A Review of Inverse Gas Chromatography and its Development as a Tool to Characterize Anisotropic Surface Properties of Pharmaceutical Solids †

Surface properties can profoundly impact the bulk and interfacial behavior of pharmaceutical solids, and also their manufacturability, processability in drug product processes, dissolution kinetics and mechanism in drug delivery. Variation in the interand intra-molecular interactions gives rise to anisotropic surface properties of crystalline solids which display direction-dependent characteristics relative to the orientation of the crystal unit structure. Despite its establishment since the 1950s, inverse gas chromatography (IGC) is still an evolving technology in the field of pharmaceutical R&D. In this review, the principles behind IGC as a physicochemical technique to measure the surface properties of solids are presented. The introduction is followed by an overview of its utility in pharmaceutical R&D, spanning a variety of applications including batch-to-batch variability, solid-solid transitions, physical stability, interfacial behavior in powder processing, and more. For anisotropic materials, IGC has been utilized to characterize the heterogeneity of materials using adsorption and energy distribution functions. Recent development and applications of IGC at finite concentration (IGC-FC) to determine the surface heterogeneity distribution of solids are presented. This methodology overcomes a number of limitations associated with traditional experiments.


Introduction
Many organic pharmaceutical cr ystalline solids exist in different solid-state forms in relation to their internal molecular conformations, resulting in differences in lattice energy and entropy.Anisotropic substances, as a result, exhibit different physical and chemical proper ties along dif ferent index cr ystal planes.For pharmaceutical solids, the properties that are of greatest interest are physical, thermodynamic, kinetic, surface, spectroscopic, optical, and electrical properties 1) , which to a different extent, display anisotropy at the molecular level.Due to the anisotropic nature of crystalline solids, it is a subject of considerable interest for researchers.A careful understanding of the anisotropic material properties plays a critical role in pharmaceutical development, formulation and manufacturing.During crystallization of the active ingredient, the crystal morphology is controlled by the solute-solvent interactions at the nuclei solid-liquid interface and the internal crystal structure anisoptropy 2) .Particle size and morphological control in mechanical size reduction processes are impacted by mechanical anisotropy which plays a fundamental role in the underlying breakage mechanism 3) .The dissolution rate of a solid form is dictated by the dissolution kinetics of individual cr ystal planes, therefore the cr ystal morphology of the drug 4) .Owing to their anisotropic optical properties, crystalline solids exhibit birefringence under polarized light microscopes -a common method to detect residual crystallinity 5) .The surface properties of a drug are influenced by its localized, facet-dependent surface chemistry, which ultimately dictates the strength and polarity of interactions at the interface 6) .For anisotropic materials, it is not uncommon to assume the dominant crystal plane to dictate its overall properties.However, with the technological improvement of physical characterization techniques and molecular modeling tools in combination with knowledge advancements, the anisotropy of the cr ystalline solids can now be more readily studied and predicted.The modern paradigm of materials science is gradually shifting away from old-school approaches based on simplified assumptions that particles are spherical and isotropic.This is evident in the growing number of research publications relating to anisotropic material properties, for instance, in the subjects of indentation hardness anisotropy 7) , impact of crystal morphology on processing 8,9) , and physical simulation modeling of crystal shape using polygonal meshing and spherical composites 10) .The surface properties, the focus of this review, are known to profoundly impact material properties including dissolution 11) , flowability 12) , cohesion and adhesion 13) , and a variety of processing behaviors including crystallization 14) , wet granulation 15) , milling 16) , drug-excipient compatibility and mixing 17) , coating, tableting 18) and aerolization performance of dry powder inhaler (DPI) formulations 19) .The experimental validation of the surface anisotropy of crystalline solids was reported by Heng et al. who conducted sessile drop contact angle measurements and X-ray photoelectron spectroscopy (XPS) analyses on indexed surfaces of macroscopic cr ystals and showed, for the first time, the facet-dependent surface chemistry of crystalline pharmaceutical solids 20) .Although the anisotropic surface properties of a crystalline solid may be dominantly attributed to individual cr ystal planes, the unique combination of impurities, growth steps, crystal edges, surface pores, surface disorders, and local degree of cr ystallinity can be important, particularly for imperfect cr ystalline systems.The anisotropic surface properties of cr ystalline solids are also referred to as heterogeneous surface properties, owing to the fact that the material exhibits a continuous distribution of surface characteristics.In this review, the principles behind inverse gas chromatography (IGC) as a tool to measure surface properties of solids are presented.This is followed by an overview of its applications in pharmaceutical research and development, and its utilization to measure the anisotropy and heterogeneity of compounds using adsorption and energy distribution functions.Finally, recent development and applications of IGC at finite concentration (IGC-FC) as a tool to determine the surface heterogeneity distribution of solids are presented.

Partition coefficient
The discovery of chromatography was made by the American petroleum chemist David T. Day and the Russian botanist Mikhail S. Tswett between 1903 and 1906.Tswett, however, was the first to recognize the sequential sorption-desorption interactions of chromatographic processes in his original experiments in which plant pigments were classified into colored bands by elution with petroleum ether through a bed of powdered calcium carbonate 21) .Since then, a highly sophisticated family of chromatographic techniques has evolved for physicochemical and analytical studies, involving gas-liquid, gas-solid, liquidliquid and liquid-solid chromatography.IGC, the inverse use of gas chromatography (GC) or gas-solid chromatography (GSC), was developed in the 1950s when the focus of physicochemical studies was directed towards the derivation of kinetic information and determination of thermodynamic quantities from sorption equilibria 22) .Much effort of this earlier work was focused on the characterization of catalytic materials such as activated carbon, alumina and silica.To date, IGC has become a wellestablished source of physicochemical data for various uses including polymers 23) , inorganic compounds and catalysts 24,25) , food substances 26) , carbon nanotubes 27) , wood composites 28) , and pharmaceuticals 29) .Its applications cover a wide spectrum including characterization of surface energetics, surface acidbase properties, solid-solid phase transitions, adsorption isotherm, energy distribution, solubility parameters 30) , Flor y-Huggins interaction parameters 23) , diffusion kinetics, and polymer cross-link density.IGC is a vapor probe technique applicable to powders and fibrous materials, and is compatible with sample porosity, irregular surface topographies, and surface heterogeneity, making it an attractive way of characterizing particulate pharmaceutical solids.In IGC, probe molecules in the vapor phase (adsorbate or solute) are passed over a sample solid (stationary phase adsorbent) which is packed in a column of inert surface via a carrier gas (mobile phase).The distribution of the solute between the stationary phase and the mobile phase at a particular temperature and pressure corresponds to an equilibrium when the solute-free energy is at a minimum.The partition coefficient (unit: unit length), K R , is related to the concentration of adsorbate in the mobile phase (unit: mass/mole per unit area), c M , and that in the stationary phase (unit: mass/mole per unit volume), c S , via: The partition coefficient is also directly related to the mass of the solid, m S , the specific surface area of the solid, σ, and the net retention volume, V N , which is defined as the volume of carrier gas required to elute the injected adsorbate through the column.Both K R and V N are indications of the interaction strength between the adsorbate and the solid sample of interest.Together, they form the fundamental basis for the derivations of equilibrium sorption thermodynamic parameters.

Pulse experiments at different surface coverage values
IGC experiments at infinite dilution are conducted via pulse injection of the probe vapors using ver y small amounts of adsorbate (<3% partial pressure typically).The use of low probe molecule concentrations results in very low surface coverage on the adsorbent and is therefore commonly referred to as adsorption at zero surface coverageʼ.Adsorption under these conditions follows Henryʼs Law, where the amount of probe molecules adsorbed is linearly dependent on the injection concentration.Chromatograms obtained under this regime are symmetrical and Gaussian in shape, and the retention time, t R , is independent of the injection size (Fig. 1a).Due to high sensitivity, this regime is most ideal for measurements of thermodynamic parameters.Experiments where the probe injection concentrations are increased beyond the Henryʼs Law region are referred to as finite concentration (IGC-FC) experiments.Chromatographic peaks under the finite concentration regime can show either tailing (Fig. 1b) or fronting (Fig. 1c).In tailingʼ, the rate of change in the amount adsorbed decreases as a function of partial pressure.This adsorption is characterized by a Type I, II, or IV mechanism (IUPAC classification) with the formation of a monolayer of adsorbate on the adsorbent.In frontingʼ, the rate of change in the adsorbed amount increases as a function of partial pressure.This adsorption is characterized by a Type III or IV mechanism, in which the binding affinity between adsorbate and adsorbent is weaker than the affinity between adsorbate molecules.Two commonly employed detectors in gas chromatography are the thermal conductivity detector (TCD) and the flame ionization detector (FID), although the latter is probably more frequently used.The TCD measures the reduction in thermal conductivity of the carrier gas in the presence of the analyte and is therefore universally suitable for the detection of both organic and inorganic vapors.Thermal conductivity is established by measuring the heating resistance of a heating filament, and the measured current is compared to a reference current in a bridge circuit.However, since the thermal conductivity difference between the carrier gas and the analyte is important to the sensitivity of this type of detector, carrier gases other than hydrogen and helium (thermal conductivities are 6-10 times higher than those of organic vapors) are in principle not suitable because the drop in thermal conductivity in the presence of the analyte can be too small (detection limit: 10 -8 g ml -1 ) 31) .The FID, on the other hand, is particularly sensitive to all compounds containing C-C and C-H bonds, and measures the change in electrical conductivity of a hydrogen flame in an electric field when feeding organic compounds.The solute gross retention time, t R , is the time required for the center of gravity of the solute band to pass completely through the column.In the infinite dilution regime where the chromatogram peak is symmetrical with a Gaussian profile, the centre of gravity is at the point of maxima of the chromato- gram and therefore t R corresponds to the time taken between injection and the peak maximum (Fig. 2).
In the finite concentration regime, the elution time at peak maximum may be substantially different to the elution time computed from the center of gravity of the chromatogram.
The retention time is a measure of the strength of molecular interactions between the probe and the solid surface of sample packing in the column, and is the key measurement parameter in IGC analysis.The dead time, t 0 , is the time required for a non-interacting, non-adsorbed (K R = 0) solute to pass through the column.It is typically determined by methane, but argon, nitrogen and hydrogen are sometimes used.Experiments are possible with the frontal technique, in which case the breakthrough point of the breakthrough cur ve corresponds to the retention time.
Due to its rarity in the literature, experiments in the frontal mode are not elaborated in this review.With the column dead-time and gross solute retention time, the net retention volume, V N, is determined via: where F c is the carrier gas flow rate in the column, and j is the James-Martin correction factor which corrects the net retention time for the pressure drop and variation in packing density of the solids within the column bed.The James-Martin correction factor, j, is defined as: where P in and P out are the inlet and outlet pressures, respectively.Since V N is dependent on the amount of the station-ary phase and the experimental temperature, T, the specific retention volume referenced to 0°C, V 0 g , is sometimes used in place of the net retention volume, where: Combining equation 2 and 4 yields:

Thermodynamic relationships
The standard Gibbs free energy change associated with the isothermal adsorption and desorption per mole of molecules, expressed respectively as ∆G 0 ad and ∆G 0 de , is related to the net retention volume and partition coefficient via: ∆G where C 1 and C 2 are constants depending on the chosen reference state 32) .Adsorption is considered as an exothermic event, whereas desorption is an endothermic event -the reason for the opposite directionality between the two ∆G 0 .Using vanʼt Hoffʼs relationship, the standard enthalpy change of adsorption, ∆H 0 ad , can be obtained by measuring V N at a range of experimental temperatures: And the standard entropy change of adsorption, ∆S 0 ad , is related to the enthalpy and free energy terms by: The net retention volume of each alkane injection is related to the standard Gibbs free energy change of adsorption, ∆G 0 ad , and the work of adhesion by: where N A is the Avogadro number and a m is the molecular cross-sectional area of the adsorbed probe molecule.Applying Fowkes principle for Lifshitz van der Waals interaction 33) , Eq. 10 leads to: Schultz et al. identified that the dispersive surface energy of solids can be obtained using a series of Fig. 2 The increase in retention time for increasing nalkane chain lengths at infinite dilution concentration. .The γ d S V of the solid is calculated from the slope of a linear regression of the n-alkane line (Fig. 3a).Alternatively, γ d S V can be determined by considering the contribution of a methylene group (CH 2 ) in the normal alkane series to the free energy of adsorption, ∆G 0 CH 2 (Fig. 3b) according to the Dorris and Gray approach 34) : where a CH 2 is the surface area occupied by a methy- lene group (6 Å 2 ) and γ CH 2 is the surface energy of a methylene group.Both methods are reported to give similar results for polycarbonates 35) .
With the use of polar solute molecules, the Gibbs free energy change of adsorption now comprises an acid-base (polar) adsorption component, ∆G 0 AB , in addition to the dispersive component, ∆G 0 D : As such, the retention volumes measured with a polar probe will consist of a dispersive and acid-base (polar) component leading to higher absolute V N values when compared with the occurrence of dispersive interactions alone using alkanes.The difference between the alkane regression line and the polar probe equates to ∆G 0 AB (Fig. 3a).By measuring the absorbent with polar probes at different temperatures following Eq.8, the acid-base (polar) component of ∆H 0 ad and ∆S 0 ad can be determined (Fig. 4a).The values of ∆H 0 AB obtained using various polar probes can then be used to evaluate the acid-base characteristics of the solid adsorbent via  the modified Gutmannʼs equation 36,37) : where DN is the electron donor number of the adsorbate and AN* is the corrected electron acceptor number of the adsorbate, and K A and K B are the acid and base numbers of the solid reflecting its surface electron accepting and donating characteristics, respectively.The acid and base numbers, K A and K B , are obtained from the slope and intercept, respectively, of a plot of ∆H 0 AB /AN * versus DN/AN* (Fig. 4b).

Batch-to-batch variability, polymorphs and optical forms
With the suitability for particulate solid materials, it is not surprising that the interest in IGC in the pharmaceutical industr y is growing.Despite the history of IGC in the studies of inorganic materials and polymers, it was not until the late 1980s that the first application of IGC on pharmaceutical solids was reported.The technique was first applied to study the bulk property of pharmaceutical solids by Phuoc et al., who determined the Hansen partial and total solubility parameters of lactose, caffeine, theophylline, methyl-p-hydroxybenzoate and microcrystalline cellulose 38,39) .The use of solubility parameters in the design of the pharmaceutical dosage form was reviewed by Hancock 30) .In 1994 and 1996, IGC was first applied to study the surface properties of pharmaceutical solids by Ticehurst et al., who differentiated chemically and structurally equivalent batches of salbutamol sulphate 40) and α-lactose monohydrate 41) , which exhibited variable processing performance.In the subject of polymorphism, Tong et al. examined the thermodynamic properties of two polymorphs of salmeterol xinafoate prepared from supercritical fluids in which the metastable polymorph was shown to exhibit a higher surface energy, surface entropy and surface polarity than the stable form 42) .The strength of dispersive and acid-base (polar) interactions of solid surfaces can be explained based on the localized surface chemical environment.In a study of DL-mannitol and the β polymorph of Dmannitol by Grimsey et al., differences in the surface properties were attributed to the surface densities of the dispersive and acidic sites from inspection of the crystallographic structure 43) .Sexena et al. combined IGC with dynamic molecular modeling to study the impact on surface energy due to factors such as surface pore, surface cavity size, functional groups and the presence of surface water molecules 44) .The modeled kinetic, structural and thermodynamic factors at the molecular level were responsible for the measured surface energetics.

Amorphous solids and glass transition
IGC has been used to study amorphous solids, including the determination of glass transition temperature (T g ), localized disorder, and structural relaxation of the amorphous glass.This capability stems from the unique sensitivity of IGC at infinite dilution in detecting subtle localized surface disorders, where alternative solid-state characterization techniques, such as conventional powder X-ray diffraction (PXRD), differential scanning calorimetry (DSC), thermal gravimetric analysis (TGA), Raman or FTIR spectroscopy, may fail.Ohta and Buckton obser ved a direct correlation between the Gutmann basicity ratio (K B /K A ) and the percentage cr ystallinity of cefditoren pivoxil from IGC measurements 45) .The dependence of γ d S V and basicity as a function of % RH was attributed to the exposure of basic carbonyl groups on the surface as cr ystallinity decreased.Hasegawa et al. applied IGC to examine the surface structural relaxation of an IMC-PVP dispersion below the T g by monitoring the retention volume of decane over time 46) .The decrease in V N was attributed to the surface structural relaxation of the solid dispersion.The kinetics of relaxation was modeled based on the rate of decrease in V N to yield the relaxation parameter, τ β , of the Kohlraush-Williams-Watts equation.The smaller τ β from IGC than those obtained from DSC was at- tributed to a faster rate of structural relaxation at the surface compared to the bulk.The physicochemical stability of granules of amorphous cefditoren pivoxil, with and without polymers, prepared from spray-dr ying and wet granulation was compared by Yokoi et al. using a variety of techniques including IGC 47) .The addition of polymers increased the physical stability of both spray-dried and wet-granulated cefditoren pivoxil.However, while the increase in physicochemical stability could be attributed to a decrease in molecular mobility (as determined by T g shift) for the spray-dried materials, the granulated materials did not exhibit any difference in T g by DSC.IGC and diffuse reflection IR revealed that for the granulated materials, the physicochemical stability was determined by molecular interactions between the drug and polymer at the surface, resulting in granules displaying a similar stability to that obtained from spray-drying.The glass transition temperature of amorphous solids can be determined from retention volume measurements using IGC.The methodology is based on the principle that amorphous materials exhibit very different sorption mechanisms above and below the T g , leading to changes in the molecular retention mechanism.As an amorphous solid is heated through the glass transition region, the glassy and rubber y states can co-exist, resulting in a retention mechanism combining surface adsorption and bulk absorption.Below the glass transition region where the glassy state dominates, surface adsorption is the dominating retention mechanism.In the rubbery state above T g , the retention mechanism is via a combination of both surface and bulk sorption.The variation in V 0 g as a function of temperature can therefore be used to determine the onset of T g .Following this approach, Surana et al. investigated the T g of amorphous sucrose and co-lyophilized sucrose-PVP mixtures using IGC, and compared the results with results obtained from DSC 48) .T g values at 0% RH obtained by IGC were in good agreement with those determined using DSC 49) .As the relative humidity increased, a progressive decrease in T g was measured as a result of the plasticizing effect of water.The predicted T g values of the plasticized materials were in very good agreement with those determined experimentally using IGC.Otte and Caravajal employed a combination of particle size, surface area, IGC, DSC and PXRD to examine the surface disorder and bulk properties of ketoconazole and griseofulvin as a function of cryomilling time 50) .A reduction in surface cr ystallinity and an increase in γ d S V and ∆G 0 AB were obser ved for both compounds upon cryomilling.Ketoconazole under went a change in surface structure, whilst a phase transition event which occurred at much lower temperature than the T g was detected by IGC for griseofulvin.The authors concluded that the mechanofusion of sur face fines and the existence of an intermediate metastable phase are the consequences of cryomilling for the two compounds, respectively.Brum and Burnett utilized IGC to quantify the amorphous content in both a model drug substance and lactose by a linear summation of the individual work of adhesion components due to the amorphous phase and crystalline phase based on their respective surface area fractions 51) .Using physical mixtures of known quantities and known amorphous surface area fraction, a calibration cur ve can be established from the measurements of surface area normalized disper-sive surface energies.By characterizing the dispersive surface energy, the fraction of amorphous content in lactose and in the model drug substance was quantified successfully from the calibration curves.

Cr ystal morphologies
The recrystallization of crystalline materials from solution can result in various forms of crystal morphologies or shapes.Storey investigated the surface properties of ibuprofen recrystallized from methanol, acetonitrile and hexane by IGC 52) .Both the γ d S V and acid-base surface properties were found to be dependent on the cr ystal shape, and could be explained based on the difference in the percentage area of polar and apolar crystal facets as a consequence of the solvent used during crystallization.Heng et al. studied the surface properties of paracetamol cr ystals grown from methanol and acetone, which resulted in crystals exhibiting prismatic and planar morphology, respectively.Although the γ d S V values were found to be similar, the two morphological entities possessed different K A and K B

53)
. It is expected that crystals with the planar morphology are dominant in facet (201), and hence will exhibit a higher degree of basicity due to the presence of carbonyl functionality.

Surface properties and powder processing
Infinite dilution IGC has also been used to measure process-related changes in surface properties and to assess the performance of drug delivery systems and powder processes.The effects of milling on the surface properties of paracetamol were investigated by Heng et al. 54) .Cr ystals may cleave across their cleavage planes upon milling with increasing tendency as the particle size is reduced.Surface property measurements were performed on unmilled and milled samples of paracetamol cr ystals of various size fractions.An increase of γ d S V with decreasing particle size was measured, revealing an increased dominance of the weakest attachment energy facet.On the other hand, the γ d S V for the unmilled fractions was independent of particle size and was reflective of the surface energies of the external facet.The effects of micronization on the γ d S V and relative basicity of salbutamol sulphate and DL-propranolol hydrochloride were investigated by Feeley et al. 55) and York et al. 56) , respectively.Both micronized materials were found to possess a higher γ d S V than the unmicronized substances, whereas the acid-base properties were slightly different.It was concluded that the change in surface properties was due to preferential cleavage along the weakly attached facet.Davies et al. com-pared surface energetics of unmilled and micronized budesonide determined from atomic force microscopy (AFM) and IGC.In AFM, a small fraction of the total surface area is measured, and the surface energies are therefore related to the location and local topography, whereas IGC at infinite dilution would overestimate the average surface energy of a material.However, because of their abilities to characterize materials at different scales (particulate level or bulk level), it was concluded that both AFM and IGC are useful complimentary tools to assess, in a quantitative manner, particulate interactions and intrinsic material properties.The performance of dry powder inhalation formulations was examined by Tong et al., who investigated the relative influence of drug-drug cohesion and drug-carrier adhesion on the in vitro performance of salmeterol xinafoate (drug) with lactose (carrier) by measuring the surface energies and solubility parameters of the components using IGC at infinite dilution 57) .The inhaler performance of salmeterol xinafoate was found to improve significantly if the drug-carrier adhesion was stronger than drug-drug cohesion.Cline and Dalby similarly obser ved that increasing surface interaction between drug and carrier, as measured by resulted in an improved fine particle fraction of the drug 58) .Das et al. found a negative correlation between the total surface energy and dispersiblity of a DPI formulation of salmeterol xinafoate and lactose 59) .The fine particle fraction of the mixture decreased significantly after storage at 75%RH.The decrease was primarily attributed to the presence of surface-adsorbed moisture after storage at 75%RH, resulting in an increase of surface energy due to the introduction of new polar sites on the surfaces of the particles.In a separate study on the suspension stability of pressurized metered dose inhalers (pMDIs), Triani et al. investigated the surface properties of salbutamol sulphate, budensonite and formoterol fumarate dihydrate in a model propellant, using both AFM and IGC 60) .The authors suggested that polar contributions to surface energy may be crucial in determining the stability of these suspensions.The effect of primary particle surface energy in fluidized bed wet granulation was studied by Thielmann et al., in which the hydrophobic particles agglomerated to become larger particles compared to the hydrophilic species 61) .Dispersive and non-dispersive surface energies of the particles, measured by IGC pre-and post-granulation, revealed that a thin binder coating layer was present on the surface of the hy-drophilic particles after granulation.It was suggested that the relatively thin binder layer was not able to dissipate the kinetic energy of granules during impact, resulting in a certain critical granule size.Fichtner et al. studied the effect of surface energy on the compactability of amorphous spray-dried lactose, with or without a low proportion of surfactant, by measuring γ d S V , K A and K B using IGC 18) .The compactability and γ d S V was found to be dependent on the composition of powders.At constant tablet porosities, the decrease of tablet strength was due to the decrease in powder surface energy.The authors concluded that the strength of bonding forces between particle contacts is controlled by surface energy which in turn can be altered by the presence of surfactants.

Existing Approaches for Characterizing Energy Distributions
Despite the sensitivity of IGC at infinite dilution, experiments at this concentration range may be criticized for their tendency to characterize higher energy sites of the solid surface.Small concentrations of adsorbates that are used in infinite dilution studies are thought to preferentially interact with the higher energy sites on the surface, and therefore the interaction with lower energy sites of the adsorbent would be limited, if not excluded 62) .As a result, this measured upper limit estimate may not be representative of the whole surface and may yet be incomparable to the surface energetics measured by alternative characterization techniques, such as sessile drop contact angle when the energy is expressed as an average of the probed area.
The possibility to characterize the surface energy distribution of a surface by IGC was recognized in the 1970s.These approaches to characterize energy heterogeneity can be categorized into pressure-or temperature-dependent adsorption and desorption methods 63) .The former method results in either an adsorption energy distribution 64) or adsorption potential distribution 63,65) , whereas the latter leads to a desorption energy distribution.More elaborated reviews on the determination of surface heterogeneity from adsorption measurements can be found elsewhere 66,67) .

Adsorption energy distribution
All approaches on the adsorption energy distribution described in the literature are based on a physical model which assume that an energetically hetero-geneous surface of a continuous distribution of adsorption energies and may be described as a superposition of a series of homogenous adsorption patches.In this model, the surface coverage, n/n m , is given by the integral equation: where θ L is the local adsorption isotherm, ∈ is the adsorption energy of a site and χ is the continuous adsorption energy distribution function.As the equation has no general solution, approaches towards its solution are not trivial and typically rely on extensive computer simulations.The task of solving the equation is additionally magnified by its ill-posed nature, which is related to the large variation in energy distribution function with just a small variation in the experimental data 64) .Since the distribution function χ(∈) is a quantitative description of energetic hetero- geneity characterizing the solid surface, extensive efforts have been made in solving Eq. 15 with respect to χ(∈) .In Eq. 15, only the experimental (global) isotherm is known, the choice of the local isotherm θ L (∈, P, T) has to be chosen according to the physical hypothesis describing the interactions between probe and adsorbent molecules, and the interactions between neighboring probe molecules.The energy distribution function χ(∈) is therefore strongly dependent on the choice of this local isotherm.A number of methods have been proposed to describe the energy distribution function 64) .The first category of methods ascribes a given analytical form to χ(∈) , but many authors who tried to develop solution methods failed to consider the shape of the distribution function.
The second category of methods assumes a discrete distribution of monoenergetic sites.With an increasing number of monoenergetic sites, the inversion of the matrix containing a large number of linear equations is not a simple task, leading to instability of the solution.The third category of methods is by solving Eq. 15 using local isotherm approximations such as condensation approximation.The final categor y of methods relies on the application of Plancherelʼ s theorem to correlate the Fourier transform of the distribution function directly to the ratio of the Fourier transforms of the experimental isotherm and the Fourier transform of the local isotherm.
Although considerable developments have been made in trying to solve the integral equation, it is conceded that the method will not allow the attainment of the absolute surface energy distribution of the solids.Sacchetti reported the use of the discretization approach in the analysis of the surface heterogeneity of α-lactose monohydrate which displays regions of higher energy and ultrahigh energy hotspotsʼ 68) .To the best of our knowledge, this is the only application of the adsorption energy distribution on an organic pharmaceutical solid.

Adsorption potential distribution
From the adsorption isotherm, the adsorption potential, A, can be related to the equilibrium partial pressure, P, the saturation pressure, P 0 , and the column temperature, T, via 69) : The adsorption potential distribution, X n , is obtained from the first derivative of the characteristic adsorption curve which is a plot of the amount adsorbed n against the adsorption potential A.
The adsorption potential distribution can be normalized by dividing Eq. 16 by the monolayer capacity.The approach of adsorption potential distribution was repor ted to be less af fected by experimental variations, to produce more reliable results, and to be much simpler 25) .Nevertheless, the validity of the adsorption potential distribution relies on the assumption of a physisorption interaction between the probe and the surface site 25) .Molecules such as some polar probes adsorb via a reversible or irreversible chemisorption, leading to a slow desorption or nondesorption of the probes.

Desorption energy distribution
In the case of reversible chemisorption, the surface heterogeneity can be determined by combining IGC with thermal desorption methods in the temperatureprogrammed desorption (TPD) technique.In this approach, a sample is heated at a defined heating rate and the partial pressure and amount of the desorbed molecules are measured.Although one can obtain the surface heterogeneity of the sample from the desorption profile (or TPD spectrum), in reality the quantitative interpretation of the desorption profiles is difficult due to diffusion and re-adsorption effects on the overall rate of desorption 70) .Characterization of the desorption energy distribution is based on the first linear Fredholm equation, similar in structure to Eq. 15, and the desorption distribution function is calculated as a kernel by inversion of the equation 71) .The solution of this ill-posed problem again requires an extensive computing algorithm which has been mainly applied in the studies of catalysts [71][72][73] .
In comparison between the temperature and pressure method in the characterization of surface heterogeneity, the former is more suitable for highly energetic surfaces, while the latter method is more appropriate for less energetic sur faces and also more sensitive to surfaces with smaller differences between energy levels.Though these methods can lead to a distribution of surface heterogeneity, the interactions of the adsorbing molecules with surface sites depend on the chemical nature of the probes.
These methods therefore only provide information concerning the relative heterogeneity and can only be used as fingerprints for comparisons between different materials.Until recently, there has been a lack of emphasis on the understanding of the surface energy heterogeneity of pharmaceutical materials, despite the apparent anisotropic properties displayed by many crystalline materials.

Theoretical aspects
We recently reported a new methodology to measure the surface energy heterogeneity of particulate solids based on IGC-FC 6, 74,75) .The advantage of this new approach not only stems from the fact that it contains fewer drawbacks compared to methodologies described in the previous section, this approach also permits attainment of an explicit γ d S V distribution from adsorption data using IGC.The measurement of dispersive and acid-base (polar) surface energy distribution relies on a series of finite concentration IGC experiments using a series of n-alkanes and polar solvents.The first step is to determine the adsorption isotherms of all probe molecules with IGC at finite concentrations.A detailed description of the determination of adsorption isotherms by IGC-FC is given by Cremer and Huber 76) .Generally, two methods can be distinguished for adsorption isotherm calculations from IGC elution chromatograms: the peak maximum method (PM) and the elution of a characteristic point (ECP) method (Fig. 5).In the PM method, an increasing concentration of probe vapor is injected into the column, and the equilibrium partial pressure for each single concentration is measured from the peak maximum of each chromatogram (Fig. 6 -Step 1).In the ECP method, it is assumed that the maxima of each individual peak from the single injections coincide with the rear of the chromatogram produced by the highest injection concentration.The ECP method is experimentally faster because the isotherm is calculated from the chromatogram of only the greatest injection concentration.The ECP method, however, relies on the adsorption mechanism being physisorption of Type II, IV or V, and that the rear of the chromatogram is entirely due to sorption effects which neglect the gas phase diffusion and which may deviate the ECP calculations from those resulting from the PM method.In the PM method which in the experience of the authors, provides more accurate adsorption isotherms, the equilibrium partial pressure, P, for each concentration of vapor in the column can be calculated from the chromatogram shape via (Fig. 6 -Step 2): where h c is the chromatogram peak height, A c is the chromatogram peak area, V Loop is the injection loop volume, T Loop the injection loop temperature and P inj is the partial pressure of the solute inside the injection loop.The corresponding retention volume, V N , for each injection concentration can be determined as usual via Eq. 2. The adsorbed amount, n, and therefore the adsorption isotherm for each probe vapor can then be obtained by integration of V N versus P (Fig. 6 -Step 3): In order to determine the surface energy distribu- tion, the retention volumes must be corrected to their corresponding surface coverage, n/n m .This is due to the fact that injections with different concentrations of adsorbate will result in different extents of surface coverage of the probe over the solid surface (Fig. 6 -Step 4).By assuming that the alkanes or weakly polar probes are absorbed in a Type II or IV adsorpmechanism, the BET model can be applied to determine the monolayer capacity, n m , from Eq. 20 provided that the specific surface area of the sample, σ , is known: The BET specific sur face area of the sample can be calculated from one of the alkane isotherms or determined separately by another technique, for instance by nitrogen adsorption at 77K.Although the adsorption mechanisms of pharmaceutical solids are predominantly Type II or IV, the measurement of a complete adsorption and desorption profile will pro-vide added assurance on the applicability of the BET model.The corresponding surface coverage, n/n m , at each injection concentration can now be calculated from the amount adsorbed n from the monolayer capacity.The process is repeated on other n-alkanes and polar probes being tested such that the net retention volumes are provided as a function of probe surface coverage values.The next step in the methodology is to determine γ d S V and ∆G 0 AB (for polar probes) at constant isosteres, i.e. constant probe surface coverage values.The solid-vapor surface energy can be determined at a number of isosteres by applying the Schultz or Dorris-Gray approach to calculate the γ d S V values, thereby resulting in a profile of γ d S V as a function of surface coverage (Fig. 6 -Step 5).Similarly for a polar probe, the profile of the acid-base (polar) Gibbs free energy change of adsorption, ∆G 0 AB , can be calculated.The regression coefficient, R 2 , for the linearity of fit of the alkane retention data was proposed as a criterion for predicting the robustness of the γ d S V profiles obtained 77) .It was proposed that the relative error in γ d S V values determined from the alkane regression line is a function of the R 2 coefficient, with an R 2 of 0.9999 providing γ d S V profiles of least error.Applying the van Oss, Chaudhur y and Good (vOCG) approach, the acid-base sur face energy profile, γ ab S V , of the solid can be obtained by using a pair of monopolar acidic and monopolar basic probes from their respective ∆G 0 AB values 78) .However, the accuracy of this approach to obtain γ ab S V is sensi- Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Fig. 6 Determination surface energy distribution by IGC-FC.
tive to the acid, γ + , and base, γ -, parameters of the probes.Oss et al. utilized water as a reference with γ + : γ -equal to unity, which is assumed to be the cause of the "basicity catastrophe" 79) .Della Volpe and Siboni, however, proposed a different ratio suggesting that water is acidic rather than amphoteric 80) .
In addition to the debate surrounding the scale of acidity to basicity, the monopolarity of the probes is a critical requirement for this approach.Hence in its current state, this reported approach to the calculation of acid-base surface energy profiles may only be applicable on relative terms to compare the relative magnitude of acid-base properties between batches.
The sur face energy profile can be fur ther processed mathematically to obtain a surface energy distribution by numerical integration of the profile across the entire range of surface coverages (0-100%) (Fig. 6 -Step 6).A surface energy distribution is analogous in principle to a particle size distribution wherein particle size is replaced by surface energy and frequency density is replaced by area increment or surface area percentage.However, to enable an integration across the full surface coverage range, the experimentally determined functions of V N versus n/n m may need to be extrapolated with best-fit curves to full surface coverage.This need is primarily driven by the fact that traditional IGC instrumentation is designed and constructed to perform small injections in the Henry region.Therefore a full coverage, especially with large molecular probes with low saturation vapor pressures, may not be achievable due to hardware limitations.

Recent applications for phar maceutical characterization and process understanding
Applying this methodology using IGC-FC, the surface energy heterogeneity due to the anisotropy of crystalline pharmaceutical solids was investigated using a pharmaceutical excipient, D-mannitol, as a model material 6) .To test the robustness of this methodology, the measured surface energetic heterogeneity profile was compared to the facet-dependent surface energetics measured by the sessile drop contact angle technique on macroscopic single crystals.The γ d S V determined from both IGC-FC and contact angle showed extremely good agreement: the lowest γ d S V measured with IGC-FC corresponds to the lowest γ d S V measured on single crystals, i.e. facet (011), and γ d S V of facet (120) and (010) both fall within the range of the measured surface energy profile.In a separate study, the sensitivity of the technique in probing subtle anisotropic variations of the crys-tals due to morphological changes was examined 74) .As the aspect ratio of D-mannitol decreased (longer cr ystals), the surface energy profile revealed a decrease in the overall contribution of the lower dispersive surface energy regions, which could be attributed to the decrease in the proportion of the lowest energy crystal facet (011).With these validational experiments, it is clear that the surface energy heterogeneity/anisotropy by IGC-FC correlates well with complimentary techniques such as sessile drop contact angle.Due to the practicality and suitability for studying particulate materials, it is logical that IGC-FC will become a more useful tool to determine the surface heterogeneity of solids.In addition to applications for heterogeneous/ anisotropic materials, this methodology was also employed to examine sur face homogeneous/isotropic solids prepared from surface modification via functionalization of methyl groups 6) .Upon surface modification, surface energy profiles of D-mannitol exhibited remarkable changes: untreated D-mannitol showed a high level of surface heterogeneity in both dispersive and polar interactions, whereas upon surface methylation, the surface energy profiles of γ d S V and ∆G 0 AB became energetically homogeneous.These studies revealed that the new approach using IGC-FC is not only sensitive to surface energetic and chemical anisotropy, but also to surface homogeneity and subtle surface chemical changes.For drug delivery systems and in the processing of crystalline solids, the utilization of surface energy heterogeneity may play an important role in understanding the underlying fundamental behavior.One such example is demonstrated by the performance of carrier-based DPI drug delivery systems which is determined to a significant degree by the preparation of the formulation 19) .It is known that the relative strength of drug-carrier adhesion and drug-drug cohesion is critical to provide an adequate fine particle dose (FPD) of the drug 81) .In this respect, the study of surface energy distributions may provide further understanding of the intrinsic material properties, in addition to the aerodynamic particle size distributions, shape distributions, bulk and true densities, surface roughness (rugosity) and electrostatic propensity which can also impact the aerolization performance of the formulation 82) .Considered from a purely surface energetic perspective, if the loaded drug dose is increased beyond a state in which the high energy sites of the carrier are fully saturatedʼ, the overall drug-excipient interaction would be weak enough to provide adequate drug detachment upon aerosolization.However, if the cohesion between drug-drug particles or excipient-excipient particles is stronger drug-excipient adhesion, a robust formulation would be difficult to achieve.Thielmann et al. employed IGC-FC to examine differences in γ d S V profiles of α-lactose monohydrate as a function of processing routes, in which untreated, recr ystallized and amorphous lactose, prepared by spray-dr ying, was investigated 83) .Recr ystallized lactose was energetically more homogeneous than milled and untreated lactose.The broader and more heterogeneous surface energy distribution displayed by milled and untreated lactose was due to surface amorphous regions and anomeric compositions, respectively.Their findings are particularly interesting, especially from a formulation development perspective, because the trend of γ d S V at low surface coverages (infinite dilution region) compared to that at high surface coverages is reversed.This study provided important insights into various aspects of the anisotropic/heterogeneous properties in drug deliver y systems, e.g.What are the relative contributions to the overall behavior?What is the dominant contribution?Which is the contribution that is dictating the process?To answer these questions, it is clear that attention should be paid to the utilization of characterization approaches, not limited to IGC, which would provide a thorough understanding of the materialʼs physical and chemical properties.In DPI formulation, the use of a ternary formulation to optimize drug delivery of the active ingredient has been reported, whereby a small amount of fine excipient particles is added to the coarse carrier and the drug blend 84,85) .Despite the evidence that the use of such methods shows an improved FPD of the active agent, the mechanism by which the fine particles alter the performance of the formulation has remained elusive 86) .A hypothetical mechanism is the passivation of high energy sites by the addition of fine excipient particles such that the drug particles are forced to bind to surfaces with lower energies during subsequent blending of the drug into the binary carrier-fine particle mixture 86) .To provide practical understanding behind the mechanism, the surface energetic profiles of different processed coarse lactose and the subsequent changes to the surface energy profiles by the blending of fine lactose particles were characterized using IGC-FC 75) .The unmilled coarse lactose sample (LH100) showed less heterogeneity than the milled sample (LH250), whilst the fine particles (LH210) exhibited a higher surface energy than both the coarse and milled lactose.Upon loading different quantities of fine fraction (LH210) to milled lactose (LH250), the surface energy heterogeneity of the blend decreased with increasing fines content.It is postulated that homogenizing the surface properties of the carrier by the addition of fine excipient particles may create more consistent drugcarrier interactions, therefore increasing the stability of the formulation.In a recent study, IGC-FC was employed to study the impact of cr ystal morphology on the milling mechanism of cr ystalline solids using D-mannitol as a model compound 87) .Upon milling D-mannitol with a needle morphology, the crystals preferentially fractured along their shor test axis, exposing the (011) plane to create particles with greater hydrophilicity and a lower aspect ratio.The results are in contrast to attachment energy modeling which predicts a fracture mechanism across the (010) plane with increased hydrophobicity and a small change in particle shape.This study is the first in literature to show that the surface energy, γ d S V , can be reduced by milling, which typically results in a higher surface energy and entropy due to surface disorders induced as a result of high-energy processing.The reduction in sur face energy was caused by a geometrically driven fracture mechanism along the axis of least distance as opposed to a thermodynamic fracture process based on intrinsic cr ystal lattice energy.The study revealed that the surface properties are directly dictated by the mechanism of milling, leading to substantial changes in the surface chemistry -increase or decrease in surface polarity are both equally viable.This work also provides a fundamental scientific basis on how to investigate the anisotropic behavior of pharmaceutical solids, and its impact on processing behavior.Besides size reduction processes, IGC-FC has also been applied to study agglomeration processes in the example of high-shear wet granulation 15) .The initial phase in wet granulation is driven by successful nucleation of agglomerates from primary particles in the granulator by the liquid binder.These nuclei granules are initiated by energetically favorable wetting of the solid particles by the liquid binder, which is governed by interfacial thermodynamics.In a study on the wet granulation of an energetically heterogeneous model excipient and an energetically homogeneous excipient, clear distinctions between the cumulative granule size distributions can be observed by changing the ratio of the two components in the mixture.The study revealed that increasing the mass of low-energy energetically homogeneous content resulted in a reduction of the granule particle size.This was explained by the lack viscous contacts between particles of poorly wettable surfaces resulting in a low probability of particle coalescence, because the collision kinetic energy cannot be dissipated sufficiently without the presence of a liquid boundary layer.Furthermore, the microstructure of the granules was revealed to differ significantly, with granules of lower mechanical integrity generated from an increase of the low surface energy component in the mixture.

Conclusions
The anisotropic properties of pharmaceutical solids pose difficult challenges to API and drug product process development.A lack of understanding in the relationship between bulk powder behavior and its anisotropic properties can lead to difficulties in assessing the effects of material properties on process performance without significant experimental effort, which is inefficient and resource-consuming.To aid the prediction of process and formulation performance from material properties, traditional approaches have been based on idealized assumptions of particles.Nevertheless, rapid advancement in the field of materials science for pharmaceutical solids together with technological improvements in characterization, processing and modeling tools are attracting growing research efforts in the field of particle engineering.As discussed in this review, the surface properties of materials are important to help predict a wide variety of material behavior, processing and product performances in pharmaceutical development and manufacturing.Although IGC has been applied widely in the study of inorganic materials and polymers for over 50 years, it is still an evolving technology in pharmaceutical R&D.As reviewed in this work, IGC is a complimentary tool to provide additional information on the solid state properties, in particular the surface properties, of materials to aid the understanding of material changes under various conditions, as well as their behavior during processing.Experiments with IGC are often conducted at infinite dilution conditions, often probing the higher energetic sites.When the surface properties of crystalline solids are anisotropic, conventional approaches for the determination of surface energetics that yield a single value can be of limited use.Recently, a novel methodology using IGC at finite concentrations has been developed for the measurement of particulate surface energetic distributions.This methodology, extensively reviewed here, allows the surface energy of particulates to be determined at different surface coverage, overcoming drawbacks and limitations to the conventional approach of measuring surface heterogeneity using energy or potential distribution functions.Its applications have been demonstrated in a number of different case studies in this review.For a pharmaceutical crystalline material, it is the relationship between its crystallographic structure, solid state properties, and bulk and interfacial behavior that will determine its performance as a drug substance or drug product.

Fig. 1
Fig. 1 IGC chromatograms at a) infinite dilution b) finite concentration with ʻtailingʼ and c) finite concentration with ʻfrontingʼ.

Fig. 4
Fig. 4 Determination of a) acid-base enthalpy and entropy of adsorption from the acid/base free energy of adsorption via temperature variation, and b) the Gutmannʼs acid and base constants.

Fig. 3
Fig. 3 Determination of solid-vapor surface free energy and acid/base free energy change of adsorption from a) the Schultz approach and b) the Dorris-Gray approach.

Fig. 5
Fig. 5 Schematic of a) the PM method and b) the ECP method.