powl1976)

(4.2) $$\begin{eqnarray}U_{10-4}(r)=\frac{10}{3}\unicode[STIX]{x1D700}\left[\frac{1}{5}\left(\frac{\unicode[STIX]{x1D70E}}{r}\right)^{10}-\frac{1}{2}\left(\frac{\unicode[STIX]{x1D70E}}{r}\right)^{4}\right],\end{eqnarray}$$

and the interaction between gas molecules and a semi-infinite slab of solid is given by (Everett & Powl Reference Everett and Powl1976)

(4.3) $$\begin{eqnarray}U_{9-3}(r)=\frac{3}{\sqrt{10}}\unicode[STIX]{x1D700}\left[\frac{2}{15}\left(\frac{\unicode[STIX]{x1D70E}}{r}\right)^{9}-\left(\frac{\unicode[STIX]{x1D70E}}{r}\right)^{3}\right].\end{eqnarray}$$

Although there is no evidence that these two potential functions are suitable for gas–particle interactions, in the present paper, we consider them as a possible choice for illustrative purposes. It is expected that the $r$ dependence for the gas–particle interaction potential is covered by these three cases (12–6, 10–4 and 9–3) or their combinations (Luo et al. Reference Luo, Wang, Xia and Li2016).

In figure 5, $\unicode[STIX]{x1D6F1}_{1}^{\ast }+\unicode[STIX]{x1D6F1}_{2}^{\ast }$ is depicted as a function of reduced Ne concentration $n_{1}/n$ in a Ne–Ar gas mixture at various temperatures, wherein (4.1)–(4.3) are employed to model the interactions between the particle and gas molecules. Being similar to the results in figure 3, for interaction potentials given by (4.1)–(4.3), the dependence of $\unicode[STIX]{x1D6F1}_{1}^{\ast }+\unicode[STIX]{x1D6F1}_{2}^{\ast }$ on the single-species concentration ratio is also nonlinear. A positive lift force can be observed at certain temperature associated with the direction change from positive to negative with increasing Ne concentration. Furthermore, at the same concentration of Ne, the temperature value corresponding to the lift force direction change (if there is a change in lift force direction) increases from 12–6, to 10–4, to 9–3 potentials. This trend may be due to the increasing attractive interactions between gas molecules and the particle. Since we can observe the direction change of the lift force with all three types of interaction potentials, it is expected that the direction change can be observed experimentally. Owing to the computational complexity of the reduced collision integrals, it is not easy to offer a rigorous treatment of the gas–particle interactions. Further intensive experimental studies and theoretical analyses are needed on this issue.

Figure 5.

Variation of $\unicode[STIX]{x1D6F1}_{1}^{\ast }+\unicode[STIX]{x1D6F1}_{2}^{\ast }$ (Ne–Ar) as a function of reduced Ne concentration $n_{1}/n$ for the different gas–particle interaction potential functions at $T=400~\text{K}$ (a), $T=600~\text{K}$ (b), $T=800~\text{K}$ (c) and $T=1000~\text{K}$ (d).