Mass Transfer Equation for Proteins in Very High-Pressure Liquid

Anal. Chem. 2009, 81, 2723–2736

Mass Transfer Equation for Proteins in Very High-Pressure Liquid Chromatography Fabrice Gritti and Georges Guiochon* Department of Chemistry, University of Tennessee Knoxville, Tennessee 37996-1600 The mass transfer kinetics of human insulin was investigated on a 50 mm × 2.1 mm column packed with 1.7 µm BEH-C18 particles, eluted with a water/acetonitrile/ trifluoroacetic acid (TFA) (68/32/0.1, v/v/v) solution. The different contributions to the mass transfer kinetics, e.g., those of longitudinal diffusion, eddy dispersion, the film mass transfer resistance, cross-particle diffusivity, adsorption-desorption kinetics, and transcolumn differential sorption, were incorporated into a general mass transfer equation designed to account for the mass transfer kinetics of proteins under high pressure. More specifically, this equation includes the effects of pore size exclusion, pressure, and temperature on the band broadening of a protein. The flow rate was first increased from 0.001 to 0.250 mL/min, the pressure drop increasing from 2 to 298 bar, and the column being placed in stagnant air at 296.5 K, in order to determine the effective diffusivity of insulin through the porous particles, the mass transfer rate constants, and the adsorption equilibrium constant in the low-pressure range. Then, the column inlet pressure was increased by using capillary flow restrictors downstream the column, at the constant flow rate of 0.03 mL/min. The column temperature was kept uniform by immersing the column in a circulating water bath thermostatted at 298.7 and 323.15 K, successively. The results showed that the surface diffusion coefficient of insulin decreases faster than its bulk diffusion coefficient with increasing average column pressure. This is consistent with the adsorption energy of insulin onto the BEH-C18 surface increasing strongly with increasing pressure. In contrast, given the precision of the height equivalent to a theoretical plate (HETP) measurement ((12%), the adsorption kinetics of insulin appears to be rather independent of the pressure. On average, the adsorption rate constant of insulin is doubled from about 40 to 80 s-1 when the temperature increases from 298.7 to 323.15 K. Purification of proteins is often carried out by ion-exchange chromatography due to the sensitivity of retention factors in this technique to the specific ionization potential of the protein. An alternative technique is the hydrophobic interaction chromatography or reversed-phase liquid chromatography (RPLC) mode, * Corresponding author. Fax: 865-974-2667. E-mail: [email protected]. 10.1021/ac8026299 CCC: $40.75  2009 American Chemical Society Published on Web 03/03/2009

in which the separation mechanism is governed by the distribution of the protein between a polar eluent and a hydrophobic surface. Pressure, temperature, and organic content of the aqueous mobile phase are all adjustable experimental parameters that drastically affect the adsorption behavior of a protein onto hydrophobic surfaces.1,2 Also critical is the rate of the mass transfer of large biomolecules in the packed bed made of porous silica based particles,3-6 which eventually determines the final performance of the chromatographic process.7 This problem is particularly important when separations are performed on columns packed with fine particles as those used in very high-pressure liquid chromatography (VHPLC). The mass transfer of bulky, slowly diffusive molecules like those of proteins in a packed column is complex. The main differences with the mass transfer of small molecules can be summarized as follows: (1) Proteins are excluded from a fraction of the pore volume by a fraction (1 - (RH/r))3 with RH the hydrodynamic radius of the protein and r the pore radius and assuming that the pores are open spheres.8 (2) The adsorption behavior of proteins depends strongly on the local pressure, due to change in the molar volume of the protein upon uptake onto hydrophobic surfaces.9,10 (3) The kinetics of adsorptiondesorption of proteins is slower. It is often modeled as a two step process, the protein being first adsorbed onto the surface, then spreading over the surface.11 The peculiarities mentioned above are rarely incorporated together into mass transfer models, for the sake of their simplicity. In nearly all cases, this has little or no practical incidence since priority is generally given to the model ability to qualitatively reproduce the experimental observations. Recently, a general height equivalent to a theoretical plate (HETP) model was successfully extended to characterize the band broadening of polystyrene standards in size-exclusion liquid chromatography,12 in which case the retention factor was strictly zero. In its classical (1) Szabelki, P.; Cavazzini, A.; Kaczmarski, K.; Liu, X.; Van Horn, J.; Guiochon, G. J. Chromatogr., A 2002, 950, 41. (2) Liu, X.; Zhou, D.; Szabelki, P.; Guiochon, G. J. Chromatogr., A 2002, 950, 41. (3) Yao, Y.; Lenhoff, A. J. Chromatogr., A 2006, 1126, 107. (4) Yang, K.; Sun, Y. J. Chromatogr., A 2006, 1126, 107. (5) Bankston, T.; Stone, M.; Carta, G. J. Chromatogr., A 2008, 1188, 242. (6) Farnan, D.; Frey, D.; Horva¨th, C. J. Chromatogr., A 2002, 959, 65. (7) Guiochon, G.; Felinger, A.; Katti, A.; Shirazi, D. Fundamentals of Preparative and Nonlinear Chromatography, 2nd ed.; Academic Press: Boston, MA, 2006. (8) Gritti, F.; Guiochon, G. J. Chromatogr., A 2007, 1176, 107. (9) McGuffin, V.; Evans, C. E. J. Microcol. Sep. 1991, 3, 513. (10) Guiochon, G.; Sepaniak, M. J. J. Chromatogr. 1992, 606, 248. (11) Haimer, E.; Tscheliessnig, A.; Hahn, R.; Jungbauer, A. J. Chromatogr., A 2007, 1139, 84. (12) Gritti, F.; Guiochon, G. Anal. Chem. 2007, 79, 3188.

Analytical Chemistry, Vol. 81, No. 7, April 1, 2009

2723

formulation, the general rate model7 of chromatography may account for the adsorption-desorption kinetics but it ignores the effects of size exclusion and of the pressure on the second central moment of large molecules. The goal of this work is to provide a similar theoretical framework accounting for the band broadening of proteins in a modern HPLC column in which the temperature and the pressure are not constant along nor across the bed. The theoretical results will be illustrated with the elution profiles of insulin bands recorded on a column packed with very fine particles (1.7 µm BEH-C18), operated at a low and a high temperatures (297.55 and 323.15 K), with an inlet pressure of 80-800 bar. The influence of a high pressure on the adsorption kinetics and the surface diffusion of insulin will be discussed.

component of the linear velocity, the general expression of the linear velocity at high pressure is14

u(T, P) )

(

)

Fν F(Tinlet, Pinlet) 1 + b + b1T c × ε πR2 F(T , P0) P + b + b1T e

ref

exp(-R[T - Tref] - β[T 2 - Tref2]) (3) The bulk molecular diffusion coefficient Dm of monomeric (i.e, not self-aggregated) insulin was estimated from a reference value measured at T0 ) 303.15 K, under P0 ) 1 bar, and pH ) 3.6. Dm(T0, P0) ) 1.63 × 10-6 cm2/s.15 The bulk phase was a mixture of acetonitrile and water (35/65, v/v) very similar to the composition used in this work (32/68, v/v) at pH ) 2.2. The general expression of Dm(T, P) is then

THEORY In this section, we propose a general HETP model that accounts for the local band broadening of a protein band migrating along a column packed with totally porous sub-2 µm particles. The model should address the following issues: (1) The partial exclusion of the protein molecules from the mesopore volume because these molecules cannot penetrate into the pores that are smaller than their gyration radius in the liquid phase. (2) The equilibrium constant K(T, P) of the protein between the liquid phase and the solid adsorbent is a function of both the local pressure and temperature. (3) The adsorption-desorption rate constant of the protein kads is finite. General HETP Equation for a Protein. Consider a column section where the average pressure is P and the average temperature T. The general reduced HETP equation valid for any type of compound is13

[

2 γe + h)

i)3

+

∑ i)1

+

]

1 - εe Ω(T, P) εe ν

ωiν ωi 1+ ν 2λi

( ) ( ) ( )( )

δ0 2 1 1 εe ν 30 1 - εe 1 + δ0 Ω(T, P)

Dm(T, P) ) Dm(T0, P0)

η(T0, P0) T η(T, P) T0

(4)

In eq 3, the values of Fν, Tinlet, and Pinlet are directly measured. Table 2 lists the parameters Tref, b, b1, c, R, and β that are necessary for the calculation of the eluent density. The viscosity η(T, P) can be estimated from

( )(1 + ζ[P - P ])

η(T, P) ) 10

A+

B T

0

(5)

The parameters A, B, and ζ are given in Table 2, according to the experimental data given in refs 16 and 17. In the right-hand-side of eq 1 are written the six independent contributions to the overall band broadening of a protein in VHPLC: longitudinal diffusion (1), eddy dispersion (2), particle mass transfer (3), film mass transfer (4), adsorption kinetic (5), and transcolumn differential sorption (6). These terms are analyzed in detail in the next six paragraphs. The first term (1) in eq 1 accounts for axial diffusion of the protein in the accessible column volume that can be fractionated into the following three locations inside the column: (i) the interparticle volume (obstruction factor γe ∼ 0.60, external porosity εe ) 0.373), in which the diffusion coefficient of the protein is equal to the bulk diffusion coefficient Dm (no hindrance). The contribution, Dext, of the external diffusion to the total longitudinal diffusion coefficient is

5⁄3

+

δ0 2 2⁄3 1 εe ν 3.27 1 - εe 1 + δ0

δ0 +2 1 + δ0 + Cm

2

Dext ) εeγeDm

kp 2 εe 1 ν 1 + kp 1 - εe DF(λ)

dc ν′ dp

(1)

The local reduced interstitial linear velocity ν is

ν)

u(T, P)dp Dm(T, P)

(2)

The local linear velocity of the mobile phase depends on the local temperature and pressure, due to the compressibility and thermal expansion of the eluent. With neglect of the radial 2724


(6)

and (ii) the internal pore volume (sum of the individual volumes of all the pores of radius r that are accessible to the protein. The particle porosity accessible to the protein is ε*p ) 0.231, a value smaller than the actual particle porosity εp). The diffusion of the protein molecules confined in these pores is hindered. First, the internal pathways are tortuous (obstruction parameter 1/τ*2 p ) 0.522, estimated from the Suzuki and Smith correlations,18 (13) Gritti, F.; Guiochon, G. Anal. Chem. 2006, 78, 5329. (14) Gritti, F.; Guiochon, G. Anal. Chem. 2008, 80, 5009. (15) Bocian, W.; Sitkowski, J.; Bednarek, E.; Tarnowska, A.; Kawecki, R.; Kozerski, L. J. Biomol. NMR 2008, 40, 55. (16) Easteal, A.; Woolf, L. J. Chem. Thermodyn. 1988, 20, 693. (17) Thompson, J.; Kaiser, T.; Jorgenson, J. J. Chromatogr., A 2006, 1134, 201. (18) Suzuki, M.; Smith, J. Chem. Eng. J. 1972, 3, 256.

Table 1. Physicochemical Properties of the BEH-C18 Column Given by the Manufacturer and Measured in Our Laboratorya neat hybrid material particle size [µm] pore diameter [Å] surface area [m2/g] pore volume [cm3/g] total carbon [%] density [g/cm3] bonded phase analysis total carbon [%] surface coverage [µmol/m2] endcapping

bridged ethylsiloxane/silica hybrid (BEH) 1.7 130 185 0.7 6.6 2.02 BEH-C18 18 3.10 proprietary

(which, admittedly, is a simplification of the problem because the channels are anastomosed in a complex network but is also more realistic that the addition of the pore resistances),3 the contribution of the pores, Dpores, to the diffusivity of the protein inside the particle is written as a function of the protein radius:

Dpores(RH) )

εp*

∫

τp*2

The total porosity was measured by pycnometry (THF-CH2Cl2). The external porosity was measured by inverse size-exclusion chromatography (polystyrene standards).

λ

εp* ¯r R*K *(T, P)DS τp*2 r 2

∫

(9)

∞

1 f(r)dr r ¯r ) ∞ 1 f(r)dr RH 4 r RH 3

(10)

∫

CH3CN/H2O, 32/68, v/v Density 298.15 K 105 Pa 946.1 kg/m3 5.528 10-4 K-1 -2.121 10-6 K-2 5.214 108 Pa -7.87 105 Pa K-1 0.144 Viscosity -2.742 808 K 1.544 10-9 Pa-1 Heat Capacity 3.669 106 J m-3 K-1 -24.806 103 J m-3 K-2 33.533 J m-3 K-3 Heat Conductivity 0.430 W m-1 K-1

and

∫

∞

1 f(r)dr r 2 r ) ∞ 1 f(r)dr RH 4 r RH 2

(11)

∫

DS is the surface diffusion coefficient and K* is the corrected Henry’s constant, the correction being due to the exclusion of the protein molecules from a fraction (1 - (ε*p/εp)) of the total mesopore volume. This corrected constant is

K *(T, P) ) with τ*p ) ε*p + 1.5(1 - ε*p) ) 1.384). Second, the confinement of the protein molecules offers an additionnal resistance to its diffusion (steric hindrance parameter F(RH/r)). Various correlation for the hindrance parameter F(RH/r) are available such as the Renkin,19 the Brenner and Gaydos,20 and the Karger and Ruthven21 equations. The Renkin correlation will be used because it applies for all ratios λ ) RH/r of the protein hydrodynamic radius to the pore radius. F(λ) ) (1 - λ)2(1 - 2.1044λ2 + 2.089λ3 - 0.948λ5)

(8)

where ¯r and r 2 are the number average pore diameter and squared pore diameter, respectively. If we assume spherical pores,

Table 2. Complete List of Parameters of the Eluent Used in the Calculation of the Temperature Profiles

cp,m a1 a2

∫

Dsurface ) 2

a

A B ζ

( )

RH 1 f(r)F dr 2 εp* r r Dm ) F(λ)Dm ∞ 1 τp*2 f(r)dr RH 2 r

and (iii) the surface area of the mesopores inside the particles, on which the adsorbed molecules can diffuse. The contribution of surface diffusion Dsurface to the particle diffusivity is written as13

packed column analysis serial number 014737108255 48 dimension (mm × mm) 2.1 × 50 total porositya 0.642 external porositya 0.373 particle porosity 0.429

Tref P0 F(P 0Tref) R β b b1 c

∞

RH

εt* k′*(T, P) 1 - εt*

(12)

The parameter R* is a function of the packing material properties,13 including the mass percent % BEH of unbonded solid BEH in the derivatized particles of density FBEH-C18, and S*p(RH) is the specific surface area of the packing material corrected for the protein exclusion:8 R* )

1 FBEH-C18(% BEH)Sp*(RH)

(13)

(7) with

If f(r) is the distribution in volume of the mesopores and if we assume a parallel contribution of all the mesopores of radius r (19) Renkin, E. J. Gen. Physiol. 1954, 38, 225. (20) Brenner, H.; Gaydos, L. J. Colloid Interface Sci. 1977, 58, 312. (21) Karger, J.; Ruthven, D. Diffusion in Zeolites and Other Microporous Solids, 5th ed.; Wiley: New York, 1992.

Sp*(RH) ) Sp(BET)

∫

∞

RH

∫

∞

RN2

(

(

)

RH 2 1 f(r)dr r r2 RN2 2 1 f(r)dr 1r r2 1-

)


(14)

2725

where RN2 is the molecular radius of adsorbed nitrogen in the low-temperature nitrogen adsorption (LTNA) measurement (∼2.5 Å). The surface diffusion coefficient DS of the protein is unknown. It will be derived from the values measured for the second central moment of the peaks of insulin and from their analysis. If we assume that the contributions (ii) and (iii) are additive,4 the effective diffusivity of the protein inside the porous particles is written as13

De ) Dpores(RH) + Dsurface ) ) Ω(T, P)Dm

εp* τp*2

[

F(λ)Dm + 2

]

¯r * * R K (T, P)DS ¯r 2 (15)

Ω(T, P) is the relative sample diffusivity through the porous particles, De, to the bulk diffusion coefficient Dm. The second term (2) in eq 1 accounts for the axial dispersion of the protein in the interstitial bulk liquid phase contained in the column and where the local eluent velocity is not uniform. Giddings treated this problem and proposed three main transfer mechanisms accounting for the differential eluent velocity:22 the transchannel, the short-range interchannel, and the long-range interchannel transfer mechanisms. Each mechanism i is characterized by a relative velocity bias (ωβ,i), a characteristic relative diffusion length with respect to the particle size dp (ωR,i), and a characteristic relative flow length (ωλ,i). These two relative distances are those over which the molecules of the sample located in the streamlines having the extreme velocities are exchanged, i.e., every ωR,i particle diameter (by lateral diffusion) or every ωλ,i particle diameter (by axial convection). The reduced HETP term associated with each one of these independent mechanisms is given in eq 1 with

ωi )

ωβ,i2ωR,i2 2

(16)

ωβ,i ωλ,i 2

(22) Giddings, J. Dynamics of Chromatography; Marcel Dekker: New York, 1965.


1 - εe [εp* + (1 - εp*)K *(T, P))] εe

(18)

The fourth term (4) in eq 1 accounts for the mass transfer resistance due to the existence of a thin stagnant film of eluent surrounding the porous particles, which the protein has to cross before it can penetrate inside the pore structure. The film mass transfer coefficient was assumed to follow the Wilson and Geankoplis correlation,23 recently validated for applications to HPLC columns.24 In eq 1, the expression of kf as a function of the interstitial linear velocity is

( )

kf ) 1.09

Dm εedp

2⁄3

u1⁄3

(19)

The fifth term (5) in eq 1 accounts for the slow kinetic of adsorption-desorption of the protein onto the silica-C18 adsorbent surface area. In the general rate model, a first-order kinetic adsorption is considered. The number D is the Damko ¨hler number, which depends on the adsorption rate constant kads. Originally, the Damko ¨hler number was derived to apply to chemical reactors, and it characterizes the ratio of the reaction rate to the convection rate. On the basis of the description of the general rate model with mesoporous particles, in liquid chromatography it measures the ratio of the adsorption rate to the diffusion rate in the bulk phase inside the particles or

(17)

In his coupling theory of eddy dispersion in packed columns, Giddings proposed quantitative estimates for the different velocity inequalities ωβ,i, radial diffusion length ωR,i, and axial convection length ωλ,i. For the transchannel mechanism, ωβ,1 ) 1, ωR,1 ) 1 /6, and ωλ,1 ) 1. For the short-range interchannel mechanism, ωβ,2 ) 0.8, ωR,2 ) 5/4, and ωλ,2 ) 3/2. For the long-range interchannel mechanism, ωβ,3 ) 0.2, ωR,3 ) 10, and ωλ,3 ) 5. Abundant details on the values of these estimates, a complete discussion of their derivation, and the rationale behind the theory of eddy diffusion were given in ref 22. Note that these parameter estimates are valid for well-packed columns having a reduced A term of the order of 2.0. Although the average size of the particles used to pack columns has considerably decreased during the last 40 years, due to the development of new packing materials, the structure of packed beds has remained the same (self-similarity). The same slurry packing method is still used, and the standard deviation of the particle

2726

δ0 )

D)

2

λi )

size distribution has decreased only slightly, remaining of the order of 20%. The values of the parameters listed, which were estimated by Giddings more than 40 years ago, are still meaningfull for today’s packed columns. The third term (3) in eq 1 accounts for the mass transfer of the protein inside the mesoporous particles. It is derived from the Laplace transform of the general rate model equations.7 The term δ0 is defined by

kadsdp2 ¯ (λ)Dm F

(20)

Large values of D (typically >10) means that the diffusion time in the pores is much longer than the reaction characteristic time (1/kads). Small values of D < 0.1 reflect the important role played by the adsorption process of the protein. In eq 1, the parameter kp is written as7 kp )

1 - εp* K *(T, P) εp*

(21)

The sixth term (6) in eq 1 accounts for the differential sorption of the compound across the column diameter. In VHPLC, the difference between the average linear velocity of the protein in the center of the column and close to its wall can be important, in which case it causes serious band deformation.25 For instance, a significant radial temperature gradient can markedly affect the (23) Wilson, E.; Geankoplis, C. J. Ind. Eng. Chem. Fundam. 1966, 5, 9. (24) Miyabe, K.; Ando, M.; Ando, N.; Guiochon, G. J. Chromatogr., A 2008, 1210, 60. (25) Gritti, F.; Guiochon G. J. Chromatogr., A 2009, 1216, 1353.

band broadening of a solute. The parameter Cm depends on the radial distribution of the temperature and the pressure. A reference value is Cm ) 1/96, when the radial velocity distribution is caused by the viscous flow of the mobile phase in a circular tube. The ratio dc/dp is the ratio of the column to the average particle diameter. In the present case, this ratio is equal to 1235. The reduced linear velocity ν′ is defined as u'dc

ν' )

(22)

Dr

with ν′ and Dr being the average reduced linear velocity of the eluent and the radial dispersion coefficient across the column diameter.26 Let x ) r/R:

u' ) 2

∫ u(x)x dx ) 2∫ 1

0

1

0

u0(x) x dx 1 + k′*(x)

(23)

The contribution of interparticle convection to the radial dispersion coefficient was estimated based on NMR studies in packed columns:27

Dr ) 2

∫ D(x)x dx ) 2∫ ([ε γ + (1 - ε )Ω(x)]D 1

0

1

e e

0

m(x) +

e

0.16dpεeu(x))x dx (24) The coefficient Cm can be calculated according to the Aris dispersion theory28-30 If we define the function φ(x) as the ratio of the axial linear velocity at reduced radius x to the cross-section average linear velocity and Ψ(x) as the ratio of the radial dispersion coefficient at reduced radius x to the bulk diffusion coefficient Dm, the coefficient Cm is written as

Cm )

I1 - 2I2 + I3 Dr 2 D0

(25)

with

I1 )

∫

1

0

Φ(x)2 dx 2xΨ(x)

(26)

I2 )

∫

1

0

xΦ(x) dx 2Ψ(x)

(27)

I3 )

∫

1

x3 dx 2Ψ(x)

(28)

0

and

Φ(x) ) (26) (27) (28) (29) (30)

∫ 2x φ(x')dx x

0

(29)

Gritti, F.; Guiochon, G. J. Chromatogr., A 2008, 1206, 113. Tallareck, U.; Klaus, A.; Bayer, E.; Guiochon, G. AIChE J. 1996, 42, 3041. Aris, R. Proc. R. Soc. London, Ser. A 1956, 235, 67. Martin, M.; Guiochon, G. Anal. Chem. 1982, 54, 1533. Gritti, F.; Martin, M.; Guiochon, G. Anal. Chem.

The combination of all these equations provides a model for the band broadening of protein molecules in a VHPLC column. EXPERIMENTAL SECTION Chemicals. The mobile phase used in this work was a solution of acetonitrile in water (32/68, v/v), containing 0.1% of trifluoroacetic acid (TFA). These two solvents were HPLC grade from Fisher Scientific (Fair Lawn, NJ). The mobile phase was filtered before use on a surfactant-free cellulose acetate filter membrane, 0.2 µm pore size (Suwannee, GA). The polystyrene standards (MW ) 590, 1 100, 3 680, 6 400, 13 200, 31 600, 90 000, 171 000, 560 900, and 900 000) were purchased from Phenomenex (Torrance, CA). Samples of human insulin used in this work were generously offered by Eli Lilly (Indianapolis, IN). Materials. The 50 mm × 2.1 mm BEH-C18 column used was a gift from the column manufacturer (Waters Inc., Milford, MA). The synthesis of the silica matrix is based on the organic (ethyl)/inorganic (silica) hybrid technology. The main characteristics of the bare porous silica and those of the derivatized packing material are summarized in Table 1. A LAUDA model M20 (Delran, NJ) water bath heater was used, with the temperature of water set at 25 and 50 °C to keep the temperature of the column wall constant. A glassware tube that fits closely the column geometry was prepared, to immerse the column into the temperature controlled liquid bath, without modifying the column connections. Apparatus. The apparatus used was a Acquity UPLC liquid chromatograph (Waters, Milford, MA). This instrument includes a quaternary solvent delivery system, an autosampler with a 10 µL sample loop, a monochromatic UV detector, a column oven, and a data station running the Empower data software from Waters. From the exit of the Rheodyne injection valve to the column inlet and from the column outlet to the detector cell, the total extra-column volume of the instrument is 13.6 µL, measured as the apparent hold-up volume of a zero-volume union connector in place of the column. A time offset of 0.58 s was measured after the zero injection time was recorded. The flow rate delivered by the high-pressure pumps of the instrument is true at the column inlet. During the HETP measurements, the inlet flow rates were fixed at 0.001, 0.002, 0.005, 0.01, 0.02, 0.03, 0.045, 0.06, 0.12, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50, 0.55, 0.60, 0.65, 0.70, 0.75, 0.80, 0.85, 0.90, 0.95, 1.00, and 1.05 mL/min when permitted. The maximum pressure allowed during a run was 992 bar at 1.10 mL/ min. The laboratory temperature was controlled at 296 ± 1 K by the laboratory temperature control system. RESULTS AND DISCUSSION In the first part of this work, we studied the accessibility of the protein within the mesopore structure of the BEH-C18 packing material. Nitrogen adsorption measurements and inverse sizeexclusion chromatography served to derive accurate estimate of the actual volume and surface area accessible to insulin molecules. In a second part of this work, we describe the complete retention map of insulin monomer at four different temperatures (297.55, 307.15, 317.15, and 323.15 K) and under average column pressures increasing from 80 to 850 bar. In the last part, we discuss the values of the second central moment of insulin profiles and the intraparticle and surface Analytical Chemistry, Vol. 81, No. 7, April 1, 2009

2727

diffusion coefficients of insulin derived from these moments within the same temperature and pressure ranges. The comparison between the theoretical (eq 1) and the experimental reduced HETP in the absence of viscous heat dissipation is finally discussed. Pore Accessibility of a Protein. The volume pore size distribution of mesoporous materials is generally reported as the ratio of the incremental pore volume dV per unit mass of adsorbent to the increase by d log D of the logarithm of the pore size, D ) 2r (r being the pore radius, given in cm3/g Å): f(r) )

dV d log(2r)

(30)

Let RH be the gyration radius of the protein molecule. In the absence of bottleneck mesopores, the pore volume Vp(RH) accessible to this molecule in the whole column can simply be estimated from8

Vp(RH) ) εp(1 - εe)Vc

∫

∞

RH

∫

∞

Rpyc

( (

) )

RH n 1 f(r)dr r r Rpyc n 1 f(r)dr 1r r 1-

(31)

where εe, εp, and Vc are the external porosity, the particle porosity, and the volume of the column tube, respectively, Rpyc is the molecular radius of the eluent (∼2.9 Å) used to measure the hold-up volume of the column by pycnometry, and n stands for the shape assumed for the mesopores, n ) 2 or n ) 3 if this shape is cylindrical or spherical, respectively. Figure 1A is a schematic of the circular section of these pores. They can be arranged in space so that they form the three-dimensional fractal structure that we may reasonably expect for mesoporous silica. The pore size distribution of the BEH-C18 packing material was computed from the experimental adsorption isotherm data of nitrogen vapor measured at 77.3 K for values of the reduced pressure PN2/P*N2 increasing from 0.015 to 1, with P*N2 the saturation vapor pressure of nitrogen at 77.3 K. The method used for the computation of the pore volume distribution is based on the Barret-Joyner-Halenda technique,31 which assumes a circular cross-section for all mesopores. The distribution function f(r) is shown in Figure 1B for the adsorption of the BET branch, with the best fit being the sum of two first-order exponential modified Gaussian functions.32 Equation 31 predicts best the inverse size-exclusion chromatography (ISEC) data when n ) 3 and the experimental pore size distribution obtained from the low-temperature adsorption data (adsorption BET branch) is used, which does not necessarily mean that all the mesopores are actually spherical. Figure 1C compares the experimental cumulative pore volume measured by ISEC and those calculated according to eq 31, assuming the experimental pore size distribution shown in Figure 1B. The two curves are in good agreement, although LTNA gives a slight overestimate with respect to the ISEC data. This probably mirrors the existence of large bottleneck pores that are inaccessible for the polystyrene standard molecules but are accessible to nitrogen (31) Barret, E.; Joyner, L. G.; Halenda, P. P. J. Am. Chem. Soc. 1951, 73, 373. (32) Felinger, A. Data Analysis and Signal Processing Chromatography; Elsevier: Amsterdam, The Netherlands, 1998.

2728


Figure 1. (A) Schematic representation of the fractal structure of porous silica with open spherical pores. Note the significant exclusion volume of globular proteins. (B) Experimental pore size distribution of the BEH-C18 packing material (lot no. 147) estimated from the adsorption data of nitrogen at 77.3 K. The solid line represents the best fit to the experimental distribution using a combination of two first-order exponential modified Gaussian functions. D is the pore size in Angstro¨m units. (C) Comparison between the experimental ISEC volumes of polystyrene standards and the calculated exclusion volumes based on the pore size distribution given in Figure 1B and using eq 31. The shape of the pores is assumed to be spherical (n ) 3). Note the good agreement between the two plots.

vapor. The pore volume, Vp(RH), that is accessible to a globular protein having a gyration radius RH can then be estimated either from the low-temperature nitrogen adsorption measurements, assuming spherical pores (n ) 3), or from ISEC, using polystyrene random coils as standards. The column volume is Vc ) 173.2 µL (manufacturer specifications), εe ) 0.373 (ISEC),

Table 3. Best Fitting Parameters of Equation 35 to the Experimental Data Given in Figure 3C parameters in eq 35 pressure [bar]

a(P)

b(P)

c(P)

100 200 300 400 500 600 700 800 900

1.761 81 2.087 87 2.401 73 2.703 6 2.993 57 3.271 69 3.538 01 3.792 53 4.035 27

-3096.206 15 -2764.407 02 -2508.847 06 -2333.973 81 -2241.930 8 -2233.876 62 -2310.493 19 -2472.209 56 -2719.310 89

-4 371 586.512 2 -6 214 906.000 14 -7 672 499.390 72 -8 730 155.296 29 -9 380 932.841 52 -9 621 060.457 95 -9 448 322.967 83 -8 861 341.697 57 -7 859 220.755 94

εp ) 0.429 (ISEC combined with pycnometry) so the total pore volume in the 50 mm × 2.1 mm i.d. BEH-C18 column that is accessible to the eluent is 46.6 µL. The highly practical eq 31 gives a rapid and fair estimate of the exclusion volume of any protein, provided its gyration radius is known. The gyration radius of insulin in solution in acetonitrile and water (32/68, v/v + 0.1% TFA, pH ) 2.1) at 1 µM was determined

Figure 3. (A) Plot of the logarithm of the corrected retention factor k* versus the uniform temperature and pressure of the BEH-C18. (B) Effect of pressure and temperature on the change of partial molar volumes of insulin at infinite dilution upon adsorption onto BEH-C18 from the bulk mobile phase. (C) Isobaric corrected retention factor versus the reciprocal temperature. Note that the retention increases with increasing temperature.

from the diffusion coefficient of insulin monomer in a mixture of comparable composition (35/65, v/v). According to ref 15, the molecular diffusivity of insulin monomer at 30 °C is 1.63 × 10-6 cm2/s, at inifnite dilution. Following the Einstein-Stokes relationship, we can estimate the gyration radius RH of monomer insulin: Figure 2. (A) Change in the column hold-up volume V0 with increasing temperature (297.45-313.15 K) and pressure (80-800 bar). Note that the largest relative variation never exceeds 1%. The compressibility of liquid octadecane was assumed in the calculation. (B) Variation of the column phase ratio as a function of the uniform temperature and pressure of the BEH-C18 column.

Dm )

kBT 6πηRH

(32)

where kB is the Boltzmann constant (1.38 × 10-23 J K-1), T the temperature (303.15 K), and η the viscosity of the solvent (0.85 Analytical Chemistry, Vol. 81, No. 7, April 1, 2009

2729

Table 4. Parameters in Equation 36 Used to Predict the Retention Factor of Insulin at Any Given Pressure and Temperature

P0 [bar] Tref [K] ln k′(Tref, P0) a1 a2 b0 [K] b1 [K] b2 [K] c0 [K2] c1 [K2] c2 [K2]

parameters in eq 36 1 323.15 1.42774 0.003 434 66 -5.9438 × 10-7 -3520.6 4.630 45 -0.004 151 5 -2 089 902.8 -24 762.3 20.377 5

× 10-3 Pa s). The gyration radius of insulin is then estimated to be 16.2 Å. According to eq 31, the pore volume accessible by insulin is 25.1 µL, representing about half the total pore volume accessible to the solvent (46.6 µL). The particle porosity of the column must then be corrected for the protein exclusion. In the next part of this work, the thermodynamic and mass transfer properties will refer to this corrected and physically meaningful particle porosity for the protein. The apparent particle porosity of the mesoporous material used in our work becomes ε*p ) 0.231 (0.429 for small eluent molecules). The apparent total porosity is also corrected to ε*t ) 0.518 (0.642 for small eluent molecules). The pore volume that insulin cannot occupy due to steric hindrance will simply be considered as an apparent stationary phase volume. Determination of the Retention Pattern of Insulin onto BEH-C18. The retention behavior of insulin was investigated previously on silica-C18 phases as a function of the temperature1 and the average column pressure.2 While the conclusions of these works are not questionable and the retention of insulin does increase with increasing pressure and temperature, the accuracy of these measurements could be improved. First, the pressure was taken as the average column pressure when the column was run at 1 mL/min and the maximum column inlet pressure was limited to 260 bar. Second, the column was placed in an air-oven compartment where the inlet temperature was certainly smaller than the temperature set by the instrument. In our work, the pressure drop was limited to 40 bar and the column was operated at the low flow rate of 0.03 mL/min. Capillary restrictors were connected between the column outlet and the detector cell in order to increase the column inlet pressure from about 80 to 800 bar at constant flow rate. The temperature of the column was fixed at 297.55, 307.15, 315.15, and 323.15 ± 0.1 K by a circulating water bath, directly in contact with the column. Since the flow rate was low, no internal heat was released by friction and the temperature remained uniform throughout the column. The retention factors k′* were measured by deriving the total porosity of the column from the pore size exclusion of insulin, εt* ) 0.518. This is necessary in order to calculate the correct mass transfer coefficients in the classical general rate model of chromatography. Before measuring these retention factors, we had to make sure that the hold-up volume of the BEH-C18 column does not vary significantly when the temperature is increased from 297.55 to 323.15 K and the pressure raised from 80 to 800 bar. The density of 2730


Figure 4. Best fit of the parameters a(P) (A), b(P) (B), and c(P) (C) in eq 35. Note that simple parabolic functions are sufficient to account for the experiments.

the C18-bonded layer was assumed to be equal to that of liquid octadecane (0.778 g/cm3). On the basis of the data given in Table 1, we derived a value of the density of the C18 chains of 0.90 g/cm3 and a carbon content of 17.3%, assuming the ligands being monomeric bonded C18 groups, one chain reacting with one single silanol group. These two estimates are in good agreement with the density of liquid octadecane and with the carbon content given by the manufacturer (18%). The slightly larger density found in the data in Table 1 is probably explained by the strained conformation of the C18 chains, the presence of a silicon atom, heavier than a carbon atom, at the chain anchoring point to the BEH surface, and/or

considered to the other (297.45 K, 840 bar to 323.15 K, 95 bar). The corresponding variations of the logarithm of the phase ratio are plotted in Figure 2B. The average slope (∂ln φ/∂P)T is equal to -2.60 × 10-5 bar-1. The very same conclusion would have been obtained had we chosen a density of the octadecyl chain equal to 0.90 g/cm3. The average slope of the plots in Figure 2B would have been -2.74 × 10-5 bar-1. After thermodynamic considerations, the variation of the retention factor with the local pressure is written as2

(

Figure 5. (A) Elevation of the column temperature with increasing the flow rate at ambient temperature and correlation with the heat power Pf released per unit length of column. (B) Example of the temperature profile inside the column (packed bed + stainless steel tube) at a flow rate of 0.25 mL/min. The radial temperature gradients (0.16 K) induces an increase of the reduced HETP by 0.53, according to the sixth term in eq 1.

the error made in the LTNA measurement of the specific surface area of the unbonded BEH adsorbent (185 m2/g). The compressibility of octadecane was derived from the Tait equation, using the correlations given in ref 33 to estimate the parameters of the Tait equation. The densities of liquid octadecane under normal pressure and at different temperatures (298.15, 323.15, 348.15, and 373.15 K) were obtained from ref 34. From the material characteristics given in Table 1, the porosity of the unbonded BEH particles is εp0 ) 0.585. After derivatization and endcapping, this porosity decreases to εp ) 0.429. Accordingly, the volume of the bonded phase inside the 50 mm × 2.1 mm i.d. BEH-C18 column is 18 µL. The density of octadecane under normal pressure and at 298.15 K is 0.7772 g/cm3, so we can estimate the mass of bonded phase to be 14 mg. The variation of the hold-up volume is directly related to the variation of the bonded phase volume when pressure or temperature change. We neglect the compressibility of the BEH hybrid silica and the deformation of the stainless steel tube. Figure 2A shows the relative variation of the hold-up column volume when the pressure increases from 80 to 800 bar and the temperature increases from 297.55 to 323.15 K. This relative variation does not exceed 1% from one extreme of the pressure-temperature range

∂ln k′* ∂P

)

T

)-

S - VmM Vm ∂ln φ + RT ∂P

(

)

(33)

T

where VSm and VM m are the partial molar volumes of insulin at infinite dilution in the stationary phase and in the mobile phase, respectively. In other words, if retention increases with pressure at constant temperature, it is because the partial molar volume decreases when the protein is transferred from the bulk mobile phase onto the bonded phase. Figure 3A shows the variation of the retention factor with the average column pressure at four different temperatures (297.55, 307.15, 315.15, and 323.15 K). Note that the corresponding column pressure drops were only 45, 37, 32, and 28 bar, respectively (the flow rate being 0.03 mL/min). The average pressure inside the column is increased by connecting flow restrictors of different lengths downstream the column. The curves are slightly convex upward and nearly parallel to each other. In the range studied, temperature does not affect the difference much between the partial molar volumes of insulin in the adsorbed and the bulk phase. However, it is obvious that this difference is lower at high than at low pressures. This is not surprising because matter is less compressible at high pressures. The average slope of these curves is 2.80 × 10-3 bar-1, 2 orders of magnitude larger than the slope measured in Figure 2B by considering the variations of the hold-up volume. The average changes in molar volume of insulin upon adsorption are -66.9, -73.4, -78.8, and -78.3 mL/mol at 297.55, 307.15, 315.15, and 323.15 K, respectively. These values are smaller than those reported in refs 2 and1 because we operated in a wider pressures range while the pressure drop was smaller. Actually, the plot of ln k′* versus the pressure is not strictly linear:

(

ln k′*(P, T) ) ln k′*(P0, T ) + p1(T )

)

(

)

2 P P - 1 + p2(T ) 0 - 1 0 P P (34)

where p1(T ) and p2(T ) are dimensionless empirical parameters, which depend on the temperature and ln k′*(P 0, T) is the extrapolated logarithm of the retention factor of the protein under normal pressure and at the temperature T. Figure 3B shows plots of the molar volume change as a function of the pressure at the four temperatures applied. As expected from the convex upward shape of the ln k′* versus P plots, the change of molar volume of insulin upon adsorption on BEH-C18 decreases with increasing pressure. It also increases with increasing temperature, up to 317 K. Figure 3B provides a more reliable picture of the change in molar volume of insulin with the local pressure and temperature. (33) Thompson, J.; Kaiser, T.; Jorgenson, J. J. Chromatogr., A 2006, 1134, 201. (34) Caudwell, D.; Trusler, J.; Vesovic, V.; Wakeham, W. Int. J. Thermophys. 2004, 25, 1339.


2731

It shows that the relative change in molar volume of insulin upon adsorption onto the hydrophobic surface from the acetonitrile-water solution is no larger than 1.5%. Figure 3C shows the effect of temperature on the retention factor of insulin at constant pressure. On the basis of the experimental curvature of the plots, the general expression of the retention factor of insulin as a function of both the temperature T and the pressure P can be written as

(

ln k′*(P, T) ) a(P) + b(P)

)

(

1 1 1 1 + c(P) T Tref T Tref

)

2

(35)

where a(P), b(P), and c(P) are functions of the pressure. Parts A-C of Figure 4 show the variation of these functions with the pressure, when the data in Figure 3C were fitted to eq 35. Table 3 lists the best fitting parameters of eq 35 to the experimental data given in Figure 3. The curves have a parabolic form, and we can write the general expression of the retention factor as

( ) ( ) ( ) ]( ) ( ) ]( ) [

2 P P - 1 + a2 0 - 1 + 0 P P 1 P P 2 1 b0 + b1 0 + b2 0 + T Tref P P 1 2 P P 2 1 c0 + c1 0 + c2 0 (36) T T P P ref

ln k′*(P, T) ) ln k′*(P0, Tref) + a1

[

Equation 36 provides a simple analytical expression for the determination of the local retention factor of insulin inside the column, as functions of the local pressure P and temperature T. Table 4 lists the parameters used in eq 36 used to predict the retention factor of insulin at any given pressure and temperature. Determination of the Mass Transfer Kinetic of Insulin. As stated in the theoretical section, six independent mass transfer terms can affect the band broadening of insulin. Three of the parameters involved are unknown: (1) The intraparticle diffusion coefficient De ) ΩDm. On the one hand, it depends on the average hindrance parameter F(λ) inside each individual accessible mesopore, which is estimated based on a parallel pore diffusion process inside the particles (see scheme of the cross-section pore structure in Figure 1A). Assuming the Renkin correlation eq 7, we estimate F(λ) ) 0.054 from eq 8. On the other hand, this coefficient also depends on the surface diffusion of the protein. The surface accessible to the protein is estimated from eq 14. It is found Sp(RH ) 16.2 Å) ) 71.3 m2/g, which represents half the total surface area measured from LTNA experiment (142.6 m2/g). The ratio of the average (in number) of the pore radius squared r2, see eq 11) to the average of the pore radius (r¯, eq 10) is also needed to estimate the contribution of surface diffusion of the protein to the particle diffusivity. We found (r2/r¯) ) 49.3 Å for RH ) 16.2 Å. Finally, the parameter R* in eq 13 is computed from the density of the packing material (FBEH-C18 ) 1.67 g/cm3) and the mass percentage of underivatized hybrid silica BEH (% BEH ) 0.86). The only remaining unknown parameter necessary to calculate the intraparticle diffusivity is the surface diffusion coefficient DS. This coefficient will be estimated from the analysis of the experimental reduced HETP of insulin recorded at different temperatures and pressures. (2) The transcolumn differential sorption term is unknown. Its effect is made negligible by applying a very low flow rate (0.03 mL/ 2732


min) for which the heat released by friction is very small. The amplitude of the radial temperature gradients formed across the column diameter and causing additional band broadening is negligible. Heat effects usually affect sample band broadening only when the power friction Pf dissipated is larger than 4 W per meter of column length.25 This power in written as35 Pf ) Fν

∆P L

(37)

where Fν is the flow rate, ∆P is the pressure drop between the column inlet and the column outlet, and L is the column length. In a first series of experiments, the maximum pressure drop was ∆P ) 276 bar. The highest applied flow rate was Fν ) 0.25 mL/ min, and the column length is L ) 5 cm.The maximum power released per unit length of column is then 2.3 W/m, smaller than the 4 W/m that is considered as the threshold value. The last term in the general reduced HETP equation will always be negligible. (3) The kinetic adsorption constant, kads (or the Damko ¨ hler number D) is unknown in our particular case. We will assume first that the rate of adsorption of insulin is much faster than the rate of diffusion through the mesopores. Then, we will check the validity of this assumption by moderately increasing the flow rate and comparing experimental and theoretical reduced HETP based on the first estimated set of DS values. The other terms (axial diffusion in the external column volume, eddy dispersion, and film mass transfer resistance) were directly calculated according to eqs 6, 16, and 18. The first moment (µ1,t,exp) and the second central moment (µ′2,t) of the insulin peak profiles (Figure 6D) corrected for the extra′ column time contribution (µ1,t,ex and µ2,t,ex ) were measured at the constant flow rate of 0.03 mL/min:

∫ tC(t)dt ) -µ ∫ C(t)dt ∞

µ1,t,exp

0

∞

1,t,ex

(38)

- µ′2,t,exp

(39)

0

µ′2,t,exp )

∫

∞

0

(t - µ1,t)2C(t)dt

∫

∞

0

C(t)dt

The measurements were repeated three times, and the results indicate that the relative errors made on the first and second central moments are always smaller than 1 and 5%, respectively. Accordingly, the maximum relative error made on the measurement of the reduced column HETP is 12%. Neglecting the Adsorption-Desorption Kinetic of Insulin. Under this assumption, the fifth term in the general HETP equation vanishes. The reduced HETP is written as

L µ′2,t,exp h) ) dp µ2,t,exp

[

2 γe +

( )

]

1 - εe Ω i)3 εe + ν i)1

∑

ωiν 1 εe + × 30 1 - εe ωi 1+ ν 2λi

( )

5⁄3 δ0 2 2⁄3 dc δ0 2 1 1 εe ν+ ν + Cm ν' (40) 1 + δ0 Ω 3.27 1 - εe 1 + δ0 dp

Figure 6. (A) Comparison between experimental (g) and calculated (b) reduced HETP of insulin assuming infinitely fast kinetics of adsorption. Note the serious disagreement in the low reduced linear velocity range. The best fit is obtained with Ω ) 0.038. (B) The same as in Figure 6A except with a finite adsorption rate constant kads. Note the excellent agreement with Ω ) 0.28 and kads ) 84.7 s-1. (C) Contribution of axial diffusion, eddy dispersion, film mass transfer, particle diffusivity, adsorption kinetics, and transcolumn differential sorption to the total reduced HETP of insulin at moderate reduced linear velocities. Room temperature, acetonitrile-water-TFA, 32/68/0.1. (D) Experimental band profiles of insulin recorded at different flow rates.

Equation 40 was solved for Ω at a flow rate of 0.25 mL/min. The optimized value was Ω ) 0.038. The molecular diffusivity of insulin, De ) ΩDm, inside the porous BEH-C18 particles was found to be about 25 times slower than in the bulk solution. This is plausible for very large proteins in porous media where a factor 20 has been reported.3 However, this is unexpectedly slow for a small globular protein like insulin. As a result, the surface diffusion coefficent of insulin, DS, would be about 1 order of magnitude smaller than its bulk diffusion coefficient. However, according to the surface diffusion model elaborated by Miyabe et al. in RPLC,36 this ratio should be close to 1 for weakly retained compounds such as insulin in the low-pressure range studied here (Henry’s constant K ) 0.3). This certainly proves wrong the initial assumption made regarding the infinitely fast adsorption-desorption of insulin onto the BEHC18 surface. Evaluation of the Adsorption-Desorption Kinetics of Insulin. In the first series of experiments, a large range of mobile phase velocity was investigated. Flow rates of 0.001, 0.002, 0.01, 0.02, 0.03, 0.06, 0.09, 0.12, 0.15, 0.20, and 0.25 mL/min were applied to (35) Lin, H.; Horva´th, C. Chem. Eng. Sci. 1981, 36, 47. (36) Miyabe, K.; Takeuchi, S. Ind. Eng. Chem. Res. 1998, 37, 1154.

the column. The column was kept under still-air condition, in order to minimize the amplitude of the radial temperature gradient formed when the flow rate increases. At 0.25 mL/min, the highest power friction released per unit length of column was 2.30 W/m and the effect of transcolumn differential sorption can be neglected. Nevertheless, the column warms up and the heat eventually produced is essentially transported along the column by convection. A weak longitudinal temperature gradient is formed along the column. We placed three surface thermocouples on the external surface of the column (at the inlet, middle, and outlet) in order to measure the average column temperature at the different flow rates. Figure 5A shows how the heat power and the temperature in the middle of the column wall are correlated. The average temperature of the packed bed increases by no more than 3.5 K. Figure 5B shows the axial temperature profile inside the packed bed at the highest flow rate. The maximum amplitude of the fully developed radial temperature gradient is 0.16 K. The Cm term was calculated according to eq 25. The maximum reduced HETP expected for the transcolumn differential sorption is equal to 0.54 at 0.25 mL/min and rapidly decreases with decreasing flow rate to become negligible below 0.06 mL/min. Analytical Chemistry, Vol. 81, No. 7, April 1, 2009

2733

Figure 7. Effect of pressure on the experimental reduced HETP (empty symbols) of insulin and comparison to the theoretical model with fixed values of Ω ) 0.28 and kads ) 84.7 s-1. Note the increasing trend in the low-pressure range (500 bar). The error bar represents the range of experimental errors made on the determination of the reduced HETP h, derived from the standard deviation of three consecutive experiments ((12%).

The experimental reduced HETPs were measured according to eqs 38 and 39and the left-hand-side term of eq 40. For the calculation of the reduced HETP, the parameter Ω determined in the previous section was used (Ω ) 0.038, flow rate rate ) 0.25 mL/min). Figure 6A clearly shows that this value of Ω is physically inconsistent with the observation. The deviation between experimental and calculated reduced HETP is especially important in the low-linear velocity range. Clearly, the values of Ω were underestimated in our first analysis because the contribution of the adsorption-desorption kinetics of insulin was neglected. The kinetic parameter kads was introduced into the reduced HETP equation. For different values of Ω, the best value of kads was optimized in order to minimize the difference between experimental and calculated HETP. Figure 6B shows plots of the sum of the relative squared residuals between experiment and calculated values as a function of the fixed parameter Ω. The best match is found for Ω ) 0.28 and kads ) 84.7 s-1. This value of Ω makes more sense than the small value of 0.038, given the large experimental reduced HETP values measured for a reduced linear velocity ν between 0.15 and 1.6, e.g., when the longitudinal diffusion of insulin controls the overall mass transfer. It is interesting to compare the characteristic time of adsorption of insulin onto the BEH-C18 particles tads ) 12 ms to the diffusion time tdiff,pores necessary to cross the average particle size in the absence of surface diffusion, e.g., if tdiff,pores ) [(d2pτ*p, 2)/2ε*pF(λ)Dm] ) 1.3 s. Without surface diffusion, the effective diffusivity of insulin through the particle would be 100 times slower than the adsorption-desorption kinetics, which then could be easily neglected in the study of the mass transfer mechanism. However, surface diffusion significantly accelerates the transport of insulin through the porous particle, and the actual characteristic time of effective diffusion through the particle is only 32 ms. Adsorptiondesorption kinetics cannot be neglected; it will affect the band broadening of insulin at high linear velocities. Figure 6C shows the different mass transfer contributions to the total reduced HETP of insulin at room temperature (∼297.5 K) and for usual average 2734


Figure 8. (A) Effect of pressure on the surface diffusion coefficient of insulin. (B) Effect of pressure on the adsorption rate constant of insulin onto BEH-C18 adsorbents. The error bar has the same meaning as in Figure 7.

columnpressures(160barmaximum).Itisnoteworthythattheadsorptiondesorption kinetics of insulin govern the mass transfer mechanism for ν > 20. Effect of High Pressures on the Reduced HETP of Insulin. In a second series of experiment, the flow rate was maintained constant at 0.03 mL/min (ν∼5) and the pressures inside the column were increased by inserting capillary flow restrictors of different lengths between the column outlet and the UV detector cell. Figure 7 compares the experimental reduced HETP to those calculated assuming constant the parameters Ω and kads for the two temperatures of 297.55 and 323.15 K. The agreement is only satisfactory at low pressures, proving that at least one of these two parameters is pressure dependent. The experiments show a steeper increase of the reduced HETP with increasing pressure than is predicted at low pressures (600 bar), the measured HETP remains nearly constant at 297.55 K and even decreases at 323.15 K while the calculated values keep increasing. Since δ0 tends toward 1 as the retention factor becomes infinite, the increase in the reduced HETP with increasing pressure was expected. This result is consistent with the convex downward shape of all the curves in Figure 7. Adjusting the parameter Ω at a constant value of kads gives values that are not physically acceptable for the surface

diffusion of insulin (DS . Dm). In contrast, adjusting the parameter kads by keeping constant the factor Ω generates meaningful surface diffusion coefficients, DS. Figure 8A shows that DS is smaller than the bulk diffusion coefficient of insulin and that its pressure dependence is much more pronounced. As the pressure increases, the adsorption strength strongly increases and the random diffusive motion of insulin at the interface between the hydrophobic layer of C18 chains and the polar eluent slows down. The adsorption rate constant kads depends on the pressure, as shown in Figure 8B. The parameter kads is unique since, when Ω is fixed, it is the only adjustable parameter left. It is provided by the exact match between the value given by the general HETP equation (eq 36) and the experimental data shown in Figure 7 (precision ±12%). Consistent with the expectations is the increase of kads with increasing temperature. The strong increase of kads observed at 323.15 K could possibly be explained by the experimental error bar on the h data. Overall, the curves in Figure 8B suggest strongly that the adsorption rate constant kads is probably independent of the local pressure P in the range from 80 to 800 bar. CONCLUSION The general reduced HETP equation derived earlier accounts well for the band broadening of a retained protein. The model includes axial diffusion, eddy dispersion, film mass transfer, particle effective diffusion, first order adsorption kinetics, and transcolumn differential sorption. To apply the model, we need (1) the complete profiles of the retention factor as a function of the temperature and the pressure in the range of experimental conditions investigated; (2) ISEC data to determine the exclusion volume of the protein; (3) the pore size distribution in order to estimate the hindrance diffusion parameter inside the porous particles; and (4) the exact temperature profile inside the column when very high inlet pressures are applied, in order to estimate the transcolumn differential sorption term. With the use of very small flow rates and flow restrictors, the effects of the friction heat on the observed band-broadening can be neglected at high pressures and high temperatures. The model predicts well the experimental increase of the reduced HETP of insulin with increasing pressure and accounts for the change in the adsorption kinetics of the protein. It is found that surface diffusion of the protein controls its diffusivity through the porous particles (pore diffusion accounts for only 3%) and decreases when the pressure increases. However, no surface diffusion model of proteins are available at present to validate or falsify these experimental conclusions. The theoretical framework presented in this work will be used to predict the apparent HETP of proteins in very high-pressure liquid chromatography. The local HETP equation derived for a constant temperature and pressure was integrated along the column, knowing the temperature and pressure profiles. In a future work, the band profiles of insulin will be measured in a high-flow rate range, with a different thermal environment of the column, and the results will be compared to those of calculations. ACKNOWLEDGMENT This work was supported in part by Grant CHE-06-08659 of the National Science Foundation and by the cooperative agreement between the University of Tennessee and the Oak Ridge National Laboratory. We thank Marianna Kele, Xu Yuehong, and Uwe Neue

(Waters, Milford) for the generous gift of the column used in this work, the complete LTNA adsorption data on the BEH-C18 packing material (lot no. 147), and for fruitful discussions. APPENDIX Roman Letters

empirical pressure dependent parameter in eq 35. empirical pressure dependent parameter in eq 35 (K). empirical pressure dependent parameter in eq 35 (K2).

a(P) b(P) c(P) a1

empirical parameter in eq 36.

a2

empirical parameter in eq 36.

b0

empirical parameter in eq 36 (K).

b1

empirical parameter in eq 36 (K).

b2

empirical parameter in eq 36 (K). empirical parameter in eq 36 (K2). empirical parameter in eq 36 (K2). empirical parameter in eq 36 (K2). empirical coefficient in eq 5. empirical coefficient in eq 5 (K). mass percent of solid matrix BEH in the packing material BEH-C18. compressibility coefficient in eq 3 (Pa). compressibility coefficient in eq 3 (Pa K-1). compressibility coefficient in eq 3. sample mobile phase concentration (kg/m3). transcolumn mass transfer coefficient due to viscous heating. Damko¨hler number. effective mesopore diffusion coefficient in the particle volume (m2/s). column cross-section average radial dispersion coefficient (m2/s). surface diffusion coefficient (m2/s). effective surface diffusion coefficient in the particle volume (m2/s).

c0 c1 c2 A B % BEH b b1 c C Cm D Dpores Dr DS Dsurface dc

column internal diameter (m).

dp

average particle size (m). De effective particle diffusivity (m2/s). Dm(T, P) bulk molecular diffusion coefficient at local temperature T and pressure P (m2/s). Dext axial diffusion coefficient along the interstitial volume of the packed column (m2/s). Dpores average restricted diffusion coefficient in the particle mesopores (m2/s). F(λ) pore steric hindrance parameter. ¯ (λ) F average particle pore steric hindrance parameter. Fν f(r) h I1 I2 I3 kB k′

inlet flow rate (m3/s). pore size distribution in volume (m3/kg Å). total reduced column HETP. integral eq 26. integral eq 27. integral eq 28. Boltzmann’s constant (J K-1). retention factor of nonexcluded analytes. Analytical Chemistry, Vol. 81, No. 7, April 1, 2009

2735

k′* K(T, P) K*(T, P) kads kf kp k*p L P Pf ∆P Pinlet p1(T) p2(T) r R RH RN2 Rpyc Sp S*p

T T0 Tinlet Tref u(T, P) u0 u′ M Vm S Vm

x

retention factor of partially excluded proteins. Henry’s constant at local pressure P and temperature T for nonexcluded analytes. Henry’s constant at local pressure P and temperature T for partially excluded proteins. -1

adsorption rate constant (s ). film mass transfer coefficient (m/s). particle retention factor for nonexcluded analytes. particle retention factor for partially excluded proteins. column length (m). pressure variable (Pa). -1

power viscous friction (W m ). column pressure drop (Pa). inlet eluent pressure (Pa). empirical temperature dependent parameter in eq 34. empirical temperature dependent parameter in eq 34. mesopore radius (m). column inner radius (m). hydrodynamic radius of the protein (m). molecular radius of adsorbed nitrogen at 77 K (m). molecular radius of the eluent molecules used in pycnometry experiments (m). specific surface area of the solid matrix before derivatization (m2/kg). specific surface area of the solid matrix before derivatization corrected for the protein partial exclusion (m2/kg). temperature (K). experimental reference temperature for the diffusion coefficient of insulin (K). inlet eluent temperature (K). reference temperature for determination of the eluent density (K). interstitial linear velocity at local temperature T and pressure P (m/s). chromatographic linear velocity (m/s). sample migration linear velocity (m/s). partial molar volume of the sample at infinite dilution in the bulk eluent (m3/mol). partial molar volume of the sample at infinite dilution in the stationary phase (m3/mol). reduced radial coordinate.

εp εp0 ε*p εt εt* φ φ(x)

Φ(x) γe λ λi µ1, t,exp µ1, t,ex

µ′2, t,exp µ′2, t,ex

ν

ν′

ωi ωR,i ωβ,i ωλ,i Ω(T, P)

Ψ(x)

F(T, P) Greek Letters

R R* β η(T, P) δ0 εe

2736

eluent expansion coefficient in eq 3 (K-1). structural parameter of the adsorbent for partially excluded proteins defined in eq 13 (m). eluent expansion coefficient in eq 3 (K-2). eluent’s viscosity at local temperature T and pressure P (Pa s). retention parameter of nonexcluded analytes defined in eq 18. external column porosity.


FBEH-C18 τp τ*p ζ

particle porosity of the endcapped BEH-C18 adsorbent for nonexcluded analytes. particle porosity of the unbonded BEH matrix for nonexcluded analytes. particle porosity for partially excluded proteins. total column porosity for nonexcluded analytes. total column porosity for partially excluded proteins. column phase ratio. ratio of the local axial migration linear velocity to the cross-section average migration linear velocity at reduced radial coordinate x. integral eq 29. external obstructive factor of the packed bed. ratio of the hydrodynamic radius of the analyte to the mesopore radius. eddy dispersion coefficient related to a flow exchange mechanism for a velocity bias of type i. experimental first moment in presence of column (s). first moment of the extra-column band profiles (column replaced with a zero volume union connector) (s). experimental second central moment in presence of the column (s2). second central moment of the extra-column band profiles (column replaced with a zero volume union connector) (s2). reduced interstitial linear velocity of the eluent to the particle diameter dp and bulk molecular diffusion coefficient Dm. reduced migration linear velocity of the sample to the column diameter dc and the radial column dispersion coefficient Dr. eddy dispersion coefficient related to a diffusion exchange mechanism for a velocity bias of type i. reduced radial diffusion length for a velocity bias of type i reported to the particle diameter dp. relative velocity inequality for a velocity bias of type i. reduced axial flow length for a velocity bias of type i reported to the particle diameter dp. ratio of the particle diffusivity to the bulk diffusion coefficient at the local temperature T and pressure P. ratio of the radial dispersion coefficient to the bulk diffusion coefficient at reduced radial coordinate x. eluent density at local temperature T and pressure P (kg/m3). density of the derivatized packing material BEH-C18 (kg/m3). particle tortuosity factor for nonexcluded analytes. particle tortuosity factor for partially excluded protein. empirical coefficient in eq 5 (Pa1-).

Received for review December 12, 2008. Accepted January 22, 2009. AC8026299

Mass Transfer Equation for Proteins in Very High-Pressure Liquid

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