On the “Boltzmann Law” in bipolar charging

Pergamon

J AerosolSct~VoL 25, No 4, pp 617-621, 1994 Copyright ~ 1994 Elsevier Science Lid Printed m Great Britain All riots mmrved 0021-8502/94 $7.00+0.00

0021-gs0204)E0e3o.2

ON THE "BOLTZMANN

LAW" IN BIPOLAR CHARGING Y. S. MAYYA

Environmental AssessmentDivision,Bhabha AtomicResearchCentre, Bombay400 085, India (First received 21 September 1993; and in final form 31 December 1993)

Almtrm--Matsoukas (1994, J. Aerosol Sci. 25, 599) presents arguments in defence of the equilibrium approach to calculating bipolar charge distributions in aerosols. Here we examine the general validity of these arguments by postulating a thermodynamic equilibrium process based on the ion adsorption-desorption idea of Fuchs, along with the correct expression for the electrostatic energy. The charge distributions for this hypothetical process are found to be at complete variance with the conventional Boltzmann law. This quantifies the classical criticism of Fuchs against the a priori use of an equilibrium hypothesis. It also strongly supports the view that bipolar charging is a nonequilibnum steady-state process in all regimes.

ARGUMENTS AGAINST THE E Q U I L I B R I U M APPROACH Matsoukas's (1994) attempt at setting up criteria for the validity of the equilibrium approach, originally postulated by Keefe et al. (1959), for the case of bipolar charging in aerosols motivates some serious thinking on this question. Contrary to his claim, the process is without doubt irreversible both in the macroscopic and in the microscopic sense of statistical mechanics. In an otherwise closed system, one requires either a constant creation of ion pairs (input of energy) or their constant influx (input of matter), in order to keep the process going. In the absence of these external inputs, sooner or later all the ions would be exhausted and the charging process would cease to exist. This is contrary to thermally activated equilibrium processes which attain stationarity in time without external inputs of matter and energy. Similarly, the microscopic irreversibility of the charging process refers to the fact that, while the charged gas molecules (ions) of either sign transfer their charges to the particles, the particles do not transfer their charges back to neutral molecules. This is symbolised by the unidirectional reaction P + M ± --, P± + M °,

(A)

where P refers to the particle and M to the molecules. This is a variant of Fuchs' criticism (1963) which states that for the equilibrium approach to be valid, ions should be adsorbed and desorbed as ions, thereby setting up an adsorption-desorption equilibrium. The second objection to the equilibrium approach concerns the energetics of the problem. Keefe et al. treat charge as a canonical degree of freedom and write the electrostatic energy of a particle of radius R carrying q units of net charge as

E= 82q2/2R (~ is the unit of electric charge),

(1)

and obtain the well-known 'Boltzmann law' of charge distribution,

f(q) =f(0) exp [ - ~2q2/2RkT-].

(2)

However, equation (1) is to be interpreted as the "self-energy" of creating a total charge 'qd on the particle and not as the energy of assembling pre-existing unit charges. It refers to the work done in building up a charge q~ by bringing infinitisimal charges from infinity, against the repulsive potential of the sphere i.e. 82q2/2R--S~(q'8/R)sdq'. For discrete charges, the integration procedure is not valid; it should be replaced by a sum and we obtain

=(8/R). ~ +(2~/R). ~ +(3~/R). ~ + . . . + {(q- 1)8/R}. 8, --82(q 2 - Iql)/2R.

(3) 617

618

Y.S. MAYYA

This formula is analogous to the Coulomb energy of protons in the nucleus (Kaplan, 1963). It suggests that the equilibrium distribution should assume the form

f(q) =f(O) exp [

-

-

fl(q2 _

Iql)/23,

(4)

where

fl=e2/RkT.

(5)

Equation (4) implies that f(1)=f(0), i.e. neutral and singly charged particles should occur with equal probabilities. Since this is not supported either by continuum kinetics or by experimental evidence, the equilibrium arguments must fail. This reasoning, while providing the basic ground for quantitative arguments against equilibrium, stands corrected by the remarkable observation (Filippov, 1994) that electrostatic energy in bipolar systems cannot be uniquely defined in terms of the net charge alone. For example, for a static distribution of a mixture of n positive and m negative point charges on the surface of a sphere, the energy is not expressible as a function of the net charge q-= (n-m) alone, but is given by (see Appendix)

E(n, ra)=(e2/2R)[(n-m)2-(n+m)],

(n,m=0, 1,2,...).

(6)

This follows from the use of the well-known formula,

E=(e2/2) ~, q,q,/Rtj

(7)

(R~= distance between charges i,j). Equation (6) reduces to equation (3) only for the special case of unipolar charging (i.e. either n or m = 0) and hence, in general, the net charge cannot be considered as the correct variable for the energy of the system. This provides a stronger argument against the energy-equilibrium approach of Keefe et al. CHARGE DISTRIBUTION OF AEROSOLS IN (HYPOTHETICAL) THERMAL ADSORPTION-DESORPTION EQUILIBRIUM Supposing we postulate a hypothetical equilibrium counterpart of the charging process, what would be the charge distribution attained by the particles? In this system, net charge cannot be treated as a canonical variable (e.g. position and momenta) but should be treated as a number variable (number of positive charges-number of negative charges). Number fluctuations are handled by grand canonical approaches (Huang, 1963) involving thermal free-energy changes. The correct equilibrium version of aerosol charging should involve the reversible model (A) and take into account the total (thermal +electrostatic) free-energy changes associated with particle-neutral molecule electron transfers, in addition to those for the particle-ion electron transfers. As this is somewhat difficult to analyse, we examine Fuchs' idea (1963) by assuming that ions are indestructible entities which are adsorbed and desorbed by the particle as ions. For such a system, an absorption-desorption equilibrium will exist at the particle surface. If C ± are the ion concentrations in the gas and if the particle contains n positive ions and m negative ions, the reaction schemes + Un,m P{n, m} , ' P { n - 1, m} + M + + gn,m Rn.m P{n,m} ~ ' P{n,m-1}+MKn,m

(g)

are micro-reversible and being activated by thermal fluctuations, they will be equilibrium processes. The electrostatic part of the free energy of the particle is as given by equation (6). The associated thermal free energy change will be -RTIn(C±/Co) per ion, where Co = Ca e x p ( -

#/kT),

(8)

On the "Boltzmann law" m bipolar charging

619

is the equilibrium concentration at which the thermal free-energy change will be zero (Cs is the inverse specific volume of the ions in the adsorbed phase and/t is the adsorption energy). In the above and in what follows, we assume that the equilibrium concentration Co is the same for both the species. Hence, the total free energy of the n, m system is

W(n, m) = - kT[n In (C+/Co) + m ln(C-/Co)] + E(n, m).

(9)

The forward and the backward reaction rates in (B) will now strictly bear the ratios dictated by the differences in the respective free energy levels. The distribution in various n, m states follows from the Boltzmann factor for W(m,n) [equation (9)]. Let us first consider unipolar charging with positive ions maintained at a concentration C. The adsorption-desorption equilibrium permits a stationary unipolar distribution. As the net charge q= n in this case, upon setting m=O in equations (9) and (6) and then applying the Boltzmann factor, one obtains an equilibrium unipolar charge distribution

f(q) =f(0) exp [ - {flq2 _ q(fl + 2 INS}/2],

q = 0, 1, 2 . . . .

(10)

where S-C/Co is the ratio of the maintained and the equilibrium ion concentrations. Significantly, f(1)/f(O)= S and not unity as expected from pure electrostatic arguments of equation (4). For S> 1, equation (10) possesses a peak at qm=(1/2)+(lnS)/fl. Next, consider the case of symmetric bipolar charging. It is assumed that both the ionic species are maintained at the same concentration C + = C - = C, and we define the ratio $ =-C/Co. Let n, m specify the number of positive and negative ions on the particle whose free energy is defined in equations (9) and (6). The Boltzmann factor for the probability P(n, m) of occupancy of this state, is P(n, m) ~ A ¢xp [ - W(n, m)/kT],

(11)

where A is a normalization constant independent of n, m. We seek a distribution function f(q) for the next charge

q=n-m;

n,m=0, 1,2 . . . . . (q=0, _+1, _+2. . . . ),

(12)

by summing P(n, m) over appropriate values of n, m. For q > 0, U

f(q)- ~ P(n,n-q),

q=0,1,2 . . . .

(13)

n=q

and a similar sum may be defined for q <0. In equation (13), N is assumed to be large, but still kept finite for reasons of summability as will be obvious from the following. From equations (6), (9), (11) and (13), we obtain, upon rearrangement. N

f(q)=exp[-(fl/2)(q2 +q)-qlnS] A ~ exp[n(fl+ 21nS)],

q>0.

(14)

nmq

The sum on the rhs (geometric series) will diverge as N~oo, if fl+21nS>0, but will remain finite if fl + 2InS < 0. The divergence is not a serious problem, for it may be absorbed in the quantity f(0) (to be renormalised later) defined as N

f(O)- A ~ exp[n(fl + 21nS)] = A exp{(N+ 1)(fl+21nS)}- 1 .=o {exp(fl+21nS)- I}

(15)

In the convergent case (fl+21nS
f(q)ffif(O)exp[-{[3q2+jql.[[3+21nSI}/2],

q=0, +1,-1-2 . . . .

(16)

The condition Y~f(q)= 1 yields the 'renormalised' neutral fraction: f(0)= 1/[1 +2 ~ exp{-(flq2 +qJfl+21nSJ)/2}]. q=l

(17)

620

Y S MAYYA

Equation (16) differs from the "Boltzmann law" of Keefe et al. in two fundamental respects. (i) It has an explicit ion concentration dependence through the ratio S ( - C / C o ) . This is a reflection of the fact that due to the additive nature of the thermal free energies of the ionic species, the effects of the positive and negative ion concentrations do not cancel out for symmetric charging occurring at equilibrium. (il) The variance of the equilibrium distribution does not indefinitely increase with the particle size (fl--*0), but attains a stationary value, 2 S / I S - I I 2. These features are not reflected in the Boltzmann law and hence it cannot be claimed to be an equilibrium distribution. Although the conventional Boltzmann law is coincidentally in agreement with the result of the well-founded kinetic approach in the limit fl<< 1, it generally underestimates the charged fractions (Hoppel and Frick, 1986). Equation (16) will, in general, underestimate them even further. This clearly demonstrates the non-equilibrium nature of the observed aerosol charging processes. One can generalise these arguments to the case of asymmetric charging (C + ~ C =) of our equilibrium system. By performing an analysis similar to the one above, we can show that f(q)=f(O)exp[-{flq2+lq[ • I t + I n S +. S - { - q l n ( S + / S - ) ~ / 2 ] , q=O, +_1, +2 . . . .

(18)

where S + - C±/Co are ratios of the maintained and the equilibrium concentrations for the two ionic species. Equation (18) goes over smoothly to the symmetric bipolar formula when S + = S- and to the one-sided unipolar formula (equation (10)) when, say, S- --*0. It does not depend solely upon the asymmetric factor S+/S - as in the case of experimentally observed asymmetric charging in aerosols, but also depends on the product S + S-. This, once again, demonstrates that in true equilibrium systems, it is the ratios S ± which independently govern the charge distribution, but not the concentration ratio, C+/C -. We have thus demonstrated that the hypothetical adsorptlon-desorption charging process occurring at true thermodynamic equilibrium leads to charge distributions which are at complete variance with those observed in aerosols as well as the conventional Boltzmann law. This conclusion quantifies the qualitative criticism of Fuchs against postulating Boltzmann equilibrium a priori in bipolar charging. Until experimental evidence confirms a spontaneous desorption of charges from particles (say, in unipolar charging), we may safely assume that the process occurs far from thermodynamic equilibrium and is kineticcontrolled in all regimes. The nomenclature 'Boltzmann law' may, however, be retained for historical reasons and for brevity, but it does not appear to have anything to do with the Gibbs-Boltzmann hypothesis of equilibrium statistical mechanics. Acknowledoements--I am thankful to Dr K S. V. Nambl and Dr D V Gopmath for their encouragement and support and to Mr S. L. Naraslmhan for assisting with this manuscript

REFERENCES Flhppov, A. (1994) J. Aerosol Scz. 25, 611. Fuchs, N. A. (1963) Geofisica Pura e Applicata 56, 185. Hoppel, W. A. and Frick, G. M. (1986) Aerosol Scl. Technol. 5, 1 Huang K. (1963) Statistical Mechanics. Wiley, New York. Kaplan, I. (1963) Nuclear Physics, p. 510. Addison Wesley, Rerding, Massachusetts. Keefe, D., Nolan, P. J. and Rich, T. A. (1959) Prec. Royal lrish Acad A 60, 27 Matsoukas, T. (1994) J. Aerosol Sc~. 25, 599.

APPENDIX Proof of E = (e2/2R) [(n - m)2 - (n + m)] Let n positive and m negative point charges be distributed randomly on the surfaze of a sphere of radim R. We need to calculate the energy of all pairs by excluding self-interaction and averaging it wrt all configurations, We first fix a positive charge i and calculate its configuration averaged energy of interaction with all other charges. This is achieved by spreading the remaining ( n - 1) positive and m negative charges uniformly over the surface. The total energy of all the positive charges is n times this. We can calculate the energy of interaction of all the negative charges m a similar manner. Let a ÷ =8(n-m-1)/4~R2(charge density excluding the ith +ve charge) t~- = e(n - m + 1)/4nRZ(charge density excluding the lth -- ve charge).

On the "Boitzmann law" in bipolar charging

621

The computations are made with reference to the following figure. The standard formula now goes over to

R

x

ith fixed charge

Fig. 1. E l ( l / 2 ) ~ q, qj/r,j (i runs from I to n a n d j from 1 to m) i,,j

--,(1/2) ~ (+,.,~+)

"

1=1

f

dS/x+(1/2)y+(-~.~-) dS/~, " f J=l

where (IS is the area of the differential ring located at a distance x from the fixed charge. From the figure dS=2xR',~nO dO, (0<0<~) and x=2Rcos(O/2). With thss,

f dS/x=27rRf~sin(O/2)dO=4~tR. Hence, E=(e2/2R) In(n-m- 1)-rn(n-rn+ 1)]

=(eZ/2R) ['(n--m) 2 - ( n + m ) ] . This proves the result.

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