PHYSICAL REVIEW B VOLUME 51, NUMBER 14 1 APRIL 1995-II Porosity evolution in Sn02 xerogels during sintering under isothermal conditions C. V. Santilli and S. H. Pulcinelli Instituto de Quimica UN—ESP, P.O. Box 355, 14800 900 A-raraquara, SVo Paulo, Brazil A. F. Craievich LNLS Con—selho Nacional de Desenvolvimento Cientij7co e Tecnologico, P.O. Box 6192, 13081 970,-Campinas, Steno Paulo, Brazii and Instituto de Fssica, Universidade de Sd'o Paulo, SNo Paulo, Brazil (Received 3 October 1994) The structural evolution during isothermal sintering (200& T~600'C) of SnO& xerogels was studied by small-angle x-ray scattering (SAXS) using synchrotron radiation. The SAXS intensity and, conse- quently, the structure function of the studied samples exhibit, at low q-wave numbers, a sharp decrease for increasing q, and a characteristic peak at larger q values. We associated these two features to the ex- istence of a bimodal size distribution of electronic density heterogeneities related to (i) interaggregate porosity and (ii) internal microporosity, respectively. The maximum of the peak increases with the sintering time in all studied samples. At 300'C the q value associated with the maximum intensity remains constant. The data analysis of the set of scattering curves for increasing time intervals at 300 C is in agreement with Cahn's theory for spinodal decomposition. At higher temperatures, 400—600 C, the maximum of the structure function increases with time, its position shifts continuously to lower q values, and the value of the integrated intensity in reciprocal space remains constant. The structure function of microporous Sn02 under isothermal treatment in the 400—600'C range exhibits the dynamical scaling property. The experimental results suggest that the microporosity coarsening is controlled by the coagu- lation mechanism. I. INTRODUCTION The kinetics aspect of phase separation has received considerable attention during recent years' due to the relevance of this phenomenon for a wide range of materi- als including polymers, glasses, metallic alloys, and ceramics. The small-angle x-ray scattering (SAXS) tech- nique is useful for studying this process. It allows, for ex- ample, a direct verification of the applicability of Cahn's theory of spinodal decomposition and the statistical dynamical scaling model. Cahn's theory for phase separation assumes that the boundary between the phases is diffuse, without sharp discontinuities. This theory is based on a Fick diffusion equation with an additional term which accounts for the surface energy associated with the incipient interphases which are formed during the first stages of phase separa- tion. The linear diffusion equation, which is valid for the first stages of the process, was solved for isotropic sys- tems by the Fourier transform method. The result indi- cates that the structure function S (q, t) and also the SAXS intensity 1(q, t) (which is proportional to the struc- ture function) exhibit an exponential growth: S(q, t)=S(q, O) exp[2Q(q)t] and I (q, t) =I(q, O) exp[2Q (q)t], was observed during the first stages of amorphous phase separation in glasses. To justify some deviations of ex- perirnents from the theory, different arguments, such as the inQuence of nonlinear terms, statistical fluctuations in composition, structural relaxation, and competitive mechanisms, have been suggested. The classic paper of Lifshitz and Slyozov proposes that the kinetics of growth of domains of the second phase during advanced stages of phase separation is con- trolled by a coarsening mechanism in which the larger "particles" grow at the expense of the smaller ones. For long periods of time, the asymptotic variation of the aver- age domain size is given by R(t) ~t', where the exponent a is equal to —,'. This type of potential behavior was observed in studies of grain growth in poly- crystalline metals and ceramics, a being dependent on the specific model assumptions and on the microscopic mech- anisms of the domain, particle, or grain growth. In analogy with statistical theories for formation of fer- romagnetic domains, theoretical studies of phase separa- tion suggest that the potential behavior [Eq. (2)] can be explained by a self-similar growth pattern, character- ized by a single length scale R (t). After an initial and transient process, R (t) also exhibits a potential behavior and the isotropic structure function S(q, t) satisfies the following scaling property: where q is the wave number of the scattering vector and "composition wave" and Q(q) is an amplification factor. A qualitative agreement between theory and experiment and S (q, t) =R "(t)F(qR (t) ) 0163-1829/95/51(14)/8801(9)/$06. 00 51 8801 1995 The American Physical Society C. V. SANTILLI, S. H. PULCINELLI, AND A. F. CRAIEVICH where d is the dimensionality of the system, q is the modulus of the scattering wave vector, and F is a scaling function. The wide applicability of the dynamical scaling model has been established by computer simulations on two- and three-dimensional models. Comparisons with experimental data are limited to studies of phase separation in some binary liquid mixtures, ' quasibinary glasses, "and binary alloys. ' This fact is partially due to the experimental difticulties of determining the structure function over wide time and length ranges. Grain and pore growths are verified during the first stages of sintering of porous ceramics such as SiOz xero- gels, ' SnOz compacted powders, ' and a-Fe203 powders. ' Sn02 ceramics exhibit large temperature and time ranges in which the volume fraction of pore and solid phase remain constant, which is a necessary condi- tion for the dynamical scaling property of the structure function. A preliminary analysis established the constan- cy of the volume fraction of the pore growth in Sn02 porous ceramics' in the temperature range 400—1500 C. This makes Sn02 ceramics good candidates for testing the applicability of the scaling theory to processes of porosity coarsening also. This investigation aims at a quantitative verification of the Cahn theory for spinodal decomposition and of the dynamical scaling properties of the structure function as- sociated with porosity evolution in SnOz porous ceramics at several temperatures. The porous samples were prepared by a hydrothermal sol-gel route. The parame- ters associated with the hydrothermal treatment were chosen to obtain amorphous samples with a specially designed texture, i.e., pore and particle size ranges ap- propriate for the SAXS technique. II. EXPERIMENTAL PROCEDURES nally the gel was freeze dried and the powdered xerogel compacted to obtain 0.2-mm-thick slides, by means of a biaxial pressure of 200 MPa. The SAXS experiments were carried out using the D24 workstation at the synchrotron x-ray source DCI at LURE, Orsay (France). The workstation provided a monochromatic (A, =1.60 A) and horizontally focused beam. Two sets of slits were used to define a pinholelike collimated beam. A high-temperature chamber, stable within 1 C, was used for in situ isothermal studies of the xerogel samples during SAXS measurements. The poros- ity evolution was investigated at 200, 300, 400, 500, and 600'C. SAXS spectra were recorded a few minutes after placing the sample into the high-temperature cell. A vertical one-dimensional position-sensitive x-ray detector and a standard multichannel analyzer were used to deter- mine the intensity function I(q), in relative units, as a function of the modulus of the scattering vector q. Parasitic air and slit scatterings were subtracted from the total intensity. An ionization chamber, placed downstream after the sample, was used to monitor the intensity decay of the transmitted beam and to determine the sample attenua- tion. The natural decay in intensity of the incident beam was monitored by also recording the electronic current in the synchrotron source. Since the electron current inten- sity as a function of time was proportional to the intensi- ty of the transmitted beam, we concluded that the inten- sity measured downstream after the sample actually mea- sures the decay of the incident beam without significant interference from scattering effects. This proportionality also indicates that the attenuation of the sample is a con- stant during the studied process. Because of the small size of the incident beam cross section at the detection plane, no mathematical desmear- ing of the experimental data was needed. Each spectrum corresponds to a data collection time interval of 300 s. Sn02 xerogel samples were prepared by the sol-gel pro- cess as described elsewhere. ' The precipitate was prepared at pH 11 by the addition of NH40H to the SnC1~ (0.25 mol dm ) aqueous solution and submitted to prolonged dialysis to eliminate remnant Cl and NH4 ions. After that, peptization of the precipitate occurred, allowing for the formation of a transparent co11oidal sus- pension ([Sn]=0.2 mol dm; [Cl ] ( 10 mol dm 3). The sol-gel transition was promoted by partial evapora- tion of water (60 C) to reach the critical concentration of sol ([Sn]=0.42 mol dm ) at which gelation occurs. This process allowed for the formation of a tin oxihydroxide gel, SnO, 2(OH), 6, constituted by primary particles presenting a single-mode size distribution and an initial average size of 3 nm. Aggregation led to 150-nm-length chains. ' This oxihydroxide was introduced into Pyrex- glass sealed tubes and submitted to hydrothermal treat- ment in subcritical conditions (2SO C, 52 Mpa) for 60 min. This treatment improves the secondary condensa- tion reactions, leading to a decrease in the number of hy- droxyl groups, the nominal formula becoming SnO& s(OH)&4. This strategy was used to improve the chemical and structural homogeneity of the samples. Fi- III. CRITERIA FOR TESTING THE THEORETICAL MODELS Equation (1) makes it evident that Cahn's theory for spinodal decomposition can be directly tested with SAXS data. Necessary conditions for the applicability of Cahn's theory are (i) constant value q (t) =q of the position of the maxi- ma in S (q, t) vs q curves; (ii) exponential variation of S (q, t) as a function of time for every q value; and (iii) single crossover at q, =V2q of the S (q, t) func- tions. In order to verify the dynamical scaling properties of the structure function, we analyzed the time dependence of the unnormalized, S„(t), and normalized, q„(t), mo- ments of the structure function, defined as follows: S„(t)=J S(q, t)q "dq, n =0, 1,2, 0 POROSITY EVOLUTION IN Sn02 XEROGELS DURING. . . 8803 The characteristic size or length R (t) is defined as the in- verse of the first normalized moment 1/q&(t). Then Eq. (3) can be written as [q, (t)]3S(q, t)=F(x) where x =q/qi . (6) This equation defines the time-independent scaling func- tion F(x). The unnormalized, S„(t), and the normalized first mo- ment, q, (t), of the structure function are related by (7) where K„=J o x "I' (x)dx. The normalized first moment, q, (t), and the maximum of the structure function, S =S(q, t), evolve with time as follows: S2=C„S q, =C2, and q~/q, =C3 . (10) where pI and p2 are the electronic densities within the phases. When the tota1 volume fractions of the phases are invariant and the difFerence in electronic density (p, —p2) remains a constant, the asymptotic value of S(q)q (q~ ~ ) yields the time variation in relative units of the interface area. IV. EXPERIMENTAL RESULTS For biphasic (two-electronic-density) systems, the time variation of the interface area S, can be estimated using Porod's law: lim S(q) ~(p, —p2) S, /q and S o(- t' where a'=3a . Because of the invariance of the total volume fraction of phases and dynamical scaling properties, the following quantities remain constant: Time evolutions of the SAXS intensity, measured in relative units during in situ isothermal heat treatment at 200, 300, 400, and 600 C, are shown in Figs. 1(a), 1(b), 1(c), and 1(d), respectively. In a previous study, the SAXS intensity functions corresponding to a sample heat treated at 500'C were determined. ' All spectra exhibit 80— Time (min) 4.5 ~~ 40— ~~ (hI 0$ 40— ~ Sha CO 07 ~gkl I' II )II(i!Id& II„ VNI CI'l 'l5.5 21.5 43.5 62.5r 200 C 300 C 0.00 kL 80— 0.10 l 0.20 l 0.30 0.00 II 200— I 0.10 0.20 q(& ) I 0.30 Time evolution of the SAXS intensity (or structure function) of Sn02 porous xero- gels during in situ isothermal sintering at 200, 300, 400, and 600'C. (0 40— ~~ CO 2 100— 0) 0.00 400 C I 0.05 q(~ ) IR II ~ I I 0.10 l 0.15 0.00 I 0.04 q(~ ) 0.08 C. V. SANTILLI, S. H. PULCINELLI, AND A. F. CRAIEVICH two q ranges with clearly difFerent behaviors. (i) In the low-q range, the SAXS spectra have a sharp decrease for increasing q. This contribution remains essentially invariant with time and temperature of heat treatment. We attributed the origin of this component of the SAXS intensity to the coarse interaggregate porosi- 16 (ii) In the high-q range, the SAXS spectra have a characteristic peak whose intensity varies with time and is highly dependent on treatment temperature. At 200'C, the scattering intensity is almost time invari- ant [Fig. 1(a)], even after 1 h of isothermal treatment. At 300'C, the value of the maximum I increases with time while the maximum position q remains constant [Fig. 1(b)]. Above 300'C, an increase in I and a clear de- crease in q are observed. The time evolution at 300 C is qualitatively similar to that predicted for spinodal decomposition associated with phase separation in binary systems. The characteristics of SAXS results for T) 300 C [Figs. 1(c) and 1(d)] are in qualitative agree- ment with those expected from the theoretical approach for advanced stages of phase separation. In this paper, we focus our attention on the varying part of the scatter- ing functions, which is associated with the structural modifications in the intra-aggregate microporosity of the studied material. Prior to the calculations of the difFerent moments of the experimental scattering functions, the contribution to SAXS from coarse porosity was subtracted by two different procedures: (i) by extrapolating the low-q range of the total scattering intensity I, (q) (produced mainly by the coarse porosity), in which the total intensity is de- scribed by a power law [I (q) o- 1/q ], and subtracting it from I, (q) over the whole experimental q range; and (ii) by suppressing the decreasing part of the total scattering function; this part being substituted by a linear extrapola- tion between a minimum value of q =q;„, selected for each curve, and q =0 assuming I (0)=0. I(q), I (q)q, and I(q)q were integrated numerically between q;„and the 0 maximum experimental value of q, q „=0.18 A, for which the intensity function obeys Porod's law. For q )q,„, the integrations were carried out analytically after extrapolations of the q dependence. The total in- tegrals yielded Sp S] and S2. The two procedures for the subtraction of I (q) led to slightly different numerical values but both exhibited essentially the same variation with time. Figure 2 shows the log-log plots of the normalized first moment q&(t) as a function of time for samples treated isothermally at 200, 300, 400, 500, and 600'C. We note a linear behavior for all temperatures. At T =200'C q& is time independent. For T =300'C, a weak decrease in q& is observed. The experimental results of Fig. 2 concerning the sam- ples heat treated about 300'C indicate that the theoreti- cal prediction specified by Eq. (8) is obeyed. The values of the exponents a are about the same for samples treated at 400 C and 500'C (a =0.15 and 0.16, respectively) and slightly lower for those treated at 600'C (a =0.13). The approximate invariance of the exponent observed Temperature ( C) 200 0.10— 300 400 500 600 10— 1/T {K ) 0.0011 0.0013 0.0015 0.01 I I I I I I I 10 Time (min) I I I I I I I I 100 FIG. 2. Log-log plots of the normalized first moment q &(t) as a function of time of sintering. The insertion shows the values of 1/q, (t) extrapolated to the origin as a function of the re- ciprocal temperature. for T ~ 400 C allowed us to determine the activation en- ergy of the growth process characterized by the length R (t)=1/q&(t) The va. lues of 1/q, extrapolated to t =0, for the difFerent temperatures, are presented in an Ar- rhenius plot in the inset of Fig. 2. A least-squares fitting of a straight line yields a value of 22 kJ mol ' for the ac- tivation energy of R (t) growth. The time variations of the maximum of the scattered intensity S for samples treated at 300, 400, 500, and 600 C are presented in Fig. 3 on logarithmic scales. The linear behavior observed is in agreement with Eq. (9). For samples treated at 400, 500, and 600'C, the slope a' is approximately equal to 3a as predicted by Eq. (9) (the exponents a' are equal to 0.34, 0.46, and 0.35 for temper- atures of 400, 500, and 600 C, respectively). The equa- tion a'=3a is approximately obeyed for samples heat treated at 500 C. The exponent. a' is somewhat lower than 3a for T=400and 600 C. Figure 4 shows the log-log plots of the moments So(t)/So(tf ), S,(t)/S, (tf ), S2(t)/S2(tf ), and q2(t)/q2(tf ) as functions of q, (t)/q, (tf ), where tf corre- sponds to the longest time at each temperature. The linear theoretical behavior predicted by Eq. (5) is ap- parent for all moments corresponding to temperatures of 400, 500, and 600'C. Data concerning samples treated at 200 and 300'C present a considerable deviation from the theoretical behavior and are not shown in the figure. The invariance of S2, observed in Fig. 4, indicates that the volume fraction and electronic density of both phases remain constant during heat treatment. Under these con- ditions, the time variation of the interface area S, is pro- portional to the asymptotic value of S(q)q of the corre- sponding SAXS curves [Eq. (11)]. In Fig. 5, the time evo- POROSITY EVOLUTION IN SnOz XEROGELS DURING. . . 8805 1000— Temperature ( C ) 600 perature ( C) 400 500 600 4 C: 100—E 500 400 300 th Ch lO CF O D GO CI (h I I I I I I 10 Time (min) I I I I I I I 100 I 20 I 40 Time (min) I 60 FIG. 3. Time dependence of the maximum of the structure function, S,on a double logarithmic scale. lution of the ratio S,(t)/S, (0) is plotted, where S,(0) is the asymptotic value of S(q)q of the sample without thermal treatment. For all temperatures, a monotonic decrease of interface area is apparent, as expected for a coarsening mechanism of porosity evolution. The plots of q2/q, and S q, as functions of time of FIG. 5. Time evolution of S,(t)/S, (0), where S,(0) is associ- ated with the asymptotic value of S(q)q4 [Eq. (11)] for the sam- ple without heat treatment. isothermal treatment at 300, 400, 500, and 600'C are shown in Figs. 6(a) and 6(b), respectively. As the intensi- ty measurements were carried out in relative units, their values were normalized, dividing by the time-independent moment S2, in the whole q range. The time invariances of qz/q, and S q, /S2 are other experimental evidences o.e— 1.4— 1.3— X X 1,2— 1 A. s~ a ~ + 0 l M +~ I W ~ I 00 C7l O -0.4— S2 S) I 20 1.4— X X X I 40 Time (min) X ~X 0 0 -0.8— S0 rature ( C) 300 0.4 I 0.0 0.2 log10[q1 (k"}} FIG. 4. Log-log plots of the moments S0( t) /S0( tf ), S)(t)/Sl(tf), S2(t)/S2(tf), and q2(t)/q2(tf) as functions of q&(t)/q&{tf), where tf is the longest time of sintering. The straight lines have the slopes predicted by the theory ( —2, —1, 0, and 2, respectively). 0.5 I 20 (b) I 40 Time (min) !. x I 4oo 500 I 80 FIG 6 Time dependences of q&/q& and S q& for samples sintered at 300, 400, 500, and 600 C. 8806 C. V. SANTILLI, S. H. PULCINELLI, AND A. F. CRAIEVICH 0.0010— 0.0005— Time (min) + 4.s e.s 15.5 21.5 C] 28.0 43.5 62.5 77.5 for the applicability of the dynamical scaling theory [Eqs. (10)] to the studied porous system heat treated above 300'C. These conditions are not fulfilled by the sample treated at 300'C, for which a clear continuous increase of S q, occurs.3 The normalized experimental structure functions at 400 and 600'C are plotted in Fig. 7 as S(q, t)q, (t)/S2 versus q/qi. With the exception of the first F(x) func- tion, corresponding to the initial states of the evolution, all others are actually time independent. This implies that, except during the very early stages of the process, the dynamical scaling model applies to the studied sys- tem. Taking into account all the above-mentioned experi- mental results, it is apparent that the dynamical scaling model does not hold for the porous samples treated at 200 and 300'C. Since, for these temperatures, the stud- ied system is in its initial stage of porosity evolution, we tried to verify the applicabihty of Cahn s theory for spi- nodal decomposition. The SAXS intensities associated with microporosity in the sample treated at 300'C are shown in Fig. 8. Due to the relatively strong contribu- tion of coarse porosity to the total scattered intensity, it was not possible to accurately determine the q and q, values for samples treated at 200 C. One of the vertical dashed lines in Fig. 8 clearly shows that q (position of the maxima) is time invariant. A crossover of experimen- tal SAXS functions, q„ is observed for t )9.5 min. The experimental intensities for t &9.5 min do not have a defined crossover. Moreover, for long treatment time the ratio q, /q is equal to 1.76 instead of 1.41 as predicted by Cahn's theory. The time evolution of the structure function at several values of the composition wave number q is shown in Fig. 9, for the sample treated at 300'C. The linear time dependence of logS(q), which is apparent for t & 10 min, agrees with Cahn's theory for spinodal decomposition [Eq. (1)]. For q &q„ the amplitude of the composition wave increases and for q & q, it decreases, in agreement with Cahn's theory. However, the experimental results for very short times deviate from linear behavior. Table I groups some structural parameters, determined by means of experimental techniques other than SAXS, for samples treated at several temperatures. These pa- rameters are the real and bulk density (determined by He and Hg pycnometry, respectively), total pore volume, average pore size, and specific surface area, determined by N2 isothermal adsorption, and average crystallite size (obtained from x-ray-diffraction peak width). It is in- teresting to note the increase in crystallite and pore sizes, and the large decrease in the surface area, for increasing sintering temperature. 0.0000 q/q& 60— 0.0010— CD D D CD 0.0005— 20— 0,0000 600 C 0.00 I 0.05 I 0.10 I 0.15 I 0.20 q/q& FICx. 7. Dynamical scaling behavior of the experimental structure function of the samples sintered at 400 and 600 C. FIG. 8. Structure function corresponding to the micropores as a function of q for the sample treated at 300'C. Vertical dashed lines indicate q and q, values. 51 S O XER~GELS D«INOSIT~ EyOLUTION IN ' ties of xerogels sintered or 10 and 30 min.f the structural characteris ics oTABLE I. Time evolution of e ( ) Bragg peaks are not wet we11 defined. T(C) Time (min) Density (g cm ) Real Bulk Pore volume (cm g ') Surface area (m g ') Average size (nm) Pore Crystallites 110 400 400 600 600 1080 10 30 10 30 5.07 6.27 6.29 6.45 6.48 4.11 4.01 4.04 4.10 4.12 8.6 9.0 9.1 8.8 9.0 184 125 104 36 36 1.7 2.9 3.5 9.6 10.6 13 14 30 33 V. DISCUSSION Low-temperature process ,300 C) of the experimental data correspond-p 'C and their comparison wit a n sing to 300 C, an f the structure func-two regimes orfor the time evolution o 100— 0.14 0.16 0.18 10— 0.20 ' all SAXS experimental resultsits show two =300'C d dCh' tho f viors: (i) at T=, a the ex erirnental results an a n s ition was observed for time of heating min the behavior observe in e eh' h han1 mi, due to a transient structura e ec, w ' times being due ' n and (ii) for the tempera-be discussed in the next section, an ii or f 400~ T~600 C a general agreement wit the statistical theory which pre ic s in roperty o e sf th structure function was observe . e recall that both theories were establishe d forshould hase-se aration phenomena u ey ~ ~Cahn's theory is an approximationsta es of the process: a ns e el holds ill discuss both sets of results (atfor advanced ones. We will discuss o 300'C and in the range 400—600'C) separately. The first one (short-time regimetion are apparent. e rs in lo S(q, t) versus t Th 1 t t d b a varying slope in og p o q g(Fi . 9). The q va u ses. For longer times a eroswhile q, decreases. of S(q) on t areand a linear dependence o ' s. 8 and 9, respectively, inobserved, as can be seen in Figs. an The only deviation fromg ' h Cahn's mode. hi her than expected tical redictions is t e v =1.76 which is hig erqe qm = (q, q/ =1.41). s ex erimental investigationp p basic Cahn equation for spino a1 d th t th b s accurately descn e eosition does not acc y the deviation, P I order to explain C h19 a modification in a n s sociated wit s ' h statistical composition uc ua ' h odified equation leads /q h' her than 1.41. Th e- The solution to the mo i e to a value of the quo ' q, q~otient /q ig er fiuctuations may explain t e exper'fore statistica uc u ' t e ex er / =1.76 which is ig er avalue of q~ q ri inal Cahn theory. m 1 fluctuations requires theanalysis of the effect of statistics uc u nt of SAXS intensity in absolute units. fl S( ) versus t plots dur-m linearity o og q v es of se aration in bora e g asg g o p ence of transient e as ic stributed to the existence les. In the disor-'th the uenching of the samples. n eated wit e q 1 t temperatures close todered matrix oof SnO xeroge s a em d (cr stalline) phasecation of an ordere crys300 C, the nuc ea ' the sample com-is is followed by a change in eoccui s. T is is 0 the loss of remnan t OH groups, and by aposition, due to the (T ble I). Doherty 1ase in real density a econsequent increa roduces elastic strainss that this transformation pro uces e a - Th-. .-- -.P-b.bly--hindering crystallization. ese s r the increase, at short times,sponsible for the increase, or and, consequent y, o1 of the slopes ofamplification factor ssive stress re-t lots. Due to a progressivlogS(q, t) versus p ld decrease ande am lification factor woulaxation, the amp ' for advanced stages ateventually become a constant or a va 300 C as shown in Fig. 9. I 20 I 40 Time (min) I 60 ion of the structure function at several in-FIG. 9. Time evolution o e 300 C in a dou-dicated q values for the saxnp le heat treated at ble logarithmic scale. High-temperature process anal sis of the experimental resu tss associated withThe y d lf h at 400 C an a ove validity of the yd namical scaling mo e or 1 . As expected, devia-function of SnOz porous xerogels. . (9)] have been ob-tions from the scaling behavior [Eq. av 8808 C. V. SANTILLI, S. H. PULCINELLI, AND A. F. CRAIEVICH served at low temperatures (T ~300 C) and during the first minutes of heat treatments at 400, 500, and 600'C. That means that (i) at 300'C, the porosity evolution cor- responds to initial stages, even for the highest time, and (ii) during the early stages at 400, 500, and 600 C, the matrices still have varying composition and/or volume fractions. A transient short-time effect is observed for the values of pore volumes and real densities (see Table I). The increase in real densities with time would indicate that nucleation of an ordered phase (crystallization) occurs during the first stages, leading to the transient in- crease in the pore volume fraction. We should recall that the dynamical scaling model applies to two-density sys- tems with constant composition and volume fractions. Additional evidence of the validity of the scaling hy- pothesis is given by the values of a and a', the time ex- ponents defined by Eqs. (8) and (9). The expected relation a ' =3a is obeyed for T =500 'C and approximately verified for T =400 and 600'C. Differences between a' and 3a observed for some metallic systems were attribut- ed to crystal anisotropy. ' The time exponent of the characteristic length R (t) is an approximately constant value close to 0.16 at 500'C and slightly lower at 400 and 600'C. This value is ex- pected for a mechanism of phase growth controlled by cluster coagulation. Computer simulations demonstrat- ed that this mechanism occurs predominantly during the last stages of phase separation in binary systems with composition near the center of the miscibility gap in which the volume fraction of phases is close to 0.50. In the studied SnO2 xerogel, we have approximately 45% of porous phase and 55% of solid phase. These values are similar to those expected for the part of the miscibility gap associated with two-phase separation. Moreover, the time evolution of the shape of the structure function (Fig. 1) and the considerable broadening of the scaled structure function (Fig. 7) are analogous to those observed for glass samples close to the center of the miscibility gap. Thus the results obtained for Sn02 porous xerogels seem to in- dicate that the validity of the dynamical scaling model is not restricted to phase separation in crystalline or amor- phous binary or quasibinary mixtures, but has a more universal character, also including structural evolutions in porous systems. Specifically concerning the microstructure evolution of the studied Sn02 ceramics during the sintering process, several mechanisms have been proposed in the past to ex- plain the observed pore and grain growth. Some authors have attributed this phenomenon to a process controlled by diffusion, ' whereas others to a nondiffusional pro- cess, like evaporation-condensation. ' The good agree- ment between experimental SAXS data and phase- separation theories allows us to conclude that the basic process for structure evolution in porous Sn02 xerogels is vacancy diffusion associated with the coagulation mecha- nism. However, the value of 22 kJmol ' for the activa- tion energy for the growth of the characteristic length, R (t) is smaller than that observed for difFusion in crystal- line oxides. This low value may be due to the high con- centration of vacant sites resulting from the evaporation of volatiles during drying and to the poor crystallinity of the samples. For TiOz „(isomorphic to SnOz), for ex- ample, the activation energy of a diffusion-controlled pro- cess changes continuously from 280 kJ mol ' for x =0 to 137 kJmol for reduced rutile with composition corre- sponding to x =0.01. Moreover, dissolved hydrogen may greatly affect the properties of rutile-type com- pounds; when hydrogen-doped rutile is heated, the loss of water molecules leads to the formation of oxygen vacan- cies. For SnOz xerogels, thermogravimetric analysis shows a small and continuous loss of water; this may con- tribute to a high vacancy concentration and explain the decrease in the activation energy of the diffusional pro- cess. VI. CONCLUSION The presented SAXS results indicate that the structur- al modifications in the microporosity of Sn02 xerogels can be described by classical models which are currently applied to phase separation in alloys, glasses, and poly- mers. After a transient behavior, the structural modifications in Sn02 xerogels, under isothermal condi- tions at different temperatures, follow two different kinet- ic models. (i) At low temperatures (T =300'C) SAXS results are in good agreement with the predictions of Cahn's theory for spinodal decomposition. (ii) At high temperatures (400~ T ~ 600 C) the experi- mental SAXS results are in accordance with the model of dynamical scaling. These results suggest that a Sn02 xerogel, heat treated in the temperature range 400—600 C, may be seen as a two-phase system composed of (i) a nearly homogeneous Sn02 matrix containing a high vacancy concentration, and (ii) empty microvoids, the total volume fraction being time constant and the structural variation governed by vacancy diffusion. Under these conditions of structural transformation, no significant densification occurs. Efficient sintering leading to the effective reduction or elimination of voids requires other mechanisms which are active at much higher temperatures (above 1500'C for Sn02). For this type of densification process, the models proposed for low temperatures (200 ~ T ~ 600'C), obviously, do not apply. 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