- Email: [email protected]
Sol-gel polymerization in alkoxysilanes: 29si nmr study and simulation of chemical kinetics
MATERIALS SCIEHCE & EMGIWEERINC
ELSEVIER
B
Materials Scienceand EngineeringB37 (1996) 197-200
Sol-gel polymerization in alkoxysilanes: 2gSi NMR study and simulation of chemical kinetics A. VainTuba,‘, F. Devreux”, J. P. Boilot”, F. Chaput”, M. Sarkarb “Lnboratoire
de Physique
de la Mat&e bICI PLC,
CondensCe, 9 Millbank,
Ecole Polyrechnique, London, SWlP3JF,
91128 UK
Palaiseau
Cedex,
France
Abstract We report on the results of 2gSi NMR monitoring of alkoxysilane/ethanol/acid water sol-gels with tetraethoxysilane (TEOS), methyltriethoxysilane (MTEOS) and vinyltriethoxysilane (VTEOS). The gel time of the TEOS solution was 49 days at the degree of condensation c = 0.81, whereas MTEOS and VTEOS do not gel in spite of the high c = 0.95 achieved. The polymerization kinetics progressively slows with the time owing to the growth of sterical restrictions. To account for the effect we developed a kinetic model based on the reaction rates which explicitly decay as a function of the degree of condensation. The model describes the complete sol-gel process in all the studied systems including a gel transition in TEOS and aging of the gel. Keyworrls:
Sol-gels; Alkoxysilanes;
Models of chemical kinetics; Nuclear magnetic resonance
Sol-gel processing of alkoxysilanes and organoalkoxysilanes is a promising technology to produce advanced bulk materials and coatings [I]. We report on the results of 2gSi NMR monitoring of alkoxysilane/ethanol/acid water = 1:6:10 sol-gels with tetraethoxysilane methyltriethoxysilane VEW, (MTEOS) and vinyltriethoxysilane (VTEOS). The water contained 8. 10e3 mol 1-l of paramagnatic Cr(NO,), .9H,O salt to fix pH = 2.5 and to make nuclear relaxation faster. The gel time of the TEOS solution was 49 days at the degree of condensation c = 0.81, whereas MTEOS and VTEOS do not gel in spite of the high c = 0.95 achieved. 2gSi NMR provides the concentration qi of Qi silicon species as a function of the reaction time where i is the number of silicon atoms attached to the siloxane bonds. Kinetic data (see Ref. [2] and later articles [3-61) are analyzed by the method of kinetic differential equations written for the concentrations of the different silicon species. The simplest statistical model [7] takes assumption of equal reaction rate KcTbbetween two each SiOH
groups a and b regardless of the type of the species they belong. More general models [8,9] use different Kab for various species. All the models assume constant rates of reaction which do not change in he course of reaction. These models fit experimental kinetics only fo very initial stage of polymerization up to the time when silicons with two siloxane bonds appear. For example, the kinetics was successfully simulated during first 10 h for TEOS systems gelling in 35 and 60 days [8]. The failure at long times arises from the following reasons. First, kinetic equations are true if the probability of collision of two reactive groups does not change in the course of reaction. In fact the collision become less probable since the growth of polymers increases sterical restrictions and also produces “dilution effect” [8]. Second, reactions of cycle and cage formation become dominant at late stages of condensation [2,3]. These intramolecular reactions need higher thermal activation energy and so are slower than intermolecular reactions of chain propagation or branching. All the mentioned effects tend to decrease the condensation rate.
’ Permanent address: Institute of Chemical Physics and Biophysics, Tallinn EEOlOO, Estonia.
2. Chemical kinetics model
1. Introduction
0921-5107/96/515.00 0 1996 - Elsevier Science S.A. All rights reserved
Slowing
of condensation
owing to the growth of
A. Vainrub et al. 1 Materials Science and Engineering B37 (1396) 197400
198
sterical hindrance in the course of reaction, dilution effect and formation of cycles is modeled in our approach by decay in time of the reaction rate coefficients. Under our reaction conditions (acid medium, high H,O/Si molar ratio) the hydrolysis is fast and completes before the beginning of condensation. Therefore reacting species are fully hydrolyzed and a set of kinetic equations for the concentration qi of Qi species in MTEOS and VTEOS which have a functionality equal to 3 (i = 0, 1, 2, 3) is:
where k&t) is the time dependent reaction rate and is a decreasing function of the time t. For TEOS with a functionality equal to 4 a set of Eq. (1) contains 5 equations (i = 0, 1...4); we omit the equations here for briefness. Appropriate decay function f(t) was found from comparison of experimental results with prediction of statistical reaction model. For this model f(t) = 1 and the rate constants are k, = ijk. Kinetic equation for the extent of reaction c = (1/3)(ql + 2q, + 3q,) is:
f(t)
dc/dt = k(1 - c)”
where 0 < s < 1. Such a choice is less justified if the rate constants k, in Eq. (1) deviate from the rates in the statistical model. However, in the next section we show that in spite of nonstatistical type of studied reactions their kinetics is successfully simulated using the decay function Eq. (7) although with the power I’ less than following from the statistical model s = l-p.
3. Kinetic simulations vs. experimental
results
Fig. 1 shows experimental concentrations 4i as a function of time. For MTEOS and VTEOS to simulate four qi curves we have to adjust eight parameters: six rates k, in Eq. (1) and also b and s in Eq. (7). For TEOS the number of the rates is ten. We divided this complicated problem into two easier feasible steps. Analysis of Eq. (1) shows that the concentration cli(C) as a function of the degree of condensation c depends only on the ratios of the rates k, but it is independent of the choice of the functionf(t). So, first, we simulated
TEOS
(2)
The solution of Eq. (2) is: 1 - c = l/(1 + kt)
(3)
We checked that the experimental curves c(t) allow a good fit by the function l-c = l/(1 + dt)p, where constants d and p are fitting parameters. We found p = 0.22 for MTEOS, 0.18 for VTEOS and 0.18 for TEOS. Hence, whereas statistical model predicts l-c N l/t at large times, experimental kinetics as expected is slower: l-c N l/P @ < 1). Thus Eq. (2) does not account for the observed kinetics. However, we noted that if the constant reaction rate k is replaced by the following decreasing function (g>O) of the extent of reaction c: k = k,(l - c)”
(4)
the model comes into accordance with the experiment. Indeed, under this condition the solution of Eq. (2) is: 1 - c = l/(1 + (1 +g)k,t)“(‘+g)
6 z” 8
(5)
in accordance with observed power law of time dependence with the power p < 1. Combining Eqs. (4) and (5) we obtain the reaction rate as a function of time: k=k,/(l f(t)
+(l
+g)&)g’(‘+g)
(6)
According to this result we choose decay function in kinetic Eqs. (1) to be:
f(t) = l/(1 + bt)
(7)
TIME
(hours)
Fig. 1. Concentration of the species q1 as a function of the time for TEOS, MTEOS and VTEOS. Experimental data (points), simulation (lines). Vertical straight line shows the gel time 49 days for TEOS whereas MTEOS and VTEOS do not gel.
A. Vaiwub
0.8
et al. J Materials
Science
and Engineering
0.6 -
Li
0.4 x x’
0.2 -
x’ 3 r;r
O1 0.2
1 0.4 DEGREE
1 0.6 OF
! 0.8
I I
CONDENSATION
q,(C) profiles by numerical solution of Eq. (1) with the simplest f(t) = 1. Statistical reaction rates k, = (F- i)(F-j) [7] and the model with multiplication factor I’ (Y< 1) [9] (F-
199
i)(F--j)r”+Jl
b s
koo km km k,, k,,
k,,
MTEOS VTEOS
0.169 1.2 0.5 0.075 0.8 0.5
Statistical TEOS
0 100 66.7 33.3 44.4 22.2 11.1 0.072 0.17 0.82 100 41.3 15.1 17.0 6.24 2.29 (k,,=4.15; k,,= 1.72; k,,=0.629; k,,=0.174)
100 11.1 2.77 3.71 0.924 0.232
4. Discussion
Fig. 2. Concentration of the species qi as a function of the degree of condensation. MTEOS (points), VTEOS (crosses).
k,=
197-200
Table 1 Reaction rate constants ok, (normalized to k,, = 100) and parameters b and s of the time decay of reactions f(t) = l/(1 + bt)” found by numerical simulation of the experimental kinetics
-
-0.2 1 0
B37 (1996)
(8)
where functionality F= 3 for MTEOS and VTEOS, F = 4 for TEOS, have been used as an initial trial set. MTEOS and VTEOS show very similar qi(c) displayed in Fig. 2. This implies the same set of k, for the both systems which we have determined by try and error. Good fitting was was achieved as Fig. 3 shows. The same kind of the fitting process was applied for TEOS. In this case good fitting was obtained already by k, expressed by Eq. (8) and hence by choosing just one parameter Y= 0.55 to evaluate all the ten rection rate constants. Secondly, we used the determined rate constants k, to fit q!(t) by variation the values b and s. One more fitting parameter t’ was used to convert the normalized rates k, (k,, = 100) into the true rates ck,. Fig, 1 shows a good agreement of the simulated qi(t) with the experimental data. All the simulation parameters together with their values in the statistical model are listed in Table 1.
Developed model account for a complete range of the condensation kinetics in MTEOS, VTEOS and TEOS solutions (Fig. 1). Remarkably, the model is also valid for the TEOS gel: it correctly describes aging of the gel from the gel time 49 days until 200 days. We also successively applied the model to the trialkoxysilanes with a large (molecular weight 300-600) polymer radical [lo]. Now we consider obtained kinetics data in Table 1. The first group of the simulation parameters, the reaction rate constants uk,, describes kinetics at the early times of condensation bt < 1. Therefore k, corresponds to the reactions between the species included in small linear oligomers. Table 1 shows that reactions between condensed species are suppressed in comparison with a random polymerization process. For example, for a statistical reaction k,,: kll:kz2 = 100:44: 11 whereas for MTEOS and VTEOS the ratio is 100:4:0.2. The same trend is observed for TEOS in accordance with reported observations [7,8] and is accounted for by the multiplication factor r = 0.55 in Eq. (8). These results are quite clear: SiOH groups do not condense at random since sterical restrictions, for example, for two Ql species in dimers are stronger than for a reaction of two monomers Q”. The other parameters b and s determine the decay of reactions at a long time bt > 1 as l/t”. The same time dependence with s= 0.5 for MTEOS and VTEOS reflects, probably, close thermal activation energy of the cycles formation as well as similar sterical restrictions for the two systems with comparable size methyl and vinyl groups. In contrast, for four functional TEOS the steric hindrance and the barrier to form cages are expected to be larger and to be responsible for a faster decay of condensation with time with s = 0.82.
5. Conclusion DEGREE
OF CONDENSATION
Fig. 3. Concentration of the species qi as a function of the degree of condensation for MTEOS. Experimental data (points), simulation (lines).
We generalized known chemical kinetics models of inorganic polymerization in alkoxysilanes by explicit including of the time decay of the reaction rates with
200
A. Vainrub et al. 1 Materials Science and Engineering B37 (1996) 197-200
the extent of reaction. We successively simulated complete range of condensation kinetics in solution for MTEOS, VTEOS and TEOS whereas known models accounts for only initial stage of the process. Our approach provides deeper understanding of complex steric effects and cycle formation. Moreover, the same kinetic model successively describes a gel transition and aging of TEOS gel.
References [l] C.J. Brinker and G.W. Sherer, Sol-Gel Science: The Physics and Chemistry of Sol-Gel Processing, Academic Press, San Diego,
1990. [2] R.A. Assink and B.D. Kay, ANTI.Reo. Mater. Sci., 21 (1991) 491. [3] F. Devreux, J.P. Boilot, F. Chaput and A, Lecomte, P/QT. Rec., A41 (1990) 6901. [4] J.J. van Beek, D. Seykens and J.B.M. Jnnsen, J. Nordryst. Solids, 146 (1992) 111. [5] Y. Sugahara, S. Okada, S. Sate, K. Kuroda and C. Kato, J. Non-Cryst. Solids, 167 (1994) 21. [6] J. Sanchez and A.V. McCormick, J. Non-Gryst. Solids, 167 (1994) 289. [7] B.D. Kay and R.A. Assink, J. lVon-Cry$f. Solids, 103 (19SS) 112. [S] J.C. Pouxviel and J.P. Boilot, J. Nou-Cryst. Solids, 94 (1987) 374. [9] C.J. Brinker and R.A. Assink, J. Non-Cryst. Solids, 111 (1989) 48. [lo] A. Vainrub, F. Devreux, M. Sarkar, P.D. Palasz and A.N. Burgess, J. Sol-Gel Sci. Tech,, to be published.
ELSEVIER
B
Materials Scienceand EngineeringB37 (1996) 197-200
Sol-gel polymerization in alkoxysilanes: 2gSi NMR study and simulation of chemical kinetics A. VainTuba,‘, F. Devreux”, J. P. Boilot”, F. Chaput”, M. Sarkarb “Lnboratoire
de Physique
de la Mat&e bICI PLC,
CondensCe, 9 Millbank,
Ecole Polyrechnique, London, SWlP3JF,
91128 UK
Palaiseau
Cedex,
France
Abstract We report on the results of 2gSi NMR monitoring of alkoxysilane/ethanol/acid water sol-gels with tetraethoxysilane (TEOS), methyltriethoxysilane (MTEOS) and vinyltriethoxysilane (VTEOS). The gel time of the TEOS solution was 49 days at the degree of condensation c = 0.81, whereas MTEOS and VTEOS do not gel in spite of the high c = 0.95 achieved. The polymerization kinetics progressively slows with the time owing to the growth of sterical restrictions. To account for the effect we developed a kinetic model based on the reaction rates which explicitly decay as a function of the degree of condensation. The model describes the complete sol-gel process in all the studied systems including a gel transition in TEOS and aging of the gel. Keyworrls:
Sol-gels; Alkoxysilanes;
Models of chemical kinetics; Nuclear magnetic resonance
Sol-gel processing of alkoxysilanes and organoalkoxysilanes is a promising technology to produce advanced bulk materials and coatings [I]. We report on the results of 2gSi NMR monitoring of alkoxysilane/ethanol/acid water = 1:6:10 sol-gels with tetraethoxysilane methyltriethoxysilane VEW, (MTEOS) and vinyltriethoxysilane (VTEOS). The water contained 8. 10e3 mol 1-l of paramagnatic Cr(NO,), .9H,O salt to fix pH = 2.5 and to make nuclear relaxation faster. The gel time of the TEOS solution was 49 days at the degree of condensation c = 0.81, whereas MTEOS and VTEOS do not gel in spite of the high c = 0.95 achieved. 2gSi NMR provides the concentration qi of Qi silicon species as a function of the reaction time where i is the number of silicon atoms attached to the siloxane bonds. Kinetic data (see Ref. [2] and later articles [3-61) are analyzed by the method of kinetic differential equations written for the concentrations of the different silicon species. The simplest statistical model [7] takes assumption of equal reaction rate KcTbbetween two each SiOH
groups a and b regardless of the type of the species they belong. More general models [8,9] use different Kab for various species. All the models assume constant rates of reaction which do not change in he course of reaction. These models fit experimental kinetics only fo very initial stage of polymerization up to the time when silicons with two siloxane bonds appear. For example, the kinetics was successfully simulated during first 10 h for TEOS systems gelling in 35 and 60 days [8]. The failure at long times arises from the following reasons. First, kinetic equations are true if the probability of collision of two reactive groups does not change in the course of reaction. In fact the collision become less probable since the growth of polymers increases sterical restrictions and also produces “dilution effect” [8]. Second, reactions of cycle and cage formation become dominant at late stages of condensation [2,3]. These intramolecular reactions need higher thermal activation energy and so are slower than intermolecular reactions of chain propagation or branching. All the mentioned effects tend to decrease the condensation rate.
’ Permanent address: Institute of Chemical Physics and Biophysics, Tallinn EEOlOO, Estonia.
2. Chemical kinetics model
1. Introduction
0921-5107/96/515.00 0 1996 - Elsevier Science S.A. All rights reserved
Slowing
of condensation
owing to the growth of
A. Vainrub et al. 1 Materials Science and Engineering B37 (1396) 197400
198
sterical hindrance in the course of reaction, dilution effect and formation of cycles is modeled in our approach by decay in time of the reaction rate coefficients. Under our reaction conditions (acid medium, high H,O/Si molar ratio) the hydrolysis is fast and completes before the beginning of condensation. Therefore reacting species are fully hydrolyzed and a set of kinetic equations for the concentration qi of Qi species in MTEOS and VTEOS which have a functionality equal to 3 (i = 0, 1, 2, 3) is:
where k&t) is the time dependent reaction rate and is a decreasing function of the time t. For TEOS with a functionality equal to 4 a set of Eq. (1) contains 5 equations (i = 0, 1...4); we omit the equations here for briefness. Appropriate decay function f(t) was found from comparison of experimental results with prediction of statistical reaction model. For this model f(t) = 1 and the rate constants are k, = ijk. Kinetic equation for the extent of reaction c = (1/3)(ql + 2q, + 3q,) is:
f(t)
dc/dt = k(1 - c)”
where 0 < s < 1. Such a choice is less justified if the rate constants k, in Eq. (1) deviate from the rates in the statistical model. However, in the next section we show that in spite of nonstatistical type of studied reactions their kinetics is successfully simulated using the decay function Eq. (7) although with the power I’ less than following from the statistical model s = l-p.
3. Kinetic simulations vs. experimental
results
Fig. 1 shows experimental concentrations 4i as a function of time. For MTEOS and VTEOS to simulate four qi curves we have to adjust eight parameters: six rates k, in Eq. (1) and also b and s in Eq. (7). For TEOS the number of the rates is ten. We divided this complicated problem into two easier feasible steps. Analysis of Eq. (1) shows that the concentration cli(C) as a function of the degree of condensation c depends only on the ratios of the rates k, but it is independent of the choice of the functionf(t). So, first, we simulated
TEOS
(2)
The solution of Eq. (2) is: 1 - c = l/(1 + kt)
(3)
We checked that the experimental curves c(t) allow a good fit by the function l-c = l/(1 + dt)p, where constants d and p are fitting parameters. We found p = 0.22 for MTEOS, 0.18 for VTEOS and 0.18 for TEOS. Hence, whereas statistical model predicts l-c N l/t at large times, experimental kinetics as expected is slower: l-c N l/P @ < 1). Thus Eq. (2) does not account for the observed kinetics. However, we noted that if the constant reaction rate k is replaced by the following decreasing function (g>O) of the extent of reaction c: k = k,(l - c)”
(4)
the model comes into accordance with the experiment. Indeed, under this condition the solution of Eq. (2) is: 1 - c = l/(1 + (1 +g)k,t)“(‘+g)
6 z” 8
(5)
in accordance with observed power law of time dependence with the power p < 1. Combining Eqs. (4) and (5) we obtain the reaction rate as a function of time: k=k,/(l f(t)
+(l
+g)&)g’(‘+g)
(6)
According to this result we choose decay function in kinetic Eqs. (1) to be:
f(t) = l/(1 + bt)
(7)
TIME
(hours)
Fig. 1. Concentration of the species q1 as a function of the time for TEOS, MTEOS and VTEOS. Experimental data (points), simulation (lines). Vertical straight line shows the gel time 49 days for TEOS whereas MTEOS and VTEOS do not gel.
A. Vaiwub
0.8
et al. J Materials
Science
and Engineering
0.6 -
Li
0.4 x x’
0.2 -
x’ 3 r;r
O1 0.2
1 0.4 DEGREE
1 0.6 OF
! 0.8
I I
CONDENSATION
q,(C) profiles by numerical solution of Eq. (1) with the simplest f(t) = 1. Statistical reaction rates k, = (F- i)(F-j) [7] and the model with multiplication factor I’ (Y< 1) [9] (F-
199
i)(F--j)r”+Jl
b s
koo km km k,, k,,
k,,
MTEOS VTEOS
0.169 1.2 0.5 0.075 0.8 0.5
Statistical TEOS
0 100 66.7 33.3 44.4 22.2 11.1 0.072 0.17 0.82 100 41.3 15.1 17.0 6.24 2.29 (k,,=4.15; k,,= 1.72; k,,=0.629; k,,=0.174)
100 11.1 2.77 3.71 0.924 0.232
4. Discussion
Fig. 2. Concentration of the species qi as a function of the degree of condensation. MTEOS (points), VTEOS (crosses).
k,=
197-200
Table 1 Reaction rate constants ok, (normalized to k,, = 100) and parameters b and s of the time decay of reactions f(t) = l/(1 + bt)” found by numerical simulation of the experimental kinetics
-
-0.2 1 0
B37 (1996)
(8)
where functionality F= 3 for MTEOS and VTEOS, F = 4 for TEOS, have been used as an initial trial set. MTEOS and VTEOS show very similar qi(c) displayed in Fig. 2. This implies the same set of k, for the both systems which we have determined by try and error. Good fitting was was achieved as Fig. 3 shows. The same kind of the fitting process was applied for TEOS. In this case good fitting was obtained already by k, expressed by Eq. (8) and hence by choosing just one parameter Y= 0.55 to evaluate all the ten rection rate constants. Secondly, we used the determined rate constants k, to fit q!(t) by variation the values b and s. One more fitting parameter t’ was used to convert the normalized rates k, (k,, = 100) into the true rates ck,. Fig, 1 shows a good agreement of the simulated qi(t) with the experimental data. All the simulation parameters together with their values in the statistical model are listed in Table 1.
Developed model account for a complete range of the condensation kinetics in MTEOS, VTEOS and TEOS solutions (Fig. 1). Remarkably, the model is also valid for the TEOS gel: it correctly describes aging of the gel from the gel time 49 days until 200 days. We also successively applied the model to the trialkoxysilanes with a large (molecular weight 300-600) polymer radical [lo]. Now we consider obtained kinetics data in Table 1. The first group of the simulation parameters, the reaction rate constants uk,, describes kinetics at the early times of condensation bt < 1. Therefore k, corresponds to the reactions between the species included in small linear oligomers. Table 1 shows that reactions between condensed species are suppressed in comparison with a random polymerization process. For example, for a statistical reaction k,,: kll:kz2 = 100:44: 11 whereas for MTEOS and VTEOS the ratio is 100:4:0.2. The same trend is observed for TEOS in accordance with reported observations [7,8] and is accounted for by the multiplication factor r = 0.55 in Eq. (8). These results are quite clear: SiOH groups do not condense at random since sterical restrictions, for example, for two Ql species in dimers are stronger than for a reaction of two monomers Q”. The other parameters b and s determine the decay of reactions at a long time bt > 1 as l/t”. The same time dependence with s= 0.5 for MTEOS and VTEOS reflects, probably, close thermal activation energy of the cycles formation as well as similar sterical restrictions for the two systems with comparable size methyl and vinyl groups. In contrast, for four functional TEOS the steric hindrance and the barrier to form cages are expected to be larger and to be responsible for a faster decay of condensation with time with s = 0.82.
5. Conclusion DEGREE
OF CONDENSATION
Fig. 3. Concentration of the species qi as a function of the degree of condensation for MTEOS. Experimental data (points), simulation (lines).
We generalized known chemical kinetics models of inorganic polymerization in alkoxysilanes by explicit including of the time decay of the reaction rates with
200
A. Vainrub et al. 1 Materials Science and Engineering B37 (1996) 197-200
the extent of reaction. We successively simulated complete range of condensation kinetics in solution for MTEOS, VTEOS and TEOS whereas known models accounts for only initial stage of the process. Our approach provides deeper understanding of complex steric effects and cycle formation. Moreover, the same kinetic model successively describes a gel transition and aging of TEOS gel.
References [l] C.J. Brinker and G.W. Sherer, Sol-Gel Science: The Physics and Chemistry of Sol-Gel Processing, Academic Press, San Diego,
1990. [2] R.A. Assink and B.D. Kay, ANTI.Reo. Mater. Sci., 21 (1991) 491. [3] F. Devreux, J.P. Boilot, F. Chaput and A, Lecomte, P/QT. Rec., A41 (1990) 6901. [4] J.J. van Beek, D. Seykens and J.B.M. Jnnsen, J. Nordryst. Solids, 146 (1992) 111. [5] Y. Sugahara, S. Okada, S. Sate, K. Kuroda and C. Kato, J. Non-Cryst. Solids, 167 (1994) 21. [6] J. Sanchez and A.V. McCormick, J. Non-Gryst. Solids, 167 (1994) 289. [7] B.D. Kay and R.A. Assink, J. lVon-Cry$f. Solids, 103 (19SS) 112. [S] J.C. Pouxviel and J.P. Boilot, J. Nou-Cryst. Solids, 94 (1987) 374. [9] C.J. Brinker and R.A. Assink, J. Non-Cryst. Solids, 111 (1989) 48. [lo] A. Vainrub, F. Devreux, M. Sarkar, P.D. Palasz and A.N. Burgess, J. Sol-Gel Sci. Tech,, to be published.