Reactions in emulsions—I

Chemical Engineering Science, 1967, Vol. 22, pp. 35-42. Pergamon Press Ltd., Oxford.

Printed in Great Britain.

Reactjons in emulsions-1 The rate of bromination of benzene droplets in a dense, vigorously agitated aqueous emulsion I. GALLILY,G. M. J. SCHMIDTand E. BERNSTEIN* Department

of X-ray Crystallography,

The Weizmann Institute of Science, Rehovoth, Israel

(Received 15 February 1966; in revised form 31 May 1966) Abstract-The rate of bromination of benzene droplets by acidified hypobromous acid in a turbulent, dense aqueous emulsion has been analysed theoretically and studied experimentally. The bromination is found to conform to a model in which the reactions are assumed to occur essentially in the continuous medium, the kinetics is chemically controlled, and the species distributed almost thermodynamically between the phases.

1. INTRODUCTION CHEMICALreactions in emulsions are important from the practical and the fundamental points of view; however, apart from some studies on the oxidation [l] and saponification of oils [2, 31 and the alkaline hydrolysis of esters [4, 51, and besides the specific though extensive research on emulsion polymerization, no general investigations in this field have as yet appeared. Reactions in emulsions, like in other heterogeneous systems, can be classified as (a) true surface reactions, (b) apparent surface reactions and (c) as combinations of (a) and (b) [6]. In a true surface reaction one at least of the reactants forms part of a surface, and the rate of the chemical reaction enters into the boundary conditions for the equations of mass and heat transfer [7]. In an apparent surface reaction the chemical process occurs near the interface on one or both of its sides. We have begun our study of reactions in emulsions by confining ourselves to apparent surface reactions that occur essentially in the continuous phase. Assuming isothermal conditions, one can define for such reactions a spatially averaged rate, a,,j(t), by Ki,j(f) = b ki,j JJsf(Ciy V *

Contributor

Ci) dX dy

dZ

(1)

where ci, Cj, the concentrations of the reactants in the continuous medium, are given with some simplifying assumptions by the solution of [6] aCi/at = div(l)ii grad

Ci - VCi) - vikl,if(ci,

Cl)

and aCj/at = div(l)jj grad

cj - VCj) - vjki,f(ci,

c,) I (2)

is the local rate of the reaction (kt,j being the relevant rate constant) [8], V the total volume of the continuous phase, and t the time. In Eqs. (2) Dii, Ojj denote the appropriate diffusion coefficients, vi and Vj are the coefficients of the ith and the jth species in the stoichiometric equation, respectively, and v stands for the hydrodynamic velocity. Usually, the analytical solution of Eqs. (2) and the subsequent calculation of &(t) are complicated if not impossible; however, when the chemical reaction is much slower than the rate of diffusion or vice versa, one can simplify these equations by neglecting either the fist or second righthand term. ki,jf(c,,cj)

2.

SPECIFXITYOF REACTIONS IN TURBULENT, DENSEEMULSIONS

The process of mass transfer from and within the droplets of a turbulent emulsion is affected by the

in part.

35

I. GALLILY,G. M. J. SCHMIDT and E. BERNSTEIN TABLE I. THE LOWERLIMIT OF h, IN PACKEDEMULSIONS

motion of the particles themselves, by their proximity to each other and, in the absence of stabilizing surfactants, by their constant collision and redis-

cc

persion. Assuming, first, the diffusional resistance to reside mainly near the interfaces, secondly, a quasi-stationary

behaviour

0*

and, thirdly, the pre-

sence of the reactantj in the continuous phase only,

0.1

we can replace Eqs. (2) by dc,,ldt

= VCci,,

-

c&t>1- vik,jf(Ci,b,cj,A

and

0.2

(3) dcj,bldt =

0.3

vjki,jf(Ci,b, cj,&

and thus estimate the influence of the various factors

0.02 0.2 2 0.013 0.13 1.3 0.026 0.26 2.6 0.05 0.5 5

S/I 2 l-3 2.6

5

of the ith species at the

surface, ci,b its time-dependent concentration in the bulk of the continuous medium, Cj,b the bulk concentration of species j, S the specific interfacial area

the

array having a volumetric packing fraction a

ventional problem of the homogeneous system; the difference between the models for the heterogeneous and the homogeneous reactions resides mainly in the fact that in the former some reactants and procontinuous and the dispersed media. Actually, the chemically controlled reaction of the heterogeneous system should be regarded as an asymptotic case in

1 typical

results of the effective mass-transfer coefficient and number,

is

ducts are distributed at every instance between the (4)

while hi is of the order of D,,/(u-d).* As an illustration we give in Table

the rate of the chemical reaction

droplets of the type of emulsion under discussion the kinetic situation must approximate to the con-

suspension to a fixed array of equal spheres of diameter d and inter-centre distance a. In such an

a/d = (~/6c#‘~

SYSTEMCHOSEN AND KINETIC EQUATIONS

Whenever

of the droplets, and hi an effective mass-transfer

diffusion, can be evaluated by approximating

3.

much lower than the rate of mass transfer from the

coefficient. For a dense emulsion the lower limit of this coefficient, conforming to quiet molecular

its related Sherwood

hi (cm set-I)

10 1 0.1 10 1 0.1 10 1 0.1 10 1 o-1

* c( = 0 refers to a single drop in an Smite medium.

on the total rate of the reaction. In Eqs. (3) c~,~is the equilibrium concentration

d(cL)

CLOSELY

Sh = hid/Dii, for

which the rate of production of materials attains its maximal possible value. Any appreciable reduction of the diffusional speed, whether through the size of

closely packed arrays and substances of low molecular weight (Dii LZ lo-’ cm2 set-‘) [IO]. The actual values of hi in turbulent emulsions are greater than the values calculated above because of the fluctuating relative motion between the droplet and the continuous medium [ 11, 121; this relative motion constantly brings new masses of fluid to the interface of the dissolving particle where saturation occurs instantaneously [13] and increases the rate of diffusion away from it. If the emulsions are not stabilized then constant collision, deformation and break-up of droplets also enhance the rate of diffusion inside the dispersed phase. * HARRIOTT[9] defines a mass-transfer coefficient for a dense array of spheres somewhat differently. 36

the droplets or the intensity of agitation, will decrease the total rate of the reaction. For various reasons we have chosen to investigate the bromination

of benzene droplets by an

aqueous solution of hypobromous and perchloric acids; as a preparatory stage we have studied the kinetics of this reaction in the homogeneous phase. Bromination in aqueous solution, consisting of two consecutive substitutions in the aromatic ring, has been shown by us to approximate, for the first half-life time, to a second-order, acid-catalyzed reaction; it therefore resembles those systems in which the bromonium-ion mechanism is assumed to operate [14-171. The rate constant of bromination to monobromobenzene, k,, was found to be

Reactions in emulsions-I.

The rate of bromination

of benzene droplets in a dense, vigorously agitated aqueous emulsion

(1340 f 32) x [H+] 1. mole-’ min-’ at 30°C for small concentrations of the perchloric acid (supposed equal to the hydrogen ion concentration [H+]) ; the Arrhenius activation energy was determined as 13.1 -+_0.33 kcal. mole-l. For the heterogeneous system we assume that the hypobromous acid is soluble only in the aqueous medium, which contines the reaction to that phase; likewise for a typical case of a dense emulsion of a = 0.3 and d = 10~ (Table I), k, = 0.268 1. mole-l min” ([H ClO,] = 2 x 10-4mole 1.-l), and an initial concentration of the hypobromous acid, [A],, , of lo-’ mole 1.-l, we find that hiS’/k,[A], 2 2 x lo6 at 30°C

phase (Appendix), one may disregard here also the second stage of the bromination as well as any effect of the products on the rate of the reaction. In this case the stoichiometric equations become (7)

CA1= [Alo - *Ccl,

(8)

and

where *[B] and *[Cl denote total number of moles of species B and C per unit volume of suspension, respectively; [A] stands for the number of moles of species A per unit volume of the aqueous phase; the subscript zero marks initial values, and a is as previously defined. The rate of production of bromobenzene is expressed by

(5)

which would indicate [Eqs. (3)] the preponderance of the diffusional rate over the reaction velocity. In addition, we assume that the products of the reaction, bromobenzene (C) and dibromobenzene (D), as well as the benzene (B) itself, are distributed between the phases almost thermodynamical1y.t The validity of this important assumption is not only implied by the condition of rapid mass transfer in the continuous medium, but depends also on a sufficiently high rate of diffusion inside the droplets. The kinetic equations for the system under study, and their solution, are set forth in the Appendix; here we treat a special case that still contains the characteristic points of our model and which can be checked experimentally. Given that the initial amount of the hypobromous acid (A) is small compared with that of benzene and that, as assumed, the species are almost thermodynamically distributed between the phases, we may neglect the effect of the dilution of the droplets with products on the rate of mass transfer to the aqueous phase. Except for a short time of concentration build-up, we can write

PI = CBI,- f(t) g PI,

*[B] = *[B],, - *[Cl

4*CCI/(1 - 4W

= k,CBI,CAl=

= UW,
(9)

which gives as a solution

*Ccl = CAM - #l 4.

- exp(- k,CB1,0~.(10)

EXPERIMENTAL

Method

The kinetics of bromination was studied in a system of droplets that were produced and kept suspended by strong mechanical stirring; we did not use emulsifying agents as we did not wish at this stage to introduce additional factors into our system. The packing fraction of benzene was 3/13 and its initial number of moles was about 300 times greater than that of the hypobromous acid, as posited in the special example discussed above. The runs comprised two sets of reactions that differed in the concentration of the perchloric acid used. To measure the progress of these reactions we determined the increase with time of the concentration of bromobenzene.

(6)

where [B] denotes the number of moles of species B in the continuous medium per unit volume of suspension, the subscript s stands for the saturation value, andf(t) expresses the small influence of the dilution of the droplets. Owing to the accumulation of the reaction products mostly in the dispersed

Apparatus The reaction apparatus consisted of a 250 ml, three-neck unbaffled flask fitted with a paddle stirrer and immersed in a thermostatic bath which was held at 30 f 0.03”C. As we wished to study the case where the intensity of agitation no longer

t See Appendix. 37

I. GALLILY, G. M. J. SCHMIDTand

influenced the rate of the reaction we did not investigate the turbulence characteristics of the stirred fluid; however, the constant frequency of revolution as well as the power ~o~umption per unit mass were noted. materials The chemical compounds employed were of analytical grade. The benzene was a BDH “Analar” product and the perchloric acid was a Fluka chemical; the water had a specific resistivity of 444 x 10% cm. The hypobromous acid was prepared according to a method described by BRANCH and JONES[18]. It was kept near 4°C over a Ccl, layer which absorbed the small amount of bromine produced during its decomposition [I 81, and was purified again by extraction with the same solvent immediately before each kinetic run. The purity of this acid was assessed by an iodimetric determination of its total oxidising power, a check of its HBr03 content [15], and by a s~ctrocolorimet~c assay of the percentage of bromine. Before the kinetic run the undiluted hypobromous acid contained 0.5-0.7 % of HBr03 and 2-2.2 % of Br,. Procedure

After temperature equilibration, the kinetic runs were started by the addition of 30 ml of benzene to 100 ml of the hypobromous-perchloric acid solution. During the course of the reaction we sampled the product by stopping the stirring and pipetting out a small portion of the rapidly creamed aromatic phase into aqueous sodium bisulp~te; this operation lasted about 30 sec. In adopting this technique we have assumed all the bromobenzene to be essentially in the dispersed phase; an independent test showed this assumption to be correct. Along with the kinetic runs we checked the spontaneous decomposition of the hypobromous acid on an undiluted solution. Under these severe contions [19] the con~ntration of bromine was found to increase to approximately 2.5 % after about one hour. The concentration of HBrO, increased correspondingly to about lx, but the total oxidizing power was practically unchanged. We have assumed [I91 that the diluted HBrO solutions used in the emulsion experiments decomposed to an even lesser extent.

E. BERNSTJXIN

Analysis

The solutions of bromobenzene in benzene, containing a weighed amount of pure p-dichlorobenzene as comparison-tracer, were analysed in a 800 Perkin Elmer Gas C~omato~aph equipped with a flame ionization detector and a 150 ft, O-01in. id., poly(propylene glycol) UC oil LB - 550 -X capillary column that was maintained isothermally at 125°C. In the preparation of the solutions for analysis the weighing was carried out with an accuracy of O-2 per cent, and care was exercised to minimize the evaporation of benzene. By the present analytical method the value of *[Cl/(1 - ar) [A], could be deter~ned with an average sensitivity of 1.13 x lo-’ and a precision of approximately c/e = 0.04 where Q is the standard deviation and f! the arithmetic mean. 5.

REsur.rs

The changes of the ~n~ntration of the bromobenzene with time are shown in Figs. 1 and 2. Figure 3 shows a typical cumulative size distribution of a sample from our emulsion photographed under the microscope; the median diameter and the semi-interquartile range of this distribution are 5.1~ and 1.6~~ respectively. Because of partial creaming that occurred during sampling these values should be regarded as upper limits. In further experiments the frequency of revolution in the stirring was lowered so that the aromatic phase disintegrated to a much lesser extent. The results of these runs are shown in Fig. 4 together with the yield of product after comparable time under conditions of fast stirring. 6.

DISCUSSION

Using the relationship between In < 1 - *[Cl/ (1 - ol) [A],, > and t [Eq. (lo)], we have determined from the results of each run the appropriate value of k,[B], and compared the curve drawn according the arithmetic average of these values with the curve based on the rate constant of the homogeneous-phase bromination at 30°C and the solubility of benzene in water [20],t which is t FRANK et al. 1211 give solubility value lower by 6.8% than the one used here. 38

Reaktions in emulsions-I.

The rate of bromi~tion

of benzene droplets in a dense, vigorously agitated aqueous emulsion

becomes better the greater the ratio hiS/~,[A]o [Eq. (5)], we may assume that the kinetics of bromination at temperatures lower than 30°C but under conditions otherwise similar to those of the present study will be described by Eq. (10) too. The rate of production of bromobenzene decreases, as expected, with the rate of stirring; this decrease shows qualitatively the role of mass transfer which could not be observed in the experiments with the vigorously agitated emulsions. NOTATION

A

k

0

I

IO

I

20

WI

the hypobromous acid species number of moles of A per unit volume of the continuous

a

inter-centre distance in fixed array of spheres

phase

I

30

Time, min FIG. 1. The change of concentration of bromobenzene with time. [A]o g lo-* mole 1.-l; [HC104] = 2 x 1O-4 moIe L-1; T = 303.2°K; a = 3/13; rotational frequency = 680 f 15 r.p.m.; power consumption per unit mass4.2 x 10-B metric h.p. g-1; E-curve deduced from present study experiments: F--curve based on homogeneous-phase and solubility data; 0, 0, V,Sexperimental points.

assumed to be indentical with that in our reaction mixture (Figs. 1 and 2). The curves for each set of experiments fit one another with a deviation of 1.2% (Fig. 1) and 3.5 % (Fig. 2); we may conclude that the model of a chemically controlled reaction is applicable to our present system and that our assumption concerning the distribution of the product between the two phases is essentially correct. This assumption could not be checked exactly because of the low solubility of bromobenzene in water and the accuracy of our analytical method which was insufficient for that purpose; however, it is supported also by the demonstration of essentially all the product in the aromatic phase. As the app~cability of the chemical control model

I

IO

I

I

20 Time,

30 min

FIG. 2. The change of the concentration of bromobenzene with time. [AJOg 10-Z mole 1.-l; [IX1041 = 10 -4 mole I.--l ; T = 303*2”K ; a = 3/l 3 ; rotational frequency = 680 + 15 r.p.m.; power consumption per unit mass4.2 x 10-s metric h.p. g-1 G--curve deduced from presentstudyexperiments; H-curvebasedonhomogeneousphase and solubility data; 0, 0, V,-experimental points.

I. GALLILY,G. M. J.

Ii

0

I 20

I

40

SCHMIDT and

I

I

60

80 d*

E.

BERNSEIN

I

KXJ

I

120

I

140

P

FIG. 3. A typical cumulative size distribution of benzene droplets. Cf fraction of droplets with diameter below d; N-total number of droplets; smallest diameter , measured-l.4 CL;precision of measurements-O.2 p. the benzene species B *[B], l1Bl. .- *. IBl . . total number of moles of B, the number of moles in the dispersed phase and the number of moles in the continuous phase, respectively, all per unit volume of suspension c The bromobenzene species *Pa l[C], [C] refer to numbers of moles of species C similarly to the case of B concentrations of species i and j in the continuous phase the dibromobenzene species D *IDI, r[D], [D] refer to numbers of moles of species D similarly to the case of B Dii, Djj diffusion coe.fficients of species i and j in the continuous phase d diameter of a droplet effective mass-transfer coefficient of species i in the hr continuous phase V-U+ hydrogen ion concentration rate constant of reaction between species i and j kr,, rate constant of bromination of benzene to bromobenkl zene in homogeneous phase (l.mole-r min-l)

kz

rate constant of bromination of bromobenzene to dibromobenzene in homogeneous phase (1. mole-l min-l)

arithmetic mean of *[Cl/(1 - u)[A]o Q I&.,(r) spatially averaged rate of reaction specific interfacial area of droplets (surface/unit volume S of suspension) Sherwood number hid/Dta S/I time t total volume of the continuous phase V hydrodynamic velocity mole fraction of species i in the dispersed phase, z (i = B, C, D) X,Y,Z space co-ordinates volumetric packing fraction of the dispersed phase cc distribution coefficient of species i between the phases ai (i = B, C, D) 1 + *IBlo, (i = B, C, D) ;: v1, 0 o

Subscripts refers to the bulk of the continuous phase b 0 denotes initial time value stands for saturation state s

111

CARLESSE.

[31 r41

NORRIS M.B. and MCBAIN J.W., J.Chem.Soc. 1922, 1362. KING A. and MUKERJEEL.N., J.Soc.Chem.Znd. 1938 57 431.

NIXON

coefficients of species i and j in the stoichiometric equation standard d;$ation of the values of *[Cl/(1 - a) 0

REFERENCES and

k&l 8db

J.R., J. Pharm. Pharma. 1960 12 348.

PI Taaun J.-P.,J. Chim. Phys. 1918 16 107.

40

Reactions in emulsions-I.

The rate of bromination

of benzene droplets in a dense vigorously agitated aqueous emulsion

A., Chem.Engng Sci. 1961 14 183. [51 VIAL.LARLI Sci. 1963 18 365. ]61 FRIEDLANDER S.K. and KBLLERK. H., Chem.Engng LEVICHV.G., Physicochemical Hydrodynamics, pp. 53-7 Prentice Hall, N.J. 1962. t;; BENSONS.W., The Foundations of Chemical Kinetics, p.11 McGraw-Hill, New York 1960. 191 HARRIO~~P., Document 6958, American Documentation Institute, Library of Congress, Washington DC. [lOI WARD W.J. and QUINN J.A., A.Z.Ch.E. Jl 1964 10 155. SK., ibid. 1957 3 381. 1111 FRIEDLANDER U21 Soo S.L., Chem.Engng Sci. 1956 5 57. r131 SHERWOODT.K., Mass Transfer Between Phases, pp. 72-6 Thirty-Third Annual Priestly Lectures, The Pennsylvania State University, 1959. [I41 SHILOVE. and KANIAEVN., C. r. (Dokl.) AkadSci., USSR 1939 24 890. DERBYSHIRB D.H. and WATERSW.A., J.Chem.Soc. 1950, 564. DE LA MAKE P.B.D. and HARVEYJ.T., J.Chem.Soc. 1956, 36; 1957, 131. t::; r171 DE LA MARE P.B.D., DUNN T.M. and HARVEYJ.T., ibid 1957, 923. 1181 BRANCHS.J. and JONESB., ibid. 1954, 2317. t191 PRUTTONC.F. and MARONS.H., J.Am.Chem.Soc. 1935 57 1652. PO1 STEPHEN H. and STBPHBN T., Eds. Solubilities of Inorganic and Organic Compounds Vol. 1, Table 1317, Pergamon Press, Oxford 1963. WI FRANKSF., GENT M. and JOHNSONH.H., J.Chem.Soc. 1963, 2716. WI OLAH G.A., Private communication. t231 JUNGERSJ.C., (Ed.), CinPtique Chimique Appliquee, pp. 171-183 SocietC de Edition Technip, Paris 1958. IA1 = [Alo - <*[Cl/(1 - a)> - 2 <*[Dl/(l - Co>,

APPENDIX We note that the solubilities of benzene, bromobenzene and dibromobenzene in water are low, and assume that their solubilities in the aqueous phase of our system are almost identical with those in the corresponding binary systems. We assume further an almost ideal behaviour in the dispersed phase. From the stoichiometry of the reactions and the two assumptions concerning the distribution of the species, we have *[Blo = *PI

+ *[Cl + *ID],

(11)

15-

(12) XB = an [BIN - a) Xc = ac [Cl/(1 - a)

(13)

and XC = ao PI/(1 - a); likewise, from mass balance, *[Bl = ‘LB1 + WI, *rc1 = ‘ICI + [Cl and (14) *[Dl= WI + Dl where *[ 1, r[ 1, [ ] denote, respectively, total number of moles, the number of moles in the dispersed phase and the number of moles in the continuous phase, all per unit volume of suspension; XB, XC and XD are the mole fractions of B, C and D in the dispersed phase; aB, ac and c(D are the appropriate distribution coefficients, and all other symbols are as previously defined. At t = 01 [B]o = *[B]o - [B]o, *[C]O = 0 and *[D]o = 0, while afterwards

IO-

l[B] = =

-%=B(*p]O
[B] - [C] - [Dl> =

-

a)>

IBI(*Pl

s- -0

d*[B]/dt = E

[Cl - IDI)

(15)

d’[B]/dt + dB]/dt z (1 +


-

X dlS]/dt = 0

LB1-

-

etc. From these relations, and because of the low concentrations of the species in the aqueous medium, we can express the relevant rates of change by

I 200

Rotational

I

I

400

600

frequency,

rev/min

d*[Cl/dt r (1 +
flB

a)>*lBlo}x d[Bl/dt,

(16)

- a)>*[Blo]x

x d[C]/dt = fit d[C]/dt

(17)

and

FIG. 4. Yield of bromobenzene vs. rate of stirring. [A]0 g 10-Z moleI.-l; [HClO] = 2 x 1O-4 molel.-l; reaction time: 20 min.; T = 303*2”K; a = 3/13.

d*[Dl/dt g 11 + <~/cl axd[D]/dt

41

a)> *[Blo]x

= fl&D]/dt.

(18)

I. GALLILY,G. M. J. SCHMIDTand E. BERNSTEIN The chemical reactions occur in the continuous phase in two steps each of which was found by us to conform to a second order kinetics. Thus, neglecting possible effects caused by unequal concentration of the bromonium-ion around the B and C molecules, which have different electrical polarities [22], we can write ,&4Bl/d~ = -[l/(1 @d]Cl/dt

- a)1 h[Al [Bl,

Equations (19-21), which are non-linear differential equations of the generalized Volterra type, can be solved by a previously employed method [23] if we neglect the small change of the packing fraction a with time. As a result, one obtains first

[Cl = ( WPlo - (M/[~IoY’~>,

(22)

(19) where y = @z/k,) and 6 = (&/p~); subsequently, utilizing the stoichiometric Eqs. (ll-12), we can transform Eq. (19) into

= [l/(1 - a)1 (20)

and

Bod[Dlld~ = [l/(1 - aI1 M41 [Cl where ka is the rate constant

(1 - CO,/~Bd[Blldt = --kl[Bl{(l

(21)

- a)[Alo - 2~4Bl0 +

+ ,&]Blo<(]BI/[Blo) - (]Bl/[Blo)y’d>l(~ - 6) + 2/3~]Bll

of the second substitution.?

(23) t Here, the positionally parallel bromination of C can be treated as a single reaction controlled by a composite rate constant.

which can be integrated immediately to give, after numerical evaluation of the resulting integral, the sought relations between [B], and *[B], *[Cl *[D] and the time.

R&nnr? - Le taux de bromuration de gouttelettes de benzene par un acide hypobromeux en milieu acide dans une emulsion aqueuse dense et turbulente, a et6 analyst theoriquement et etudie experimentalement. On trouva que la bromuration se conforme a la formulation dans laquelle les reactions sont supposQs se dbouler essentiellement dans le milieu continu, les vitesses dans un milieu chimiquement control& et les esp&ces d&rib&es presque thermodynamiquement entre les phases. Znsamrnenfassnng - Die Geschwindigkeit der Bromierung von Benzoltropfchen durch angesluerte unterbromige SIure in einer turbulenten, dichten wlssrigen Emulsion wurde theoretisch analysiert und experimentell untersucht. Es wurde gefunden, dass die Bromierung einem Model1 entspricht, in welchem angenommen wird, dass die Reaktionen im wesentlichen im kontinuierlichen Medium verlaufen, die Kinetik chemisch gesteuert, und das Material beinahe thermodynamisch zwischen den Phasen verteilt ist.

42