Combined solvent- and non-uniform temperature-programmed gradient liquid chromatography. I – A theoretical investigation

Accepted Manuscript Title: Combined solvent- and non-uniform temperature-programmed gradient liquid chromatography I- A theoretical investigation Author: Fabrice Gritti PII: DOI: Reference:

S0021-9673(16)31212-2 http://dx.doi.org/doi:10.1016/j.chroma.2016.09.026 CHROMA 357898

To appear in:

Journal of Chromatography A

Received date: Revised date: Accepted date:

1-7-2016 26-8-2016 13-9-2016

Please cite this article as: Fabrice Gritti, Combined solvent- and nonuniform temperature-programmed gradient liquid chromatography I- A theoretical investigation, (2016), http://dx.doi.org/10.1016/j.chroma.2016.09.026 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Non-uniform temperature and solvent gradients are combined in liquid chromatography The fundamental gradient equations are solved for retention and peak width Combined temperature and solvent gradients are equivalent to an apparent solvent gradient

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The benefit of such gradient is assessed for the resolution of protein digests.

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cr

A maximum gain of 30% is expected for the peak capacity per unit time.

Page 1 of 41

Combined solvent- and non-uniform

2

temperature-programmed gradient liquid

3

chromatography

4

I- A theoretical investigation

cr

us

Fabrice Gritti∗

5

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1

an

Waters Corporation, Instrument/Core Research/Fundamental

M

Milford, MA 01757, USA

Abstract

d

6

An new class of gradient liquid chromatography (GLC) is proposed and its performance

8

is analyzed from a theoretical viewpoint. During the course of such gradients, both the sol-

9

vent strength and the column temperature are simultaneously changed in time and space.

10

The solvent and temperature gradients propagate along the chromatographic column at their

11

own and independent linear velocity. This class of gradient is called combined solvent- and

12

temperature-programmed gradient liquid chromatography (CST-GLC). The general expressions

13

of the retention time, retention factor, and of the temporal peak width of the analytes at elu-

14

tion in CST-GLC are derived for linear solvent strength (LSS) retention models, modified van’t

15

Hoff retention behavior, linear and non-distorted solvent gradients, and for linear temperature

16

gradients. In these conditions, the theory predicts that CST-GLC is equivalent to a unique and

17

apparent dynamic solvent gradient. The apparent solvent gradient steepness is the sum of the

18

solvent and temperature steepness. The apparent solvent linear velocity is the reciprocal of

19

the steepness-averaged sum of the reciprocal of the actual solvent and temperature linear ve-

20

locities. The advantage of CST-GLC over conventional GLC is demonstrated for the resolution

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7

∗

Corresponding author: (E-mail) Fabrice [email protected]; (Tel) 508-482-2311; (Fax) 508-482-3625.

Page 2 of 41

of protein digests (peptide mapping) when applying smooth, retained, and linear acetonitrile

22

gradients in combination with a linear temperature gradient (from 20o C to 90o C) using 300

23

µm × 150 mm capillary columns packed with sub-2 µm particles. The benefit of CST-GLC is

24

demonstrated when the temperature gradient propagates at the same velocity as the chromato-

25

graphic speed. The experimental proof-of-concept for the realization of temperature

26

ramps propagating at a finite and constant linear velocity is also briefly described.

27

Keywords: Combined solvent- and temperature-programmed gradient liquid chromatography;

28

gradient retention time; band compression factor; gradient peak width; peak capacity per unit

29

time.

30

* Tel: 508-482-2311; Fax: 508-482-3625; E-mail: Fabrice [email protected]

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21

2

Page 3 of 41

31

Contents

32

1 Introduction

4

33

2 Theory

6

2.1

35

Definition of combined solvent- and temperature-programmed gradient liquid chro-

ip t

34

matography (CST-GLC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.2

Solvent-temperature retention model . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

37

2.3

Expression for the retention time and local retention factor in CST-GLC . . . . . . .

8

38

2.4

Expression for the band compression factor and temporal peak width in CST-GLC . 10

39

2.5

Expression for the peak capacity in CST-GLC . . . . . . . . . . . . . . . . . . . . . . 12

41

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40

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36

3 Results and Discussion 3.1

16

Advantages and limitations of CST-GLC . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1.1

Advantages of CST-GLC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

43

3.1.2

Limitations of CST-GLC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

44

3.1.3

Limitations and approximations of the derived CST-GLC model . . . . . . . 18

3.3

Predicted performance of CST-GLC : fixing the linear velocity of the temperature gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

48

49

d

independent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

46

47

Predicted performance of CST-GLC : keeping the temperature and solvent steepness

te

3.2

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45

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42

3.4

Realization of traversing temperature ramps for CST-GLC . . . . . . . . . . . . . . . 22

50

4 Conclusion

25

51

5 Acknowledgements

27

52

6 Figures

34

3

Page 4 of 41

53

1

Introduction

54

The basic theory of gradient liquid chromatography (GLC) was established half a century ago

55

Due to its ceaseless and widespread application in the field of separation sciences, it has been contin-

56

uously reviewed, discussed, and refined until today 3–16 . For the sake of finding analytical solutions,

57

most theories of gradient liquid chromatography predicting retention and peak width assume that

58

the sample retention obeys the linear solvation strength (LSS) model

59

dient remains linear and not retained upon migration along the chromatographic columns. These

60

assumptions are excellent approximations when narrow ranges of solvent composition and shallow

61

gradients are experimentally applied.

ip t

and that the solvent gra-

us

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17

1, 2 .

In fact, they often remain inaccurate when applying fast and steep gradients and analyzing

63

complex high-molecular compounds (peptides, intact proteins, IgG, etc...). For such applications,

64

gradients can be retained and distorted while the retention behavior may be strongly non-linear

65

18, 19 .

66

to distorted gradients

67

phase

68

to cope with the disagreement between experimental and theoretical retention 18, 25, 26 . However, in

69

most cases, the chromatographer can only rely on solving the gradient problem numerically 10 . New

70

predictive tools for the retention time and the peak width are still needed in order to improve sample

71

quantification in LC. Experimental protocols aiming at measuring accurately the band compression

72

factor have also been proposed 27–29 . Yet, the results are scarce and appear uncertain for the lack of

73

understanding of the phenomenon of peak retention and band compression for complex gradients.

74

In a recent report, the prediction of the retention factor at elution and band compression factor was

75

investigated from a theoretical viewpoint for a large variety of gradients

76

and linear solvent gradients combined with a LSS retention model, non-uniform columns under

77

stationary regime

an

62

11, 20

and

(due to the actual adsorption of the strong solvent onto the stationary

d

have been investigated. Additionally, non-LSS retention models have been proposed

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te

21–24 )

14–16

M

Therefore, extensions of the standard theory of GLC to retained linear gradients

31, 32 ,

30 :

they include retained

and temperature-programmed gradients.

78

In this work, a new class of LC gradient is analyzed from a theoretical perspective: not only is

79

the solvent strength increasing with time and decreasing with increasing the axial posi-

80

tion along the column length (as it is for conventional solvent-programmed gradients) 4

Page 5 of 41

81

but so does the local temperature along the chromatographic column simultaneously.

82

Note that the combination of temperature and solvent gradient was already reported

83

33–38 .

84

creased along the whole column body. Therefore, no additional thermodynamic band

85

compression (to that of the solvent gradient alone) is observed. In contrast, in this

86

study, the temperature gradient is not uniform along the column length providing

87

additional band compression to that of the solvent gradient. This study concerns pri-

88

marily small i.d. columns such as capillary columns for which the local temperature

89

can be changed quasi-instantaneously across its diameter. The solvent and temperature

90

gradients act independently. They propagate along the column at their own velocity. In the first

91

part of this work, the general expression of the retention factor and temporal peak

92

width at elution are derived from the general equations of gradient chromatography

ip t

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an

1, 30

for such specific solvent-temperature gradients. In the second part, the expected

M

93

But, in these previous cases, the temperature of the column is uniformly in-

advantages and the limitations of this new class of gradient are discussed. In the

95

third part and based on the theoretical results, the benefits of combining solvent and

96

non-uniform temperature gradients are predicted theoretically for peptide mapping.

97

Finally, the first prototype device designed to generate temperature gradients that can

98

propagate at a finite linear velocity along the separation medium is briefly presented.

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5

Page 6 of 41

99

2

Theory

100

The purpose of this section is to derive, at any axial position z along the column, the exact

101

expressions of the local retention factor, k (z), of the band compression factor, G212 (z), and of the

102

spatial and temporal peak widths, σz (z) and σt (z), respectively, for LC gradients involving the

103

simultaneous change in solvent strength and temperature with time and distance along the

104

column.

105

2.1

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0

us

Definition of combined solvent- and temperature-programmed gradient liquid chromatography (CST-GLC)

106

The particularity of combined solvent- and temperature-programmed gradients is the simultaneous

108

changes (in time and space) of the solvent strength and column temperature. Such gradients

109

offer new options for the analysts: the characteristics of the solvent and temperature gradients

110

(steepness and linear velocity) can be independently set by the operator. For instance, a smooth

111

and fast solvent gradient can be combined with a steep and slow temperature gradients or vice-versa.

112

Issues regarding the potential advantage of combining the properties of solvent and temperature

113

gradients will be answered quantitatively in the next sections in terms of peak capacity per unit

114

time.

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107

On one hand, the solvent-programmed gradient remains classical: the composition of the strong

116

solvent is assumed to increase linearly with increasing time at the column inlet. Additionally, it

117

is not distorted during its progression along the chromatographic column, so, it propagates at a

118

constant linear velocity, uA , which is given by:

Ac ce p

115

uA =

u0 0 1 + kA

(1) 0

119

where u0 is the chromatographic linear velocity and kA is the constant retention factor of the

120

strong solvent on the stationary phase for any mobile phase composition and any temperature

121

applied during the gradient.

122

The variation of the volume fraction, ϕ(z, t), of the strong solvent as a function of the elapsed

6

Page 7 of 41

123

gradient time, t, and of the axial position, z, along the column is then written:   z ϕ(z, t) = ϕ0 + β t − uA

(2)

where t = 0 when the solvent gradient starts at the column inlet (z = 0), ϕ0 is the initial volume

125

fraction of the strong solvent in the eluent mixture, and β is the temporal steepness of the solvent

126

gradient.

cr

ip t

124

On the other hand, the temperature-programmed gradient is not conventional: it is not about

128

simply imposing a stationary (zero linear velocity), spatial temperature gradient along the column

129

length. Also, it does not consist in changing uniformly the temperature of the column with increas-

130

ing time. It is fundamental and important to note that the proposed CST-GLC is based

131

on the delivery of a temperature ramp that can propagate at a finite and constant

132

linear velocity along the chromatographic column. In fact, similarly to the above-mentioned

133

solvent gradient, it is about preparing a temperature gradient characterized by its temporal steep-

134

ness (τ ) and its own linear velocity (uT ). Therefore, uT can be fixed independently of uA within

135

practical limits. The temperature profile along the column is then written:

te

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127

(3)

Ac ce p

  z T (z, t) = T0 + τ t − uT

136

where T0 is the initial temperature before the temperature gradient starts at t = 0.

137

2.2

138

For the sake of didactics and finding a solution for the retention time and temporal peak width in

139

CST-GLC, the retention factor k of the analytes is assumed to follow the linear solvent strength

140

(LSS) retention model at a fixed temperature, T . Additionally, at constant mobile phase compo-

141

sition, ϕ, the variation of the retention factor with changing temperature obeys a modified van’t

142

Hoff relationship as reported in a previous publication

143

for compounds characterized by a relatively small isosteric heat of adsorption Qst (<25 kJ/mol)

144

and for a moderate amplitude of the temperature gradient. According to such retention model,

Solvent-temperature retention model

0

30 .

Essentially, this relationship applies well

7

Page 8 of 41

145

there is neither quadratic terms (ϕ2 and T 2 ) nor coupling term (ϕT ) involved. Therefore, at a first

146

approximation, the general expression of k (ϕ, T ) can be written:

0

0

0

0

−

e

Qst (T −T0 ) RT02

(4)

ip t

k (ϕ, T ) = k (0)e

−S(ϕ−ϕ0 )

0

where k (0) = k (ϕ0 , T0 ) is the initial retention factor of the compound before the gradient starts

148

and S is the analyte parameter in the LSS retention model. Note that S and Qst are assumed

149

to be independent of the temperature and of the mobile phase composition, respectively. Such

150

retention behavior was observed for a series of n-alkanophenones in acetonitrile/water

151

mixtures and from ambient to 60o C

152

2.3

153

The general fundamental equation used for the determination of the gradient retention time, tG , of

154

the analyte is written

us

cr

147

18, 20 .

1, 2 :

tG − uz

0

dts z = 0 u0 k (z)

(5)

d

Z

M

an

Expression for the retention time and local retention factor in CST-GLC

0

In Eq. 5, the dummy variable ts = t− uz0 is the time spent by the analyte in the stationary phase

156

when the center of its concentration profile has reached the axial position z along the column. In its

157

original form, the integration of Eq. 5 is problematic because the expression of the local retention

158

factor, k (z), is not an explicit function of ts . Based on the above definition of CST-GLC, it is first

159

expressed as a function of two time variables, ts,A and ts,T , which are defined by:

Ac ce p

te

155

0

ts,A = ts,T

=

ϕ(z, t) − ϕ0 =t− β T (z, t) − T0 =t− τ

z uA z uT

(6) (7) 0

160

After substitution of Eq. 6 and 7 into Eq. 4, the expression of k is written:

8

Page 9 of 41

0

Qst τ ts,T RT02

−

0

= k (0)e−Sβts,A e 0

= k (0)e

e

A

0

RT0

A

  −S β+ 0

  t−z

 

= β+

uW

=

‘ kW

=

+

(13)

Qst τ uT SRT02

u0 −1 uW

d

(14)

A new time variable, ts,W , naturally appears in Eq. 11. It is written:

Ac ce p

z uW

(15)

The local retention factor can now be expressed as a function of the sole time variable ts,W :

0

0

The relationship between the differentials dts and dts,W is written 0

dts = 166

167

(11)

(12)

M

β uA

Qst τ SRT02 βW

k (ts,W ) = k (0)e−SβW ts,W

165

(10)

us

βW

ts,W = t −

164



an

factor, kW , as:

te

163

Qst τ β uA + u SRT 2 T 0 Q τ β+ st 2 SRT0

Let us define the apparent solvent steepness, βW , the linear velocity, uW , and the retention 0

162

Qst τ uT RT02



cr

= k (0)e

Qst τ SRT02

(9)

0

   Q τ − t Sβ+ st2 −z uSβ +

= k (0)e

161

(8)

    Q τ − st2 t− uz −Sβ t− uz RT T

ip t

k

k 0 ‘ k − kW

(16) 30 :

! dts,W

(17)

Finally, the general gradient Eq. 5 is rewritten after changing the dummy variable ts to the new and relevant variable ts,W . Accordingly,

9

Page 10 of 41

Z

tG − uz

W

k

0 168

0

dts,W −Sβ W ts,W (0)e

−

=

‘ kW

z u0

(18)

Eq. 18 can now be integrated with respect to the dummy variable ts,W . This was done in 30 ).

a previous report (see details in the supplementary material of reference

170

gradient retention time, tG , of the center of the sample zone at the axial position z along the

171

column is written:

cr

h 0 i z  ‘ 0 −SβW kW ‘ u0 k (0) − k (0) − kW e  ln  ‘ kW

tG (z) =

z 1 + uW SβW

(19)

us



Finally, the retention factor at elution is given from the substitution of Eq. 19 into Eq. 16:

an

172

Accordingly, the

ip t

169

0

‘ k (0)kW k (z) =
−SβW k‘ z ‘ W u0 k 0 (0) − k 0 (0) − kW e

M

0

(20)

173

It is important to keep in mind here that these solutions in Eqs 19 and 20 are only valid if the

174

initial retention factor, k (0), of the analyte is larger than kW . Otherwise, the compound would

175

elute isocratically along a certain fraction of the column length. This issue was not considered in

176

this work, so, the apparent solvent gradient of steepness βW and linear velocity (uW ) encountered

177

in CST-GLC always catches the least retained analyte at z = 0.

178

2.4

180

te

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179

0

d

0

Expression for the band compression factor and temporal peak width in CST-GLC

The definition of the spatial band variance, σz2 (z), at position z in gradient elution is

σz2 (z) = G212 (z)Hz

39, 40 :

(21)

181

where H is the plate height of the column which is assumed to be independent of the mobile phase

182

composition, of the temperature, and of the axial position z. This assumption is often met when

183

running fast gradients with columns packed with fine particles for which the HETP curves are

184

usually flat over a wide range of reduced velocity above the optimum velocity. Temperature and 10

Page 11 of 41

185

eluent composition have a minimum impact on the column plate height. In Eq. 21, G212 (z) is the

186

band compression factor unambiguously defined in the space domain

4, 30 .

In a previous report, the general expression of G212 (z) was derived for any gradient (either

188

stationary or dynamic gradients, LSS or non-LSS retention models, solvent or temperature gra-

189

dients) provided that the plate height, H, and the chromatographic linear velcoicty, u0 , remain

190

constant along the column 4, 30 . Accordingly, the general expression of the band compression factor

191

in continuous gradient liquid chromatography is written Rz 0

2k‘ (u)a(u) du 1+k‘ (u)

Z

z Ru

e

0

0

cr

1 − = e z

30 :

2k‘ (t)a(t) dt 1+k‘ (t)

du

us

G212 (z)

ip t

187

(22)

In order to derive the exact expression of G212 (z), one needs first to compute the intensity of the

193

coefficient a(z) in Eq. 22 as a function of the axial position, z, where the sample zone is located

194

during the gradient. a(z) is defined by

an

192

M

4, 30 :

(23)

d

 ‘ 1 ∂k a(z) = ‘ k (z) ∂x z,t

where x represents the distance deviation from the fixed position z where the center of the sample

196

zone is located.

198

Since the retention factor is a function of both the mobile phase composition, ϕ, and of the

Ac ce p

197

te

195

temperature, T , the coefficient a(z) is written:

1 k‘

0

dk dx

!

=

z,t

= =

=⇒

199

a(z) =

   ! ∂k ‘ dT + ∂T ϕ dx z,t T z,t h      i β Qst 0 1 τ 0 −Sk − + − k − uA uT k‘ RT02   β Qst τ S + uA uT SRT02 SβW = aW uW 1 k‘



∂k ‘ ∂ϕ

 

dϕ dx





(24) (25) (26) (27)

It is noteworthy that a(z) is independent of z because both the apparent solvent steepness, βW ,

11

Page 12 of 41

200

and the apparent linear velocity, uW , of the solvent gradient are constant. 0

Secondly, for k (0) 6=0, the integrand in the general Eq. 22 can be rewritten in a simpler form

201

202

30 :

s= 204

‘ 1 + kW ‘ aW kW

and d =

‘ k ‘ (0) − kW ‘ k ‘ (0) aW k W

cr

where

us

203

ip t

2k ‘ (z)a(z) 2 = z ‘ 1 + k (z) s − de− s

(28)

(29)

It is noteworthy that s is constant while d is compound-dependent through the variable k ‘ (0). After basic algebra (see details in the supplementary material of reference

206

for the band compression factor in CST-GLC is written:

=

30 ),

the final expression

(30)

(31)

te

=

−2 Z z  u 2 1 z s se − d se s − d du z 0 z

e 2zs −1 2 s d e −1 d − 2 z 2z s s + s s  z 2 d es − s

d

G212 (z)

M

an

205

The temporal peak width, σt (z), can then be derived:

Ac ce p

207

σt (z) =

208

2.5

209

The definition of the peak capacity is:

√ 1 + k ‘ (z) G12 (z) Hz u0

(32)

Expression for the peak capacity in CST-GLC

Z

tF

Pc = 1 + tI

dt 4σt (L)

(33)

210

where t is the dummy time variable, tI and tF are the gradient retention times of the first and

211

last retained compounds, respectively, and L is the column length. Let us define the new dummy

12

Page 13 of 41

variable k as:

k= 213

where t0 =

L u0

t − t0 0 kW t0

0

so dt = kW t0 dk

(34)

is the hold-up time of the column.

ip t

212

0

215

It is noteworthy that k is only a function of the analyte parameter k(0) = the constant intrinsic gradient steepness, G as:

us

SβW 0 k L u0 W   SβW u0 = −1 L u0 uW   SβW uW = 1− L uW u0 ! 0 kW = aW L 0 1 + kW

219

Ac ce p

218

(36) (37) (38) (39)


1 ln k(0) − [k(0) − 1] e−G G 0

(40)

0

Note that k=1 for the least retained compound when k0 = kW , e.g., when k(0)=1. The general relationship between k(0) and k is then:

k(0) =

eGk − 1 eG − 1

(41)

The temporal peak width is written from Eq. 32 at z = L r h i H ‘ σt (L) = t0 1 + k (L) G12 (L) L

220

(35)

From Eq. 19, the relationship between the variables k and k(0) is given by:

k =1+

217

L s

te

216

Let us define

d

=

M

an

G =

k (0) . 0 kW

cr

214

(42)

where the retention factor at elution, k ‘ (L), is given by Eq. 20:

13

Page 14 of 41

0

k (L) k(0) = ‘ k(0) − [k(0) − 1] e−G kW

222

Substitution of Eq. 41 into Eq. 43 gives the expression of the retention factor at elution as a function of the dummy variable k:

0

The band compression factor, G212 (L), at the column outlet is expressed as a function of the analyte variable g =

d s

in Eq. 31. After a few algebra, the expression of g(k) is:

us

224

eGk − eG 1 ‘ eGk − 1 1 + kW

(45)

(46)

an

g(k) = 225

and

G

G212 (g)

(g − eG )2

d

as a function of the peak gradient retention parameter k >1 is written:

G212 (k)

228

e2G −1 2G

te

227

=

g 2 − 2 e G−1 g +

By substituting Eq. 45 into Eq. 46, the final expression of the band compression factor, G212 (k),

Ac ce p

226

(44)

M

223

1 − e−Gk 1 − e−G

cr

‘ k (k) = kW

ip t

221

(43)

=

eGk − eG


2

G

‘ ) e −1 eGk − eG − 2(1 + kW G






‘ )2 e2G −1 eGk − 1 2 eGk − 1 + (1 + kW 2G 2

2

‘ )eG (eGk − eG ) (eGk − 1) + (1 + k ‘ )2 e2G (eGk − 1) (eGk − eG ) − 2(1 + kW W

(47)

Finally, the expression of the peak capacity is:

Pc = 1 +

L H

‘ kW

√

N 4

Z

kF

kI

dk h

−Gk

‘ 1−e G12 (k) 1 + kW 1−e−G

229

where N =

230

tI and tF , respectively, of the first and last eluted compounds:

i

(48)

is the column efficiency, and kI and kF are given from the gradient retention times,

kI =

tG,I − t0 0 kW t0

and kF =

tG,F − t0 0 kW t0

(49)

14

Page 15 of 41

231

The integral in Eq. 48 has no analytical solution and can only be computed numerically.

232

The peak capacity per unit time, Pc , is defined by:

0

0

or Pc =

ln



SβW Pc  kF (0)−[kF (0)−1]e−G kI (0)−[kI (0)−1]e−G

(50)

Ac ce p

te

d

M

an

us

cr

Pc tG,F − tG,I

ip t

0

Pc =

15

Page 16 of 41

3

Results and Discussion

234

In this section, the advantage and limitations of CST-GLC are first discussed. The

235

relative performance of CST-GLC over classical solvent-programmed GLC is investi-

236

gated for the resolution of protein digests (peptide mapping) using capillary columns.

237

The analysis of the peak capacity per unit time (Pc ) is reported for any arbitrary and

238

physically possible combination of solvent and temperature gradient steepness. The

239

peak capacity per unit time was preferred to the usual peak capacity because both

240

gradient steepness and temperature steepness affect the gradient time window. In the

241

end, the chromatographer is always interested in finding the analytical method that

242

can deliver the largest number of resolved peaks per unit time. A fair comparison

243

regarding the intrinsic advantage of CST-GLC over standard GLC is then performed.

244

The CST-GLC gradient obeys a single constraint : the end values (the highest content

245

of the strong solvent in the eluent mixture and the highest applied temperature) of

246

the two gradients are simultaneously observed at the column outlet which defines the

247

retention time of the last eluted peptide in the sample mixture. The same investiga-

248

tions were carried out but for an additional second constraint: the linear velocity of

249

the temperature gradient is equal to the chromatographic velocity in which case the

250

two gradient steepness are no longer independent. Finally, the first prototype device,

251

which is designed to generate temperature gradients that can propagate at a finite

252

linear velocity along a separation medium, is also briefly presented.

253

3.1

254

3.1.1

255

The main advantage of the proposed CST-GLC is the additional band compression

256

that it can generate in addition to that of the solvent gradient alone. It is important to

257

recall that the traditional combination of temperature and solvent gradients previously

258

reported in the literature

ip t

233

Ac ce p

te

d

M

an

us

cr

0

Advantages and limitations of CST-GLC Advantages of CST-GLC

33–38

enables to shorten analysis times and to speed up mass

16

Page 17 of 41

transfer. However, it cannot provide any additional compression of the sample zone

260

because the temperature remains uniform along the entire column length at any time

261

during the gradient. The front and rear parts of the sample zone are always at the

262

same temperature. In contrast, in the proposed CST-GLC, the temperature profile

263

along the column is also not uniform: it is designed in practice (see next section

264

??) so that 1) the rear part of the peak remains constantly at a higher temperature

265

(and propagates faster) than that of the front part and 2) the temperature at a given

266

location is continuously increasing with time. In other words, the intrinsic advantage

267

of CST-GLC is that a temperature ramp can propagate at a finite linear velocity along

268

the separation medium.

269

3.1.2

270

CST-GLC can only be applied to thin capillary or microfluidic separation devices

271

because the temperature profile across the column diameter has to remain uniform

272

when the temperature program is running. This will ensure the integrity of the peak

273

shape in CST-GLC. A sub-250 µm transverse dimension for the silica-based separation

274

bed is definitely suitable: the specific heat of solid silica is about 1.5 106 J/m3 -K and

275

its thermal conductivity is 1.4 W/m-K. Overall, the thermal diffusivity (DT =

276

silica particles is typically of the order of 10−6 m2 /s. The characteristic time, tc , for

277

the relaxation of thermal gradients across a cylindrical tube is given by:

cr

us

an

te

d

M

Limitations of CST-GLC

Ac ce p

278

ip t

259

d2 = 4DT tc

λ Cp )

of

(51)

where d is the transverse dimension of the chromatographic bed.

279

According to Eq. 51, if d <250 µm, then, the relaxation time tc is smaller than

280

about 0.01 second which is three orders of magnitude faster than a fast ten seconds

281

separation run. Definitely, CST-GLC is not suitable for 4.6 mm i.d. columns because

282

the characteristic time tc would become comparable to the analysis time. This would

283

cause a serious distortion of the peak shape due to the transient radial temperature 17

Page 18 of 41

284

gradients across the column i.d.

285

Additionally, this additional band compression can only be expected if the retention

286

factor of the analytes is decreasing significantly with increasing temperature. This is

287

most often the case for compounds which retention behavior is enthalpy-driven. How-

288

ever, there may be some exceptions

289

Hoff plots is found negative. Therefore, CST-GLC is limited to sample mixtures for

290

which 1) temperature affects significantly their retention and 2) most of the analytes

291

in the sample mixture follow a classical van’t Hoff behavior over the investigated range

292

of temperature.

293

3.1.3

294

The model of CST-GLC derived in this work is limited to the assumptions expressed by

295

Eq. 4 for the effect of solvent composition and temperature on the retention factor of

296

the analytes and by Eq. 22 for the calculation of the band compression factor

297

retention model in Eq. 4 describes qualitatively well the retention behavior of most

298

low- to medium-molecular-weight analytes (classical LSSM and modified van’t Hoff

299

temperature behaviors apply) over a moderate range of temperature from ambient to

300

90o C. This was experimentally confirmed for a series of n-alkanophenones

301

are definitely some exceptions

302

apply strictly. The expression of the band compression factor in Eq. 22 assumes that

303

the chromatographic linear velocity (u0 ) and the local plate height (H) do not change

304

during the gradient. This is true for u0 in LC but only approximate for H. However,

305

if the gradient is operated above the optimum velocity and the column is packed with

306

fine sub-2 µm particles (flat Cu term in the van Deemter equation), the impact of the

307

temperature change from ambient to 90o C on the plate height will remain small and

308

the prediction of the derived CST-GLC model will still remain relevant.

ip t

for which the experimental slope of the van’t

us

cr

35

30 .

The

Ac ce p

te

d

M

an

Limitations and approximations of the derived CST-GLC model

35

18, 20 .

There

in which case the derived model of CST-GLC will not

18

Page 19 of 41

309

3.2

Predicted performance of CST-GLC : keeping the temperature and solvent steepness independent

310

0

In this section, the prediction of the peak capacity per unit time (Pc ) in CST-GLC is reported

312

for the resolution of complex mixtures of peptides originating from a typical protein digest. It is

313

assumed that these small biomolecules are characterized by a constant retention parameter S=20 9

and a constant isosteric heat of adsorption Qst =25 kJ/mol for peptide backbone made of about 41 .

0

315

a dozen residues

316

kF (0) for the least and most retained peptide, respectively.

cr

314

ip t

311

0

The retention parameter, k (0), is peptide-dependent and varies from kI (0) to

us

0

0

The intensity of kI (0) is such that the CST gradient immediately catches the least retained

318

peptide at the column inlet at z=0. As a result, not a single peptide in the mixture is eluted at

319

either constant temperature or constant eluent strength during the gradient. The derived expression

320

(Eq. 18) for the elution time during CST-GLC can then be applied for all peptides present in the

321

mixture. This condition implies that k (0) ≥ kW , so, kI (0) = kW . Therefore, by definition,

322

kI (0)=1 and, according to Eq. 40, the lower integration limit, kI , in Eq. 48 is also equal to unity.

323

The intensity of kF (0) or that of the upper integration limit in Eq. 48, kF , is unambiguously

324

determined so that the most retained peptide is eluted when the solvent and temperature gradients

325

are reaching simultaneously the column outlet at z = L. kF will be derived in the next paragraphs.

0

0

0

M

0

an

317

te

d

0

The experimental conditions during the combined gradient are as follows:

327

- A capillary column (L=150 mm, 300 µm i.d., total porosity t =0.65, external porosity e =0.38)

328

packed with dp =1.8 µm particles (H=3.6 µm, N =41667) is used. It is run at a constant flow rate

329

above the optimum velocity (flow rate Fv =10µL/min, hold-up time t0 =41.4 s).

Ac ce p

326

330

- The solvent gradient consists in increasing the volume fraction of acetonitrile in water from

331

ϕ0 =5% to ϕF =45%. The retention factor of the strong solvent is set to be constant at kA =0.3,

332

which is typical for shallow acetonitrile gradients over this range of eluent composition

333

propagation velocity of the solvent gradient is then assumed to remain constant at uA =0.28 cm/s.

334

The temperature gradient is set to increase from T0 =293 K to TF =363 K. For instance, in

335

practice, high strength silica - stable bond - C18 endcapped particles can be safely

336

operated up to 120o C at 1 kbar pressure drop.

0

14 .

The

19

Page 20 of 41

337

- The steepness, β and τ , of the solvent and temperature gradients, respectively, are arbitrarily

338

and independently set within physically acceptable limits. The solvent gradient time may increase

339

from tg,A,min = t0 to tg,A,max = 25t0 . The temperature gradient may increase from tg,T,min = t0

340

(steep) to tg,T,max = 25t0 (smooth).

ip t

342

Once β (or tg,A ) is arbitrarily fixed, the gradient elution time, tG,F , of the most retained peptide is imposed from the above-mentioned and single gradient constraint. According to Eq. 2:

344

(52)

Once τ (or tg,V ) is independently chosen, the linear velocity, uT , of the temperature gradient is also imposed from the same constraint. According to Eq. 3:

an

343

L ϕF − ϕ0 L + tg,A = + uA uA β

us

tG,F (β) =

cr

341

L L = tG,F (β) − tg,T tG,F (β) −

TF −T0 τ

(53)

M

uT (β, τ ) =

Consistent with the gradient constraint, the duration of temperature gradient, tg,T , cannot be

346

larger than the gradient elution time, tG,F , of the most retained peptide because uT has to be

347

positive.

te

349

The intrinsic gradient steepness, G(β, τ ) =

L s(β,τ ) ,

of the CST gradient is then fixed from the

expression of s(β, τ ) in Eq. 29:

Ac ce p

348

d

345

s(β, τ ) =

‘ (β, τ ) uW (β, τ ) 1 + kW ‘ (β, τ ) SβW (β, τ ) kW

(54)

350

‘ as a function of β, τ , and u are given by Eqs 12, 13, where the expressions of βW , uW , and kW T

351

and 14, respectively.

352

Finally, the upper integration limit, kF , in Eq. 48 is directly given by Eq. 49:

kF (β, τ ) =

tG,F (β) − t0 0 kW (β, τ )t0

(55)

0

353

The peak capacity per unit time, Pc (β, τ ), is predicted by numerical calculation of the integral

354

in Eq. 50 for each realizable and arbitrary pair {β;τ }. The results are summarized as contour plots

20

Page 21 of 41

in Figure 1. The gradient resolution power is increasing from the regions in purple color to those

356

in dark red color. First and foremost, the heat map confirms that, when temperature and solvent

357

gradients are combined altogether, increasing either the temperature steepness at constant solvent

358

steepness or vice-versa is nearly always beneficial in terms of gradient resolution per unit time.

359

The far bottom-left region of the heat map shows some rare exceptions for the smoothest solvent

360

gradients (β < 0.001 s−1 ): the peak capacity per unit time may slightly decrease when increasing

361

the temperature steepness since the dotted black contour lines, β = f (τ ), exhibit a minimum. 0

cr

ip t

355

Figure 2 compares the predicted plots of the realizable Pc versus the solvent gradient steepness

363

β for 4 different temperature steepness: τ =0.05 (solid red line), 0.1 (solid green line), 0.2 (solid

364

blue line), and 0.6 K.s−1 (solid dark yellow line). Additionally, for the sake of comparison, Figure 2

365

represents the same plot but in absence of temperature gradient (T = T0 =293 K, solid black line).

366

Remarkably, for any arbitrary temperature steepness, the relative gain in peak capacity expected

367

in CST-GLC with respect to conventional GLC shows either one or two maxima for one or two

368

particular solvent steepness. The locations of these optima, βopt (τ ), can be visualized in Figure 3,

369

which plots the contour lines of these relative gain in peak capacity per unit time. Accordingly, the

370

gains are maximum (+30−40%) for smooth realizable solvent gradients (at β =0.001 and 0.002 s−1 )

371

and for most of temperature steepness (0.2< τ <1.3 K/s). As the solvent steepness is increasing, the

372

gains are decreasing but they remain significant. In contrast, note that the smoothest temperature

373

gradients (τ <0.2 K/s) combined with the steepest solvent gradients are found ineffective with

374

slightly negative gain values (down to -5%). Under such specific conditions, the calculations

375

predict a slightly lower gradient performance than the standard solvent gradient. The

376

reason is related to the selection of the peak capacity per unit time as the reported

377

gradient performance. A steep temperature gradient combined with a smooth solvent

378

gradient accidentally affects the retention space in a way that differs from that for the

379

same steep temperature gradient but combined with a steeper solvent gradient.

Ac ce p

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362

21

Page 22 of 41

380

3.3

Predicted performance of CST-GLC : fixing the linear velocity of the temperature gradient

381

In this section, the CST gradient is such that the linear velocity, uT , of the temperature gradient is

383

equal to the chromatographic linear velocity, u0 . Because of this additional constraint, the solvent

384

and temperature steepness are no longer independent. Once β is arbitrarily chosen, the elution

385

time of the most retained peptide is fixed (see Eq. 52), and the imposed temperature gradient

386

steepness, τ (β), is written: TF − T0 kA uL0 + ϕF β−ϕ0 0

us

τ (β) =

cr

ip t

382

(56)

The expressions of G(β), s(β), and kF (β) are the same as those used in the previous section. In

388

Figure 4, the calculated peak capacities are represented as a function of either the solvent gradient

389

steepness (bottom x-axis) or of the temperature gradient steepness (top x-axis) in CST-GLC (solid

390

red line), solvent-programmed GLC (solid black line), and temperature-programmed GLC (solid

391

blue line). These curves confirm that combining solvent and temperature gradient dynamically

392

is always advantageous with respect to any of the two other gradients. Figure 5 shows that the

393

relative gains in peak capacity per unit time may be as large as 30% for τ =0.24 K.s−1 with respect

394

to the traditional solvent gradient for β=0.0017 s−1 .

395

3.4

396

CST-GLC requires the fabrication of a device whose temperature can be controlled

397

locally in time and space. The top picture in Figure 6 shows the initial development

398

of the first prototype device to be used in CST-GLC: a 10 cm long heated ceramic tile

399

(see the white material) was used for the proof of concept. The capillary column or

400

the separation channel (represented by the solid red line) is located in between two

401

horizontal, thin, metal resistance heaters, which appear in grey color in the picture.

402

The column inlet is located on the left. Two additional heaters were placed perpendic-

403

ular to the flow direction about 1 cm apart form the inlet and outlet of the separation

Ac ce p

te

d

M

an

387

Realization of traversing temperature ramps for CST-GLC

22

Page 23 of 41

channel. The extremities of the four metal resistance heaters are all connected to an

405

adjustable power supply. Changing continuously the voltage difference between the

406

ends of the metal heaters enables the experimenter to control the amount of Joule

407

heating (RI 2 , I is the electrical current) transferred locally from the heaters to the

408

ceramic tile.

The electrical resistance, R, of any metal conductor depends its cross-section area

409

R=

ρl S

cr

S:

us

410

ip t

404

(57)

where ρ is the resistivity of the material used, l its length, and S is its cross-section

412

area.

413

an

411

It is noteworthy that the bottom horizontal heater has a constant cross-section area: therefore, the amount of Joule heating is uniform along its length.

415

voltage difference is continuously increased with time between its two ends, the tem-

416

perature of the tile would increase uniformly along the tile length in absence of the

417

top horizontal heater. In fact, at the same time, a second and constant voltage differ-

418

ence is applied at both ends of the top horizontal heater. It is important to observe

419

that the cross-section area of this heater is increasing linearly from left (inlet) to right

420

(outlet): the amount of Joule heating is then increasing from the inlet to the outlet

421

of the separation device which generates a non-uniform temperature gradient along

422

the tile length. Overall, the sum of the Joule heating delivered by the two horizon-

423

tal heaters enable the experimenter to generate temperature ramp propagating at

424

constant linear velocity along the column length. This is exactly what is needed to

425

perform CST-GLC. For the sake of evidence, at a given time, the bottom picture in

426

Figure 6 shows the experimental temperature profile recorded by an infra-red camera

427

along the solid red line drawn drawn in the top picture. The video shows that, at any

428

given time, the temperature always varies quasi-linearly from the inlet to the outlet

429

of the separation channel. Note that the sudden drops in temperature observed at

As the

Ac ce p

te

d

M

414

23

Page 24 of 41

both ends of the device are only apparent because they are related to the difference

431

between the emissivity coefficients of the ceramic plate and that of the vertical metal

432

heaters. In conclusion, it is demonstrated experimentally that temperature ramps

433

propagating at a finite linear velocity along a separation micro-channel can definitely

434

be achieved.

Ac ce p

te

d

M

an

us

cr

ip t

430

24

Page 25 of 41

4

Conclusion

436

In this work, the fundamental gradient equations have been derived for the retention time, the

437

retention factor, the temporal peak width, and the peak capacity for a new class of dynamic gradi-

438

ents in liquid chromatography. The gradient model assumes a LSS retention model (the S

439

parameter is kept constant for all analytes), a linearized van’t Hoff retention behavior

440

(the isosteric heat of adssorption Qst is assumed to be constant for all compounds),

441

and a constant local plate height along the column during the gradient. The speci-

442

ficity of this gradient class is the simultaneous application or combination of dynamic

443

(non-stationary) temperature and solvent gradients, which may remain independent

444

from each other. The combined solvent and temperature gradients have their own

445

separate temporal steepness and propagation velocity along the column.

an

us

cr

ip t

435

The results of the theoretical investigations demonstrate that CST-GLC is equivalent to an

447

apparent solvent gradient with the following two properties: 1) the apparent solvent gradient

448

steepness is the sum of the actual solvent and temperature steepness; 2) the apparent solvent

449

linear velocity is the reciprocal of the steepness-averaged sum of the reciprocal of the actual solvent

450

and temperature linear velocities.

te

d

M

446

The numerical calculation of the peak capacity per unit time reveals that CST-GLC can be

452

advantageous in liquid chromatography with respect to conventional solvent gradients. The maxi-

453

mum gain in resolution per unit time is expected to be around 30% for the separation of peptides

454

in protein digests. Most likely, CST-GLC will be most suitable for low thermal mass separation

455

systems such as capillary columns. In order to neglect potential radial temperature gradients, the

456

time required to heat up the whole mass of packing material across the column diameter should

457

be a few orders of magnitude smaller than the chromatographic hold-up time. The amount of heat

458

delivered to the external surface of the capillary column will have to be meticulously adjusted.

Ac ce p

451

459

On a more practical level, it was demonstrated that a temperature ramp propagat-

460

ing at a finite linear velocity along a microfluidic device can be realized in the lab by

461

combining judiciously two different metallic resistance heaters. The amount of Joule

462

heating delivered by the first heater is uniform along the device and it increases with 25

Page 26 of 41

increasing time. Simultaneously, the second heater is designed so that the amount of

464

heat delivered decreases linearly along the device length by increasing the cross-section

465

area of the heaters along its length. The observed temperature profile results from the

466

sum of heat delivered at a given time by both heaters. Eventually, the temperature

467

ramp remains quasi-linear from the inlet to the outlet of the column with increas-

468

ing time. Efforts are currently in progress to implement this heating device around

469

a capillary column or a microfluidic separation channel for the measurement of the

470

peak capacity per unit time as predicted by the proposed model of CST-GLC. The

471

experimental conditions will be selected so that the model assumptions are respected

472

to the best of our possibilities.

Ac ce p

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M

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cr

ip t

463

26

Page 27 of 41

5

Acknowledgements

474

The authors would like to acknowledge and thank Joseph Michienzi and Michael Fogwill (Waters,

475

Milford, MA, USA) for fruitful discussions regarding the design of traversing temperature ramps

476

on microfluidic device.

Ac ce p

te

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473

27

Page 28 of 41

480

481

482

483

Amsterdam, 1986. [2] P. Jandera, J. Churacek, Gradient Elution in Column Liquid Chromatography-Theory and

ip t

479

[1] L. Snyder, High Performance Liquid Chromatography - Advances and Perspectives, Elsevier,

Practice, Elsevier, Amsterdam, 1985.

cr

478

References

[3] L. Snyder, J. Dolan, High Performance Gradient Elution - The Practical Application of the Linear-Solvent-Strength Model, Wiley, Hoboken, 2007.

us

477

[4] H. Poppe, J. Paanakker, J. Bronckhorst, Peak width in solvent-programmed chromatography

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: I. general description of peak broadening in solvent-programmed elution, J. Chromatogr. 204

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(1981) 77–84.

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[6] L. Blumberg, Variance of a zone migrating in a linear medium: Ii. time-varying non-uniform

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Chromatogr. A 596 (1992) 1–13.

medium, J. Chromatogr. A 637 (1993) 119–128.

te

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[5] L. Blumberg, Variance of a zone migrating in a non-uniform time-invariant linear medium, J.

[7] U. D. Neue, Theory of peak capacity in gradient elution, J. Chromatogr. A 1079 (2005) 153–

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[8] U. Neue, Peak capacity in unidimensional chromatography, J. Chromatogr. A 1184 (2008) 107–130.

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[9] M. Gilar, U. Neue, Peak capacity in gradient reversed-phase liquid chromatography of biopoly-

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mers. theoretical and practical implications for the separation of oligonucleotides, J. Chro-

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matogr. A 1169 (2007) 139–150.

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[10] P. Nikitas, A. Pappa-Louisi, Expressions of the fundamental equation of gradient elution and

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a numerical solution of these equations under any gradient profile, Anal. Chem. 77 (2005)

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[12] K. Lan, J. Jorgenson, Theoretical investigation of the spatial progression of temporal statistical moments in linear chromatography, Anal. Chem. 72 (2000) 1555–1563.

ip t

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organic modifier, J. Chromatogr. A 1145 (2007) 67–82.

[13] F. Gritti, G. Guiochon, The ultimate band compression factor in gradient elution chromatography, J. Chromatogr. A 1178 (2007) 79–91.

cr

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[11] F. Gritti, G. Guiochon, The bandwidth in gradient elution chromatography with a retained

[14] F. Gritti, G. Guiochon, The distortion of gradient profiles in reversed-phase liquid chromatography, J. Chromatogr. A 1340 (2014) 50–58.

us

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[15] F. Gritti, G. Guiochon, Separations by gradient elution: Why are steep gradient profiles

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distorted and what is their impact on resolution in reversed-phase liquid chromatography, J.

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and non-linear solvation strength retention models, J. Chromatogr. A 1356 (2014) 96–104.

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[17] L. Snyder, J. Dolan, J. Gant, Gradient elution in high-performance liquid chromatography, J. Chromatogr. 165 (1979) 3–30.

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in gradient reversed-phase liquid chromatography, J. Chromatogr. A 1390 (2015) 86–94.

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[19] F. Gritti, G. Guiochon, Exact peak compression factor in linear gradient elution, J. Chro-

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matogr. A 1212 (2008) 35–40.

[20] M. Martin, On the fundamental retention equation in gradient elution liquid chromatography, J. Liq. Chromatogr. 11 (1988) 1809–1816. [21] Y. V. Kazakhevich, H. M. McNair, Thermodynamic definition of hplc dead volume, J. Chromatogr. Sci. 31 (1993) 317–322. [22] Y. V. Kazakhevich, H. M. McNair, Study of the excess adsorption of the eluent components on different reversed-phase adsorbents, J. Chromatogr. Sci. 33 (1995) 321–327. 29

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[23] F. Gritti, G. Guiochon, Thermodynamics of adsorption of binary aqueous organic liquid mixtures on a rplc adsorbent, J. Chromatogr. A 1155 (2007) 85–99. [24] F. Gritti, Y. Kazakhevich, G. Guiochon, Effect of the surface coverage of endcapped c18-silica

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on the excess adsorption isotherms of commonly used organic solvents from water in reversed

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phase liquid chromatography, J. Chromatogr. A 1169 (2007) 111–124.

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[27] U. Neue, D. Marchand, L. Snyder, Peak compression in reversed-phase gradient elution, J. Chromatogr. A 1111 (2006) 32–39.

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[26] U. Neue, H.-J. Kuss, Improved reversed-phase gradient retention modeling, J. Chromatogr. A

[28] F. Gritti, G. Guiochon, Experimental band compression factor of a neutral compound under high pressure gradient elution, J. Chromatogr. A 1215 (2008) 64–73.

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[29] F. Gritti, G. Guiochon, Peak compression factor of proteins, J. Chromatogr. A 1216 (2009) 6124–6133.

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[25] U. Neue, Nonlinear retention relationships in reversed-phase chromatography, Chro-

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1433 (2016) 114–122.

543

[31] J. Giddings, Dynamics of Chromatography, Marcel Dekker, New York, NY, 1965.

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[32] F. Gritti, G. Guiochon, Band broadening along gradient reversed phase columns: A potential

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gain in resolution factor, J. Chromatogr. A 1342 (2014) 24–29. [33] L. Snyder, J. Dolan, Reversed-phase separation of achiral isomers by varying temperature and either gradient time or solvent strength, J. Chromatogr. A 892 (2000) 107–121.

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[34] T. Welerowicz, P. Jandera, K. Novotna, B. Buszewski, Solvent and temperature gradients in

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separation of synthetic oxyethylene-oxypropylene block (co)polymers using high-temperature

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liquid chromatography, J. Sep. Sci. 29 (2006) 1155–1165. 30

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[35] P. Jandera, K. Krupczynska, K. Novotna, B. Buszewski, Combined effects of mobile phase

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composition and temperature on the retention of homologous and polar test compounds on

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polydentate c8 column, J. Chromatogr. A 1217 (2010) 6052–6060. [36] P. Jandera, K. Vynuchalova, K. Necilova, Combined effects of mobile phase composition and

555

temperature on the retention of phenolic antioxidants on an octylsilica polydentate column, J.

556

Chromatogr. A 1317 (2013) 49–58.

cr

558

[37] K. Monks, H. Rieger, I. Molnar, Expanding the term ”design space in high performance liquid chromatography (i), J. Pharm. Biomed. Anal. 56 (2011) 874–879.

us

557

ip t

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[38] R. Kormany, I. Molnar, , H. Rieger, Exploring better column selectivity choices in ultra-high

560

performance liquid chromatography using quality by design principles, J. Pharm. Biomed.

561

Anal. 80 (2013) 79–88.

565

566

567

M

d

564

161.

[40] U. D. Neue, Peak capacity in unidimensional chromatography, J. Chromatogr. A 1184 (2008) 107–130.

te

563

[39] U. D. Neue, Theory of peak capacity in gradient elution, J. Chromatogr. A 1079 (2005) 153–

[41] S.-H. Chen, C.-W. Li, Thermodynamic studies of pressure-induced retention of peptides in

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reversed-phase liquid chromatography, J. Chromatogr. A 1023 (2004) 41–47.

31

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569

Figure captions 1

0

Calculated (Eq. 50) contour plot of the peak capacity per unit time, Pc , as a function of the physically realizable pair of solvent and temperature gradient, {β;τ }, in CST-

571

GLC for the resolution of complex mixtures of peptides (S=20, Qst =25 kJ/mol).

572

The solvent and temperature gradients have to reach the column outlet at the same

573

time, which defines the elution time of the most retained peptide. All other rele-

574

vant experimental parameters during the CST gradient are given in the results and

575

discussion section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

cr

2

us

576

ip t

570

Plots of the peak capacity per unit time versus the solvent gradient steepness for four particular temperature gradient steepness as indicated in the legend. The solid

578

black line serves as a reference curve for the conventional solvent gradient. See text

579

for more details on the calculation of these curves. . . . . . . . . . . . . . . . . . . . 36

unit time with respect to conventional solvent gradients . . . . . . . . . . . . . . . . 37

581

582

Same representation as in Figure 1, except for the relative gain in peak capacity per

M

3

4

0

Calculated (Eq. 50) plot of the peak capacity per unit time, Pc , as a function of the

d

580

an

577

solvent gradient β represented along the bottom x-axis. The temperature gradient

584

steepness τ , represented along the top x-axis), is such that the temperature gradient

585

propagates at a velocity uT equal to the chromatographic linear velocity u0 . CSTGLC applied for the resolution of complex mixtures of peptides (S=20, Qst =25

586

kJ/mol). All other relevant experimental parameters during the CST gradient are

587

given in the results and discussion section. . . . . . . . . . . . . . . . . . . . . . . . . 38

588

589

590

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583

5

Same as in Figure 4, except for the relative gain in peak capacity per unit time with respect to conventional solvent gradients. . . . . . . . . . . . . . . . . . . . . . . . . 39

32

Page 33 of 41

591

6

(Top graph) Realization of dynamic temperature gradients on a ceramic tile (white color) by utilizing two horizontal and two vertical heaters (grey color). The bottom

593

horizontal heater contributes to generate a dynamic and uniform temperature gradi-

594

ent by increasing the voltage difference between its extremeties. The top horizontal

595

heater contribule to generate a non-uniform temperature gradient along the length

596

of the separation device due to progressive increase of its cross-section area. (Bottom

597

graph) Observation by IR camera of the quasi-linear temperature profle (red signal)

598

at a given time along the solid red line shown in the top graph. The temperature drop

599

observed at both ends is only apparent and due to different emissivity coefficients of

600

the ceramic tile and metal heaters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

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592

33

Page 34 of 41

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Figures

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6

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34

Page 35 of 41

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Ac ce p

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d

M

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us

cr

 

0

Figure 1: Calculated (Eq. 50) contour plot of the peak capacity per unit time, Pc , as a function of the physically realizable pair of solvent and temperature gradient, {β;τ }, in CST-GLC for the resolution of complex mixtures of peptides (S=20, Qst =25 kJ/mol). The solvent and temperature gradients have to reach the column outlet at the same time, which defines the elution time of the most retained peptide. All other relevant experimental parameters during the CST gradient are given in the results and discussion section.

35

Page 36 of 41

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2.0  

M

1.5

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0.5

0.0 0.000

No temperature gradient Temperature steepness,=0.05 K.s-1 Temperature steepness,=0.10 K.s-1 Temperature steepness,=0.20 K.s-1 Temperature steepness, =0.60 K.s-1

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1.0

Ac ce p

Peak capacity per unit time P'c [s-1]

cr

Figure 2: Plots of the peak capacity per unit time versus the solvent gradient steepness for four particular temperature gradient steepness as indicated in the legend. The solid black line serves as a reference curve for the conventional solvent gradient. See text for more details on the calculation of these curves.

0.002

0.004

0.006

0.008

0.010

Solvent gradient steepness  [s ] -1

36

Page 37 of 41

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Figure 3: Same representation as in Figure 1, except for the relative gain in peak capacity per unit time with respect to conventional solvent gradients

.

Ac ce p

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d

M

an

 

37

Page 38 of 41

ip t 0

us

cr

Figure 4: Calculated (Eq. 50) plot of the peak capacity per unit time, Pc , as a function of the solvent gradient β represented along the bottom x-axis. The temperature gradient steepness τ , represented along the top x-axis), is such that the temperature gradient propagates at a velocity uT equal to the chromatographic linear velocity u0 . CST-GLC applied for the resolution of complex mixtures of peptides (S=20, Qst =25 kJ/mol). All other relevant experimental parameters during the CST gradient are given in the results and discussion section.

Temperature gradient steepness  [K.s-1] 0.2

0.4

0.6

0.8

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1.2

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2.0

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1.5

1.0

1.0

M

2.5

Ac ce p

Peak capacity per unit time P'c [s-1]

 

GLC (no temperature gradient)

0.5

CST-GLC (uT=u0) Temperature gradient only

0.0

0.002

0.004

0.006

0.008

Solvent gradient steepness  [s-1]

38

Page 39 of 41

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1.29

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Relative gain peak capacity per unit time CST-GLC/GLC

Figure 5: Same as in Figure 4, except for the relative gain in peak capacity per unit time with respect to conventional solvent gradients.

 

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1.28

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1.27

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1.24

uT = u0

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1.23 0.000

0.002

0.004

0.006

0.008

0.010

Solvent gradient steepness  [s ] -1

39

Page 40 of 41

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Figure 6: (Top graph) Realization of dynamic temperature gradients on a ceramic tile (white color) by utilizing two horizontal and two vertical heaters (grey color). The bottom horizontal heater contributes to generate a dynamic and uniform temperature gradient by increasing the voltage difference between its extremeties. The top horizontal heater contribule to generate a non-uniform temperature gradient along the length of the separation device due to progressive increase of its cross-section area. (Bottom graph) Observation by IR camera of the quasi-linear temperature profle (red signal) at a given time along the solid red line shown in the top graph. The temperature drop observed at both ends is only apparent and due to different emissivity coefficients of the ceramic tile and metal heaters.

Ac ce p

te

d

M

 

Inlet 

Outlet

40

Page 41 of 41