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Adsorbed layer thickness determination for convective based media from pressure drop data Ales Podgornik, and Mark Etzel Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.7b04156 • Publication Date (Web): 27 Mar 2018 Downloaded from http://pubs.acs.org on March 28, 2018

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Analytical Chemistry

Adsorbed layer thickness determination for convective based media from pressure drop data

Aleš Podgornik, †,‡,* Mark R. Etzel,††

†

University of Ljubljana, Faculty of Chemistry and Chemical Technology, Večna pot 113, Ljubljana, Slovenia COBIK, Tovarniška 26, 5270 Ajdovščina, Slovenia †† University of Wisconsin, Department of Chemical and Biological Engineering, 1605 Linden Drive, Madison, WI 53706, USA ‡

E-mail addresses: [email protected]; [email protected] *

Corresponding author: (e-mail) [email protected]; (tel) +386 1 4798584; (fax) +386 1 2419144.

1 ACS Paragon Plus Environment

Analytical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract. In the present work, we derive a theoretical framework to determine the adsorbed layer thickness from pressure drop measurements for convective monolithic media without any assumptions about the geometry of the pore structure of the stationary phase matrix. Equations are presented to calculate accuracy of the estimated adsorbed layer thickness as a consequence of measurement error and approximations of the mathematical model. We discovered that there is a minimum in the error for certain pressure drops that results in optimal experimental conditions for determining the adsorbed layer thickness. We demonstrate that the adsorbed layer thickness can be determined with less than 10% error using a wide range of experimental conditions simply from pressure drop data. By careful selection of porous bed dimensions and flow rates, the adsorbed layer thicknesses from sub-nanometer dimensions up to several hundred nanometers can be determined by measurement of the pressure drop in a range of several bars. The method was experimentally tested on methacrylate monolithic columns using monodisperse latex nanoparticles as a reference standard and two different proteins as unknowns demonstrating close agreement with calculations.


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1. INTRODUCTION Convective based media are the chromatographic matrix of choice in many fields of separation science, especially when high volumetric throughput is needed.1-6 The main advantage is open flow channels (further named pores) through which the mobile phase flows, whereby all molecules are transported by convection rather than diffusion.7, 8 This results in flow unaffected properties such as resolution and dynamic binding capacity even for macromolecules.9 However, as there is a convective flow through all of the pores, adsorption taking place on walls of the same pores results in an increase of pressure drop due to a decrease in pore size and consequently also porosity. Based on this observation, a novel application of convective media is presently described. By comparing the pressure drop generated by a mobile phase passing through an original convective matrix and same matrix saturated with adsorbed molecules, one can estimate thickness of the layer formed by the adsorbed molecules.10-13 This method is attractive because it can be implemented on any convective based porous material without any particular pretreatment, using any mobile phase enabling adsorption, and pressure drop measurements are easy to interpret and are obtained almost instantaneously during the experiment. The method was originally developed for matrixes for which pressure drop can be described by the Kozeny-Carman equation (KC model) and an analytical equation to estimate adsorbed layer thickness was derived.10 Since for beds of porosity above 0.7, the KC model is not appropriate, this approach was recently extended to beds that can be described by Happel, i.e. a tetrahedral skeleton column (TSC), or a representative unit cell (RUC) pressure drop model.14 It was shown that when same pressure drop model was used for estimation of pore size and adsorbed layer thickness, an error of only about 10% was made, regardless which pressure drop model was used. Unfortunately, in the equations of Happel and of the RUC model, the estimated layer thickness cannot be explicitly written, therefore solution has to be obtained numerically. In addition, for a porous matrix structure where the pressure drop cannot be described by any of the previously analyzed models, application of proposed method is questionable, limiting its general applicability. It is therefore important to derive methodology to determine adsorbed layer thickness without any information about the matrix structure. In this work we propose a novel approach to determine adsorbed layer thickness without any information on matrix structure together with accuracy analysis of thickness estimation.

2. THEORETICAL SECTION In order to derive a model to estimate the adsorbed layer thickness without using any structural information on the matrix, the general, non-variant features of any convective matrix were contemplated. Every convective matrix consists of a solid part either of physically connected particles or of a single porous skeleton, and of an open porous part where flow occurs. If the convective matrix has a uniform structure, then it can be assumed to consist of certain connected repetitive domains of solids and voids (e.g. spheres, cubes, icosahedron, tetrahedrons or any other more irregular structures). Depending on the symmetry and orientation of the domains, the solids and voids can form an isotropic bed, typical for chromatographic stationary phases, however it can also be an anisotropic bed, e.g. when consisting of elongated fibers.15,16 or flat sheets.1,17,18 Using 3D printing, almost any combination of solids and voids is possible.19-21 Three representative structures are schematically presented in Figure 1a. a)

b)



Page 4 of 15

L h

L

h

flow direction h L

flow direction h

h h

flow direction

Figure 1: Representative structures of porous beds: a) flat sheets (upper), tubes (middle), spheres (bottom); L represents characteristic bed dimension; b) pore size change due to adsorption (thickness h) for corresponding structure: in 1 dimension - upper, in 2 dimensions - middle, in 3 dimensions - bottom. Structural differences are important to properly describe adsorption of molecules. When molecules adsorb to the pore wall they occupy a certain portion of the pore volume. For isotropic beds, pore volume is proportional to the cube of its characteristic dimension, and adsorption causes a decrease of the pore in all three dimensions (Figure 1b, bottom). If the bed consists of long fibers, then the pores are actually the voids between the fibers (or inside the fibers when the fibers are hollow), and can extend practically over the entire bed. Pore volume decreases due to adsorption, and is proportional to the square of a pore size (Figure 1b, middle) multiplied by characteristic bed dimension L (Figure 1a, middle). For a bed consisting of stacked planar sheets, adsorption volume is proportional to pore size to the first power (Figure 1b, top) multiplied by the square of the characteristic bed dimension L (Figure 1a, top). For real beds, one cannot expect any of these idealized structures but rather some combination. Because of that, pore volume is not necessarily proportional to the pore size raised to an integer exponent, but it can be also any intermediate power between 1 and 3. As this exponent represents bed isotropy toward adsorption, we call the exponent the isotropy constant, n. Bed pore volume can then be described as: ∙ ∙ = ∙

(1)

where m is the total number of pores in bed, GV is a constant reflecting pore geometry, dV is the characteristic linear pore dimension (further referred as a pore size) and V is the bed volume. For example, GV = 1 for cubical shaped voids and GV = π/6 for spherical shaped voids. In both cases n = 3 (Figure 1b, bottom). For tubular shaped voids (Figure 1b, middle) GV = π/4·L and n = 2. Finally, for a bed consisting of flat sheets (Figure 1, top) GV = L2 and n = 1. The pore shape and the number of pores is preserved during formation of the adsorbed layer if the total bed volume does not change during adsorption of molecules (reasonable assumption for rigid beds) and the thickness of adsorbed layer is small in comparison to a pore size. By writing Eq. 1 as a ratio of the bed saturated with adsorbed molecules to the original bed, we obtain: 4 ACS Paragon Plus Environment

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∙ ∙

∙ ∙

or

=

=

∙ ∙

(2)

where A indicates the bed having adsorbed molecules while o indicates the original bed. To implement Eq. 2 for prediction of the adsorbed layer thickness, a correlation between pressure drop and a pore size is used:22 ∆ = ∗ ∗ ∗ ∗

∗

(3)

where kV is structural constant of the bed, v is the linear velocity and µ is the viscosity of the mobile phase. Similarly to Eq. 1, also Eq. 3 is used for description of bed before and after adsorption: ∆ = ∗ ∗ ∗ ∗

∗

(4)

∆ = ∗ ∗ ∗ ∗ ∗

(5)

The parameter kV remains constant during adsorption when the thickness of the adsorbed layer is small in comparison to the pore size. Taking the ratio of Eqs. 4 and 5, and expressing the bed porosity in terms of the pore size (Eq. 2), we obtain: =

∆

∆

!

= 1 −

!$

!

(6)

or when rewritten in a logarithmic form: %&

∆

∆

= (& + 2) ∗ %& 1 − !$

(7)

where h is the thickness of adsorbed layer defined as h=(dVo-dVA)/2. The adsorbed layer thickness h is calculated using Eq. 7 and two parameters characterizing bed: the pore size dVo and the isotropy constant n. Due to lack of information about the value of n for particular bed, a method was developed to eliminate n in the calculation of h. In this approach, the bed pressure drop is measured when saturated with molecules forming layer of a known thickness (named reference standard) and when saturated with a sample molecule, and a ratio taken. Eq. 7 for a reference standard can be written as: %&

∆

∆+

= (& + 2) ∗ %& 1 −

!$+

where R indicates reference standard. By dividing equations 8 and 7 we get: 5 ACS Paragon Plus Environment

(8)


∆. / ∆. + ∆. ,- / ∆.

,-

=

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1+ / 2 1 / ,-0 2

,-0

(9)

The isotropy constant n was eliminated by introduction of a reference standard. L'Hôpital’s rule is used to simplify Eq. 9 for the case where the adsorbed layer thickness h is small compared to the pore size (2hR/dVo) and therefore ln(1-x) ≈ -x when x is small: lim →7

1 ,-02 + /

1 / ,-0 2

=

$+ $

(10)

Interestingly, this asymptotic expression has the advantage that it is a simple ratio of the calculated layer thickness h for the reference standard and a sample, and is independent on the pore size. 3. RESULTS SECTION The new relationship between the adsorbed layer thickness h and the pressure drop does not require any information about the matrix pore geometry (n) or the pore size (dVo). This powerful conclusion was investigated for accuracy using an error analysis. We investigated for which pore size and adsorbed layer thickness h does the exchange of the exact expression with the limiting one acceptable. The relative error (further referred as limit error 8,9 ) caused by such substitution is defined as: 8,9 [%] = 100 ∙

1+ / 2 > 1 ?-@A 1 / 2 1 ?-@A + / 2 1 ?-@A / 2

1 > +0

?-@A

(11)

Equation 11 contains three parameters, namely hR, h and dVo, but it can be expressed using only two dimensionless parameters, H and R, defined as B=

$+ $

and

in the following form: 8,9 [%] = 100 ∙

C=

!$+

?(@A+) + D ?@A F ?(@A+) + ?@A F

DE0

(12)

Because the thickness of the adsorbed layer cannot be larger than a half of the pore size, we know that 0 < R < 1 and R < H. The value of R value must be much smaller than this to fulfill the assumption of preserved pore geometry during adsorption. When the adsorbed layer thickness is at most 10% of the pore size, then R < 0.2. When the size of the reference standard matches precisely the size of the sample, H = 1, and the limit error 8,9 = 0 from Eq. 12. However, error in the pressure drop measurement is minimized when the difference 6 ACS Paragon Plus Environment

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between the size of the reference standard and sample is greatest. Therefore, the limit error 8,9 for values of R between 0 and 0.2, and wide range of H values was calculated (Table 1). Table 1: Limit error 8,9 [%] calculated from Eq. 12 for a wide range of R and H values. H\R

0.001

0.018

0.025

0.045

0.080

0.100

0.110

0.120

0.140

0.160

0.180

0.200

5 3 2.5 2.4 2.2 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

0.04 0.03 0.03 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.02 0.01 0.01 0.01 0.00 0.00 0.01 0.01 0.02 0.03 0.05 0.08 0.12 0.20

0.72 0.60 0.54 0.53 0.50 0.45 0.43 0.40 0.37 0.34 0.30 0.26 0.21 0.15 0.08 0.00 0.10 0.23 0.39 0.61 0.93 1.40 2.19 3.84

1.01 0.84 0.76 0.74 0.69 0.63 0.60 0.56 0.52 0.48 0.42 0.36 0.29 0.21 0.12 0.00 0.14 0.32 0.55 0.86 1.30 1.97 3.10 5.48

1.82 1.53 1.38 1.34 1.25 1.15 1.09 1.02 0.95 0.87 0.77 0.66 0.54 0.39 0.21 0.00 0.26 0.59 1.02 1.59 2.41 3.68 5.89 10.72

3.28 2.75 2.49 2.42 2.27 2.08 1.98 1.86 1.72 1.57 1.40 1.21 0.98 0.71 0.39 0.00 0.48 1.09 1.88 2.97 4.55 7.05 11.59 22.53

4.13 3.47 3.14 3.05 2.86 2.63 2.50 2.35 2.18 1.99 1.78 1.53 1.24 0.90 0.49 0.00 0.61 1.39 2.42 3.83 5.90 9.22 15.45

4.55 3.83 3.47 3.38 3.17 2.91 2.76 2.60 2.41 2.21 1.97 1.69 1.37 1.00 0.55 0.00 0.68 1.55 2.69 4.27 6.60 10.38 17.59

4.98 4.20 3.80 3.70 3.47 3.19 3.03 2.85 2.65 2.42 2.16 1.86 1.51 1.10 0.60 0.00 0.75 1.71 2.98 4.73 7.34 11.61 19.88

5.85 4.94 4.48 4.36 4.09 3.77 3.58 3.37 3.13 2.86 2.55 2.20 1.79 1.30 0.71 0.00 0.89 2.04 3.57 5.70 8.90 14.25 25.04

6.73 5.69 5.16 5.03 4.72 4.35 4.14 3.89 3.62 3.31 2.96 2.55 2.07 1.51 0.83 0.00 1.04 2.39 4.19 6.73 10.60 17.19 31.14

7.62 6.46 5.87 5.72 5.37 4.95 4.71 4.44 4.13 3.78 3.38 2.91 2.37 1.73 0.95 0.00 1.20 2.75 4.85 7.84 12.44 20.50 38.52

8.53 7.24 6.58 6.42 6.03 5.57 5.29 4.99 4.65 4.25 3.81 3.29 2.68 1.95 1.08 0.00 1.36 3.14 5.55 9.02 14.46 24.25

As seen in Table 1, as H departs from 1 the relative error 8,9 increases. More importantly, there is a wide range of values of H and R where 8,9 is less than 5%. Therefore, we can rewrite Eq. 9 without substantial error using Eq. 10, and determine the adsorbed layer thickness h without information on the convective matrix pore geometry (n) or the pore size (dVo): ℎ=

$+

∆. / ∆.+ ∆. ?/ ∆.

?-

=

∆. / ∆. ∆. ,-∆. / +

$+ ∗,-

(13)

The remaining source of error in Eq. 13 is the pressure drop measurement. The effect of experimental error on the calculation of the adsorbed layer thickness h was determined by error propagation23 based on Eq. 13. In this analysis, pressure drop was assumed to be measured using a differential manometer, where there was no correlation between pressure drop measurements on the original bed and the bed saturated with adsorbed molecules, and that the relative accuracy of the manometer (RSD) was constant over the entire measurement range. For a variable that is a function of three independent variables y=f(x1, x2, x3), an error of y due to a limited accuracy of x1, x2 and x3, can be estimated as:23 7 ACS Paragon Plus Environment


I(J) = K or

I(J) = K

LM

LN@

LM

LN@

!

Page 8 of 15

!

I(O ) + LN I(O! ) + LN I(OQ )

!

LM

LM

P

!

!

∙ O ∙ R(O ) + LN ∙ O! ∙ R(O! ) + LN ∙ OQ ∙ R(OQ ) LM

LM

P

(14) !

(15)

where I (J) is an absolute error of y, I(O) is an absolute error of x, I (O ) = O ∙ R(O), and R(O) is a relative error, that when constant, allows Eq. 15 to be written as: I(J) = R(O)K

!

!

∙ O + LN ∙ O! + LN ∙ OQ

LM

LN@

LM

LM

P

!

(16)

and R (J ) =

S(M) M

T(N)U

=

VW ∙N VX@ @

VW ∙N VX

M

VW ∙N VXP P

(17)

where w(y) is a relative error of y. In the present analysis, the independent experimental measurements are ΔPo, ΔPA and ΔPAR. The thickness of the adsorbed layer for the reference standard is known. Therefore, using Eq. 16 to calculate the propagated error in the adsorbed layer thickness h from Eq. 13 gives: I(ℎ) = or

T(∆)

-,

∆.Y / ∆.+

I(ℎ) =

∆. ∆. +

$+ ∙,

UZ

∆Y

$+ ∙T(∆)

-,

∆. Y / ∆.+

!

$+ ∙,

∙ ∆[ \ + Z

K%&

∆

∆+

∆.Y ∆. +

∆

!

+ %&

!

∙ ∆ \ + Z

∆Y

∆+

!

+ %&

$+ ∙,

∆. Y ∆.

∆+

∆Y !

∆

!

∙ ∆] \

(18)

and for Eq. 17: R (ℎ ) =

T(∆)

∆. ∆. , Y ∙, Y ∆.+ ∆.

K%&

∆

∆+

!

+ %&

∆Y

∆+

!

+ %&

∆Y !

∆

(19)

where I(ℎ) is the absolute error of h, R(ℎ) is the relative error of h, and R(∆) is the relative error of the pressure drop difference measurement. By inspection of Eq. 19, we see that R(ℎ) is independent of absolute pressure values as only ratios of these values appear. Pressure drop ratios

∆Y

∆+

can be expressed in terms of R using Eq. 8, rewritten as: ∆

∆+

= (1 − C)

!


(20)

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We limit our analysis to the working range of R between 0 and 0.2 as discussed previously. The lower limit ∆ ∆ is Y = 1 when R = 0, while the upper limit is Y = 0.8n+2 when R = 0.2. By definition, the value of n is ∆+

∆+

expected to be between 2 and 3, and probably closer to 3. For n = 2, To analyze the broadest likely range, we set

∆Y

∆+

∆Y

∆+

= 0.41 and for n = 3,

∆Y

∆+

= 0.33.

to the range between 1 and 0.33. By analogy, this range

also applies to the pressure drop measurement ratio of the sample

∆Y

∆

.

Experimental error in the adsorbed layer thickness R(ℎ) according Eq. 19, normalized by the relative measurement error R (∆), is plotted in Figure 2. As shown in Figure 2, the normalized experimental error R(ℎ) ∆ 8^N_ = R(∆) rapidly increases as ∆ decreases towards 1. This makes sense because when the adsorbed

layer thickness is very small, then the pressure drop ratio is too small to measure. Interestingly, there is ∆ ∆ minimum value of the normalized experimental error 8^N_ versus for each fixed value of +, and after

that point the value of 8^N_ does not increase significantly as

∆ ∆ ∆

∆

increases. The position of the minimum

was determined by setting the derivative of Eq. 19 to zero, resulting in: (∆] )! = ∆[ ∙ ∆

(21)

20 18 16 H = 0.113

H = 0.146

H = 0.242

14

σexp = w(h)/w(∆P)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


H = 0.096

ΔPAR/ΔPo = 1.1

12

ΔPAR/ΔPo = 1.3

10

ΔPAR/ΔPo = 1.9 8

ΔPAR/ΔPo = 3.0

6

H = 0.656

4

H = 1.547

2

H = 2.532

H = 0.395

H = 0.305

H = 0.259

H = 0.931

H = 0.719

H = 0.611

H = 1.178

H = 1.000

H = 1.524

0 1

1.5

2

2.5

3

Figure 2: Normalized experimental error 8^N_ of the adsorbed layer thickness h. H is calculated according ∆PA/ΔPo

Eq. 23 for

∆ ∆

= 1.5, 2, 2.5, 3, and for

∆+ ∆

= 1.1, 1.3, 1.9, 3.0 when the isotropy constant n = 3.

When Eq. 21 is inserted in Eq. 19, the minimal experimental error for a given 8^N_, 9

=

K3b ∆

2

%& ∆cC d

∆Y

∆+

ratio is:

(22)



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Equations 21 and 22 enable calculation of the optimal measuring conditions for the least error in determination of the adsorbed layer thickness h. As seen from Eq. 22, as that once

∆ ∆

∆+ ∆

increases, 8^N_, 9 decreases. However, it is important to notice from Figure 2

exceeds about 1.5, the value of 8^N_ is not appreciably different than 8^N_, 9 . Little is gained

in going to pressure drop ratios higher than this in terms of decreasing the value of 8^N_ . We know from Eq. 21 that the values of the pressure drop ratio at the point of minimum error are related by the ∆ ∆ ∆ ∆ expression = ( + )!. Therefore, once exceeds about 1.5, and + exceeds about 1.22, then the ∆

∆

∆

∆

normalized experimental error in the adsorbed layer thickness is essentially at the minimum value. The R < ∆ ∆ 0.2 approximation sets the upper limit for the pressure drop ratio such that < 3 and + < 3. Because

of that, any value of

∆ ∆

between 1.5 and 3.0, and value of

∆+ ∆

∆Y

∆

between 1.22 and 3.0, is appropriate

experimentally for estimation of adsorbed layer thickness h with low experimental error 8^N_ . At the upper limit, when

∆+ ∆

=

∆ ∆Y

ℎ = 3, then 8^N_ = RR(∆ = 1.3, which means that the relative error in the ) ( )

determination of h, is only 30% greater than the relative error in measurement of the pressure drop difference. Manometer measurement error is commonly about 2% RSD, which makes it possible to determine the adsorbed layer thickness to about 2.6% error using the method described in the present work. Accuracy of h estimation is combination of both errors, and can be defined as σtot !

! =K8,9

+ 8^N_ ∙ R(∆) , where σtot is total error and R(∆) has % units. This requires to correlate

besides R also H value to a pressure drop ratio

taking into account definition of H (Eq. 12):

∆

∆+

B=

and

∆

∆

. This can be done by combining Eqs. 7 and 8,

@ ∆. g / ∆.+ @ ∆. g 0- / ∆.

0-

(23)

As for R, also to calculate H, value of isotropy constant is required. Again we assume that the value n is close to 3, therefore we set n =3 and calculate H values for points indicated in Figure 2. As shown in Figure 3, there is a broad range of experimental conditions where total error is below 10%. ∆ ∆ The total error is symmetric for and + as expected. Even higher accuracy can be achieved if ∆

∆

measurement range is narrowed. In presented calculations R(∆) of 2% was assumed for measuring instrument, therefore values of total error σtot would differ for different measurement accuracy but trends would be the same. This analysis demonstrates that accurate determination of adsorbed layer thickness h is possible, making the proposed method attractive for different applications. It is therefore interesting to estimate what thickness of adsorbed layer can be actually determined with high accuracy. Figure 3 shows that for low σtot, the pressure drop ratio between original and bed with adsorbed molecules should be in the range between 1.5 and 3, as mentioned previously.


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Figure 3: Total experimental error σtot as a function of

∆ ∆Y

and

∆+ ∆Y

for an isotropy constant n = 3.

According to Eqs. 6 and 8, the pressure drop ratio is defined by the layer thickness and the pore diameter. We therefore can estimate what pore size is required to allow accurate determination of a certain adsorbed layer thickness h according Eq. 24. =

!$

@ ∆. g h0- / i ∆.

(24)

Although pressure drop of porous bed depends on several parameters (see Eq. 3) we made an estimation for methacrylate monoliths having porosity of 0.6 and structural constant kV of 0.013.24 Results are presented in Table 2. Table 2: Estimated pressure drop of the original bed ΔPo according Eq. 3 as a function of pore size together with the minimal and maximal adsorbed layer thickness h that can be estimated with defined accuracy and maximal pressure drop after adsorption. Relative measurement accuracy R(∆) was assumed to be 2% and the isotropy constant n=3. Parameters for pressure drop calculation are: bed thickness 1 mm; bed porosity 0.6; bed structural constant kv = 0.013; mobile phase linear velocity = 40 cm/h; mobile phase viscosity = 1 mPas. hmin [nm] dvo [nm] 50 100 200 300 400 500 600 700 800

∆P0 [bar] 56.980 14.245 3.561 1.583 0.890 0.570 0.396 0.291 0.223

20% 0.69 1.38 2.76 4.13 5.51 6.89 8.27 9.65 11.03

10% 1.28 2.56 5.11 7.67 10.22 12.78 15.34 17.89 20.45 11

5% 2.52 5.03 10.07 15.10 20.14 25.17 30.21 35.24 40.28

ACS Paragon Plus Environment

hmax [nm] ∆PA,max [bar] 4.93 170.940 9.86 42.735 19.73 10.684 29.59 4.748 39.45 2.671 49.31 1.709 59.18 1.187 69.04 0.872 78.90 0.668


900 1100 1300 1400 1600 1800 2100 2500 3000 3500 4000 4500 5000 6000 7000 8000 9000 10000

0.176 0.118 0.084 0.073 0.056 0.044 0.032 0.023 0.016 0.012 0.009 0.007 0.006 0.004 0.003 0.002 0.002 0.001

12.40 15.16 17.92 19.30 22.05 24.81 28.94 34.46 41.35 48.24 55.13 62.02 68.91 82.70 96.48 110.26 124.04 137.83

23.00 28.12 33.23 35.78 40.90 46.01 53.68 63.90 76.68 89.46 102.24 115.02 127.80 153.36 178.92 204.48 230.04 255.60

45.31 55.38 65.45 70.48 80.55 90.62 105.72 125.86 151.03 176.20 201.38 226.55 251.72 302.07 352.41 402.75 453.10 503.44

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88.77 108.49 128.22 138.08 157.81 177.53 207.12 246.57 295.89 345.20 394.52 443.83 493.15 591.78 690.40 789.03 887.66 986.29

0.528 0.353 0.253 0.218 0.167 0.132 0.097 0.068 0.047 0.035 0.027 0.021 0.017 0.012 0.009 0.007 0.005 0.004

Based on Table 2, determination of an adsorbed layer thickness h below 1 nm seems possible to be determined using a very low pore size below 100 nm. This small pore size can be easily obtained for e.g. methacrylate monoliths by adjusting properly polymerization mixture composition and polymerization temperature.4 Whether it is indeed possible to measure such a small thickness is still to be confirmed, as other surface phenomena such as proton transfer and electric double layer can shield the results.25-27 Additional sources of error, not considered in our estimation, might be change of surface properties after adsorption that might occur with molecules having heterogeneous surface properties e.g. polar and nonpolar domains, such as detergents or certain membrane proteins. In this case drag on the mobile phase might be different possibly resulting in inaccurate estimation of the adsorbed layer thickness. Again, this effect would be more pronounced for small molecules. On the other hand, there is no upper limit of the measured thickness because with an increase of the pore size, also the measured adsorbed layer thickness can be larger. Table 2 covers thicknesses in the range from peptides (few nanometers) up to large viruses or plasmid DNA (several hundred nanometers). It is important to emphasize that although values of the pore size are presented in Table 3, there is no need for precise determination or the pore size to accurately estimate adsorbed layer thickness h. Approximate estimation of the pore size is required only to keep R value within required range (R < 0.2) or avoid e.g. blocking of the bed by adsorbing molecules28 and can be determined via mercury porosimetry4 or pressure drop data combined with porosity14 when needed. The implementation of the theoretical method for the determination of the adsorbed layer thickness from pressure drop data was examined using mono-disperse latex spheres as the reference standard and 2 different proteins of known size as the adsorbed layer thickness to be determined. CIM QA disk monolithic column (340 µL, 3 mm thick and 12 mm diameter, pore diameter 1300 nm), obtained from BIA Separations (Ajdovščina, Slovenia) was connected to an HPLC pump. The pressure drop on the monolithic column was measured using differential manometer (Mid-West Instruments, Sterling Heights, MI, USA). Mono-disperse carboxylated polymeric spheres of 25 nm diameter (Micromod Partikeltechnologie GmbH, Rostock, Germany) were used as the reference standard, and thyroglobulin (THY) and bovine serum albumin (BSA) (both Sigma-Aldrich, St. Lois, MO) were used for form test adsorbed layers. Pressure drops of original column and column saturated with adsorbed molecules were recorded 12 ACS Paragon Plus Environment

Page 13 of 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


and used in Eq. 13 to calculate the adsorbed layer thickness using the average value of the latex spheres as the reference standard.

∆+ ∆

= 1.63 bar for

Results are shown in Table 3. The layer thickness calculated using Eq. 13 matched literature values for the hydrodynamic radius of BSA (7.2 nm) and THY (20.2 nm). The pressure drop ratios were all within the limit ∆ of the approximation R < 0.2 which requires that < 3. However, to reach the minimum error requires that

∆ ∆

> 1.5 and

∆+ ∆

∆Y

> 1.22. This was true for the latex spheres reference standard and for the THY test

protein, but not true for BSA. Nevertheless, the confidence intervals were quite small and the values of the layer thicknesses were quite reasonable. This means that the method of the present work is robust and may be useful for conditions that violate the restrictions calculated from the propagation of error analysis. Table 3. Illustration of the determination of the adsorbed layer thickness from pressure drop data using polystyrene latex spheres of 25 nm diameter as the reference standard, and thyroglobulin (THY) and bovine serum albumin (BSA) as the unknown adsorbed layers. Different flow rates were used.

∆Po (bar) ∆PA (bar) ∆PA/∆Po h (nm) Eq. 13 BSA

2.1 1.5 1.3

2.4 1.7 1.5

1.14 1.13 1.15

6.8 6.4 7.3 6.9 (0.3)*

THY

2.5 1.3

3.6 2.0

1.44 1.54

18.7 22.1 20.4 (1.7)

Latex spheres

3.5 2.85 2.35 5.5

5.7 4.3 3.9 9.5

1.63 1.51 1.66 1.72

25.0 21.1 26.0 27.7 24.9 (1.4)

*average (95% CI)

4. CONCLUSIONS The proposed methodology demonstrates that robust measurement of adsorbed layer thickness can be made based on pressure drop measurements using very simple experimental set-up. No information about the adsorption bed pore size or morphology is needed, and any convective based matrix can be used. The proposed method requires no particular pretreatment of sample or matrix, and can be performed on-line, therefore one can envision its application in monitoring and study of various processes involving peptides, proteins, DNA or viruses while adsorbed on surfaces, and the method may find use in the field of polymers and nanoparticles. Under a wide range of experimental conditions, error of the thickness determination was below 10%, which makes it feasible to study various surface adsorption phenomena such as molecular orientation or conformational changes. The method was illustrated using experimental data, and the mathematical method and experimental results were in close agreement. We believe this novel method 13 ACS Paragon Plus Environment


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offers a low-cost way to characterize molecules on surfaces, and to study bi-molecular interactions on surfaces. ACKNOWLEDGEMENTS This work was partly supported by Slovenian Research Agency (ARRS) through programme P1-0153 and project L4-7628. Study was also supported by European Regional Development Fund and Slovenian Ministry of Education, Science and Sport (project BioPharm.Si). The authors equally contributed to the work. The authors declare no competing financial interest.

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