Novel concepts for characterisation of heterogeneous particulate surfaces

Applied Surface Science 196 (2002) 30–40

Novel concepts for characterisation of heterogeneous particulate surfaces Wolfgang Peukert*, Carsten Mehler, Martin Go¨tzinger Institute of Particle Technology, Technische Universita¨t Mu¨nchen, Boltzmannstrasse 15, 85748 Garching, Mu¨nchen, Germany

Abstract Surface properties of fine particles are of high relevance in several technical applications, such as nanotechnology, adsorption or crystallisation. Particulate surfaces are not well-defined, may contain flaws and dislocations, internal stresses and all kinds of impurities. Thus, these surfaces are structurally and energetically heterogeneous. Their interactions are, therefore, poorly understood. Adsorption equilibria, adhesion forces or contact angles reflect the interaction of molecules and particles with surfaces and fluid phases. Modelling of these elementary processes leads to mean or distributed surface properties of technical surfaces and small particles. The dispersive interactions of heterogeneous surfaces are characterised by mean parameters like a Hamaker constant. Henry coefficients of gases on activated carbon, graphite and alumina have been quantitatively described as a function of the critical temperature and pressure of the respective adsorptive and the Hamaker constant of the solid. Conversely, Hamaker constants determined from adsorption experiments are used to predict measured adhesion forces of alumina particles. Polar interactions are described by a new approach derived from continuum models of liquids. The particle surface is characterised by a density distribution of partial charges on small surface patches. By means of a statistical mechanical approach Henry coefficients both in the gas and liquid phase can be described quantitatively. # 2002 Published by Elsevier Science B.V. Keywords: Particle surfaces; Characterisation; Adsorption; Adhesion

1. Introduction Interactions of particles with gaseous, liquid or solid phases are of high relevance in many applications. Gas or water cleaning by means of adsorption, catalysis, crystallisation, interaction of biomolecules with organic or inorganic interfaces, agglomeration, dispersion technology, as well as nanotechnology are examples where surface properties of fine particles are important. The underlying fundamental physical–chemical processes of adsorption, wetting and adhesion *

Corresponding author. Tel.: þ4989-289-15652; fax: þ49-89289-15674. E-mail address: [email protected] (W. Peukert).

are poorly understood. The methods of surface physics can often not be applied to technical solids since these methods need well-defined crystalline surfaces and work often under UHV conditions. Technical surfaces are not well-defined, may contain flaws and dislocations, internal stresses and all kinds of impurities. Adsorption equilibria can be described by a vast number of different isotherm equations [1]. Usually, the adsorption equilibrium of a specific component has to be measured first and then the parameters of a certain adsorption isotherm are fitted to the experimental data. This approach is being followed both in the gas and liquid phases. In most adsorption isotherm equations, the influence of the solid does only appear implicitly in model parameters. This holds true both for relatively

0169-4332/02/$ – see front matter # 2002 Published by Elsevier Science B.V. PII: S 0 1 6 9 - 4 3 3 2 ( 0 2 ) 0 0 0 4 0 - 5

W. Peukert et al. / Applied Surface Science 196 (2002) 30–40

Nomenclature ai A b0 c Ci d fi(s) He Heaqu,i k Mi ~ M N pc p(s) R SBET T Tc z

regression parameter of the ith moment ˚ 2i mol1 ei) (J A Hamaker constant (J) constant (J mol1) concentration (mol l1) van der Waals parameter (J m6) atomic diameter (m) profile-function of the ith moment ˚ 2i) (ei A Henry coefficient (mol m2 Pa1) Henry coefficient from aqueous solutions (m3 kg1) Boltzmann constant (J m6) ˚ 22i) ith moment (ei A molecular mass (g mol1) number of patches critical pressure ( 105 Pa) frequency distribution of charges ˚ 4 e1) (A gas constant (J K1 mol1) BET surface area (m2 g1) temperature (K) critical temperature (K) surface distance (m)

Greek letters g surface tension (J m2) D distance between two crystal layers (m) e dielectric constant (C2 N1 m2) y coverage r density (m3) ˚ 2) s charge (e A j surface potential (J) simple adsorption isotherms like Langmuir or BET models but also for more sophisticated models based on potential energy approaches. Even the most advanced theories using Monte-Carlo methods, molecular dynamics or density functional theory do mostly include adjusted adsorbate–adsorptive interactions as mixed parameters. This paper shows methods how parameters of the solid phase can be determined and used to describe adsorption equilibria both in the gas and liquid phases by using material properties of the adsorbent. In addition, these parameters are used to

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predict also particle adhesion establishing a connection between adsorption and adhesion. The surface region is in contrast to the well-known Gibbs dividing surface extending a few atomic layers into the solid. This region is characterised by a deformation of the lattice parameters (surface relaxation and/or surface reconstruction) as well as by a deformation of the electronic band structure even without impurities, defects and/or adsorbate layers being present. Since the atomic composition and atomic structure is generally not known, it is not possible to calculate the electronic structure and bond energies using the methods of computational quantum chemistry and physics. In order to characterise these technical surfaces, as shown in Fig. 1, mean or distributed properties have to be used. Mean properties might be a rms value of surface roughness surface or a mean surface energy. By probing the particle surface by means of adsorption, wetting or adhesion measurements we get information on the properties of the solid surface which is relatively close to it’s state in a technical process. In an adsorption experiment, for instance, the adsorptive acts as probe molecule. By measuring the adsorption equilibrium at low concentrations in the Henry regime, i.e. by neglecting lateral interactions between the adsorbing molecules, we are able to extract information on the surface properties by adequate statistical modelling. The obtained data are sampled over all particles and represent, therefore, mean surface properties of the particle ensemble. This mean property, e.g. surface energy or roughness may be either described by one single parameter or by a distribution function. Of course, models using distribution functions contain much more information than the one parameter models. The obtained surface properties can be understood as equivalent surface properties. Equivalent surface properties of non-ideal surfaces are determined by means of a physical model of a well-defined surface. The non-ideal surface is then characterised by the model surface with similar properties. Equivalent properties are well-known in particle technology, e.g. for particle size characterisation [2]. Since it is extremely difficult to describe the size of a real nonspherical particle, say a grain of sand, the concept of equivalent diameters is used relating the diameter of the real particle under investigation to the diameter of a reference particle, for instance, a sphere with the

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W. Peukert et al. / Applied Surface Science 196 (2002) 30–40

Fig. 1. Structure of surface science.

same settling velocity (and the same density). For graphite, for instance, the model surface may consist of ideal layers of carbon atoms (see Section 3.1). The equivalent surface energy of a heterogeneous carbon surface shows the same physical behaviour as the well-defined model surface. Other equivalent properties may be surface roughness, surface charge or group density. These equivalent properties can be single values or distributions of physical parameters. The introduction of equivalent mean or distributed quantities simplifies a complex physical situation with many dependent variables in order to enable a mathematical treatment of the problem.

has a BET surface of 123 m2 g1 and a density of 3.41 g cm3. The a-alumina was produced by heating aluminium hydroxide (hydrargillite, Merck) to 700 8C for 5 h and subsequently heating to 1200 8C for 7 h. This alumina has a BET surface of 8.14 m2 g1 and a density of 3.98 g cm3. Third, the fumed Alu C, was heated to 1200 8C for 4 h, dissolved in pure water, dried and finally heated to 250 8C for 4 h. This sample has a BET surface of 75 m2 g1 and a density of 3.42 g cm3. The crystal structures were obtained from XRD using Debye Scherrer method and a Philips PW 1730/1316/19 unit, BET surface was measured with a volumetric sorption analyser Nova 2000, Quantachrome and density was determined using a He-pycnometer AccuPyc 1330, Micromeretics.

2. Experimental 2.2. Procedure 2.1. Adsorption in the gas phase 2.1.1. Materials The gaseous probe molecules (argon 4.6, nitrogen 5.0, methane 5.0 and propane 3.6) were obtained from Messer Griesheim GmbH, Germany. Different alumina adsorbents were characterised in the gas phase. The commercial alumina ‘‘aluminium oxide 90 active, neutral’’, obtained from Merck Eurolab GmbH, Darmstadt, Germany, a home-made a-alumina and a modified d-alumina (Alu C), Degussa (purity 99.6%) Germany, were used. The amorphous Merck-alumina

Single component adsorption equilibria of gases were measured by a gravimetric method. The amount of adsorbed gas or vapour was determined from the variation of the sample weight due to rising pressure in a sealed volume. The so-called static method was applied, this means that pressure was increased stepwise by opening and closing an inlet valve. After reaching equilibrium, the new sample weight was recorded. To correct buoyancy effects, the sample volume was determined by the Helium method. Essential part of the adsorption setup is a Satorius ultramicro

W. Peukert et al. / Applied Surface Science 196 (2002) 30–40

balance (Satorius GmbH, type 7014, Go¨ ttingen, Germany), an accuracy of 1 mg can be obtained in the measuring range of 0–100 mg and in the range of 0–10 mg, an accuracy of 0.2 mg is possible. This high resolution is achieved by the electromagnetic force compensation method. In the sealed balance housing, wires are hooked to the suspension of a balance beam on both sides, a pan with the sample hangs on one wire, a reference weight on the other one. To reduce thermal influences on the results, the upper housing is thermostatted to 302 K. The possible pressures range is from <104 mbar to 150 bar. For sample preparation, the upper balance housing is flushed overnight with helium for 12 h, while the lower housing containing the sample is heated to a given preparatory temperature. After that, the pressure is reduced to 50 mbar using a turbo molecular pump and a throttled valve. Then the lower housing is thermostatted to the given temperature after 15 min, the housing is evacuated completely for 2 h to a pressure smaller than 104 mbar. To measure an adsorption isotherm, the pressure is raised stepwise by an inlet valve. After reaching thermal equilibrium, weight and pressure are recorded. Thermal equilibrium is defined as constant weight for 3 min. Argon, nitrogen, methane, propane and pentane are used as probe gases. The smaller the used molecules, the faster equilibrium is reached, for instance argon needs 15 min, propane 1 h. 2.3. Adsorption from aqueous solutions 2.3.1. Materials All chemicals used were obtained by Merck and were high-purity grade reagents with a stated purity greater than 99.9%. The activated carbon C 40/1 from CarboTech with a measured BET surface of 1500 m2 g1 was used as adsorbent. To remove adsorbed compounds

33

from the surface, the activated carbon was conditioned at T ¼ 200 8C for 7 days. The water was cleaned by molecular sieves and reverse osmosis and had a conductivity of 0.05 mS cm1 and a TOC content <30 ppb. 2.3.2. Procedure Erlenmeyer flasks (V ¼ 120 ml) were filled with solutions of different initial concentrations and a constant amount of adsorbent. The concentration range and the mass of adsorbent were dependent of the equilibrium constant, which was estimated either by literature data or by the size of the adsorptive molecule. To remove volatile components from the glass surface, Erlenmeyer flasks were cleaned before the experiment with pure water and regenerated at 200 8C for 2 days. The adsorptives were cooled down before mixing with water to reduce the errors by evaporation. Equilibrium was established within 4 days in an incubator shaker at T ¼ 25 8C. To calculate the adsorbed amount, the concentration in equilibrium was measured. As the concentration of adsorptive was below the detection limit of analysis by GC or HPLC, the components have been enriched before the analysis with the solid phase micro-extraction (SMPE) method. For the experiments, the AS 8200 (SPME) from Varian was used. The schematic principle is shown in Fig. 2. The solution (1.2 ml) is filled by a syringe in the sample vessel and a specially-coated fibre is inserted. The enrichment is caused by adsorption from the solution on the fibre. To accelerate the establishment of the equilibrium the fibre is agitated; the adsorption time was 28 min. As various fibres are available the specific analysis of different groups of substances is possible. After the adsorption the fibre is removed from the solution and inserted into the GC injector. During the injection time of 2 min, the adsorbed components are thermally desorbed and can be analysed by the GC (CP 9000 from Chrompack). Due to the complete removal

Fig. 2. Schematic principle of the SPME.

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W. Peukert et al. / Applied Surface Science 196 (2002) 30–40

of all volatile compounds during the injection, the fibre is also regenerated and can be used immediately for the next analysis. By use of an autosampler constant conditions and an automation were realised. The detection limit of SPME is about 10–100 ppb. Due to the automation and the fast analysis, the measurement of an isotherm can be realised within 2 days. The concentration is determined by the use of calibration curves which have to be measured before the analysis. The analysis is reproducible with an accuracy better than 10–15%, depending on the analysed component. 2.4. Atomic force microscope (AFM) For AFM measurements, a commercial AFM (Nanoscope IIIa, Digital Instruments, Veeco, Santa Babara, USA) was used. Sample scans in contact mode were performed with standard V-shaped silicon nitride cantilevers (Olympus, OTR8), the same cantilevers were used for particle adhesion measurements. For this task, spherical alumina particles were glued to the cantilevers. The adhesion force can be obtained from the necessary deflection of the cantilever to pull off the particle from the surface and the force constant of the cantilever, applying Hook’s law.

3. Results and discussion 3.1. Characterisation of particle surface properties by means of one parameter models So far, models for adsorption equilibria do not include measured or measurable properties of the solid. Maurer and co-workers [3,4], however, have shown that Henry coefficients for adsorption of various gaseous components on activated carbon and on graphite can be described as a function of the critical temperature (Tc) and critical pressure (pc) of the adsorptives and the Hamaker constant of the solid. The term Tc/p0:5 c is a measure of the van der Waals interaction parameter of the gaseous components. The Henry coefficients can be described as a function of an interaction potential (ji,S) according to Z zmax    ji;S  1 Hei ¼ exp   1 dz (1) RT zmin kT

where z is the distance from the solid surface, T the temperature, R the gas constant and k the Boltzmann factor. The interaction potential depends on the nature of the solid, particle density and on the Hamaker constant of the solid. For activated carbon, a dusty gas approach was used describing the solid by a cloud of carbon particles. Graphite was modelled by ideal carbon layers. Interestingly, the Hamaker constants were determined by contact angle measurements of carbon fibres. These fibres were fairly smooth so that the contact angle hysteresis was not unacceptable high. This mean field model describes the experimental data within a range of 50%, which is surprisingly good taking into account that different solids are characterised by only one mean Hamaker constant for activated carbon (A ¼ 6:0  1020 J) and graphite (A ¼ 23:8  1020 J), respectively. The experimental data cover seven orders of magnitude. A better accuracy is achieved when the Hamaker constants of the individual solids are used. Taking only the measured data from Maurer into account, the accuracy can be improved for activated carbon at 300 K by a factor of 3 to 16%, for instance. By measuring the adsorption equilibrium, it is possible to deduce Hamaker constants and surface energies by self-consistently solving Eq. (1) with an appropriate interaction potential derived for a model surface. The Hamaker constants represent the dispersive interaction of the surface layers and not necessarily the Hamaker constant of the bulk solid. They include all dispersive contributions of the surface including impurities. Thus, we obtain equivalent Hamaker constants for an appropriately defined model surface (i.e. a carbon particle for activated carbon, carbon sheets for graphite and layers of oxygen and aluminium atoms for alumina). For alumina, the model surface shown in Fig. 3 was chosen, neglecting different crystalline faces. Without any problems, this approach can be extended by including all relevant

Fig. 3. Model [0 0 0 1]—surface of alumina, oxgen-terminated.

W. Peukert et al. / Applied Surface Science 196 (2002) 30–40

crystalline surfaces. It is shown, however, that our simplified method is quite sufficient. The dispersive interactions (approximated by a Sutherland potential, j / z6 ) decline rapidly with the distance z. The potential energy function for a molecule above a spherical adsorbent particle is given for activated carbon by ji;S ¼ 

4pCi;S rS 3ðdi;S þ zÞ3

(2)

The interaction potential for a molecule at distance z of a flat surfaces (carbon, alumina) is given by ji;S ðzÞ ¼

1 X g¼0



prS;2D Ci;S 2ðdi;S þ gD þ zÞ4

where g is the number of the layer, D the distance between two layers and rS,2D the two-dimensional density of a layer. In case of alumina, mean values for these parameters were used. The parameter Ci,S is a mean parameter for oxygen and aluminium ion. The Hamaker constant is calculated from the dispersive interaction parameter Ci,S between gas (i) and solid (S) according to A ¼ AS;S0 ¼ p2 r2S CS;S0

(3)

and from the well-known mixing rules of Bertholet. It follows after some rearrangements pffiffiffiffiffi!2 pc 16rS Ci;S 3 3 A ¼ 32p dS (4) 3 T 9kðdi þ dS Þ c

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tures, the Hamaker constant can be predicted with a correlation coefficient of R2 ¼ 0:999, to 6:32 1020 J. The investigations of Maurer indicate, therefore, that surface properties determined from contact angle measurements (A ¼ 5:26:2 1020 J) can be used for the prediction of adsorption equilibria and adhesive interactions, at least in the case of carbon materials. However, these data were taken mostly from literature for different solids. Therefore, Go¨ tzinger [5] is studying adsorption and adhesion for alumina. Different alumina samples are analysed with one probe molecule (argon) and one solid material (a-Al2O3) has been studied with different probe molecules. Fig. 4 shows the adsorption of argon on different alumina samples in the Henry range. Before adsorption measurement, all samples were pre-treated identically according to the procedure given in Section 2. It is clearly shown that different alumina samples show significantly different adsorption behaviour even for argon which interacts only via non-specific van der Waals forces. It is important to notice that the alumina surface is covered with OH-groups and possibly with adsorbed molecules like water, carbon dioxide and hydrocarbons at ambient conditions. Different thermal treatment leads to different surface properties even for the same solid. Table 1 shows the influence of the heat treatment of the solid which influences the hydroxylgroup density on the surface and thus, the adsorption equilibrium. The hydroxyl-group densities on a-alumina as a function of the thermal history were reported

where di and dS are the molecular diameters of the gas and solid molecules, respectively. The Hamaker constants can be converted in surface energies gdS using Eq. (5). gdS ¼

A 24pz20

(5)

where z0 is the contact distance estimated to z0 ¼ 0:165 nm according to Israelachvili [15]. Hamaker constants were calculated for 17 different activated carbons. They differ by about 50% between 5:0  1020 and 7:8  1020 J. The differences are caused by surface heterogeneity and by the pre-treatment of the solids. With the use of the experimentally determined Henry’s law coefficients of N2, CH4 and CHF3 on Lurgi Supersorbon WS at different tempera-

Fig. 4. Adsorption isotherms of argon on different type of alumina, measured at 296 K. The samples were degassed at a temperature of 100 8C for 12 h.

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W. Peukert et al. / Applied Surface Science 196 (2002) 30–40

Table 1 Calculated Hamaker constants from adsorption data as function of preparatory degassing temperaturea Degassing temperature (8C)

OH-group density (y)

Hamaker constant ( 1020 J)

100 250 450

1.00 0.68 0.32

13.4  0.19 14.5  0.23 15.2  0.24

a BET surface area ¼ 8:1 m2 g1; a-alumina; adsorption temperature T ¼ 294 K.

by Wagner [13]. On a surface with low hydroxyl-group density, a value for the Hamaker constant of 15:2  1020 J can be calculated for the a-alumina being in excellent agreement with the Hamaker constant calculated from Lifshitz theory (A ¼ 1516 1020 J) [14].

The Hamaker constants of the other samples at 100 8C are A ¼ 10:61020 J for d-alumina and A ¼ 11:9  1020 J for the amorphous alumina. Besides argon, other probe molecules were used to characterise a-alumina. Altogether, Ar, N2, CH4, C3H8 and C5H12 were used. These data indicate that experimental results of the adsorption of non-polar molecules are described satisfactorily by Eq. (1). The measured Henry coefficients on a-alumina can best be described by a mean Hamaker constant of A ¼ 11 1020 J for 100 8C pre-treatment (Fig. 5; LHS). However, the Henry coefficients for C5H12 are leading to consistently too small Hamaker values which may be due to the assumption that all adsorptives (including pentane) are modelled as spheres. Neglecting the value for pentane the Hamaker constants in Table 1 have been

Fig. 5. Calculated Hamaker constant from alumina adsorption data and measured adhesion force of alumina spheres.

W. Peukert et al. / Applied Surface Science 196 (2002) 30–40

obtained. Further informations can be obtained from the comparison between the adsorption equilibria between individual adsorptives reflecting additional interactions. For instance, the somewhat larger Henry coefficient of N2 in comparison to Ar indicates an influence of the quadrupole moment of N2. Another way to probe unspecific van der Waals interactions is the adhesion of particles. An AFM is used for the adhesion measurements which are complemented by thorough characterisation of the roughness of alumina substrates and particles. The adhesion force is plotted as a function of the particle radius which was glued to the cantilever of the AFM (Fig. 5, RHS). The measured forces at 20 8C in pure nitrogen can be described by a Hamaker constant of 5:3 1020 J for interactions of alumina across adsorbed water layers [5]. Although the data are consistent, further investigations are necessary in order to elucidate the influence of other parameters like the mechanical behaviour (deformation) of the materials, the formation of liquid bridges and surface chemistry. Details of the evaluation will be published elsewhere. 3.2. Characterisation of particle surface energy by means of distribution functions Adsorption equilibria can also be used to derive distributed surface energies. The adsorption isotherm can be written as the following Fredholm integral equation [6,7]: Z jmax q¼ qloc f ðjÞ dj (6) jmin

where q is the measured loading on the solid surface (global isotherm), qloc the local adsorption isotherm

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on adsorption sites with interaction energy j with the adsorptive and f(j) the distribution function of the interaction of the adsorptive with the surface. The solution depends on the functional form of qloc which is not a priori known. In the literature, various functionals have been used, a common one is the Langmuir isotherm, for instance. The function f(j) contains the interaction energy which depends both on the solid and on the respective adsorbent. It is thus not representing a ‘‘pure’’ property of the solid surface. The following approach which is derived from continuum models for liquids overcomes this disadvantage leading to distributed surface properties. A molecule (or a particle) is embedded in a continuous ideal conductor. The particle surface is divided into small patches. Based on density functional theory, the electron density distribution on the surface can be calculated if the molecular conformation is known [8,9] (Fig. 6). As a boundary condition an ideal conductor is used. This simplification allows a faster solution of the underlying Poisson equation. The error scales with ðe  1Þ=ðe þ 0:5Þ which is acceptable for water, e is the dielectric constant which is e 78 for water. For solids with unknown surface composition and structure this approach is, of course, not feasible on the basis of quantum mechanics. However, knowing equilibrium data, it is possible to extract the required distributed surface properties by solving the inverse problem. By this procedure, we are again defining an equivalent surface property. Thus, distribution functions of the partial charge density p(s) or of the surface energy distribution of the patches may be obtained which reflect the possibility of a molecule or a particle to interact with other molecules. It may interpreted as an equivalent partial surface charge in an ideal solvent.

Fig. 6. General idea of the continuum model, example: vanillin molecule.

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W. Peukert et al. / Applied Surface Science 196 (2002) 30–40

This idea which was originally developed for molecules in solvents and is now applied to macroscopic particles being in interaction with molecules in the liquid and gas phases. In a first approach to test the model, we calculated Henry coefficients in dependence of a series of so-called s-moments Mi [10,11]: kT ln Heaqu;i ¼ Mi ¼ N

Z

2 X

ai M i þ b0

i¼2 þ1

fi ðsÞpðsÞ ds

(7)

calculated Henry coefficients both in the gas phase and in the liquid phase is quite good [11,12], indicating that the continuum model is a promising method (Fig. 7). In order to extract surface parameters, the following equation has to be solved:

Heaqu;i 1 ¼ ln ~ S rS kT M  Z  CS;S0 þ px ðsÞðmS0 ðsÞmS ðsÞÞ ds

1

(8)

with fi ðsÞ ¼ si

for i 0

and



f2=1 ðsÞ ¼ facc=don ðsÞ ffi

0

if  s < shb

s þ shb

if  s > shb

where ai and b0 are, for the time being, free adjustable parameters. They contain information on the influence of the solid surface. The function f2/1 defines a threshold value of s for which hydrogen bonds are included. The comparison between measured and

~ S and rS are the molecular weight and density of the M solid, mS0 ðsÞ the hypothetical chemical potential of the solvent (e.g. water) which can directly be calculated if the frequency charge distribution is known [9]. The combinatorial contribution CS;S0 takes into account size and shape effects of solute and solvent and is direct proportional to b0 of Eq. (7). mS(s) denotes a hypothetical chemical potentials of the solids’ surface patches which may be converted into a partial charge distribution. These distributed surface properties are new descriptors of particulate surfaces. They are a

Fig. 7. Comparison between measured and predicted Henry coefficients both in the gas and liquid phases.

W. Peukert et al. / Applied Surface Science 196 (2002) 30–40

measure for possible interactions of the surface patches with the patches of other molecules. Eq. (8) is a Fredholm integral, which is an ill-posed problem. If mS(s) is calculated by direct inversion, it leads to an oscillating function, which is not physically interpretable. This is caused by the fact, that the influence of experimental errors on ill-posed problems is very high. The resulting function can either be smoothed by regularisation methods or mS(s) can be calculated by the help of physical interpretation of the parameters ai and b. With the former way it is tried to reduce the influence of experimental errors by the use of a regularisation matrix. As the value of the regularisation parameter (which is a measure for the influence of the regularisation matrix) has to be estimated, this fact is a source of errors. If this parameter is too high, the result is mainly based on the regularisation and if it is to low, the influence of the experimental errors is not sufficiently reduced. Another way uses the physical interpretation of the s-moments. As M0 represents the surface of the molecule, it is a measure of dispersive interactions. M1 is the negative of the total charge and vanishes if the molecules are neutral. M2 is a measure of the polarity of the solute, b0 represents the integration constant. Thus, each moment can be expressed by a

39

function which includes the physical background. With Eqs. (8) and (9),

Heaqu;i 1 ¼ ln ~ kT M S rS  Z  CS;S0 þ pðsÞðmS0 ðsÞ mS ðsÞÞ ds " # m X 1 ffi C 0þ ai M i kT S;S i¼2 " # Z m X 1 C 0 þ pðsÞ ¼ ai fi ðsÞ ds kT S;S i¼2 (9) the influence of the different moments is weighted by the value of ai and mS(s) can be calculated as a composite of these functions (Eq. (10); Fig. 8). mS ðsÞ ¼ mS0 ðsÞ 

m X

ai fi ðsÞ

(10)

i¼2

In the range of low charges, the interactions of dry activated carbon with non-polar patches are fairly strong. The calculated curves show low interactions for dry activated carbons with higher charged patches which may be due to the fact that the measured

Fig. 8. Distributed surface energies.

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W. Peukert et al. / Applied Surface Science 196 (2002) 30–40

adsorptives in the gas phase were mostly non-polar. For a better characterisation in the range of high charges, more measurements with polar adsorptives are necessary. As expected, the co-adsorption of water lowers the ability of wet activated carbon to interact with low charged patches. The hydrogen donor-affinity, which is the ability to act as hydrogen acceptor, is almost similar to that of water—thus, donors like hydroxylgroups are only weakly adsorbed. This may be explained by hydrogen-bonded water clusters on donor sites so that the probe molecules ‘‘recognise’’ the carbon surface as similar to water. Due to the lower acceptor affinity of wet activated carbon compared to water, the interactions of acceptor groups interact stronger with water and thus, decrease the adsorption affinity. As a next step, we will perform DFT simulations of model surfaces with differing pertubations. This will lead to a deeper physical understanding of heterogeneous particle surfaces.

4. Conclusions Heterogeneous particle surfaces are compositional and structural heterogeneous so that their surface properties can in general not be predicted from first principles. Therefore, we are using adsorption and adhesion measurements to predict properties of heterogeneous surfaces using the concept of equivalent surface properties. It has been shown that Henry coefficients can be predicted using Hamaker constants of the solid and critical temperature and pressure of the adsorptive. Alternatively, Hamaker constants or dispersive surface energies can be calculated using measured Henry coefficients in the gas phase. Polar interactions including hydrogen bonds are described

by a new continuum model. With this approach Henry coefficients both in the liquid and phases are described very well. Using this approach, the solid surface can be described by a distribution of patch energies.

Acknowledgement Financial support of Deutsche Forschungsgemeinschaft is gratefully acknowledged. References [1] D.D. Do, Adsorption Analysis: Equilibria and Kinetics, Imperial College Press, London, Great Britain, 1998. [2] K. Leschonski, Particle Size Analysis and Characterization of a Classification Process, Vol. B2, VCH Publishers, Ullmann, 1988. [3] S. Maurer, A. Mersmann, W. Peukert, Chem. Eng. Sci. 56 (2001) 3443–3453. [4] S. Maurer, Ph.D. thesis, Technische Universita¨ t Mu¨ nchen, Germany, 2000. [5] M. Go¨ tzinger, W. Peukert, in: Proceedings of the Seventh International Symposium on Agglomeration, Albi, France, 2001. [6] W. Rudzinsky, D.H. Everett, Adsorption of Gases on Heterogeneous Surfaces, Academic Press, New York, 1992. [7] M. Janoniec, R. Madey, Physical Adsorption on Heterogeneous Solids: Studies in Physical and Theoretical Chemistry, Vol. 59, Elsevier, Amsterdam, 1988. [8] C. Cramer, D. Truhlar, Chem. Rev. 94 (1999) 2027–2094. [9] A. Klamt, J. Phys. Chem. 103 (1995) 9312–9320. [10] C. Mehler, A. Klamt, W. Peukert, AIChE J., accepted. [11] C. Mehler, W. Peukert, Chem. Ing. Techn. 72 (8) (2000) 822– 826. [12] C. Mehler, W. Peukert, Chem. Eng. Techn. 8 (2001) 789– 794. [13] R. Wagner, Ph.D. thesis, Du¨ sseldorf, 1987. [14] L. Bergstro¨ m, A. Meurk, H. Arwin, A.D. Rowcliff, J. Am. Ceram. Soc. 79 (2) (1996) 339. [15] J. Israelachvili, Surface and Intermolecular Forces, Academic Press, New York, 1997.