mechanisms of retention in hplc - uv

1

Mechanisms of retention in HPLCMechanisms of retention in HPLC

María Celia García-Álvarez-Coque

Department of Analytical Chemistry

University of Valencia

Valencia, Spain

https://sites.google.com/site/fuschrom/

HPLC’2013 (Amsterdam)

Part 6Part 6

2


1. Retention in reversed-phase, normal-phase and HILIC

2. Secondary equilibria in reversed-phase liquid chromatography: Part A

3. Secondary equilibria in reversed-phase liquid chromatography: Part B

4. Retention modelling (quantification or prediction): Part A

5. Retention modelling (quantification or prediction): Part B

6. Gradient elution

7. Peak profile and peak purity

8. Computer simulation

Index

Mechanisms of retention in HPLC

3


6.1. Previous considerations

6.2. General equation for gradient elution

6.3. Integration of the general equation

6.3.1. Equations based on integrable models: Solvent strength model

6.3.2. Non-integrable retention models

6.4. Multi-linear gradients

6.5. Recommended literature

6. Gradient elution

4



6. Gradient elution

Isocratic elution: advantages

● Greater simplicity, simpler instrumentation,

lower cost, and no need of column re-

equilibration between consecutive injections.

● Allows the maximal capability of the system:

it expands the separation among peaks as

much as possible.

HPLC separations can be approached from two

perspectives: isocratic and gradient elution.

Isocratic elution

Gradient elution

5

HPLC’2013 (Amsterdam)6. Gradient elution

Isocratic elution

6


Isocratic elution: disadvantage

6. Gradient elution

● Owing to the dependence between the retention and solute polarity:

if the polarity range is too wide, it will be difficult to find a set of chromatographic

conditions able to balance a satisfactory separation power for the least retained

solutes, and a reasonable retention time for the most retained ones.

o/w10 loglog Pcck (6.1)

General elution problem of chromatography: The utility of

isocratic elution is seriously restricted to sets of compounds in

a relatively narrow range of polarities.

7


General elution problem of chromatography

6. Gradient elution

A logical solution is gradient elution, where the elution strength is increased

gradually by altering at least one experimental factor (organic solvent

content, pH, temperature or flow rate) as the analysis progresses, so that the

elution of the most retained compounds is expedited.

Retention time

8


The effect of gradient elution on retention can be graphically expressed as

closing an accordion, or contracting an elastic rubber hold by only one extreme.

Accordion effect

9


Gradients of organic modifier

6. Gradient elution

Transference of data between elution modes

● The prediction of retention under gradient conditions is usually made from training

sets involving gradient experiments.

● Crossing the information between both elution modes is also possible:

data coming from isocratic experiments can be used to anticipate the retention in

gradient runs, and vice versa.

In RPLC, the use of gradients of organic modifier has experienced a large

development in the separation of solutes in complex samples.

Modellin

gst

ep

(sourc

e) Experimentaldesign

Predict

ion

step

(targ

et) Prediction

GradientIsocratic

GradientIsocratic

10


● In gradient elution, there is always an underlying model isocratically relating retention to

mobile phase composition.

● Modelling gradient retention from wisely planned isocratic experiments may be safer and

also faster, since no re-equilibration step is needed.

● Isocratic data may be available from the literature, or from previous data sets, and the

chromatographer may wish to explore whether a gradient separation can give satisfactory

results before carrying out any gradient experiment.

Isocratic

Gradient

Isocratic versus gradient prediction

OPA derivatives proteic primary amino acids

0 10 20 30 40 50 60 70 80

2 3 4 5 6 7

Ala

Gly

Thr

Arg

His

Ser+Gln

Asn

Glu+Asp

LysLeu

TrpIleMet Phe

TyrCys

Val Ac

eto

nit

rile

, %

Time, min

t0

Asp

Glu

AsnSer

Gln

HisThr

Gly

Arg

AlaTyr

CysVal

MetIle Lys

Trp

Phe

0 10 20 30 40 50 60 70 80

Time, min

11


● The quadratic logarithmic model describes wide ranges of organic solvent concentrations

in both isocratic-to-isocratic and isocratic-to-gradient predictions. This contrasts with the

results offered by the linear model, which yields unsatisfactory predictions.

● The picture in gradient-to-gradient predictions may be different: both retention models

may perform similarly, unless the modifier concentration range is significantly wide.

2210log ccck

10log cck


GradientIsocratic

GradientIsocratic

2210log ccck

12


● Gradient elution concerns narrower solvent concentration domains compared

to the isocratic case.

● The migration of each solute under gradient elution is affected by a rather

narrow fraction of the concentrations scanned by the gradient program, which

has been called

solvent concentration informative range

the solute starts to migrate significantly once a certain modifier concentration

is reached, and can leave the column before the completion of the gradient.

● This explains the poor results yielded in gradient-to-isocratic predictions, which

may imply extrapolations.

Isocratic experiments are considerably richer in information !!!


Gradient

Isocratic

13


Solvent concentration informative range

6. Gradient elution

14









6. Gradient elution

15



6. Gradient elution

Steps along a gradient: organic solvent concentration at the column inlet

%B

16


Dwell time

6. Gradient elution

For simplicity, we will first consider:

● linear gradients

● the gradient reaches instantaneously the column inlet

(i.e. there is no delay associated to the dwell time)

17


)),((R ltt

L

dt

dlu

(6.2)

● When a solute is eluted in the gradient mode, it can be considered that it moves

isocratically a dl distance inside the column in an infinitesimal time range, dt.

u : mobile phase linear velocity

t : time from the beginning of the gradient

l : distance from the column inlet to the position where the solute is at time t

L : total column length

tR : retention time that the solute would present if it were eluted isocratically along the

whole column, using the instantaneous mobile phase composition

: instantaneous mobile phase composition (organic solvent content at the column

point where the solute is located), which depends on t and l

L

l

t

Solute linear velocity

dl

dt

isocratically

18


t0,l = l u−1 : time needed by the gradient to go through the distance l to reach

the position where the solute is at time t

(i.e. the dead time for the corresponding column segment)

u

lmt

u

lmmt

u

ltmlt l

)(),( 000

φ = φ0 + m tc

tc

t

φ

td = 0

φl=0 = φ0 + m t

l

L

)),((R ltt

L

dt

dlu

(6.3)

Gtm

(6.4)

Nm10log Pcck

nbak

1

Skk wlnln

2wlnln bakk

Note that for positive gradients, the

organic solvent content at the solute

position is smaller than at the column inlet.

19


0

0R

t

ttk

)),((R ltt

L

dt

dlu

))),((1(0 ltkt

Lu

dt

ltkt

Ldtudl

)),((10

g

0 00)),((1

t

dtltkt

LLdl

L

tg : retention time under gradient conditions

(time a solute requires to reach the column outlet during the gradient)

(6.2) (6.5)

(6.6)

(6.7)

(6.8)

20


g

0 00)),((1

t

dtltkt

LLdl

L

g

0

0)),((1

t

ltk

dtt

u

ltt c

φ = φ0 + m (t – l/u) = φ0 + m tc

(6.8)

(6.9)

(6.10)

corrected time at the point where

the solute is located inside the

column at time t

lφ

u

dldtdt c (6.11)

Change of time variable

tc

φ

td = 0 t

21


g

0

0)),((1

t

ltk

dtt

dt

tk

tkdt

tkdt

tkt

uLdtdt

))((1

))((

))((1

11

))((1

/

c

c

cc0c

0g0g

0 c

cc

0 c

cc0

))(())((1

))((/))((1tttt

tk

dtdt

tk

tktkt

u

dldtdt c u

Lt 0

dtltkt

Ldtudl

)),((10

Eq. (6.13) has been called fundamental or general equation for gradient elution, and

is most extensively used, since it describes in a practical way the retention times for

a solute eluted under a given gradient program. The retention time can be calculated

provided the composite function k((tc)) is known.

(6.12)

(6.13)

tg – t0 replaces tg

General equation for gradient elution

cc

c

))((

))((1dt

tk

tkdt

22









6. Gradient elution

23



6.3.1. Equations based on integrable models: Linear solvent strength theory

6. Gradient elution

● Two nested equations:

retention model k(φ) : dependence of the retention factor on the

concentration of organic solvent

gradient program φ(tc) : change in the concentration of organic solvent with time

● The integral can be solved analytically only in a few cases.

the most simple case: linear log k(φ) and φ(tc) linear solvent strength model

● This approach is the most widely used and can be easily extended to the construction of

several consecutive linear segments (multi-linear gradients).

0g

0 c

c0

))((

tt

tk

dtt

(6.13)

24


kw : retention factor in water

S : elution strength

φ0 : initial concentration of modifier at the column inlet

m : gradient slope

Δφ : difference between final and initial concentrations of organic modifier in the gradient

tG : total gradient time (time at which the gradient program reaches the final composition)

Skk wlnln

Gtm

cc0c00w

)(ww

tmStmSStmSS ekeekekekk

(6.14) (6.15) (6.4)

(6.16)

0g

0 c

c0

))((

tt

tk

dtt

Linear solvent strength model

c0 tm

25


d00

0d0

d00g )3.21(log

)(1lnttbk

b

ttt

mS

ttkmSt

(6.13)

(6.16)

(6.17)

(6.18)

(6.19)

dwell time term

td : dwell time

k0 : retention factor for the initial gradient conditions

b = S m t0

0g

0 c

c0

))((

tt

tk

dtt

cc0c00w

)(ww

tmStmSStmSS ekeekekekk

11 )(

00

d)(

0

c

0 0

c0

d0g

0g

d

dc

d

tttmS

tt

tttmS

t

ekmSk

t

ek

dt

k

dtt

f

tmS

fktmS

t

t

tmS

ttkmSk

0

00

0

d

0

d00g

)(1ln)(1ln

gradient time

retention factor

Analytical integration of the linear models

26


Jandera’s model

6. Gradient elution

a, b, n : fitting parameters

The analytical integration can be applied to other equations, as that proposed by

Jandera in NPLC, also useful for RPLC.

n

bak

1

00 baa

mbn

a

mbn

tttmba

k

t

ttmba

dt

k

dtt

nntt

tn

t

)1()1()(

10

1

d0g0

0

d

dc0

c

0 0

c0

0g

d

d

d0

0

1/1

10

0

d0

g

1

ttmb

aambnk

tt

t

n

n

(6.13) (6.20)(6.15)

(6.21)

(6.22)

(6.23)

0g

0 c

c0

))((

tt

tk

dtt

c0 tm

27



6. Gradient elution

● A drawback of the quadratic model is that the

analytical integration is not possible, or at least

not straightforward.

● Independently of the retention model, the

analytical integration cannot be applied when the

change in the concentration of organic modifier

with time is non-linear (non-linear gradients).

The linear retention model describes the retention accurately only in narrow

concentration ranges of organic modifier. For wider ranges, a quadratic term should

be added to make a proper description of the retention.

2wlnln bakk

Skk wlnln

0g

0 c

c0

))((

tt

tk

dtt

28


Solution for non-linear gradients

6. Gradient elution

Non-linear gradients can be satisfactorily solved by approximating

them as summations of several linear segments.

29


Numerical integration

6. Gradient elution

● k(t) is assumed to be constant inside each step.

● The precision in tg can be increased to any desired level, just narrowing the time

intervals (ti+1 − ti ).

● The higher the precision, the longer the computation time.

● Limitation associated with the pumping system: gradients are generated by the

instrument by approximating changes in composition in small steps. Therefore,

investing more effort in increasing the precision is useless.

1

1

2

1

10g

)()(...

)()()(00

0

i

i

i

i

t

t

t

t

t

t

ttt

tk

dt

tk

dt

tk

dt

tk

dt

tk

dtt

1,

1

,1

1

2,1

12

1,0

10 ...

ii

ii

ii

ii

k

tt

k

tt

k

tt

k

tt

2

)()( 11,

ii

ii

tktkk

(6.25)

(6.26)

(6.27)

to consider !!

Gradient retention times are calculated by splitting the integral in small isocratic steps.

30


In any case, the numerical integration is suitable. It is a fully

flexible universal approach, and therefore, it is preferred by

some analysts. However, its computation is somewhat slower

and requires programming skills. In contrast, the analytical

integration (when possible) is rapid, and the mathematical

expressions are relatively simple, being relatively easy to be

applied by any analyst.

31

HPLC’2013 (Amsterdam)4. Retention modelling (quantification or prediction): Part A

2210log ccck

10log cck

32 36 40 44 48 52

-1

0

1

2

3

4

ln k

Acetonitrile (%, v/v)

Another alternative is possible !!!

Retention in RPLC using linear gradients can be predicted assuming that

the non-linear retention behaviour is not far from the linear behaviour.

Therefore, it is possible to describe the retention based on the integral

equation obtained for the linear model.

32


Alternative approach: assumptions

6. Gradient elution

● The retention behaviour under gradient elution is formally similar to that for the linear

model.

● Since S is the minus slope of the logarithmic linear model, for the non-linear model it is

the minus derivative of the quadratic function at the initial gradient condition:

)2(ln

0

0

bad

kdS

f

tmba

fktmbaf

tmd

kd

fktmd

kd

k

00

000

00

000

g)2(

)()2(1ln

ln

)(ln

1ln

linear model quadratic model

(6.198)

(6.29)

(6.28)

linear model

quadratic model

2wlnln bakk Skk wlnln

wlnln kk

S

f

tmS

fktmSk

0

00g

)(1lnlinear solvent strength model

33


200 )341.11(

068.2ln

a

d

kdS

linear model polarity model

(6.19)

(6.31)

(6.30)

linear model

polarity model

Skk wlnln

341.11

068.2lnln 0

akk

ftm

a

fktma

k

020

0020

g

)341.11(

068.2

)()341.11(

068.21ln

0

0

341.11

068.2ln

0

akw

ek(6.32)

wlnln kk

S

f

tmS

fktmSk

0

00g

)(1ln

Alternative approach: polarity model

34


6.4. Representation of retention factors: isocratic elution

6. Gradient elution

The plot of log k as a function of the organic solvent content is

conventionally used to illustrate isocratic elution.

32 36 40 44 48 52

-1

0

1

2

3

4

ln k

Acetonitrile (%, v/v)

35


6.4. Representation of retention factors: triangular elution plots

6. Gradient elution

● The diagram is similar to the conventional diagram in isocratic elution.

● Upper line: isocratic elution. Other lines: different gradient slopes (m).

● The lines delimit all possible retention factors and define a triangular region.

● The effect of the gradient is larger for the most retained solutes, and practically null for

those weakly retained.

20 30 40 50 60

-1

0

1

2

3

4

5

lnk

Initial acetonitrile, % v/v

m = 0

m = 1.2

The plot of log k for different gradients versus initial organic solvent content allows a

convenient visualization of gradient elution.

a solute under

several gradients

three solutes under

extreme gradients

Gtm

5

20 30 40 50 60

1

2

3

4

lnk

m = 0

m = 0.2

m = 0.4

m = 0.6

m = 0.8

m = 1.0 m = 1.2

fastest gradient

Initial acetonitrile, % v/v

36









6. Gradient elution

37



6. Gradient elution

● Gradients consist usually of simple linear increases of an experimental factor.

● Chromatographs generate gradients by performing consecutive small isocratic steps, usually

at increasing modifier concentration.

● Gradients can also be approximated to a few large isocratic steps (stepwise or multi-isocratic

gradients).

The number of gradient programs that can be implemented is virtually unlimited,

and the success of the separation depends critically on the selected profile.

tR

AC

N,

%


● In multi-linear gradients, the modifier concentration is linearly increased, but the gradient has

two or more segments of different slope.

● To speed up the elution, the gradient slopes should be always positive, but occasionally, one or

more isocratic steps can be inserted, or the slope is decreased with respect to previous

segments to get optimal separations.

Several segments of different slope, linear (multi-linear gradients)

or curvilinear, can be also implemented.

38

39


● The simplest multi-linear gradient approach consists of inserting, in a trial-

and-error fashion, one or more nodes where the slope of the gradient is

changed:

one node will give rise to a two-segment gradient

(bilinear gradient)

two nodes will yield a three-segment gradient

(trilinear gradient), etc.

● The slope of the gradient can be particularised to each solute cluster in the

eluted mixture.

Number of nodes

40


Trial-and-error

Retention time, min

The trial-and-error approach is appropriate for cases with

well-resolved peaks at the end of the chromatogram: the

location of the node or nodes is non-critical and the

experience of the chromatographer is enough to find a

satisfactory gradient.

41


● The selection of the location of the nodes and gradient slopes is optimal.

● To increase the possibilities of success, gradient interpretive optimisation can

be adapted to the particularities of the sample by inserting a large number of

nodes.

● There is no limit in the number of nodes considered in the optimisation, but

when n > 3 the required computation time begins to constitute a limit that

constraints the practical application of the algorithms.

● The architecture of the software can expedite the process.

● It is worth to check if the complexity of a gradient program deserves be

increased to enhance resolution by inserting more linear segments, or on the

contrary, no significant improvements are expected under more complex

gradients.

Computer-assisted approach

42


o-Phthalaldehyde

derivatives of

proteic primary

amino acids

Time, min

0 10 20 30 40 50 60 70 80

5

10

15

20

25

30

t0

12

34 5 6 7 89

10 1113

12 14 15

1617

18+19

time, min

AC

N,

%

0 10 20 30 40 50 60 70 80

5

10

15

20

25

30

t0

11AC

N,

%

1

2 3

4

567 8 9

1012

13

14 15

16

17

18 19

0 10 20 30 40 50 60 70 80

5

10

15

20

25

30

t0

1

2

3

456

7 8 9

1011 13

1214

1516

1718

time, min

AC

N,

%

19

0 10 20 30 40 50 60 70 80

5

10

15

20

25

30

t0

1

2

3

4

568 9 7 10

12 13

11 14 15 1718

19 16

AC

N,

%

mechanisms of retention in hplc - uv

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