Langmuir 1987, 3,968-973
968
On the Conditions of Thermodynamic Stability of Sols toward Coagulation+ E. D. Shchukin* and V. V. Yaminsky The Institute of Physical Chemistry, USSR Academy of Sciences, Moscow 117915, USSR Received June 30, 1987 The reversible coagulation + peptization transition in organophilic and hydrophilic colloid dispersions is analyzed from the standpoint of the results of direct measurements of interaction forces between molecularly smooth macroscopic solids in the relevant model systems. The conditions of this thermodynamic transition are related to both the short-range adhesion (surface energy) and the long-range repulsive (electric double layer) contributions to the binding energy of colloid particles in the primary potential minimum. The general thermodynamic reason for the instability of disperse systems toward recondensation, coalescence, or coagulation is the excess free energy of interfaces, characterized-per unit area-by the value of interfacial tension. With respect to the stability of dispersions of solids in liquids toward coagulation, the role of interfacial tension is immediately manifested while considering adhesion of particles in contact. The value of solid-medium interfacial tension u is related to the value of the specific (per unit area) free energy of adhesion F (1) l/F = u - 727 where y, the free energy of the interface between solids in equilibrium contact, is generally a non-zero quantity even for similar solids. The thermodynamic eq 1 actually defines F as the difference between the tensions of the infinitely thick ( 2 0 ) and of the thin (y)films. Specific Gibbs free energy of adhesion F corresponds to the depth of a minimum of a potential energy vs. distance curve F(h) of plane surfaces at their equilibrium separation ho. (Though the signs of the function F(h) are considered according to thermodynamic convention, F = IF(ho)l is implied to avoid new symbols; i.e., specific free energy of adhesion F is assumed positive (is described by its modulus).) In the treatment of colloid stability in therms of disjoining pressure II(h),which is defined as the force per unit area between plane parallel surfaces, F(h) appears to be the first integral of II(h)with respect to h:
F ( h ) = - s h I I ( h )dh
(2)
IIih) is measured directly in experiments with liquid films, the surfaces of the film being plane parallel. The force p ( h ) between solid particles whose surfaces are generally nonplanar turns out to be the result of the twofold integration of II(h) over the gap between the particles. According to the well-known Derjaguin theorem,’ this integration can be substituted for the multiplication of F ( h ) by some effective curvature radius R p ( h ) = rRF(h) (3) provided h >> R . For two equal spheres, R simply equals their radius R. Particularly, the force of adhesion p,23i.e., the maximum pull-off force necessary to disrupt the contact, is given by p = aRF (4) Additional specifications of eq 4 taking into account finite elasticity of solid particles are considered in ref 2. The +Presenteda t the “VIIIth Conference on Surface Forces”, Dec 3-5, 1985, Moscow; Professor B. V. Derjaguin, Chairman.
0743-7463/87/2403-0968$01.50/0
Table I. Specific Free Energy of Adhesion, 1/2F (erg/cm2), of Hydrophobic (Methylated) and Hydrophilic (Nonmodified) Silica Glass Surfaces in Polar and Nonpolar Mediae medium air water n-heptane
methylated surfaces
18-22 35-40
510-2
nonmodified surfaces about 40 510-2 about 20
role of adhesion forces becomes rather important while considering the ability of concentrated dispersions to withstand applied mechanical stress, i.e., the rheology of dispersions. Adhesion forces are the directly measured quantity in model experiments using molecularly smooth macroscopic elastic solids. The third integration of n(h)with respect to h results in the free energy of interaction between the particles h
w(h) = -rRSm F ( h ) dh
(5)
According to the integral mean theorem, the binding energy of the particles in equilibrium contact, w = Iw(ho)l, i.e., the depth of the minimum of potential curve w(h),can be related to the force of adhesion p = nRF as
w = rRFh,ff
(6)
For the effective decay distance heffof interaction forces, eq 6 can be taken as ita operational definition. The value of w is especially important since it determines the conditions of thermodynamic stability toward coagulation when the DLVO barrier is overcome or absent. The well-known expressions for n(h)(dyn/cm2),F(h) (dyn/cm = erg/cm2), p ( h ) (dyn), and w(h) (erg) in the case of nonretarded dispersion forces are
A II(h) = -6rh3
A F(h) = -12nh2 AR AR w(h) = -p ( h ) = rRF(h) = -12h2 12h Adhesion force subsequently completed by force-distance measurements between glass and mica surfaces since the pioneering works of Derjaguin and co-workers,6were performed in a number of laboratories (1) Derjaguin, B. V. Kolloid Z. 1934, 69, 155. (2) Muller, V. M.; Yushchenko, V. S.; Derjaguin, B. V. J. Colloid Interface Sci. 1980, 77, 91. (3) Tomlinson, G. Philos. Mag. 1928, 6 , 695. (4) Bradley, R. S. Philos. Mag. 1932, 13, 853. (5) Malkina, A. D.; Derjaguin, V. B. Kolloidn. Zh.1950, 12,431.
0 1987 American Chemical Society
Thermodynamic Stability of Sols
Langmuir, Vol. 3, No. 6,1987 969
Table 11. Specific Free E n e r g y of Adhesion of N o n p o l a r (Methylated) Silica Glass S u r f a c e s in Liquids of D i f f e r e n t Polarity' medium water ethylene glycol ethanol 1-propanol n-heptane
1/2F, erg/cm2 1/2Fd,
35-40 16 1.6 0.1 50.01
- ALT,erg/cm2 39 16 51
51 51
b
-4
4 lg C ( M )
Figure 2. Specific free energy of adhesion, '12F,of methylated surfaces vs. concentration, C, of surfactants sodium lauryl sulfate (l),cetylpyridinium bromide (2), and poly(oxyethy1ene) dodecyl ether (3); closed circles represent the values of '/~FH*o - Au for aqueous solutions of sodium lauryl ~ u l f a t e . ~
adhesion between hydrophilic and hydrophobic surfaces in air and in liquid polar (water) and nonpolar (n-heptane) media, demonstrating extreme cases of lyophilicity (high affinity) and lyophobicity (poor affinity) between the surfaces and the medium. In lyophilic cases of nonpolar surfaces in a nonpolar medium and of polar surfaces in a polar medium the reduced values of interfacial tension u result in low values of 1/2F.When the surfaces and the medium are strongly different in polarity, interfacial tension is increased and the values of l/$ are high, the difference between lyophilic and lyophobic cases reaching 4 orders of magnitude. This wide range can be covered in steps (Table 11) as well as gradually (Figures 1and 2). Table I1 shows the data for nonpolar surfaces in liquids of different polarity. The values of 1/2F= p/(27rR) obtained in adhesion force measurements are compared to interfacial tension changes Au obtained in independent contact angle (e) measurements, that is, to the free energy of wetting A u = us - usL = uL cos 0 + TS. In this case the surface pressure 7rs of the vapor of polar liquids on a nonpolar solid of low surface energy can be neglected. The initial value of l12Fair for
adhesion in air, in accordance with the experimental results, equals 22 erg/cm2, which also corresponds to the critical surface tension of wetting (a,) currently considered as the value representative of the surface energy of nonpolar solids use The correspondence between A1/2Fand ACTindicates that for the systems under consideration the changes AT are low. This means that the medium is squeezed away from the gap so that the equilibrium distance between the surfaces in the primary potential minimum corresponds to their immediate contact (ho, equals a few angstroms). Figure 1presents lj2Fisotherms for hydrophobic surfaces in aqueous solutions of low molecular weight alcohols. Also in this case, the values of virtually coincide with - uL cos 6 (open and closed circles in the values of Figure 1,respectively, curve 2 for water-ethanol mixtures). Similarly, the reduction of 1 / 2 Fon addition of sodium lauryl sulfate as compared to the initial maximum value of of about 40 erg/cm2 in pure water equals the reduction of interfacial tension Au, obtained in the present case as the interfacial pressure of the surfactant from independent adsorption measurements using Gibbs equation (Figure 2). These data demonstrate that the immediate relation between the reduction of contact strength p = 7rRF and the reduction of cr can be considered as the direct quantitative manifestation of the Rehbinder effect.'O Let us proceed now from the lyophilicity of an interface characterized by the values of '18to the concept of lyophilicity of disperse systems. This concept, dating back to Volmer," was further developed in the 1950s by Shchukin and Rehbinder.I2 This approach, currently adopted in the microemulsions field,13J4was also applied to the stability of colloid dispersions toward coagulation.l"18 Our main idea is that in the overall balance of
(6) Derjaguin, B. V.; Abrikosova, I. I. J. Exper. Theor. Phys. (USSR) 1951,2I, 945; Discuss. Faraday SOC. 1954,18, 182. Rabinovich, Ya. I. Kolloidn. Zh. 1977,39, 1094; Rabinovich, Ya. I.; Derjaguin, B. V.; Churaev, N. V. A d a Colloid Interface Sci. 1984, 16, 63. (7) Yaminsky, V. V.; Pchelin, V. A.; Amelina, E. A.; Shchukin, E. D. Coagulation Contacts in Disperse Systems; Nauka: Moskow, 1982 (in Russian; for brief review in English, see: Shchukin, E. D.; Amelina, E. A.; Yaminsky, V. V. Colloids Surf. 1981, 2, 221). (8) Israelachvili, J. N.; Pashley, R. M. J. Colloid Interface Sci. 1984, 98,590. Pashley, R. M.; McGuiggan, P. M.; Ninham, B. W.; Evans, D. F. Science (Washington, DC) 1985,229, 1088. (9) Amelina, E. A.; Yaminsky, V. V.; Shchukin, E. D. Kolloidn. Zh. 1975,37,536. Yaminsky, V. V.; Yusupov, R. K.; Amelina, E. A,; Pchelin, V. A.; Shchukin, E. D. Kolloidn. Zh. 1975, 37, 918.
(10) Rehbinder, P. A.; Shchukin, E. D. Prog. Surf. Sci. 1972, 3, 97. (11) Volmer, M. Z. Phys. Chem. 1927,151, 125; 1931, 155, 281. (12) Shchukin, E. D.; Rehbinder, P. A. Kolloidn. Zh. 1958, 20, 645. (13) Shchukin, E. D.; Fedoseeva, N. P.; Kochanova, L. A.; Rehbinder, P. A. Dokl. Akad. Nauk SSSR 1969,189,123. Shchukin, E. D.; Kochanova, L. A.; Pertsov, A. V. In Physicochemical Mechanics and Lyophilicity of Disperse Systems; Naukova Dumka: Kiev, 1979; Vol. 11, p 15 (In Russian). (14) Ruckenstein, E.; Krishnan, R. J. Colloid Interface Sci. 1980, 76, 188, 201. (15) Shchukin, E. D. Abstracts of Papers, VIth All-Union Colloid Conference, Voronezh, 1968; p 27. Shchukin, E. D. In Advances in Colloid Chemistry; Nauka: Moscow, 1973; p 169 (in Russian). (16) Martynov, G. A,; Muller, V. M. Kolloidn. Zh. 1973, 36, 687.
c
I
0.5
%OH
Figure 1. Specific free energy of adhesion, I/& of macroscopic methylated silica glass surfaces vs. volume fraction, cpaoH,of methyl (11,ethyl (2), n-propyl(3), and n-butyl(4) alcohols in water; closed circles represent the values of 'lzF- Au for water-ethanol mixtures.',g using both hydrophilic and hydrophobic surfaces.-
Table
I lists the results of our early measurements (19759)of
970 Langmuir, Vol. 3, No. 6,1987
Shchukin and Yaminsky
the free energy of the system, 4, the work necessary to create a new interface (during dispersion of an initially compact phase as well as due to the rupture of contacts between particles in a porous disperse structure), can be compensated by the entropy gain due to the involvement of separating particles in heat motion as kinetically independent units. For the dispersion of a coagulate
4 = n,1/zw - n,kT@
1
@' e
0
(7)
where z is the coordination number of a particle, i.e., the number of its neighbors in the coagulate, and n, is the concentration of particles in the sol. The dimensionless value B = In (u,/ug) is the dilution term which depends on the ratio of free volumes per particle in the sol, u, = l / n , , and in the coagulate (gel), ug, respectively. For common moderately diluted sols, B is of the order of 10. The condition of the free energy minimum, a4/an, = 0, corresponds to sol-gel equilibrium, i.e., the dynamic equilibrium between coagulation and peptization. According to eq 7, the concentration of sol particles in equilibrium with the coagulate is given by
n, = l/o, exp[-'/zw/(kT)J
(8)
This equilibrium value n,, which we define as colloid solubility, is strongly dependent on the ratio 1/2zw/(kT) under the exponent. The transition from extremely low n, values to high ones according to the actually observed peptization + coagulation transition should correspond to a relatively narrow interval of the values of this ratio, about 10-20. At ambient temperatures (T 300 K)for loosely packed coagulate ( z not much above its minimum value zmin= 2), the corresponding values of the binding energy of the particles in the primary potential minimum w are in the range (4-8) X erg. This means that the particles should be small enough and the value of 1/2F should be sufficiently low. That is, according to eq 6 for the particles 100 A in diameter, i.e., R = 5 nm, the "critical" value (1/2F(heff)), = w / (27rR) is (approximately) several units of dyn. Let the effective decay distance heflfor the particles in the immediate adhesive contact be of the order of usual molecular dimensions, e.g., about 1nm. In such a case the '/@',critical values (corresponding to the coagulation == peptization transition) may prove to be several units of erg/cm2. These values are well in the range observed for the macroscopic surfaces in the experiments described above. Thus it is possible to compare 1/2F= p / ( 2 ~ Rvalues ) for model (macroscopic) surfaces to the critical w and accordingly l/$heff = w/(27rR)values correspondingto the observed transition from coagulation to thermodynamic stability (peptization), that is, the transition from a lyophobic to a lyophilic colloid system. The results of our model experiments using methylated glass spheres, together with the data on coagulation stability transitions in methylated silica dispersions in surfactant solutions observed in our e~periments'~ and in alcohol-water mixtures according to Benitez et al.,19 indicate that this transition actually occurs a t 1/2Fvalues of several units of erg/cm2. The next important problem concerns the role of electrolyte and DLVO barriers on the lyophilicity + lyophobicity transition in disperse systems. This implies their influence on the depth of the primary minimum, which
. a
c.2
c
C.2
c.?
c c
r.-
'p Figure 3. Reduced optical density,D (corrected for the variations of the refractive index of the medium),of methylated colloid silica dispersions vs. volume fraction of water, cp , in ethyl alcohol in the absence of added salt (1) and in the presence of lo-' M NaCl (2).
-
(17)Shchukin, E. D.; Amelina, E. A,; Yaminsky, V. V. J. Colloid Interface Sci. 1982, 90,137. (18) Yaminsky, V. V.;Shchukin, E. D. Kolloidn. Zh. 1985,47, 1211. (19) Benitez, R.;Contreras, S.;Goldfarb, J. J. Colloid Interface Sci. 1971,36, 146.
c p 2
n.2
?.A.
c 6
C.F
'p
Figure 4. Specific free energy of adhesion, '/$, of methylated glass surfaces vs. volume fraction of water, cp, in ethyl (1) and n-propyl (2) alcohols.
determines thermodynamic aspects of this transition, apart from the kinetics of overcoming the barrier. We shall first consider the experiments performed with hydrophobic (methylated) surfaceslSand then will proceed to similar results obtained with the initially hydrophilic (nonmodified) silica surfaces.20 Figure 3 shows the enhancement of coagulation marked by a strong increase of turbidity on addition of water to the stable sol of methylated aerosil in ethanol (the diameter of the primary particles is about 100 A). In the case of distilled water with the residual electrolyte concentration less than lo-* M (conductivity is 5 x lo4 W cm-'), when the extended electric double layers produce a prolonged electrostatic barrier coagulation starts when the volume fraction, cp, of water is increased to 0.4. According to Figure 4, the corresponding value of 'l2Fis 5-6 erg/cm2. In the presence of lo-' M NaC1, when the barrier is suppressed, the critical coagulation concentration of water, (20) Yaminsky, V.V.;Nanikaachvili, P. M.; Shchukin, E. D. Kolloidn. Zh.1985, 47, 1214. (21) Derjaguin, B. V.;Churaev, N. V.; Muller, V. M Surfaces Forces; Nauka: Moscow, 1985.
Langmuir, Vol. 3, No. 6,1987 971
Thermodynamic Stability of Sols
c
0.2
c.4
r.r
0.f
cp Figure 5. Reduced optical density, D, of methylated silica dispersions vs. volume fraction of water, cp, in n-propyl alcohol in the absence of added salt (1)and in the presence of (2) and lo-' M KI (3).
0
1 2
0
4
6
F
SF, ergs/"
cpo is shifted to the left and equals 0.1. The corresponding
'12F value is 2.5-3 erg/cm2. Similarly, cp, values for 1propanol-water mixtures are 0.6 in the absence of added salt and are reduced to 0.2 in 10-' M KI (Figure 5). According to Figure 4 this again corresponds to the reduction of '12F from 4-5 to 2.5-3 erg/cm2. At the same time it is important to stress that the '13 curves in Figure 4 obtained in the absence of added salt and in lo-' M electrolyte virtually coincide. Thus, the values of lI2F that are in the range of ones and tens of ergs/cm2 prove to be insensitive to the presence or absence of an electrostatic barrier, at least within the accuracy of fractions of 1 erg/cm2. In other words, the initial experimental value, the force of adhesion of macroscopic surfaces, 'does not feel" the repulsive electrostatic interaction within the accuracy of several percent. This is in agreement with the low value of the force barrier, corresponding to about lo-' erg/cm2 when converted to specific energy F. Thus, electrolyte has a minor effect on the force of adhesion p and accordingly on 'I.$'although it increases substantially the depth w (with respect to the zero level at h =J) of the primary minimum of the interaction energy of colloid particles; that is, it influences significantly this main thermodynamic parameter of the disperse system. Figure 6 brings together the data of Figures 3-5 the values of lI2F along the x-axis measure the lyophobicity of the interface; the values of optical density, which increase along the y-axis due to the transition from stability to coagulation, stand for the increased lyophobicity of the entire disperse system on addition of water to organophilic sol. Finally, we report one more set of experiments using initially hydrophilic (nonmodified) macroscopic glass surfaces and colloid silica dispersions in aqueous cetylpyridinium bromide (CPB) solutions. In this case l / z F increases (from zero in pure water) on increasing CPB concentration; accordingly, the sol becomes progressively lyophobic, due to adsorption of CPB. In Figure 7, as in Figures 3 and 5 , the increase of turbidity at some definite concentration of "lyophobizer" (CPB in the present case) corresponds to the enhancement of coagulation. In the absence of foreign salt, when the DLVO barrier is present, this critical value C, equals approximately 2 X 10" M,
-
Figure 6. Reduced optical density, D, of methylated silica dispersions (scaled by the optical density of the dispersion in pure alcohol, Do)w. specific free energy of adhesion of methylated silica glass surfaces in ethanol-water (circles)and 1-propanol-water (quadrates)mixutres in the absence of added salt (open symbols) and in lo-' M electrolyte (closed symbols). I .c
I D
0.P
0.6
0.4
0.2
0
IO
0
20
-----0
I 2
30
c,,
Io-F,
I -
3
'I I7 c , I C - ~M Figure 7. Optical density, D, of nonmodified silica dispersions vs. initial, C,, and equilibrium, C, concentrations of cetylpyridinium bromide in the absence of foreign salt (1)and in the presence of lo-' M NaCl (2).
while in 10-1 M NaCl the value of Ccis only about 5 X lo+ M. The results of lI2F = p / ( 2 & ) measurements for the initially (in the absence of CPB) hydrophilic macroscopic surfaces in the same media are shown in Figure 8. Both in the absence of added salt and in lo-' M NaCl the data on lI2F practically coincide. The values of ' l Z F corresponding to two C, values are about 4 and 2 erg/cm2, respectively. The explanation of this set of the data is readily illustrated when the depth and the width of the adhesion minimum in relation t o the height and the width of the electrostatic barrier for the F(h) curves are plotted on a
Shchukin and Yaminsky
972 Langmuir, Vol. 3, No. 6, 1987
0
0 00
'1
IC
0 0
-1
* O
5
0
00 0.
'o1b , &
-5
0.8
0*0°
5
O
1
1
-5
-7
0
-3 lg
c (r)
-5
Figure 8. Specific free energy of adhesion, ' / p ,of nonmodified silica glass surfaces in water vs. concentration, C, of cetylpyridinium bromide.
-IO 3
true scale. Such a curve at the top of Figure 9 corresponds to the results of force-distance measurements (performed in collaboration with Sonntag and Gotze9 for the initially hydrophilic surfaces in lo* M CPB, before and after addition of NaC1. The barrier at the "force curve" '/,F(h) = p(h)/(2&) without electrolyte is only about 0.05 dyn/cm = erg/cm2, though it is extended for several hundred angstroms (in the presence of 10-1 M NaCl it is totally suppressed). By use of this force curve, ll2F(h), and eq 5 for the 10-nm particles, the integral value w(+), which is approximately equal to the barrier height for the "energy curve" w ( h ) , can be estimated: it turns out to be about 8-10 kT. The depth of the primary minimum '12F = -1/2F(ho)for the same force curve is about 1 erg/cm2 both in the absence and in the presence of electrolyte. Multiplication of the latter value according to eq 6 by 2rR (R = 50 A) and by h,E 1 nm (i.e., integration with respect to distance) gives for the depth of the minimum of the energy curve w(-) 5 kT. These and subsequent data are schematically plotted in Figure 9. In the above case, corresponding to a CPB concentration of 1 X lo* M, the repulsive (DLVO barrier) w(+)and attractive (primary minimum) w(-) contributions to the "energy curve" w ( h ) at h ho are comparable in value. In the absence of added electrolyte, the short-distance minimum of the w ( h ) curve is situated in the positive energy region: w(h0) = w(+)- w(-) > 0. Accordingly, the sol is stable. The suppression of the electrostatic barrier shifts the minimum downward, and it is situated now in the negative energy region. Nevertheless, its depth at the given extent of lyophobization (hydrophobization) is still too low to violate the thermodynamic stability (lyophilicity) of the system. On subsequent lyophobization of the interfaces, when going to the critical (in 10-1 M NaC1) CPB concentration C, = 5 X lo4 M, the repulsive energy contribution of the barrier remains approximately of the same order, with w(+) about 5-7 k T (the reduction of surface charge due to the adsorption of CPB at this concentration is still relatively small). A t the same time the depth of the primary energy
-
-
-
(22) Yaminsky, V. V.; Nanikachvili, P. M.; Gotze, Th.; Sonntag, H.; Shchukin, E. D. Kolloidn. Zh. 1986, 48,1205. (23) In a series of our previous p a p e r ~ ~ the ~ ~ letter J * ~ ~N w a ~used instead of p to designate the force of adhesion according to Derjaguin's original work.' In the current literature different symbols are used to designate the same qimntity;Le., the letter F is adopted by the Australian groups for the interaction force.
c -6
-In -15 -? 0
Figure 9. "Energy" curves, w ( h ) , (schematic)for nonmodified silica articlea in water at cetylpyridiniumbromide concentrations of 10 (a),5 X lo4 (b),and 2 X lod M (c) in the absence of added salt (1)and in lo-' M NaCl(2). *Force"curves '/&h) at a CPB concentration of lo4 M, corresponding to "energy" curves in the case a, are also shown (a').
P
minimum, w@),reaches, according to '/,F measurements, about 10 kT. The difference w = w(-) - w(+)is sufficiently small so as not to violate the stability of the sol in the absence of added salt. However, on removal of the DLVO barrier the energy minimum is deep enough, w F= w(-) = 10 k T , to enhance coagulation. At this point the critical condition of the transition from a lyophilic system to a lyophobic one is fulfilled. On additional increase of the concentration of CPB to 2X M (and more) the system becomes strongly lyophobic, and the negative attractive term w(-) (about 15-20 kT and greater) appears to be high enough for the condition of thermodynamic instability ( w > 10 k T ) to be fulfilled, also in the absence of foreign salt (with the energy barrier w(+)of a few k T still present). That is, coagulation becomes thermodynamically predetermined independently of the electrolyte content. In conclusion, we consider once more the initial data concerning the variations of the specific free energy of adhesion '/,F. Both for the initially hydrophobic (methylated) surfaces as the polarity of the medium is increased (on addition of water to alcohol) and for the initially hydrophilic surfaces as the concentration of the solution of the surfactant hydrophobizing the surfaces is increased, similar critical values of llzF= p/(2?rR) corresponding to stability coagulation transition in the relevant colloid dispersions are observed. In both cases these critical values are reduced on addition of salt (for particles 100 A in diameter) from 4-6 erg/cm2when the additional electrolyte is absent to 2-3 erg/cm2 in 10-1 M electrolyte solution. According to the above results this is because the area of the barrier and that of the primary minimum at the "force"
Langmuir 1987, 3, 973-975 interaction plots l/$(h) are of the same order of magnitude. Accordingly, in all these cases the values of the effective decay length of the hydrophobic forces, or, in the other terms, of the attractive structural forces21responsible for the adhesion of particles in the primary minimum, are approximately the same (heffis about 1 nm). Thus the comparison of two sets of independent data on (y2Fh)c from the stability experiments and on l/$ = p l ( 2 r R ) from
973
adhesion force measurements leads to this typical value of the effective width of the primary minimum. This result is in line with the results of Derjaguin and co-workers, Israelachvili, Pashley and co-workers in Australia, Stenius and co-workers in Sweden, and our colleagues in several other research groups obtained by using other combinations of methods.8.21 Registry No. CPB, 140-72-7;silica, 7631-86-9.
Kinetic Analysis of Temperature-Programmed Desorption Curves: Application of the Freeman and Carroll Method J. M. Criado, P. Malet,* and G. Munuera Instituto de Ciencia de Materiales, C.S.I.C., Universidad de Sevilla, Sevilla, Spain Received December 5, 1986. In Final Form: April 13, 1987 The aim of this paper is to test the validity of the Freeman and Carroll method,’ widely used in the kinetic analysis of solid-state reactions under temperature program, when applied to temperature-programmed desorption curves. Conclusions achieved are confirmed by applying the method to simulated and experimental temperature-programmed desorption curves.
Introduction It is well-known that the Freeman and Carroll method has been widely used in the literatweld for determining simultaneously both the activation energy and the “order” of solid-state reactions fulfilling a “ n order” kinetic law. Moreover, it has been shown in a recent paper5 that the Freeman and Carroll equation is fitted by whatever would be the kinetic law obeyed by the reaction. The main advantage of this method is that the experimental data of the TPD profile are transformed into a straight line with slope -E/R, where E is the activation energy of the process, and an intercept n characteristic of the reaction kinetics, resulting in a useful kinetic analysis method. Taking into account the similarity between the formal expressions of the kinetic equations of both solid-state reactions and desorption of gases from solid surfaces, this paper explores if the above method can be applied to the kinetic analysis of temperature-programmed desorption (TPD) curves and if it is able to discern between pure desorption kinetics (no readsorption of previously desorbed species) and desorption under an equilibrium between the adsorbed and gas phases along the T P D run (free readsorption conditions).
Theoretical Section Desorption kinetics considered in the literaturefor the analysis of TPD profiles include first- and second-order desorption from energetically homogeneoussurfaces, assuming two limiting cases: no readsorption of the previously desorbed species (desorption without readsorption) or equilibrium conditions between the adsorbed and the gas phases during the TPD run (desorptionwith (1)Freeman, E.S.;Carrol, B. J. Phys. Chem. 1958, 62, 394. (2)Wendlandt, W.W. Thermal Methods of Analysis; Wiley: New
York. - _._ -1974. - (3) Chen, D. T. Y.; Fong, P. H. I
J. Therm. Anal. 1976, 8, 295. (4)Johnson, D.W.; Gallagher, P. K. J. Phys. Chem. 1972, 76, 1474. (5) Criado, J. M.; Dollimore, D.; Heal, G. R. Thermochim. Acta 1982, 54, 159. 0743-7463/87/2403-0973$01.50/0
free readsorption). The desorption rate can be expressed by the general relationship (-dO/dt) = A exp(-E/RT) f ( 8 ) (1) where A is the Arrhenius preexponential factor, E the activation energy of the desorption process in the nonreadsorption case (or the adsorption enthalpy in the free readsorption limit), 8 the surface coverage, and f ( 8 ) a function depending on the desorption kinetics. f ( 8 ) expressions for different desorption kinetics are collected in Table I. Under nonreadsorption experimental conditions eq 1can be written in the form (-dO/dt) = A exp(-E/Rr)
On
(2)
By differentiating the logarithmic form of eq 2 with respect to In 8, we obtain
d In (-de/dt) d(l/T) = (-E/R)d In 8 d In 8 + n
(3)
A In (-dO/dt) A(1 / r ) = (-E/R)+n A In 8 A In 8
(4)
or
Therefore, the plots of the left-hand side of eq 3 and 4 vs. d(l/T)/d In 8 or A ( l / T ) / A In 8, respectively, should yield a straight line with a slope -E/R and an intercept equal to the desorption order n. This method, equivalent to that proposed by Freeman and Carrol’ in order to analyze nonisothermal solid decomposition traces, could be a useful and easy way ta perform line shape analysis of TPD curves in experimental conditions of nonreadsorption. However, experimental conditions in TPD make feasible the readsorption of previously desorbed species, and it is not evident that the former plots will fit straight lines with a characteristic apparent order n if these free readsorption kinetics apply. If we differentiate the logarithmic form of eq 1 with respect to In 8, we get d In (-de/dt) d(l/T) d le f ( 0 ) (5) = (-E/R)d In 8 dln8 dln8 +
0 1987 American Chemical Society
