8 CHARACTERIZATION OF PAPER PROPERTIES
8.3 Porosity
Figure 7.46. Gas adsorption isotherms according to the IUPAC system. (Taken from ref. (64). Reprinted with permission from Elsevier Science)
standard means by which to determine the surface area of rough and porous solids, and is commonly used in the paper industry to investigate smaller pores in paper coatings, and also the intra-fibre pores in the base sheet.
Performing a BET surface area measurement involves determination of the adsorption isotherm, and then the fitting of the theory to the linear (multilayer adsorp-tion) part of the isotherm by using an established pro-cedure (65). The measurement principle itself can be manometric (gas pressure measurement), gravimetric (measurement of mass change of the sample), or mea-surement of gas flow. An inert gas (e.g. He, N2 or Ar) is commonly used at a fixed temperature (usually 77 K).
This is often done discontinuously. For example, in the gravimetric method, the pressure is ramped by intro-duction of the gas, equilibrium obtained at the required p, and the mass change recorded. The specific BET surface area a (BET) is then given through the relation-ship between the monolayer capacity nm (the amount required to cover the surface with a monolayer), which appears in the theoretical BET isotherm and is deter-mined by fitting to the measurement, and the average area per molecule a:
a (BET) = nmNAcr (7.49) where NA is the Avogadro number.
Gas adsorption can also be used in systems with mesopores to measure pore size distributions. In this range of pore sizes, the surface energy of the pore walls causes a condensation of the gas (usually N2 in practice) at pressures where it would remain in the gas state if not confined; an interface forms with surface tension y and the reduced vapour pressure on the convex side of the meniscus, as expected from the Laplace equation, explains the condensation at equilibrium. Equating the
chemical potentials of the gas and liquid phases, and application of Laplace's equation:
Ap =
_2ycosl
( 7 5 0 )rp
where Ap is the pressure difference across the interface, 0 is the contact angle of the liquid-vapour-wall, and rp is the radius of the pore wall, leads to the Kelvin equation, as follows:
l n- = - ^ ( 7-5 1 )
Po rKRT
where r& is the mean radius of curvature of the condensate, vl is the molar volume of liquid, and the contact angle is now assumed to be 0.
As stated above, a Type IV isotherm is indicative of capillary condensation and provides a means of deter-mining the pore size distribution of mesoporous materi-als through gas adsorption. The hysteresis often seen in these isotherms is due to the establishment of metastable states, more commonly a feature of adsorption rather than desorption. Moreover, on desorption from a full pore, the inner layers of the condensed phase tend to be removed first, so that the contact angle 6 remains 0. In the Barrett, Joyner and Halenda (BJH) method, a stepwise procedure is used to determine the pore size distribution by following the desorption branch of the isotherm, whereby the differential volume of removed gas at each pressure step is related to the differential volume of pores emptied in that step, through the Kelvin equation. Consideration has to be given to the fact that not only is the condensate being removed from pores above a certain size, determined by the Kelvin equation for a given pressure, but also layers of adsorbed gas are being stripped away from pores emptied of liquid, at the same time (as is done in a BET experiment). Mea-surement of the amount of nitrogen desorbed at each reduction in pressure gives the pore size distribution.
Some subtleties must be dealt with in connection with the assumption of pore shape and the range of validity of the Kelvin equation (65). The theoretical and practical complications for the interpretation of adsorption to both micro- and macroporous systems effectively restrict the method to the mesoporous regime.
8.3.2 Mercury porosimetry
As stated above, for many systems of interest, including paper coatings, the majority of pores lie somewhere near to or in the macroporous regime, where gas adsorption Relative pressure
Amount adsorbed
becomes less effective. The technique commonly used to investigate these larger pores is mercury porosimetry.
For a non-wetting fluid (90° < 6 < 180°) like mer-cury, the Laplace equation predicts that the liquid will recede from the pore. Hence, Hg must be forced hydraulically into the pore space. The principle of mer-cury porosimetry uses this fact to provide a simple means to determine pore size distributions: The amount of non-wetting liquid that intrudes into pores as a func-tion of applied pressure is monitored. Equafunc-tion (7.50) provides a relationship between the pressure and the size of the smallest pore intruded. Some artefacts have to be considered, such as ink-bottle effects, where pores with small opening "necks" fill only when the minimum pressure is sufficient to force the liquid past the open-ing. Moreover, at the lower end of the mesopore regime, very large pressures are required (207 MPa at 7.2 nm) to access the pores, thus causing deformation of the sample which must be taken into account. However, the method has proven accurate (and popular) in the measurement of pore size distributions of larger mesoporous, and macro-porous, systems. By using Fick's law to describe fluid diffusion through cylindrical paths, it is even possible to deduce the tortuosity of the pore network, which is a measure of the deviation of diffusion paths away from aligned straight cylinders.
Modern mercury porosimeters can reach pressures up to ~ 400 MPa, and perform intrusion and extrusion measurements automatically. This is carried out in either a step wise or scanning mode. Recently, there has been a move towards the scanning mode method, as the step-ping mode approach has several drawbacks, including the possibility of missing important information dur-ing the filldur-ing of smaller mesopores. Other issues of
importance are accurate determination of the contact angle of mercury on solid surfaces, the use of mercury of high purity, and the control of deformation effects by running scans on a non-porous sample of similar com-pressibility and comparing the results to those obtained from analysis on the real sample. The technique has been used on a wide variety of porous materials, from porous rocks to paper. An example for a comparative measurement on several ink-jet coating grades is given in Figure 7.47 (66).
Thermoporosimetry (cryoporometry)
Thermoporosimetry (67-69) is another technique where, similarly to gas adsorption, a capillary-induced phase transition is exploited as a pore characterization method.
However, in this case, it is the liquid-to-solid, rather than vapour-to-liquid, transition that forms the basis of the method. As described by Gibbs and Thompson (Kelvin) over a century ago, the melting (and freezing) temper-ature of a crystalline solid in a pore is lowered relative to the bulk melting temperature, due to the limitation on crystal size and the large surface-to-volume ratio in a con-fined region. The Gibbs-Thompson equation describing the melting temperature suppression in a porous mate-rial can be derived by applying the Laplace equation at the solid-liquid interface and equilibrating the chemical potentials of the two phases, thus leading to the suppres-sion in the melting temperature of magnitude ATm:
ATm = Tm- Tm(rp) = - ^ - = - (7.52) where Tm is the melting temperature in the bulk, Tm(rp) is the melting temperature in a pore of radius rp, AHf is Pore diameter/^m
Figure 7.47. Comparison of pore size distributions obtained by using Hg porosimetry. (Taken from ref. (66))
Log differential intrusion/ml g 1
the bulk enthalpy of fusion of the solid-liquid transition, and p is the density of the solid. The physical constants for a given material can therefore be grouped as k. In practice, performing experiments during melting rather than freezing is preferable, as supercooling and contact angle effects are avoided.
The thermoporosimetry measurement exploits the Gibbs-Thompson phenomenon as follows: A sample is saturated with a wetting liquid (usually purified water or cyclohexane, depending on the material), and the sample quenched to well below the bulk melting temperature, such that liquid in pores of all feasible sizes is frozen. (A temperature of — 60°C is sufficient to ensure this for water.) Equation (7.52) in fact predicts that ice in successively larger pores will be melted as the temperature is increased. Thus, to obtain the pore size distribution, all that is needed is a probe that can either distinguish between the relative amounts of liquid and solid in the pores, or monitor the progressive transition between the two phases. Nuclear magnetic resonance (NMR) spectroscopy can be used in the former case, where the two widely different spin-spin relaxation times of the liquid (long) and the solid (short) are used to determine the liquid and solid fractions of the substance imbibed in a porous sample. Alternatively, differential scanning calorimetry (DSC) can be used to monitor the progressive phase transition as the temperature is continuously raised.
The technique has proven successful in measuring the pore size distribution in the mesopore regime, for a variety of model (67-69) and practical (67) systems.
An example of a measurement on a glossy coated paper is given in Figure 7.48 (67). For pores with rp >
50 nm, instrument resolution, temperature control and
defect formation in the pore ice restrict the usefulness of the technique. Its advantage for mesopores over mercury porosimetry is in significantly reduced material deformation, whereas the detection method, especially in the NMR implementation, is more direct and less model-based than the procedure for gas desorption. However, issues regarding the pore damage caused by the density changes during the freezing and melting of the probe fluid must be borne in mind.
8.3.3 Microscopy
The methods described above are generically intrusive:
they require the invasion of the pore space by either a gas or liquid. Non-intrusive methods for studying pore structure are also commonly used, with microscopy being the main such method. With the lengths of interest often residing at the nanometre to micron scale, it is not surprising that scanning electron microscopy (SEM) has been widely used to image the internal pore structure of paper and paper coatings. A concise review is given in ref. (70).
The sample preparation procedure for SEM imaging of paper involves the embedding of small cuttings (e.g. cm width) of sheets into epoxy resin and either microtoming, or as has recently been introduced (70), polishing and chemical removal of a thin (urn) resin layer. These are then mounted in the SEM unit. Both secondary and back-scattered electron images can be used to investigate the cross-sections. In secondary electron mode, SEM relies on differences is surface topography to provide images, and hence the new trend towards exposure of the section surface by removal of the resin. Computer-based image analysis can then be applied to identify the distribution of different components (fibres, fillers, pigments and binders) and to quantitatively evaluate the distribution of pores. An example of the SEM imaging of paper cross-sections is shown in Figure 7.49. Advantages of this approach chiefly centre on the ability to identify the transverse distribution of individual components and to obtain a direct image of the sub-surface environment, including the pore structure. The disadvantages lie mainly in the fact that one is attempting to construct a three-dimensional overview of the sample from a series of two-dimensional images. This is a classic issue in microscopy, tomography and imaging. It is hoped that new methods, such as confocal microscopy and X-ray tomography, will provide more direct means of imaging the sub-surface zone of paper and coating layers.
Pore size/nm
Figure 7.48. Pore size distribution of pores up to 80 nm in diameter of a glossy coated paper. (Taken from ref. (67))
Relative abundance