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Kinetics of slow collapse process: Thermodynamic description of rate constants
Applied Surface Science 253 (2006) 1469–1472 www.elsevier.com/locate/apsusc
Kinetics of slow collapse process: Thermodynamic description of rate constants M. Weis * Faculty of Electrical Engineering and Information Technology SUT, Ilkovicˇova 3, 812 19 Bratislava 1, Slovakia Received 6 February 2006; accepted 14 February 2006 Available online 11 April 2006
Abstract Insoluble monolayer formed at the air/water interface compressed at a surface pressure above the equilibrium spreading pressure is unstable. In presented paper the classical theory for homogeneous nucleation is modified adding the activation energy term. It allows quantitative thermodynamic interpretation of the slow collapse phase transformation. The collapse of stearic acid Langmuir films has been carried out by systematic measurements of the area loss–time isobaric dependencies at various temperatures and isothermal dependencies at various pressures. Activation energies (activation enthalpies) and activation entropies have been evaluated for the nucleation (Ea = 1.32 eV) and the growth (Eb = 1.55 eV) processes. The experimental data for various pressures are discussed on the basis of the Gibbs energy analysis. # 2006 Elsevier B.V. All rights reserved. PACS: 68.18.Fg; 64.70.Nd; 68.35.Rh Keywords: Langmuir–Blodgett films on liquids—Structure: measurements and simulations; Structural transitions in nanoscale materials; Phase transitions and critical phenomena
1. Introduction Monolayers spontaneously formed at the air/water interface represents a two-dimensional system with various potential applications [1,2]. More recently, with increasing interest in the deposition technique on the solid substrate (Langmuir–Blodgett technique), the applications have appeared in the field of physics [3,4], chemistry, or biology. However, the monolayer instability [5] is one of the main problems, which obstructs its practical application. Over the past decades the collapse phenomena in Langmuir films have been a subject of numerous experimental studies in physics or chemistry (e.g. [6]). In general, there are two classes of structural changes in the Langmuir film, transitions between different two-dimensional phases [7], and transitions in which dimensionality changes from the degree of two to three [8,9]. We will now deal only with the latter class of transitions, which is usually termed as a monolayer collapse. Different basic types of monolayer
* Tel.: +421 2 602 91 846; fax: +421 2 654 27 427. E-mail address: [email protected]. 0169-4332/$ – see front matter # 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2006.02.029
collapse can be distinguished according to their different mechanisms. At higher surface pressures the monolayer structure is abolished by fracture collapse, sometimes also called catastrophic collapse. There are several mechanisms, which can likely contribute to the process, but the final effect is easy to observe as a sudden pressure fall in the surface pressure–area isotherm. At lower pressures, the degradation of the monolayer is caused by so-called ‘‘slow collapse’’. Also here the final effect can be realized through various microscopic processes—buckling instability, folding of the monolayer, or the formation and growth of cracks [10–12]. Therefore, various models were proposed for the description of the experiment [13–15]. These theories are in excellent agreement with many direct and indirect experimental results, the choice of the theory usually varies with the used surfactant and the dominated degradation process. 2. Theory Gaines [16] proposed that there might be a monolayer stability limit, which he defined as the pressure of equilibrium between the film and the freshly collapsed material.
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The three-dimensional phase in this case would have a degree of stability lying between that of the monolayer and the equilibrium bulk phase. Degradation of the monolayer in the pressure range between the equilibrium spreading pressure and the fracture pressure was a center of interest in several experimental works (e.g. [17]). Smith and Berg [18] were the first who described the slow collapse mechanism by homogenous nucleation and growth. It is possible to express the change of molecular area (relative area loss) as a dependence on time: A ¼ expðat bt2 Þ A0
(1)
where A0 is the initial area and parameters a and b are characteristics of the nucleation and the growth processes, respectively. A more complex view on the nucleation phenomena was presented by Vollhardt and Retter [19], who considered shape effects of the critical nuclei. Later works showed that on the basis of these nucleation-growth theories important parameters of the classical nucleation theory such as, critical size of nuclei and free energy for the formation of the critical nuclei can be determined [13–15]. In general, parameters a and b are considered only as experimental constants. However, if we describe the slow collapse of the monolayer as a phase transition, it is possible to assign the activation energy Ea to this process:
Ea a ¼ aT;0 exp kB T
kB T DSa DHa exp ¼ exp h R RT (2)
or to express the activation enthalpy DHa and the activation entropy DSa by the Eyring equation, where kB is the Boltzmann constant and h is the Planck constant. A formally similar relation to Eq. (2) can be applied for parameter b. Rate parameters a and b are proportional to the rate of formation of critical nuclei J what can be expressed [18] for the Langmuir film in the form: k2 J ¼ k1 p exp 2 (3) ln ðp=pE Þ
Fig. 1. Area loss–time dependencies at surface pressure of 30 mN/m for various temperatures.
and in analogy a formally similar relation to Eqs. (4) and (5) can be applied for rate parameter b. Constants k and pE are the same for the both rate parameters. 3. Experimental Surface pressure–area isotherms were measured in a Langmuir trough (Type 611, NIMA Technology Ltd., UK). The surface pressure measurements were carried out with a filter-paper Wilhelmy plate. The Langmuir monolayer of stearic acid (purchased from Fluka, Switzerland) was formed by careful casting a 1 mg/ml chloroform solution to the water surface (bidistilled deionized water, 15 MV/cm). The solvent was allowed to evaporate for at least 15 min prior to compressing the surface. Before a collapse measurement the monolayer situated on the surface of water was firstly two times slowly compressed up to the pressure of 10 mN/m and subsequently released for monolayer homogenization. All the chemicals used were of spectroscopic grade purity.
where k1 and k2 are constants and pE is the equilibrium spreading pressure. Therefore, the pressure dependence of the rate parameter can be written: a ¼ ap;0 p exp
k ln2 ðp=pE Þ
(4)
Using Eqs. (2) and (4) the behaviour of the Gibbs energy can be described by the relationship: ap;0 hp k 2 DGa ¼ RT ln kB T ln ðp=pE Þ
(5) Fig. 2. The Arrhenius plot for experimentally obtained parameters a and b.
M. Weis / Applied Surface Science 253 (2006) 1469–1472
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4. Results and discussion
Fig. 3. Area loss–time dependencies at temperature 24 8C for various surface pressures.
Fig. 4. Experimentally obtained rate parameters a and b vs. surface pressure. Solid line represents theoretical results of Eq. (4) for k = 40, ap,0 = 70 s1 and bp,0 = 8 s2. The equilibrium spreading pressure is pE = 5 mN/m [9].
The stability of stearic acid monolayers was studied by monitoring their relative area loss (A/A0) with respect to time at a constant surface pressure of 30 mN/m. The temperatures were varied in the range from 10 to 26 8C. The obtained experimental results are shown in Fig. 1. As shown in [18] for the stearic acid monolayer collapse, the description by Eq. (1) is sufficient. The parabolic part of ln(A/ A0) was fitted by a polynomial of the second degree. The obtained values of parameters a and b from Fig. 1 for each temperature are shown in the Arrhenius plot in Fig. 2. The activation energies (activation enthalpies) of nucleation and growth are Ea = 1.32 eV (DHa = 127.4 kJ/mol) and Eb = 1.55 eV (DHb = 150 kJ/mol), respectively. The difference between the activation energy and enthalpy (DH = E RT) is lesser than the error of experimental data. The activation entropies are positive, DSa = 1.83 eV/K (176.5 J/K mol) and DSb = 2.43 eV/K (234.4 J/K mol), which agrees with theoretical predictions [14]. According to this fact a new three-dimensional
Fig. 5. The Gibbs energy vs. surface pressure. Solid line represents theoretical results of Eq. (5) for k = 40, ap,0 = 70 s1 and bp,0 = 8 s2. The equilibrium spreading pressure is pE = 5 mN/m [9].
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phase is generated, providing a more disordered system in comparison with monolayer. The surface pressure dependencies of slow collapse mechanism of stearic acid monolayers was studied by their relative area loss (A/A0) versus time functions at a constant temperature of 24 8C. The surface pressures were varied from 25 to 30 mN/m. The recorded experimental results are shown in Fig. 3. The values of rate parameters a and b from Fig. 3 for each surface pressure were compared with theoretical results of Eq. (4) for k = 40, ap,0 = 70 s1 and bp,0 = 8 s2 in Fig. 4. The value of the equilibrium spreading pressure was taken pE = 5 mN/m [9]. More general view provides development of the Gibbs energy with respect to surface pressure (Fig. 5), where the values of the Gibbs energy were obtained from Eq. (2). 5. Conclusions The stability of stearic acid monolayers at the air/water interface was evaluated by their area loss on time at a constant surface pressure for various temperatures and at a constant temperature for various surface pressures. The experimental data can be interpreted by the theory of homogenous nucleation and growth [18] as the phase transition between two and threedimensional molecular systems. After initial regular collapse of the monolayer as described by Eq. (1), the dependence ln(A/A0) versus time became more linear. This effect is more pronounced at higher temperatures. This is probably caused by a mutual contact of growing threedimensional aggregates, which results in stopping its growth. Our course of approach in the quantitative interpretation is appropriate to evaluate the activation energies and the Gibbs
energies of nucleation and growth, which (according to [18,20]) control the slow collapse process. Acknowledgement The work was supported by the Slovak grant agency VEGA, project no. 1/0277/03. References [1] M.C. Petty, Thin Solid Films 210/211 (1992) 417. [2] D. Barancˇok, J. Cira´k, P. Tomcˇ´ık, J. Electr. Eng. 45 (1994) 41. [3] D. Barancˇok, J. Cira´k, P. Tomcˇ´ık, J. Vajda, Phys. Stat. Sol. A 169 (1998) 297. [4] J. Cira´k, D. Barancˇok, P. Tomcˇ´ık, J. Vajda, Mater. Sci. Eng. C 8/9 (1999) 13. ˇ ervenˇ, O. Foltı´n, J. Electr. Eng. 47 [5] D. Barancˇok, J. Cira´k, P. Tomcˇ´ık, I. C (1996) 209. [6] L.V.N. Avila, S.M. Saraive, J.F. Oliveira, Colloids Surf. A 154 (1999) 209. [7] D. Vollhardt, V. Fainerman, Colloids Surf. A 176 (2001) 117. [8] D. Vollhardt, Colloids Surf. A 143 (1998) 185. [9] D. Vollhardt, Colloids Surf. A 156 (1999) 79. [10] E. Hatta, D. Suzuki, J. Nagao, Eur. Phys. J. B 11 (1999) 609. [11] A. Gopal, K.Y.C. Lee, J. Phys. Chem. B 105 (2001) 10348. [12] C. Ybert, W. Lu, G. Mo¨ller, C.M. Knobler, J. Phys. Chem. B 106 (2002) 2004. [13] D. Vollhardt, M. Ziller, U. Retter, Langmuir 9 (1993) 3208. [14] U. Retter, D. Vollhardt, Langmuir 9 (1993) 2478. [15] U. Retter, K. Siegler, D. Vollhardt, Langmuir 12 (1996) 3976. [16] G.L. Gaines, Insoluble Monolayers at Liquid/Gas Interfaces, Wiley– Interscience, New York, 1966. [17] P.D. Keyser, P. Joos, J. Phys. Chem. 88 (1984) 274. [18] R.D. Smith, J.C. Berg, J. Colloid Interf. Sci. 74 (1980) 273. [19] D. Vollhardt, U. Retter, J. Phys. Chem. 95 (1991) 3723. [20] Y.B. Vysotsky, V.S. Bryantsev, D. Vollhardt, R. Miller, V.B. Fainerman, Colloids Surf. A 239 (2004) 135.
Kinetics of slow collapse process: Thermodynamic description of rate constants M. Weis * Faculty of Electrical Engineering and Information Technology SUT, Ilkovicˇova 3, 812 19 Bratislava 1, Slovakia Received 6 February 2006; accepted 14 February 2006 Available online 11 April 2006
Abstract Insoluble monolayer formed at the air/water interface compressed at a surface pressure above the equilibrium spreading pressure is unstable. In presented paper the classical theory for homogeneous nucleation is modified adding the activation energy term. It allows quantitative thermodynamic interpretation of the slow collapse phase transformation. The collapse of stearic acid Langmuir films has been carried out by systematic measurements of the area loss–time isobaric dependencies at various temperatures and isothermal dependencies at various pressures. Activation energies (activation enthalpies) and activation entropies have been evaluated for the nucleation (Ea = 1.32 eV) and the growth (Eb = 1.55 eV) processes. The experimental data for various pressures are discussed on the basis of the Gibbs energy analysis. # 2006 Elsevier B.V. All rights reserved. PACS: 68.18.Fg; 64.70.Nd; 68.35.Rh Keywords: Langmuir–Blodgett films on liquids—Structure: measurements and simulations; Structural transitions in nanoscale materials; Phase transitions and critical phenomena
1. Introduction Monolayers spontaneously formed at the air/water interface represents a two-dimensional system with various potential applications [1,2]. More recently, with increasing interest in the deposition technique on the solid substrate (Langmuir–Blodgett technique), the applications have appeared in the field of physics [3,4], chemistry, or biology. However, the monolayer instability [5] is one of the main problems, which obstructs its practical application. Over the past decades the collapse phenomena in Langmuir films have been a subject of numerous experimental studies in physics or chemistry (e.g. [6]). In general, there are two classes of structural changes in the Langmuir film, transitions between different two-dimensional phases [7], and transitions in which dimensionality changes from the degree of two to three [8,9]. We will now deal only with the latter class of transitions, which is usually termed as a monolayer collapse. Different basic types of monolayer
* Tel.: +421 2 602 91 846; fax: +421 2 654 27 427. E-mail address: [email protected]. 0169-4332/$ – see front matter # 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2006.02.029
collapse can be distinguished according to their different mechanisms. At higher surface pressures the monolayer structure is abolished by fracture collapse, sometimes also called catastrophic collapse. There are several mechanisms, which can likely contribute to the process, but the final effect is easy to observe as a sudden pressure fall in the surface pressure–area isotherm. At lower pressures, the degradation of the monolayer is caused by so-called ‘‘slow collapse’’. Also here the final effect can be realized through various microscopic processes—buckling instability, folding of the monolayer, or the formation and growth of cracks [10–12]. Therefore, various models were proposed for the description of the experiment [13–15]. These theories are in excellent agreement with many direct and indirect experimental results, the choice of the theory usually varies with the used surfactant and the dominated degradation process. 2. Theory Gaines [16] proposed that there might be a monolayer stability limit, which he defined as the pressure of equilibrium between the film and the freshly collapsed material.
1470
M. Weis / Applied Surface Science 253 (2006) 1469–1472
The three-dimensional phase in this case would have a degree of stability lying between that of the monolayer and the equilibrium bulk phase. Degradation of the monolayer in the pressure range between the equilibrium spreading pressure and the fracture pressure was a center of interest in several experimental works (e.g. [17]). Smith and Berg [18] were the first who described the slow collapse mechanism by homogenous nucleation and growth. It is possible to express the change of molecular area (relative area loss) as a dependence on time: A ¼ expðat bt2 Þ A0
(1)
where A0 is the initial area and parameters a and b are characteristics of the nucleation and the growth processes, respectively. A more complex view on the nucleation phenomena was presented by Vollhardt and Retter [19], who considered shape effects of the critical nuclei. Later works showed that on the basis of these nucleation-growth theories important parameters of the classical nucleation theory such as, critical size of nuclei and free energy for the formation of the critical nuclei can be determined [13–15]. In general, parameters a and b are considered only as experimental constants. However, if we describe the slow collapse of the monolayer as a phase transition, it is possible to assign the activation energy Ea to this process:
Ea a ¼ aT;0 exp kB T
kB T DSa DHa exp ¼ exp h R RT (2)
or to express the activation enthalpy DHa and the activation entropy DSa by the Eyring equation, where kB is the Boltzmann constant and h is the Planck constant. A formally similar relation to Eq. (2) can be applied for parameter b. Rate parameters a and b are proportional to the rate of formation of critical nuclei J what can be expressed [18] for the Langmuir film in the form: k2 J ¼ k1 p exp 2 (3) ln ðp=pE Þ
Fig. 1. Area loss–time dependencies at surface pressure of 30 mN/m for various temperatures.
and in analogy a formally similar relation to Eqs. (4) and (5) can be applied for rate parameter b. Constants k and pE are the same for the both rate parameters. 3. Experimental Surface pressure–area isotherms were measured in a Langmuir trough (Type 611, NIMA Technology Ltd., UK). The surface pressure measurements were carried out with a filter-paper Wilhelmy plate. The Langmuir monolayer of stearic acid (purchased from Fluka, Switzerland) was formed by careful casting a 1 mg/ml chloroform solution to the water surface (bidistilled deionized water, 15 MV/cm). The solvent was allowed to evaporate for at least 15 min prior to compressing the surface. Before a collapse measurement the monolayer situated on the surface of water was firstly two times slowly compressed up to the pressure of 10 mN/m and subsequently released for monolayer homogenization. All the chemicals used were of spectroscopic grade purity.
where k1 and k2 are constants and pE is the equilibrium spreading pressure. Therefore, the pressure dependence of the rate parameter can be written: a ¼ ap;0 p exp
k ln2 ðp=pE Þ
(4)
Using Eqs. (2) and (4) the behaviour of the Gibbs energy can be described by the relationship: ap;0 hp k 2 DGa ¼ RT ln kB T ln ðp=pE Þ
(5) Fig. 2. The Arrhenius plot for experimentally obtained parameters a and b.
M. Weis / Applied Surface Science 253 (2006) 1469–1472
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4. Results and discussion
Fig. 3. Area loss–time dependencies at temperature 24 8C for various surface pressures.
Fig. 4. Experimentally obtained rate parameters a and b vs. surface pressure. Solid line represents theoretical results of Eq. (4) for k = 40, ap,0 = 70 s1 and bp,0 = 8 s2. The equilibrium spreading pressure is pE = 5 mN/m [9].
The stability of stearic acid monolayers was studied by monitoring their relative area loss (A/A0) with respect to time at a constant surface pressure of 30 mN/m. The temperatures were varied in the range from 10 to 26 8C. The obtained experimental results are shown in Fig. 1. As shown in [18] for the stearic acid monolayer collapse, the description by Eq. (1) is sufficient. The parabolic part of ln(A/ A0) was fitted by a polynomial of the second degree. The obtained values of parameters a and b from Fig. 1 for each temperature are shown in the Arrhenius plot in Fig. 2. The activation energies (activation enthalpies) of nucleation and growth are Ea = 1.32 eV (DHa = 127.4 kJ/mol) and Eb = 1.55 eV (DHb = 150 kJ/mol), respectively. The difference between the activation energy and enthalpy (DH = E RT) is lesser than the error of experimental data. The activation entropies are positive, DSa = 1.83 eV/K (176.5 J/K mol) and DSb = 2.43 eV/K (234.4 J/K mol), which agrees with theoretical predictions [14]. According to this fact a new three-dimensional
Fig. 5. The Gibbs energy vs. surface pressure. Solid line represents theoretical results of Eq. (5) for k = 40, ap,0 = 70 s1 and bp,0 = 8 s2. The equilibrium spreading pressure is pE = 5 mN/m [9].
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phase is generated, providing a more disordered system in comparison with monolayer. The surface pressure dependencies of slow collapse mechanism of stearic acid monolayers was studied by their relative area loss (A/A0) versus time functions at a constant temperature of 24 8C. The surface pressures were varied from 25 to 30 mN/m. The recorded experimental results are shown in Fig. 3. The values of rate parameters a and b from Fig. 3 for each surface pressure were compared with theoretical results of Eq. (4) for k = 40, ap,0 = 70 s1 and bp,0 = 8 s2 in Fig. 4. The value of the equilibrium spreading pressure was taken pE = 5 mN/m [9]. More general view provides development of the Gibbs energy with respect to surface pressure (Fig. 5), where the values of the Gibbs energy were obtained from Eq. (2). 5. Conclusions The stability of stearic acid monolayers at the air/water interface was evaluated by their area loss on time at a constant surface pressure for various temperatures and at a constant temperature for various surface pressures. The experimental data can be interpreted by the theory of homogenous nucleation and growth [18] as the phase transition between two and threedimensional molecular systems. After initial regular collapse of the monolayer as described by Eq. (1), the dependence ln(A/A0) versus time became more linear. This effect is more pronounced at higher temperatures. This is probably caused by a mutual contact of growing threedimensional aggregates, which results in stopping its growth. Our course of approach in the quantitative interpretation is appropriate to evaluate the activation energies and the Gibbs
energies of nucleation and growth, which (according to [18,20]) control the slow collapse process. Acknowledgement The work was supported by the Slovak grant agency VEGA, project no. 1/0277/03. References [1] M.C. Petty, Thin Solid Films 210/211 (1992) 417. [2] D. Barancˇok, J. Cira´k, P. Tomcˇ´ık, J. Electr. Eng. 45 (1994) 41. [3] D. Barancˇok, J. Cira´k, P. Tomcˇ´ık, J. Vajda, Phys. Stat. Sol. A 169 (1998) 297. [4] J. Cira´k, D. Barancˇok, P. Tomcˇ´ık, J. Vajda, Mater. Sci. Eng. C 8/9 (1999) 13. ˇ ervenˇ, O. Foltı´n, J. Electr. Eng. 47 [5] D. Barancˇok, J. Cira´k, P. Tomcˇ´ık, I. C (1996) 209. [6] L.V.N. Avila, S.M. Saraive, J.F. Oliveira, Colloids Surf. A 154 (1999) 209. [7] D. Vollhardt, V. Fainerman, Colloids Surf. A 176 (2001) 117. [8] D. Vollhardt, Colloids Surf. A 143 (1998) 185. [9] D. Vollhardt, Colloids Surf. A 156 (1999) 79. [10] E. Hatta, D. Suzuki, J. Nagao, Eur. Phys. J. B 11 (1999) 609. [11] A. Gopal, K.Y.C. Lee, J. Phys. Chem. B 105 (2001) 10348. [12] C. Ybert, W. Lu, G. Mo¨ller, C.M. Knobler, J. Phys. Chem. B 106 (2002) 2004. [13] D. Vollhardt, M. Ziller, U. Retter, Langmuir 9 (1993) 3208. [14] U. Retter, D. Vollhardt, Langmuir 9 (1993) 2478. [15] U. Retter, K. Siegler, D. Vollhardt, Langmuir 12 (1996) 3976. [16] G.L. Gaines, Insoluble Monolayers at Liquid/Gas Interfaces, Wiley– Interscience, New York, 1966. [17] P.D. Keyser, P. Joos, J. Phys. Chem. 88 (1984) 274. [18] R.D. Smith, J.C. Berg, J. Colloid Interf. Sci. 74 (1980) 273. [19] D. Vollhardt, U. Retter, J. Phys. Chem. 95 (1991) 3723. [20] Y.B. Vysotsky, V.S. Bryantsev, D. Vollhardt, R. Miller, V.B. Fainerman, Colloids Surf. A 239 (2004) 135.