**5. Theoretical aspects**

two SPE frits. It is quite unfortunate that the authors did not publish the resultant SPE device otherwise it would have contributed towards a better understanding of the behaviour of the sorbent bed under compression. Similar to conventional MEPS [77] (see Fig.12.) procedure that proceeds via a draw eject cycle, a variable speed stirring motor was employed to drive solvents through the electrospun nanofiber MEPS device. Despite the fact that the authors did not publish their fabricated device, their contribution was invaluable as they were the first to

In our group, we fabricated a miniaturised version of an electrospun nylon 6 nanofiber based disk SPE device (Fig.13.a). The device was fabricated by cutting out circular portions of 5 mm diameter from the nanofiber sheet (Figure.13.b) and stacking them up to an optimal sorbent

**Figure 13.** (a) Photograph of disk (II) SPE device, (b) photograph of an electrospun nylon 6 nanofiber mat showing

Operationally, solvents have been driven through the SPE sorbent bed manually by a micro‐ pipette, syringe or by a vacuum manifold. All these approaches seem to be tedious particularly

regions from which disks were cut and (c) SEM image of sorbent bed zoomed in top surface [59].

demonstrate an electrospun nanofiber based MEPS device.

**Figure 12.** Representation of a MEPS syringe from SGE Analytical Science [77]

mass of 4.6 mg (5× 350 µm disks).

18 Advances in Nanofibers

An optimal sorbent could provide a platform for fast analyte mass-transfer kinetics, which depends on the physicochemical properties of the sorbent (surface area, pore structure and surface chemistry). Although the physicochemical properties of the sorbent may be used to predict SPE performance, bias towards sorbent characterization in SPE-method development is more effective, as the contribution of physicochemical properties and flow rate of solution to analyze mass transfer kinetics is significant. Initial evaluation of new sorbent materials for SPE application can be achieved without complete characterization of the physicochemical properties of the sorbent. Progress could be made towards a better understanding of new sorbent materials by relying on the theoretical prediction of the physicochemical properties of the sorbent.

<sup>2</sup> \* *B R <sup>N</sup> V V*

*B Ro* 2 *a a V Va*

number of theoretical plates Eq.3.

Where *ao* =(1 - *<sup>b</sup>*)<sup>2</sup>

expressed as;

teristics.

through the sorbent bed).

flow characteristics of the sample phase.

data points do not always follow a smooth pattern.

*N* æ ö - <sup>=</sup> ç ÷ ç ÷ è ø

1

, *a*1 and *a*2 are complex functions of *b* evaluated from the tabular data [80]

*VV k R M* = + (1 ) (4)


<sup>2</sup> 1 2

*N N*

and *b* is the breakthorugh level (the fraction of the total mass of analyte which has passed

The retention factor (*k*) is calculated from the fundamental equation of chromatography

Given that the equations for calculating (*N* ) and (*k*) have been applied to microparticle based sorbents (commercially available sorbents), a question raised would be whether they would be suitable for nanofibrous sorbent bed material. Moreso, bearing in mind that the research work on electrospun nanofiber based SPE sorbents is a step towards the development of electrospun nanofiber based HPLC stationary phases, it is necessary to have a fundamental understanding of their kinetic and thermodynamic properties in relation to retention charac‐

Theoretical and experimental methods have been proposed for determining breakthrough curves [81, 82]. Although experimental breakthrough curve determination by frontal chroma‐ tography (continuously added sample to sorbent bed see Fig.16.) is more tedious, it is more useful for SPE device fabrication. This is due to the fact that it serves as a guide for under‐ standing the effect of sorbent packing format, packing density and sorbent morphology on the

Under ideal conditions, the breakthrough curve forms a smooth sigmoid shape (see Fig.17.)

However, experimentally the breakthrough curve is plotted by using a line of best fit as the

The correct determination of breakthrough parameters is subject to debate as several methods have been proposed [83, 84]. Nevertheless, a mathematical modeling approach may be said to

with the specific shape depending on the sorbent retention characteristics

æ ö = ++ ç ÷ è ø

While the second method was proposed by Lövkvist and Jönsson [80] who adopted a differ‐ ential numerical solution to the breakthrough curve for analytes on a sorbent bed with a low

(2)

21

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(3)

For sorbent screening, does one start off with batch experiments and then transfer to packed sorbent formats? Batch experiments are important as they give information about sorbent adsorption characteristics, in particular adsorption kinetics derived from the equilibrium isotherms (Freundlich and Langmuir) [78]. Although batch and flow through SPE rely on the distribution or partition coefficient, thus directly linked, a flow through based screening experimental approach that relies on breakthrough profiles could be employed.

Given that the major sorption parameters characterizing a sorbent are analyte recovery efficiency and breakthrough volume [21], the experimental design for fabrication and evalu‐ ation of SPE devices could proceed in two steps:


One of the most important characteristic parameters in establishing the suitability of a SPE sorbent bed for extracting target analytes is the breakthrough volume (*VB*) as it gives an indication of the sorbent's loading capacity. Assuming that there is a measurable analyte retention, the breakthrough curve forms a sigmoid shape (see Fig.17.) that gives an indication of the analyte mass transfer kinetics as a function of the sorbent retention characteristics [21]. In addition to the breakthrough volume, two important parameters that are obtained from the breakthrough curve are holdup volume (*VM* ) and retention volume (*V <sup>R</sup>*).

These three parameters: (*VB*), (*V <sup>R</sup>*), (*VM* ) can be defined as points corresponding (on the breakthrough curve) to 1%, 99% and 50% of the maximum concentration of analyte in the eluate. From these parameters, two important chromatographic characteristics of a sorbent bed; theoretical plates (*N* ) and retention factor (*k*) are calculated.

The number of theoretical plates corresponds to the extraction efficiency and it depends on the physicochemical properties of the sorbent, particularly the available surface area for analyte retention. While the retention factor gives an indication of the quantity of sorbed analyte and it is directly related to the sorbent recovery efficiency.

Two methods have been employed for the calculation of theoretical plates for conventional SPE sorbents, the first was proposed by Werkhoheve-Goëwie and co-workers [79] Eq.2.

Electrospun Nanofiber Based Solid Phase Extraction http://dx.doi.org/10.5772/57100 21

$$V\_B = V\_R \, ^\star \left(\frac{\sqrt{N} - 2}{\sqrt{N}}\right) \tag{2}$$

While the second method was proposed by Lövkvist and Jönsson [80] who adopted a differ‐ ential numerical solution to the breakthrough curve for analytes on a sorbent bed with a low number of theoretical plates Eq.3.

surface chemistry). Although the physicochemical properties of the sorbent may be used to predict SPE performance, bias towards sorbent characterization in SPE-method development is more effective, as the contribution of physicochemical properties and flow rate of solution to analyze mass transfer kinetics is significant. Initial evaluation of new sorbent materials for SPE application can be achieved without complete characterization of the physicochemical properties of the sorbent. Progress could be made towards a better understanding of new sorbent materials by relying on the theoretical prediction of the physicochemical properties of

For sorbent screening, does one start off with batch experiments and then transfer to packed sorbent formats? Batch experiments are important as they give information about sorbent adsorption characteristics, in particular adsorption kinetics derived from the equilibrium isotherms (Freundlich and Langmuir) [78]. Although batch and flow through SPE rely on the distribution or partition coefficient, thus directly linked, a flow through based screening

Given that the major sorption parameters characterizing a sorbent are analyte recovery efficiency and breakthrough volume [21], the experimental design for fabrication and evalu‐

**i.** The first involves establishing an optimal sorbent mass, packing format and SPE

**ii.** The second involves determining breakthrough curves from which sorbent retention

One of the most important characteristic parameters in establishing the suitability of a SPE sorbent bed for extracting target analytes is the breakthrough volume (*VB*) as it gives an indication of the sorbent's loading capacity. Assuming that there is a measurable analyte retention, the breakthrough curve forms a sigmoid shape (see Fig.17.) that gives an indication of the analyte mass transfer kinetics as a function of the sorbent retention characteristics [21]. In addition to the breakthrough volume, two important parameters that are obtained from the

These three parameters: (*VB*), (*V <sup>R</sup>*), (*VM* ) can be defined as points corresponding (on the breakthrough curve) to 1%, 99% and 50% of the maximum concentration of analyte in the eluate. From these parameters, two important chromatographic characteristics of a sorbent

The number of theoretical plates corresponds to the extraction efficiency and it depends on the physicochemical properties of the sorbent, particularly the available surface area for analyte retention. While the retention factor gives an indication of the quantity of sorbed

Two methods have been employed for the calculation of theoretical plates for conventional SPE sorbents, the first was proposed by Werkhoheve-Goëwie and co-workers [79] Eq.2.

method at which quantitative recoveries (preferably above 80%) are achieved; and

experimental approach that relies on breakthrough profiles could be employed.

breakthrough curve are holdup volume (*VM* ) and retention volume (*V <sup>R</sup>*).

bed; theoretical plates (*N* ) and retention factor (*k*) are calculated.

analyte and it is directly related to the sorbent recovery efficiency.

ation of SPE devices could proceed in two steps:

characteristics are derived.

the sorbent.

20 Advances in Nanofibers

$$V\_B = V\_R \left( a\_o + \frac{a\_1}{N} + \frac{a\_2}{N^2} \right)^{-1} \tag{3}$$

Where *ao* =(1 - *<sup>b</sup>*)<sup>2</sup> , *a*1 and *a*2 are complex functions of *b* evaluated from the tabular data [80] and *b* is the breakthorugh level (the fraction of the total mass of analyte which has passed through the sorbent bed).

The retention factor (*k*) is calculated from the fundamental equation of chromatography expressed as;

$$V\_R = V\_M \left(1 + k\right) \tag{4}$$

Given that the equations for calculating (*N* ) and (*k*) have been applied to microparticle based sorbents (commercially available sorbents), a question raised would be whether they would be suitable for nanofibrous sorbent bed material. Moreso, bearing in mind that the research work on electrospun nanofiber based SPE sorbents is a step towards the development of electrospun nanofiber based HPLC stationary phases, it is necessary to have a fundamental understanding of their kinetic and thermodynamic properties in relation to retention charac‐ teristics.

Theoretical and experimental methods have been proposed for determining breakthrough curves [81, 82]. Although experimental breakthrough curve determination by frontal chroma‐ tography (continuously added sample to sorbent bed see Fig.16.) is more tedious, it is more useful for SPE device fabrication. This is due to the fact that it serves as a guide for under‐ standing the effect of sorbent packing format, packing density and sorbent morphology on the flow characteristics of the sample phase.

Under ideal conditions, the breakthrough curve forms a smooth sigmoid shape (see Fig.17.) with the specific shape depending on the sorbent retention characteristics

However, experimentally the breakthrough curve is plotted by using a line of best fit as the data points do not always follow a smooth pattern.

The correct determination of breakthrough parameters is subject to debate as several methods have been proposed [83, 84]. Nevertheless, a mathematical modeling approach may be said to

( ) <sup>1</sup> <sup>1</sup> *<sup>x</sup> Y x e* - <sup>=</sup> <sup>+</sup>

exponential decay is observed.

Where the variable *Y* might be considered to denote analyte concentration, *e* is Euler's number and the variable *x* might be thought of as volume. For values of *x* in a defined range of real numbers from -∞ to +∞, exponential growth which slows to linear growth and progresses to

The logistic function is the solution of the simple first order non-linear differential equation

The qualitative behavior may be understood as follows; the derivative is 0 at *Y* = 0 or 1 and the derivative is positive for *Y* between 0 and 1, and negative for *Y* above 1 or less than 0. Thus

1

*xc x*

Besides the logistic function, sigmoid functions include the ordinary arctangent, the hyperbolic tangent and the error function. However, for this book contribution, the logistic function was employed as the fundamental equation upon which curve fitting models were developed. The curve fitting mathematical model depends on the "S" shape of the data points and the selection

The Boltzmann model has been used to fit experimental data for conventional SPE sorbents as it has generally been accepted as an accurate method [84]. Thus it was the first model that was

1 2

*e* -

is the volume of sample flowing through the sorbent, *A*1 and *A*<sup>2</sup> are two regression parameters.

) is (*A*2) and it is obtained when *x* →*∞*, while the minimum value

+

*<sup>o</sup> x x dx*

2 1

Where Y represents the ratio of the eluted (*Ce*) to the inlet (*Ci*

*Ci*

*A A Y A*


*ee e*- = = + +

*dx* = - (6)

Electrospun Nanofiber Based Solid Phase Extraction

http://dx.doi.org/10.5772/57100

( ) ( )(<sup>1</sup> ( )) *<sup>d</sup> Yx Yx Yx*

( ) <sup>1</sup>

*x*

for any value of *Y* greater than 0 and less than 1, *Y* grows to 1.

Where 1 is chosen as the constant of integration (*e <sup>c</sup>* =1)

is guided by by the resulting regression constants.

explored (see equation 8).

The maximum value of ( *Ce*

Alternatively, the logistic function may be expressed symbolically as;

*<sup>e</sup> Y x*

(5)

23

(7)

(8)

*Ci* ), *x*

) analyte concentration ( *Ce*

**Figure 16.** Representation of a (a) online (b) offline frontal chromatography set-up

**Figure 17.** Mathematical simulations of breakthrough curves that can be obtained

be more preferable as it takes into consideration the actual shape of the breakthrough curve. In addition, interpreting observed experimental phenomena from the mathematical modeling perspective is invaluable as it provides a platform for linking up the various parameters influencing any system [85-89].

Our research group [68] and that of Susan Olesik were the first to report chromatographic parameters of electrospun nanofiber sorbent beds and chromatographic parameters of thin layer chromatography (TLC) plates [90], respectively. Therefore, an approach adopted in our group and results obtained will be discussed. Due to the importance of the accurate determi‐ nation of breakthrough parameters, it was imperative for us to compare different mathematical models for each set of data points obtained in the experimental procedure.

A breakthrough curve typically follows a form of an "S" shape (Fig.18.). Therefore, it may be considered to be a form of a logistic function. A simple logistic function may be defined by the formula;

Electrospun Nanofiber Based Solid Phase Extraction http://dx.doi.org/10.5772/57100 23

$$Y(\mathbf{x}) = \frac{1}{1 + e^{-\mathbf{x}}} \tag{5}$$

Where the variable *Y* might be considered to denote analyte concentration, *e* is Euler's number and the variable *x* might be thought of as volume. For values of *x* in a defined range of real numbers from -∞ to +∞, exponential growth which slows to linear growth and progresses to exponential decay is observed.

The logistic function is the solution of the simple first order non-linear differential equation

$$\frac{d}{dx}Y(x) = Y(x)(1 - Y(x))\tag{6}$$

The qualitative behavior may be understood as follows; the derivative is 0 at *Y* = 0 or 1 and the derivative is positive for *Y* between 0 and 1, and negative for *Y* above 1 or less than 0. Thus for any value of *Y* greater than 0 and less than 1, *Y* grows to 1.

Alternatively, the logistic function may be expressed symbolically as;

$$Y(\mathbf{x}) = \frac{e^{\mathbf{x}^{\times}}}{e^{\mathbf{x}^{\times}} + e^{\mathbf{c}^{\times}}} = \frac{1}{1 + e^{-\mathbf{x}}} \tag{7}$$

Where 1 is chosen as the constant of integration (*e <sup>c</sup>* =1)

be more preferable as it takes into consideration the actual shape of the breakthrough curve. In addition, interpreting observed experimental phenomena from the mathematical modeling perspective is invaluable as it provides a platform for linking up the various parameters

Our research group [68] and that of Susan Olesik were the first to report chromatographic parameters of electrospun nanofiber sorbent beds and chromatographic parameters of thin layer chromatography (TLC) plates [90], respectively. Therefore, an approach adopted in our group and results obtained will be discussed. Due to the importance of the accurate determi‐ nation of breakthrough parameters, it was imperative for us to compare different mathematical

A breakthrough curve typically follows a form of an "S" shape (Fig.18.). Therefore, it may be considered to be a form of a logistic function. A simple logistic function may be defined by the

models for each set of data points obtained in the experimental procedure.

**Figure 16.** Representation of a (a) online (b) offline frontal chromatography set-up

**Figure 17.** Mathematical simulations of breakthrough curves that can be obtained

influencing any system [85-89].

22 Advances in Nanofibers

formula;

Besides the logistic function, sigmoid functions include the ordinary arctangent, the hyperbolic tangent and the error function. However, for this book contribution, the logistic function was employed as the fundamental equation upon which curve fitting models were developed. The curve fitting mathematical model depends on the "S" shape of the data points and the selection is guided by by the resulting regression constants.

The Boltzmann model has been used to fit experimental data for conventional SPE sorbents as it has generally been accepted as an accurate method [84]. Thus it was the first model that was explored (see equation 8).

$$Y = A\_2 + \frac{A\_1 - A\_2}{1 + e^{\frac{x - x\_o}{dx}}} \tag{8}$$

Where Y represents the ratio of the eluted (*Ce*) to the inlet (*Ci* ) analyte concentration ( *Ce Ci* ), *x* is the volume of sample flowing through the sorbent, *A*1 and *A*<sup>2</sup> are two regression parameters. The maximum value of ( *Ce Ci* ) is (*A*2) and it is obtained when *x* →*∞*, while the minimum value of ( *Ce Ci* ) is approximately *A*1, obtained for *x* →0. The retention volume which corresponds to the point of inflexion on the breakthrough curve was obtained from the model and it corre‐ sponds to the ( *Ce Ci* ) <sup>=</sup> *<sup>A</sup>*<sup>1</sup> <sup>+</sup> *<sup>A</sup>*<sup>2</sup> 2

$$\mathbf{V}\_R = \mathbf{x}\_o \tag{9}$$

Where *<sup>F</sup>* <sup>=</sup> <sup>1</sup>

expression below

1 + *e* - ( *x* -*x o*) *b*

If the maximum value of ( *Ce*

tively.

and solving for *x* (corresponding to either *VM* or *VB*) resulted in the

<sup>1</sup> - *<sup>F</sup>* (16)

Electrospun Nanofiber Based Solid Phase Extraction

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25

(18)

*<sup>x</sup>* <sup>=</sup> *xo* <sup>+</sup> *<sup>b</sup>*ln *<sup>F</sup>*

*VM* and *VB* were obtained from equation 16 at *F* values of 0.99 and 0.01 respectively.

*<sup>Y</sup>* <sup>=</sup>*Yo* <sup>+</sup> *<sup>a</sup>* <sup>1</sup> - *<sup>e</sup>*-

corresponds to the point of inflexion while *b* and *c* are regression parameters.

) is expressed as

*<sup>F</sup>* \*(*Yo* <sup>+</sup> *<sup>a</sup>* ) <sup>=</sup>*Yo* <sup>+</sup> *<sup>a</sup>* <sup>1</sup> - *<sup>e</sup>*-

1

*<sup>c</sup>* <sup>+</sup> *<sup>b</sup>* ln ( *Yo*

Therefore, *VM* and *VB* can be calculated at *F* values corresponding to <sup>99</sup>

breakthrough curve can be expressed as a fraction (*<sup>F</sup>* ) of the maximum value of ( *Ce*

As *<sup>x</sup>* <sup>→</sup>*∞*, *Y* approaches the maximum value of ( *Ce*

*Ci*

*x* = *xo* - *b*ln 2

bed at 0.1 mlmin-1 with subsequent HPLC-DAD detection.

Solving for *x* results in the expression below;

The third model that was explored is the Weibull five parameter model (see equation 17)

( *<sup>x</sup>*-*xo*+*b*ln <sup>2</sup> 1 *c b* ) *c*

*Yo* and *a* correspond to the maximum and minimum points on the breakthrough curve, *xo*

*Ci*

( *<sup>x</sup>*-*xo*+*b*ln <sup>2</sup> 1 *c b* ) *c*

*<sup>a</sup>* <sup>+</sup> 1)-1

An example in which we compared the Boltzmann model and the sigmoid Weibull five parameter model for the evaluation of a microcolumn SPE device. The experimental approach to determine the breakthrough curves involved the use of a syringe pump (see Fig.14) to drive a sample of 500 ngml-1 corticosteroids through a 10 mg electrospun polystyrene fiber sorbent

(1 - *F* )-1

1

) <sup>=</sup>*Yo* <sup>+</sup> *<sup>a</sup>* thus, ( *Ce*

*Ci*

) at any point of the

*Ci* ).

*<sup>c</sup>* (19)

100 and <sup>1</sup>

<sup>100</sup> respec‐

The hold-up volume (*VM* ) and the breakthrough volume (*VB*) were calculated as the *x* values obtained from equating the following;

$$(\frac{99}{100})^\* A\_2 = A\_2 + \frac{A\_1 \cdot A\_2}{1 + e^{\frac{x \cdot x\_s}{\alpha}}} \tag{10}$$

$$A \left(\frac{1}{100}\right) \* A\_2 = A\_2 + \frac{A\_1 \cdot A\_2}{1 + e^{\frac{\cdot \cdot x\_s}{\alpha}}} \tag{11}$$

And by solving these equations, the formulae below were derived;

$$V\_B = \chi\_o + (d\chi)^\ast \text{ln} \left(\frac{100}{99} \left(1 - \frac{A\_1}{A\_2}\right) \cdot 1\right) \tag{12}$$

$$V\_M = \mathbf{x}\_o + (d\mathbf{x})^\ast \text{ln} \tag{99 - 100^\ast \frac{A\_1}{A\_2}} \tag{13}$$

The second mathematical model employed is the sigmoid three parameter model (see equation 14)

$$Y = \frac{a}{1 + e^{-\frac{\{x \colon x\_o\}}{b}}} \tag{14}$$

Similar to the Boltzmann model, Y represents the ratio of the eluted (*Ce*) to the inlet (*Ci* ) analyte concentration ( *Ce Ci* ), *x* is the volume of sample flowing through the sorbent, *a* and *b* are two regression parameters. The maximum value of ( *Ce Ci* ) is (*a*). Calculation of *VM* and *VB* was achieved by expressing ( *Ce Ci* ) as a fraction of *a* as follows;

$$F\left(\frac{C\_s}{C\_i}\right) = F^\* a \tag{15}$$

Where *<sup>F</sup>* <sup>=</sup> <sup>1</sup> 1 + *e* - ( *x* -*x o*) *b* and solving for *x* (corresponding to either *VM* or *VB*) resulted in the expression below

$$\mathbf{x} = \mathbf{x}\_o + b \ln \frac{F}{1 - F} \tag{16}$$

*VM* and *VB* were obtained from equation 16 at *F* values of 0.99 and 0.01 respectively.

The third model that was explored is the Weibull five parameter model (see equation 17)

$$Y = Y\_o + a \mathbf{1} \cdot e^{-\left(\frac{x - x\_o + b \ln \frac{1}{2}}{b}\right)^k} \tag{17}$$

*Yo* and *a* correspond to the maximum and minimum points on the breakthrough curve, *xo* corresponds to the point of inflexion while *b* and *c* are regression parameters.

As *<sup>x</sup>* <sup>→</sup>*∞*, *Y* approaches the maximum value of ( *Ce Ci* ) <sup>=</sup>*Yo* <sup>+</sup> *<sup>a</sup>* thus, ( *Ce Ci* ) at any point of the breakthrough curve can be expressed as a fraction (*<sup>F</sup>* ) of the maximum value of ( *Ce Ci* ).

If the maximum value of ( *Ce Ci* ) is expressed as

of ( *Ce Ci*

14)

analyte concentration ( *Ce*

achieved by expressing ( *Ce*

*Ci*

two regression parameters. The maximum value of ( *Ce*

*Ci*

sponds to the ( *Ce*

24 Advances in Nanofibers

*Ci*

) <sup>=</sup> *<sup>A</sup>*<sup>1</sup> <sup>+</sup> *<sup>A</sup>*<sup>2</sup> 2

obtained from equating the following;

) is approximately *A*1, obtained for *x* →0. The retention volume which corresponds to

*V x R o* = (9)

) - 1) (12)

) (13)

) is (*a*). Calculation of *VM* and *VB* was

) = *F* \**a* (15)

(10)

(11)

(14)

)

the point of inflexion on the breakthrough curve was obtained from the model and it corre‐

The hold-up volume (*VM* ) and the breakthrough volume (*VB*) were calculated as the *x* values

*A*<sup>1</sup> - *A*<sup>2</sup>

*A*<sup>1</sup> - *A*<sup>2</sup>

1 + *e x* -*x o dx*

1 + *e x* -*x o dx*

<sup>99</sup> (1 - *<sup>A</sup>*<sup>1</sup> *A*2

The second mathematical model employed is the sigmoid three parameter model (see equation

Similar to the Boltzmann model, Y represents the ratio of the eluted (*Ce*) to the inlet (*Ci*

*A*1 *A*2

), *x* is the volume of sample flowing through the sorbent, *a* and *b* are

*Ci*

( <sup>99</sup>

( 1

And by solving these equations, the formulae below were derived;

*VB* <sup>=</sup> *xo* <sup>+</sup> (*dx*)\*ln ( <sup>100</sup>

*VM* = *xo* + (*dx*)\*ln (99 - 100\*

*<sup>Y</sup>* <sup>=</sup> *<sup>a</sup>* 1 + *e* - ( *x* -*x o*) *b*

) as a fraction of *a* as follows;

( *Ce Ci*

<sup>100</sup> )\**A*<sup>2</sup> = *A*<sup>2</sup> +

<sup>100</sup> )\**A*<sup>2</sup> = *A*<sup>2</sup> +

$$F^\*(Y\_o + a\_o) = Y\_o + a \mathbf{1} \cdot e^{\left[\underbrace{x \cdot x\_o \times b \ln\frac{1}{a^\*}}\_{b}\right]^\lambda} \tag{18}$$

Solving for *x* results in the expression below;

$$\mathbf{x} = \mathbf{x}\_o - b \ln 2^{\frac{1}{e^-}} + b \left[ \ln \left[ \left( \frac{\mathbf{y}\_o}{a} + 1 \right)^{\cdot 1} (1 - F)^{\cdot 1} \right] \right]^{\frac{1}{e^-}} \tag{19}$$

Therefore, *VM* and *VB* can be calculated at *F* values corresponding to <sup>99</sup> 100 and <sup>1</sup> <sup>100</sup> respec‐ tively.

An example in which we compared the Boltzmann model and the sigmoid Weibull five parameter model for the evaluation of a microcolumn SPE device. The experimental approach to determine the breakthrough curves involved the use of a syringe pump (see Fig.14) to drive a sample of 500 ngml-1 corticosteroids through a 10 mg electrospun polystyrene fiber sorbent bed at 0.1 mlmin-1 with subsequent HPLC-DAD detection.


a larger number of theoretical plates and consequently a large retention capacity as mass transfer kinetics would be enhanced. Therefore, a larger number of theoretical plates would correspond to a steeper slope of the breakthrough curve as a result of fast mass transfer kinetics.

**Figure 19.** Breakthrough curves determined for two steroids on electrospun polystyrene fibers packed in microcol‐

**Coefficient**

879140.39 (7.94) 3088.50 (40.73) -0.0143 (0.0094) 0.9953

Electrospun Nanofiber Based Solid Phase Extraction

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27

23676.51 (7.37) 3032.19 (34.33) -0.0159 (0.0110) 0.9943

*VM* **(µl)** *VR***(µl)** *Ncal Nobs k*

**R2**

umn sorbent bed format presented as lines of best fit using the Weibull five parameter model.

(0.30)

15042770.75 (8.91)

> *VB*(**µl**) Calculated

Betamethasone 1400 901 4250 3088.5 7.98 13.38 -0.27 Dexamethasone 1400 1020 4240 3032.2 9.1 13.80 -0.28

**Table 4.** Chromatographic parameters of the electrospun polystyrene fiber based micro column sorbent bed format

Given the fact that the feasibility of electrospun nanofiber based SPE has been successfully demonstrated, it is proposed that future research efforts should be channelled towards

Betamethasone 1.0396 (0.01) 545252504.47

(0.02)

**Table 3.** Regression parameters for the Weibull five parameter model

Observed

derived from the Weibull five parameter model.

Dexamethasone 1.0528

**Analyte** *VB*(**µl**)

**6. Conclusions**

**Analyte a b c xo yo**

**Table 2.** Chromatographic parameters of the electrospun polystyrene fiber based micro colum sorbent bed format derived from the sigmoid Boltzmann model

**Figure 18.** Breakthrough curves determined for two steroids on electrospun polystyrene fibers packed in micro col‐ umn sorbent bed format presented as lines of best fit using the sigmoid Boltzmann model.

The breakthrough volumes, equilibrium volumes, retention volumes, theoretical plates and retention factor values for betamethasone and dexamethasone were very close which was consistent with the closeness of their logP values. This consistency was observed for the breakthrough curves fitted using both the Boltzmann and the Weibull five parameter model. On that basis, it may be concluded that these mathematical models would be suitable for characterisation of any electrospun fiber sorbent-analyte system that follows a similar profile.

Comparison of the calculated theoretical plates for the disk (1.39-2.82) and the micro column (7.98-9.1) SPE devices revealed that the shape of the breakthrough curve could be related to the theoretical plates [80, 91]. In addition, the observed breakthrough volumes for the disk (400-500 µl) and micro column (1400 µl) respectively were consistent with the theoretical plates. This may mean that theoretical plates could be important for the retention character‐ istics of a SPE sorbent bed.

Given the fact that theoretical plates are a function of the available surface area for analyte interaction, it could be concluded that a sorbent material with a larger surface area may exhibit a larger number of theoretical plates and consequently a large retention capacity as mass transfer kinetics would be enhanced. Therefore, a larger number of theoretical plates would correspond to a steeper slope of the breakthrough curve as a result of fast mass transfer kinetics.

**Figure 19.** Breakthrough curves determined for two steroids on electrospun polystyrene fibers packed in microcol‐ umn sorbent bed format presented as lines of best fit using the Weibull five parameter model.


**Table 3.** Regression parameters for the Weibull five parameter model


**Table 4.** Chromatographic parameters of the electrospun polystyrene fiber based micro column sorbent bed format derived from the Weibull five parameter model.

## **6. Conclusions**

**Analyte**

26 Advances in Nanofibers

*VB*(µ**l**) Observed

derived from the sigmoid Boltzmann model

istics of a SPE sorbent bed.

*VB*(µ**l**) Calculated

Betamethasone 1400 848.18 5185.39 3044.89 7.69 13.71 -0.41 Dexamethasone 1400 924.53 5189.44 3100.98 8.12 13.29 -0.40

**Table 2.** Chromatographic parameters of the electrospun polystyrene fiber based micro colum sorbent bed format

**Figure 18.** Breakthrough curves determined for two steroids on electrospun polystyrene fibers packed in micro col‐

The breakthrough volumes, equilibrium volumes, retention volumes, theoretical plates and retention factor values for betamethasone and dexamethasone were very close which was consistent with the closeness of their logP values. This consistency was observed for the breakthrough curves fitted using both the Boltzmann and the Weibull five parameter model. On that basis, it may be concluded that these mathematical models would be suitable for characterisation of any electrospun fiber sorbent-analyte system that follows a similar profile.

Comparison of the calculated theoretical plates for the disk (1.39-2.82) and the micro column (7.98-9.1) SPE devices revealed that the shape of the breakthrough curve could be related to the theoretical plates [80, 91]. In addition, the observed breakthrough volumes for the disk (400-500 µl) and micro column (1400 µl) respectively were consistent with the theoretical plates. This may mean that theoretical plates could be important for the retention character‐

Given the fact that theoretical plates are a function of the available surface area for analyte interaction, it could be concluded that a sorbent material with a larger surface area may exhibit

umn sorbent bed format presented as lines of best fit using the sigmoid Boltzmann model.

*VM* **(**µ**l)** *VR***(**µ**l)** *Ncal Nobs k*

Given the fact that the feasibility of electrospun nanofiber based SPE has been successfully demonstrated, it is proposed that future research efforts should be channelled towards simplifying SPE device fabrication and SPE procedures. In addition, more research efforts should be channelled towards physicochemical characterisation of electrospun nanofibers and relating to retention characteristics of the electrospun nanofiber based sorbent beds. Therefore, it is expected that with a better control of nanofiber orientation, packing procedures and pore structure, analytically useful electrospun nanofiber based chromatographic sorbent beds for HPLC could be fabricated.

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By describing the wide nanofiber sorbent range of chemistries, functionalities and morpholo‐ gies, it is hoped that this book chapter has provided enough evidence to support the hypothesis that electrospun nanofibers will be an effective class of SPE sorbents.

However several questions relating to the role of electrospinning, surface area to volume ratio, sorptive capacity and the miniaturisation route (should it be SPE or SPME variations?) still need to be answered. Looking into the future one wonders if nanostructured materials will mark the end of the sorbent technology development chain. In conclusion one can say with some degree of confidence that through electrospinning, a new class of sorbent based techni‐ ques/devices will eventually find their way into the analytical process either as a low or high resolution separation step.
