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Langmuir 2000, 16, 4374-4379
Determining the Contact Angle of a Nonwetting Liquid in Pores by Liquid Intrusion Calorimetry F. Gomez, R. Denoyel,* and J. Rouquerol Centre de Thermodynamique et Microcalorime´ trie, 26 rue du e` me RIA, 13331 Marseille Cedex 03, France Received October 29, 1999. In Final Form: January 21, 2000 A microcalorimetric method is proposed to determine the contact angle of a nonwetting liquid within a mesoporous solid. It is based on the simultaneous measurement of the heat and work of wetting, as the pressure is raised. The method simply requires a previous determination of the surface area to be wetted. It allows one to carry out liquid intrusion porosimetry without any a priori assumption on the contact angle. This method is, in principle, applicable to mercury porosimetry. Its feasibility is shown here for water in porous hydrophobic silicas.
1. Introduction Nonwetting liquid/solid systems (also called “lyophobic”1), especially when the solid is porous, have a number of interesting applications: mercury intrusion porosimetry, reverse phase chromatography, water treatment, and, more recently, energy storage and damping.2 Since the distinct behavior of these systems lies in great part on their large liquid/solid contact angle, it makes sense of trying to determine this angle within the pores themselves. For instance, the contact angle measured for water provides a quantitative way of determining the hydrophobicity of the surface. It is common to determine the contact angle on a flat surface. In the present case, this means compressing and flattening the porous solid, with the expectation that this newly processed external surface is representative of the internal surface of the pores. This may be true in a number of cases but deserves being checked. Immersion calorimetry was proposed to characterize the surface energy of solids 3,4 and even to evaluate the contact angle.5 Nevertheless, for contact angles higher than 90°, one may question about a safe and reproducible wetting on immersion under atmospheric pressure: Spagnolo et al.5 used water/alcohol mixtures of various compositions to overcome this difficulty. This is why we found it interesting to assess the contact angle in the mesopores through the simultaneous measurement of the work and heat of intrusion, following the reasoning given hereafter. We also wish to derive the pore-size distribution of the porous adsorbent (with no assumption on the value of the contact angle, contrary to what is usually done in mercury porosimetry), by using an analysis close to that proposed by Gusev.1 * To whom correspondence should be addressed. E-mail: rdenoy@ ctm.cnrs-mrs.fr. (1) Gusev, V. Y. On Thermodynamics of Permanent Hysteresis in Capillary Lyophobic Systems and Interface Characterization. Langmuir 1994, 10, 235-240. (2) Fadeev, A. Y.; Eroshenko, V. A. Study of Penetration of Water into Hydrophobized Porous Silicas. J. Colloid Interface Sci. 1997, 187, 275-282. (3) Douillard, J. M.; Elwafir, M.; Partyka S. J. Colloid Interface Sci. 1994, 164, 238. (4) Denoyel, R.; Fernandez-Colinas, J.; Grillet, Y.; Rouquerol, J. Langmuir 1993, 9, 515. (5) Spagnolo, D. A.; Maham, Y.; Chuang, K. T. J. Phys. Chem. 1996, 100, 6626.
In the following, we shall successively present the principle of the method, the setup, and the experimental results and we shall end with a comparison with two other methods for the determination of the pore-size distribution (namely, benzene thermoporometry and the BJH analysis of the N2 adsorption-desorption isotherm). The contact angles will also be compared to the values determined on flat surfaces of the same chemical nature and to that calculated by the method of Spagnolo et al.5 since an energy of wetting is also determined in the proposed experiment. 2. Principle of the Determination of the Contact Angle and Pore-Size Distribution Let us consider the general case of a liquid/lyophobic solid system where intrusion and extrusion of the liquid are obtained at constant temperature by a step by step modification of the imposed pressure. The principle of the determination is simply to make use of two independent relationships where the pore size r and the contact angle θ are the two unknowns. The first route directly derives from the Laplace equation, which gives the pressure drop ∆p through a curved fluid/fluid interface whose radii of curvature are R1 and R2:
∆p ) γ
(
)
1 1 + R1 R2
(1)
Here γ is the interfacial tension. Assuming (i) that the interface has the shape of a spherical meniscus of radius R (so that 1/R1 + 1/R2 becomes 2/R), (ii) that it is located in a cylindrical pore of radius r, where the contact angle, larger than π/2, is θ (so that r ) -R cos θ), and (iii) that the porous solid is initially evacuated and that the saturating vapor pressure of the liquid can be neglected (so that the external pressure p applied on intrusion is equal to ∆p), we arrive at the usual Washburn equation:
p)-
2γLG cos θ r
(2)
Provided the mechanical equilibrium is reached, this equation can be applied to a simple intrusion experiment where the pressure p is continuously monitored, together with the intrusion volume Vcum. Since γLG, the surface
10.1021/la9914256 CCC: $19.00 © 2000 American Chemical Society Published on Web 03/28/2000
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tension of the liquid/vapor interface, is known from the tables, the unknowns are r and θ. It is only after assuming a value for θ (as done in conventional mercury porosimetry) that one can derive a pore-size distribution, for instance in the form dV/dr ) fa(r). The second route considers that the isothermal change dUT,V in internal energy of the capillary system (liquid + porous solid) can be split into a bulk and an interfacial contribution and focuses the attention on the second term. Elimination of the bulk contribution is made by calculation of the work of compression of the 3D phases
Wcomp )
∫p dV
(3)
and of the corresponding heats of compression
Qcomp )
∫mh dp
∑(
∂U ∂A
)
T,V
dA
(5)
where ∑ encompasses the various types of interfaces (i.e. solid-vapor, solid-liquid, and liquid-vapor). To get the relation between the interfacial internal energy and the main parameters of the intrusion process (temperature, contact angle, and interfacial tensions), let us consider the equation defining the interfacial tension:
∂F γ) ∂A
∂U ) T,V ∂A
( )
∂S -T T,V ∂A
( )
( )
(6)
T,V
( )
and since S ) -(∂F/∂T)V
T,V
T∂2F ∂γ )γ-T ∂A∂T ∂T
( )
(7)
A,V
We can introduce (7) in (5), and after assuming that the total area of the liquid/vapor interface does not change on intrusion and that only the solid/vapor interface is converted into a solid/liquid interface of identical area (dA ) dASL ) -dASV), eq 5 becomes
[
dU ) γSL - T
( )
( )]
∂γSL ∂γSV - γSV + T ∂T ∂T
dA
(8)
Now, taking into account the Young-Dupre´ equation
γSV ) γSL + γLV cos θ we get
(
dU ) T
∆U ) β(∆Atot)
)
(10)
where β is the term in parentheses in eq 9. This term can be derived from ∆Utot, which is the total internal energy change on wetting (assessed from the measurements of pressure, volume, and heat), and from ∆Atot, which is the total internal area of the pores accessible to the wetting liquid. In case the wetting liquid is water and the adsorbent does not have any microporosity, ∆Atot is conveniently assessed by applying the BET method to nitrogen adsorption. Therefore
β ) ∆Utot/A (BET, N2) Equation 9 therefore simplifies into
dA )
dU β
(11)
whereas the pore radius can be introduced from the variation of dA with dV in cylindrical pores:
2dV 2∆V in a real step by or r ) dA ∆A
(
)
step experiment (12)
∂U ∂S )γ+T ∂A ∂A
)γ-
)
∂γLV ∂ cos θ + TγLV - γLV cos θ dA (9) ∂T ∂T
This equation can be applied along a reversible path. It shows that the interfacial internal energy is proportional to the wetted surface area, provided the interfacial tensions are considered independent of pressure in the pressure range scanned during the intrusion-extrusion process. If we examine the literature in the field, this is a reasonable approximation.6,7 The surface must also be homogeneous; i.e., the contact angle is assumed to be the same everywhere in the solid. Then, between two equilibrium states, one can write
r)
Hence
(∂U ∂A )
(
dU ) T cos θ
(4)
where m is the mass of the bulk phase and h its specific enthalpy of compression, as known from the tables. Preliminary evacuation of the porous solid eliminates any heat of compression of a gas phase. In the frequent case when the heat of compression of the solid is negligible, eq 4 will therefore have to be applied only to the liquid phase. After correcting the experimental work and heat of intrusion from these bulk contributions, we can consider that the remaining change in internal energy only results from an interfacial contribution:
δW + δQ ) dU )
or
∂(γLV cos θ) - γLV cos θ dA ∂T
The consequence is that the simultaneous measurement of heat, pressure, and volume allows to assess a pore-size distribution in terms of wetted surface area as a function of radius, for instance dA/dr ) f(r). This surface area distribution can be transformed into a volume distribution (called fb(r) in the following) by using again eq 12 in order to be compared with the volume distribution directly obtained from the Washburn equation fa(r). The value of the contact angle used in eq 2 is selected (by trail and error) to provide the best fit between the two pore-size distributions. The derivation of eq 10 through the integration of eq 9 is correct only along a reversible transformation. Nevertheless, the integration can be done along an imaginary reversible path between two real equilibrium states. In the case of the lyophobic systems which present permanent hysteresis,2 it is not possible to know which branch of the hysteresis is at equilibrium, if any, but it is possible to use eq 10 between two closure points of the hysteresis, corresponding to equilibrium. This allows us to determine the parameter β without any assumption. Our proposal, following Gusev1 but not discussed in his paper, of applying eq 10 even along a nonreversible path is possible only if (6) Hansen, G.; Denoyel, R.; Hamouda, A. Colloids Surf., A 1999, 154, 353. (7) Jennings, H. Y. J. Colloid Interface Sci. 1967, 34, 323.
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Gomez et al. κ)-
Figure 1. Experimental setup used for intrusion/extrusion experiments. The system can be evacuated before the experiment. Table 1. Physical Properties of the Samples Used particle size/µm
pore width/nm
15
30
PEP300 0.60
10
PEP100 0.71
15
porous vol/cm3‚g-1
specific surface/m2‚g-1 50.5 132
it is assumed that internal surface energy is still proportional to the wetted area with the same coefficient on the chosen branch of the hystereris; i.e., the interfacial tensions are equal to those at equilibrium. In this case the reason for the irreversibility giving rise to hysteresis ought to be found in the mechanism of filling, the pore shape, the connectivity, or the contact angle. Depending on the branch used for the calculation, advancing or receding contact angles are derived. 3. Experimental Section 3.1. Samples. Two commercial hydrophobic porous silicas supplied by Hypersil, namely PEP300 and PEP100, were selected. Both samples are grafted with C18 chains and end-capped using trimethylchlorosilane. The particle size, pore width, and porous volume indicated by the manufacturer are given in Table 1. 3.2. Setup for Water Intrusion and Calorimetry. An experimental setup (Figure 1) was built to determine the work and heat exchanged during experiments of intrusion/extrusion of water in hydrophobic porous solids. It therefore allows one to simultaneously measure, at any time, the quasi-equilibrium pressure (in the 0.1-50 MPa range), the volume intruded, and the corresponding heat exchanged (by means of a Tian-Calvet microcalorimeter). About 0.5 g of solid was introduced into the high-pressure stainless steel cell of the microcalorimeter at 25 °C. The sample was then evacuated (10-2 mbar) during ca. 10 h before water was introduced by means of a HPLC pump (Jasco PU-980). Intrusion was carried out in steps of ca. 20 to 50 µL, after which a duration of ca. 1 h was left for the complete recording of the heat effect. The full intrusion was completed in ca. 20 steps. By adjustment of a calibrated compensation volume, the pressure was then decreased by steps in the system with simultaneous recording of extruded volume, pressure, and heat flow. Raw data had to be corrected from the effects of liquid compressibility. The corrected volume is indeed the intruded volume minus the compression volume of the whole. The corrected heat is obtained by subtracting the heat of compression of the liquid present in the detection volume of the microcalorimeter from the net heat measured. The heat of compression was calculated from the coefficient of compressibility (κ) and the specific heat of compression h of the intrusion liquid which, for water at 25 °C, are8 (8) Handbook of Physical Chemistry, 75th ed.; Lide, D. R., Ed.; CRC Press: London, 1995.
1 ∂ν ν ∂p
( )
T
) 0.457 GPa-1
h ) -78 mJ‚g-1‚MPa-1
These values were also used to validate the calibration of our setup by carrying out blank compression experiments in a cell with no adsorbent. 3.3. Determination of Surface Area and Pore-Size Distribution by Nitrogen Adsorption. Physisorption of nitrogen was carried out at 77 K using an ASAP 2010 (Micromeritics) apparatus. Before adsorption, samples were evacuated 10 h at 150 °C. The measured BET surface areas are reported in Table 1. The mesopore size distribution was calculated by the Barrett, Joyner, and Halenda (BJH) method9 applied to the desorption branch of the adsorption isotherm. The calculation was done by the Micromeritics software which uses the recurrent method and applies the Harkins and Jura equation for the multilayer thickness. 3.4. Thermoporometry. Due to the hydrophobic character of the samples, benzene was chosen as the test liquid for thermoporometry. Samples were evacuated up to 150 °C before their immersion into benzene, in the pan of a differential scanning calorimeter. This was of a low-temperature type (DSC-92 from Setaram) and was used in the 173-280 K temperature range to determine the enthalpy of freezing in the pores. To avoid any supercooling, a first solidification/melting run was carried out and was stopped at the onset of melting of the bulk liquid phase.10 The solidification was then studied at a cooling rate of 1 K‚min-1. This method is based on the relation between the lowering of the triple point temperature of a condensate inside a porous solid and its pore radius.11 It can be shown that
1 1 ) Rn 2σLS
∫
T∆fusSm
T0
νL
dT
Rn ) curvature radius of the liquid-solid interface σLS ) surface tension of the liquid-solid interface T - T0 ) ∆T ) depression of the triple point temperature ∆fusSm ) molar entropy of melting νL ) molar volume of the liquid phase Solidification of the liquid may proceed in two different ways: either from a nucleation occurring inside the pores or by penetration of the external solid phase (“plastic-ice” model10-12) The respective part of these mechanisms is still not well defined, but both of them give rise, during solidification, to a spherical liquid-solid interface (where the solid is the condensate and not the adsorbent) whose curvature is CLS:
CLS )
2 2 ) Rp - t Rn
Here Rp is the pore radius and t is the thickness of a layer of condensate which, on the pore walls, does not experience any phase change in the scanned temperature range. Brun et al.10 developed a comprehensive thermodynamical analysis of the phenomenon which allowed them to derive, for a given liquid, two equations which are in the case of benzene
Rp (nm) ) -
131.6 + 0.54 ∆T
(9) Barrett, E. P.; Joyner, L. G.; Halenda, P. H. J. Am. Chem. Soc. 1951, 73, 373. (10) Quinson, J. F.; Brun, M. In Characterization of porous solids; Unger, K. K., Ed.; Elsevier Science Publishers: Amsterdam, 1988; p 307. (11) Brun, M.; Lallemand, A.; Quinson, J. F.; Eyraud, C. Thermochim. Acta 1977, 21, 59-88. (12) Enu¨stu¨n, B. V.; Gunnink, B. W.; Demirel, T. J. Colloid Interface Sci. 1990, 134, 264.
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Figure 2. Intrusion/extrusion of water in PEP300 at 25 °C: pressure/volume data.
Figure 4. Integral heat measured during intrusion of water into porous sample PEP300, at 25 °C.
Figure 3. Intrusion/extrusion of water in PEP100 at 25 °C: pressure/volume data.
Figure 5. Integral heat measured during intrusion of water into porous sample PEP100, at 25 °C.
which relates the pore radius to the freezing temperature of the liquid, and
Qa (J‚g-1) ) -8.87 × 10-3∆T2 - 1.76∆T - 127 which gives the corresponding heat measured. This heat contains both solidification and interfacial contributions. At a given temperature, on the calorimetric recording, the heat flow P is proportional to the derivative dV/dt of the pore volume in which the solidification has occurred with a specific apparent heat of solidification Qa(T). Therefore
P dt ) FQa dV, where F is the mass density. On the other hand, by differentiating eq 3, we get
dRp )
A d (∆T) (∆T)2
(where A ) -131.6), which we associate with eq 5 to obtain 2
(∆T) dV )k P dRp Qa where k is a constant only depending on experimental parameters.
4. Results and Discussion Experimental results of intrusion/extrusion of water for the two samples PEP300 and PEP100 are shown in Figures 2 and 3. Pressure is plotted vs the volume of water intruded per gram of solid, prior to any correction for fluid compression. For the first sample (PEP300), no extrusion of liquid is observed when the pressure is lowered (Figure 2). Only small volume variations due to liquid compressibility appear for further intrusion/extrusion runs with the same sample. On the contrary, complete extrusion takes place for PEP100, with a hysteresis (Figure 3). The curves are superimposed for all successive intrusion/extrusion runs.
Figure 6. Cumulative distributions of pore volume obtained from experimental intruded volumes (a) and from calculation using work and heat evolved (b), for PEP300.
Nevertheless, the time needed for extrusion is quite long when pressure is lower than 0.3 MPa. An explanation may be the presence of “ink bottle” pores. In both cases, intrusion is an exothermal phenomenon. Integral heats measured during this process are shown in Figures 4 and 5, which take into account the corrections for the compression of the liquid. Using intruded volume, heat, and pressure data, the two distributions fa(r) and fb(r) may be constructed. For this purpose, we have to know the surface area of the solids. BET areas calculated from nitrogen adsorption data correspond to 50.5 and 132 m2‚g-1 for PEP300 and PEP100, respectively. Then, after correction for the compressibility of the liquid, the variation of superficial internal energy is calculated by adding the measured heat of intrusion and the calculated work of intrusion. The values of β are 0.0180 and 0.0147 J‚m-2 for PEP300 and PEP100, respectively. As indicated in the description of the procedure, the value of the solid/liquid contact angle is determined by an iterative calculation where the value providing the best consistency between the two distributions is retained. The corresponding pore volume distributions appear in Figures 6-9. The more striking result is that the two distributions are well superimposed. It
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Langmuir, Vol. 16, No. 9, 2000
Figure 7. Differential distributions of pore volume obtained from experimental intruded volumes (a) and from calculation using work and heat evolved (b), for PEP300.
Figure 8. Cumulative distributions of pore volume obtained from experimental intruded volumes (a) and from calculation using work and heat evolved (b), for PEP100.
Figure 9. Differential distributions of pore volume obtained from experimental intruded volumes (a) and from calculation using work and heat evolved (b), for PEP100.
means that all the porosity can be described with only one value of the contact angle, which is an assessment of the homogeneity required to derive the equations of part 2. For both samples, the calculated contact angle between water and the hydrophobized silica is (126 ( 5)°. This result could indicate that the two samples have the same surface properties, i.e., the same hydrophobic character. This is in good agreement with the chemical treatment experienced by the solids. Both PEP300 and PEP100 correspond to silicas which have been grafted with C18 chains and afterward end-capped. Only their pore size distributions are different. Eroshenko and Fadeev2,13 determined comparable values of the advancing solid/liquid contact angle from intrusion of water in chemically modified porous silicas. In their calculations, they only considered the WashburnLaplace equation. A mean pore radius was independently determined for each silica (from adsorption isotherm of benzene at room temperature). They found that θ varies from 129 to 143° according to the nature of the grafted (13) Eroshenko, V. A.; Fadeev, A. Y. Colloid J. 1995, 57 (4), 446449.
Gomez et al.
Figure 10. Cumulative distributions of pore volumes obtained for the sample PEP300 using intrusion experiment, thermoporometry, and BJH calculations.
Figure 11. Differential distributions of pore volumes obtained for the sample PEP300 using intrusion experiment, thermoporometry, and BJH calculations.
Figure 12. Cumulative distributions of pore volumes obtained for the sample PEP100 using intrusion experiment, thermoporometry, and BJH calculations.
Figure 13. Differential distributions of pore volumes obtained for the sample PEP100 using intrusion experiment, thermoporometry, and BJH calculations.
alkylsilane. On another hand, values determined on grafted flat surfaces by direct observation of drops14-16 (14) Chen, Y. L.; Helm, C. A.; Israelachvili, J. N. J. Phys. Chem. 1991, 95, 10736. (15) Horn, T. J.; Ralston, J.; Smart, R. C. Colloids Surf., A 1995, 97, 183. (16) Silberzan, P.; Le´ger, L.; Benattar, J. J. Langmuir 1991, 7, 1647.
Contact Angle of a Nonwetting Liquid
range between 110 and 135°. We then consider that our hypothesis for the calculation of a solid/liquid contact angle from intrusion measurements are validated. The contact angle calculated by the equation given by Spagnolo et al.5 is
cos θ )
-0.07T - ∆Himm σLV
To apply this equation to the case of our experiment, immersion enthalpy is calculated by adding the immersion enthalpy obtained by the standard method17 (immersion enthalpies of the solids initially under vacuum, which are -25.8 and -17.5 mJ‚m-2 for PEP100 and PEP300, respectively) to the enthalpy variation during the intrusion; the latter is simply equal to the interfacial internal energy variation given by β. One gets -10.5 and 0.5 mJ‚m-2 for PEP 100 and PEP300, respectively. The calculated contact angle are then 98 and 107°, respectively, which are much smaller values than those measured above. Nevertheless, the meaning of the immersion enthalpy in the case of grafted surfaces is not clear. Indeed, starting from vacuum, the immersion of the solid under atmospheric pressure does not lead to the filling of the porosity. The measured value probably corresponds to the adsorption of water molecules on some hydrophilic sites between the grafted layer and the silica surface (for example nonmodified silanols), whereas, during the intrusion, the liquid water wets the grafted layer. If the contact angle is now calculated by considering only the energy β exchanged during the intrusion 123 and 119.5° are obtained for PEP300 and PEP100, respectively. This is in better agreement with the present results. The contact angle given by Spagnolo’s equation is theoretically an equilibrium contact angle, which is, as expected, lower than the advancing contact angle. (17) Partyka, S.; Rouquerol, F.; Rouquerol, J. J. Colloid Surf. Sci. 1979, 68, 21.
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The difference observed between the β values (which are also expected to be equal) can have several origins: The first is an unknown overestimation of the surface wetted during the intrusion experiment. (There is indeed a part of the external surface area which is already wetted when the intrusion is started. This point will be improved in the future by a better control of the heat exchanged during the initial filling of the cell.) The second is an error on the BET surface area. (An error of 10% in the BET surface area leads to 5° error in the contact angle.) Another possibility is that the interfacial internal energy, which is assumed independent of the pressure, actually slightly depends on the pressure. The difference between the D values of PEP100 and PEP300 could then be due to the fact that they are not analyzed in the same pressure range. To confirm the validity of the pore volume distributions obtained from intrusion experiments, we compared these results to nitrogen physisorption and thermoporometry calculations (Figures 10-13). The volume distributions obtained by the three independent methods are in reasonable agreement at least for the total pore volume and the mean pore size, which is also close to that communicated by the manufacturer. It validates the thermodynamic calculation described for the calculation of porous distributions using intrusion experiments and, therefore, the determination of the contact angle. Nevertheless some details in the differential pore size distributions are different. In particular there is a peak in the distribution obtained by water intrusion, which does not appear with the other methods in the case of PEP100. This could show a difference in filling mechanism between the various approaches. Acknowledgment. The authors are indebted to V. Eroshenko and J.F. Gobin and F. Rouquerol for fruitful and inspiring discussions, and they acknowledge financial support from the French ministries of Defense and of Education, Research, and Technology (Contract DUAL No. 98B0312). LA9914256
