Powder Technology 247 (2013) 270–278

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An analysis of mass transport fluxes in titania-based mesoporous ceramic matrices

Anuradha Subramanian ⁎, Hemanth KaligotlaDepartment of Chemical Engineering, University of Nebraska, Lincoln, NE 68588-0643, USA

⁎ Corresponding author at: 207L Othmer Hall, Unive68588-0643, USA. Tel.: +1 402 472 3463; fax: +1 402

E-mail address: asubramanian2@unl.edu (A. Subram

0032-5910/$ – see front matter. Published by Elsevier Bhttp://dx.doi.org/10.1016/j.powtec.2012.08.007

a b s t r a c t

a r t i c l e i n f o

Available online 21 August 2012

Keywords:TitaniaMesoporous ceramic matricesHydrogel supportsPulse injectionHETPMass transfer resistances

Titania particles, with a particle size of 40±5microns, having pore sizes of 300 °A, 1000 °A and 2000 °A weremodified with Ethylenediamine-N,N′-tetra (methyl phosphonic) acid (EDTPA), to yield mesoporous ceramiccationic-exchange supports for potential use in bioseparations. Our objective is to understand the influence ofthe pore size on solute transport, and the solute binding phenomena in mesoporous titania based supports.Pulse injection techniques were used to elucidate and estimate the individual solute transfer parameters inmatrices with different pore sizes. Elution peaks were approximated to a Gaussian distribution and the cor-responding height-equivalent-of a theoretical-plate (HETP) was calculated. Elution profiles obtained underretained and unretained conditions were used to estimate the corresponding HETP contribution. Using equa-tions in the literature, the theoretical numbers of transfer units (dimensionless groups) were estimated inorder to identify the dominant transport mechanism for the adsorption processes.In the case ofmesoporous ceramicmaterials, pore diffusionwas established to be the rate limiting process for theadsorption of the solute. When the pore size nearly equaled the solute size, the diffusion phenomena seemed tobe hindered; as pore size increased, there seemed to be an enhancedmode of solute transport in these matrices.

Published by Elsevier B.V.

1. Introduction

Design of supports and the separation methodologies require abetter understanding of the static and dynamic binding capacities ofthe target protein(s) and their mass transfer properties within thesupport particles [1–4]. Knowledge of the binding capacities, adsorp-tion rates and solute mass transfer characteristics is often essential forthe optimization of chromatographic processes [5–8]. Chromato-graphic supports and their transport features have been extensivelydiscussed in the literature and numerous rate, transport and masstransfer models have been developed [8–10]. In interactive chroma-tography, it is necessary to use supports with pores large enough toallow the non-restricted access of proteins, to avoid or minimize dif-fusional limitations [11–15]. The accessibility of pore volume and in-ternal surface by a solute molecule and the free diffusion ofmacromolecules are some of the features that characterize a goodsupport and are the dominant factors that contribute to the bindingof the solute [15,16]. Optimization of a chromatographic process re-quires a compromise between particles with high surface area andgood mass transfer or fluid-flow properties of the support matrices[17,18]. There is always a tradeoff between the area and volume inorder to achieve the proper surface area-to-volume ratio [6].Reducing resistance to mass transfer by using various types ofchromatographic particles modified through their structure, shape,

rsity of Nebraska, Lincoln, NE472 6989.anian).

.V.

and size, as well as eliminating diffusional limitations is therefore arelevant objective in chromatographic optimization.

Traditionally, mass transfer studies have been carried out by frontalchromatography where the estimation of mass transfer resistance in-volves elaborate measurements and a prior knowledge of the adsorp-tion isotherm, and is usually based on the numerical interpretation ofbreakthrough curves [19–29]. Alternatively, in elution chromatographythe determination of mass transfer parameters based onmoment theo-ry in combination with pulse analysis has also been employed[10,30,31]. To derive the rate parameters from pulse injections, equa-tions relatingheight-equivalent-of-a-theoretical-plate (HETP) to opera-tional parameters are needed. Pulse injections under retained andunretained conditions are employed to estimate the transport proper-ties for different types of stationary phase materials. Dimensionlessgroups are then constructed to evaluate the relative contributions ofthe various transport mechanisms.

We are interested in better understanding the rate-limiting mecha-nisms that govern the transport of biomolecules in meso- and macro-porous supports and further research the concept of pore-to-solutesize ratio. Titania particles, with a uniform particle size of 40 micronsand with pore sizes of 300 °A, 1000 °A and 2000 °A, were modifiedwith ethylenediamine-N,N′-tetra (methyl phosphonic) acid (EDTPA)to yield mesoporous ceramic supports, designated in this study asR-PET. An effort was made to understand the influence of the poresize on themode of solute transport and the solute binding phenomenain these supports. Cytochrome C (Cyt-C) and human immunoglobulin Gwere the two test proteins used in the present study. In this researchstudy, the contributions to the mass transfer mechanisms that occur

271A. Subramanian, H. Kaligotla / Powder Technology 247 (2013) 270–278

during the adsorption of HIgG and Cyt C to R-PET have been investigat-ed by pulse injection techniques. The experimental setup was tailoredto operate in the linear adsorption regime of the dynamic adsorptionisotherm. This research examines the extent to which mass transfercan be regulated by available pore volume. It is postulated that the com-bination of convective and diffusive transport modes increases masstransfer rates and causes rapid solute transport in porous chromato-graphic sorbents. Hence, solute transport and themodeling of the trans-port phenomena in different matrices are primary requirements toachieve mass transfer performance enhancement.

2. Materials and methods

All chemicals were of analytical grade or better. Sodium chloridewas purchased from Fischer Scientific (Hanover Park, IL, USA). N, N,N′, N′-Ethylenediaminetetra methylenephosphonic acid (EDTPA)was purchased from TCI America (Portland, OR, USA). 2-morpholinoethanesulfonic acid monohydrate was purchased fromSigma-Aldrich (St. Louis, MO, USA). Potassium phosphate monobasic(KH2PO4) and potassium phosphate dibasic anhydrous (K2HPO4) werepurchased from Fisher Scientific (Hanover Park, IL, USA). Bovine serumalbumin (BSA), pure human immunoglobulin G, cytochrome c wasobtained from Sigma Chemical Company (St. Louis, MO, USA). Allproteins and reagents were usedwithout further purification. An Agilent1100 series™ model from Agilent Technologies Instruments UV-visiblespectrophotometer was used to record the adsorption measurements.A bench top microcentrifuge (Eppendorf Centrifuge 5415C) was usedto sediment the r-PET particles for batch experiments. The equationsused to model and validate various parameters are listed in the theorysection.

EDTPA modified titania particles were supplied by Zirchrom Inc.,(Anoka, MN) and were further used in this study without furthermodification. The supports were 40 microns in size and with poresizes of 300 °A, 1000 °A, and 2000 °A. Titania particles were packedinto a 0.46 cm (i.d.)×5.0 cm (height) column HPLC column byZirchrom Inc (Anoka, MN) and the volume of the packed columnwas estimated to be 0.83 ml.

2.1. Adsorption isotherms

Batch experiments (static isotherms) were conducted in order todetermine the maximum binding capacity and the equilibrium disso-ciation constants, and the procedure detailed elsewhere was followedto carry out the experiments [29,30].

The procedure detailed elsewhere [29,30]was followed to obtain thedynamic binding capacity at various feed concentrations, at a linear ve-locity of 0.2 cm/s. Briefly, a solution of pure protein at various concentra-tions ranging from 0.5 mg/ml to 10 mg/ml in loading buffer (LB, 20 mMMES, 0.025 MNaCl, 4 mM EDTPA, pH 5.5) was column-fed until the ab-sorbance of the effluent reached 80% of the inlet concentration. The col-umn was then washed with loading buffer until the absorbance at280 nmreached the baseline. The adsorbed proteinwas elutedwith elu-tion buffer (20 mMMES, 1.0 MNaCl, 4 mMEDTPA, pH5.5) (EB). For thelinear velocity the dynamic capacity of the column was determined asthe amount of protein adsorbed per milliliter of bead.

2.2. Chromatography

For all samples, 50-μl pulse injections were made manually to thechromatographic system. The system consisted of an Agilent 1100 se-ries. Injected Protein was monitored at 280 nm by a online spectrom-eter. Sodium Nitrate and Blue Dextran were monitored at anabsorbance of 300 nm and 280 nm respectively. The absorbance ofthe feed and fractions were also measured at 280 nm using the spec-trophotometer. All buffer solutions were filtered through ChromTech's Metal-Free solvent (type A-427) 10 μm UHMWPE (Ultra High

Molecular Weight Polyethylene) membrane filter during the time ofuse. Elution of bound protein and regeneration of the column wascarried out using elution buffer.

2.3. Interstitial, intraparticle porosity and extra-columncontribution determination

Pulse injections of 50-μl were made with blue dextran at a concen-tration of 0.5 mg/ml to estimate the packed bed or interstitial poros-ity under unretained conditions (i.e. dissolved in EB). To determinethe intra-particle porosity, sodium nitrate at a concentration of0.01 M was pulsed into the system. Interstitial porosity and intra-particle porosity were determined from the first moments obtainedunder various flow rates using blue dextran and sodium nitrate, re-spectively, using the equations listed elsewhere [30]. In order to de-termine the extra column contributions to HETP, pulse injections ofprotein dissolved in EB were made at flow rates of 0.025, 0.05, 0.1,0.2 and 0.4 cm/s with the column off-line, by connecting the up-stream and downstream tubing with a coupling unit. The first andsecond moments of the resultant peaks were calculated and theHETP contribution of the system estimated by equations listed inthe theory section and also reported elsewhere [30].

2.4. Modeling and simulation to obtain retained and unretained HETP

The first moments for the elution peaks obtained under unretainedand retained conditions are important as they determine the residencetimes (tr). Briefly, HIgG or Cyt-C was dissolved in Loading Buffer, 4 mMEDTPA, 20 mMMES; (further referred to as LB) with various concentra-tions of salt. Salt concentrations of 0.04, 0.05, 0.075, 0.1, 0.1025, 0.15 and1 M were used. Pulse injections were made at superficial linear veloci-ties of 0.025, 0.05, 0.1, 0.2 and 0.4 cm/s. Sample bound were elutedusing EB and the profiles recorded. The first and second moments ofthe eluted profileswere estimated from thefit of their Gaussian profiles.The total HETP of the eluted peakwas determined by using Eq. (12). TheHETP contribution by the column alone was obtained after eliminatingextra column effects, H'=Htot−Hec.

A plot of H' verses linear velocity under unretained conditions per-mits the calculation of Dp and kf using Eq. (7) and the values of εi andεp obtained from the porosity studies. In order to do this, the equationdefining unretained HETP was fit to the data by a program written inthe MATLAB environment. In this method, the intercept of the dataplotwas initially estimated by simple linear regression and subsequent-ly kept constant as the constraint in the optimization routine. Values ofb0 were determined analytically using Eq. (11) using the first momentsof the elution peaks that were recorded earlier. The values of Dp and kfobtained from unretained HETP, were assumed not to vary with con-centration and were used to curve fit Eq. (9) for the retained peaks.

For retained peaks, the actual HETP contribution was determinedas Hactual=H'−Hfilm, where Hfilm was determined as an averagevalue from Eq. (3). An approach similar to the unretained data wastaken for the retained data. Namely, the intercepts of the plots werekept as the constraints. After performing constrained optimizationusing Eq. (9), the values of r and kdes were obtained.

3. Results and discussion

Titanium dioxide is virtually insoluble in acids as well as bases,thus sorbents based on titania enable separations to be performedat extreme pH values and the ability to withstand cleaning in place(CIP) procedures with 1 M NaOH is of special interest for preparativeHPLC. The surface chemistry of titania is quite different from the moretraditional silica and polymeric stationary phases, but is closely relat-ed to zirconia. The surface concentration of Lewis acid sites on bare ti-tania is estimated to be 4 to 6 μmol groups per m2 [32], and ifunmodified- or untitrated-titania surfaces were used, non-specific

272 A. Subramanian, H. Kaligotla / Powder Technology 247 (2013) 270–278

adsorptions and peak-tailing was typically observed. EDTPA is a spe-cial chelator that renders Lewis acid deactivated on both titania andzirconia particles [32,33]. Additionally, our previous work hasshown that the modification of porous particles with EPTPA doesnot impact the average pore diameter [30]. Thus in the presentstudy, 40 μm titania particles modified with EDPTA, to yield a station-ary phase with cation-exchange properties, were used. As noted inthe methods section EDTPA modified titania particles were providedto us by Zirchrom Inc and used without any further modification. Ti-tania sorbents (unmodified) and EDPTAmodified titania sorbents (i.e.R-PET) exhibited a unimodal pore size distribution with pore diame-ter distributions similar to that published elsewhere [32]. BET surfaceareas of 20.68, 10.06 and 5.53 (m2/g) were obtained for 300 °A,1000 °A and 2000 °A beads, respectively. Additionally, 4 mM EDTPAwas always included in the buffers to ensure that the surface concen-tration was maintained. The physical properties of the base titaniaparticles are detailed elsewhere [33].

3.0.1. Chromatogram of a Dextran pulse and Gaussian fit to experimentalpeak profile

Elution peaks were approximated to a Gaussian distribution [30]and the respective first and second moments were obtained by usinga curve fitting routine from MATLAB. The interstitial porosities andintra-particle porosities were estimated from the first moments of thepeaks obtained with dextran and sodium nitrate injections, respective-ly. A column porosity of 0.39 was calculated, which was in good agree-mentwith the values provided to us by Zirchrom Inc., (Anoka, MN). Theintraparticle porosity values of 0.47 for 300 °A, 0.55 for 1000 °A, and0.62 for 2000 °A, respectively, were also estimated.

3.0.2. Static binding isotherms and estimation of specific binding capacityLigand binding isotherms were observed to follow the saturation

pattern as predicted by the pseudo-Langmuir model and the bindingparameters were obtained by theoretically fitting the adsorption curveto the Langmuir equation. Static binding capacities of 70.09±8 mg ofCyt-C/ml of beads, 39.71±5 mg of cyt-C/ml of beads and 28.35±3mg of Cyt-C/ml of bead, were obtained for matrices with pore sizes300 °A, 1000 °A and 2000 °A, respectively. Static binding capacities of63.22±5 mg of HIgG/ml of beads, 47.05±3 mg of HIgG/ml of beadsand 31.91±4 mg of HIgG/ml of beads were obtained for matriceswith pore sizes 300 °A, 1000 °A and 2000 °A, respectively. For the bind-ing of Cyt-C to R-PET, the equilibrium-binding constant (Kd_static) wasfound to range from 5.77E−05 to 1.37E−05M. For the binding ofHIgG to R-PET, the equilibrium-binding constant (Kd_static) was foundto range from 9.34E−06 to 3.00E−06M. A relatively larger bindingcapacity was obtained for Cyt-C when compared to HIgG on the nega-tively charged R-PET matrices, as at a pH of 5.5 the net charge onCyt-C was probably higher due to the differences of the pI CytochromeC (pI 10.7) and HIgG (pI 7.1). The static binding capacities seemed to

Table 1Specific binding capacity (S) on R-PET.

Pore size (°A) Specific binding capacity (SBCa) Total s

Cytochrome c HIgG Area (

300 0.273 0.020 20.6801000 0.318 0.031 10.0622000 0.414 0.038 5.528

SCyt-C'=0.366 SHIgG'=0.0345

a Specific binding capacity (SBC) in micromol/m2 was calculated by dividing static bindinthe support. Total surface area was obtained from BET analysis of the supports and static c

b Kac: Protein‐specific surface accessibility coefficient is defined as the space within the mausing equation; Kac=S/S′. S′ is the specific protein binding capacity in a non‐size discriminatKac*At, where At is the total surface area obtained from BET measurements.

decreasewith the increase in pore size for both the proteins, suggestingthe decrease of total available surface area. In order to determinewhether there were any cooperative effects due to protein–protein in-teractions during the adsorption process, isotherm data (from static ex-periments) were analyzed by a Hill-plot using the transformedLangmuir equation [33]. A co-operativity coefficient of unity indicatedno cooperativity. Analysis of isotherms for both Cyt-C and HIgG onR-PET matrices indicated n values of unity, which suggests that thereis no cooperativity due to protein–protein interactions.

3.0.3. Influence of pore diameter on specific binding capacity and acces-sible surface area

Specific binding capacity values were calculated from the staticisotherm data to better relate protein accessibility of the internal sur-face area of the R-PET supports using the equations elsewhere[18,34]. Specific binding capacity of Cyt-C increased from a value of0.273 obtained for 300 °A to a value of 0.414 obtained for 2000 °Abeads. Similarly, specific binding capacity of HIgG increased from avalue of 0.020 obtained for 300 °A beads to a value of 0.038 obtainedfor 2000 °A beads. The capacity increased from 300 °A to 2000 °A forboth Cyt-C and HIgG. When compared individually, Cyt-C had morespecific binding than HIgG in all the supports evaluated.

Protein accessibility coefficients and total surface areas (At) wereused to calculate the surface area accessible (Aac) to these two pro-teins using the equations listed elsewhere [18,34]. A summary ofthe values is listed in Table 1. The accessible surface area of Cyt-C ishigher than HIgG, suggesting the importance of size in the accessibil-ity of the interior loading space of the r-PET supports. The r-PET sup-ports of pore diameter 1000 °A, 2000 °A have similar Aac for bothCyt-C and HIgG. The supports having≥1000 °A pore diameter donot seem to distinguish proteins on the basis of size in the range test-ed, but supports with 300 °A pore diameter clearly are not equally ac-cessible to both the proteins. With Cyt-C, a protein-specific surfaceaccessibility coefficient, which denotes complete internal accessibilityof the matrix to the target protein for binding, of 1.0 was obtained forthe three matrices evaluated. With HIgG, Kac values ranging from 0.8to 1.0 were obtained with 1000 °A and 2000 °A matrices, except inthe case of 300 °A in HIgGwhere a Kac 0.5 was obtained, indicating re-duced accessibility of the matrix. The accessible surface area for Cyt-Cdecreased from 16.54 in 300 °A to 5.53 in 2000 °A, and in the case ofHIgG it decreased from 10.33 in 300 °A to 5.53 in 2000 °A.

3.0.4. Pulse analysis of CYT-C and HIgG on R-PET matricesWe have employed pulse analysis in the estimation of HETP

values, an approach analogous to many papers in the literature[30,31]. Prior to pulse analyses, adsorption isotherms were obtainedin column-flow throughmode (i.e. dynamic isotherms) for the uptakeof Cyt-C and HIgG and experiments were carried out at a feed concen-tration of 0.5 mg/ml.

urface Accessible surface area(Ac)

Protein-specific surfaceaccessibility coefficient,(Kac

b)

m2/g) (m2/g) Cyt C HIgG

Cyt C HIgG

16.54 10.33 0.745 0.5809.055 9.055 0.869 0.8985.528 5.528 1.0 1.0

g capacity (expressed as micromol protein/g of support) by total surface area (m2/g) ofapacity was obtained from fitting the isotherm data o a pseudo‐Langmuir model.trix which is completely accessible to the target protein for binding, and was calculateding system (under saturating conditions). Accessible surface area (Ac) was calculated as

273A. Subramanian, H. Kaligotla / Powder Technology 247 (2013) 270–278

3.0.5. Determination of unretained HETP on R-PET with varyingpore sizes

The peak profiles obtained with pulse injections of Cyt-C and HIgGunder unretained conditions (i.e. buffer with 0.5 M NaCl) were ap-proximated by the Gaussian equation and the corresponding HETPwas then calculated using Eqs. (7) and (12) listed in Theory section.The dependence of HETP values obtained under unretained condi-tions on the linear velocity, for Cyt-C is shown in Fig. 1a. For Cyt-Cunder unretained conditions, an intercept value of 0.235 and 0.307,and a slope value of 1.522 and 0.982 were noted for 300 °A and

0.0

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A

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pore size 300 A

pore size 2000 A

R² = 0.9079

R² = 0.9766

R² = 0.8622

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2000 °A, respectively. Additionally, similar slopes and interceptswere noted for supports with 1000 °A and 2000 °A when testedwith Cyt-C under unretained conditions. The dependence of HETPvalues obtained under unretained conditions on the linear velocity,for IgG is shown in Fig. 2a. The intercept values for HIgG from300 °A to 2000 °A were 0.168, 0.231, and 0.181 and the slopes were3.699, 2.755 and 2.111. All values were determined by linear regres-sion analysis and lines depicted are the best fit to the data. The initialslope of this plot is related with intraparticle diffusivity, increase inthe slope to pore diffusivity, and formation of the plateau to finallyconvection-controlled phenomena. The rate-limiting mass transportstep, which is diffusion in most of the porous media, seems to bethe dominant mechanism here. An additional thrust arises from theease of transport due to the size of pore and protein. This additionalthrust [35,36] seems to be attributed to enhancement of diffusivityby convection called the “augmented” diffusivity. The augmented dif-fusivity is the sum of effective diffusivity from diffusion (pore or sur-face) and a convection term. The presence of this fact is furtherelucidated from the values of the pore diffusion coefficients of cyto-chrome c, which increased from 1.45E−07 to 2.58E−07.

Curve fitting with the SOLVER optimization routine using Eq. (7) inthe theory section was performed by keeping the intercept as a fixedvalue (as obtained from the regression model) and as a constraint.The values of the various parameters were then optimized. The axialdispersion parameter (Da), the film mass transfer coefficient (kf), andthe pore diffusion coefficient (Dp) were then calculated for Cyt-C andHIgG on the R-PET supports of different pore sizes. A summary of allthe parameters obtained is listed in Table 2. Table 2 (columns 4 and 5)lists the diffusion coefficients of both the proteins as function of poresize of the matrices, and shows an increase in value with the poresize. Hfilm for the system were also calculated. The values obtained inthese steps were used in the subsequent calculations.

In determining the parameters in Table 2, it was assumed that porediffusive flux was independent of feed concentration. The axial disper-sion parameter ξ from the intercept values obtained from the linearHETP plot and tortuosity factor τ from the values of the pore diffusioncoefficientwere calculated. The dispersion parameter and the tortuositywere more or less the same in all the supports with both the proteins.The value of kf was estimated and approximated as detailed elsewhere[37], but the value was further optimized by the optimization routinerepeatedly until the error function was minimized, as the value wasvery approximate. The linear dependence of the kf was taken into con-sideration and the film mass transfer resistances were calculated, asthey contributed to the intraparticle mass transfer resistance. Thevalues were in the same range for both the proteins, as the size of theprotein does not contribute to the film mass transfer.

The size of the two model protein molecules is an important con-sideration on the binding capacity. If each protein had equal access tothe internal surface area of all thematrices, therewould be aminimaldifference in accessible surface area or in protein-specific surface

Fig. 1. A. HETP of the R-PET column for Cytochrome C under unretained conditions as afunction of linear velocity. The values of Dp and kf were calculated and the intercept ofthe data plot is found by linear regression and subsequently kept constant. The valueswere used for the curve fitting of HETP profiles under retained conditions. The mobilephase used is 4 mM EDTPA, 20 mM MES, 1 M NaCl at pH 7 for unretained conditions. B.RetainedHETP for Cyt C, Pulse injections aremadeunder various combinations of salt con-centrations and superficial velocities and further different superficial velocities wereexperimented at a given salt concentration. This is repeated for several salt concentra-tions. Gaussian peaks are fitted to the elution profiles and the HETP calculated with theslope of the plot recorded. The first moments of the resultant peaks are employed to es-timate the b0 for a given salt concentration. Cytochrome Cwas injected into the R-PETcolumn and the salt concentration changed by the Loading buffer salt. C. Variation ofthe slope of the HETP curves (S) as a function of bo on the R-PET supports of differentpore sizes. The evaluation of the slope, S, at the various salt concentrations can beused to readily calculate r and kdes. The shape of the curve S vs. bo provides a signifi-cant insight into the importance of surface diffusion for a material. Cytochrome Cplots for the supports of 300 and 2000A pore size.

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50Linear Velocity (cm/s)

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R² = 0.9775

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274 A. Subramanian, H. Kaligotla / Powder Technology 247 (2013) 270–278

accessibility coefficients for a given protein on different pore diame-ter matrices. While the protein-specific surface accessibility coeffi-cients for the smaller protein, Cyt-C with a hydrodynamic diameterof 39 °A, are reasonably independent of pore size, for HIgG that hasa molecular dimension of 110 °A, it was observed to vary with thepore diameter. In the literature, as predicted by the Renkin's equa-tion, the optimum value for ratio of the pore diameter (Dpore) to sol-ute diameter (Dsolute) is around 5 [17]. In the case of HIgG, the ratioDpore/Dsolute was 10 and 20 for matrices having pore sizes of1000 °A and 2000 °A, respectively, thus similar values of Dp wereobtained. In the case of 300 °A support, the ratio was 3, hencethree-fold lower values of Dp were obtained (Table 2). The resultssuggest that large pore supports are required for the uniform andcomplete accessibility of proteins. Consequently this requirestradeoff on the total available surface area for binding.

3.0.6. Determination of retained HETP on R-PET with varying pore sizes,as a function of salt concentration

The peak profiles obtained with pulse injections of Cyt-C and HIgGunder retained conditions were approximated by the Gaussian equa-tion and modeled with the curve fitting toolbox in MATLAB to obtainthe first and second moments of the elution peaks. The correspondingHETP was calculated using Eqs. (1)–(3) listed in Theory section. Pulseinjections were made under various combinations of salt concentra-tions and superficial velocities. It was assumed that the variance in theHETP contribution due to film mass transfer was negligible under therange of the linear velocities of operation. The variance of HETP forCyt-C and HIgG, on R-PET particles with a pore size of 300 °A, with su-perficial linear velocity and salt concentration is shown in Figs. 1B and2B, respectively.While select curves are shown for the 300 °A supports,HETP under retained conditions were obtained at a variety of salt con-centrations ranging from 0 mM to 150 mM. HETP computed underretained conditions was observed to increase with an increase in veloc-ity at a given salt concentration for Cyt-C on matrices with pore sizes of300 °A and 2000 °A. HETP plots computed under retained conditionsfor HIgG onmatrices with pore sizes of 1000 °A and 2000 °A were sim-ilar to Fig. 2B. A summary of the relevant parameters obtained fromdatamodeling and analyses are shown in Table 3.

3.0.7. Variation of the slope of the retained HETP plots as a function of boThe slope, S, of the HETPret versus the u plot, as a function of salt

concentration, for Cyt-C and HIgG (except for 300 °A), is shown inFigs. 1B and 2B, respectively. Thus, by evaluating the slope, S, at thevarious salt concentrations and the first moments of the peaks forb0 for a given salt concentration using Eqs. (10) and (11), we canreadily calculate r and kdes, as per Eq. (9). The values obtained fromr and kdes were used to determine an overall plot of slope and its var-iance with respect to bo. The S versus bo plots for Cyt-C and HIgG, onmatrices with different pore sizes are shown in Figs. 1C and 2C, re-spectively. The model fit is shown by solid or broken lines and

Fig. 2. A. HETP of the R_PET column for HIgG under unretained conditions as a functionof linear velocity. The values of Dp and Kf were calculated and the intercept of the dataplot is found by linear regression and subsequently kept constant. The values wereused for the curve fitting of HETP profiles under retained conditions. The mobilephase used is 4 mM EDTPA, 20 mM MES, 1 M NaCl at pH 7 for unretained conditions.B. Retained HETP for HIgG, Pulse injections are made under various combinations ofsalt concentrations and superficial velocities and further different superficial velocitieswere experimented at a given salt concentration. This is repeated for several salt con-centrations. Gaussian peaks are fitted to the elution profiles and the HETP calculatedwith the slope of the plot recorded. The firstmoments of the resultant peaks are employedto estimate the b0 for a given salt concentration. HIgGwas injected into the R-PET columnand the salt concentration changed by the Loading buffer salt. C. Variation of the slope ofthe HETP curves (S) as a function of bo on the R-PET supports of different pore sizes. Theevaluation of the slope, S, at the various salt concentrations can be used to readily calculater and kdes. The shape of the curve S vs. bo provides a significant insight into the importanceof surface diffusion for amaterial. HIgG plots for the supports of 300, 1000 and 2000Aporesize.

Table 2Results from the analyses of the chromatographic data obtained under unretained conditions.

Pore size (°A) Slope of HETPplot (s)a

Dpb (cm²/s) ζc (cm) Γd kfilm

e (cm/s) Hfilmf (cm)

Cyt C HIgG Cyt C HIgG Cyt C HIgG Cyt C HIgG Cyt C HIgG Cyt C HIgG

300 1.523 3.699 1.453E−07 5.751E−08 0.5 0.42 3.91 3.43 1.57E−03 1.44E−03 2.58E−03 4.76E−031000 – 2.755 – 8.275E−08 – 0.578 – 2.79 – 1.42E−03 – 4.85E−032000 0.982 2.111 2.586E−07 1.158E−07 0.767 0.452 2.87 2.24 2.68E−03 1.42E−03 2.64E−03 4.87E−03

a Value obtained from the linear regression analysis of the HETP vs. Linear velocity plot from unretained conditions (Eq. (7) in Appendix).b Value obtained from the value of the slope from linear regression analysis and with the constants (Eq. (7) in Appendix).c Value obtained from the linear regression analysis of the HETP vs. Linear velocity plot from unretained conditions (Eqs. (7) and (17) in Appendix).d Value obtained from the linear regression analysis of the HETP vs. Linear velocity plot from unretained conditions (Eqs. (7) and (16) in Appendix).e Value obtained from the calculation from (Eq. (14) in Appendix).f Value obtained from (Eq. (3) in Appendix).

A

B

0

50

100

150

200

250

300

350

Peclet Number

Red

uced

HE

TP

300 A

2000 A

600

0 200 400 600 800 1000 1200

275A. Subramanian, H. Kaligotla / Powder Technology 247 (2013) 270–278

open/filled circles, squares and triangles represent the experimentallyobtained data points. A summary of the parameters obtained bymodel analyses and curve fitting to Eq. (9) are shown in Table 3.The concentration of the salt in the buffer is an important consider-ation; with increasing salt concentrations adsorption process is clear-ly inhibited. At low solute concentrations, protein molecules disperseminimally in the axial direction and contribute more toward convec-tive effects. As the concentration increases, the parameter depen-dence on the concentration increases, making the estimation ofparameters complex. According to the equations, the axial dispersionterm is independent of bo. The values of r and kdes obtained for thetwo proteins on the R-PET different pore size supports are presentedin Table 3. The ratio of pore to surface diffusion, r, has a significant ef-fect on the shape of the S vs. bo curve (Figs. 1C and 2C). The curvesmonotonically increases and then flatten out, signifying the absenceof surface diffusion or, if present, its negligibility (i.e. Ds=0).

3.1. A plot of reduced plate height with Peclet number

Fig. 3A andb show the plot between reduced plate heights versus re-duced velocities for cytochrome C and HIgG, respectively. The plot has areduced plate height on its Y-axis which has been calculated from themoments obtained for the Gaussian fit to the experimental peakprofile [36]. At lower Peclet numbers, linearity exists in the curve onall the supports with cytochrome c and human IgG. As the Pecletnumber increases the curves tend to flatten and a plateau region isformed, except for the 300 °A pore size support where the curve islinear throughout. Thesemesoporous R-PET supports exhibit diffusionalmass transfer at lower pore sizes. As the pore size starts increasing from300 °A to 2000 °A, the particles seem to behave like gigaporous sup-ports exhibiting both diffusion and convection in the pore volume. If sig-nificant convection occurs within these particles, an increase in theapparent diffusivity would be realized at high Peclet numbers (Table. 4).

Dimensionless parameters were estimated from the several param-eters calculated to ascertain the importance of the various transport

Table 3Results from the analyses of the chromatographic data obtained under retainedconditions.

Pore size (A) ra kdesb

Cyt C HIgG Cyt C HIgG

300 0.004 0.092 1537 19751000 ND 0.005 – 20592000 0.0025 0.004 5119 2345

ND, not determined.a Value calculated from the regression analysis of Eq. (9) in Appendix.b Value calculated from the regression analysis of Eq. (9) in Appendix.

mechanisms. Table 3 presents the values of dimensionless groups forthe two model proteins. The dimensionless groups relate the rates ofthe various transport mechanisms to the convective transport rate.The lowest value of the dimensionless group gives the dominantnon-ideality of the system. From the values of the dimensionlessgroups, it can be seen that the pore diffusion process is the rate limitingstep for this system. The dimensionless groups have the linear velocityin the data with the exception of the Np to enable direct comparisonof the data at any velocity. The values of Np increased with the poresize, and they differ in many orders of magnitude from Ndes. Ns wasnot calculated as there is very little or no surface diffusion. Surface dif-fusion and desorption kinetics are too rapid to be measured at highsalt concentrations. This analysis helps in establishing a transportmodel for the given system with the dominant transport mechanismbeing the pore diffusion.

4. Conclusions

A simple methodology was used in estimating themass transfer pa-rameters and the coefficients. The peak profiles obtained with pulse

0

100

200

300

400

500

0 500 1000 1500 2000 2500

Peclet Number

Red

uced

HE

TP

300 A

1000 A

2000 A

Fig. 3. A. Variation of the reduced plate height with Peclet number for Cytochrome C onthe 300 and 2000A R-PET supports. B. Variation of the reduced plate height with Pecletnumber for HIgG on the 300 1000 and 2000A R-PET supports.

Table 4NTU contributions for cytochrome-c and HIgG on R_PET matrices.

Pore size (A) Np Ndes Nfilm Npe

Cyt C HIgG Cyt C HIgG Cyt C HIgG Cyt C HIgG

300 0.32/u 0.127/u 7685/u 9875/u 15.7/u 14.4/u 10 11.51000 – 0.1838/u – 10295/u – 14.2/u – 8.652000 0.57/u 0.257/u 25595/u 11725/u 26.8/u 14.2/u 6.5 11.66

Dimensionless groups (NTU) Description

Np=(DpL)/(R2u) Pore diffusion: convective transferNfilm=(3kfL)/(Ru) Film transport: convective transfer1/Npe=(Da)/(Lu) Axial dispersion: convective transferNdes=(kdesL)/(u) Desorption kinetics: convective transfer

276 A. Subramanian, H. Kaligotla / Powder Technology 247 (2013) 270–278

injections under unretained and retained conditions were approximat-ed by the Gaussian equation and modeled with the Curve fitting toolbox in MATLAB for parameters (first and second moments of thepeak) as explained and the corresponding HETP was calculated. Byfitting the plots using the developed equations, one could then estimatethe relevant transport properties for a given protein on a specific sup-port system. Pore diffusion was determined as the rate limiting masstransfermechanism in the R-PET supports and the diffusion coefficientsprovided insight into the phenomena of augmented diffusivity. The factwas further established by the magnitude of pore diffusion coefficientsincreasing with the pore size, in the same order as bulk diffusioncoefficients.

Estimation of solute transport parameters by pulse analysis haspaved a way in understanding the importance of pore size and masstransfer coefficients in supports during preparative separations. Sol-ute transport and the modeling of the transport phenomena have fa-cilitated scale-up and design. A clear understanding in this area alongwith the particle size of the support gives us a powerful tool inselecting the right matrix with the right properties for better separa-tion efficiency, resolution, capacity and mass transfer phenomena.

4.1. Theory section

The HETP contribution by the column alone (H') was obtainedafter eliminating extra column effects,

H′ ¼ Htot−Hec: ð1Þ

For retained peaks, the actual HETP contribution was determined as

Hactual ¼ H′−Hfilm ð2Þ

Where Hfilm was determined as (where kf values were determinedanalytically from experimental data of the unretained elution pro-files).

Hfilm ¼ 2 1−εið Þεpuεi þ 1−εið Þεph i R

3kf

� �ð3Þ

In this paper the reaction-dispersive model was investigated. Thefollowing equation relates the effect of salt concentration and linear

velocity to the total HETP (without extra column HETP contribution)[31]:

H ¼ 2Da

uL

þ 2 1−εið Þεpb02uεi þ 1−εið Þεpb0

n o2

R3kf

þ R2

15Dp 1þ b0−1f grð Þ þb0−1ð Þb0

2kdes

" #ð4Þ

Where εi is the intra-particle porosity, R the radius of the matrixparticle, Dp the pore diffusivity, kdes is the desorption rate constantand r and b0 are defined as

r ¼ Dp

Dsð5Þ

Ds is the surface diffusion coefficient.

b0 ¼ 1þ k0 ð6Þ

And k' is the mass distribution ratio. Determination of k' values forthe system have been discussed in the later part of this section.

Under unretained conditions, b0 is equal to 1 as no adsorption ofsolute to the matrix occurs (i.e. k'=0) and Eq. (1) simplifies to [31]:

H ¼ 2Da

uLþ 2 1−εið Þεpu

εi þ 1−εið Þεpn o2

R3kf

þ R2

15Dp

" #ð7Þ

For retained conditions, subtracting the HETP contributed by filmmass transfer, Eq. (3) becomes [31]:

H ¼ 2Da

uLþ 2 1−εið Þεpb02u

εi þ 1−εið Þεpb0n o2

R2

15Dp 1þ b0−1f grð Þ þb0−1ð Þb0

2kdes

" #ð8Þ

The slope of Eq. (8) is a function of b0, which maybe written afterdifferentiating it with respect to u as,

S ¼ 2 1−εið Þεpb02

εi þ 1−εið Þεpb0n o2

R2

15Dp 1þ b0−1f grð Þ þb0−1ð Þb0

2kdes

" #ð9Þ

4.2. Porosity determination

The porosity of the column is related to the first moment and lin-ear velocity as

μ1 ¼ Lu

εi þ 1−εið Þεpbo� �

ð10Þ

277A. Subramanian, H. Kaligotla / Powder Technology 247 (2013) 270–278

Rearrangement of Eq. (10) allows the calculation of b0 as follows:

b0 ¼ 11−εið Þεp

μ1uL−εi

h ið11Þ

Where L the length of the column, u is the linear velocity, εi is theinterstitial porosity and εp is the intra-particle porosity and bo is theparameter reflecting retention factor. Under unretained conditionsbo is equal to 1 by definition.

4.3. HETP determination

The elution profiles obtained were approximated with a Gaussianprofile and the first and second moments were determined. The totalHETP of the Gaussian profile was determined using the followingequation

Htot ¼L

5:54tw;1=2

tr

� �2ð12Þ

Where tw,1/2 is the width of the Gaussian profile at half height andtr is the retention time.

The extra column contribution was determined by the followingequation:

Hec ¼ LσecFVobo

� �2ð13Þ

Where σec is the second moment of the resultant peak, Vo is thecolumn dead volume, bo is the mass partition coefficient (in thiscase equal to one as all species are non‐binding) and F is the flow rate.

The film mass transfer coefficient (kf) was estimated using thewell-established correlation

Sh ¼ 2þ 1:45Re1=2Sc1=3 ð14Þ

The molecular diffusivity of the proteins was determined from thesemi-empirical equation of Polson based on the Stokes–Einsteinequation.

Dm ¼ 9:4� 10−15 T=μMw1=3

h ið15Þ

The following relation between Dp and Dm was used in calculations:

Dp¼εp Dm=τtor ð16Þ

The following relation between axial dispersion parameter

Da ð17Þ

and u was used in calculations Da=ζ(u)

List of symbols used

b0 mass partition coefficientdp diameter of the particle, cmDa axial dispersion co-efficient, cm2/sDm molecular dispersion co-efficient, cm2/sDp pore diffusion co-efficient, cm2/sDs surface diffusion coefficient, cm2/sDsolute diameter of the solute biomolecule.F flow rate, ml/minHtot total height equivalent to a theoretical plate (HETP) of the

system, cmHec extra column contribution to the HETP, cmH' HETP of the column, cm

Hfilm HETP contribution from film mass transfer, cmkdes desorption rate constant,kf film transport coefficient, cm/sL length of column, cmPe Peclet number (dp×u/Dm)r ratio of surface to pore diffusionR particle radius, mS slope of HETP vs u plots, stw,1/2 width at half height, mintr retention time, minu superficial velocity, cm/sV0 column dead volume, mlKd-static Equilibrium disassociation constant, measured under static

binding conditions

Greek charactersεi interstitial porosityεp particle porosityeεt total porosityη kinematic viscosity (cm2/s)μ1 first momentσec square root of variance (min)τtor tortuosity factorζ axial dispersion parameter (cm)

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