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Particulate processes in freshly prepared silver iodide hydrosols part II. Sedimentation analysis of the sols
Technology. 31 (1982) 75 - 84 0 Elsevier Sequoia S.A, Iausanne -Printed
in The Netherlands
Particulate Processes in Freshly Part II. Sedimentation Analysis
Prepared Silver Iodide of the Sols
BORIS SUBOTIC Irzstifute Ruder Bo;koviE.
Box 1016. Croatia (Yugoslavia)
Powder
4100 Zagreb. P-0.
75
Hydrosols
(Received April 24.1981)
Negative stable silver iodide hydrosols prepared in statu nascendi were investigated by fhe method of sedimentation analysis_ This investigaeion has been carried out in order to evaluate the assumption that the primary formed particles of stable, negative silver iodide hydrosols prepared in statu nuscendi are monodispersed, and that the primary particles aggregate into secondary monodispersed particles by a second-order kinetic process. On the basis of experimental results, by measuring unsettled fractions, fg’, of AgI after centrifugal-ion as a funcCion of w’t, and ageing time tA , it is proved that the primary formed particles of so& are really monodisversed and that their aggregation into monodispersed secondary particles is a secondarder kinetic process. A radiotracer technique for the determination of fractions fE was used.
sols during their ageing. Existence of a particle size bimodal distribution at ageing times 0 < tA < tA, (where tA, corresponds to the ageing time at which the first ageing process is completed), contrary to monomodal distributions at rA = 0 and tA = tA,, points to the possibility that the aggregation of primary particles into secondary ones is a second-order kinetic process_ Further, the analysis of possible dispersities at the beginning and at the end of the fast ageing process* in the sense of weillmown laws of coagulation processes [l, 21 led to the conclusion that the changes observed in particle size distribution during the ageing of sols (see Figs. 10 and 11 in Part I) _ are possible only in the case when monodispersed prim-a particles aggregate into secondary ones by a second-order kinetic process_ A sedimentation analysis of sols is performed with the intention of proving or disproving the stated assumptions.
LNTRODUCTION
THEORETICAL
Zn the first work of this series (Part I), analysis of the heterogeneous exchange processes in silver iodide hydrosols established that the ageing of silver iodide sols takes place through two different processes: fast, with duration A tA , inversely proportional to the molar conce&ation Co of the sol, and slow, whose occurrence can be observed a long time after the &st process is finished. Comparative analysis of heterogeneous exchange processes in stable and fishly coagulated labelled sols led to the conclusion that the particles of the stable sols double their volume during the fast process of ageing and simultaneously change their basic structural characteristics. The first stated conclusion (doubling of volume) was proved by electin microscopy of the stable
Elementary principles of sedimentation analysis The behaviour of dispersed particles under the action of centrifugal force can be generally described by the Lamm differential equation for the ultracentrifuge [ 33 :
SUMMARY
where C1 represents the concentration of dispersed particles of uniform shape and size at radial distance x from the axis of rotation, t, (i) primary particles are monodispersed and particles are also monodispersed, (ii) primary particles are monodispersed, secondary particles are polydkpersed etc. *ie.
secondary
76
duration of. centrifugation, D, and s1 are the diffusion and sedimentation coefficients of the dispersed phase, and w is the angular speed of the centrifuge rotor (radians per second)_ D, and sr can be expressed as
is the
(2)
Sl
=
WP
(3)
-PO)/%
where k is the gas constant, T is absolute temperature, q and p. are the viscosity and density of dispersing medium, and r, and p are the radius and density, respectively, of the dispersed particles. For a sufficiently large particle size, the diffusion coefficient becomes insignificant in comparison with the sedimentation coefficient_ This case is mathematically equivalent to treating the Lamm equation with D, = O_ For D, = 0, the Lamm equation [2] reduce to
-aCr =-_ ah
i
x
a(X2&1c1) ax
(4)
= c
exp(-2s,w2t,)
where CT is the concentration particles before centrifugation is valid for x1 exp(slW2tc)
< x < x2
(5) of the dispersed Equation (5) (6)
x-
K
RADIAL
DLSTANCE.
X
Fig. l_ Concentration Cl of a monodkpened particle system after centrifugation under a given L?C, value, plotted against the radial distance x from the axis of spinning. Cy is the concentration of particles before cenkifugation. x1 and ~2 are the distance of the system level and the centrifuge tube bottom respectively from the axis of spinning, and I* is defined as given in eqn. (7).
C,
= kCi i=l
Therefore, the Lamm equation for the case D, = 0 seems not only interesting mathematicaJ.ly but also useful for the sedimentation analysis of molecules of large sizes such as high polymeric substances and colloids 143 _ The solution of the partial differential equation (4) is C,
0
= gC4exp(-2s,W2t,)
(8)
i=l
where Cp are the concentrations of separate dispersed groups, with the particle sizes within the interval r and r + Ar. before centrifugation, and Ci are the concentrations of the same dispersed groups which after centrifugation remain within the distance region between xfand x2, where Xi* is defined as
xi’ =
x1
exp(siw2t,)
(9)
Cz is then the total concentration of the dispersed phase which after centrrfugation remains witbin the distance region between X: and x2 as shown in Fig. 2. Analogous to eqn. (9). the value of x,’ is *ex xP
1 exp(spw2k)
(10)
as can be seen in Fig_ 1. The magnitudes x1 and x2 in the relation (6) represent the least radial distance of the centrifuging system (i-e. the liquid level) and the radial distance of the centrifuge tube bottom or/and wall (depending upon the tube position with respect to the axis of rotation). respectively, both from the axis of spinning. The value of X* in Fig. 1 may be expressed as Xf =
Xl
exP(slw*G)
(7)
which means that X* move towards the tube bottom (and/or wall) with time t, in accordance with eqn (7)_ For a polydispersed system, eqn (5) can be expressed in the following form:
Fig. 2- Concentration Cr of a polydispersed particle system after centxifugation under a given 02t, value, plotted against the radial distance x from tbe axis of spinning. CO, x1 and ~2 have the Same meaning as in Fig. 1. xp’ is defmed as given in eqn. (lo)_
77
where s, is the sedimentation coefficient of the particles having the largest particle size. The total concentration of the dispersed particles at each position within the distance region between x1 and x2, before centrifugation, can be expressed as
Dividing fx
eqn. (8) by eqn. (ll),
= Cz/c”x
= 2
we obtain
(Cp/cOx) exp(-2si&&)
i=l
= fglfo
eXp(-2S+J2tc)
(12)
where fx and fro represent the total and the separate particle fractions respectively, which after centrifugation remain within the distance region between xz and the distance defmed by eqns. (9) and (10) respectively_ For i = 1, eqn. (12) results in
fi = exp(-2s,w2t,)
(13)
which is equivalent to the form obtained dividing eqn. (5) by e_ Putting exp(-_2sio2t,)
= Ei
by (14)
and exp(+?s,
w2&) = El
eqns. (12) fZ
=
5
and (13)
(15) can be rewritten
f6i
as (16)
i=l
and
fi = El
(17)
From eqn. (13) for a monodispersed system, ln fi is a linear function of w2t, within the distance region deGned by the relation (6). h-l fi = -2s,w=t,
(18)
For a polydispersed ln
fx
= ln 5 f:
system
exp(-2sJW2t,)
(19)
i=l
where In fx is not in linear relationship with w2tc within the distance region between x1 exp sp(w2G),, and x2, where (w2G),,, represents the maximal value of w2t, during the analysis.
Use of the radiotracer technique in sedimentation analysis for the evaluation of monodispersity (or polydispersity) of dispersed particle systems On the basis of eqns. (18) and (19), by analysing In fi and ln fx values as functions of W2tc values, monodispersity or polydispersity of systems can be verified. For this purpose it is necessary to determine the concentrations of the dispersed phase within the determined distance regions, before and after centrifugation, for different values of m2t,. The mass fraction of dispersed particles can be determined by various methods (i-e_ gravimetric, calorimetric, complexometric, etc.), but each of the methods mentioned requires special treatment after sampling which can cause numerous errors. Moreover, the abovementioned methods of analysis are often limited by the available quantity of the samples. These problems may be elegantly solved by radiometric analysis. Besides the fact that radiometric analysis is an accurate method [S] , it enables the measurement of extremely small quantities of materials [7] _ In radiometric analysis the measured values are expressed by the ratioactivities which are, rela, tive to their mass or surface of determined dispersed groups, proportional to their concentrations. Analysed particles can be made radioactive in three ways: (i) by radioactive labelling of precipitating components prior to the formation of particles, (ii) reaching the equilibrium state in a radioactive-non-radioactive heterogeneous exchange process, (iii) by specific adsorption of radioactive materials on the surface of the particles. In the mt two cases the radioactivity of particles is proportional to their mass and in the third case the radioactivity of particles is proportional to their surface. It is easy to show that both the mass and the surface fraction are in linear relationship with w2_f, in the case of monodispersed systems If the radioisotope is uniformly distributed over the whole mass of the dispersed phase, then the radioactivity ai of each particle in a determined disperse group will be proportional to its mass mi. ai =7=m,
(29)
where T= is the specific radioactivity OS the dispersed phase (radioactivity per unit of
i6
mass)_ Then the radioactivity A? of all particles of the determined dispersed group contained in the sample of volume u is Aq = YmmiRi
(21)
where Ri represents in volume u_
the number
of particles
ni=cLf
(22)
Now, _a:=
eqn_ (21)
may be rewritten
as
LJ-rmrniCP
(23)
=u-ym
gm,Co
(24)
i=l
in
Combining eqns. (S), (14), (23) and (24), radioactivities Ai of the particles contained in volume u of the sample, inside the distance re-g-ion between xi* and/or x,* and xa, after centrifugation, can be expressed as A, = uy,m,Ci
= oy,miCj’Ei
(25)
and hence A x = ~Ai
= UT,
gmiC:Ei
(25)
i=l
i=l
Since
om,C”
(27)
and = ME
(28)
i=l
where M$’ represents the mass of separate dispersed group and Mg is the total mass of the dispersed phase, contained in volume u before centrifugationDividing eqn. (26) by eqn. (24) leads to the following expressicnr
y,U
Ax/A%
5
= r,u
2 MPEi
t7liCFEi
i=l
gmiCp
=
i=l
Me
(29)
i=l
Introducing Mj’/Mg
=
eqn_ (29) f2
= 2 i=l
(f2r)o
(30)
can be rewritten (fy)oEi
= Ax/As
=cl =A,fA’:
in the form
(321
It is evident, in accordance with eqns. (18) and (19), that ln f,” is in linear reIationship with lr (and thus with a*&), and that ln fp is not a linear function of o’t,_ In a similar way it can be proved that (33)
= Arsl(AS10
and E
= f;
= rzl = A,Z/(Ag,
(34)
where As , (A%),-, , A; and (AZ ). are the surface radioactivities of polydispesed and monodispersed systems after and before centrifuging, and E and R correspond to the surface fractions_ In this way, a simple radioactivity determination of the systems. within the determined distance regions (see eqns- (7), (9) and (10)) prior to and aftercentrifugation, reveals monodispersity or polydispersity of dispersed particle systems. Application of the radiotracer technique for the determination of parameters in the second-order kinetic process by sedimentation analysis The
=Mio
v gmiC!p
f”,’ = f”
E = ig (FloEi
The total radioactivity Ag of all particles contained in volume u is then A% =gAy
Fori=l
theoretical
consideration
and
conclu-
sion given in the former section, concerning the theoretical part, can also be applied to the study of interactions between the particles themselves during the ageing of the systemsNaturally, this can be used only in the case of a simple interaction, like second-order kinetics- Otherwise radiosedimentation analysis of complex interactions requires a theoretical and experimental treaturent which is not the purpose of this work_ Let us assume that at f, = 0 the entire dispersed phase consists of monodispersed primary particle:, and that the primary particles aggregate into secondary ones by a second-order kinetic process. If the reaction rate between the primary particles is larger than the others, i.e. secondary-secondary and primary-secondary particles reaction rates, then the following reaction is valid: n = rlr + Ry =n,+(np+-n,)=(n~+n,)/2 (35)
(31)
where n is the total number of particles contained in the volume u of the system at time
79
tx , n, and n2 are numbers of the primary and the secondary (formed by interaction of two primary particles) particles in the same volume and in the same tune, and rzy is the number of primary particles at t, = 0. If the radioactivity of a primary particle is a, = rmml (see eqn. (ZO)), where m, is the mass of the primary particle, then in accordance with eqn. (35), the radioactivity a2 of the secondary particle is a2
=
(39)
%,m,
In accordance with eqn_ (21), the radioactivity At of all primary particles contained in volume u of the system is A:
= -rmmln,
(37)
and the radioactivity A: of all secondary particles contained in volume u of the system is A:
= 27=mIn2
(33)
The total radioactivity A% of all particles contained in volume 0 is A:
= A? + A;
= ymmlrzl
= ymmlnl
+ -rmm,(n!?
+ By,m,n., -h)
=
= YmmlnY
fp
= 2n,/nf
(40)
= AZ/A0 I%
(41)
and thus
flM = n$‘A~/A~
(42)
n2 = ny
fp
(43)
In accordance that
(f29o=f,”
with eqn. (35)
+fF=l
it is evident (44)
and that (fg), is independent of ageing time in the absence of centrifugal force. fp and fp represent the mass ikactions of the primary and secondary particles in relation to the total mass of the dispersed phase. Lf systems with differently aged particles are centrifuged under the same conditions (constant w’t,), in accordance with eqns. (5) and (22), n=(c)
= n1 exp(-2slw2tC)
n(c) = n,(c)
= n,e,
+ n2(c) = nlel
(46)
+ n2e2
and in accordance with eqns. their radioactivity As is A\- = 7mml(nIE1
+
(37)
(47) and (38) (48)
2n2~2)
Substituting the expression n2 in eqn. (48) with the equivalent expression (ny - n1)/2, and dividing eqn. (48) by eqn_ (39), one can obtain
f”,’ = AZ/A:
=
= (f%J -
f,“el
+ (1 - f,“)e
(49)
2
(45)
E2)/(%
-
(50)
E2)
where f, ” is the mass fraction of the particles which after centrifugation remain inside the distance region between x1 exp(s2w2t,) and x2. The disappearance of the primary particles during the ageing process of the system can be obtained as a solution of the differential equation of the second-order kinetic process: dn,/dt,
n, = ny
J2 = nyAg/2Ag
= n2e2
where nl(c) and n,(c) are the numbers of primary and secondary particles contained in volume u under the distance region between JC~ exp(s&t,) and x2, after centrifugation, coeffiand s1 and ss are the sedimentation cients of the primary and secondary particles. The total number of particles n(c), contained in volume u, under the distance region between x1 exp(s2w2t,) and xp , after centrifugation, is then
f”
and it is independent of ageing time. Dividing eqns. (37) and (38) by eqn. (39), one can obtain
= Af/A”r
exp(-2s2w2tc)
and hence (39)
f1”’ = n1 /II:
r&c) = rz.2
= -k,nf
whose analytical n1 = lf(k,t,
(51) solution
is
+ I/n:)
(52)
where k2 is a constant of the second-order kinetic process. Division of eqn. (52) by ny results in
f”
= n,/n y = l/(n:k,t,
+ 1)
(53)
Substituting: ny = k,C,-, , where Cc is the molar concentration of the dispersed phase and k, is a proportionality constant, and combining eqns_ (50) and (53), we have l/f?’
= (6
-
E&/(fF
-
E2) = Cok&.J*
+ 1
(54)
where k, = k, k, _ It is evident from eqn- (54) that the inverse mass fraction of primary particles, experimentally determined by mea-
80
suring the fx values as a function of the ageing time, under the given experimental conditions (constant and known values of e1 and aa), must be in linear relationship with the ageing time tA, if primary particles ziggregate into secondary ones by the secondorder kinetic process.
EXPERMEhTAI.
Preparation of siluer iodide hydrosols Silver iodide hydrosols were prepared and treated prior to the experimental procedures as described in the first part of this series (Part I)_ For radiosedimentation analysis, prior to the precipitation of silver iodide hydrosols, the solution of NaI was labelled with 1311- _ Addition of AgNOa gave radioactive sols of uniform specific radioactivity.
Experimental
procedures
-4. In order to evaluate
the monodispersity (or polydispersity) of dispersed silver iodide particle systems, the fractions fg were determined as a function of w2tc- For this purpose, aliquots (8 ml) of the_ system aged for chosen times tA were centrifuged at different values of w’t, in a Sorvall RCB-B centrifuge. The w2C, values were varied by changing the angular speed revolution. The time of centrifugation t, , at constaut speed revolutions (and the maximal at the same time) was always the same, kept at 5 min. The time of centrifugation tc at a specified rate was corrected for the run-up and slow-down periods of the centrifugeFor this purpose, Marshall [S J derived the following formula for the overall equivalent time t, of centaifugationr
tc = tb
-
t, +
t&w’
+ (tt - t&&w*
where tr is the run-up time, tb is the running time up to the start of braking, tt is the total running time, W: is the mean square of the run-up revolutions, wb2 .is the mean square of the braking revolutions, and w is the number of constant speed revolutions. For the HB4 high-speed swinging bucket centrifuge rotor used, and the RCB-B SorvaIl centrifuge, the corrected centrifugal time t, calculated by Marshall’s formluIa may be expressed as a function of constant speed revolutions and time t, of cons’;ant speed revolutions 0nIyr
tc = tl + w/A
+ w~‘~/B
where A and B are constants. B. In order to evaluate whether the aggregation of primary particles of the analysed sols 12 a second-order kinetic process or not, and for a possible determination of kinetic parameters, the fe values of differently aged sols were deter&ed for a chosen constant value of w*t, _ After centrifugation, 0.2 ml aliquots of centrifuged systems were used for the determination of radioactivity A-. The samples were always taken from the bottom of the centrifuge tube. In this way the samples were-taken inside the distance region between x1 exp(+w*t,) and x2 (see Fig. 2). The radioactivity A-’ of samples taken after centrifugation may be expressed as A=+=A,L+A
-=Ak+ff$A$
(55)
where A: is the equilibrium the liquid phase- Hence fp
= (A”=
-
Ak))lA$
radioactivity
of (56)
Since A‘$
= A,,
-Ak
(57)
where A, is the total radioactivity of the system, including the radioactivity of the liquid phase and the radioactivity of the solid phase (AgI), fz
= (A”=
-
A:)/(Ao
-
Ak)
(58)
The experimental procedures for the determination of radioactivities A,-, and A2 have been described earlier [9] _ Ail systems examined were prepared in triplicate, and the reported results represent the average values. Maximal relative errors were ?7% in all cases.
RESULTS AND DISCUSSION Figures 8 - 6 give the log fg values of differently aged O-01, O-002 and 0.0002 molar silver iodide sols, plotted against w2tc_ In order to show the influence of Nd ions on the aggregation rate of AgI, the 0.002 molar sols were prepared in such a way that the concentration of Na* ions in the systems is 0.004 and 0.035 mol elm-‘, respectively. In all other sols analysed, the molar concentration of Nd ions is two times greater than the molar concentration of AgI. The chosen ageing times r,
81
c4Illmoldd
Fig. 3. Logarithm of rnsss fraction # of silver iodide r.emainingaIter centrifugation of 0.5, 15 and 300 min aged 0.01 molar silver iodide sol, inside a distance region between -rcl erp(sz&t,) and .T*, plotted as a function of w%e.
= 0.002
CNdllmold+
= ODD2
[Naq I moldm-3
= 0.035
Fig. 5. Logarithm of rns fraction log @ of silver iodide remaining after centrifugation of 2.5, 60 and 1500 min aged 0.002 molar silver iodide sol, inside the distance region between rl exp(sgw2tc) and x2, plotted as a function of w2tc. Concentration of Na’ ions in the system is 0.035 ma1 dmm3.
CAglllmoldm-== 00042
Fig_ 4. Logarithm of mass fraction ry of silver iodide remaining after centrifugation of 5. 100, 1450 and 2300 min aged 0.002 molar silver iodide sol, inside a distance region between x1 exp(s2w2t,) and I~, plotted ss a function of w2t,.
approximately correspond to the beginning of the time interval At-, (Le. tA = 0, see Part I), to the end of the time interval At,, (i-e. r* = tA,), and to the intermediate times between = Oand tA < tAz _ For instance, 5 minute tA aged silver iodide sol is a good approximation for the zero minute aged (tA = 0) 0.002 molar sol, with regard to tbe duration of the time = 500 mininntervalAtA, for this sol (At,, utes; see Fig. 3 in Part I)_ It is evident that in this case only a small percentage of primary particles is aggregated into secondary ones during the 5 minutes of ageing, and that the system practicaIIy consists only of primary particles. The time difference between tbe
Fig. 6. Logarithm of mars fraction log fg of silver iodide remaining after centrifugation of 5, 850 and 15000 min aged O_OOOZ moizu silver iodide sol, inside the distance region between x1 exp(szw2t,) and 12, plotted as a function of w2tc.
true start of the ageing process (tA = 0) and its experimental approximation, and the errors involved, cannot be avoided during the preparation of the systems for centrifugation (filling the centrifuge tubes). For the same reason, in order to check the aggregation process, 0.01 molar AgI sol (At, = 100 min), aged for 0.5, 15 and 300 mm, was diluted with double-distilled water to a 0.002 moIar sol, just before centrifugation. The possible errors caused by the time difference between the true start of the aggregation process and its experimental approximation
82
become negligible as the time interval atA, increases_ In this way, the systems aged for the shortest times (see Figs. 3 - 6) correspond approximately to the systems containing only primary particles of silver iodide. As is shown in Figs_ 3 - 6, the values of log f? of these systems are in linear relationship with w2 t, _ In accordance with eqn_ (18), these results confium the assumption stated earlier (Part I) that primary particles of freshly prepared silver iodide hydrosols are monodispersed. The fg values of sols aged for 0 < tA < tA,, whose electron micrographs show a bimodal particle size distribution (Part I), are not in linear relationship with w2rc_ These results are in accordance with an earlier assumption, founded on election-microscopic and radiometric investigations of silver iodide sols (-Part I), that the particles of sols aged for 0 < tA < tA, are possibly bidispersed- The fz values of sols aged for fAE min, where t_%,is taken from the heterogeneous exchange data and defmed (Part I), are in linear relationas tx. = l/C, ship with w’f, only for the lower values of w2t,, although from the heterogeneous exchange data it was expected that the sol particles are monodispersed after tA = tA,_ However, as the ageing time increases, the linear part of log fg uersus w2tc function is longer and longer, and at ageing times sufficiently longer than t, the linearity is retained up to the w2tc values corresponding to log fz = -0.8 to -1 (see Fig. 4). This means that nonlinearity of the log fg versus w2t, frxnAions be,$ns when approximately 10 - 15% of the mass of silver iodide is unsettled. Even for the 0.002 molar silver iodide sol, containing 0.035 mol dm-” of Na’ ions, aged for 1500 min (see Fig_ 5), log _fg is in line= relationship with w2tc for all w2t, values used_ The partial nonlinearity of log fy versns w2t, functions for sols aged.for tA Z tn. min may be explained by the fact that the experimental time tA, cannot be exactly determined, and that some priary particles also exist after this time has elapsed_ In reality, if the aggregation of primary particles into secondary ones is a second-order kinetic process, a number of primary particles show a tendency to approach zero when tA tends to go to infinity_ For this reason, the time tA, has not a theoretical but only an experimental character, and determines the point at which the specific influence of primary particles on
the experimental results becomes negligible. Sedimentation analysis is more sensitive to the composition of dispersed particles than is beterogeneous exchange analysis. Thus, the specific influence of primary particles (even at low concentrations of primary particles in comparison with the concentration of secondary particles) is much more reflected in the results obtained by sedimentation analysis than in the results obtained by heterogeneous exchange analysis. The sedimentation coefficient of secondary particles is considerably larger than the sedimentation coefficient of primary particles. Therefore the former particles settle at the bottom of the centrifuge tube faster than the latter, during centrifugation. Hence, as the 02t, value increases, the number of unsettled primary particles related to the unsettled secondary ones increases. For this reason, for sufficiently large w2rC values, the unsettled part of particles ought to be treated as a bidispersed particle system, and then its log fy values are not in linear relationship with w2tC _ Taking into consideration this explanation, the linear part of log f”,’ versus r_d2tc functions of sols aged for tx 2 tn, represents the log f? versus u2t, function characteristic for monodispersed secondary particles_ In this way, the results represented in Figs- 3 - 6, and in accordance with eqns. (18) and (19), show that the primarily formed particles of silver iodide sols prepared in statu nascendi are monodispersed, and that during ageing they aggregate into systems of monodispersed secondary particles. This is clear experimental verification of the assumption stated in the first part of this series (Part I). Experimental evaluation of the assumption that the aggregation of primary particles of silver iodide into secondary ones is a secondorder kinetic process is performed by measuring the f? values of analysed sols as a function of the -eeing time tA . Experimentally obtained Pg values are via eqns. (50) and (54) transformed into the corresponding flM and l/f? values. In accordance with eqns (17), (45) and (46), values Q and e2 in eqns. (50) and (54) correspond to the diction fy of the primary particles and to the &action fp of the secondary particles, respectively, remaining inside the distance region between x1 exp(s2w2t,) and x2 after centrifugation at a given w2tc value. For practical purposes, e1 and e2 of analysed systems and the w2tC used
83
10
08
Fig_ 7. A. Mass fractions fg and fy of silver iodide remaining after centrifugation of O.Ol molar silver iodide sol under w’t, = 1.2 x 10’ rad s-l, inside the distance region between x1 exp(s2w2t,) and x2, pIotted as a function of the ageing time tA_ B. Inverse maLs fraction l/f:’ of primary particles, plotted as a function of the ageing time t,.
h
CAgll/m~Mm-~ = 0002 CNaIllmcldrr-3= 0.002 CNa'llmddmJ= O-(135
Fig. 9. A. Mass fractions fy and f:’ of silver iodide remaining after centrifugation of O-002 molar silver iodide sol under w2 t, = 1.2 x 10’ rad s-l, inside the distance region between x1 exp(S2w2t,) and x2, plotted as a function of the ageing time t_4_Concentration of Na+ ions in the system is O-035 mol dm-l_ B. Inverse mass fraction l/f,” of primary particles, plotted as a function of the ageing time tA_
CAglllmoIdm~3 = 0002
10 t
CNalllmaldm-3 =I3002
CAgIl/moIdrn-3= OOOCQ CNaIllmoldm~" =a0002
Fig. 8_ A_ Mass fractions fF and f? of silver iodide remaining after centifuga%on of O-002 molar silver iodide sol under w2tc = 1.2 x lo7 rad s-1, inside the distance region between x1 exp(s2w*t,) and x2, plotted as a function of the ageing time tA_ B. Inverse mass fraction l/f? of primary particles, plotted as a function of the ageing time CA_
may be determined as the fg value of primary and secondary particles respectively (linear functions of fg uersus w2tc), corresponding to the experimentally used w2tc value, as shown in Fig. 3. The results, plotted against
t,1lnin Fig. 10. A. Mass fractions fg and fl” of silver iodide remaining after centrifugation of 0_0002 molar silver iodide sol under w2t, = 1.2 X 10’ rad s-l, inside the distance region between x1 exp(szw2tc) and 3~2, plotted as a function of the ageing time tA_ B. Inverse mass fraction l/f,” of primary particks, plotted as a function of the ageing time tA_
time t, , are represented as fy and flM - 10(A), and as l/f? in Figs_ 7(B) - 10(B). The w2t, value used was 1.2 X 10’ rad s-l in all cases. The inverse fraction
ageing
in Figs. 7(A)
84
of the primary particles is, under the l/f,” ageing times examined, in linear relationship with the ageing time for ah sols examined. In accordance with eqn (54), this result represents an experimental verification of the idea that the disappearance of primary particles of siIver iodide is a second-order kinetic process. The constant k,, proportionaI to the constant fia of the second-order kinetic process for the sols examined, can be calculated by dividing the slope of the linear function l/f,” uersus t_%by the molar concentration Ce of the sol (see Figs. 7B - 10B): k, =
4(1/f:‘)/c,Ma
Constvlts h of the systems examined do not differ significantly- Thus, it is clear that the absolute aggregation rate of primary particles into secondary on= is more strongly influenced by the molar concentration of sols than by the composition of the liquid phase. In this sense may be &so explained the fact that the experimentally determined time period t+ is approximately inversely proportional to the molar concentration of soIs_ The small difference in k, values determined for 0.002 molar silver iodide sols of different molar Na‘ ion concentration (see Figs. 8 and 9) could be explained by electrostatic interaction of Nd ions with the negatively charged colloidal silver iodide particles. When an increrfied concentration of coagulating ions is present, then the potential barrier decreases around the colloidal particles and lowers the activation energy of their interaction, and hence also increases the aggregation rate.
CONCLUSION
Applying radiosedimentation analysis of freshly prepared negative, stable silver iodide hydrosols, the idea is verified that the primary particles of the sols are monodispersed. It is also proved that the aggregation of primary sol particles into secondary ones is a secondorder kinetic process, and that secondary particles are also monodispersed. The results obtained are in good agreement with the conclusion obtained by radiomelxic and electronmicroscopic investigations of silver iodide sols and give a basis for further investigations on phase transformations during the ageing of silver iodide hydrosols_
REFERENCES M. Smoluchowski, Phys. 2.. 17 (1916) 557_ H. R. Crayt. Col!oid Science, Vol. I, Elsevier, Amsterdam. 1952_ 0. Lamm. 2. Phys. Chem_ (Leipzig). A134 (1929) 177. H. Fuji& Mathematical Theoofsedimentation Analysis. Academic Press. New York, 1962_ T. Allen. ParlXe Ske Measurement. Chapman and Hall Ltd.. London, 1974. B. Subotie, Powder TechnoL, 24 (1979) 40_ Ft. Despotovii et aL, CroaL Chem_ Acta, 51 (1978) 113. C. E. Marsha& Rot_ R. Sot., 126A (1930) 427. R. Despotovie and B. SubotiC. J. Ino%_ Nucl. Chem., 38 (1976) 1317.
in The Netherlands
Particulate Processes in Freshly Part II. Sedimentation Analysis
Prepared Silver Iodide of the Sols
BORIS SUBOTIC Irzstifute Ruder Bo;koviE.
Box 1016. Croatia (Yugoslavia)
Powder
4100 Zagreb. P-0.
75
Hydrosols
(Received April 24.1981)
Negative stable silver iodide hydrosols prepared in statu nascendi were investigated by fhe method of sedimentation analysis_ This investigaeion has been carried out in order to evaluate the assumption that the primary formed particles of stable, negative silver iodide hydrosols prepared in statu nuscendi are monodispersed, and that the primary particles aggregate into secondary monodispersed particles by a second-order kinetic process. On the basis of experimental results, by measuring unsettled fractions, fg’, of AgI after centrifugal-ion as a funcCion of w’t, and ageing time tA , it is proved that the primary formed particles of so& are really monodisversed and that their aggregation into monodispersed secondary particles is a secondarder kinetic process. A radiotracer technique for the determination of fractions fE was used.
sols during their ageing. Existence of a particle size bimodal distribution at ageing times 0 < tA < tA, (where tA, corresponds to the ageing time at which the first ageing process is completed), contrary to monomodal distributions at rA = 0 and tA = tA,, points to the possibility that the aggregation of primary particles into secondary ones is a second-order kinetic process_ Further, the analysis of possible dispersities at the beginning and at the end of the fast ageing process* in the sense of weillmown laws of coagulation processes [l, 21 led to the conclusion that the changes observed in particle size distribution during the ageing of sols (see Figs. 10 and 11 in Part I) _ are possible only in the case when monodispersed prim-a particles aggregate into secondary ones by a second-order kinetic process_ A sedimentation analysis of sols is performed with the intention of proving or disproving the stated assumptions.
LNTRODUCTION
THEORETICAL
Zn the first work of this series (Part I), analysis of the heterogeneous exchange processes in silver iodide hydrosols established that the ageing of silver iodide sols takes place through two different processes: fast, with duration A tA , inversely proportional to the molar conce&ation Co of the sol, and slow, whose occurrence can be observed a long time after the &st process is finished. Comparative analysis of heterogeneous exchange processes in stable and fishly coagulated labelled sols led to the conclusion that the particles of the stable sols double their volume during the fast process of ageing and simultaneously change their basic structural characteristics. The first stated conclusion (doubling of volume) was proved by electin microscopy of the stable
Elementary principles of sedimentation analysis The behaviour of dispersed particles under the action of centrifugal force can be generally described by the Lamm differential equation for the ultracentrifuge [ 33 :
SUMMARY
where C1 represents the concentration of dispersed particles of uniform shape and size at radial distance x from the axis of rotation, t, (i) primary particles are monodispersed and particles are also monodispersed, (ii) primary particles are monodispersed, secondary particles are polydkpersed etc. *ie.
secondary
76
duration of. centrifugation, D, and s1 are the diffusion and sedimentation coefficients of the dispersed phase, and w is the angular speed of the centrifuge rotor (radians per second)_ D, and sr can be expressed as
is the
(2)
Sl
=
WP
(3)
-PO)/%
where k is the gas constant, T is absolute temperature, q and p. are the viscosity and density of dispersing medium, and r, and p are the radius and density, respectively, of the dispersed particles. For a sufficiently large particle size, the diffusion coefficient becomes insignificant in comparison with the sedimentation coefficient_ This case is mathematically equivalent to treating the Lamm equation with D, = O_ For D, = 0, the Lamm equation [2] reduce to
-aCr =-_ ah
i
x
a(X2&1c1) ax
(4)
= c
exp(-2s,w2t,)
where CT is the concentration particles before centrifugation is valid for x1 exp(slW2tc)
< x < x2
(5) of the dispersed Equation (5) (6)
x-
K
RADIAL
DLSTANCE.
X
Fig. l_ Concentration Cl of a monodkpened particle system after centrifugation under a given L?C, value, plotted against the radial distance x from the axis of spinning. Cy is the concentration of particles before cenkifugation. x1 and ~2 are the distance of the system level and the centrifuge tube bottom respectively from the axis of spinning, and I* is defined as given in eqn. (7).
C,
= kCi i=l
Therefore, the Lamm equation for the case D, = 0 seems not only interesting mathematicaJ.ly but also useful for the sedimentation analysis of molecules of large sizes such as high polymeric substances and colloids 143 _ The solution of the partial differential equation (4) is C,
0
= gC4exp(-2s,W2t,)
(8)
i=l
where Cp are the concentrations of separate dispersed groups, with the particle sizes within the interval r and r + Ar. before centrifugation, and Ci are the concentrations of the same dispersed groups which after centrifugation remain within the distance region between xfand x2, where Xi* is defined as
xi’ =
x1
exp(siw2t,)
(9)
Cz is then the total concentration of the dispersed phase which after centrrfugation remains witbin the distance region between X: and x2 as shown in Fig. 2. Analogous to eqn. (9). the value of x,’ is *ex xP
1 exp(spw2k)
(10)
as can be seen in Fig_ 1. The magnitudes x1 and x2 in the relation (6) represent the least radial distance of the centrifuging system (i-e. the liquid level) and the radial distance of the centrifuge tube bottom or/and wall (depending upon the tube position with respect to the axis of rotation). respectively, both from the axis of spinning. The value of X* in Fig. 1 may be expressed as Xf =
Xl
exP(slw*G)
(7)
which means that X* move towards the tube bottom (and/or wall) with time t, in accordance with eqn (7)_ For a polydispersed system, eqn (5) can be expressed in the following form:
Fig. 2- Concentration Cr of a polydispersed particle system after centxifugation under a given 02t, value, plotted against the radial distance x from tbe axis of spinning. CO, x1 and ~2 have the Same meaning as in Fig. 1. xp’ is defmed as given in eqn. (lo)_
77
where s, is the sedimentation coefficient of the particles having the largest particle size. The total concentration of the dispersed particles at each position within the distance region between x1 and x2, before centrifugation, can be expressed as
Dividing fx
eqn. (8) by eqn. (ll),
= Cz/c”x
= 2
we obtain
(Cp/cOx) exp(-2si&&)
i=l
= fglfo
eXp(-2S+J2tc)
(12)
where fx and fro represent the total and the separate particle fractions respectively, which after centrifugation remain within the distance region between xz and the distance defmed by eqns. (9) and (10) respectively_ For i = 1, eqn. (12) results in
fi = exp(-2s,w2t,)
(13)
which is equivalent to the form obtained dividing eqn. (5) by e_ Putting exp(-_2sio2t,)
= Ei
by (14)
and exp(+?s,
w2&) = El
eqns. (12) fZ
=
5
and (13)
(15) can be rewritten
f6i
as (16)
i=l
and
fi = El
(17)
From eqn. (13) for a monodispersed system, ln fi is a linear function of w2t, within the distance region deGned by the relation (6). h-l fi = -2s,w=t,
(18)
For a polydispersed ln
fx
= ln 5 f:
system
exp(-2sJW2t,)
(19)
i=l
where In fx is not in linear relationship with w2tc within the distance region between x1 exp sp(w2G),, and x2, where (w2G),,, represents the maximal value of w2t, during the analysis.
Use of the radiotracer technique in sedimentation analysis for the evaluation of monodispersity (or polydispersity) of dispersed particle systems On the basis of eqns. (18) and (19), by analysing In fi and ln fx values as functions of W2tc values, monodispersity or polydispersity of systems can be verified. For this purpose it is necessary to determine the concentrations of the dispersed phase within the determined distance regions, before and after centrifugation, for different values of m2t,. The mass fraction of dispersed particles can be determined by various methods (i-e_ gravimetric, calorimetric, complexometric, etc.), but each of the methods mentioned requires special treatment after sampling which can cause numerous errors. Moreover, the abovementioned methods of analysis are often limited by the available quantity of the samples. These problems may be elegantly solved by radiometric analysis. Besides the fact that radiometric analysis is an accurate method [S] , it enables the measurement of extremely small quantities of materials [7] _ In radiometric analysis the measured values are expressed by the ratioactivities which are, rela, tive to their mass or surface of determined dispersed groups, proportional to their concentrations. Analysed particles can be made radioactive in three ways: (i) by radioactive labelling of precipitating components prior to the formation of particles, (ii) reaching the equilibrium state in a radioactive-non-radioactive heterogeneous exchange process, (iii) by specific adsorption of radioactive materials on the surface of the particles. In the mt two cases the radioactivity of particles is proportional to their mass and in the third case the radioactivity of particles is proportional to their surface. It is easy to show that both the mass and the surface fraction are in linear relationship with w2_f, in the case of monodispersed systems If the radioisotope is uniformly distributed over the whole mass of the dispersed phase, then the radioactivity ai of each particle in a determined disperse group will be proportional to its mass mi. ai =7=m,
(29)
where T= is the specific radioactivity OS the dispersed phase (radioactivity per unit of
i6
mass)_ Then the radioactivity A? of all particles of the determined dispersed group contained in the sample of volume u is Aq = YmmiRi
(21)
where Ri represents in volume u_
the number
of particles
ni=cLf
(22)
Now, _a:=
eqn_ (21)
may be rewritten
as
LJ-rmrniCP
(23)
=u-ym
gm,Co
(24)
i=l
in
Combining eqns. (S), (14), (23) and (24), radioactivities Ai of the particles contained in volume u of the sample, inside the distance re-g-ion between xi* and/or x,* and xa, after centrifugation, can be expressed as A, = uy,m,Ci
= oy,miCj’Ei
(25)
and hence A x = ~Ai
= UT,
gmiC:Ei
(25)
i=l
i=l
Since
om,C”
(27)
and = ME
(28)
i=l
where M$’ represents the mass of separate dispersed group and Mg is the total mass of the dispersed phase, contained in volume u before centrifugationDividing eqn. (26) by eqn. (24) leads to the following expressicnr
y,U
Ax/A%
5
= r,u
2 MPEi
t7liCFEi
i=l
gmiCp
=
i=l
Me
(29)
i=l
Introducing Mj’/Mg
=
eqn_ (29) f2
= 2 i=l
(f2r)o
(30)
can be rewritten (fy)oEi
= Ax/As
=cl =A,fA’:
in the form
(321
It is evident, in accordance with eqns. (18) and (19), that ln f,” is in linear reIationship with lr (and thus with a*&), and that ln fp is not a linear function of o’t,_ In a similar way it can be proved that (33)
= Arsl(AS10
and E
= f;
= rzl = A,Z/(Ag,
(34)
where As , (A%),-, , A; and (AZ ). are the surface radioactivities of polydispesed and monodispersed systems after and before centrifuging, and E and R correspond to the surface fractions_ In this way, a simple radioactivity determination of the systems. within the determined distance regions (see eqns- (7), (9) and (10)) prior to and aftercentrifugation, reveals monodispersity or polydispersity of dispersed particle systems. Application of the radiotracer technique for the determination of parameters in the second-order kinetic process by sedimentation analysis The
=Mio
v gmiC!p
f”,’ = f”
E = ig (FloEi
The total radioactivity Ag of all particles contained in volume u is then A% =gAy
Fori=l
theoretical
consideration
and
conclu-
sion given in the former section, concerning the theoretical part, can also be applied to the study of interactions between the particles themselves during the ageing of the systemsNaturally, this can be used only in the case of a simple interaction, like second-order kinetics- Otherwise radiosedimentation analysis of complex interactions requires a theoretical and experimental treaturent which is not the purpose of this work_ Let us assume that at f, = 0 the entire dispersed phase consists of monodispersed primary particle:, and that the primary particles aggregate into secondary ones by a second-order kinetic process. If the reaction rate between the primary particles is larger than the others, i.e. secondary-secondary and primary-secondary particles reaction rates, then the following reaction is valid: n = rlr + Ry =n,+(np+-n,)=(n~+n,)/2 (35)
(31)
where n is the total number of particles contained in the volume u of the system at time
79
tx , n, and n2 are numbers of the primary and the secondary (formed by interaction of two primary particles) particles in the same volume and in the same tune, and rzy is the number of primary particles at t, = 0. If the radioactivity of a primary particle is a, = rmml (see eqn. (ZO)), where m, is the mass of the primary particle, then in accordance with eqn. (35), the radioactivity a2 of the secondary particle is a2
=
(39)
%,m,
In accordance with eqn_ (21), the radioactivity At of all primary particles contained in volume u of the system is A:
= -rmmln,
(37)
and the radioactivity A: of all secondary particles contained in volume u of the system is A:
= 27=mIn2
(33)
The total radioactivity A% of all particles contained in volume 0 is A:
= A? + A;
= ymmlrzl
= ymmlnl
+ -rmm,(n!?
+ By,m,n., -h)
=
= YmmlnY
fp
= 2n,/nf
(40)
= AZ/A0 I%
(41)
and thus
flM = n$‘A~/A~
(42)
n2 = ny
fp
(43)
In accordance that
(f29o=f,”
with eqn. (35)
+fF=l
it is evident (44)
and that (fg), is independent of ageing time in the absence of centrifugal force. fp and fp represent the mass ikactions of the primary and secondary particles in relation to the total mass of the dispersed phase. Lf systems with differently aged particles are centrifuged under the same conditions (constant w’t,), in accordance with eqns. (5) and (22), n=(c)
= n1 exp(-2slw2tC)
n(c) = n,(c)
= n,e,
+ n2(c) = nlel
(46)
+ n2e2
and in accordance with eqns. their radioactivity As is A\- = 7mml(nIE1
+
(37)
(47) and (38) (48)
2n2~2)
Substituting the expression n2 in eqn. (48) with the equivalent expression (ny - n1)/2, and dividing eqn. (48) by eqn_ (39), one can obtain
f”,’ = AZ/A:
=
= (f%J -
f,“el
+ (1 - f,“)e
(49)
2
(45)
E2)/(%
-
(50)
E2)
where f, ” is the mass fraction of the particles which after centrifugation remain inside the distance region between x1 exp(s2w2t,) and x2. The disappearance of the primary particles during the ageing process of the system can be obtained as a solution of the differential equation of the second-order kinetic process: dn,/dt,
n, = ny
J2 = nyAg/2Ag
= n2e2
where nl(c) and n,(c) are the numbers of primary and secondary particles contained in volume u under the distance region between JC~ exp(s&t,) and x2, after centrifugation, coeffiand s1 and ss are the sedimentation cients of the primary and secondary particles. The total number of particles n(c), contained in volume u, under the distance region between x1 exp(s2w2t,) and xp , after centrifugation, is then
f”
and it is independent of ageing time. Dividing eqns. (37) and (38) by eqn. (39), one can obtain
= Af/A”r
exp(-2s2w2tc)
and hence (39)
f1”’ = n1 /II:
r&c) = rz.2
= -k,nf
whose analytical n1 = lf(k,t,
(51) solution
is
+ I/n:)
(52)
where k2 is a constant of the second-order kinetic process. Division of eqn. (52) by ny results in
f”
= n,/n y = l/(n:k,t,
+ 1)
(53)
Substituting: ny = k,C,-, , where Cc is the molar concentration of the dispersed phase and k, is a proportionality constant, and combining eqns_ (50) and (53), we have l/f?’
= (6
-
E&/(fF
-
E2) = Cok&.J*
+ 1
(54)
where k, = k, k, _ It is evident from eqn- (54) that the inverse mass fraction of primary particles, experimentally determined by mea-
80
suring the fx values as a function of the ageing time, under the given experimental conditions (constant and known values of e1 and aa), must be in linear relationship with the ageing time tA, if primary particles ziggregate into secondary ones by the secondorder kinetic process.
EXPERMEhTAI.
Preparation of siluer iodide hydrosols Silver iodide hydrosols were prepared and treated prior to the experimental procedures as described in the first part of this series (Part I)_ For radiosedimentation analysis, prior to the precipitation of silver iodide hydrosols, the solution of NaI was labelled with 1311- _ Addition of AgNOa gave radioactive sols of uniform specific radioactivity.
Experimental
procedures
-4. In order to evaluate
the monodispersity (or polydispersity) of dispersed silver iodide particle systems, the fractions fg were determined as a function of w2tc- For this purpose, aliquots (8 ml) of the_ system aged for chosen times tA were centrifuged at different values of w’t, in a Sorvall RCB-B centrifuge. The w2C, values were varied by changing the angular speed revolution. The time of centrifugation t, , at constaut speed revolutions (and the maximal at the same time) was always the same, kept at 5 min. The time of centrifugation tc at a specified rate was corrected for the run-up and slow-down periods of the centrifugeFor this purpose, Marshall [S J derived the following formula for the overall equivalent time t, of centaifugationr
tc = tb
-
t, +
t&w’
+ (tt - t&&w*
where tr is the run-up time, tb is the running time up to the start of braking, tt is the total running time, W: is the mean square of the run-up revolutions, wb2 .is the mean square of the braking revolutions, and w is the number of constant speed revolutions. For the HB4 high-speed swinging bucket centrifuge rotor used, and the RCB-B SorvaIl centrifuge, the corrected centrifugal time t, calculated by Marshall’s formluIa may be expressed as a function of constant speed revolutions and time t, of cons’;ant speed revolutions 0nIyr
tc = tl + w/A
+ w~‘~/B
where A and B are constants. B. In order to evaluate whether the aggregation of primary particles of the analysed sols 12 a second-order kinetic process or not, and for a possible determination of kinetic parameters, the fe values of differently aged sols were deter&ed for a chosen constant value of w*t, _ After centrifugation, 0.2 ml aliquots of centrifuged systems were used for the determination of radioactivity A-. The samples were always taken from the bottom of the centrifuge tube. In this way the samples were-taken inside the distance region between x1 exp(+w*t,) and x2 (see Fig. 2). The radioactivity A-’ of samples taken after centrifugation may be expressed as A=+=A,L+A
-=Ak+ff$A$
(55)
where A: is the equilibrium the liquid phase- Hence fp
= (A”=
-
Ak))lA$
radioactivity
of (56)
Since A‘$
= A,,
-Ak
(57)
where A, is the total radioactivity of the system, including the radioactivity of the liquid phase and the radioactivity of the solid phase (AgI), fz
= (A”=
-
A:)/(Ao
-
Ak)
(58)
The experimental procedures for the determination of radioactivities A,-, and A2 have been described earlier [9] _ Ail systems examined were prepared in triplicate, and the reported results represent the average values. Maximal relative errors were ?7% in all cases.
RESULTS AND DISCUSSION Figures 8 - 6 give the log fg values of differently aged O-01, O-002 and 0.0002 molar silver iodide sols, plotted against w2tc_ In order to show the influence of Nd ions on the aggregation rate of AgI, the 0.002 molar sols were prepared in such a way that the concentration of Na* ions in the systems is 0.004 and 0.035 mol elm-‘, respectively. In all other sols analysed, the molar concentration of Nd ions is two times greater than the molar concentration of AgI. The chosen ageing times r,
81
c4Illmoldd
Fig. 3. Logarithm of rnsss fraction # of silver iodide r.emainingaIter centrifugation of 0.5, 15 and 300 min aged 0.01 molar silver iodide sol, inside a distance region between -rcl erp(sz&t,) and .T*, plotted as a function of w%e.
= 0.002
CNdllmold+
= ODD2
[Naq I moldm-3
= 0.035
Fig. 5. Logarithm of rns fraction log @ of silver iodide remaining after centrifugation of 2.5, 60 and 1500 min aged 0.002 molar silver iodide sol, inside the distance region between rl exp(sgw2tc) and x2, plotted as a function of w2tc. Concentration of Na’ ions in the system is 0.035 ma1 dmm3.
CAglllmoldm-== 00042
Fig_ 4. Logarithm of mass fraction ry of silver iodide remaining after centrifugation of 5. 100, 1450 and 2300 min aged 0.002 molar silver iodide sol, inside a distance region between x1 exp(s2w2t,) and I~, plotted ss a function of w2t,.
approximately correspond to the beginning of the time interval At-, (Le. tA = 0, see Part I), to the end of the time interval At,, (i-e. r* = tA,), and to the intermediate times between = Oand tA < tAz _ For instance, 5 minute tA aged silver iodide sol is a good approximation for the zero minute aged (tA = 0) 0.002 molar sol, with regard to tbe duration of the time = 500 mininntervalAtA, for this sol (At,, utes; see Fig. 3 in Part I)_ It is evident that in this case only a small percentage of primary particles is aggregated into secondary ones during the 5 minutes of ageing, and that the system practicaIIy consists only of primary particles. The time difference between tbe
Fig. 6. Logarithm of mars fraction log fg of silver iodide remaining after centrifugation of 5, 850 and 15000 min aged O_OOOZ moizu silver iodide sol, inside the distance region between x1 exp(szw2t,) and 12, plotted as a function of w2tc.
true start of the ageing process (tA = 0) and its experimental approximation, and the errors involved, cannot be avoided during the preparation of the systems for centrifugation (filling the centrifuge tubes). For the same reason, in order to check the aggregation process, 0.01 molar AgI sol (At, = 100 min), aged for 0.5, 15 and 300 mm, was diluted with double-distilled water to a 0.002 moIar sol, just before centrifugation. The possible errors caused by the time difference between the true start of the aggregation process and its experimental approximation
82
become negligible as the time interval atA, increases_ In this way, the systems aged for the shortest times (see Figs. 3 - 6) correspond approximately to the systems containing only primary particles of silver iodide. As is shown in Figs_ 3 - 6, the values of log f? of these systems are in linear relationship with w2 t, _ In accordance with eqn_ (18), these results confium the assumption stated earlier (Part I) that primary particles of freshly prepared silver iodide hydrosols are monodispersed. The fg values of sols aged for 0 < tA < tA,, whose electron micrographs show a bimodal particle size distribution (Part I), are not in linear relationship with w2rc_ These results are in accordance with an earlier assumption, founded on election-microscopic and radiometric investigations of silver iodide sols (-Part I), that the particles of sols aged for 0 < tA < tA, are possibly bidispersed- The fz values of sols aged for fAE min, where t_%,is taken from the heterogeneous exchange data and defmed (Part I), are in linear relationas tx. = l/C, ship with w’f, only for the lower values of w2t,, although from the heterogeneous exchange data it was expected that the sol particles are monodispersed after tA = tA,_ However, as the ageing time increases, the linear part of log fg uersus w2tc function is longer and longer, and at ageing times sufficiently longer than t, the linearity is retained up to the w2tc values corresponding to log fz = -0.8 to -1 (see Fig. 4). This means that nonlinearity of the log fg versus w2t, frxnAions be,$ns when approximately 10 - 15% of the mass of silver iodide is unsettled. Even for the 0.002 molar silver iodide sol, containing 0.035 mol dm-” of Na’ ions, aged for 1500 min (see Fig_ 5), log _fg is in line= relationship with w2tc for all w2t, values used_ The partial nonlinearity of log fy versns w2t, functions for sols aged.for tA Z tn. min may be explained by the fact that the experimental time tA, cannot be exactly determined, and that some priary particles also exist after this time has elapsed_ In reality, if the aggregation of primary particles into secondary ones is a second-order kinetic process, a number of primary particles show a tendency to approach zero when tA tends to go to infinity_ For this reason, the time tA, has not a theoretical but only an experimental character, and determines the point at which the specific influence of primary particles on
the experimental results becomes negligible. Sedimentation analysis is more sensitive to the composition of dispersed particles than is beterogeneous exchange analysis. Thus, the specific influence of primary particles (even at low concentrations of primary particles in comparison with the concentration of secondary particles) is much more reflected in the results obtained by sedimentation analysis than in the results obtained by heterogeneous exchange analysis. The sedimentation coefficient of secondary particles is considerably larger than the sedimentation coefficient of primary particles. Therefore the former particles settle at the bottom of the centrifuge tube faster than the latter, during centrifugation. Hence, as the 02t, value increases, the number of unsettled primary particles related to the unsettled secondary ones increases. For this reason, for sufficiently large w2rC values, the unsettled part of particles ought to be treated as a bidispersed particle system, and then its log fy values are not in linear relationship with w2tC _ Taking into consideration this explanation, the linear part of log f”,’ versus r_d2tc functions of sols aged for tx 2 tn, represents the log f? versus u2t, function characteristic for monodispersed secondary particles_ In this way, the results represented in Figs- 3 - 6, and in accordance with eqns. (18) and (19), show that the primarily formed particles of silver iodide sols prepared in statu nascendi are monodispersed, and that during ageing they aggregate into systems of monodispersed secondary particles. This is clear experimental verification of the assumption stated in the first part of this series (Part I). Experimental evaluation of the assumption that the aggregation of primary particles of silver iodide into secondary ones is a secondorder kinetic process is performed by measuring the f? values of analysed sols as a function of the -eeing time tA . Experimentally obtained Pg values are via eqns. (50) and (54) transformed into the corresponding flM and l/f? values. In accordance with eqns (17), (45) and (46), values Q and e2 in eqns. (50) and (54) correspond to the diction fy of the primary particles and to the &action fp of the secondary particles, respectively, remaining inside the distance region between x1 exp(s2w2t,) and x2 after centrifugation at a given w2tc value. For practical purposes, e1 and e2 of analysed systems and the w2tC used
83
10
08
Fig_ 7. A. Mass fractions fg and fy of silver iodide remaining after centrifugation of O.Ol molar silver iodide sol under w’t, = 1.2 x 10’ rad s-l, inside the distance region between x1 exp(s2w2t,) and x2, pIotted as a function of the ageing time tA_ B. Inverse maLs fraction l/f:’ of primary particles, plotted as a function of the ageing time t,.
h
CAgll/m~Mm-~ = 0002 CNaIllmcldrr-3= 0.002 CNa'llmddmJ= O-(135
Fig. 9. A. Mass fractions fy and f:’ of silver iodide remaining after centrifugation of O-002 molar silver iodide sol under w2 t, = 1.2 x 10’ rad s-l, inside the distance region between x1 exp(S2w2t,) and x2, plotted as a function of the ageing time t_4_Concentration of Na+ ions in the system is O-035 mol dm-l_ B. Inverse mass fraction l/f,” of primary particles, plotted as a function of the ageing time tA_
CAglllmoIdm~3 = 0002
10 t
CNalllmaldm-3 =I3002
CAgIl/moIdrn-3= OOOCQ CNaIllmoldm~" =a0002
Fig. 8_ A_ Mass fractions fF and f? of silver iodide remaining after centifuga%on of O-002 molar silver iodide sol under w2tc = 1.2 x lo7 rad s-1, inside the distance region between x1 exp(s2w*t,) and x2, plotted as a function of the ageing time tA_ B. Inverse mass fraction l/f? of primary particles, plotted as a function of the ageing time CA_
may be determined as the fg value of primary and secondary particles respectively (linear functions of fg uersus w2tc), corresponding to the experimentally used w2tc value, as shown in Fig. 3. The results, plotted against
t,1lnin Fig. 10. A. Mass fractions fg and fl” of silver iodide remaining after centrifugation of 0_0002 molar silver iodide sol under w2t, = 1.2 X 10’ rad s-l, inside the distance region between x1 exp(szw2tc) and 3~2, plotted as a function of the ageing time tA_ B. Inverse mass fraction l/f,” of primary particks, plotted as a function of the ageing time tA_
time t, , are represented as fy and flM - 10(A), and as l/f? in Figs_ 7(B) - 10(B). The w2t, value used was 1.2 X 10’ rad s-l in all cases. The inverse fraction
ageing
in Figs. 7(A)
84
of the primary particles is, under the l/f,” ageing times examined, in linear relationship with the ageing time for ah sols examined. In accordance with eqn (54), this result represents an experimental verification of the idea that the disappearance of primary particles of siIver iodide is a second-order kinetic process. The constant k,, proportionaI to the constant fia of the second-order kinetic process for the sols examined, can be calculated by dividing the slope of the linear function l/f,” uersus t_%by the molar concentration Ce of the sol (see Figs. 7B - 10B): k, =
4(1/f:‘)/c,Ma
Constvlts h of the systems examined do not differ significantly- Thus, it is clear that the absolute aggregation rate of primary particles into secondary on= is more strongly influenced by the molar concentration of sols than by the composition of the liquid phase. In this sense may be &so explained the fact that the experimentally determined time period t+ is approximately inversely proportional to the molar concentration of soIs_ The small difference in k, values determined for 0.002 molar silver iodide sols of different molar Na‘ ion concentration (see Figs. 8 and 9) could be explained by electrostatic interaction of Nd ions with the negatively charged colloidal silver iodide particles. When an increrfied concentration of coagulating ions is present, then the potential barrier decreases around the colloidal particles and lowers the activation energy of their interaction, and hence also increases the aggregation rate.
CONCLUSION
Applying radiosedimentation analysis of freshly prepared negative, stable silver iodide hydrosols, the idea is verified that the primary particles of the sols are monodispersed. It is also proved that the aggregation of primary sol particles into secondary ones is a secondorder kinetic process, and that secondary particles are also monodispersed. The results obtained are in good agreement with the conclusion obtained by radiomelxic and electronmicroscopic investigations of silver iodide sols and give a basis for further investigations on phase transformations during the ageing of silver iodide hydrosols_
REFERENCES M. Smoluchowski, Phys. 2.. 17 (1916) 557_ H. R. Crayt. Col!oid Science, Vol. I, Elsevier, Amsterdam. 1952_ 0. Lamm. 2. Phys. Chem_ (Leipzig). A134 (1929) 177. H. Fuji& Mathematical Theoofsedimentation Analysis. Academic Press. New York, 1962_ T. Allen. ParlXe Ske Measurement. Chapman and Hall Ltd.. London, 1974. B. Subotie, Powder TechnoL, 24 (1979) 40_ Ft. Despotovii et aL, CroaL Chem_ Acta, 51 (1978) 113. C. E. Marsha& Rot_ R. Sot., 126A (1930) 427. R. Despotovie and B. SubotiC. J. Ino%_ Nucl. Chem., 38 (1976) 1317.