1416
Anal. Chem. 1990, 62, 1416-1423
compd
9i response with respect to glucose
are the unusual stability and the constant-potential operation. The utility of RuOz CMEs for the electrocatalytic detection of other organic compounds and for a nonenzymatic glucose sensing is under investigation in this laboratory.
glucose galactose fructose ribose deoxyribose lactose maltose sucrose gluconic acid tartaric acid glycerol
100
LITERATURE CITED
Table I. FIA Response of RuOz CMEs to Various Polyhydroxy1 Compoundsa
87 78 96 20 64
53 20 88 30 22
Based on response to 5 mM solutions a t +0.4 V.
structural requirements for obtaining catalytic response remain to be elucidated, the characteristic common of the compounds tested in Table I is that they all contain multiple hydroxyl groups. (In the case of glycerol, the only functional group is the hydroxyl moiety.) Analogous observations were reported for Cu CMEs (4). In summary, we have demonstrated the utility of RuOz CMEs for amperometric detection. These electrocatalytic surfaces offer a marked decrease in overpotentials for the oxidation of carbohydrate compounds. Particularly attractive
(1) Hughes, S.; Johnson, D. C. Anal. Chim. Acta 1981, 732, 11. Rocklin, R. D.; Pohl, C. A. J. Liq. Chromatogr. 1983, 6 , 1577. Santos, L. M.: Baldwin, R. P. Anal. Chem. 1987, 59, 1766. Prabhu, S.V.; Baldwin. R . P. Anal. Chem. 1989, 67, 2258. Cox, J. A.; Kulesza, P. J. Anal. Chem. 1984, 56, 1021. (6)Cox. J. A.; Gray, T.; Kulkarni, K. Anal. Chem. 1988. 60, 1710. (7) Kulesza, P. J.; Faulkner, L. R. J. Electroanal. Chem. 1988, 248, 305. (8)Trasatti, S.; O'Grady, W. E. I n Advances in Electrochemistry and Electrochemical Engineering; Gerischer, H., Tobias, C. W..Eds.; Wiley: New York, 1981; Vol. 12, p 179. (9) Burke, L. D.;Whelan, D. J . Electroanal. Chem. 1979, 703, 179. (10)Kleijn, M.; Van Leeuwen, H. J. Necfroanal. Chem. 1988, 247, 253. (11) Burke, L. D.; Murphy, 0. J . Electroenal. Chem. 1979, 707,351. (12) Laule, G.;Hawk, R.; Miller, D. J. Nectroanal. Chem. 1986, 273, 329. (13) Burke, L. D.; Healy, J. F. J. Necfroanal. Chem. 1981, 724, 327. (14) Kulesza, P. J.; Mlcdnicka, T.; Haber, J. J . Electroanal. Chem. 1988,
(2) (3) (4) (5)
257,167. (15) Kutner, W.; Gilbert, J. A.; Tomaszewski, A.; Meyer, T. J.; Murray, R. W. J. Electroanal. Chem. 1986, 205, 185. (16) Prabhu, S.V.; Baldwin, R. P. Anal. Chem. 1969, 67,852.
RECEIVEDfor review January 22, 1990. Accepted March 30, 1990. This work was supported by the donors of the Petroleum Research Fund, administrated by the American Chemical Society, and by the National Institutes of Health (Grant GM 30913-06).
Solvent Modulation in Liquid Chromatography: General Concept and Theory Jon H. Wahl, Christie G . Enke, and Victoria L. McGuffin*
Department of Chemistry, Michigan State Uniuersity, East Lansing, Michigan 48824
A novel solvent dellvery method, called solvent modulation, Is described as an alternatlve to premixed mobile phases in iiquld chromatography. I n solvent modulation, Individual solvent zones are introduced onto the chromatographk column In a varylng or repeating sequence. Because the solvents are of constant composttion and the zones are spatially separated from one another, solute retentlon should be controlled Independently In each zone. Thus, the overall retention of the solute Is a readlly predictable, linear combinatlon of the capacity factors In the lndlvldual solvent zones it encounters. I n this paper, the general concept and theory of solute retention are developed and posslble devlatlons from ideal behavlor are discussed. The extent of solvent zone dispersion by the chromabgraphk column, and the resulting impreclsion in solvent zone length, Is shown to be a primary contributlon to lmpreclsion in solute retention. Based on this inherent llmitatlon, the experimental conditlons suitable for solvent modulation are examined. Under these conditions, solvent modulation is shown to offer a simple, versatile, and accurately modeled means to control and predlct solute retention.
INTRODUCTION In liquid chromatography, solute retention and resolution are controlled primarily by modification of the mobile-phase composition. Many methods have been developed to optimize
* Author to whom correspondence should be addressed.
the strength and selectivity of the mobile phase, most of which are based on the use of a mixed solvent system (1). In general, these methods utilize a preliminary series of experiments to determine solute retention in solvents of known composition, from which the optimum solvent mixture is predicted. Unfortunately, molecular interactions are not completely independent of one another within a solvent mixture, especially for the polar, highly interacting solvents of interest in reversed-phase liquid chromatography. Because of the deviations from regular solution behavior, many physical properties such as density, viscosity, dielectric constant, compressibility, etc. are not a linear function of the solvent composition ( 2 ) . These physical properties directly influence hydrodynamic and physicochemical processes in chromatographic separations; hence, solute retention is generally not a simple, linear function of the mobile-phase composition (3-5). Various models have been developed to predict the effect of mobile-phase composition on chromatographic retention (6). Theoretical treatments such as the Hildebrand solubility parameter ( 7 , 8 ) ,the solvophobic theory of Horvath and coworkers (9), and the lattice model of Martire and Boehm (10, 11) have been employed with some success for qualitative predictions. In addition, empirical and semiempirical treatments have been somewhat successful, including chromatographically based parametric models such as the polarity (P') and solvent strength (S) indices developed by Snyder (12-14), spectroscopically based parametric models such as the solvatochromic scales (15-18), a kinetically based parametric model (191,and others (4,20, 21). In each of these models,
0003-2700/90/0362-1416$02.50/00 1990 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 62, NO. 14, JULY 15, 1990
however, it is assumed that solute-solvent interactions in mixed mobile phases are independent and simply additive. As previously stated, this assumption is often not true in polar solvents due to nonideal solvent-solvent interactions. In order to eliminate such nonideal interactions, we have explored solvent modulation as an alternative to premixed solvent systems in liquid chromatography. In solvent modulation, the individual components of the mobile phase are never physically combined instead, solvent segments or zones of predetermined composition are introduced sequentially onto the chromatographic column. Because the solvent segments are spatially and temporally separated from one another, solute retention should be controlled independently within each zone. The overall solute retention should, therefore, be a simple and predictable function of the capacity factor within each solvent and the fractional time the solute is exposed to that solvent zone. If so, the solvent modulation technique should allow existing optimization strategies, such as the window-diagram method (22),overlapping resolution mapping (3),and computer simulation methods (23),to be implemented with much greater accuracy and precision than with premixed mobile phases. Although the concept of solvent modulation is novel, the use of individual solvent pulses in liquid chromatography has been demonstrated previously. By varying the strength and selectivity of the sample matrix, Yang (24)showed that a single solvent pulse could be used to influence solute retention. Berry (25)developed a technique in which one or more pulses of a weak eluent were strategically timed to arrive at a set of unresolved peaks and alter abruptly the local mobile-phase composition. This scheme resulted in improved resolution between the poorly separated solute peaks but had little or no effect on neighboring solutes. More recently, Gluckman and co-workers (26) have applied this method with acid-base, ion-pairing, and hydrophobic interactions to elute organic dyes selectively according to their chemical properties. In addition, the use of individual solvent zones has been examined in a computer simulation of whole-column detection by Gelderloos and co-workers (27). From these related studies, the experimental feasibility and potential impact of the solvent modulation technique are evident. In this paper, the conceptual basis of solvent modulation is broadly developed and the theories of solute retention and solvent zone dispersion are derived. The implications of these theoretical models are used to establish the practical range of experimental conditions for solvent modulation in liquid chromatography.
GENERAL CONCEPT In the solvent modulation technique, individual solvent zones are introduced sequentially onto the chromatographic column. Each solvent is delivered for a specified time period, which determines the fractional length along the column. This timed sequence of individual solvent segments may be repeated once or many times to achieve the chromatographic separation. Solute retention under the conditions of solvent modulation may be systematically controlled through four experimental parameters. First, the number of solvents selected for the modulated mobile phase is determined by the complexity of the sample to be separated. A sample with only a few components may be separated with only two solvents, while a highly complex sample may require many more. Secondly, the composition of the individual solvents is chosen to adjust the strength and selectivity of the modulated mobile phase to provide differential retention for the solutes of interest. These individual solvents may conceivably vary in type and concentration of organic component, buffer, ion-pair agent, or other modifier as well as in bulk properties such as pH, ionic strength, etc. The third parameter, termed proportion, rep-
e
B
25-75
B
4
53-53
3
A
A
1417
B
A
75-25
A
4
B
3
B
i
3
3
e
135-3
This concept is illustrated in Figure 1 (left), where solvents A and B are modulated in proportions ranging from 0% to 100% of the modulation cycle. It may be noted that the proportion, which is a length ratio, is analogous to the volumetric ratio used in premixed solvent systems. The fourth parameter, termed repetition rate, represents the real or integer number of modulation cycles that may reside on a chromatographic column of length (L). For two representative solvents, the repetition rate (RR) is given by
L RR = -
(2) + xB This concept is illustrated in Figure 1 (right), where a modulation sequence with a constant proportion of % % A-75% B is shown at varying integer repetition rates. It may be noted that a premixed mobile phase which adheres to regular solution theory is conceptually identical with a modulated mobile phase at infinite repetition rate. From the preceding discussion, it is apparent that the number of possible modulation sequences increases dramatically with the number of solvents. The permutations in solvent order increase in a factorial manner, so that a twosolvent modulation sequence will have two possible permutations (A-B and B-A), a three-solvent system will have six permutations (A-B-C, A-C-B, B-A-C, B-C-A, C-A-B, and C-B-A), and so on. Furthermore, the proportion of each solvent and the repetition rate are independently and, within experimental constraints, infinitely variable. The selected solvent sequence may be repeated without alteration, which is analogous to isocratic elution with premixed mobile phases. Alternatively, the proportion or repetition rate may be changed continuously or discontinuously during the analysis, comparable to linear or stepwise gradient elution. Thus, solvent modulation offers an extremely versatile means to control both solvent strength and selectivity in liquid chromatography. Because of the many possible variations, the theoretical basis of solute retention under the conditions of solvent modulation must be well established in order to control and optimize separations. THEORY OF SOLUTE RETENTION The fundamental assumption underlying the theory of solvent modulation is that solute retention is controlled inxA
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ANALYTICAL CHEMISTRY, VOL. 62. NO. 14, JULY 15, 1990
dependently within each solvent zone. It is, therefore, necessary that the solvent segments be of known sequence and length and not be intermixed a t the boundary. Another assumption implicit in this model is that rapid equilibrium and mass transfer exist a t the interface of the mobile and stationary phases. Therefore, the column is assumed to achieve steady-state conditions and behavior very quickly after each change in solvent composition. This assumption is reasonable when solvents of similar composition are modulated but may not be valid for solvents of significantly different strength. Finally, it is assumed that the phase ratio and the separation mechanism, such as partition or adsorption. do not change with the solvent composition. The solute capacity factor ( k ) , which represents the partition or adsorption coefficient ( K )divided by the volumetric ratio of the mobile and stationary phases (p),may be expressed as follows:
(3) where t R and to are the elution times of a retained and a nonretained solute, respectively. Under the conditions of solvent modulation, the solute retention time is the summation of residence times in the individual solvent segments ( t j ) (4) The time spent by the solute in a solvent segment of length x i is given by
(5) where u is the linear velocity of the modulated mobile phase. By substitution in the above equations, the overall capacity factor (k) may be expressed as the time-weighted average of the capacity factors in the individual solvent segments (k,)
The limit of the summation index ( n ) ,which represents the total number of solvent segments required to elute the solute from a column of length L , is determined by evaluating the expression n Xj
E-=L
j=o kj
(7)
If this limit has a noninteger value, the summation in eq 6 is performed in the normal manner for solvent segments 0 to n, and the remainder is treated as a fractional multiplier for the last ( n 1) solvent segment. Prediction and Control of Solute Retention. Practical application of the solvent modulation technique requires accurate knowledge of the solute capacity factor in each solvent in order to predict the overall solute retention using eq 6. The magnitude of these individual solute capacity factors determines the theoretical limits of retention in solvent modulation. Within these limits, any solute capacity factor is theoretically achievable by adjustment of the order, the repetition rate, or the proportion of each solvent. For the simple case of two solvents, the effect of solvent order is shown in Figure 2 for a solute with approximately equal capacity factors in each solvent ( k A = 1.00, kB = 1.01) at a proportion of 50% A-50% B. If the solute i s eluted within the first solvent zone, it will retain an overall capacity factor equal to
+
'0'00
,,
,
,
,
,
,
,
,
-
Figure 2. Effect of solvent order on the overall solute retention as a function of repetition rate: (A) sotvent order A-B; (B) solvent order B-A; k , = 1.00, k , = 1.01; solvent proportion 50% A-50% 8. The periodicity of these functions arises because the solute is exposed to fractional portions of individual solvent zones. An overall capacity factor of approximately 1.005 is approached asymptotically as repetition rate increases, regardless of solvent order.
that in the individual solvent. In Figure 2, this occurs at repetition rates less than approximately 0.500 for the solvent orders of either A-B or B-A. At repetition rates greater than this value, the solute is exposed to fractional lengths of the second and subsequent solvent zones, which will increase or decrease the retention of the solute accordingly. Thus, the overall solute capacity factor will exhibit a periodic dependence on the repetition rate, as shown in Figure 2. The amplitude and frequency of the observed waveform are controlled by the solvent proportion and the individual solute capacity factor values. Whenever the solute is eluted with a complete modulation cycle or integer multiples thereof, a unique capacity factor is attained that is identical regardless of solvent order. In Figure 2, this occurs a t repetition rates that are integer multiples of 1.00 (RR = PA/kA + PB/kB), where the overall capacity factor is approximately equal to 1.005. Provided that the solute is exposed to a statistically large number of solvent zones (RR 2 lo), the overall capacity factor asymptotically approaches that in an ideal premixed solvent system. For Figure 2, this is analogous to a premixed solvent system composed of 50% A-50% B. It is apparent from eq 6, however, that the overall solute capacity factor does not vary linearly between those in the individual solvents, but rather is proportional to the factor (1 + k j ) / k j . Shown in Figure 3 are examples of this nonlinear relationship where k A is given a constant value of 1.000 while kB is systematically increased from 1.010 to 100.0. On examination of the cases where the solvent proportion is 50% A-50% B, when k A = 1.000 and kB = 1.010 (Figure 3A), the overall solute capacity factor attains a maximum value of approximately 1.005 as the repetition rate increases. When k B = 1.100 (Figure 3B), a maximum capacity factor of 1.048 is attained, whereas when kB = 2.000 (Figure 3C), the maximum value is 1.333. This nonlinear behavior in solute retention becomes more dramatic when the difference between the individual capacity factors is increased. For example, when k B = 10.00 (Figure 3D), a maximum capacity factor of 1.818 is attained, and when kB = 100.0 (Figure 3E), the maximum value is 1.980. It may thus be deduced from eq 6 that, with further increase in kB, a limiting capacity factor of 2.000 is approached at a proportion of 50% A-50% B. For the more general case, the limiting capacity factor as k B approaches infinity is given by lim k = k A / P A (8) kB-RR-m
so that solute capacity factor values of 1.333, 2.000, and 4.000
ANALYTICAL CHEMISTRY, VOL. 62, NO. 14, JULY 15, 1990
1419
At proportions greater than 75% A, any small change in composition causes a dramatic change in retention, thereby requiring precise control of the solvent proportion and repetition rate. Thus, accurate knowledge of the individual solute capacity factors, repetition rate, and solvent proportion is essential to the control and prediction of retention in solvent modulation. Precision of Predicted Solute Retention. In order to evaluate the practical utility of the solvent modulation technique, it is necessary to determine the expected precision of solute retention predicted by using eq 6. The uncertainty in the predicted capacity factor may be readily estimated by applying the method of propagation of errors, assuming that all errors are random in nature. If all variables are assumed to be independent, the precision of the predicted capacity factor (as measured by the variance, uk2)for a two-solvent modulation sequence may be expressed as a sum of the individual contributions
1.010
: ID33
1.075-
1OM-
47i D
3
i
2w
1
0.W.l
~~,
I . _
E M,O-
where u ~and~ u , (~~ represent ) ~) ~ the uncertainty (variance) in the solvent segment lengths, and (rk(B)' express the variance in the capacity factors in the individual solvents, and aL2 is the variance in the column length. In order to simplify this calculation, it is assumed that the intermixed region between adjacent solvent zones contributes only to the uncertainty in the estimation of the solvent zone length, not to the uncertainty in the individual capacity factors. Under these conditions, application of the equation for error propagation to eq 6 results in the following expression:
MO-
M.020.0-
00
Repetition Rote (RR)
Figure 3. Effect of the individual solute capacity factors on the overall solute retention as a function of repetition rate: (A) k A = 1.000, k, = 1.010; (8) kA = 1.000, kB = 1.100; IC)kA = 1.000,k, = 2.000; (D) k, = 1.000, k, = 10.00(E) kA = 1.000,k, = 100.0 Sobent order A-B with solvent proportions of 25% A-75% B, 50% A-50% B, and 7 5 % A-25% B. The nonlinear relationship of the overall solute retention is a function of the individual capacity factors and the solvent proportion as shown in eqs 6 and 8. Thus, as the difference between the indMdual capacity factors increases, the range of accessible overall solute capacity factors decreases.
are approached asymptotically for the solvent proportions of 75% A-25% B, 50% A-5070 B, and 25% A-75% B, respectively, as shown in Figure 3. As a result of this nonlinear relationship of solute retention, the full range of solute capacity factors between k A and kB may not be readily attainable. For example, when the individual capacity factors are of comparable value (Figure 3A-C), as may be expected for solvent-selectivity modulation, the overall solute capacity factor is nearly a linear combination. Under these conditions, all capacity factors between k A and k B can be attained with equal ease. As the difference between the individual capacity factors becomes greater (Figure 3D-E), as may be expected for solvent-strength modulation, the range of readily accessible capacity factors decreases. For example, in Figure 3E, where k A = 1.000 and kB = 100.0, only overall solute capacity factors in the range from 1.00 to 4.00 can be attained within the proportion of solvent A from 0% to 75%.
where nAand nB are the number of zones of solvent A and B, respectively, encountered by the solute. By examination of eq 10, it is apparent that the first and second terms and the third and fourth terms are equivalent mathematically, differing only in the solvent they represent, and, therefore, have identical behavior. In general, the overall importance of each term in eq 10 is controlled by a coefficient related to the individual variables and by the magnitude of the variance of each variable. The effect of each variable and its corresponding variance is examined sequentially in the following discussion. Shown in Table I is the effect of varying the solvent zone length ( x j ) and its associated variance (uxbT)upon the precision of the predicted capacity factor, all other parameters remaining constant. In practice, the solvent zone length is adjusted by means of the solvent proportion (eq 1)and repetition rate (eq 2). The variance of the solvent zone length may be influenced by fluctuations in the mobile-phase linear velocity, which cause relative errors in x j , and by chromatographic dispersion processes (both on-column and extracolumn), which cause constant errors in xj. For the purpose
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ANALYTICAL CHEMISTRY, VOL. 62, NO. 14, JULY 15, 1990
Table I. Effect of Solvent Zone Length (x,)on the Overall Precision of Solute Retentionn
relative error in xib
constant error in xic
kA
kB
%A
% B
RR
nA
nB
L cm
k
ah2
1.00 1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00 1.00
25 50 75 50 50
75 50 25 50 50
1.0 1.0 1.0 5.0 10.0
1.00 1.00 1.00 5.00 10.00
1.00 1.00 1.00 5.00 10.00
25.0 25.0 25.0 25.0 25.0
1.00 1.00 1.00 1.00 1.00
7.125 X 6.500 X 10"' 7.125 X 10"' 6.500 X 6.500 X
1.00 1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00 1.00
25 50 75 50 50
75 50 25 50 50
1.0 1.o 1.0 5.0 10.0
1.00 1.00 1.00 5.00 10.00
1.00 1.00 1.00 5.00 10.00
25.0 25.0 25.0 25.0 25.0
1.00
6.625 X lo4 6.500 X lo-' 6.625 X 5.450 X 2.045 X
Relative error assumed for all variables except x,: ukO)= 0.01k,, uL = 0.01L. Relative error in x,: O.Ol(P,L/RR),where L = 25 cm, RR = 1.0, P , = 0.5.
1.00 1.00 1.00 1.00
uxo)=
O.Olx,.
Constant error in x,:
uz,,, =
Table 11. Effect of Individual Solute Capacity Factors on the Overall Precision of Solute Retention" kA
kB
1.00 1.00 1.oo 1.00 1.00
0.10 1.01 1.10 10.00 100.00
Yo
A
50 50 50 50 50
"Relative error assumed for all variables:
% B
RR
nA
nB
L, cm
k
50 50 50 50 50
1.00 1.00 1.00 1.oo 1 00
1.000 1.010 1.091 1.900 1.990
0.100 1.000 1.000 1.000 1.000
25.0 25.0 25.0 25.0 25.0
0.550 1.005 1.046 1.450 1.495
uX0)=
O.Olx,,
b&)
uh2
4.205 x 6.530 X 6.789 X 1.082 X 1.143 x
10-4 lo-' lo-' 10-3
= 0.01k,, uL = 0.01L.
Table 111. Effect of Column Length on the Overall Precision of Solute Retention"
relative error in xJb
constant error in xic
kA
kB
% A
% B
RR
nA
nB
L, cm
k
1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00
50 50 50 50
50 50 50 50
1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00
5.0 10.0 25.0 100.0
1.00 1.00 1.00 1.00
6.500 X 6.500 X 6.500 X 6.500 X
1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00
50 50 50 50
50 50 50 50
1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00
5.0 10.0 25.0 100.0
1.00 1.00 1.00
5.450 X low3 1.700 X 6.500 X IO4 4.625 X
Relative error assumed for all variables except x : bk0) = 0.01k,, uL = 0.01L. Relative error in x,: O.Ol(P,L/RR), where L = 25 cm, RR = 1.0, b, = 0.5.
ux,,) =
1.00 O.Olx,.
6k2
Constant error in xi:
urljl=
of this discussion, relative errors of 1%are assumed for all variables unless otherwise stated, while constant errors are calculated based on a solvent proportion of 50% A-50% B at 1%relative error. When the solvent proportion is altered at a constant repetition rate, as shown in Table I, the variance in the predicted capacity factor reaches a minimum when the solvent zone lengths are equal. This minimum in the retention variance is observed for both relative and constant sources of error in xi. As may be noted in eq 10, the overall variance in solute retention results from a sum of squares of the error in xA and XB (terms 1 and 2 ) , as well as from a sum of squares of the magnitude of xA and xB (terms 3 and 4). When a relative source of error is assumed, the errors in XA and XB vary inversely with one another, as do the individual solvent zone lengths, xA and xg. Since the effect of varying solvent zone length is canceled in term 5, a minimum in the overall capacity factor variance is expected as the solvent proportion is altered. When the repetition rate is altered at a constant proportion, the precision of solute retention is dependent upon the predominant source of error in solvent zone length. If a relative error is x, is presumed, the variance of the predicted capacity factor remains constant with increasing repetition rate (Table I). Since the number of solvent zones (nj)required to elute the solute increases as the solvent zone length (xi)decreases, this constant precision is expected from eq 10. In contrast, when a constant error in xi is assumed, terms 1 and 2 increase with nj,resulting in an increase in the variance of the predicted solute retention for increasing repetition rate.
Shown in Table I1 is the effect of varying the individual capacity factors ( k j )and their associated variance (uhbj) upon the precision of the predicted capacity factor, all other parameters remaining constant. In general, the variance of the individual capacity factors is influenced by fluctuations in the mobile-phase linear velocity and fluctuations in temperature (28),both of which cause relative errors in kj. As the individual capacity factor increases, the precision in the predicted overall capacity factor decreases (Table 11). This result may seem contradictory to eq 10, since every term is inversely related to the individual capacity factors, so that precision might be expected to increase as retention increases. However, as kB increases, a greater number of solvent A zones is required to elute the solute. Thus, the terms containing kB in the denominator become negligible, and the increase in the variance of the predicted solute capacity factors is a direct result of the increase in nA. Shown in Table I11 is the effect of varying the column length ( L )and its associated variance ((rL2) upon the precision of the predicted capacity factor, all other parameters remaining constant. It can be seen from Table I11 that the variance in the predicted capacity factor is constant with increasing column length, for a relative error in the solvent zone length. This result may seem surprising, since every term in eq 10 is inversely related to the column length. However, the solvent zone length is directly proportional to the column length for a specified repetition rate and solvent proportion. Consequently, for a relative error in both solvent zone length and
ANALYTICAL CHEMISTRY, VOL. 62, NO. 14, JULY 15, 1990
column length, all terms in eq 10 remain constant. In contrast, the variance in the predicted capacity factor decreases as the column length increases, for a constant error in x j . In this case, only the first and second terms contribute to the decrease in the variance of the overall solute capacity factor with increasing column length. It is apparent that the precision of the predicted solute capacity factor in solvent modulation is dependent on the absolute magnitude of the experimental parameters as well as the error in each parameter. Assuming the predominant sources of error are random in nature, several recommendations can be made concerning the selection of experimental operating conditions. As derived in the propagation of error analysis, the precision of the predicted capacity factor will be maximized with solvent proportions near 50% A-50% B and with low repetition rates. Whenever possible, the magnitude of the individual solute capacity factors should be minimized as well. Although the column length is often not an adjustable parameter, it is recommended that the maximum practical length be utilized. Thus, it is preferable to use a strong solvent with a long column under the conditions of solvent modulation, rather than a weak solvent with a short column, as commonly practiced with premixed mobile phases. In addition to the experimental parameters, the error in each parameter also contributes to the imprecision in the predicted capacity factor. It is essential to identify and to minimize the largest and most influential of these sources of error. Thus, the ultimate goal in solvent modulation is to control all experimental parameters and errors which influence the accuracy and the precision in the predicted capacity factor.
THEORY OF SOLVENT ZONE DISPERSION Although the individual capacity factors and the column length may be determined precisely and accurately, the measurement of solvent zone length is more difficult. Moreover, the precision in the overall capacity factor is affected substantially by the error in the solvent zone length (Table I). Even when zones of known length are introduced onto the column under optimal experimental conditions, the broadening processes inherent in chromatographic separations will ultimately limit the error in solvent zone length. Thus, the precise prediction of the overall capacity factor requires a detailed understanding of the dispersion of solvent zones on a chromatographic column. By evaluation of the effect of column dispersion on the solvent modulated zones, the practical limits of the solvent modulation technique may be determined under common chromatographic conditions. In order to accomplish this goal, the length of the initial solvent zone must be maximized relative to the broadening introduced by the chromatographic column. This problem is the exact reverse of that usually encountered in the evaluation of extra-column dispersion, where the goal is to minimize external sources of variance relative to the column variance. Nevertheless, the mathematical approach developed by Martin, Eon, and Guiochon (29)is directly applicable. In the case of solvent modulation, however, the initial solvent zone variance (u:) must be expressed as a multiple (e2), rather than a fraction, of the column variance (ucoL2) If there are no additional sources of dispersion, and if the variance contributions from the initial solvent zone and the column are independent, then the final zone variance (ut)may be expressed as uf2
=
UCOL2
+ (Ti2 = UCOL2(1 + 02)
(12)
Thus, we can evaluate the extent of mixing introduced by the chromatographic column in terms of the initial and final zone variances. This measure of the solvent zone purity may be
1421
defined by the following ratio:
which has a value of unity when the initial zone is unaltered by column dispersion processes. Determination of the solvent zone purity using eq 13 requires explicit mathematical functions for the initial and final zone variances. For a single rectangular zone a t a solvent proportion of 50% A-50% B, the initial concentration profile Ci(x) may be described by a normalized impulse function of finite length L/2RR 2RR Ci(X) = 7 L
where -L/4RR Ix 5 L4RR. The variance of the initial zone, determined by calculation of the second statistical moment (30), is given by 2
Mathematical expressions for the final concentration profile of the solvent zone may be derived by the method of convolution of integrals (30). In this technique, the initial concentration profile (eq 14) is convolved with a normalized Gaussian operator, descriptive of the chromatographic dispersion processes
where ucoL2 is the column variance in length units. The final profile (C,(x)) resulting from the convolution of an individual solvent zone with the Gaussian function is
The variance of this final solvent zone ( u t ) is determined by calculation of the second statistical moment to be
By substitution of eqs 15 and 18 into eq 13, the solvent zone purity may be calculated. The corresponding final concentration profile, evaluated by using eq 17, is schematically represented as a function of the solvent zone purity in Figure 4. As the zone purity is varied from an unperturbed ( O z / ( l + 02) = 1.00) to a Gaussian-modified zone (02/(1 + 02) = 0.70), the detrimental effect of the column dispersion processes becomes apparent. If the maximum permissible column variance is arbitrarily defined as 5% of the initial zone variance (Oz/(l+ On) = 0.95), the experimental conditions necessary to fulfill this requirement may be determined. This may be accomplished by selecting representative chromatographic conditions and subsequently determining the corresponding range of permissible solvent modulation conditions. The dispersion contributed by the chromatographic column is usually expressed in terms of the plate height (H).Although many theoretical equations for the plate height can be found
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ANALYTICAL CHEMISTRY, VOL. 62, NO. 14, JULY 15, 1990
d
I
\
-1
-
^^
3c
2 00
1s 3G
R 32
6 30
3 e p e - tion ? a t e (RR)
Figure 5. Solvent zone purity (02/(1 + 02)) as a function of repetition rate for chromatographic columns of constant length ( L = 25 cm) and varying particle size: (A) d , = 3 km; (B) d , = 5 km; (C) d , = 10 Mm; operated at optimum velocity (H = oCa2/L = 2d,). With increasing particle size, the solvent zone purity and the maximum permissible repetition rate are reduced.
I
I__-
Flgure 4. Schematic representation of solvent zone profiles according to eq 17. As the solvent zone is broadened by chromatographic dispersion processes, the zone purity (02/(1 + 8')) decreases as predicted by eq 13. in the literature, most represent a sum of the contributions resulting from multiple paths ( A ) ,longitudinal diffusion ( B ) , and mass transfer processes (C)
H = - UCOL2 -A L
+ B-U + CU
An equation of this form predicts a minimum in the plate height as a function of the mobile-phase linear velocity ( u ) . At optimum velocity, the column variance is approximately equal to 2 d 4 . Thus, the column variance is a function of both the particle diameter (d,) and column length ( L ) ,while the initial zone variance is defined by the repetition rate and column length (eq 15). For a chromatographic column of specified particle diameter and column length, the zone purity decreases nonlinearly with repetition rate as shown in Figures 5 and 6. As the repetition rate increases, the initial solvent zone variance becomes a successively smaller fraction of the final zone variance. Eventually, the solvent zone purity is reduced below the permissible level (0.95), which defines the maximum practical repetition rate for that chromatographic column at optimum linear velocity. A t velocities above or below the optimum value, the column variance increases according to eq 19 and the maximum repetition rate is commensurately reduced. The effect of particle diameter on the solvent zone purity is illustrated in Figure 5. An increase in particle size increases the column variance, thereby decreasing the maximum repetition rate. Conversely, at a constant repetition rate, an increase in particle size reduces the solvent zone purity. Thus, small particle diameters are recommended because the small column variance will maintain the solvent zone integrity and, concurrently, will improve the chromatographic efficiency. The effect of column length on the solvent zone purity is illustrated in Figure 6. Since column variance increases with
3+----, r 3c
i
\
,
I
i
cc
L
30
,
,
5
,
co
, 800
.
4 13 00
R e p e t i t on R a t e (RR)
Figure 6 . Solvent zone purity (02/(1 4- 02))as a function of repetition rate for chromatographic columns of constant particle size ( d p = 3 pm) and varying length: (A) L = 100 cm; (B) L = 25 cm; (C) L = 10 cm; operated at optimum velocity (H = uca2/L = 2d,). with increasing column length, the solvent zone purAy and the maximum permissible repetition rate are increased. length, it is somewhat surprising that the maximum repetition rate and solvent zone purity increase as well. These unexpected results arise because the column variance increases linearly with column length, while the initial solvent zone variance increases with the square of column length at constant repetition rate. Consequently, the resultant solvent zone purity and the maximum repetition rate we improved by using the longest possible column length consistent with reasonable analysis time. Based on this theoretical analysis, the range of practical operating conditions which meets the specified requirements for solvent zone purity may be established for any chromatographic column. For example, both an inefficient column (d, = 10 pm, L = 10 cm) and a highly efficient column (d, = 3 pm, L = 100 cm) may be used effectively under the conditions of solvent modulation. However, to achieve the same level of accuracy and precision in predicting solute retention, the former column requires that the repetition rate be less than 2.3, while the latter allow repetition rates as high as 13.5. Thus, the more efficient column makes possible a wider range of operating conditions in solvent modulation for greater versatility in optimizing separations.
CONCLUSIONS In solvent modulation, individual solvent zones are introduced sequentially onto the chromatographic column. Because
ANALYTICAL CHEMISTRY, VOL. 62, NO. 14, JULY 15, 1990
the solvent zones are separated both spatially and temporally, solute retention is controlled independently within each zone and may be accurately predicted. By simple adjustment of the solvent proportion, the repetition rate, and the solute capacity factors in the individual solvent zones, the overall solute capacity factor may be reliably controlled. Many optimization techniques utilized in liquid chromatography are based on the assumption of linear additivity of solute retention. Because this presumption is rigorously correct under the conditions of solvent modulation, these optimization methods can be implemented more accurately and precisely than with premixed mobile phases. Moreover, solute retention may be readily predicted for any modulation sequence based on knowledge of the capacity factors in the individual solvents. Hence, the number of characterizing experiments necessary to optimize the separation of complex mixtures may be drastically reduced. While this advantage is important for isocratic separations, it becomes essential for the optimization of gradient elution and multidimensional separations. The conceptual and theoretical basis of solvent modulation may be readily extended to include these more complicated separation systems. Solvent modulation appears to offer an extremely versatile and powerful alternative to premixed mobile phases. In future studies, we will examine the validity of the theoretical models derived herein (31,32),compare solvent modulation with the equivalent premixed mobile phases (32),and develop optimization strategies for the separation of complex samples (33).
ACKNOWLEDGMENT The authors wish to thank Christine Evans for assistance in the preparation of the manuscript and for many helpful discussions.
LITERATURE CITED (1) Balke, S . T. Quantitative Column Liqukl Chromatography: a survey of chemometric methods ; Journal of Chromatography Library; Elsevier Science Publishing Co., Inc.: New York, 1984; Volume 29, Chapters 2 and 3. (2) Janz, G. J.; Tomkins, R. P. T. Nonaqueous E k t r d y e Hsnd600k, Volume 1 ; Academic Press: New York, 1972; pp 85-1 18.
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(3) Glajch, J. L.; Kirkland, J. J.; Squire, K. M.; Minor, J. M. J. C h r m t o g r . 1980, 199. 57. (4) Colin, H.; Guiochon, G.; Jandera, P. Anal. Chem. 1983, 55, 442. (5) Drouen, A. C. J. H.; Billit, H. A. H.; de Galan, L. J. Chromatog. 1986, 352, 127. (6) Jandera, P.; Churacek, J. &adient Elution in Column LlquM Chromatogfaphy: theory and practice; Journal of Chromatography Library; Elsevier Science Publishing Co., Inc.: New York, 1985; Volume 31. (7) Hildebrand, J. H.; Scott, R. L. The Solu6ility of None/ectro&tes; Reinhold: New York, 1950; Chapter 23. (8) Barton, A. F. M. Chem. Rev. 1975, 75, 731. (9) Horvath, Cs.; Meiander, W.; Molnar, I.J . Chromatogr. 1978, 125, 129. (10) Martire, D. E.; Boehm, R. E. J . Liq. Chromatcgr. 1980, 3 , 753. (11) Martire, D. E.; Boehm, R. E. J. Phys. Chem. 1983, 87, 1045. (12) Snyder, L. R. J. Chromatogr. 1974, 9 2 , 223. (13) Snyder, L. R. J. Chromatogr. Sci. 1978, 16, 223. (14) Snyder, L. R.; Kirkland. J. J. Introduction to Modern Liquld Chromatography Zed.; Wiley: New York, 1979; Chapter 6. (15) Johnson, B. P.; Khaledi, M. G.; Dorsey, J. G. Anal. Chem. 1986, 58, 2354. (16) Koppel, I.; Koppel, J. Org. React. (Tartu) 1983, 2 0 , 523. (17) Reichardt, C.; Dimroth, K. Fortschr. Chem. Fwsch. 1968, 1 1 , 1. (18) Sadek, P. C.;Carr, P. W.; Doherty. R. M.; Kamlet, M. J.; Taft, R. W.; Abraham, M. H. Anal. Chem. 1985, 57, 2971. (19) Grunwald, E.; Winstein, S. J. Am. Chem. SOC. 1948, 70, 848. (20) Jandera, P.; Colin, H.; Guiochon, G. Anal. Ch8m. 1982, 54, 435. (21) Colin, H.; Guiochon, G.; Jandera, P. Chromatographis 1983, 17, 83. (22) Laub, R. J.; Purneil, J. H. J. Chromatogr. 1975, 112, 71. (23) Dolan, J. W.; Snyder, L. R.; Quarry, M. A. Chromatographis 1967, 24, 261. (24) Yang, F. J. J. Chromatogr. 1982, 236, 265. (25) Berry, V. V. J. Chromatogr. 1985, 321, 33. (26) Gluckman, J. C.; Slais, K.; Brinkman, U. A. Th.; Frei, R. W. Anal. Chem. 1987, 5 9 , 79. (27) Geklerloos, D. G.; Rowlen, K. L.; Birks, J. W.; Avery, J. P.; Enke, C. G. Anal. Chem. 1986, 58, 900. (28) Grushka, E.; Zamir, I. Chem. Anal. 1989, 9 8 , 529. (29) Martin, M.; Eon, C.; Guiochon, G. J. Chromatogr. 1875, 108, 229. (30) Sternberg. J. C. Adv. Chromatogr. 1966, 2 , 205. (31) Wahi, J. H.; Enke, C. G.; McGuffin, V. L. HRC C C , J . High Resolut. Chromatogr. Chromatogr. Commun. 1968, 1 1 , 858. (32) Wahl, J. H.; Enke, C. G.; McGuffin, V. L., unpublished results. (33) Wahl, J. H.; McGuffin, V. L. J. Chromatogr. 1989, 485, 541.
RECEIVED for review December 11,1989. Accepted March 23, 1990. This research was supported in part by the Michigan State University Foundation, Eli Lilly and Company, and the Dow Chemical Company. Preliminary results were presented at the Ninth International Symposium on Capillary Chromatography, Monterey, CA, 1988.
