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Particulate processes in freshly prepared silver iodide hydrosols part III. Disagreement between experimental results and the existing models of silver iodide hydrosols
PowderTechnology.34(1983)9-
9
1S
Particulate Processes in Freshly Prepared Silver Iodide Hydrosols Part III. Disagreement between Experimental Results and the Existing Models of Silver Iodide Hydrosols BORIS SUBOTIC Institute "RuderBo&otG",
41OGI Zagreb. P-0. Box 1016. Croatia{ITxgosZaria)
(Received April %X,1981)
SUhlMARY Silver iodide hydrosols prepared in statu nascendi exhibit some specific effects that cannot be explained on the basis of the e_xisting models of silver iodide hydrosolsThese effects are acceleration of a heterogeneous exchange process caused by coagulation of the system, and different aggregation number obtained in the same system applying two different exchange methods. In order to explain these effects, a comparison has been made of the differences between the data calculated on the basis of the ezxisting model of silver iodide hydrosols and the values determined by radiometric, optic, and sedimentation analysis. The previous assumption on the existence of a new structural form of silver iodide. characteristic of stable silver iodide hydrosols, has been confirmed_ This new structural form of silver iodide transforms into known hexagonal and cubic modifications during the aggregation of primary particles of silver iodide to secondac ones. The properties of the new structural form of silver iodide are predicted_
mising of precipitating components and whose duration is approsimately inversely proportional to the molar concentration of silver iodide, and a slow process, whose occurrence can be indicated after the rapid process is practically finished. (ii) Primarily formed hydrosol particles axe monodisperse and aggregate into secondary ones by a second-order kinetic process occurring during the rapid process of sol ageing. (iii) The results of comparative radiometric analysis of silver iodide hydrosols indicate that particles of the hydrosols change their structural characteristics during ageing- This conclusion is corroborated by the eariier observed effect of acceleration of the heterogeneous eschange process by addition of a coagulating electrolyte to the system [3] _ This cannot be explained by the well-known eschange mechanisms in systems with crystallographically defined silver iodide. Utilizing these findings, the present work was undertaken to determine whether the substantial and structural characteristics of primary sol particles differ from the substantial and structural characteristics of secondary particles and to determine the character of possible differences.
INTRODUCTION In earlier papers of this series (Part I, Part II), several features of processes occurring during the ageing of siiver iodide hydrosols prepared in statu nascendi Cl] were established by optic, radiometric, and sedimentation analysis methods. These properties may be briefly summarized as follows: (i) The ageing of silver iodide hydrosols prepared in statu nascendi talxzs place through two processes of differing rates: a rapid process, which starts immediately after the 003%591OJS3JOOOO-OOOOJSO3-00
EXPERIbIENTAL The materials used and the preparation of sols as well as their treatment prior to further esperimextal procedures, were described in the first paper of this series (Part I)_ The intensity of light scattered on sol particles was measured as a function of ageing time tn by means of a Karl Zeiss Spekol ZV spectrophotometer with a nephelometric device, combined with an automatic recorder_ 0 Elsevier Sequoiaprinted in The Setherlands
10
Depending on the sol concentration, the instrument was standardized using latex suspensions of different concentrations. The intensity of light scattered in the supernatant was used as a correction factor. The measured values correspond to the relative intensities I SCrjtof light scattered on sol particles at an angle of 135” relative to the incident beam with a wavelength of 546 nm. In order to explain the observed effects, the values calculated on the basis of the existing model of silver iodide hydrosois were compared with the data obtained by heterogeneous exchange (see Part I) and with those obtained by sedimentation analysis of silver iodide hydrosols (see Part II).
RESULTS
AND
DISCUSSION
The existing modeIs of silver iodide hydrosols [3 - 91 assume that dispersed silver iodide particles are composed of the crystallographitally defined solid phase only (most frequently as a mixture of cubic and hexagonal crystallographic modifications of siIver iodide 110 - 171). Such a model allows structural changes only in relation to a spontaneous transformation from a thermodynamically unstable hexagonal modification to a thermodynamically more stable cubic modification. Most of the X-ray experiments performed with freshly prepared silver iodide hydrosols prepared in statu nascendi, have shown that the transformation from a hexagonal to a cubic crystalIographic modification is a very slow (or negligible) process compared with some other features of sol ageing, e.g. the particle aggregation process [lS, 19]_ For instance, the ratio of cubic to hexagonal modification, C/H, (measured by X-ray diffractometry) remains practically constant during the rapid ageing process (during a time interval AtA,) (Part I) of negative stable silver iodide hydrosols prepared in statu nascendi 1191. In this case, the dispersed particles are iso-structural during the observed period of sol ageing, and therefore we shall call such a model ‘the iso-structural in time model’ (IST-model). On the other hand, the results of radiometric investigations of stable silver iodide hydrosols prepared in statu nascendi indicate that some specific effects (such as acceleration of the heterogeneous
exchange process caused by coagulation of systems, and different aggregation number obtained for the systems applying two different exchange methods) and the change of their quantitative values during the ageing of systems cannot be expIa.ined on the basis of the IST-model (Part I, l,2,20). The results of radiometric (Part I), electron-micrographic (Part I), and sedimentation (Part II) analysis of stable silver iodide hydrosols indicate that primarily formed sol particles markedly change their structural characteristics during the process of aggregation_ This conclusion is in disagreement with the IST-model predictions and the resuhs obtained by the usual method of X-ray ana3ysis for sta3le silver iodide hydrosols (solid-liquid phase separation by centrifuging or coagulation and X-ray analysis of the solid phase released from the Iiquid phase) [17, 191. Now, what is true? Does the rapid ageing process of stabIe siiver iodide hydrosols manifest itself only as an aggregation of primary particles to isostructural secondary particles (IST-model), or is the aggregation process accompanied by drastic structural changes which are different from the transformation of a hexagonal to a cubic crystallographic modification of silver iodide? Or, to pose the same question in another way: Is there a structural form of silver iodide hitherto unknown which is characteristic only of freshly prepared, stable silver iodide hydrosols? Unfortunately, this basic probZn cannot be solved in a direct way. In fact, a direct X-ray analysis of stable sols that would indicate all structural forms of silver iodide cannot be performed because the molar ratio [Hz01 /[silver iodide] is too high; the diffraction maxima of the liquid phase ‘cover’ all diffraction maxima of crystalIographica.lly known and possibly unknown modifications of silver iodide. The usual experimental procedure of X-ray analysis of silver iodide hydrosols, including the solid-liquid phase separation [17,19], is not usefu1, because the possible unknown structural form is characteristic only for stable sols, and the solid-liquid separation (centrifuging, coagulation) causes a rapid transformation of the unknown structures to the well-known cubic and hexagonal modifications [l, 2, 15, 203 . For this reason, we shall try to solve the problem in an indirect way. The first step in this attempt is to over-
11
come a shortcoming of the IST-model: The IST-model of silver iodide does not provide a satisfactory interpretation of the experimental results obtained for stable silver iodide ~01s. In order to find a form of IST-model suitable for comparative analysis between experimental facts, and the values calculated on the basis of such a model, we start from the experimentally proved fact that during the rapid ageing process, primarily formed monodispersed particles of silver iodide aggregate to secondary ones by a second-order kinetic process (Part II), and that the C/H ratio of silver iodide remains constant_ Such a synthetic model is schematically represented in Fig. 1. For the sake of simplicity, primary particles are represented as spheres of volume V, and diameter dl. Secondary particles of volume V, = 2X7, can be formed via total coalescence (Fig. 1A) or via partial coalescence (Fig. X3), after t-he collision of two primary particles. In both cases the C/H ratio of silver iodide is assumed to remain constant. In accordance with the experimental results described in the second part of this series (Part II), it is assumed that the interaction between primary particles and newly formed secondary particles and the interaction between secondary particles themselves are negligible in comparison with the interaction between primary particles_ In this case, the number n, of primary particles can be expressed as a function of ageing time tA, by the following equation
n1 = l/(n,“k,t,
(1)
Here, nlo is the number of primary particles at tA = 0 and 12, is the rate constant of the second-order kinetic aggregation processSuch a simple model of aggregation of primary silver iodide particles to secondm ones, based on the IST-model of colloidal silver iodide sol, enables one to predict the behaviour of the system during the rapid process of sol ageing (during the time interval .M,J). For instance, applying this model, it is possible to calculate the change in the heterogeneous exchange fraction F during the time interval coefficients AfA, 3 the ratio of sedimentation sz/sl of secondary and primary particles, and the change of relative intensity of light scattered on sol particles, as a function of rhe ageing time tA_ Figure 2 represents the F versus t_., functions calculated on the basis of the IST-model (broken lines) compared with the functions obtained experimentally by Procedure I (see Part I, ESPERIMESTXL) in 0.05, 0.01, 0.005, and 0.002 molar silver
l.,.‘.,,:..
0
200
600 ------iEO 800
L!JO
p&noIdm-3=c!-005
o-71
Fig. 1. Schematical representation of the aggregation of two primary particles having volume VI and diameter dl to a secondary particle on the basis of the IST-model. A secondary particle is formed by total (A) or partial (B) coalescence after collision of two primary particles_ In both cases, the structural characteristics of secondary particles are the same as those of primary particles.
+ 1)
,I,,
:.t’.
0
10
.
80'
,I,
'120
,
'160.
,I.
200
Fig. 2_ Fractions of exchange F, calculated on the basis of the IST-model (broken turuzs) and those determined experimentally (solid curves) in silver iodide h_vdrosols, plotted against ageing time t_=,_ The time of exchange TV is 5 min in all experiments_
12
iodide soIs. The calculation on the basis of the IST-model has been made for the case of total coalescence (see Fig. 1A). In this case,
the difference in particle surface area betlveen primary and secondary particles is larger than in all other cases (partial coalescence). Thus, in the cases of partial coalescence (see Fig. lB), the differences in F values calculated for primary and secondary particles are smaller than in the case of total coalescence. The method of calculating F values on the basis of the IST-model, represented by broken lines in Fig. 2, is explained in the APPENDIX. As shown in Fig. 2, the exchange fraction F, measured by Procedure I, during an exchange time tz = 5 min, decreases with ageing time more rapidly than the exchange fraction calculated on the basis of the IST-model. Taking into account that the heterogeneous exchange process in the examined sols is governed by self-diffusion of I- ions inside colloidal particles of silver iodide (Part I), the difference between the F versus t=, function calculated on the basis of the IST-model and the F versus t, curve obtained by Procedure I in real systems may be due to the following: (i) increase in particle size of the examined sols during sol ageing proceeds more rapdily than the increase in particle size caused by the second-order kinetic aggregation process, and (ii) contrary to the IST-model, the selfdiffusion coefficient of I- ions inside colloidal particles of silver iodide decreases considerably during the sol ageing. It has been proved esperimentally that the aggregation of primary to secondary particles is a second-order kinetic process (Part II), and that the volume of secondary particles is approximately twice as large as the volume of primary particles (Part I) (and thus d, = 1.26 d,).Hence, the difference between the F values calculated on the basis of the IST-model and those obtained experimentally in the real system is due only
to the decrease of the self-diffusion coefficient of I- ions inside colloidal particles of silver iodide during the ageing of ~01s. Since the self-diffusion coefficient is closely related to the structure of colloidal particles, it may be concluded that the structural characterlstics of secondary particles are essentially different from the structural characteristics of p-in-my particles. Thus, the structural changes influencing the change of the self-diffusion coefficient of I- ions inside colloidal silver
iodide particles during the sol ageing, are the consequence of collision of primary particles during the same ageing time. Taking into account that the C/H ratio of silver iodide particles remains constant during the ageing time used, the change of structural characteristics of sol particles cannot be explained on tine basis of the transformation of silver iodide from a hexagonal to a cubic form. Even if one assumes that primary particles are all hexagonal and that each collision of two primary particles causes complete transformation
from
the hesagonal
to the cubic
form
(SO
that secondary particles are all cubic), the eschange fractions would increase with ageing time (see APPENDIX). In this case the influence of the increase of the self-diffusion coefficient of I- ions on the exchange rate is more pronounced than that of the decrease of particle surface area during the same ageing time_ This observation is directly opposite to the results calculated on the basis of the IST-model and the results obtained experimentally (see Fig. 2). These conclusions support the earlier assumption on the existence of a new structural form of silver iodide, characteristic of stable sols, and its transformation to the known crystallographic forms (cubic and hexagonal) during the ageing of stable silver iodide hydrosols (Part I, 1,
2,20). A further support for this assumption is the difference between the sedimentation coefficient ratio s&5,, calculated on the basis of the IST-model, and the ratio sz/s, calculated on the basis of experimental results. The sedimentation coefficient s, of primary particles and the sedimentation coefficient s2 of secondary particles can be calculated from the values-e, and ez respectively, obtained by sedimentation analysis of the sols (Part II). e1 and e2 correspond to the mass fraction f,“’ of primary particles and to the mass fraction
of secondary particles, respectively, remaining inside the distance region between x1 exp(s2&t,) and x2 after centrifuging of sols at given 02t, value (see Figs. 3 - 6 in Part II). The values ei and ~2 can be expressed as f2 ”
cl
=
exp(-2slo2t,)
E2
=
exp(-2s2w2t,)
where xi and x1 represent the last radial distance of the centrifuging system (liquid
(3)
13
level) and the radial distance of the centrifuge tube bottom, respectively, both from the asis of spinning; ct) is the angular speed of the centrifuge rotor (radians per second) and t, is the duration of centrifuging_ From eqns. (2) and (3) it follows
s, = --In er/2-~*t,
(4)
s2 = --In e2/2- ant,
(5)
and their ratio is s+,
= In e2/ln e1
(6)
For O-01 molar silver iodide sol, the experimentally determined s&r ratio is 3-75 (see Fig. 3 in Part II), for O-002 molar sol, s2/s1 = 6.67 (see Fig. 4 in Part II), and for 0.0902 molar sol, s&r = 20.33 (see Fig. 6 in Part II). The sedimentation coefficients of primary and secondary particles are functions of their sizes r, and r2, respectively, specific densities and also of the ~1 and PZ. respectively, viscosity TZand the specific density p. of the dispersing media. The sedimentation coefficients s, and s2 can be espressed in the form
1211 Sl = W2(P,
-PO)/%
(7)
s2 = 2r2TPz
-PO)/%
(S)
Applying the IST-model to the second-order kinetic aggregation process (see Fig. I), one obtains r2 = 1.26 r, and p2 = p,, and thus It is evident that the S,lS 1 = r22/r,2 = 1.588. s2/si ratios determined experimentally depend on the sol concentration Co, and that these values are considerably larger than s2/s, values calculated on the basis of the IST-model. As in the analysis of heterogeneous exchange data, it may be assumed that in an extreme but unreal case, primary particles of silver iodide are all hexagonal, and secondary particles all cubic. Under this assumption SJS, = 1.588(p, - po)/(ph - po) = 1.588(6.01 - l)/ (5.683 - 1) = 1.7. This value is only slightly larger than the s2/sI ratio calculated on the basis of IST-model. Here, pc = 6.01 is the specific density of the cubic form of silver iodide, ph = 5.683 is the specific density of the hexagonal form, and p. = 1. As expressed in eqns. (2) and (3), the values e1 and e2 are
functions cf the sedimentation coefficients s1 and s2 respectively, for a given 02t, value. Hence, in accordance with eqns. (7) and (S), e1 and ez are functions of the corresponding particle radii rl and r2 respectively, of the
specific densities p, and p2 respectively, of the primary and secondary particles, and the viscosity n and specific density p. of the dispersing media. The values of viscosity and specific density of the dispersing media do not change during the aggregation process, and the experimentally determined rr!r, ratio of the esamined sols corresponds to the r,jr, ratio (r,/r, = l-26) calculated on the basis of the IST-model (Part I)_ Hence, the larger values of the experimentally determined +is, ratios may be a consequence of the difference between p, and pz of real systems (p, -: pz)Such large differences cannot be esplained bthe difference between the specific densities of the cubic and hesagonal forms of silver iodide even if primary particles are all hesagonal and secondary particles all cubic (in this case sr/s, = 1.7). This means that the specific density of secondary sol particles is considerably larger than the specific density of primary particles, which may be a consequence of a drastic structural difference between primary and secondary sol particles_ This conclusion is in disagreement with the existing IST-model of silver iodide h?;drosols. but it corroborates the assumption of a new structural form of silver iodide, characterstic of stable hydrosols only_ The relative intensity of light scattered on sol particles may be espressed in accordance with Rayleigh’s theory of light scattering, provided that the particle size of the esamined systems does not esceed the size dt allowed for in Rayleigh scattering and espressed as dL = X120, where X is the wavelength of the light beam used. In our case, X = 546 nm (see ESPERI,\IEXTI\L) and thus dL = 27.3 nm. Electron micrographs of 0.01 and 0.002 molar silver iodide sols show that the particle diameters of secondary particles are less than 2’7.3 nm in both sols examined (d2 =$, = 22.5 nm in 0.01 molar sol and d2 = L2 = 22.3 nm in O-002 molar sol) (Fart 1). For this reason, the relative intensity Zscr,of light scattered on particles of the esamined sols can be expressed in accordance with Rayleighk scattering theory as I s(,, = Z09RZ[(m2 X
i)/(m"
(1 + cos 0 )mv’/4;lr’
-
Z)] ' X
(91
Here, lo is the intensity of the incident beam, x is the distance from the light source to the sample, rra is the relative refractive indes of
the dispersed phnsc, 0 is the angle of scnttering, 11is tho particic conccntrntion (numbQr of particles per unity volume) and u is the particle volume. Under constant experimental conditions (lo, h, x and 0 constant), Isc, depends on m, n, and u only. I=(r,= KM,zvZ
(10)
where K = 9n’1,(1 + cos 0)/4h~’ and ICI= [(rn - l)/(m - 2)] ‘. Applying Rayleigh’s scattering theory to the IST-model of silver iodide particles aggregation, schematically represented in Fig. 1, the relative intensity of light L(r), , scattered on primary particles at times t, > 0 is IJW‘ = Khl,nlv,l
(11)
and at tA = 0 1SW,, = KAllrz,OuIZ (12) The relative intensity of light Ise,, , scattered on isostructural (rn? = rn , , hf, = hfl) secondary particles, is I S(r): = h’nf 1n 2v22
(13)
where nz is the concentration of secondary particles and v2 is their volume. As for the second-order kinetic process, represented by eqn. (l), nz = (nlo - ni)/2 and v1 = 2v, IS(r): = 2KhZlv,‘(n,”
-n,)
(14)
The total relative intensity of light Is(r,lr scattered on all particles contained in the systems at times tA > 0, is
I w(r),
=
Is(r), -+ Is(r), =
h’Mau,%2n1“ - n 1)
(15)
Replacing ?a1 in cqn. (15) by the equivalent value, expressed by eqn. (l), and then. dividing eqn. (15) by cqn. (ll), one obtains I E(r),/&,,
= 2 -
l/(Cch’,fl\
+ 1)
(16)
= ColzJl\ + 1
(17)
and thus I/(2
- &,,lIIi(r,,)
where rS, = 12i/5? and k I = n I”& = constant. The results expressed by eqn. (17) show that the value l/(2 - Jse,,lJ.c,,,) ought to be in linear relationship with t,, during the fast ageing process (during the time interval AtA,), if the IST-model is valid, i.e. if M2 = M, and v2 = 2ui. Contrary to this result, the values of l/(2 - Izcr,,/l,c,,,) calculated from the measured relative intensities of light Isfr,, scattered on particles of differently aged 0.01 and 0.002 molar silver iodide sois, are not linear functions of t, , as shown in Figs. 3 and 4. The values for Isej, are obtained by graphic extrapolation of Ise,( versus tA curves to zero ageing time (t, = 0). The functions represesented in Figs. 3 and 4 have an interruption point after the l/(2 - Is(r,Jlsc,J values change their sign end tend asymptotically to zero. At the interruption point, Isc,JlscjO = 2. In accordance with eqn. (16), calculated on the basis of the IST-model, such a ratio can be attained for fA + 00. This means that in real systems the increase of Is(r)tis stronger during the ageing of systems than is predicted on the
Fig. 3. Values of l/(2 - Zs(r)f/Zs(I)O) plotted as a function of ageing time tA_ I,.,), represents the intensity of light scattered on particles of 0.01 molar silver iodide sols aged for tA > 0, and Zs(,)0 is the intensity of light scattered on the Same particles at tA = 0.
LAgll/moldm3-0002 CNolUmddm;r
-40
-
-60
-
=a002
/I,(r),,) plotted as a function of ageing timrn t,,_ I,(r), rqwesrnts thr intcvkty c)i Fig. 4. Values of l/(2 -I,(r), liaht scattered on particles of 0.002 molar silver iodide sols aged for I,, > 0, and f,(r), is thr intensity of Ii&r
s&ttered on the&w
particles atl~ = 0.
basis of the XT-model and expressed by eqns. (16) and (17) respectively. Since it has been proved esperimentally that the aggregation kinetics of real systems is identical with that assumed for the ET-model, the stronger increase of the experimentally determined I s(r>r&L ratio may be accounted for either by u2 > 2u, or Mz > M1 (and thus mz > m,). Since the sedimentation analys% of the sols show that p2 > p, , the volume z+ of secondary particles cannot be larger than 2u,, which is also in agreement with electron-microscopic data (Part I). In this case, for fiZ2 = M,, additional (&r)*/&(r), It, - = = 2. Hence, increase of the IsCr,I/lsCr,, ratio ought to be caused only by the increase in the relative refractive index of the dispersed phase during aggregation_ This indicates that the relative refractive index of secondary particles is larger than the relative refractive index of primary silver iodide particles_ Such a conclusion supports the assumption on structural changes of silver iodide hydrosol particles during aggregation, and thus it speaks in favor of the existence of a new structural form of silver iodide characteristic of freshly prepared, stable hydrosols.
CONCLUSION A comparative analysis has been made of the results predicted on the basis of :he ISTmodel and the data obtained by the methods
of heterogeneous exchange. sedimentation, and light-scattering analysis. This analysis confirms the assumption on a considerable structural difference between primary and secondary particles of stable silver iodide sols prepared in statu nascendi, and the idea of a new structural form of silver iodide characteristics of stable scls. Such analysis also yields information on possible characteristics of the new structural form of silver iodide. The large self-diffusion coefficient for I- ions indicates that the constituents inside the new structural form are very mobile, and that the forces between them are not as strong as the forces between the constituents (_.g’. 1. ) of known structural modifications. It is in this sense that very rapid eschange process can be esplained in the absence of the potential barrier around sol particles (the acceleration of the eschange process caused by addition of a coagulating electrolyte to the system)_ The consequences of weak attractive forces between the constituents are increase in interionic distances and thus decrease of specific density, as well as lowering of the relative refractive indes of the new structural form relative to the compact structures of cubic ard hexagonal forms. The masimum quantitative values of specific effects, characteristic of the new structan-a.l form of silver iodide (for instance. quantitative values of the exchange effect caused by coagulation), were obtained immediately after precipitation of sols- This led to the conclusion that primary
16 particles are probably completely built of a new structural form of silver iodide. The lowering of the quantitative values of the effect characteristic of the new structural form during the sol ageing (see Fig. 2) indicate that the aggregation process causes the transformation of the new structural form to well-known structural forms of silver iodide. However, the measurable effect of coagulating electrolyte on the heterogeneous exchange process after the end of the rapid ageing process (tA > t4, ) indicates that, in comparison with primary particles, secondary particles constitute a considerably smaller but not negligible quantity of the new structural form of silver iodide. Thus, secondary particles constitute a mixture of crystallographitally well-defined solid silver iodide and the new structural form of silver iodide.
APPENDIX
For a heterogeneous eschange process governed by self-diffusion in a well-stirred solid-liquid system, the interrelationship between the exchange fraction F, reached at an exchange time tx, and the parameters of the system is expressed by Wagner’s solution of the differential equation for diffusion 1223 : l-F=
s
2(1 -f CX)
rn= 1 m2ir2/3 + 3(o~ + or’)
X exp(-m%r2DStJf2)
that the ratio of the imaginary particle sizes (*Fz/*F,),, determined by the analysis of experimentally measured F values, applying the graphical form of eqn_ (X3), is equal to the ratio (f2/F1)c of particle sizes before coagulation_ In other words, if fractions of exchange are measured by Procedure I in two sols with different starting particle sizes (F,)c and (Fz),, (particle sizes prior to the coagulation and the exchange process), then the following relation is valid: (*~,l*F,),,(,,
=
(*F2/*p,&,,
=
(*~2/*Qh&n,
=
(*r2l*F,&2,
=
=
VzF,)o
---
WV
to (*F.‘/*f,)lE(m, correspond Here, (*F2/*~dtEco, to the ratios of imaginary particle sizes *fz and *F1 throughout to the heterogeneous exchange process. This estimate was checked experimentally for silver iodide hydrosols (Part I, 24). Using eqn. (19), we calculated the change of exchange fractions for systems having the properties predicted by the ISTmodel, and compared it in Fig. 2 with the results obtained esperimentally. Applying the ET-model, the mean particle size (fi),, of sol aged for time tA(l) before coagulation is
= r,(ni
+ 1_26n&(n,
+ nz)
(26)
where ri and r2 are the radii of primary and secondary particles respectively. Since nI = l/(h2ta + l/n,“) and nz = (RI0 - n,)/Z,
X
(13)
Here, Q! is molar ratio of exchangeable ions between the solid and the liquid phase, 0, is a self-diffusion coefficient of exchangeable ions in the solid phase, f is the mean radius of particles and m represents an integer, a natural number that has all values from 1 to inf5iity. The original form of eqn. (18) may be issed for the analysis of a self-diffusiongoverned exchange process only when D,, f, and Q(are known and constant. In our case, direct analysis based on eqn. (18) cannot be applied because the results represented in Fig. 2 are obtained by Procedure I (Part I), inclclding the change of particle size during the exchange process. However, calculations derived on the basis of Rajagopal’s investigations of Brownian coagulation [23,24] show
(f,)o =
0.74r,/(JzztA(1)
+ lln,“)
nzo + l/(iz2tn(l)
Dividing
eqn. (21)
(F1)o/nl”-=
by nIo gives
0_74r,/(C,k,t,(l) 1 +
(21)
-i- l/nIo) -+ 1)
W! 1/(COk$A(1)
+
1)
The ratio for a sol aged for time tA(2) can be
calculated by an analogous expression: (f2)o!n i” = [O-74r,l(CoUA(2)
f1 +
1/(COh,tA(2)
+ +
III/ (23)
l)j
and thus (f*)o/(F,)o=
(~22/~1)0
0.37 =
[ COh,tA(2)
+
1
0.37 COk-rtA(l)
I[
1
l+
CO%C~A(~)
l+ +
1
+
1 I
+
f
I Co&t,@)
1
(24)
OL-
1-F 03-
l-F2
M
l
-
1 Fig. 5_ Graphical representation of Wagner’s solution of the differential equation for diffusion under conditions of heterogeneous exchange process governed by self-diffusion_ F represents the fraction of exchange attained for the exchange time in in the system having solid particles of mean radius ? and self-diffusion coefficient for exchangeable ions D,, dispersed in the liquid phase. x1 and _r, are numerical values of ii’D,t~/f= corresponding to the val&s 1 - F1 and lL Fz_
Assuming that primary particles of both the real system and that represented in Fig. 1 (ET-model) have identical structure and size, i-e_ if the initial conditions for both systems are the same, the fraction of exchange F at
t, = 0 must be identical for both systems. Since the measurement of the exchange fractions at tA = 0 is associated with experimental difficulties, l-minute aged sols were used as a good approximation for t, = 0. Such an approximation is suitable, because at tA = 1 min all examined sols consist practically of primary particles only_ However, the n,/nz ratio (nl measured at t, = 1 min) decreases with increasing concentration of sol and the error due to the assumption used is larger for sols of higher concentration (Ce). Thus, the differences between the measured and calculated F values decrease with increasing concentration of ~01s. To the value F, determined in l-minute aged sol, corresponds the numerical value x1 cf the quantity sr’D,t,/F* in the graphical presentation of eqn. (18) (see Fig. 5). The value of r, corresponding to the F value experimentally determined by Procedure I, should have not a real but an imaginary value *f: f0 < *f < FEE, where f, and FEare the real mean radii of sol particles at the beginning (tE = 0) and at the ;zccr.ge time FE= tE(experimental) [ 24]_ 9
F,2(det)
= *f,’
= A~D~?~/x,
(25)
where f,‘(det) is a value of F,’ corresponding to the experimentally determined F value for 5, = t,(experimental)_ In accordance with eqns. (19) and (22), the ratio of both the real (r21f,), and imaginary (*fll*fi)tE
1 =[c::L]ll+ :.i7f::i (26) O-37
I
Cok,t,f2)
+ 1 II
Since the k, values for analysed sols are determined by sedimentation analysis (Part II), the numerical value of p(2,l) for 2 given time t*(2) can be calculated using the righthand side of eqn. (26) Then *fz2 = Cp(2,1)]
‘*F1’ = Ln(2,1)]
‘ii’D&x, (27)
In accordance with eqn (25), Fzz can also be espressed as *f-z! _ = x’D,& j-r2 (28) Hence, (29)
x2 = 7~~D&/*i=~~ =x,&42,
3
l)]’
The numerical value -r2, calculated from the known numerical values x1 and ~(2, l), has the corresponding value of 1 - F2 and 1 - F, respectively, as shown in Fig. 5. The value F2 corresponds to a fraction of exchange that would be measured by Procedure I in a real system if the IST-model were applicable to it. For the assumed case when primary particles of silver iodide hydrosol are all of the hexagonal form and secondary particles all cubic, the following relations are valid: x x2
I-
- rrZD,(H)&s/7,2 =
n*D,(C)t,&*
(31) (32)
-I 5 6
7 S
FoZge.
of the Stability
13
1-Z
16
and
1s
Fz)FI
19
which is in disagreement with the experimentally obtained results and the data calculated on the basis of the IST-model.
20 21
REFERENCES 1 R_ Despot&if and B. Sub&f. Croat. Chenx_ Acta. 43 (19il) 153. 2 R. Despotox% and B. Subotie, J. Inorg. ~VucZ. Chem. 38 (1976) 1317_
of Lyophobic
CoZZoids.
Theory
Elsevier.
Amsterdam. 19-4s. _ _ E. Cohen and W. J. D. Van Dobbenburgh, 2. Physih. Chem., A137 (1928) 289. 11 R. Bioch and H. hIBIer, 2. Physik. Chem., A 1.52 (1931) 245. 12 G. Burley and H. E. Kissinger, HZf. Bar. Standards.
rz>_r,
(33)
240.
10
14 15
= 1O-*5
49 (1966)
9 E. J \V_ Verwey~and J. Th. G. Overbeek,
cm2 s-i is the self-diffusion coefficient of I- ions inside the crystal space of hexagonal modification of silver iodide and D 5(C) = 2 X lo-i2 cm2 s-’ is - the corresponding self-diffusion coefficient for cubic modification of silver iodide [25] _ It is evident that in this case the following relations hold:
D,(H)
H. R. Cruyt, CoZZoZrZScience. Vol. I, Elsevier, Amsterdam, 1952. B. H. Bijsterbosch and d. Lyklema, A&_ CoZZoid Interface Sci.. 9 (1978) l-47. B. Tezak. Croat. Chem. Accta. 42 (1970) 81. AI. AIirnik, Croat. Chem. Acta. 42 (1970) 161. A. Watanabe, BUZZ. Inst. Chem. Res.. Kyoto University, 1960, p_ 2. R. Matejec and R. Meyer, 2. Phys. Chem. Neue
22 23 21 25
64.~ (1960) 403. S. N. Chatterjee, Intern. Kongr. Elektronen Uikroskopie. 4. Berlin 1958. Verhander 1. (1960)
453. ii. Yamada, BUZZ.Sot. Phot. Japan, I1 (1961) 1. R. DespotoviS and S. Popovie, Croat. Chem. Actc. 38 (1966) 321. R. Despotovie, 2. Despotox%, &I_ Jajetie, 2. Teligman and S. Popovie, Croat. Chem. Acta. 42 (1970) 145. R. Despotovie, 2. Despotovie, 31. Mirnik, B. SubotiE, Croat. Chem. Acta, 42 (1950) 445. R. Despotovic, Tenside-Detergents, 10 (1973) 297. R. Despotovic, N. Filipovi&Vincekovie and B. Subolie. CoZZoid and Interface Science, Vol. IV, Academic Press, New York-San FranciscoLondon. 1976, p- 297. R. Despotox% and B. Subotic, Croat. Chem. Acta. 45 (1973) 37-Z_ Z. K. Jelinek. PurticZe Size Analysis, John Wiley C Sons, New York-London-Sydney-Toronto. 1974. K. E. Zimens, Arkiu Kemi. Mineral. GeoL, 3OA (1915) 1. E_ S_ Rajagopal, KoZZoid Z_. I6i (1960) 17. B. Subotie, Powder TechnoL. 24 (1979) 35. B. Jordan and M. Pochon, HeZc. Phys. Acta. 30 (1957)_33.
9
1S
Particulate Processes in Freshly Prepared Silver Iodide Hydrosols Part III. Disagreement between Experimental Results and the Existing Models of Silver Iodide Hydrosols BORIS SUBOTIC Institute "RuderBo&otG",
41OGI Zagreb. P-0. Box 1016. Croatia{ITxgosZaria)
(Received April %X,1981)
SUhlMARY Silver iodide hydrosols prepared in statu nascendi exhibit some specific effects that cannot be explained on the basis of the e_xisting models of silver iodide hydrosolsThese effects are acceleration of a heterogeneous exchange process caused by coagulation of the system, and different aggregation number obtained in the same system applying two different exchange methods. In order to explain these effects, a comparison has been made of the differences between the data calculated on the basis of the ezxisting model of silver iodide hydrosols and the values determined by radiometric, optic, and sedimentation analysis. The previous assumption on the existence of a new structural form of silver iodide. characteristic of stable silver iodide hydrosols, has been confirmed_ This new structural form of silver iodide transforms into known hexagonal and cubic modifications during the aggregation of primary particles of silver iodide to secondac ones. The properties of the new structural form of silver iodide are predicted_
mising of precipitating components and whose duration is approsimately inversely proportional to the molar concentration of silver iodide, and a slow process, whose occurrence can be indicated after the rapid process is practically finished. (ii) Primarily formed hydrosol particles axe monodisperse and aggregate into secondary ones by a second-order kinetic process occurring during the rapid process of sol ageing. (iii) The results of comparative radiometric analysis of silver iodide hydrosols indicate that particles of the hydrosols change their structural characteristics during ageing- This conclusion is corroborated by the eariier observed effect of acceleration of the heterogeneous eschange process by addition of a coagulating electrolyte to the system [3] _ This cannot be explained by the well-known eschange mechanisms in systems with crystallographically defined silver iodide. Utilizing these findings, the present work was undertaken to determine whether the substantial and structural characteristics of primary sol particles differ from the substantial and structural characteristics of secondary particles and to determine the character of possible differences.
INTRODUCTION In earlier papers of this series (Part I, Part II), several features of processes occurring during the ageing of siiver iodide hydrosols prepared in statu nascendi Cl] were established by optic, radiometric, and sedimentation analysis methods. These properties may be briefly summarized as follows: (i) The ageing of silver iodide hydrosols prepared in statu nascendi talxzs place through two processes of differing rates: a rapid process, which starts immediately after the 003%591OJS3JOOOO-OOOOJSO3-00
EXPERIbIENTAL The materials used and the preparation of sols as well as their treatment prior to further esperimextal procedures, were described in the first paper of this series (Part I)_ The intensity of light scattered on sol particles was measured as a function of ageing time tn by means of a Karl Zeiss Spekol ZV spectrophotometer with a nephelometric device, combined with an automatic recorder_ 0 Elsevier Sequoiaprinted in The Setherlands
10
Depending on the sol concentration, the instrument was standardized using latex suspensions of different concentrations. The intensity of light scattered in the supernatant was used as a correction factor. The measured values correspond to the relative intensities I SCrjtof light scattered on sol particles at an angle of 135” relative to the incident beam with a wavelength of 546 nm. In order to explain the observed effects, the values calculated on the basis of the existing model of silver iodide hydrosois were compared with the data obtained by heterogeneous exchange (see Part I) and with those obtained by sedimentation analysis of silver iodide hydrosols (see Part II).
RESULTS
AND
DISCUSSION
The existing modeIs of silver iodide hydrosols [3 - 91 assume that dispersed silver iodide particles are composed of the crystallographitally defined solid phase only (most frequently as a mixture of cubic and hexagonal crystallographic modifications of siIver iodide 110 - 171). Such a model allows structural changes only in relation to a spontaneous transformation from a thermodynamically unstable hexagonal modification to a thermodynamically more stable cubic modification. Most of the X-ray experiments performed with freshly prepared silver iodide hydrosols prepared in statu nascendi, have shown that the transformation from a hexagonal to a cubic crystalIographic modification is a very slow (or negligible) process compared with some other features of sol ageing, e.g. the particle aggregation process [lS, 19]_ For instance, the ratio of cubic to hexagonal modification, C/H, (measured by X-ray diffractometry) remains practically constant during the rapid ageing process (during a time interval AtA,) (Part I) of negative stable silver iodide hydrosols prepared in statu nascendi 1191. In this case, the dispersed particles are iso-structural during the observed period of sol ageing, and therefore we shall call such a model ‘the iso-structural in time model’ (IST-model). On the other hand, the results of radiometric investigations of stable silver iodide hydrosols prepared in statu nascendi indicate that some specific effects (such as acceleration of the heterogeneous
exchange process caused by coagulation of systems, and different aggregation number obtained for the systems applying two different exchange methods) and the change of their quantitative values during the ageing of systems cannot be expIa.ined on the basis of the IST-model (Part I, l,2,20). The results of radiometric (Part I), electron-micrographic (Part I), and sedimentation (Part II) analysis of stable silver iodide hydrosols indicate that primarily formed sol particles markedly change their structural characteristics during the process of aggregation_ This conclusion is in disagreement with the IST-model predictions and the resuhs obtained by the usual method of X-ray ana3ysis for sta3le silver iodide hydrosols (solid-liquid phase separation by centrifuging or coagulation and X-ray analysis of the solid phase released from the Iiquid phase) [17, 191. Now, what is true? Does the rapid ageing process of stabIe siiver iodide hydrosols manifest itself only as an aggregation of primary particles to isostructural secondary particles (IST-model), or is the aggregation process accompanied by drastic structural changes which are different from the transformation of a hexagonal to a cubic crystallographic modification of silver iodide? Or, to pose the same question in another way: Is there a structural form of silver iodide hitherto unknown which is characteristic only of freshly prepared, stable silver iodide hydrosols? Unfortunately, this basic probZn cannot be solved in a direct way. In fact, a direct X-ray analysis of stable sols that would indicate all structural forms of silver iodide cannot be performed because the molar ratio [Hz01 /[silver iodide] is too high; the diffraction maxima of the liquid phase ‘cover’ all diffraction maxima of crystalIographica.lly known and possibly unknown modifications of silver iodide. The usual experimental procedure of X-ray analysis of silver iodide hydrosols, including the solid-liquid phase separation [17,19], is not usefu1, because the possible unknown structural form is characteristic only for stable sols, and the solid-liquid separation (centrifuging, coagulation) causes a rapid transformation of the unknown structures to the well-known cubic and hexagonal modifications [l, 2, 15, 203 . For this reason, we shall try to solve the problem in an indirect way. The first step in this attempt is to over-
11
come a shortcoming of the IST-model: The IST-model of silver iodide does not provide a satisfactory interpretation of the experimental results obtained for stable silver iodide ~01s. In order to find a form of IST-model suitable for comparative analysis between experimental facts, and the values calculated on the basis of such a model, we start from the experimentally proved fact that during the rapid ageing process, primarily formed monodispersed particles of silver iodide aggregate to secondary ones by a second-order kinetic process (Part II), and that the C/H ratio of silver iodide remains constant_ Such a synthetic model is schematically represented in Fig. 1. For the sake of simplicity, primary particles are represented as spheres of volume V, and diameter dl. Secondary particles of volume V, = 2X7, can be formed via total coalescence (Fig. 1A) or via partial coalescence (Fig. X3), after t-he collision of two primary particles. In both cases the C/H ratio of silver iodide is assumed to remain constant. In accordance with the experimental results described in the second part of this series (Part II), it is assumed that the interaction between primary particles and newly formed secondary particles and the interaction between secondary particles themselves are negligible in comparison with the interaction between primary particles_ In this case, the number n, of primary particles can be expressed as a function of ageing time tA, by the following equation
n1 = l/(n,“k,t,
(1)
Here, nlo is the number of primary particles at tA = 0 and 12, is the rate constant of the second-order kinetic aggregation processSuch a simple model of aggregation of primary silver iodide particles to secondm ones, based on the IST-model of colloidal silver iodide sol, enables one to predict the behaviour of the system during the rapid process of sol ageing (during the time interval .M,J). For instance, applying this model, it is possible to calculate the change in the heterogeneous exchange fraction F during the time interval coefficients AfA, 3 the ratio of sedimentation sz/sl of secondary and primary particles, and the change of relative intensity of light scattered on sol particles, as a function of rhe ageing time tA_ Figure 2 represents the F versus t_., functions calculated on the basis of the IST-model (broken lines) compared with the functions obtained experimentally by Procedure I (see Part I, ESPERIMESTXL) in 0.05, 0.01, 0.005, and 0.002 molar silver
l.,.‘.,,:..
0
200
600 ------iEO 800
L!JO
p&noIdm-3=c!-005
o-71
Fig. 1. Schematical representation of the aggregation of two primary particles having volume VI and diameter dl to a secondary particle on the basis of the IST-model. A secondary particle is formed by total (A) or partial (B) coalescence after collision of two primary particles_ In both cases, the structural characteristics of secondary particles are the same as those of primary particles.
+ 1)
,I,,
:.t’.
0
10
.
80'
,I,
'120
,
'160.
,I.
200
Fig. 2_ Fractions of exchange F, calculated on the basis of the IST-model (broken turuzs) and those determined experimentally (solid curves) in silver iodide h_vdrosols, plotted against ageing time t_=,_ The time of exchange TV is 5 min in all experiments_
12
iodide soIs. The calculation on the basis of the IST-model has been made for the case of total coalescence (see Fig. 1A). In this case,
the difference in particle surface area betlveen primary and secondary particles is larger than in all other cases (partial coalescence). Thus, in the cases of partial coalescence (see Fig. lB), the differences in F values calculated for primary and secondary particles are smaller than in the case of total coalescence. The method of calculating F values on the basis of the IST-model, represented by broken lines in Fig. 2, is explained in the APPENDIX. As shown in Fig. 2, the exchange fraction F, measured by Procedure I, during an exchange time tz = 5 min, decreases with ageing time more rapidly than the exchange fraction calculated on the basis of the IST-model. Taking into account that the heterogeneous exchange process in the examined sols is governed by self-diffusion of I- ions inside colloidal particles of silver iodide (Part I), the difference between the F versus t=, function calculated on the basis of the IST-model and the F versus t, curve obtained by Procedure I in real systems may be due to the following: (i) increase in particle size of the examined sols during sol ageing proceeds more rapdily than the increase in particle size caused by the second-order kinetic aggregation process, and (ii) contrary to the IST-model, the selfdiffusion coefficient of I- ions inside colloidal particles of silver iodide decreases considerably during the sol ageing. It has been proved esperimentally that the aggregation of primary to secondary particles is a second-order kinetic process (Part II), and that the volume of secondary particles is approximately twice as large as the volume of primary particles (Part I) (and thus d, = 1.26 d,).Hence, the difference between the F values calculated on the basis of the IST-model and those obtained experimentally in the real system is due only
to the decrease of the self-diffusion coefficient of I- ions inside colloidal particles of silver iodide during the ageing of ~01s. Since the self-diffusion coefficient is closely related to the structure of colloidal particles, it may be concluded that the structural characterlstics of secondary particles are essentially different from the structural characteristics of p-in-my particles. Thus, the structural changes influencing the change of the self-diffusion coefficient of I- ions inside colloidal silver
iodide particles during the sol ageing, are the consequence of collision of primary particles during the same ageing time. Taking into account that the C/H ratio of silver iodide particles remains constant during the ageing time used, the change of structural characteristics of sol particles cannot be explained on tine basis of the transformation of silver iodide from a hexagonal to a cubic form. Even if one assumes that primary particles are all hexagonal and that each collision of two primary particles causes complete transformation
from
the hesagonal
to the cubic
form
(SO
that secondary particles are all cubic), the eschange fractions would increase with ageing time (see APPENDIX). In this case the influence of the increase of the self-diffusion coefficient of I- ions on the exchange rate is more pronounced than that of the decrease of particle surface area during the same ageing time_ This observation is directly opposite to the results calculated on the basis of the IST-model and the results obtained experimentally (see Fig. 2). These conclusions support the earlier assumption on the existence of a new structural form of silver iodide, characteristic of stable sols, and its transformation to the known crystallographic forms (cubic and hexagonal) during the ageing of stable silver iodide hydrosols (Part I, 1,
2,20). A further support for this assumption is the difference between the sedimentation coefficient ratio s&5,, calculated on the basis of the IST-model, and the ratio sz/s, calculated on the basis of experimental results. The sedimentation coefficient s, of primary particles and the sedimentation coefficient s2 of secondary particles can be calculated from the values-e, and ez respectively, obtained by sedimentation analysis of the sols (Part II). e1 and e2 correspond to the mass fraction f,“’ of primary particles and to the mass fraction
of secondary particles, respectively, remaining inside the distance region between x1 exp(s2&t,) and x2 after centrifuging of sols at given 02t, value (see Figs. 3 - 6 in Part II). The values ei and ~2 can be expressed as f2 ”
cl
=
exp(-2slo2t,)
E2
=
exp(-2s2w2t,)
where xi and x1 represent the last radial distance of the centrifuging system (liquid
(3)
13
level) and the radial distance of the centrifuge tube bottom, respectively, both from the asis of spinning; ct) is the angular speed of the centrifuge rotor (radians per second) and t, is the duration of centrifuging_ From eqns. (2) and (3) it follows
s, = --In er/2-~*t,
(4)
s2 = --In e2/2- ant,
(5)
and their ratio is s+,
= In e2/ln e1
(6)
For O-01 molar silver iodide sol, the experimentally determined s&r ratio is 3-75 (see Fig. 3 in Part II), for O-002 molar sol, s2/s1 = 6.67 (see Fig. 4 in Part II), and for 0.0902 molar sol, s&r = 20.33 (see Fig. 6 in Part II). The sedimentation coefficients of primary and secondary particles are functions of their sizes r, and r2, respectively, specific densities and also of the ~1 and PZ. respectively, viscosity TZand the specific density p. of the dispersing media. The sedimentation coefficients s, and s2 can be espressed in the form
1211 Sl = W2(P,
-PO)/%
(7)
s2 = 2r2TPz
-PO)/%
(S)
Applying the IST-model to the second-order kinetic aggregation process (see Fig. I), one obtains r2 = 1.26 r, and p2 = p,, and thus It is evident that the S,lS 1 = r22/r,2 = 1.588. s2/si ratios determined experimentally depend on the sol concentration Co, and that these values are considerably larger than s2/s, values calculated on the basis of the IST-model. As in the analysis of heterogeneous exchange data, it may be assumed that in an extreme but unreal case, primary particles of silver iodide are all hexagonal, and secondary particles all cubic. Under this assumption SJS, = 1.588(p, - po)/(ph - po) = 1.588(6.01 - l)/ (5.683 - 1) = 1.7. This value is only slightly larger than the s2/sI ratio calculated on the basis of IST-model. Here, pc = 6.01 is the specific density of the cubic form of silver iodide, ph = 5.683 is the specific density of the hexagonal form, and p. = 1. As expressed in eqns. (2) and (3), the values e1 and e2 are
functions cf the sedimentation coefficients s1 and s2 respectively, for a given 02t, value. Hence, in accordance with eqns. (7) and (S), e1 and ez are functions of the corresponding particle radii rl and r2 respectively, of the
specific densities p, and p2 respectively, of the primary and secondary particles, and the viscosity n and specific density p. of the dispersing media. The values of viscosity and specific density of the dispersing media do not change during the aggregation process, and the experimentally determined rr!r, ratio of the esamined sols corresponds to the r,jr, ratio (r,/r, = l-26) calculated on the basis of the IST-model (Part I)_ Hence, the larger values of the experimentally determined +is, ratios may be a consequence of the difference between p, and pz of real systems (p, -: pz)Such large differences cannot be esplained bthe difference between the specific densities of the cubic and hesagonal forms of silver iodide even if primary particles are all hesagonal and secondary particles all cubic (in this case sr/s, = 1.7). This means that the specific density of secondary sol particles is considerably larger than the specific density of primary particles, which may be a consequence of a drastic structural difference between primary and secondary sol particles_ This conclusion is in disagreement with the existing IST-model of silver iodide h?;drosols. but it corroborates the assumption of a new structural form of silver iodide, characterstic of stable hydrosols only_ The relative intensity of light scattered on sol particles may be espressed in accordance with Rayleigh’s theory of light scattering, provided that the particle size of the esamined systems does not esceed the size dt allowed for in Rayleigh scattering and espressed as dL = X120, where X is the wavelength of the light beam used. In our case, X = 546 nm (see ESPERI,\IEXTI\L) and thus dL = 27.3 nm. Electron micrographs of 0.01 and 0.002 molar silver iodide sols show that the particle diameters of secondary particles are less than 2’7.3 nm in both sols examined (d2 =$, = 22.5 nm in 0.01 molar sol and d2 = L2 = 22.3 nm in O-002 molar sol) (Fart 1). For this reason, the relative intensity Zscr,of light scattered on particles of the esamined sols can be expressed in accordance with Rayleighk scattering theory as I s(,, = Z09RZ[(m2 X
i)/(m"
(1 + cos 0 )mv’/4;lr’
-
Z)] ' X
(91
Here, lo is the intensity of the incident beam, x is the distance from the light source to the sample, rra is the relative refractive indes of
the dispersed phnsc, 0 is the angle of scnttering, 11is tho particic conccntrntion (numbQr of particles per unity volume) and u is the particle volume. Under constant experimental conditions (lo, h, x and 0 constant), Isc, depends on m, n, and u only. I=(r,= KM,zvZ
(10)
where K = 9n’1,(1 + cos 0)/4h~’ and ICI= [(rn - l)/(m - 2)] ‘. Applying Rayleigh’s scattering theory to the IST-model of silver iodide particles aggregation, schematically represented in Fig. 1, the relative intensity of light L(r), , scattered on primary particles at times t, > 0 is IJW‘ = Khl,nlv,l
(11)
and at tA = 0 1SW,, = KAllrz,OuIZ (12) The relative intensity of light Ise,, , scattered on isostructural (rn? = rn , , hf, = hfl) secondary particles, is I S(r): = h’nf 1n 2v22
(13)
where nz is the concentration of secondary particles and v2 is their volume. As for the second-order kinetic process, represented by eqn. (l), nz = (nlo - ni)/2 and v1 = 2v, IS(r): = 2KhZlv,‘(n,”
-n,)
(14)
The total relative intensity of light Is(r,lr scattered on all particles contained in the systems at times tA > 0, is
I w(r),
=
Is(r), -+ Is(r), =
h’Mau,%2n1“ - n 1)
(15)
Replacing ?a1 in cqn. (15) by the equivalent value, expressed by eqn. (l), and then. dividing eqn. (15) by cqn. (ll), one obtains I E(r),/&,,
= 2 -
l/(Cch’,fl\
+ 1)
(16)
= ColzJl\ + 1
(17)
and thus I/(2
- &,,lIIi(r,,)
where rS, = 12i/5? and k I = n I”& = constant. The results expressed by eqn. (17) show that the value l/(2 - Jse,,lJ.c,,,) ought to be in linear relationship with t,, during the fast ageing process (during the time interval AtA,), if the IST-model is valid, i.e. if M2 = M, and v2 = 2ui. Contrary to this result, the values of l/(2 - Izcr,,/l,c,,,) calculated from the measured relative intensities of light Isfr,, scattered on particles of differently aged 0.01 and 0.002 molar silver iodide sois, are not linear functions of t, , as shown in Figs. 3 and 4. The values for Isej, are obtained by graphic extrapolation of Ise,( versus tA curves to zero ageing time (t, = 0). The functions represesented in Figs. 3 and 4 have an interruption point after the l/(2 - Is(r,Jlsc,J values change their sign end tend asymptotically to zero. At the interruption point, Isc,JlscjO = 2. In accordance with eqn. (16), calculated on the basis of the IST-model, such a ratio can be attained for fA + 00. This means that in real systems the increase of Is(r)tis stronger during the ageing of systems than is predicted on the
Fig. 3. Values of l/(2 - Zs(r)f/Zs(I)O) plotted as a function of ageing time tA_ I,.,), represents the intensity of light scattered on particles of 0.01 molar silver iodide sols aged for tA > 0, and Zs(,)0 is the intensity of light scattered on the Same particles at tA = 0.
LAgll/moldm3-0002 CNolUmddm;r
-40
-
-60
-
=a002
/I,(r),,) plotted as a function of ageing timrn t,,_ I,(r), rqwesrnts thr intcvkty c)i Fig. 4. Values of l/(2 -I,(r), liaht scattered on particles of 0.002 molar silver iodide sols aged for I,, > 0, and f,(r), is thr intensity of Ii&r
s&ttered on the&w
particles atl~ = 0.
basis of the XT-model and expressed by eqns. (16) and (17) respectively. Since it has been proved esperimentally that the aggregation kinetics of real systems is identical with that assumed for the ET-model, the stronger increase of the experimentally determined I s(r>r&L ratio may be accounted for either by u2 > 2u, or Mz > M1 (and thus mz > m,). Since the sedimentation analys% of the sols show that p2 > p, , the volume z+ of secondary particles cannot be larger than 2u,, which is also in agreement with electron-microscopic data (Part I). In this case, for fiZ2 = M,, additional (&r)*/&(r), It, - = = 2. Hence, increase of the IsCr,I/lsCr,, ratio ought to be caused only by the increase in the relative refractive index of the dispersed phase during aggregation_ This indicates that the relative refractive index of secondary particles is larger than the relative refractive index of primary silver iodide particles_ Such a conclusion supports the assumption on structural changes of silver iodide hydrosol particles during aggregation, and thus it speaks in favor of the existence of a new structural form of silver iodide characteristic of freshly prepared, stable hydrosols.
CONCLUSION A comparative analysis has been made of the results predicted on the basis of :he ISTmodel and the data obtained by the methods
of heterogeneous exchange. sedimentation, and light-scattering analysis. This analysis confirms the assumption on a considerable structural difference between primary and secondary particles of stable silver iodide sols prepared in statu nascendi, and the idea of a new structural form of silver iodide characteristics of stable scls. Such analysis also yields information on possible characteristics of the new structural form of silver iodide. The large self-diffusion coefficient for I- ions indicates that the constituents inside the new structural form are very mobile, and that the forces between them are not as strong as the forces between the constituents (_.g’. 1. ) of known structural modifications. It is in this sense that very rapid eschange process can be esplained in the absence of the potential barrier around sol particles (the acceleration of the eschange process caused by addition of a coagulating electrolyte to the system)_ The consequences of weak attractive forces between the constituents are increase in interionic distances and thus decrease of specific density, as well as lowering of the relative refractive indes of the new structural form relative to the compact structures of cubic ard hexagonal forms. The masimum quantitative values of specific effects, characteristic of the new structan-a.l form of silver iodide (for instance. quantitative values of the exchange effect caused by coagulation), were obtained immediately after precipitation of sols- This led to the conclusion that primary
16 particles are probably completely built of a new structural form of silver iodide. The lowering of the quantitative values of the effect characteristic of the new structural form during the sol ageing (see Fig. 2) indicate that the aggregation process causes the transformation of the new structural form to well-known structural forms of silver iodide. However, the measurable effect of coagulating electrolyte on the heterogeneous exchange process after the end of the rapid ageing process (tA > t4, ) indicates that, in comparison with primary particles, secondary particles constitute a considerably smaller but not negligible quantity of the new structural form of silver iodide. Thus, secondary particles constitute a mixture of crystallographitally well-defined solid silver iodide and the new structural form of silver iodide.
APPENDIX
For a heterogeneous eschange process governed by self-diffusion in a well-stirred solid-liquid system, the interrelationship between the exchange fraction F, reached at an exchange time tx, and the parameters of the system is expressed by Wagner’s solution of the differential equation for diffusion 1223 : l-F=
s
2(1 -f CX)
rn= 1 m2ir2/3 + 3(o~ + or’)
X exp(-m%r2DStJf2)
that the ratio of the imaginary particle sizes (*Fz/*F,),, determined by the analysis of experimentally measured F values, applying the graphical form of eqn_ (X3), is equal to the ratio (f2/F1)c of particle sizes before coagulation_ In other words, if fractions of exchange are measured by Procedure I in two sols with different starting particle sizes (F,)c and (Fz),, (particle sizes prior to the coagulation and the exchange process), then the following relation is valid: (*~,l*F,),,(,,
=
(*F2/*p,&,,
=
(*~2/*Qh&n,
=
(*r2l*F,&2,
=
=
VzF,)o
---
WV
to (*F.‘/*f,)lE(m, correspond Here, (*F2/*~dtEco, to the ratios of imaginary particle sizes *fz and *F1 throughout to the heterogeneous exchange process. This estimate was checked experimentally for silver iodide hydrosols (Part I, 24). Using eqn. (19), we calculated the change of exchange fractions for systems having the properties predicted by the ISTmodel, and compared it in Fig. 2 with the results obtained esperimentally. Applying the ET-model, the mean particle size (fi),, of sol aged for time tA(l) before coagulation is
= r,(ni
+ 1_26n&(n,
+ nz)
(26)
where ri and r2 are the radii of primary and secondary particles respectively. Since nI = l/(h2ta + l/n,“) and nz = (RI0 - n,)/Z,
X
(13)
Here, Q! is molar ratio of exchangeable ions between the solid and the liquid phase, 0, is a self-diffusion coefficient of exchangeable ions in the solid phase, f is the mean radius of particles and m represents an integer, a natural number that has all values from 1 to inf5iity. The original form of eqn. (18) may be issed for the analysis of a self-diffusiongoverned exchange process only when D,, f, and Q(are known and constant. In our case, direct analysis based on eqn. (18) cannot be applied because the results represented in Fig. 2 are obtained by Procedure I (Part I), inclclding the change of particle size during the exchange process. However, calculations derived on the basis of Rajagopal’s investigations of Brownian coagulation [23,24] show
(f,)o =
0.74r,/(JzztA(1)
+ lln,“)
nzo + l/(iz2tn(l)
Dividing
eqn. (21)
(F1)o/nl”-=
by nIo gives
0_74r,/(C,k,t,(l) 1 +
(21)
-i- l/nIo) -+ 1)
W! 1/(COk$A(1)
+
1)
The ratio for a sol aged for time tA(2) can be
calculated by an analogous expression: (f2)o!n i” = [O-74r,l(CoUA(2)
f1 +
1/(COh,tA(2)
+ +
III/ (23)
l)j
and thus (f*)o/(F,)o=
(~22/~1)0
0.37 =
[ COh,tA(2)
+
1
0.37 COk-rtA(l)
I[
1
l+
CO%C~A(~)
l+ +
1
+
1 I
+
f
I Co&t,@)
1
(24)
OL-
1-F 03-
l-F2
M
l
-
1 Fig. 5_ Graphical representation of Wagner’s solution of the differential equation for diffusion under conditions of heterogeneous exchange process governed by self-diffusion_ F represents the fraction of exchange attained for the exchange time in in the system having solid particles of mean radius ? and self-diffusion coefficient for exchangeable ions D,, dispersed in the liquid phase. x1 and _r, are numerical values of ii’D,t~/f= corresponding to the val&s 1 - F1 and lL Fz_
Assuming that primary particles of both the real system and that represented in Fig. 1 (ET-model) have identical structure and size, i-e_ if the initial conditions for both systems are the same, the fraction of exchange F at
t, = 0 must be identical for both systems. Since the measurement of the exchange fractions at tA = 0 is associated with experimental difficulties, l-minute aged sols were used as a good approximation for t, = 0. Such an approximation is suitable, because at tA = 1 min all examined sols consist practically of primary particles only_ However, the n,/nz ratio (nl measured at t, = 1 min) decreases with increasing concentration of sol and the error due to the assumption used is larger for sols of higher concentration (Ce). Thus, the differences between the measured and calculated F values decrease with increasing concentration of ~01s. To the value F, determined in l-minute aged sol, corresponds the numerical value x1 cf the quantity sr’D,t,/F* in the graphical presentation of eqn. (18) (see Fig. 5). The value of r, corresponding to the F value experimentally determined by Procedure I, should have not a real but an imaginary value *f: f0 < *f < FEE, where f, and FEare the real mean radii of sol particles at the beginning (tE = 0) and at the ;zccr.ge time FE= tE(experimental) [ 24]_ 9
F,2(det)
= *f,’
= A~D~?~/x,
(25)
where f,‘(det) is a value of F,’ corresponding to the experimentally determined F value for 5, = t,(experimental)_ In accordance with eqns. (19) and (22), the ratio of both the real (r21f,), and imaginary (*fll*fi)tE
1 =[c::L]ll+ :.i7f::i (26) O-37
I
Cok,t,f2)
+ 1 II
Since the k, values for analysed sols are determined by sedimentation analysis (Part II), the numerical value of p(2,l) for 2 given time t*(2) can be calculated using the righthand side of eqn. (26) Then *fz2 = Cp(2,1)]
‘*F1’ = Ln(2,1)]
‘ii’D&x, (27)
In accordance with eqn (25), Fzz can also be espressed as *f-z! _ = x’D,& j-r2 (28) Hence, (29)
x2 = 7~~D&/*i=~~ =x,&42,
3
l)]’
The numerical value -r2, calculated from the known numerical values x1 and ~(2, l), has the corresponding value of 1 - F2 and 1 - F, respectively, as shown in Fig. 5. The value F2 corresponds to a fraction of exchange that would be measured by Procedure I in a real system if the IST-model were applicable to it. For the assumed case when primary particles of silver iodide hydrosol are all of the hexagonal form and secondary particles all cubic, the following relations are valid: x x2
I-
- rrZD,(H)&s/7,2 =
n*D,(C)t,&*
(31) (32)
-I 5 6
7 S
FoZge.
of the Stability
13
1-Z
16
and
1s
Fz)FI
19
which is in disagreement with the experimentally obtained results and the data calculated on the basis of the IST-model.
20 21
REFERENCES 1 R_ Despot&if and B. Sub&f. Croat. Chenx_ Acta. 43 (19il) 153. 2 R. Despotox% and B. Subotie, J. Inorg. ~VucZ. Chem. 38 (1976) 1317_
of Lyophobic
CoZZoids.
Theory
Elsevier.
Amsterdam. 19-4s. _ _ E. Cohen and W. J. D. Van Dobbenburgh, 2. Physih. Chem., A137 (1928) 289. 11 R. Bioch and H. hIBIer, 2. Physik. Chem., A 1.52 (1931) 245. 12 G. Burley and H. E. Kissinger, HZf. Bar. Standards.
rz>_r,
(33)
240.
10
14 15
= 1O-*5
49 (1966)
9 E. J \V_ Verwey~and J. Th. G. Overbeek,
cm2 s-i is the self-diffusion coefficient of I- ions inside the crystal space of hexagonal modification of silver iodide and D 5(C) = 2 X lo-i2 cm2 s-’ is - the corresponding self-diffusion coefficient for cubic modification of silver iodide [25] _ It is evident that in this case the following relations hold:
D,(H)
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