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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
HPLC’2013 (Amsterdam)
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
HPLC’2013 (Amsterdam)
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
HPLC’2013 (Amsterdam)
6.1. Previous considerations
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
HPLC’2013 (Amsterdam)
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
HPLC’2013 (Amsterdam)
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
HPLC’2013 (Amsterdam)6. Gradient elution
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
HPLC’2013 (Amsterdam)
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
HPLC’2013 (Amsterdam)6. Gradient elution
● 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
HPLC’2013 (Amsterdam)6. Gradient elution
● 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
Isocratic versus gradient prediction
GradientIsocratic
GradientIsocratic
2210log ccck
12
HPLC’2013 (Amsterdam)6. Gradient elution
● 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 !!!
Isocratic versus gradient prediction
Gradient
Isocratic
13
HPLC’2013 (Amsterdam)
Solvent concentration informative range
6. Gradient elution
14
HPLC’2013 (Amsterdam)
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
15
HPLC’2013 (Amsterdam)
6.2. General equation for gradient elution
6. Gradient elution
Steps along a gradient: organic solvent concentration at the column inlet
%B
16
HPLC’2013 (Amsterdam)
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
HPLC’2013 (Amsterdam)6. Gradient elution
)),((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
HPLC’2013 (Amsterdam)6. Gradient elution
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
HPLC’2013 (Amsterdam)6. Gradient elution
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
HPLC’2013 (Amsterdam)6. Gradient elution
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
HPLC’2013 (Amsterdam)6. Gradient elution
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
HPLC’2013 (Amsterdam)
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
23
HPLC’2013 (Amsterdam)
6.3. Integration of the general equation
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
HPLC’2013 (Amsterdam)6. Gradient elution
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
HPLC’2013 (Amsterdam)6. Gradient elution
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
HPLC’2013 (Amsterdam)
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
HPLC’2013 (Amsterdam)
6.3.2. Non-integrable retention models
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
HPLC’2013 (Amsterdam)
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
HPLC’2013 (Amsterdam)
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
HPLC’2013 (Amsterdam)6. Gradient elution
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
HPLC’2013 (Amsterdam)
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
HPLC’2013 (Amsterdam)6. Gradient elution
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
HPLC’2013 (Amsterdam)
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
HPLC’2013 (Amsterdam)
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
HPLC’2013 (Amsterdam)
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
37
HPLC’2013 (Amsterdam)
6.4. Multi-linear gradients
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,
%
HPLC’2013 (Amsterdam)6. Gradient elution
● 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
HPLC’2013 (Amsterdam)6. Gradient elution
● 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
HPLC’2013 (Amsterdam)6. Gradient elution
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
HPLC’2013 (Amsterdam)6. Gradient elution
● 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
HPLC’2013 (Amsterdam)6. Gradient elution
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,
%
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