Calculation of collision frequency function for aerosol particles in free molecule regime in presence of force fields

Abstract

The collision frequency function for aerosol particles has already been calculated for the free molecule regime and for the continuum range. The present work, taking into account the influence of internal force fields such as magnetic force, electric force and molecular forces, created by particles themselves, recalculated the collision frequency in the case of particles much smaller than the mean free path of the gas (free molecule regime). Attractive forces increase naturally the collision frequency, while repulsive forces decrease it. The calculation was performed for all types of central forces deriving from a potential, including Coulomb forces and Van der Waals forces.

Keywords

aerosol particles / collision frequency function / coagulation

Introduction

Coagulation is a process whereby dispersed particles collide with one another due to their relative motion, and agglomerate. For an aerosol system, it is one of the fundamental physical phenomena. The coagulation processes increase indeed the average size of the aerosol particles. Prediction of the size distribution of a coagulating aerosol has been a continuing interest in aerosol physics. The basic theoretical results for coagulation were presented by Friedlander [1]. More recently, some factors still needed to be considered were detailed by Benichou [2], among which the influence of force fields is the most important. The collision frequency function for aerosol particles was formulized for the free molecule regime and for the continuum range in Friedlander [1]. Fuchs [3] gave a general interpolation formula for the transition range, and the recent experiments of Kim et al. [4] accorded with the theoretical results.

The above equations for collision frequency are suitable for the instance that there is no internal force among particles. But, the fact is that the internal forces such as van der Waals force exist among aerosol particles. Marlow [5] derivated an aerosol aollosion rate for singular attractive contanct potentials. Alam [6] studeied the effect of viscous forces on the aerosol coagulation. The present paper gives an analytical result of the influence of central forces fields on the free molecular regime, including the most relevant types of forces—van der Waals forces, which are always present and Coulomb forces, which appear when the particles are charged.

Collision frequency

The definition of the collision frequency was given by Friedlander [1] as

(1)N_{ab}=\beta (a,b)n_a{n}_b,

where N_ab is the number of collisions occurring per unit time per unit volume between the two classes of particles a and b, a and b are the volumes of the two classes of particles, n_a and n_b are the concentrations of the two classes of particles a and b, and \beta (a,b) is the collision frequency which depends on the sizes of the colliding particles and on such properties of the system as temperature and pressure. The expression of collision frequency in the free molecule regime can be expressed as

(2)\beta (a,b)={\left({\displaystyle \frac{\mathrm{3}}{\mathrm{4}\pi }}\right)}^{\mathrm{1}/\mathrm{6}}{\left({\displaystyle \frac{\mathrm{6}kT}{{\rho }_p}}\right)}^{\mathrm{1}/\mathrm{2}}{\left({\displaystyle \frac{\mathrm{1}}{v_a}}+{\displaystyle \frac{\mathrm{1}}{v_b}}\right)}^{\mathrm{1}/\mathrm{2}}{(v_a^{\mathrm{1}/\mathrm{3}}+v_b^{\mathrm{1}/\mathrm{3}})}^{\mathrm{2}},

where k is Boltzmann constant, T is the system temperature, {\rho }_pis the particle density, and v_a and v_b are the volumes of the two classes of particles a and b.

Physical model

For the collision of particles with internal central forces, the following assumptions are adopted:

1) The particles are spherical.

2) The particles are uniformly distributed over the space; there is neither concentration gradient nor temperature gradient.

3) The force field generated by a particle has a limited radius: beyond a given distance of r₀, its influence can be neglected. This defines an influence sphere with a radius of r₀. The concentration of particles is tenuous, so that the average distance between particles is greater than this radius.

4) The distribution of the velocity of particles is given by the statistic law of Maxwell-Boltzmann [7] when they are outside the force field influence sphere of other particles.

5) When a particle enters the influence sphere, its movement is completely ruled by the interaction between two particles with the central forces.

6) The collisions are assumed to occur only between two particles, the collisions between more particles are not taken into account.

Critical radius

Figure 1 shows the analysis model of the collision of particles. It is assumed that when one particle with a radius of r_a is fixed, a particle with a radiusof r_b has an initial velocity of V which is equal to the initial relative velocity of two particles. When particle b enters the influencing sphere of particle a with a radius of r_{\mathrm{0}}, its movement is ruled by the theory of central forces. The radius of r_{\max }, depending on the intensity and the attractive or repulsive character of the interforce as well as the relative velocity V, is the maximal distance of particle b from the axis while particle b collides with particle a. The axis was designated to be parallel to the relative velocity of the two particles and pass through the center of the fixed particle a. The radius r_{\max } is called a critical radius. Particles which are further away from the axis than r_{\max }do not collide, while those which are nearer do. Therefore, it is evident that the number of particles which collide during dt is the same as the number of particles which enter the influencing sphere within a distance to the axis equalling to the critical radius duringdt. This corresponds here also to a cylinder with a height of Vdt and a radius of r_{\max }. This cylinder is referred to as collision cylinder.

The aim is, therefore, to calculate the critical radius r_{\max }. The critical radius r_{\max } is obviously related to the initial velocity V by the following equations.

There are two extreme cases for an attractive force

(3)r_{\max }(\mathrm{0})=r_{\mathrm{0}},

(4)r_{\max }(\infty )=r_a+r_b.

If the initial velocity is almost equal to zero, the incident particle has no other choice than being irremediably attracted by the other particle. On the contrary, if the initial velocity is extremely high, the force field will not be strong enough to modify the trajectory of the particle.

There are two extreme cases for a repulsive force

(5)r_{\max }(\infty )=r_a+r_b,

(6)

If the velocity of the incident particle is greatly high, it is the same case as that described just now. But if the velocity is not sufficient, even if the direction of the particle is just along the axis, it will not be strong enough to cross the repulsive force field, and collision will not occur. This corresponds to a minimal value of the velocity, V_{\min }.

The movement of the accident particle in the influence sphere must be submitted to the conservation of angular momentum and total energy. The main properties of central forces are to be always in the direction of the two particles, and their intensity depends only on the distance between particles. The central force can, therefore, be expressed as

(7){\overrightarrow{\mathit{F}}}_{a\to b}=f(r)\overrightarrow{{\mathit{u}}_r},

where \overrightarrow{u_r}represents the unity vector in the direction from particle a to particle b.

As dipicted in Fig. 2, the movement of particle b begins with the radius r_{\mathrm{0}} and the angle {\theta }_{\mathrm{0}} when it enters the influence sphere. The angle \theta and the radius r are main parameters to describe the movement of the particle. The movement trajectory of particle b have only three different types [8]: elliptic, parabolic or hyperbolic.

It is assumed that the collision occurs only if the minimal distance between particles reaches the value of the sum of the radii of particles, which means

(8)collision\iff r_{\min }\le r_a+r_b,

where r_{\min } represents the minimal distance between the two particles which will be reached during the movement of centroid.

The angular momentum is conserved during the movement,

(9)L=\mu rV\sin \theta =\mu r_{\mathrm{0}}V\sin {\theta }_{\mathrm{0}},

where \mu represents the reduced mass of the system, which is expressed as

(10)\mu ={\displaystyle \frac{m_a{m}_b}{m_a+m_b}}.

The theorem of energy conservation gives

(11)E=E_c+U=E_c+{U|}_{t=\mathrm{0}},

where E_crepresents the kinetic energy, and U is the potential related to the force field.

(12)E_c={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\mu {\Vert \left(\begin{array}{l}\dot{r}\\ r\dot{\theta }\end{array}\right)\Vert }^{\mathrm{2}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\mu {\dot{r}}^{\mathrm{2}}+{\displaystyle \frac{L^{\mathrm{2}}}{\mathrm{2}\mu r^{\mathrm{2}}}}.

At r=r_{\min },\dot{r}=\mathrm{0}:

(13){\displaystyle \frac{L^{\mathrm{2}}}{\mathrm{2}\mu r_{\min }^{\mathrm{2}}}}+U(r_{\min })={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\mu V^{\mathrm{2}}+U(r_{\mathrm{0}}).

By replacing r_{\min } by s=r_a+r_b , an expression of the limit condition for collision can be obtained, in the form of a relation between V and {\theta }_{\mathrm{0}}. However, the relation does not exist that in the case of a too intense repulsive force, r_{\min } is always larger than s, and in the case of too intense an attractive force, r_{\min } is always smaller than s. The latter case should be avoided by an adapted choice of the radius of the influence sphere.

Let \mathrm{\Delta }U=U(r_{\mathrm{0}})-U(s), and s=r_a+r_b, Eq. (13) is rewritten as

(14)r_{\max }(V)=s\sqrt{\mathrm{1}+{\displaystyle \frac{\mathrm{2}\mathrm{\Delta }U}{\mu V^{\mathrm{2}}}}}.

From the physical processes, it is known that r_{\max }cannot be greater than r_{\mathrm{0}}. When the relative velocity is too low, or the intensity of the attractive force field too high, r_{\max } should be equal tor_{\mathrm{0}}. In this case, the particles which enter the influencing sphere will collide. The minimum velocity is

(15)V_{\min }^A=\sqrt{{\displaystyle \frac{\mathrm{2}\mathrm{\Delta }U}{\mu (r_{\mathrm{0}}^{\mathrm{2}}/s^{\mathrm{2}}-\mathrm{1})}}}.

This velocity is obviously dependant on the intensity of the potential.

In Eq. (14), when V tends to be 0, the critical radius r_{\max } tends to be infinity. To be relevant to the hypothesis of a limited influencing radius, r_{\max } should be equal tor_{\mathrm{0}}.

Eq. (14) can be solved only if

\mathrm{1}+{\displaystyle \frac{\mathrm{2}\mathrm{\Delta }U}{\mu V^{\mathrm{2}}}}>\mathrm{0.}

When \mathrm{\Delta }U>\mathrm{0}, which is the case of an attractive force, this is always effective. But, when , which is the case of a repulsive force, the relative velocity has to be larger than the limit velocity

V_{\min }=\sqrt{-{\displaystyle \frac{\mathrm{2}\mathrm{\Delta }U}{\mu }}}.

This is the minimal velocity to reach, in order to overpass the repulsive force field.

(16)

Number of collisions

The number of collisions which occur during the interval of time dt between particles with volume v_a and particles with volume v_b having a relative velocity V is now given by

(17)V_b^{\mathrm{2}}=V_{bx}^{\mathrm{2}}+V_{by}^{\mathrm{2}}+V_{bz}^{\mathrm{2}},

(18)d{N}_{ab}=d{n}_{a}d{n}_b\pi r_{\max }{(V)}^{\mathrm{2}}V,

where V_b^{} is the relative velocity of particle b to particle a; V_{bx}^{}, V_{by} and V_{bz} are respectively the relative velocity components along directions x, y and z; d{n}_a and d{n}_b are the number of particles a and b which are comprised respectively between V_{bx}^{} and V_{bx}+d{V}_{bx}, V_{by}^{} and V_{by}+d{V}_{by}, V_{bz}^{} and V_{bz}+d{V}_{bz}. Their expressions are given [7] for type “b” particles as

(19)d{n}_b=n_b{\left({\displaystyle \frac{m_b}{\mathrm{2}\pi kT}}\right)}^{\mathrm{3}/\mathrm{2}}\exp \left(-{\displaystyle \frac{m_b{V}_b^{\mathrm{2}}}{\mathrm{2}kT}}\right)d{V}_{bx}d{V}_{by}d{V}_{bz}.

The combination of Eq. (1) with Eqs. (18) and (19) gives the expression of \beta (a,b),

(20)$$\begin{gathered} \beta (a,b)={\displaystyle \frac{\pi }{\mathrm{8}}}{\displaystyle \frac{{(m_a{m}_b)}^{\mathrm{3}/\mathrm{2}}}{{(\pi kT)}^{\mathrm{3}}}}{\displaystyle \int {\displaystyle \int {\displaystyle \int {\displaystyle \int {\displaystyle \int {\displaystyle \int \exp \left(-{\displaystyle \frac{m_a{V}_a^{\mathrm{2}}+m_b{V}_b^{\mathrm{2}}}{\mathrm{2}kT}}\right)}}}}}}V{r}_{\max }{(V)}^{\mathrm{2}}. \\ \cdot d{V}_{ax}d{V}_{ay}d{V}_{az}d{V}_{bx}d{V}_{by}d{V}_{bz}. \end{gathered}$$

The final result of the calculation, which is different for attractive and repulsive forces, is presented here.

For a repulsive force:

(21)

For an attractive force:

(22)\underset{\mathrm{\Delta }U>\mathrm{0}}{\beta (a,b)}={\left({\displaystyle \frac{\mathrm{3}}{\mathrm{4}\pi }}\right)}^{\mathrm{1}/\mathrm{6}}\sqrt{{\displaystyle \frac{\mathrm{6}kT}{{\rho }_p}}}{\left({\displaystyle \frac{\mathrm{1}}{v_a}}+{\displaystyle \frac{\mathrm{1}}{v_b}}\right)}^{\mathrm{1}/\mathrm{2}}{(v_a^{\mathrm{1}/\mathrm{3}}+v_b^{\mathrm{1}/\mathrm{3}})}^{\mathrm{2}}\left[{\displaystyle \frac{r_{\mathrm{0}}^{\mathrm{2}}}{s^{\mathrm{2}}}}-\left({\displaystyle \frac{r_{\mathrm{0}}^{\mathrm{2}}}{s^{\mathrm{2}}}}-\mathrm{1}\right)\exp \left(-{\displaystyle \frac{\mathrm{\Delta }U}{kT(r_{\mathrm{0}}^{\mathrm{2}}/s^{\mathrm{2}}-\mathrm{1})}}\right)\right].

When \mathrm{\Delta }U=\mathrm{0}, which is the case when there is no force, the result is the same as that of Eq. (2).

Comments

The results are presented separately, distinguishing the cases of attractive and repulsive forces. In fact, this is not totally exact. The two cases are \mathrm{\Delta }U>\mathrm{0} and , where \mathrm{\Delta }U=U(r_{\mathrm{0}})-U(s) is the difference of the potential between the infinitely far (if r_{\mathrm{0}} is correctly chosen) and the collision point. This can not reflect the variations of the force field between these two points. The whole calculation, according to the theory of central forces, does not depend on the value of the potential at each point but only at these two places. This means if the differential of potential \mathrm{\Delta }Uis positive, the force action will be considered as attractive, otherwise it will be seen as repulsive. The force action does not depend on the property of the force (attractive or repulsive) at each point. This is important, for instance, for the potential of Lennard-Jones which is repulsive at very short distance but attractive otherwise

Comparison

It is possible to compare the results with the case without forces. Equations (20) and (21) can be simplified as

(23)

(24)\underset{\mathrm{\Delta }U>\mathrm{0}}{\beta (a,b)}=\left[{\displaystyle \frac{r_{\mathrm{0}}^{\mathrm{2}}}{s^{\mathrm{2}}}}-\left({\displaystyle \frac{r_{\mathrm{0}}^{\mathrm{2}}}{s^{\mathrm{2}}}}-\mathrm{1}\right)\exp \left({\displaystyle \frac{-\mathrm{\Delta }U}{kT(r_{\mathrm{0}}^{\mathrm{2}}/s^{\mathrm{2}}-\mathrm{1})}}\right)\right]\underset{\mathrm{\Delta }U=\mathrm{0}}{\beta (a,b)}.

In the cases^{\mathrm{2}}/r_{\mathrm{0}}^{\mathrm{2}}\ll \mathrm{1}, which corresponds to a radius of the influence sphere much larger than the diameters of particles, which is generally the case, the final expression is simplified as

(25)\underset{\mathrm{\Delta }U>\mathrm{0}}{\beta (a,b)}=\mathrm{1}+{\displaystyle \frac{r_{\mathrm{0}}^{\mathrm{2}}}{s^{\mathrm{2}}}}\left[\mathrm{1}-\exp \left({\displaystyle \frac{-\mathrm{\Delta }U{s}^{\mathrm{2}}}{kT{r}_{\mathrm{0}}^{\mathrm{2}}}}\right)\right]\underset{\mathrm{\Delta }U=\mathrm{0}}{\beta (a,b)}

When r_{\mathrm{0}}/s is infinite, Eq. (25) becomes

(26)\underset{\mathrm{\Delta }U>\mathrm{0}}{\beta (a,b)}=\left(\mathrm{1}+{\displaystyle \frac{\mathrm{\Delta }U}{kT}}\right){\underset{\mathrm{\Delta }u=\mathrm{0}}{\beta (a,b)}|}_{\frac{r_{\mathrm{0}}}{s}\to \infty }

Figure 3 presents the variation of the ratio between the collision frequency function with consideration of force fields and without consideration of them. In Fig. 3, the r_{\mathrm{0}}/s is set at 10. The results show that the ratio varies in a very larger range. This reveals that the influence of the force field is significant compared with that of thermal energy of particles. The collision frequency varies greatly due to existence of force fields.

Conclusion

The collision rate for aerosols particle in the free molecule range was analyzed in the presence of particle force fields. The influence of attractive or repulsive forces were quantified, which could now be used to better describe the processes of collision, and one step further, of coagulation of aerosols particles. These results can be applied to different cases, by considering Coulomb forces or Van der Waals forces, which has not been done yet in the free molecule range.

References

[1] Friedlander S K. Smoke, Dust and Haze – Fundamentals of Aerosol Behavior. New York: Wiley-Interscience, 1977

[2] Benichou Y, Luo X W, Yu S Y. Several factors to be considered in aerosol particle coagulation. Journal of Wuhan University of Technology, 2007, 29(10): 93–95 (in Chinese)

[3] Fuchs N A. The Mechanics of Aerosols. London: Pergamon Press, 1964

[4] Kim D S, Park S H, Song Y M, Kim D H, Lee K W. Brownian coagulation of polydisperse aerosols in the transition regime. Journal of Aerosol Science, 2003, 34(7): 859-868

[5] Marlow W H. Derivation of aerosol collision rates for singular attractive contact potentials. Journal of Chemical Physics, 1980, 73(12): 6284-6287

[6] Alam M K. The effect of van der waals and viscous forces on the aersol coagulation. Aerosol Science and Technology, 1987, 6(1): 41-52

[7] Schroeder D V. Thermal Physics. San Francisco: Addison-Wesley Pub. Co, 2000

[8] Goldstein H. Classical Mechanics. 2nd. ed. San Francisco: Addison-Wesley Pub. Co, 1980

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