Thermodynamic Analysis of Brownian Motion-Induced Particle Agglomeration Using the Taylor-Series Expansion Method of Moments

Abstract

On the basis of binary perfectly inelastic collision theory, the time evolutions of kinetic energy and surface area for a particle agglomerate system, due to Brownian motion, are investigated by using the Taylor series expansion technology. The asymptotic behaviors over a long time period show a significantly negative power function of time. The thermodynamic constraints of this system are then obtained according to the principle of maximum entropy, which establishes a relationship of inequality between the first three particle moments and some physical parameters (i.e., surface tension and temperature). In the thermodynamic equilibrium state, this function provides a new approach for estimating the effect of molecular structure on surface tension of liquid polymers.

1. Introduction

Particle agglomeration is a common phenomenon in both nature and industrial applications, such as particle synthesis and soot formation processes. It plays a significant role in these aerosol processes by profoundly affecting the size distribution of a particle system [1], which strongly determines the physical properties of aerosol particles, such as light scattering, toxicity, deposition rate and diffusion. Nowadays with the escalation of fine particle pollution, the agglomeration processes are also widely used in the field of contamination control to improve removal efficiency, especially for the particles whose diameters are less than 2.5 μm [2,3]. The main principle is that through physical or chemical action, particles can coagulate with each other to form particles with larger particle size and then be removed efficiently. An appropriate approach for investigating the time evolution of particle size distribution (PSD) due to agglomeration is typically called the population balance equation (PBE) or the classic Smoluchowski equation (SE), which can be expressed as the following form [4]:

$$\frac{\partial n(\upsilon ,t)}{\partial t}=\frac{1}{2}{\displaystyle {\int }_0^{\upsilon }\beta ({\upsilon }_1,\upsilon -{\upsilon }_1)}n({\upsilon }_1,t)n(\upsilon -{\upsilon }_1,t)d{\upsilon }_1-{\displaystyle {\int }_0^{\infty }\beta ({\upsilon }_1,\upsilon )n(\upsilon ,t)}n({\upsilon }_1,t)d{\upsilon }_1$$

(1)

where n(υ, t) is the number density function of the particles with volume from υ to υ + dυ at time t; β(υ, υ₁) is the collision frequency function between particles with volume υ and υ₁.

Due to the strong non-linear integro-differential structure, the PBE is difficult to solve analytically. By trading off between accuracy and computational cost, three main numerical methods are proposed and developed, including the method of moments (MOM) [5,6], sectional method (SM) [7] and Monte Carlo method (MCM) [8]. It can’t be ignored that the analytical solutions show great merit in computational cost and direct physical insights into agglomeration mechanisms. Thus, some researchers focus on the asymptotic or analytical solutions of the moments of PSD by converting the original PBE to a system of ordinary differential equations (ODEs). Due to the complexity of its kernel function and the universality of Brownian motion, the study on the solution of the PBE for Brownian agglomeration is considered to be important but one of the most difficulties. Mainly using the log-normal method of moments (LG-MOM) [6] or the Taylor series expansion method of moments (TEMOM) [9], the asymptotic behavior of moments, due to Brownian coagulation (for spherical particles) and agglomeration (for agglomerates) over the entire particle size regimes [10,11,12,13], the analytical solution for Brownian coagulation in the free-molecule and the continuum regime [14], and so on, are obtained. These articles reveal that the geometric standard deviation will reach a constant for a long period of time, namely, the self-preserving size distribution theory [15].

Particle coalescence upon collisions subject to conservation of mass and momentum is called ballistic aggregation, but it is well known that the kinetic energy of this system decreases with time [16]. However, the loss of particle kinetic energy after collisions is rarely taken into account in the framework of PBE. Nowadays, with an assumption of a perfectly inelastic collision process, the rate of change for kinetic energy is correlated with that of particle number density, and the relationship of inequality between particle moments and some physical parameters (i.e., surface tension and temperature) for Brownian coagulation have been firstly proposed by Xie and Yu based on the principle of maximum entropy [17,18]. In this paper, we will extend their efforts to Brownian agglomeration, and the asymptotic behaviors of kinetic energy, surface area and entropy over a long period of time are obtained.

2. Theory and Model

2.1. Brownian Agglomeration

Particle agglomeration due to thermal motion is called Brownian agglomeration. Unlike spherical particles, agglomerates are not rigid structures and can be described as fractal morphology statistically. They are clusters of primary particles, which are ideally considered to be spherical with point contacts and uniform size. Considering the case of monodisperse primary particles, which form power law agglomerates, the Brownian agglomeration kernels β are represented as [1]:

$${\beta }_{FM}=B_1{({\upsilon }_i^{-1}+{\upsilon }_j^{-1})}^{1/2}{({\upsilon }_i^{1/D_f}+{\upsilon }_j^{1/D_f})}^2$$

(2)

$${\beta }_{CR}=B_2({\upsilon }_i^{1/D_f}+{\upsilon }_j^{1/D_f})({\upsilon }_i^{-1/D_f}+{\upsilon }_j^{-1/D_f})$$

(3)

Here, the subscripts FM and CR stand for agglomeration in the free molecular and continuum regimes, respectively; the constants $B_1={\left(\frac{6k_{\mathrm{B}}T}{{\rho }_p}\right)}^{\frac{1}{2}}{\left(3/4\pi \right)}^{2/D_f-1/2}{a}_{p0}^{2-6/D_f}$ and B₂ = 2k_BT/3μ, with k_B the Boltzmann’s constant; T is the temperature; μ is the gas viscosity; ρ_p is the particle density; a_p0 is the radius of a primary particle; υ is the particle volume; D_f is called the fractal dimension, which can be related to the arrangement of the primary particles within an agglomerate. It should be noted that D_f < 2 is not applicable for Equation (2) in physics [1], thus the following discussions are limited to a range of 2 ≤ D_f ≤ 3.

2.2. Taylor Series Expansion Method of Moments

With the definition of k-th order moment M_k,

$$M_k={\displaystyle {\int }_0^{\infty }{\upsilon }^{k}n(\upsilon )d}\upsilon$$

(4)

Equation (1) can be converted into a system of original differential equations by multiplying both sides with ν^k and then integrating over all particle sizes:

$$\frac{d{M}_k}{dt}=\frac{1}{2}{\displaystyle {\int }_0^{\infty }{\displaystyle {\int }_0^{\infty }\left[{\left(\upsilon +{\upsilon }_1\right)}^k-{\upsilon }^k-{\upsilon }_1{}^k\right]\beta (\upsilon ,{\upsilon }_1)}n(\upsilon ,t)n({\upsilon }_1,t)d\upsilon d{\upsilon }_1}$$

(5)

The main objective of all MOMs is to achieve the closure of Equation (5). In the classic TEMOM, this is accomplished in two procedures [9,19]: (1) the collision kernel is directly approximated by a two-variable third-order Taylor series expansion, for example, the power function ${({\upsilon }_i^{-1}+{\upsilon }_j^{-1})}^{1/2}$ in Equation (2) can be expanded with respect to mean volume u = M₁/M₀; (2) and all the higher and fractional moments are approximated by the polynomial equation with respect to the first three moments:

$$M_k=\frac{M_1^k}{M_0^{k-1}}\left[1+\frac{k(k-1)(M_C-1)}{2}\right]$$

(6)

Here the dimensionless moment M_C = M₀M₂/M₁² is the function of geometric standard deviation σ which can be noted as $\ln \left(M_C\right)/9={\ln }^2\sigma$ [6]. The moment equations based on TEMOM in the free molecule regime are obtained [11]:

$$\begin{array}{c}{\left.\frac{d{M}_0}{dt}\right|}_{FM}=-\frac{\sqrt{2}{B}_1{M}_0{}^2}{64D_f^4}{\left(\frac{M_1}{M_0}\right)}^{\frac{4-D_f}{2D_f}}(a_1{M}_C{}^2+a_2{M}_C+a_3)\\ {\left.\frac{d{M}_1}{dt}\right|}_{FM}=0\\ {\left.\frac{d{M}_2}{dt}\right|}_{FM}=-\frac{\sqrt{2}{B}_1{M}_1{}^2}{32D_f^4}{\left(\frac{M_1}{M_0}\right)}^{\frac{4-D_f}{2D_f}}(b_1{M}_C{}^2+b_2{M}_C+b_3)\end{array}$$

(7)

where the coefficients a₁, a₂, a₃, b₁, b₂, b₃ are:

$$\begin{array}{c}{a}_1=D_f^4-24D_f^3+70D_f^2-48D_f+16\\ a_2=54D_f^4-144D_f^3+52D_f^2+96D_f-32\\ a_3=73D_f^4+168D_f^3-122D_f^2-48D_f+16\\ b_1=3D_f^4+16D_f^3+10D_f^2-16D_f-16\\ b_2=2D_f^4-96D_f^3-212D_f^2+32D_f+32\\ b_3=-133D_f^4+80D_f^3+202D_f^2-16D_f-16\end{array}$$

(8)

Now the most important moments for describing the particle dynamics, namely, the particle number density M₀, total particle volume M₁ and a polydispersity variable M₂, can be obtained. Here, M₁ remains constant due to the rigorous mass conservation requirement. The corresponding moment equations in the continuum regime are:

$$\begin{array}{c}{\left.\frac{d{M}_0}{dt}\right|}_{CR}=-\frac{B_2{M}_0{}^2}{4D_f^4}(p_1{M}_C{}^2+p_2{M}_C+p_3)\\ {\left.\frac{d{M}_1}{dt}\right|}_{CR}=0\\ {\left.\frac{d{M}_2}{dt}\right|}_{CR}=\frac{B_2{M}_1{}^2}{2D_f^4}(p_1{M}_C{}^2+p_2{M}_C+p_3)\end{array}$$

(9)

where the coefficients p₁, p₂, p₃ are:

$$p_1=1-D_f^2;p_2=-2+6D_f^2;p_3=1-5D_f^2+8D_f^4$$

(10)

2.3. Principle of Maximum Entropy

As a characteristic function composed of internal energy U, total particle volume M₁, and particle number M₀, the rate of change for entropy S of a disperse system can be expressed as:

$$\frac{dS}{dt}=\frac{\partial S}{\partial M_0}\frac{d{M}_0}{dt}+\frac{\partial S}{\partial U}\frac{dU}{dt}+\frac{\partial S}{\partial M_1}\frac{d{M}_1}{dt}$$

(11)

According to the thermodynamic analysis [18], the rate of change for S can be arranged and then correlated with that of M₀, the particle kinetic energy k_e, and the particle specific surface area s:

$$\frac{dS}{dt}=-k_B\ln (M_0{\lambda }_{th}^3)\frac{d{M}_0}{dt}+\frac{1}{T}\left(\frac{d{k}_e}{dt}+\gamma \frac{ds}{dt}\right)$$

(12)

in which λ_th is the thermal wavelength and γ is the surface tension. Thus, the focal point is to determine dk_e/dt and ds/dt. With the assumption of simplified physical model according to the binary perfectly inelastic collision theory, the loss of particle kinetic energy after collision for two colliding particles and the whole system are [18]:

$$\Delta k_e=-\frac{k_{b}T}{2}\left(1-\frac{2\sqrt{{\upsilon }_1{\upsilon }_2}}{{\upsilon }_1+{\upsilon }_2}\right)\le 0$$

(13)

$$\frac{d{k}_e}{dt}=-\frac{k_{B}T}{4}{\displaystyle {\int }_0^{\infty }{\displaystyle {\int }_0^{\infty }\left(1-\frac{2\sqrt{{\upsilon }_1{\upsilon }_2}}{{\upsilon }_1+{\upsilon }_2}\right)\beta ({\upsilon }_1,{\upsilon }_2)}n({\upsilon }_1,t)n({\upsilon }_2,t)d{\upsilon }_{2}d{\upsilon }_1}$$

(14)

Assuming that υ₁ is the larger particle, the relative loss of k_e increases with a larger ratio of υ₁ to υ₂, which is illustrated in Figure 1. This shows that a wider range of PSD, namely, a larger M_C, would lead to a more rapid reduction in k_e. Substituting Equation (2) into the above equation and then using the Taylor series expansion technology, we can get the rate of change for kinetic energy in the free molecular regime:

$${\left.\frac{d{k}_e}{dt}\right|}_{FM}=k_{B}T\frac{x_1{M}_C^2+x_2{M}_C+x_3}{a_1{M}_C{}^2+a_2{M}_C+a_3}\frac{d{M}_0}{dt}$$

(15)

in which x₁, x₂, x₃ are:

$$\begin{array}{c}{x}_1=\left(D_f^4-12D_f^3+8D_f^2\right)/2\\ x_2=15D_f^4+12D_f^3-8D_f^2\\ x_3=\left(-31D_f^4-12D_f^3+8D_f^2\right)/2\end{array}$$

(16)

As the structure of agglomerate is complex, modeling of its surface area is even more difficult, given the scarcity of experimental data. It is also very difficult to numerically determine which part of primary particles are the boundary particles and which part of the surface of these boundary particles forms the agglomerate surface. For an ideal case that k primary particles agglomerate with point contacts, its surface area equals $4k\mathsf{\pi }{a}_{p0}^2$, where a_p0 is the radius of primary particle [20]. Obviously, it is more suitable for chain-like structures with D_f→1 but not compact aggregates with D_f →3. In a statistical sense, the collision radius of agglomerates composed of k monomer is [1]:

$$r=A{a}_{p0}{\left(\frac{\upsilon }{{\upsilon }_0}\right)}^{1/D_f}=A{a}_{p0}{k}^{1/D_f}$$

(17)

where A is the dimensionless proportionality constant and can be assumed to be in unity to simplify calculations. In this paper, we will use this collision radius to calculate the surface area approximately:

$$s=4\pi r^2=4\pi k^{2/D_f}{a}_{p0}^2=B_3{\upsilon }^{2/D_f}$$

(18)

in which the constant $B_3=4\pi {\left(3/4\pi \right)}^{2/D_f}{a}_{p0}^{2-6/D_f}$. Thus, for chain-like structures with D_f = 2, s = $4k\mathsf{\pi }{a}_{p0}^2$ equals to that of an agglomerate without necking and for compact aggregates with D_f = 3, s = (36π)^1/3υ^2/3 equals to that of a spherical particle. Apparently, the agglomerates composed of the same number of primary particles with smaller D_f would have larger specific surface area and collision radius. Now the total surface area of this system can be expressed as:

$$s={\displaystyle {\int }_0^{\infty }s}n(\upsilon ,t)d\upsilon =B_3{M}_{2/D_f}$$

(19)

and the rate of change for s can be written as:

$$\frac{ds}{dt}=B_3\frac{d{M}_{2/D_f}}{dt}$$

(20)

$$\frac{ds}{sdt}=\frac{d{M}_{2/D_f}}{M_{2/D_f}dt}$$

(21)

where the fractional moment M_2/Df is approximated by using Equation (6):

$$\frac{ds}{sdt}=\frac{d{M}_{2/D_f}}{M_{2/D_f}dt}$$

(22)

and its derivative with time t can be achieved:

$$\frac{d{M}_{2/D_f}}{dt}=\frac{(1-2/D_f)}{D_f}\frac{M_1^{2/D_f}}{M_0^{2/D_f}}\left[(1/D_f-1)(2M_C-2-D_f)\frac{d{M}_0}{dt}-\frac{M_0^{2}d{M}_2}{M_1^{2}dt}\right]$$

(23)

Combining the first and third equations in Equation (7) gives:

$${\left.\frac{d{M}_2}{dt}\right|}_{FM}=\frac{2M_1^2}{M_0^2}\frac{b_1{M}_C{}^2+b_2{M}_C+b_3}{a_1{M}_C{}^2+a_2{M}_C+a_3}\frac{d{M}_0}{dt}$$

(24)

Then Equation (23) can be rearranged as:

$${\left.\frac{d{M}_{2/D_f}}{dt}\right|}_{FM}=\frac{(1-2/D_f)}{D_f}\frac{M_1^{2/D_f}}{M_0^{2/D_f}}\left[\begin{array}{l}(1/D_f-1)(2M_C-2-D_f)-\\ \frac{2(b_1{M}_C{}^2+b_2{M}_C+b_3)}{a_1{M}_C{}^2+a_2{M}_C+a_3}\end{array}\right]\frac{d{M}_0}{dt}$$

(25)

Substituting the above equation into Equation (21), the rate of change for s in the free molecular regime has the following form:

$${\left.\frac{ds}{dt}\right|}_{FM}=\frac{(1-2/D_f)\left[(1/D_f-1)(2M_C-2-D_f)-\frac{2(b_1{M}_C{}^2+b_2{M}_C+b_3)}{a_1{M}_C{}^2+a_2{M}_C+a_3}\right]}{D_f+(2/D_f-1)(M_C-1)}\frac{sd{M}_0}{M_{0}dt}$$

(26)

Analogously, the rate of change for particle kinetic energy and surface area in the continuum regime can be calculated as:

$${\left.\frac{d{k}_e}{dt}\right|}_{CR}=k_{B}T\frac{q_1{M}_C{}^2+q_2{M}_C+q_3}{p_1{M}_C{}^2+p_2{M}_C+p_3}\frac{d{M}_0}{dt}$$

(27)

$${\left.\frac{ds}{dt}\right|}_{CR}=\frac{(1-2/D_f)\left[(1/D_f-1)(2M_C-2-D_f)+2\right]}{D_f+(2/D_f-1)(M_C-1)}\frac{sd{M}_0}{M_{0}dt}$$

(28)

where q₁, q₂, q₃ are noted as:

$$\begin{array}{c}{q}_1=(-9D_f^4+12D_f^2)/16\\ q_2=(34D_f^4-24D_f^2)/16\\ q_3=(-25D_f^4+12D_f^2)/16\end{array}$$

(29)

Finally, the rate of change for S can be found:

$$\frac{dS}{dt}=\frac{1}{T}\left(-k_{B}T\ln (M_0{\lambda }_{th}^3)+k_{B}T{C}_2+\frac{\gamma s}{M_0}{C}_1\right)\frac{d{M}_0}{dt}$$

(30)

in which C₁, C₂ are functions of the dimensionless moment M_C and fractal dimension D_f:

$${\left.C_1\right|}_{FM}=\frac{(1-2/D_f)\left[(1/D_f-1)(2M_C-2-D_f)-\frac{2(b_1{M}_C{}^2+b_2{M}_C+b_3)}{a_1{M}_C{}^2+a_2{M}_C+a_3}\right]}{D_f+(2/D_f-1)(M_C-1)}$$

(31)

$${\left.C_1\right|}_{CR}=\frac{(1-2/D_f)\left[(1/D_f-1)(2M_C-2-D_f)+2\right]}{D_f+(2/D_f-1)(M_C-1)}$$

(32)

$${\left.C_2\right|}_{FM}=\frac{x_1{M}_C^2+x_2{M}_C+x_3}{a_1{M}_C{}^2+a_2{M}_C+a_3}$$

(33)

$${\left.C_2\right|}_{CR}=\frac{q_1{M}_C{}^2+q_2{M}_C+q_3}{p_1{M}_C{}^2+p_2{M}_C+p_3}$$

(34)

From the viewpoint of the second law of thermodynamics, the entropy of an isolated system will never decrease: dS/dt ≥ 0. Moreover, the total particle number M₀ will decrease with time due to agglomeration: dM₀/dt < 0. Thus, the thermodynamics constraints for Brownian agglomeration at a certain temperature and pressure can be obtained:

$$\frac{\gamma s}{k_{B}T}\le \frac{M_0(\ln (M_0{\lambda }_{th}^3)-C_2)}{C_1}$$

(35)

The equality would hold in the thermodynamic equilibrium state, and the critical time to reach this state can be determined. Moreover, the growth of the mean particle size M₁/M₀ will tend to a limit depending on the operating temperature and specific surface energy for thermal agglomeration technology.

3. Results

According to the self-preserving size distribution theory, the dimensionless particle moment M_C will tend to a constant at long time periods, and the asymptotic solutions of particle moments based on the TEMOM model can be found [11]. Here, the results are listed in the Appendix A. In the free molecular regime, the asymptotic solution of kinetic energy can be solved by directly integrating Equation (15) with respect to t:

$${\left.k_e\right|}_{FM}=C_3+k_{b}T{C}_2{M}_0\to k_{b}T{C}_2{[g_{1}t]}^{-\frac{2D_f}{3D_f-4}}$$

(36)

where C₃ = k_e(t₁) − k_eTC₂(t₁)M₀(t₁) is the integral constant, t₁ is the critical time in which the particle size distribution approaches self-preserving and the definition of g₁ is shown as Equation (A7). In our previous work [21], a criterion to calculate this critical time has been given based on the asymptotic solution of M₀ in the continuum regime, which can also be available in the free molecular regime. Now the effect of primary particle size a_p0 on k_e can be obtained, which is as the same as that on M₀:

$${\left.k_e\right|}_{FM}\propto a_{p0}{}^{\frac{12-4D_f}{3D_f-4}}$$

(37)

Thus, its relative dissipative rate becomes:

$${\left.\frac{d{k}_e}{k_{e}dt}\right|}_{FM}=\frac{d{M}_0}{M_{0}dt}=-\frac{2D_f}{3D_f-4}{t}^{-1}$$

(38)

The asymptotic solution of surface area and the effect of primary particle size can be expressed as the following forms after substitution of Equations (A6) and (22) into Equation (19):

$${\left.s\right|}_{FM}\to B_3{M}_1^{2/D_f}\left[1+\frac{(2/D_f-1)(M_C-1)}{D_f}\right]{[g_{1}t]}^{-\frac{2D_f-4}{3D_f-4}}$$

(39)

$${\left.s\right|}_{FM}\propto a_{p0}{}^{-\frac{6-2D_f}{3D_f-4}}$$

(40)

and its relative dissipative rate becomes:

$${\left.\frac{ds}{sdt}\right|}_{FM}=\frac{D_f-2}{D_f}\frac{d{M}_0}{M_{0}dt}=-\frac{2D_f-4}{3D_f-4}{t}^{-1}$$

(41)

For simplification and without loss of generality, the calculation can be non-dimensionalized through the following relations: M₀ * = M₀ /M₀₀, M₁ * = M₁/M₁₀, M₂ * = M_C0M₂/M₂₀, $t^{*}=t{B}_1{M}_{00}^{\left(3Df-4\right)/2Df}{M}_{10}^{\left(4-Df\right)/2Df}$, k_e * = k_e/(M₀₀k_bT), $s^{*}=s{M}_{00}^{2/Df-1}{M}_{10}^{-2Df}/B_3$. Then the Equation (7) coupling with Equations (15) and (19) can be solved numerically by means of fourth-order Runge–Kutta method with the initial dimensionless conditions set as M₀₀ = 1, M₁₀ = 1, M₂₀ = 4/3, k_e0 = 1/2 (the star symbol ‘*’ is omitted thereafter). The numerical and asymptotic solutions of kinetic energy and surface area are shown in Figure 2. According to the principle of equipartition of energy, the agglomerates share the molecular thermal motion of the fluid and have the same initial kinetic energy, thus the dissipative rate of k_e strongly depends on the collision rate. The evolutions of kinetic energy with time show the larger descent at lower fractal dimension because of the larger collision radius, which result in a more rapid agglomerated rate [22]. Oppositely, the contacting surface between primary particles in an agglomerate with smaller fractal dimension is less than that in an agglomerate with larger fractal dimension, thus the decay of surface area shows the reverse trend in the dual role of the higher specific surface area and more rapid agglomerated rate.

Analogously, the asymptotic solutions of kinetic energy and surface area, as well as their relative dissipative rates, in the continuum regime can be expressed as:

$${\left.k_e\right|}_{CR}\to k_{b}T{C}_2{M}_0=k_{b}T{C}_2{g}_2^{-1}{t}^{-1}$$

(42)

$${\left.\frac{d{k}_e}{k_{e}dt}\right|}_{CR}=\frac{d{M}_0}{M_{0}dt}=-t^{-1}$$

(43)

$${\left.s\right|}_{CR}\to B_3{M}_1^{2/D_f}\left[1+\frac{(2/D_f-1)(M_C-1)}{D_f}\right]{[g_{2}t]}^{2/D_f-1}$$

(44)

$${\left.\frac{ds}{sdt}\right|}_{CR}=\frac{D_f-2}{D_f}\frac{d{M}_0}{M_{0}dt}=-\frac{D_f-2}{D_f}{t}^{-1}$$

(45)

where g₂ is a function of D_f showed as Equation (A12). The results are showed in Figure 3. These allow us to simplify the rate of change for S as the asymptotic form in both the free molecular and continuum regime:

$$\frac{dS}{dt}=\frac{1}{T}\left(-k_{B}T{M}_0\ln (M_0{\lambda }_{th}^3)+k_e+\frac{D_f-2}{D_f}\gamma s\right)\frac{d{M}_0}{M_{0}dt}$$

(46)

And the corresponding thermodynamics constraints are:

$$\frac{\gamma s}{k_{B}T}\le \frac{M_0(\ln (M_0{\lambda }_{th}^3)-C_2)}{1-2/D_f}$$

(47)

4. Discussion

The above equation establishes an inequality relationship between moments and some physical parameters, such as temperature and specific surface energy, and the equality holds if and only if the system reaches the thermodynamic equilibrium. Obviously, increasing temperature leads to decreasing particle number density and greater mean volume, which can be useful for dust collection efficiency. It also shows the effect of molecular structure on surface tension. By substituting the expression of surface area into this equation, we can get the following formula in the thermal equilibrium state:

$$\gamma B_3\frac{M_1^{2/D_f}}{M_0^{2/D_f}}\left[1+\frac{(2/D_f-1)(M_C-1)}{D_f}\right]=\frac{\left(\ln (M_0{\lambda }_{th}^3)-C_2\right)}{1-2/D_f}{k}_{b}T$$

(48)

For liquid pure substance, M₀λ_th³ usually takes the value as V_m in the free molecular regime and M₁/M₀ = V_m /N_A, where V_m is the molar volume and N_A is the Avogadro constant. A modification coefficient γ_∞, which is equal to the value of surface tension at infinite molar volume, should be introduced because the surface tension decreases almost linearly with the increase of temperature. Then a correction function of molar volume can be constructed:

$$\gamma ={\gamma }_{\infty }-\frac{k_1}{V_m{}^{2/3}}$$

(49)

where k₁ is a function of the temperature and fractal dimension:

$$k_1=\frac{\left(\ln V_m-C_2\right)}{V_m{}^{2/D_f-2/3}}\frac{N_A^{2/D_f}{D}_f}{B_3\left(1-2/D_f\right)\left[D_f+(2/D_f-1)(M_C-1)\right]}{k}_{b}T$$

(50)

The surface tension increases as molar volume increases and tends toward the constant γ_∞ at infinite molar volume, and it decreases monotonously with increasing temperature and tends to zero at the critical temperature. Unfortunately, it should be noted that some important factors, i.e., the effect of end groups, cannot be considered. It also can be written as the molecular weight-surface tension relationship with V_m = M/ρ,

$$\gamma ={\gamma }_{\infty }-\frac{k_2}{M^{2/3}}$$

(51)

$$k_2=\frac{\left(\ln (M/\rho )-C_2\right)}{M^{2/D_f-2/3}}\frac{{(\rho N_A)}^{2/D_f}{D}_f}{B_3\left(1-2/D_f\right)\left[D_f+(2/D_f-1)(M_C-1)\right]}{k}_{b}T$$

(52)

where M is the molecular weight and ρ is the density. Some research shows that the correlation with molecular weight is better than that with molar volume for alkanes and perfluoro alkanes, but the discrepancy can be ignored for the siloxanes [23]. The effect of fractal dimension on the slope k₂ is illustrated in Figure 4 for n-alkanes at 0 °C, where a_p0 = 0.2 nm is equal to the radius of methane and n is the number of carbons. Compared to bulks, the surface tension of molecules with long-chain structure generally increases due to the large contact areas and intermolecular forces, which leads to a small slope k₂. In the range of 2.7 to 2.8, the result is mostly close to the value k₂ = 360 of least-squares fitting based on experimental data [24].

5. Conclusions

On the basis of the theory of maximum entropy and binary perfectly inelastic collision, the thermodynamic constraints of Brownian coagulation for spherical particles are extended to agglomerates, and a relationship of inequality between particle moments and some physical parameters is established using the TEMOM. Meanwhile, the evolutions of kinetic energy and surface area with time are presented, as well as their asymptotic behaviors. While some of our present simplifying assumptions will have to be relaxed, even our present results are of potential interest for a number of applications, for example, the estimation of surface tension of liquid polymers and the enhancement of dust collection efficiency.